# Autoconverted from set.mm by xlat_mm.py and gen_set_mm.py import (PROP pax/prop () "") import (SET_MM_AX zfc/set_mm_ax (PROP) "") # set.mm - Version of 17-Oct-2006 # # PUBLIC DOMAIN # # This file (specifically, the version of this file with the above date) # has been released into the Public Domain per the Creative Commons Public # Domain Dedication. http://creativecommons.org/licenses/publicdomain/ # # Norman Megill - email: nm(at)alum(dot)mit(dot)edu # var (wff ph) var (wff ps) var (wff ch) var (wff th) var (wff ta) thm (dummylink () ((dummylink.1 ph) (dummylink.2 ps)) ph (dummylink.1)) thm (a1i () ((a1i.1 ph)) (-> ps ph) (a1i.1 ph ps ax-1 ax-mp)) thm (a2i () ((a2i.1 (-> ph (-> ps ch)))) (-> (-> ph ps) (-> ph ch)) (a2i.1 ph ps ch ax-2 ax-mp)) thm (syl () ((syl.1 (-> ph ps)) (syl.2 (-> ps ch))) (-> ph ch) (syl.1 syl.2 ph a1i a2i ax-mp)) thm (com12 () ((com12.1 (-> ph (-> ps ch)))) (-> ps (-> ph ch)) (ps ph ax-1 com12.1 a2i syl)) thm (a1d () ((a1d.1 (-> ph ps))) (-> ph (-> ch ps)) (a1d.1 ps ch ax-1 syl)) thm (a2d () ((a2d.1 (-> ph (-> ps (-> ch th))))) (-> ph (-> (-> ps ch) (-> ps th))) (a2d.1 ps ch th ax-2 syl)) thm (imim2 () () (-> (-> ph ps) (-> (-> ch ph) (-> ch ps))) ((-> ph ps) ch ax-1 a2d)) thm (imim1 () () (-> (-> ph ps) (-> (-> ps ch) (-> ph ch))) (ps ch ph imim2 com12)) thm (imim1i () ((imim1i.1 (-> ph ps))) (-> (-> ps ch) (-> ph ch)) (imim1i.1 ph ps ch imim1 ax-mp)) thm (imim2i () ((imim1i.1 (-> ph ps))) (-> (-> ch ph) (-> ch ps)) (imim1i.1 ch a1i a2i)) thm (imim12i () ((imim12i.1 (-> ph ps)) (imim12i.2 (-> ch th))) (-> (-> ps ch) (-> ph th)) (imim12i.2 ps imim2i imim12i.1 th imim1i syl)) thm (3syl () ((3syl.1 (-> ph ps)) (3syl.2 (-> ps ch)) (3syl.3 (-> ch th))) (-> ph th) (3syl.1 3syl.2 syl 3syl.3 syl)) thm (syl5 () ((syl5.1 (-> ph (-> ps ch))) (syl5.2 (-> th ps))) (-> ph (-> th ch)) (syl5.1 syl5.2 ch imim1i syl)) thm (syl6 () ((syl6.1 (-> ph (-> ps ch))) (syl6.2 (-> ch th))) (-> ph (-> ps th)) (syl6.1 syl6.2 ps imim2i syl)) thm (syl7 () ((syl7.1 (-> ph (-> ps (-> ch th)))) (syl7.2 (-> ta ch))) (-> ph (-> ps (-> ta th))) (syl7.1 syl7.2 th imim1i syl6)) thm (syl8 () ((syl8.1 (-> ph (-> ps (-> ch th)))) (syl8.2 (-> th ta))) (-> ph (-> ps (-> ch ta))) (syl8.1 syl8.2 ch imim2i syl6)) thm (imim2d () ((imim1d.1 (-> ph (-> ps ch)))) (-> ph (-> (-> th ps) (-> th ch))) (imim1d.1 th a1d a2d)) thm (syld () ((syld.1 (-> ph (-> ps ch))) (syld.2 (-> ph (-> ch th)))) (-> ph (-> ps th)) (syld.1 syld.2 ps imim2d a2i ax-mp)) thm (imim1d () ((imim2d.1 (-> ph (-> ps ch)))) (-> ph (-> (-> ch th) (-> ps th))) (imim2d.1 ps ch th imim1 syl)) thm (imim12d () ((imim12d.1 (-> ph (-> ps ch))) (imim12d.2 (-> ph (-> th ta)))) (-> ph (-> (-> ch th) (-> ps ta))) (imim12d.1 th imim1d imim12d.2 ps imim2d syld)) thm (pm2.04 () () (-> (-> ph (-> ps ch)) (-> ps (-> ph ch))) (ph ps ch ax-2 ps ph ax-1 syl5)) thm (pm2.83 () () (-> (-> ph (-> ps ch)) (-> (-> ph (-> ch th)) (-> ph (-> ps th)))) (ps ch th imim1 ph imim2i a2d)) thm (com23 () ((com3.1 (-> ph (-> ps (-> ch th))))) (-> ph (-> ch (-> ps th))) (com3.1 ps ch th pm2.04 syl)) thm (com13 () ((com3.1 (-> ph (-> ps (-> ch th))))) (-> ch (-> ps (-> ph th))) (com3.1 com12 com23 com12)) thm (com3l () ((com3.1 (-> ph (-> ps (-> ch th))))) (-> ps (-> ch (-> ph th))) (com3.1 com23 com13)) thm (com3r () ((com3.1 (-> ph (-> ps (-> ch th))))) (-> ch (-> ph (-> ps th))) (com3.1 com3l com3l)) thm (com34 () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ph (-> ps (-> th (-> ch ta)))) (com4.1 ch th ta pm2.04 syl6)) thm (com24 () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ph (-> th (-> ch (-> ps ta)))) (com4.1 com34 com23 com34)) thm (com14 () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> th (-> ps (-> ch (-> ph ta)))) (com4.1 com34 com13 com34)) thm (com4l () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ps (-> ch (-> th (-> ph ta)))) (com4.1 com14 com3l)) thm (com4t () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ch (-> th (-> ph (-> ps ta)))) (com4.1 com4l com4l)) thm (com4r () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> th (-> ph (-> ps (-> ch ta)))) (com4.1 com4t com4l)) thm (a1dd () ((a1dd.1 (-> ph (-> ps ch)))) (-> ph (-> ps (-> th ch))) (a1dd.1 th a1d com23)) thm (mp2 () ((mp2.1 ph) (mp2.2 ps) (mp2.3 (-> ph (-> ps ch)))) ch (mp2.2 mp2.1 mp2.3 ax-mp ax-mp)) thm (mpd () ((mpd.1 (-> ph ps)) (mpd.2 (-> ph (-> ps ch)))) (-> ph ch) (mpd.1 mpd.2 a2i ax-mp)) thm (mpi () ((mpi.1 ps) (mpi.2 (-> ph (-> ps ch)))) (-> ph ch) (mpi.1 ph a1i mpi.2 mpd)) thm (mpii () ((mpii.1 ch) (mpii.2 (-> ph (-> ps (-> ch th))))) (-> ph (-> ps th)) (mpii.1 mpii.2 com23 mpi)) thm (mpdd () ((mpdd.1 (-> ph (-> ps ch))) (mpdd.2 (-> ph (-> ps (-> ch th))))) (-> ph (-> ps th)) (mpdd.1 mpdd.2 a2d mpd)) thm (mpid () ((mpid.1 (-> ph ch)) (mpid.2 (-> ph (-> ps (-> ch th))))) (-> ph (-> ps th)) (mpid.1 ps a1d mpid.2 mpdd)) thm (mpdi () ((mpdi.1 (-> ps ch)) (mpdi.2 (-> ph (-> ps (-> ch th))))) (-> ph (-> ps th)) (mpdi.1 mpdi.2 com12 mpid com12)) thm (mpcom () ((mpcom.1 (-> ps ph)) (mpcom.2 (-> ph (-> ps ch)))) (-> ps ch) (mpcom.1 mpcom.2 com12 mpd)) thm (syldd () ((syldd.1 (-> ph (-> ps (-> ch th)))) (syldd.2 (-> ph (-> ps (-> th ta))))) (-> ph (-> ps (-> ch ta))) (syldd.1 syldd.2 th ta ch imim2 syl6 mpdd)) thm (sylcom () ((sylcom.1 (-> ph (-> ps ch))) (sylcom.2 (-> ps (-> ch th)))) (-> ph (-> ps th)) (sylcom.1 sylcom.2 a2i syl)) thm (syl5com () ((syl5com.2 (-> ph (-> ps ch))) (syl5com.1 (-> th ps))) (-> th (-> ph ch)) (syl5com.1 ph a1d syl5com.2 sylcom)) thm (syl6com () ((syl6com.1 (-> ph (-> ps ch))) (syl6com.2 (-> ch th))) (-> ps (-> ph th)) (syl6com.1 syl6com.2 syl6 com12)) thm (syli () ((syli.1 (-> ps (-> ph ch))) (syli.2 (-> ch (-> ph th)))) (-> ps (-> ph th)) (syli.1 syli.2 com12 sylcom)) thm (syl5d () ((syl5d.1 (-> ph (-> ps (-> ch th)))) (syl5d.2 (-> ph (-> ta ch)))) (-> ph (-> ps (-> ta th))) (syl5d.2 ps a1d syl5d.1 syldd)) thm (syl6d () ((syl6d.1 (-> ph (-> ps (-> ch th)))) (syl6d.2 (-> ph (-> th ta)))) (-> ph (-> ps (-> ch ta))) (syl6d.1 syl6d.2 ps a1d syldd)) thm (syl9 () ((syl9.1 (-> ph (-> ps ch))) (syl9.2 (-> th (-> ch ta)))) (-> ph (-> th (-> ps ta))) (syl9.2 ph a1i syl9.1 syl5d)) thm (syl9r () ((syl9r.1 (-> ph (-> ps ch))) (syl9r.2 (-> th (-> ch ta)))) (-> th (-> ph (-> ps ta))) (syl9r.1 syl9r.2 syl9 com12)) thm (id () () (-> ph ph) (ph ph ax-1 ph (-> ph ph) ax-1 mpd)) thm (id1 () () (-> ph ph) (ph ph ax-1 ph (-> ph ph) ax-1 ph (-> ph ph) ph ax-2 ax-mp ax-mp)) thm (idd () () (-> ph (-> ps ps)) (ps id ph a1i)) thm (pm2.27 () () (-> ph (-> (-> ph ps) ps)) ((-> ph ps) id com12)) thm (pm2.43 () () (-> (-> ph (-> ph ps)) (-> ph ps)) (ph ps pm2.27 a2i)) thm (pm2.43i () ((pm2.43i.1 (-> ph (-> ph ps)))) (-> ph ps) (pm2.43i.1 ph ps pm2.43 ax-mp)) thm (pm2.43d () ((pm2.43d.1 (-> ph (-> ps (-> ps ch))))) (-> ph (-> ps ch)) (ph ps idd pm2.43d.1 mpdd)) thm (pm2.43a () ((pm2.43a.1 (-> ps (-> ph (-> ps ch))))) (-> ps (-> ph ch)) (ps ph ax-1 pm2.43a.1 mpdd)) thm (pm2.43b () ((pm2.43a.1 (-> ps (-> ph (-> ps ch))))) (-> ph (-> ps ch)) (pm2.43a.1 pm2.43a com12)) thm (sylc () ((sylc.1 (-> ph (-> ps ch))) (sylc.2 (-> th ph)) (sylc.3 (-> th ps))) (-> th ch) (sylc.3 sylc.2 sylc.1 syl mpd)) thm (pm2.86 () () (-> (-> (-> ph ps) (-> ph ch)) (-> ph (-> ps ch))) (ps ph ax-1 (-> ph ch) imim1i com23)) thm (pm2.86i () ((pm2.86i.1 (-> (-> ph ps) (-> ph ch)))) (-> ph (-> ps ch)) (pm2.86i.1 ph ps ch pm2.86 ax-mp)) thm (pm2.86d () ((pm2.86d.1 (-> ph (-> (-> ps ch) (-> ps th))))) (-> ph (-> ps (-> ch th))) (pm2.86d.1 ps ch th pm2.86 syl)) thm (loolin () () (-> (-> (-> ph ps) (-> ps ph)) (-> ps ph)) (ps ph ax-1 (-> ps ph) imim1i pm2.43d)) thm (loowoz () () (-> (-> (-> ph ps) (-> ph ch)) (-> (-> ps ph) (-> ps ch))) (ps ph ax-1 (-> ph ch) imim1i a2d)) thm (a3i () ((a3i.1 (-> (-. ph) (-. ps)))) (-> ps ph) (a3i.1 ph ps ax-3 ax-mp)) thm (a3d () ((a3d.1 (-> ph (-> (-. ps) (-. ch))))) (-> ph (-> ch ps)) (a3d.1 ps ch ax-3 syl)) thm (pm2.21 () () (-> (-. ph) (-> ph ps)) ((-. ph) (-. ps) ax-1 a3d)) thm (pm2.24 () () (-> ph (-> (-. ph) ps)) (ph ps pm2.21 com12)) thm (pm2.21i () ((pm2.21i.1 (-. ph))) (-> ph ps) (pm2.21i.1 (-. ps) a1i a3i)) thm (pm2.21d () ((pm2.21d.1 (-> ph (-. ps)))) (-> ph (-> ps ch)) (pm2.21d.1 (-. ch) a1d a3d)) thm (pm2.18 () () (-> (-> (-. ph) ph) ph) (ph (-. (-> (-. ph) ph)) pm2.21 a2i a3d pm2.43i)) thm (peirce () () (-> (-> (-> ph ps) ph) ph) (ph ps pm2.21 ph imim1i ph pm2.18 syl)) thm (looinv () () (-> (-> (-> ph ps) ps) (-> (-> ps ph) ph)) ((-> ph ps) ps ph imim1 ph ps peirce syl6)) thm (nega () () (-> (-. (-. ph)) ph) ((-. ph) ph pm2.21 ph pm2.18 syl)) thm (negb () () (-> ph (-. (-. ph))) ((-. ph) nega a3i)) thm (pm2.01 () () (-> (-> ph (-. ph)) (-. ph)) (ph nega (-. ph) imim1i (-. ph) pm2.18 syl)) thm (pm2.01d () ((pm2.01d.1 (-> ph (-> ps (-. ps))))) (-> ph (-. ps)) (pm2.01d.1 ps pm2.01 syl)) thm (con2 () () (-> (-> ph (-. ps)) (-> ps (-. ph))) (ph nega (-. ps) imim1i a3d)) thm (con2d () ((con2d.1 (-> ph (-> ps (-. ch))))) (-> ph (-> ch (-. ps))) (con2d.1 ps ch con2 syl)) thm (con1 () () (-> (-> (-. ph) ps) (-> (-. ps) ph)) (ps negb (-. ph) imim2i a3d)) thm (con1d () ((con1d.1 (-> ph (-> (-. ps) ch)))) (-> ph (-> (-. ch) ps)) (con1d.1 ps ch con1 syl)) thm (con3 () () (-> (-> ph ps) (-> (-. ps) (-. ph))) (ps negb ph imim2i con2d)) thm (con3d () ((con3d.1 (-> ph (-> ps ch)))) (-> ph (-> (-. ch) (-. ps))) (con3d.1 ps ch con3 syl)) thm (con1i () ((con1.a (-> (-. ph) ps))) (-> (-. ps) ph) (con1.a ps negb syl a3i)) thm (con2i () ((con2.a (-> ph (-. ps)))) (-> ps (-. ph)) (ph nega con2.a syl a3i)) thm (con3i () ((con3.a (-> ph ps))) (-> (-. ps) (-. ph)) (ph nega con3.a syl con1i)) thm (pm2.36 () () (-> (-> ps ch) (-> (-> (-. ph) ps) (-> (-. ch) ph))) (ps ch (-. ph) imim2 ph ch con1 syl6)) thm (pm2.37 () () (-> (-> ps ch) (-> (-> (-. ps) ph) (-> (-. ph) ch))) (ps ph con1 ch imim1d com12)) thm (pm2.38 () () (-> (-> ps ch) (-> (-> (-. ps) ph) (-> (-. ch) ph))) (ps ch con3 ph imim1d)) thm (pm2.5 () () (-> (-. (-> ph ps)) (-> (-. ph) ps)) (ph ps pm2.21 con3i ps pm2.21d)) thm (pm2.51 () () (-> (-. (-> ph ps)) (-> ph (-. ps))) (ps ph ax-1 con3i ph a1d)) thm (pm2.52 () () (-> (-. (-> ph ps)) (-> (-. ph) (-. ps))) (ps ph ax-1 con3i (-. ph) a1d)) thm (pm2.521 () () (-> (-. (-> ph ps)) (-> ps ph)) (ph ps pm2.52 a3d)) thm (pm2.21ni () ((pm2.21ni.1 ph)) (-> (-. ph) ps) (pm2.21ni.1 (-. ps) a1i con1i)) thm (pm2.21nd () ((pm2.21nd.1 (-> ph ps))) (-> ph (-> (-. ps) ch)) (pm2.21nd.1 (-. ch) a1d con1d)) thm (mto () ((mto.1 (-. ps)) (mto.2 (-> ph ps))) (-. ph) (mto.1 mto.2 con3i ax-mp)) thm (mtoi () ((mtoi.1 (-. ch)) (mtoi.2 (-> ph (-> ps ch)))) (-> ph (-. ps)) (mtoi.1 mtoi.2 con3d mpi)) thm (mtod () ((mtod.1 (-> ph (-. ch))) (mtod.2 (-> ph (-> ps ch)))) (-> ph (-. ps)) (mtod.1 mtod.2 con3d mpd)) thm (mt2 () ((mt2.1 ps) (mt2.2 (-> ph (-. ps)))) (-. ph) (mt2.1 mt2.2 con2i ax-mp)) thm (mt2i () ((mt2i.1 ch) (mt2i.2 (-> ph (-> ps (-. ch))))) (-> ph (-. ps)) (mt2i.1 mt2i.2 con2d mpi)) thm (mt2d () ((mt2d.1 (-> ph ch)) (mt2d.2 (-> ph (-> ps (-. ch))))) (-> ph (-. ps)) (mt2d.1 mt2d.2 con2d mpd)) thm (mt3 () ((mt3.1 (-. ps)) (mt3.2 (-> (-. ph) ps))) ph (mt3.1 mt3.2 con1i ax-mp)) thm (mt3i () ((mt3i.1 (-. ch)) (mt3i.2 (-> ph (-> (-. ps) ch)))) (-> ph ps) (mt3i.1 mt3i.2 con1d mpi)) thm (mt3d () ((mt3d.1 (-> ph (-. ch))) (mt3d.2 (-> ph (-> (-. ps) ch)))) (-> ph ps) (mt3d.1 mt3d.2 con1d mpd)) thm (mt4d () ((mt4d.1 (-> ph ps)) (mt4d.2 (-> ph (-> (-. ch) (-. ps))))) (-> ph ch) (mt4d.1 mt4d.2 a3d mpd)) thm (nsyl () ((nsyl.1 (-> ph (-. ps))) (nsyl.2 (-> ch ps))) (-> ph (-. ch)) (nsyl.1 nsyl.2 con3i syl)) thm (nsyld () ((nsyld.1 (-> ph (-> ps (-. ch)))) (nsyld.2 (-> ph (-> ta ch)))) (-> ph (-> ps (-. ta))) (nsyld.1 nsyld.2 con3d syld)) thm (nsyl2 () ((nsyl2.1 (-> ph (-. ps))) (nsyl2.2 (-> (-. ch) ps))) (-> ph ch) (nsyl2.1 nsyl2.2 con1i syl)) thm (nsyl3 () ((nsyl3.1 (-> ph (-. ps))) (nsyl3.2 (-> ch ps))) (-> ch (-. ph)) (nsyl3.2 nsyl3.1 con2i syl)) thm (nsyl4 () ((nsyl4.1 (-> ph ps)) (nsyl4.2 (-> (-. ph) ch))) (-> (-. ch) ps) (nsyl4.2 con1i nsyl4.1 syl)) thm (nsyli () ((nsyli.1 (-> ph (-> ps ch))) (nsyli.2 (-> th (-. ch)))) (-> ph (-> th (-. ps))) (nsyli.1 con3d nsyli.2 syl5)) thm (pm3.2im () () (-> ph (-> ps (-. (-> ph (-. ps))))) (ph (-. ps) pm2.27 con2d)) thm (mth8 () () (-> ph (-> (-. ps) (-. (-> ph ps)))) (ph ps pm2.27 con3d)) thm (pm2.61 () () (-> (-> ph ps) (-> (-> (-. ph) ps) ps)) (ph ps (-. ps) imim2 ps pm2.18 syl6 ph ps con1 syl5)) thm (pm2.61i () ((pm2.61i.1 (-> ph ps)) (pm2.61i.2 (-> (-. ph) ps))) ps (pm2.61i.1 pm2.61i.2 ph ps pm2.61 mp2)) thm (pm2.61d () ((pm2.61d.1 (-> ph (-> ps ch))) (pm2.61d.2 (-> ph (-> (-. ps) ch)))) (-> ph ch) (pm2.61d.1 com12 pm2.61d.2 com12 pm2.61i)) thm (pm2.61d1 () ((pm2.61d1.1 (-> ph (-> ps ch))) (pm2.61d1.2 (-> (-. ps) ch))) (-> ph ch) (pm2.61d1.1 pm2.61d1.2 ph a1i pm2.61d)) thm (pm2.61d2 () ((pm2.61d2.1 (-> ph (-> (-. ps) ch))) (pm2.61d2.2 (-> ps ch))) (-> ph ch) (pm2.61d2.2 ph a1i pm2.61d2.1 pm2.61d)) thm (pm2.61ii () ((pm2.61ii.1 (-> (-. ph) (-> (-. ps) ch))) (pm2.61ii.2 (-> ph ch)) (pm2.61ii.3 (-> ps ch))) ch (pm2.61ii.2 pm2.61ii.1 pm2.61ii.3 pm2.61d2 pm2.61i)) thm (pm2.61nii () ((pm2.61nii.1 (-> ph (-> ps ch))) (pm2.61nii.2 (-> (-. ph) ch)) (pm2.61nii.3 (-> (-. ps) ch))) ch (pm2.61nii.1 com12 pm2.61nii.2 pm2.61d1 pm2.61nii.3 pm2.61i)) thm (pm2.61iii () ((pm2.61iii.1 (-> (-. ph) (-> (-. ps) (-> (-. ch) th)))) (pm2.61iii.2 (-> ph th)) (pm2.61iii.3 (-> ps th)) (pm2.61iii.4 (-> ch th))) th (pm2.61iii.2 (-. ch) a1d (-. ps) a1d pm2.61iii.1 pm2.61i pm2.61iii.3 pm2.61iii.4 pm2.61ii)) thm (pm2.6 () () (-> (-> (-. ph) ps) (-> (-> ph ps) ps)) (ph ps pm2.61 com12)) thm (pm2.65 () () (-> (-> ph ps) (-> (-> ph (-. ps)) (-. ph))) (ph ps pm3.2im a2i con2d)) thm (pm2.65i () ((pm2.65i.1 (-> ph ps)) (pm2.65i.2 (-> ph (-. ps)))) (-. ph) (pm2.65i.2 pm2.65i.1 nsyl ph pm2.01 ax-mp)) thm (pm2.65d () ((pm2.65d.1 (-> ph (-> ps ch))) (pm2.65d.2 (-> ph (-> ps (-. ch))))) (-> ph (-. ps)) (ps ch pm2.65 pm2.65d.1 pm2.65d.2 sylc)) thm (ja () ((ja.1 (-> (-. ph) ch)) (ja.2 (-> ps ch))) (-> (-> ph ps) ch) (ph ps pm2.27 ja.2 syl6 ja.1 (-> ph ps) a1d pm2.61i)) thm (jc () ((jc.1 (-> ph ps)) (jc.2 (-> ph ch))) (-> ph (-. (-> ps (-. ch)))) (ps ch pm3.2im jc.1 jc.2 sylc)) thm (pm3.26im () () (-> (-. (-> ph (-. ps))) ph) (ph (-. ps) pm2.21 con1i)) thm (pm3.27im () () (-> (-. (-> ph (-. ps))) ps) ((-. ps) ph ax-1 con1i)) thm (impt () () (-> (-> ph (-> ps ch)) (-> (-. (-> ph (-. ps))) ch)) (ps ch con3 ph imim2i com23 con1d)) thm (expt () () (-> (-> (-. (-> ph (-. ps))) ch) (-> ph (-> ps ch))) (ph ps pm3.2im ch imim1d com12)) thm (impi () ((impi.1 (-> ph (-> ps ch)))) (-> (-. (-> ph (-. ps))) ch) (impi.1 ph ps ch impt ax-mp)) thm (expi () ((expi.1 (-> (-. (-> ph (-. ps))) ch))) (-> ph (-> ps ch)) (expi.1 ph ps ch expt ax-mp)) thm (bijust () () (-. (-> (-> ph ph) (-. (-> ph ph)))) (ph id (-> ph ph) pm2.01 mt2)) thm (bi1 () () (-> (<-> ph ps) (-> ph ps)) (ph ps df-bi (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps)) pm3.26im ax-mp (-> ph ps) (-> ps ph) pm3.26im syl)) thm (bi2 () () (-> (<-> ph ps) (-> ps ph)) (ph ps df-bi (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps)) pm3.26im ax-mp (-> ph ps) (-> ps ph) pm3.27im syl)) thm (bi3 () () (-> (-> ph ps) (-> (-> ps ph) (<-> ph ps))) (ph ps df-bi (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps)) pm3.27im ax-mp expi)) thm (biimp () ((biimp.1 (<-> ph ps))) (-> ph ps) (biimp.1 ph ps bi1 ax-mp)) thm (biimpr () ((biimpr.1 (<-> ph ps))) (-> ps ph) (biimpr.1 ph ps bi2 ax-mp)) thm (biimpd () ((biimpd.1 (-> ph (<-> ps ch)))) (-> ph (-> ps ch)) (biimpd.1 ps ch bi1 syl)) thm (biimprd () ((biimpd.1 (-> ph (<-> ps ch)))) (-> ph (-> ch ps)) (biimpd.1 ps ch bi2 syl)) thm (biimpcd () ((biimpd.1 (-> ph (<-> ps ch)))) (-> ps (-> ph ch)) (biimpd.1 biimpd com12)) thm (biimprcd () ((biimpd.1 (-> ph (<-> ps ch)))) (-> ch (-> ph ps)) (biimpd.1 biimprd com12)) thm (impbi () ((impbi.1 (-> ph ps)) (impbi.2 (-> ps ph))) (<-> ph ps) (impbi.1 impbi.2 ph ps bi3 mp2)) thm (bii () () (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (ph ps bi1 ph ps bi2 jc ph ps bi3 impi impbi)) thm (biigb () () (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (ph ps df-bi ch th ax-1 (-. (-> (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps))))) (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))))) (-> (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps)))))) ax-1 (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph)))) df-bi (-. (-> (-> (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps)))))) (-. (-> (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps))))) (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))))))) (-. (-. (-> ch (-> th ch)))) ax-1 ax-mp (-. (-> ch (-> th ch))) (-> (-> (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps)))))) (-. (-> (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps))))) (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph)))))))) ax-3 ax-mp (-> (-> (-> (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps)))))) (-. (-> (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps))))) (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph)))))))) (-. (-> ch (-> th ch)))) (-. (-> (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps))))) (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))))) ax-1 ax-mp (-. (-> (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps))))) (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))))) (-> (-> (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps)))))) (-. (-> (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps))))) (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph)))))))) (-. (-> ch (-> th ch))) ax-2 ax-mp ax-mp (-> (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps))))) (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph)))))) (-> ch (-> th ch)) ax-3 ax-mp ax-mp ax-mp)) thm (bi2.04 () () (<-> (-> ph (-> ps ch)) (-> ps (-> ph ch))) (ph ps ch pm2.04 ps ph ch pm2.04 impbi)) thm (pm4.13 () () (<-> ph (-. (-. ph))) (ph negb ph nega impbi)) thm (pm4.8 () () (<-> (-> ph (-. ph)) (-. ph)) (ph pm2.01 (-. ph) ph ax-1 impbi)) thm (pm4.81 () () (<-> (-> (-. ph) ph) ph) (ph pm2.18 ph ph pm2.24 impbi)) thm (pm4.1 () () (<-> (-> ph ps) (-> (-. ps) (-. ph))) (ph ps con3 ps ph ax-3 impbi)) thm (bi2.03 () () (<-> (-> ph (-. ps)) (-> ps (-. ph))) (ph ps con2 ps ph con2 impbi)) thm (bi2.15 () () (<-> (-> (-. ph) ps) (-> (-. ps) ph)) (ph ps con1 ps ph con1 impbi)) thm (pm5.4 () () (<-> (-> ph (-> ph ps)) (-> ph ps)) (ph ps pm2.43 (-> ph ps) ph ax-1 impbi)) thm (imdi () () (<-> (-> ph (-> ps ch)) (-> (-> ph ps) (-> ph ch))) (ph ps ch ax-2 ph ps ch pm2.86 impbi)) thm (pm5.41 () () (<-> (-> (-> ph ps) (-> ph ch)) (-> ph (-> ps ch))) (ph ps ch pm2.86 ph ps ch ax-2 impbi)) thm (pm4.2 () () (<-> ph ph) (ph id ph id impbi)) thm (pm4.2i () () (-> ph (<-> ps ps)) (ps pm4.2 ph a1i)) thm (bicomi () ((bicom.1 (<-> ph ps))) (<-> ps ph) (bicom.1 biimpr bicom.1 biimp impbi)) thm (bitr () ((bitr.1 (<-> ph ps)) (bitr.2 (<-> ps ch))) (<-> ph ch) (bitr.1 biimp bitr.2 biimp syl bitr.2 biimpr bitr.1 biimpr syl impbi)) thm (bitr2 () ((bitr2.1 (<-> ph ps)) (bitr2.2 (<-> ps ch))) (<-> ch ph) (bitr2.1 bitr2.2 bitr bicomi)) thm (bitr3 () ((bitr3.1 (<-> ps ph)) (bitr3.2 (<-> ps ch))) (<-> ph ch) (bitr3.1 bicomi bitr3.2 bitr)) thm (bitr4 () ((bitr4.1 (<-> ph ps)) (bitr4.2 (<-> ch ps))) (<-> ph ch) (bitr4.1 bitr4.2 bicomi bitr)) thm (3bitr () ((3bitr.1 (<-> ph ps)) (3bitr.2 (<-> ps ch)) (3bitr.3 (<-> ch th))) (<-> ph th) (3bitr.1 3bitr.2 3bitr.3 bitr bitr)) thm (3bitrr () ((3bitr.1 (<-> ph ps)) (3bitr.2 (<-> ps ch)) (3bitr.3 (<-> ch th))) (<-> th ph) (3bitr.3 3bitr.1 3bitr.2 bitr2 bitr3)) thm (3bitr2 () ((3bitr2.1 (<-> ph ps)) (3bitr2.2 (<-> ch ps)) (3bitr2.3 (<-> ch th))) (<-> ph th) (3bitr2.1 3bitr2.2 bitr4 3bitr2.3 bitr)) thm (3bitr2r () ((3bitr2.1 (<-> ph ps)) (3bitr2.2 (<-> ch ps)) (3bitr2.3 (<-> ch th))) (<-> th ph) (3bitr2.1 3bitr2.2 bitr4 3bitr2.3 bitr2)) thm (3bitr3 () ((3bitr3.1 (<-> ph ps)) (3bitr3.2 (<-> ph ch)) (3bitr3.3 (<-> ps th))) (<-> ch th) (3bitr3.2 3bitr3.1 bitr3 3bitr3.3 bitr)) thm (3bitr3r () ((3bitr3.1 (<-> ph ps)) (3bitr3.2 (<-> ph ch)) (3bitr3.3 (<-> ps th))) (<-> th ch) (3bitr3.3 3bitr3.1 3bitr3.2 bitr3 bitr3)) thm (3bitr4 () ((3bitr4.1 (<-> ph ps)) (3bitr4.2 (<-> ch ph)) (3bitr4.3 (<-> th ps))) (<-> ch th) (3bitr4.2 3bitr4.1 3bitr4.3 bitr4 bitr)) thm (3bitr4r () ((3bitr4.1 (<-> ph ps)) (3bitr4.2 (<-> ch ph)) (3bitr4.3 (<-> th ps))) (<-> th ch) (3bitr4.2 3bitr4.1 3bitr4.3 bitr4 bitr2)) thm (imbi2i () ((bi.a (<-> ph ps))) (<-> (-> ch ph) (-> ch ps)) (bi.a biimp ch imim2i bi.a biimpr ch imim2i impbi)) thm (imbi1i () ((bi.a (<-> ph ps))) (<-> (-> ph ch) (-> ps ch)) (bi.a biimpr ch imim1i bi.a biimp ch imim1i impbi)) thm (negbii () ((bi.a (<-> ph ps))) (<-> (-. ph) (-. ps)) (bi.a biimpr con3i bi.a biimp con3i impbi)) thm (imbi12i () ((imbi12i.1 (<-> ph ps)) (imbi12i.2 (<-> ch th))) (<-> (-> ph ch) (-> ps th)) (imbi12i.2 ph imbi2i imbi12i.1 th imbi1i bitr)) thm (mpbi () ((mpbi.min ph) (mpbi.maj (<-> ph ps))) ps (mpbi.min mpbi.maj biimp ax-mp)) thm (mpbir () ((mpbir.min ps) (mpbir.maj (<-> ph ps))) ph (mpbir.min mpbir.maj biimpr ax-mp)) thm (mtbi () ((mtbi.1 (-. ph)) (mtbi.2 (<-> ph ps))) (-. ps) (mtbi.1 mtbi.2 negbii mpbi)) thm (mtbir () ((mtbir.1 (-. ps)) (mtbir.2 (<-> ph ps))) (-. ph) (mtbir.1 mtbir.2 negbii mpbir)) thm (mpbii () ((mpbii.min ps) (mpbii.maj (-> ph (<-> ps ch)))) (-> ph ch) (mpbii.min mpbii.maj biimpd mpi)) thm (mpbiri () ((mpbiri.min ch) (mpbiri.maj (-> ph (<-> ps ch)))) (-> ph ps) (mpbiri.min mpbiri.maj biimprd mpi)) thm (mpbid () ((mpbid.min (-> ph ps)) (mpbid.maj (-> ph (<-> ps ch)))) (-> ph ch) (mpbid.min mpbid.maj biimpd mpd)) thm (mpbird () ((mpbird.min (-> ph ch)) (mpbird.maj (-> ph (<-> ps ch)))) (-> ph ps) (mpbird.min mpbird.maj biimprd mpd)) thm (a1bi () ((a1bi.1 ph)) (<-> ps (-> ph ps)) (ps ph ax-1 a1bi.1 ph ps pm2.27 ax-mp impbi)) thm (sylib () ((sylib.1 (-> ph ps)) (sylib.2 (<-> ps ch))) (-> ph ch) (sylib.1 sylib.2 biimp syl)) thm (sylbi () ((sylbi.1 (<-> ph ps)) (sylbi.2 (-> ps ch))) (-> ph ch) (sylbi.1 biimp sylbi.2 syl)) thm (sylibr () ((sylibr.1 (-> ph ps)) (sylibr.2 (<-> ch ps))) (-> ph ch) (sylibr.1 sylibr.2 biimpr syl)) thm (sylbir () ((sylbir.1 (<-> ps ph)) (sylbir.2 (-> ps ch))) (-> ph ch) (sylbir.1 biimpr sylbir.2 syl)) thm (sylibd () ((sylibd.1 (-> ph (-> ps ch))) (sylibd.2 (-> ph (<-> ch th)))) (-> ph (-> ps th)) (sylibd.1 sylibd.2 biimpd syld)) thm (sylbid () ((sylbid.1 (-> ph (<-> ps ch))) (sylbid.2 (-> ph (-> ch th)))) (-> ph (-> ps th)) (sylbid.1 biimpd sylbid.2 syld)) thm (sylibrd () ((sylibrd.1 (-> ph (-> ps ch))) (sylibrd.2 (-> ph (<-> th ch)))) (-> ph (-> ps th)) (sylibrd.1 sylibrd.2 biimprd syld)) thm (sylbird () ((sylbird.1 (-> ph (<-> ch ps))) (sylbird.2 (-> ph (-> ch th)))) (-> ph (-> ps th)) (sylbird.1 biimprd sylbird.2 syld)) thm (syl5ib () ((syl5ib.1 (-> ph (-> ps ch))) (syl5ib.2 (<-> th ps))) (-> ph (-> th ch)) (syl5ib.1 syl5ib.2 biimp syl5)) thm (syl5ibr () ((syl5ibr.1 (-> ph (-> ps ch))) (syl5ibr.2 (<-> ps th))) (-> ph (-> th ch)) (syl5ibr.1 syl5ibr.2 biimpr syl5)) thm (syl5bi () ((syl5bi.1 (-> ph (<-> ps ch))) (syl5bi.2 (-> th ps))) (-> ph (-> th ch)) (syl5bi.1 biimpd syl5bi.2 syl5)) thm (syl5bir () ((syl5bir.1 (-> ph (<-> ps ch))) (syl5bir.2 (-> th ch))) (-> ph (-> th ps)) (syl5bir.1 biimprd syl5bir.2 syl5)) thm (syl6ib () ((syl6ib.1 (-> ph (-> ps ch))) (syl6ib.2 (<-> ch th))) (-> ph (-> ps th)) (syl6ib.1 syl6ib.2 biimp syl6)) thm (syl6ibr () ((syl6ibr.1 (-> ph (-> ps ch))) (syl6ibr.2 (<-> th ch))) (-> ph (-> ps th)) (syl6ibr.1 syl6ibr.2 biimpr syl6)) thm (syl6bi () ((syl6bi.1 (-> ph (<-> ps ch))) (syl6bi.2 (-> ch th))) (-> ph (-> ps th)) (syl6bi.1 biimpd syl6bi.2 syl6)) thm (syl6bir () ((syl6bir.1 (-> ph (<-> ch ps))) (syl6bir.2 (-> ch th))) (-> ph (-> ps th)) (syl6bir.1 biimprd syl6bir.2 syl6)) thm (syl7ib () ((syl7ib.1 (-> ph (-> ps (-> ch th)))) (syl7ib.2 (<-> ta ch))) (-> ph (-> ps (-> ta th))) (syl7ib.1 syl7ib.2 biimp syl7)) thm (syl8ib () ((syl8ib.1 (-> ph (-> ps (-> ch th)))) (syl8ib.2 (<-> th ta))) (-> ph (-> ps (-> ch ta))) (syl8ib.1 syl8ib.2 biimp syl8)) thm (3imtr3 () ((3imtr3.1 (-> ph ps)) (3imtr3.2 (<-> ph ch)) (3imtr3.3 (<-> ps th))) (-> ch th) (3imtr3.2 3imtr3.1 sylbir 3imtr3.3 sylib)) thm (3imtr4 () ((3imtr4.1 (-> ph ps)) (3imtr4.2 (<-> ch ph)) (3imtr4.3 (<-> th ps))) (-> ch th) (3imtr4.2 3imtr4.1 sylbi 3imtr4.3 sylibr)) thm (con1bii () ((con1bii.1 (<-> (-. ph) ps))) (<-> (-. ps) ph) (con1bii.1 biimp con1i con1bii.1 biimpr con2i impbi)) thm (con2bii () ((con2bii.1 (<-> ph (-. ps)))) (<-> ps (-. ph)) (con2bii.1 bicomi con1bii bicomi)) thm (pm4.64 () () (<-> (-> (-. ph) ps) (\/ ph ps)) (ph ps df-or bicomi)) thm (pm2.54 () () (-> (-> (-. ph) ps) (\/ ph ps)) (ph ps df-or biimpr)) thm (pm4.63 () () (<-> (-. (-> ph (-. ps))) (/\ ph ps)) (ph ps df-an bicomi)) thm (dfor2 () () (<-> (\/ ph ps) (-> (-> ph ps) ps)) (ph ps df-or ph ps pm2.6 ph ps pm2.21 ps imim1i impbi bitr)) thm (ori () ((ori.1 (\/ ph ps))) (-> (-. ph) ps) (ori.1 ph ps df-or mpbi)) thm (orri () ((orri.1 (-> (-. ph) ps))) (\/ ph ps) (orri.1 ph ps df-or mpbir)) thm (ord () ((ord.1 (-> ph (\/ ps ch)))) (-> ph (-> (-. ps) ch)) (ord.1 ps ch df-or sylib)) thm (orrd () ((orrd.1 (-> ph (-> (-. ps) ch)))) (-> ph (\/ ps ch)) (orrd.1 ps ch df-or sylibr)) thm (imor () () (<-> (-> ph ps) (\/ (-. ph) ps)) (ph pm4.13 ps imbi1i (-. ph) ps df-or bitr4)) thm (pm4.62 () () (<-> (-> ph (-. ps)) (\/ (-. ph) (-. ps))) (ph (-. ps) imor)) thm (pm4.66 () () (<-> (-> (-. ph) (-. ps)) (\/ ph (-. ps))) (ph (-. ps) pm4.64)) thm (iman () () (<-> (-> ph ps) (-. (/\ ph (-. ps)))) (ps pm4.13 ph imbi2i ph (-. ps) df-an con2bii bitr)) thm (annim () () (<-> (/\ ph (-. ps)) (-. (-> ph ps))) (ph ps iman con2bii)) thm (pm4.61 () () (<-> (-. (-> ph ps)) (/\ ph (-. ps))) (ph ps annim bicomi)) thm (pm4.65 () () (<-> (-. (-> (-. ph) ps)) (/\ (-. ph) (-. ps))) ((-. ph) ps pm4.61)) thm (pm4.67 () () (<-> (-. (-> (-. ph) (-. ps))) (/\ (-. ph) ps)) ((-. ph) ps pm4.63)) thm (imnan () () (<-> (-> ph (-. ps)) (-. (/\ ph ps))) (ph ps df-an con2bii)) thm (oridm () () (<-> (\/ ph ph) ph) (ph ph df-or ph ph pm2.24 ph pm2.18 impbi bitr4)) thm (pm4.25 () () (<-> ph (\/ ph ph)) (ph oridm bicomi)) thm (pm1.2 () () (-> (\/ ph ph) ph) (ph oridm biimp)) thm (orcom () () (<-> (\/ ph ps) (\/ ps ph)) (ph ps bi2.15 ph ps df-or ps ph df-or 3bitr4)) thm (pm1.4 () () (-> (\/ ph ps) (\/ ps ph)) (ph ps orcom biimp)) thm (pm2.62 () () (-> (\/ ph ps) (-> (-> ph ps) ps)) (ph ps dfor2 biimp)) thm (pm2.621 () () (-> (-> ph ps) (-> (\/ ph ps) ps)) (ph ps pm2.62 com12)) thm (pm2.68 () () (-> (-> (-> ph ps) ps) (\/ ph ps)) (ph ps dfor2 biimpr)) thm (orel1 () () (-> (-. ph) (-> (\/ ph ps) ps)) (ph ps df-or biimp com12)) thm (orel2 () () (-> (-. ph) (-> (\/ ps ph) ps)) (ph ps orel1 ps ph orcom syl5ib)) thm (pm2.25 () () (\/ ph (-> (\/ ph ps) ps)) (ph ps orel1 orri)) thm (pm2.53 () () (-> (\/ ph ps) (-> (-. ph) ps)) (ph ps orel1 com12)) thm (orbi2i () ((orbi2i.1 (<-> ph ps))) (<-> (\/ ch ph) (\/ ch ps)) (orbi2i.1 (-. ch) imbi2i ch ph df-or ch ps df-or 3bitr4)) thm (orbi1i () ((orbi2i.1 (<-> ph ps))) (<-> (\/ ph ch) (\/ ps ch)) (ph ch orcom orbi2i.1 ch orbi2i ch ps orcom 3bitr)) thm (orbi12i () ((orbi12i.1 (<-> ph ps)) (orbi12i.2 (<-> ch th))) (<-> (\/ ph ch) (\/ ps th)) (orbi12i.2 ph orbi2i orbi12i.1 th orbi1i bitr)) thm (or12 () () (<-> (\/ ph (\/ ps ch)) (\/ ps (\/ ph ch))) ((-. ps) (-. ph) ch bi2.04 ph ch df-or (-. ps) imbi2i ps ch df-or (-. ph) imbi2i 3bitr4r ph (\/ ps ch) df-or ps (\/ ph ch) df-or 3bitr4)) thm (pm1.5 () () (-> (\/ ph (\/ ps ch)) (\/ ps (\/ ph ch))) (ph ps ch or12 biimp)) thm (orass () () (<-> (\/ (\/ ph ps) ch) (\/ ph (\/ ps ch))) (ch ph ps or12 (\/ ph ps) ch orcom ps ch orcom ph orbi2i 3bitr4)) thm (pm2.31 () () (-> (\/ ph (\/ ps ch)) (\/ (\/ ph ps) ch)) (ph ps ch orass biimpr)) thm (pm2.32 () () (-> (\/ (\/ ph ps) ch) (\/ ph (\/ ps ch))) (ph ps ch orass biimp)) thm (or23 () () (<-> (\/ (\/ ph ps) ch) (\/ (\/ ph ch) ps)) (ps ch orcom ph orbi2i ph ps ch orass ph ch ps orass 3bitr4)) thm (or4 () () (<-> (\/ (\/ ph ps) (\/ ch th)) (\/ (\/ ph ch) (\/ ps th))) (ps ch th or12 ph orbi2i ph ps (\/ ch th) orass ph ch (\/ ps th) orass 3bitr4)) thm (or42 () () (<-> (\/ (\/ ph ps) (\/ ch th)) (\/ (\/ ph ch) (\/ th ps))) (ph ps ch th or4 ps th orcom (\/ ph ch) orbi2i bitr)) thm (orordi () () (<-> (\/ ph (\/ ps ch)) (\/ (\/ ph ps) (\/ ph ch))) (ph oridm (\/ ps ch) orbi1i ph ph ps ch or4 bitr3)) thm (orordir () () (<-> (\/ (\/ ph ps) ch) (\/ (\/ ph ch) (\/ ps ch))) (ch oridm (\/ ph ps) orbi2i ph ps ch ch or4 bitr3)) thm (olc () () (-> ph (\/ ps ph)) (ph (-. ps) ax-1 orrd)) thm (pm2.07 () () (-> ph (\/ ph ph)) (ph ph olc)) thm (orc () () (-> ph (\/ ph ps)) (ph ps pm2.24 orrd)) thm (orci () ((orci.1 (-> (\/ ph ps) ch))) (-> ph ch) (ph ps orc orci.1 syl)) thm (olci () ((olci.1 (-> (\/ ph ps) ch))) (-> ps ch) (ps ph olc olci.1 syl)) thm (pm2.45 () () (-> (-. (\/ ph ps)) (-. ph)) (ph ps orc con3i)) thm (pm2.46 () () (-> (-. (\/ ph ps)) (-. ps)) (ps ph olc con3i)) thm (pm2.47 () () (-> (-. (\/ ph ps)) (\/ (-. ph) ps)) (ph ps pm2.45 (-. ph) ps orc syl)) thm (pm2.48 () () (-> (-. (\/ ph ps)) (\/ ph (-. ps))) (ph ps pm2.46 (-. ph) a1d orrd)) thm (pm2.49 () () (-> (-. (\/ ph ps)) (\/ (-. ph) (-. ps))) (ph ps pm2.46 (-. (-. ph)) a1d orrd)) thm (pm2.67 () () (-> (-> (\/ ph ps) ps) (-> ph ps)) (ph ps orc ps imim1i)) thm (pm3.2 () () (-> ph (-> ps (/\ ph ps))) (ph ps df-an biimpr expi)) thm (pm3.21 () () (-> ph (-> ps (/\ ps ph))) (ps ph pm3.2 com12)) thm (pm3.2i () ((pm3.2i.1 ph) (pm3.2i.2 ps)) (/\ ph ps) (pm3.2i.1 pm3.2i.2 ph ps pm3.2 mp2)) thm (pm3.37 () () (-> (-> (/\ ph ps) ch) (-> (/\ ph (-. ch)) (-. ps))) (ps ph pm3.21 ch imim1d com12 ph ch iman syl6ib con2d)) thm (pm3.43i () () (-> (-> ph ps) (-> (-> ph ch) (-> ph (/\ ps ch)))) (ps ch pm3.2 ph imim2i a2d)) thm (jca () ((jca.1 (-> ph ps)) (jca.2 (-> ph ch))) (-> ph (/\ ps ch)) (jca.1 jca.2 jc ps ch df-an sylibr)) thm (jcai () ((jcai.1 (-> ph ps)) (jcai.2 (-> ph (-> ps ch)))) (-> ph (/\ ps ch)) (jcai.1 jcai.1 jcai.2 mpd jca)) thm (jctl () ((jctl.1 ps)) (-> ph (/\ ps ph)) (jctl.1 ph a1i ph id jca)) thm (jctr () ((jctr.1 ps)) (-> ph (/\ ph ps)) (ph id jctr.1 ph a1i jca)) thm (jctil () ((jctil.1 (-> ph ps)) (jctil.2 ch)) (-> ph (/\ ch ps)) (jctil.2 ph a1i jctil.1 jca)) thm (jctir () ((jctir.1 (-> ph ps)) (jctir.2 ch)) (-> ph (/\ ps ch)) (jctir.1 jctir.2 ph a1i jca)) thm (ancl () () (-> (-> ph ps) (-> ph (/\ ph ps))) (ph ps pm3.2 a2i)) thm (ancr () () (-> (-> ph ps) (-> ph (/\ ps ph))) (ph ps pm3.21 a2i)) thm (ancli () ((ancli.1 (-> ph ps))) (-> ph (/\ ph ps)) (ph id ancli.1 jca)) thm (ancri () ((ancri.1 (-> ph ps))) (-> ph (/\ ps ph)) (ancri.1 ph id jca)) thm (ancld () ((ancld.1 (-> ph (-> ps ch)))) (-> ph (-> ps (/\ ps ch))) (ancld.1 ps ch ancl syl)) thm (ancrd () ((ancrd.1 (-> ph (-> ps ch)))) (-> ph (-> ps (/\ ch ps))) (ancrd.1 ps ch ancr syl)) thm (anc2l () () (-> (-> ph (-> ps ch)) (-> ph (-> ps (/\ ph ch)))) (ph ch pm3.2 ps imim2d a2i)) thm (anc2r () () (-> (-> ph (-> ps ch)) (-> ph (-> ps (/\ ch ph)))) (ph ch pm3.21 ps imim2d a2i)) thm (anc2li () ((anc2li.1 (-> ph (-> ps ch)))) (-> ph (-> ps (/\ ph ch))) (anc2li.1 ph ps ch anc2l ax-mp)) thm (anc2ri () ((anc2ri.1 (-> ph (-> ps ch)))) (-> ph (-> ps (/\ ch ph))) (anc2ri.1 ph ps ch anc2r ax-mp)) thm (anor () () (<-> (/\ ph ps) (-. (\/ (-. ph) (-. ps)))) (ph ps df-an ph (-. ps) imor negbii bitr)) thm (ianor () () (<-> (-. (/\ ph ps)) (\/ (-. ph) (-. ps))) (ph ps anor negbii (\/ (-. ph) (-. ps)) pm4.13 bitr4)) thm (ioran () () (<-> (-. (\/ ph ps)) (/\ (-. ph) (-. ps))) (ph pm4.13 ps pm4.13 orbi12i negbii (-. ph) (-. ps) anor bitr4)) thm (pm4.52 () () (<-> (/\ ph (-. ps)) (-. (\/ (-. ph) ps))) (ph (-. ps) anor ps pm4.13 (-. ph) orbi2i negbii bitr4)) thm (pm4.53 () () (<-> (-. (/\ ph (-. ps))) (\/ (-. ph) ps)) (ph ps pm4.52 con2bii bicomi)) thm (pm4.54 () () (<-> (/\ (-. ph) ps) (-. (\/ ph (-. ps)))) ((-. ph) ps anor ph pm4.13 (-. ps) orbi1i negbii bitr4)) thm (pm4.55 () () (<-> (-. (/\ (-. ph) ps)) (\/ ph (-. ps))) (ph ps pm4.54 con2bii bicomi)) thm (pm4.56 () () (<-> (/\ (-. ph) (-. ps)) (-. (\/ ph ps))) (ph ps ioran bicomi)) thm (oran () () (<-> (\/ ph ps) (-. (/\ (-. ph) (-. ps)))) ((\/ ph ps) pm4.13 ph ps ioran negbii bitr)) thm (pm4.57 () () (<-> (-. (/\ (-. ph) (-. ps))) (\/ ph ps)) (ph ps oran bicomi)) thm (pm3.1 () () (-> (/\ ph ps) (-. (\/ (-. ph) (-. ps)))) (ph ps anor biimp)) thm (pm3.11 () () (-> (-. (\/ (-. ph) (-. ps))) (/\ ph ps)) (ph ps anor biimpr)) thm (pm3.12 () () (\/ (\/ (-. ph) (-. ps)) (/\ ph ps)) (ph ps pm3.11 orri)) thm (pm3.13 () () (-> (-. (/\ ph ps)) (\/ (-. ph) (-. ps))) (ph ps pm3.11 con1i)) thm (pm3.14 () () (-> (\/ (-. ph) (-. ps)) (-. (/\ ph ps))) (ph ps pm3.1 con2i)) thm (pm3.26 () () (-> (/\ ph ps) ph) (ph ps df-an ph ps pm3.26im sylbi)) thm (pm3.26i () ((pm3.26i.1 (/\ ph ps))) ph (pm3.26i.1 ph ps pm3.26 ax-mp)) thm (pm3.26d () ((pm3.26d.1 (-> ph (/\ ps ch)))) (-> ph ps) (pm3.26d.1 ps ch pm3.26 syl)) thm (pm3.26bd () ((pm3.26bd.1 (<-> ph (/\ ps ch)))) (-> ph ps) (pm3.26bd.1 biimp pm3.26d)) thm (pm3.27 () () (-> (/\ ph ps) ps) (ph ps df-an ph ps pm3.27im sylbi)) thm (pm3.27i () ((pm3.27i.1 (/\ ph ps))) ps (pm3.27i.1 ph ps pm3.27 ax-mp)) thm (pm3.27d () ((pm3.27d.1 (-> ph (/\ ps ch)))) (-> ph ch) (pm3.27d.1 ps ch pm3.27 syl)) thm (pm3.27bd () ((pm3.27bd.1 (<-> ph (/\ ps ch)))) (-> ph ch) (pm3.27bd.1 biimp pm3.27d)) thm (pm3.41 () () (-> (-> ph ch) (-> (/\ ph ps) ch)) (ph ps pm3.26 ch imim1i)) thm (pm3.42 () () (-> (-> ps ch) (-> (/\ ph ps) ch)) (ph ps pm3.27 ch imim1i)) thm (anclb () () (<-> (-> ph ps) (-> ph (/\ ph ps))) (ph ps ancl ph ps pm3.27 ph imim2i impbi)) thm (ancrb () () (<-> (-> ph ps) (-> ph (/\ ps ph))) (ph ps ancr ps ph pm3.26 ph imim2i impbi)) thm (pm3.4 () () (-> (/\ ph ps) (-> ph ps)) (ph ps pm3.27 ph a1d)) thm (pm4.45im () () (<-> ph (/\ ph (-> ps ph))) (ph ps ax-1 ancli ph (-> ps ph) pm3.26 impbi)) thm (anim12i () ((anim12i.1 (-> ph ps)) (anim12i.2 (-> ch th))) (-> (/\ ph ch) (/\ ps th)) (ph ch pm3.26 anim12i.1 syl ph ch pm3.27 anim12i.2 syl jca)) thm (anim1i () ((anim1i.1 (-> ph ps))) (-> (/\ ph ch) (/\ ps ch)) (anim1i.1 ch id anim12i)) thm (anim2i () ((anim1i.1 (-> ph ps))) (-> (/\ ch ph) (/\ ch ps)) (ch id anim1i.1 anim12i)) thm (orim12i () ((orim12i.1 (-> ph ps)) (orim12i.2 (-> ch th))) (-> (\/ ph ch) (\/ ps th)) (orim12i.1 con3i orim12i.2 con3i anim12i con3i ph ch oran ps th oran 3imtr4)) thm (orim1i () ((orim1i.1 (-> ph ps))) (-> (\/ ph ch) (\/ ps ch)) (orim1i.1 ch id orim12i)) thm (orim2i () ((orim1i.1 (-> ph ps))) (-> (\/ ch ph) (\/ ch ps)) (ch id orim1i.1 orim12i)) thm (pm2.3 () () (-> (\/ ph (\/ ps ch)) (\/ ph (\/ ch ps))) (ps ch pm1.4 ph orim2i)) thm (jao () () (-> (-> ph ps) (-> (-> ch ps) (-> (\/ ph ch) ps))) (ph ps con3 (-. ps) (-. ph) (-. ch) pm3.43i ps (/\ (-. ph) (-. ch)) con1 syl6 ph ch oran syl7ib ch ps con3 syl5 syl)) thm (jaoi () ((jaoi.1 (-> ph ps)) (jaoi.2 (-> ch ps))) (-> (\/ ph ch) ps) (jaoi.1 jaoi.2 ph ps ch jao mp2)) thm (pm2.41 () () (-> (\/ ps (\/ ph ps)) (\/ ph ps)) (ps ph olc (\/ ph ps) id jaoi)) thm (pm2.42 () () (-> (\/ (-. ph) (-> ph ps)) (-> ph ps)) (ph ps pm2.21 (-> ph ps) id jaoi)) thm (pm2.4 () () (-> (\/ ph (\/ ph ps)) (\/ ph ps)) (ph ps orc (\/ ph ps) id jaoi)) thm (pm4.44 () () (<-> ph (\/ ph (/\ ph ps))) (ph (/\ ph ps) orc ph id ph ps pm3.26 jaoi impbi)) thm (pm5.63 () () (<-> (\/ ph ps) (\/ ph (/\ (-. ph) ps))) (ph ps pm2.53 ancld orrd ph ps pm2.24 (-. ph) ps pm3.4 jaoi orrd impbi)) thm (impexp () () (<-> (-> (/\ ph ps) ch) (-> ph (-> ps ch))) (ph ps df-an ch imbi1i ph ps ch expt ph ps ch impt impbi bitr)) thm (pm3.3 () () (-> (-> (/\ ph ps) ch) (-> ph (-> ps ch))) (ph ps ch impexp biimp)) thm (pm3.31 () () (-> (-> ph (-> ps ch)) (-> (/\ ph ps) ch)) (ph ps ch impexp biimpr)) thm (imp () ((imp.1 (-> ph (-> ps ch)))) (-> (/\ ph ps) ch) (ph ps df-an imp.1 impi sylbi)) thm (impcom () ((imp.1 (-> ph (-> ps ch)))) (-> (/\ ps ph) ch) (imp.1 com12 imp)) thm (pm4.14 () () (<-> (-> (/\ ph ps) ch) (-> (/\ ph (-. ch)) (-. ps))) (ph ps ch impexp ph ps ch bi2.04 bitr ph ch iman ps imbi2i ps (/\ ph (-. ch)) bi2.03 3bitr)) thm (pm4.15 () () (<-> (-> (/\ ph ps) (-. ch)) (-> (/\ ps ch) (-. ph))) (ph ps (-. ch) impexp ps ch imnan ph imbi2i ph (/\ ps ch) bi2.03 3bitr)) thm (pm4.78 () () (<-> (\/ (-> ph ps) (-> ph ch)) (-> ph (\/ ps ch))) (ph (-. ps) (-> ph ch) impexp ph ps annim (-> ph ch) imbi1i (-. ps) ph ch bi2.04 ph imbi2i ph (-> (-. ps) ch) pm5.4 bitr 3bitr3 (-> ph ps) (-> ph ch) df-or ps ch df-or ph imbi2i 3bitr4)) thm (pm4.79 () () (<-> (\/ (-> ps ph) (-> ch ph)) (-> (/\ ps ch) ph)) ((-. ph) (-. ps) (-. ch) pm4.78 ps ch ianor (-. ph) imbi2i bitr4 ps ph pm4.1 ch ph pm4.1 orbi12i (/\ ps ch) ph pm4.1 3bitr4)) thm (pm4.87 () () (/\ (/\ (<-> (-> (/\ ph ps) ch) (-> ph (-> ps ch))) (<-> (-> ph (-> ps ch)) (-> ps (-> ph ch)))) (<-> (-> ps (-> ph ch)) (-> (/\ ps ph) ch))) (ph ps ch impexp ph ps ch bi2.04 pm3.2i ps ph ch pm3.31 ps ph ch pm3.3 impbi pm3.2i)) thm (pm3.33 () () (-> (/\ (-> ph ps) (-> ps ch)) (-> ph ch)) (ph ps ch imim1 imp)) thm (pm3.34 () () (-> (/\ (-> ps ch) (-> ph ps)) (-> ph ch)) (ps ch ph imim2 imp)) thm (pm3.35 () () (-> (/\ ph (-> ph ps)) ps) (ph ps pm2.27 imp)) thm (pm5.31 () () (-> (/\ ch (-> ph ps)) (-> ph (/\ ps ch))) (ch ps pm3.21 ph imim2d imp)) thm (imp3a () ((imp3.1 (-> ph (-> ps (-> ch th))))) (-> ph (-> (/\ ps ch) th)) (imp3.1 ps ch th impexp sylibr)) thm (imp31 () ((imp3.1 (-> ph (-> ps (-> ch th))))) (-> (/\ (/\ ph ps) ch) th) (imp3.1 imp imp)) thm (imp32 () ((imp3.1 (-> ph (-> ps (-> ch th))))) (-> (/\ ph (/\ ps ch)) th) (imp3.1 imp3a imp)) thm (imp4a () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ph (-> ps (-> (/\ ch th) ta))) (imp4.1 ch th ta impexp syl6ibr)) thm (imp4b () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ ph ps) (-> (/\ ch th) ta)) (imp4.1 imp4a imp)) thm (imp4c () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ph (-> (/\ (/\ ps ch) th) ta)) (imp4.1 imp3a imp3a)) thm (imp4d () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ph (-> (/\ ps (/\ ch th)) ta)) (imp4.1 imp4a imp3a)) thm (imp41 () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ (/\ (/\ ph ps) ch) th) ta) (imp4.1 imp imp31)) thm (imp42 () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ (/\ ph (/\ ps ch)) th) ta) (imp4.1 imp32 imp)) thm (imp43 () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ (/\ ph ps) (/\ ch th)) ta) (imp4.1 imp4b imp)) thm (imp44 () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ ph (/\ (/\ ps ch) th)) ta) (imp4.1 imp4c imp)) thm (imp45 () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ ph (/\ ps (/\ ch th))) ta) (imp4.1 imp4d imp)) thm (ex () ((exp.1 (-> (/\ ph ps) ch))) (-> ph (-> ps ch)) (ph ps df-an exp.1 sylbir expi)) thm (expcom () ((exp.1 (-> (/\ ph ps) ch))) (-> ps (-> ph ch)) (exp.1 ex com12)) thm (exp3a () ((exp3a.1 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> ps (-> ch th))) (exp3a.1 ps ch th impexp sylib)) thm (exp31 () ((exp31.1 (-> (/\ (/\ ph ps) ch) th))) (-> ph (-> ps (-> ch th))) (exp31.1 ex ex)) thm (exp32 () ((exp32.1 (-> (/\ ph (/\ ps ch)) th))) (-> ph (-> ps (-> ch th))) (exp32.1 ex exp3a)) thm (exp4a () ((exp4a.1 (-> ph (-> ps (-> (/\ ch th) ta))))) (-> ph (-> ps (-> ch (-> th ta)))) (exp4a.1 ch th ta impexp syl6ib)) thm (exp4b () ((exp4b.1 (-> (/\ ph ps) (-> (/\ ch th) ta)))) (-> ph (-> ps (-> ch (-> th ta)))) (exp4b.1 exp3a ex)) thm (exp4c () ((exp4c.1 (-> ph (-> (/\ (/\ ps ch) th) ta)))) (-> ph (-> ps (-> ch (-> th ta)))) (exp4c.1 exp3a exp3a)) thm (exp4d () ((exp4d.1 (-> ph (-> (/\ ps (/\ ch th)) ta)))) (-> ph (-> ps (-> ch (-> th ta)))) (exp4d.1 exp3a exp4a)) thm (exp41 () ((exp41.1 (-> (/\ (/\ (/\ ph ps) ch) th) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (exp41.1 ex exp31)) thm (exp42 () ((exp42.1 (-> (/\ (/\ ph (/\ ps ch)) th) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (exp42.1 exp31 exp3a)) thm (exp43 () ((exp43.1 (-> (/\ (/\ ph ps) (/\ ch th)) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (exp43.1 ex exp4b)) thm (exp44 () ((exp44.1 (-> (/\ ph (/\ (/\ ps ch) th)) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (exp44.1 exp32 exp3a)) thm (exp45 () ((exp45.1 (-> (/\ ph (/\ ps (/\ ch th))) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (exp45.1 exp32 exp4a)) thm (impac () ((impac.1 (-> ph (-> ps ch)))) (-> (/\ ph ps) (/\ ch ps)) (impac.1 ancrd imp)) thm (adantl () ((adantl.1 (-> ph ps))) (-> (/\ ch ph) ps) (adantl.1 ch a1i imp)) thm (adantr () ((adantr.1 (-> ph ps))) (-> (/\ ph ch) ps) (adantr.1 ch a1d imp)) thm (adantld () ((adantld.1 (-> ph (-> ps ch)))) (-> ph (-> (/\ th ps) ch)) (adantld.1 th a1d imp3a)) thm (adantrd () ((adantrd.1 (-> ph (-> ps ch)))) (-> ph (-> (/\ ps th) ch)) (adantrd.1 ps th pm3.26 syl5)) thm (adantll () ((adant2.1 (-> (/\ ph ps) ch))) (-> (/\ (/\ th ph) ps) ch) (adant2.1 ex th adantl imp)) thm (adantlr () ((adant2.1 (-> (/\ ph ps) ch))) (-> (/\ (/\ ph th) ps) ch) (adant2.1 ex th adantr imp)) thm (adantrl () ((adant2.1 (-> (/\ ph ps) ch))) (-> (/\ ph (/\ th ps)) ch) (adant2.1 ex th adantld imp)) thm (adantrr () ((adant2.1 (-> (/\ ph ps) ch))) (-> (/\ ph (/\ ps th)) ch) (adant2.1 ex th adantrd imp)) thm (adantlll () ((adantl2.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ (/\ ta ph) ps) ch) th) (adantl2.1 exp31 ta a1i imp41)) thm (adantllr () ((adantl2.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ (/\ ph ta) ps) ch) th) (adantl2.1 exp31 ta a1d imp41)) thm (adantlrl () ((adantl2.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ ph (/\ ta ps)) ch) th) (adantl2.1 exp31 ta a1d imp42)) thm (adantlrr () ((adantl2.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ ph (/\ ps ta)) ch) th) (adantl2.1 exp31 ta a1dd imp42)) thm (adantrll () ((adantr2.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\ (/\ ta ps) ch)) th) (adantr2.1 exp32 ta a1d imp44)) thm (adantrlr () ((adantr2.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\ (/\ ps ta) ch)) th) (adantr2.1 exp32 ta a1dd imp44)) thm (adantrrl () ((adantr2.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\ ps (/\ ta ch))) th) (adantr2.1 exp32 ta a1dd imp45)) thm (adantrrr () ((adantr2.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\ ps (/\ ch ta))) th) (adantr2.1 ta a1d exp32 imp45)) thm (ad2antrr () ((ad2ant.1 (-> ph ps))) (-> (/\ (/\ ph ch) th) ps) (ad2ant.1 ch adantr th adantr)) thm (ad2antlr () ((ad2ant.1 (-> ph ps))) (-> (/\ (/\ ch ph) th) ps) (ad2ant.1 ch adantl th adantr)) thm (ad2antrl () ((ad2ant.1 (-> ph ps))) (-> (/\ ch (/\ ph th)) ps) (ad2ant.1 th adantr ch adantl)) thm (ad2antll () ((ad2ant.1 (-> ph ps))) (-> (/\ ch (/\ th ph)) ps) (ad2ant.1 th adantl ch adantl)) thm (ad2ant2l () ((ad2ant2.1 (-> (/\ ph ps) ch))) (-> (/\ (/\ th ph) (/\ ta ps)) ch) (ad2ant2.1 ta adantrl th adantll)) thm (ad2ant2r () ((ad2ant2.1 (-> (/\ ph ps) ch))) (-> (/\ (/\ ph th) (/\ ps ta)) ch) (ad2ant2.1 ta adantrr th adantlr)) thm (biimpa () ((biimpa.1 (-> ph (<-> ps ch)))) (-> (/\ ph ps) ch) (biimpa.1 biimpd imp)) thm (biimpar () ((biimpa.1 (-> ph (<-> ps ch)))) (-> (/\ ph ch) ps) (biimpa.1 biimprd imp)) thm (biimpac () ((biimpa.1 (-> ph (<-> ps ch)))) (-> (/\ ps ph) ch) (biimpa.1 biimpcd imp)) thm (biimparc () ((biimpa.1 (-> ph (<-> ps ch)))) (-> (/\ ch ph) ps) (biimpa.1 biimprcd imp)) thm (jaob () () (<-> (-> (\/ ph ch) ps) (/\ (-> ph ps) (-> ch ps))) (ph ch orc ps imim1i ch ph olc ps imim1i jca ph ps ch jao imp impbi)) thm (pm4.77 () () (<-> (/\ (-> ps ph) (-> ch ph)) (-> (\/ ps ch) ph)) (ps ch ph jaob bicomi)) thm (jaod () ((jaod.1 (-> ph (-> ps ch))) (jaod.2 (-> ph (-> th ch)))) (-> ph (-> (\/ ps th) ch)) (ps ch th jao jaod.1 jaod.2 sylc)) thm (jaoian () ((jaoian.1 (-> (/\ ph ps) ch)) (jaoian.2 (-> (/\ th ps) ch))) (-> (/\ (\/ ph th) ps) ch) (jaoian.1 ex jaoian.2 ex jaoi imp)) thm (jaodan () ((jaodan.1 (-> (/\ ph ps) ch)) (jaodan.2 (-> (/\ ph th) ch))) (-> (/\ ph (\/ ps th)) ch) (jaodan.1 ex jaodan.2 ex jaod imp)) thm (jaao () ((jaao.1 (-> ph (-> ps ch))) (jaao.2 (-> th (-> ta ch)))) (-> (/\ ph th) (-> (\/ ps ta) ch)) (jaao.1 th adantr jaao.2 ph adantl jaod)) thm (pm2.63 () () (-> (\/ ph ps) (-> (\/ (-. ph) ps) ps)) (ph ps pm2.53 (\/ ph ps) ps idd jaod)) thm (pm2.64 () () (-> (\/ ph ps) (-> (\/ ph (-. ps)) ph)) (ph (\/ ph ps) ax-1 ps ph orel2 jaoi com12)) thm (pm3.44 () () (-> (/\ (-> ps ph) (-> ch ph)) (-> (\/ ps ch) ph)) (ps ch ph jaob biimpr)) thm (pm4.43 () () (<-> ph (/\ (\/ ph ps) (\/ ph (-. ps)))) (ph ps orc ph (-. ps) orc jca ph ps pm2.64 imp impbi)) thm (anidm () () (<-> (/\ ph ph) ph) (ph ph pm3.26 ph ph pm3.2 pm2.43i impbi)) thm (pm4.24 () () (<-> ph (/\ ph ph)) (ph anidm bicomi)) thm (anidms () ((anidms.1 (-> (/\ ph ph) ps))) (-> ph ps) (anidms.1 ex pm2.43i)) thm (ancom () () (<-> (/\ ph ps) (/\ ps ph)) (ph ps pm3.27 ph ps pm3.26 jca ps ph pm3.27 ps ph pm3.26 jca impbi)) thm (ancoms () ((ancoms.1 (-> (/\ ph ps) ch))) (-> (/\ ps ph) ch) (ps ph ancom ancoms.1 sylbi)) thm (ancomsd () ((ancomsd.1 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> (/\ ch ps) th)) (ancomsd.1 ch ps ancom syl5ib)) thm (pm3.22 () () (-> (/\ ph ps) (/\ ps ph)) (ph ps ancom biimp)) thm (anass () () (<-> (/\ (/\ ph ps) ch) (/\ ph (/\ ps ch))) (ph ps (-. ch) impexp ps ch imnan ph imbi2i bitr negbii (/\ ph ps) ch df-an ph (/\ ps ch) df-an 3bitr4)) thm (anasss () ((anasss.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ ph (/\ ps ch)) th) (anasss.1 exp31 imp32)) thm (anassrs () ((anassrs.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ (/\ ph ps) ch) th) (anassrs.1 exp32 imp31)) thm (imdistan () () (<-> (-> ph (-> ps ch)) (-> (/\ ph ps) (/\ ph ch))) (ph ps ch anc2l imp3a ph ch pm3.27 (/\ ph ps) imim2i exp3a impbi)) thm (imdistani () ((imdistani.1 (-> ph (-> ps ch)))) (-> (/\ ph ps) (/\ ph ch)) (imdistani.1 anc2li imp)) thm (imdistanri () ((imdistanri.1 (-> ph (-> ps ch)))) (-> (/\ ps ph) (/\ ch ph)) (imdistanri.1 com12 impac)) thm (imdistand () ((imdistand.1 (-> ph (-> ps (-> ch th))))) (-> ph (-> (/\ ps ch) (/\ ps th))) (imdistand.1 ps ch th imdistan sylib)) thm (pm5.3 () () (<-> (-> (/\ ph ps) ch) (-> (/\ ph ps) (/\ ph ch))) (ph ps ch pm3.3 imdistand ph ch pm3.27 (/\ ph ps) imim2i impbi)) thm (pm5.61 () () (<-> (/\ (\/ ph ps) (-. ps)) (/\ ph (-. ps))) (ps ph orel2 imdistanri ph ps orc (-. ps) anim1i impbi)) thm (sylan () ((sylan.1 (-> (/\ ph ps) ch)) (sylan.2 (-> th ph))) (-> (/\ th ps) ch) (sylan.2 sylan.1 ex syl imp)) thm (sylanb () ((sylan.1 (-> (/\ ph ps) ch)) (sylanb.2 (<-> th ph))) (-> (/\ th ps) ch) (sylan.1 sylanb.2 biimp sylan)) thm (sylanbr () ((sylan.1 (-> (/\ ph ps) ch)) (sylanbr.2 (<-> ph th))) (-> (/\ th ps) ch) (sylan.1 sylanbr.2 biimpr sylan)) thm (sylan2 () ((sylan.1 (-> (/\ ph ps) ch)) (sylan2.2 (-> th ps))) (-> (/\ ph th) ch) (sylan.1 ex sylan2.2 syl5 imp)) thm (sylan2b () ((sylan.1 (-> (/\ ph ps) ch)) (sylan2b.2 (<-> th ps))) (-> (/\ ph th) ch) (sylan.1 sylan2b.2 biimp sylan2)) thm (sylan2br () ((sylan.1 (-> (/\ ph ps) ch)) (sylan2br.2 (<-> ps th))) (-> (/\ ph th) ch) (sylan.1 sylan2br.2 biimpr sylan2)) thm (syl2an () ((sylan.1 (-> (/\ ph ps) ch)) (syl2an.2 (-> th ph)) (syl2an.3 (-> ta ps))) (-> (/\ th ta) ch) (sylan.1 syl2an.2 sylan syl2an.3 sylan2)) thm (syl2anb () ((sylan.1 (-> (/\ ph ps) ch)) (syl2anb.2 (<-> th ph)) (syl2anb.3 (<-> ta ps))) (-> (/\ th ta) ch) (sylan.1 syl2anb.2 sylanb syl2anb.3 sylan2b)) thm (syl2anbr () ((sylan.1 (-> (/\ ph ps) ch)) (syl2anbr.2 (<-> ph th)) (syl2anbr.3 (<-> ps ta))) (-> (/\ th ta) ch) (sylan.1 syl2anbr.2 sylanbr syl2anbr.3 sylan2br)) thm (syland () ((syland.1 (-> ph (-> (/\ ps ch) th))) (syland.2 (-> ph (-> ta ps)))) (-> ph (-> (/\ ta ch) th)) (syland.2 syland.1 exp3a syld imp3a)) thm (sylan2d () ((sylan2d.1 (-> ph (-> (/\ ps ch) th))) (sylan2d.2 (-> ph (-> ta ch)))) (-> ph (-> (/\ ps ta) th)) (sylan2d.1 ancomsd sylan2d.2 syland ancomsd)) var (wff et) thm (syl2and () ((syl2and.1 (-> ph (-> (/\ ps ch) th))) (syl2and.2 (-> ph (-> ta ps))) (syl2and.3 (-> ph (-> et ch)))) (-> ph (-> (/\ ta et) th)) (syl2and.1 syl2and.3 sylan2d syl2and.2 syland)) thm (sylanl1 () ((sylanl1.1 (-> (/\ (/\ ph ps) ch) th)) (sylanl1.2 (-> ta ph))) (-> (/\ (/\ ta ps) ch) th) (sylanl1.1 sylanl1.2 ps anim1i sylan)) thm (sylanl2 () ((sylanl2.1 (-> (/\ (/\ ph ps) ch) th)) (sylanl2.2 (-> ta ps))) (-> (/\ (/\ ph ta) ch) th) (sylanl2.1 sylanl2.2 ph anim2i sylan)) thm (sylanr1 () ((sylanr1.1 (-> (/\ ph (/\ ps ch)) th)) (sylanr1.2 (-> ta ps))) (-> (/\ ph (/\ ta ch)) th) (sylanr1.1 sylanr1.2 ch anim1i sylan2)) thm (sylanr2 () ((sylanr2.1 (-> (/\ ph (/\ ps ch)) th)) (sylanr2.2 (-> ta ch))) (-> (/\ ph (/\ ps ta)) th) (sylanr2.1 sylanr2.2 ps anim2i sylan2)) thm (sylani () ((sylani.1 (-> ph (-> (/\ ps ch) th))) (sylani.2 (-> ta ps))) (-> ph (-> (/\ ta ch) th)) (sylani.1 sylani.2 ph a1i syland)) thm (sylan2i () ((sylan2i.1 (-> ph (-> (/\ ps ch) th))) (sylan2i.2 (-> ta ch))) (-> ph (-> (/\ ps ta) th)) (sylan2i.1 sylan2i.2 ph a1i sylan2d)) thm (syl2ani () ((syl2ani.1 (-> ph (-> (/\ ps ch) th))) (syl2ani.2 (-> ta ps)) (syl2ani.3 (-> et ch))) (-> ph (-> (/\ ta et) th)) (syl2ani.1 syl2ani.3 sylan2i syl2ani.2 sylani)) thm (syldan () ((syldan.1 (-> (/\ ph ps) ch)) (syldan.2 (-> (/\ ph ch) th))) (-> (/\ ph ps) th) (syldan.1 ex syldan.2 ex syld imp)) thm (sylan9 () ((sylan9.1 (-> ph (-> ps ch))) (sylan9.2 (-> th (-> ch ta)))) (-> (/\ ph th) (-> ps ta)) (sylan9.1 th adantr sylan9.2 ph adantl syld)) thm (sylan9r () ((sylan9r.1 (-> ph (-> ps ch))) (sylan9r.2 (-> th (-> ch ta)))) (-> (/\ th ph) (-> ps ta)) (sylan9r.1 sylan9r.2 syl9r imp)) thm (sylanc () ((sylanc.1 (-> (/\ ph ps) ch)) (sylanc.2 (-> th ph)) (sylanc.3 (-> th ps))) (-> th ch) (sylanc.1 ex sylanc.2 sylanc.3 sylc)) thm (sylancb () ((sylancb.1 (-> (/\ ph ps) ch)) (sylancb.2 (<-> th ph)) (sylancb.3 (<-> th ps))) (-> th ch) (sylancb.1 sylancb.2 sylancb.3 syl2anb anidms)) thm (sylancbr () ((sylancbr.1 (-> (/\ ph ps) ch)) (sylancbr.2 (<-> ph th)) (sylancbr.3 (<-> ps th))) (-> th ch) (sylancbr.1 sylancbr.2 sylancbr.3 syl2anbr anidms)) thm (pm2.61ian () ((pm2.61ian.1 (-> (/\ ph ps) ch)) (pm2.61ian.2 (-> (/\ (-. ph) ps) ch))) (-> ps ch) (pm2.61ian.1 ex pm2.61ian.2 ex pm2.61i)) thm (pm2.61dan () ((pm2.61dan.1 (-> (/\ ph ps) ch)) (pm2.61dan.2 (-> (/\ ph (-. ps)) ch))) (-> ph ch) (pm2.61dan.1 ex pm2.61dan.2 ex pm2.61d)) thm (condan () ((condan.1 (-> (/\ ph (-. ps)) ch)) (condan.2 (-> (/\ ph (-. ps)) (-. ch)))) (-> ph ps) (condan.1 ex condan.2 ex pm2.65d ps nega syl)) thm (abai () () (<-> (/\ ph ps) (/\ ph (-> ph ps))) (ph ps pm3.26 ph ps pm3.4 jca ph (-> ph ps) pm3.26 ph ps pm3.35 jca impbi)) thm (anbi2i () ((bi.aa (<-> ph ps))) (<-> (/\ ch ph) (/\ ch ps)) (bi.aa biimp ch anim2i bi.aa biimpr ch anim2i impbi)) thm (anbi1i () ((bi.aa (<-> ph ps))) (<-> (/\ ph ch) (/\ ps ch)) (ph ch ancom bi.aa ch anbi2i ch ps ancom 3bitr)) thm (anbi12i () ((bian12.1 (<-> ph ps)) (bian12.2 (<-> ch th))) (<-> (/\ ph ch) (/\ ps th)) (bian12.1 ch anbi1i bian12.2 ps anbi2i bitr)) thm (pm5.53 () () (<-> (-> (\/ (\/ ph ps) ch) th) (/\ (/\ (-> ph th) (-> ps th)) (-> ch th))) ((\/ ph ps) ch th jaob ph ps th jaob (-> ch th) anbi1i bitr)) thm (an12 () () (<-> (/\ ph (/\ ps ch)) (/\ ps (/\ ph ch))) (ph ps ancom ch anbi1i ph ps ch anass ps ph ch anass 3bitr3)) thm (an23 () () (<-> (/\ (/\ ph ps) ch) (/\ (/\ ph ch) ps)) (ps ch ancom ph anbi2i ph ps ch anass ph ch ps anass 3bitr4)) thm (an1s () ((an1s.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ps (/\ ph ch)) th) (ps ph ch an12 an1s.1 sylbi)) thm (ancom2s () ((an1s.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\ ch ps)) th) (an1s.1 exp32 com23 imp32)) thm (ancom13s () ((an1s.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ch (/\ ps ph)) th) (an1s.1 exp32 com13 imp32)) thm (an1rs () ((an1rs.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ ph ch) ps) th) (ph ch ps an23 an1rs.1 sylbi)) thm (ancom1s () ((an1rs.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ ps ph) ch) th) (an1rs.1 exp31 com12 imp31)) thm (ancom31s () ((an1rs.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ ch ps) ph) th) (an1rs.1 exp31 com13 imp31)) thm (anabs1 () () (<-> (/\ (/\ ph ps) ph) (/\ ph ps)) ((/\ ph ps) ph pm3.26 ph ps pm3.26 ancli impbi)) thm (anabs5 () () (<-> (/\ ph (/\ ph ps)) (/\ ph ps)) ((/\ ph ps) ph ancom ph ps anabs1 bitr3)) thm (anabs7 () () (<-> (/\ ps (/\ ph ps)) (/\ ph ps)) (ps (/\ ph ps) pm3.27 ph ps pm3.27 ancri impbi)) thm (anabsi5 () ((anabsi5.1 (-> ph (-> (/\ ph ps) ch)))) (-> (/\ ph ps) ch) (anabsi5.1 ps adantr pm2.43i)) thm (anabsi6 () ((anabsi6.1 (-> ph (-> (/\ ps ph) ch)))) (-> (/\ ph ps) ch) (anabsi6.1 ancomsd anabsi5)) thm (anabsi7 () ((anabsi7.1 (-> ps (-> (/\ ph ps) ch)))) (-> (/\ ph ps) ch) (anabsi7.1 exp3a pm2.43b imp)) thm (anabsi8 () ((anabsi8.1 (-> ps (-> (/\ ps ph) ch)))) (-> (/\ ph ps) ch) (anabsi8.1 anabsi5 ancoms)) thm (anabss1 () ((anabss1.1 (-> (/\ (/\ ph ps) ph) ch))) (-> (/\ ph ps) ch) (anabss1.1 ps adantrr anidms)) thm (anabss3 () ((anabss3.1 (-> (/\ (/\ ph ps) ps) ch))) (-> (/\ ph ps) ch) (anabss3.1 ph adantrl anidms)) thm (anabss4 () ((anabss4.1 (-> (/\ (/\ ps ph) ps) ch))) (-> (/\ ph ps) ch) (anabss4.1 anabss1 ancoms)) thm (anabss5 () ((anabss5.1 (-> (/\ ph (/\ ph ps)) ch))) (-> (/\ ph ps) ch) (anabss5.1 ps adantlr anidms)) thm (anabss7 () ((anabss7.1 (-> (/\ ps (/\ ph ps)) ch))) (-> (/\ ph ps) ch) (anabss7.1 ex anabsi7)) thm (anabsan () ((anabsan.1 (-> (/\ (/\ ph ph) ps) ch))) (-> (/\ ph ps) ch) (anabsan.1 an1rs anabss1)) thm (anabsan2 () ((anabsan2.1 (-> (/\ ph (/\ ps ps)) ch))) (-> (/\ ph ps) ch) (anabsan2.1 anassrs anabss3)) thm (an4 () () (<-> (/\ (/\ ph ps) (/\ ch th)) (/\ (/\ ph ch) (/\ ps th))) (ps ch th an12 ph anbi2i ph ps (/\ ch th) anass ph ch (/\ ps th) anass 3bitr4)) thm (an42 () () (<-> (/\ (/\ ph ps) (/\ ch th)) (/\ (/\ ph ch) (/\ th ps))) (ph ps ch th an4 ps th ancom (/\ ph ch) anbi2i bitr)) thm (an4s () ((an4s.1 (-> (/\ (/\ ph ps) (/\ ch th)) ta))) (-> (/\ (/\ ph ch) (/\ ps th)) ta) (ph ch ps th an4 an4s.1 sylbi)) thm (an42s () ((an41r3s.1 (-> (/\ (/\ ph ps) (/\ ch th)) ta))) (-> (/\ (/\ ph ch) (/\ th ps)) ta) (ph ps ch th an42 an41r3s.1 sylbir)) thm (anandi () () (<-> (/\ ph (/\ ps ch)) (/\ (/\ ph ps) (/\ ph ch))) (ph anidm (/\ ps ch) anbi1i ph ph ps ch an4 bitr3)) thm (anandir () () (<-> (/\ (/\ ph ps) ch) (/\ (/\ ph ch) (/\ ps ch))) (ch anidm (/\ ph ps) anbi2i ph ps ch ch an4 bitr3)) thm (anandis () ((anandis.1 (-> (/\ (/\ ph ps) (/\ ph ch)) ta))) (-> (/\ ph (/\ ps ch)) ta) (anandis.1 an4s anabsan)) thm (anandirs () ((anandirs.1 (-> (/\ (/\ ph ch) (/\ ps ch)) ta))) (-> (/\ (/\ ph ps) ch) ta) (anandirs.1 an4s anabsan2)) thm (bi () () (<-> (<-> ph ps) (/\ (-> ph ps) (-> ps ph))) (ph ps bii (-> ph ps) (-> ps ph) df-an bitr4)) thm (impbid () ((impbid.1 (-> ph (-> ps ch))) (impbid.2 (-> ph (-> ch ps)))) (-> ph (<-> ps ch)) (impbid.1 impbid.2 jca ps ch bi sylibr)) thm (bicom () () (<-> (<-> ph ps) (<-> ps ph)) ((-> ph ps) (-> ps ph) ancom ph ps bi ps ph bi 3bitr4)) thm (bicomd () ((bicomd.1 (-> ph (<-> ps ch)))) (-> ph (<-> ch ps)) (bicomd.1 ps ch bicom sylib)) thm (pm4.11 () () (<-> (<-> ph ps) (<-> (-. ph) (-. ps))) (ph ps pm4.1 ps ph pm4.1 anbi12i ph ps bi (-. ps) (-. ph) bi 3bitr4 (-. ps) (-. ph) bicom bitr)) thm (con4bii () ((con4bii.1 (<-> (-. ph) (-. ps)))) (<-> ph ps) (con4bii.1 ph ps pm4.11 mpbir)) thm (con4bid () ((con4bid.1 (-> ph (<-> (-. ps) (-. ch))))) (-> ph (<-> ps ch)) (con4bid.1 ps ch pm4.11 sylibr)) thm (con2bi () () (<-> (<-> ph (-. ps)) (<-> ps (-. ph))) (ph ps bi2.03 ps ph bi2.15 anbi12i ph (-. ps) bi ps (-. ph) bi 3bitr4)) thm (con2bid () ((con2bid.1 (-> ph (<-> ps (-. ch))))) (-> ph (<-> ch (-. ps))) (con2bid.1 ch ps con2bi sylibr)) thm (con1bid () ((con1bid.1 (-> ph (<-> (-. ps) ch)))) (-> ph (<-> (-. ch) ps)) (con1bid.1 bicomd con2bid bicomd)) thm (bitrd () ((bitrd.1 (-> ph (<-> ps ch))) (bitrd.2 (-> ph (<-> ch th)))) (-> ph (<-> ps th)) (bitrd.1 biimpd bitrd.2 sylibd bitrd.2 biimprd bitrd.1 sylibrd impbid)) thm (bitr2d () ((bitr2d.1 (-> ph (<-> ps ch))) (bitr2d.2 (-> ph (<-> ch th)))) (-> ph (<-> th ps)) (bitr2d.1 bitr2d.2 bitrd bicomd)) thm (bitr3d () ((bitr3d.1 (-> ph (<-> ps ch))) (bitr3d.2 (-> ph (<-> ps th)))) (-> ph (<-> ch th)) (bitr3d.1 bicomd bitr3d.2 bitrd)) thm (bitr4d () ((bitr4d.1 (-> ph (<-> ps ch))) (bitr4d.2 (-> ph (<-> th ch)))) (-> ph (<-> ps th)) (bitr4d.1 bitr4d.2 bicomd bitrd)) thm (syl5bb () ((syl5bb.1 (-> ph (<-> ps ch))) (syl5bb.2 (<-> th ps))) (-> ph (<-> th ch)) (syl5bb.2 ph a1i syl5bb.1 bitrd)) thm (syl5rbb () ((syl5rbb.1 (-> ph (<-> ps ch))) (syl5rbb.2 (<-> th ps))) (-> ph (<-> ch th)) (syl5rbb.1 syl5rbb.2 syl5bb bicomd)) thm (syl5bbr () ((syl5bbr.1 (-> ph (<-> ps ch))) (syl5bbr.2 (<-> ps th))) (-> ph (<-> th ch)) (syl5bbr.1 syl5bbr.2 bicomi syl5bb)) thm (syl5rbbr () ((syl5rbbr.1 (-> ph (<-> ps ch))) (syl5rbbr.2 (<-> ps th))) (-> ph (<-> ch th)) (syl5rbbr.1 syl5rbbr.2 bicomi syl5rbb)) thm (syl6bb () ((syl6bb.1 (-> ph (<-> ps ch))) (syl6bb.2 (<-> ch th))) (-> ph (<-> ps th)) (syl6bb.1 syl6bb.2 ph a1i bitrd)) thm (syl6rbb () ((syl6rbb.1 (-> ph (<-> ps ch))) (syl6rbb.2 (<-> ch th))) (-> ph (<-> th ps)) (syl6rbb.1 syl6rbb.2 syl6bb bicomd)) thm (syl6bbr () ((syl6bbr.1 (-> ph (<-> ps ch))) (syl6bbr.2 (<-> th ch))) (-> ph (<-> ps th)) (syl6bbr.1 syl6bbr.2 bicomi syl6bb)) thm (syl6rbbr () ((syl6rbbr.1 (-> ph (<-> ps ch))) (syl6rbbr.2 (<-> th ch))) (-> ph (<-> th ps)) (syl6rbbr.1 syl6rbbr.2 bicomi syl6rbb)) thm (sylan9bb () ((sylan9bb.1 (-> ph (<-> ps ch))) (sylan9bb.2 (-> th (<-> ch ta)))) (-> (/\ ph th) (<-> ps ta)) (sylan9bb.1 th adantr sylan9bb.2 ph adantl bitrd)) thm (sylan9bbr () ((sylan9bbr.1 (-> ph (<-> ps ch))) (sylan9bbr.2 (-> th (<-> ch ta)))) (-> (/\ th ph) (<-> ps ta)) (sylan9bbr.1 sylan9bbr.2 sylan9bb ancoms)) thm (3imtr3d () ((3imtr3d.1 (-> ph (-> ps ch))) (3imtr3d.2 (-> ph (<-> ps th))) (3imtr3d.3 (-> ph (<-> ch ta)))) (-> ph (-> th ta)) (3imtr3d.2 3imtr3d.1 3imtr3d.3 sylibd sylbird)) thm (3imtr4d () ((3imtr4d.1 (-> ph (-> ps ch))) (3imtr4d.2 (-> ph (<-> th ps))) (3imtr4d.3 (-> ph (<-> ta ch)))) (-> ph (-> th ta)) (3imtr4d.2 3imtr4d.1 3imtr4d.3 sylibrd sylbid)) thm (3bitrd () ((3bitrd.1 (-> ph (<-> ps ch))) (3bitrd.2 (-> ph (<-> ch th))) (3bitrd.3 (-> ph (<-> th ta)))) (-> ph (<-> ps ta)) (3bitrd.1 3bitrd.2 bitrd 3bitrd.3 bitrd)) thm (3bitrrd () ((3bitrd.1 (-> ph (<-> ps ch))) (3bitrd.2 (-> ph (<-> ch th))) (3bitrd.3 (-> ph (<-> th ta)))) (-> ph (<-> ta ps)) (3bitrd.3 3bitrd.1 3bitrd.2 bitr2d bitr3d)) thm (3bitr2d () ((3bitr2d.1 (-> ph (<-> ps ch))) (3bitr2d.2 (-> ph (<-> th ch))) (3bitr2d.3 (-> ph (<-> th ta)))) (-> ph (<-> ps ta)) (3bitr2d.1 3bitr2d.2 bitr4d 3bitr2d.3 bitrd)) thm (3bitr2rd () ((3bitr2d.1 (-> ph (<-> ps ch))) (3bitr2d.2 (-> ph (<-> th ch))) (3bitr2d.3 (-> ph (<-> th ta)))) (-> ph (<-> ta ps)) (3bitr2d.1 3bitr2d.2 bitr4d 3bitr2d.3 bitr2d)) thm (3bitr3d () ((3bitr3d.1 (-> ph (<-> ps ch))) (3bitr3d.2 (-> ph (<-> ps th))) (3bitr3d.3 (-> ph (<-> ch ta)))) (-> ph (<-> th ta)) (3bitr3d.2 3bitr3d.1 bitr3d 3bitr3d.3 bitrd)) thm (3bitr3rd () ((3bitr3d.1 (-> ph (<-> ps ch))) (3bitr3d.2 (-> ph (<-> ps th))) (3bitr3d.3 (-> ph (<-> ch ta)))) (-> ph (<-> ta th)) (3bitr3d.3 3bitr3d.1 3bitr3d.2 bitr3d bitr3d)) thm (3bitr4d () ((3bitr4d.1 (-> ph (<-> ps ch))) (3bitr4d.2 (-> ph (<-> th ps))) (3bitr4d.3 (-> ph (<-> ta ch)))) (-> ph (<-> th ta)) (3bitr4d.2 3bitr4d.1 3bitr4d.3 bitr4d bitrd)) thm (3bitr4rd () ((3bitr4d.1 (-> ph (<-> ps ch))) (3bitr4d.2 (-> ph (<-> th ps))) (3bitr4d.3 (-> ph (<-> ta ch)))) (-> ph (<-> ta th)) (3bitr4d.3 3bitr4d.1 bitr4d 3bitr4d.2 bitr4d)) thm (3imtr3g () ((3imtr3g.1 (-> ph (-> ps ch))) (3imtr3g.2 (<-> ps th)) (3imtr3g.3 (<-> ch ta))) (-> ph (-> th ta)) (3imtr3g.1 imp 3imtr3g.2 ph anbi2i 3imtr3g.3 3imtr3 ex)) thm (3imtr4g () ((3imtr4g.1 (-> ph (-> ps ch))) (3imtr4g.2 (<-> th ps)) (3imtr4g.3 (<-> ta ch))) (-> ph (-> th ta)) (3imtr4g.1 3imtr4g.2 bicomi 3imtr4g.3 bicomi 3imtr3g)) thm (3bitr3g () ((3bitr3g.1 (-> ph (<-> ps ch))) (3bitr3g.2 (<-> ps th)) (3bitr3g.3 (<-> ch ta))) (-> ph (<-> th ta)) (3bitr3g.1 3bitr3g.2 syl5bbr 3bitr3g.3 syl6bb)) thm (3bitr4g () ((3bitr4g.1 (-> ph (<-> ps ch))) (3bitr4g.2 (<-> th ps)) (3bitr4g.3 (<-> ta ch))) (-> ph (<-> th ta)) (3bitr4g.1 3bitr4g.2 syl5bb 3bitr4g.3 syl6bbr)) thm (prth () () (-> (/\ (-> ph ps) (-> ch th)) (-> (/\ ph ch) (/\ ps th))) (ps th pm3.2 ch imim2d ph imim2i com23 imp4b)) thm (pm3.48 () () (-> (/\ (-> ph ps) (-> ch th)) (-> (\/ ph ch) (\/ ps th))) ((-> ph ps) (-> ch th) pm3.26 con3d (-> ph ps) (-> ch th) pm3.27 imim12d ph ch df-or ps th df-or 3imtr4g)) thm (anim12d () ((anim12d.1 (-> ph (-> ps ch))) (anim12d.2 (-> ph (-> th ta)))) (-> ph (-> (/\ ps th) (/\ ch ta))) (ps ch th ta prth anim12d.1 anim12d.2 sylanc)) thm (anim1d () ((anim1d.1 (-> ph (-> ps ch)))) (-> ph (-> (/\ ps th) (/\ ch th))) (anim1d.1 ph th idd anim12d)) thm (anim2d () ((anim1d.1 (-> ph (-> ps ch)))) (-> ph (-> (/\ th ps) (/\ th ch))) (ph th idd anim1d.1 anim12d)) thm (pm3.45 () () (-> (-> ph ps) (-> (/\ ph ch) (/\ ps ch))) ((-> ph ps) id ch anim1d)) thm (im2anan9 () ((im2an9.1 (-> ph (-> ps ch))) (im2an9.2 (-> th (-> ta et)))) (-> (/\ ph th) (-> (/\ ps ta) (/\ ch et))) (im2an9.1 th adantr im2an9.2 ph adantl anim12d)) thm (im2anan9r () ((im2an9.1 (-> ph (-> ps ch))) (im2an9.2 (-> th (-> ta et)))) (-> (/\ th ph) (-> (/\ ps ta) (/\ ch et))) (im2an9.1 th adantl im2an9.2 ph adantr anim12d)) thm (orim12d () ((orim12d.1 (-> ph (-> ps ch))) (orim12d.2 (-> ph (-> th ta)))) (-> ph (-> (\/ ps th) (\/ ch ta))) (ps ch th ta pm3.48 orim12d.1 orim12d.2 sylanc)) thm (orim1d () ((orim1d.1 (-> ph (-> ps ch)))) (-> ph (-> (\/ ps th) (\/ ch th))) (orim1d.1 ph th idd orim12d)) thm (orim2d () ((orim1d.1 (-> ph (-> ps ch)))) (-> ph (-> (\/ th ps) (\/ th ch))) (ph th idd orim1d.1 orim12d)) thm (orim2 () () (-> (-> ps ch) (-> (\/ ph ps) (\/ ph ch))) ((-> ps ch) id ph orim2d)) thm (pm2.73 () () (-> (-> ph ps) (-> (\/ (\/ ph ps) ch) (\/ ps ch))) (ph ps pm2.621 ch orim1d)) thm (pm2.74 () () (-> (-> ps ph) (-> (\/ (\/ ph ps) ch) (\/ ph ch))) (ps ph orel2 ch orim1d ph ch orc (\/ (\/ ph ps) ch) a1d ja)) thm (pm2.75 () () (-> (\/ ph ps) (-> (\/ ph (-> ps ch)) (\/ ph ch))) (ph ch orc (\/ ph (-> ps ch)) a1d ps ch pm2.27 ph orim2d jaoi)) thm (pm2.76 () () (-> (\/ ph (-> ps ch)) (-> (\/ ph ps) (\/ ph ch))) (ph ps ch pm2.75 com12)) thm (pm2.8 () () (-> (\/ ps ch) (-> (\/ (-. ch) th) (\/ ps th))) (ps th orc (\/ (-. ch) th) a1d ch ps pm2.24 th orim1d jaoi)) thm (pm2.81 () () (-> (-> ps (-> ch th)) (-> (\/ ph ps) (-> (\/ ph ch) (\/ ph th)))) (ps (-> ch th) ph orim2 ph ch th pm2.76 syl6)) thm (pm2.82 () () (-> (\/ (\/ ph ps) ch) (-> (\/ (\/ ph (-. ch)) th) (\/ (\/ ph ps) th))) ((\/ ph ps) (\/ ph (-. ch)) ax-1 ch ps pm2.24 ph orim2d jaoi th orim1d)) thm (pm2.85 () () (-> (-> (\/ ph ps) (\/ ph ch)) (\/ ph (-> ps ch))) ((\/ ph ps) (\/ ph ch) imor ph ps pm2.48 (\/ ph ch) orim1i sylbi ps ch imor ph orbi2i ph (-. ps) ch orordi bitr2 sylib)) thm (pm3.2ni () ((pm3.2ni.1 (-. ph)) (pm3.2ni.2 (-. ps))) (-. (\/ ph ps)) (pm3.2ni.1 pm3.2ni.2 pm3.2i ph ps ioran mpbir)) thm (orabs () () (<-> ph (/\ (\/ ph ps) ph)) (ph ps orc ancri (\/ ph ps) ph pm3.27 impbi)) thm (oranabs () () (<-> (/\ (\/ ph (-. ps)) ps) (/\ ph ps)) (ph ps pm3.2 ps (/\ ph ps) pm2.21 jaoi imp ph (-. ps) orc ps anim1i impbi)) thm (pm5.74 () () (<-> (-> ph (<-> ps ch)) (<-> (-> ph ps) (-> ph ch))) (ps ch bi1 ph imim2i a2d ps ch bi2 ph imim2i a2d impbid (-> ph ps) (-> ph ch) bi1 pm2.86d (-> ph ps) (-> ph ch) bi2 pm2.86d anim12d ph pm4.24 ps ch bi 3imtr4g impbi)) thm (pm5.74i () ((pm5.74i.1 (-> ph (<-> ps ch)))) (<-> (-> ph ps) (-> ph ch)) (pm5.74i.1 ph ps ch pm5.74 mpbi)) thm (pm5.74d () ((pm5.74d.1 (-> ph (-> ps (<-> ch th))))) (-> ph (<-> (-> ps ch) (-> ps th))) (pm5.74d.1 ps ch th pm5.74 sylib)) thm (pm5.74ri () ((pm5.74ri.1 (<-> (-> ph ps) (-> ph ch)))) (-> ph (<-> ps ch)) (pm5.74ri.1 ph ps ch pm5.74 mpbir)) thm (pm5.74rd () ((pm5.74rd.1 (-> ph (<-> (-> ps ch) (-> ps th))))) (-> ph (-> ps (<-> ch th))) (pm5.74rd.1 ps ch th pm5.74 sylibr)) thm (mpbidi () ((mpbidi.min (-> th (-> ph ps))) (mpbidi.maj (-> ph (<-> ps ch)))) (-> th (-> ph ch)) (mpbidi.min mpbidi.maj pm5.74i sylib)) thm (ibib () () (<-> (-> ph ps) (-> ph (<-> ph ps))) (ph ps pm3.4 ph ps pm3.26 ps a1d impbid ex ph ps bi1 com12 impbid pm5.74i)) thm (ibibr () () (<-> (-> ph ps) (-> ph (<-> ps ph))) (ph ps ibib ph ps bicom ph imbi2i bitr)) thm (ibi () ((ibi.1 (-> ph (<-> ph ps)))) (-> ph ps) (ibi.1 biimpd pm2.43i)) thm (ibir () ((ibir.1 (-> ph (<-> ps ph)))) (-> ph ps) (ibir.1 bicomd ibi)) thm (ibd () ((ibd.1 (-> ph (-> ps (<-> ps ch))))) (-> ph (-> ps ch)) (ibd.1 ps ch ibib sylibr)) thm (pm5.501 () () (-> ph (<-> ps (<-> ph ps))) (ph ps ibib pm5.74ri)) thm (ordi () () (<-> (\/ ph (/\ ps ch)) (/\ (\/ ph ps) (\/ ph ch))) (ps ch pm3.26 ph orim2i ps ch pm3.27 ph orim2i jca ph ps df-or (-. ph) ps ch pm3.43i ph ch df-or ph (/\ ps ch) df-or 3imtr4g sylbi imp impbi)) thm (ordir () () (<-> (\/ (/\ ph ps) ch) (/\ (\/ ph ch) (\/ ps ch))) (ch ph ps ordi (/\ ph ps) ch orcom ph ch orcom ps ch orcom anbi12i 3bitr4)) thm (jcab () () (<-> (-> ph (/\ ps ch)) (/\ (-> ph ps) (-> ph ch))) ((-. ph) ps ch ordi ph (/\ ps ch) imor ph ps imor ph ch imor anbi12i 3bitr4)) thm (pm4.76 () () (<-> (/\ (-> ph ps) (-> ph ch)) (-> ph (/\ ps ch))) (ph ps ch jcab bicomi)) thm (jcad () ((jcad.1 (-> ph (-> ps ch))) (jcad.2 (-> ph (-> ps th)))) (-> ph (-> ps (/\ ch th))) (jcad.1 imp jcad.2 imp jca ex)) thm (jctild () ((jctild.1 (-> ph (-> ps ch))) (jctild.2 (-> ph th))) (-> ph (-> ps (/\ th ch))) (jctild.2 ps a1d jctild.1 jcad)) thm (jctird () ((jctird.1 (-> ph (-> ps ch))) (jctird.2 (-> ph th))) (-> ph (-> ps (/\ ch th))) (jctird.1 jctird.2 ps a1d jcad)) thm (pm3.43 () () (-> (/\ (-> ph ps) (-> ph ch)) (-> ph (/\ ps ch))) (ph ps ch jcab biimpr)) thm (andi () () (<-> (/\ ph (\/ ps ch)) (\/ (/\ ph ps) (/\ ph ch))) ((-. ph) (-. ps) (-. ch) ordi ps ch ioran (-. ph) orbi2i ph ps ianor ph ch ianor anbi12i 3bitr4 negbii ph (\/ ps ch) anor (/\ ph ps) (/\ ph ch) oran 3bitr4)) thm (andir () () (<-> (/\ (\/ ph ps) ch) (\/ (/\ ph ch) (/\ ps ch))) (ch ph ps andi (\/ ph ps) ch ancom ph ch ancom ps ch ancom orbi12i 3bitr4)) thm (orddi () () (<-> (\/ (/\ ph ps) (/\ ch th)) (/\ (/\ (\/ ph ch) (\/ ph th)) (/\ (\/ ps ch) (\/ ps th)))) (ph ps (/\ ch th) ordir ph ch th ordi ps ch th ordi anbi12i bitr)) thm (anddi () () (<-> (/\ (\/ ph ps) (\/ ch th)) (\/ (\/ (/\ ph ch) (/\ ph th)) (\/ (/\ ps ch) (/\ ps th)))) (ph ps (\/ ch th) andir ph ch th andi ps ch th andi orbi12i bitr)) thm (bibi2i () ((bibi.a (<-> ph ps))) (<-> (<-> ch ph) (<-> ch ps)) (ch ph bi bibi.a ch imbi1i (-> ch ph) anbi2i bibi.a ch imbi2i (-> ps ch) anbi1i ch ps bi bitr4 3bitr)) thm (bibi1i () ((bibi.a (<-> ph ps))) (<-> (<-> ph ch) (<-> ps ch)) (ph ch bicom bibi.a ch bibi2i ch ps bicom 3bitr)) thm (bibi12i () ((bibi.a (<-> ph ps)) (bibi12.2 (<-> ch th))) (<-> (<-> ph ch) (<-> ps th)) (bibi12.2 ph bibi2i bibi.a th bibi1i bitr)) thm (negbid () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (-. ps) (-. ch))) (bid.1 ps ch pm4.11 sylib)) thm (imbi2d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (-> th ps) (-> th ch))) (bid.1 th a1d pm5.74d)) thm (imbi1d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (-> ps th) (-> ch th))) (bid.1 negbid (-. th) imbi2d ps th pm4.1 ch th pm4.1 3bitr4g)) thm (orbi2d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (\/ th ps) (\/ th ch))) (bid.1 (-. th) imbi2d th ps df-or th ch df-or 3bitr4g)) thm (orbi1d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (\/ ps th) (\/ ch th))) (bid.1 th orbi2d ps th orcom ch th orcom 3bitr4g)) thm (anbi2d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (/\ th ps) (/\ th ch))) (bid.1 biimpd th anim2d bid.1 biimprd th anim2d impbid)) thm (anbi1d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (/\ ps th) (/\ ch th))) (bid.1 th anbi2d ps th ancom ch th ancom 3bitr4g)) thm (bibi2d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (<-> th ps) (<-> th ch))) (bid.1 th imbi2d (-> ps th) anbi1d bid.1 th imbi1d (-> th ch) anbi2d bitrd th ps bi th ch bi 3bitr4g)) thm (bibi1d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (<-> ps th) (<-> ch th))) (bid.1 th bibi2d ps th bicom ch th bicom 3bitr4g)) thm (orbi1 () () (-> (<-> ph ps) (<-> (\/ ph ch) (\/ ps ch))) ((<-> ph ps) id ch orbi1d)) thm (anbi1 () () (-> (<-> ph ps) (<-> (/\ ph ch) (/\ ps ch))) ((<-> ph ps) id ch anbi1d)) thm (pm4.22 () () (-> (/\ (<-> ph ps) (<-> ps ch)) (<-> ph ch)) ((<-> ph ps) id ch bibi1d biimpar)) thm (imbi1 () () (-> (<-> ph ps) (<-> (-> ph ch) (-> ps ch))) ((<-> ph ps) id ch imbi1d)) thm (imbi2 () () (-> (<-> ph ps) (<-> (-> ch ph) (-> ch ps))) ((<-> ph ps) ch ax-1 pm5.74d)) thm (bibi1 () () (-> (<-> ph ps) (<-> (<-> ph ch) (<-> ps ch))) ((<-> ph ps) id ch bibi1d)) thm (imbi12d () ((bi12d.1 (-> ph (<-> ps ch))) (bi12d.2 (-> ph (<-> th ta)))) (-> ph (<-> (-> ps th) (-> ch ta))) (bi12d.1 th imbi1d bi12d.2 ch imbi2d bitrd)) thm (orbi12d () ((bi12d.1 (-> ph (<-> ps ch))) (bi12d.2 (-> ph (<-> th ta)))) (-> ph (<-> (\/ ps th) (\/ ch ta))) (bi12d.1 th orbi1d bi12d.2 ch orbi2d bitrd)) thm (anbi12d () ((bi12d.1 (-> ph (<-> ps ch))) (bi12d.2 (-> ph (<-> th ta)))) (-> ph (<-> (/\ ps th) (/\ ch ta))) (bi12d.1 th anbi1d bi12d.2 ch anbi2d bitrd)) thm (bibi12d () ((bi12d.1 (-> ph (<-> ps ch))) (bi12d.2 (-> ph (<-> th ta)))) (-> ph (<-> (<-> ps th) (<-> ch ta))) (bi12d.1 th bibi1d bi12d.2 ch bibi2d bitrd)) thm (pm4.39 () () (-> (/\ (<-> ph ch) (<-> ps th)) (<-> (\/ ph ps) (\/ ch th))) ((<-> ph ch) (<-> ps th) pm3.26 (<-> ph ch) (<-> ps th) pm3.27 orbi12d)) thm (pm4.38 () () (-> (/\ (<-> ph ch) (<-> ps th)) (<-> (/\ ph ps) (/\ ch th))) ((<-> ph ch) (<-> ps th) pm3.26 (<-> ph ch) (<-> ps th) pm3.27 anbi12d)) thm (bi2anan9 () ((bi2an9.1 (-> ph (<-> ps ch))) (bi2an9.2 (-> th (<-> ta et)))) (-> (/\ ph th) (<-> (/\ ps ta) (/\ ch et))) (bi2an9.1 ta anbi1d bi2an9.2 ch anbi2d sylan9bb)) thm (bi2anan9r () ((bi2an9.1 (-> ph (<-> ps ch))) (bi2an9.2 (-> th (<-> ta et)))) (-> (/\ th ph) (<-> (/\ ps ta) (/\ ch et))) (bi2an9.1 bi2an9.2 bi2anan9 ancoms)) thm (bi2bian9 () ((bi2an9.1 (-> ph (<-> ps ch))) (bi2an9.2 (-> th (<-> ta et)))) (-> (/\ ph th) (<-> (<-> ps ta) (<-> ch et))) (bi2an9.1 th adantr bi2an9.2 ph adantl bibi12d)) thm (pm4.71 () () (<-> (-> ph ps) (<-> ph (/\ ph ps))) (ph ps ancl ph ps pm3.26 (-> ph ps) a1i impbid ph (/\ ph ps) bi1 ph ps pm3.27 syl6 impbi)) thm (pm4.71r () () (<-> (-> ph ps) (<-> ph (/\ ps ph))) (ph ps pm4.71 ph ps ancom ph bibi2i bitr)) thm (pm4.71i () ((pm4.71i.1 (-> ph ps))) (<-> ph (/\ ph ps)) (pm4.71i.1 ph ps pm4.71 mpbi)) thm (pm4.71ri () ((pm4.71ri.1 (-> ph ps))) (<-> ph (/\ ps ph)) (pm4.71ri.1 ph ps pm4.71r mpbi)) thm (pm4.71rd () ((pm4.71rd.1 (-> ph (-> ps ch)))) (-> ph (<-> ps (/\ ch ps))) (pm4.71rd.1 ps ch pm4.71r sylib)) thm (pm4.45 () () (<-> ph (/\ ph (\/ ph ps))) (ph ps orc pm4.71i)) thm (pm4.72 () () (<-> (-> ph ps) (<-> ps (\/ ph ps))) (ps ph olc (-> ph ps) a1i ph ps pm2.621 impbid ps (\/ ph ps) bi2 ph ps pm2.67 syl impbi)) thm (iba () () (-> ph (<-> ps (/\ ps ph))) (ph ps ancrb pm5.74ri)) thm (ibar () () (-> ph (<-> ps (/\ ph ps))) (ph ps anclb pm5.74ri)) thm (pm5.32 () () (<-> (-> ph (<-> ps ch)) (<-> (/\ ph ps) (/\ ph ch))) (ps ch pm4.11 ph imbi2i ph (-. ps) (-. ch) pm5.74 (-> ph (-. ps)) (-> ph (-. ch)) pm4.11 3bitr ph ps df-an ph ch df-an bibi12i bitr4)) thm (pm5.32i () ((pm5.32i.1 (-> ph (<-> ps ch)))) (<-> (/\ ph ps) (/\ ph ch)) (pm5.32i.1 ph ps ch pm5.32 mpbi)) thm (pm5.32ri () ((pm5.32i.1 (-> ph (<-> ps ch)))) (<-> (/\ ps ph) (/\ ch ph)) (pm5.32i.1 pm5.32i ps ph ancom ch ph ancom 3bitr4)) thm (pm5.32d () ((pm5.32d.1 (-> ph (-> ps (<-> ch th))))) (-> ph (<-> (/\ ps ch) (/\ ps th))) (pm5.32d.1 ps ch th pm5.32 sylib)) thm (pm5.32rd () ((pm5.32d.1 (-> ph (-> ps (<-> ch th))))) (-> ph (<-> (/\ ch ps) (/\ th ps))) (pm5.32d.1 pm5.32d ch ps ancom th ps ancom 3bitr4g)) thm (pm5.33 () () (<-> (/\ ph (-> ps ch)) (/\ ph (-> (/\ ph ps) ch))) (ph ps ibar ch imbi1d pm5.32i)) thm (pm5.36 () () (<-> (/\ ph (<-> ph ps)) (/\ ps (<-> ph ps))) ((<-> ph ps) id pm5.32ri)) thm (pm5.42 () () (<-> (-> ph (-> ps ch)) (-> ph (-> ps (/\ ph ch)))) (ph ch ibar ps imbi2d pm5.74i)) thm (oibabs () () (<-> (<-> ph ps) (-> (\/ ph ps) (<-> ph ps))) ((<-> ph ps) (\/ ph ps) ax-1 ph ps orc (<-> ph ps) imim1i ibd ps ph olc (<-> ph ps) imim1i ps ph ibibr sylibr impbid impbi)) thm (exmid () () (\/ ph (-. ph)) ((-. ph) id orri)) thm (pm2.1 () () (\/ (-. ph) ph) (ph nega orri)) thm (pm2.13 () () (\/ ph (-. (-. (-. ph)))) ((-. ph) negb orri)) thm (pm3.24 () () (-. (/\ ph (-. ph))) ((-. ph) exmid ph (-. ph) ianor mpbir)) thm (pm2.26 () () (\/ (-. ph) (-> (-> ph ps) ps)) (ph nega ps imim1i com12 orri)) thm (pm5.18 () () (<-> (<-> ph ps) (-. (<-> ph (-. ps)))) (ph ps bicom ph (-. ps) bicom ps ph pm2.61 ps ph pm2.65 ph ps con2 syl5 anim12d (-. ps) ph bi syl5ib ph ps annim syl6ib com12 (-> ps ph) (-> ph ps) imnan sylib ps ph bi negbii sylibr ps ph bi negbii ps ph pm2.5 ps ph annim ph (-. ps) pm2.21 ps adantl sylbir jca ph ps annim ph (-. ps) ax-1 (-. ps) adantr sylbir ph ps pm2.51 jca jaoi (-> ps ph) (-> ph ps) ianor (-. ps) ph bi 3imtr4 sylbi impbi bitr con2bii bitr)) thm (nbbn () () (<-> (<-> (-. ph) ps) (-. (<-> ph ps))) ((-. ph) ps bicom ph ps bicom ps ph pm5.18 bitr con2bii bitr)) thm (pm5.11 () () (\/ (-> ph ps) (-> (-. ph) ps)) (ph ps pm2.5 orri)) thm (pm5.12 () () (\/ (-> ph ps) (-> ph (-. ps))) (ph ps pm2.51 orri)) thm (pm5.13 () () (\/ (-> ph ps) (-> ps ph)) (ph ps pm2.521 orri)) thm (pm5.14 () () (\/ (-> ph ps) (-> ps ch)) (ps ph ax-1 con3i ch pm2.21d orri)) thm (pm5.15 () () (\/ (<-> ph ps) (<-> ph (-. ps))) (ph ps pm5.18 biimpr con1i orri)) thm (pm5.16 () () (-. (/\ (<-> ph ps) (<-> ph (-. ps)))) (ph ps pm5.18 biimp (<-> ph ps) (<-> ph (-. ps)) pm4.62 mpbi (<-> ph ps) (<-> ph (-. ps)) ianor mpbir)) thm (pm5.17 () () (<-> (/\ (\/ ph ps) (-. (/\ ph ps))) (<-> ph (-. ps))) (ph ps orcom ps ph df-or bitr ph ps imnan bicomi anbi12i (-. ps) ph bi (-. ps) ph bicom 3bitr2)) thm (pm5.19 () () (-. (<-> ph (-. ph))) (ph pm4.2 ph ph pm5.18 mpbi)) thm (dfbi () () (<-> (<-> ph ps) (\/ (/\ ph ps) (/\ (-. ph) (-. ps)))) (ph ps pm5.18 ph ps imnan ps ph bi2.15 (-. ph) ps iman bitr anbi12i ph (-. ps) bi (/\ ph ps) (/\ (-. ph) (-. ps)) ioran 3bitr4r con1bii bitr)) thm (xor () () (<-> (-. (<-> ph ps)) (\/ (/\ ph (-. ps)) (/\ ps (-. ph)))) ((-. ph) ps dfbi ph ps nbbn ps (-. ph) ancom ph pm4.13 (-. ps) anbi1i orbi12i (/\ ps (-. ph)) (/\ ph (-. ps)) orcom bitr3 3bitr3)) thm (pm5.24 () () (<-> (-. (\/ (/\ ph ps) (/\ (-. ph) (-. ps)))) (\/ (/\ ph (-. ps)) (/\ ps (-. ph)))) (ph ps dfbi negbii ph ps xor bitr3)) thm (xor2 () () (<-> (-. (<-> ph ps)) (/\ (\/ ph ps) (-. (/\ ph ps)))) (ph ps xor (/\ ph ps) (/\ (-. ph) (-. ps)) ioran ph ps pm5.24 ph ps oran (-. (/\ ph ps)) anbi2i (-. (/\ ph ps)) (\/ ph ps) ancom bitr3 3bitr3 bitr)) thm (xor3 () () (<-> (-. (<-> ph ps)) (<-> ph (-. ps))) (ph ps pm5.18 con2bii bicomi)) thm (pm5.55 () () (\/ (<-> (\/ ph ps) ph) (<-> (\/ ph ps) ps)) (ps ph pm5.13 ps ph pm4.72 ps ph orcom ph bibi2i ph (\/ ph ps) bicom 3bitr ph ps pm4.72 ps (\/ ph ps) bicom bitr orbi12i mpbi)) thm (pm5.1 () () (-> (/\ ph ps) (<-> ph ps)) (ph ps pm5.501 biimpa)) thm (pm5.21 () () (-> (/\ (-. ph) (-. ps)) (<-> ph ps)) ((-. ph) (-. ps) pm5.1 con4bid)) thm (pm5.21ni () ((pm5.21ni.1 (-> ph ps)) (pm5.21ni.2 (-> ch ps))) (-> (-. ps) (<-> ph ch)) (ph ch pm5.21 pm5.21ni.1 con3i pm5.21ni.2 con3i sylanc)) thm (pm5.21nii () ((pm5.21ni.1 (-> ph ps)) (pm5.21ni.2 (-> ch ps)) (pm5.21nii.3 (-> ps (<-> ph ch)))) (<-> ph ch) (pm5.21nii.3 pm5.21ni.1 pm5.21ni.2 pm5.21ni pm2.61i)) thm (pm5.21nd () ((pm5.21nd.1 (-> (/\ ph ps) th)) (pm5.21nd.2 (-> (/\ ph ch) th)) (pm5.21nd.3 (-> th (<-> ps ch)))) (-> ph (<-> ps ch)) (pm5.21nd.1 ex con3d pm5.21nd.2 ex con3d jcad ps ch pm5.21 syl6 pm5.21nd.3 pm2.61d2)) thm (pm5.35 () () (-> (/\ (-> ph ps) (-> ph ch)) (-> ph (<-> ps ch))) ((-> ph ps) (-> ph ch) pm5.1 pm5.74rd)) thm (pm5.54 () () (\/ (<-> (/\ ph ps) ph) (<-> (/\ ph ps) ps)) ((/\ ph ps) ph pm5.1 anabss1 ps ph iba bicomd pm5.21ni orri)) thm (elimant () () (-> (/\ (-> ph ps) (-> (-> ps ch) (-> ph th))) (-> ph (-> ch th))) ((-> ph ps) (-> (-> ps ch) (-> ph th)) pm3.27 ch ps ax-1 syl5 com23)) thm (baib () ((baib.1 (<-> ph (/\ ps ch)))) (-> ps (<-> ph ch)) (ps ch ibar baib.1 syl6rbbr)) thm (baibr () ((baibr.1 (<-> ph (/\ ps ch)))) (-> ps (<-> ch ph)) (baibr.1 baib bicomd)) thm (pm5.44 () () (-> (-> ph ps) (<-> (-> ph ch) (-> ph (/\ ps ch)))) (ph ps ch jcab baibr)) thm (orcana () () (<-> (-> ph (\/ ps ch)) (-> (/\ ph (-. ps)) ch)) (ps ch df-or ph imbi2i ph (-. ps) ch impexp bitr4)) thm (pm5.6 () () (<-> (-> (/\ ph (-. ps)) ch) (-> ph (\/ ps ch))) (ph ps ch orcana bicomi)) thm (nan () () (<-> (-> ph (-. (/\ ps ch))) (-> (/\ ph ps) (-. ch))) (ph ps (-. ch) impexp ps ch imnan ph imbi2i bitr2)) thm (orcanai () ((orcanai.1 (-> ph (\/ ps ch)))) (-> (/\ ph (-. ps)) ch) (orcanai.1 ord imp)) thm (intnan () ((intnan.1 (-. ph))) (-. (/\ ps ph)) (intnan.1 ps ph pm3.27 mto)) thm (intnanr () ((intnan.1 (-. ph))) (-. (/\ ph ps)) (intnan.1 ph ps pm3.26 mto)) thm (intnand () ((intnand.1 (-> ph (-. ps)))) (-> ph (-. (/\ ch ps))) (intnand.1 ch ps pm3.27 nsyl)) thm (intnanrd () ((intnand.1 (-> ph (-. ps)))) (-> ph (-. (/\ ps ch))) (intnand.1 ps ch pm3.26 nsyl)) thm (mpan () ((mpan.1 ph) (mpan.2 (-> (/\ ph ps) ch))) (-> ps ch) (mpan.1 mpan.2 ex ax-mp)) thm (mpan2 () ((mpan2.1 ps) (mpan2.2 (-> (/\ ph ps) ch))) (-> ph ch) (mpan2.1 mpan2.2 ex mpi)) thm (mp2an () ((mp2an.1 ph) (mp2an.2 ps) (mp2an.3 (-> (/\ ph ps) ch))) ch (mp2an.2 mp2an.1 mp2an.3 mpan ax-mp)) thm (mpani () ((mpani.1 ps) (mpani.2 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> ch th)) (mpani.1 mpani.2 exp3a mpi)) thm (mpan2i () ((mpan2i.1 ch) (mpan2i.2 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> ps th)) (mpan2i.1 mpan2i.2 exp3a mpii)) thm (mp2ani () ((mp2ani.1 ps) (mp2ani.2 ch) (mp2ani.3 (-> ph (-> (/\ ps ch) th)))) (-> ph th) (mp2ani.2 mp2ani.1 mp2ani.3 mpani mpi)) thm (mpand () ((mpand.1 (-> ph ps)) (mpand.2 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> ch th)) (mpand.1 mpand.2 exp3a mpd)) thm (mpan2d () ((mpan2d.1 (-> ph ch)) (mpan2d.2 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> ps th)) (mpan2d.1 mpan2d.2 exp3a mpid)) thm (mp2and () ((mp2and.1 (-> ph ps)) (mp2and.2 (-> ph ch)) (mp2and.3 (-> ph (-> (/\ ps ch) th)))) (-> ph th) (mp2and.2 mp2and.1 mp2and.3 mpand mpd)) thm (mpdan () ((mpdan.1 (-> ph ps)) (mpdan.2 (-> (/\ ph ps) ch))) (-> ph ch) (mpdan.1 mpdan.2 ex mpd)) thm (mpancom () ((mpancom.1 (-> ps ph)) (mpancom.2 (-> (/\ ph ps) ch))) (-> ps ch) (mpancom.1 mpancom.2 ancoms mpdan)) thm (mpanl1 () ((mpanl1.1 ph) (mpanl1.2 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ ps ch) th) (mpanl1.1 mpanl1.2 ex mpan imp)) thm (mpanl2 () ((mpanl2.1 ps) (mpanl2.2 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ ph ch) th) (mpanl2.1 mpanl2.2 ex mpan2 imp)) thm (mpanl12 () ((mpanl12.1 ph) (mpanl12.2 ps) (mpanl12.3 (-> (/\ (/\ ph ps) ch) th))) (-> ch th) (mpanl12.2 mpanl12.1 mpanl12.3 mpanl1 mpan)) thm (mpanr1 () ((mpanr1.1 ps) (mpanr1.2 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph ch) th) (mpanr1.1 mpanr1.2 ex mpani imp)) thm (mpanr2 () ((mpanr2.1 ch) (mpanr2.2 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph ps) th) (mpanr2.1 mpanr2.2 ex mpan2i imp)) thm (mpanlr1 () ((mpanlr1.1 ps) (mpanlr1.2 (-> (/\ (/\ ph (/\ ps ch)) th) ta))) (-> (/\ (/\ ph ch) th) ta) (mpanlr1.1 mpanlr1.2 ex mpanr1 imp)) thm (mtt () () (-> (-. ph) (<-> (-. ps) (-> ps ph))) (ps ph pm2.21 (-. ph) a1i ps ph con3 com12 impbid)) thm (mt2bi () ((mt2bi.1 ph)) (<-> (-. ps) (-> ps (-. ph))) (ps (-. ph) pm2.21 mt2bi.1 ps ph con2 mpi impbi)) thm (mtbid () ((mtbid.min (-> ph (-. ps))) (mtbid.maj (-> ph (<-> ps ch)))) (-> ph (-. ch)) (mtbid.min mtbid.maj biimprd mtod)) thm (mtbird () ((mtbird.min (-> ph (-. ch))) (mtbird.maj (-> ph (<-> ps ch)))) (-> ph (-. ps)) (mtbird.min mtbird.maj biimpd mtod)) thm (mtbii () ((mtbii.min (-. ps)) (mtbii.maj (-> ph (<-> ps ch)))) (-> ph (-. ch)) (mtbii.min mtbii.maj biimprd mtoi)) thm (mtbiri () ((mtbiri.min (-. ch)) (mtbiri.maj (-> ph (<-> ps ch)))) (-> ph (-. ps)) (mtbiri.min mtbiri.maj biimpd mtoi)) thm (2th () ((2th.1 ph) (2th.2 ps)) (<-> ph ps) (2th.2 ph a1i 2th.1 ps a1i impbi)) thm (2false () ((2false.1 (-. ph)) (2false.2 (-. ps))) (<-> ph ps) (2false.1 2false.2 ph ps pm5.21 mp2an)) thm (tbt () ((tbt.1 ph)) (<-> ps (<-> ps ph)) (tbt.1 ph ps pm5.501 ax-mp ph ps bicom bitr)) thm (nbn2 () () (-> (-. ph) (<-> (-. ps) (<-> ph ps))) (ph ps pm5.21 ex ph ps pm4.11 biimp biimpcd impbid)) thm (nbn () ((nbn.1 (-. ph))) (<-> (-. ps) (<-> ps ph)) (nbn.1 ph ps nbn2 ax-mp ph ps bicom bitr)) thm (nbn3 () ((nbn3.1 ph)) (<-> (-. ps) (<-> ps (-. ph))) (nbn3.1 ph negb ax-mp ps nbn)) thm (biantru () ((biantru.1 ph)) (<-> ps (/\ ps ph)) (biantru.1 ph ps iba ax-mp)) thm (biantrur () ((biantrur.1 ph)) (<-> ps (/\ ph ps)) (biantrur.1 ph ps ibar ax-mp)) thm (biantrud () ((biantrud.1 (-> ph ps))) (-> ph (<-> ch (/\ ch ps))) (biantrud.1 ch anim2i expcom ch ps pm3.26 ph a1i impbid)) thm (biantrurd () ((biantrud.1 (-> ph ps))) (-> ph (<-> ch (/\ ps ch))) (biantrud.1 ch biantrud ch ps ancom syl6bb)) thm (mpbiran () ((mpbiran.1 (<-> ph (/\ ps ch))) (mpbiran.2 ps)) (<-> ph ch) (mpbiran.1 mpbiran.2 ch biantrur bitr4)) thm (mpbiran2 () ((mpbiran.1 (<-> ph (/\ ps ch))) (mpbiran2.2 ch)) (<-> ph ps) (mpbiran.1 mpbiran2.2 ps biantru bitr4)) thm (mpbir2an () ((mpbiran.1 (<-> ph (/\ ps ch))) (mpbir2an.2 ps) (mpbir2an.3 ch)) ph (mpbir2an.3 mpbiran.1 mpbir2an.2 mpbiran mpbir)) thm (biimt () () (-> ph (<-> ps (-> ph ps))) (ps ph ax-1 ph a1i ph ps pm2.27 impbid)) thm (pm5.5 () () (-> ph (<-> (-> ph ps) ps)) (ph ps biimt bicomd)) thm (pm5.62 () () (<-> (\/ (/\ ph ps) (-. ps)) (\/ ph (-. ps))) (ph ps (-. ps) ordir ps exmid mpbiran2)) thm (biort () () (-> ph (<-> ph (\/ ph ps))) (ph ps orc ph a1d ph (\/ ph ps) ax-1 impbid)) thm (biorf () () (-> (-. ph) (<-> ps (\/ ph ps))) ((-. ph) ps biimt ph ps df-or syl6bbr)) thm (biorfi () ((biorfi.1 (-. ph))) (<-> ps (\/ ps ph)) (biorfi.1 ph ps biorf ax-mp ph ps orcom bitr)) thm (bianfi () ((bianfi.1 (-. ph))) (<-> ph (/\ ps ph)) (bianfi.1 (/\ ps ph) pm2.21i ps ph pm3.27 impbi)) thm (bianfd () ((bianfd.1 (-> ph (-. ps)))) (-> ph (<-> ps (/\ ps ch))) (bianfd.1 (/\ ps ch) pm2.21d ps ch pm3.26 ph a1i impbid)) thm (pm4.82 () () (<-> (/\ (-> ph ps) (-> ph (-. ps))) (-. ph)) (ph ps pm2.65 imp ph ps pm2.21 ph (-. ps) pm2.21 jca impbi)) thm (pm4.83 () () (<-> (/\ (-> ph ps) (-> (-. ph) ps)) ps) (ph exmid ps a1bi ph (-. ph) ps jaob bitr2)) thm (pclem6 () () (-> (<-> ph (/\ ps (-. ph))) (-. ps)) (ph (/\ ps (-. ph)) bi1 ps (-. ph) pm3.27 syl6 pm2.01d ph (/\ ps (-. ph)) bi2 exp3a com23 ps ph con3 syli mpd)) thm (biantr () () (-> (/\ (<-> ph ps) (<-> ch ps)) (<-> ph ch)) ((<-> ch ps) id ph bibi2d biimparc)) thm (orbidi () () (<-> (\/ ph (<-> ps ch)) (<-> (\/ ph ps) (\/ ph ch))) (ph ch orc (\/ ph ps) a1d ph ps orc (\/ ph ch) a1d impbid (<-> ps ch) id ph orbi2d jaoi ph ps ch pm2.85 ph ch ps pm2.85 anim12i (\/ ph ps) (\/ ph ch) bi ps ch bi ph orbi2i ph (-> ps ch) (-> ch ps) ordi bitr 3imtr4 impbi)) thm (biass () () (<-> (<-> (<-> ph ps) ch) (<-> ph (<-> ps ch))) (ph ps pm5.501 ch bibi1d ph (<-> ps ch) pm5.501 bitr3d ps ch nbbn (-. ph) a1i ph ps nbn2 ch bibi1d ph (<-> ps ch) nbn2 3bitr3d pm2.61i)) thm (biluk () () (<-> (<-> ph ps) (<-> (<-> ch ps) (<-> ph ch))) (ph ps bicom ch bibi1i ps ph ch biass bitr (<-> ph ps) ch (<-> ps (<-> ph ch)) biass mpbi ch ps (<-> ph ch) biass bitr4)) thm (pm5.7 () () (<-> (<-> (\/ ph ch) (\/ ps ch)) (\/ ch (<-> ph ps))) (ch ph ps orbidi ch ph orcom ch ps orcom bibi12i bitr2)) thm (bigolden () () (<-> (<-> (/\ ph ps) ph) (<-> ps (\/ ph ps))) (ph ps pm4.71 ph ps pm4.72 ph (/\ ph ps) bicom 3bitr3r)) thm (pm5.71 () () (-> (-> ps (-. ch)) (<-> (/\ (\/ ph ps) ch) (/\ ph ch))) (ps ph orel2 ph ps orc (-. ps) a1i impbid ch anbi1d ch (<-> (\/ ph ps) ph) pm2.21 pm5.32rd ja)) thm (pm5.75 () () (-> (/\ (-> ch (-. ps)) (<-> ph (\/ ps ch))) (<-> (/\ ph (-. ps)) ch)) (ph (\/ ps ch) bi1 ph ps ch pm5.6 sylibr (-> ch (-. ps)) adantl ph (\/ ps ch) bi2 ch ps olc ph ps olc imim12i syl ch ps ph pm5.6 sylibr exp3a a2d impcom (-> ch (-. ps)) (<-> ph (\/ ps ch)) pm3.26 jcad impbid)) thm (bimsc1 () () (-> (/\ (-> ph ps) (<-> ch (/\ ps ph))) (<-> ch ph)) ((<-> ch (/\ ps ph)) id ph ps pm4.71r biimp bicomd sylan9bbr)) thm (ecase2d () ((ecase2d.1 (-> ph ps)) (ecase2d.2 (-> ph (-. (/\ ps ch)))) (ecase2d.3 (-> ph (-. (/\ ps th)))) (ecase2d.4 (-> ph (\/ ta (\/ ch th))))) (-> ph ta) (ecase2d.1 ecase2d.2 ps ch imnan sylibr mpd ecase2d.1 ecase2d.3 ps th imnan sylibr mpd jca ch th ioran sylibr ecase2d.4 ta (\/ ch th) orcom sylib ord mpd)) thm (ecase3 () ((ecase3.1 (-> ph ch)) (ecase3.2 (-> ps ch)) (ecase3.3 (-> (-. (\/ ph ps)) ch))) ch (ph ps ioran ecase3.3 sylbir ex ecase3.1 ecase3.2 pm2.61ii)) thm (ecase () ((ecase.1 (-> (-. ph) ch)) (ecase.2 (-> (-. ps) ch)) (ecase.3 (-> (/\ ph ps) ch))) ch (ecase.3 ex ecase.1 ecase.2 pm2.61nii)) thm (ecase3d () ((ecase3d.1 (-> ph (-> ps th))) (ecase3d.2 (-> ph (-> ch th))) (ecase3d.3 (-> ph (-> (-. (\/ ps ch)) th)))) (-> ph th) (ecase3d.1 com12 ecase3d.2 com12 ecase3d.3 com12 ecase3)) thm (caselem () () (<-> (/\ (\/ ph ps) (\/ ch th)) (\/ (\/ (/\ ph ch) (/\ ps ch)) (\/ (/\ ph th) (/\ ps th)))) ((\/ ph ps) ch th andi ph ps ch andir ph ps th andir orbi12i bitr)) thm (ccase () ((ccase.1 (-> (/\ ph ps) ta)) (ccase.2 (-> (/\ ch ps) ta)) (ccase.3 (-> (/\ ph th) ta)) (ccase.4 (-> (/\ ch th) ta))) (-> (/\ (\/ ph ch) (\/ ps th)) ta) (ph ch ps th caselem ccase.1 ccase.2 jaoi ccase.3 ccase.4 jaoi jaoi sylbi)) thm (ccased () ((ccased.1 (-> ph (-> (/\ ps ch) et))) (ccased.2 (-> ph (-> (/\ th ch) et))) (ccased.3 (-> ph (-> (/\ ps ta) et))) (ccased.4 (-> ph (-> (/\ th ta) et)))) (-> ph (-> (/\ (\/ ps th) (\/ ch ta)) et)) (ccased.1 ccased.2 jaod ccased.3 ccased.4 jaod jaod ps th ch ta caselem syl5ib)) thm (ccase2 () ((ccase2.1 (-> (/\ ph ps) ta)) (ccase2.2 (-> ch ta)) (ccase2.3 (-> th ta))) (-> (/\ (\/ ph ch) (\/ ps th)) ta) (ccase2.1 ccase2.2 ps adantr ccase2.3 ph adantl ccase2.3 ch adantl ccase)) thm (4cases () ((4cases.1 (-> (/\ ph ps) ch)) (4cases.2 (-> (/\ ph (-. ps)) ch)) (4cases.3 (-> (/\ (-. ph) ps) ch)) (4cases.4 (-> (/\ (-. ph) (-. ps)) ch))) ch (4cases.1 4cases.3 pm2.61ian 4cases.2 4cases.4 pm2.61ian pm2.61i)) thm (niabn () ((niabn.1 ph)) (-> (-. ps) (<-> (/\ ch ps) (-. ph))) (ch ps pm3.27 niabn.1 ps pm2.21ni pm5.21ni)) thm (dedlem0a () () (-> ph (<-> ps (-> (-> ch ph) (/\ ps ph)))) (ps (-> ch ph) ax-1 ph a1i ph ch ax-1 ps imim1i com12 impbid ph ps iba (-> ch ph) imbi2d bitrd)) thm (dedlem0b () () (-> (-. ph) (<-> ps (-> (-> ps ph) (/\ ch ph)))) (ph (/\ ch ph) pm2.21 ps imim2d com23 ps ph pm2.21 ch ph pm3.27 imim12i con1d com12 impbid)) thm (dedlema () () (-> ph (<-> ps (\/ (/\ ps ph) (/\ ch (-. ph))))) (ps (/\ ch (-. ph)) orc ph a1i ph ps idd ph ps pm2.24 ch adantld jaod impbid ph ps iba (/\ ch (-. ph)) orbi1d bitrd)) thm (dedlemb () () (-> (-. ph) (<-> ch (\/ (/\ ps ph) (/\ ch (-. ph))))) ((/\ ch (-. ph)) (/\ ps ph) olc expcom ph (-> ps ch) pm2.21 com23 imp3a ch (-. ph) pm3.26 (-. ph) a1i jaod impbid)) thm (elimh () ((elimh.1 (-> (<-> ph (\/ (/\ ph ch) (/\ ps (-. ch)))) (<-> ch ta))) (elimh.2 (-> (<-> ps (\/ (/\ ph ch) (/\ ps (-. ch)))) (<-> th ta))) (elimh.3 th)) ta (ch ph ps dedlema elimh.1 syl ibi elimh.3 ch ps ph dedlemb elimh.2 syl mpbii pm2.61i)) thm (dedt () ((dedt.1 (-> (<-> ph (\/ (/\ ph ch) (/\ ps (-. ch)))) (<-> th ta))) (dedt.2 ta)) (-> ch th) (ch ph ps dedlema dedt.2 dedt.1 mpbiri syl)) thm (con3th () () (-> (-> ph ps) (-> (-. ps) (-. ph))) ((<-> ps (\/ (/\ ps (-> ph ps)) (/\ ph (-. (-> ph ps))))) id negbid (-. ph) imbi1d (<-> ps (\/ (/\ ps (-> ph ps)) (/\ ph (-. (-> ph ps))))) id ph imbi2d (<-> ph (\/ (/\ ps (-> ph ps)) (/\ ph (-. (-> ph ps))))) id ph imbi2d ph id elimh con3i dedt)) thm (consensus () () (<-> (\/ (\/ (/\ ph ps) (/\ (-. ph) ch)) (/\ ps ch)) (\/ (/\ ph ps) (/\ (-. ph) ch))) ((\/ (/\ ph ps) (/\ (-. ph) ch)) id ph ps ch dedlema biimpd ch adantrd ph ch ps dedlemb biimpd ps adantld pm2.61i ps ph ancom ch (-. ph) ancom orbi12i sylib jaoi (\/ (/\ ph ps) (/\ (-. ph) ch)) (/\ ps ch) orc impbi)) thm (pm4.42 () () (<-> ph (\/ (/\ ph ps) (/\ ph (-. ps)))) (ps ph ph dedlema ps ph ph dedlemb pm2.61i)) thm (ninba () ((ninba.1 ph)) (-> (-. ps) (<-> (-. ph) (/\ ch ps))) (ninba.1 ps ch niabn bicomd)) thm (prlem1 () ((prlem1.1 (-> ph (<-> et ch))) (prlem1.2 (-> ps (-. th)))) (-> ph (-> ps (-> (\/ (/\ ps ch) (/\ th ta)) et))) (prlem1.1 biimprcd ps adantl ps a1dd th et pm2.24 prlem1.2 syl5 ta adantr ph a1d jaoi com3l)) thm (prlem2 () () (<-> (\/ (/\ ph ps) (/\ ch th)) (/\ (\/ ph ch) (\/ (/\ ph ps) (/\ ch th)))) (ph ch orabs ps anbi1i (\/ ph ch) ph ps anass bitr ch ph orabs ch ph orcom ch anbi1i bitr th anbi1i (\/ ph ch) ch th anass bitr orbi12i (\/ ph ch) (/\ ph ps) (/\ ch th) andi bitr4)) thm (oplem1 () ((oplem1.1 (-> ph (\/ ps ch))) (oplem1.2 (-> ph (\/ th ta))) (oplem1.3 (<-> ps th)) (oplem1.4 (-> ch (<-> th ta)))) (-> ph ps) (oplem1.1 ord oplem1.2 ord oplem1.3 negbii syl5ib jcad oplem1.4 oplem1.3 syl5bb biimpar syl6 ps pm2.18 syl)) thm (rnlem () () (<-> (/\ (/\ ph ps) (/\ ch th)) (/\ (/\ (/\ ph ch) (/\ ps th)) (/\ (/\ ph th) (/\ ps ch)))) (ph ps (/\ ch th) anandir ph ch th anandi ps ch th anandi anbi12i (/\ ps ch) (/\ ps th) ancom (/\ (/\ ph ch) (/\ ph th)) anbi2i (/\ ph ch) (/\ ph th) (/\ ps th) (/\ ps ch) an4 bitr 3bitr)) thm (3orass () () (<-> (\/\/ ph ps ch) (\/ ph (\/ ps ch))) (ph ps ch df-3or ph ps ch orass bitr)) thm (3anass () () (<-> (/\/\ ph ps ch) (/\ ph (/\ ps ch))) (ph ps ch df-3an ph ps ch anass bitr)) thm (3anrot () () (<-> (/\/\ ph ps ch) (/\/\ ps ch ph)) (ph (/\ ps ch) ancom ph ps ch 3anass ps ch ph df-3an 3bitr4)) thm (3orrot () () (<-> (\/\/ ph ps ch) (\/\/ ps ch ph)) (ph (\/ ps ch) orcom ph ps ch 3orass ps ch ph df-3or 3bitr4)) thm (3ancoma () () (<-> (/\/\ ph ps ch) (/\/\ ps ph ch)) (ph ps ancom ch anbi1i ph ps ch df-3an ps ph ch df-3an 3bitr4)) thm (3ancomb () () (<-> (/\/\ ph ps ch) (/\/\ ph ch ps)) (ph ps ch 3ancoma ps ph ch 3anrot bitr)) thm (3anrev () () (<-> (/\/\ ph ps ch) (/\/\ ch ps ph)) (ph ps ch 3ancoma ch ps ph 3anrot bitr4)) thm (3simpa () () (-> (/\/\ ph ps ch) (/\ ph ps)) (ph ps ch df-3an pm3.26bd)) thm (3simpb () () (-> (/\/\ ph ps ch) (/\ ph ch)) (ph ps ch 3ancomb ph ch ps 3simpa sylbi)) thm (3simpc () () (-> (/\/\ ph ps ch) (/\ ps ch)) (ph ps ch 3anrot ps ch ph 3simpa sylbi)) thm (3simp1 () () (-> (/\/\ ph ps ch) ph) (ph ps ch 3simpa pm3.26d)) thm (3simp2 () () (-> (/\/\ ph ps ch) ps) (ph ps ch 3simpa pm3.27d)) thm (3simp3 () () (-> (/\/\ ph ps ch) ch) (ph ps ch 3simpc pm3.27d)) thm (3simp1i () ((3simp1i.1 (/\/\ ph ps ch))) ph (3simp1i.1 ph ps ch 3simp1 ax-mp)) thm (3simp2i () ((3simp1i.1 (/\/\ ph ps ch))) ps (3simp1i.1 ph ps ch 3simp2 ax-mp)) thm (3simp3i () ((3simp1i.1 (/\/\ ph ps ch))) ch (3simp1i.1 ph ps ch 3simp3 ax-mp)) thm (3simp1d () ((3simp1d.1 (-> ph (/\/\ ps ch th)))) (-> ph ps) (3simp1d.1 ps ch th 3simp1 syl)) thm (3simp2d () ((3simp1d.1 (-> ph (/\/\ ps ch th)))) (-> ph ch) (3simp1d.1 ps ch th 3simp2 syl)) thm (3simp3d () ((3simp1d.1 (-> ph (/\/\ ps ch th)))) (-> ph th) (3simp1d.1 ps ch th 3simp3 syl)) thm (3adant1 () ((3adant.1 (-> (/\ ph ps) ch))) (-> (/\/\ th ph ps) ch) (th ph ps 3simpc 3adant.1 syl)) thm (3adant2 () ((3adant.1 (-> (/\ ph ps) ch))) (-> (/\/\ ph th ps) ch) (ph th ps 3simpb 3adant.1 syl)) thm (3adant3 () ((3adant.1 (-> (/\ ph ps) ch))) (-> (/\/\ ph ps th) ch) (ph ps th 3simpa 3adant.1 syl)) thm (3ad2ant1 () ((3ad2ant.1 (-> ph ch))) (-> (/\/\ ph ps th) ch) (3ad2ant.1 th adantr ps 3adant2)) thm (3ad2ant2 () ((3ad2ant.1 (-> ph ch))) (-> (/\/\ ps ph th) ch) (3ad2ant.1 th adantr ps 3adant1)) thm (3ad2ant3 () ((3ad2ant.1 (-> ph ch))) (-> (/\/\ ps th ph) ch) (3ad2ant.1 th adantl ps 3adant1)) thm (3adantl1 () ((3adantl.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\/\ ta ph ps) ch) th) (3adantl.1 ex ta 3adant1 imp)) thm (3adantl2 () ((3adantl.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\/\ ph ta ps) ch) th) (3adantl.1 ex ta 3adant2 imp)) thm (3adantl3 () ((3adantl.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\/\ ph ps ta) ch) th) (3adantl.1 ex ta 3adant3 imp)) thm (3adantr1 () ((3adantr.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\/\ ta ps ch)) th) (3adantr.1 ancoms ta 3adantl1 ancoms)) thm (3adantr2 () ((3adantr.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\/\ ps ta ch)) th) (3adantr.1 ancoms ta 3adantl2 ancoms)) thm (3adantr3 () ((3adantr.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\/\ ps ch ta)) th) (3adantr.1 ancoms ta 3adantl3 ancoms)) thm (3mix1 () () (-> ph (\/\/ ph ps ch)) (ph (\/ ps ch) orc ph ps ch 3orass sylibr)) thm (3mix2 () () (-> ph (\/\/ ps ph ch)) (ph ch ps 3mix1 ps ph ch 3orrot sylibr)) thm (3mix3 () () (-> ph (\/\/ ps ch ph)) (ph ps ch 3mix1 ph ps ch 3orrot sylib)) thm (3pm3.2i () ((3pm3.2i.1 ph) (3pm3.2i.2 ps) (3pm3.2i.3 ch)) (/\/\ ph ps ch) (ph ps ch df-3an 3pm3.2i.1 3pm3.2i.2 pm3.2i 3pm3.2i.3 mpbir2an)) thm (3jca () ((3jca.1 (-> ph ps)) (3jca.2 (-> ph ch)) (3jca.3 (-> ph th))) (-> ph (/\/\ ps ch th)) (3jca.1 3jca.2 jca 3jca.3 jca ps ch th df-3an sylibr)) thm (3jcad () ((3jcad.1 (-> ph (-> ps ch))) (3jcad.2 (-> ph (-> ps th))) (3jcad.3 (-> ph (-> ps ta)))) (-> ph (-> ps (/\/\ ch th ta))) (3jcad.1 imp 3jcad.2 imp 3jcad.3 imp 3jca ex)) thm (3anim123i () ((3anim123i.1 (-> ph ps)) (3anim123i.2 (-> ch th)) (3anim123i.3 (-> ta et))) (-> (/\/\ ph ch ta) (/\/\ ps th et)) (3anim123i.1 3anim123i.2 anim12i 3anim123i.3 anim12i ph ch ta df-3an ps th et df-3an 3imtr4)) thm (3anbi123i () ((bi3.1 (<-> ph ps)) (bi3.2 (<-> ch th)) (bi3.3 (<-> ta et))) (<-> (/\/\ ph ch ta) (/\/\ ps th et)) (bi3.1 bi3.2 anbi12i bi3.3 anbi12i ph ch ta df-3an ps th et df-3an 3bitr4)) thm (3orbi123i () ((bi3.1 (<-> ph ps)) (bi3.2 (<-> ch th)) (bi3.3 (<-> ta et))) (<-> (\/\/ ph ch ta) (\/\/ ps th et)) (bi3.1 bi3.2 orbi12i bi3.3 orbi12i ph ch ta df-3or ps th et df-3or 3bitr4)) thm (3anbi1i () ((3anbi1i.1 (<-> ph ps))) (<-> (/\/\ ph ch th) (/\/\ ps ch th)) (3anbi1i.1 ch pm4.2 th pm4.2 3anbi123i)) thm (3anbi2i () ((3anbi1i.1 (<-> ph ps))) (<-> (/\/\ ch ph th) (/\/\ ch ps th)) (ch pm4.2 3anbi1i.1 th pm4.2 3anbi123i)) thm (3imp () ((3imp.1 (-> ph (-> ps (-> ch th))))) (-> (/\/\ ph ps ch) th) (ph ps ch df-3an 3imp.1 imp31 sylbi)) thm (3impa () ((3impa.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\/\ ph ps ch) th) (3impa.1 exp31 3imp)) thm (3impb () ((3impb.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\/\ ph ps ch) th) (3impb.1 exp32 3imp)) thm (3impia () ((3impia.1 (-> (/\ ph ps) (-> ch th)))) (-> (/\/\ ph ps ch) th) (3impia.1 ex 3imp)) thm (3impib () ((3impib.1 (-> ph (-> (/\ ps ch) th)))) (-> (/\/\ ph ps ch) th) (3impib.1 exp3a 3imp)) thm (3exp () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> ph (-> ps (-> ch th))) (ph ps ch df-3an 3exp.1 sylbir exp31)) thm (3expa () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\ (/\ ph ps) ch) th) (3exp.1 3exp imp31)) thm (3expb () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\ ph (/\ ps ch)) th) (3exp.1 3exp imp32)) thm (3com12 () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ps ph ch) th) (3exp.1 3exp com12 3imp)) thm (3com13 () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ch ps ph) th) (ch ps ph 3anrev 3exp.1 sylbi)) thm (3com23 () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ph ch ps) th) (3exp.1 3exp com23 3imp)) thm (3coml () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ps ch ph) th) (3exp.1 3com23 3com13)) thm (3comr () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ch ph ps) th) (3exp.1 3coml 3coml)) thm (3imp1 () ((3imp1.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ (/\/\ ph ps ch) th) ta) (3imp1.1 3imp imp)) thm (3exp1 () ((3exp1.1 (-> (/\ (/\/\ ph ps ch) th) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (3exp1.1 ex 3exp)) thm (3adant1l () ((3adant1l.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ (/\ ta ph) ps ch) th) (3adant1l.1 3expb ta adantll 3impb)) thm (3adant1r () ((3adant1l.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ (/\ ph ta) ps ch) th) (3adant1l.1 3expb ta adantlr 3impb)) thm (3adant2l () ((3adant1l.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ph (/\ ta ps) ch) th) (3adant1l.1 3com12 ta 3adant1l 3com12)) thm (3adant2r () ((3adant1l.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ph (/\ ps ta) ch) th) (3adant1l.1 3com12 ta 3adant1r 3com12)) thm (3adant3l () ((3adant1l.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ph ps (/\ ta ch)) th) (3adant1l.1 3com13 ta 3adant1l 3com13)) thm (3adant3r () ((3adant1l.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ph ps (/\ ch ta)) th) (3adant1l.1 3com13 ta 3adant1r 3com13)) var (wff ze) thm (syl3an1 () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an1.2 (-> ta ph))) (-> (/\/\ ta ps ch) th) (syl3an.1 3expb syl3an1.2 sylan 3impb)) thm (syl3an2 () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an2.2 (-> ta ps))) (-> (/\/\ ph ta ch) th) (syl3an.1 3exp syl3an2.2 syl5 3imp)) thm (syl3an3 () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an3.2 (-> ta ch))) (-> (/\/\ ph ps ta) th) (syl3an.1 3exp syl3an3.2 syl7 3imp)) thm (syl3an1b () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an1b.2 (<-> ta ph))) (-> (/\/\ ta ps ch) th) (syl3an.1 syl3an1b.2 biimp syl3an1)) thm (syl3an2b () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an2b.2 (<-> ta ps))) (-> (/\/\ ph ta ch) th) (syl3an.1 syl3an2b.2 biimp syl3an2)) thm (syl3an3b () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an3b.2 (<-> ta ch))) (-> (/\/\ ph ps ta) th) (syl3an.1 syl3an3b.2 biimp syl3an3)) thm (syl3an1br () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an1br.2 (<-> ph ta))) (-> (/\/\ ta ps ch) th) (syl3an.1 syl3an1br.2 biimpr syl3an1)) thm (syl3an2br () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an2br.2 (<-> ps ta))) (-> (/\/\ ph ta ch) th) (syl3an.1 syl3an2br.2 biimpr syl3an2)) thm (syl3an3br () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an3br.2 (<-> ch ta))) (-> (/\/\ ph ps ta) th) (syl3an.1 syl3an3br.2 biimpr syl3an3)) thm (syl3an () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an.2 (-> ta ph)) (syl3an.3 (-> et ps)) (syl3an.4 (-> ze ch))) (-> (/\/\ ta et ze) th) (syl3an.2 syl3an.3 syl3an.4 3anim123i syl3an.1 syl)) thm (syl3anb () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3anb.2 (<-> ta ph)) (syl3anb.3 (<-> et ps)) (syl3anb.4 (<-> ze ch))) (-> (/\/\ ta et ze) th) (syl3anb.2 syl3anb.3 syl3anb.4 3anbi123i syl3an.1 sylbi)) thm (syl3anl1 () ((syl3anl1.1 (-> (/\ (/\/\ ph ps ch) th) ta)) (syl3an11.2 (-> et ph))) (-> (/\ (/\/\ et ps ch) th) ta) (syl3anl1.1 ex syl3an11.2 syl3an1 imp)) thm (syl3anl2 () ((syl3anl1.1 (-> (/\ (/\/\ ph ps ch) th) ta)) (syl3an12.2 (-> et ps))) (-> (/\ (/\/\ ph et ch) th) ta) (syl3anl1.1 ex syl3an12.2 syl3an2 imp)) thm (syl3anl3 () ((syl3anl1.1 (-> (/\ (/\/\ ph ps ch) th) ta)) (syl3an13.2 (-> et ch))) (-> (/\ (/\/\ ph ps et) th) ta) (syl3anl1.1 ex syl3an13.2 syl3an3 imp)) thm (syl3anc () ((syl3anc.1 (-> (/\/\ ph ps ch) th)) (syl3anc.2 (-> ta ph)) (syl3anc.3 (-> ta ps)) (syl3anc.4 (-> ta ch))) (-> ta th) (syl3anc.2 syl3anc.3 syl3anc.4 3jca syl3anc.1 syl)) thm (3impdi () ((3impdi.1 (-> (/\ (/\ ph ps) (/\ ph ch)) th))) (-> (/\/\ ph ps ch) th) (3impdi.1 anandis 3impb)) thm (3impdir () ((3impdir.1 (-> (/\ (/\ ph ps) (/\ ch ps)) th))) (-> (/\/\ ph ch ps) th) (3impdir.1 anandirs 3impa)) thm (3ori () ((3ori.1 (\/\/ ph ps ch))) (-> (/\ (-. ph) (-. ps)) ch) (ph ps ioran 3ori.1 ph ps ch df-3or mpbi ori sylbir)) thm (3jao () () (-> (/\/\ (-> ph ps) (-> ch ps) (-> th ps)) (-> (\/\/ ph ch th) ps)) (ph ps ch jao (\/ ph ch) ps th jao syl6 3imp ph ch th df-3or syl5ib)) thm (3jaoi () ((3jaoi.1 (-> ph ps)) (3jaoi.2 (-> ch ps)) (3jaoi.3 (-> th ps))) (-> (\/\/ ph ch th) ps) (3jaoi.1 3jaoi.2 3jaoi.3 3pm3.2i ph ps ch th 3jao ax-mp)) thm (3jaod () ((3jaod.1 (-> ph (-> ps ch))) (3jaod.2 (-> ph (-> th ch))) (3jaod.3 (-> ph (-> ta ch)))) (-> ph (-> (\/\/ ps th ta) ch)) (ps ch th ta 3jao 3jaod.1 3jaod.2 3jaod.3 syl3anc)) thm (3jaoian () ((3jaoian.1 (-> (/\ ph ps) ch)) (3jaoian.2 (-> (/\ th ps) ch)) (3jaoian.3 (-> (/\ ta ps) ch))) (-> (/\ (\/\/ ph th ta) ps) ch) (3jaoian.1 ex 3jaoian.2 ex 3jaoian.3 ex 3jaoi imp)) thm (3jaodan () ((3jaodan.1 (-> (/\ ph ps) ch)) (3jaodan.2 (-> (/\ ph th) ch)) (3jaodan.3 (-> (/\ ph ta) ch))) (-> (/\ ph (\/\/ ps th ta)) ch) (3jaodan.1 ex 3jaodan.2 ex 3jaodan.3 ex 3jaod imp)) thm (syl3an9b () ((syl3an9b.1 (-> ph (<-> ps ch))) (syl3an9b.2 (-> th (<-> ch ta))) (syl3an9b.3 (-> et (<-> ta ze)))) (-> (/\/\ ph th et) (<-> ps ze)) (syl3an9b.1 syl3an9b.2 sylan9bb syl3an9b.3 sylan9bb 3impa)) thm (3orbi123d () ((bi3d.1 (-> ph (<-> ps ch))) (bi3d.2 (-> ph (<-> th ta))) (bi3d.3 (-> ph (<-> et ze)))) (-> ph (<-> (\/\/ ps th et) (\/\/ ch ta ze))) (bi3d.1 bi3d.2 orbi12d bi3d.3 orbi12d ps th et df-3or ch ta ze df-3or 3bitr4g)) thm (3anbi123d () ((bi3d.1 (-> ph (<-> ps ch))) (bi3d.2 (-> ph (<-> th ta))) (bi3d.3 (-> ph (<-> et ze)))) (-> ph (<-> (/\/\ ps th et) (/\/\ ch ta ze))) (bi3d.1 bi3d.2 anbi12d bi3d.3 anbi12d ps th et df-3an ch ta ze df-3an 3bitr4g)) thm (3anbi12d () ((3anbi12d.1 (-> ph (<-> ps ch))) (3anbi12d.2 (-> ph (<-> th ta)))) (-> ph (<-> (/\/\ ps th et) (/\/\ ch ta et))) (3anbi12d.1 3anbi12d.2 ph et pm4.2i 3anbi123d)) thm (3anbi13d () ((3anbi12d.1 (-> ph (<-> ps ch))) (3anbi12d.2 (-> ph (<-> th ta)))) (-> ph (<-> (/\/\ ps et th) (/\/\ ch et ta))) (3anbi12d.1 ph et pm4.2i 3anbi12d.2 3anbi123d)) thm (3anbi23d () ((3anbi12d.1 (-> ph (<-> ps ch))) (3anbi12d.2 (-> ph (<-> th ta)))) (-> ph (<-> (/\/\ et ps th) (/\/\ et ch ta))) (ph et pm4.2i 3anbi12d.1 3anbi12d.2 3anbi123d)) thm (3anbi1d () ((3anbi1d.1 (-> ph (<-> ps ch)))) (-> ph (<-> (/\/\ ps th ta) (/\/\ ch th ta))) (3anbi1d.1 ph th pm4.2i ta 3anbi12d)) thm (3anbi3d () ((3anbi1d.1 (-> ph (<-> ps ch)))) (-> ph (<-> (/\/\ th ta ps) (/\/\ th ta ch))) (ph th pm4.2i 3anbi1d.1 ta 3anbi13d)) thm (3anim123d () ((3anim123d.1 (-> ph (-> ps ch))) (3anim123d.2 (-> ph (-> th ta))) (3anim123d.3 (-> ph (-> et ze)))) (-> ph (-> (/\/\ ps th et) (/\/\ ch ta ze))) (3anim123d.1 3anim123d.2 anim12d 3anim123d.3 anim12d ps th et df-3an ch ta ze df-3an 3imtr4g)) thm (3orim123d () ((3anim123d.1 (-> ph (-> ps ch))) (3anim123d.2 (-> ph (-> th ta))) (3anim123d.3 (-> ph (-> et ze)))) (-> ph (-> (\/\/ ps th et) (\/\/ ch ta ze))) (3anim123d.1 3anim123d.2 orim12d 3anim123d.3 orim12d ps th et df-3or ch ta ze df-3or 3imtr4g)) thm (an6 () () (<-> (/\ (/\/\ ph ps ch) (/\/\ th ta et)) (/\/\ (/\ ph th) (/\ ps ta) (/\ ch et))) (ph ps ch df-3an th ta et df-3an anbi12i (/\ ph ps) ch (/\ th ta) et an4 ph ps th ta an4 (/\ ch et) anbi1i 3bitr (/\ ph th) (/\ ps ta) (/\ ch et) df-3an bitr4)) thm (mp3an1 () ((mp3an1.1 ph) (mp3an1.2 (-> (/\/\ ph ps ch) th))) (-> (/\ ps ch) th) (mp3an1.1 mp3an1.2 3expb mpan)) thm (mp3an2 () ((mp3an2.1 ps) (mp3an2.2 (-> (/\/\ ph ps ch) th))) (-> (/\ ph ch) th) (mp3an2.1 mp3an2.2 3expa mpanl2)) thm (mp3an3 () ((mp3an3.1 ch) (mp3an3.2 (-> (/\/\ ph ps ch) th))) (-> (/\ ph ps) th) (mp3an3.1 mp3an3.2 3expa mpan2)) thm (mp3an12 () ((mp3an12.1 ph) (mp3an12.2 ps) (mp3an12.3 (-> (/\/\ ph ps ch) th))) (-> ch th) (mp3an12.2 mp3an12.1 mp3an12.3 mp3an1 mpan)) thm (mp3an13 () ((mp3an13.1 ph) (mp3an13.2 ch) (mp3an13.3 (-> (/\/\ ph ps ch) th))) (-> ps th) (mp3an13.1 mp3an13.2 mp3an13.3 mp3an3 mpan)) thm (mp3an23 () ((mp3an23.1 ps) (mp3an23.2 ch) (mp3an23.3 (-> (/\/\ ph ps ch) th))) (-> ph th) (mp3an23.1 mp3an23.2 mp3an23.3 mp3an3 mpan2)) thm (mp3an1i () ((mp3an1i.1 ps) (mp3an1i.2 (-> ph (-> (/\/\ ps ch th) ta)))) (-> ph (-> (/\ ch th) ta)) (mp3an1i.1 mp3an1i.2 com12 mp3an1 com12)) thm (mp3anl1 () ((mp3anl1.1 ph) (mp3anl1.2 (-> (/\ (/\/\ ph ps ch) th) ta))) (-> (/\ (/\ ps ch) th) ta) (mp3anl1.1 mp3anl1.2 ex mp3an1 imp)) thm (mp3anl2 () ((mp3anl2.1 ps) (mp3anl2.2 (-> (/\ (/\/\ ph ps ch) th) ta))) (-> (/\ (/\ ph ch) th) ta) (mp3anl2.1 mp3anl2.2 ex mp3an2 imp)) thm (mp3anl3 () ((mp3anl3.1 ch) (mp3anl3.2 (-> (/\ (/\/\ ph ps ch) th) ta))) (-> (/\ (/\ ph ps) th) ta) (mp3anl3.1 mp3anl3.2 ex mp3an3 imp)) thm (mp3an () ((mp3an.1 ph) (mp3an.2 ps) (mp3an.3 ch) (mp3an.4 (-> (/\/\ ph ps ch) th))) th (mp3an.2 mp3an.3 mp3an.1 mp3an.4 mp3an1 mp2an)) thm (biimp3a () ((biimp3a.1 (-> (/\ ph ps) (<-> ch th)))) (-> (/\/\ ph ps ch) th) (biimp3a.1 biimpa 3impa)) thm (3anandirs () ((3anandirs.1 (-> (/\/\ (/\ ph ta) (/\ ps ta) (/\ ch ta)) th))) (-> (/\ (/\/\ ph ps ch) ta) th) (3anandirs.1 ph ps ch 3simp1 ta anim1i ph ps ch 3simp2 ta anim1i ph ps ch 3simp3 ta anim1i syl3anc)) thm (ecase23d () ((ecase23d.1 (-> ph (-. ch))) (ecase23d.2 (-> ph (-. th))) (ecase23d.3 (-> ph (\/\/ ps ch th)))) (-> ph ps) (ecase23d.1 ecase23d.2 jca ch th ioran sylibr ecase23d.3 ps ch th 3orass sylib ord mt3d)) thm (3ecase () ((3ecase.1 (-> (-. ph) th)) (3ecase.2 (-> (-. ps) th)) (3ecase.3 (-> (-. ch) th)) (3ecase.4 (-> (/\/\ ph ps ch) th))) th (3ecase.4 3exp 3ecase.1 ch a1d ps a1d pm2.61i 3ecase.2 3ecase.3 pm2.61nii)) thm (meredith () () (-> (-> (-> (-> (-> ph ps) (-> (-. ch) (-. th))) ch) ta) (-> (-> ta ph) (-> th ph))) (ch (-> (-. ch) (-> (-. ph) (-. th))) ax-3 ph ps pm2.21 (-> (-. ch) (-. th)) imim1i com23 syl5 ta imim1i con3d (-. ch) (-> (-. ph) (-. th)) pm2.27 impi com12 (-. ta) imim2d com12 a2d ta ph con3 syl5 ph th ax-3 syl6 syl)) thm (merlem1 () () (-> (-> (-> ch (-> (-. ph) ps)) ta) (-> ph ta)) ((-. ph) ps (-> (-. ta) (-. ch)) (-. (-> (-. ph) ps)) ta meredith (-> (-. ph) ps) (-> (-. (-> (-. ta) (-. ch))) (-. (-. (-> (-. ph) ps)))) ta ch (-> (-> ta (-. ph)) (-> (-. (-> (-. ph) ps)) (-. ph))) meredith ax-mp ta (-. ph) (-> (-. ph) ps) ph (-> ch (-> (-. ph) ps)) meredith ax-mp)) thm (merlem2 () () (-> (-> (-> ph ph) ch) (-> th ch)) ((-> ch ch) ph (-. th) ph merlem1 ch ch ph th (-> ph ph) meredith ax-mp)) thm (merlem3 () () (-> (-> (-> ps ch) ph) (-> ch ph)) ((-. ch) (-> (-. ch) (-. ch)) (-> ph ph) merlem2 (-> (-. ch) (-. ch)) (-> (-> ph ph) (-> (-. ch) (-. ch))) (-> (-> (-> ch ph) (-> (-. ps) (-. ps))) ps) merlem2 ax-mp ch ph ps ps (-> (-> ph ph) (-> (-. ch) (-. ch))) meredith ax-mp ph ph ch ch (-> ps ch) meredith ax-mp)) thm (merlem4 () () (-> ta (-> (-> ta ph) (-> th ph))) (ph ph th th ta meredith (-> (-> (-> ph ph) (-> (-. th) (-. th))) th) ta (-> (-> ta ph) (-> th ph)) merlem3 ax-mp)) thm (merlem5 () () (-> (-> ph ps) (-> (-. (-. ph)) ps)) (ps ps ps ps ps meredith ps ps ps (-. (-. ph)) ph meredith (-> ph ps) (-. ph) ps (-. (-> (-> (-> (-> (-> ps ps) (-> (-. ps) (-. ps))) ps) ps) (-> (-> ps ps) (-> ps ps)))) merlem1 (-> (-> (-> (-> ph ps) (-> (-. (-. ph)) ps)) (-. (-> (-> (-> (-> (-> ps ps) (-> (-. ps) (-. ps))) ps) ps) (-> (-> ps ps) (-> ps ps))))) (-> (-. ph) (-. (-> (-> (-> (-> (-> ps ps) (-> (-. ps) (-. ps))) ps) ps) (-> (-> ps ps) (-> ps ps)))))) ph (-> (-> (-> ps ps) (-> (-. ps) (-. (-. (-. ph))))) ps) merlem4 ax-mp (-> (-> ph ps) (-> (-. (-. ph)) ps)) (-. (-> (-> (-> (-> (-> ps ps) (-> (-. ps) (-. ps))) ps) ps) (-> (-> ps ps) (-> ps ps)))) ph (-> (-> (-> (-> (-> ps ps) (-> (-. ps) (-. ps))) ps) ps) (-> (-> ps ps) (-> ps ps))) (-> (-> (-> (-> ps ps) (-> (-. ps) (-. (-. (-. ph))))) ps) ph) meredith ax-mp ax-mp ax-mp)) thm (merlem6 () () (-> ch (-> (-> (-> ps ch) ph) (-> th ph))) ((-> ps ch) ph th merlem4 ps ch (-> (-> (-> ps ch) ph) (-> th ph)) merlem3 ax-mp)) thm (merlem7 () () (-> ph (-> (-> (-> ps ch) th) (-> (-> (-> ch ta) (-> (-. th) (-. ps))) th))) ((-> ps ch) th (-> (-> ch ta) (-> (-. th) (-. ps))) merlem4 (-> (-> (-> ch ta) (-> (-. th) (-. ps))) th) (-> (-> ps ch) th) (-. ph) (-. ch) merlem6 ch ta th ps (-> (-> (-> (-> (-> ps ch) th) (-> (-> (-> ch ta) (-> (-. th) (-. ps))) th)) (-. ph)) (-> (-. ch) (-. ph))) meredith ax-mp (-> (-> (-> ps ch) th) (-> (-> (-> ch ta) (-> (-. th) (-. ps))) th)) (-. ph) ch ph (-> ps ch) meredith ax-mp ax-mp)) thm (merlem8 () () (-> (-> (-> ps ch) th) (-> (-> (-> ch ta) (-> (-. th) (-. ps))) th)) (ph ph ph ph ph meredith (-> (-> (-> (-> (-> ph ph) (-> (-. ph) (-. ph))) ph) ph) (-> (-> ph ph) (-> ph ph))) ps ch th ta merlem7 ax-mp)) thm (merlem9 () () (-> (-> (-> ph ps) (-> ch (-> th (-> ps ta)))) (-> et (-> ch (-> th (-> ps ta))))) ((-> th (-> ps ta)) ch (-. et) (-. ps) merlem6 th (-> ps ta) (-> (-> (-> ch (-> th (-> ps ta))) (-. et)) (-> (-. ps) (-. et))) (-> (-. (-> (-. (-> (-> (-> ch (-> th (-> ps ta))) (-. et)) (-> (-. ps) (-. et)))) (-. th))) (-. ph)) merlem8 ax-mp ps ta (-> (-. (-> (-> (-> ch (-> th (-> ps ta))) (-. et)) (-> (-. ps) (-. et)))) (-. th)) ph (-> (-> (-> ch (-> th (-> ps ta))) (-. et)) (-> (-. ps) (-. et))) meredith ax-mp (-> ch (-> th (-> ps ta))) (-. et) ps et (-> ph ps) meredith ax-mp)) thm (merlem10 () () (-> (-> ph (-> ph ps)) (-> th (-> ph ps))) (ph ph ph ph ph meredith (-> ph ps) ph ph th ph meredith (-> (-> (-> (-> ph ps) ph) (-> (-. ph) (-. th))) ph) ph (-> ph (-> ph ps)) th ps (-> (-> (-> (-> (-> ph ph) (-> (-. ph) (-. ph))) ph) ph) (-> (-> ph ph) (-> ph ph))) merlem9 ax-mp ax-mp)) thm (merlem11 () () (-> (-> ph (-> ph ps)) (-> ph ps)) (ph ph ph ph ph meredith ph ps (-> ph (-> ph ps)) merlem10 (-> ph (-> ph ps)) (-> ph ps) (-> (-> (-> (-> (-> ph ph) (-> (-. ph) (-. ph))) ph) ph) (-> (-> ph ph) (-> ph ph))) merlem10 ax-mp ax-mp)) thm (merlem12 () () (-> (-> (-> th (-> (-. (-. ch)) ch)) ph) ph) (ch ch merlem5 ch (-> (-. (-. ch)) ch) th merlem2 ax-mp (-> th (-> (-. (-. ch)) ch)) ph (-> (-> th (-> (-. (-. ch)) ch)) ph) merlem4 ax-mp (-> (-> th (-> (-. (-. ch)) ch)) ph) ph merlem11 ax-mp)) thm (merlem13 () () (-> (-> ph ps) (-> (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph))) ps)) (th ch (-. (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph)))) merlem12 th ch (-. (-. ph)) merlem12 (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph))) (-. (-. ph)) merlem5 ax-mp (-> (-. (-. (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph))))) (-. (-. ph))) (-> (-. (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph)))) ps) (-. (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph)))) (-> th (-> (-. (-. ch)) ch)) merlem6 ax-mp (-. (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph)))) ps (-. (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph)))) (-. ph) (-> (-> th (-> (-. (-. ch)) ch)) (-. (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph))))) meredith ax-mp ax-mp (-> (-. ph) (-. (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph))))) (-> ps ps) ph (-> (-> (-> ps ps) (-> (-. ph) (-. (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph)))))) ph) merlem6 ax-mp (-> (-> (-> ps ps) (-> (-. ph) (-. (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph)))))) ph) ph merlem11 ax-mp ps ps ph (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph))) ph meredith ax-mp)) thm (luk-1 () () (-> (-> ph ps) (-> (-> ps ch) (-> ph ch))) (ch ch (-. (-. ph)) ph ps meredith ph ps (-> ch ch) (-. ph) merlem13 (-> ph ps) (-> (-> (-> (-> ch ch) (-> (-. (-. (-. ph))) (-. ph))) (-. (-. ph))) ps) (-> (-> (-> ps ch) (-> ph ch)) ph) (-. (-> ph ps)) merlem13 ax-mp (-> (-> ps ch) (-> ph ch)) ph (-. (-. (-> ph ps))) (-> ph ps) (-> (-> (-> (-> ch ch) (-> (-. (-. (-. ph))) (-. ph))) (-. (-. ph))) ps) meredith ax-mp ax-mp)) thm (luk-2 () () (-> (-> (-. ph) ph) ph) (ph (-. (-> (-. ph) ph)) merlem5 (-> (-> ph (-. (-> (-. ph) ph))) (-> (-. (-. ph)) (-. (-> (-. ph) ph)))) (-. ph) (-> (-> (-> ph (-. (-> (-. ph) ph))) (-> (-. (-. ph)) (-. (-> (-. ph) ph)))) (-. ph)) merlem4 ax-mp (-> (-> (-> ph (-. (-> (-. ph) ph))) (-> (-. (-. ph)) (-. (-> (-. ph) ph)))) (-. ph)) (-. ph) merlem11 ax-mp ph (-. (-> (-. ph) ph)) (-. ph) (-> (-. ph) ph) (-. ph) meredith ax-mp (-> (-. ph) ph) ph merlem11 ax-mp)) thm (luk-3 () () (-> ph (-> (-. ph) ps)) ((-. ph) ps merlem11 (-. ph) ph ps (-> (-. ph) ps) merlem1 ax-mp)) thm (luklem1 () ((luklem1.1 (-> ph ps)) (luklem1.2 (-> ps ch))) (-> ph ch) (luklem1.2 luklem1.1 ph ps ch luk-1 ax-mp ax-mp)) thm (luklem2 () () (-> (-> ph (-. ps)) (-> (-> (-> ph ch) th) (-> ps th))) (ph (-. ps) ch luk-1 ps ch luk-3 ps (-> (-. ps) ch) (-> ph ch) luk-1 ax-mp luklem1 ps (-> ph ch) th luk-1 luklem1)) thm (luklem3 () () (-> ph (-> (-> (-> (-. ph) ps) ch) (-> th ch))) (ph (-. th) luk-3 (-. ph) th ps ch luklem2 luklem1)) thm (luklem4 () () (-> (-> (-> (-> (-. ph) ph) ph) ps) ps) ((-> (-> (-. ph) ph) ph) luk-2 ph luk-2 (-> (-> (-. ph) ph) ph) (-> (-> (-. ph) ph) ph) (-> (-> (-. ph) ph) ph) (-. ps) luklem3 ax-mp ax-mp (-. ps) (-> (-> (-. ph) ph) ph) ps luk-1 ax-mp ps luk-2 luklem1)) thm (luklem5 () () (-> ph (-> ps ph)) (ph ph ph ps luklem3 ph (-> ps ph) luklem4 luklem1)) thm (luklem6 () () (-> (-> ph (-> ph ps)) (-> ph ps)) (ph (-> ph ps) ps luk-1 (-. (-> ph ps)) (-. ps) luklem5 (-. ps) (-> ph ps) ps ps luklem2 ps (-> (-> ph ps) ps) luklem4 luklem1 luklem1 (-. (-> ph ps)) (-> (-> ph ps) ps) (-> ph ps) luk-1 ax-mp (-> (-> (-> ph ps) ps) (-> ph ps)) (-> (-. (-> ph ps)) (-> ph ps)) (-> ph ps) luk-1 ax-mp (-> ph ps) (-> (-> (-> (-> ph ps) ps) (-> ph ps)) (-> ph ps)) luklem4 ax-mp luklem1)) thm (luklem7 () () (-> (-> ph (-> ps ch)) (-> ps (-> ph ch))) (ph (-> ps ch) ch luk-1 ps (-> ps ch) luklem5 (-> ps ch) ps ch luk-1 luklem1 (-> ps ch) ch luklem6 luklem1 ps (-> (-> ps ch) ch) (-> ph ch) luk-1 ax-mp luklem1)) thm (luklem8 () () (-> (-> ph ps) (-> (-> ch ph) (-> ch ps))) (ch ph ps luk-1 (-> ch ph) (-> ph ps) (-> ch ps) luklem7 ax-mp)) thm (ax1 () () (-> ph (-> ps ph)) (ph ps luklem5)) thm (ax2 () () (-> (-> ph (-> ps ch)) (-> (-> ph ps) (-> ph ch))) (ph ps ch luklem7 ps (-> ph ch) ph luklem8 ph ch luklem6 (-> ph (-> ph ch)) (-> ph ch) (-> ph ps) luklem8 ax-mp luklem1 luklem1)) thm (ax3 () () (-> (-> (-. ph) (-. ps)) (-> ps ph)) ((-. ph) ps ph ph luklem2 ph (-> ps ph) luklem4 luklem1)) var (set x) var (set y) var (set z) var (set w) var (set v) var (set u) thm (ax46 () () (-> (-> (A. x (-. (A. x ph))) (A. x ph)) ph) (x ph ax-6 x ph ax-4 ja)) thm (ax4 () () (-> (A. x ph) ph) ((A. x ph) (A. x (-. (A. x ph))) ax-1 x ph ax46 syl)) thm (ax6 () () (-> (-. (A. x (-. (A. x ph)))) ph) ((A. x (-. (A. x ph))) (A. x ph) pm2.21 x ph ax46 syl)) thm (a4i () ((a4i.1 (A. x ph))) ph (a4i.1 x ph ax-4 ax-mp)) thm (gen2 () ((gen2.1 ph)) (A. x (A. y ph)) (gen2.1 y ax-gen x ax-gen)) thm (a4s () ((a4s.1 (-> ph ps))) (-> (A. x ph) ps) (x ph ax-4 a4s.1 syl)) thm (a4sd () ((a4sd.1 (-> ph (-> ps ch)))) (-> ph (-> (A. x ps) ch)) (a4sd.1 x ps ax-4 syl5)) thm (mpg () ((mpg.1 (-> (A. x ph) ps)) (mpg.2 ph)) ps (mpg.2 x ax-gen mpg.1 ax-mp)) thm (mpgbi () ((mpgbi.1 (<-> (A. x ph) ps)) (mpgbi.2 ph)) ps (mpgbi.1 biimp mpgbi.2 mpg)) thm (mpgbir () ((mpgbir.1 (<-> ph (A. x ps))) (mpgbir.2 ps)) ph (mpgbir.1 biimpr mpgbir.2 mpg)) thm (a5i () ((a5i.1 (-> (A. x ph) ps))) (-> (A. x ph) (A. x ps)) (x ph ps ax-5 a5i.1 mpg)) thm (a6e () () (-> (E. x (A. x ph)) ph) (x (A. x ph) df-ex x ph ax-6 sylbi)) thm (a7s () ((a7s.1 (-> (A. x (A. y ph)) ps))) (-> (A. y (A. x ph)) ps) (y x ph ax-7 a7s.1 syl)) thm (19.20 () () (-> (A. x (-> ph ps)) (-> (A. x ph) (A. x ps))) (x ph ax-4 ps imim1i x a4s a5i x ph ps ax-5 syl)) thm (19.20i () ((19.20i.1 (-> ph ps))) (-> (A. x ph) (A. x ps)) (19.20i.1 x a4s a5i)) thm (19.20i2 () ((19.20i.1 (-> ph ps))) (-> (A. x (A. y ph)) (A. x (A. y ps))) (19.20i.1 y 19.20i x 19.20i)) thm (19.20ii () ((19.20ii.1 (-> ph (-> ps ch)))) (-> (A. x ph) (-> (A. x ps) (A. x ch))) (19.20ii.1 x 19.20i x ps ch 19.20 syl)) thm (19.20d () ((19.20d.1 (-> ph (A. x ph))) (19.20d.2 (-> ph (-> ps ch)))) (-> ph (-> (A. x ps) (A. x ch))) (19.20d.1 19.20d.2 x 19.20ii syl)) thm (19.15 () () (-> (A. x (<-> ph ps)) (<-> (A. x ph) (A. x ps))) (ph ps bi1 x 19.20ii ph ps bi2 x 19.20ii impbid)) thm (19.21ai () ((19.21ai.1 (-> ph (A. x ph))) (19.21ai.2 (-> ph ps))) (-> ph (A. x ps)) (19.21ai.1 19.21ai.2 x 19.20i syl)) thm (albii () ((albii.1 (<-> ph ps))) (<-> (A. x ph) (A. x ps)) (x ph ps 19.15 albii.1 mpg)) thm (2albii () ((albii.1 (<-> ph ps))) (<-> (A. x (A. y ph)) (A. x (A. y ps))) (albii.1 y albii x albii)) thm (hbth () ((hbth.1 ph)) (-> ph (A. x ph)) (hbth.1 x ax-gen ph a1i)) thm (hbnt () () (-> (A. x (-> ph (A. x ph))) (-> (-. ph) (A. x (-. ph)))) (ph (A. x ph) con3 x 19.20ii x ph ax-6 con1i syl5)) thm (hbnd () ((hbnd.1 (-> ph (A. x ph))) (hbnd.2 (-> ph (-> ps (A. x ps))))) (-> ph (-> (-. ps) (A. x (-. ps)))) (hbnd.1 hbnd.2 19.21ai x ps hbnt syl)) thm (hbimd () ((hbimd.1 (-> ph (A. x ph))) (hbimd.2 (-> ph (-> ps (A. x ps)))) (hbimd.3 (-> ph (-> ch (A. x ch))))) (-> ph (-> (-> ps ch) (A. x (-> ps ch)))) (hbimd.1 hbimd.2 hbnd ps ch pm2.21 x 19.20i syl6com hbimd.3 ch ps ax-1 x 19.20i syl6com ja com12)) thm (hbald () ((hbald.1 (-> ph (A. y ph))) (hbald.2 (-> ph (-> ps (A. x ps))))) (-> ph (-> (A. y ps) (A. x (A. y ps)))) (hbald.1 hbald.2 19.20d y x ps ax-7 syl6)) thm (hba1 () () (-> (A. x ph) (A. x (A. x ph))) ((A. x ph) id a5i)) thm (hbne () ((hb.1 (-> ph (A. x ph)))) (-> (-. ph) (A. x (-. ph))) (x ph hbnt hb.1 mpg)) thm (hbal () ((hb.1 (-> ph (A. x ph)))) (-> (A. y ph) (A. x (A. y ph))) (hb.1 y 19.20i y x ph ax-7 syl)) thm (hbex () ((hb.1 (-> ph (A. x ph)))) (-> (E. y ph) (A. x (E. y ph))) (hb.1 hbne y hbal hbne y ph df-ex y ph df-ex x albii 3imtr4)) thm (hbim () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps)))) (-> (-> ph ps) (A. x (-> ph ps))) (ph id ph id x hbth hb.1 (-> ph ph) a1i hb.2 (-> ph ph) a1i hbimd ax-mp)) thm (hbor () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps)))) (-> (\/ ph ps) (A. x (\/ ph ps))) (hb.1 hbne hb.2 hbim ph ps df-or ph ps df-or x albii 3imtr4)) thm (hban () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps)))) (-> (/\ ph ps) (A. x (/\ ph ps))) (hb.1 hb.2 hbne hbim hbne ph ps df-an ph ps df-an x albii 3imtr4)) thm (hbbi () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps)))) (-> (<-> ph ps) (A. x (<-> ph ps))) (hb.1 hb.2 hbim hb.2 hb.1 hbim hban ph ps bi ph ps bi x albii 3imtr4)) thm (hb3or () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps))) (hb.3 (-> ch (A. x ch)))) (-> (\/\/ ph ps ch) (A. x (\/\/ ph ps ch))) (hb.1 hb.2 hbor hb.3 hbor ph ps ch df-3or ph ps ch df-3or x albii 3imtr4)) thm (hb3an () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps))) (hb.3 (-> ch (A. x ch)))) (-> (/\/\ ph ps ch) (A. x (/\/\ ph ps ch))) (hb.1 hb.2 hban hb.3 hban ph ps ch df-3an ph ps ch df-3an x albii 3imtr4)) thm (hbn1 () () (-> (-. (A. x ph)) (A. x (-. (A. x ph)))) (x ph hba1 hbne)) thm (hbe1 () () (-> (E. x ph) (A. x (E. x ph))) (x (-. ph) hbn1 x ph df-ex x ph df-ex x albii 3imtr4)) thm (ax6-2 () () (-> (-. (A. x ph)) (A. x (-. (A. x ph)))) (x ph hbn1)) thm (modal-5 () () (-> (-. (A. x (-. ph))) (A. x (-. (A. x (-. ph))))) (x (-. ph) hbn1)) thm (modal-b () () (-> ph (A. x (-. (A. x (-. ph))))) (x (-. ph) ax-6 a3i)) thm (19.8a () () (-> ph (E. x ph)) (x (-. ph) ax-4 con2i x ph df-ex sylibr)) thm (19.2 () () (-> (A. x ph) (E. x ph)) (ph x 19.8a x a4s)) thm (19.3r () ((19.3r.1 (-> ph (A. x ph)))) (<-> ph (A. x ph)) (19.3r.1 x ph ax-4 impbi)) thm (alcom () () (<-> (A. x (A. y ph)) (A. y (A. x ph))) (x y ph ax-7 y x ph ax-7 impbi)) thm (alnex () () (<-> (A. x (-. ph)) (-. (E. x ph))) (x ph df-ex con2bii)) thm (alex () () (<-> (A. x ph) (-. (E. x (-. ph)))) (ph pm4.13 x albii x (-. ph) alnex bitr)) thm (19.9r () ((19.9r.1 (-> ph (A. x ph)))) (<-> ph (E. x ph)) (ph x 19.8a x ph df-ex 19.9r.1 con3i x 19.20i con3i x ph ax-6 syl sylbi impbi)) thm (19.9t () () (-> (A. x (-> ph (A. x ph))) (-> (E. x ph) ph)) (x ph hbnt con1d x ph df-ex syl5ib)) thm (19.9d () ((19.9d.1 (-> ps (A. x ps))) (19.9d.2 (-> ps (-> ph (A. x ph))))) (-> ps (-> (E. x ph) ph)) (19.9d.1 19.9d.2 x 19.20i x ph 19.9t 3syl)) thm (exnal () () (<-> (E. x (-. ph)) (-. (A. x ph))) (x ph alex con2bii)) thm (19.22 () () (-> (A. x (-> ph ps)) (-> (E. x ph) (E. x ps))) (ph ps con3 x 19.20ii con3d x ph df-ex x ps df-ex 3imtr4g)) thm (19.22i () ((19.22i.1 (-> ph ps))) (-> (E. x ph) (E. x ps)) (x ph ps 19.22 19.22i.1 mpg)) thm (19.22i2 () ((19.22i.1 (-> ph ps))) (-> (E. x (E. y ph)) (E. x (E. y ps))) (19.22i.1 y 19.22i x 19.22i)) thm (alinexa () () (<-> (A. x (-> ph (-. ps))) (-. (E. x (/\ ph ps)))) (ph ps imnan x albii x (/\ ph ps) alnex bitr)) thm (exanali () () (<-> (E. x (/\ ph (-. ps))) (-. (A. x (-> ph ps)))) (ph ps iman x albii x (/\ ph (-. ps)) alnex bitr con2bii)) thm (alexn () () (<-> (A. x (E. y (-. ph))) (-. (E. x (A. y ph)))) (y ph exnal x albii x (A. y ph) alnex bitr)) thm (excomim () () (-> (E. x (E. y ph)) (E. y (E. x ph))) (ph x 19.8a x y 19.22i2 x ph hbe1 y hbex 19.9r sylibr)) thm (excom () () (<-> (E. x (E. y ph)) (E. y (E. x ph))) (x y ph excomim y x ph excomim impbi)) thm (19.12 () () (-> (E. x (A. y ph)) (A. y (E. x ph))) (y ph hba1 x hbex y ph ax-4 x 19.22i y 19.20i syl)) thm (19.16 () ((19.16.1 (-> ph (A. x ph)))) (-> (A. x (<-> ph ps)) (<-> ph (A. x ps))) (x ph ps 19.15 19.16.1 19.3r syl5bb)) thm (19.17 () ((19.17.1 (-> ps (A. x ps)))) (-> (A. x (<-> ph ps)) (<-> (A. x ph) ps)) (x ph ps 19.15 19.17.1 19.3r syl6bbr)) thm (19.18 () () (-> (A. x (<-> ph ps)) (<-> (E. x ph) (E. x ps))) (ph ps bi1 x 19.20i x ph ps 19.22 syl ph ps bi2 x 19.20i x ps ph 19.22 syl impbid)) thm (exbii () ((exbii.1 (<-> ph ps))) (<-> (E. x ph) (E. x ps)) (x ph ps 19.18 exbii.1 mpg)) thm (2exbii () ((2exbii.1 (<-> ph ps))) (<-> (E. x (E. y ph)) (E. x (E. y ps))) (2exbii.1 y exbii x exbii)) thm (3exbi () ((3exbii.1 (<-> ph ps))) (<-> (E. x (E. y (E. z ph))) (E. x (E. y (E. z ps)))) (3exbii.1 z exbii x y 2exbii)) thm (exancom () () (<-> (E. x (/\ ph ps)) (E. x (/\ ps ph))) (ph ps ancom x exbii)) thm (19.19 () ((19.19.1 (-> ph (A. x ph)))) (-> (A. x (<-> ph ps)) (<-> ph (E. x ps))) (x ph ps 19.18 19.19.1 19.9r syl5bb)) thm (19.21 () ((19.21.1 (-> ph (A. x ph)))) (<-> (A. x (-> ph ps)) (-> ph (A. x ps))) (x ph ps 19.20 19.21.1 syl5 19.21.1 x ps hba1 hbim x ps ax-4 ph imim2i x 19.20i syl impbi)) thm (19.21-2 () ((19.21-2.1 (-> ph (A. x ph))) (19.21-2.2 (-> ph (A. y ph)))) (<-> (A. x (A. y (-> ph ps))) (-> ph (A. x (A. y ps)))) (19.21-2.2 ps 19.21 x albii 19.21-2.1 (A. y ps) 19.21 bitr)) thm (stdpc5 () ((stdpc5.1 (-> ph (A. x ph)))) (-> (A. x (-> ph ps)) (-> ph (A. x ps))) (stdpc5.1 ps 19.21 biimp)) thm (19.21ad () ((19.21ad.1 (-> ph (A. x ph))) (19.21ad.2 (-> ps (A. x ps))) (19.21ad.3 (-> ph (-> ps ch)))) (-> ph (-> ps (A. x ch))) (19.21ad.1 19.21ad.2 hban 19.21ad.3 imp 19.21ai ex)) thm (19.21bi () ((19.21bi.1 (-> ph (A. x ps)))) (-> ph ps) (19.21bi.1 x ps ax-4 syl)) thm (19.21bbi () ((19.21bbi.1 (-> ph (A. x (A. y ps))))) (-> ph ps) (19.21bbi.1 19.21bi 19.21bi)) thm (19.22d () ((19.22d.1 (-> ph (A. x ph))) (19.22d.2 (-> ph (-> ps ch)))) (-> ph (-> (E. x ps) (E. x ch))) (19.22d.1 19.22d.2 19.21ai x ps ch 19.22 syl)) thm (19.23 () ((19.23.1 (-> ps (A. x ps)))) (<-> (A. x (-> ph ps)) (-> (E. x ph) ps)) (x ph ps 19.22 19.23.1 19.9r syl6ibr x ph hbe1 19.23.1 hbim ph x 19.8a ps imim1i 19.21ai impbi)) thm (19.23ai () ((19.23ai.1 (-> ps (A. x ps))) (19.23ai.2 (-> ph ps))) (-> (E. x ph) ps) (19.23ai.2 x 19.22i 19.23ai.1 19.9r sylibr)) thm (19.23bi () ((19.23bi.1 (-> (E. x ph) ps))) (-> ph ps) (ph x 19.8a 19.23bi.1 syl)) thm (19.23ad () ((19.23ad.1 (-> ph (A. x ph))) (19.23ad.2 (-> ch (A. x ch))) (19.23ad.3 (-> ph (-> ps ch)))) (-> ph (-> (E. x ps) ch)) (19.23ad.1 19.23ad.3 19.21ai 19.23ad.2 ps 19.23 sylib)) thm (19.26 () () (<-> (A. x (/\ ph ps)) (/\ (A. x ph) (A. x ps))) (ph ps pm3.26 x 19.20i ph ps pm3.27 x 19.20i jca ph ps pm3.2 x 19.20ii imp impbi)) thm (19.26-2 () () (<-> (A. x (A. y (/\ ph ps))) (/\ (A. x (A. y ph)) (A. x (A. y ps)))) (y ph ps 19.26 x albii x (A. y ph) (A. y ps) 19.26 bitr)) thm (19.27 () ((19.27.1 (-> ps (A. x ps)))) (<-> (A. x (/\ ph ps)) (/\ (A. x ph) ps)) (x ph ps 19.26 19.27.1 19.3r (A. x ph) anbi2i bitr4)) thm (19.28 () ((19.28.1 (-> ph (A. x ph)))) (<-> (A. x (/\ ph ps)) (/\ ph (A. x ps))) (x ph ps 19.26 19.28.1 19.3r (A. x ps) anbi1i bitr4)) thm (19.29 () () (-> (/\ (A. x ph) (E. x ps)) (E. x (/\ ph ps))) (x ph (-. ps) 19.20 x ps alnex syl6ib con3i (A. x ph) (E. x ps) df-an x (-> ph (-. ps)) exnal 3imtr4 ph ps df-an x exbii sylibr)) thm (19.29r () () (-> (/\ (E. x ph) (A. x ps)) (E. x (/\ ph ps))) (x ps ph 19.29 (E. x ph) (A. x ps) ancom x ph ps exancom 3imtr4)) thm (19.29r2 () () (-> (/\ (E. x (E. y ph)) (A. x (A. y ps))) (E. x (E. y (/\ ph ps)))) (x (E. y ph) (A. y ps) 19.29r y ph ps 19.29r x 19.22i syl)) thm (19.29x () () (-> (/\ (E. x (A. y ph)) (A. x (E. y ps))) (E. x (E. y (/\ ph ps)))) (x (A. y ph) (E. y ps) 19.29r y ph ps 19.29 x 19.22i syl)) thm (19.35 () () (<-> (E. x (-> ph ps)) (-> (A. x ph) (E. x ps))) (x ph (-. ps) 19.26 ph ps annim x albii (A. x ph) (A. x (-. ps)) df-an 3bitr3 con2bii x ps df-ex (A. x ph) imbi2i x (-> ph ps) df-ex 3bitr4r)) thm (19.35i () ((19.35i.1 (E. x (-> ph ps)))) (-> (A. x ph) (E. x ps)) (19.35i.1 x ph ps 19.35 mpbi)) thm (19.35ri () ((19.35ri.1 (-> (A. x ph) (E. x ps)))) (E. x (-> ph ps)) (19.35ri.1 x ph ps 19.35 mpbir)) thm (19.36 () ((19.36.1 (-> ps (A. x ps)))) (<-> (E. x (-> ph ps)) (-> (A. x ph) ps)) (x ph ps 19.35 19.36.1 19.9r (A. x ph) imbi2i bitr4)) thm (19.36i () ((19.36i.1 (-> ps (A. x ps))) (19.36i.2 (E. x (-> ph ps)))) (-> (A. x ph) ps) (19.36i.2 19.36i.1 ph 19.36 mpbi)) thm (19.37 () ((19.37.1 (-> ph (A. x ph)))) (<-> (E. x (-> ph ps)) (-> ph (E. x ps))) (x ph ps 19.35 19.37.1 19.3r (E. x ps) imbi1i bitr4)) thm (19.38 () () (-> (-> (E. x ph) (A. x ps)) (A. x (-> ph ps))) (x ph hbe1 x ps hba1 hbim ph x 19.8a x ps ax-4 imim12i 19.21ai)) thm (19.39 () () (-> (-> (E. x ph) (E. x ps)) (E. x (-> ph ps))) (x ph 19.2 (E. x ps) imim1i x ph ps 19.35 sylibr)) thm (19.24 () () (-> (-> (A. x ph) (A. x ps)) (E. x (-> ph ps))) (x ps 19.2 (A. x ph) imim2i x ph ps 19.35 sylibr)) thm (19.25 () () (-> (A. y (E. x (-> ph ps))) (-> (E. y (A. x ph)) (E. y (E. x ps)))) (x ph ps 19.35 biimp y 19.20i y (A. x ph) (E. x ps) 19.22 syl)) thm (19.30 () () (-> (A. x (\/ ph ps)) (\/ (A. x ph) (E. x ps))) (x (-. ps) ph 19.20 ph ps orcom ps ph df-or bitr x albii (A. x ph) (-. (A. x (-. ps))) orcom x ps df-ex (A. x ph) orbi2i (A. x (-. ps)) (A. x ph) imor 3bitr4 3imtr4)) thm (19.32 () ((19.32.1 (-> ph (A. x ph)))) (<-> (A. x (\/ ph ps)) (\/ ph (A. x ps))) (19.32.1 hbne ps 19.21 ph ps df-or x albii ph (A. x ps) df-or 3bitr4)) thm (19.31 () ((19.31.1 (-> ps (A. x ps)))) (<-> (A. x (\/ ph ps)) (\/ (A. x ph) ps)) (19.31.1 ph 19.32 ph ps orcom x albii (A. x ph) ps orcom 3bitr4)) thm (19.43 () () (<-> (E. x (\/ ph ps)) (\/ (E. x ph) (E. x ps))) (ph ps ioran x albii x (-. ph) (-. ps) 19.26 x ph alnex x ps alnex anbi12i 3bitr negbii x (\/ ph ps) df-ex (E. x ph) (E. x ps) oran 3bitr4)) thm (19.44 () ((19.44.1 (-> ps (A. x ps)))) (<-> (E. x (\/ ph ps)) (\/ (E. x ph) ps)) (x ph ps 19.43 19.44.1 19.9r (E. x ph) orbi2i bitr4)) thm (19.45 () ((19.45.1 (-> ph (A. x ph)))) (<-> (E. x (\/ ph ps)) (\/ ph (E. x ps))) (x ph ps 19.43 19.45.1 19.9r (E. x ps) orbi1i bitr4)) thm (19.33 () () (-> (\/ (A. x ph) (A. x ps)) (A. x (\/ ph ps))) (ph ps orc x 19.20i ps ph olc x 19.20i jaoi)) thm (19.33b () () (-> (-. (/\ (E. x ph) (E. x ps))) (<-> (A. x (\/ ph ps)) (\/ (A. x ph) (A. x ps)))) ((E. x ph) (E. x ps) ianor x ph alnex x ps alnex orbi12i bitr4 ph ps biorf x 19.20i x ps (\/ ph ps) 19.15 syl (A. x ps) (A. x ph) olc syl6bir ps ph biorf ps ph orcom syl6bb x 19.20i x ph (\/ ph ps) 19.15 syl (A. x ph) (A. x ps) orc syl6bir jaoi sylbi x ph ps 19.33 (-. (/\ (E. x ph) (E. x ps))) a1i impbid)) thm (19.34 () () (-> (\/ (A. x ph) (E. x ps)) (E. x (\/ ph ps))) (x ph 19.2 (E. x ps) orim1i x ph ps 19.43 sylibr)) thm (19.40 () () (-> (E. x (/\ ph ps)) (/\ (E. x ph) (E. x ps))) (ph ps pm3.26 x 19.22i ph ps pm3.27 x 19.22i jca)) thm (19.41 () ((19.41.1 (-> ps (A. x ps)))) (<-> (E. x (/\ ph ps)) (/\ (E. x ph) ps)) (x (/\ ph ps) df-ex 19.41.1 hbne (-. ph) 19.31 ph ps ianor x albii (E. x ph) ps ianor x ph alnex (-. ps) orbi1i bitr4 3bitr4 con2bii bitr4)) thm (19.42 () ((19.42.1 (-> ph (A. x ph)))) (<-> (E. x (/\ ph ps)) (/\ ph (E. x ps))) (19.42.1 ps 19.41 x ph ps exancom ph (E. x ps) ancom 3bitr4)) thm (alrot4 () () (<-> (A. x (A. y (A. z (A. w ph)))) (A. z (A. w (A. x (A. y ph))))) (y z (A. w ph) alcom y w ph alcom z albii bitr x albii x z (A. w (A. y ph)) alcom x w (A. y ph) alcom z albii 3bitr)) thm (excom13 () () (<-> (E. x (E. y (E. z ph))) (E. z (E. y (E. x ph)))) (x y (E. z ph) excom x z ph excom y exbii y z (E. x ph) excom 3bitr)) thm (exrot3 () () (<-> (E. x (E. y (E. z ph))) (E. y (E. z (E. x ph)))) (x y z ph excom13 z y (E. x ph) excom bitr)) thm (exrot4 () () (<-> (E. x (E. y (E. z (E. w ph)))) (E. z (E. w (E. x (E. y ph))))) (y z w ph excom13 x exbii x w z (E. y ph) excom13 bitr)) thm (nex () ((nex.1 (-. ph))) (-. (E. x ph)) (x ph alnex nex.1 mpgbi)) thm (nexd () ((nexd.1 (-> ph (A. x ph))) (nexd.2 (-> ph (-. ps)))) (-> ph (-. (E. x ps))) (nexd.1 nexd.2 19.21ai x ps alnex sylib)) thm (hbim1 () ((hbim1.1 (-> ph (A. x ph))) (hbim1.2 (-> ph (-> ps (A. x ps))))) (-> (-> ph ps) (A. x (-> ph ps))) (hbim1.2 a2i hbim1.1 ps 19.21 sylibr)) thm (albid () ((albid.1 (-> ph (A. x ph))) (albid.2 (-> ph (<-> ps ch)))) (-> ph (<-> (A. x ps) (A. x ch))) (albid.1 albid.2 19.21ai x ps ch 19.15 syl)) thm (exbid () ((exbid.1 (-> ph (A. x ph))) (exbid.2 (-> ph (<-> ps ch)))) (-> ph (<-> (E. x ps) (E. x ch))) (exbid.1 exbid.2 19.21ai x ps ch 19.18 syl)) thm (exan () ((exan.1 (/\ (E. x ph) ps))) (E. x (/\ ph ps)) (x ph hbe1 ps 19.27 exan.1 (E. x ph) ps ancom mpbi mpgbi x ps ph 19.29 ax-mp x ps ph exancom mpbi)) thm (albi () () (<-> (A. x (<-> ph ps)) (/\ (A. x (-> ph ps)) (A. x (-> ps ph)))) (ph ps bi x albii x (-> ph ps) (-> ps ph) 19.26 bitr)) thm (2albi () () (<-> (A. x (A. y (<-> ph ps))) (/\ (A. x (A. y (-> ph ps))) (A. x (A. y (-> ps ph))))) (y ph ps albi x albii x (A. y (-> ph ps)) (A. y (-> ps ph)) 19.26 bitr)) thm (hband () ((hband.1 (-> ph (-> ps (A. x ps)))) (hband.2 (-> ph (-> ch (A. x ch))))) (-> ph (-> (/\ ps ch) (A. x (/\ ps ch)))) (hband.1 hband.2 anim12d x ps ch 19.26 syl6ibr)) thm (hbbid () ((hbbid.1 (-> ph (A. x ph))) (hbbid.2 (-> ph (-> ps (A. x ps)))) (hbbid.3 (-> ph (-> ch (A. x ch))))) (-> ph (-> (<-> ps ch) (A. x (<-> ps ch)))) (hbbid.1 hbbid.2 hbbid.3 hbimd hbbid.1 hbbid.3 hbbid.2 hbimd anim12d ps ch bi x ps ch albi 3imtr4g)) thm (hbexd () ((hbexd.1 (-> ph (A. y ph))) (hbexd.2 (-> ph (-> ps (A. x ps))))) (-> ph (-> (E. y ps) (A. x (E. y ps)))) (hbexd.1 hbexd.2 19.22d y x ps 19.12 syl6)) thm (19.21g () () (-> (A. x (-> ph (A. x ph))) (<-> (A. x (-> ph ps)) (-> ph (A. x ps)))) (x ph ps 19.20 ph imim2d com12 x a4s x (-> ph (A. x ph)) hba1 x (-> ph (A. x ph)) ax-4 x ps hba1 (A. x (-> ph (A. x ph))) a1i hbimd x ps ax-4 ph imim2i x 19.20i syl6 impbid)) thm (19.23g () () (-> (A. x (-> ps (A. x ps))) (<-> (A. x (-> ph ps)) (-> (E. x ph) ps))) (x (-> ps (A. x ps)) hba1 x ps ax-4 (A. x (-> ps (A. x ps))) a1i x (-> ps (A. x ps)) ax-4 impbid ph imbi2d albid x ps ax-4 (A. x (-> ps (A. x ps))) a1i x (-> ps (A. x ps)) ax-4 impbid (E. x ph) imbi2d x ps hba1 ph 19.23 syl5bb bitr3d)) thm (exintr () () (-> (A. x (-> ph ps)) (-> (E. x ph) (E. x (/\ ph ps)))) (x (-> ph ps) hba1 ph ps ancl x a4s 19.22d)) thm (exintrbi () () (-> (A. x (-> ph ps)) (<-> (E. x ph) (E. x (/\ ph ps)))) (ph ps pm4.71 x albii x ph (/\ ph ps) 19.18 sylbi)) thm (aaan () ((aaan.1 (-> ph (A. y ph))) (aaan.2 (-> ps (A. x ps)))) (<-> (A. x (A. y (/\ ph ps))) (/\ (A. x ph) (A. y ps))) (aaan.1 ps 19.28 x albii aaan.2 y hbal ph 19.27 bitr)) thm (eeor () ((eeor.1 (-> ph (A. y ph))) (eeor.2 (-> ps (A. x ps)))) (<-> (E. x (E. y (\/ ph ps))) (\/ (E. x ph) (E. y ps))) (eeor.1 ps 19.45 x exbii eeor.2 y hbex ph 19.44 bitr)) thm (qexmid () () (E. x (-> ph (A. x ph))) ((A. x ph) x 19.8a 19.35ri)) var (class A) var (class B) thm (ax9 () () (-> (A. x (-> (= (cv x) (cv y)) (A. x ph))) ph) (x y ax-9 x (= (cv x) (cv y)) df-ex mpbir x (= (cv x) (cv y)) (A. x ph) 19.22 mpi x ph a6e syl)) thm (ax9a () () (-. (A. x (-. (= (cv x) (cv y))))) (x y (-. (A. x (-. (= (cv x) (cv y))))) ax9 (= (cv x) (cv y)) x modal-b mpg)) thm (a9e () () (E. x (= (cv x) (cv y))) (x y ax9a x (= (cv x) (cv y)) df-ex mpbir)) thm (equid () () (= (cv x) (cv x)) (x x x ax-12 pm2.43i x 19.20i x x (= (cv x) (cv x)) ax9 syl x (= (cv x) (cv x)) ax-6 pm2.61i)) thm (stdpc6 () () (A. x (= (cv x) (cv x))) (x equid x ax-gen)) thm (equcomi () () (-> (= (cv x) (cv y)) (= (cv y) (cv x))) (x equid x y x ax-8 mpi)) thm (equcom () () (<-> (= (cv x) (cv y)) (= (cv y) (cv x))) (x y equcomi y x equcomi impbi)) thm (equcoms () ((equcoms.1 (-> (= (cv x) (cv y)) ph))) (-> (= (cv y) (cv x)) ph) (y x equcomi equcoms.1 syl)) thm (equtr () () (-> (= (cv x) (cv y)) (-> (= (cv y) (cv z)) (= (cv x) (cv z)))) (y x z ax-8 equcoms)) thm (equtrr () () (-> (= (cv x) (cv y)) (-> (= (cv z) (cv x)) (= (cv z) (cv y)))) (z x y equtr com12)) thm (equtr2 () () (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv z))) (= (cv x) (cv y))) (x z y equtr y z equcomi syl5 imp)) thm (equequ1 () () (-> (= (cv x) (cv y)) (<-> (= (cv x) (cv z)) (= (cv y) (cv z)))) (x y z ax-8 x y z equtr impbid)) thm (equequ2 () () (-> (= (cv x) (cv y)) (<-> (= (cv z) (cv x)) (= (cv z) (cv y)))) (x y z equtrr y x z equtrr equcoms impbid)) thm (elequ1 () () (-> (= (cv x) (cv y)) (<-> (e. (cv x) (cv z)) (e. (cv y) (cv z)))) (x y z ax-13 y x z ax-13 equcoms impbid)) thm (elequ2 () () (-> (= (cv x) (cv y)) (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y)))) (x y z ax-14 y x z ax-14 equcoms impbid)) thm (alequcom () () (-> (A. x (= (cv x) (cv y))) (A. y (= (cv y) (cv x)))) (x y (= (cv x) (cv y)) ax-10 pm2.43i x y equcomi y 19.20i syl)) thm (alequcoms () ((alequcoms.1 (-> (A. x (= (cv x) (cv y))) ph))) (-> (A. y (= (cv y) (cv x))) ph) (y x alequcom alequcoms.1 syl)) thm (nalequcoms () ((nalequcoms.1 (-> (-. (A. x (= (cv x) (cv y)))) ph))) (-> (-. (A. y (= (cv y) (cv x)))) ph) (x y alequcom nalequcoms.1 nsyl4 con1i)) thm (hbae () () (-> (A. x (= (cv x) (cv y))) (A. z (A. x (= (cv x) (cv y))))) (z x y ax-12 x (= (cv x) (cv y)) ax-4 syl7 x z (= (cv x) (cv y)) ax-10 alequcoms y z (= (cv x) (cv y)) ax-10 x y (= (cv x) (cv y)) ax-10 pm2.43i syl5 alequcoms pm2.61ii a5i x z (= (cv x) (cv y)) ax-7 syl)) thm (hbaes () ((hbalequs.1 (-> (A. z (A. x (= (cv x) (cv y)))) ph))) (-> (A. x (= (cv x) (cv y))) ph) (x y z hbae hbalequs.1 syl)) thm (hbnae () () (-> (-. (A. x (= (cv x) (cv y)))) (A. z (-. (A. x (= (cv x) (cv y)))))) (x y z hbae hbne)) thm (hbnaes () ((hbnalequs.1 (-> (A. z (-. (A. x (= (cv x) (cv y))))) ph))) (-> (-. (A. x (= (cv x) (cv y)))) ph) (x y z hbnae hbnalequs.1 syl)) thm (equs3 () () (<-> (E. x (/\ (= (cv x) (cv y)) ph)) (-. (A. x (-> (= (cv x) (cv y)) (-. ph))))) (x (= (cv x) (cv y)) ph alinexa con2bii)) thm (equs4 () () (-> (A. x (-> (= (cv x) (cv y)) ph)) (E. x (/\ (= (cv x) (cv y)) ph))) ((A. x (-> (= (cv x) (cv y)) ph)) (= (cv x) (cv y)) pm3.27 x (-> (= (cv x) (cv y)) ph) ax-4 imp jc x (-> (= (cv x) (cv y)) (-. ph)) ax-4 nsyl ex x (-> (= (cv x) (cv y)) (-. ph)) hbn1 syl6 a5i x y (-. (A. x (-> (= (cv x) (cv y)) (-. ph)))) ax9 syl x y ph equs3 sylibr)) thm (equs5 () () (-> (-. (A. x (= (cv x) (cv y)))) (-> (E. x (/\ (= (cv x) (cv y)) ph)) (A. x (-> (= (cv x) (cv y)) ph)))) (x y x hbnae x (-> (= (cv x) (cv y)) (-. ph)) hbn1 x y (-. ph) ax-11 ph (A. x (-> (= (cv x) (cv y)) (-. ph))) con1 syl6 com23 19.21ad x y ph equs3 syl5ib)) thm (equsal () ((equsal.1 (-> ps (A. x ps))) (equsal.2 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A. x (-> (= (cv x) (cv y)) ph)) ps) (equsal.2 equsal.1 19.3r syl6bb pm5.74i x albii equsal.1 (A. x ps) (= (cv x) (cv y)) ax-1 a5i syl x y ps ax9 impbi bitr4)) thm (equsex () ((equsex.1 (-> ps (A. x ps))) (equsex.2 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E. x (/\ (= (cv x) (cv y)) ph)) ps) (x (-> (= (cv x) (cv y)) (-. ph)) exnal (= (cv x) (cv y)) ph df-an x exbii equsex.1 hbne equsex.2 negbid equsal con2bii 3bitr4)) thm (dral1 () ((dral1.1 (-> (A. x (= (cv x) (cv y))) (<-> ph ps)))) (-> (A. x (= (cv x) (cv y))) (<-> (A. x ph) (A. y ps))) (dral1.1 biimpd x 19.20ii hbaes x y ps ax-10 syld dral1.1 biimprd y 19.20ii hbaes y x ph ax-10 alequcoms syld impbid)) thm (dral2 () ((dral2.1 (-> (A. x (= (cv x) (cv y))) (<-> ph ps)))) (-> (A. x (= (cv x) (cv y))) (<-> (A. z ph) (A. z ps))) (x y z hbae dral2.1 albid)) thm (drex1 () ((drex1.1 (-> (A. x (= (cv x) (cv y))) (<-> ph ps)))) (-> (A. x (= (cv x) (cv y))) (<-> (E. x ph) (E. y ps))) (drex1.1 negbid dral1 negbid x ph df-ex y ps df-ex 3bitr4g)) thm (drex2 () ((drex2.1 (-> (A. x (= (cv x) (cv y))) (<-> ph ps)))) (-> (A. x (= (cv x) (cv y))) (<-> (E. z ph) (E. z ps))) (x y z hbae drex2.1 exbid)) thm (a4a () ((a4a.1 (-> ps (A. x ps))) (a4a.2 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> (A. x ph) ps) (a4a.2 a4a.1 syl6com x 19.20i x y ps ax9 syl)) thm (a4c () ((a4c.1 (-> ph (A. x ph))) (a4c.2 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> ph (E. x ps)) (a4c.1 hbne a4c.2 con3d a4a con2i x ps df-ex sylibr)) thm (a4c1 () ((a4c1.1 (-> ch (A. x ch))) (a4c1.2 (-> ch (-> ph (A. x ph)))) (a4c1.3 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> ch (-> ph (E. x ps))) (a4c1.1 ph adantr a4c1.2 imp jca x ch ph 19.26 sylibr a4c1.3 ch adantld a4c ex)) thm (cbv1 () ((cbv1.1 (-> ph (-> ps (A. y ps)))) (cbv1.2 (-> ph (-> ch (A. x ch)))) (cbv1.3 (-> ph (-> (= (cv x) (cv y)) (-> ps ch))))) (-> (A. x (A. y ph)) (-> (A. x ps) (A. y ch))) (cbv1.1 y a4s x 19.20ii x y ps ax-7 syl6 cbv1.3 com23 cbv1.2 syl6d x 19.20ii x y ch ax9 syl6 y 19.20ii a7s syld)) thm (cbv2 () ((cbv2.1 (-> ph (-> ps (A. y ps)))) (cbv2.2 (-> ph (-> ch (A. x ch)))) (cbv2.3 (-> ph (-> (= (cv x) (cv y)) (<-> ps ch))))) (-> (A. x (A. y ph)) (<-> (A. x ps) (A. y ch))) (cbv2.1 cbv2.2 cbv2.3 ps ch bi1 syl6 cbv1 cbv2.2 cbv2.1 cbv2.3 ps ch bi2 syl6 y x equcomi syl5 cbv1 a7s impbid)) thm (cbv3 () ((cbv3.1 (-> ph (A. y ph))) (cbv3.2 (-> ps (A. x ps))) (cbv3.3 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> (A. x ph) (A. y ps)) (cbv3.1 ph imim2i cbv3.2 (-> ph ph) a1i cbv3.3 (-> ph ph) a1i cbv1 ph id y ax-gen mpg)) thm (cbval () ((cbval.1 (-> ph (A. y ph))) (cbval.2 (-> ps (A. x ps))) (cbval.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A. x ph) (A. y ps)) (cbval.1 ph imim2i cbval.2 (-> ph ph) a1i cbval.3 (-> ph ph) a1i cbv2 ph id y ax-gen mpg)) thm (cbvex () ((cbvex.1 (-> ph (A. y ph))) (cbvex.2 (-> ps (A. x ps))) (cbvex.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E. x ph) (E. y ps)) (cbvex.1 hbne cbvex.2 hbne cbvex.3 negbid cbval negbii x ph df-ex y ps df-ex 3bitr4)) thm (chvar () ((chv2.1 (-> ps (A. x ps))) (chv2.2 (-> (= (cv x) (cv y)) (<-> ph ps))) (chv2.3 ph)) ps (chv2.1 chv2.2 biimpd a4a chv2.3 mpg)) thm (equvini () () (-> (= (cv x) (cv y)) (E. z (/\ (= (cv x) (cv z)) (= (cv z) (cv y))))) (z y a9e z equid (= (cv z) (cv y)) jctl z 19.22i ax-mp z (= (cv z) (cv x)) hba1 z x z ax-8 z a4s (= (cv z) (cv y)) anim1d 19.22d mpi z x a9e z x equcomi z equid jctir z 19.22i ax-mp z (= (cv z) (cv y)) hba1 z y z equtrr z a4s (= (cv x) (cv z)) anim2d 19.22d mpi jaoi (= (cv x) (cv y)) a1d (A. z (= (cv z) (cv x))) (A. z (= (cv z) (cv y))) ioran z x z hbnae z y z hbnae hban z x y ax-12 imp x z y ax-8 anc2li equcoms a4c1 sylbi pm2.61i)) thm (hbequid () () (-> (= (cv x) (cv x)) (A. x (= (cv x) (cv x)))) (x x x ax-12 pm2.43i x a4s x (-. (A. x (= (cv x) (cv x)))) hbn1 x (= (cv x) (cv x)) ax-6 19.21ai (= (cv x) (cv x)) a1d pm2.61i)) thm (ax11b () () (-> (/\ (-. (A. x (= (cv x) (cv y)))) (= (cv x) (cv y))) (<-> ph (A. x (-> (= (cv x) (cv y)) ph)))) (x y ph ax-11 imp x (-> (= (cv x) (cv y)) ph) ax-4 com12 (-. (A. x (= (cv x) (cv y)))) adantl impbid)) thm (sbimi () ((sbimi.1 (-> ph ps))) (-> ([/] (cv y) x ph) ([/] (cv y) x ps)) (sbimi.1 (= (cv x) (cv y)) imim2i sbimi.1 (= (cv x) (cv y)) anim2i x 19.22i anim12i y x ph df-sb y x ps df-sb 3imtr4)) thm (sbbii () ((sbbii.1 (<-> ph ps))) (<-> ([/] (cv y) x ph) ([/] (cv y) x ps)) (sbbii.1 biimp y x sbimi sbbii.1 biimpr y x sbimi impbi)) thm (drsb1 () () (-> (A. x (= (cv x) (cv y))) (<-> ([/] (cv z) x ph) ([/] (cv z) y ph))) (x y z equequ1 x a4s ph imbi1d x y z equequ1 x a4s ph anbi1d drex1 anbi12d z x ph df-sb z y ph df-sb 3bitr4g)) thm (sb1 () () (-> ([/] (cv y) x ph) (E. x (/\ (= (cv x) (cv y)) ph))) (y x ph df-sb pm3.27bd)) thm (sb2 () () (-> (A. x (-> (= (cv x) (cv y)) ph)) ([/] (cv y) x ph)) (x (-> (= (cv x) (cv y)) ph) ax-4 x y ph equs4 jca y x ph df-sb sylibr)) thm (sb3 () () (-> (-. (A. x (= (cv x) (cv y)))) (-> (E. x (/\ (= (cv x) (cv y)) ph)) ([/] (cv y) x ph))) (x y ph equs5 x y ph sb2 syl6)) thm (sb4 () () (-> (-. (A. x (= (cv x) (cv y)))) (-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ph)))) (x y ph equs5 y x ph sb1 syl5)) thm (sb4b () () (-> (-. (A. x (= (cv x) (cv y)))) (<-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ph)))) (x y ph sb4 x y ph sb2 (-. (A. x (= (cv x) (cv y)))) a1i impbid)) thm (sbequ1 () () (-> (= (cv x) (cv y)) (-> ph ([/] (cv y) x ph))) ((= (cv x) (cv y)) ph pm3.4 (/\ (= (cv x) (cv y)) ph) x 19.8a jca y x ph df-sb sylibr ex)) thm (sbequ2 () () (-> (= (cv x) (cv y)) (-> ([/] (cv y) x ph) ph)) ((-> (= (cv x) (cv y)) ph) (E. x (/\ (= (cv x) (cv y)) ph)) pm3.26 com12 y x ph df-sb syl5ib)) thm (dfsb2 () () (<-> ([/] (cv y) x ph) (\/ (/\ (= (cv x) (cv y)) ph) (A. x (-> (= (cv x) (cv y)) ph)))) (x y ph sbequ2 x a4s x (= (cv x) (cv y)) ax-4 jctild (/\ (= (cv x) (cv y)) ph) (A. x (-> (= (cv x) (cv y)) ph)) orc syl6 x y ph sb4 (A. x (-> (= (cv x) (cv y)) ph)) (/\ (= (cv x) (cv y)) ph) olc syl6 pm2.61i x y ph sbequ1 imp x y ph sb2 jaoi impbi)) thm (stdpc7 () () (-> (= (cv x) (cv y)) (-> ([/] (cv x) y ph) ph)) (y x ph sbequ2 equcoms)) thm (sbequ12 () () (-> (= (cv x) (cv y)) (<-> ph ([/] (cv y) x ph))) (x y ph sbequ1 x y ph sbequ2 impbid)) thm (sbequ12r () () (-> (= (cv x) (cv y)) (<-> ([/] (cv x) y ph) ph)) (y x ph sbequ12 x y equcom ([/] (cv x) y ph) ph bicom 3imtr4)) thm (sbequ12a () () (-> (= (cv x) (cv y)) (<-> ([/] (cv y) x ph) ([/] (cv x) y ph))) (x y ph sbequ12 y x ph sbequ12 equcoms bitr3d)) thm (sbid () () (<-> ([/] (cv x) x ph) ph) (x equid x x ph sbequ12 ax-mp bicomi)) thm (stdpc4 () () (-> (A. x ph) ([/] (cv y) x ph)) (ph (= (cv x) (cv y)) ax-1 x 19.20i x y ph sb2 syl)) thm (sbf () ((sbf.1 (-> ph (A. x ph)))) (<-> ([/] (cv y) x ph) ph) (y x ph sb1 sbf.1 (= (cv x) (cv y)) 19.41 sylib pm3.27d sbf.1 x ph y stdpc4 syl impbi)) thm (sbf2 () () (<-> ([/] (cv y) x (A. x ph)) (A. x ph)) (x ph hba1 y sbf)) thm (sb6x () ((sb6x.1 (-> ph (A. x ph)))) (<-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ph))) (sb6x.1 y sbf sb6x.1 ph (= (cv x) (cv y)) ax-1 19.21ai sylbi x y ph sb2 impbi)) thm (sb6y () ((sb6y.1 (-> ph (A. y ph)))) (<-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ph))) (x y ph sbequ2 x a4s y x ph ax-10 alequcoms ph (= (cv x) (cv y)) ax-1 x 19.20i syl6 sb6y.1 syl5 syld x y ph sb4 pm2.61i x y ph sb2 impbi)) thm (hbsb2 () () (-> (-. (A. x (= (cv x) (cv y)))) (-> ([/] (cv y) x ph) (A. x ([/] (cv y) x ph)))) (x y ph sb4 x y ph sb2 a5i syl6)) thm (hbs1f () ((hbs1f.1 (-> ph (A. x ph)))) (-> ([/] (cv y) x ph) (A. x ([/] (cv y) x ph))) (y x ph sb1 hbs1f.1 (= (cv x) (cv y)) 19.41 sylib pm3.27d hbs1f.1 x ph y stdpc4 a5i 3syl)) thm (hbsb3 () ((hbsb3.1 (-> ph (A. y ph)))) (-> ([/] (cv y) x ph) (A. x ([/] (cv y) x ph))) (x y ph sbequ2 x a4s y x ph ax-10 alequcoms hbsb3.1 syl5 syld x y ph sbequ1 x 19.20ii syld x y ph hbsb2 pm2.61i)) thm (sbequi () () (-> (= (cv x) (cv y)) (-> ([/] (cv x) z ph) ([/] (cv y) z ph))) (z x ph hbsb2 x y z equvini x z ph stdpc7 z y ph sbequ1 sylan9 z 19.22i syl z ([/] (cv x) z ph) ([/] (cv y) z ph) 19.35 sylib sylan9 z y z hbnae z y ph hbsb2 19.9d syl9 ex com23 z x ph sbequ2 z a4s (= (cv x) (cv y)) adantr x y ph sbequ1 x z y ph drsb1 biimpd alequcoms sylan9r syld ex z y x ph drsb1 biimpd x y ph stdpc7 sylan9 z y ph sbequ1 z a4s (= (cv x) (cv y)) adantr syld ex pm2.61ii)) thm (sbequ () () (-> (= (cv x) (cv y)) (<-> ([/] (cv x) z ph) ([/] (cv y) z ph))) (x y z ph sbequi y x z ph sbequi equcoms impbid)) thm (drsb2 () () (-> (A. x (= (cv x) (cv y))) (<-> ([/] (cv x) z ph) ([/] (cv y) z ph))) (x y z ph sbequi x a4s y x z ph sbequi equcoms x a4s impbid)) thm (sbn1 () () (-> ([/] (cv y) x (-. ph)) (-. ([/] (cv y) x ph))) (x y (-. ph) sbequ2 x y ph sbequ2 nsyld x a4s x y (-. ph) sb4 y x ph sb1 x y ph equs3 sylib con2i syl6 pm2.61i)) thm (sbn2 () () (-> (-. ([/] (cv y) x ph)) ([/] (cv y) x (-. ph))) (x y ph sbequ1 con3d com12 x y (-. (-. ph)) sb2 ph pm4.13 y x sbbii sylibr con3i x y (-. ph) equs3 sylibr jca y x (-. ph) df-sb sylibr)) thm (sbn () () (<-> ([/] (cv y) x (-. ph)) (-. ([/] (cv y) x ph))) (y x ph sbn1 y x ph sbn2 impbi)) thm (sb5y () ((sb5y.1 (-> ph (A. y ph)))) (<-> ([/] (cv y) x ph) (E. x (/\ (= (cv x) (cv y)) ph))) (sb5y.1 hbne x sb6y y x ph sbn bitr3 con2bii x y ph equs3 bitr4)) thm (sbi1 () () (-> ([/] (cv y) x (-> ph ps)) (-> ([/] (cv y) x ph) ([/] (cv y) x ps))) (x y (-> ph ps) sbequ2 x y ph sbequ2 syl5d x y ps sbequ1 syl6d x a4s x y (-> ph ps) sb4 (= (cv x) (cv y)) ph ps ax-2 x 19.20ii x y ps sb2 syl6 syl6 x y ph sb4 syl5d pm2.61i)) thm (sbi2 () () (-> (-> ([/] (cv y) x ph) ([/] (cv y) x ps)) ([/] (cv y) x (-> ph ps))) (y x ph sbn ph ps pm2.21 y x sbimi sylbir ps ph ax-1 y x sbimi ja)) thm (sbim () () (<-> ([/] (cv y) x (-> ph ps)) (-> ([/] (cv y) x ph) ([/] (cv y) x ps))) (y x ph ps sbi1 y x ph ps sbi2 impbi)) thm (sbor () () (<-> ([/] (cv y) x (\/ ph ps)) (\/ ([/] (cv y) x ph) ([/] (cv y) x ps))) (y x (-. ph) ps sbim y x ph sbn ([/] (cv y) x ps) imbi1i bitr ph ps df-or y x sbbii ([/] (cv y) x ph) ([/] (cv y) x ps) df-or 3bitr4)) thm (sb19.21 () ((sb19.21.1 (-> ph (A. x ph)))) (<-> ([/] (cv y) x (-> ph ps)) (-> ph ([/] (cv y) x ps))) (y x ph ps sbim sb19.21.1 y sbf ([/] (cv y) x ps) imbi1i bitr)) thm (sban () () (<-> ([/] (cv y) x (/\ ph ps)) (/\ ([/] (cv y) x ph) ([/] (cv y) x ps))) (y x (-> ph (-. ps)) sbn y x ph (-. ps) sbim y x ps sbn ([/] (cv y) x ph) imbi2i bitr negbii bitr ph ps df-an y x sbbii ([/] (cv y) x ph) ([/] (cv y) x ps) df-an 3bitr4)) thm (sbbi () () (<-> ([/] (cv y) x (<-> ph ps)) (<-> ([/] (cv y) x ph) ([/] (cv y) x ps))) (ph ps bi y x sbbii y x ph ps sbim y x ps ph sbim anbi12i y x (-> ph ps) (-> ps ph) sban ([/] (cv y) x ph) ([/] (cv y) x ps) bi 3bitr4 bitr)) thm (sblbis () ((sblbis.1 (<-> ([/] (cv y) x ph) ps))) (<-> ([/] (cv y) x (<-> ch ph)) (<-> ([/] (cv y) x ch) ps)) (y x ch ph sbbi sblbis.1 ([/] (cv y) x ch) bibi2i bitr)) thm (sbrbis () ((sbrbis.1 (<-> ([/] (cv y) x ph) ps))) (<-> ([/] (cv y) x (<-> ph ch)) (<-> ps ([/] (cv y) x ch))) (y x ph ch sbbi sbrbis.1 ([/] (cv y) x ch) bibi1i bitr)) thm (sbrbif () ((sbrbif.1 (-> ch (A. x ch))) (sbrbif.2 (<-> ([/] (cv y) x ph) ps))) (<-> ([/] (cv y) x (<-> ph ch)) (<-> ps ch)) (sbrbif.2 ch sbrbis sbrbif.1 y sbf ps bibi2i bitr)) thm (sbea4 () () (-> ([/] (cv y) x ph) (E. x ph)) (x (-. ph) y stdpc4 y x ph sbn sylib con2i x ph df-ex sylibr)) thm (sbia4 () () (-> (A. x (-> ph ps)) (-> ([/] (cv y) x ph) ([/] (cv y) x ps))) (x (-> ph ps) y stdpc4 y x ph ps sbim sylib)) thm (sbba4 () () (-> (A. x (<-> ph ps)) (<-> ([/] (cv y) x ph) ([/] (cv y) x ps))) (x (<-> ph ps) y stdpc4 y x ph ps sbbi sylib)) thm (sbbid () ((sbbid.1 (-> ph (A. x ph))) (sbbid.2 (-> ph (<-> ps ch)))) (-> ph (<-> ([/] (cv y) x ps) ([/] (cv y) x ch))) (sbbid.1 sbbid.2 19.21ai x ps ch y sbba4 syl)) thm (sbequ5 () () (<-> ([/] (cv w) z (A. x (= (cv x) (cv y)))) (A. x (= (cv x) (cv y)))) (x y z hbae w sbf)) thm (sbequ6 () () (<-> ([/] (cv w) z (-. (A. x (= (cv x) (cv y))))) (-. (A. x (= (cv x) (cv y))))) (x y z hbnae w sbf)) thm (sbt () ((sbt.1 ph)) ([/] (cv y) x ph) (x y ph sb2 sbt.1 (= (cv x) (cv y)) a1i mpg)) thm (equsb1 () () ([/] (cv y) x (= (cv x) (cv y))) (x y (= (cv x) (cv y)) sb2 (= (cv x) (cv y)) id mpg)) thm (equsb2 () () ([/] (cv y) x (= (cv y) (cv x))) (x y (= (cv y) (cv x)) sb2 x y equcomi mpg)) thm (sbequ8 () () (<-> ([/] (cv y) x ph) ([/] (cv y) x (-> (= (cv x) (cv y)) ph))) (y x equsb1 ([/] (cv y) x ph) a1bi y x (= (cv x) (cv y)) ph sbim bitr4)) thm (sbied () ((sbied.1 (-> ph (A. x ph))) (sbied.2 (-> ph (-> ch (A. x ch)))) (sbied.3 (-> ph (-> (= (cv x) (cv y)) (<-> ps ch))))) (-> ph (<-> ([/] (cv y) x ps) ch)) (sbied.1 sbied.3 ps ch bi1 syl6 imp3a x 19.20i x (/\ (= (cv x) (cv y)) ps) ch 19.22 syl y x ps sb1 syl5 sbied.2 x 19.20i x ch hba1 ch 19.23 sylib x ch ax-4 syl6 syld syl sbied.1 sbied.2 x a4s sbied.3 ps ch bi2 syl6 com23 x 19.20ii x y ps sb2 syl6 syld syl impbid)) thm (sbie () ((sbie.1 (-> ps (A. x ps))) (sbie.2 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> ([/] (cv y) x ph) ps) (ph id ph id x hbth sbie.1 (-> ph ph) a1i sbie.2 (-> ph ph) a1i sbied ax-mp)) thm (hbsb4 () ((hbsb4.1 (-> ph (A. z ph)))) (-> (-. (A. z (= (cv z) (cv y)))) (-> ([/] (cv y) x ph) (A. z ([/] (cv y) x ph)))) (z x y equequ1 z a4s dral1 negbid x y ph hbsb2 x z ([/] (cv y) x ph) ax-10 alequcoms syl9r sylbid x y z hbae x (= (cv x) (cv y)) ax-4 z 19.20i x y ph sbequ2 z a4s x y ph sbequ1 z 19.20ii hbsb4.1 syl5 syld 3syl (/\ (-. (A. z (= (cv z) (cv x)))) (-. (A. z (= (cv z) (cv y))))) a1d x y ph sb4 z x x hbnae z y x hbnae hban z x z hbnae z y z hbnae hban z x y ax-12 imp hbsb4.1 (/\ (-. (A. z (= (cv z) (cv x)))) (-. (A. z (= (cv z) (cv y))))) a1i hbimd 19.20d x y ph sb2 z 19.20i a7s syl6 syl9 pm2.61i ex pm2.61i)) thm (hbsb4t () () (-> (A. x (A. z (-> ph (A. z ph)))) (-> (-. (A. z (= (cv z) (cv y)))) (-> ([/] (cv y) x ph) (A. z ([/] (cv y) x ph))))) (z ph ax-4 (-> ph (A. z ph)) biantru ph (A. z ph) bi bitr4 x z 2albii x ph (A. z ph) y sbba4 z a4s z (A. x (<-> ph (A. z ph))) hba1 x ph (A. z ph) y sbba4 z a4s albid imbi12d a7s sylbi z ph hba1 y x hbsb4 syl5bir)) thm (dvelimf2 () ((dvelimf2.1 (-> ph (A. x ph))) (dvelimf2.2 (-> ps (A. z ps))) (dvelimf2.3 (-> (= (cv z) (cv y)) (<-> ph ps)))) (-> (-. (A. x (= (cv x) (cv y)))) (-> ps (A. x ps))) (z x (A. z (-> (= (cv z) (cv y)) ph)) ax-10 alequcoms z (-> (= (cv z) (cv y)) ph) hba1 syl5 (-. (A. x (= (cv x) (cv y)))) a1d x z z hbnae x y z hbnae hban x z x hbnae x y x hbnae hban x z y ax-12 imp dvelimf2.1 (/\ (-. (A. x (= (cv x) (cv z)))) (-. (A. x (= (cv x) (cv y))))) a1i hbimd hbald ex pm2.61i dvelimf2.2 dvelimf2.3 equsal dvelimf2.2 dvelimf2.3 equsal x albii 3imtr3g)) thm (dvelimf () ((dvelimf.1 (-> ph (A. x ph))) (dvelimf.2 (-> ps (A. z ps))) (dvelimf.3 (-> (= (cv z) (cv y)) (<-> ph ps)))) (-> (-. (A. x (= (cv x) (cv y)))) (-> ps (A. x ps))) (dvelimf.1 y z hbsb4 dvelimf.2 dvelimf.3 sbie dvelimf.2 dvelimf.3 sbie x albii 3imtr3g)) thm (dvelimdf () ((dvelimdf.1 (-> ph (A. x ph))) (dvelimdf.2 (-> ph (A. z ph))) (dvelimdf.3 (-> ph (-> ps (A. x ps)))) (dvelimdf.4 (-> ph (-> ch (A. z ch)))) (dvelimdf.5 (-> ph (-> (= (cv z) (cv y)) (<-> ps ch))))) (-> ph (-> (-. (A. x (= (cv x) (cv y)))) (-> ch (A. x ch)))) (dvelimdf.2 dvelimdf.1 19.21ai dvelimdf.3 z x 19.20i2 z x ps y hbsb4t 3syl imp dvelimdf.2 dvelimdf.4 dvelimdf.5 sbied (-. (A. x (= (cv x) (cv y)))) adantr dvelimdf.1 dvelimdf.2 dvelimdf.4 dvelimdf.5 sbied albid (-. (A. x (= (cv x) (cv y)))) adantr 3imtr3d ex)) thm (sbco () () (<-> ([/] (cv y) x ([/] (cv x) y ph)) ([/] (cv y) x ph)) (y x equsb2 y x ph sbequ12 bicomd y x sbimi ax-mp y x ([/] (cv x) y ph) ph sbbi mpbi)) thm (sbid2 () ((sbid2.1 (-> ph (A. x ph)))) (<-> ([/] (cv y) x ([/] (cv x) y ph)) ph) (y x ph sbco sbid2.1 y sbf bitr)) thm (sbidm () () (<-> ([/] (cv y) x ([/] (cv y) x ph)) ([/] (cv y) x ph)) (x y ([/] (cv y) x ph) sbequ12 bicomd x a4s x y x hbnae x y ph hbsb2 (= (cv x) (cv y)) ([/] (cv y) x ph) pm4.2i (-. (A. x (= (cv x) (cv y)))) a1i sbied pm2.61i)) thm (sbco2 () ((sbco2.1 (-> ph (A. z ph)))) (<-> ([/] (cv y) z ([/] (cv z) x ph)) ([/] (cv y) x ph)) (x y z ([/] (cv z) x ph) sbequ sbco2.1 x sbid2 syl5bbr x y ph sbequ12 bitr3d x a4s x y x hbnae sbco2.1 x hbsb3 y z hbsb4 x y z ([/] (cv z) x ph) sbequ sbco2.1 x sbid2 syl5bbr (-. (A. x (= (cv x) (cv y)))) a1i sbied bicomd pm2.61i)) thm (sbco2d () ((sbco2d.1 (-> ph (A. x ph))) (sbco2d.2 (-> ph (A. z ph))) (sbco2d.3 (-> ph (-> ps (A. z ps))))) (-> ph (<-> ([/] (cv y) z ([/] (cv z) x ps)) ([/] (cv y) x ps))) (sbco2d.2 sbco2d.3 hbim1 y x sbco2 sbco2d.1 z ps sb19.21 y z sbbii sbco2d.2 y ([/] (cv z) x ps) sb19.21 bitr sbco2d.1 y ps sb19.21 3bitr3 pm5.74ri)) thm (sbco3 () () (<-> ([/] (cv z) y ([/] (cv y) x ph)) ([/] (cv z) x ([/] (cv x) y ph))) (x y z ([/] (cv y) x ph) drsb1 x y ph sbequ12a x 19.20i x ([/] (cv y) x ph) ([/] (cv x) y ph) z sbba4 syl bitr3d x y y hbnae x y x hbnae x y ph hbsb2 z sbco2d x y ph sbco z x sbbii syl5rbbr pm2.61i)) thm (sbcom () () (<-> ([/] (cv y) z ([/] (cv y) x ph)) ([/] (cv y) x ([/] (cv y) z ph))) (x z y ([/] (cv y) x ph) drsb1 x z x hbae x z y ph drsb1 y sbbid bitr3d (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. z (= (cv z) (cv y))))) adantr x z z hbnae x y z hbnae hban x z x hbnae x y x hbnae hban x z y ax-12 imp x 19.20i x (= (cv z) (cv y)) (-> (= (cv x) (cv y)) ph) 19.21g 3syl albid (-. (A. z (= (cv z) (cv y)))) adantrr x z x hbnae z y x hbnae hban x z z hbnae z y z hbnae hban z x y ax-12 nalequcoms imp z 19.20i z (= (cv x) (cv y)) (-> (= (cv z) (cv y)) ph) 19.21g 3syl (= (cv z) (cv y)) (= (cv x) (cv y)) ph bi2.04 z albii syl5bb albid z x (-> (= (cv z) (cv y)) (-> (= (cv x) (cv y)) ph)) alcom syl5bb (-. (A. x (= (cv x) (cv y)))) adantrl bitr3d z y ([/] (cv y) x ph) sb4b x y z hbnae x y ph sb4b (= (cv z) (cv y)) imbi2d albid sylan9bbr (-. (A. x (= (cv x) (cv z)))) adantl x y ([/] (cv y) z ph) sb4b z y x hbnae z y ph sb4b (= (cv x) (cv y)) imbi2d albid sylan9bb (-. (A. x (= (cv x) (cv z)))) adantl 3bitr4d pm2.61ian ex x y z hbae x y ph sbequ12 x a4s y sbbid x y ([/] (cv y) z ph) sbequ12 x a4s bitr3d z y ([/] (cv y) x ph) sbequ12 z a4s z y x hbae z y ph sbequ12 z a4s y sbbid bitr3d pm2.61ii)) thm (sb5rf () ((sb5rf.1 (-> ph (A. y ph)))) (<-> ph (E. y (/\ (= (cv y) (cv x)) ([/] (cv y) x ph)))) (sb5rf.1 x sbid2 x y ([/] (cv y) x ph) sb1 sylbir sb5rf.1 y x ph sbequ12r biimpa 19.23ai impbi)) thm (sb6rf () ((sb5rf.1 (-> ph (A. y ph)))) (<-> ph (A. y (-> (= (cv y) (cv x)) ([/] (cv y) x ph)))) (sb5rf.1 x y ph sbequ1 equcoms com12 19.21ai y x ([/] (cv y) x ph) sb2 x y ph sbco sylib sb5rf.1 x sbf sylib impbi)) thm (sb8 () ((sb8.1 (-> ph (A. y ph)))) (<-> (A. x ph) (A. y ([/] (cv y) x ph))) (sb8.1 x hbal x ph y stdpc4 19.21ai sb8.1 x hbsb3 y hbal y ([/] (cv y) x ph) x stdpc4 sb8.1 x sbid2 sylib 19.21ai impbi)) thm (sb8e () ((sb8e.1 (-> ph (A. y ph)))) (<-> (E. x ph) (E. y ([/] (cv y) x ph))) (sb8e.1 hbne x sb8 y x ph sbn y albii bitr negbii x ph df-ex y ([/] (cv y) x ph) df-ex 3bitr4)) thm (sb9i () () (-> (A. x ([/] (cv x) y ph)) (A. y ([/] (cv y) x ph))) (y x y ph drsb1 y x y ph drsb2 bitr3d dral1 biimprd y x ph hbsb2 x 19.20ii hbnaes x ([/] (cv x) y ph) y stdpc4 y x ph sbco sylib y 19.20i a7s syl6 pm2.61i)) thm (sb9 () () (<-> (A. x ([/] (cv x) y ph)) (A. y ([/] (cv y) x ph))) (x y ph sb9i y x ph sb9i impbi)) thm (ax16 ((x y)) ((ax16.1 (-. (A. x (= (cv x) (cv y)))))) (-> (A. x (= (cv x) (cv y))) (-> ph (A. x ph))) (ax16.1 (-> ph (A. x ph)) pm2.21i)) thm (ax17eq ((x z) (y z)) () (-> (= (cv x) (cv y)) (A. z (= (cv x) (cv y)))) (z x y ax-12 z x (= (cv x) (cv y)) ax-16 z y (= (cv x) (cv y)) ax-16 pm2.61ii)) thm (ax17el ((x z) (y z)) () (-> (e. (cv x) (cv y)) (A. z (e. (cv x) (cv y)))) (z x y ax-15 z x (e. (cv x) (cv y)) ax-16 z y (e. (cv x) (cv y)) ax-16 pm2.61ii)) thm (ax11a ((x y)) () (-> (= (cv x) (cv y)) (-> ph (A. x (-> (= (cv x) (cv y)) ph)))) (x y (-> (= (cv x) (cv y)) ph) ax-16 ph (= (cv x) (cv y)) ax-1 syl5 (= (cv x) (cv y)) a1d x y ph ax-11 pm2.61i)) thm (equid2 ((x y)) () (= (cv x) (cv x)) (y x a9e (= (cv x) (cv x)) y ax-17 y x x ax-8 pm2.43i 19.23ai ax-mp)) thm (a4b ((ps x)) ((a4b.1 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> (A. x ph) ps) (ps x ax-17 a4b.1 a4a)) thm (a4b1 ((ps x)) ((a4b1.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (-> (A. x ph) ps) (a4b1.1 biimpd a4b)) thm (a4w ((ph x)) ((a4w.1 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> ph (E. x ps)) (ph x ax-17 a4w.1 a4c)) thm (a4w1 ((ps x)) ((a4w1.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (a4w1.2 ps)) (E. x ph) (a4w1.2 a4w1.1 biimprd a4w ax-mp)) thm (equvin ((x z) (y z)) () (<-> (= (cv x) (cv y)) (E. z (/\ (= (cv x) (cv z)) (= (cv z) (cv y))))) (x y z equvini (= (cv x) (cv y)) z ax-17 x z y equtr imp 19.23ai impbi)) thm (a16g ((x y)) () (-> (A. x (= (cv x) (cv y))) (-> ph (A. z ph))) (x y z hbae x z ax9a x y (-. (= (cv x) (cv z))) ax-16 mt3i x z equcomi syl 19.21ai x y ph ax-16 (A. z (= (cv z) (cv x))) ph pm4.2i dral1 biimprd syl9r mpcom)) thm (a16gb ((x y)) () (-> (A. x (= (cv x) (cv y))) (<-> ph (A. z ph))) (x y ph z a16g z ph ax-4 (A. x (= (cv x) (cv y))) a1i impbid)) thm (albidv ((ph x)) ((albidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (A. x ps) (A. x ch))) (ph x ax-17 albidv.1 albid)) thm (exbidv ((ph x)) ((albidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E. x ps) (E. x ch))) (ph x ax-17 albidv.1 exbid)) thm (2albidv ((ph x) (ph y)) ((2albidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (A. x (A. y ps)) (A. x (A. y ch)))) (2albidv.1 y albidv x albidv)) thm (2exbidv ((ph x) (ph y)) ((2albidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E. x (E. y ps)) (E. x (E. y ch)))) (2albidv.1 y exbidv x exbidv)) thm (3exbidv ((ph x) (ph y) (ph z)) ((3exbidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E. x (E. y (E. z ps))) (E. x (E. y (E. z ch))))) (3exbidv.1 z exbidv x y 2exbidv)) thm (4exbidv ((ph x) (ph y) (ph z) (ph w)) ((4exbidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E. x (E. y (E. z (E. w ps)))) (E. x (E. y (E. z (E. w ch)))))) (4exbidv.1 z w 2exbidv x y 2exbidv)) thm (19.9rv ((ph x)) () (<-> ph (E. x ph)) (ph x ax-17 19.9r)) thm (19.21v ((ph x)) () (<-> (A. x (-> ph ps)) (-> ph (A. x ps))) (ph x ax-17 ps 19.21)) thm (19.21aiv ((ph x)) ((19.21aiv.1 (-> ph ps))) (-> ph (A. x ps)) (ph x ax-17 19.21aiv.1 19.21ai)) thm (19.21aivv ((ph x) (ph y)) ((19.21aivv.1 (-> ph ps))) (-> ph (A. x (A. y ps))) (19.21aivv.1 y 19.21aiv x 19.21aiv)) thm (19.21adv ((ph x) (ps x)) ((19.21adv.1 (-> ph (-> ps ch)))) (-> ph (-> ps (A. x ch))) (ph x ax-17 ps x ax-17 19.21adv.1 19.21ad)) thm (19.20dv ((ph x)) ((19.20dv.1 (-> ph (-> ps ch)))) (-> ph (-> (A. x ps) (A. x ch))) (ph x ax-17 19.20dv.1 19.20d)) thm (19.22dv ((ph x)) ((19.20dv.1 (-> ph (-> ps ch)))) (-> ph (-> (E. x ps) (E. x ch))) (ph x ax-17 19.20dv.1 19.22d)) thm (19.20dvv ((ph x) (ph y)) ((19.20dvv.1 (-> ph (-> ps ch)))) (-> ph (-> (A. x (A. y ps)) (A. x (A. y ch)))) (19.20dvv.1 y 19.20dv x 19.20dv)) thm (19.22dvv ((ph x) (ph y)) ((19.20dvv.1 (-> ph (-> ps ch)))) (-> ph (-> (E. x (E. y ps)) (E. x (E. y ch)))) (19.20dvv.1 y 19.22dv x 19.22dv)) thm (19.23v ((ps x)) () (<-> (A. x (-> ph ps)) (-> (E. x ph) ps)) (ps x ax-17 ph 19.23)) thm (19.23vv ((ps x) (ps y)) () (<-> (A. x (A. y (-> ph ps))) (-> (E. x (E. y ph)) ps)) (y ph ps 19.23v x albii x (E. y ph) ps 19.23v bitr)) thm (19.23aiv ((ps x)) ((19.23aiv.1 (-> ph ps))) (-> (E. x ph) ps) (ps x ax-17 19.23aiv.1 19.23ai)) thm (19.23aivv ((ps x) (ps y)) ((19.23aivv.1 (-> ph ps))) (-> (E. x (E. y ph)) ps) (19.23aivv.1 y 19.23aiv x 19.23aiv)) thm (19.23adv ((ch x) (ph x)) ((19.23adv.1 (-> ph (-> ps ch)))) (-> ph (-> (E. x ps) ch)) (ph x ax-17 ch x ax-17 19.23adv.1 19.23ad)) thm (19.23advv ((ch x) (ph x) (ch y) (ph y)) ((19.23advv.1 (-> ph (-> ps ch)))) (-> ph (-> (E. x (E. y ps)) ch)) (19.23advv.1 y 19.23adv x 19.23adv)) thm (19.27v ((ps x)) () (<-> (A. x (/\ ph ps)) (/\ (A. x ph) ps)) (ps x ax-17 ph 19.27)) thm (19.28v ((ph x)) () (<-> (A. x (/\ ph ps)) (/\ ph (A. x ps))) (ph x ax-17 ps 19.28)) thm (19.36v ((ps x)) () (<-> (E. x (-> ph ps)) (-> (A. x ph) ps)) (ps x ax-17 ph 19.36)) thm (19.36aiv ((ps x)) ((19.36aiv.1 (E. x (-> ph ps)))) (-> (A. x ph) ps) (ps x ax-17 19.36aiv.1 19.36i)) thm (19.12vv ((x y) (ps x) (ph y)) () (<-> (E. x (A. y (-> ph ps))) (A. y (E. x (-> ph ps)))) (y ph ps 19.21v x exbii x ph (A. y ps) 19.36v x ph ps 19.36v y albii y (A. x ph) ps 19.21v bitr2 3bitr)) thm (19.37v ((ph x)) () (<-> (E. x (-> ph ps)) (-> ph (E. x ps))) (ph x ax-17 ps 19.37)) thm (19.37aiv ((ph x)) ((19.37aiv.1 (E. x (-> ph ps)))) (-> ph (E. x ps)) (19.37aiv.1 x ph ps 19.37v mpbi)) thm (19.41v ((ps x)) () (<-> (E. x (/\ ph ps)) (/\ (E. x ph) ps)) (ps x ax-17 ph 19.41)) thm (19.41vv ((ps x) (ps y)) () (<-> (E. x (E. y (/\ ph ps))) (/\ (E. x (E. y ph)) ps)) (y ph ps 19.41v x exbii x (E. y ph) ps 19.41v bitr)) thm (19.41vvv ((ps x) (ps y) (ps z)) () (<-> (E. x (E. y (E. z (/\ ph ps)))) (/\ (E. x (E. y (E. z ph))) ps)) (y z ph ps 19.41vv x exbii x (E. y (E. z ph)) ps 19.41v bitr)) thm (19.42v ((ph x)) () (<-> (E. x (/\ ph ps)) (/\ ph (E. x ps))) (ph x ax-17 ps 19.42)) thm (exdistr ((ph y)) () (<-> (E. x (E. y (/\ ph ps))) (E. x (/\ ph (E. y ps)))) (y ph ps 19.42v x exbii)) thm (19.42vv ((ph x) (ph y)) () (<-> (E. x (E. y (/\ ph ps))) (/\ ph (E. x (E. y ps)))) (x y ph ps exdistr x ph (E. y ps) 19.42v bitr)) thm (exdistr2 ((ph y) (ph z)) () (<-> (E. x (E. y (E. z (/\ ph ps)))) (E. x (/\ ph (E. y (E. z ps))))) (y z ph ps 19.42vv x exbii)) thm (3exdistr ((ph y) (ph z) (ps z)) () (<-> (E. x (E. y (E. z (/\/\ ph ps ch)))) (E. x (/\ ph (E. y (/\ ps (E. z ch)))))) (ph ps ch 3anass z exbii z ph (/\ ps ch) 19.42v z ps ch 19.42v ph anbi2i 3bitr y exbii y ph (/\ ps (E. z ch)) 19.42v bitr x exbii)) thm (4exdistr ((ph y) (ph z) (ph w) (ps z) (ps w) (ch w)) () (<-> (E. x (E. y (E. z (E. w (/\ (/\ ph ps) (/\ ch th)))))) (E. x (/\ ph (E. y (/\ ps (E. z (/\ ch (E. w th)))))))) (ph ps (/\ ch th) anass w exbii w ph (/\ ps (/\ ch th)) 19.42v w ps (/\ ch th) 19.42v ph anbi2i w ch th 19.42v ps anbi2i ph anbi2i 3bitr bitr z exbii z ph (/\ ps (/\ ch (E. w th))) 19.42v z ps (/\ ch (E. w th)) 19.42v ph anbi2i 3bitr y exbii y ph (/\ ps (E. z (/\ ch (E. w th)))) 19.42v bitr x exbii)) thm (cbvalv ((ph y) (ps x)) ((cbvalv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A. x ph) (A. y ps)) (ph y ax-17 ps x ax-17 cbvalv.1 cbval)) thm (cbvexv ((ph y) (ps x)) ((cbvalv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E. x ph) (E. y ps)) (ph y ax-17 ps x ax-17 cbvalv.1 cbvex)) thm (cbval2 ((x y) (y z) (w x) (w z)) ((cbval2.1 (-> ph (A. z ph))) (cbval2.2 (-> ph (A. w ph))) (cbval2.3 (-> ps (A. x ps))) (cbval2.4 (-> ps (A. y ps))) (cbval2.5 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (<-> (A. x (A. y ph)) (A. z (A. w ps))) (cbval2.1 y hbal cbval2.3 w hbal (= (cv x) (cv z)) w ax-17 cbval2.2 hban (= (cv x) (cv z)) y ax-17 cbval2.4 hban cbval2.5 expcom pm5.32d cbval (= (cv x) (cv z)) y ax-17 ph 19.28 (= (cv x) (cv z)) w ax-17 ps 19.28 3bitr3 (= (cv x) (cv z)) (A. y ph) (A. w ps) pm5.32 mpbir cbval)) thm (cbvex2 ((x y) (y z) (w x) (w z)) ((cbval2.1 (-> ph (A. z ph))) (cbval2.2 (-> ph (A. w ph))) (cbval2.3 (-> ps (A. x ps))) (cbval2.4 (-> ps (A. y ps))) (cbval2.5 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (<-> (E. x (E. y ph)) (E. z (E. w ps))) (cbval2.1 y hbex cbval2.3 w hbex (= (cv x) (cv z)) w ax-17 cbval2.2 hban (= (cv x) (cv z)) y ax-17 cbval2.4 hban cbval2.5 expcom pm5.32d cbvex (= (cv x) (cv z)) y ax-17 ph 19.42 (= (cv x) (cv z)) w ax-17 ps 19.42 3bitr3 (= (cv x) (cv z)) (E. y ph) (E. w ps) pm5.32 mpbir cbvex)) thm (cbval2v ((w z) (ph z) (ph w) (x y) (ps x) (ps y) (w x) (y z)) ((cbval2v.1 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (<-> (A. x (A. y ph)) (A. z (A. w ps))) (ph z ax-17 ph w ax-17 ps x ax-17 ps y ax-17 cbval2v.1 cbval2)) thm (cbvex2v ((w z) (ph z) (ph w) (x y) (ps x) (ps y) (w x) (y z)) ((cbval2v.1 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (<-> (E. x (E. y ph)) (E. z (E. w ps))) (ph z ax-17 ph w ax-17 ps x ax-17 ps y ax-17 cbval2v.1 cbvex2)) thm (cbvald ((ph x) (ch x)) ((cbvald.1 (-> ph (A. y ph))) (cbvald.2 (-> ph (-> ps (A. y ps)))) (cbvald.3 (-> ph (-> (= (cv x) (cv y)) (<-> ps ch))))) (-> ph (<-> (A. x ps) (A. y ch))) (cbvald.1 cbvald.2 hbim1 (-> ph ch) x ax-17 cbvald.3 com12 pm5.74d cbval x ph ps 19.21v cbvald.1 ch 19.21 3bitr3 pm5.74ri)) thm (cbvexd ((ph x) (ch x)) ((cbvald.1 (-> ph (A. y ph))) (cbvald.2 (-> ph (-> ps (A. y ps)))) (cbvald.3 (-> ph (-> (= (cv x) (cv y)) (<-> ps ch))))) (-> ph (<-> (E. x ps) (E. y ch))) (cbvald.1 cbvald.1 cbvald.2 hbnd cbvald.3 ps ch pm4.11 syl6ib cbvald negbid x ps df-ex y ch df-ex 3bitr4g)) var (set f) var (set g) thm (cbvex4v ((u v) (ph v) (ph u) (f g) (f ph) (g ph) (x y) (ch x) (ch y) (w z) (ch z) (ch w) (ps x) (ps y) (f ps) (g ps) (x z) (w x) (u x) (u z) (u w) (y z) (w y) (v y) (v z) (v w) (g z) (f w)) ((cbvex4v.1 (-> (/\ (= (cv x) (cv v)) (= (cv y) (cv u))) (<-> ph ps))) (cbvex4v.2 (-> (/\ (= (cv z) (cv f)) (= (cv w) (cv g))) (<-> ps ch)))) (<-> (E. x (E. y (E. z (E. w ph)))) (E. v (E. u (E. f (E. g ch))))) (cbvex4v.1 z w 2exbidv cbvex2v cbvex4v.2 cbvex2v v u 2exbii bitr)) thm (eeanv ((ph y) (ps x)) () (<-> (E. x (E. y (/\ ph ps))) (/\ (E. x ph) (E. y ps))) (x y ph ps exdistr ps x ax-17 y hbex ph 19.41 bitr)) thm (eeeanv ((y z) (ph y) (ph z) (x z) (ps x) (ps z) (x y) (ch x) (ch y)) () (<-> (E. x (E. y (E. z (/\/\ ph ps ch)))) (/\/\ (E. x ph) (E. y ps) (E. z ch))) (y z ph (/\ ps ch) 19.42vv y z ps ch eeanv ph anbi2i bitr x exbii x ph (/\ (E. y ps) (E. z ch)) 19.41v bitr ph ps ch 3anass x y z 3exbi (E. x ph) (E. y ps) (E. z ch) 3anass 3bitr4)) thm (ee4anv ((ph z) (ph w) (ps x) (ps y) (y z) (w x)) () (<-> (E. x (E. y (E. z (E. w (/\ ph ps))))) (/\ (E. x (E. y ph)) (E. z (E. w ps)))) (y z (E. w (/\ ph ps)) excom x exbii y w ph ps eeanv x z 2exbii x z (E. y ph) (E. w ps) eeanv 3bitr)) thm (nexdv ((ph x)) ((nexdv.1 (-> ph (-. ps)))) (-> ph (-. (E. x ps))) (ph x ax-17 nexdv.1 nexd)) thm (chvarv ((ps x)) ((chv.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (chv.2 ph)) ps (chv.1 a4b1 chv.2 mpg)) thm (cleljust ((x z) (y z)) () (<-> (e. (cv x) (cv y)) (E. z (/\ (= (cv z) (cv x)) (e. (cv z) (cv y))))) ((e. (cv x) (cv y)) z ax-17 z x y elequ1 equsex bicomi)) thm (equsb3lem ((y z) (x y)) () (<-> ([/] (cv x) y (= (cv y) (cv z))) (= (cv x) (cv z))) (x y equsb2 x y z equequ1 x y sbimi ax-mp x y (= (cv x) (cv z)) (= (cv y) (cv z)) sbbi mpbi (= (cv x) (cv z)) y ax-17 x sbf bitr3)) thm (equsb3 ((w y) (w z) (y z) (w x)) () (<-> ([/] (cv x) y (= (cv y) (cv z))) (= (cv x) (cv z))) (w y z equsb3lem x w sbbii (= (cv y) (cv z)) w ax-17 x y sbco2 x w z equsb3lem 3bitr3)) thm (hbs1 ((x y)) () (-> ([/] (cv y) x ph) (A. x ([/] (cv y) x ph))) (x y ([/] (cv y) x ph) ax-16 x y ph hbsb2 pm2.61i)) thm (hbsb ((y z)) ((hbsb.1 (-> ph (A. z ph)))) (-> ([/] (cv y) x ph) (A. z ([/] (cv y) x ph))) (z y ([/] (cv y) x ph) ax-16 hbsb.1 y x hbsb4 pm2.61i)) thm (sbcom2 ((x z) (w x) (y z)) () (<-> ([/] (cv w) z ([/] (cv y) x ph)) ([/] (cv y) x ([/] (cv w) z ph))) (z x (-> (= (cv x) (cv y)) (-> (= (cv z) (cv w)) ph)) alcom (= (cv x) (cv y)) (= (cv z) (cv w)) ph bi2.04 x albii x (= (cv z) (cv w)) (-> (= (cv x) (cv y)) ph) 19.21v bitr z albii z (= (cv x) (cv y)) (-> (= (cv z) (cv w)) ph) 19.21v x albii 3bitr3 (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. z (= (cv z) (cv w))))) a1i z w ([/] (cv y) x ph) sb4b x y ph sb4b (= (cv z) (cv w)) imbi2d z albidv sylan9bbr x y ([/] (cv w) z ph) sb4b z w ph sb4b (= (cv x) (cv y)) imbi2d x albidv sylan9bb 3bitr4d ex x y z hbae x y ph sbequ12 x a4s w sbbid x y ([/] (cv w) z ph) sbequ12 x a4s bitr3d z w ([/] (cv y) x ph) sbequ12 z a4s z w x hbae z w ph sbequ12 z a4s y sbbid bitr3d pm2.61ii)) thm (sb5 ((x y)) () (<-> ([/] (cv y) x ph) (E. x (/\ (= (cv x) (cv y)) ph))) (y x ph hbs1 x y ph sbequ12 equsex bicomi)) thm (sb6 ((x y)) () (<-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ph))) (y x ph hbs1 x y ph sbequ12 equsal bicomi)) thm (2sb5 ((x y) (x z) (y z) (w y)) () (<-> ([/] (cv z) x ([/] (cv w) y ph)) (E. x (E. y (/\ (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) ph)))) (z x ([/] (cv w) y ph) sb5 y (= (cv x) (cv z)) (/\ (= (cv y) (cv w)) ph) 19.42v (= (cv x) (cv z)) (= (cv y) (cv w)) ph anass y exbii w y ph sb5 (= (cv x) (cv z)) anbi2i 3bitr4r x exbii bitr)) thm (2sb6 ((x y) (x z) (y z) (w y)) () (<-> ([/] (cv z) x ([/] (cv w) y ph)) (A. x (A. y (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) ph)))) (z x ([/] (cv w) y ph) sb6 y (= (cv x) (cv z)) (-> (= (cv y) (cv w)) ph) 19.21v (= (cv x) (cv z)) (= (cv y) (cv w)) ph impexp y albii w y ph sb6 (= (cv x) (cv z)) imbi2i 3bitr4r x albii bitr)) thm (sb6a ((x y)) () (<-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ([/] (cv x) y ph)))) (y x ph sb6 y x ph sbequ12 equcoms pm5.74i x albii bitr)) thm (2sb5rf ((x y) (w x) (y z) (w z)) ((2sb5rf.1 (-> ph (A. z ph))) (2sb5rf.2 (-> ph (A. w ph)))) (<-> ph (E. z (E. w (/\ (/\ (= (cv z) (cv x)) (= (cv w) (cv y))) ([/] (cv z) x ([/] (cv w) y ph)))))) (2sb5rf.1 x sb5rf w (= (cv z) (cv x)) (/\ (= (cv w) (cv y)) ([/] (cv w) y ([/] (cv z) x ph))) 19.42v z x w y ph sbcom2 (/\ (= (cv z) (cv x)) (= (cv w) (cv y))) anbi2i (= (cv z) (cv x)) (= (cv w) (cv y)) ([/] (cv w) y ([/] (cv z) x ph)) anass bitr w exbii 2sb5rf.2 z x hbsb y sb5rf (= (cv z) (cv x)) anbi2i 3bitr4r z exbii bitr)) thm (2sb6rf ((x y) (w x) (y z) (w z)) ((2sb5rf.1 (-> ph (A. z ph))) (2sb5rf.2 (-> ph (A. w ph)))) (<-> ph (A. z (A. w (-> (/\ (= (cv z) (cv x)) (= (cv w) (cv y))) ([/] (cv z) x ([/] (cv w) y ph)))))) (2sb5rf.1 x sb6rf w (= (cv z) (cv x)) (-> (= (cv w) (cv y)) ([/] (cv w) y ([/] (cv z) x ph))) 19.21v z x w y ph sbcom2 (/\ (= (cv z) (cv x)) (= (cv w) (cv y))) imbi2i (= (cv z) (cv x)) (= (cv w) (cv y)) ([/] (cv w) y ([/] (cv z) x ph)) impexp bitr w albii 2sb5rf.2 z x hbsb y sb6rf (= (cv z) (cv x)) imbi2i 3bitr4r z albii bitr)) thm (sb7 ((x z) (y z) (ph z)) () (<-> ([/] (cv y) x ph) (E. z (/\ (= (cv z) (cv y)) (E. x (/\ (= (cv x) (cv z)) ph))))) (z x ph sb5 y z sbbii ph z ax-17 y x sbco2 y z (E. x (/\ (= (cv x) (cv z)) ph)) sb5 3bitr3)) thm (sb7f ((w x) (w z) (x z) (w y) (y z) (ph w)) ((sb7f.1 (-> ph (A. z ph)))) (<-> ([/] (cv y) x ph) (E. z (/\ (= (cv z) (cv y)) (E. x (/\ (= (cv x) (cv z)) ph))))) (y x ph w sb7 (= (cv w) (cv y)) z ax-17 (= (cv x) (cv w)) z ax-17 sb7f.1 hban x hbex hban (/\ (= (cv z) (cv y)) (E. x (/\ (= (cv x) (cv z)) ph))) w ax-17 w z y equequ1 w z x equequ2 ph anbi1d x exbidv anbi12d cbvex bitr)) thm (sb10f ((x y)) ((sb10f.1 (-> ph (A. x ph)))) (<-> ([/] (cv y) z ph) (E. x (/\ (= (cv x) (cv y)) ([/] (cv x) z ph)))) (sb10f.1 y z hbsb x y z ph sbequ equsex bicomi)) thm (sbid2v ((ph x)) () (<-> ([/] (cv y) x ([/] (cv x) y ph)) ph) (ph x ax-17 y sbid2)) thm (sbelx ((x y) (ph x)) () (<-> ph (E. x (/\ (= (cv x) (cv y)) ([/] (cv x) y ph)))) (y x ph sbid2v y x ([/] (cv x) y ph) sb5 bitr3)) thm (sbel2x ((x y) (x z) (w x) (y z) (w y) (w z) (ph x) (ph y)) () (<-> ph (E. x (E. y (/\ (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) ([/] (cv y) w ([/] (cv x) z ph)))))) (([/] (cv x) z ph) y w sbelx (= (cv x) (cv z)) anbi2i x exbii ph x z sbelx x y (= (cv x) (cv z)) (/\ (= (cv y) (cv w)) ([/] (cv y) w ([/] (cv x) z ph))) exdistr 3bitr4 (= (cv x) (cv z)) (= (cv y) (cv w)) ([/] (cv y) w ([/] (cv x) z ph)) anass x y 2exbii bitr4)) thm (sbal1 ((x y)) () (-> (-. (A. x (= (cv x) (cv z)))) (<-> ([/] (cv z) y (A. x ph)) (A. x ([/] (cv z) y ph)))) (y z (A. x ph) sbequ12 y a4s y z ph sbequ12 y a4s x dral2 bitr3d (-. (A. x (= (cv x) (cv z)))) a1d x ph hba1 z y hbsb4 x ph ax-4 z y sbimi x 19.20i syl6 (-. (A. y (= (cv y) (cv z)))) adantl y z ph sb4 x 19.20ii hbnaes x y (-> (= (cv y) (cv z)) ph) ax-7 syl6 x y (= (cv y) (cv z)) ax-16 (-. (A. x (= (cv x) (cv z)))) a1d x y z ax-12 pm2.61i x (= (cv y) (cv z)) ph 19.20 syl9 y 19.20ii y z (A. x ph) sb2 syl6 hbnaes sylan9 impbid ex pm2.61i)) thm (sbal ((x y) (x z)) () (<-> ([/] (cv z) y (A. x ph)) (A. x ([/] (cv z) y ph))) (x z ph x a16gb z y sbimi z y x z sbequ5 z y ph (A. x ph) sbbi 3imtr3 x z ([/] (cv z) y ph) x a16gb bitr3d x z y ph sbal1 pm2.61i)) thm (sbex ((x y) (x z)) () (<-> ([/] (cv z) y (E. x ph)) (E. x ([/] (cv z) y ph))) (z y (A. x (-. ph)) sbn z y x (-. ph) sbal z y ph sbn x albii bitr negbii bitr x ph df-ex z y sbbii x ([/] (cv z) y ph) df-ex 3bitr4)) thm (sbalv ((x z) (y z)) ((sbalv.1 (<-> ([/] (cv y) x ph) ps))) (<-> ([/] (cv y) x (A. z ph)) (A. z ps)) (y x z ph sbal sbalv.1 z albii bitr)) thm (exsb ((x y) (ph y)) () (<-> (E. x ph) (E. y (A. x (-> (= (cv x) (cv y)) ph)))) (ph y ax-17 x sb8e y x ph sb6 y exbii bitr)) thm (2exsb ((x y) (x z) (y z) (w y) (w z) (ph z) (ph w)) () (<-> (E. x (E. y ph)) (E. z (E. w (A. x (A. y (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) ph)))))) (y ph w exsb x exbii x w (A. y (-> (= (cv y) (cv w)) ph)) excom bitr x (A. y (-> (= (cv y) (cv w)) ph)) z exsb (= (cv x) (cv z)) (= (cv y) (cv w)) ph impexp y albii y (= (cv x) (cv z)) (-> (= (cv y) (cv w)) ph) 19.21v bitr2 x albii z exbii bitr w exbii w z (A. x (A. y (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) ph))) excom 3bitr)) thm (dvelim ((ps z)) ((dvelim.1 (-> ph (A. x ph))) (dvelim.2 (-> (= (cv z) (cv y)) (<-> ph ps)))) (-> (-. (A. x (= (cv x) (cv y)))) (-> ps (A. x ps))) (dvelim.1 ps z ax-17 dvelim.2 dvelimf)) thm (dveeq1 ((w z) (w x) (x z) (w y)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (= (cv y) (cv z)) (A. x (= (cv y) (cv z))))) ((= (cv w) (cv z)) x ax-17 w y z equequ1 dvelim)) thm (dveeq2 ((w z) (w x) (x z) (w y)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (= (cv z) (cv y)) (A. x (= (cv z) (cv y))))) ((= (cv z) (cv w)) x ax-17 w y z equequ2 dvelim)) thm (dveel1 ((w z) (w x) (x z) (w y)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (e. (cv y) (cv z)) (A. x (e. (cv y) (cv z))))) ((e. (cv w) (cv z)) x ax-17 w y z elequ1 dvelim)) thm (dveel2 ((w z) (w x) (x z) (w y)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (e. (cv z) (cv y)) (A. x (e. (cv z) (cv y))))) ((e. (cv z) (cv w)) x ax-17 w y z elequ2 dvelim)) thm (sbal2 ((y z) (x z)) () (-> (-. (A. x (= (cv x) (cv y)))) (<-> ([/] (cv z) y (A. x ph)) (A. x ([/] (cv z) y ph)))) (x y y hbnae x y z dveeq1 x 19.20i hbnaes x (= (cv y) (cv z)) ph 19.21g syl albid y x (-> (= (cv y) (cv z)) ph) alcom syl5rbbr z y (A. x ph) sb6 z y ph sb6 x albii 3bitr4g)) thm (ax15 ((w y) (w z) (w x)) () (-> (-. (A. z (= (cv z) (cv x)))) (-> (-. (A. z (= (cv z) (cv y)))) (-> (e. (cv x) (cv y)) (A. z (e. (cv x) (cv y)))))) (z (= (cv z) (cv y)) hbn1 z y w dveel2 hbim1 w x y elequ1 (-. (A. z (= (cv z) (cv y)))) imbi2d dvelim z (= (cv z) (cv y)) hbn1 (e. (cv x) (cv y)) 19.21 syl6ib pm2.86d)) thm (ax11el ((x z) (y z)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (= (cv x) (cv y)) (-> (e. (cv x) (cv y)) (A. x (-> (= (cv x) (cv y)) (e. (cv x) (cv y))))))) (x y y elequ1 (-. (A. x (= (cv x) (cv y)))) adantl (e. (cv z) (cv z)) x ax-17 (e. (cv y) (cv y)) z ax-17 z y z elequ1 z y y elequ2 bitrd dvelimf2 x y y elequ1 biimprcd x 19.20i syl6 (= (cv x) (cv y)) adantr sylbid ex)) thm (euf ((x y) (x z) (y z) (ph z)) ((euf.1 (-> ph (A. y ph)))) (<-> (E! x ph) (E. y (A. x (<-> ph (= (cv x) (cv y)))))) (x ph z df-eu euf.1 (= (cv x) (cv z)) y ax-17 hbbi x hbal (A. x (<-> ph (= (cv x) (cv y)))) z ax-17 z y x equequ2 ph bibi2d x albidv cbvex bitr)) thm (eubid ((x y) (ph y) (ps y) (ch y)) ((eubid.1 (-> ph (A. x ph))) (eubid.2 (-> ph (<-> ps ch)))) (-> ph (<-> (E! x ps) (E! x ch))) (eubid.1 eubid.2 (= (cv x) (cv y)) bibi1d albid y exbidv x ps y df-eu x ch y df-eu 3bitr4g)) thm (eubidv ((ph x)) ((eubidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E! x ps) (E! x ch))) (ph x ax-17 eubidv.1 eubid)) thm (eubii () ((eubii.1 (<-> ph ps))) (<-> (E! x ph) (E! x ps)) (x equid x hbequid eubii.1 (= (cv x) (cv x)) a1i eubid ax-mp)) thm (hbeu1 ((x y) (ph y)) () (-> (E! x ph) (A. x (E! x ph))) (x (<-> ph (= (cv x) (cv y))) hba1 y hbex x ph y df-eu x ph y df-eu x albii 3imtr4)) thm (hbeu ((y z) (x z) (ph z)) ((hbeu.1 (-> ph (A. x ph)))) (-> (E! y ph) (A. x (E! y ph))) (y x (A. y (<-> ph (= (cv y) (cv z)))) ax-10 alequcoms y (<-> ph (= (cv y) (cv z))) hba1 syl5 x y y hbnae x y x hbnae hbeu.1 (-. (A. x (= (cv x) (cv y)))) a1i x y z dveeq1 hbbid hbald pm2.61i z hbex y ph z df-eu y ph z df-eu x albii 3imtr4)) thm (sb8eu ((x y) (x z) (y z) (ph z)) ((sb8eu.1 (-> ph (A. y ph)))) (<-> (E! x ph) (E! y ([/] (cv y) x ph))) (sb8eu.1 (= (cv x) (cv z)) y ax-17 hbbi x sb8 (= (cv y) (cv z)) x ax-17 x y z equequ1 sbie ph sblbis y albii bitr z exbii x ph z df-eu y ([/] (cv y) x ph) z df-eu 3bitr4)) thm (cbveu ((x y)) ((cbveu.1 (-> ph (A. y ph))) (cbveu.2 (-> ps (A. x ps))) (cbveu.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E! x ph) (E! y ps)) (cbveu.1 x sb8eu cbveu.2 cbveu.3 sbie y eubii bitr)) thm (eu1 ((x y)) ((eu1.1 (-> ph (A. y ph)))) (<-> (E! x ph) (E. x (/\ ph (A. y (-> ([/] (cv y) x ph) (= (cv x) (cv y))))))) (y x ph hbs1 y euf eu1.1 x sb8eu x y equcom ([/] (cv y) x ph) imbi2i y albii eu1.1 x sb6rf anbi12i ph (A. y (-> ([/] (cv y) x ph) (= (cv x) (cv y)))) ancom y ([/] (cv y) x ph) (= (cv y) (cv x)) albi 3bitr4 x exbii 3bitr4)) thm (mo ((x y) (x z) (y z) (ph z)) ((mo.1 (-> ph (A. y ph)))) (<-> (E. y (A. x (-> ph (= (cv x) (cv y))))) (A. x (A. y (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y)))))) (mo.1 (= (cv x) (cv z)) y ax-17 hbim x hbal (A. x (-> ph (= (cv x) (cv y)))) z ax-17 z y x equequ2 ph imbi2d x albidv cbvex mo.1 (= (cv x) (cv z)) y ax-17 hbim y x ph hbs1 (= (cv y) (cv z)) x ax-17 hbim x y ph sbequ2 x y z ax-8 imim12d cbv3 ancli mo.1 (= (cv x) (cv z)) y ax-17 hbim y x ph hbs1 (= (cv y) (cv z)) x ax-17 hbim aaan sylibr ph (= (cv x) (cv z)) ([/] (cv y) x ph) (= (cv y) (cv z)) prth x z y equtr2 syl6 x y 19.20i2 syl z 19.23aiv sylbir x ([/] (cv y) x ph) (-> ph (= (cv x) (cv y))) 19.20 y 19.20i a7s y (A. x ([/] (cv y) x ph)) (A. x (-> ph (= (cv x) (cv y)))) 19.22 syl mo.1 x hbsb3 y 19.22i syl5com ph ([/] (cv y) x ph) (= (cv x) (cv y)) impexp ph ([/] (cv y) x ph) (= (cv x) (cv y)) bi2.04 bitr x y 2albii syl5ib y ([/] (cv y) x ph) alnex mo.1 x hbsb3 hbne mo.1 hbne x y ph sbequ1 equcoms con3d cbv3 ph (= (cv x) (cv y)) pm2.21 x 19.20i (A. x (-> ph (= (cv x) (cv y)))) y 19.8a 3syl sylbir (A. x (A. y (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y))))) a1d pm2.61i impbi)) thm (euex ((x y) (ph y)) () (-> (E! x ph) (E. x ph)) (ph y ax-17 x eu1 x ph (A. y (-> ([/] (cv y) x ph) (= (cv x) (cv y)))) 19.40 sylbi pm3.26d)) thm (eumo0 ((x y)) ((eumo0.1 (-> ph (A. y ph)))) (-> (E! x ph) (E. y (A. x (-> ph (= (cv x) (cv y)))))) (eumo0.1 x euf ph (= (cv x) (cv y)) bi1 x 19.20i y 19.22i sylbi)) thm (eu2 ((x y)) ((eu2.1 (-> ph (A. y ph)))) (<-> (E! x ph) (/\ (E. x ph) (A. x (A. y (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y))))))) (x ph euex eu2.1 x eumo0 eu2.1 x mo sylib jca x ph (A. y (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y)))) 19.29r ph ([/] (cv y) x ph) (= (cv x) (cv y)) impexp y albii eu2.1 (-> ([/] (cv y) x ph) (= (cv x) (cv y))) 19.21 bitr ph anbi2i ph (A. y (-> ([/] (cv y) x ph) (= (cv x) (cv y)))) abai bitr4 x exbii sylib eu2.1 x eu1 sylibr impbi)) thm (eu3 ((x y)) ((eu3.1 (-> ph (A. y ph)))) (<-> (E! x ph) (/\ (E. x ph) (E. y (A. x (-> ph (= (cv x) (cv y))))))) (eu3.1 x eu2 eu3.1 x mo (E. x ph) anbi2i bitr4)) thm (euor () ((euor.1 (-> ph (A. x ph)))) (-> (/\ (-. ph) (E! x ps)) (E! x (\/ ph ps))) (euor.1 hbne ph ps biorf eubid biimpa)) thm (euorv ((ph x)) () (-> (/\ (-. ph) (E! x ps)) (E! x (\/ ph ps))) (ph x ax-17 ps euor)) thm (mo2 ((x y)) ((mo2.1 (-> ph (A. y ph)))) (<-> (E* x ph) (E. y (A. x (-> ph (= (cv x) (cv y)))))) (x ph df-mo x ph alnex ph (= (cv x) (cv y)) pm2.21 x 19.20i (A. x (-> ph (= (cv x) (cv y)))) y 19.8a syl sylbir mo2.1 x eumo0 ja mo2.1 x eu3 biimpr expcom impbi bitr)) thm (mo3 ((x y)) ((mo3.1 (-> ph (A. y ph)))) (<-> (E* x ph) (A. x (A. y (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y)))))) (mo3.1 x mo2 mo3.1 x mo bitr)) thm (mo4f ((x y) (ph y)) ((mo4f.1 (-> ps (A. x ps))) (mo4f.2 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E* x ph) (A. x (A. y (-> (/\ ph ps) (= (cv x) (cv y)))))) (ph y ax-17 x mo3 mo4f.1 mo4f.2 sbie ph anbi2i (= (cv x) (cv y)) imbi1i x y 2albii bitr)) thm (mo4 ((x y) (ph y) (ps x)) ((mo4.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E* x ph) (A. x (A. y (-> (/\ ph ps) (= (cv x) (cv y)))))) (ps x ax-17 mo4.1 mo4f)) thm (mobid () ((mobid.1 (-> ph (A. x ph))) (mobid.2 (-> ph (<-> ps ch)))) (-> ph (<-> (E* x ps) (E* x ch))) (mobid.1 mobid.2 exbid mobid.1 mobid.2 eubid imbi12d x ps df-mo x ch df-mo 3bitr4g)) thm (mobii () ((mobii.1 (<-> ps ch))) (<-> (E* x ps) (E* x ch)) (x equid x hbequid mobii.1 (= (cv x) (cv x)) a1i mobid ax-mp)) thm (hbmo1 () () (-> (E* x ph) (A. x (E* x ph))) (x ph hbe1 x ph hbeu1 hbim x ph df-mo x ph df-mo x albii 3imtr4)) thm (hbmo () ((hbmo.1 (-> ph (A. x ph)))) (-> (E* y ph) (A. x (E* y ph))) (hbmo.1 y hbex hbmo.1 y hbeu hbim y ph df-mo y ph df-mo x albii 3imtr4)) thm (cbvmo ((x y)) ((cbvmo.1 (-> ph (A. y ph))) (cbvmo.2 (-> ps (A. x ps))) (cbvmo.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E* x ph) (E* y ps)) (cbvmo.1 cbvmo.2 cbvmo.3 cbvex cbvmo.1 cbvmo.2 cbvmo.3 cbveu imbi12i x ph df-mo y ps df-mo 3bitr4)) thm (eu5 ((x y) (ph y)) () (<-> (E! x ph) (/\ (E. x ph) (E* x ph))) (ph y ax-17 x eu3 ph y ax-17 x mo2 (E. x ph) anbi2i bitr4)) thm (eu4 ((x y) (ph y) (ps x)) ((eu4.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E! x ph) (/\ (E. x ph) (A. x (A. y (-> (/\ ph ps) (= (cv x) (cv y))))))) (x ph eu5 eu4.1 mo4 (E. x ph) anbi2i bitr)) thm (eumo () () (-> (E! x ph) (E* x ph)) (x ph eu5 pm3.27bd)) thm (eumoi () ((eumoi.1 (E! x ph))) (E* x ph) (eumoi.1 x ph eumo ax-mp)) thm (exmoeu () () (<-> (E. x ph) (-> (E* x ph) (E! x ph))) (x ph df-mo biimp com12 x ph df-mo biimpr x ph euex imim12i (E. x ph) (E! x ph) peirce syl impbi)) thm (exmoeu2 () () (-> (E. x ph) (<-> (E* x ph) (E! x ph))) (x ph eu5 baibr)) thm (moabs () () (<-> (E* x ph) (-> (E. x ph) (E* x ph))) ((E. x ph) (E! x ph) pm5.4 x ph df-mo (E. x ph) imbi2i x ph df-mo 3bitr4r)) thm (exmo () () (\/ (E. x ph) (E* x ph)) ((E. x ph) (E! x ph) pm2.21 x ph df-mo sylibr orri)) thm (immo ((x y) (ph y) (ps y)) () (-> (A. x (-> ph ps)) (-> (E* x ps) (E* x ph))) (ph ps (= (cv x) (cv y)) imim1 x 19.20ii y 19.22dv ps y ax-17 x mo2 ph y ax-17 x mo2 3imtr4g)) thm (immoi () ((immoi.1 (-> ph ps))) (-> (E* x ps) (E* x ph)) (x ph ps immo immoi.1 mpg)) thm (moimv ((x y) (ph x) (ph y) (ps y)) () (-> (E* x (-> ph ps)) (-> ph (E* x ps))) (ps ph ax-1 ph a1i (= (cv x) (cv y)) imim1d x 19.20dv y 19.22dv (-> ph ps) y ax-17 x mo2 ps y ax-17 x mo2 3imtr4g com12)) thm (euimmo () () (-> (A. x (-> ph ps)) (-> (E! x ps) (E* x ph))) (x ph ps immo x ps eumo syl5)) thm (euim () () (-> (/\ (E. x ph) (A. x (-> ph ps))) (-> (E! x ps) (E! x ph))) (x ph ps euimmo (E. x ph) adantl x ph exmoeu2 (A. x (-> ph ps)) adantr sylibd)) thm (moan () () (-> (E* x ph) (E* x (/\ ps ph))) (ps ph pm3.27 x immoi)) thm (moani () ((moani.1 (E* x ph))) (E* x (/\ ps ph)) (moani.1 x ph ps moan ax-mp)) thm (moor () () (-> (E* x (\/ ph ps)) (E* x ph)) (ph ps orc x immoi)) thm (mooran1 () () (-> (\/ (E* x ph) (E* x ps)) (E* x (/\ ph ps))) (x ph ps moan ps ph ancom x mobii sylib x ps ph moan jaoi)) thm (mooran2 () () (-> (E* x (\/ ph ps)) (/\ (E* x ph) (E* x ps))) (x ph ps moor ph ps orcom x mobii x ps ph moor sylbi jca)) thm (moanim ((x y) (ph y) (ps y)) ((moanim.1 (-> ph (A. x ph)))) (<-> (E* x (/\ ph ps)) (-> ph (E* x ps))) (ph ps (= (cv x) (cv y)) impexp x albii moanim.1 (-> ps (= (cv x) (cv y))) 19.21 bitr y exbii (/\ ph ps) y ax-17 x mo2 ps y ax-17 x mo2 ph imbi2i y ph (A. x (-> ps (= (cv x) (cv y)))) 19.37v bitr4 3bitr4)) thm (euan () ((moanim.1 (-> ph (A. x ph)))) (<-> (E! x (/\ ph ps)) (/\ ph (E! x ps))) (moanim.1 ps 19.42 moanim.1 ps moanim ph anbi2i ph (E* x ps) abai bitr4 anbi12i (E. x (/\ ph ps)) ph (E* x (/\ ph ps)) anass ph ph (E. x ps) (E* x ps) an4 3bitr4 x (/\ ph ps) eu5 ph ps anabs1 x exbii moanim.1 (/\ ph ps) 19.41 bitr3 (E* x (/\ ph ps)) anbi1i bitr ph pm4.24 x ps eu5 anbi12i 3bitr4)) thm (moanimv ((ph x)) () (<-> (E* x (/\ ph ps)) (-> ph (E* x ps))) (ph x ax-17 ps moanim)) thm (moaneu () () (E* x (/\ ph (E! x ph))) (x ph eumo x ph hbeu1 ph moanim mpbir ph (E! x ph) ancom x mobii mpbir)) thm (moanmo () () (E* x (/\ ph (E* x ph))) ((E* x ph) id x ph hbmo1 ph moanim mpbir ph (E* x ph) ancom x mobii mpbir)) thm (euanv ((ph x)) () (<-> (E! x (/\ ph ps)) (/\ ph (E! x ps))) (ph x ax-17 ps euan)) thm (mopick ((x y) (ph y) (ps y)) () (-> (/\ (E* x ph) (E. x (/\ ph ps))) (-> ph ps)) ((/\ ph ps) y ax-17 y x ph hbs1 y x ps hbs1 hban x y ph sbequ12 x y ps sbequ12 anbi12d cbvex x y ps sbequ2 (/\ ph ([/] (cv y) x ph)) imim2i exp3a com4t imp ph y ax-17 x mo3 y (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y))) ax-4 x a4s sylbi syl5 y 19.23aiv sylbi impcom)) thm (eupick () () (-> (/\ (E! x ph) (E. x (/\ ph ps))) (-> ph ps)) (x ph ps mopick x ph eumo sylan)) thm (eupickb () () (-> (/\/\ (E! x ph) (E! x ps) (E. x (/\ ph ps))) (<-> ph ps)) (x ph ps eupick (E! x ps) 3adant2 (E! x ph) (E! x ps) (E. x (/\ ph ps)) 3simpc ph ps pm3.22 x 19.22i (E! x ps) anim2i x ps ph eupick 3syl impbid)) thm (mopick2 () () (-> (/\/\ (E* x ph) (E. x (/\ ph ps)) (E. x (/\ ph ch))) (E. x (/\/\ ph ps ch))) (ph ps pm3.26 x 19.22i (E* x ph) (E. x (/\ ph ch)) 3ad2ant2 x ph hbmo1 x (/\ ph ps) hbe1 x (/\ ph ch) hbe1 hb3an x ph ps mopick x ph ch mopick anim12i (E* x ph) (E. x (/\ ph ps)) (E. x (/\ ph ch)) 3anass (E* x ph) (E. x (/\ ph ps)) (E. x (/\ ph ch)) anandi bitr ph ps ch jcab 3imtr4 ancld ph ps ch 3anass syl6ibr 19.22d mpd)) thm (euor2 () () (-> (-. (E. x ph)) (<-> (E! x (\/ ph ps)) (E! x ps))) (x (\/ ph ps) euex x ph ps 19.43 sylib ord com12 x (\/ ph ps) eumo ph ps orcom x mobii x ps ph moor sylbi syl (-. (E. x ph)) a1i jcad x ps eu5 syl6ibr x ph hbe1 ps euor x ps euex ps ph olc x 19.22i ph x 19.8a ps orim1i x ax-gen x (\/ ph ps) (\/ (E. x ph) ps) euim mpan2 3syl (-. (E. x ph)) adantl mpd ex impbid)) thm (moexex () ((moexex.1 (-> ph (A. y ph)))) (-> (/\ (E* x ph) (A. x (E* y ps))) (E* y (E. x (/\ ph ps)))) (x ph hbmo1 x (E* y ps) hba1 x (/\ ph ps) hbe1 y hbmo hbim hbim moexex.1 moexex.1 x hbmo x ph ps mopick ex com3r 19.21ad y (E. x (/\ ph ps)) ps immo x a4sd syl6 19.23ai moexex.1 x hbex ph ps pm3.26 x 19.22i 19.23ai con3i y (E. x (/\ ph ps)) exmo ori syl (A. x (E* y ps)) a1d (E* x ph) a1d pm2.61i imp)) thm (moexexv ((ph y)) () (-> (/\ (E* x ph) (A. x (E* y ps))) (E* y (E. x (/\ ph ps)))) (ph y ax-17 x ps moexex)) thm (2moex () () (-> (E* x (E. y ph)) (A. y (E* x ph))) (y ph hbe1 x hbmo ph y 19.8a x immoi 19.21ai)) thm (2euex () () (-> (E! x (E. y ph)) (E. y (E! x ph))) (x (E. y ph) eu5 y ph hbe1 x hbmo (E. x ph) 19.41 biimpr x y ph excom sylanb x y ph 2moex 19.21bi (E. x ph) anim2i x ph eu5 sylibr y 19.22i syl sylbi)) thm (2eumo () () (-> (E! x (E* y ph)) (E* x (E! y ph))) (x (E! y ph) (E* y ph) euimmo y ph eumo mpg)) thm (2eu2ex () () (-> (E! x (E! y ph)) (E. x (E. y ph))) (x (E! y ph) euex y ph euex x 19.22i syl)) thm (2moswap () () (-> (A. x (E* y ph)) (-> (E* x (E. y ph)) (E* y (E. x ph)))) (y ph hbe1 x ph moexex expcom ph y 19.8a pm4.71ri x exbii y mobii syl6ibr)) thm (2euswap () () (-> (A. x (E* y ph)) (-> (E! x (E. y ph)) (E! y (E. x ph)))) (x y ph excomim (A. x (E* y ph)) a1i x y ph 2moswap anim12d x (E. y ph) eu5 y (E. x ph) eu5 3imtr4g)) thm (2exeu () () (-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E! x (E! y ph))) (x ph hbe1 y hbmo (E. y ph) 19.41 ph x 19.8a y immoi (E. y ph) anim2i x 19.22i sylbir y x ph excom sylanb (E. y ph) (E* y ph) pm3.26 x immoi (E. x (E. y ph)) adantl anim12i ancoms x (E. y ph) eu5 y (E. x ph) eu5 anbi12i x (E! y ph) eu5 y ph eu5 x exbii y ph eu5 x mobii anbi12i bitr 3imtr4)) thm (2mo ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph z) (ph w) (ph v) (ph u)) () (<-> (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))) (A. x (A. y (A. z (A. w (-> (/\ ph ([/] (cv z) x ([/] (cv w) y ph))) (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))))) (v z x equequ2 u w y equequ2 bi2anan9 ph imbi2d x y 2albidv cbvex2v (-> ph (/\ (= (cv x) (cv v)) (= (cv y) (cv u)))) z ax-17 (-> ph (/\ (= (cv x) (cv v)) (= (cv y) (cv u)))) w ax-17 z x ([/] (cv w) y ph) hbs1 (/\ (= (cv z) (cv v)) (= (cv w) (cv u))) x ax-17 hbim w y ph hbs1 z x hbsb (/\ (= (cv z) (cv v)) (= (cv w) (cv u))) y ax-17 hbim y w ph sbequ12 x z ([/] (cv w) y ph) sbequ12 sylan9bbr x z v equequ1 y w u equequ1 bi2anan9 imbi12d cbval2 biimp ancli y z (A. w (/\ (-> ph (/\ (= (cv x) (cv v)) (= (cv y) (cv u)))) (-> ([/] (cv z) x ([/] (cv w) y ph)) (/\ (= (cv z) (cv v)) (= (cv w) (cv u)))))) alcom (-> ph (/\ (= (cv x) (cv v)) (= (cv y) (cv u)))) w ax-17 w y ph hbs1 z x hbsb (/\ (= (cv z) (cv v)) (= (cv w) (cv u))) y ax-17 hbim aaan z albii bitr x albii (A. y (-> ph (/\ (= (cv x) (cv v)) (= (cv y) (cv u))))) z ax-17 z x ([/] (cv w) y ph) hbs1 (/\ (= (cv z) (cv v)) (= (cv w) (cv u))) x ax-17 hbim w hbal aaan bitr sylibr ph (/\ (= (cv x) (cv v)) (= (cv y) (cv u))) ([/] (cv z) x ([/] (cv w) y ph)) (/\ (= (cv z) (cv v)) (= (cv w) (cv u))) prth x v z equtr2 y u w equtr2 anim12i an4s syl6 z w 19.20i2 x y 19.20i2 syl v u 19.23aivv sylbir x y z w (-> ([/] (cv z) x ([/] (cv w) y ph)) (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))) alrot4 y ([/] (cv z) x ([/] (cv w) y ph)) (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))) 19.20 x 19.20ii z w 19.20i2 sylbi w (A. x (A. y ([/] (cv z) x ([/] (cv w) y ph)))) (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))) 19.22 z 19.20i z (E. w (A. x (A. y ([/] (cv z) x ([/] (cv w) y ph))))) (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))) 19.22 3syl z x ([/] (cv w) y ph) hbs1 w y ph hbs1 z x hbsb 19.21ai z w 19.22i2 syl5com ph ([/] (cv z) x ([/] (cv w) y ph)) (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) impexp ph ([/] (cv z) x ([/] (cv w) y ph)) (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) bi2.04 bitr z w 2albii x y 2albii syl5ib w ([/] (cv z) x ([/] (cv w) y ph)) alnex z albii z (E. w ([/] (cv z) x ([/] (cv w) y ph))) alnex bitr (-. ph) z ax-17 (-. ph) w ax-17 z x ([/] (cv w) y ph) hbs1 hbne w y ph hbs1 z x hbsb hbne y w ph sbequ12 x z ([/] (cv w) y ph) sbequ12 sylan9bbr negbid cbval2 biimpr ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) pm2.21 x y 19.20i2 (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))) z 19.8a 19.23bi 3syl sylbir (A. x (A. y (A. z (A. w (-> (/\ ph ([/] (cv z) x ([/] (cv w) y ph))) (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))) a1d pm2.61i impbi)) thm (2mos ((w z) (ph z) (ph w) (x y) (ps x) (ps y) (x z) (w x) (y z) (w y)) ((2mos.1 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (<-> (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))) (A. x (A. y (A. z (A. w (-> (/\ ph ps) (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))))) (z w x y ph 2mo ps x ax-17 (= (cv x) (cv z)) y ax-17 w ph sb19.21 (-> (= (cv x) (cv z)) ps) y ax-17 2mos.1 expcom pm5.74d sbie bitr3 pm5.74ri sbie ph anbi2i (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) imbi1i z w 2albii x y 2albii bitr)) thm (2eu1 () () (-> (A. x (E* y ph)) (<-> (E! x (E! y ph)) (/\ (E! x (E. y ph)) (E! y (E. x ph))))) (x (E! y ph) eu5 y ph eu5 x exbii y ph eu5 x mobii anbi12i bitr pm3.27bd x (E* y ph) ax-4 (E. y ph) anim2i ancoms x immoi x (E* y ph) hba1 (E. y ph) moanim sylib ancrd x y ph 2moswap com12 imdistani syl6 syl x y ph 2eu2ex x y ph 2eu2ex x y ph excom sylib jca jctild x (E. y ph) eu5 y (E. x ph) eu5 anbi12i (E. x (E. y ph)) (E* x (E. y ph)) (E. y (E. x ph)) (E* y (E. x ph)) an4 bitr syl6ibr com12 x y ph 2exeu (A. x (E* y ph)) a1i impbid)) thm (2eu2 () () (-> (E! y (E. x ph)) (<-> (E! x (E! y ph)) (E! x (E. y ph)))) (y (E. x ph) eumo y x ph 2moex x y ph 2eu1 (E! x (E. y ph)) (E! y (E. x ph)) pm3.26 syl6bi 3syl x y ph 2exeu expcom impbid)) thm (2eu3 () () (-> (A. x (A. y (\/ (E* x ph) (E* y ph)))) (<-> (/\ (E! x (E! y ph)) (E! y (E! x ph))) (/\ (E! x (E. y ph)) (E! y (E. x ph))))) (y ph hbmo1 (E* x ph) 19.31 x albii x ph hbmo1 y hbal (E* y ph) 19.32 bitr y x ph 2eu1 biimpd (E! y (E. x ph)) (E! x (E. y ph)) ancom syl6ib (E! x (E! y ph)) adantld x y ph 2eu1 biimpd (E! y (E! x ph)) adantrd jaoi x y ph 2exeu y x ph 2exeu ancoms jca (\/ (A. y (E* x ph)) (A. x (E* y ph))) a1i impbid sylbi)) thm (2eu4 ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (/\ (E. x (E. y ph)) (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))))) ((E. y ph) z ax-17 x eu3 (E. x ph) w ax-17 y eu3 anbi12i (E. x (E. y ph)) (E. z (A. x (-> (E. y ph) (= (cv x) (cv z))))) (E. y (E. x ph)) (E. w (A. y (-> (E. x ph) (= (cv y) (cv w))))) an4 y x ph excom (E. x (E. y ph)) anbi2i (E. x (E. y ph)) anidm bitr x (A. y (-> ph (= (cv y) (cv w)))) hba1 19.3r (A. x (A. y (-> ph (= (cv x) (cv z))))) anbi2i ph (= (cv x) (cv z)) (= (cv y) (cv w)) jcab y albii y (-> ph (= (cv x) (cv z))) (-> ph (= (cv y) (cv w))) 19.26 bitr x albii x (A. y (-> ph (= (cv x) (cv z)))) (A. y (-> ph (= (cv y) (cv w)))) 19.26 bitr x (A. y (-> ph (= (cv x) (cv z)))) (A. x (A. y (-> ph (= (cv y) (cv w))))) 19.26 3bitr4 y (A. y (-> ph (= (cv x) (cv z)))) (A. x (-> ph (= (cv y) (cv w)))) 19.26 y (A. y (-> ph (= (cv x) (cv z)))) ax-4 y (-> ph (= (cv x) (cv z))) hba1 impbi y x (-> ph (= (cv y) (cv w))) alcom anbi12i bitr x albii bitr4 y ph (= (cv x) (cv z)) 19.23v x ph (= (cv y) (cv w)) 19.23v anbi12i x y 2albii y ph hbe1 (= (cv x) (cv z)) y ax-17 hbim x ph hbe1 (= (cv y) (cv w)) x ax-17 hbim aaan 3bitr z w 2exbii z w (A. x (-> (E. y ph) (= (cv x) (cv z)))) (A. y (-> (E. x ph) (= (cv y) (cv w)))) eeanv bitr2 anbi12i 3bitr)) thm (2eu5 ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (<-> (/\ (E! x (E! y ph)) (A. x (E* y ph))) (/\ (E. x (E. y ph)) (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))))) (x y ph 2eu1 pm5.32ri y (E. x ph) eumo (E! x (E. y ph)) adantl y x ph 2moex syl pm4.71i x y ph z w 2eu4 3bitr2)) thm (2eu6 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph z) (ph w) (ph v) (ph u)) () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E. z (E. w (A. x (A. y (<-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))))) (x y ph z w 2eu4 z w ([/] (cv z) x ([/] (cv w) y ph)) (A. v (A. u (-> (/\ ([/] (cv z) x ([/] (cv w) y ph)) ([/] (cv v) z ([/] (cv u) w ([/] (cv z) x ([/] (cv w) y ph))))) (/\ (= (cv z) (cv v)) (= (cv w) (cv u)))))) 19.29r2 ph z ax-17 ph w ax-17 z x ([/] (cv w) y ph) hbs1 w y ph hbs1 z x hbsb y w ph sbequ12 x z ([/] (cv w) y ph) sbequ12 sylan9bbr cbvex2 z v x equequ2 w u y equequ2 bi2anan9 ph imbi2d x y 2albidv cbvex2v (-> ph (/\ (= (cv x) (cv v)) (= (cv y) (cv u)))) z ax-17 (-> ph (/\ (= (cv x) (cv v)) (= (cv y) (cv u)))) w ax-17 z x ([/] (cv w) y ph) hbs1 (/\ (= (cv z) (cv v)) (= (cv w) (cv u))) x ax-17 hbim w y ph hbs1 z x hbsb (/\ (= (cv z) (cv v)) (= (cv w) (cv u))) y ax-17 hbim y w ph sbequ12 x z ([/] (cv w) y ph) sbequ12 sylan9bbr x z v equequ1 y w u equequ1 bi2anan9 imbi12d cbval2 v u 2exbii v u z w ([/] (cv z) x ([/] (cv w) y ph)) 2mo 3bitr anbi12i x y ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) 2albi (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))) (A. x (A. y (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) ph))) ancom bitr z w 2exbii z x equcom w y equcom anbi12i (/\ ([/] (cv z) x ([/] (cv w) y ph)) ph) imbi2i ([/] (cv z) x ([/] (cv w) y ph)) ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) impexp bitr x y 2albii (-> (/\ ([/] (cv z) x ([/] (cv w) y ph)) ph) (/\ (= (cv z) (cv x)) (= (cv w) (cv y)))) v ax-17 (-> (/\ ([/] (cv z) x ([/] (cv w) y ph)) ph) (/\ (= (cv z) (cv x)) (= (cv w) (cv y)))) u ax-17 z x ([/] (cv w) y ph) hbs1 z x ([/] (cv w) y ph) hbs1 u w hbsb v z hbsb hban (/\ (= (cv z) (cv v)) (= (cv w) (cv u))) x ax-17 hbim w y ph hbs1 z x hbsb w y ph hbs1 z x hbsb u w hbsb v z hbsb hban (/\ (= (cv z) (cv v)) (= (cv w) (cv u))) y ax-17 hbim y u ph sbequ12 x v ([/] (cv u) y ph) sbequ12 sylan9bbr ([/] (cv u) w ([/] (cv w) y ph)) z ax-17 v x sbco2 z x u w ([/] (cv w) y ph) sbcom2 v z sbbii ph w ax-17 u y sbco2 v x sbbii 3bitr3r syl6bb ([/] (cv z) x ([/] (cv w) y ph)) anbi2d x v z equequ2 y u w equequ2 bi2anan9 imbi12d cbval2 z x ([/] (cv w) y ph) hbs1 w y ph hbs1 z x hbsb (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))) 19.21-2 3bitr3 ([/] (cv z) x ([/] (cv w) y ph)) anbi2i ([/] (cv z) x ([/] (cv w) y ph)) (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))) abai z x w y ph 2sb6 (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))) anbi1i 3bitr2 z w 2exbii bitr4 3imtr4 ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) bi2 x y 19.20i2 z w 19.22i2 x y ph z w 2exsb sylibr ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) bi1 x y 19.20i2 z w 19.22i2 jca impbi bitr)) thm (2eu7 () () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E! x (E! y (/\ (E. x ph) (E. y ph))))) (x ph hbe1 y hbeu (E. y ph) euan (E. x ph) (E. y ph) ancom y eubii y ph hbe1 (E. x ph) euan (E. y ph) (E! y (E. x ph)) ancom 3bitr x eubii (E! x (E. y ph)) (E! y (E. x ph)) ancom 3bitr4r)) thm (2eu8 () () (<-> (E! x (E! y (/\ (E. x ph) (E. y ph)))) (E! x (E! y (/\ (E! x ph) (E. y ph))))) (x y ph 2eu2 pm5.32i x ph hbeu1 y hbeu (E. y ph) euan (E! x ph) (E. y ph) ancom y eubii y ph hbe1 (E! x ph) euan (E. y ph) (E! y (E! x ph)) ancom 3bitr x eubii (E! x (E. y ph)) (E! y (E! x ph)) ancom 3bitr4r x y ph 2eu7 3bitr3r)) thm (exists1 ((x y)) () (<-> (E! x (= (cv x) (cv x))) (A. x (= (cv x) (cv y)))) (x equid (= (cv x) (cv y)) tbt (= (cv x) (cv y)) (= (cv x) (cv x)) bicom bitr x albii y exbii x y y hbae 19.9r x (= (cv x) (cv x)) y df-eu 3bitr4r)) thm (exists2 ((x y)) () (-> (/\ (E. x ph) (E. x (-. ph))) (-. (E! x (= (cv x) (cv x))))) (x y exists1 ph pm3.24 x y ph ax-16 a5i x ph 19.9t syl x y (-. ph) ax-16 a5i x (-. ph) 19.9t syl anim12d mtoi sylbi con2i)) var (set t) thm (axext ((x y) (x z) (y z)) () (E. z (-> (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y))) (= (cv x) (cv y)))) (z x y ax-ext z (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y))) (= (cv x) (cv y)) 19.36v mpbir)) thm (zfext2 ((x z) (w z) (w x) (y z) (w y)) () (-> (A. z (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y)))) (= (cv x) (cv y))) (w x a9e z w y ax-ext w x z elequ2 (e. (cv z) (cv y)) bibi1d z albidv w x y equequ1 imbi12d mpbii w 19.23aiv ax-mp)) thm (bm1.1 ((x y) (x z) (y z) (ph z)) ((bm1.1.1 (-> ph (A. x ph)))) (-> (E. x (A. y (<-> (e. (cv y) (cv x)) ph))) (E! x (A. y (<-> (e. (cv y) (cv x)) ph)))) (y (<-> (e. (cv y) (cv x)) ph) (<-> (e. (cv y) (cv z)) ph) 19.26 (e. (cv y) (cv x)) ph (e. (cv y) (cv z)) biantr y 19.20i y x z ax-ext syl sylbir (e. (cv y) (cv z)) x ax-17 bm1.1.1 hbbi y hbal x z y elequ2 ph bibi1d y albidv sbie sylan2b x z gen2 (E. x (A. y (<-> (e. (cv y) (cv x)) ph))) jctr (A. y (<-> (e. (cv y) (cv x)) ph)) z ax-17 x eu2 sylibr)) var (class C) var (class D) var (class P) var (class Q) var (class R) var (class S) var (class T) var (class U) thm (abid () () (<-> (e. (cv x) ({|} x ph)) ph) (x x ph df-clab x ph sbid bitr)) thm (hbab1 ((x y)) () (-> (e. (cv y) ({|} x ph)) (A. x (e. (cv y) ({|} x ph)))) (y x ph hbs1 y x ph df-clab y x ph df-clab x albii 3imtr4)) thm (hbab ((x z)) ((hbab.1 (-> ph (A. x ph)))) (-> (e. (cv z) ({|} y ph)) (A. x (e. (cv z) ({|} y ph)))) (x z ([/] (cv z) y ph) ax-16 hbab.1 z y hbsb4 pm2.61i z y ph df-clab z y ph df-clab x albii 3imtr4)) thm (hbabd ((x z)) ((hbabd.1 (-> ph (A. x (A. y ph)))) (hbabd.2 (-> ph (-> ps (A. x ps))))) (-> ph (-> (e. (cv z) ({|} y ps)) (A. x (e. (cv z) ({|} y ps))))) (hbabd.1 x y ph ax-7 syl hbabd.2 y x 19.20i2 y x ps z hbsb4t 3syl x z ([/] (cv z) y ps) ax-16 pm2.61d2 z y ps df-clab z y ps df-clab x albii 3imtr4g)) thm (dfcleq ((A x) (B x) (x y) (x z) (y z)) () (<-> (= A B) (A. x (<-> (e. (cv x) A) (e. (cv x) B)))) (x y z ax-ext A B df-cleq)) thm (eqrdv ((A x) (B x) (ph x)) ((eqrd.1 (-> ph (<-> (e. (cv x) A) (e. (cv x) B))))) (-> ph (= A B)) (eqrd.1 x 19.21aiv A B x dfcleq sylibr)) thm (eqriv ((A x) (B x)) ((eqri.1 (<-> (e. (cv x) A) (e. (cv x) B)))) (= A B) (A B x dfcleq eqri.1 mpgbir)) thm (eqid ((A x)) () (= A A) ((e. (cv x) A) pm4.2 eqriv)) thm (eqcom ((A x) (B x)) () (<-> (= A B) (= B A)) ((e. (cv x) A) (e. (cv x) B) bicom x albii A B x dfcleq B A x dfcleq 3bitr4)) thm (eqcoms () ((eqcoms.1 (-> (= A B) ph))) (-> (= B A) ph) (B A eqcom eqcoms.1 sylbi)) thm (eqcomi () ((eqcomi.1 (= A B))) (= B A) (eqcomi.1 A B eqcom mpbi)) thm (eqcomd () ((eqcomd.1 (-> ph (= A B)))) (-> ph (= B A)) (eqcomd.1 A B eqcom sylib)) thm (eqeq1 ((A x) (B x) (C x)) () (-> (= A B) (<-> (= A C) (= B C))) (A B x dfcleq biimp 19.21bi (e. (cv x) C) bibi1d x albidv A C x dfcleq B C x dfcleq 3bitr4g)) thm (eqeq1i () ((eqeq1i.1 (= A B))) (<-> (= A C) (= B C)) (eqeq1i.1 A B C eqeq1 ax-mp)) thm (eqeq1d () ((eqeq1d.1 (-> ph (= A B)))) (-> ph (<-> (= A C) (= B C))) (eqeq1d.1 A B C eqeq1 syl)) thm (eqeq2 () () (-> (= A B) (<-> (= C A) (= C B))) (A B C eqeq1 C A eqcom C B eqcom 3bitr4g)) thm (eqeq2i () ((eqeq2i.1 (= A B))) (<-> (= C A) (= C B)) (eqeq2i.1 A B C eqeq2 ax-mp)) thm (eqeq2d () ((eqeq2d.1 (-> ph (= A B)))) (-> ph (<-> (= C A) (= C B))) (eqeq2d.1 A B C eqeq2 syl)) thm (eqeq12 () () (-> (/\ (= A B) (= C D)) (<-> (= A C) (= B D))) (A B C eqeq1 C D B eqeq2 sylan9bb)) thm (eqeq12i () ((eqeq12i.1 (= A B)) (eqeq12i.2 (= C D))) (<-> (= A C) (= B D)) (eqeq12i.1 C eqeq1i eqeq12i.2 B eqeq2i bitr)) thm (eqeq12d () ((eqeq12d.1 (-> ph (= A B))) (eqeq12d.2 (-> ph (= C D)))) (-> ph (<-> (= A C) (= B D))) (eqeq12d.1 C eqeq1d eqeq12d.2 B eqeq2d bitrd)) thm (eqeqan12d () ((eqeqan12d.1 (-> ph (= A B))) (eqeqan12d.2 (-> ps (= C D)))) (-> (/\ ph ps) (<-> (= A C) (= B D))) (eqeqan12d.1 ps adantr eqeqan12d.2 ph adantl eqeq12d)) thm (eqeqan12rd () ((eqeqan12rd.1 (-> ph (= A B))) (eqeqan12rd.2 (-> ps (= C D)))) (-> (/\ ps ph) (<-> (= A C) (= B D))) (eqeqan12rd.1 eqeqan12rd.2 eqeqan12d ancoms)) thm (eqtrt () () (-> (/\ (= A B) (= B C)) (= A C)) (A B C eqeq1 biimpar)) thm (eqtr2t () () (-> (/\ (= A B) (= A C)) (= B C)) (B A C eqtrt A B eqcom sylanb)) thm (eqtr3t () () (-> (/\ (= A C) (= B C)) (= A B)) (A C B eqtrt B C eqcom sylan2b)) thm (eqtr () ((eqtr.1 (= A B)) (eqtr.2 (= B C))) (= A C) (eqtr.1 eqtr.2 A eqeq2i mpbi)) thm (eqtr2 () ((eqtr2.1 (= A B)) (eqtr2.2 (= B C))) (= C A) (eqtr2.1 eqtr2.2 eqtr eqcomi)) thm (eqtr3 () ((eqtr3.1 (= A B)) (eqtr3.2 (= A C))) (= B C) (eqtr3.1 eqcomi eqtr3.2 eqtr)) thm (eqtr4 () ((eqtr4.1 (= A B)) (eqtr4.2 (= C B))) (= A C) (eqtr4.1 eqtr4.2 eqcomi eqtr)) thm (3eqtr () ((3eqtr.1 (= A B)) (3eqtr.2 (= B C)) (3eqtr.3 (= C D))) (= A D) (3eqtr.1 3eqtr.2 3eqtr.3 eqtr eqtr)) thm (3eqtrr () ((3eqtr.1 (= A B)) (3eqtr.2 (= B C)) (3eqtr.3 (= C D))) (= D A) (3eqtr.3 3eqtr.1 3eqtr.2 eqtr2 eqtr3)) thm (3eqtr2 () ((3eqtr2.1 (= A B)) (3eqtr2.2 (= C B)) (3eqtr2.3 (= C D))) (= A D) (3eqtr2.1 3eqtr2.2 eqtr4 3eqtr2.3 eqtr)) thm (3eqtr2r () ((3eqtr2.1 (= A B)) (3eqtr2.2 (= C B)) (3eqtr2.3 (= C D))) (= D A) (3eqtr2.1 3eqtr2.2 eqcomi 3eqtr2.3 3eqtrr)) thm (3eqtr3 () ((3eqtr3.1 (= A B)) (3eqtr3.2 (= A C)) (3eqtr3.3 (= B D))) (= C D) (3eqtr3.2 3eqtr3.1 3eqtr3.3 eqtr eqtr3)) thm (3eqtr3r () ((3eqtr3.1 (= A B)) (3eqtr3.2 (= A C)) (3eqtr3.3 (= B D))) (= D C) (3eqtr3.3 3eqtr3.1 3eqtr3.2 eqtr3 eqtr3)) thm (3eqtr4 () ((3eqtr4.1 (= A B)) (3eqtr4.2 (= C A)) (3eqtr4.3 (= D B))) (= C D) (3eqtr4.2 3eqtr4.1 3eqtr4.3 eqtr4 eqtr)) thm (3eqtr4r () ((3eqtr4.1 (= A B)) (3eqtr4.2 (= C A)) (3eqtr4.3 (= D B))) (= D C) (3eqtr4.2 3eqtr4.1 3eqtr4.3 eqtr4 eqtr2)) thm (eqtrd () ((eqtrd.1 (-> ph (= A B))) (eqtrd.2 (-> ph (= B C)))) (-> ph (= A C)) (eqtrd.1 eqtrd.2 A eqeq2d mpbid)) thm (eqtr2d () ((eqtr2d.1 (-> ph (= A B))) (eqtr2d.2 (-> ph (= B C)))) (-> ph (= C A)) (eqtr2d.1 eqtr2d.2 eqtrd eqcomd)) thm (eqtr3d () ((eqtr3d.1 (-> ph (= A B))) (eqtr3d.2 (-> ph (= A C)))) (-> ph (= B C)) (eqtr3d.1 eqcomd eqtr3d.2 eqtrd)) thm (eqtr4d () ((eqtr4d.1 (-> ph (= A B))) (eqtr4d.2 (-> ph (= C B)))) (-> ph (= A C)) (eqtr4d.1 eqtr4d.2 eqcomd eqtrd)) thm (3eqtrd () ((3eqtrd.1 (-> ph (= A B))) (3eqtrd.2 (-> ph (= B C))) (3eqtrd.3 (-> ph (= C D)))) (-> ph (= A D)) (3eqtrd.1 3eqtrd.2 3eqtrd.3 eqtrd eqtrd)) thm (3eqtrrd () ((3eqtrd.1 (-> ph (= A B))) (3eqtrd.2 (-> ph (= B C))) (3eqtrd.3 (-> ph (= C D)))) (-> ph (= D A)) (3eqtrd.3 3eqtrd.1 3eqtrd.2 eqtr2d eqtr3d)) thm (3eqtr2d () ((3eqtr2d.1 (-> ph (= A B))) (3eqtr2d.2 (-> ph (= C B))) (3eqtr2d.3 (-> ph (= C D)))) (-> ph (= A D)) (3eqtr2d.1 3eqtr2d.2 eqtr4d 3eqtr2d.3 eqtrd)) thm (3eqtr2rd () ((3eqtr2d.1 (-> ph (= A B))) (3eqtr2d.2 (-> ph (= C B))) (3eqtr2d.3 (-> ph (= C D)))) (-> ph (= D A)) (3eqtr2d.1 3eqtr2d.2 eqtr4d 3eqtr2d.3 eqtr2d)) thm (3eqtr3d () ((3eqtr3d.1 (-> ph (= A B))) (3eqtr3d.2 (-> ph (= A C))) (3eqtr3d.3 (-> ph (= B D)))) (-> ph (= C D)) (3eqtr3d.2 3eqtr3d.1 3eqtr3d.3 eqtrd eqtr3d)) thm (3eqtr3rd () ((3eqtr3d.1 (-> ph (= A B))) (3eqtr3d.2 (-> ph (= A C))) (3eqtr3d.3 (-> ph (= B D)))) (-> ph (= D C)) (3eqtr3d.3 3eqtr3d.1 3eqtr3d.2 eqtr3d eqtr3d)) thm (3eqtr4d () ((3eqtr4d.1 (-> ph (= A B))) (3eqtr4d.2 (-> ph (= C A))) (3eqtr4d.3 (-> ph (= D B)))) (-> ph (= C D)) (3eqtr4d.2 3eqtr4d.1 3eqtr4d.3 eqtr4d eqtrd)) thm (3eqtr4rd () ((3eqtr4d.1 (-> ph (= A B))) (3eqtr4d.2 (-> ph (= C A))) (3eqtr4d.3 (-> ph (= D B)))) (-> ph (= D C)) (3eqtr4d.3 3eqtr4d.1 eqtr4d 3eqtr4d.2 eqtr4d)) thm (syl5eq () ((syl5eq.1 (-> ph (= A B))) (syl5eq.2 (= C A))) (-> ph (= C B)) (syl5eq.2 ph a1i syl5eq.1 eqtrd)) thm (syl5req () ((syl5req.1 (-> ph (= A B))) (syl5req.2 (= C A))) (-> ph (= B C)) (syl5req.1 syl5req.2 syl5eq eqcomd)) thm (syl5eqr () ((syl5eqr.1 (-> ph (= A B))) (syl5eqr.2 (= A C))) (-> ph (= C B)) (syl5eqr.1 syl5eqr.2 eqcomi syl5eq)) thm (syl5reqr () ((syl5reqr.1 (-> ph (= A B))) (syl5reqr.2 (= A C))) (-> ph (= B C)) (syl5reqr.1 syl5reqr.2 eqcomi syl5req)) thm (syl6eq () ((syl6eq.1 (-> ph (= A B))) (syl6eq.2 (= B C))) (-> ph (= A C)) (syl6eq.1 syl6eq.2 ph a1i eqtrd)) thm (syl6req () ((syl6req.1 (-> ph (= A B))) (syl6req.2 (= B C))) (-> ph (= C A)) (syl6req.1 syl6req.2 syl6eq eqcomd)) thm (syl6eqr () ((syl6eqr.1 (-> ph (= A B))) (syl6eqr.2 (= C B))) (-> ph (= A C)) (syl6eqr.1 syl6eqr.2 eqcomi syl6eq)) thm (syl6reqr () ((syl6reqr.1 (-> ph (= A B))) (syl6reqr.2 (= C B))) (-> ph (= C A)) (syl6reqr.1 syl6reqr.2 eqcomi syl6req)) thm (sylan9eq () ((sylan9eq.1 (-> ph (= A B))) (sylan9eq.2 (-> ps (= B C)))) (-> (/\ ph ps) (= A C)) (sylan9eq.1 ps adantr sylan9eq.2 ph adantl eqtrd)) thm (sylan9eqr () ((sylan9eqr.1 (-> ph (= A B))) (sylan9eqr.2 (-> ps (= B C)))) (-> (/\ ps ph) (= A C)) (sylan9eqr.1 sylan9eqr.2 sylan9eq ancoms)) thm (3eqtr3g () ((3eqtr3g.1 (-> ph (= A B))) (3eqtr3g.2 (= A C)) (3eqtr3g.3 (= B D))) (-> ph (= C D)) (3eqtr3g.1 3eqtr3g.2 syl5eqr 3eqtr3g.3 syl6eq)) thm (3eqtr4g () ((3eqtr4g.1 (-> ph (= A B))) (3eqtr4g.2 (= C A)) (3eqtr4g.3 (= D B))) (-> ph (= C D)) (3eqtr4g.1 3eqtr4g.2 syl5eq 3eqtr4g.3 syl6eqr)) var (class F) var (class G) thm (eq2tr () ((eq2tr.1 (-> (= A C) (= D F))) (eq2tr.2 (-> (= B D) (= C G)))) (<-> (/\ (= A C) (= B F)) (/\ (= B D) (= A G))) ((= A C) (= B D) ancom eq2tr.1 B eqeq2d pm5.32i eq2tr.2 A eqeq2d pm5.32i 3bitr3)) thm (eleq1 ((A x) (B x) (C x)) () (-> (= A B) (<-> (e. A C) (e. B C))) (A B (cv x) eqeq2 (e. (cv x) C) anbi1d x exbidv A C x df-clel B C x df-clel 3bitr4g)) thm (eleq2 ((A x) (B x) (C x)) () (-> (= A B) (<-> (e. C A) (e. C B))) (A B x dfcleq biimp 19.21bi (= (cv x) C) anbi2d x exbidv C A x df-clel C B x df-clel 3bitr4g)) thm (eleq12 () () (-> (/\ (= A B) (= C D)) (<-> (e. A C) (e. B D))) (A B C eleq1 C D B eleq2 sylan9bb)) thm (eleq1i () ((eleq1i.1 (= A B))) (<-> (e. A C) (e. B C)) (eleq1i.1 A B C eleq1 ax-mp)) thm (eleq2i () ((eleq1i.1 (= A B))) (<-> (e. C A) (e. C B)) (eleq1i.1 A B C eleq2 ax-mp)) thm (eleq12i () ((eleq1i.1 (= A B)) (eleq12i.2 (= C D))) (<-> (e. A C) (e. B D)) (eleq12i.2 A eleq2i eleq1i.1 D eleq1i bitr)) thm (eleq1d () ((eleq1d.1 (-> ph (= A B)))) (-> ph (<-> (e. A C) (e. B C))) (eleq1d.1 A B C eleq1 syl)) thm (eleq2d () ((eleq1d.1 (-> ph (= A B)))) (-> ph (<-> (e. C A) (e. C B))) (eleq1d.1 A B C eleq2 syl)) thm (eleq12d () ((eleq1d.1 (-> ph (= A B))) (eleq12d.2 (-> ph (= C D)))) (-> ph (<-> (e. A C) (e. B D))) (eleq12d.2 A eleq2d eleq1d.1 D eleq1d bitrd)) thm (eleq1a () () (-> (e. A B) (-> (= C A) (e. C B))) (C A B eleq1 biimprcd)) thm (eqeltr () ((eqeltr.1 (= A B)) (eqeltr.2 (e. B C))) (e. A C) (eqeltr.2 eqeltr.1 C eleq1i mpbir)) thm (eqeltrr () ((eqeltrr.1 (= A B)) (eqeltrr.2 (e. A C))) (e. B C) (eqeltrr.1 eqcomi eqeltrr.2 eqeltr)) thm (eleqtr () ((eleqtr.1 (e. A B)) (eleqtr.2 (= B C))) (e. A C) (eleqtr.1 eleqtr.2 A eleq2i mpbi)) thm (eleqtrr () ((eleqtrr.1 (e. A B)) (eleqtrr.2 (= C B))) (e. A C) (eleqtrr.1 eleqtrr.2 eqcomi eleqtr)) thm (eqeltrd () ((eqeltrd.1 (-> ph (= A B))) (eqeltrd.2 (-> ph (e. B C)))) (-> ph (e. A C)) (eqeltrd.2 eqeltrd.1 C eleq1d mpbird)) thm (eqeltrrd () ((eqeltrrd.1 (-> ph (= A B))) (eqeltrrd.2 (-> ph (e. A C)))) (-> ph (e. B C)) (eqeltrrd.1 eqcomd eqeltrrd.2 eqeltrd)) thm (eleqtrd () ((eleqtrd.1 (-> ph (e. A B))) (eleqtrd.2 (-> ph (= B C)))) (-> ph (e. A C)) (eleqtrd.1 eleqtrd.2 A eleq2d mpbid)) thm (eleqtrrd () ((eleqtrrd.1 (-> ph (e. A B))) (eleqtrrd.2 (-> ph (= C B)))) (-> ph (e. A C)) (eleqtrrd.1 eleqtrrd.2 eqcomd eleqtrd)) thm (syl5eqel () ((syl5eqel.1 (-> ph (e. A B))) (syl5eqel.2 (= C A))) (-> ph (e. C B)) (syl5eqel.2 ph a1i syl5eqel.1 eqeltrd)) thm (syl5eqelr () ((syl5eqelr.1 (-> ph (e. A B))) (syl5eqelr.2 (= A C))) (-> ph (e. C B)) (syl5eqelr.1 syl5eqelr.2 eqcomi syl5eqel)) thm (syl5eleq () ((syl5eleq.1 (-> ph (= A B))) (syl5eleq.2 (e. C A))) (-> ph (e. C B)) (syl5eleq.2 ph a1i syl5eleq.1 eleqtrd)) thm (syl5eleqr () ((syl5eleqr.1 (-> ph (= B A))) (syl5eleqr.2 (e. C A))) (-> ph (e. C B)) (syl5eleqr.1 eqcomd syl5eleqr.2 syl5eleq)) thm (syl6eqel () ((syl6eqel.1 (-> ph (= A B))) (syl6eqel.2 (e. B C))) (-> ph (e. A C)) (syl6eqel.1 syl6eqel.2 ph a1i eqeltrd)) thm (syl6eqelr () ((syl6eqelr.1 (-> ph (= B A))) (syl6eqelr.2 (e. B C))) (-> ph (e. A C)) (syl6eqelr.1 eqcomd syl6eqelr.2 syl6eqel)) thm (syl6eleq () ((syl6eleq.1 (-> ph (e. A B))) (syl6eleq.2 (= B C))) (-> ph (e. A C)) (syl6eleq.1 syl6eleq.2 ph a1i eleqtrd)) thm (syl6eleqr () ((syl6eleqr.1 (-> ph (e. A B))) (syl6eleqr.2 (= C B))) (-> ph (e. A C)) (syl6eleqr.1 syl6eleqr.2 eqcomi syl6eleq)) thm (cleqf ((A y) (B y) (x y)) ((cleqf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (cleqf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (<-> (= A B) (A. x (<-> (e. (cv x) A) (e. (cv x) B)))) (A B y dfcleq (<-> (e. (cv x) A) (e. (cv x) B)) y ax-17 cleqf.1 cleqf.2 hbbi (cv x) (cv y) A eleq1 (cv x) (cv y) B eleq1 bibi12d cbval bitr4)) thm (nelneq () () (-> (/\ (e. A C) (-. (e. B C))) (-. (= A B))) (A B C eleq1 biimpcd con3d imp)) thm (nelneq2 () () (-> (/\ (e. A B) (-. (e. A C))) (-. (= B C))) (B C A eleq2 biimpcd con3d imp)) thm (hblem ((A y) (x y) (x z) (y z)) ((hblem.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv z) A) (A. x (e. (cv z) A))) ((cv y) (cv z) A eleq1 (cv y) (cv z) A eleq1 x albidv imbi12d hblem.1 chvarv)) thm (hbeq ((A y) (w y) (A w) (B z) (w z) (B w) (x y) (w x) (x z)) ((hbeq.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbeq.2 (-> (e. (cv z) B) (A. x (e. (cv z) B))))) (-> (= A B) (A. x (= A B))) (hbeq.1 w hblem hbeq.2 w hblem hbbi w hbal A B w dfcleq A B w dfcleq x albii 3imtr4)) thm (hbel ((A y) (w y) (A w) (B z) (w z) (B w) (x y) (w x) (x z)) ((hbel.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbel.2 (-> (e. (cv z) B) (A. x (e. (cv z) B))))) (-> (e. A B) (A. x (e. A B))) ((e. (cv y) (cv w)) x ax-17 hbel.1 hbeq hbel.2 w hblem hban w hbex A B w df-clel A B w df-clel x albii 3imtr4)) thm (hbeleq ((x y) (x z) (y z) (A y) (A z)) ((hbeleq.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (= (cv y) A) (A. x (= (cv y) A))) ((e. (cv z) (cv y)) x ax-17 (e. (cv y) (cv z)) x ax-17 hbeleq.1 hbel hbeq)) thm (abeq2 ((A x) (x y) (A y) (ph y)) () (<-> (= A ({|} x ph)) (A. x (<-> (e. (cv x) A) ph))) ((e. (cv y) A) x ax-17 y x ph hbab1 cleqf x ph abid (e. (cv x) A) bibi2i x albii bitr)) thm (abeq1 ((A x)) () (<-> (= ({|} x ph) A) (A. x (<-> ph (e. (cv x) A)))) (A x ph abeq2 ({|} x ph) A eqcom ph (e. (cv x) A) bicom x albii 3bitr4)) thm (abeq2i () ((abeqi.1 (= A ({|} x ph)))) (<-> (e. (cv x) A) ph) (abeqi.1 (cv x) eleq2i x ph abid bitr)) thm (abeq1i () ((abeqri.1 (= ({|} x ph) A))) (<-> ph (e. (cv x) A)) (x ph abid abeqri.1 (cv x) eleq2i bitr3)) thm (abeq2d () ((abeqd.1 (-> ph (= A ({|} x ps))))) (-> ph (<-> (e. (cv x) A) ps)) (abeqd.1 (cv x) eleq2d x ps abid syl6bb)) thm (eq2ab ((ph y) (ps y) (x y)) () (<-> (= ({|} x ph) ({|} x ps)) (A. x (<-> ph ps))) (y x ph hbab1 y x ps hbab1 cleqf x ph abid x ps abid bibi12i x albii bitr)) thm (abbi2i ((A x)) ((abbiri.1 (<-> (e. (cv x) A) ph))) (= A ({|} x ph)) (A x ph abeq2 abbiri.1 mpgbir)) thm (abbii () ((abbii.1 (<-> ph ps))) (= ({|} x ph) ({|} x ps)) (x ph ps eq2ab abbii.1 mpgbir)) thm (abbid () ((abbid.1 (-> ph (A. x ph))) (abbid.2 (-> ph (<-> ps ch)))) (-> ph (= ({|} x ps) ({|} x ch))) (abbid.1 abbid.2 19.21ai x ps ch eq2ab sylibr)) thm (abbidv ((ph x)) ((abbidv.1 (-> ph (<-> ps ch)))) (-> ph (= ({|} x ps) ({|} x ch))) (ph x ax-17 abbidv.1 abbid)) thm (abbi2dv ((A x) (ph x)) ((abbirdv.1 (-> ph (<-> (e. (cv x) A) ps)))) (-> ph (= A ({|} x ps))) (abbirdv.1 x 19.21aiv A x ps abeq2 sylibr)) thm (abbi1dv ((A x) (ph x)) ((abbildv.1 (-> ph (<-> ps (e. (cv x) A))))) (-> ph (= ({|} x ps) A)) (abbildv.1 x 19.21aiv x ps A abeq1 sylibr)) thm (abid2 ((A x)) () (= ({|} x (e. (cv x) A)) A) ((e. (cv x) A) pm4.2 abbi2i eqcomi)) thm (clelab ((A x) (x y) (A y) (ph y)) () (<-> (e. A ({|} x ph)) (E. x (/\ (= (cv x) A) ph))) (y x ph df-clab (= (cv y) A) anbi2i y exbii A ({|} x ph) y df-clel (/\ (= (cv x) A) ph) y ax-17 (= (cv y) A) x ax-17 y x ph hbs1 hban (cv x) (cv y) A eqeq1 x y ph sbequ12 anbi12d cbvex 3bitr4)) thm (clabel ((A y) (ph y) (x y)) () (<-> (e. ({|} x ph) A) (E. y (/\ (e. (cv y) A) (A. x (<-> (e. (cv x) (cv y)) ph))))) (({|} x ph) A y df-clel (cv y) x ph abeq2 (e. (cv y) A) anbi1i (A. x (<-> (e. (cv x) (cv y)) ph)) (e. (cv y) A) ancom bitr y exbii bitr)) thm (sbab ((A z) (x z) (y z)) () (-> (= (cv x) (cv y)) (= A ({|} z ([/] (cv y) x (e. (cv z) A))))) (x y (e. (cv z) A) sbequ12 abbi2dv)) thm (sbabel ((v w) (A w) (A v) (w x) (x z) (v x) (v z) (y z) (v y) (ph v)) ((sbabel.1 (-> (e. (cv w) A) (A. x (e. (cv w) A))))) (<-> ([/] (cv y) x (e. ({|} z ph) A)) (e. ({|} z ([/] (cv y) x ph)) A)) (y x v (/\ (= (cv v) ({|} z ph)) (e. (cv v) A)) sbex y x (= (cv v) ({|} z ph)) (e. (cv v) A) sban y x z (<-> (e. (cv z) (cv v)) ph) sbal (e. (cv z) (cv v)) x ax-17 y sbf ph sbrbis z albii bitr (cv v) z ph abeq2 y x sbbii (cv v) z ([/] (cv y) x ph) abeq2 3bitr4 (e. (cv w) (cv v)) x ax-17 sbabel.1 hbel y sbf anbi12i bitr v exbii bitr ({|} z ph) A v df-clel y x sbbii ({|} z ([/] (cv y) x ph)) A v df-clel 3bitr4)) thm (neeq1 () () (-> (= A B) (<-> (=/= A C) (=/= B C))) (A B C eqeq1 negbid A C df-ne B C df-ne 3bitr4g)) thm (neeq2 () () (-> (= A B) (<-> (=/= C A) (=/= C B))) (A B C eqeq2 negbid C A df-ne C B df-ne 3bitr4g)) thm (neeq1i () ((neeq1i.1 (= A B))) (<-> (=/= A C) (=/= B C)) (neeq1i.1 A B C neeq1 ax-mp)) thm (neeq2i () ((neeq1i.1 (= A B))) (<-> (=/= C A) (=/= C B)) (neeq1i.1 A B C neeq2 ax-mp)) thm (neeq1d () ((neeq1d.1 (-> ph (= A B)))) (-> ph (<-> (=/= A C) (=/= B C))) (neeq1d.1 A B C neeq1 syl)) thm (neeq2d () ((neeq1d.1 (-> ph (= A B)))) (-> ph (<-> (=/= C A) (=/= C B))) (neeq1d.1 A B C neeq2 syl)) thm (eqneqi () ((eqneqi.1 (<-> (= A B) (= C D)))) (<-> (=/= A B) (=/= C D)) (eqneqi.1 negbii A B df-ne C D df-ne 3bitr4)) thm (eqneqd () ((eqneqd.1 (-> ph (<-> (= A B) (= C D))))) (-> ph (<-> (=/= A B) (=/= C D))) (eqneqd.1 negbid A B df-ne C D df-ne 3bitr4g)) thm (necon3d () ((necon3d.1 (-> ph (-> (= A B) (= C D))))) (-> ph (-> (=/= C D) (=/= A B))) (necon3d.1 con3d C D df-ne A B df-ne 3imtr4g)) thm (necon3i () ((necon3i.1 (-> (= A B) (= C D)))) (-> (=/= C D) (=/= A B)) (necon3i.1 (-> (= A B) (= C D)) id necon3d ax-mp)) thm (necom () () (<-> (=/= A B) (=/= B A)) (A B eqcom eqneqi)) thm (nne () () (<-> (-. (=/= A B)) (= A B)) (A B df-ne con2bii bicomi)) thm (pm2.61ne () ((pm2.61ne.1 (-> (= A B) (<-> ps ch))) (pm2.61ne.2 (-> (/\ ph (=/= A B)) ps)) (pm2.61ne.3 (-> ph ch))) (-> ph ps) (pm2.61ne.1 pm2.61ne.3 syl5bir impcom pm2.61ne.2 A B df-ne sylan2br pm2.61dan)) thm (neleq1 () () (-> (= A B) (<-> (e/ A C) (e/ B C))) (A B C eleq1 negbid A C df-nel B C df-nel 3bitr4g)) thm (neleq2 () () (-> (= A B) (<-> (e/ C A) (e/ C B))) (A B C eleq2 negbid C A df-nel C B df-nel 3bitr4g)) thm (ralnex () () (<-> (A.e. x A (-. ph)) (-. (E.e. x A ph))) (x (e. (cv x) A) ph alinexa x A (-. ph) df-ral x A ph df-rex negbii 3bitr4)) thm (rexnal () () (<-> (E.e. x A (-. ph)) (-. (A.e. x A ph))) (x (e. (cv x) A) ph exanali x A (-. ph) df-rex x A ph df-ral negbii 3bitr4)) thm (dfral2 () () (<-> (A.e. x A ph) (-. (E.e. x A (-. ph)))) (x A ph rexnal con2bii)) thm (dfrex2 () () (<-> (E.e. x A ph) (-. (A.e. x A (-. ph)))) (x A ph ralnex con2bii)) thm (ralbida () ((ralbida.1 (-> ph (A. x ph))) (ralbida.2 (-> (/\ ph (e. (cv x) A)) (<-> ps ch)))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))) (ralbida.1 ralbida.2 ex pm5.74d albid x A ps df-ral x A ch df-ral 3bitr4g)) thm (rexbida () ((ralbida.1 (-> ph (A. x ph))) (ralbida.2 (-> (/\ ph (e. (cv x) A)) (<-> ps ch)))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))) (ralbida.1 ralbida.2 ex pm5.32d exbid x A ps df-rex x A ch df-rex 3bitr4g)) thm (ralbidva ((ph x)) ((ralbidva.1 (-> (/\ ph (e. (cv x) A)) (<-> ps ch)))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))) (ph x ax-17 ralbidva.1 ralbida)) thm (rexbidva ((ph x)) ((ralbidva.1 (-> (/\ ph (e. (cv x) A)) (<-> ps ch)))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))) (ph x ax-17 ralbidva.1 rexbida)) thm (ralbid () ((ralbid.1 (-> ph (A. x ph))) (ralbid.2 (-> ph (<-> ps ch)))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))) (ralbid.1 ralbid.2 (e. (cv x) A) adantr ralbida)) thm (rexbid () ((ralbid.1 (-> ph (A. x ph))) (ralbid.2 (-> ph (<-> ps ch)))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))) (ralbid.1 ralbid.2 (e. (cv x) A) adantr rexbida)) thm (ralbidv ((ph x)) ((ralbidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))) (ph x ax-17 ralbidv.1 A ralbid)) thm (rexbidv ((ph x)) ((ralbidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))) (ph x ax-17 ralbidv.1 A rexbid)) thm (ralbidv2 ((ph x)) ((ralbidv2.1 (-> ph (<-> (-> (e. (cv x) A) ps) (-> (e. (cv x) B) ch))))) (-> ph (<-> (A.e. x A ps) (A.e. x B ch))) (ralbidv2.1 x albidv x A ps df-ral x B ch df-ral 3bitr4g)) thm (rexbidv2 ((ph x)) ((rexbidv2.1 (-> ph (<-> (/\ (e. (cv x) A) ps) (/\ (e. (cv x) B) ch))))) (-> ph (<-> (E.e. x A ps) (E.e. x B ch))) (rexbidv2.1 x exbidv x A ps df-rex x B ch df-rex 3bitr4g)) thm (ralbii () ((ralbii.1 (<-> ph ps))) (<-> (A.e. x A ph) (A.e. x A ps)) (ph pm4.2 ph pm4.2 x hbth ralbii.1 (<-> ph ph) a1i A ralbid ax-mp)) thm (rexbii () ((ralbii.1 (<-> ph ps))) (<-> (E.e. x A ph) (E.e. x A ps)) (ph pm4.2 ph pm4.2 x hbth ralbii.1 (<-> ph ph) a1i A rexbid ax-mp)) thm (2ralbii () ((ralbii.1 (<-> ph ps))) (<-> (A.e. x A (A.e. y B ph)) (A.e. x A (A.e. y B ps))) (ralbii.1 y B ralbii x A ralbii)) thm (2rexbii () ((ralbii.1 (<-> ph ps))) (<-> (E.e. x A (E.e. y B ph)) (E.e. x A (E.e. y B ps))) (ralbii.1 y B rexbii x A rexbii)) thm (ralbii2 () ((ralbii2.1 (<-> (-> (e. (cv x) A) ph) (-> (e. (cv x) B) ps)))) (<-> (A.e. x A ph) (A.e. x B ps)) (ralbii2.1 x albii x A ph df-ral x B ps df-ral 3bitr4)) thm (rexbii2 () ((rexbii2.1 (<-> (/\ (e. (cv x) A) ph) (/\ (e. (cv x) B) ps)))) (<-> (E.e. x A ph) (E.e. x B ps)) (rexbii2.1 x exbii x A ph df-rex x B ps df-rex 3bitr4)) thm (ralbiia () ((ralbiia.1 (-> (e. (cv x) A) (<-> ph ps)))) (<-> (A.e. x A ph) (A.e. x A ps)) (ralbiia.1 pm5.74i ralbii2)) thm (rexbiia () ((ralbiia.1 (-> (e. (cv x) A) (<-> ph ps)))) (<-> (E.e. x A ph) (E.e. x A ps)) (ralbiia.1 pm5.32i rexbii2)) thm (2rexbiia ((x y) (A y)) ((2rexbiia.1 (-> (/\ (e. (cv x) A) (e. (cv y) B)) (<-> ph ps)))) (<-> (E.e. x A (E.e. y B ph)) (E.e. x A (E.e. y B ps))) (2rexbiia.1 rexbidva rexbiia)) thm (r2al ((x y) (A y)) () (<-> (A.e. x A (A.e. y B ph)) (A. x (A. y (-> (/\ (e. (cv x) A) (e. (cv y) B)) ph)))) (x A (A.e. y B ph) df-ral y (e. (cv x) A) (-> (e. (cv y) B) ph) 19.21v (e. (cv x) A) (e. (cv y) B) ph impexp y albii y B ph df-ral (e. (cv x) A) imbi2i 3bitr4 x albii bitr4)) thm (2ralbida ((x y) (A y)) ((2ralbida.1 (-> ph (A. x ph))) (2ralbida.2 (-> ph (A. y ph))) (2ralbida.3 (-> (/\ ph (/\ (e. (cv x) A) (e. (cv y) B))) (<-> ps ch)))) (-> ph (<-> (A.e. x A (A.e. y B ps)) (A.e. x A (A.e. y B ch)))) (2ralbida.1 2ralbida.2 (e. (cv x) A) y ax-17 hban 2ralbida.3 anassrs ralbida ralbida)) thm (2ralbidva ((x y) (ph x) (ph y) (A y)) ((2ralbidva.1 (-> (/\ ph (/\ (e. (cv x) A) (e. (cv y) B))) (<-> ps ch)))) (-> ph (<-> (A.e. x A (A.e. y B ps)) (A.e. x A (A.e. y B ch)))) (ph x ax-17 ph y ax-17 2ralbidva.1 2ralbida)) thm (2rexbidva ((x y) (ph x) (ph y) (A y)) ((2ralbidva.1 (-> (/\ ph (/\ (e. (cv x) A) (e. (cv y) B))) (<-> ps ch)))) (-> ph (<-> (E.e. x A (E.e. y B ps)) (E.e. x A (E.e. y B ch)))) (2ralbidva.1 anassrs rexbidva rexbidva)) thm (2ralbidv ((x y) (ph x) (ph y)) ((2ralbidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (A.e. x A (A.e. y B ps)) (A.e. x A (A.e. y B ch)))) (2ralbidv.1 y B ralbidv x A ralbidv)) thm (2rexbidv ((x y) (ph x) (ph y)) ((2ralbidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E.e. x A (E.e. y B ps)) (E.e. x A (E.e. y B ch)))) (2ralbidv.1 y B rexbidv x A rexbidv)) thm (rexralbidv ((x y) (ph x) (ph y)) ((2ralbidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E.e. x A (A.e. y B ps)) (E.e. x A (A.e. y B ch)))) (2ralbidv.1 y B ralbidv x A rexbidv)) thm (ralinexa () () (<-> (A.e. x A (-> ph (-. ps))) (-. (E.e. x A (/\ ph ps)))) (ph ps imnan x A ralbii x A (/\ ph ps) ralnex bitr)) thm (rexanali () () (<-> (E.e. x A (/\ ph (-. ps))) (-. (A.e. x A (-> ph ps)))) (ph ps annim x A rexbii x A (-> ph ps) rexnal bitr)) thm (risset ((A x) (B x)) () (<-> (e. A B) (E.e. x B (= (cv x) A))) (x (e. (cv x) B) (= (cv x) A) exancom x B (= (cv x) A) df-rex A B x df-clel 3bitr4r)) thm (hbral ((x y)) ((hbral.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbral.2 (-> ph (A. x ph)))) (-> (A.e. y A ph) (A. x (A.e. y A ph))) (hbral.1 hbral.2 hbim y hbal y A ph df-ral y A ph df-ral x albii 3imtr4)) thm (hbra1 () () (-> (A.e. x A ph) (A. x (A.e. x A ph))) (x (-> (e. (cv x) A) ph) hba1 x A ph df-ral x A ph df-ral x albii 3imtr4)) thm (hbrex ((x y)) ((hbrex.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbrex.2 (-> ph (A. x ph)))) (-> (E.e. y A ph) (A. x (E.e. y A ph))) (hbrex.1 hbrex.2 hban y hbex y A ph df-rex y A ph df-rex x albii 3imtr4)) thm (hbre1 () () (-> (E.e. x A ph) (A. x (E.e. x A ph))) (x (/\ (e. (cv x) A) ph) hbe1 x A ph df-rex x A ph df-rex x albii 3imtr4)) thm (r3al ((x y) (x z) (y z) (A y) (A z) (B z)) () (<-> (A.e. x A (A.e. y B (A.e. z C ph))) (A. x (A. y (A. z (-> (/\/\ (e. (cv x) A) (e. (cv y) B) (e. (cv z) C)) ph))))) (x A (A. y (A. z (-> (/\ (e. (cv y) B) (e. (cv z) C)) ph))) df-ral y B z C ph r2al x A ralbii (e. (cv x) A) (e. (cv y) B) (e. (cv z) C) 3anass ph imbi1i (e. (cv x) A) (/\ (e. (cv y) B) (e. (cv z) C)) ph impexp bitr z albii z (e. (cv x) A) (-> (/\ (e. (cv y) B) (e. (cv z) C)) ph) 19.21v bitr y albii y (e. (cv x) A) (A. z (-> (/\ (e. (cv y) B) (e. (cv z) C)) ph)) 19.21v bitr x albii 3bitr4)) thm (r2ex ((x y) (A y)) () (<-> (E.e. x A (E.e. y B ph)) (E. x (E. y (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph)))) (x A (E.e. y B ph) df-rex y (e. (cv x) A) (/\ (e. (cv y) B) ph) 19.42v (e. (cv x) A) (e. (cv y) B) ph anass y exbii y B ph df-rex (e. (cv x) A) anbi2i 3bitr4 x exbii bitr4)) thm (rexex () () (-> (E.e. x A ph) (E. x ph)) (x A ph df-rex (e. (cv x) A) ph pm3.27 x 19.22i sylbi)) thm (ra4 () () (-> (A.e. x A ph) (-> (e. (cv x) A) ph)) (x A ph df-ral x (-> (e. (cv x) A) ph) ax-4 sylbi)) thm (ra4e () () (-> (/\ (e. (cv x) A) ph) (E.e. x A ph)) ((/\ (e. (cv x) A) ph) x 19.8a x A ph df-rex sylibr)) thm (ra42 () () (-> (A.e. x A (A.e. y B ph)) (-> (/\ (e. (cv x) A) (e. (cv y) B)) ph)) (x A (A.e. y B ph) ra4 y B ph ra4 syl6 imp3a)) thm (rspec () ((rspec.1 (A.e. x A ph))) (-> (e. (cv x) A) ph) (rspec.1 x A ph ra4 ax-mp)) thm (rgen () ((rgen.1 (-> (e. (cv x) A) ph))) (A.e. x A ph) (x A ph df-ral rgen.1 mpgbir)) thm (rgen2 ((y z) (A y) (A z) (x z)) ((rgen2.1 (-> (/\ (e. (cv x) A) (e. (cv y) A)) ph))) (A.e. x A (A.e. y A ph)) (rgen2.1 ex y ax-gen (cv y) (cv x) A eleq1 y a4s (-> (e. (cv y) A) ph) imbi1d y dral2 mpbiri (e. (cv y) A) ph pm2.43 y 19.20i (A. y (-> (e. (cv y) A) ph)) (e. (cv x) A) ax-1 3syl (e. (cv z) A) y ax-17 (cv z) (cv x) A eleq1 dvelim rgen2.1 ex y 19.20i syl6 pm2.61i y A ph df-ral sylibr rgen)) thm (mprg () ((mprg.1 (-> (A.e. x A ph) ps)) (mprg.2 (-> (e. (cv x) A) ph))) ps (mprg.2 rgen mprg.1 ax-mp)) thm (mprgbir () ((mprgbir.1 (<-> ph (A.e. x A ps))) (mprgbir.2 (-> (e. (cv x) A) ps))) ph (mprgbir.2 rgen mprgbir.1 mpbir)) thm (r19.20 () () (-> (A.e. x A (-> ph ps)) (-> (A.e. x A ph) (A.e. x A ps))) (x A (-> ph ps) df-ral (e. (cv x) A) ph ps ax-2 x 19.20ii sylbi x A ph df-ral x A ps df-ral 3imtr4g)) thm (r19.20i2 () ((r19.20i2.1 (-> (-> (e. (cv x) A) ph) (-> (e. (cv x) B) ps)))) (-> (A.e. x A ph) (A.e. x B ps)) (r19.20i2.1 x 19.20i x A ph df-ral x B ps df-ral 3imtr4)) thm (r19.20i () ((r19.20i.1 (-> (e. (cv x) A) (-> ph ps)))) (-> (A.e. x A ph) (A.e. x A ps)) (r19.20i.1 a2i r19.20i2)) thm (r19.20si () ((r19.20si.1 (-> ph ps))) (-> (A.e. x A ph) (A.e. x A ps)) (r19.20si.1 (e. (cv x) A) a1i r19.20i)) thm (r19.20da () ((r19.20da.1 (-> ph (A. x ph))) (r19.20da.2 (-> (/\ ph (e. (cv x) A)) (-> ps ch)))) (-> ph (-> (A.e. x A ps) (A.e. x A ch))) (r19.20da.1 r19.20da.2 ex a2d 19.20d x A ps df-ral x A ch df-ral 3imtr4g)) thm (r19.20dva ((ph x)) ((r19.20dva.1 (-> (/\ ph (e. (cv x) A)) (-> ps ch)))) (-> ph (-> (A.e. x A ps) (A.e. x A ch))) (ph x ax-17 r19.20dva.1 r19.20da)) thm (r19.20sdv ((ph x)) ((r19.20sdv.1 (-> ph (-> ps ch)))) (-> ph (-> (A.e. x A ps) (A.e. x A ch))) (r19.20sdv.1 (e. (cv x) A) adantr r19.20dva)) thm (r19.20dv2 ((ph x)) ((r19.20dv2.1 (-> ph (-> (-> (e. (cv x) A) ps) (-> (e. (cv x) B) ch))))) (-> ph (-> (A.e. x A ps) (A.e. x B ch))) (r19.20dv2.1 x 19.20dv x A ps df-ral x B ch df-ral 3imtr4g)) thm (r19.21ai () ((r19.21ai.1 (-> ph (A. x ph))) (r19.21ai.2 (-> ph (-> (e. (cv x) A) ps)))) (-> ph (A.e. x A ps)) (r19.21ai.1 r19.21ai.2 19.21ai x A ps df-ral sylibr)) thm (r19.21aiv ((ph x)) ((r19.21aiv.1 (-> ph (-> (e. (cv x) A) ps)))) (-> ph (A.e. x A ps)) (ph x ax-17 r19.21aiv.1 r19.21ai)) thm (r19.21aiva ((ph x)) ((r19.21aiva.1 (-> (/\ ph (e. (cv x) A)) ps))) (-> ph (A.e. x A ps)) (r19.21aiva.1 ex r19.21aiv)) thm (r19.21v ((ph x)) () (<-> (A.e. x A (-> ph ps)) (-> ph (A.e. x A ps))) ((e. (cv x) A) ph ps bi2.04 x albii x ph (-> (e. (cv x) A) ps) 19.21v bitr x A (-> ph ps) df-ral x A ps df-ral ph imbi2i 3bitr4)) thm (r19.21ad () ((r19.21ad.1 (-> ph (A. x ph))) (r19.21ad.2 (-> ps (A. x ps))) (r19.21ad.3 (-> ph (-> ps (-> (e. (cv x) A) ch))))) (-> ph (-> ps (A.e. x A ch))) (r19.21ad.1 r19.21ad.2 r19.21ad.3 19.21ad x A ch df-ral syl6ibr)) thm (r19.21adv ((ph x) (ps x)) ((r19.21adv.1 (-> ph (-> ps (-> (e. (cv x) A) ch))))) (-> ph (-> ps (A.e. x A ch))) (ph x ax-17 ps x ax-17 r19.21adv.1 r19.21ad)) thm (r19.21aivv ((x y) (ph x) (ph y) (A y)) ((r19.21aivv.1 (-> ph (-> (/\ (e. (cv x) A) (e. (cv y) B)) ps)))) (-> ph (A.e. x A (A.e. y B ps))) (r19.21aivv.1 exp3a r19.21adv r19.21aiv)) thm (r19.21advv ((x y) (ph x) (ph y) (ps x) (ps y) (A y)) ((r19.21advv.1 (-> ph (-> ps (-> (/\ (e. (cv x) A) (e. (cv y) B)) ch))))) (-> ph (-> ps (A.e. x A (A.e. y B ch)))) (r19.21advv.1 imp r19.21aivv ex)) thm (rgen2a ((x y) (A y)) ((rgen2a.1 (-> (/\ (e. (cv x) A) (e. (cv y) B)) ph))) (A.e. x A (A.e. y B ph)) (rgen2a.1 r19.21aiva rgen)) thm (rgen3 ((y z) (A y) (A z) (x z)) ((rgen3.1 (-> (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A)) ph))) (A.e. x A (A.e. y A (A.e. z A ph))) (rgen3.1 3exp imp r19.21aiv rgen2)) thm (r19.21bi () ((r19.21bi.1 (-> ph (A.e. x A ps)))) (-> (/\ ph (e. (cv x) A)) ps) (r19.21bi.1 x A ps df-ral sylib 19.21bi imp)) thm (rspec2 () ((rspec2.1 (A.e. x A (A.e. y B ph)))) (-> (/\ (e. (cv x) A) (e. (cv y) B)) ph) (rspec2.1 rspec r19.21bi)) thm (rspec3 () ((rspec3.1 (A.e. x A (A.e. y B (A.e. z C ph))))) (-> (/\/\ (e. (cv x) A) (e. (cv y) B) (e. (cv z) C)) ph) (rspec3.1 rspec2 r19.21bi 3impa)) thm (r19.21be () ((r19.21be.1 (-> ph (A.e. x A ps)))) (A.e. x A (-> ph ps)) (r19.21be.1 r19.21bi expcom rgen)) thm (nrex () ((nrex.1 (-> (e. (cv x) A) (-. ps)))) (-. (E.e. x A ps)) (nrex.1 rgen x A ps ralnex mpbi)) thm (nrexdv ((ph x)) ((nrexdv.1 (-> (/\ ph (e. (cv x) A)) (-. ps)))) (-> ph (-. (E.e. x A ps))) (nrexdv.1 r19.21aiva x A ps ralnex sylib)) thm (r19.22 () () (-> (A.e. x A (-> ph ps)) (-> (E.e. x A ph) (E.e. x A ps))) ((e. (cv x) A) ph ps imdistan x albii x (/\ (e. (cv x) A) ph) (/\ (e. (cv x) A) ps) 19.22 sylbi x A (-> ph ps) df-ral x A ph df-rex x A ps df-rex imbi12i 3imtr4)) thm (r19.22i () ((r19.22i.1 (-> (e. (cv x) A) (-> ph ps)))) (-> (E.e. x A ph) (E.e. x A ps)) (x A ph ps r19.22 r19.22i.1 mprg)) thm (r19.22i2 () ((r19.22i2.1 (-> (/\ (e. (cv x) A) ph) (/\ (e. (cv x) B) ps)))) (-> (E.e. x A ph) (E.e. x B ps)) (r19.22i2.1 x 19.22i x A ph df-rex x B ps df-rex 3imtr4)) thm (r19.22si () ((r19.22si.1 (-> ph ps))) (-> (E.e. x A ph) (E.e. x A ps)) (r19.22si.1 (e. (cv x) A) a1i r19.22i)) thm (r19.22d () ((r19.22d.1 (-> ph (A. x ph))) (r19.22d.2 (-> ph (-> (e. (cv x) A) (-> ps ch))))) (-> ph (-> (E.e. x A ps) (E.e. x A ch))) (r19.22d.1 r19.22d.2 r19.21ai x A ps ch r19.22 syl)) thm (r19.22dv2 ((ph x)) ((r19.22dv2.1 (-> ph (-> (/\ (e. (cv x) A) ps) (/\ (e. (cv x) B) ch))))) (-> ph (-> (E.e. x A ps) (E.e. x B ch))) (r19.22dv2.1 x 19.22dv x A ps df-rex x B ch df-rex 3imtr4g)) thm (r19.22dv ((ph x)) ((r19.22dv.1 (-> ph (-> (e. (cv x) A) (-> ps ch))))) (-> ph (-> (E.e. x A ps) (E.e. x A ch))) (ph x ax-17 r19.22dv.1 r19.22d)) thm (r19.22sdv ((ph x)) ((r19.22sdv.1 (-> ph (-> ps ch)))) (-> ph (-> (E.e. x A ps) (E.e. x A ch))) (r19.22sdv.1 (e. (cv x) A) a1d r19.22dv)) thm (r19.22dva ((ph x)) ((r19.22dva.1 (-> (/\ ph (e. (cv x) A)) (-> ps ch)))) (-> ph (-> (E.e. x A ps) (E.e. x A ch))) (r19.22dva.1 ex r19.22dv)) thm (r19.12 ((x y) (A y) (B x)) () (-> (E.e. x A (A.e. y B ph)) (A.e. y B (E.e. x A ph))) ((e. (cv x) A) y ax-17 y B ph hbra1 hbrex (E.e. x A (A.e. y B ph)) (e. (cv y) B) ax-1 r19.21ai y B ph ra4 com12 (e. (cv x) A) a1d r19.22dv r19.20i syl)) thm (r19.23v ((ps x)) () (<-> (A.e. x A (-> ph ps)) (-> (E.e. x A ph) ps)) ((e. (cv x) A) ph ps impexp x albii x A ph df-rex ps imbi1i x (/\ (e. (cv x) A) ph) ps 19.23v bitr4 x A (-> ph ps) df-ral 3bitr4r)) thm (r19.23ai () ((r19.23ai.1 (-> ps (A. x ps))) (r19.23ai.2 (-> (e. (cv x) A) (-> ph ps)))) (-> (E.e. x A ph) ps) (x A ph df-rex r19.23ai.1 r19.23ai.2 imp 19.23ai sylbi)) thm (r19.23aiv ((ps x)) ((r19.23aiv.1 (-> (e. (cv x) A) (-> ph ps)))) (-> (E.e. x A ph) ps) (ps x ax-17 r19.23aiv.1 r19.23ai)) thm (r19.23ad () ((r19.23ad.1 (-> ph (A. x ph))) (r19.23ad.2 (-> ch (A. x ch))) (r19.23ad.3 (-> ph (-> (e. (cv x) A) (-> ps ch))))) (-> ph (-> (E.e. x A ps) ch)) (r19.23ad.1 r19.23ad.2 r19.23ad.3 imp3a 19.23ad x A ps df-rex syl5ib)) thm (r19.23adv ((ph x) (ch x)) ((r19.23adv.1 (-> ph (-> (e. (cv x) A) (-> ps ch))))) (-> ph (-> (E.e. x A ps) ch)) (r19.23adv.1 imp3a x 19.23adv x A ps df-rex syl5ib)) thm (r19.23aivv ((x y) (ps x) (ps y) (A y)) ((r19.23aivv.1 (-> (/\ (e. (cv x) A) (e. (cv y) B)) (-> ph ps)))) (-> (E.e. x A (E.e. y B ph)) ps) (r19.23aivv.1 ex r19.23adv r19.23aiv)) thm (r19.23advv ((x y) (ph x) (ph y) (ch x) (ch y) (A y)) ((r19.23advv.1 (-> ph (-> (/\ (e. (cv x) A) (e. (cv y) B)) (-> ps ch))))) (-> ph (-> (E.e. x A (E.e. y B ps)) ch)) (r19.23advv.1 exp3a imp r19.23adv ex r19.23adv)) thm (r19.26 () () (<-> (A.e. x A (/\ ph ps)) (/\ (A.e. x A ph) (A.e. x A ps))) ((e. (cv x) A) ph ps jcab x albii x (-> (e. (cv x) A) ph) (-> (e. (cv x) A) ps) 19.26 bitr x A (/\ ph ps) df-ral x A ph df-ral x A ps df-ral anbi12i 3bitr4)) thm (r19.26-2 () () (<-> (A.e. x A (A.e. y B (/\ ph ps))) (/\ (A.e. x A (A.e. y B ph)) (A.e. x A (A.e. y B ps)))) (y B ph ps r19.26 x A ralbii x A (A.e. y B ph) (A.e. y B ps) r19.26 bitr)) thm (r19.26m () () (<-> (A. x (/\ (-> (e. (cv x) A) ph) (-> (e. (cv x) B) ps))) (/\ (A.e. x A ph) (A.e. x B ps))) (x (-> (e. (cv x) A) ph) (-> (e. (cv x) B) ps) 19.26 x A ph df-ral x B ps df-ral anbi12i bitr4)) thm (r19.15 () () (-> (A.e. x A (<-> ph ps)) (<-> (A.e. x A ph) (A.e. x A ps))) (x A (<-> ph ps) hbra1 x A (<-> ph ps) ra4 imp ralbida)) thm (r19.27av ((ps x)) () (-> (/\ (A.e. x A ph) ps) (A.e. x A (/\ ph ps))) ((e. (cv x) A) ph pm2.27 ps anim1d com12 x 19.20i x A ph df-ral ps anbi1i x (-> (e. (cv x) A) ph) ps 19.27v bitr4 x A (/\ ph ps) df-ral 3imtr4)) thm (r19.28av ((ph x)) () (-> (/\ ph (A.e. x A ps)) (A.e. x A (/\ ph ps))) (x A ps ph r19.27av ph (A.e. x A ps) ancom ph ps ancom x A ralbii 3imtr4)) thm (r19.29 () () (-> (/\ (A.e. x A ph) (E.e. x A ps)) (E.e. x A (/\ ph ps))) (x (-> (e. (cv x) A) ph) (/\ (e. (cv x) A) ps) 19.29 (e. (cv x) A) ph ps anandi (e. (cv x) A) ph abai (/\ (e. (cv x) A) ps) anbi1i (e. (cv x) A) (-> (e. (cv x) A) ph) ps anandi bitr4 (e. (cv x) A) (-> (e. (cv x) A) ph) ps an12 3bitr x exbii sylibr x A ph df-ral x A ps df-rex anbi12i x A (/\ ph ps) df-rex 3imtr4)) thm (r19.29r () () (-> (/\ (E.e. x A ph) (A.e. x A ps)) (E.e. x A (/\ ph ps))) (x A ps ph r19.29 (E.e. x A ph) (A.e. x A ps) ancom ph ps ancom x A rexbii 3imtr4)) thm (r19.32v ((ph x)) () (<-> (A.e. x A (\/ ph ps)) (\/ ph (A.e. x A ps))) (x A (-. ph) ps r19.21v ph ps df-or x A ralbii ph (A.e. x A ps) df-or 3bitr4)) thm (r19.35 () () (<-> (E.e. x A (-> ph ps)) (-> (A.e. x A ph) (E.e. x A ps))) (x A ph (-. ps) r19.26 ph ps annim x A ralbii (A.e. x A ph) (A.e. x A (-. ps)) df-an 3bitr3 con2bii x A ps dfrex2 (A.e. x A ph) imbi2i x A (-> ph ps) dfrex2 3bitr4r)) thm (r19.36av ((ps x)) () (-> (E.e. x A (-> ph ps)) (-> (A.e. x A ph) ps)) (x A ph ps r19.35 (e. (cv x) A) ps idd r19.23aiv (A.e. x A ph) imim2i sylbi)) thm (r19.37av ((ph x)) () (-> (E.e. x A (-> ph ps)) (-> ph (E.e. x A ps))) (x A ph ps r19.35 ph (e. (cv x) A) ax-1 r19.21aiv (E.e. x A ps) imim1i sylbi)) thm (r19.40 () () (-> (E.e. x A (/\ ph ps)) (/\ (E.e. x A ph) (E.e. x A ps))) (ph ps pm3.26 x A r19.22si ph ps pm3.27 x A r19.22si jca)) thm (r19.41v ((ps x)) () (<-> (E.e. x A (/\ ph ps)) (/\ (E.e. x A ph) ps)) ((e. (cv x) A) ph ps anass x exbii x (/\ (e. (cv x) A) ph) ps 19.41v bitr3 x A (/\ ph ps) df-rex x A ph df-rex ps anbi1i 3bitr4)) thm (r19.42v ((ph x)) () (<-> (E.e. x A (/\ ph ps)) (/\ ph (E.e. x A ps))) (x A ps ph r19.41v ph ps ancom x A rexbii ph (E.e. x A ps) ancom 3bitr4)) thm (r19.43 () () (<-> (E.e. x A (\/ ph ps)) (\/ (E.e. x A ph) (E.e. x A ps))) ((e. (cv x) A) ph ps andi x exbii x (/\ (e. (cv x) A) ph) (/\ (e. (cv x) A) ps) 19.43 bitr x A (\/ ph ps) df-rex x A ph df-rex x A ps df-rex orbi12i 3bitr4)) thm (r19.44av ((ps x)) () (-> (E.e. x A (\/ ph ps)) (\/ (E.e. x A ph) ps)) (x A ph ps r19.43 (e. (cv x) A) ps idd r19.23aiv (E.e. x A ph) orim2i sylbi)) thm (r19.45av ((ph x)) () (-> (E.e. x A (\/ ph ps)) (\/ ph (E.e. x A ps))) (x A ph ps r19.43 (e. (cv x) A) ph idd r19.23aiv (E.e. x A ps) orim1i sylbi)) thm (hbreu1 () () (-> (E!e. x A ph) (A. x (E!e. x A ph))) (x (/\ (e. (cv x) A) ph) hbeu1 x A ph df-reu x A ph df-reu x albii 3imtr4)) thm (rabid () () (<-> (e. (cv x) ({e.|} x A ph)) (/\ (e. (cv x) A) ph)) (x A ph df-rab abeq2i)) thm (rabid2 ((A x)) () (<-> (= A ({e.|} x A ph)) (A.e. x A ph)) ((e. (cv x) A) ph pm4.71 x albii A x (/\ (e. (cv x) A) ph) abeq2 bitr4 x A ph df-ral x A ph df-rab A eqeq2i 3bitr4r)) thm (rabswap () () (= ({e.|} x A (e. (cv x) B)) ({e.|} x B (e. (cv x) A))) ((e. (cv x) A) (e. (cv x) B) ancom x abbii x A (e. (cv x) B) df-rab x B (e. (cv x) A) df-rab 3eqtr4)) thm (hbrab1 ((x y)) () (-> (e. (cv y) ({e.|} x A ph)) (A. x (e. (cv y) ({e.|} x A ph)))) (y x (/\ (e. (cv x) A) ph) hbab1 x A ph df-rab (cv y) eleq2i x A ph df-rab (cv y) eleq2i x albii 3imtr4)) thm (hbrab ((x y) (x z) (y z) (A z)) ((hbrab.1 (-> ph (A. x ph))) (hbrab.2 (-> (e. (cv z) A) (A. x (e. (cv z) A))))) (-> (e. (cv z) ({e.|} y A ph)) (A. x (e. (cv z) ({e.|} y A ph)))) ((e. (cv z) (cv y)) x ax-17 hbrab.2 hbel hbrab.1 hban z y hbab y A ph df-rab (cv z) eleq2i y A ph df-rab (cv z) eleq2i x albii 3imtr4)) thm (ralcom ((x y) (B x) (A y)) () (<-> (A.e. x A (A.e. y B ph)) (A.e. y B (A.e. x A ph))) ((e. (cv x) A) (e. (cv y) B) ancom ph imbi1i x y 2albii x y (-> (/\ (e. (cv y) B) (e. (cv x) A)) ph) alcom bitr x A y B ph r2al y B x A ph r2al 3bitr4)) thm (rexcom ((x y) (B x) (A y)) () (<-> (E.e. x A (E.e. y B ph)) (E.e. y B (E.e. x A ph))) ((e. (cv x) A) (e. (cv y) B) ancom ph anbi1i x y 2exbii x y (/\ (/\ (e. (cv y) B) (e. (cv x) A)) ph) excom bitr x A y B ph r2ex y B x A ph r2ex 3bitr4)) thm (ralcom2 ((y z) (A y) (A z) (x z) (A x)) () (-> (A.e. x A (A.e. y A ph)) (A.e. y A (A.e. x A ph))) ((A. x (-> (e. (cv x) A) (A. x (-> (e. (cv x) A) ph)))) id (cv x) (cv y) A eleq1 x a4s ph imbi1d dral1 (e. (cv x) A) imbi2d x dral2 (cv x) (cv y) A eleq1 x a4s (A. x (-> (e. (cv x) A) ph)) imbi1d dral1 imbi12d mpbii x y x hbnae y hbal x y y hbnae (e. (cv z) A) y ax-17 (cv z) (cv x) A eleq1 dvelim nalequcoms y (-> (e. (cv y) A) ph) hba1 (-. (A. x (= (cv x) (cv y)))) a1i hbimd y a4s hbald (e. (cv z) A) x ax-17 (cv z) (cv y) A eleq1 dvelim y (-> (e. (cv y) A) ph) ax-4 (e. (cv x) A) imim2i com23 x 19.20ii syl9 y 19.20ii syld hbnaes pm2.61i x A (A.e. y A ph) df-ral y A ph df-ral (e. (cv x) A) imbi2i x albii bitr y A (A.e. x A ph) df-ral x A ph df-ral (e. (cv y) A) imbi2i y albii bitr 3imtr4)) thm (ralcom3 () () (<-> (A.e. x A (-> (e. (cv x) B) ph)) (A.e. x B (-> (e. (cv x) A) ph))) ((e. (cv x) A) (e. (cv x) B) ph pm2.04 r19.20i2 (e. (cv x) B) (e. (cv x) A) ph pm2.04 r19.20i2 impbi)) thm (reeanv ((ph y) (ps x) (x y) (A y) (B x)) () (<-> (E.e. x A (E.e. y B (/\ ph ps))) (/\ (E.e. x A ph) (E.e. y B ps))) ((e. (cv x) A) (e. (cv y) B) (/\ ph ps) anass (e. (cv x) A) (e. (cv y) B) ph ps an4 bitr3 x y 2exbii x y (e. (cv x) A) (/\ (e. (cv y) B) (/\ ph ps)) exdistr x y (/\ (e. (cv x) A) ph) (/\ (e. (cv y) B) ps) eeanv 3bitr3 y B (/\ ph ps) df-rex x A rexbii x A (E. y (/\ (e. (cv y) B) (/\ ph ps))) df-rex bitr x A ph df-rex y B ps df-rex anbi12i 3bitr4)) thm (reubidva ((ph x)) ((reubidva.1 (-> (/\ ph (e. (cv x) A)) (<-> ps ch)))) (-> ph (<-> (E!e. x A ps) (E!e. x A ch))) (reubidva.1 ex pm5.32d x eubidv x A ps df-reu x A ch df-reu 3bitr4g)) thm (reubidv ((ph x)) ((reubidv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E!e. x A ps) (E!e. x A ch))) (reubidv.1 (e. (cv x) A) adantr reubidva)) thm (reubiia () ((reubiia.1 (-> (e. (cv x) A) (<-> ph ps)))) (<-> (E!e. x A ph) (E!e. x A ps)) (reubiia.1 pm5.32i x eubii x A ph df-reu x A ps df-reu 3bitr4)) thm (reubii () ((reubii.1 (<-> ph ps))) (<-> (E!e. x A ph) (E!e. x A ps)) (reubii.1 (e. (cv x) A) a1i reubiia)) thm (raleq1f ((A y) (B y) (x y)) ((raleq1f.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (raleq1f.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (= A B) (<-> (A.e. x A ph) (A.e. x B ph))) (raleq1f.1 raleq1f.2 hbeq A B (cv x) eleq2 ph imbi1d albid x A ph df-ral x B ph df-ral 3bitr4g)) thm (rexeq1f ((A y) (B y) (x y)) ((raleq1f.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (raleq1f.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (= A B) (<-> (E.e. x A ph) (E.e. x B ph))) (raleq1f.1 raleq1f.2 hbeq A B (cv x) eleq2 ph anbi1d exbid x A ph df-rex x B ph df-rex 3bitr4g)) thm (reueq1f ((A y) (B y) (x y)) ((raleq1f.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (raleq1f.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (= A B) (<-> (E!e. x A ph) (E!e. x B ph))) (raleq1f.1 raleq1f.2 hbeq A B (cv x) eleq2 ph anbi1d eubid x A ph df-reu x B ph df-reu 3bitr4g)) thm (raleq1 ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (<-> (A.e. x A ph) (A.e. x B ph))) ((e. (cv y) A) x ax-17 (e. (cv y) B) x ax-17 ph raleq1f)) thm (rexeq1 ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (<-> (E.e. x A ph) (E.e. x B ph))) ((e. (cv y) A) x ax-17 (e. (cv y) B) x ax-17 ph rexeq1f)) thm (reueq1 ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (<-> (E!e. x A ph) (E!e. x B ph))) ((e. (cv y) A) x ax-17 (e. (cv y) B) x ax-17 ph reueq1f)) thm (raleq1d ((A x) (B x)) ((raleq1d.1 (-> ph (= A B)))) (-> ph (<-> (A.e. x A ps) (A.e. x B ps))) (raleq1d.1 A B x ps raleq1 syl)) thm (raleqd ((A x) (B x)) ((raleqd.1 (-> (= A B) (<-> ph ps)))) (-> (= A B) (<-> (A.e. x A ph) (A.e. x B ps))) (A B x ph raleq1 raleqd.1 x B ralbidv bitrd)) thm (rexeqd ((A x) (B x)) ((raleqd.1 (-> (= A B) (<-> ph ps)))) (-> (= A B) (<-> (E.e. x A ph) (E.e. x B ps))) (A B x ph rexeq1 raleqd.1 x B rexbidv bitrd)) thm (reueqd ((A x) (B x)) ((raleqd.1 (-> (= A B) (<-> ph ps)))) (-> (= A B) (<-> (E!e. x A ph) (E!e. x B ps))) (A B x ph reueq1 raleqd.1 x B reubidv bitrd)) thm (cbvralf ((A z) (x y) (x z) (y z)) ((cbvralf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (cbvralf.2 (-> (e. (cv z) A) (A. y (e. (cv z) A)))) (cbvralf.3 (-> ph (A. y ph))) (cbvralf.4 (-> ps (A. x ps))) (cbvralf.5 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A.e. x A ph) (A.e. y A ps)) ((e. (cv z) (cv x)) y ax-17 cbvralf.2 hbel cbvralf.3 hbim (e. (cv z) (cv y)) x ax-17 cbvralf.1 hbel cbvralf.4 hbim (cv x) (cv y) A eleq1 cbvralf.5 imbi12d cbval x A ph df-ral y A ps df-ral 3bitr4)) thm (cbvral ((x y) (x z) (A x) (y z) (A y) (A z)) ((cbvral.1 (-> ph (A. y ph))) (cbvral.2 (-> ps (A. x ps))) (cbvral.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A.e. x A ph) (A.e. y A ps)) ((e. (cv z) A) x ax-17 (e. (cv z) A) y ax-17 cbvral.1 cbvral.2 cbvral.3 cbvralf)) thm (cbvrex ((x y) (A x) (A y)) ((cbvral.1 (-> ph (A. y ph))) (cbvral.2 (-> ps (A. x ps))) (cbvral.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E.e. x A ph) (E.e. y A ps)) ((e. (cv x) A) y ax-17 cbvral.1 hban (e. (cv y) A) x ax-17 cbvral.2 hban (cv x) (cv y) A eleq1 cbvral.3 anbi12d cbvex x A ph df-rex y A ps df-rex 3bitr4)) thm (cbvralv ((ph y) (ps x) (x y) (A x) (A y)) ((cbvralv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A.e. x A ph) (A.e. y A ps)) (ph y ax-17 ps x ax-17 cbvralv.1 A cbvral)) thm (cbvrexv ((ph y) (ps x) (x y) (A x) (A y)) ((cbvralv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E.e. x A ph) (E.e. y A ps)) (ph y ax-17 ps x ax-17 cbvralv.1 A cbvrex)) thm (cbvreuv ((ph y) (ps x) (x y) (A x) (A y)) ((cbvralv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E!e. x A ph) (E!e. y A ps)) ((/\ (e. (cv x) A) ph) y ax-17 (/\ (e. (cv y) A) ps) x ax-17 (cv x) (cv y) A eleq1 cbvralv.1 anbi12d cbveu x A ph df-reu y A ps df-reu 3bitr4)) thm (cbvral2v ((x z) (A x) (A z) (x y) (w x) (B x) (y z) (w y) (B y) (w z) (B z) (B w) (ph z) (ps y) (ch x) (ch w)) ((cbvral2v.1 (-> (= (cv x) (cv z)) (<-> ph ch))) (cbvral2v.2 (-> (= (cv y) (cv w)) (<-> ch ps)))) (<-> (A.e. x A (A.e. y B ph)) (A.e. z A (A.e. w B ps))) (cbvral2v.1 y B ralbidv A cbvralv cbvral2v.2 B cbvralv z A ralbii bitr)) thm (cbvral3v ((ph w) (ps z) (v x) (ch x) (ch v) (u y) (th y) (th u) (w x) (A x) (A w) (x y) (B x) (w y) (v y) (B y) (v w) (B w) (B v) (x z) (u x) (C x) (y z) (C y) (w z) (v z) (u z) (C z) (u w) (C w) (u v) (C v) (C u)) ((cbvral3v.1 (-> (= (cv x) (cv w)) (<-> ph ch))) (cbvral3v.2 (-> (= (cv y) (cv v)) (<-> ch th))) (cbvral3v.3 (-> (= (cv z) (cv u)) (<-> th ps)))) (<-> (A.e. x A (A.e. y B (A.e. z C ph))) (A.e. w A (A.e. v B (A.e. u C ps)))) (cbvral3v.1 y B z C 2ralbidv A cbvralv cbvral3v.2 cbvral3v.3 B C cbvral2v w A ralbii bitr)) thm (reurex () () (-> (E!e. x A ph) (E.e. x A ph)) (x (/\ (e. (cv x) A) ph) euex x A ph df-reu x A ph df-rex 3imtr4)) thm (reu2 ((x y) (A x) (A y) (ph y)) () (<-> (E!e. x A ph) (/\ (E.e. x A ph) (A.e. x A (A.e. y A (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y))))))) ((/\ (e. (cv x) A) ph) y ax-17 x eu2 x A ph df-reu x A ph df-rex x A (A.e. y A (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y)))) df-ral y (e. (cv x) A) (-> (e. (cv y) A) (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y)))) 19.21v (e. (cv y) A) x ax-17 y x ph hbs1 hban (cv x) (cv y) A eleq1 x y ph sbequ12 anbi12d sbie (/\ (e. (cv x) A) ph) anbi2i (e. (cv x) A) ph (e. (cv y) A) ([/] (cv y) x ph) an4 bitr (= (cv x) (cv y)) imbi1i (/\ (e. (cv x) A) (e. (cv y) A)) (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y)) impexp (e. (cv x) A) (e. (cv y) A) (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y))) impexp 3bitr y albii y A (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y))) df-ral (e. (cv x) A) imbi2i 3bitr4 x albii bitr4 anbi12i 3bitr4)) thm (reu5 () () (<-> (E!e. x A ph) (/\ (E.e. x A ph) (E* x (/\ (e. (cv x) A) ph)))) (x (/\ (e. (cv x) A) ph) eu5 x A ph df-reu x A ph df-rex (E* x (/\ (e. (cv x) A) ph)) anbi1i 3bitr4)) thm (reu4 ((x y) (A x) (A y) (ph y) (ps x)) ((reu4.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E!e. x A ph) (/\ (E.e. x A ph) (A.e. x A (A.e. y A (-> (/\ ph ps) (= (cv x) (cv y))))))) (x A ph y reu2 ps x ax-17 reu4.1 sbie ph anbi2i (= (cv x) (cv y)) imbi1i x A y A 2ralbii (E.e. x A ph) anbi2i bitr)) thm (2reuswap ((x y) (A x) (A y)) () (-> (A.e. x A (E* y (/\ (e. (cv y) A) ph))) (-> (E!e. x A (E.e. y A ph)) (E!e. y A (E.e. x A ph)))) (x A (E* y (/\ (e. (cv y) A) ph)) df-ral y (e. (cv x) A) (/\ (e. (cv y) A) ph) moanimv x albii bitr4 x y (/\ (e. (cv x) A) (/\ (e. (cv y) A) ph)) 2euswap x A (E.e. y A ph) df-reu y A (/\ (e. (cv x) A) ph) df-rex y A (e. (cv x) A) ph r19.42v (e. (cv y) A) (e. (cv x) A) ph an12 y exbii 3bitr3 x eubii bitr y A (E.e. x A ph) df-reu x A (e. (cv y) A) ph r19.42v x A (/\ (e. (cv y) A) ph) df-rex bitr3 y eubii bitr 3imtr4g sylbi)) thm (rabbii () ((rabbii.1 (-> (e. (cv x) A) (<-> ps ch)))) (= ({e.|} x A ps) ({e.|} x A ch)) (rabbii.1 pm5.32i x abbii x A ps df-rab x A ch df-rab 3eqtr4)) thm (rabbidv ((ph x)) ((rabbidv.1 (-> ph (-> (e. (cv x) A) (<-> ps ch))))) (-> ph (= ({e.|} x A ps) ({e.|} x A ch))) (rabbidv.1 pm5.32d x abbidv x A ps df-rab x A ch df-rab 3eqtr4g)) thm (rabbisdv ((ph x)) ((rabbisdv.1 (-> ph (<-> ps ch)))) (-> ph (= ({e.|} x A ps) ({e.|} x A ch))) (rabbisdv.1 (e. (cv x) A) a1d rabbidv)) thm (rabeqf ((A y) (B y) (x y)) ((rabeqf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (rabeqf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (= A B) (= ({e.|} x A ph) ({e.|} x B ph))) (rabeqf.1 rabeqf.2 hbeq A B (cv x) eleq2 ph anbi1d abbid x A ph df-rab x B ph df-rab 3eqtr4g)) thm (rabeq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (= ({e.|} x A ph) ({e.|} x B ph))) ((e. (cv y) A) x ax-17 (e. (cv y) B) x ax-17 ph rabeqf)) thm (rabeq2i () ((rabeqi.1 (= A ({e.|} x B ph)))) (<-> (e. (cv x) A) (/\ (e. (cv x) B) ph)) (rabeqi.1 (cv x) eleq2i x B ph rabid bitr)) thm (visset () () (e. (cv x) (V)) ((cv x) eqid x df-v abeq2i mpbir)) thm (isset ((A x)) () (<-> (e. A (V)) (E. x (= (cv x) A))) (A (V) x df-clel x visset (= (cv x) A) biantru x exbii bitr4)) thm (isseti ((A x)) ((isseti.1 (e. A (V)))) (E. x (= (cv x) A)) (isseti.1 A x isset mpbi)) thm (issetri ((A x)) ((issetri.1 (E. x (= (cv x) A)))) (e. A (V)) (issetri.1 A x isset mpbir)) thm (elisset ((A x) (B x)) () (-> (e. A B) (e. A (V))) (A B x df-clel x (= (cv x) A) (e. (cv x) B) 19.40 sylbi pm3.26d A x isset sylibr)) thm (elisseti () ((elisseti.1 (e. A B))) (e. A (V)) (elisseti.1 A B elisset ax-mp)) thm (elex ((A x)) () (-> (e. A B) (E. x (= (cv x) A))) (A B elisset A x isset sylib)) thm (ralv () () (<-> (A.e. x (V) ph) (A. x ph)) (x (V) ph df-ral x visset ph a1bi x albii bitr4)) thm (rexv () () (<-> (E.e. x (V) ph) (E. x ph)) (x (V) ph df-rex x visset ph biantrur x exbii bitr4)) thm (rabab () () (= ({e.|} x (V) ph) ({|} x ph)) (x (V) ph df-rab (e. (cv x) (V)) ph pm3.27 x visset ph jctl impbi x abbii eqtr)) thm (ralcom4 ((x y) (A y)) () (<-> (A.e. x A (A. y ph)) (A. y (A.e. x A ph))) (y (V) x A ph ralcom y (A.e. x A ph) ralv y ph ralv x A ralbii 3bitr3r)) thm (rexcom4 ((x y) (A y)) () (<-> (E.e. x A (E. y ph)) (E. y (E.e. x A ph))) (y (V) x A ph rexcom y (E.e. x A ph) rexv y ph rexv x A rexbii 3bitr3r)) thm (ceqsalg ((A x)) ((ceqsalg.1 (-> ps (A. x ps))) (ceqsalg.2 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (A. x (-> (= (cv x) A) ph)) ps)) (ceqsalg.2 biimpd a2i x 19.20i ceqsalg.1 (= (cv x) A) 19.23 sylib A B x elex syl5com ceqsalg.1 ceqsalg.2 biimprcd 19.21ai (e. A B) a1i impbid)) thm (ceqsal ((A x)) ((ceqsal.1 (-> ps (A. x ps))) (ceqsal.2 (e. A (V))) (ceqsal.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (A. x (-> (= (cv x) A) ph)) ps) (ceqsal.2 ceqsal.1 ceqsal.3 (V) ceqsalg ax-mp)) thm (ceqsalv ((A x) (ps x)) ((ceqsalv.1 (e. A (V))) (ceqsalv.2 (-> (= (cv x) A) (<-> ph ps)))) (<-> (A. x (-> (= (cv x) A) ph)) ps) (ps x ax-17 ceqsalv.1 ceqsalv.2 ceqsal)) thm (gencl ((ps x)) ((gencl.1 (<-> th (E. x (/\ ch (= A B))))) (gencl.2 (-> (= A B) (<-> ph ps))) (gencl.3 (-> ch ph))) (-> th ps) (gencl.1 gencl.2 gencl.3 syl5bi impcom x 19.23aiv sylbi)) thm (2gencl ((x y) (R x) (ps x) (C y) (S y) (ch y)) ((2gencl.1 (<-> (e. C S) (E. x (/\ (e. (cv x) R) (= A C))))) (2gencl.2 (<-> (e. D S) (E. y (/\ (e. (cv y) R) (= B D))))) (2gencl.3 (-> (= A C) (<-> ph ps))) (2gencl.4 (-> (= B D) (<-> ps ch))) (2gencl.5 (-> (/\ (e. (cv x) R) (e. (cv y) R)) ph))) (-> (/\ (e. C S) (e. D S)) ch) (2gencl.2 2gencl.4 (e. C S) imbi2d 2gencl.1 2gencl.3 (e. (cv y) R) imbi2d 2gencl.5 ex gencl com12 gencl impcom)) thm (3gencl ((x y) (x z) (y z) (D y) (D z) (F z) (R x) (R y) (S y) (S z) (ps x) (ch y) (th z)) ((3gencl.1 (<-> (e. D S) (E. x (/\ (e. (cv x) R) (= A D))))) (3gencl.2 (<-> (e. F S) (E. y (/\ (e. (cv y) R) (= B F))))) (3gencl.3 (<-> (e. G S) (E. z (/\ (e. (cv z) R) (= C G))))) (3gencl.4 (-> (= A D) (<-> ph ps))) (3gencl.5 (-> (= B F) (<-> ps ch))) (3gencl.6 (-> (= C G) (<-> ch th))) (3gencl.7 (-> (/\/\ (e. (cv x) R) (e. (cv y) R) (e. (cv z) R)) ph))) (-> (/\/\ (e. D S) (e. F S) (e. G S)) th) (3gencl.3 3gencl.6 (/\ (e. D S) (e. F S)) imbi2d 3gencl.1 3gencl.2 3gencl.4 (e. (cv z) R) imbi2d 3gencl.5 (e. (cv z) R) imbi2d 3gencl.7 3exp imp 2gencl com12 gencl com12 3impia)) thm (cgsex2g ((x y) (ps x) (ps y) (A x) (A y) (B x) (B y)) ((cgsex2g.1 (-> (/\ (= (cv x) A) (= (cv y) B)) ch)) (cgsex2g.2 (-> ch (<-> ph ps)))) (-> (/\ (e. A C) (e. B D)) (<-> (E. x (E. y (/\ ch ph))) ps)) (cgsex2g.2 biimpa x y 19.23aivv (/\ (e. A C) (e. B D)) a1i cgsex2g.2 biimprcd ancld x y 19.22dvv A C x elex B D y elex anim12i x y (= (cv x) A) (= (cv y) B) eeanv sylibr cgsex2g.1 x y 19.22i2 syl syl5com impbid)) thm (cgsex4g ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w) (ps x) (ps y) (ps z) (ps w)) ((cgsex4g.1 (-> (/\ (/\ (= (cv x) A) (= (cv y) B)) (/\ (= (cv z) C) (= (cv w) D))) ch)) (cgsex4g.2 (-> ch (<-> ph ps)))) (-> (/\ (/\ (e. A R) (e. B S)) (/\ (e. C R) (e. D S))) (<-> (E. x (E. y (E. z (E. w (/\ ch ph))))) ps)) (cgsex4g.2 biimpa z w 19.23aivv x y 19.23aivv (/\ (/\ (e. A R) (e. B S)) (/\ (e. C R) (e. D S))) a1i cgsex4g.2 biimprcd ancld z w 19.22dvv x y 19.22dvv A R x elex B S y elex anim12i x y (= (cv x) A) (= (cv y) B) eeanv sylibr C R z elex D S w elex anim12i z w (= (cv z) C) (= (cv w) D) eeanv sylibr anim12i x y z w (/\ (= (cv x) A) (= (cv y) B)) (/\ (= (cv z) C) (= (cv w) D)) ee4anv sylibr cgsex4g.1 z w 19.22i2 x y 19.22i2 syl syl5com impbid)) thm (ceqsex ((A x)) ((ceqsex.1 (-> ps (A. x ps))) (ceqsex.2 (e. A (V))) (ceqsex.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (E. x (/\ (= (cv x) A) ph)) ps) (ceqsex.1 ceqsex.3 biimpa 19.23ai ceqsex.2 x isseti ceqsex.1 ceqsex.3 biimprcd 19.21ai x (= (cv x) A) ph exintr syl mpi impbi)) thm (ceqsexv ((A x) (ps x)) ((ceqsexv.1 (e. A (V))) (ceqsexv.2 (-> (= (cv x) A) (<-> ph ps)))) (<-> (E. x (/\ (= (cv x) A) ph)) ps) (ps x ax-17 ceqsexv.1 ceqsexv.2 ceqsex)) thm (ceqsex2 ((x y) (A x) (A y) (B x) (B y)) ((ceqsex2.1 (-> ps (A. x ps))) (ceqsex2.2 (-> ch (A. y ch))) (ceqsex2.3 (e. A (V))) (ceqsex2.4 (e. B (V))) (ceqsex2.5 (-> (= (cv x) A) (<-> ph ps))) (ceqsex2.6 (-> (= (cv y) B) (<-> ps ch)))) (<-> (E. x (E. y (/\ (/\ (= (cv x) A) (= (cv y) B)) ph))) ch) ((= (cv x) A) (= (cv y) B) ph anass y exbii y (= (cv x) A) (/\ (= (cv y) B) ph) 19.42v bitr x exbii (= (cv y) B) x ax-17 ceqsex2.1 hban y hbex ceqsex2.3 ceqsex2.5 (= (cv y) B) anbi2d y exbidv ceqsex ceqsex2.2 ceqsex2.4 ceqsex2.6 ceqsex 3bitr)) thm (ceqsex2v ((x y) (A x) (A y) (B x) (B y) (ch y) (ps x)) ((ceqsex2v.1 (e. A (V))) (ceqsex2v.2 (e. B (V))) (ceqsex2v.3 (-> (= (cv x) A) (<-> ph ps))) (ceqsex2v.4 (-> (= (cv y) B) (<-> ps ch)))) (<-> (E. x (E. y (/\ (/\ (= (cv x) A) (= (cv y) B)) ph))) ch) (ps x ax-17 ch y ax-17 ceqsex2v.1 ceqsex2v.2 ceqsex2v.3 ceqsex2v.4 ceqsex2)) thm (gencbvex ((ps x) (ph y) (th x) (ch y) (A y)) ((gencbvex.1 (e. A (V))) (gencbvex.2 (-> (= A (cv y)) (<-> ph ps))) (gencbvex.3 (-> (= A (cv y)) (<-> ch th))) (gencbvex.4 (<-> th (E. x (/\ ch (= A (cv y))))))) (<-> (E. x (/\ ch ph)) (E. y (/\ th ps))) (x y (/\ (= (cv y) A) (/\ th ps)) excom gencbvex.1 gencbvex.3 gencbvex.2 anbi12d bicomd eqcoms ceqsexv x exbii (E. x (= (cv y) A)) th ps anass gencbvex.4 gencbvex.3 pm5.32i (= A (cv y)) ch ancom A (cv y) eqcom th anbi1i 3bitr3 x exbii x (= (cv y) A) th 19.41v 3bitr ps anbi1i x (= (cv y) A) (/\ th ps) 19.41v 3bitr4r y exbii 3bitr3)) thm (gencbval ((ps x) (ph y) (th x) (ch y) (A y)) ((gencbval.1 (e. A (V))) (gencbval.2 (-> (= A (cv y)) (<-> ph ps))) (gencbval.3 (-> (= A (cv y)) (<-> ch th))) (gencbval.4 (<-> th (E. x (/\ ch (= A (cv y))))))) (<-> (A. x (-> ch ph)) (A. y (-> th ps))) (gencbval.1 gencbval.2 negbid gencbval.3 gencbval.4 gencbvex x ch ph exanali y th ps exanali 3bitr3 con4bii)) thm (vtoclf ((A x)) ((vtoclf.1 (-> ps (A. x ps))) (vtoclf.2 (e. A (V))) (vtoclf.3 (-> (= (cv x) A) (<-> ph ps))) (vtoclf.4 ph)) ps (vtoclf.1 vtoclf.2 x isseti vtoclf.3 biimpd x 19.22i ax-mp 19.36i vtoclf.4 mpg)) thm (vtocl ((A x) (ps x)) ((vtocl.1 (e. A (V))) (vtocl.2 (-> (= (cv x) A) (<-> ph ps))) (vtocl.3 ph)) ps (ps x ax-17 vtocl.1 vtocl.2 vtocl.3 vtoclf)) thm (vtocl2 ((x y) (A x) (A y) (B x) (B y) (ps x) (ps y)) ((vtocl2.1 (e. A (V))) (vtocl2.2 (e. B (V))) (vtocl2.3 (-> (/\ (= (cv x) A) (= (cv y) B)) (<-> ph ps))) (vtocl2.4 ph)) ps (vtocl2.1 x isseti vtocl2.2 y isseti x y (= (cv x) A) (= (cv y) B) eeanv vtocl2.3 biimpd x y 19.22i2 sylbir mp2an y ph ps 19.36v x exbii x (A. y ph) ps 19.36v bitr mpbi vtocl2.4 y ax-gen mpg)) thm (vtocl3 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (ps x) (ps y) (ps z)) ((vtocl3.1 (e. A (V))) (vtocl3.2 (e. B (V))) (vtocl3.3 (e. C (V))) (vtocl3.4 (-> (/\/\ (= (cv x) A) (= (cv y) B) (= (cv z) C)) (<-> ph ps))) (vtocl3.5 ph)) ps (vtocl3.1 x isseti vtocl3.2 y isseti vtocl3.3 z isseti x y z (= (cv x) A) (= (cv y) B) (= (cv z) C) eeeanv vtocl3.4 biimpd z 19.22i x y 19.22i2 sylbir mp3an z ph ps 19.36v y exbii y (A. z ph) ps 19.36v bitr x exbii x (A. y (A. z ph)) ps 19.36v bitr mpbi vtocl3.5 y z gen2 mpg)) thm (vtoclb ((A x) (ch x) (th x)) ((vtoclb.1 (e. A (V))) (vtoclb.2 (-> (= (cv x) A) (<-> ph ch))) (vtoclb.3 (-> (= (cv x) A) (<-> ps th))) (vtoclb.4 (<-> ph ps))) (<-> ch th) (vtoclb.1 vtoclb.2 vtoclb.3 bibi12d vtoclb.4 vtocl)) thm (vtoclgf ((A y) (x y)) ((vtoclgf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (vtoclgf.2 (-> ps (A. x ps))) (vtoclgf.3 (-> (= (cv x) A) (<-> ph ps))) (vtoclgf.4 ph)) (-> (e. A B) ps) (A B elisset A y isset vtoclgf.1 hbeleq (= (cv x) A) y ax-17 (cv y) (cv x) A eqeq1 cbvex bitr vtoclgf.2 vtoclgf.4 vtoclgf.3 mpbii 19.23ai sylbi syl)) thm (vtoclg ((x y) (A x) (A y) (ps x)) ((vtoclg.1 (-> (= (cv x) A) (<-> ph ps))) (vtoclg.2 ph)) (-> (e. A B) ps) ((e. (cv y) A) x ax-17 ps x ax-17 vtoclg.1 vtoclg.2 B vtoclgf)) thm (vtoclbg ((A x) (ch x) (th x)) ((vtoclbg.1 (-> (= (cv x) A) (<-> ph ch))) (vtoclbg.2 (-> (= (cv x) A) (<-> ps th))) (vtoclbg.3 (<-> ph ps))) (-> (e. A B) (<-> ch th)) (vtoclbg.1 vtoclbg.2 bibi12d vtoclbg.3 B vtoclg)) thm (vtocl2gf ((A x) (y z) (A y) (A z) (B y) (B z) (x z)) ((vtocl2gf.1 (-> ps (A. x ps))) (vtocl2gf.2 (-> ch (A. y ch))) (vtocl2gf.3 (-> (= (cv x) A) (<-> ph ps))) (vtocl2gf.4 (-> (= (cv y) B) (<-> ps ch))) (vtocl2gf.5 ph)) (-> (/\ (e. A C) (e. B D)) ch) ((e. (cv z) B) y ax-17 (e. A (V)) y ax-17 vtocl2gf.2 hbim vtocl2gf.4 (e. A (V)) imbi2d (e. (cv z) A) x ax-17 vtocl2gf.1 vtocl2gf.3 vtocl2gf.5 (V) vtoclgf D vtoclgf impcom A C elisset sylan)) thm (vtocl2g ((A x) (A y) (B y) (ps x) (ch y)) ((vtocl2g.1 (-> (= (cv x) A) (<-> ph ps))) (vtocl2g.2 (-> (= (cv y) B) (<-> ps ch))) (vtocl2g.3 ph)) (-> (/\ (e. A C) (e. B D)) ch) (ps x ax-17 ch y ax-17 vtocl2g.1 vtocl2g.2 vtocl2g.3 C D vtocl2gf)) thm (vtoclgaf ((A y) (x y) (B x) (B y)) ((vtoclgaf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (vtoclgaf.2 (-> ps (A. x ps))) (vtoclgaf.3 (-> (= (cv x) A) (<-> ph ps))) (vtoclgaf.4 (-> (e. (cv x) B) ph))) (-> (e. A B) ps) (vtoclgaf.1 vtoclgaf.1 (e. (cv y) B) x ax-17 hbel vtoclgaf.2 hbim (cv x) A B eleq1 vtoclgaf.3 imbi12d vtoclgaf.4 B vtoclgf pm2.43i)) thm (vtoclga ((x y) (A x) (A y) (B x) (B y) (ps x)) ((vtoclga.1 (-> (= (cv x) A) (<-> ph ps))) (vtoclga.2 (-> (e. (cv x) B) ph))) (-> (e. A B) ps) ((e. (cv y) A) x ax-17 ps x ax-17 vtoclga.1 vtoclga.2 vtoclgaf)) thm (vtocl2ga ((x y) (A x) (A y) (B y) (C x) (C y) (D x) (D y) (ps x) (ch y)) ((vtocl2ga.1 (-> (= (cv x) A) (<-> ph ps))) (vtocl2ga.2 (-> (= (cv y) B) (<-> ps ch))) (vtocl2ga.3 (-> (/\ (e. (cv x) C) (e. (cv y) D)) ph))) (-> (/\ (e. A C) (e. B D)) ch) (vtocl2ga.2 (e. A C) imbi2d vtocl2ga.1 (e. (cv y) D) imbi2d vtocl2ga.3 ex vtoclga com12 vtoclga impcom)) thm (vtocl3ga ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z) (C z) (D x) (D y) (D z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch y) (th z)) ((vtocl3ga.1 (-> (= (cv x) A) (<-> ph ps))) (vtocl3ga.2 (-> (= (cv y) B) (<-> ps ch))) (vtocl3ga.3 (-> (= (cv z) C) (<-> ch th))) (vtocl3ga.4 (-> (/\/\ (e. (cv x) D) (e. (cv y) R) (e. (cv z) S)) ph))) (-> (/\/\ (e. A D) (e. B R) (e. C S)) th) (vtocl3ga.2 (e. A D) imbi2d vtocl3ga.3 (e. A D) imbi2d vtocl3ga.1 (/\ (e. (cv y) R) (e. (cv z) S)) imbi2d vtocl3ga.4 3exp imp3a vtoclga com12 vtocl2ga ex com3r 3imp)) thm (vtocleg ((A x) (ph x)) ((vtocleg.1 (-> (= (cv x) A) ph))) (-> (e. A B) ph) (A B x elex vtocleg.1 x 19.23aiv syl)) thm (vtoclefg ((A x)) () (-> (/\/\ (e. A B) (A. x (-> ph (A. x ph))) (A. x (-> (= (cv x) A) ph))) ph) (x ph (= (cv x) A) 19.23g (e. A B) adantl A B x elex (E. x (= (cv x) A)) ph pm2.27 syl (A. x (-> ph (A. x ph))) adantr sylbid 3impia)) thm (vtoclef ((A x)) ((vtoclef.1 (-> ph (A. x ph))) (vtoclef.2 (e. A (V))) (vtoclef.3 (-> (= (cv x) A) ph))) ph (vtoclef.2 x isseti vtoclef.1 vtoclef.3 19.23ai ax-mp)) thm (vtocle ((A x) (ph x)) ((vtocle.1 (e. A (V))) (vtocle.2 (-> (= (cv x) A) ph))) ph (vtocle.1 vtocle.2 (V) vtocleg ax-mp)) thm (vtoclri ((A x) (B x) (ps x)) ((vtoclri.1 (-> (= (cv x) A) (<-> ph ps))) (vtoclri.2 (A.e. x B ph))) (-> (e. A B) ps) (vtoclri.1 vtoclri.2 rspec vtoclga)) thm (cla4gf ((x y) (A y)) ((cla4gf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (cla4gf.2 (-> ps (A. x ps))) (cla4gf.3 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> (A. x ph) ps)) (A B elisset A y isset cla4gf.1 hbeleq (= (cv x) A) y ax-17 (cv y) (cv x) A eqeq1 cbvex bitr cla4gf.3 biimpd x 19.22i sylbi cla4gf.2 ph 19.36 sylib syl)) thm (cla4egf ((x y) (A y)) ((cla4gf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (cla4gf.2 (-> ps (A. x ps))) (cla4gf.3 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> ps (E. x ph))) (cla4gf.1 cla4gf.2 hbne cla4gf.3 negbid B cla4gf con2d x ph df-ex syl6ibr)) thm (cla4gv ((ps x) (x y) (A x) (A y)) ((cla4gv.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> (A. x ph) ps)) ((e. (cv y) A) x ax-17 ps x ax-17 cla4gv.1 B cla4gf)) thm (cla4egv ((ps x) (x y) (A x) (A y)) ((cla4gv.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> ps (E. x ph))) ((e. (cv y) A) x ax-17 ps x ax-17 cla4gv.1 B cla4egf)) thm (cla4e2gv ((x y) (A x) (A y) (B x) (B y) (ps x) (ps y)) ((cla4e2gv.1 (-> (/\ (= (cv x) A) (= (cv y) B)) (<-> ph ps)))) (-> (/\ (e. A C) (e. B D)) (-> ps (E. x (E. y ph)))) (cla4e2gv.1 biimprcd x y 19.22dvv A C x elex B D y elex anim12i x y (= (cv x) A) (= (cv y) B) eeanv sylibr syl5com)) thm (cla42gv ((x y) (A x) (A y) (B x) (B y) (ps x) (ps y)) ((cla4e2gv.1 (-> (/\ (= (cv x) A) (= (cv y) B)) (<-> ph ps)))) (-> (/\ (e. A C) (e. B D)) (-> (A. x (A. y ph)) ps)) (cla4e2gv.1 negbid C D cla4e2gv y ph exnal x exbii x (A. y ph) exnal bitr2 syl6ibr a3d)) thm (cla4v ((A x) (ps x)) ((cla4v.1 (e. A (V))) (cla4v.2 (-> (= (cv x) A) (<-> ph ps)))) (-> (A. x ph) ps) (cla4v.1 cla4v.2 (V) cla4gv ax-mp)) thm (cla4ev ((A x) (ps x)) ((cla4v.1 (e. A (V))) (cla4v.2 (-> (= (cv x) A) (<-> ph ps)))) (-> ps (E. x ph)) (cla4v.1 cla4v.2 negbid cla4v con2i x ph df-ex sylibr)) thm (cla4e2v ((x y) (A x) (A y) (B x) (B y) (ps x) (ps y)) ((cla4e2v.1 (e. A (V))) (cla4e2v.2 (e. B (V))) (cla4e2v.3 (-> (/\ (= (cv x) A) (= (cv y) B)) (<-> ph ps)))) (-> ps (E. x (E. y ph))) (cla4e2v.1 cla4e2v.2 cla4e2v.3 (V) (V) cla4e2gv mp2an)) thm (rcla4 ((x y) (A x) (A y) (B x)) ((rcla4.1 (-> ps (A. x ps))) (rcla4.2 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> (A.e. x B ph) ps)) (x B ph df-ral (e. (cv y) A) x ax-17 (e. A B) x ax-17 rcla4.1 hbim (cv x) A B eleq1 rcla4.2 imbi12d B cla4gf pm2.43b sylbi com12)) thm (rcla4v ((A x) (B x) (ps x)) ((rcla4v.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> (A.e. x B ph) ps)) (ps x ax-17 rcla4v.1 B rcla4)) thm (rcla4cv ((A x) (B x) (ps x)) ((rcla4v.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (A.e. x B ph) (-> (e. A B) ps)) (rcla4v.1 B rcla4v com12)) thm (rcla4va ((A x) (B x) (ps x)) ((rcla4v.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (/\ (e. A B) (A.e. x B ph)) ps) (rcla4v.1 B rcla4v imp)) thm (rcla4cva ((A x) (B x) (ps x)) ((rcla4v.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (/\ (A.e. x B ph) (e. A B)) ps) (rcla4v.1 B rcla4va ancoms)) thm (rcla4ev ((A x) (B x) (ps x)) ((rcla4v.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (/\ (e. A B) ps) (E.e. x B ph)) ((cv x) A B eleq1 rcla4v.1 anbi12d B cla4egv anabsi5 x B ph df-rex sylibr)) thm (rcla42v ((x y) (A x) (A y) (C x) (D x) (B y) (D y) (ch x) (ps y)) ((rcla42v.1 (-> (= (cv x) A) (<-> ph ch))) (rcla42v.2 (-> (= (cv y) B) (<-> ch ps)))) (-> (/\ (e. A C) (e. B D)) (-> (A.e. x C (A.e. y D ph)) ps)) (rcla42v.1 y D ralbidv C rcla4v rcla42v.2 D rcla4v sylan9)) thm (rcla42ev ((x y) (A x) (A y) (C x) (D x) (B y) (D y) (ch x) (ps y)) ((rcla42v.1 (-> (= (cv x) A) (<-> ph ch))) (rcla42v.2 (-> (= (cv y) B) (<-> ch ps)))) (-> (/\ (/\ (e. A C) (e. B D)) ps) (E.e. x C (E.e. y D ph))) (rcla42v.2 D rcla4ev (e. A C) anim2i anassrs rcla42v.1 y D rexbidv C rcla4ev syl)) thm (rcla43v ((ps z) (ch x) (th y) (x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z) (C z) (R x) (S x) (S y) (T x) (T y) (T z)) ((rcla43v.1 (-> (= (cv x) A) (<-> ph ch))) (rcla43v.2 (-> (= (cv y) B) (<-> ch th))) (rcla43v.3 (-> (= (cv z) C) (<-> th ps)))) (-> (/\/\ (e. A R) (e. B S) (e. C T)) (-> (A.e. x R (A.e. y S (A.e. z T ph))) ps)) (rcla43v.1 z T ralbidv rcla43v.2 z T ralbidv R S rcla42v rcla43v.3 T rcla4v sylan9 3impa)) thm (eqvinc ((A x) (x y) (x z) (A y) (A z) (y z) (B x) (B y) (B z)) ((eqvinc.1 (e. A (V)))) (<-> (= A B) (E. x (/\ (= (cv x) A) (= (cv x) B)))) (eqvinc.1 A B (V) eleq1 mpbii x visset (cv x) B (V) eleq1 mpbii (= (cv x) A) adantl x 19.23aiv (cv z) B A eqeq2 (cv z) B (cv x) eqeq2 (= (cv x) A) anbi2d x exbidv eqvinc.1 (cv y) A (cv z) eqeq1 (cv y) A (cv x) eqeq1 A (cv x) eqcom syl6bb (= (cv x) (cv z)) anbi1d x exbidv y z x equvin vtoclb (V) vtoclbg pm5.21nii)) thm (eqvincf ((A y) (B y) (x y)) ((eqvincf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (eqvincf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (eqvincf.3 (e. A (V)))) (<-> (= A B) (E. x (/\ (= (cv x) A) (= (cv x) B)))) (eqvincf.3 B y eqvinc eqvincf.1 hbeleq eqvincf.2 hbeleq hban (/\ (= (cv x) A) (= (cv x) B)) y ax-17 (cv y) (cv x) A eqeq1 (cv y) (cv x) B eqeq1 anbi12d cbvex bitr)) thm (alexeq ((A x) (x y) (A y) (ph y)) ((alexeq.1 (e. A (V)))) (<-> (A. x (-> (= (cv x) A) ph)) (E. x (/\ (= (cv x) A) ph))) (alexeq.1 (cv y) A (cv x) eqeq2 ph imbi1d x albidv (cv y) A (cv x) eqeq2 ph anbi1d x exbidv x y ph equs4 x y x hbae x (-> (= (cv x) (cv y)) ph) hba1 x y (-> (= (cv x) (cv y)) ph) ax-16 (= (cv x) (cv y)) ph pm3.4 syl5 19.23ad x y ph equs5 pm2.61i impbi vtoclb)) thm (ceqex ((A x) (x y) (A y) (ph y)) () (-> (= (cv x) A) (<-> ph (E. x (/\ (= (cv x) A) ph)))) ((= (cv x) A) x 19.8a A x isset sylibr (cv y) A (cv x) eqeq2 (cv y) A (cv x) eqeq2 ph anbi1d x exbidv ph bibi2d imbi12d (/\ (= (cv x) (cv y)) ph) x 19.8a ex x (-> (= (cv x) (cv y)) ph) ax-4 com12 y visset x ph alexeq syl5ibr impbid (V) vtoclg mpcom)) thm (ceqsexg ((x y) (A x) (A y)) ((ceqsexg.1 (-> ps (A. x ps))) (ceqsexg.2 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (E. x (/\ (= (cv x) A) ph)) ps)) ((e. (cv y) A) x ax-17 x (/\ (= (cv x) A) ph) hbe1 ceqsexg.1 hbbi x A ph ceqex ceqsexg.2 bibi12d ph pm4.2 B vtoclgf)) thm (ceqsexgv ((A x) (ps x)) ((ceqsexgv.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (E. x (/\ (= (cv x) A) ph)) ps)) (ps x ax-17 ceqsexgv.1 B ceqsexg)) thm (ceqsrexv ((A x) (B x) (ps x)) ((ceqsrexv.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (E.e. x B (/\ (= (cv x) A) ph)) ps)) ((cv x) A B eleq1 ceqsrexv.1 anbi12d B ceqsexgv (e. A B) ps ibar bitr4d x B (/\ (= (cv x) A) ph) df-rex (= (cv x) A) (e. (cv x) B) ph an12 x exbii bitr4 syl5bb)) thm (ceqsrex2v ((x y) (A x) (A y) (B x) (B y) (C x) (D x) (D y) (ps x) (ch y)) ((ceqsrex2v.1 (-> (= (cv x) A) (<-> ph ps))) (ceqsrex2v.2 (-> (= (cv y) B) (<-> ps ch)))) (-> (/\ (e. A C) (e. B D)) (<-> (E.e. x C (E.e. y D (/\ (/\ (= (cv x) A) (= (cv y) B)) ph))) ch)) (ceqsrex2v.1 (= (cv y) B) anbi2d y D rexbidv C ceqsrexv (= (cv x) A) (= (cv y) B) ph anass y D rexbii y D (= (cv x) A) (/\ (= (cv y) B) ph) r19.42v bitr x C rexbii syl5bb ceqsrex2v.2 D ceqsrexv sylan9bb)) thm (clel2 ((A x) (B x)) ((clel2.1 (e. A (V)))) (<-> (e. A B) (A. x (-> (= (cv x) A) (e. (cv x) B)))) (clel2.1 (cv x) A B eleq1 ceqsalv bicomi)) thm (clel3g ((A x) (B x)) () (-> (e. B C) (<-> (e. A B) (E. x (/\ (= (cv x) B) (e. A (cv x)))))) ((cv x) B A eleq2 C ceqsexgv bicomd)) thm (clel3 ((A x) (B x)) ((clel3.1 (e. B (V)))) (<-> (e. A B) (E. x (/\ (= (cv x) B) (e. A (cv x))))) (clel3.1 B (V) A x clel3g ax-mp)) thm (clel4 ((A x) (B x)) ((clel4.1 (e. B (V)))) (<-> (e. A B) (A. x (-> (= (cv x) B) (e. A (cv x))))) (clel4.1 (cv x) B A eleq2 ceqsalv bicomi)) thm (elabt ((x y) (A x) (A y) (ph y) (ps x)) () (-> (/\ (e. A B) (A. x (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({|} x ph)) ps)) ((e. (cv y) A) x ax-17 y x ph hbab1 hbel ps x ax-17 hbbi x ax-gen A B x (<-> (e. A ({|} x ph)) ps) vtoclefg mp3an2 (cv x) A ({|} x ph) eleq1 x ph abid syl5rbbr ps bibi1d biimprd a2i x 19.20i sylan2)) thm (elabf ((ps y) (A x) (x y) (A y) (ph y)) ((elabf.1 (-> ps (A. x ps))) (elabf.2 (e. A (V))) (elabf.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({|} x ph)) ps) ((e. (cv y) A) x ax-17 y x ph hbab1 hbel elabf.1 hbbi elabf.2 (cv x) A ({|} x ph) eleq1 x ph abid syl5bbr elabf.3 bitr3d vtoclef)) thm (elab ((ps x) (A x)) ((elab.1 (e. A (V))) (elab.2 (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({|} x ph)) ps) (ps x ax-17 elab.1 elab.2 elabf)) thm (elabgf ((A y) (ph z) (x y) (x z) (y z)) ((elabgf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (elabgf.2 (-> ps (A. x ps))) (elabgf.3 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (e. A ({|} x ph)) ps)) (elabgf.1 elabgf.1 z x ph hbab1 hbel elabgf.2 hbbi (cv x) A ({|} x ph) eleq1 elabgf.3 bibi12d x ph abid B vtoclgf)) thm (elabg ((ps x) (x y) (A x) (A y)) ((elabg.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (e. A ({|} x ph)) ps)) ((e. (cv y) A) x ax-17 ps x ax-17 elabg.1 B elabgf)) thm (elab2g ((ps x) (A x)) ((elab2g.1 (-> (= (cv x) A) (<-> ph ps))) (elab2g.2 (= B ({|} x ph)))) (-> (e. A C) (<-> (e. A B) ps)) (elab2g.1 C elabg elab2g.2 A eleq2i syl5bb)) thm (elab2 ((ps x) (A x)) ((elab2.1 (e. A (V))) (elab2.2 (-> (= (cv x) A) (<-> ph ps))) (elab2.3 (= B ({|} x ph)))) (<-> (e. A B) ps) (elab2.1 elab2.2 elab2.3 (V) elab2g ax-mp)) thm (elab3g ((ps x) (A x)) ((elab3g.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (-> ps (e. A B)) (<-> (e. A ({|} x ph)) ps)) (elab3g.1 ({|} x ph) elabg ibi (-> ps (e. A B)) a1i elab3g.1 B elabg ps imim2i ps (e. A ({|} x ph)) ibibr sylibr impbid)) thm (elab3 ((ps x) (A x)) ((elab3.1 (-> ps (e. A (V)))) (elab3.2 (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({|} x ph)) ps) (elab3.1 elab3.2 (V) elab3g ax-mp)) thm (elrabf ((x y) (A y) (B y)) ((elrabf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (elrabf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (elrabf.3 (-> ps (A. x ps))) (elrabf.4 (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({e.|} x B ph)) (/\ (e. A B) ps)) (A ({e.|} x B ph) elisset A B elisset ps adantr elrabf.1 elrabf.1 elrabf.2 hbel elrabf.3 hban (cv x) A B eleq1 elrabf.4 anbi12d (V) elabgf x B ph df-rab A eleq2i syl5bb pm5.21nii)) thm (elrab ((ps x) (x y) (A x) (A y) (B x) (B y)) ((elrab.1 (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({e.|} x B ph)) (/\ (e. A B) ps)) ((e. (cv y) A) x ax-17 (e. (cv y) B) x ax-17 ps x ax-17 elrab.1 elrabf)) thm (elrab2 ((ps x) (A x) (B x)) ((elrab.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (e. A ({e.|} x B ph)) ps)) (elrab.1 B elrab baib)) thm (cbvab ((x y) (x z) (y z) (ph z) (ps z)) ((cbvab.1 (-> ph (A. y ph))) (cbvab.2 (-> ps (A. x ps))) (cbvab.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (= ({|} x ph) ({|} y ps)) (cbvab.1 z x hbab z y ps hbab1 cleqf cbvab.2 y visset cbvab.3 elabf y ps abid bitr4 mpgbir)) thm (cbvabv ((x y) (ph y) (ps x)) ((cbvabv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (= ({|} x ph) ({|} y ps)) (ph y ax-17 ps x ax-17 cbvabv.1 cbvab)) thm (cbvrab ((x y) (x z) (y z) (A z)) ((cbvrab.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (cbvrab.2 (-> (e. (cv z) A) (A. y (e. (cv z) A)))) (cbvrab.3 (-> ph (A. y ph))) (cbvrab.4 (-> ps (A. x ps))) (cbvrab.5 (-> (= (cv x) (cv y)) (<-> ph ps)))) (= ({e.|} x A ph) ({e.|} y A ps)) ((e. (cv z) (cv x)) y ax-17 cbvrab.2 hbel cbvrab.3 hban (e. (cv z) (cv y)) x ax-17 cbvrab.1 hbel cbvrab.4 hban (cv x) (cv y) A eleq1 cbvrab.5 anbi12d cbvab x A ph df-rab y A ps df-rab 3eqtr4)) thm (cbvrabv ((x y) (x z) (A x) (y z) (A y) (A z) (ph y) (ps x)) ((cbvrabv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (= ({e.|} x A ph) ({e.|} y A ps)) ((e. (cv z) A) x ax-17 (e. (cv z) A) y ax-17 ph y ax-17 ps x ax-17 cbvrabv.1 cbvrab)) thm (abidhb ((y z) (A y) (A z) (x y) (x z)) () (-> (A. y (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (= ({|} z (A. x (e. (cv z) A))) A)) (y (-> (e. (cv y) A) (A. x (e. (cv y) A))) hba1 x (e. (cv y) A) ax-4 (-> (e. (cv y) A) (A. x (e. (cv y) A))) a1i (-> (e. (cv y) A) (A. x (e. (cv y) A))) id impbid y a4s abbid y A abid2 syl6eq (cv z) (cv y) A eleq1 x albidv cbvabv syl5eq)) thm (hbeqd ((y z) (A y) (A z) (B y) (B z) (ph y) (x y) (x z)) ((hbeqd.1 (-> ph (A. x ph))) (hbeqd.2 (-> ph (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (hbeqd.3 (-> ph (-> (e. (cv y) B) (A. x (e. (cv y) B)))))) (-> ph (-> (= A B) (A. x (= A B)))) (x (e. (cv z) A) hba1 y z hbab x (e. (cv z) B) hba1 y z hbab hbeq ph a1i hbeqd.2 y 19.21aiv y A x z abidhb syl hbeqd.3 y 19.21aiv y B x z abidhb syl eqeq12d hbeqd.1 hbeqd.2 y 19.21aiv y A x z abidhb syl hbeqd.3 y 19.21aiv y B x z abidhb syl eqeq12d albid 3imtr3d)) thm (hbeld ((y z) (A y) (A z) (B y) (B z) (ph y) (x y) (x z)) ((hbeld.1 (-> ph (A. x ph))) (hbeld.2 (-> ph (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (hbeld.3 (-> ph (-> (e. (cv y) B) (A. x (e. (cv y) B)))))) (-> ph (-> (e. A B) (A. x (e. A B)))) (x (e. (cv z) A) hba1 y z hbab x (e. (cv z) B) hba1 y z hbab hbel ph a1i hbeld.2 y 19.21aiv y A x z abidhb syl hbeld.3 y 19.21aiv y B x z abidhb syl eleq12d hbeld.1 hbeld.2 y 19.21aiv y A x z abidhb syl hbeld.3 y 19.21aiv y B x z abidhb syl eleq12d albid 3imtr3d)) thm (eueq ((x y) (A x) (A y)) () (<-> (e. A (V)) (E! x (= (cv x) A))) ((cv x) A (cv y) eqtr3t x y gen2 (E. x (= (cv x) A)) biantru A x isset (cv x) (cv y) A eqeq1 eu4 3bitr4)) thm (eueq1 ((A x)) ((eueq1.1 (e. A (V)))) (E! x (= (cv x) A)) (eueq1.1 A x eueq mpbi)) thm (eueq2 ((ph x) (A x) (B x)) ((eueq2.1 (e. A (V))) (eueq2.2 (e. B (V)))) (E! x (\/ (/\ ph (= (cv x) A)) (/\ (-. ph) (= (cv x) B)))) ((-. ph) x (/\ ph (= (cv x) A)) euorv ph negb eueq2.1 x eueq1 x ph (= (cv x) A) euanv biimpr mpan2 sylanc ph negb (= (cv x) B) bianfd (/\ ph (= (cv x) A)) orbi2d (-. ph) (/\ ph (= (cv x) A)) orcom syl5bb x eubidv mpbid eueq2.2 x eueq1 x (-. ph) (= (cv x) B) euanv biimpr mpan2 ph x (/\ (-. ph) (= (cv x) B)) euorv mpdan (-. ph) id (= (cv x) A) bianfd (/\ (-. ph) (= (cv x) B)) orbi1d x eubidv mpbid pm2.61i)) thm (eueq3 ((ph x) (ps x) (A x) (B x) (C x)) ((eueq3.1 (e. A (V))) (eueq3.2 (e. B (V))) (eueq3.3 (e. C (V))) (eueq3.4 (-. (/\ ph ps)))) (E! x (\/\/ (/\ ph (= (cv x) A)) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C)))) ((\/ (-. (\/ ph ps)) ps) x (/\ ph (= (cv x) A)) euorv ph ps pm2.45 eueq3.4 ph ps imnan mpbir con2i jaoi con2i eueq3.1 x eueq1 x ph (= (cv x) A) euanv biimpr mpan2 sylanc (\/ ph ps) negb orci (= (cv x) B) bianfd eueq3.4 ph ps imnan mpbir (= (cv x) C) bianfd orbi12d (/\ ph (= (cv x) A)) orbi2d (\/ (-. (\/ ph ps)) ps) (/\ ph (= (cv x) A)) orcom (/\ ph (= (cv x) A)) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C)) 3orass 3bitr4g x eubidv mpbid (\/ ph (-. (\/ ph ps))) x (/\ ps (= (cv x) C)) euorv eueq3.4 ph ps imnan mpbir ph ps pm2.46 jaoi con2i eueq3.3 x eueq1 x ps (= (cv x) C) euanv biimpr mpan2 sylanc eueq3.4 ph ps imnan mpbir con2i (= (cv x) A) bianfd (\/ ph ps) negb olci (= (cv x) B) bianfd orbi12d (/\ ps (= (cv x) C)) orbi1d (/\ ph (= (cv x) A)) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C)) df-3or syl6bbr x eubidv mpbid eueq3.2 x eueq1 x (-. (\/ ph ps)) (= (cv x) B) euanv biimpr mpan2 (\/ ph ps) x (/\ (-. (\/ ph ps)) (= (cv x) B)) euorv mpdan ph ps pm2.45 (= (cv x) A) bianfd ph ps pm2.46 (= (cv x) C) bianfd (/\ (-. (\/ ph ps)) (= (cv x) B)) orbi1d (/\ ps (= (cv x) C)) (/\ (-. (\/ ph ps)) (= (cv x) B)) orcom syl6bb orbi12d ph ps (/\ (-. (\/ ph ps)) (= (cv x) B)) orass (/\ ph (= (cv x) A)) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C)) 3orass 3bitr4g x eubidv mpbid ecase3)) thm (moeq ((A x)) () (E* x (= (cv x) A)) (A x isset A x eueq bitr3 biimp x (= (cv x) A) df-mo mpbir)) thm (moeq3 ((x y) (ph x) (ph y) (ps x) (ps y) (A x) (A y) (B x) (B y) (C x) (C y)) ((moeq3.1 (e. B (V))) (moeq3.2 (e. C (V))) (moeq3.3 (-. (/\ ph ps)))) (E* x (\/\/ (/\ ph (= (cv x) A)) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C)))) ((cv y) A (cv x) eqeq2 ph anbi2d (= (cv y) A) (/\ (-. (\/ ph ps)) (= (cv x) B)) pm4.2i (= (cv y) A) (/\ ps (= (cv x) C)) pm4.2i 3orbi123d x eubidv y visset moeq3.1 moeq3.2 moeq3.3 x eueq3 (V) vtoclg x (\/\/ (/\ ph (= (cv x) A)) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C))) eumo syl y visset moeq3.1 moeq3.2 moeq3.3 x eueq3 (e. A (V)) (= (cv x) (cv y)) pm2.21 x visset (cv x) A (V) eleq1 mpbii syl5 ph anim2d (\/ (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C))) orim1d (/\ ph (= (cv x) A)) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C)) 3orass (/\ ph (= (cv x) (cv y))) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C)) 3orass 3imtr4g x 19.21aiv x (\/\/ (/\ ph (= (cv x) A)) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C))) (\/\/ (/\ ph (= (cv x) (cv y))) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C))) euimmo syl mpi pm2.61i)) thm (mosub ((x y) (A x) (A y)) ((mosub.1 (E* x ph))) (E* x (E. y (/\ (= (cv y) A) ph))) (y A moeq mosub.1 y ax-gen y (= (cv y) A) x ph moexexv mp2an)) thm (mo2icl ((x y) (A x) (A y) (ph y)) () (-> (A. x (-> ph (= (cv x) A))) (E* x ph)) ((cv y) A (cv x) eqeq2 ph imbi2d x albidv (E* x ph) imbi1d (A. x (-> ph (= (cv x) (cv y)))) y 19.8a ph y ax-17 x mo2 sylibr (V) vtoclg x visset (cv x) A (V) eleq1 mpbii ph imim2i con3d com12 x 19.20dv x ph alnex x ph exmo ori sylbi syl6 pm2.61i)) thm (moi ((x y) (A x) (A y) (B x) (B y) (ch x) (ch y) (ph y) (ps x) (ps y)) ((moi.1 (-> (= (cv x) A) (<-> ph ps))) (moi.2 (-> (= (cv x) B) (<-> ph ch)))) (-> (/\/\ (/\ (e. A C) (e. B D)) (E* x ph) (/\ ps ch)) (= A B)) ((e. (cv y) A) x ax-17 ps x ax-17 y x ph hbs1 hban (= A (cv y)) x ax-17 hbim y hbal moi.1 ([/] (cv y) x ph) anbi1d (cv x) A (cv y) eqeq1 imbi12d y albidv C cla4gf y visset B x eqvinc y x ph hbs1 ch x ax-17 hbbi x y ph sbequ12 bicomd moi.2 sylan9bb 19.23ai sylbi ps anbi2d (cv y) B A eqeq2 imbi12d D cla4gv sylan9 y x ph hbs1 x y ph sbequ12 mo4f syl5ib 3imp)) thm (euxfr2 ((ph x) (A x) (x y)) ((euxfr2.1 (e. A (V))) (euxfr2.2 (E* y (= (cv x) A)))) (<-> (E! x (E. y (/\ (= (cv x) A) ph))) (E! y ph)) (x y (/\ (= (cv x) A) ph) 2euswap euxfr2.2 ph moani ph (= (cv x) A) ancom y mobii mpbi mpg y x (/\ (= (cv x) A) ph) 2euswap x A moeq ph moani ph (= (cv x) A) ancom x mobii mpbi mpg impbi euxfr2.1 (= (cv x) A) ph pm4.2i ceqsexv y eubii bitr)) thm (euxfr ((ps x) (ph y) (A x) (x y)) ((euxfr.1 (e. A (V))) (euxfr.2 (E! y (= (cv x) A))) (euxfr.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (E! x ph) (E! y ps)) (euxfr.2 y (= (cv x) A) euex ax-mp ph biantrur y (= (cv x) A) ph 19.41v euxfr.3 pm5.32i y exbii 3bitr2 x eubii euxfr.1 euxfr.2 eumoi ps euxfr2 bitr)) thm (ru ((x y)) () (e/ ({|} x (e/ (cv x) (cv x))) (V)) ((e. (cv y) (cv y)) pm5.19 (cv x) (cv y) (cv y) eleq1 (= (cv x) (cv y)) id (= (cv x) (cv y)) id eleq12d negbid (cv x) (cv x) df-nel syl5bb bibi12d a4b1 mto (cv y) x (e/ (cv x) (cv x)) abeq2 mtbir y nex ({|} x (e/ (cv x) (cv x))) y isset mtbir ({|} x (e/ (cv x) (cv x))) (V) df-nel mpbir)) thm (vsbcint ((ps x) (ps y) (x y) (A x)) ((vsbcint.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (= (cv y) A) (<-> ([/] (cv y) x ph) ps)) (y visset (cv x) (cv y) A eqeq1 ceqsexv y x ph hbs1 ps x ax-17 hbbi x y ph sbequ12 bicomd vsbcint.1 sylan9bb 19.23ai sylbir)) thm (sbralie ((x y) (x z) (y z) (ph y) (ph z) (ps x) (ps z)) ((sbralie.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> ([/] (cv x) y (A.e. x (cv y) ph)) (A.e. y (cv x) ps)) (ph z ax-17 z x ph hbs1 x z ph sbequ12 (cv y) cbvral x y sbbii (A.e. z (cv x) ([/] (cv z) x ph)) y ax-17 (cv y) (cv x) z ([/] (cv z) x ph) raleq1 sbie bitr ([/] (cv z) x ph) y ax-17 y z ([/] (cv z) x ph) hbs1 z y ([/] (cv z) x ph) sbequ12 (cv x) cbvral ph z ax-17 y x sbco2 ps x ax-17 sbralie.1 sbie bitr y (cv x) ralbii 3bitr)) thm (dfsbcq () () (-> (= A B) (<-> ([/] A x ph) ([/] B x ph))) (A B ({|} x ph) eleq1 A x ph df-sbc B x ph df-sbc 3bitr4g)) thm (sbceq1a () () (-> (= (cv x) A) (<-> ph ([/] A x ph))) ((cv x) A x ph dfsbcq x ph sbid syl5bbr)) thm (a4sbc ((ph y) (A y) (x y)) () (-> (e. A B) (-> (A. x ph) ([/] A x ph))) ((cv y) A x ph dfsbcq x ph y stdpc4 syl5bi B vtocleg)) thm (sbcth () ((sbcth.1 ph)) (-> (e. A B) ([/] A x ph)) (sbcth.1 x ax-gen A B x ph a4sbc mpi)) thm (sbcthdv ((ph x)) ((sbcthdv.1 (-> ph ps))) (-> (/\ ph (e. A B)) ([/] A x ps)) (sbcthdv.1 x 19.21aiv (e. A B) adantr A B x ps a4sbc ph adantl mpd)) thm (hbsbc1g ((x y) (x z) (y z) (A y) (A z) (ph z) (B y)) ((hbsbc1g.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. A B) (-> ([/] A x ph) (A. x ([/] A x ph)))) ((cv z) A x ph dfsbcq (e. (cv y) (cv z)) x ax-17 hbsbc1g.1 hbeq (cv z) A x ph dfsbcq albid z x ph hbs1 syl5bi sylbird B vtocleg)) thm (hbsbc1 ((B x) (x y) (A y) (B y)) ((hbsbc1g.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (-> (e. A B) ([/] A x ph)) (A. x (-> (e. A B) ([/] A x ph)))) (hbsbc1g.1 (e. (cv y) B) x ax-17 hbel hbsbc1g.1 B ph hbsbc1g hbim1)) thm (hbsbc1v ((A x) (x y) (A y)) ((hbsbcv.1 (e. A (V)))) (-> ([/] A x ph) (A. x ([/] A x ph))) ((e. (cv y) A) x ax-17 (V) ph hbsbc1 hbsbcv.1 ([/] A x ph) a1bi hbsbcv.1 ([/] A x ph) a1bi x albii 3imtr4)) thm (hbsbcg ((w z) (A w) (A z) (w y) (x z) (w x) (ph w)) ((hbsbcg.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (hbsbcg.2 (-> ph (A. x ph)))) (-> (e. A B) (-> ([/] A y ph) (A. x ([/] A y ph)))) ((e. (cv z) A) w ax-17 (-> ([/] A y ph) (A. x ([/] A y ph))) w ax-17 (cv w) A y ph dfsbcq (e. (cv z) (cv w)) x ax-17 hbsbcg.1 hbeq (cv w) A y ph dfsbcq albid imbi12d hbsbcg.2 w y hbsb B vtoclgf)) thm (sbccog ((x z) (A z) (y z) (ph y) (ph z)) () (-> (e. A B) (<-> ([/] A y ([/] (cv y) x ph)) ([/] A x ph))) ((cv z) A y ([/] (cv y) x ph) dfsbcq (cv z) A x ph dfsbcq ph y ax-17 z x sbco2 B vtoclbg)) thm (sbcco2 ((x y) (ph y) (A y)) ((sbcco2.1 (-> (= (cv x) (cv y)) (= A B)))) (<-> ([/] (cv x) y ([/] B x ph)) ([/] A x ph)) (([/] A x ph) y ax-17 sbcco2.1 (cv y) (cv x) eqcom B A eqcom 3imtr4 B A x ph dfsbcq syl sbie)) thm (sbc5g ((x y) (A x) (A y)) () (-> (e. A B) (<-> ([/] A x ph) (E. x (/\ (= (cv x) A) ph)))) ((e. A (V)) ph biimt (= (cv x) A) anbi2d x exbidv (e. A (V)) (E. x (/\ (= (cv x) A) ph)) biimt (e. (cv y) A) x ax-17 (V) ph hbsbc1 x A ph sbceq1a (e. A (V)) imbi2d (V) ceqsexg 3bitr3rd pm5.74rd A B elisset A B elisset sylc)) thm (sbc6g ((x y) (A x) (A y)) () (-> (e. A B) (<-> ([/] A x ph) (A. x (-> (= (cv x) A) ph)))) ((e. A (V)) ph biimt (= (cv x) A) imbi2d x albidv (e. A (V)) (A. x (-> (= (cv x) A) ph)) biimt (e. (cv y) A) x ax-17 (V) ph hbsbc1 x A ph sbceq1a (e. A (V)) imbi2d (V) ceqsalg 3bitr3rd pm5.74rd A B elisset A B elisset sylc)) thm (sbc5 ((A x)) ((sbc5.1 (e. A (V)))) (<-> ([/] A x ph) (E. x (/\ (= (cv x) A) ph))) (sbc5.1 A (V) x ph sbc5g ax-mp)) thm (sbc6 ((A x)) ((sbc6.1 (e. A (V)))) (<-> ([/] A x ph) (A. x (-> (= (cv x) A) ph))) (sbc6.1 x ph hbsbc1v sbc6.1 x A ph sbceq1a ceqsal bicomi)) thm (sbc2or ((A x)) () (\/ (<-> ([/] A x ph) (E. x (/\ (= (cv x) A) ph))) (<-> ([/] A x ph) (A. x (-> (= (cv x) A) ph)))) (A (V) x ph sbc5g (<-> ([/] A x ph) (E. x (/\ (= (cv x) A) ph))) (<-> ([/] A x ph) (A. x (-> (= (cv x) A) ph))) orc syl ([/] A x ph) (E. x (/\ (= (cv x) A) ph)) pm5.15 (-. (E. x (/\ (= (cv x) A) ph))) (A. x (-> (= (cv x) A) ph)) pm5.1 x visset (cv x) A (V) eleq1 mpbii ph adantr con3i x nexdv x visset (cv x) A (V) eleq1 mpbii con3i ph pm2.21d x 19.21aiv sylanc ([/] A x ph) bibi2d (<-> ([/] A x ph) (E. x (/\ (= (cv x) A) ph))) orbi2d mpbii pm2.61i)) thm (sbciet ((A x)) () (-> (/\/\ (e. A B) (A. x (-> ps (A. x ps))) (A. x (-> (= (cv x) A) (<-> ph ps)))) (<-> ([/] A x ph) ps)) (A B x ph sbc5g (A. x (-> ps (A. x ps))) (A. x (-> (= (cv x) A) (<-> ph ps))) 3ad2ant1 x ps (/\ (= (cv x) A) ph) 19.23g biimpa ph ps bi1 (= (cv x) A) imim2i imp3a x 19.20i sylan2 (e. A B) 3adant1 sylbid x ps (-> (= (cv x) A) ph) 19.21g biimpa ph ps bi2 (= (cv x) A) imim2i com23 x 19.20i sylan2 (e. A B) 3adant1 A B x ph sbc6g (A. x (-> ps (A. x ps))) (A. x (-> (= (cv x) A) (<-> ph ps))) 3ad2ant1 sylibrd impbid)) thm (sbciegf ((A x) (B x)) ((sbciegf.1 (-> (e. A B) (-> ps (A. x ps)))) (sbciegf.2 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> ([/] A x ph) ps)) (sbciegf.1 x 19.21aiv sbciegf.2 x ax-gen A B x ps ph sbciet 3exp mpii mpd)) thm (sbcieg ((A x) (ps x)) ((sbcieg.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> ([/] A x ph) ps)) (A B elisset ps x ax-17 (e. A (V)) a1i sbcieg.1 sbciegf syl)) thm (sbcie ((A x) (ps x)) ((sbcie.1 (e. A (V))) (sbcie.2 (-> (= (cv x) A) (<-> ph ps)))) (<-> ([/] A x ph) ps) (sbcie.1 sbcie.2 (V) sbcieg ax-mp)) thm (elrabsf ((w z) (A z) (A w) (y z) (w y) (B y) (B z) (B w) (ph z) (x y) (x z) (w x)) ((elrabsf.1 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (<-> (e. A ({e.|} x B ph)) (/\ (e. A B) ([/] A x ph))) (elrabsf.1 (e. (cv y) B) z ax-17 ph z ax-17 z x ph hbs1 x z ph sbequ12 cbvrab A eleq2i (e. (cv w) A) z ax-17 (e. (cv w) B) z ax-17 (e. (cv w) A) z ax-17 (V) ([/] (cv z) x ph) hbsbc1 z A ([/] (cv z) x ph) sbceq1a (= (cv z) A) z 19.8a A z isset sylibr (e. A (V)) ([/] A z ([/] (cv z) x ph)) biimt syl bitrd elrabf A B elisset (e. A (V)) ([/] A z ([/] (cv z) x ph)) biimt syl pm5.32i bitr4 A B z x ph sbccog pm5.32i 3bitr)) thm (elabs2 ((x y)) () (<-> (e. A ({|} x ph)) (/\ (e. A (V)) ([/] A x ph))) (x ph rabab A eleq2i (e. (cv y) (V)) x ax-17 A ph elrabsf bitr3)) thm (elabsg () () (-> (e. A B) (<-> (e. A ({|} x ph)) ([/] A x ph))) (A B elisset A x ph elabs2 baib syl)) thm (elabs () ((elabs.1 (e. A (V)))) (<-> (e. A ({|} x ph)) ([/] A x ph)) (A x ph elabs2 elabs.1 mpbiran)) thm (cbvsbcv ((x y) (A x) (A y) (ph y)) () (<-> (A.e. x A ph) (A.e. y A ([/] (cv y) x ph))) (ph y ax-17 y x ph hbs1 x y ph sbequ12 A cbvral)) thm (sbcng ((x y) (A y) (ph y)) () (-> (e. A B) (<-> ([/] A x (-. ph)) (-. ([/] A x ph)))) ((cv y) A x (-. ph) dfsbcq (cv y) A x ph dfsbcq negbid y x ph sbn B vtoclbg)) thm (sbcimg ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (-> ph ps)) (-> ([/] A x ph) ([/] A x ps)))) ((cv y) A x (-> ph ps) dfsbcq (cv y) A x ph dfsbcq (cv y) A x ps dfsbcq imbi12d y x ph ps sbim B vtoclbg)) thm (sbcang ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (/\ ph ps)) (/\ ([/] A x ph) ([/] A x ps)))) ((cv y) A x (/\ ph ps) dfsbcq (cv y) A x ph dfsbcq (cv y) A x ps dfsbcq anbi12d y x ph ps sban B vtoclbg)) thm (sbcorg ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (\/ ph ps)) (\/ ([/] A x ph) ([/] A x ps)))) ((cv y) A x (\/ ph ps) dfsbcq (cv y) A x ph dfsbcq (cv y) A x ps dfsbcq orbi12d y x ph ps sbor B vtoclbg)) thm (sbcbidig ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (<-> ph ps)) (<-> ([/] A x ph) ([/] A x ps)))) ((cv y) A x (<-> ph ps) dfsbcq (cv y) A x ph dfsbcq (cv y) A x ps dfsbcq bibi12d y x ph ps sbbi B vtoclbg)) thm (sbcalg ((x z) (A x) (A z) (x y) (y z) (ph z)) () (-> (e. A B) (<-> ([/] A y (A. x ph)) (A. x ([/] A y ph)))) ((cv z) A y (A. x ph) dfsbcq (cv z) A y ph dfsbcq x albidv z y x ph sbal B vtoclbg)) thm (sbcexg ((x z) (A x) (A z) (x y) (y z) (ph z)) () (-> (e. A B) (<-> ([/] A y (E. x ph)) (E. x ([/] A y ph)))) ((cv z) A y (E. x ph) dfsbcq (cv z) A y ph dfsbcq x exbidv z y x ph sbex B vtoclbg)) thm (sbcel1gv ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A C) (<-> ([/] A x (e. (cv x) B)) (e. A B))) (A C elisset A (V) x elex (e. (cv y) A) x ax-17 (V) (e. (cv x) B) hbsbc1 (-> (e. A (V)) (e. A B)) x ax-17 hbbi x A (e. (cv x) B) sbceq1a (cv x) A B eleq1 bitr3d (e. A (V)) imbi2d 19.23ai pm5.74rd mpcom syl)) thm (sbcel2gv ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B C) (<-> ([/] B x (e. A (cv x))) (e. A B))) (B C elisset B (V) x elex (e. (cv y) B) x ax-17 (V) (e. A (cv x)) hbsbc1 (-> (e. B (V)) (e. A B)) x ax-17 hbbi x B (e. A (cv x)) sbceq1a (cv x) B A eleq2 bitr3d (e. B (V)) imbi2d 19.23ai pm5.74rd mpcom syl)) thm (sbcbid () ((sbcbid.1 (-> ph (A. x ph))) (sbcbid.2 (-> ph (<-> ps ch)))) (-> (/\ ph (e. A B)) (<-> ([/] A x ps) ([/] A x ch))) (A B x (<-> ps ch) a4sbc sbcbid.1 sbcbid.2 19.21ai syl5 A B x ps ch sbcbidig sylibd impcom)) thm (sbcbidv ((ph x)) ((sbcbidv.1 (-> ph (<-> ps ch)))) (-> (/\ ph (e. A B)) (<-> ([/] A x ps) ([/] A x ch))) (A B x (<-> ps ch) a4sbc sbcbidv.1 x 19.21aiv syl5 A B x ps ch sbcbidig sylibd impcom)) thm (sbcbii () ((sbcbii.1 (<-> ph ps))) (-> (e. A B) (<-> ([/] A x ph) ([/] A x ps))) ((V) eqid sbcbii.1 (= (V) (V)) a1i A B x sbcbidv mpan)) thm (hbsbc1gd ((y z) (A y) (A z) (ph y) (x y) (x z)) ((hbsbc1gd.1 (-> ph (A. x ph))) (hbsbc1gd.2 (-> ph (-> (e. (cv y) A) (A. x (e. (cv y) A)))))) (-> (/\ ph (e. A B)) (-> ([/] A x ps) (A. x ([/] A x ps)))) (x (e. (cv y) A) ax-4 ph a1i hbsbc1gd.2 impbid y abbidv (cv y) (cv z) A eleq1 x albidv cbvabv y A abid2 3eqtr3g (V) eleq1d biimpar x (e. (cv z) A) hba1 y z hbab (V) ps hbsbc1g syl hbsbc1gd.2 y 19.21aiv y A x z abidhb ({|} z (A. x (e. (cv z) A))) A x ps dfsbcq 3syl (e. A (V)) adantr hbsbc1gd.1 ph a1d hbsbc1gd.1 hbsbc1gd.2 (e. (cv y) (V)) x ax-17 ph a1i hbeld hband anabsi5 hbsbc1gd.2 y 19.21aiv y A x z abidhb ({|} z (A. x (e. (cv z) A))) A x ps dfsbcq 3syl (e. A (V)) adantr albid 3imtr3d A B elisset sylan2)) thm (hbsbcgd ((w z) (A w) (A z) (ph z) (w x) (x z)) ((hbsbcgd.1 (-> ph (A. x ph))) (hbsbcgd.2 (-> ph (A. y ph))) (hbsbcgd.3 (-> ph (-> (e. (cv z) A) (A. x (e. (cv z) A))))) (hbsbcgd.4 (-> ph (-> ps (A. x ps))))) (-> (/\ ph (e. A C)) (-> ([/] A y ps) (A. x ([/] A y ps)))) (x (e. (cv z) A) ax-4 ph a1i hbsbcgd.3 impbid z abbidv (cv z) (cv w) A eleq1 x albidv cbvabv z A abid2 3eqtr3g (V) eleq1d biimpar x (e. (cv w) A) hba1 z w hbab x ps hba1 (V) y hbsbcg syl hbsbcgd.3 z 19.21aiv z A x w abidhb ({|} w (A. x (e. (cv w) A))) A y (A. x ps) dfsbcq 3syl (e. A (V)) adantr hbsbcgd.2 x ps ax-4 ph a1i hbsbcgd.4 impbid A (V) sbcbid bitrd hbsbcgd.1 ph a1d hbsbcgd.1 hbsbcgd.3 (e. (cv z) (V)) x ax-17 ph a1i hbeld hband anabsi5 hbsbcgd.3 z 19.21aiv z A x w abidhb ({|} w (A. x (e. (cv w) A))) A y (A. x ps) dfsbcq 3syl (e. A (V)) adantr hbsbcgd.2 x ps ax-4 ph a1i hbsbcgd.4 impbid A (V) sbcbid bitrd albid 3imtr3d A C elisset sylan2)) thm (sbc19.20dv ((ph x)) ((sbc19.20dv.1 (-> ph (-> ps ch)))) (-> (/\ ph (e. A B)) (-> ([/] A x ps) ([/] A x ch))) (A B x (-> ps ch) a4sbc sbc19.20dv.1 x 19.21aiv syl5 A B x ps ch sbcimg sylibd impcom)) thm (sbcgf ((A y) (ph y) (x y)) ((sbcgf.1 (-> ph (A. x ph)))) (-> (e. A B) (<-> ([/] A x ph) ph)) (A B y x ph sbccog sbcgf.1 y sbf A B y sbcbii A B y ph sbc5g A B y elex ph biantrurd y (= (cv y) A) ph 19.41v syl6rbbr 3bitrd bitr3d)) thm (sbc19.21g () ((sbc19.21g.1 (-> ph (A. x ph)))) (-> (e. A B) (<-> ([/] A x (-> ph ps)) (-> ph ([/] A x ps)))) (A B x ph ps sbcimg sbc19.21g.1 A B sbcgf ([/] A x ps) imbi1d bitrd)) thm (sbccomglem ((x y) (A x) (A y) (B x) (B y) (C y) (D x)) () (-> (/\ (e. A C) (e. B D)) (<-> ([/] A x ([/] B y ph)) ([/] B y ([/] A x ph)))) (B D y ph sbc5g A C x sbcbidv ancoms A C x (E. y (/\ (= (cv y) B) ph)) sbc5g (e. B D) adantr A C x ph sbc5g B D y sbcbidv B D y (E. x (/\ (= (cv x) A) ph)) sbc5g (e. A C) adantl bitr2d x y (/\ (= (cv x) A) (/\ (= (cv y) B) ph)) excom x y (= (cv x) A) (/\ (= (cv y) B) ph) exdistr (= (cv x) A) (= (cv y) B) ph an12 x exbii x (= (cv y) B) (/\ (= (cv x) A) ph) 19.42v bitr y exbii 3bitr3 syl5bb 3bitrd)) thm (sbccomg ((w y) (w z) (A w) (y z) (A y) (A z) (w x) (B w) (x z) (B x) (B z) (ph w) (ph z) (x y)) () (-> (/\ (e. A C) (e. B D)) (<-> ([/] A x ([/] B y ph)) ([/] B y ([/] A x ph)))) (A (V) z x ([/] B w ([/] (cv w) y ph)) sbccog (e. B (V)) adantr z visset B (V) (cv z) (V) w x ([/] (cv w) y ph) sbccomglem mpan2 z visset w visset (cv z) (V) (cv w) (V) x y ph sbccomglem mp2an B (V) w sbcbii bitr3d A (V) z sbcbidv ancoms A (V) B (V) z w ([/] (cv w) y ([/] (cv z) x ph)) sbccomglem w visset A (V) (cv w) (V) z y ([/] (cv z) x ph) sbccomglem mpan2 B (V) w sbcbidv 3bitrd B (V) w y ([/] A z ([/] (cv z) x ph)) sbccog (e. A (V)) adantl A (V) z x ph sbccog B (V) y sbcbidv 3bitrd (e. B (V)) ([/] B w ([/] (cv w) y ph)) pm4.2i B (V) w y ph sbccog bitrd A (V) x sbcbidv ancoms 3bitr3rd A C elisset B D elisset syl2an)) thm (sbcral ((y z) (A y) (A z) (x z) (B x) (B z) (ph z) (x y)) () (-> (e. A C) (<-> ([/] A x (A.e. y B ph)) (A.e. y B ([/] A x ph)))) (A C elisset A (V) z (A.e. y B ([/] (cv z) x ph)) sbc6g y B z (-> (= (cv z) A) ([/] (cv z) x ph)) ralcom4 y B (= (cv z) A) ([/] (cv z) x ph) r19.21v z albii bitr2 syl6bb z visset (cv z) (V) x (A.e. y B ph) sbc6g y B x (-> (= (cv x) (cv z)) ph) ralcom4 y B (= (cv x) (cv z)) ph r19.21v x albii bitr2 syl6bb (cv z) (V) x ph sbc6g y B ralbidv bitr4d ax-mp A (V) z sbcbii A (V) z ([/] (cv z) x ph) sbc6g y B ralbidv 3bitr4d A (V) z x (A.e. y B ph) sbccog A (V) z x ph sbccog y B ralbidv 3bitr3d syl)) thm (sbcrex ((A y) (B x) (x y)) () (-> (e. A C) (<-> ([/] A x (E.e. y B ph)) (E.e. y B ([/] A x ph)))) (A C elisset y B ph dfrex2 A (V) x sbcbii A (V) x (A.e. y B (-. ph)) sbcng A (V) x y B (-. ph) sbcral A (V) x ph sbcng y B ralbidv bitrd negbid 3bitrd y B ([/] A x ph) dfrex2 syl6bbr syl)) thm (sbcabel ((w y) (A y) (A w) (w z) (B w) (B z) (ph w) (x y) (w x) (x z)) ((sbcabel.1 (-> (e. (cv z) B) (A. x (e. (cv z) B))))) (-> (e. A C) (<-> ([/] A x (e. ({|} y ph) B)) (e. ({|} y ([/] A x ph)) B))) (A C elisset ({|} y ph) B w df-clel A (V) x sbcbii A (V) x w (/\ (= (cv w) ({|} y ph)) (e. (cv w) B)) sbcexg A (V) x (= (cv w) ({|} y ph)) (e. (cv w) B) sbcang (cv w) y ph abeq2 A (V) x sbcbii A (V) x y (<-> (e. (cv y) (cv w)) ph) sbcalg A (V) x (e. (cv y) (cv w)) ph sbcbidig (e. (cv y) (cv w)) x ax-17 A (V) sbcgf ([/] A x ph) bibi1d bitrd y albidv 3bitrd (cv w) y ([/] A x ph) abeq2 syl6bbr (e. (cv z) (cv w)) x ax-17 sbcabel.1 hbel A (V) sbcgf anbi12d bitrd w exbidv 3bitrd ({|} y ([/] A x ph)) B w df-clel syl6bbr syl)) thm (ra4sbcf ((A y) (x y) (B x) (B y)) ((ra4sbcf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. A B) (-> (A.e. x B ph) ([/] A x ph))) (x B ph df-ral ra4sbcf.1 ra4sbcf.1 B ph hbsbc1 (cv x) A B eleq1 x A ph sbceq1a imbi12d B cla4gf pm2.43b sylbi com12)) thm (ra4sbca ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A B) (A.e. x B ph)) ([/] A x ph)) ((e. (cv y) A) x ax-17 B ph ra4sbcf imp)) thm (ra4sbcgf ((y z) (A y) (A z) (x y) (B x) (B y) (x z)) () (-> (/\ (A. y (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (e. A B)) (-> (A.e. x B ph) ([/] A x ph))) (y (-> (e. (cv y) A) (A. x (e. (cv y) A))) hba1 y (-> (e. (cv y) A) (A. x (e. (cv y) A))) ax-4 x (e. (cv y) A) ax-4 (A. y (-> (e. (cv y) A) (A. x (e. (cv y) A)))) a1i impbid abbid y A abid2 (cv y) (cv z) A eleq1 x albidv cbvabv 3eqtr3g B eleq1d y (-> (e. (cv y) A) (A. x (e. (cv y) A))) hba1 y (-> (e. (cv y) A) (A. x (e. (cv y) A))) ax-4 x (e. (cv y) A) ax-4 (A. y (-> (e. (cv y) A) (A. x (e. (cv y) A)))) a1i impbid abbid y A abid2 (cv y) (cv z) A eleq1 x albidv cbvabv 3eqtr3g A ({|} z (A. x (e. (cv z) A))) x ph dfsbcq syl (A.e. x B ph) imbi2d x (e. (cv z) A) hba1 y z hbab B ph ra4sbcf syl5bir sylbid imp)) thm (ra5 () ((ra5.1 (-> ph (A. x ph)))) (-> (A.e. x A (-> ph ps)) (-> ph (A.e. x A ps))) (x A (-> ph ps) df-ral (e. (cv x) A) ph ps bi2.04 x albii bitr ra5.1 (-> (e. (cv x) A) ps) stdpc5 sylbi x A ps df-ral syl6ibr)) thm (csbeq1 ((x y) (A y) (B y) (C y)) () (-> (= A B) (= ([_/]_ A x C) ([_/]_ B x C))) (A B x (e. (cv y) C) dfsbcq y abbidv A x C y df-csb B x C y df-csb 3eqtr4g)) thm (csbeq1d () ((csbeq1d.1 (-> ph (= A B)))) (-> ph (= ([_/]_ A x C) ([_/]_ B x C))) (csbeq1d.1 A B x C csbeq1 syl)) thm (csbid ((x y) (A y)) () (= ([_/]_ (cv x) x A) A) ((cv x) x A y df-csb x (e. (cv y) A) sbid y abbii y A abid2 3eqtr)) thm (csbeq1a () () (-> (= (cv x) A) (= B ([_/]_ A x B))) ((cv x) A x B csbeq1 x B csbid syl5eqr)) thm (csbcog ((A z) (y z) (B y) (B z) (C z) (x z)) () (-> (e. A C) (= ([_/]_ A y ([_/]_ (cv y) x B)) ([_/]_ A x B))) ((cv y) x B z df-csb abeq2i A C y sbcbii A C y x (e. (cv z) B) sbccog bitrd z abbidv A y ([_/]_ (cv y) x B) z df-csb A x B z df-csb 3eqtr4g)) thm (csbexg ((A y) (B y) (x y)) () (-> (/\ (e. A C) (A. x (e. B D))) (e. ([_/]_ A x B) (V))) (A C x (e. ({|} y (e. (cv y) B)) (V)) a4sbc B D elisset y B abid2 syl5eqel x 19.20i syl5 imp (e. (cv y) (V)) x ax-17 A C y (e. (cv y) B) sbcabel (A. x (e. B D)) adantr mpbid A x B y df-csb syl5eqel)) thm (csbconstgf ((A y) (B y) (C y) (x y)) ((csbconstgf.1 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. A C) (= ([_/]_ A x B) B)) (csbconstgf.1 A C sbcgf y abbidv y B abid2 syl6eq A x B y df-csb syl5eq)) thm (sbcel12g ((x y) (x z) (y z) (A y) (A z) (B y) (B z) (C y) (C z)) () (-> (e. A D) (<-> ([/] A x (e. B C)) (e. ([_/]_ A x B) ([_/]_ A x C)))) (A D elisset A (V) x z (/\ (= (cv z) B) (e. (cv z) C)) sbcexg B C z df-clel A (V) x sbcbii (cv z) B y dfcleq A (V) x sbcbii A (V) x y (<-> (e. (cv y) (cv z)) (e. (cv y) B)) sbcalg A (V) x (e. (cv y) (cv z)) (e. (cv y) B) sbcbidig (e. (cv y) (cv z)) x ax-17 A (V) sbcgf ([/] A x (e. (cv y) B)) bibi1d bitrd y albidv 3bitrd (cv z) y ([/] A x (e. (cv y) B)) abeq2 syl6rbbr (cv y) (cv z) C eleq1 A (V) x sbcbidv expcom y 19.21aiv z visset (cv z) (V) y ([/] A x (e. (cv y) C)) ([/] A x (e. (cv z) C)) elabt mpan syl anbi12d A (V) x (= (cv z) B) (e. (cv z) C) sbcang bitr4d z exbidv 3bitr4d ({|} y ([/] A x (e. (cv y) B))) ({|} y ([/] A x (e. (cv y) C))) z df-clel syl6bbr syl A x B y df-csb A x C y df-csb eleq12i syl6bbr)) thm (sbceqdig ((x y) (x z) (y z) (A y) (A z) (B y) (B z) (C y) (C z)) () (-> (e. A D) (<-> ([/] A x (= B C)) (= ([_/]_ A x B) ([_/]_ A x C)))) (A D elisset A (V) x z (<-> (e. (cv z) B) (e. (cv z) C)) sbcalg B C z dfcleq A (V) x sbcbii (cv y) (cv z) B eleq1 A (V) x sbcbidv expcom y 19.21aiv z visset (cv z) (V) y ([/] A x (e. (cv y) B)) ([/] A x (e. (cv z) B)) elabt mpan syl (cv y) (cv z) C eleq1 A (V) x sbcbidv expcom y 19.21aiv z visset (cv z) (V) y ([/] A x (e. (cv y) C)) ([/] A x (e. (cv z) C)) elabt mpan syl bibi12d A (V) x (e. (cv z) B) (e. (cv z) C) sbcbidig bitr4d z albidv 3bitr4d ({|} y ([/] A x (e. (cv y) B))) ({|} y ([/] A x (e. (cv y) C))) z dfcleq syl6bbr syl A x B y df-csb A x C y df-csb eqeq12i syl6bbr)) thm (sbcel1g ((A y) (x y) (C x) (C y) (D y)) () (-> (e. A D) (<-> ([/] A x (e. B C)) (e. ([_/]_ A x B) C))) (A D x B C sbcel12g (e. (cv y) C) x ax-17 A D csbconstgf ([_/]_ A x B) eleq2d bitrd)) thm (sbceq1dig ((A y) (x y) (C x) (C y) (D y)) () (-> (e. A D) (<-> ([/] A x (= B C)) (= ([_/]_ A x B) C))) (A D x B C sbceqdig (e. (cv y) C) x ax-17 A D csbconstgf ([_/]_ A x B) eqeq2d bitrd)) thm (sbcel2g ((A y) (x y) (B x) (B y) (D y)) () (-> (e. A D) (<-> ([/] A x (e. B C)) (e. B ([_/]_ A x C)))) (A D x B C sbcel12g (e. (cv y) B) x ax-17 A D csbconstgf ([_/]_ A x C) eleq1d bitrd)) thm (sbceq2dig ((A y) (x y) (B x) (B y) (D y)) () (-> (e. A D) (<-> ([/] A x (= B C)) (= B ([_/]_ A x C)))) (A D x B C sbceqdig (e. (cv y) B) x ax-17 A D csbconstgf ([_/]_ A x C) eqeq1d bitrd)) thm (csbcomg ((y z) (A y) (A z) (x z) (B x) (B z) (C z) (x y)) () (-> (/\ (e. A R) (e. B S)) (= ([_/]_ A x ([_/]_ B y C)) ([_/]_ B y ([_/]_ A x C)))) (A (V) B (V) x y (e. (cv z) C) sbccomg B (V) y (cv z) C sbcel2g A (V) x sbcbidv ancoms A (V) x (cv z) C sbcel2g B (V) y sbcbidv 3bitr3d A (V) x (cv z) ([_/]_ B y C) sbcel2g (e. B (V)) adantr B (V) y (cv z) ([_/]_ A x C) sbcel2g (e. A (V)) adantl 3bitr3d eqrdv A R elisset B S elisset syl2an)) thm (csbeq2d () ((csbeq2d.1 (-> ph (A. x ph))) (csbeq2d.2 (-> ph (= B C)))) (-> (/\ ph (e. A D)) (= ([_/]_ A x B) ([_/]_ A x C))) (A D x (= B C) a4sbc csbeq2d.1 csbeq2d.2 19.21ai syl5 A D x B C sbceqdig sylibd impcom)) thm (csbeq2dv ((ph x)) ((csbeq2dv.1 (-> ph (= B C)))) (-> (/\ ph (e. A D)) (= ([_/]_ A x B) ([_/]_ A x C))) (ph x ax-17 csbeq2dv.1 A D csbeq2d)) thm (csbeq2i () ((csbeq2i.1 (= B C))) (-> (e. A D) (= ([_/]_ A x B) ([_/]_ A x C))) ((V) eqid csbeq2i.1 (= (V) (V)) a1i A D x csbeq2dv mpan)) thm (csbvarg ((y z) (A y) (A z) (x y) (x z)) () (-> (e. A B) (= ([_/]_ A x (cv x)) A)) (A B elisset y visset (cv y) (V) x (cv z) sbcel2gv abbi1dv (cv y) x (cv x) z df-csb syl5eq ax-mp A (V) y csbeq2i A (V) y x (cv x) csbcog A (V) y (cv z) sbcel2gv abbi1dv A y (cv y) z df-csb syl5eq 3eqtr3d syl)) thm (sbccsbg () () (-> (e. A B) (<-> ([/] A x ph) (e. A ([_/]_ A x ({|} x ph))))) (A B x (cv x) ({|} x ph) sbcel12g x ph abid A B x sbcbii A B x csbvarg ([_/]_ A x ({|} x ph)) eleq1d 3bitr3d)) thm (hbcsb1g ((y z) (A y) (A z) (B z) (x y) (x z)) ((hbcsb1g.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. A C) (-> (e. (cv y) ([_/]_ A x B)) (A. x (e. (cv y) ([_/]_ A x B))))) (A C elisset hbcsb1g.1 (e. (cv y) (V)) x ax-17 hbel (e. A (V)) z ax-17 19.21ai hbcsb1g.1 (V) (e. (cv z) B) hbsbc1g y hbabd A x B z df-csb (cv y) eleq2i A x B z df-csb (cv y) eleq2i x albii 3imtr4g syl)) thm (hbcsb1 ((A y) (x y)) ((hbcsb1.1 (e. A (V))) (hbcsb1.2 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) ([_/]_ A x B)) (A. x (e. (cv y) ([_/]_ A x B)))) (hbcsb1.1 hbcsb1.2 (V) B hbcsb1g ax-mp)) thm (hbcsbg ((w z) (A w) (A z) (B w) (B z) (w x) (x z) (w y)) ((hbcsbg.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (hbcsbg.2 (-> (e. (cv z) B) (A. x (e. (cv z) B))))) (-> (e. A C) (-> (e. (cv z) ([_/]_ A y B)) (A. x (e. (cv z) ([_/]_ A y B))))) (A C elisset hbcsbg.1 (e. (cv z) (V)) x ax-17 hbel (e. A (V)) w ax-17 19.21ai hbcsbg.1 (e. (cv z) (cv w)) x ax-17 hbcsbg.2 hbel (V) y hbsbcg z hbabd A y B w df-csb (cv z) eleq2i A y B w df-csb (cv z) eleq2i x albii 3imtr4g syl)) thm (hbcsb1gd ((y z) (A y) (A z) (B z) (ph y) (ph z) (x y) (x z)) ((hbcsb1gd.1 (-> ph (A. x ph))) (hbcsb1gd.2 (-> ph (-> (e. (cv y) A) (A. x (e. (cv y) A)))))) (-> (/\ ph (e. A C)) (-> (e. (cv y) ([_/]_ A x B)) (A. x (e. (cv y) ([_/]_ A x B))))) (hbcsb1gd.1 ph a1d hbcsb1gd.1 hbcsb1gd.2 (e. (cv y) (V)) x ax-17 ph a1i hbeld hband anabsi5 (e. (cv z) (cv y)) x ax-17 (/\ ph (e. A (V))) a1i hbcsb1gd.1 hbcsb1gd.2 (V) (e. (cv z) B) hbsbc1gd A (V) x (cv z) B sbcel2g ph adantl hbcsb1gd.1 ph a1d hbcsb1gd.1 hbcsb1gd.2 (e. (cv y) (V)) x ax-17 ph a1i hbeld hband anabsi5 A (V) x (cv z) B sbcel2g ph adantl albid 3imtr3d hbeld A C elisset sylan2)) thm (hbcsbgd ((w z) (A w) (A z) (B w) (B z) (ph w) (ph z) (w x) (x z) (w y)) ((hbcsbgd.1 (-> ph (A. x ph))) (hbcsbgd.2 (-> ph (A. y ph))) (hbcsbgd.3 (-> ph (-> (e. (cv z) A) (A. x (e. (cv z) A))))) (hbcsbgd.4 (-> ph (-> (e. (cv z) B) (A. x (e. (cv z) B)))))) (-> (/\ ph (e. A C)) (-> (e. (cv z) ([_/]_ A y B)) (A. x (e. (cv z) ([_/]_ A y B))))) (hbcsbgd.1 ph a1d hbcsbgd.1 hbcsbgd.3 (e. (cv z) (V)) x ax-17 ph a1i hbeld hband anabsi5 (e. (cv w) (cv z)) x ax-17 (/\ ph (e. A (V))) a1i hbcsbgd.1 hbcsbgd.2 hbcsbgd.3 hbcsbgd.1 (e. (cv z) (cv w)) x ax-17 ph a1i hbcsbgd.4 hbeld (V) hbsbcgd A (V) y (cv w) B sbcel2g ph adantl hbcsbgd.1 ph a1d hbcsbgd.1 hbcsbgd.3 (e. (cv z) (V)) x ax-17 ph a1i hbeld hband anabsi5 A (V) y (cv w) B sbcel2g ph adantl albid 3imtr3d hbeld A C elisset sylan2)) thm (csbiet ((x z) (A x) (A z) (B z) (y z) (C y) (C z) (D z) (x y)) () (-> (/\/\ (e. A D) (A. x (A. y (-> (e. (cv y) C) (A. x (e. (cv y) C))))) (A. x (-> (= (cv x) A) (= B C)))) (= ([_/]_ A x B) C)) (A D x (e. (cv z) C) (e. (cv z) B) sbciet (e. A D) id z visset (cv y) (cv z) C eleq1 (cv y) (cv z) C eleq1 x albidv imbi12d cla4v x 19.20i B C (cv z) eleq2 (= (cv x) A) imim2i x 19.20i syl3an abbi1dv A x B z df-csb syl5eq)) thm (csbiegf ((x y) (A x) (A y) (C y) (D x) (D y)) ((csbiegf.1 (-> (e. A D) (-> (e. (cv y) C) (A. x (e. (cv y) C))))) (csbiegf.2 (-> (= (cv x) A) (= B C)))) (-> (e. A D) (= ([_/]_ A x B) C)) (csbiegf.1 x y 19.21aivv csbiegf.2 x ax-gen jctir A D x y C B csbiet 3expb mpdan)) thm (csbief ((x y) (A x) (A y) (C y)) ((csbief.1 (e. A (V))) (csbief.2 (-> (e. (cv y) C) (A. x (e. (cv y) C)))) (csbief.3 (-> (= (cv x) A) (= B C)))) (= ([_/]_ A x B) C) (csbief.1 csbief.2 (e. A (V)) a1i csbief.3 csbiegf ax-mp)) thm (csbnestglem ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z) (C x) (C z) (R x) (R y) (R z) (S y) (S z)) () (-> (/\ (e. A R) (A. x (e. B S))) (= ([_/]_ A x ([_/]_ B y C)) ([_/]_ ([_/]_ A x B) y C))) (A R x z ([_/]_ ([_/]_ A x B) y C) ([_/]_ B y C) csbiet (e. A R) (A. x (e. B S)) pm3.26 (e. A R) x ax-17 x (e. B S) hba1 hban A R x B S csbexg (e. A R) x ax-17 x (e. B S) hba1 hban (e. A R) y ax-17 (A. x (e. B S)) y ax-17 hban (e. (cv z) A) x ax-17 R B hbcsb1g (A. x (e. B S)) adantr (e. (cv z) C) x ax-17 (/\ (e. A R) (A. x (e. B S))) a1i (V) hbcsbgd mpdan z 19.21aiv 19.21ai x A B csbeq1a y C csbeq1d x ax-gen (/\ (e. A R) (A. x (e. B S))) a1i syl3anc)) thm (csbnestg ((w z) (A w) (A z) (B w) (B z) (w x) (C w) (x z) (C x) (C z) (w y) (x y)) () (-> (/\ (e. A R) (A. x (e. B S))) (= ([_/]_ A x ([_/]_ B y C)) ([_/]_ ([_/]_ A x B) y C))) (A (V) w x ([_/]_ B z ([_/]_ (cv z) y C)) csbcog (A. x (e. B (V))) adantr w visset (cv w) (V) x B (V) z ([_/]_ (cv z) y C) csbnestglem mpan A (V) w csbeq2dv ancoms A (V) w ([_/]_ (cv w) x B) (V) z ([_/]_ (cv z) y C) csbnestglem w visset (cv w) (V) x B (V) csbexg mpan w 19.21aiv sylan2 A (V) w x B csbcog z ([_/]_ (cv z) y C) csbeq1d (A. x (e. B (V))) adantr 3eqtrd eqtr3d x (e. B (V)) hba1 B (V) z y C csbcog x a4s A (V) csbeq2d ancoms A (V) x B (V) csbexg ([_/]_ A x B) (V) z y C csbcog syl 3eqtr3d A R elisset B S elisset x 19.20i syl2an)) thm (csbnest1g ((x y) (A x) (A y) (B y) (C y) (S y)) () (-> (/\ (e. A R) (A. x (e. B S))) (= ([_/]_ A x ([_/]_ B x C)) ([_/]_ ([_/]_ A x B) x C))) (A (V) x y ([_/]_ ([_/]_ A x B) x C) ([_/]_ B x C) csbiet (e. A (V)) (A. x (e. B S)) pm3.26 (e. A (V)) x ax-17 x (e. B S) hba1 hban A (V) x B S csbexg (e. A (V)) x ax-17 (e. (cv y) A) x ax-17 (V) B hbcsb1g (V) C hbcsb1gd syldan y 19.21aiv 19.21ai x A B csbeq1a x C csbeq1d x ax-gen (/\ (e. A (V)) (A. x (e. B S))) a1i syl3anc A R elisset sylan)) thm (sbcnestg ((ph x) (x y)) () (-> (/\ (e. A R) (A. x (e. B S))) (<-> ([/] A x ([/] B y ph)) ([/] ([_/]_ A x B) y ph))) (x (e. B (V)) hba1 B (V) y ph sbccsbg x a4s A R sbcbid ancoms A R x B ([_/]_ B y ({|} y ph)) sbcel12g (A. x (e. B (V))) adantr A R x B (V) y ({|} y ph) csbnestg ([_/]_ A x B) eleq2d A R x B (V) csbexg ([_/]_ A x B) (V) y ph sbccsbg syl bitr4d 3bitrd B S elisset x 19.20i sylan2)) thm (csbco3g ((A x) (C x) (x z) (D x) (D z) (x y)) ((csbco3g.1 (-> (= (cv x) A) (= B D)))) (-> (/\ (e. A R) (A. x (e. B S))) (= ([_/]_ A x ([_/]_ B y C)) ([_/]_ D y C))) (A R x B S y C csbnestg (e. (cv z) D) x ax-17 x z gen2 csbco3g.1 x ax-gen A R x z D B csbiet mp3an23 y C csbeq1d (A. x (e. B S)) adantr eqtrd)) thm (sbcco3g ((A x) (ph x) (x z) (C x) (C z) (x y)) ((sbcco3g.1 (-> (= (cv x) A) (= B C)))) (-> (/\ (e. A R) (A. x (e. B S))) (<-> ([/] A x ([/] B y ph)) ([/] C y ph))) (A R x B S y ph sbcnestg (e. (cv z) C) x ax-17 x z gen2 sbcco3g.1 x ax-gen A R x z C B csbiet mp3an23 ([_/]_ A x B) C y ph dfsbcq syl (A. x (e. B S)) adantr bitrd)) thm (ra4csbela ((A x) (B x) (D x)) () (-> (/\ (e. A B) (A.e. x B (e. C D))) (e. ([_/]_ A x C) D)) (A B x (e. C D) ra4sbca ex A B x C D sbcel1g sylibd imp)) thm (csbabg ((y z) (A y) (A z) (ph z) (x y) (x z)) () (-> (e. A B) (= ([_/]_ A x ({|} y ph)) ({|} y ([/] A x ph)))) (A B elisset z visset (cv z) (V) A (V) y x ph sbccomg mpan z y ph df-clab A (V) x sbcbii bitr4d z abbidv ([/] A x ph) z ax-17 z y ([/] A x ph) hbs1 y z ([/] A x ph) sbequ12 cbvab syl5eq A x ({|} y ph) z df-csb syl6reqr syl)) thm (dfin5 ((A x) (B x)) () (= (i^i A B) ({e.|} x A (e. (cv x) B))) (A B x df-in x A (e. (cv x) B) df-rab eqtr4)) thm (dfss () () (<-> (C_ A B) (= A (i^i A B))) (A B df-ss (i^i A B) A eqcom bitr)) thm (dfdif2 ((A x) (B x)) () (= (\ A B) ({e.|} x A (-. (e. (cv x) B)))) (A B x df-dif x A (-. (e. (cv x) B)) df-rab eqtr4)) thm (eldif ((A x) (B x) (C x)) () (<-> (e. A (\ B C)) (/\ (e. A B) (-. (e. A C)))) (A (\ B C) elisset A B elisset (-. (e. A C)) adantr (cv x) A B eleq1 (cv x) A C eleq1 negbid anbi12d B C x df-dif (V) elab2g pm5.21nii)) thm (dfss2 ((A x) (B x)) () (<-> (C_ A B) (A. x (-> (e. (cv x) A) (e. (cv x) B)))) (A B dfss A B x df-in A eqeq2i A x (/\ (e. (cv x) A) (e. (cv x) B)) abeq2 3bitr (e. (cv x) A) (e. (cv x) B) pm4.71 x albii bitr4)) thm (dfss3 ((A x) (B x)) () (<-> (C_ A B) (A.e. x A (e. (cv x) B))) (A B x dfss2 x A (e. (cv x) B) df-ral bitr4)) thm (dfss2f ((A y) (B y) (x y)) ((dfss2f.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (dfss2f.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (<-> (C_ A B) (A. x (-> (e. (cv x) A) (e. (cv x) B)))) (A B y dfss2 (-> (e. (cv x) A) (e. (cv x) B)) y ax-17 dfss2f.1 dfss2f.2 hbim (cv x) (cv y) A eleq1 (cv x) (cv y) B eleq1 imbi12d cbval bitr4)) thm (dfss3f ((A y) (B y) (x y)) ((dfss2f.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (dfss2f.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (<-> (C_ A B) (A.e. x A (e. (cv x) B))) (dfss2f.1 dfss2f.2 dfss2f x A (e. (cv x) B) df-ral bitr4)) thm (hbss ((A y) (B y) (x y)) ((dfss2f.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (dfss2f.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (C_ A B) (A. x (C_ A B))) (x (-> (e. (cv x) A) (e. (cv x) B)) hba1 dfss2f.1 dfss2f.2 dfss2f dfss2f.1 dfss2f.2 dfss2f x albii 3imtr4)) thm (ssel ((A x) (B x) (C x)) () (-> (C_ A B) (-> (e. C A) (e. C B))) (A B x dfss2 biimp 19.21bi (= (cv x) C) anim2d x 19.22dv C A x df-clel C B x df-clel 3imtr4g)) thm (ssel2 () () (-> (/\ (C_ A B) (e. C A)) (e. C B)) (A B C ssel imp)) thm (sseli () ((sseli.1 (C_ A B))) (-> (e. C A) (e. C B)) (sseli.1 A B C ssel ax-mp)) thm (sselii () ((sseli.1 (C_ A B)) (sselii.2 (e. C A))) (e. C B) (sselii.2 sseli.1 C sseli ax-mp)) thm (sseld () ((sseld.1 (-> ph (C_ A B)))) (-> ph (-> (e. C A) (e. C B))) (sseld.1 A B C ssel syl)) thm (sseldd () ((sseld.1 (-> ph (C_ A B))) (sseldd.2 (-> ph (e. C A)))) (-> ph (e. C B)) (sseldd.2 sseld.1 C sseld mpd)) thm (ssriv ((A x) (B x)) ((ssriv.1 (-> (e. (cv x) A) (e. (cv x) B)))) (C_ A B) (A B x dfss2 ssriv.1 mpgbir)) thm (ssrdv ((A x) (B x) (ph x)) ((ssrdv.1 (-> ph (-> (e. (cv x) A) (e. (cv x) B))))) (-> ph (C_ A B)) (ssrdv.1 x 19.21aiv A B x dfss2 sylibr)) thm (sstr2 ((A x) (B x) (C x)) () (-> (C_ A B) (-> (C_ B C) (C_ A C))) (A B x dfss2 (e. (cv x) A) (e. (cv x) B) (e. (cv x) C) imim1 x 19.20ii B C x dfss2 A C x dfss2 3imtr4g sylbi)) thm (sstr () () (-> (/\ (C_ A B) (C_ B C)) (C_ A C)) (A B C sstr2 imp)) thm (sstri () ((sstri.1 (C_ A B)) (sstri.2 (C_ B C))) (C_ A C) (sstri.1 sstri.2 A B C sstr2 mp2)) thm (sstrd () ((sstrd.1 (-> ph (C_ A B))) (sstrd.2 (-> ph (C_ B C)))) (-> ph (C_ A C)) (A B C sstr sstrd.1 sstrd.2 sylanc)) thm (sylan9ss () ((sylan9ss.1 (-> ph (C_ A B))) (sylan9ss.2 (-> ps (C_ B C)))) (-> (/\ ph ps) (C_ A C)) (sylan9ss.1 ps adantr sylan9ss.2 ph adantl sstrd)) thm (sylan9ssr () ((sylan9ssr.1 (-> ph (C_ A B))) (sylan9ssr.2 (-> ps (C_ B C)))) (-> (/\ ps ph) (C_ A C)) (sylan9ssr.1 sylan9ssr.2 sylan9ss ancoms)) thm (eqss ((A x) (B x)) () (<-> (= A B) (/\ (C_ A B) (C_ B A))) (x (e. (cv x) A) (e. (cv x) B) albi A B x dfcleq A B x dfss2 B A x dfss2 anbi12i 3bitr4)) thm (eqssi () ((eqssi.1 (C_ A B)) (eqssi.2 (C_ B A))) (= A B) (A B eqss eqssi.1 eqssi.2 mpbir2an)) thm (eqssd () ((eqssd.1 (-> ph (C_ A B))) (eqssd.2 (-> ph (C_ B A)))) (-> ph (= A B)) (eqssd.1 eqssd.2 jca A B eqss sylibr)) thm (ssid () () (C_ A A) (A eqid A A eqss mpbi pm3.26i)) thm (ssv ((A x)) () (C_ A (V)) ((cv x) A elisset ssriv)) thm (sseq1 () () (-> (= A B) (<-> (C_ A C) (C_ B C))) (B A C sstr2 A B C sstr2 anim12i B A eqss (C_ A C) (C_ B C) bi 3imtr4 eqcoms)) thm (sseq2 () () (-> (= A B) (<-> (C_ C A) (C_ C B))) (C A B sstr2 com12 C B A sstr2 com12 anim12i A B eqss (C_ C A) (C_ C B) bi 3imtr4)) thm (sseq12 () () (-> (/\ (= A B) (= C D)) (<-> (C_ A C) (C_ B D))) (A B C sseq1 C D B sseq2 sylan9bb)) thm (sseq1i () ((sseq1i.1 (= A B))) (<-> (C_ A C) (C_ B C)) (sseq1i.1 A B C sseq1 ax-mp)) thm (sseq2i () ((sseq1i.1 (= A B))) (<-> (C_ C A) (C_ C B)) (sseq1i.1 A B C sseq2 ax-mp)) thm (sseq12i () ((sseq1i.1 (= A B)) (sseq12i.2 (= C D))) (<-> (C_ A C) (C_ B D)) (sseq1i.1 C sseq1i sseq12i.2 B sseq2i bitr)) thm (sseq1d () ((sseq1d.1 (-> ph (= A B)))) (-> ph (<-> (C_ A C) (C_ B C))) (sseq1d.1 A B C sseq1 syl)) thm (sseq2d () ((sseq1d.1 (-> ph (= A B)))) (-> ph (<-> (C_ C A) (C_ C B))) (sseq1d.1 A B C sseq2 syl)) thm (sseq12d () ((sseq1d.1 (-> ph (= A B))) (sseq12d.2 (-> ph (= C D)))) (-> ph (<-> (C_ A C) (C_ B D))) (sseq1d.1 C sseq1d sseq12d.2 B sseq2d bitrd)) thm (eqsstr () ((eqsstr.1 (= A B)) (eqsstr.2 (C_ B C))) (C_ A C) (eqsstr.2 eqsstr.1 C sseq1i mpbir)) thm (eqsstr3 () ((eqsstr3.1 (= B A)) (eqsstr3.2 (C_ B C))) (C_ A C) (eqsstr3.1 eqcomi eqsstr3.2 eqsstr)) thm (sseqtr () ((sseqtr.1 (C_ A B)) (sseqtr.2 (= B C))) (C_ A C) (sseqtr.1 sseqtr.2 A sseq2i mpbi)) thm (sseqtr4 () ((sseqtr4.1 (C_ A B)) (sseqtr4.2 (= C B))) (C_ A C) (sseqtr4.1 sseqtr4.2 eqcomi sseqtr)) thm (eqsstrd () ((eqsstrd.1 (-> ph (= A B))) (eqsstrd.2 (-> ph (C_ B C)))) (-> ph (C_ A C)) (eqsstrd.2 eqsstrd.1 C sseq1d mpbird)) thm (eqsstr3d () ((eqsstr3d.1 (-> ph (= B A))) (eqsstr3d.2 (-> ph (C_ B C)))) (-> ph (C_ A C)) (eqsstr3d.1 eqcomd eqsstr3d.2 eqsstrd)) thm (sseqtrd () ((sseqtrd.1 (-> ph (C_ A B))) (sseqtrd.2 (-> ph (= B C)))) (-> ph (C_ A C)) (sseqtrd.1 sseqtrd.2 A sseq2d mpbid)) thm (sseqtr4d () ((sseqtr4d.1 (-> ph (C_ A B))) (sseqtr4d.2 (-> ph (= C B)))) (-> ph (C_ A C)) (sseqtr4d.1 sseqtr4d.2 eqcomd sseqtrd)) thm (3sstr3 () ((3sstr3.1 (C_ A B)) (3sstr3.2 (= A C)) (3sstr3.3 (= B D))) (C_ C D) (3sstr3.1 3sstr3.2 B sseq1i 3sstr3.3 C sseq2i bitr2 mpbir)) thm (3sstr4 () ((3sstr4.1 (C_ A B)) (3sstr4.2 (= C A)) (3sstr4.3 (= D B))) (C_ C D) (3sstr4.1 3sstr4.2 eqcomi 3sstr4.3 eqcomi 3sstr3)) thm (3sstr3g () ((3sstr3g.1 (-> ph (C_ A B))) (3sstr3g.2 (= A C)) (3sstr3g.3 (= B D))) (-> ph (C_ C D)) (3sstr3g.1 3sstr3g.2 3sstr3g.3 sseq12i sylib)) thm (3sstr4g () ((3sstr4g.1 (-> ph (C_ A B))) (3sstr4g.2 (= C A)) (3sstr4g.3 (= D B))) (-> ph (C_ C D)) (3sstr4g.1 3sstr4g.2 eqcomi 3sstr4g.3 eqcomi 3sstr3g)) thm (3sstr3d () ((3sstr3d.1 (-> ph (C_ A B))) (3sstr3d.2 (-> ph (= A C))) (3sstr3d.3 (-> ph (= B D)))) (-> ph (C_ C D)) (3sstr3d.1 3sstr3d.2 3sstr3d.3 sseq12d mpbid)) thm (3sstr4d () ((3sstr4d.1 (-> ph (C_ A B))) (3sstr4d.2 (-> ph (= C A))) (3sstr4d.3 (-> ph (= D B)))) (-> ph (C_ C D)) (3sstr4d.1 3sstr4d.2 eqcomd 3sstr4d.3 eqcomd 3sstr3d)) thm (syl5ss () ((syl5ss.1 (-> ph (C_ A B))) (syl5ss.2 (= C A))) (-> ph (C_ C B)) (syl5ss.1 syl5ss.2 B sseq1i sylibr)) thm (syl5ssr () ((syl5ssr.1 (-> ph (C_ A B))) (syl5ssr.2 (= A C))) (-> ph (C_ C B)) (syl5ssr.1 syl5ssr.2 eqcomi syl5ss)) thm (syl6ss () ((syl6ss.1 (-> ph (C_ A B))) (syl6ss.2 (= B C))) (-> ph (C_ A C)) (syl6ss.1 syl6ss.2 A sseq2i sylib)) thm (syl6ssr () ((syl6ssr.1 (-> ph (C_ A B))) (syl6ssr.2 (= C B))) (-> ph (C_ A C)) (syl6ssr.1 syl6ssr.2 eqcomi syl6ss)) thm (eqimss () () (-> (= A B) (C_ A B)) (A ssid A B A sseq2 mpbii)) thm (eqimss2 () () (-> (= B A) (C_ A B)) (A B eqimss eqcoms)) thm (nss ((A x) (B x)) () (<-> (-. (C_ A B)) (E. x (/\ (e. (cv x) A) (-. (e. (cv x) B))))) (x (-> (e. (cv x) A) (e. (cv x) B)) exnal (e. (cv x) A) (e. (cv x) B) annim x exbii A B x dfss2 negbii 3bitr4r)) thm (ssralv ((A x) (B x)) () (-> (C_ A B) (-> (A.e. x B ph) (A.e. x A ph))) (A B (cv x) ssel ph imim1d r19.20dv2)) thm (ss2ab ((ph y) (ps y) (x y)) () (<-> (C_ ({|} x ph) ({|} x ps)) (A. x (-> ph ps))) (y x ph hbab1 y x ps hbab1 dfss2f x ph abid x ps abid imbi12i x albii bitr)) thm (abss ((A x)) () (<-> (C_ ({|} x ph) A) (A. x (-> ph (e. (cv x) A)))) (x A abid2 ({|} x ph) sseq2i x ph (e. (cv x) A) ss2ab bitr3)) thm (ssab ((A x)) () (<-> (C_ A ({|} x ph)) (A. x (-> (e. (cv x) A) ph))) (x A abid2 ({|} x ph) sseq1i x (e. (cv x) A) ph ss2ab bitr3)) thm (ssabral ((A x)) () (<-> (C_ A ({|} x ph)) (A.e. x A ph)) (A x ph ssab x A ph df-ral bitr4)) thm (ss2abi () ((ss2abi.1 (-> ph ps))) (C_ ({|} x ph) ({|} x ps)) (x ph ps ss2ab ss2abi.1 mpgbir)) thm (abssdv ((ph x) (A x)) ((abssdv.1 (-> ph (-> ps (e. (cv x) A))))) (-> ph (C_ ({|} x ps) A)) (abssdv.1 x 19.21aiv x ps A abss sylibr)) thm (abssi ((A x)) ((abssi.1 (-> ph (e. (cv x) A)))) (C_ ({|} x ph) A) (abssi.1 x ss2abi x A abid2 sseqtr)) thm (ss2rab () () (<-> (C_ ({e.|} x A ph) ({e.|} x A ps)) (A.e. x A (-> ph ps))) (x A ph df-rab x A ps df-rab sseq12i x (/\ (e. (cv x) A) ph) (/\ (e. (cv x) A) ps) ss2ab x A (-> ph ps) df-ral (e. (cv x) A) ph ps imdistan x albii bitr2 3bitr)) thm (rabss ((B x)) () (<-> (C_ ({e.|} x A ph) B) (A.e. x A (-> ph (e. (cv x) B)))) (x A ph df-rab B sseq1i x (/\ (e. (cv x) A) ph) B abss (e. (cv x) A) ph (e. (cv x) B) impexp x albii x A (-> ph (e. (cv x) B)) df-ral bitr4 3bitr)) thm (ssrab ((A x) (B x)) () (<-> (C_ B ({e.|} x A ph)) (/\ (C_ B A) (A.e. x B ph))) (x A ph df-rab B sseq2i B x (/\ (e. (cv x) A) ph) ssab B A x dfss3 (A.e. x B ph) anbi1i x B (e. (cv x) A) ph r19.26 x B (/\ (e. (cv x) A) ph) df-ral 3bitr2r 3bitr)) thm (ssrabdv ((A x) (B x) (ph x)) ((ssrabdv.1 (-> ph (C_ B A))) (ssrabdv.2 (-> (/\ ph (e. (cv x) B)) ps))) (-> ph (C_ B ({e.|} x A ps))) (ssrabdv.1 ssrabdv.2 r19.21aiva jca B x A ps ssrab sylibr)) thm (ss2rabdv ((ph x)) ((ss2rabdv.1 (-> (/\ ph (e. (cv x) A)) (-> ps ch)))) (-> ph (C_ ({e.|} x A ps) ({e.|} x A ch))) (ss2rabdv.1 r19.21aiva x A ps ch ss2rab sylibr)) thm (ss2rabi () ((ss2rabi.1 (-> (e. (cv x) A) (-> ph ps)))) (C_ ({e.|} x A ph) ({e.|} x A ps)) (x A ph ps ss2rab ss2rabi.1 mprgbir)) thm (rabss2 ((A x) (B x)) () (-> (C_ A B) (C_ ({e.|} x A ph) ({e.|} x B ph))) ((e. (cv x) A) (e. (cv x) B) ph pm3.45 x 19.20i x (/\ (e. (cv x) A) ph) (/\ (e. (cv x) B) ph) ss2ab sylibr A B x dfss2 x A ph df-rab x B ph df-rab sseq12i 3imtr4)) thm (ssab2 ((A x)) () (C_ ({|} x (/\ (e. (cv x) A) ph)) A) ((e. (cv x) A) ph pm3.26 abssi)) thm (ssrab2 ((A x)) () (C_ ({e.|} x A ph) A) (x A ph df-rab x A ph ssab2 eqsstr)) thm (uniiunlem ((w y) (w z) (A w) (y z) (A y) (A z) (B w) (B y) (B z) (w x) (C w) (x z) (C x) (C z) (D w) (x y)) () (-> (A.e. x A (e. B D)) (<-> (A.e. x A (e. B C)) (C_ ({|} y (E.e. x A (= (cv y) B))) C))) (x A (e. B C) hbra1 (e. (cv z) C) x ax-17 x A (e. B C) ra4 B C (cv z) eleq1a syl6 r19.23ad z 19.21aiv (A.e. x A (e. B D)) a1i x A (= (cv z) B) hbre1 (e. (cv z) C) x ax-17 hbim z hbal x (cv w) B csbeq1a eqcoms eqcomd B eqeq1d B eqid a4w1 w visset (e. (cv z) (cv w)) x ax-17 B hbcsb1 hbeleq (cv z) ([_/]_ (cv w) x B) B eqeq1 A rexbid (cv z) ([_/]_ (cv w) x B) C eleq1 imbi12d D cla4gv x A (= ([_/]_ (cv w) x B) B) ra4e syl7 exp4a com4r ([_/]_ (cv w) x B) B D eleq1 ([_/]_ (cv w) x B) B C eleq1 (e. (cv x) A) imbi2d (A. z (-> (E.e. x A (= (cv z) B)) (e. (cv z) C))) imbi2d 3imtr3d w 19.23aiv ax-mp imp3a com12 r19.20da com12 impbid z (E.e. x A (= (cv z) B)) C abss (cv z) (cv y) B eqeq1 x A rexbidv cbvabv C sseq1i bitr3 syl6bb)) thm (dfpss2 () () (<-> (C: A B) (/\ (C_ A B) (-. (= A B)))) (A B df-pss A B df-ne (C_ A B) anbi2i bitr)) thm (dfpss3 () () (<-> (C: A B) (/\ (C_ A B) (-. (C_ B A)))) (A B eqss negbii (C_ A B) anbi2i A B dfpss2 (C_ A B) (C_ B A) anclb (C_ A B) (C_ B A) iman (C_ A B) (/\ (C_ A B) (C_ B A)) iman 3bitr3 con4bii 3bitr4)) thm (psseq1 () () (-> (= A B) (<-> (C: A C) (C: B C))) (A B C sseq1 A B C neeq1 anbi12d A C df-pss B C df-pss 3bitr4g)) thm (psseq2 () () (-> (= A B) (<-> (C: C A) (C: C B))) (A B C sseq2 A B C neeq2 anbi12d C A df-pss C B df-pss 3bitr4g)) thm (psseq1i () ((psseq1i.1 (= A B))) (<-> (C: A C) (C: B C)) (psseq1i.1 A B C psseq1 ax-mp)) thm (psseq2i () ((psseq1i.1 (= A B))) (<-> (C: C A) (C: C B)) (psseq1i.1 A B C psseq2 ax-mp)) thm (psseq12i () ((psseq1i.1 (= A B)) (psseq12i.2 (= C D))) (<-> (C: A C) (C: B D)) (psseq1i.1 C psseq1i psseq12i.2 B psseq2i bitr)) thm (psseq1d () ((psseq1d.1 (-> ph (= A B)))) (-> ph (<-> (C: A C) (C: B C))) (psseq1d.1 A B C psseq1 syl)) thm (psseq2d () ((psseq1d.1 (-> ph (= A B)))) (-> ph (<-> (C: C A) (C: C B))) (psseq1d.1 A B C psseq2 syl)) thm (psseq12d () ((psseq1d.1 (-> ph (= A B))) (psseq12d.2 (-> ph (= C D)))) (-> ph (<-> (C: A C) (C: B D))) (psseq1d.1 C psseq1d psseq12d.2 B psseq2d bitrd)) thm (pssss () () (-> (C: A B) (C_ A B)) (A B df-pss pm3.26bd)) thm (pssssd () ((pssssd.1 (-> ph (C: A B)))) (-> ph (C_ A B)) (pssssd.1 A B pssss syl)) thm (sspss () () (<-> (C_ A B) (\/ (C: A B) (= A B))) (A B dfpss2 biimpr ex con1d orrd A B pssss A B eqimss jaoi impbi)) thm (pssirr () () (-. (C: A A)) ((C_ A A) pm3.24 A A dfpss3 mtbir)) thm (pssn2lp () () (-. (/\ (C: A B) (C: B A))) ((/\ (C_ A B) (C_ B A)) pm3.24 A B dfpss3 B A dfpss3 anbi12i (C_ A B) (-. (C_ B A)) (C_ B A) (-. (C_ A B)) an42 bitr (-. (C_ A B)) (-. (C_ B A)) orc (-. (C_ B A)) adantr (C_ A B) (C_ B A) ianor sylibr (/\ (C_ A B) (C_ B A)) anim2i sylbi mto)) thm (sspsstri () () (<-> (\/ (C_ A B) (C_ B A)) (\/\/ (C: A B) (= A B) (C: B A))) (A B sspss B A sspss B A eqcom (C: B A) orbi2i bitr orbi12i (C: A B) (C: B A) (= A B) orordir (C: A B) (C: B A) (= A B) or23 (C: A B) (= A B) (C: B A) df-3or bitr4 3bitr2)) thm (ssnpss () () (-> (C_ A B) (-. (C: B A))) (A B sspss A B pssn2lp (C: A B) (C: B A) imnan mpbir A pssirr A B A psseq1 mtbii jaoi sylbi)) thm (psstr () () (-> (/\ (C: A B) (C: B C)) (C: A C)) (A B pssss B C pssss sylan9ss C B pssn2lp A C B psseq1 (C: B C) anbi1d mtbiri con2i jca A C dfpss2 sylibr)) thm (sspsstr () () (-> (/\ (C_ A B) (C: B C)) (C: A C)) (A B C psstr ex A B C psseq1 biimprd jaoi imp A B sspss sylanb)) thm (psssstr () () (-> (/\ (C: A B) (C_ B C)) (C: A C)) (A B C psstr ex B C A psseq2 biimpcd jaod imp B C sspss sylan2b)) thm (difeq1 ((A x) (B x) (C x)) () (-> (= A B) (= (\ A C) (\ B C))) (A B (cv x) eleq2 (-. (e. (cv x) C)) anbi1d x abbidv A C x df-dif B C x df-dif 3eqtr4g)) thm (difeq2 ((A x) (B x) (C x)) () (-> (= A B) (= (\ C A) (\ C B))) (A B (cv x) eleq2 negbid (e. (cv x) C) anbi2d x abbidv C A x df-dif C B x df-dif 3eqtr4g)) thm (difeq1i () ((difeq1i.1 (= A B))) (= (\ A C) (\ B C)) (difeq1i.1 A B C difeq1 ax-mp)) thm (difeq2i () ((difeq1i.1 (= A B))) (= (\ C A) (\ C B)) (difeq1i.1 A B C difeq2 ax-mp)) thm (difeq12i () ((difeq1i.1 (= A B)) (difeq12i.2 (= C D))) (= (\ A C) (\ B D)) (difeq1i.1 C difeq1i difeq12i.2 B difeq2i eqtr)) thm (difeq1d () ((difeq1d.1 (-> ph (= A B)))) (-> ph (= (\ A C) (\ B C))) (difeq1d.1 A B C difeq1 syl)) thm (difeq2d () ((difeq1d.1 (-> ph (= A B)))) (-> ph (= (\ C A) (\ C B))) (difeq1d.1 A B C difeq2 syl)) thm (difeqri ((A x) (B x) (C x)) ((difeqri.1 (<-> (/\ (e. (cv x) A) (-. (e. (cv x) B))) (e. (cv x) C)))) (= (\ A B) C) (A B x df-dif difeqri.1 bicomi abbi2i eqtr4)) thm (hbdif ((A y) (B y) (x y)) ((hbdif.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbdif.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (\ A B)) (A. x (e. (cv y) (\ A B)))) (hbdif.1 hbdif.2 hbne hban (cv y) A B eldif (cv y) A B eldif x albii 3imtr4)) thm (eldifi () () (-> (e. A (\ B C)) (e. A B)) (A B C eldif pm3.26bd)) thm (eldifn () () (-> (e. A (\ B C)) (-. (e. A C))) (A B C eldif pm3.27bd)) thm (elndif () () (-> (e. A B) (-. (e. A (\ C B)))) (A C B eldifn con2i)) thm (neldif () () (-> (/\ (e. A B) (-. (e. A (\ B C)))) (e. A C)) (A B C eldif biimpr ex con1d imp)) thm (difdif ((A x) (B x)) () (= (\ A (\ B A)) A) ((cv x) B A eldif negbii (e. (cv x) B) (e. (cv x) A) iman bitr4 (e. (cv x) A) anbi2i (e. (cv x) A) (e. (cv x) B) pm4.45im bitr4 difeqri)) thm (difss ((A x) (B x)) () (C_ (\ A B) A) ((cv x) A B eldifi ssriv)) thm (ddif ((A x)) () (= (\ (V) (\ (V) A)) A) ((cv x) (V) A eldif x visset mpbiran con2bii x visset (-. (e. (cv x) (\ (V) A))) biantrur bitr2 difeqri)) thm (ssconb ((A x) (B x) (C x)) () (-> (/\ (C_ A C) (C_ B C)) (<-> (C_ A (\ C B)) (C_ B (\ C A)))) ((-> (e. (cv x) A) (e. (cv x) C)) (-> (e. (cv x) B) (e. (cv x) C)) pm5.1 A C (cv x) ssel B C (cv x) ssel syl2an (e. (cv x) A) (e. (cv x) B) bi2.03 (/\ (C_ A C) (C_ B C)) a1i anbi12d (e. (cv x) A) (e. (cv x) C) (-. (e. (cv x) B)) jcab (e. (cv x) B) (e. (cv x) C) (-. (e. (cv x) A)) jcab 3bitr4g (cv x) C B eldif (e. (cv x) A) imbi2i (cv x) C A eldif (e. (cv x) B) imbi2i 3bitr4g x albidv A (\ C B) x dfss2 B (\ C A) x dfss2 3bitr4g)) thm (sscon ((A x) (B x) (C x)) () (-> (C_ A B) (C_ (\ C B) (\ C A))) (A B (cv x) ssel con3d (e. (cv x) C) anim2d (cv x) C B eldif (cv x) C A eldif 3imtr4g ssrdv)) thm (ssdif ((A x) (B x) (C x)) () (-> (C_ A B) (C_ (\ A C) (\ B C))) (A B (cv x) ssel (-. (e. (cv x) C)) anim1d (cv x) A C eldif (cv x) B C eldif 3imtr4g ssrdv)) thm (elun ((A x) (B x) (C x)) () (<-> (e. A (u. B C)) (\/ (e. A B) (e. A C))) (A (u. B C) elisset A B elisset A C elisset jaoi (cv x) A B eleq1 (cv x) A C eleq1 orbi12d B C x df-un (V) elab2g pm5.21nii)) thm (uneqri ((A x) (B x) (C x)) ((uneqri.1 (<-> (\/ (e. (cv x) A) (e. (cv x) B)) (e. (cv x) C)))) (= (u. A B) C) ((cv x) A B elun uneqri.1 bitr eqriv)) thm (unidm ((A x)) () (= (u. A A) A) ((e. (cv x) A) oridm uneqri)) thm (uncom ((A x) (B x)) () (= (u. A B) (u. B A)) ((e. (cv x) A) (e. (cv x) B) orcom (cv x) A B elun (cv x) B A elun 3bitr4 eqriv)) thm (uneq1 ((A x) (B x) (C x)) () (-> (= A B) (= (u. A C) (u. B C))) (A B (cv x) eleq2 (e. (cv x) C) orbi1d (cv x) A C elun (cv x) B C elun 3bitr4g eqrdv)) thm (uneq2 () () (-> (= A B) (= (u. C A) (u. C B))) (A B C uneq1 C A uncom C B uncom 3eqtr4g)) thm (uneq1i () ((uneq1i.1 (= A B))) (= (u. A C) (u. B C)) (uneq1i.1 A B C uneq1 ax-mp)) thm (uneq2i () ((uneq1i.1 (= A B))) (= (u. C A) (u. C B)) (uneq1i.1 A B C uneq2 ax-mp)) thm (uneq12i () ((uneq1i.1 (= A B)) (uneq12i.2 (= C D))) (= (u. A C) (u. B D)) (uneq1i.1 C uneq1i uneq12i.2 B uneq2i eqtr)) thm (uneq1d () ((uneq1d.1 (-> ph (= A B)))) (-> ph (= (u. A C) (u. B C))) (uneq1d.1 A B C uneq1 syl)) thm (uneq2d () ((uneq1d.1 (-> ph (= A B)))) (-> ph (= (u. C A) (u. C B))) (uneq1d.1 A B C uneq2 syl)) thm (uneq12d () ((uneq1d.1 (-> ph (= A B))) (uneq12d.2 (-> ph (= C D)))) (-> ph (= (u. A C) (u. B D))) (uneq1d.1 C uneq1d uneq12d.2 B uneq2d eqtrd)) thm (uneq12 () () (-> (/\ (= A B) (= C D)) (= (u. A C) (u. B D))) (A B C uneq1 C D B uneq2 sylan9eq)) thm (hbun ((A y) (B y) (x y)) ((hbun.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbun.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (u. A B)) (A. x (e. (cv y) (u. A B)))) (hbun.1 hbun.2 hbor (cv y) A B elun (cv y) A B elun x albii 3imtr4)) thm (unass ((A x) (B x) (C x)) () (= (u. (u. A B) C) (u. A (u. B C))) ((e. (cv x) A) (e. (cv x) B) (e. (cv x) C) orass (cv x) A B elun (e. (cv x) C) orbi1i (cv x) B C elun (e. (cv x) A) orbi2i 3bitr4 (cv x) A (u. B C) elun bitr4 uneqri)) thm (un12 () () (= (u. A (u. B C)) (u. B (u. A C))) (A B uncom C uneq1i A B C unass B A C unass 3eqtr3)) thm (un23 () () (= (u. (u. A B) C) (u. (u. A C) B)) (B C uncom A uneq2i A B C unass A C B unass 3eqtr4)) thm (un4 () () (= (u. (u. A B) (u. C D)) (u. (u. A C) (u. B D))) (B C D un12 A uneq2i A B (u. C D) unass A C (u. B D) unass 3eqtr4)) thm (unundi () () (= (u. A (u. B C)) (u. (u. A B) (u. A C))) (A unidm (u. B C) uneq1i A A B C un4 eqtr3)) thm (unundir () () (= (u. (u. A B) C) (u. (u. A C) (u. B C))) (C unidm (u. A B) uneq2i A B C C un4 eqtr3)) thm (ssun1 ((A x) (B x)) () (C_ A (u. A B)) ((e. (cv x) A) (e. (cv x) B) orc (cv x) A B elun sylibr ssriv)) thm (ssun2 () () (C_ A (u. B A)) (A B ssun1 A B uncom sseqtr)) thm (ssun3 () () (-> (C_ A B) (C_ A (u. B C))) (B C ssun1 A B (u. B C) sstr2 mpi)) thm (ssun4 () () (-> (C_ A B) (C_ A (u. C B))) (B C ssun2 A B (u. C B) sstr2 mpi)) thm (elun1 () () (-> (e. A B) (e. A (u. B C))) (B C ssun1 A sseli)) thm (elun2 () () (-> (e. A B) (e. A (u. C B))) (B C ssun2 A sseli)) thm (unss1 ((A x) (B x) (C x)) () (-> (C_ A B) (C_ (u. A C) (u. B C))) ((-> (e. (cv x) A) (e. (cv x) B)) id (e. (cv x) C) orim1d (cv x) A C elun (cv x) B C elun 3imtr4g x 19.20i A B x dfss2 (u. A C) (u. B C) x dfss2 3imtr4)) thm (ssequn1 ((A x) (B x)) () (<-> (C_ A B) (= (u. A B) B)) (A B x df-un B eqeq2i (u. A B) B eqcom (e. (cv x) A) (e. (cv x) B) pm4.72 x albii A B x dfss2 B x (\/ (e. (cv x) A) (e. (cv x) B)) abeq2 3bitr4 3bitr4r)) thm (unss2 () () (-> (C_ A B) (C_ (u. C A) (u. C B))) (A B C unss1 C A uncom C B uncom 3sstr4g)) thm (unss12 () () (-> (/\ (C_ A B) (C_ C D)) (C_ (u. A C) (u. B D))) (A B C unss1 C D B unss2 sylan9ss)) thm (ssequn2 () () (<-> (C_ A B) (= (u. B A) B)) (A B ssequn1 A B uncom B eqeq1i bitr)) thm (unss ((A x) (B x) (C x)) () (<-> (/\ (C_ A C) (C_ B C)) (C_ (u. A B) C)) ((cv x) A B elun (e. (cv x) C) imbi1i (e. (cv x) A) (e. (cv x) B) (e. (cv x) C) jaob bitr x albii (u. A B) C x dfss2 A C x dfss2 B C x dfss2 anbi12i x (-> (e. (cv x) A) (e. (cv x) C)) (-> (e. (cv x) B) (e. (cv x) C)) 19.26 bitr4 3bitr4r)) thm (unssi () ((unssi.1 (C_ A C)) (unssi.2 (C_ B C))) (C_ (u. A B) C) (unssi.1 unssi.2 pm3.2i A C B unss mpbi)) thm (ssun () () (-> (\/ (C_ A B) (C_ A C)) (C_ A (u. B C))) (A B C ssun3 A C B ssun4 jaoi)) thm (elin ((A x) (B x) (C x)) () (<-> (e. A (i^i B C)) (/\ (e. A B) (e. A C))) (A (i^i B C) elisset A C elisset (e. A B) adantl (cv x) A B eleq1 (cv x) A C eleq1 anbi12d B C x df-in (V) elab2g pm5.21nii)) thm (incom ((A x) (B x)) () (= (i^i A B) (i^i B A)) ((e. (cv x) A) (e. (cv x) B) ancom (cv x) A B elin (cv x) B A elin 3bitr4 eqriv)) thm (ineqri ((A x) (B x) (C x)) ((ineqri.1 (<-> (/\ (e. (cv x) A) (e. (cv x) B)) (e. (cv x) C)))) (= (i^i A B) C) ((cv x) A B elin ineqri.1 bitr eqriv)) thm (ineq1 ((A x) (B x) (C x)) () (-> (= A B) (= (i^i A C) (i^i B C))) (A B (cv x) eleq2 (e. (cv x) C) anbi1d (cv x) A C elin (cv x) B C elin 3bitr4g eqrdv)) thm (ineq2 () () (-> (= A B) (= (i^i C A) (i^i C B))) (A B C ineq1 C A incom C B incom 3eqtr4g)) thm (ineq12 () () (-> (/\ (= A B) (= C D)) (= (i^i A C) (i^i B D))) (A B C ineq1 C D B ineq2 sylan9eq)) thm (ineq1i () ((ineq1i.1 (= A B))) (= (i^i A C) (i^i B C)) (ineq1i.1 A B C ineq1 ax-mp)) thm (ineq2i () ((ineq1i.1 (= A B))) (= (i^i C A) (i^i C B)) (ineq1i.1 A B C ineq2 ax-mp)) thm (ineq12i () ((ineq1i.1 (= A B)) (ineq12i.2 (= C D))) (= (i^i A C) (i^i B D)) (ineq1i.1 C ineq1i ineq12i.2 B ineq2i eqtr)) thm (ineq1d () ((ineq1d.1 (-> ph (= A B)))) (-> ph (= (i^i A C) (i^i B C))) (ineq1d.1 A B C ineq1 syl)) thm (ineq2d () ((ineq1d.1 (-> ph (= A B)))) (-> ph (= (i^i C A) (i^i C B))) (ineq1d.1 A B C ineq2 syl)) thm (ineq12d () ((ineq1d.1 (-> ph (= A B))) (ineq12d.2 (-> ph (= C D)))) (-> ph (= (i^i A C) (i^i B D))) (ineq1d.1 C ineq1d ineq12d.2 B ineq2d eqtrd)) thm (hbin ((A y) (B y) (x y)) ((hbin.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbin.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (i^i A B)) (A. x (e. (cv y) (i^i A B)))) (hbin.1 hbin.2 hban (cv y) A B elin (cv y) A B elin x albii 3imtr4)) thm (rabbirdv ((ph x) (A x) (B x)) ((rabbirdv.1 (-> ph (-> (e. (cv x) B) (<-> (e. (cv x) A) ch))))) (-> ph (= (i^i B A) ({e.|} x B ch))) (rabbirdv.1 pm5.32d (cv x) B A elin syl5bb abbi2dv x B ch df-rab syl6eqr)) thm (inidm ((A x)) () (= (i^i A A) A) ((e. (cv x) A) anidm ineqri)) thm (inass ((A x) (B x) (C x)) () (= (i^i (i^i A B) C) (i^i A (i^i B C))) ((e. (cv x) A) (e. (cv x) B) (e. (cv x) C) anass (cv x) B C elin (e. (cv x) A) anbi2i bitr4 (cv x) A B elin (e. (cv x) C) anbi1i (cv x) A (i^i B C) elin 3bitr4 ineqri)) thm (in12 () () (= (i^i A (i^i B C)) (i^i B (i^i A C))) (A B incom C ineq1i A B C inass B A C inass 3eqtr3)) thm (in23 () () (= (i^i (i^i A B) C) (i^i (i^i A C) B)) (B C incom A ineq2i A B C inass A C B inass 3eqtr4)) thm (in4 () () (= (i^i (i^i A B) (i^i C D)) (i^i (i^i A C) (i^i B D))) (B C D in12 A ineq2i A B (i^i C D) inass A C (i^i B D) inass 3eqtr4)) thm (inindi () () (= (i^i A (i^i B C)) (i^i (i^i A B) (i^i A C))) (A inidm (i^i B C) ineq1i A A B C in4 eqtr3)) thm (inindir () () (= (i^i (i^i A B) C) (i^i (i^i A C) (i^i B C))) (C inidm (i^i A B) ineq2i A B C C in4 eqtr3)) thm (sseqin2 () () (<-> (C_ A B) (= (i^i B A) A)) (A B df-ss A B incom A eqeq1i bitr)) thm (inss1 ((A x) (B x)) () (C_ (i^i A B) A) ((cv x) A B elin pm3.26bd ssriv)) thm (inss2 () () (C_ (i^i A B) B) (B A incom B A inss1 eqsstr3)) thm (ssin () () (<-> (/\ (C_ A B) (C_ A C)) (C_ A (i^i B C))) ((i^i A B) A (i^i A C) A ineq12 A B C inindi syl5eq A inidm syl6eq A B df-ss A C df-ss anbi12i A (i^i B C) df-ss 3imtr4 B C inss1 A (i^i B C) B sstr2 mpi B C inss2 A (i^i B C) C sstr2 mpi jca impbi)) thm (ssini () ((ssini.1 (C_ A B)) (ssini.2 (C_ A C))) (C_ A (i^i B C)) (ssini.1 ssini.2 pm3.2i A B C ssin mpbi)) thm (ssrin ((A x) (B x) (C x)) () (-> (C_ A B) (C_ (i^i A C) (i^i B C))) ((e. (cv x) A) (e. (cv x) B) (e. (cv x) C) pm3.45 (cv x) A C elin (cv x) B C elin 3imtr4g x 19.20i A B x dfss2 (i^i A C) (i^i B C) x dfss2 3imtr4)) thm (sslin () () (-> (C_ A B) (C_ (i^i C A) (i^i C B))) (A B C ssrin C A incom C B incom 3sstr4g)) thm (ss2in () () (-> (/\ (C_ A B) (C_ C D)) (C_ (i^i A C) (i^i B D))) (A B C ssrin C D B sslin sylan9ss)) thm (ssinss1 () () (-> (C_ A C) (C_ (i^i A B) C)) (A B inss1 (i^i A B) A C sstr2 ax-mp)) thm (unabs () () (= (u. A (i^i A B)) A) (A B inss1 (i^i A B) A ssequn2 mpbi)) thm (inabs () () (= (i^i A (u. A B)) A) (A B ssun1 A (u. A B) df-ss mpbi)) thm (nssinpss () () (<-> (-. (C_ A B)) (C: (i^i A B) A)) (A B inss1 (-. (C_ A (i^i A B))) biantrur A ssid (C_ A B) biantrur A A B ssin bitr negbii (i^i A B) A dfpss3 3bitr4)) thm (nsspssun () () (<-> (-. (C_ A B)) (C: B (u. A B))) (B A ssun2 (-. (C_ (u. A B) B)) biantrur B ssid (C_ A B) biantru A B B unss bitr negbii B (u. A B) dfpss3 3bitr4)) thm (dfss4 ((A x) (B x)) () (<-> (C_ A B) (= (\ B (\ B A)) A)) (A B sseqin2 (e. (cv x) B) (e. (cv x) A) abai (e. (cv x) B) (e. (cv x) A) iman (e. (cv x) B) anbi2i bitr (cv x) B A elin (cv x) B (\ B A) eldif (cv x) B A eldif negbii (e. (cv x) B) anbi2i bitr 3bitr4 eqriv A eqeq1i bitr)) thm (dfun2 ((A x) (B x)) () (= (u. A B) (\ (V) (\ (\ (V) A) B))) ((cv x) (V) A eldif x visset mpbiran (-. (e. (cv x) B)) anbi1i (cv x) (\ (V) A) B eldif (e. (cv x) A) (e. (cv x) B) ioran 3bitr4 con2bii (cv x) A B elun (cv x) (V) (\ (\ (V) A) B) eldif x visset mpbiran 3bitr4 eqriv)) thm (dfin2 ((A x) (B x)) () (= (i^i A B) (\ A (\ (V) B))) ((cv x) (V) B eldif x visset mpbiran con2bii (e. (cv x) A) anbi2i (cv x) A B elin (cv x) A (\ (V) B) eldif 3bitr4 eqriv)) thm (difin ((A x) (B x)) () (= (\ A (i^i A B)) (\ A B)) ((e. (cv x) A) (-. (e. (cv x) B)) abai (e. (cv x) A) (e. (cv x) B) imnan (e. (cv x) A) anbi2i bitr (cv x) A B eldif (cv x) A (i^i A B) eldif (cv x) A B elin negbii (e. (cv x) A) anbi2i bitr 3bitr4r eqriv)) thm (dfun3 () () (= (u. A B) (\ (V) (i^i (\ (V) A) (\ (V) B)))) (A B dfun2 (\ (V) A) (\ (V) B) dfin2 B ddif (\ (V) A) difeq2i eqtr2 (V) difeq2i eqtr)) thm (dfin3 () () (= (i^i A B) (\ (V) (u. (\ (V) A) (\ (V) B)))) ((\ A (\ (V) B)) ddif (\ (V) A) (\ (V) B) dfun2 A ddif (\ (V) B) difeq1i (V) difeq2i eqtr (V) difeq2i A B dfin2 3eqtr4r)) thm (dfin4 () () (= (i^i A B) (\ A (\ A B))) (A B inss1 (i^i A B) A dfss4 mpbi A B difin A difeq2i eqtr3)) thm (invdif () () (= (i^i A (\ (V) B)) (\ A B)) (A (\ (V) B) dfin2 B ddif A difeq2i eqtr)) thm (indif () () (= (i^i A (\ A B)) (\ A B)) (A (\ A B) dfin4 A B dfin4 A difeq2i A B difin 3eqtr2)) thm (indi ((A x) (B x) (C x)) () (= (i^i A (u. B C)) (u. (i^i A B) (i^i A C))) ((e. (cv x) A) (e. (cv x) B) (e. (cv x) C) andi (cv x) B C elun (e. (cv x) A) anbi2i (cv x) A B elin (cv x) A C elin orbi12i 3bitr4 (cv x) A (u. B C) elin (cv x) (i^i A B) (i^i A C) elun 3bitr4 eqriv)) thm (undi ((A x) (B x) (C x)) () (= (u. A (i^i B C)) (i^i (u. A B) (u. A C))) ((e. (cv x) A) (e. (cv x) B) (e. (cv x) C) ordi (cv x) B C elin (e. (cv x) A) orbi2i (cv x) A B elun (cv x) A C elun anbi12i 3bitr4 (cv x) A (i^i B C) elun (cv x) (u. A B) (u. A C) elin 3bitr4 eqriv)) thm (indir () () (= (i^i (u. A B) C) (u. (i^i A C) (i^i B C))) (C A B indi (u. A B) C incom A C incom B C incom uneq12i 3eqtr4)) thm (undir () () (= (u. (i^i A B) C) (i^i (u. A C) (u. B C))) (C A B undi (i^i A B) C uncom A C uncom B C uncom ineq12i 3eqtr4)) thm (unineq ((A x) (B x) (C x)) () (<-> (/\ (= (u. A C) (u. B C)) (= (i^i A C) (i^i B C))) (= A B)) ((e. (cv x) C) (e. (cv x) A) iba (e. (cv x) C) (e. (cv x) B) iba bibi12d (i^i A C) (i^i B C) (cv x) eleq2 (cv x) A C elin (cv x) B C elin 3bitr3g syl5bir (= (u. A C) (u. B C)) adantld (e. (cv x) C) (e. (cv x) A) biorf (e. (cv x) C) (e. (cv x) B) biorf bibi12d A C uncom B C uncom eqeq12i (u. C A) (u. C B) (cv x) eleq2 sylbi (cv x) C A elun (cv x) C B elun 3bitr3g syl5bir (= (i^i A C) (i^i B C)) adantrd pm2.61i eqrdv A B C uneq1 A B C ineq1 jca impbi)) thm (uneqin () () (<-> (= (u. A B) (i^i A B)) (= A B)) ((u. A B) (i^i A B) eqimss A (i^i A B) B unss A A B ssin A A B sstr sylbir B A B ssin (C_ B A) (C_ B B) pm3.26 sylbir anim12i sylbir A B eqss sylibr syl A B A uneq2 A B A ineq2 A inidm syl5eqr A unidm syl5eq eqtr3d impbi)) thm (difundi () () (= (\ A (u. B C)) (i^i (\ A B) (\ A C))) (B C dfun3 A difeq2i A (\ (V) B) (\ (V) C) inindi A (i^i (\ (V) B) (\ (V) C)) dfin2 A B invdif A C invdif ineq12i 3eqtr3 eqtr)) thm (difundir () () (= (\ (u. A B) C) (u. (\ A C) (\ B C))) (A B (\ (V) C) indir (u. A B) C invdif A C invdif B C invdif uneq12i 3eqtr3)) thm (difindi () () (= (\ A (i^i B C)) (u. (\ A B) (\ A C))) (B C dfin3 A difeq2i A (\ (V) B) (\ (V) C) indi A (u. (\ (V) B) (\ (V) C)) dfin2 A B invdif A C invdif uneq12i 3eqtr3 eqtr)) thm (difindir () () (= (\ (i^i A B) C) (i^i (\ A C) (\ B C))) (A B (\ (V) C) inindir (i^i A B) C invdif A C invdif B C invdif ineq12i 3eqtr3)) thm (undm () () (= (\ (V) (u. A B)) (i^i (\ (V) A) (\ (V) B))) ((V) A B difundi)) thm (indm () () (= (\ (V) (i^i A B)) (u. (\ (V) A) (\ (V) B))) ((V) A B difindi)) thm (difun1 () () (= (\ A (u. B C)) (\ (\ A B) C)) (A (\ (V) B) (\ (V) C) inass (i^i A (\ (V) B)) C invdif B C undm A ineq2i A (u. B C) invdif eqtr3 3eqtr3 A B invdif C difeq1i eqtr3)) thm (dif23 () () (= (\ (\ A B) C) (\ (\ A C) B)) (B C uncom A difeq2i A B C difun1 A C B difun1 3eqtr3)) thm (symdif1 () () (= (u. (\ A B) (\ B A)) (\ (u. A B) (i^i A B))) (A B (i^i A B) difundir A B difin A B incom B difeq2i B A difin eqtr uneq12i eqtr2)) thm (symdif2 ((A x) (B x)) () (= (u. (\ A B) (\ B A)) ({|} x (-. (<-> (e. (cv x) A) (e. (cv x) B))))) ((cv x) (\ A B) (\ B A) elun (cv x) A B eldif (e. (cv x) A) pm4.13 (-. (e. (cv x) B)) anbi1i bitr (cv x) B A eldif (e. (cv x) B) (-. (e. (cv x) A)) ancom bitr orbi12i (/\ (-. (-. (e. (cv x) A))) (-. (e. (cv x) B))) (/\ (-. (e. (cv x) A)) (e. (cv x) B)) orcom (-. (e. (cv x) A)) (e. (cv x) B) dfbi (e. (cv x) A) (e. (cv x) B) nbbn 3bitr2 3bitr abbi2i)) thm (unab ((x y) (ph y) (ps y)) () (= (u. ({|} x ph) ({|} x ps)) ({|} x (\/ ph ps))) (y x ph df-clab y x ps df-clab orbi12i y x ph ps sbor bitr4 (cv y) ({|} x ph) ({|} x ps) elun y x (\/ ph ps) df-clab 3bitr4 eqriv)) thm (inab ((x y) (ph y) (ps y)) () (= (i^i ({|} x ph) ({|} x ps)) ({|} x (/\ ph ps))) (y x ph df-clab y x ps df-clab anbi12i y x ph ps sban bitr4 (cv y) ({|} x ph) ({|} x ps) elin y x (/\ ph ps) df-clab 3bitr4 eqriv)) thm (difab ((x y) (ph y) (ps y)) () (= (\ ({|} x ph) ({|} x ps)) ({|} x (/\ ph (-. ps)))) (y x ps sbn y x (-. ps) df-clab y x ps df-clab negbii 3bitr4 y visset (-. (e. (cv y) ({|} x ps))) biantrur bitr2 difeqri ({|} x ph) ineq2i ({|} x ph) ({|} x ps) invdif x ph (-. ps) inab 3eqtr3)) thm (unrab () () (= (u. ({e.|} x A ph) ({e.|} x A ps)) ({e.|} x A (\/ ph ps))) (x (/\ (e. (cv x) A) ph) (/\ (e. (cv x) A) ps) unab (e. (cv x) A) ph ps andi x abbii eqtr4 x A ph df-rab x A ps df-rab uneq12i x A (\/ ph ps) df-rab 3eqtr4)) thm (inrab () () (= (i^i ({e.|} x A ph) ({e.|} x A ps)) ({e.|} x A (/\ ph ps))) (x (/\ (e. (cv x) A) ph) (/\ (e. (cv x) A) ps) inab (e. (cv x) A) ph ps anandi x abbii eqtr4 x A ph df-rab x A ps df-rab ineq12i x A (/\ ph ps) df-rab 3eqtr4)) thm (difrab () () (= (\ ({e.|} x A ph) ({e.|} x A ps)) ({e.|} x A (/\ ph (-. ps)))) (x (/\ (e. (cv x) A) ph) (/\ (e. (cv x) A) ps) difab (e. (cv x) A) ph (-. ps) anass (e. (cv x) A) ps pm3.27 con3i (/\ (e. (cv x) A) ph) anim2i (e. (cv x) A) ps pm3.2 ph adantr con3d imdistani impbi bitr3 x abbii eqtr4 x A ph df-rab x A ps df-rab difeq12i x A (/\ ph (-. ps)) df-rab 3eqtr4)) thm (dfrab2 ((A x)) () (= ({e.|} x A ph) (i^i ({|} x ph) A)) (x A ph df-rab x (e. (cv x) A) ph inab x A abid2 ({|} x ph) ineq1i eqtr3 A ({|} x ph) incom 3eqtr)) thm (reuss2 ((A x) (B x)) () (-> (/\ (/\ (C_ A B) (A.e. x A (-> ph ps))) (/\ (E.e. x A ph) (E!e. x B ps))) (E!e. x A ph)) ((e. (cv x) A) (e. (cv x) B) ph ps prth A B (cv x) ssel sylan exp4b com23 a2d imp4a x 19.20dv imp x A (-> ph ps) df-ral sylan2b x (/\ (e. (cv x) A) ph) (/\ (e. (cv x) B) ps) euimmo syl x (/\ (e. (cv x) A) ph) eu5 biimpr ex syl9 imp32 x A ph df-reu sylibr x A ph df-rex x B ps df-reu anbi12i sylan2b)) thm (reuss ((A x) (B x)) () (-> (/\/\ (C_ A B) (E.e. x A ph) (E!e. x B ph)) (E!e. x A ph)) ((e. (cv x) A) ph idd rgen A B x ph ph reuss2 mpanl2 3impb)) thm (reuun1 ((A x) (B x)) () (-> (/\ (E.e. x A ph) (E!e. x (u. A B) (\/ ph ps))) (E!e. x A ph)) (A B ssun1 ph ps orc (e. (cv x) A) a1i rgen A (u. A B) x ph (\/ ph ps) reuss2 mpanl12)) thm (reuun2 ((A x) (B x)) () (-> (-. (E.e. x B ph)) (<-> (E!e. x (u. A B) ph) (E!e. x A ph))) (x B ph df-rex negbii x (/\ (e. (cv x) B) ph) (/\ (e. (cv x) A) ph) euor2 sylbi x (u. A B) ph df-reu (cv x) A B elun ph anbi1i (e. (cv x) A) (e. (cv x) B) ph andir (/\ (e. (cv x) A) ph) (/\ (e. (cv x) B) ph) orcom 3bitr x eubii bitr x A ph df-reu 3bitr4g)) thm (reupick ((A x) (B x)) () (-> (/\ (/\ (C_ A B) (/\ (E.e. x A ph) (E!e. x B ph))) ph) (<-> (e. (cv x) A) (e. (cv x) B))) (A B (cv x) ssel (/\ (E.e. x A ph) (E!e. x B ph)) ph ad2antrr A B (cv x) ssel ancrd ph anim1d (e. (cv x) B) (e. (cv x) A) ph an23 syl6ib x 19.22dv x (/\ (e. (cv x) B) ph) (e. (cv x) A) eupick ex syl9 com23 imp32 x A ph df-rex x B ph df-reu anbi12i sylan2b exp3a com23 imp impbid)) thm (dfnul2 () () (= ({/}) ({|} x (-. (= (cv x) (cv x))))) (df-nul (cv x) eleq2i (cv x) (V) (V) eldif (cv x) eqid (e. (cv x) (V)) pm3.24 2th con2bii 3bitr abbi2i)) thm (dfnul3 () () (= ({/}) ({e.|} x A (-. (e. (cv x) A)))) ((e. (cv x) A) pm3.24 (cv x) eqid 2th con1bii x abbii x dfnul2 x A (-. (e. (cv x) A)) df-rab 3eqtr4)) thm (noel ((A x)) () (-. (e. A ({/}))) ((cv x) eqid x dfnul2 abeq2i con2bii mpbi (cv x) A ({/}) eleq1 mtbii (V) vtocleg A ({/}) elisset con3i pm2.61i)) thm (n0i () () (-> (e. B A) (-. (= A ({/})))) (B noel A ({/}) B eleq2 mtbiri con2i)) thm (ne0i () () (-> (e. B A) (=/= A ({/}))) (B A n0i A ({/}) df-ne sylibr)) thm (n0f ((x y) (A y)) ((nnullf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (<-> (-. (= A ({/}))) (E. x (e. (cv x) A))) (nnullf.1 (e. (cv y) ({/})) x ax-17 cleqf (cv x) noel (e. (cv x) A) nbn x albii bitr4 negbii x (e. (cv x) A) df-ex bitr4)) thm (n0 ((x y) (A x) (A y)) () (<-> (-. (= A ({/}))) (E. x (e. (cv x) A))) ((e. (cv y) A) x ax-17 n0f)) thm (ne0 ((A x)) () (<-> (=/= A ({/})) (E. x (e. (cv x) A))) (A ({/}) df-ne A x n0 bitr)) thm (abn0 ((x y) (ph y)) () (<-> (-. (= ({|} x ph) ({/}))) (E. x ph)) (({|} x ph) y n0 y x ph hbab1 (e. (cv x) ({|} x ph)) y ax-17 (cv y) (cv x) ({|} x ph) eleq1 cbvex x ph abid x exbii 3bitr)) thm (rex0 () () (-. (E.e. x ({/}) ph)) ((cv x) noel (-. ph) pm2.21i nrex)) thm (rabn0 () () (<-> (-. (= ({e.|} x A ph) ({/}))) (E.e. x A ph)) (x (/\ (e. (cv x) A) ph) abn0 x A ph df-rab ({/}) eqeq1i negbii x A ph df-rex 3bitr4)) thm (rab0 () () (= ({e.|} x ({/}) ph) ({/})) ((cv x) noel ph intnanr x nex x ({/}) ph rabn0 x ({/}) ph df-rex bitr con1bii mpbi)) thm (eq0 ((A x)) () (<-> (= A ({/})) (A. x (-. (e. (cv x) A)))) (A x n0 x (e. (cv x) A) df-ex bitr con4bii)) thm (eqv ((A x)) () (<-> (= A (V)) (A. x (e. (cv x) A))) (A (V) x dfcleq x visset (e. (cv x) A) tbt x albii bitr4)) thm (0el ((x y) (A x) (A y)) () (<-> (e. ({/}) A) (E.e. x A (A. y (-. (e. (cv y) (cv x)))))) (({/}) A x risset (cv x) y eq0 x A rexbii bitr)) thm (un0 ((A x)) () (= (u. A ({/})) A) ((cv x) noel (e. (cv x) A) biorfi bicomi uneqri)) thm (in0 ((A x)) () (= (i^i A ({/})) ({/})) ((cv x) noel (e. (cv x) A) bianfi bicomi ineqri)) thm (inv () () (= (i^i A (V)) A) (A (V) inss1 A ssid A ssv ssini eqssi)) thm (unv () () (= (u. A (V)) (V)) ((u. A (V)) ssv (V) A ssun2 eqssi)) thm (0ss ((A x)) () (C_ ({/}) A) ((cv x) noel (e. (cv x) A) pm2.21i ssriv)) thm (ss0b () () (<-> (C_ A ({/})) (= A ({/}))) (A 0ss A ({/}) eqss biimpr mpan2 A ({/}) eqimss impbi)) thm (ss0 () () (-> (C_ A ({/})) (= A ({/}))) (A ss0b biimp)) thm (un00 () () (<-> (/\ (= A ({/})) (= B ({/}))) (= (u. A B) ({/}))) (A ({/}) B ({/}) uneq12 ({/}) un0 syl6eq A B ssun1 (u. A B) ({/}) A sseq2 mpbii A ss0b sylib B A ssun2 (u. A B) ({/}) B sseq2 mpbii B ss0b sylib jca impbi)) thm (vss () () (<-> (C_ (V) A) (= A (V))) (A ssv (C_ (V) A) jctl A (V) eqss sylibr A (V) eqimss2 impbi)) thm (0pss () () (<-> (C: ({/}) A) (-. (= A ({/})))) (({/}) A dfpss2 A 0ss mpbiran ({/}) A eqcom negbii bitr)) thm (npss0 () () (-. (C: A ({/}))) (({/}) eqid A ({/}) pssss A ss0 A ({/}) ({/}) psseq1 3syl ibi ({/}) 0pss sylib mt2)) thm (pssv () () (<-> (C: A (V)) (-. (= A (V)))) (A (V) dfpss2 A ssv mpbiran)) thm (disj ((A x) (B x)) () (<-> (= (i^i A B) ({/})) (A.e. x A (-. (e. (cv x) B)))) (A B x df-in ({/}) eqeq1i x (/\ (e. (cv x) A) (e. (cv x) B)) ({/}) abeq1 (e. (cv x) A) (e. (cv x) B) imnan (cv x) noel (/\ (e. (cv x) A) (e. (cv x) B)) nbn bitr2 x albii 3bitr x A (-. (e. (cv x) B)) df-ral bitr4)) thm (disj1 ((A x) (B x)) () (<-> (= (i^i A B) ({/})) (A. x (-> (e. (cv x) A) (-. (e. (cv x) B))))) (A B x disj x A (-. (e. (cv x) B)) df-ral bitr)) thm (disj2 ((A x) (B x)) () (<-> (= (i^i A B) ({/})) (C_ A (\ (V) B))) (x visset (-. (e. (cv x) B)) biantrur (cv x) (V) B eldif bitr4 (e. (cv x) A) imbi2i x albii A B x disj1 A (\ (V) B) x dfss2 3bitr4)) thm (disj3 ((A x) (B x)) () (<-> (= (i^i A B) ({/})) (= A (\ A B))) ((e. (cv x) A) (-. (e. (cv x) B)) pm4.71 (cv x) A B eldif (e. (cv x) A) bibi2i bitr4 x albii A B x disj1 A (\ A B) x dfcleq 3bitr4)) thm (disj4 () () (<-> (= (i^i A B) ({/})) (-. (C: (\ A B) A))) (A B disj3 A (\ A B) eqcom (\ A B) A dfpss2 A B difss mpbiran con2bii 3bitr)) thm (disjpss () () (-> (/\ (= (i^i A B) ({/})) (-. (= B ({/})))) (C: A (u. A B))) ((i^i A B) ({/}) B sseq2 B ssid (C_ B A) biantru B A B ssin bitr syl5bb B ss0 syl6bi con3d imp B A nsspssun B A uncom A psseq2i bitr sylib)) thm (undisj1 () () (<-> (/\ (= (i^i A C) ({/})) (= (i^i B C) ({/}))) (= (i^i (u. A B) C) ({/}))) ((i^i A C) (i^i B C) un00 A B C indir ({/}) eqeq1i bitr4)) thm (undisj2 () () (<-> (/\ (= (i^i A B) ({/})) (= (i^i A C) ({/}))) (= (i^i A (u. B C)) ({/}))) ((i^i A B) (i^i A C) un00 A B C indi ({/}) eqeq1i bitr4)) thm (ssindif0 () () (<-> (C_ A B) (= (i^i A (\ (V) B)) ({/}))) (A (\ (V) B) disj2 B ddif A sseq2i bitr2)) thm (inelcm () () (-> (/\ (e. A B) (e. A C)) (-. (= (i^i B C) ({/})))) (A B C elin A (i^i B C) n0i sylbir)) thm (minel () () (-> (/\ (e. A B) (= (i^i C B) ({/}))) (-. (e. A C))) (A C B inelcm con2i (e. A C) (e. A B) imnan sylibr con2d impcom)) thm (undif4 ((A x) (B x) (C x)) () (-> (= (i^i A C) ({/})) (= (u. A (\ B C)) (\ (u. A B) C))) ((e. (cv x) A) (-. (e. (cv x) C)) pm2.61 (-. (e. (cv x) C)) (-. (e. (cv x) A)) ax-1 (-> (e. (cv x) A) (-. (e. (cv x) C))) a1i impbid (e. (cv x) A) (-. (e. (cv x) C)) df-or syl5bb (\/ (e. (cv x) A) (e. (cv x) B)) anbi2d (cv x) B C eldif (e. (cv x) A) orbi2i (e. (cv x) A) (e. (cv x) B) (-. (e. (cv x) C)) ordi bitr (cv x) A B elun (-. (e. (cv x) C)) anbi1i 3bitr4g (cv x) A (\ B C) elun (cv x) (u. A B) C eldif 3bitr4g x 19.20i A C x disj1 (u. A (\ B C)) (\ (u. A B) C) x dfcleq 3imtr4)) thm (disjssun ((A x) (B x) (C x)) () (-> (= (i^i A B) ({/})) (<-> (C_ A (u. B C)) (C_ A C))) (A B x disj1 x (-> (e. (cv x) A) (-. (e. (cv x) B))) ax-4 sylbi imp (e. (cv x) B) (e. (cv x) C) biorf syl (cv x) B C elun syl6rbbr ralbidva A (u. B C) x dfss3 A C x dfss3 3bitr4g)) thm (ssdif0 ((A x) (B x)) () (<-> (C_ A B) (= (\ A B) ({/}))) ((e. (cv x) A) (e. (cv x) B) iman (cv x) A B eldif negbii bitr4 x albii A B x dfss2 (\ A B) x eq0 3bitr4)) thm (vdif0 () () (<-> (= A (V)) (= (\ (V) A) ({/}))) (A vss (V) A ssdif0 bitr3)) thm (pssdifn0 () () (-> (/\ (C_ A B) (-. (= A B))) (-. (= (\ B A) ({/})))) (A B eqss biimpr ex B A ssdif0 syl5ibr con3d imp)) thm (ssnelpss () () (-> (C_ A B) (-> (/\ (e. C B) (-. (e. C A))) (C: A B))) (A B dfpss2 baibr C B A nelneq2 B A eqcom negbii sylib syl5bi)) thm (pssnel ((A x) (B x)) () (-> (C: A B) (E. x (/\ (e. (cv x) B) (-. (e. (cv x) A))))) (A B dfpss2 A B pssdifn0 sylbi (\ B A) x n0 sylib (cv x) B A eldif x exbii sylib)) thm (difin0ss ((A x) (B x) (C x)) () (-> (= (i^i (\ A B) C) ({/})) (-> (C_ C A) (C_ C B))) ((i^i (\ A B) C) x eq0 (e. (cv x) A) (e. (cv x) B) annim (e. (cv x) C) anbi2i (e. (cv x) C) (/\ (e. (cv x) A) (-. (e. (cv x) B))) ancom bitr3 negbii (e. (cv x) C) (-> (e. (cv x) A) (e. (cv x) B)) iman (cv x) (\ A B) C elin (cv x) A B eldif (e. (cv x) C) anbi1i bitr negbii 3bitr4 (e. (cv x) C) (e. (cv x) A) (e. (cv x) B) ax-2 sylbir x 19.20ii sylbi C A x dfss2 C B x dfss2 3imtr4g)) thm (inssdif0 ((A x) (B x) (C x)) () (<-> (C_ (i^i A B) C) (= (i^i A (\ B C)) ({/}))) ((e. (cv x) A) (e. (cv x) B) (e. (cv x) C) impexp (e. (cv x) B) (e. (cv x) C) iman (e. (cv x) A) imbi2i (e. (cv x) A) (/\ (e. (cv x) B) (-. (e. (cv x) C))) imnan 3bitr (cv x) A B elin (e. (cv x) C) imbi1i (cv x) A (\ B C) elin (cv x) B C eldif (e. (cv x) A) anbi2i bitr negbii 3bitr4 x albii (i^i A B) C x dfss2 (i^i A (\ B C)) x eq0 3bitr4)) thm (difid () () (= (\ A A) ({/})) (A ssid A A ssdif0 mpbi)) thm (dif0 () () (= (\ A ({/})) A) (A difid A difeq2i A A difdif eqtr3)) thm (0dif () () (= (\ ({/}) A) ({/})) (({/}) A difss (\ ({/}) A) ss0 ax-mp)) thm (difdisj () () (= (i^i A (\ B A)) ({/})) (A B inss1 A B A inssdif0 mpbi)) thm (difin0 () () (= (\ (i^i A B) B) ({/})) (A B B difindir B difid (\ A B) ineq2i (\ A B) in0 3eqtr)) thm (undifv () () (= (u. A (\ (V) A)) (V)) (A (\ (V) A) dfun3 (\ (V) A) (V) difdisj (V) difeq2i (V) dif0 3eqtr)) thm (undif1 () () (= (u. (\ A B) B) (u. A B)) (A B invdif B uneq1i A (\ (V) B) B undir (\ (V) B) B uncom B undifv eqtr (u. A B) ineq2i (u. A B) inv 3eqtr eqtr3)) thm (undif2 () () (= (u. A (\ B A)) (u. A B)) (A (\ B A) uncom B A undif1 B A uncom 3eqtr)) thm (difun2 () () (= (\ (u. A B) B) (\ A B)) (A B B difundir B difid (\ A B) uneq2i (\ A B) un0 3eqtr)) thm (ssundif () () (<-> (C_ A B) (= (u. A (\ B A)) B)) (A B ssequn1 A B undif2 B eqeq1i bitr4)) thm (difdifdir () () (= (\ (\ A B) C) (\ (\ A C) (\ B C))) (C A difdisj C (\ A C) incom eqtr3 (i^i (\ A C) (\ (V) B)) uneq2i (\ A C) B invdif (i^i (\ A C) (\ (V) B)) un0 A B C dif23 3eqtr4r (\ A C) (\ (V) B) C indi 3eqtr4 B (\ (V) C) indm B C invdif (V) difeq2i C ddif (\ (V) B) uneq2i 3eqtr3r (\ A C) ineq2i (\ A C) (\ B C) invdif 3eqtr)) thm (r19.3rzv ((A x) (ph x)) () (-> (=/= A ({/})) (<-> ph (A.e. x A ph))) (A x ne0 (E. x (e. (cv x) A)) ph biimt sylbi x A ph df-ral x (e. (cv x) A) ph 19.23v bitr syl6bbr)) thm (r19.9rzv ((A x) (ph x)) () (-> (=/= A ({/})) (<-> ph (E.e. x A ph))) (A (-. ph) x r19.3rzv bicomd con2bid x A ph dfrex2 syl6bbr)) thm (r19.28zv ((A x) (ph x)) () (-> (=/= A ({/})) (<-> (A.e. x A (/\ ph ps)) (/\ ph (A.e. x A ps)))) (A ph x r19.3rzv (A.e. x A ps) anbi1d x A ph ps r19.26 syl6rbbr)) thm (r19.45zv ((A x) (ph x)) () (-> (=/= A ({/})) (<-> (E.e. x A (\/ ph ps)) (\/ ph (E.e. x A ps)))) (A ph x r19.9rzv (E.e. x A ps) orbi1d x A ph ps r19.43 syl6rbbr)) thm (r19.27zv ((A x) (ps x)) () (-> (=/= A ({/})) (<-> (A.e. x A (/\ ph ps)) (/\ (A.e. x A ph) ps))) (A ps x r19.3rzv (A.e. x A ph) anbi2d x A ph ps r19.26 syl6rbbr)) thm (r19.36zv ((A x) (ps x)) () (-> (=/= A ({/})) (<-> (E.e. x A (-> ph ps)) (-> (A.e. x A ph) ps))) (A ps x r19.9rzv (A.e. x A ph) imbi2d x A ph ps r19.35 syl6rbbr)) thm (r19.2zOLD ((A x)) () (-> (-. (= A ({/}))) (-> (A.e. x A ph) (E.e. x A ph))) (x A ph df-ral x (e. (cv x) A) ph exintr sylbi A x n0 x A ph df-rex 3imtr4g com12)) thm (r19.3rzvOLD ((A x) (ph x)) () (-> (-. (= A ({/}))) (<-> ph (A.e. x A ph))) (A x n0 (E. x (e. (cv x) A)) ph biimt sylbi x A ph df-ral x (e. (cv x) A) ph 19.23v bitr syl6bbr)) thm (r19.9rzvOLD ((A x) (ph x)) () (-> (-. (= A ({/}))) (<-> ph (E.e. x A ph))) (A (-. ph) x r19.3rzvOLD bicomd con2bid x A ph dfrex2 syl6bbr)) thm (r19.28zvOLD ((A x) (ph x)) () (-> (-. (= A ({/}))) (<-> (A.e. x A (/\ ph ps)) (/\ ph (A.e. x A ps)))) (A ph x r19.3rzvOLD (A.e. x A ps) anbi1d x A ph ps r19.26 syl6rbbr)) thm (r19.45zvOLD ((A x) (ph x)) () (-> (-. (= A ({/}))) (<-> (E.e. x A (\/ ph ps)) (\/ ph (E.e. x A ps)))) (A ph x r19.9rzvOLD (E.e. x A ps) orbi1d x A ph ps r19.43 syl6rbbr)) thm (r19.27zvOLD ((A x) (ps x)) () (-> (-. (= A ({/}))) (<-> (A.e. x A (/\ ph ps)) (/\ (A.e. x A ph) ps))) (A ps x r19.3rzvOLD (A.e. x A ph) anbi2d x A ph ps r19.26 syl6rbbr)) thm (r19.36zvOLD ((A x) (ps x)) () (-> (-. (= A ({/}))) (<-> (E.e. x A (-> ph ps)) (-> (A.e. x A ph) ps))) (A ps x r19.9rzvOLD (A.e. x A ph) imbi2d x A ph ps r19.35 syl6rbbr)) thm (rzal ((A x)) () (-> (= A ({/})) (A.e. x A ph)) (A ({/}) (cv x) eleq2 (cv x) noel ph pm2.21i syl6bi r19.21aiv)) thm (ralidm ((A x)) () (<-> (A.e. x A (A.e. x A ph)) (A.e. x A ph)) ((A.e. x A (A.e. x A ph)) (A.e. x A ph) pm5.1 A x (A.e. x A ph) rzal A x ph rzal sylanc A x n0 (E. x (e. (cv x) A)) (A.e. x A ph) biimt x A (A.e. x A ph) df-ral x A ph hbra1 (e. (cv x) A) 19.23 bitr syl6rbbr sylbi pm2.61i)) thm (ral0 () () (A.e. x ({/}) ph) ((cv x) noel ph pm2.21i rgen)) thm (ralf0 ((A x)) ((ralf0.1 (-. ph))) (<-> (A.e. x A ph) (= A ({/}))) (ralf0.1 (e. (cv x) A) ph con3 mpi x 19.20i x A ph df-ral A x eq0 3imtr4 A x ph rzal impbi)) thm (raaan ((ph y) (ps x) (x y) (A x) (A y)) () (<-> (A.e. x A (A.e. y A (/\ ph ps))) (/\ (A.e. x A ph) (A.e. y A ps))) ((A.e. x A (A.e. y A (/\ ph ps))) (/\ (A.e. x A ph) (A.e. y A ps)) pm5.1 A x (A.e. y A (/\ ph ps)) rzal A x ph rzal A y ps rzal jca sylanc A ph y r19.3rzvOLD (A.e. y A ps) anbi1d y A ph ps r19.26 syl6rbbr x A ralbidv A (A.e. y A ps) x r19.3rzvOLD (A.e. x A ph) anbi2d x A ph (A.e. y A ps) r19.26 syl6rbbr bitrd pm2.61i)) thm (dfif2 ((ph x) (A x) (B x)) () (= (if ph A B) ({|} x (-> (-> (e. (cv x) B) ph) (/\ (e. (cv x) A) ph)))) (ph A B x df-if (/\ (e. (cv x) B) (-. ph)) (/\ (e. (cv x) A) ph) df-or (/\ (e. (cv x) A) ph) (/\ (e. (cv x) B) (-. ph)) orcom (e. (cv x) B) ph iman (/\ (e. (cv x) A) ph) imbi1i 3bitr4 x abbii eqtr)) thm (ifeq1 ((ph x) (A x) (B x) (C x)) () (-> (= A B) (= (if ph A C) (if ph B C))) (A B (cv x) eleq2 ph anbi1d (/\ (e. (cv x) C) (-. ph)) orbi1d x abbidv ph A C x df-if ph B C x df-if 3eqtr4g)) thm (ifeq2 ((ph x) (A x) (B x) (C x)) () (-> (= A B) (= (if ph C A) (if ph C B))) (A B (cv x) eleq2 (-. ph) anbi1d (/\ (e. (cv x) C) ph) orbi2d x abbidv ph C A x df-if ph C B x df-if 3eqtr4g)) thm (iftrue ((ph x) (A x) (B x)) () (-> ph (= (if ph A B) A)) (ph (e. (cv x) A) (e. (cv x) B) dedlema abbi2dv ph A B x df-if syl6reqr)) thm (iffalse ((ph x) (A x) (B x)) () (-> (-. ph) (= (if ph A B) B)) (ph (e. (cv x) B) (e. (cv x) A) dedlemb abbi2dv ph A B x df-if syl6reqr)) thm (ifeq12 () () (-> (/\ (= A B) (= C D)) (= (if ph A C) (if ph B D))) (A B ph C ifeq1 C D ph B ifeq2 sylan9eq)) thm (ifeq1d () ((ifeq1d.1 (-> ph (= A B)))) (-> ph (= (if ps A C) (if ps B C))) (ifeq1d.1 A B ps C ifeq1 syl)) thm (ifeq2d () ((ifeq1d.1 (-> ph (= A B)))) (-> ph (= (if ps C A) (if ps C B))) (ifeq1d.1 A B ps C ifeq2 syl)) thm (ifbi () () (-> (<-> ph ps) (= (if ph A B) (if ps A B))) (ph ps dfbi ph A B iftrue ps A B iftrue eqcomd sylan9eq ph A B iffalse ps A B iffalse eqcomd sylan9eq jaoi sylbi)) thm (ifbid () ((ifbid.1 (-> ph (<-> ps ch)))) (-> ph (= (if ps A B) (if ch A B))) (ifbid.1 ps ch A B ifbi syl)) thm (hbif ((x y) (x z) (y z) (A y) (A z) (B y) (B z) (ph z)) ((hbif.1 (-> ph (A. x ph))) (hbif.2 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbif.3 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (if ph A B)) (A. x (e. (cv y) (if ph A B)))) ((e. (cv y) (cv z)) x ax-17 hbif.2 hbel hbif.1 hban (e. (cv y) (cv z)) x ax-17 hbif.3 hbel hbif.1 hbne hban hbor y z hbab ph A B z df-if (cv y) eleq2i ph A B z df-if (cv y) eleq2i x albii 3imtr4)) thm (elimif () ((sbif.1 (-> (= (if ph A B) A) (<-> ps ch))) (sbif.2 (-> (= (if ph A B) B) (<-> ps th)))) (<-> ps (\/ (/\ ph ch) (/\ (-. ph) th))) (ph exmid ps biantrur ph (-. ph) ps andir ph A B iftrue sbif.1 syl pm5.32i ph A B iffalse sbif.2 syl pm5.32i orbi12i 3bitr)) thm (ifboth () ((ifboth.1 (-> (= A (if ph A B)) (<-> ps th))) (ifboth.2 (-> (= B (if ph A B)) (<-> ch th)))) (-> (/\ ps ch) th) (ph A B iftrue eqcomd ifboth.1 syl biimpa ch adantrr ph A B iffalse eqcomd ifboth.2 bicomd syl biimpar ps adantrl pm2.61ian)) thm (ifid () () (= (if ph A A) A) (ph A A iftrue ph A A iffalse pm2.61i)) thm (eqif () () (<-> (= A (if ph B C)) (\/ (/\ ph (= A B)) (/\ (-. ph) (= A C)))) ((if ph B C) B A eqeq2 (if ph B C) C A eqeq2 elimif)) thm (elif () () (<-> (e. A (if ph B C)) (\/ (/\ ph (e. A B)) (/\ (-. ph) (e. A C)))) ((if ph B C) B A eleq2 (if ph B C) C A eleq2 elimif)) thm (ifel () () (<-> (e. (if ph A B) C) (\/ (/\ ph (e. A C)) (/\ (-. ph) (e. B C)))) ((if ph A B) A C eleq1 (if ph A B) B C eleq1 elimif)) thm (ifcl () () (-> (/\ (e. A C) (e. B C)) (e. (if ph A B) C)) (A (if ph A B) C eleq1 B (if ph A B) C eleq1 ifboth)) thm (dedth () ((dedth.1 (-> (= A (if ph A B)) (<-> ps ch))) (dedth.2 ch)) (-> ph ps) (dedth.2 ph A B iftrue eqcomd dedth.1 syl mpbiri)) thm (dedth2v () ((dedth2v.1 (-> (= A (if ph A C)) (<-> ps ch))) (dedth2v.2 (-> (= B (if ph B D)) (<-> ch th))) (dedth2v.3 th)) (-> ph ps) (dedth2v.3 ph A C iftrue eqcomd dedth2v.1 syl ph B D iftrue eqcomd dedth2v.2 syl bitrd mpbiri)) thm (dedth3v () ((dedth3v.1 (-> (= A (if ph A D)) (<-> ps ch))) (dedth3v.2 (-> (= B (if ph B R)) (<-> ch th))) (dedth3v.3 (-> (= C (if ph C S)) (<-> th ta))) (dedth3v.4 ta)) (-> ph ps) (dedth3v.4 ph A D iftrue eqcomd dedth3v.1 syl ph B R iftrue eqcomd dedth3v.2 syl ph C S iftrue eqcomd dedth3v.3 syl 3bitrd mpbiri)) thm (dedth2h () ((dedth2h.1 (-> (= A (if ph A C)) (<-> ch th))) (dedth2h.2 (-> (= B (if ps B D)) (<-> th ta))) (dedth2h.3 ta)) (-> (/\ ph ps) ch) (dedth2h.1 ps imbi2d dedth2h.2 dedth2h.3 dedth dedth imp)) thm (dedth3h () ((dedth3h.1 (-> (= A (if ph A D)) (<-> th ta))) (dedth3h.2 (-> (= B (if ps B R)) (<-> ta et))) (dedth3h.3 (-> (= C (if ch C S)) (<-> et ze))) (dedth3h.4 ze)) (-> (/\/\ ph ps ch) th) (dedth3h.1 (/\ ps ch) imbi2d dedth3h.2 dedth3h.3 dedth3h.4 dedth2h dedth 3impib)) var (wff si) var (wff rh) thm (dedth4h () ((dedth4h.1 (-> (= A (if ph A R)) (<-> ta et))) (dedth4h.2 (-> (= B (if ps B S)) (<-> et ze))) (dedth4h.3 (-> (= C (if ch C F)) (<-> ze si))) (dedth4h.4 (-> (= D (if th D G)) (<-> si rh))) (dedth4h.5 rh)) (-> (/\ (/\ ph ps) (/\ ch th)) ta) (dedth4h.1 (/\ ch th) imbi2d dedth4h.2 (/\ ch th) imbi2d dedth4h.3 dedth4h.4 dedth4h.5 dedth2h dedth2h imp)) thm (elimhyp () ((elimhyp.1 (-> (= A (if ph A B)) (<-> ph ps))) (elimhyp.2 (-> (= B (if ph A B)) (<-> ch ps))) (elimhyp.3 ch)) ps (ph A B iftrue eqcomd elimhyp.1 syl ibi elimhyp.3 ph A B iffalse eqcomd elimhyp.2 syl mpbii pm2.61i)) thm (elimhyp2v () ((elimhyp2v.1 (-> (= A (if ph A C)) (<-> ph ch))) (elimhyp2v.2 (-> (= B (if ph B D)) (<-> ch th))) (elimhyp2v.3 (-> (= C (if ph A C)) (<-> ta et))) (elimhyp2v.4 (-> (= D (if ph B D)) (<-> et th))) (elimhyp2v.5 ta)) th (ph A C iftrue eqcomd elimhyp2v.1 syl ph B D iftrue eqcomd elimhyp2v.2 syl bitrd ibi elimhyp2v.5 ph A C iffalse eqcomd elimhyp2v.3 syl ph B D iffalse eqcomd elimhyp2v.4 syl bitrd mpbii pm2.61i)) thm (elimhyp3v () ((elimhyp3v.1 (-> (= A (if ph A D)) (<-> ph ch))) (elimhyp3v.2 (-> (= B (if ph B R)) (<-> ch th))) (elimhyp3v.3 (-> (= C (if ph C S)) (<-> th ta))) (elimhyp3v.4 (-> (= D (if ph A D)) (<-> et ze))) (elimhyp3v.5 (-> (= R (if ph B R)) (<-> ze si))) (elimhyp3v.6 (-> (= S (if ph C S)) (<-> si ta))) (elimhyp3v.7 et)) ta (ph A D iftrue eqcomd elimhyp3v.1 syl ph B R iftrue eqcomd elimhyp3v.2 syl ph C S iftrue eqcomd elimhyp3v.3 syl 3bitrd ibi elimhyp3v.7 ph A D iffalse eqcomd elimhyp3v.4 syl ph B R iffalse eqcomd elimhyp3v.5 syl ph C S iffalse eqcomd elimhyp3v.6 syl 3bitrd mpbii pm2.61i)) thm (elimhyp4v () ((elimhyp4v.1 (-> (= A (if ph A D)) (<-> ph ch))) (elimhyp4v.2 (-> (= B (if ph B R)) (<-> ch th))) (elimhyp4v.3 (-> (= C (if ph C S)) (<-> th ta))) (elimhyp4v.4 (-> (= F (if ph F G)) (<-> ta ps))) (elimhyp4v.5 (-> (= D (if ph A D)) (<-> et ze))) (elimhyp4v.6 (-> (= R (if ph B R)) (<-> ze si))) (elimhyp4v.7 (-> (= S (if ph C S)) (<-> si rh))) (elimhyp4v.8 (-> (= G (if ph F G)) (<-> rh ps))) (elimhyp4v.9 et)) ps (ph A D iftrue eqcomd elimhyp4v.1 syl ph B R iftrue eqcomd elimhyp4v.2 syl bitrd ph C S iftrue eqcomd elimhyp4v.3 syl ph F G iftrue eqcomd elimhyp4v.4 syl 3bitrd ibi elimhyp4v.9 ph A D iffalse eqcomd elimhyp4v.5 syl ph B R iffalse eqcomd elimhyp4v.6 syl bitrd ph C S iffalse eqcomd elimhyp4v.7 syl ph F G iffalse eqcomd elimhyp4v.8 syl 3bitrd mpbii pm2.61i)) thm (elimel () ((elimel.1 (e. B C))) (e. (if (e. A C) A B) C) (A (if (e. A C) A B) C eleq1 B (if (e. A C) A B) C eleq1 elimel.1 elimhyp)) thm (keephyp () ((keephyp.1 (-> (= A (if ph A B)) (<-> ps th))) (keephyp.2 (-> (= B (if ph A B)) (<-> ch th))) (keephyp.3 ps) (keephyp.4 ch)) th (keephyp.3 keephyp.4 keephyp.1 keephyp.2 ifboth mp2an)) thm (keephyp2v () ((keephyp2v.1 (-> (= A (if ph A C)) (<-> ps ch))) (keephyp2v.2 (-> (= B (if ph B D)) (<-> ch th))) (keephyp2v.3 (-> (= C (if ph A C)) (<-> ta et))) (keephyp2v.4 (-> (= D (if ph B D)) (<-> et th))) (keephyp2v.5 ps) (keephyp2v.6 ta)) th (keephyp2v.5 ph A C iftrue eqcomd keephyp2v.1 syl ph B D iftrue eqcomd keephyp2v.2 syl bitrd mpbii keephyp2v.6 ph A C iffalse eqcomd keephyp2v.3 syl ph B D iffalse eqcomd keephyp2v.4 syl bitrd mpbii pm2.61i)) thm (keephyp3v () ((keephyp3v.1 (-> (= A (if ph A D)) (<-> rh ch))) (keephyp3v.2 (-> (= B (if ph B R)) (<-> ch th))) (keephyp3v.3 (-> (= C (if ph C S)) (<-> th ta))) (keephyp3v.4 (-> (= D (if ph A D)) (<-> et ze))) (keephyp3v.5 (-> (= R (if ph B R)) (<-> ze si))) (keephyp3v.6 (-> (= S (if ph C S)) (<-> si ta))) (keephyp3v.7 rh) (keephyp3v.8 et)) ta (keephyp3v.7 ph A D iftrue eqcomd keephyp3v.1 syl ph B R iftrue eqcomd keephyp3v.2 syl ph C S iftrue eqcomd keephyp3v.3 syl 3bitrd mpbii keephyp3v.8 ph A D iffalse eqcomd keephyp3v.4 syl ph B R iffalse eqcomd keephyp3v.5 syl ph C S iffalse eqcomd keephyp3v.6 syl 3bitrd mpbii pm2.61i)) thm (keepel () ((keepel.1 (e. A C)) (keepel.2 (e. B C))) (e. (if ph A B) C) (A (if ph A B) C eleq1 B (if ph A B) C eleq1 keepel.1 keepel.2 keephyp)) thm (ifex () ((dedex.1 (e. A (V))) (dedex.2 (e. B (V)))) (e. (if ph A B) (V)) (dedex.1 dedex.2 ph keepel)) thm (pweq ((A x) (B x)) () (-> (= A B) (= (P~ A) (P~ B))) (A B (cv x) sseq2 x abbidv A x df-pw B x df-pw 3eqtr4g)) thm (elpw ((A x) (B x)) ((elpw.1 (e. A (V)))) (<-> (e. A (P~ B)) (C_ A B)) (elpw.1 (cv x) A (P~ B) eleq1 (cv x) A B sseq1 B x df-pw abeq2i vtoclb)) thm (elpwg ((A x) (B x)) () (-> (e. A C) (<-> (e. A (P~ B)) (C_ A B))) ((cv x) A (P~ B) eleq1 (cv x) A B sseq1 bibi12d x visset B elpw C vtoclg)) thm (hbpw ((y z) (A y) (A z) (x y) (x z)) ((hbpw.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (P~ A)) (A. x (e. (cv y) (P~ A)))) ((e. (cv z) (cv y)) x ax-17 (e. (cv y) (cv z)) x ax-17 hbpw.1 hbel hbss y visset A elpw y visset A elpw x albii 3imtr4)) thm (pwid () ((pwid.1 (e. A (V)))) (e. A (P~ A)) (A ssid pwid.1 A elpw mpbir)) thm (sneq ((A x) (B x)) () (-> (= A B) (= ({} A) ({} B))) (A B (cv x) eqeq2 x abbidv A x df-sn B x df-sn 3eqtr4g)) thm (sneqi () ((sneqi.1 (= A B))) (= ({} A) ({} B)) (sneqi.1 A B sneq ax-mp)) thm (sneqd () ((sneqd.1 (-> ph (= A B)))) (-> ph (= ({} A) ({} B))) (sneqd.1 A B sneq syl)) thm (dfsn2 () () (= ({} A) ({,} A A)) (A A df-pr ({} A) unidm eqtr2)) thm (elsn ((A x)) () (<-> (e. (cv x) ({} A)) (= (cv x) A)) (A x df-sn abeq2i)) thm (dfpr2 ((A x) (B x)) () (= ({,} A B) ({|} x (\/ (= (cv x) A) (= (cv x) B)))) (A B df-pr (cv x) ({} A) ({} B) elun x A elsn x B elsn orbi12i bitr abbi2i eqtr)) thm (elprg ((A x) (B x) (C x)) () (-> (e. A D) (<-> (e. A ({,} B C)) (\/ (= A B) (= A C)))) ((cv x) A B eqeq1 (cv x) A C eqeq1 orbi12d B C x dfpr2 D elab2g)) thm (elpr () ((elpr.1 (e. A (V)))) (<-> (e. A ({,} B C)) (\/ (= A B) (= A C))) (elpr.1 A (V) B C elprg ax-mp)) thm (elpr2 () ((elpr2.1 (e. B (V))) (elpr2.2 (e. C (V)))) (<-> (e. A ({,} B C)) (\/ (= A B) (= A C))) (A ({,} B C) B C elprg ibi elpr2.1 A B (V) eleq1 mpbiri elpr2.2 A C (V) eleq1 mpbiri jaoi A (V) B C elprg syl ibir impbi)) thm (hbpr ((A y) (B y) (x y)) ((hbpr.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hppr.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) ({,} A B)) (A. x (e. (cv y) ({,} A B)))) (hbpr.1 hbeleq hppr.2 hbeleq hbor y visset A B elpr y visset A B elpr x albii 3imtr4)) thm (elsncg () () (-> (e. A C) (<-> (e. A ({} B)) (= A B))) (A C B B elprg B dfsn2 eqcomi A eleq2i (= A B) oridm 3bitr3g)) thm (elsnc () ((elsnc.1 (e. A (V)))) (<-> (e. A ({} B)) (= A B)) (elsnc.1 A (V) B elsncg ax-mp)) thm (elsni () () (-> (e. A ({} B)) (= A B)) (A ({} B) B elsncg ibi)) thm (snidg () () (-> (e. A B) (e. A ({} A))) (A eqid A B A elsncg mpbiri)) thm (snidb () () (<-> (e. A (V)) (e. A ({} A))) (A (V) snidg A ({} A) elisset impbi)) thm (snid () ((snid.1 (e. A (V)))) (e. A ({} A)) (snid.1 A snidb mpbi)) thm (elsnc2g () () (-> (e. B C) (<-> (e. A ({} B)) (= A B))) (A B elsni (e. B C) a1i A B ({} B) eleq1 B C snidg syl5bir com12 impbid)) thm (elsnc2 () ((elsnc2.1 (e. B (V)))) (<-> (e. A ({} B)) (= A B)) (elsnc2.1 B (V) A elsnc2g ax-mp)) thm (hbsn ((A y) (x y)) ((hbsn.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) ({} A)) (A. x (e. (cv y) ({} A)))) (hbsn.1 hbsn.1 hbpr A dfsn2 (cv y) eleq2i A dfsn2 (cv y) eleq2i x albii 3imtr4)) thm (eltp () ((eltp.1 (e. A (V)))) (<-> (e. A ({,,} B C D)) (\/\/ (= A B) (= A C) (= A D))) (B C D df-tp A eleq2i A ({,} B C) ({} D) elun eltp.1 B C elpr eltp.1 D elsnc orbi12i 3bitr (= A B) (= A C) (= A D) df-3or bitr4)) thm (dftp2 ((A x) (B x) (C x)) () (= ({,,} A B C) ({|} x (\/\/ (= (cv x) A) (= (cv x) B) (= (cv x) C)))) (x visset A B C eltp abbi2i)) thm (disjsn ((A x) (B x)) () (<-> (= (i^i A ({} B)) ({/})) (-. (e. B A))) (B noel (i^i A ({} B)) ({/}) B eleq2 mtbiri B A snidg ancli B A ({} B) elin sylibr nsyl (cv x) B A eleq1 biimpcd x B elsn syl5ib con3d com12 x 19.21aiv A ({} B) x disj1 sylibr impbi)) thm (disjsn2 () () (-> (-. (= A B)) (= (i^i ({} A) ({} B)) ({/}))) (B A elsni eqcomd con3i ({} A) B disjsn sylibr)) thm (snprc ((A x)) () (<-> (-. (e. A (V))) (= ({} A) ({/}))) (x A elsn x exbii ({} A) x n0 A x isset 3bitr4 con1bii)) thm (rabsn ((A x) (B x)) () (-> (e. B A) (= ({e.|} x A (= (cv x) B)) ({} B))) ((cv x) B A eleq1 pm5.32ri baib x abbidv x A (= (cv x) B) df-rab B x df-sn 3eqtr4g)) thm (eusn ((x y) (x z) (y z) (ph y)) () (<-> (E! x ph) (E. x (= ({|} x ph) ({} (cv x))))) (x ph ({} (cv y)) abeq1 x (cv y) elsn ph bibi2i x albii bitr y exbii (= ({|} x ph) ({} (cv x))) y ax-17 y x ph hbab1 (e. (cv z) ({} (cv y))) x ax-17 hbeq (cv x) (cv y) sneq ({|} x ph) eqeq2d cbvex x ph y df-eu 3bitr4r)) thm (prcom () () (= ({,} A B) ({,} B A)) (({} A) ({} B) uncom A B df-pr B A df-pr 3eqtr4)) thm (preq1 () () (-> (= A B) (= ({,} A C) ({,} B C))) (A B sneq ({} C) uneq1d A C df-pr B C df-pr 3eqtr4g)) thm (preq2 () () (-> (= A B) (= ({,} C A) ({,} C B))) (A B C preq1 C A prcom C B prcom 3eqtr4g)) thm (pri1 () ((pri1.1 (e. A (V)))) (e. A ({,} A B)) (A eqid (= A A) (= A B) orc ax-mp pri1.1 A B elpr mpbir)) thm (pri2 () ((pri2.1 (e. B (V)))) (e. B ({,} A B)) (pri2.1 A pri1 B A prcom eleqtr)) thm (prprc1 () () (-> (-. (e. A (V))) (= ({,} A B) ({} B))) (A snprc ({} A) ({/}) ({} B) uneq1 A B df-pr ({/}) ({} B) uncom ({} B) un0 eqtr2 3eqtr4g sylbi)) thm (prprc2 () () (-> (-. (e. B (V))) (= ({,} A B) ({} A))) (B A prprc1 A B prcom syl5eq)) thm (prprc () () (-> (/\ (-. (e. A (V))) (-. (e. B (V)))) (= ({,} A B) ({/}))) (A B prprc1 B snprc biimp sylan9eq)) thm (tpi1 () ((tpi1.1 (e. A (V)))) (e. A ({,,} A B C)) (tpi1.1 B pri1 A ({,} A B) ({} C) elun1 A B C df-tp syl6eleqr ax-mp)) thm (tpi2 () ((tpi2.1 (e. B (V)))) (e. B ({,,} A B C)) (tpi2.1 A pri2 B ({,} A B) ({} C) elun1 A B C df-tp syl6eleqr ax-mp)) thm (tpi3 () ((tpi3.1 (e. C (V)))) (e. C ({,,} A B C)) (tpi3.1 snid C ({} C) ({,} A B) elun2 A B C df-tp syl6eleqr ax-mp)) thm (snnz () ((snnz.1 (e. A (V)))) (-. (= ({} A) ({/}))) (snnz.1 snid A ({} A) n0i ax-mp)) thm (prnz () ((prnz.1 (e. A (V)))) (-. (= ({,} A B) ({/}))) (prnz.1 B pri1 A ({,} A B) n0i ax-mp)) thm (tpnz () ((tpnz.1 (e. A (V)))) (-. (= ({,,} A B C) ({/}))) (tpnz.1 B C tpi1 A ({,,} A B C) n0i ax-mp)) thm (snss ((A x) (B x)) ((snss.1 (e. A (V)))) (<-> (e. A B) (C_ ({} A) B)) (x A elsn (e. (cv x) B) imbi1i x albii ({} A) B x dfss2 snss.1 B x clel2 3bitr4r)) thm (snssg ((A x) (B x)) () (-> (e. A C) (<-> (e. A B) (C_ ({} A) B))) ((cv x) A B eleq1 (cv x) A sneq B sseq1d x visset B snss C vtoclbg)) thm (difsn ((A x) (B x)) () (-> (-. (e. A B)) (= (\ B ({} A)) B)) (B ({} A) difss (-. (e. A B)) a1i (cv x) A B eleq1 biimpcd (cv x) A elsni syl5 con3d com12 ancld (cv x) B ({} A) eldif syl6ibr ssrdv eqssd)) thm (difprsn ((A x) (B x)) () (C_ (\ ({,} A B) ({} A)) ({} B)) ((= (cv x) B) (-. (= (cv x) A)) pm3.26 (cv x) ({,} A B) ({} A) eldif x visset A B elpr (= (cv x) A) (= (cv x) B) orcom bitr x A elsn negbii anbi12i (= (cv x) B) (= (cv x) A) pm5.61 3bitr x B elsn 3imtr4 ssriv)) thm (snssi () () (-> (e. A B) (C_ ({} A) B)) (A B B snssg ibi)) thm (difsnid () () (-> (e. B A) (= (u. (\ A ({} B)) ({} B)) A)) (B A snssi ({} B) A ssundif sylib (\ A ({} B)) ({} B) uncom syl5eq)) thm (pw0 () () (= (P~ ({/})) ({} ({/}))) (({/}) x df-pw abeq2i (cv x) ({/}) eqss (cv x) 0ss mpbiran2 bitr4 abbi2i ({/}) x df-sn eqtr4)) thm (pwpw0 ((x y)) () (= (P~ ({} ({/}))) ({,} ({/}) ({} ({/})))) ((cv x) ({} ({/})) y dfss2 y ({/}) elsn (e. (cv y) (cv x)) imbi2i y albii bitr y (e. (cv y) (cv x)) (= (cv y) ({/})) exintr (cv x) y n0 syl5ib y (e. (cv y) (cv x)) (= (cv y) ({/})) exancom ({/}) (cv x) y df-clel bitr4 ({/}) (cv x) snssi sylbi syl6 sylbi anc2li (cv x) ({} ({/})) eqss syl6ibr orrd ({} ({/})) 0ss (cv x) ({/}) ({} ({/})) sseq1 mpbiri (cv x) ({} ({/})) eqimss jaoi impbi x abbii ({} ({/})) x df-pw ({/}) ({} ({/})) x dfpr2 3eqtr4)) thm (snsspr () () (C_ ({} A) ({,} A B)) (A eqid (= A B) pm2.21ni orri A (V) A B elprg mpbiri A ({,} A B) snssi syl A snprc biimp ({,} A B) 0ss (-. (e. A (V))) a1i eqsstrd pm2.61i)) thm (prss ((A x) (B x) (C x)) ((prss.1 (e. A (V))) (prss.2 (e. B (V)))) (<-> (/\ (e. A C) (e. B C)) (C_ ({,} A B) C)) (A C (cv x) eleq1a B C (cv x) eleq1a jaao x visset A B elpr syl5ib ssrdv prss.1 B pri1 ({,} A B) C A ssel mpi prss.2 A pri2 ({,} A B) C B ssel mpi jca impbi)) thm (prssg ((x y) (A x) (A y) (B y) (C x) (C y)) () (-> (/\ (e. A R) (e. B S)) (<-> (/\ (e. A C) (e. B C)) (C_ ({,} A B) C))) ((cv x) A C eleq1 (e. (cv y) C) anbi1d (cv x) A (cv y) preq1 C sseq1d bibi12d (cv y) B C eleq1 (e. A C) anbi2d (cv y) B A preq2 C sseq1d bibi12d x visset y visset C prss R S vtocl2g)) thm (sssn ((A x) (B x)) () (<-> (C_ A ({} B)) (\/ (= A ({/})) (= A ({} B)))) (A ({} B) (cv x) ssel (cv x) B elsni syl6 (cv x) B A eleq1 syl6 ibd x 19.23adv A x n0 syl5ib B A snssi syl6 anc2li A ({} B) eqss syl6ibr orrd ({} B) 0ss A ({/}) ({} B) sseq1 mpbiri A ({} B) eqimss jaoi impbi)) thm (sspr () () (<-> (C_ A ({,} B C)) (\/ (\/ (= A ({/})) (= A ({} B))) (\/ (= A ({} C)) (= A ({,} B C))))) ((C_ A ({,} B C)) (-. (\/ (= A ({/})) (= A ({} B)))) pm3.26 (-. (= A ({} C))) adantr B A C A A prssg ibi B A difsn (C_ A ({,} B C)) adantl A ({,} B C) ({} B) ssdif B C difprsn (C_ A ({,} B C)) a1i sstrd (-. (e. B A)) adantr eqsstr3d ex A C sssn syl6ib con1d imp (= A ({/})) (= A ({} B)) pm2.45 (-. (= A ({} C))) anim1i (= A ({/})) (= A ({} C)) ioran sylibr sylan2 anassrs C A difsn (C_ A ({,} B C)) adantl B C prcom A sseq2i A ({,} C B) ({} C) ssdif sylbi (-. (e. C A)) adantr C B difprsn (/\ (C_ A ({,} B C)) (-. (e. C A))) a1i sstrd eqsstr3d ex A B sssn syl6ib con1d imp (-. (= A ({} C))) adantr sylanc eqssd ex orrd ex orrd ({,} B C) 0ss A ({/}) ({,} B C) sseq1 mpbiri B C snsspr A ({} B) ({,} B C) sseq1 mpbiri jaoi C B snsspr A ({} C) ({,} C B) sseq1 mpbiri C B prcom syl6ss A ({,} B C) eqimss jaoi jaoi impbi)) thm (tpss ((A x) (B x) (C x) (D x)) ((tpss.1 (e. A (V))) (tpss.2 (e. B (V))) (tpss.3 (e. C (V)))) (<-> (/\/\ (e. A D) (e. B D) (e. C D)) (C_ ({,,} A B C) D)) ((= (cv x) A) (e. (cv x) D) (= (cv x) B) (= (cv x) C) 3jao A D (cv x) eleq1a B D (cv x) eleq1a C D (cv x) eleq1a syl3an x visset A B C eltp syl5ib ssrdv tpss.1 B C tpi1 ({,,} A B C) D A ssel mpi tpss.2 A C tpi2 ({,,} A B C) D B ssel mpi tpss.3 A B tpi3 ({,,} A B C) D C ssel mpi 3jca impbi)) thm (sneqr () ((sneqr.1 (e. A (V)))) (-> (= ({} A) ({} B)) (= A B)) (sneqr.1 snid ({} A) ({} B) A eleq2 mpbii sneqr.1 B elsnc sylib)) thm (snsssn () ((sneqr.1 (e. A (V)))) (-> (C_ ({} A) ({} B)) (= A B)) (({} A) B sssn sneqr.1 snnz (= A B) pm2.21i sneqr.1 B sneqr jaoi sylbi)) thm (snsspw ((A x)) () (C_ ({} A) (P~ A)) ((cv x) A eqimss x A elsn A x df-pw abeq2i 3imtr4 ssriv)) thm (prsspw ((A x) (B x) (C x)) ((prsspw.1 (e. A (V))) (prsspw.2 (e. B (V)))) (<-> (C_ ({,} A B) (P~ C)) (/\ (C_ A C) (C_ B C))) (({,} A B) (P~ C) x dfss2 x visset A B elpr x visset C elpw imbi12i (= (cv x) A) (= (cv x) B) (C_ (cv x) C) jaob bitr x albii x (-> (= (cv x) A) (C_ (cv x) C)) (-> (= (cv x) B) (C_ (cv x) C)) 19.26 prsspw.1 (cv x) A C sseq1 ceqsalv prsspw.2 (cv x) B C sseq1 ceqsalv anbi12i bitr 3bitr)) thm (preqr1 () ((preqr1.1 (e. A (V))) (preqr1.2 (e. B (V)))) (-> (= ({,} A C) ({,} B C)) (= A B)) (preqr1.1 C pri1 ({,} A C) ({,} B C) A eleq2 mpbii preqr1.1 B C elpr sylib preqr1.2 C pri1 ({,} A C) ({,} B C) B eleq2 mpbiri preqr1.2 A C elpr sylib A B eqcom A C B eqeq2 oplem1)) thm (prer2 () ((prer2.1 (e. A (V))) (prer2.2 (e. B (V)))) (-> (= ({,} C A) ({,} C B)) (= A B)) (C A prcom C B prcom eqeq12i prer2.1 prer2.2 C preqr1 sylbi)) thm (preq12b () ((preq12b.1 (e. A (V))) (preq12b.2 (e. B (V))) (preq12b.3 (e. C (V))) (preq12b.4 (e. D (V)))) (<-> (= ({,} A B) ({,} C D)) (\/ (/\ (= A C) (= B D)) (/\ (= A D) (= B C)))) (preq12b.1 B pri1 ({,} A B) ({,} C D) A eleq2 mpbii preq12b.1 C D elpr sylib A C B preq1 ({,} C D) eqeq1d preq12b.2 preq12b.4 C prer2 syl6bi com12 ancld C D prcom ({,} A B) eqeq2i A D B preq1 ({,} D C) eqeq1d preq12b.2 preq12b.3 D prer2 syl6bi com12 sylbi ancld orim12d mpd A C B preq1 B D C preq2 sylan9eq A D B preq1 D B prcom syl6eq B C D preq1 sylan9eq jaoi impbi)) thm (prel12 () ((preq12b.1 (e. A (V))) (preq12b.2 (e. B (V))) (preq12b.3 (e. C (V))) (preq12b.4 (e. D (V)))) (-> (-. (= A B)) (<-> (= ({,} A B) ({,} C D)) (/\ (e. A ({,} C D)) (e. B ({,} C D))))) (preq12b.1 B pri1 ({,} A B) ({,} C D) A eleq2 mpbii preq12b.2 A pri2 ({,} A B) ({,} C D) B eleq2 mpbii jca (-. (= A B)) a1i B D A eqeq2 negbid (= A D) (= A C) orel2 syl6bi com3l imp ancrd B C A eqeq2 negbid (= A C) (= A D) orel1 syl6bi com3l imp ancrd orim12d preq12b.2 C D elpr (= B C) (= B D) orcom bitr preq12b.1 preq12b.2 preq12b.3 preq12b.4 preq12b 3imtr4g ex preq12b.1 C D elpr syl5ib imp3a impbid)) thm (opeq1 () () (-> (= A B) (= (<,> A C) (<,> B C))) (A B C preq1 ({,} A C) ({,} B C) ({} A) preq2 syl A B sneq ({} A) ({} B) ({,} B C) preq1 syl eqtrd A C df-op B C df-op 3eqtr4g)) thm (opeq2 () () (-> (= A B) (= (<,> C A) (<,> C B))) (A B C preq2 ({,} C A) ({,} C B) ({} C) preq2 syl C A df-op C B df-op 3eqtr4g)) thm (opeq12 () () (-> (/\ (= A C) (= B D)) (= (<,> A B) (<,> C D))) (A C B opeq1 B D C opeq2 sylan9eq)) thm (hbop ((A y) (B y) (x y)) ((hbop.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbop.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (<,> A B)) (A. x (e. (cv y) (<,> A B)))) (hbop.1 hbsn hbop.1 hbop.2 hbpr hbpr A B df-op (cv y) eleq2i A B df-op (cv y) eleq2i x albii 3imtr4)) thm (opprc1 () () (-> (-. (e. A (V))) (= (<,> A B) ({,} ({/}) ({} B)))) (A snprc ({} A) ({/}) ({,} A B) preq1 sylbi A B prprc1 ({,} A B) ({} B) ({/}) preq2 syl eqtrd A B df-op syl5eq)) thm (opprc2 () () (-> (-. (e. B (V))) (= (<,> A B) (<,> A A))) (B A prprc1 B A prcom A dfsn2 3eqtr3g ({,} A B) ({,} A A) ({} A) preq2 syl A B df-op A A df-op 3eqtr4g)) thm (pwsn ((A x)) () (= (P~ ({} A)) ({,} ({/}) ({} A))) ((cv x) A sssn x abbii ({} A) x df-pw ({/}) ({} A) x dfpr2 3eqtr4)) thm (pwv () () (= (P~ (V)) (V)) ((cv x) ssv x visset (V) elpw mpbir x visset 2th eqriv)) thm (dfuni2 ((x y) (A x) (A y)) () (= (U. A) ({|} x (E.e. y A (e. (cv x) (cv y))))) (A x y df-uni y (e. (cv x) (cv y)) (e. (cv y) A) exancom y A (e. (cv x) (cv y)) df-rex bitr4 x abbii eqtr)) thm (eluni ((A x) (x y) (A y) (B x) (B y)) () (<-> (e. A (U. B)) (E. x (/\ (e. A (cv x)) (e. (cv x) B)))) (A (U. B) elisset A (cv x) elisset (e. (cv x) B) adantr x 19.23aiv (cv y) A (cv x) eleq1 (e. (cv x) B) anbi1d x exbidv B y x df-uni (V) elab2g pm5.21nii)) thm (eluni2 ((A x) (B x)) () (<-> (e. A (U. B)) (E.e. x B (e. A (cv x)))) (x (e. A (cv x)) (e. (cv x) B) exancom A B x eluni x B (e. A (cv x)) df-rex 3bitr4)) thm (elunii ((A x) (B x) (C x)) () (-> (/\ (e. A B) (e. B C)) (e. A (U. C))) ((cv x) B A eleq2 (cv x) B C eleq1 anbi12d C cla4egv anabsi7 A C x eluni sylibr)) thm (hbuni ((y z) (A y) (A z) (x y) (x z)) ((hbuni.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (U. A)) (A. x (e. (cv y) (U. A)))) ((e. (cv y) (cv z)) x ax-17 (e. (cv y) (cv z)) x ax-17 hbuni.1 hbel hban z hbex (cv y) A z eluni (cv y) A z eluni x albii 3imtr4)) thm (unieq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (= (U. A) (U. B))) (A B (cv y) eleq2 (e. (cv x) (cv y)) anbi2d y exbidv x abbidv A x y df-uni B x y df-uni 3eqtr4g)) thm (unieqi () ((unieqi.1 (= A B))) (= (U. A) (U. B)) (unieqi.1 A B unieq ax-mp)) thm (unieqd () ((unieqd.1 (-> ph (= A B)))) (-> ph (= (U. A) (U. B))) (unieqd.1 A B unieq syl)) thm (eluniab ((A x) (x y) (A y) (ph y)) () (<-> (e. A (U. ({|} x ph))) (E. x (/\ (e. A (cv x)) ph))) (A ({|} x ph) y eluni (e. A (cv y)) x ax-17 y x ph hbab1 hban (/\ (e. A (cv x)) ph) y ax-17 (cv y) (cv x) A eleq2 (cv y) (cv x) ({|} x ph) eleq1 anbi12d x ph abid (e. A (cv x)) anbi2i syl6bb cbvex bitr)) thm (elunirab ((A x)) () (<-> (e. A (U. ({e.|} x B ph))) (E.e. x B (/\ (e. A (cv x)) ph))) (A x (/\ (e. (cv x) B) ph) eluniab x B ph df-rab unieqi A eleq2i x B (/\ (e. A (cv x)) ph) df-rex (e. (cv x) B) (e. A (cv x)) ph an12 x exbii bitr 3bitr4)) thm (unipr ((x y) (A x) (A y) (B x) (B y)) ((unipr.1 (e. A (V))) (unipr.2 (e. B (V)))) (= (U. ({,} A B)) (u. A B)) (y (/\ (e. (cv x) (cv y)) (= (cv y) A)) (/\ (e. (cv x) (cv y)) (= (cv y) B)) 19.43 y visset A B elpr (e. (cv x) (cv y)) anbi2i (e. (cv x) (cv y)) (= (cv y) A) (= (cv y) B) andi bitr y exbii unipr.1 (cv x) y clel3 y (= (cv y) A) (e. (cv x) (cv y)) exancom bitr unipr.2 (cv x) y clel3 y (= (cv y) B) (e. (cv x) (cv y)) exancom bitr orbi12i 3bitr4r x abbii A B x df-un ({,} A B) x y df-uni 3eqtr4r)) thm (uniprg ((x y) (A x) (A y) (B y)) () (-> (/\ (e. A C) (e. B D)) (= (U. ({,} A B)) (u. A B))) ((cv x) A (cv y) preq1 unieqd (cv x) A (cv y) uneq1 eqeq12d (cv y) B A preq2 unieqd (cv y) B A uneq2 eqeq12d x visset y visset unipr C D vtocl2g)) thm (unisn () ((unisn.1 (e. A (V)))) (= (U. ({} A)) A) (A dfsn2 unieqi unisn.1 unisn.1 unipr A unidm 3eqtr)) thm (unisng ((A x)) () (-> (e. A B) (= (U. ({} A)) A)) ((cv x) A sneq unieqd (= (cv x) A) id eqeq12d x visset unisn B vtoclg)) thm (uniun ((x y) (A x) (A y) (B x) (B y)) () (= (U. (u. A B)) (u. (U. A) (U. B))) (y (/\ (e. (cv x) (cv y)) (e. (cv y) A)) (/\ (e. (cv x) (cv y)) (e. (cv y) B)) 19.43 (cv y) A B elun (e. (cv x) (cv y)) anbi2i (e. (cv x) (cv y)) (e. (cv y) A) (e. (cv y) B) andi bitr y exbii (cv x) A y eluni (cv x) B y eluni orbi12i 3bitr4 (cv x) (u. A B) y eluni (cv x) (U. A) (U. B) elun 3bitr4 eqriv)) thm (uniin ((x y) (A x) (A y) (B x) (B y)) () (C_ (U. (i^i A B)) (i^i (U. A) (U. B))) (y (/\ (e. (cv x) (cv y)) (e. (cv y) A)) (/\ (e. (cv x) (cv y)) (e. (cv y) B)) 19.40 (cv x) (i^i A B) y eluni (cv y) A B elin (e. (cv x) (cv y)) anbi2i (e. (cv x) (cv y)) (e. (cv y) A) (e. (cv y) B) anandi bitr y exbii bitr (cv x) (U. A) (U. B) elin (cv x) A y eluni (cv x) B y eluni anbi12i bitr 3imtr4 ssriv)) thm (uniss ((x y) (A x) (A y) (B x) (B y)) () (-> (C_ A B) (C_ (U. A) (U. B))) (A B (cv y) ssel (e. (cv x) (cv y)) anim2d y 19.22dv x 19.21aiv x (E. y (/\ (e. (cv x) (cv y)) (e. (cv y) A))) (E. y (/\ (e. (cv x) (cv y)) (e. (cv y) B))) ss2ab sylibr A x y df-uni B x y df-uni 3sstr4g)) thm (ssuni ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (C_ A B) (e. B C)) (C_ A (U. C))) ((cv x) B A sseq2 (C_ A (U. C)) imbi1d (/\ (e. (cv y) (cv x)) (e. (cv x) C)) x 19.8a expcom (cv y) C x eluni syl6ibr (e. (cv y) A) imim2d y 19.20dv A (cv x) y dfss2 A (U. C) y dfss2 3imtr4g vtoclga impcom)) thm (uni0b ((x y) (A x) (A y)) () (<-> (= (U. A) ({/})) (C_ A ({} ({/})))) (x ({/}) elsn x A ralbii A ({} ({/})) x dfss3 (U. A) y n0 x A y (e. (cv y) (cv x)) rexcom4 (cv x) y n0 x A rexbii (cv y) A x eluni2 y exbii 3bitr4r x A (= (cv x) ({/})) rexnal 3bitr con4bii 3bitr4r)) thm (elssuni () () (-> (e. A B) (C_ A (U. B))) (A ssid A A B ssuni mpan)) thm (unissel () () (-> (/\ (C_ (U. A) B) (e. B A)) (= (U. A) B)) ((C_ (U. A) B) (e. B A) pm3.26 B A elssuni (C_ (U. A) B) adantl eqssd)) thm (unissb ((x y) (A x) (A y) (B x) (B y)) () (<-> (C_ (U. A) B) (A.e. x A (C_ (cv x) B))) ((U. A) B y dfss2 (cv y) A x eluni (e. (cv y) B) imbi1i x (/\ (e. (cv y) (cv x)) (e. (cv x) A)) (e. (cv y) B) 19.23v bitr4 y albii y x (-> (/\ (e. (cv y) (cv x)) (e. (cv x) A)) (e. (cv y) B)) alcom y (e. (cv x) A) (-> (e. (cv y) (cv x)) (e. (cv y) B)) 19.21v (e. (cv y) (cv x)) (e. (cv x) A) (e. (cv y) B) impexp (e. (cv y) (cv x)) (e. (cv x) A) (e. (cv y) B) bi2.04 bitr y albii (cv x) B y dfss2 (e. (cv x) A) imbi2i 3bitr4 x albii bitr 3bitr x A (C_ (cv x) B) df-ral bitr4)) thm (uniss2 ((x y) (A x) (A y) (B x) (B y)) () (-> (A.e. x A (E.e. y B (C_ (cv x) (cv y)))) (C_ (U. A) (U. B))) ((cv x) (cv y) B ssuni expcom r19.23aiv x A r19.20si A (U. B) x unissb sylibr)) thm (unidif ((x y) (A x) (A y) (B x) (B y)) () (-> (A.e. x A (E.e. y (\ A B) (C_ (cv x) (cv y)))) (= (U. (\ A B)) (U. A))) (x A y (\ A B) uniss2 A B difss (\ A B) A uniss ax-mp jctil (U. (\ A B)) (U. A) eqss sylibr)) thm (ssunieq ((A x) (B x)) () (-> (/\ (e. A B) (A.e. x B (C_ (cv x) A))) (= A (U. B))) (A B elssuni B A x unissb biimpr anim12i A (U. B) eqss sylibr)) thm (unimax ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A B) (= (U. ({e.|} x B (C_ (cv x) A))) A)) (A ssid (cv x) A A sseq1 B elrab2 mpbiri (cv x) (cv y) A sseq1 B elrab pm3.27bd rgen A ({e.|} x B (C_ (cv x) A)) y ssunieq eqcomd mpan2 syl)) thm (dfint2 ((x y) (A x) (A y)) () (= (|^| A) ({|} x (A.e. y A (e. (cv x) (cv y))))) (A x y df-int y A (e. (cv x) (cv y)) df-ral x abbii eqtr4)) thm (inteq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (= (|^| A) (|^| B))) (A B y (e. (cv x) (cv y)) raleq1 x abbidv A x y dfint2 B x y dfint2 3eqtr4g)) thm (inteqi () ((inteqi.1 (= A B))) (= (|^| A) (|^| B)) (inteqi.1 A B inteq ax-mp)) thm (inteqd () ((inteqd.1 (-> ph (= A B)))) (-> ph (= (|^| A) (|^| B))) (inteqd.1 A B inteq syl)) thm (elint ((A x) (x y) (A y) (B x) (B y)) ((elint.1 (e. A (V)))) (<-> (e. A (|^| B)) (A. x (-> (e. (cv x) B) (e. A (cv x))))) (elint.1 (cv y) A (cv x) eleq1 (e. (cv x) B) imbi2d x albidv B y x df-int elab2)) thm (elint2 ((A x) (B x)) ((elint2.1 (e. A (V)))) (<-> (e. A (|^| B)) (A.e. x B (e. A (cv x)))) (elint2.1 B x elint x B (e. A (cv x)) df-ral bitr4)) thm (elintg ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A C) (<-> (e. A (|^| B)) (A.e. x B (e. A (cv x))))) ((cv y) A (|^| B) eleq1 (cv y) A (cv x) eleq1 x B ralbidv bibi12d y visset B x elint2 C vtoclg)) thm (elinti ((x y) (A x) (A y) (B x) (B y) (C y)) () (-> (e. A (|^| B)) (-> (e. C B) (e. A C))) ((cv x) A (cv y) eleq1 (e. (cv y) B) imbi2d y albidv x visset B y elint biimp vtoclga (cv y) C B eleq1 (cv y) C A eleq2 imbi12d B cla4gv pm2.43b syl)) thm (hbint ((y z) (A y) (A z) (x y) (x z)) ((hbint.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (|^| A)) (A. x (e. (cv y) (|^| A)))) ((e. (cv y) (cv z)) x ax-17 hbint.1 hbel (e. (cv y) (cv z)) x ax-17 hbim z hbal y visset A z elint y visset A z elint x albii 3imtr4)) thm (elintab ((A x) (A y) (x y) (ph y)) ((inteqab.1 (e. A (V)))) (<-> (e. A (|^| ({|} x ph))) (A. x (-> ph (e. A (cv x))))) (inteqab.1 ({|} x ph) y elint y x ph hbab1 (e. A (cv y)) x ax-17 hbim (-> ph (e. A (cv x))) y ax-17 (cv y) (cv x) ({|} x ph) eleq1 x ph abid syl6bb (cv y) (cv x) A eleq2 imbi12d cbval bitr)) thm (elintrab ((A x)) ((inteqab.1 (e. A (V)))) (<-> (e. A (|^| ({e.|} x B ph))) (A.e. x B (-> ph (e. A (cv x))))) (inteqab.1 x (/\ (e. (cv x) B) ph) elintab (e. (cv x) B) ph (e. A (cv x)) impexp x albii bitr x B ph df-rab inteqi A eleq2i x B (-> ph (e. A (cv x))) df-ral 3bitr4)) thm (int0 ((x y)) () (= (|^| ({/})) (V)) ((cv y) noel (e. (cv x) (cv y)) pm2.21i y ax-gen (cv x) eqid 2th x abbii ({/}) x y df-int x df-v 3eqtr4)) thm (intss1 ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A B) (C_ (|^| B) A)) ((cv y) A B eleq1 (cv y) A (cv x) eleq2 imbi12d B cla4gv pm2.43a x visset B y elint syl5ib ssrdv)) thm (ssint ((x y) (A x) (A y) (B x) (B y)) () (<-> (C_ A (|^| B)) (A.e. x B (C_ A (cv x)))) (A (|^| B) y dfss3 y visset B x elint2 y A ralbii y A x B (e. (cv y) (cv x)) ralcom A (cv x) y dfss3 x B ralbii bitr4 3bitr)) thm (ssintab ((x y) (A x) (A y) (ph y)) () (<-> (C_ A (|^| ({|} x ph))) (A. x (-> ph (C_ A (cv x))))) (A ({|} x ph) y ssint y ({|} x ph) (C_ A (cv y)) df-ral y x ph hbab1 (C_ A (cv y)) x ax-17 hbim (-> (e. (cv x) ({|} x ph)) (C_ A (cv x))) y ax-17 (cv y) (cv x) ({|} x ph) eleq1 (cv y) (cv x) A sseq2 imbi12d cbval x ph abid (C_ A (cv x)) imbi1i x albii bitr 3bitr)) thm (ssintub ((x y) (A x) (A y) (B x) (B y)) () (C_ A (|^| ({e.|} x B (C_ A (cv x))))) (A ({e.|} x B (C_ A (cv x))) y ssint (cv x) (cv y) A sseq2 B elrab pm3.27bd mprgbir)) thm (ssmin ((A x)) () (C_ A (|^| ({|} x (/\ (C_ A (cv x)) ph)))) (A x (/\ (C_ A (cv x)) ph) ssintab (C_ A (cv x)) ph pm3.26 mpgbir)) thm (intmin ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A B) (= (|^| ({e.|} x B (C_ A (cv x)))) A)) (A ssid (cv x) A A sseq2 (cv x) A (cv y) eleq2 imbi12d B rcla4v mpii y visset x B (C_ A (cv x)) elintrab syl5ib ssrdv A x B ssintub jctir (|^| ({e.|} x B (C_ A (cv x)))) A eqss sylibr)) thm (intss ((x y) (A x) (A y) (B x) (B y)) () (-> (C_ A B) (C_ (|^| B) (|^| A))) ((e. (cv y) A) (e. (cv y) B) (e. (cv x) (cv y)) imim1 y 19.20ii x visset B y elint x visset A y elint 3imtr4g x 19.21aiv A B y dfss2 (|^| B) (|^| A) x dfss2 3imtr4)) thm (intssuni ((x y) (A x) (A y)) () (-> (-. (= A ({/}))) (C_ (|^| A) (U. A))) (A y (e. (cv x) (cv y)) r19.2zOLD x visset A y elint2 (cv x) A y eluni2 3imtr4g ssrdv)) thm (intssuni2 () () (-> (/\ (C_ A B) (-. (= A ({/})))) (C_ (|^| A) (U. B))) (A intssuni A B uniss sylan9ssr)) thm (intmin3 ((A x) (ps x)) ((intmin3.2 (-> (= (cv x) A) (<-> ph ps))) (intmin3.3 ps)) (-> (e. A B) (C_ (|^| ({|} x ph)) A)) (intmin3.3 intmin3.2 B elabg mpbiri A ({|} x ph) intss1 syl)) thm (intmin4 ((x y) (A x) (A y) (ph y)) () (-> (C_ A (|^| ({|} x ph))) (= (|^| ({|} x (/\ (C_ A (cv x)) ph))) (|^| ({|} x ph)))) (A x ph ssintab (C_ A (cv x)) ph pm3.27 (-> ph (C_ A (cv x))) a1i ph (C_ A (cv x)) ancr impbid (e. (cv y) (cv x)) imbi1d x 19.20i x (-> (/\ (C_ A (cv x)) ph) (e. (cv y) (cv x))) (-> ph (e. (cv y) (cv x))) 19.15 syl sylbi y visset x (/\ (C_ A (cv x)) ph) elintab y visset x ph elintab 3bitr4g eqrdv)) thm (intab ((x z) (A x) (A z) (ph x) (ph z) (x y) (y z)) ((intab.1 (e. A (V))) (intab.2 (e. ({|} x (E. y (/\ ph (= (cv x) A)))) (V)))) (= (|^| ({|} x (A. y (-> ph (e. A (cv x)))))) ({|} x (E. y (/\ ph (= (cv x) A))))) ((cv z) (cv x) A eqeq1 ph anbi2d y exbidv cbvabv intab.2 eqeltr y (/\ ph (= (cv z) A)) hbe1 x z hbab hbeleq (cv x) ({|} z (E. y (/\ ph (= (cv z) A)))) A eleq2 ph imbi2d albid sbcie intab.1 ph z ax-17 A (V) sbcgf ax-mp biimpr intab.1 A (V) z csbvarg ax-mp intab.1 A (V) z (cv z) A sbceq1dig ax-mp mpbir jctir intab.1 A (V) z ph (= (cv z) A) sbcang ax-mp sylibr intab.1 (/\ ph (= (cv z) A)) y 19.8a z ax-gen A (V) z (-> (/\ ph (= (cv z) A)) (E. y (/\ ph (= (cv z) A)))) a4sbc mp2 intab.1 A (V) z (/\ ph (= (cv z) A)) (E. y (/\ ph (= (cv z) A))) sbcimg ax-mp mpbi syl intab.1 z (E. y (/\ ph (= (cv z) A))) elabs sylibr mpgbir (cv z) (cv x) A eqeq1 ph anbi2d y exbidv cbvabv intab.2 eqeltr x (A. y (-> ph (e. A (cv x)))) elabs mpbir ({|} z (E. y (/\ ph (= (cv z) A)))) ({|} x (A. y (-> ph (e. A (cv x))))) intss1 ax-mp y (-> ph (e. A (cv x))) hba1 z x hbab hbint y (-> ph (e. A (cv x))) ax-4 com12 (= (cv z) A) adantr (cv z) A (cv x) eleq1 ph adantl sylibrd x 19.21aiv z visset x (A. y (-> ph (e. A (cv x)))) elintab sylibr 19.23ai abssi eqssi (cv z) (cv x) A eqeq1 ph anbi2d y exbidv cbvabv eqtr)) thm (int0el () () (-> (e. ({/}) A) (= (|^| A) ({/}))) (({/}) A intss1 (|^| A) 0ss (e. ({/}) A) a1i eqssd)) thm (intun ((x y) (A x) (A y) (B x) (B y)) () (= (|^| (u. A B)) (i^i (|^| A) (|^| B))) (y (-> (e. (cv y) A) (e. (cv x) (cv y))) (-> (e. (cv y) B) (e. (cv x) (cv y))) 19.26 (cv y) A B elun (e. (cv x) (cv y)) imbi1i (e. (cv y) A) (e. (cv y) B) (e. (cv x) (cv y)) jaob bitr y albii x visset A y elint x visset B y elint anbi12i 3bitr4 x visset (u. A B) y elint (cv x) (|^| A) (|^| B) elin 3bitr4 eqriv)) thm (intpr ((x y) (A x) (A y) (B x) (B y)) ((intpr.1 (e. A (V))) (intpr.2 (e. B (V)))) (= (|^| ({,} A B)) (i^i A B)) (y (-> (= (cv y) A) (e. (cv x) (cv y))) (-> (= (cv y) B) (e. (cv x) (cv y))) 19.26 y visset A B elpr (e. (cv x) (cv y)) imbi1i (= (cv y) A) (= (cv y) B) (e. (cv x) (cv y)) jaob bitr y albii intpr.1 (cv x) y clel4 intpr.2 (cv x) y clel4 anbi12i 3bitr4 x visset ({,} A B) y elint (cv x) A B elin 3bitr4 eqriv)) thm (intsn () ((intsn.1 (e. A (V)))) (= (|^| ({} A)) A) (A dfsn2 inteqi intsn.1 intsn.1 intpr A inidm 3eqtr)) thm (intunsn () ((intunsn.1 (e. B (V)))) (= (|^| (u. A ({} B))) (i^i (|^| A) B)) (A ({} B) intun intunsn.1 intsn (|^| A) ineq2i eqtr)) thm (eliun ((x y) (A x) (A y) (B y) (C y)) () (<-> (e. A (U_ x B C)) (E.e. x B (e. A C))) (A (U_ x B C) elisset A C elisset (e. (cv x) B) a1i r19.23aiv (cv y) A C eleq1 x B rexbidv x B C y df-iun (V) elab2g pm5.21nii)) thm (eliin ((x y) (A x) (A y) (B y) (C y)) () (-> (e. A D) (<-> (e. A (|^|_ x B C)) (A.e. x B (e. A C)))) ((cv y) A C eleq1 x B ralbidv x B C y df-iin D elab2g)) thm (iunconst ((x y) (A x) (A y) (B x) (B y)) () (-> (-. (= A ({/}))) (= (U_ x A B) B)) (A x n0 (E. x (e. (cv x) A)) (e. (cv y) B) ibar sylbi (cv y) x A B eliun x A (e. (cv y) B) df-rex x (e. (cv x) A) (e. (cv y) B) 19.41v 3bitr syl6rbbr eqrdv)) thm (iuniin ((x y) (y z) (A y) (A z) (x z) (B x) (B z) (C z)) () (C_ (U_ x A (|^|_ y B C)) (|^|_ y B (U_ x A C))) (x A y B (e. (cv z) C) r19.12 (cv z) x A (|^|_ y B C) eliun z visset (cv z) (V) y B C eliin ax-mp x A rexbii bitr z visset (cv z) (V) y B (U_ x A C) eliin ax-mp (cv z) x A C eliun y B ralbii bitr 3imtr4 ssriv)) thm (iunss1 ((x y) (A x) (A y) (B x) (B y) (C y)) () (-> (C_ A B) (C_ (U_ x A C) (U_ x B C))) (A B (cv x) ssel (e. (cv y) C) anim1d r19.22dv2 y 19.21aiv y (E.e. x A (e. (cv y) C)) (E.e. x B (e. (cv y) C)) ss2ab sylibr x A C y df-iun x B C y df-iun 3sstr4g)) thm (iuneq1 ((A x) (B x)) () (-> (= A B) (= (U_ x A C) (U_ x B C))) (A B x C iunss1 B A x C iunss1 anim12i A B eqss (U_ x A C) (U_ x B C) eqss 3imtr4)) thm (iineq1 ((x y) (A x) (A y) (B x) (B y) (C y)) () (-> (= A B) (= (|^|_ x A C) (|^|_ x B C))) (A B x (e. (cv y) C) raleq1 y abbidv x A C y df-iin x B C y df-iin 3eqtr4g)) thm (ss2iun ((x y) (A y) (B y) (C y)) () (-> (A.e. x A (C_ B C)) (C_ (U_ x A B) (U_ x A C))) (x A (C_ B C) hbra1 x A (C_ B C) ra4 B C (cv y) ssel syl6 r19.22d (cv y) x A B eliun (cv y) x A C eliun 3imtr4g ssrdv)) thm (iuneq2 () () (-> (A.e. x A (= B C)) (= (U_ x A B) (U_ x A C))) (x A B C ss2iun x A C B ss2iun anim12i B C eqss x A ralbii x A (C_ B C) (C_ C B) r19.26 bitr (U_ x A B) (U_ x A C) eqss 3imtr4)) thm (iineq2 ((x y) (A y) (B y) (C y)) () (-> (A.e. x A (= B C)) (= (|^|_ x A B) (|^|_ x A C))) (x A (= B C) hbra1 x A (= B C) ra4 B C (cv y) eleq2 syl6 imp ralbida y abbidv x A B y df-iin x A C y df-iin 3eqtr4g)) thm (iuneq2i () ((iuneq2i.1 (-> (e. (cv x) A) (= B C)))) (= (U_ x A B) (U_ x A C)) (x A B C iuneq2 iuneq2i.1 mprg)) thm (iineq2i () ((iuneq2i.1 (-> (e. (cv x) A) (= B C)))) (= (|^|_ x A B) (|^|_ x A C)) (x A B C iineq2 iuneq2i.1 mprg)) thm (iuneq2dv ((ph x)) ((iuneq2dv.1 (-> (/\ ph (e. (cv x) A)) (= B C)))) (-> ph (= (U_ x A B) (U_ x A C))) (iuneq2dv.1 r19.21aiva x A B C iuneq2 syl)) thm (hbiu1 ((y z) (A y) (A z) (B y) (B z) (x y) (x z)) () (-> (e. (cv y) (U_ x A B)) (A. x (e. (cv y) (U_ x A B)))) (x A (e. (cv z) B) hbre1 y z hbab x A B z df-iun (cv y) eleq2i x A B z df-iun (cv y) eleq2i x albii 3imtr4)) thm (hbii1 ((y z) (A y) (A z) (B y) (B z) (x y) (x z)) () (-> (e. (cv y) (|^|_ x A B)) (A. x (e. (cv y) (|^|_ x A B)))) (x A (e. (cv z) B) hbra1 y z hbab x A B z df-iin (cv y) eleq2i x A B z df-iin (cv y) eleq2i x albii 3imtr4)) thm (dfiun2g ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C w)) () (-> (A.e. x A (e. B C)) (= (U_ x A B) (U. ({|} y (E.e. x A (= (cv y) B)))))) (x A (e. B C) hbra1 x A (e. B C) ra4 B C (cv w) z clel3g z (= (cv z) B) (e. (cv w) (cv z)) exancom syl6bb syl6 pm5.32d z (e. (cv x) A) (/\ (e. (cv w) (cv z)) (= (cv z) B)) 19.42v syl6bbr exbid x z (/\ (e. (cv x) A) (/\ (e. (cv w) (cv z)) (= (cv z) B))) excom x (e. (cv w) (cv z)) (/\ (e. (cv x) A) (= (cv z) B)) 19.42v (e. (cv x) A) (e. (cv w) (cv z)) (= (cv z) B) an12 x exbii z visset (cv y) (cv z) B eqeq1 x A rexbidv elab x A (= (cv z) B) df-rex bitr (e. (cv w) (cv z)) anbi2i 3bitr4 z exbii bitr syl6bb x A (e. (cv w) B) df-rex syl5bb w abbidv x A B w df-iun ({|} y (E.e. x A (= (cv y) B))) w z df-uni 3eqtr4g)) thm (dfiun2 ((x y) (A x) (A y) (B y)) ((dfiun2.1 (e. B (V)))) (= (U_ x A B) (U. ({|} y (E.e. x A (= (cv y) B))))) (x A B (V) y dfiun2g dfiun2.1 (e. (cv x) A) a1i mprg)) thm (dfiin2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w)) ((dfiun2.1 (e. B (V)))) (= (|^|_ x A B) (|^| ({|} y (E.e. x A (= (cv y) B))))) (x A (e. (cv w) B) df-ral dfiun2.1 (cv w) z clel4 (e. (cv x) A) imbi2i z (e. (cv x) A) (-> (= (cv z) B) (e. (cv w) (cv z))) 19.21v bitr4 x albii x z (-> (e. (cv x) A) (-> (= (cv z) B) (e. (cv w) (cv z)))) alcom bitr z visset (cv y) (cv z) B eqeq1 x A rexbidv elab x A (= (cv z) B) df-rex bitr (e. (cv w) (cv z)) imbi1i x (/\ (e. (cv x) A) (= (cv z) B)) (e. (cv w) (cv z)) 19.23v (e. (cv x) A) (= (cv z) B) (e. (cv w) (cv z)) impexp x albii 3bitr2r z albii 3bitr w abbii x A B w df-iin ({|} y (E.e. x A (= (cv y) B))) w z df-int 3eqtr4)) thm (cbviun ((x y) (x z) (A x) (y z) (A y) (A z) (B z) (C z)) ((cbviun.1 (-> (e. (cv z) B) (A. y (e. (cv z) B)))) (cbviun.2 (-> (e. (cv z) C) (A. x (e. (cv z) C)))) (cbviun.3 (-> (= (cv x) (cv y)) (= B C)))) (= (U_ x A B) (U_ y A C)) (cbviun.1 cbviun.2 cbviun.3 (cv z) eleq2d A cbvrex z abbii x A B z df-iun y A C z df-iun 3eqtr4)) thm (cbviunv ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z) (C x) (C z)) ((cbviunv.1 (-> (= (cv x) (cv y)) (= B C)))) (= (U_ x A B) (U_ y A C)) ((e. (cv z) B) y ax-17 (e. (cv z) C) x ax-17 cbviunv.1 A cbviun)) thm (iunss ((x y) (C x) (C y) (A y) (B y)) () (<-> (C_ (U_ x A B) C) (A.e. x A (C_ B C))) (B C y dfss2 x A ralbii x A (A. y (-> (e. (cv y) B) (e. (cv y) C))) df-ral (e. (cv x) A) (e. (cv y) B) (e. (cv y) C) impexp y albii y (e. (cv x) A) (-> (e. (cv y) B) (e. (cv y) C)) 19.21v bitr2 x albii 3bitr x (/\ (e. (cv x) A) (e. (cv y) B)) (e. (cv y) C) 19.23v (cv y) x A B eliun x A (e. (cv y) B) df-rex bitr (e. (cv y) C) imbi1i bitr4 y albii x y (-> (/\ (e. (cv x) A) (e. (cv y) B)) (e. (cv y) C)) alcom (U_ x A B) C y dfss2 3bitr4 bitr2)) thm (ssiun ((x y) (C x) (C y) (A y) (B y)) () (-> (E.e. x A (C_ C B)) (C_ C (U_ x A B))) (x A (C_ C B) df-rex (e. (cv y) C) (e. (cv y) B) pm3.35 (e. (cv x) A) anim2i exp32 com23 imp C B (cv y) ssel sylan2 x 19.22i y 19.21aiv (cv y) x A B eliun x A (e. (cv y) B) df-rex bitr2 (e. (cv y) C) imbi2i y albii x (e. (cv y) C) (/\ (e. (cv x) A) (e. (cv y) B)) 19.37v y albii C (U_ x A B) y dfss2 3bitr4 sylib sylbi)) thm (ssiun2 ((A y) (B y) (x y)) () (-> (e. (cv x) A) (C_ B (U_ x A B))) (x A (e. (cv y) B) ra4e (cv y) x A B eliun sylibr ex ssrdv)) thm (ssiun2s ((x y) (A x) (A y) (B y) (C x) (C y) (D x) (D y)) ((ssiun2s.1 (-> (= (cv x) C) (= B D)))) (-> (e. C A) (C_ D (U_ x A B))) ((e. (cv y) C) x ax-17 (e. C A) x ax-17 (e. (cv y) D) x ax-17 y x A B hbiu1 hbss hbim (cv x) C A eleq1 ssiun2s.1 (U_ x A B) sseq1d imbi12d x A B ssiun2 A vtoclgf pm2.43i)) thm (iunss2 ((x y) (B x) (C y) (D x)) () (-> (A.e. x A (E.e. y B (C_ C D))) (C_ (U_ x A C) (U_ y B D))) (y B C D ssiun x A r19.20si x A C (U_ y B D) iunss sylibr)) thm (iunrab ((y z) (A y) (A z) (x y) (x z) (B x) (B y) (B z) (ph z)) () (= (U_ x A ({e.|} y B ph)) ({e.|} y B (E.e. x A ph))) (z B (E.e. x A ([/] (cv z) y ph)) df-rab (e. (cv x) B) y ax-17 (e. (cv x) B) z ax-17 (E.e. x A ph) z ax-17 (e. (cv x) A) y ax-17 z y ph hbs1 hbrex y z ph sbequ12 x A rexbidv cbvrab (cv z) x A ({e.|} y B ph) eliun (e. (cv x) B) y ax-17 (cv z) ph elrabsf x A rexbii x A (e. (cv z) B) ([/] (cv z) y ph) r19.42v 3bitr abbi2i 3eqtr4r)) thm (iunab ((A y) (x y)) () (= (U_ x A ({|} y ph)) ({|} y (E.e. x A ph))) (x A y (V) ph iunrab y ph rabab (e. (cv x) A) a1i iuneq2i y (E.e. x A ph) rabab 3eqtr3)) thm (iunxdif2 ((x y) (A x) (A y) (B x) (B y) (C y) (D x)) ((iunxdif2.1 (-> (= (cv x) (cv y)) (= C D)))) (-> (A.e. x A (E.e. y (\ A B) (C_ C D))) (= (U_ y (\ A B) D) (U_ x A C))) (x A y (\ A B) C D iunss2 A B difss (\ A B) A y D iunss1 ax-mp iunxdif2.1 A cbviunv sseqtr4 jctil (U_ y (\ A B) D) (U_ x A C) eqss sylibr)) thm (ssiin ((x y) (C x) (C y) (A x) (A y) (B y)) () (<-> (C_ C (|^|_ x A B)) (A.e. x A (C_ C B))) (y visset (cv y) (V) x A B eliin ax-mp (e. (cv y) C) imbi2i x A (e. (cv y) C) (e. (cv y) B) r19.21v x A (-> (e. (cv y) C) (e. (cv y) B)) df-ral 3bitr2 y albii C (|^|_ x A B) y dfss2 C B y dfss2 x A ralbii x A (A. y (-> (e. (cv y) C) (e. (cv y) B))) df-ral y (e. (cv x) A) (-> (e. (cv y) C) (e. (cv y) B)) 19.21v x albii x y (-> (e. (cv x) A) (-> (e. (cv y) C) (e. (cv y) B))) alcom bitr3 3bitr 3bitr4)) thm (iinss ((x y) (C x) (C y) (A x) (A y) (B y)) () (-> (E.e. x A (C_ B C)) (C_ (|^|_ x A B) C)) (x y (/\ (e. (cv x) A) (-> (e. (cv y) B) (e. (cv y) C))) 19.12 x A (A. y (-> (e. (cv y) B) (e. (cv y) C))) df-rex y (e. (cv x) A) (-> (e. (cv y) B) (e. (cv y) C)) 19.28v x exbii bitr4 x A (-> (e. (cv y) B) (e. (cv y) C)) df-rex y albii 3imtr4 x A (e. (cv y) B) (e. (cv y) C) r19.36av y visset (cv y) (V) x A B eliin ax-mp syl5ib y 19.20i syl B C y dfss2 x A rexbii (|^|_ x A B) C y dfss2 3imtr4)) thm (uniiun ((x y) (A x) (A y)) () (= (U. A) (U_ x A (cv x))) (A y x dfuni2 x A (cv x) y df-iun eqtr4)) thm (intiin ((x y) (A x) (A y)) () (= (|^| A) (|^|_ x A (cv x))) (A y x dfint2 x A (cv x) y df-iin eqtr4)) thm (iunid ((x y) (A x) (A y)) () (= (U_ x A ({} (cv x))) A) ((cv y) x A ({} (cv x)) eliun x A (e. (cv y) ({} (cv x))) df-rex (e. (cv x) A) (e. (cv y) ({} (cv x))) ancom y (cv x) elsn y x equcom bitr (e. (cv x) A) anbi1i bitr x exbii (e. (cv y) A) x ax-17 (cv x) (cv y) A eleq1 equsex 3bitr bitr eqriv)) thm (iun0 ((x y) (A y)) () (= (U_ x A ({/})) ({/})) ((cv y) x A ({/}) eliun (cv y) noel (e. (cv x) A) a1i nrex (cv y) noel 2false bitr eqriv)) thm (0iun ((x y) (A y)) () (= (U_ x ({/}) A) ({/})) ((cv y) x ({/}) A eliun x (e. (cv y) A) rex0 (cv y) noel 2false bitr eqriv)) thm (0iin ((x y) (A y)) () (= (|^|_ x ({/}) A) (V)) (x ({/}) A y df-iin y visset x (e. (cv y) A) ral0 2th abbi2i eqtr4)) thm (iunn0 ((x y) (A x) (A y) (B y)) () (<-> (E.e. x A (-. (= B ({/})))) (-. (= (U_ x A B) ({/})))) (B y n0 x A rexbii x A (E. y (e. (cv y) B)) df-rex x y (/\ (e. (cv x) A) (e. (cv y) B)) excom x y (e. (cv x) A) (e. (cv y) B) exdistr (cv y) x A B eliun x A (e. (cv y) B) df-rex bitr2 y exbii 3bitr3 (U_ x A B) y n0 bitr4 3bitr)) thm (iunin2 ((x y) (A x) (A y) (B x) (B y) (C y)) () (= (U_ x A (i^i B C)) (i^i B (U_ x A C))) (x A (e. (cv y) B) (e. (cv y) C) r19.42v (cv y) B C elin x A rexbii (cv y) x A C eliun (e. (cv y) B) anbi2i 3bitr4 (cv y) x A (i^i B C) eliun (cv y) B (U_ x A C) elin 3bitr4 eqriv)) thm (iinun2 ((x y) (A x) (A y) (B x) (B y) (C y)) () (= (|^|_ x A (u. B C)) (u. B (|^|_ x A C))) (x A (e. (cv y) B) (e. (cv y) C) r19.32v (cv y) B C elun x A ralbii y visset (cv y) (V) x A C eliin ax-mp (e. (cv y) B) orbi2i 3bitr4 y visset (cv y) (V) x A (u. B C) eliin ax-mp (cv y) B (|^|_ x A C) elun 3bitr4 eqriv)) thm (iundif2 ((x y) (A x) (A y) (B x) (B y) (C y)) () (= (U_ x A (\ B C)) (\ B (|^|_ x A C))) (x A (e. (cv y) B) (-. (e. (cv y) C)) r19.42v (cv y) B C eldif x A rexbii y visset (cv y) (V) x A C eliin ax-mp negbii x A (e. (cv y) C) rexnal bitr4 (e. (cv y) B) anbi2i 3bitr4 (cv y) x A (\ B C) eliun (cv y) B (|^|_ x A C) eldif 3bitr4 eqriv)) thm (iindif2OLD ((x y) (A x) (A y) (B x) (B y) (C y)) () (-> (-. (= A ({/}))) (= (|^|_ x A (\ B C)) (\ B (U_ x A C)))) (A x (e. (cv y) B) (-. (e. (cv y) C)) r19.28zvOLD (cv y) B C eldif x A ralbii (cv y) x A C eliun negbii x A (e. (cv y) C) ralnex bitr4 (e. (cv y) B) anbi2i 3bitr4g y visset (cv y) (V) x A (\ B C) eliin ax-mp (cv y) B (U_ x A C) eldif 3bitr4g eqrdv)) thm (iindif2 ((x y) (A x) (A y) (B x) (B y) (C y)) () (-> (=/= A ({/})) (= (|^|_ x A (\ B C)) (\ B (U_ x A C)))) (A x (e. (cv y) B) (-. (e. (cv y) C)) r19.28zv (cv y) B C eldif x A ralbii (cv y) x A C eliun negbii x A (e. (cv y) C) ralnex bitr4 (e. (cv y) B) anbi2i 3bitr4g y visset (cv y) (V) x A (\ B C) eliin ax-mp (cv y) B (U_ x A C) eldif 3bitr4g eqrdv)) thm (iunxsn ((x y) (A x) (A y) (B y) (C x) (C y)) ((iunxsn.1 (e. A (V))) (iunxsn.2 (-> (= (cv x) A) (= B C)))) (= (U_ x ({} A) B) C) ((cv y) x ({} A) B eliun x ({} A) (e. (cv y) B) df-rex x A elsn (e. (cv y) B) anbi1i x exbii iunxsn.1 iunxsn.2 (cv y) eleq2d ceqsexv bitr 3bitr eqriv)) thm (iunun ((x y) (A y) (B y) (C y)) () (= (U_ x A (u. B C)) (u. (U_ x A B) (U_ x A C))) ((cv y) B C elun x A rexbii x A (e. (cv y) B) (e. (cv y) C) r19.43 bitr y abbii y (E.e. x A (e. (cv y) B)) (E.e. x A (e. (cv y) C)) unab eqtr4 x A (u. B C) y df-iun x A B y df-iun x A C y df-iun uneq12i 3eqtr4)) thm (iunxun ((x y) (A y) (B y) (C y)) () (= (U_ x (u. A B) C) (u. (U_ x A C) (U_ x B C))) (x (u. A B) (e. (cv y) C) df-rex (cv x) A B elun (e. (cv y) C) anbi1i (e. (cv x) A) (e. (cv x) B) (e. (cv y) C) andir bitr x exbii x (/\ (e. (cv x) A) (e. (cv y) C)) (/\ (e. (cv x) B) (e. (cv y) C)) 19.43 (cv y) x A C eliun x A (e. (cv y) C) df-rex bitr (cv y) x B C eliun x B (e. (cv y) C) df-rex bitr orbi12i bitr4 3bitr (cv y) x (u. A B) C eliun (cv y) (U_ x A C) (U_ x B C) elun 3bitr4 eqriv)) thm (iinuni ((x y) (A x) (A y) (B x) (B y)) () (= (u. A (|^| B)) (|^|_ x B (u. A (cv x)))) (x B (e. (cv y) A) (e. (cv y) (cv x)) r19.32v (cv y) A (cv x) elun x B ralbii y visset B x elint2 (e. (cv y) A) orbi2i 3bitr4r (cv y) A (|^| B) elun y visset (cv y) (V) x B (u. A (cv x)) eliin ax-mp 3bitr4 eqriv)) thm (iununi ((x y) (A x) (A y) (B x) (B y)) () (<-> (-> (= B ({/})) (= A ({/}))) (= (u. A (U. B)) (U_ x B (u. A (cv x))))) ((= B ({/})) (= A ({/})) imor B x (e. (cv y) A) (e. (cv y) (cv x)) r19.45zvOLD (cv y) A n0i con2i (e. (cv y) A) (e. (cv y) (cv x)) biorf x B rexbidv (e. (cv y) A) (E.e. x B (e. (cv y) (cv x))) biorf bitr3d syl jaoi bicomd sylbi (cv y) A (cv x) elun x B rexbii syl6bbr (cv y) A (U. B) elun (cv y) B x eluni2 (e. (cv y) A) orbi2i bitr (cv y) x B (u. A (cv x)) eliun 3bitr4g eqrdv (u. A (U. B)) (U_ x B (u. A (cv x))) (cv y) eleq2 (cv y) B x eluni (e. (cv y) A) orbi2i (cv y) A (U. B) elun (e. (cv y) A) x ax-17 (/\ (e. (cv y) (cv x)) (e. (cv x) B)) 19.45 3bitr4 (cv y) x B (u. A (cv x)) eliun x B (e. (cv y) (u. A (cv x))) df-rex bitr 3bitr3g biimpd x (\/ (e. (cv y) A) (/\ (e. (cv y) (cv x)) (e. (cv x) B))) (/\ (e. (cv x) B) (e. (cv y) (u. A (cv x)))) 19.39 (e. (cv y) A) (/\ (e. (cv y) (cv x)) (e. (cv x) B)) orc (e. (cv x) B) (e. (cv y) (u. A (cv x))) pm3.26 imim12i x 19.22i 3syl x (e. (cv y) A) (e. (cv x) B) 19.37v sylib y 19.23adv A y n0 B x n0 3imtr4g a3d impbi)) thm (iinpw ((x y) (A x) (A y)) () (= (P~ (|^| A)) (|^|_ x A (P~ (cv x)))) ((cv y) A x ssint y visset (cv x) elpw x A ralbii bitr4 y visset (|^| A) elpw y visset (cv y) (V) x A (P~ (cv x)) eliin ax-mp 3bitr4 eqriv)) thm (iunpwss ((x y) (A x) (A y)) () (C_ (U_ x A (P~ (cv x))) (P~ (U. A))) (x A (cv y) (cv x) ssiun (cv y) x A (P~ (cv x)) eliun y visset (cv x) elpw x A rexbii bitr y visset (U. A) elpw A x uniiun (cv y) sseq2i bitr 3imtr4 ssriv)) thm (breq () () (-> (= R S) (<-> (br A R B) (br A S B))) (R S (<,> A B) eleq2 A R B df-br A S B df-br 3bitr4g)) thm (breq1 () () (-> (= A B) (<-> (br A R C) (br B R C))) (A B C opeq1 R eleq1d A R C df-br B R C df-br 3bitr4g)) thm (breq2 () () (-> (= A B) (<-> (br C R A) (br C R B))) (A B C opeq2 R eleq1d C R A df-br C R B df-br 3bitr4g)) thm (breq12 () () (-> (/\ (= A B) (= C D)) (<-> (br A R C) (br B R D))) (A B R C breq1 C D B R breq2 sylan9bb)) thm (breqi () ((breqi.1 (= R S))) (<-> (br A R B) (br A S B)) (breqi.1 R S A B breq ax-mp)) thm (breq1i () ((breq1i.1 (= A B))) (<-> (br A R C) (br B R C)) (breq1i.1 A B R C breq1 ax-mp)) thm (breq2i () ((breq1i.1 (= A B))) (<-> (br C R A) (br C R B)) (breq1i.1 A B C R breq2 ax-mp)) thm (breq12i () ((breq1i.1 (= A B)) (breq12i.2 (= C D))) (<-> (br A R C) (br B R D)) (breq1i.1 R C breq1i breq12i.2 B R breq2i bitr)) thm (breq1d () ((breq1d.1 (-> ph (= A B)))) (-> ph (<-> (br A R C) (br B R C))) (breq1d.1 A B R C breq1 syl)) thm (breq2d () ((breq1d.1 (-> ph (= A B)))) (-> ph (<-> (br C R A) (br C R B))) (breq1d.1 A B C R breq2 syl)) thm (breq12d () ((breq1d.1 (-> ph (= A B))) (breq12d.2 (-> ph (= C D)))) (-> ph (<-> (br A R C) (br B R D))) (breq1d.1 R C breq1d breq12d.2 B R breq2d bitrd)) thm (breqan12d () ((breq1d.1 (-> ph (= A B))) (breqan12i.2 (-> ps (= C D)))) (-> (/\ ph ps) (<-> (br A R C) (br B R D))) (A B C D R breq12 breq1d.1 breqan12i.2 syl2an)) thm (breqan12rd () ((breq1d.1 (-> ph (= A B))) (breqan12i.2 (-> ps (= C D)))) (-> (/\ ps ph) (<-> (br A R C) (br B R D))) (breq1d.1 breqan12i.2 R breqan12d ancoms)) thm (eqbrtr () ((eqbrtr.1 (= A B)) (eqbrtr.2 (br B R C))) (br A R C) (eqbrtr.2 eqbrtr.1 R C breq1i mpbir)) thm (eqbrtrd () ((eqbrtrd.1 (-> ph (= A B))) (eqbrtrd.2 (-> ph (br B R C)))) (-> ph (br A R C)) (eqbrtrd.2 eqbrtrd.1 R C breq1d mpbird)) thm (eqbrtrr () ((eqbrtrr.1 (= A B)) (eqbrtrr.2 (br A R C))) (br B R C) (eqbrtrr.1 eqcomi eqbrtrr.2 eqbrtr)) thm (eqbrtrrd () ((eqbrtrrd.1 (-> ph (= A B))) (eqbrtrrd.2 (-> ph (br A R C)))) (-> ph (br B R C)) (eqbrtrrd.1 eqcomd eqbrtrrd.2 eqbrtrd)) thm (breqtr () ((breqtr.1 (br A R B)) (breqtr.2 (= B C))) (br A R C) (breqtr.1 breqtr.2 A R breq2i mpbi)) thm (breqtrd () ((breqtrd.1 (-> ph (br A R B))) (breqtrd.2 (-> ph (= B C)))) (-> ph (br A R C)) (breqtrd.1 breqtrd.2 A R breq2d mpbid)) thm (breqtrr () ((breqtrr.1 (br A R B)) (breqtrr.2 (= C B))) (br A R C) (breqtrr.1 breqtrr.2 eqcomi breqtr)) thm (breqtrrd () ((breqtrrd.1 (-> ph (br A R B))) (breqtrrd.2 (-> ph (= C B)))) (-> ph (br A R C)) (breqtrrd.1 breqtrrd.2 eqcomd breqtrd)) thm (3brtr3 () ((3brtr3.1 (br A R B)) (3brtr3.2 (= A C)) (3brtr3.3 (= B D))) (br C R D) (3brtr3.2 3brtr3.1 eqbrtrr 3brtr3.3 breqtr)) thm (3brtr4 () ((3brtr4.1 (br A R B)) (3brtr4.2 (= C A)) (3brtr4.3 (= D B))) (br C R D) (3brtr4.2 3brtr4.1 eqbrtr 3brtr4.3 breqtrr)) thm (3brtr3d () ((3brtr3d.1 (-> ph (br A R B))) (3brtr3d.2 (-> ph (= A C))) (3brtr3d.3 (-> ph (= B D)))) (-> ph (br C R D)) (3brtr3d.1 3brtr3d.2 3brtr3d.3 R breq12d mpbid)) thm (3brtr4d () ((3brtr4d.1 (-> ph (br A R B))) (3brtr4d.2 (-> ph (= C A))) (3brtr4d.3 (-> ph (= D B)))) (-> ph (br C R D)) (3brtr4d.1 3brtr4d.2 3brtr4d.3 R breq12d mpbird)) thm (3brtr3g () ((3brtr3g.1 (-> ph (br A R B))) (3brtr3g.2 (= A C)) (3brtr3g.3 (= B D))) (-> ph (br C R D)) (3brtr3g.1 3brtr3g.2 3brtr3g.3 R breq12i sylib)) thm (3brtr4g () ((3brtr4g.1 (-> ph (br A R B))) (3brtr4g.2 (= C A)) (3brtr4g.3 (= D B))) (-> ph (br C R D)) (3brtr4g.1 3brtr4g.2 3brtr4g.3 R breq12i sylibr)) thm (syl5eqbr () ((syl5eqbr.1 (-> ph (br A R B))) (syl5eqbr.2 (= C A))) (-> ph (br C R B)) (syl5eqbr.1 syl5eqbr.2 B eqid 3brtr4g)) thm (syl5eqbrr () ((syl5eqbrr.1 (-> ph (br A R B))) (syl5eqbrr.2 (= A C))) (-> ph (br C R B)) (syl5eqbrr.1 syl5eqbrr.2 B eqid 3brtr3g)) thm (syl5breq () ((syl5breq.1 (-> ph (= A B))) (syl5breq.2 (br C R A))) (-> ph (br C R B)) (syl5breq.2 ph a1i syl5breq.1 breqtrd)) thm (syl5breqr () ((syl5breqr.1 (-> ph (= B A))) (syl5breqr.2 (br C R A))) (-> ph (br C R B)) (syl5breqr.1 eqcomd syl5breqr.2 syl5breq)) thm (syl6eqbr () ((syl6eqbr.1 (-> ph (= A B))) (syl6eqbr.2 (br B R C))) (-> ph (br A R C)) (syl6eqbr.2 syl6eqbr.1 R C breq1d mpbiri)) thm (syl6eqbrr () ((syl6eqbrr.1 (-> ph (= B A))) (syl6eqbrr.2 (br B R C))) (-> ph (br A R C)) (syl6eqbrr.1 eqcomd syl6eqbrr.2 syl6eqbr)) thm (syl6breq () ((syl6breq.1 (-> ph (br A R B))) (syl6breq.2 (= B C))) (-> ph (br A R C)) (syl6breq.1 A eqid syl6breq.2 3brtr3g)) thm (syl6breqr () ((syl6breqr.1 (-> ph (br A R B))) (syl6breqr.2 (= C B))) (-> ph (br A R C)) (syl6breqr.1 syl6breqr.2 eqcomi syl6breq)) thm (ssbrd () ((ssbrd.1 (-> ph (C_ A B)))) (-> ph (-> (br C A D) (br C B D))) (ssbrd.1 (<,> C D) sseld C A D df-br C B D df-br 3imtr4g)) thm (hbbr ((A y) (B y) (R y) (x y)) ((hbbr.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbbr.2 (-> (e. (cv y) R) (A. x (e. (cv y) R)))) (hbbr.3 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (br A R B) (A. x (br A R B))) (hbbr.1 hbbr.3 hbop hbbr.2 hbel A R B df-br A R B df-br x albii 3imtr4)) thm (hbbrd ((y z) (A y) (A z) (B y) (B z) (R y) (R z) (x y) (x z) (ph y)) ((hbbrd.1 (-> ph (A. x ph))) (hbbrd.2 (-> ph (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (hbbrd.3 (-> ph (-> (e. (cv y) R) (A. x (e. (cv y) R))))) (hbbrd.4 (-> ph (-> (e. (cv y) B) (A. x (e. (cv y) B)))))) (-> ph (-> (br A R B) (A. x (br A R B)))) (x (e. (cv z) A) hba1 y z hbab x (e. (cv z) R) hba1 y z hbab x (e. (cv z) B) hba1 y z hbab hbbr ph a1i hbbrd.2 y 19.21aiv y A x z abidhb syl hbbrd.4 y 19.21aiv y B x z abidhb syl ({|} z (A. x (e. (cv z) R))) breq12d hbbrd.3 y 19.21aiv y R x z abidhb syl ({|} z (A. x (e. (cv z) R))) R A B breq syl bitrd hbbrd.1 hbbrd.2 y 19.21aiv y A x z abidhb syl hbbrd.4 y 19.21aiv y B x z abidhb syl ({|} z (A. x (e. (cv z) R))) breq12d hbbrd.3 y 19.21aiv y R x z abidhb syl ({|} z (A. x (e. (cv z) R))) R A B breq syl bitrd albid 3imtr3d)) thm (brab1 ((x z) (y z) (R z)) () (<-> (br (cv x) R (cv y)) (e. (cv x) ({|} z (br (cv z) R (cv y))))) (x visset (cv z) (cv x) R (cv y) breq1 elab bicomi)) thm (brprc () () (-> (-. (e. B (V))) (<-> (br A R B) (br A R A))) (B A opprc2 R eleq1d A R B df-br A R A df-br 3bitr4g)) thm (sbcbrg ((y z) (A y) (A z) (B y) (B z) (C y) (C z) (D y) (D z) (R y) (R z) (x y) (x z)) () (-> (e. A D) (<-> ([/] A x (br B R C)) (br ([_/]_ A x B) ([_/]_ A x R) ([_/]_ A x C)))) ((e. A D) y ax-17 (e. (cv z) A) y ax-17 D ([_/]_ (cv y) x B) hbcsb1g (e. (cv z) A) y ax-17 D ([_/]_ (cv y) x R) hbcsb1g (e. (cv z) A) y ax-17 D ([_/]_ (cv y) x C) hbcsb1g hbbrd x y a9e y visset (e. (cv z) (cv y)) x ax-17 (V) (br B R C) hbsbc1g ax-mp y visset (e. (cv z) (cv y)) x ax-17 B hbcsb1 y visset (e. (cv z) (cv y)) x ax-17 R hbcsb1 y visset (e. (cv z) (cv y)) x ax-17 C hbcsb1 hbbr hbbi x (cv y) B csbeq1a x (cv y) C csbeq1a R breq12d x (cv y) (br B R C) sbceq1a x (cv y) R csbeq1a R ([_/]_ (cv y) x R) ([_/]_ (cv y) x B) ([_/]_ (cv y) x C) breq syl 3bitr3d 19.23ai ax-mp (= (cv y) A) a1i y A ([_/]_ (cv y) x B) csbeq1a y A ([_/]_ (cv y) x C) csbeq1a ([_/]_ (cv y) x R) breq12d y A ([_/]_ (cv y) x R) csbeq1a ([_/]_ (cv y) x R) ([_/]_ A y ([_/]_ (cv y) x R)) ([_/]_ A y ([_/]_ (cv y) x B)) ([_/]_ A y ([_/]_ (cv y) x C)) breq syl 3bitrd sbciegf A D y x (br B R C) sbccog A D y x B csbcog A D y x C csbcog ([_/]_ A y ([_/]_ (cv y) x R)) breq12d A D y x R csbcog ([_/]_ A y ([_/]_ (cv y) x R)) ([_/]_ A x R) ([_/]_ A x B) ([_/]_ A x C) breq syl bitrd 3bitr3d)) thm (sbcbr12g ((A y) (C y) (D y) (x y) (R x) (R y)) () (-> (e. A D) (<-> ([/] A x (br B R C)) (br ([_/]_ A x B) R ([_/]_ A x C)))) (A D x B R C sbcbrg (e. (cv y) R) x ax-17 A D csbconstgf ([_/]_ A x R) R ([_/]_ A x B) ([_/]_ A x C) breq syl bitrd)) thm (sbcbr1g ((A y) (x y) (C x) (C y) (D y) (R x) (R y)) () (-> (e. A D) (<-> ([/] A x (br B R C)) (br ([_/]_ A x B) R C))) (A D x B R C sbcbr12g (e. (cv y) C) x ax-17 A D csbconstgf ([_/]_ A x B) R breq2d bitrd)) thm (sbcbr2g ((A y) (x y) (B x) (B y) (D y) (R x) (R y)) () (-> (e. A D) (<-> ([/] A x (br B R C)) (br B R ([_/]_ A x C)))) (A D x B R C sbcbr12g (e. (cv y) B) x ax-17 A D csbconstgf R ([_/]_ A x C) breq1d bitrd)) thm (opabss ((x z) (R x) (R z) (y z) (R y)) () (C_ ({<,>|} x y (br (cv x) R (cv y))) R) (x y (br (cv x) R (cv y)) z df-opab (cv z) (<,> (cv x) (cv y)) R eleq1 biimpar (cv x) R (cv y) df-br sylan2b x y 19.23aivv z ss2abi eqsstr z R abid2 sseqtr)) thm (opabbid ((x z) (y z) (ph z) (ps z) (ch z)) ((opabbid.1 (-> ph (A. x ph))) (opabbid.2 (-> ph (A. y ph))) (opabbid.3 (-> ph (<-> ps ch)))) (-> ph (= ({<,>|} x y ps) ({<,>|} x y ch))) (ph z ax-17 opabbid.1 opabbid.2 opabbid.3 (= (cv z) (<,> (cv x) (cv y))) anbi2d exbid exbid abbid x y ps z df-opab x y ch z df-opab 3eqtr4g)) thm (opabbidv ((x y) (ph x) (ph y)) ((opabbidv.1 (-> ph (<-> ps ch)))) (-> ph (= ({<,>|} x y ps) ({<,>|} x y ch))) (ph x ax-17 ph y ax-17 opabbidv.1 opabbid)) thm (opabbii ((x y) (x z) (y z) (ph z) (ps z)) ((opabbii.1 (<-> ph ps))) (= ({<,>|} x y ph) ({<,>|} x y ps)) ((cv z) eqid opabbii.1 (= (cv z) (cv z)) a1i x y opabbidv ax-mp)) thm (cbvopab ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph v) (ps v)) ((cbvopab.1 (-> ph (A. z ph))) (cbvopab.2 (-> ph (A. w ph))) (cbvopab.3 (-> ps (A. x ps))) (cbvopab.4 (-> ps (A. y ps))) (cbvopab.5 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (= ({<,>|} x y ph) ({<,>|} z w ps)) ((= (cv v) (<,> (cv x) (cv y))) z ax-17 cbvopab.1 hban (= (cv v) (<,> (cv x) (cv y))) w ax-17 cbvopab.2 hban (= (cv v) (<,> (cv z) (cv w))) x ax-17 cbvopab.3 hban (= (cv v) (<,> (cv z) (cv w))) y ax-17 cbvopab.4 hban (cv x) (cv z) (cv y) (cv w) opeq12 (cv v) eqeq2d cbvopab.5 anbi12d cbvex2 v abbii x y ph v df-opab z w ps v df-opab 3eqtr4)) thm (cbvopabv ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w) (ps x) (ps y)) ((cbvopabv.1 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (= ({<,>|} x y ph) ({<,>|} z w ps)) (ph z ax-17 ph w ax-17 ps x ax-17 ps y ax-17 cbvopabv.1 cbvopab)) thm (cbvopab1 ((x y) (x z) (w x) (y z) (w y) (w z) (ph w) (ps w)) ((cbvopab1.1 (-> ph (A. z ph))) (cbvopab1.2 (-> ps (A. x ps))) (cbvopab1.3 (-> (= (cv x) (cv z)) (<-> ph ps)))) (= ({<,>|} x y ph) ({<,>|} z y ps)) ((= (cv w) (<,> (cv x) (cv y))) z ax-17 cbvopab1.1 hban y hbex (= (cv w) (<,> (cv z) (cv y))) x ax-17 cbvopab1.2 hban y hbex (cv x) (cv z) (cv y) opeq1 (cv w) eqeq2d cbvopab1.3 anbi12d y exbidv cbvex w abbii x y ph w df-opab z y ps w df-opab 3eqtr4)) thm (cbvopab1s ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (= ({<,>|} x y ph) ({<,>|} z y ([/] (cv z) x ph))) ((E. y (/\ (= (cv w) (<,> (cv x) (cv y))) ph)) z ax-17 (= (cv w) (<,> (cv z) (cv y))) x ax-17 z x ph hbs1 hban y hbex (cv x) (cv z) (cv y) opeq1 (cv w) eqeq2d x z ph sbequ12 anbi12d y exbidv cbvex w abbii x y ph w df-opab z y ([/] (cv z) x ph) w df-opab 3eqtr4)) thm (cbvopab1v ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w) (ps x) (ps w)) ((cbvopab1v.1 (-> (= (cv x) (cv z)) (<-> ph ps)))) (= ({<,>|} x y ph) ({<,>|} z y ps)) ((cv x) (cv z) (cv y) opeq1 (cv w) eqeq2d cbvopab1v.1 anbi12d y exbidv cbvexv w abbii x y ph w df-opab z y ps w df-opab 3eqtr4)) thm (cbvopab2v ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w) (ps y) (ps w)) ((cbvopab2v.1 (-> (= (cv y) (cv z)) (<-> ph ps)))) (= ({<,>|} x y ph) ({<,>|} x z ps)) ((cv y) (cv z) (cv x) opeq2 (cv w) eqeq2d cbvopab2v.1 anbi12d cbvexv x exbii w abbii x y ph w df-opab x z ps w df-opab 3eqtr4)) thm (unopab ((x z) (y z) (ph z) (ps z)) () (= (u. ({<,>|} x y ph) ({<,>|} x y ps)) ({<,>|} x y (\/ ph ps))) (z (E. x (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ph))) (E. x (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ps))) unab x (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ph)) (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ps)) 19.43 (= (cv z) (<,> (cv x) (cv y))) ph ps andi y exbii y (/\ (= (cv z) (<,> (cv x) (cv y))) ph) (/\ (= (cv z) (<,> (cv x) (cv y))) ps) 19.43 bitr2 x exbii bitr3 z abbii eqtr x y ph z df-opab x y ps z df-opab uneq12i x y (\/ ph ps) z df-opab 3eqtr4)) thm (dftr2 ((x y) (A x) (A y)) () (<-> (Tr A) (A. x (A. y (-> (/\ (e. (cv x) (cv y)) (e. (cv y) A)) (e. (cv x) A))))) ((U. A) A x dfss2 A df-tr y (/\ (e. (cv x) (cv y)) (e. (cv y) A)) (e. (cv x) A) 19.23v (cv x) A y eluni (e. (cv x) A) imbi1i bitr4 x albii 3bitr4)) thm (dftr5 ((x y) (A x) (A y)) () (<-> (Tr A) (A.e. x A (A.e. y (cv x) (e. (cv y) A)))) (A y x dftr2 y x (-> (/\ (e. (cv y) (cv x)) (e. (cv x) A)) (e. (cv y) A)) alcom (e. (cv y) (cv x)) (e. (cv x) A) (e. (cv y) A) impexp y albii y (cv x) (-> (e. (cv x) A) (e. (cv y) A)) df-ral y (cv x) (e. (cv x) A) (e. (cv y) A) r19.21v 3bitr2 x albii x A (A.e. y (cv x) (e. (cv y) A)) df-ral bitr4 3bitr)) thm (dftr3 ((x y) (A x) (A y)) () (<-> (Tr A) (A.e. x A (C_ (cv x) A))) (A x y dftr5 (cv x) A y dfss3 x A ralbii bitr4)) thm (dftr4 ((A x)) () (<-> (Tr A) (C_ A (P~ A))) (x visset A elpw x A ralbii A (P~ A) x dfss3 A x dftr3 3bitr4r)) thm (treq () () (-> (= A B) (<-> (Tr A) (Tr B))) (A B unieq A sseq1d A B (U. B) sseq2 bitrd A df-tr B df-tr 3bitr4g)) thm (trel ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (Tr A) (-> (/\ (e. B C) (e. C A)) (e. B A))) ((cv x) B C eleq1 (cv x) B A eleq1 (e. C A) imbi2d imbi12d (Tr A) imbi2d (cv y) C (cv x) eleq2 (cv y) C A eleq1 (e. (cv x) A) imbi1d imbi12d (Tr A) imbi2d A x y dftr2 biimp 19.21bbi exp3a A vtoclg com4l (e. C A) (e. (cv x) A) pm2.43 syl6 C vtoclg pm2.43b imp3a)) thm (trel3 () () (-> (Tr A) (-> (/\/\ (e. B C) (e. C D) (e. D A)) (e. B A))) (A C D trel (e. B C) anim2d (e. B C) (e. C D) (e. D A) 3anass syl5ib A B C trel syld)) thm (trss ((A x) (B x)) () (-> (Tr A) (-> (e. B A) (C_ B A))) ((cv x) B A eleq1 (cv x) B A sseq1 imbi12d (Tr A) imbi2d A x dftr3 x A (C_ (cv x) A) ra4 sylbi A vtoclg pm2.43b)) thm (trin ((A x) (B x)) () (-> (/\ (Tr A) (Tr B)) (Tr (i^i A B))) (A (cv x) trss B (cv x) trss im2anan9 (cv x) A B elin syl5ib (cv x) A B ssin syl6ib r19.21aiv (i^i A B) x dftr3 sylibr)) thm (tr0 () () (Tr ({/})) ((P~ ({/})) 0ss ({/}) dftr4 mpbir)) thm (trv () () (Tr (V)) ((U. (V)) ssv (V) df-tr mpbir)) thm (axrep1 ((w y) (ph w) (ph y) (w x) (w z) (x y) (x z) (y z)) () (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv y)) ph)))))) (w y x elequ2 (A. y ph) anbi1d x exbidv (e. (cv z) (cv x)) bibi2d z albidv x exbidv (A. x (E. y (A. z (-> ph (= (cv z) (cv y)))))) imbi2d y ph ax-4 (= (cv z) (cv y)) imim1i z 19.20i y 19.22i x 19.20i x y z ph w ax-rep syl (e. (cv z) (cv y)) x ax-17 x (/\ (e. (cv x) (cv w)) (A. y ph)) hbe1 hbbi z hbal (e. (cv z) (cv x)) y ax-17 (e. (cv x) (cv w)) y ax-17 y ph hba1 hban x hbex hbbi z hbal y x z elequ2 (E. x (/\ (e. (cv x) (cv w)) (A. y ph))) bibi1d z albidv cbvex sylib chvarv 19.35ri y ph ax-4 ph y ax-17 impbi (e. (cv x) (cv y)) anbi2i x exbii (e. (cv z) (cv x)) bibi2i z albii (E. y (A. z (-> ph (= (cv z) (cv y))))) imbi2i x exbii mpbi)) thm (axrep2 ((ph w) (w x) (w y) (w z) (x y) (x z) (y z)) () (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv y)) (A. y ph))))))) (w (A. z (-> (A. y ph) (= (cv z) (cv w)))) hbe1 (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv y)) (A. y ph))))) w ax-17 hbim x hbex w y x elequ2 (A. y ph) anbi1d x exbidv (e. (cv z) (cv x)) bibi2d z albidv (E. w (A. z (-> (A. y ph) (= (cv z) (cv w))))) imbi2d x exbidv x w z (A. y ph) axrep1 chvar y ph ax-4 (= (cv z) (cv y)) imim1i z 19.20i y 19.22i (A. z (-> (A. y ph) (= (cv z) (cv y)))) w ax-17 y ph hba1 (= (cv z) (cv w)) y ax-17 hbim z hbal y w z equequ2 (A. y ph) imbi2d z albidv cbvex sylib (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv y)) (A. y ph))))) imim1i x 19.22i ax-mp)) thm (axrep3 ((w x) (w y) (w z) (x y) (x z) (y z)) () (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv w)) (A. y ph))))))) (y (A. z (-> ph (= (cv z) (cv y)))) hbe1 (e. (cv z) (cv x)) y ax-17 (e. (cv x) (cv w)) y ax-17 y ph hba1 hban x hbex hbbi z hbal hbim x hbex y w x elequ2 (A. y ph) anbi1d x exbidv (e. (cv z) (cv x)) bibi2d z albidv (E. y (A. z (-> ph (= (cv z) (cv y))))) imbi2d x exbidv x y z ph axrep2 chvar)) thm (axrep4 ((x y) (x z) (w x) (y z) (w y) (w z)) ((axrep4.1 (-> ph (A. z ph)))) (-> (A. x (E. z (A. y (-> ph (= (cv y) (cv z)))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) (cv w)) ph)))))) (x z y ph w axrep3 19.35i (e. (cv y) (cv x)) z ax-17 (e. (cv x) (cv w)) z ax-17 z ph hba1 hban x hbex hbbi y hbal (e. (cv y) (cv z)) x ax-17 x (/\ (e. (cv x) (cv w)) ph) hbe1 hbbi y hbal (= (cv x) (cv z)) y ax-17 x z y elequ2 z ph ax-4 axrep4.1 impbi (e. (cv x) (cv w)) anbi2i x exbii (= (cv x) (cv z)) a1i bibi12d albid cbvex sylib)) thm (axrep5 ((x y) (x z) (w x) (y z) (w y) (w z)) ((axrep5.1 (-> ph (A. z ph)))) (-> (A. x (-> (e. (cv x) (cv w)) (E. z (A. y (-> ph (= (cv y) (cv z))))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) (cv w)) ph)))))) (z (e. (cv x) (cv w)) (A. y (-> ph (= (cv y) (cv z)))) 19.37v (e. (cv x) (cv w)) ph (= (cv y) (cv z)) impexp y albii y (e. (cv x) (cv w)) (-> ph (= (cv y) (cv z))) 19.21v bitr2 z exbii bitr3 x albii (e. (cv x) (cv w)) z ax-17 axrep5.1 hban x y w axrep4 sylbi (e. (cv x) (cv w)) ph anabs5 x exbii (e. (cv y) (cv z)) bibi2i y albii z exbii sylib)) thm (zfrepclf ((y z) (A y) (v y) (A z) (v z) (A v) (ph z) (ph v) (A w) (v w) (x y) (x z) (v x) (w x)) ((zfrepclf.1 (-> (e. (cv w) A) (A. x (e. (cv w) A)))) (zfrepclf.2 (e. A (V))) (zfrepclf.3 (-> (e. (cv x) A) (E. z (A. y (-> ph (= (cv y) (cv z)))))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) A) ph))))) (zfrepclf.2 (e. (cv w) (cv v)) x ax-17 zfrepclf.1 hbeq (cv v) A (cv x) eleq2 zfrepclf.3 syl6bi 19.21ai ph z ax-17 x v y axrep5 syl (e. (cv w) (cv v)) x ax-17 zfrepclf.1 hbeq (cv v) A (cv x) eleq2 ph anbi1d exbid (e. (cv y) (cv z)) bibi2d y albidv z exbidv mpbid vtocle)) thm (zfrep3cl ((x y) (x z) (A x) (y z) (A y) (A z) (ph z)) ((zfrep3cl.1 (e. A (V))) (zfrep3cl.2 (-> (e. (cv x) A) (E. z (A. y (-> ph (= (cv y) (cv z)))))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) A) ph))))) ((e. (cv y) A) x ax-17 zfrep3cl.1 zfrep3cl.2 zfrepclf)) thm (zfrep4 ((ph y) (ph z) (y z) (ps z) (x y) (x z)) ((zfrep4.1 (e. ({|} x ph) (V))) (zfrep4.2 (-> ph (E. z (A. y (-> ps (= (cv y) (cv z)))))))) (e. ({|} y (E. x (/\ ph ps))) (V)) (x ph abid ps anbi1i x exbii y abbii y x ph hbab1 zfrep4.1 x ph abid zfrep4.2 sylbi zfrepclf (cv z) y (E. x (/\ (e. (cv x) ({|} x ph)) ps)) abeq2 z exbii mpbir issetri eqeltrr)) thm (zfaus ((x y) (x z) (w x) (y z) (w y) (w z) (ph y) (ph w)) () (E. y (A. x (<-> (e. (cv x) (cv y)) (/\ (e. (cv x) (cv z)) ph)))) (y w a9e y w x equtr y x equcomi syl6 ph adantrd x 19.21aiv y 19.22i ax-mp (e. (cv w) (cv z)) a1i w ax-gen (/\ (= (cv w) (cv x)) ph) y ax-17 w z x axrep5 ax-mp (= (cv w) (cv x)) (e. (cv w) (cv z)) ph an12 w exbii (/\ (e. (cv x) (cv z)) ph) w ax-17 w x z elequ1 ph anbi1d equsex bitr3 (e. (cv x) (cv y)) bibi2i x albii y exbii mpbi)) thm (zfauscl ((x y) (A x) (x z) (A y) (y z) (A z) (ph y) (ph z)) ((zfauscl.1 (e. A (V)))) (E. y (A. x (<-> (e. (cv x) (cv y)) (/\ (e. (cv x) A) ph)))) (zfauscl.1 (cv z) A (cv x) eleq2 ph anbi1d (e. (cv x) (cv y)) bibi2d x albidv y exbidv y x z ph zfaus vtocl)) thm (bm1.3ii ((ph x) (x z) (ph z) (x y) (y z)) ((bm1.3ii.1 (E. x (A. y (-> ph (e. (cv y) (cv x))))))) (E. x (A. y (<-> (e. (cv y) (cv x)) ph))) (bm1.3ii.1 x z y elequ2 ph imbi2d y albidv cbvexv mpbi x y z ph zfaus pm3.2i exan x (A. y (-> ph (e. (cv y) (cv z)))) (A. y (<-> (e. (cv y) (cv x)) (/\ (e. (cv y) (cv z)) ph))) 19.42v y (-> ph (e. (cv y) (cv z))) (<-> (e. (cv y) (cv x)) (/\ (e. (cv y) (cv z)) ph)) 19.26 ph (e. (cv y) (cv z)) (e. (cv y) (cv x)) bimsc1 y 19.20i sylbir x 19.22i sylbir z 19.23aiv ax-mp)) thm (zfnul ((x y) (x z) (y z)) () (E. x (A. y (-. (e. (cv y) (cv x))))) (x y z (-. (e. (cv y) (cv z))) zfaus (e. (cv y) (cv z)) pm3.24 (<-> (e. (cv y) (cv x)) (/\ (e. (cv y) (cv z)) (-. (e. (cv y) (cv z))))) id mtbiri y 19.20i x 19.22i ax-mp)) thm (zfnuleu ((x y)) () (E! x (A. y (-. (e. (cv y) (cv x))))) (x y zfnul y equid (e. (cv y) (cv x)) nbn3 y albii x exbii mpbi (-. (= (cv y) (cv y))) x ax-17 y bm1.1 ax-mp y equid (e. (cv y) (cv x)) nbn3 y albii x eubii mpbir)) thm (0ex ((x y)) () (e. ({/}) (V)) (x y zfnuleu (cv x) y eq0 x eubii mpbir ({/}) x eueq mpbir)) thm (nalset ((x y) (x z) (y z)) () (-. (E. x (A. y (e. (cv y) (cv x))))) (x y (e. (cv y) (cv x)) alexn x visset y z (-. (e. (cv z) (cv z))) zfauscl z y y elequ1 z y x elequ1 z y z elequ1 z y y elequ2 bitrd negbid anbi12d bibi12d a4b1 (e. (cv y) (cv y)) (e. (cv y) (cv x)) pclem6 syl y 19.22i ax-mp mpgbi)) thm (nvelv ((x y)) () (-. (e. (V) (V))) (x y nalset y visset (e. (cv y) (cv x)) tbt y albii (cv x) (V) y dfcleq bitr4 x exbii mtbi (V) x isset mtbir)) thm (vnex () () (-. (E. x (= (cv x) (V)))) (nvelv (V) x isset mtbi)) thm (inex1 ((A x) (A y) (x y) (B x) (B y)) ((inex1.1 (e. A (V)))) (e. (i^i A B) (V)) (inex1.1 x y (e. (cv y) B) zfauscl (cv x) (i^i A B) y dfcleq (cv y) A B elin (e. (cv y) (cv x)) bibi2i y albii bitr x exbii mpbir issetri)) thm (inex2 () ((inex2.1 (e. A (V)))) (e. (i^i B A) (V)) (B A incom inex2.1 B inex1 eqeltr)) thm (inex1g ((A x) (B x)) () (-> (e. A C) (e. (i^i A B) (V))) ((cv x) A B ineq1 (V) eleq1d x visset B inex1 C vtoclg)) thm (ssex () ((ssex.1 (e. B (V)))) (-> (C_ A B) (e. A (V))) (A B df-ss ssex.1 A inex2 (i^i A B) A (V) eleq1 mpbii sylbi)) thm (ssexi () ((ssexi.1 (e. B (V))) (ssexi.2 (C_ A B))) (e. A (V)) (ssexi.2 ssexi.1 A ssex ax-mp)) thm (ssexg ((A x) (B x)) () (-> (/\ (C_ A B) (e. B C)) (e. A (V))) ((cv x) B A sseq2 (e. A (V)) imbi1d x visset A ssex C vtoclg impcom)) thm (difexg () () (-> (e. A C) (e. (\ A B) (V))) (A B difss (\ A B) A C ssexg mpan)) thm (zfausab ((A x)) ((zfausab.1 (e. A (V)))) (e. ({|} x (/\ (e. (cv x) A) ph)) (V)) (zfausab.1 x A ph ssab2 ssexi)) thm (rabexg ((A x)) () (-> (e. A B) (e. ({e.|} x A ph) (V))) (x A ph ssrab2 ({e.|} x A ph) A B ssexg mpan)) thm (rabex ((A x)) ((rabex.1 (e. A (V)))) (e. ({e.|} x A ph) (V)) (rabex.1 A (V) x ph rabexg ax-mp)) thm (elssab ((A x) (B x) (ps x)) ((elssab.1 (e. B (V))) (elssab.2 (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({|} x (/\ (C_ (cv x) B) ph))) (/\ (C_ A B) ps)) (elssab.1 A ssex ps adantr (cv x) A B sseq1 elssab.2 anbi12d elab3)) thm (elpw2g () () (-> (e. B C) (<-> (e. A (P~ B)) (C_ A B))) (A (P~ B) B elpwg ibi (e. B C) a1i A B C ssexg A (V) B elpwg biimparc syldan expcom impbid)) thm (intex ((A x)) () (<-> (-. (= A ({/}))) (e. (|^| A) (V))) (A x n0 (cv x) A intss1 x visset (|^| A) ssex syl x 19.23aiv sylbi nvelv A ({/}) inteq int0 syl6eq (V) eleq1d mtbiri con2i impbi)) thm (intnex () () (<-> (-. (e. (|^| A) (V))) (= (|^| A) (V))) (A intex con1bii A ({/}) inteq int0 syl6eq sylbi nvelv (|^| A) (V) (V) eleq1 mtbiri impbi)) thm (intexab () () (<-> (E. x ph) (e. (|^| ({|} x ph)) (V))) (x ph abn0 ({|} x ph) intex bitr3)) thm (intexrab () () (<-> (E.e. x A ph) (e. (|^| ({e.|} x A ph)) (V))) (x (/\ (e. (cv x) A) ph) intexab x A ph df-rex x A ph df-rab inteqi (V) eleq1i 3bitr4)) thm (intabs ((x y) (A x) (ph y) (ps x) (ch x)) ((intabs.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (intabs.2 (-> (= (cv x) (|^| ({|} y ps))) (<-> ph ch))) (intabs.3 (/\ (C_ (|^| ({|} y ps)) A) ch))) (= (|^| ({|} x (/\ (C_ (cv x) A) ph))) (|^| ({|} x ph))) ((cv x) (|^| ({|} y ps)) A sseq1 intabs.2 anbi12d intabs.3 (V) intmin3 ({|} y ps) intnex (|^| ({|} x (/\ (C_ (cv x) A) ph))) ssv (|^| ({|} y ps)) (V) (|^| ({|} x (/\ (C_ (cv x) A) ph))) sseq2 mpbiri sylbi pm2.61i intabs.1 cbvabv inteqi sseqtr4 (C_ (cv x) A) ph pm3.27 x ss2abi ({|} x (/\ (C_ (cv x) A) ph)) ({|} x ph) intss ax-mp eqssi)) thm (class2set ((A x)) () (e. ({e.|} x A (e. A (V))) (V)) (A (V) x (e. A (V)) rabexg (-. (e. A (V))) (e. (cv x) A) pm3.26 nrexdv x A (e. A (V)) rabn0 con1bii sylib 0ex syl6eqel pm2.61i)) thm (class2seteq ((A x)) () (-> (e. A B) (= ({e.|} x A (e. A (V))) A)) (A B elisset (e. A (V)) (e. (cv x) A) ax-1 r19.21aiv A x (e. A (V)) rabid2 sylibr eqcomd syl)) thm (0nep0 () () (-. (= ({/}) ({} ({/})))) (0ex snnz ({} ({/})) ({/}) eqcom mtbi)) thm (0inp0 () () (-> (= A ({/})) (-. (= A ({} ({/}))))) (0nep0 A ({/}) ({} ({/})) eqeq1 mtbiri)) thm (uni0 () () (= (U. ({/})) ({/})) (({} ({/})) 0ss ({/}) ({} ({/})) uniss ax-mp 0ex unisn sseqtr (U. ({/})) ss0 ax-mp)) thm (unidif0 () () (= (U. (\ A ({} ({/})))) (U. A)) ((\ A ({} ({/}))) ({} ({/})) uniun A ({} ({/})) undif1 A ({} ({/})) uncom eqtr2 unieqi 0ex unisn (U. (\ A ({} ({/})))) uneq2i (U. (\ A ({} ({/})))) un0 eqtr2 3eqtr4r ({} ({/})) A uniun 0ex unisn (U. A) uneq1i 3eqtr ({/}) (U. A) uncom (U. A) un0 3eqtr)) thm (iin0 ((x y) (A x) (A y)) () (<-> (-. (= A ({/}))) (= (|^|_ x A ({/})) ({/}))) (A (e. (cv y) ({/})) x r19.3rzvOLD abbi2dv x A ({/}) y df-iin syl6reqr 0ex ({/}) (V) n0i ax-mp x ({/}) 0iin ({/}) eqeq1i mtbir A ({/}) x ({/}) iineq1 ({/}) eqeq1d mtbiri con2i impbi)) thm (notzfaus ((x y) (A x) (A y)) ((notzfaus.1 (= A ({} ({/})))) (notzfaus.2 (<-> ph (-. (e. (cv x) (cv y)))))) (-. (E. y (A. x (<-> (e. (cv x) (cv y)) (/\ (e. (cv x) A) ph))))) (0ex snnz notzfaus.1 ({/}) eqeq1i mtbir A x n0 mpbi (e. (cv x) A) (e. (cv x) (cv y)) biimt (e. (cv x) A) (e. (cv x) (cv y)) iman notzfaus.2 (e. (cv x) A) anbi2i negbii bitr4 syl6bb (e. (cv x) (cv y)) (/\ (e. (cv x) A) ph) xor3 sylibr x 19.22i ax-mp x (<-> (e. (cv x) (cv y)) (/\ (e. (cv x) A) ph)) exnal mpbi y nex)) thm (axpow ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (A. y (-> (A. x (-> (e. (cv x) (cv y)) (e. (cv x) (cv z)))) (e. (cv y) (cv x))))) (x y w z ax-pow w x y elequ1 w x z elequ1 imbi12d cbvalv (e. (cv y) (cv x)) imbi1i y albii x exbii mpbi)) thm (axpow2 ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (A. z (<-> (e. (cv z) (cv y)) (A. w (-> (e. (cv w) (cv z)) (e. (cv w) (cv x))))))) (y z w x ax-pow bm1.3ii)) thm (pwex ((A x) (A y) (A z) (x y) (x z) (y z)) ((zfpowcl.1 (e. A (V)))) (e. (P~ A) (V)) (zfpowcl.1 (cv z) A pweq (V) eleq1d x y z axpow (cv y) (cv z) x dfss2 (e. (cv y) (cv x)) imbi1i y albii x exbii mpbir bm1.3ii (cv z) y df-pw (cv x) eqeq2i (cv x) y (C_ (cv y) (cv z)) abeq2 bitr x exbii mpbir issetri vtocl)) thm (pwexg ((A x)) () (-> (e. A B) (e. (P~ A) (V))) ((cv x) A pweq (V) eleq1d x visset pwex B vtoclg)) thm (abssexg ((A x)) () (-> (e. A B) (e. ({|} x (/\ (C_ (cv x) A) ph)) (V))) (A B pwexg (P~ A) (V) x ph rabexg syl x (P~ A) ph df-rab x visset A elpw ph anbi1i x abbii eqtr2 syl5eqel)) thm (snex ((A x)) () (e. ({} A) (V)) ((cv x) A sneq (V) eleq1d x visset pwex (cv x) snsspw ssexi (V) vtoclg A snprc 0ex ({} A) ({/}) (V) eleq1 mpbiri sylbi pm2.61i)) thm (el ((x y)) () (E. y (e. (cv x) (cv y))) (x visset snid (cv x) snex (cv y) ({} (cv x)) (cv x) eleq2 cla4ev ax-mp)) thm (snelpw () ((snelpw.1 (e. A (V)))) (<-> (e. A B) (e. ({} A) (P~ B))) (snelpw.1 B snss A snex B elpw bitr4)) thm (sbcsng ((A x)) () (-> (e. A B) (<-> ([/] A x ph) (A.e. x ({} A) ph))) (A B x ph sbc6g x ({} A) ph df-ral x A elsn ph imbi1i x albii bitr2 syl6bb)) thm (rext ((x y) (x z) (y z)) () (-> (A. z (-> (e. (cv x) (cv z)) (e. (cv y) (cv z)))) (= (cv x) (cv y))) (x visset snid (cv x) snex (cv z) ({} (cv x)) (cv x) eleq2 (cv z) ({} (cv x)) (cv y) eleq2 imbi12d cla4v mpi y (cv x) elsn y x equcomi sylbi syl)) thm (sspwb ((A x) (B x)) () (<-> (C_ A B) (C_ (P~ A) (P~ B))) ((cv x) A B sstr2 com12 x visset A elpw x visset B elpw 3imtr4g ssrdv (P~ A) (P~ B) ({} (cv x)) ssel (cv x) snex A elpw x visset A snss bitr4 (cv x) snex B elpw x visset B snss bitr4 3imtr3g ssrdv impbi)) thm (unipw ((A x) (A y) (x y)) () (= (U. (P~ A)) A) ((cv x) (P~ A) y eluni y visset A elpw (cv y) A (cv x) ssel sylbi impcom y 19.23aiv sylbi ssriv x visset snid (cv x) snex (cv y) ({} (cv x)) (cv x) eleq2 (cv y) ({} (cv x)) (P~ A) eleq1 anbi12d cla4ev mpan x visset A snelpw (cv x) (P~ A) y eluni 3imtr4 ssriv eqssi)) thm (pwuni ((A x)) () (C_ A (P~ (U. A))) ((cv x) A elssuni x visset (U. A) elpw sylibr ssriv)) thm (sspwuni () () (<-> (C_ A (P~ B)) (C_ (U. A) B)) (A (P~ B) uniss B unipw syl6ss (U. A) B sspwb A pwuni A (P~ (U. A)) (P~ B) sstr mpan sylbi impbi)) thm (ssextss ((A x) (B x)) () (<-> (C_ A B) (A. x (-> (C_ (cv x) A) (C_ (cv x) B)))) (A B sspwb (P~ A) (P~ B) x dfss2 x visset A elpw x visset B elpw imbi12i x albii 3bitr)) thm (ssext ((A x) (B x)) () (<-> (= A B) (A. x (<-> (C_ (cv x) A) (C_ (cv x) B)))) (A B x ssextss B A x ssextss anbi12i A B eqss x (C_ (cv x) A) (C_ (cv x) B) albi 3bitr4)) thm (nssss ((A x) (B x)) () (<-> (-. (C_ A B)) (E. x (/\ (C_ (cv x) A) (-. (C_ (cv x) B))))) (x (-> (C_ (cv x) A) (C_ (cv x) B)) exnal (C_ (cv x) A) (C_ (cv x) B) annim x exbii A B x ssextss negbii 3bitr4r)) thm (pweqb () () (<-> (= A B) (= (P~ A) (P~ B))) (A B sspwb B A sspwb anbi12i A B eqss (P~ A) (P~ B) eqss 3bitr4)) thm (moabex ((x y) (ph y)) () (-> (E* x ph) (e. ({|} x ph) (V))) (ph y ax-17 x mo2 x (-> ph (= (cv x) (cv y))) hba1 ph (= (cv x) (cv y)) pm4.71 biimp x a4s ph (= (cv x) (cv y)) ancom syl6bb abbid (cv y) x df-sn (cv y) snex eqeltrr (= (cv x) (cv y)) ph pm3.26 x ss2abi ssexi syl6eqel y 19.23aiv sylbi)) thm (euabex () () (-> (E! x ph) (e. ({|} x ph) (V))) (x ph eumo x ph moabex syl)) thm (nnullss ((x y) (A x) (A y)) () (-> (-. (= A ({/}))) (E. x (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))))) (A y n0 y visset A snss y visset snnz (cv y) snex (cv x) ({} (cv y)) A sseq1 (cv x) ({} (cv y)) ({/}) eqeq1 negbid anbi12d cla4ev mpan2 sylbi y 19.23aiv sylbi)) thm (exss ((x y) (x z) (A x) (y z) (A y) (A z) (ph y) (ph z)) () (-> (E.e. x A ph) (E. y (/\ (C_ (cv y) A) (E.e. x (cv y) ph)))) (x A ph df-rab ({/}) eqeq1i negbii x A ph rabn0 ({|} x (/\ (e. (cv x) A) ph)) z n0 3bitr3 (cv z) snex (cv y) ({} (cv z)) A sseq1 (cv y) ({} (cv z)) x ph rexeq1 anbi12d cla4ev z visset ({|} x (/\ (e. (cv x) A) ph)) snss x A ph ssab2 ({} (cv z)) ({|} x (/\ (e. (cv x) A) ph)) A sstr2 mpi sylbi ([/] (cv z) x (e. (cv x) A)) ([/] (cv z) x ph) pm3.27 z x equsb1 x (cv z) elsn z x sbbii mpbir jctil z x (/\ (e. (cv x) A) ph) df-clab z x (e. (cv x) A) ph sban bitr x ({} (cv z)) ph df-rab (cv z) eleq2i z x (/\ (e. (cv x) ({} (cv z))) ph) df-clab z x (e. (cv x) ({} (cv z))) ph sban 3bitr 3imtr4 (cv z) ({e.|} x ({} (cv z)) ph) n0i syl x ({} (cv z)) ph rabn0 sylib sylanc z 19.23aiv sylbi)) thm (p0ex () () (e. ({} ({/})) (V)) (({/}) snex)) thm (pp0ex () () (e. ({,} ({/}) ({} ({/}))) (V)) (pwpw0 p0ex pwex eqeltrr)) thm (dtru ((x y)) () (-. (A. x (= (cv x) (cv y)))) ((cv y) 0inp0 p0ex (cv x) ({} ({/})) (cv y) eqeq2 negbid cla4ev syl 0ex (cv x) ({/}) (cv y) eqeq2 negbid cla4ev pm2.61i x (= (cv y) (cv x)) exnal (cv y) (cv x) eqcom x albii negbii bitr mpbi)) thm (dtrucor ((x y)) ((dtrucor.1 (= (cv x) (cv y)))) (=/= (cv x) (cv y)) (x y dtru (=/= (cv x) (cv y)) pm2.21i dtrucor.1 mpg)) thm (zfpair ((x z) (w x) (v x) (w z) (v z) (v w) (y z) (w y) (v y)) () (e. ({,} (cv x) (cv y)) (V)) ((cv x) (cv y) w dfpr2 z (/\ (= (cv z) ({/})) (= (cv w) (cv x))) (/\ (= (cv z) ({} ({/}))) (= (cv w) (cv y))) 19.43 (= (cv z) ({/})) (= (cv w) (cv x)) (= (cv z) ({} ({/}))) (= (cv w) (cv y)) prlem2 z exbii z (= (cv z) ({/})) (= (cv w) (cv x)) 19.41v 0ex z isseti mpbiran z (= (cv z) ({} ({/}))) (= (cv w) (cv y)) 19.41v p0ex z isseti mpbiran orbi12i 3bitr3r w abbii ({/}) ({} ({/})) z dfpr2 pp0ex eqeltrr v x w equequ2 (cv z) 0inp0 (= (cv w) (cv y)) prlem1 w 19.21adv a4w v y w equequ2 (cv z) 0inp0 con2i (= (cv w) (cv x)) prlem1 (/\ (= (cv z) ({/})) (= (cv w) (cv x))) (/\ (= (cv z) ({} ({/}))) (= (cv w) (cv y))) orcom syl7ib w 19.21adv a4w jaoi zfrep4 eqeltr eqeltr)) thm (prex ((x y) (A x) (A y) (B x) (B y)) () (e. ({,} A B) (V)) ((cv x) A B preq1 (V) eleq1d (cv y) B (cv x) preq2 (V) eleq1d x y zfpair (V) vtoclg syl5bi (V) vtocleg A B prprc1 B snex syl6eqel B A prprc2 A snex syl6eqel pm2.61nii)) thm (opex () () (e. (<,> A B) (V)) (A B df-op ({} A) ({,} A B) prex eqeltr)) thm (elop () ((elop.1 (e. A (V)))) (<-> (e. A (<,> B C)) (\/ (= A ({} B)) (= A ({,} B C)))) (B C df-op A eleq2i elop.1 ({} B) ({,} B C) elpr bitr)) thm (opi1 () () (e. ({} A) (<,> A B)) (A snex ({,} A B) pri1 A B df-op eleqtrr)) thm (opi2 () () (e. ({,} A B) (<,> A B)) (A B prex ({} A) pri2 A B df-op eleqtrr)) thm (opth () ((opth.1 (e. A (V))) (opth.2 (e. B (V))) (opth.3 (e. D (V)))) (<-> (= (<,> A B) (<,> C D)) (/\ (= A C) (= B D))) (C D opi1 (<,> A B) (<,> C D) ({} C) eleq2 mpbiri C snex A B elop sylib opth.1 snid ({} C) ({} A) A eleq2 mpbiri opth.1 B pri1 ({} C) ({,} A B) A eleq2 mpbiri jaoi syl opth.1 C elsnc sylib (<,> A B) (<,> C D) (<,> C B) eqeq1 A C B opeq1 syl5bi C D df-op C B df-op eqeq12i C D prex C B prex ({} C) prer2 sylbi opth.3 opth.2 C prer2 eqcomd syl syl6 jcai A C B D opeq12 impbi)) thm (opthg ((A x) (B x) (C x) (D x)) ((opthg.1 (e. A (V))) (opthg.2 (e. B (V)))) (-> (e. D R) (<-> (= (<,> A B) (<,> C D)) (/\ (= A C) (= B D)))) ((cv x) D C opeq2 (<,> A B) eqeq2d (cv x) D B eqeq2 (= A C) anbi2d opthg.1 opthg.2 x visset C opth R vtoclbg)) thm (opthgg ((x y) (A x) (A y) (B y) (C x) (C y) (D x) (D y) (T x) (T y)) () (-> (/\/\ (e. A R) (e. B S) (e. D T)) (<-> (= (<,> A B) (<,> C D)) (/\ (= A C) (= B D)))) ((cv x) A (cv y) opeq1 (<,> C D) eqeq1d (cv x) A C eqeq1 (= (cv y) D) anbi1d bibi12d (e. D T) imbi2d (cv y) B A opeq2 (<,> C D) eqeq1d (cv y) B D eqeq1 (= A C) anbi2d bibi12d (e. D T) imbi2d x visset y visset D T C opthg R S vtocl2g 3impia)) thm (otthg () ((otthg.1 (e. A (V))) (otthg.2 (e. B (V))) (otthg.3 (e. R (V)))) (-> (/\ (e. D F) (e. S G)) (<-> (= (<,> (<,> A B) R) (<,> (<,> C D) S)) (/\/\ (= A C) (= B D) (= R S)))) (A B opex otthg.3 S G (<,> C D) opthg otthg.1 otthg.2 D F C opthg (= R S) anbi1d sylan9bbr (= A C) (= B D) (= R S) df-3an syl6bbr)) thm (eqvinop ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) ((eqvinop.1 (e. B (V))) (eqvinop.2 (e. C (V)))) (<-> (= A (<,> B C)) (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) (= (<,> (cv x) (cv y)) (<,> B C)))))) (x visset y visset eqvinop.2 B opth (= A (<,> (cv x) (cv y))) anbi2i (= A (<,> (cv x) (cv y))) (/\ (= (cv x) B) (= (cv y) C)) ancom (= (cv x) B) (= (cv y) C) (= A (<,> (cv x) (cv y))) anass 3bitr y exbii y (= (cv x) B) (/\ (= (cv y) C) (= A (<,> (cv x) (cv y)))) 19.42v eqvinop.2 (cv y) C (cv x) opeq2 A eqeq2d ceqsexv (= (cv x) B) anbi2i 3bitr x exbii eqvinop.1 (cv x) B C opeq1 A eqeq2d ceqsexv bitr2)) thm (copsexg ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (ph z) (ph w)) () (-> (= A (<,> (cv x) (cv y))) (<-> ph (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) ph))))) (x visset y visset A z w eqvinop (<,> (cv z) (cv w)) (<,> (cv x) (cv y)) eqcom x visset y visset w visset (cv z) opth bitr y (cv w) ph ceqex x (cv z) (E. y (/\ (= (cv y) (cv w)) ph)) ceqex sylan9bbr sylbi (<,> (cv z) (cv w)) (<,> (cv x) (cv y)) eqcom x visset y visset w visset (cv z) opth bitr ph anbi1i (= (cv x) (cv z)) (= (cv y) (cv w)) ph anass bitr y exbii y (= (cv x) (cv z)) (/\ (= (cv y) (cv w)) ph) 19.42v bitr x exbii syl6bbr A (<,> (cv z) (cv w)) (<,> (cv x) (cv y)) eqeq1 A (<,> (cv z) (cv w)) (<,> (cv x) (cv y)) eqeq1 ph anbi1d x y 2exbidv ph bibi2d imbi12d mpbiri (= (<,> (cv z) (cv w)) (<,> (cv x) (cv y))) adantr z w 19.23aivv sylbi pm2.43i)) thm (copsex2g ((x y) (ps x) (ps y) (A x) (A y) (B x) (B y)) ((copsex2g.1 (-> (/\ (= (cv x) A) (= (cv y) B)) (<-> ph ps)))) (-> (/\ (e. A C) (e. B D)) (<-> (E. x (E. y (/\ (= (<,> A B) (<,> (cv x) (cv y))) ph))) ps)) (x y (= (cv x) A) (= (cv y) B) eeanv x (E. y (/\ (= (<,> A B) (<,> (cv x) (cv y))) ph)) hbe1 ps x ax-17 hbbi y (/\ (= (<,> A B) (<,> (cv x) (cv y))) ph) hbe1 x hbex ps y ax-17 hbbi (cv x) A (cv y) B opeq12 (<,> A B) x y ph copsexg eqcoms syl copsex2g.1 bitr3d 19.23ai 19.23ai sylbir A C x elex B D y elex syl2an)) thm (copsex4g ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w) (ps x) (ps y) (ps z) (ps w) (R x) (R y) (R z) (R w) (S x) (S y) (S z) (S w)) ((copsex4g.1 (-> (/\ (/\ (= (cv x) A) (= (cv y) B)) (/\ (= (cv z) C) (= (cv w) D))) (<-> ph ps)))) (-> (/\ (/\ (e. A R) (e. B S)) (/\ (e. C R) (e. D S))) (<-> (E. x (E. y (E. z (E. w (/\ (/\ (= (<,> A B) (<,> (cv x) (cv y))) (= (<,> C D) (<,> (cv z) (cv w)))) ph))))) ps)) (x visset y visset B S A opthg (<,> A B) (<,> (cv x) (cv y)) eqcom syl5bb (e. A R) adantl z visset w visset D S C opthg (<,> C D) (<,> (cv z) (cv w)) eqcom syl5bb (e. C R) adantl bi2anan9 ph anbi1d x y z w 4exbidv (/\ (/\ (= (cv x) A) (= (cv y) B)) (/\ (= (cv z) C) (= (cv w) D))) id copsex4g.1 R S cgsex4g bitrd)) thm (opnz () () (-. (= (<,> A B) ({/}))) (A B opi1 ({} A) (<,> A B) n0i ax-mp)) thm (opprc1b ((A x) (B x)) () (<-> (-. (e. A (V))) (e. ({/}) (<,> A B))) (A B opprc1 0ex ({} B) pri1 syl5eleqr (cv x) A B opeq1 ({/}) eleq2d negbid x visset snnz ({} (cv x)) ({/}) eqcom mtbi x visset B prnz ({,} (cv x) B) ({/}) eqcom mtbi pm3.2ni 0ex (cv x) B elop mtbir (V) vtoclg con2i impbi)) thm (opprc3 () () (<-> (/\ (-. (e. A (V))) (-. (e. B (V)))) (= (<,> A B) ({} ({/})))) (B A opprc2 A A opprc1 A snprc ({} A) ({/}) ({/}) preq2 sylbi eqtrd sylan9eqr ({/}) dfsn2 syl6eqr 0ex snid (<,> A B) ({} ({/})) ({/}) eleq2 mpbiri A B opprc1b sylibr A B opprc1 ({} ({/})) eqeq1d B snex 0ex ({/}) prer2 ({/}) dfsn2 ({,} ({/}) ({} B)) eqeq2i B snprc 3imtr4 syl6bi anc2li mpcom impbi)) thm (opth2 ((A x) (B x) (C x) (D x)) ((opth2.1 (e. B (V))) (opth2.2 (e. D (V)))) (-> (= (<,> A B) (<,> C D)) (= B D)) ((cv x) A B opeq1 (<,> C D) eqeq1d (= B D) imbi1d x visset opth2.1 opth2.2 C opth pm3.27bd (V) vtoclg ({/}) (<,> A B) (<,> C D) nelneq2 A B opprc1b C D opprc1b con1bii bicomi syl2anb (= B D) pm2.21d A B opprc1 C D opprc1 eqeqan12d B snex D snex ({/}) prer2 opth2.1 D sneqr syl syl6bi pm2.61dan pm2.61i)) thm (moop2 ((x y) (w x) (A x) (w y) (A y) (A w) (y z) (B y) (w z) (B z) (B w) (x z)) () (E* x (= A (<,> B (cv x)))) (A (<,> B (cv x)) (<,> ({|} z ([/] (cv y) x (e. (cv z) B))) (cv y)) eqtr2t x visset y visset B ({|} z ([/] (cv y) x (e. (cv z) B))) opth2 syl x y gen2 (e. (cv w) A) x ax-17 y x (e. (cv z) B) hbs1 w z hbab (e. (cv w) (cv y)) x ax-17 hbop hbeq x y B z sbab B ({|} z ([/] (cv y) x (e. (cv z) B))) (cv x) opeq1 syl (cv x) (cv y) ({|} z ([/] (cv y) x (e. (cv z) B))) opeq2 eqtrd A eqeq2d mo4f mpbir)) thm (mosubop ((x y) (x z) (A x) (y z) (A y) (A z)) ((mosubop.1 (E* x ph))) (E* x (E. y (E. z (/\ (= A (<,> (cv y) (cv z))) ph)))) (y (E. z (/\ (= A (<,> (cv y) (cv z))) ph)) hbe1 x hbmo z (/\ (= A (<,> (cv y) (cv z))) ph) hbe1 y hbex x hbmo mosubop.1 (= A (<,> (cv y) (cv z))) x ax-17 A y z ph copsexg mobid mpbii 19.23ai 19.23ai (= A (<,> (cv y) (cv z))) ph pm3.26 y z 19.22i2 x 19.23aiv con3i x (E. y (E. z (/\ (= A (<,> (cv y) (cv z))) ph))) exmo ori syl pm2.61i)) thm (euop2 ((ph x) (A x) (x y)) () (<-> (E! x (E. y (/\ (= (cv x) (<,> A (cv y))) ph))) (E! y ph)) (A (cv y) opex y (cv x) A moop2 ph euxfr2)) thm (opthwiener () ((opthw.1 (e. A (V))) (opthw.2 (e. B (V)))) (<-> (= ({,} ({,} ({} A) ({/})) ({} ({} B))) ({,} ({,} ({} C) ({/})) ({} ({} D)))) (/\ (= A C) (= B D))) ((= ({,} ({,} ({} A) ({/})) ({} ({} B))) ({,} ({,} ({} C) ({/})) ({} ({} D)))) id ({} B) snex ({,} ({} A) ({/})) pri2 ({,} ({,} ({} A) ({/})) ({} ({} B))) ({,} ({,} ({} C) ({/})) ({} ({} D))) ({} ({} B)) eleq2 mpbii ({} B) snex ({,} ({} C) ({/})) ({} ({} D)) elpr sylib 0ex ({} C) pri2 opthw.2 snnz 0ex ({} B) elsnc ({/}) ({} B) eqcom bitr mtbir ({/}) ({,} ({} C) ({/})) ({} ({} B)) nelneq2 mp2an ({,} ({} C) ({/})) ({} ({} B)) eqcom mtbi (= ({} ({} B)) ({,} ({} C) ({/}))) (= ({} ({} B)) ({} ({} D))) biorf ax-mp sylibr ({} ({} B)) ({} ({} D)) ({,} ({} C) ({/})) preq2 syl eqtr4d ({} A) ({/}) prex ({} C) ({/}) prex ({} ({} B)) preqr1 syl A snex C snex ({/}) preqr1 syl opthw.1 C sneqr syl ({} B) snex ({,} ({} A) ({/})) pri2 ({,} ({,} ({} A) ({/})) ({} ({} B))) ({,} ({,} ({} C) ({/})) ({} ({} D))) ({} ({} B)) eleq2 mpbii ({} B) snex ({,} ({} C) ({/})) ({} ({} D)) elpr sylib 0ex ({} C) pri2 opthw.2 snnz 0ex ({} B) elsnc ({/}) ({} B) eqcom bitr mtbir ({/}) ({,} ({} C) ({/})) ({} ({} B)) nelneq2 mp2an ({,} ({} C) ({/})) ({} ({} B)) eqcom mtbi (= ({} ({} B)) ({,} ({} C) ({/}))) (= ({} ({} B)) ({} ({} D))) biorf ax-mp sylibr B snex ({} D) sneqr opthw.2 D sneqr 3syl jca A C sneq ({} A) ({} C) ({/}) preq1 ({,} ({} A) ({/})) ({,} ({} C) ({/})) ({} ({} B)) preq1 3syl B D sneq ({} B) ({} D) sneq ({} ({} B)) ({} ({} D)) ({,} ({} C) ({/})) preq2 3syl sylan9eq impbi)) thm (uniop () () (= (U. (<,> A B)) ({,} A B)) (A B df-op unieqi A snex A B prex unipr A B snsspr ({} A) ({,} A B) ssequn1 mpbi 3eqtr)) thm (opabid ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph z) (ph w) (ph v)) () (<-> (e. (<,> (cv x) (cv y)) ({<,>|} x y ph)) ph) ((<,> (cv x) (cv y)) z (E. x (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ph))) clelab x y ph z df-opab (<,> (cv x) (cv y)) eleq2i z (/\ (= (cv z) (<,> (cv x) (cv y))) (= (cv z) (<,> (cv w) (cv v)))) ([/] (cv v) y ([/] (cv w) x ph)) 19.41v (= (cv z) (<,> (cv x) (cv y))) (= (cv z) (<,> (cv w) (cv v))) ([/] (cv v) y ([/] (cv w) x ph)) anass z exbii (<,> (cv x) (cv y)) (<,> (cv w) (cv v)) eqcom (cv x) (cv y) opex (<,> (cv w) (cv v)) z eqvinc w visset v visset y visset (cv x) opth 3bitr3 ([/] (cv v) y ([/] (cv w) x ph)) anbi1i 3bitr3r w v 2exbii ph w v x y sbel2x z w (E. v (/\ (= (cv z) (<,> (cv x) (cv y))) (/\ (= (cv z) (<,> (cv w) (cv v))) ([/] (cv v) y ([/] (cv w) x ph))))) excom z w v (= (cv z) (<,> (cv x) (cv y))) (/\ (= (cv z) (<,> (cv w) (cv v))) ([/] (cv v) y ([/] (cv w) x ph))) exdistr2 z v (/\ (= (cv z) (<,> (cv x) (cv y))) (/\ (= (cv z) (<,> (cv w) (cv v))) ([/] (cv v) y ([/] (cv w) x ph)))) excom w exbii 3bitr3 3bitr4 (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ph)) w ax-17 (= (cv z) (<,> (cv w) (cv y))) x ax-17 w x ph hbs1 hban y hbex (cv x) (cv w) (cv y) opeq1 (cv z) eqeq2d x w ph sbequ12 anbi12d y exbidv cbvex (/\ (= (cv z) (<,> (cv w) (cv y))) ([/] (cv w) x ph)) v ax-17 (= (cv z) (<,> (cv w) (cv v))) y ax-17 v y ([/] (cv w) x ph) hbs1 hban (cv y) (cv v) (cv w) opeq2 (cv z) eqeq2d y v ([/] (cv w) x ph) sbequ12 anbi12d cbvex w exbii bitr (= (cv z) (<,> (cv x) (cv y))) anbi2i z exbii bitr4 3bitr4)) thm (elopab ((x y) (x z) (A x) (y z) (A y) (A z) (ph z)) () (<-> (e. A ({<,>|} x y ph)) (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) ph)))) (A ({<,>|} x y ph) elisset (cv x) (cv y) opex A (<,> (cv x) (cv y)) (V) eleq1 mpbiri ph adantr x y 19.23aivv (cv z) A ({<,>|} x y ph) eleq1 (cv z) A (<,> (cv x) (cv y)) eqeq1 ph anbi1d x y 2exbidv x y ph z df-opab abeq2i (V) vtoclbg pm5.21nii)) thm (hbopab ((x y) (x z) (w x) (y z) (w y) (w z)) ((hbopab.1 (-> ph (A. z ph)))) (-> (e. (cv w) ({<,>|} x y ph)) (A. z (e. (cv w) ({<,>|} x y ph)))) ((= (cv w) (<,> (cv x) (cv y))) z ax-17 hbopab.1 hban y hbex x hbex (cv w) x y ph elopab (cv w) x y ph elopab z albii 3imtr4)) thm (hbopab1 ((x y) (w x) (w y) (x z) (ph w)) () (-> (e. (cv z) ({<,>|} x y ph)) (A. x (e. (cv z) ({<,>|} x y ph)))) (x (E. y (/\ (= (cv w) (<,> (cv x) (cv y))) ph)) hbe1 z w hbab x y ph w df-opab (cv z) eleq2i x y ph w df-opab (cv z) eleq2i x albii 3imtr4)) thm (hbopab2 ((x y) (w x) (w y) (y z) (ph w)) () (-> (e. (cv z) ({<,>|} x y ph)) (A. y (e. (cv z) ({<,>|} x y ph)))) (y (/\ (= (cv w) (<,> (cv x) (cv y))) ph) hbe1 x hbex z w hbab x y ph w df-opab (cv z) eleq2i x y ph w df-opab (cv z) eleq2i y albii 3imtr4)) thm (opabsb ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph v)) () (<-> (e. (<,> (cv z) (cv w)) ({<,>|} x y ph)) ([/] (cv w) y ([/] (cv z) x ph))) (y w a9e (e. (cv v) (<,> (cv z) (cv w))) y ax-17 v x y ph hbopab2 hbel w y ([/] (cv z) x ph) hbs1 hbbi x z a9e (= (cv y) (cv w)) x ax-17 (e. (cv v) (<,> (cv z) (cv w))) x ax-17 v x y ph hbopab1 hbel z x ph hbs1 w y hbsb hbbi hbim (cv x) (cv z) (cv y) (cv w) opeq12 ({<,>|} x y ph) eleq1d x y ph opabid syl5bbr x z ph sbequ12 y w ([/] (cv z) x ph) sbequ12 sylan9bb bitr3d ex 19.23ai ax-mp 19.23ai ax-mp)) thm (opelopabg ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (ch x) (ch y) (ph z)) ((opelopabg.1 (-> (= (cv x) A) (<-> ph ps))) (opelopabg.2 (-> (= (cv y) B) (<-> ps ch)))) (-> (/\ (e. A C) (e. B D)) (<-> (e. (<,> A B) ({<,>|} x y ph)) ch)) (A C x elex B D y elex anim12i x y (= (cv x) A) (= (cv y) B) eeanv sylibr (e. (cv z) (<,> A B)) x ax-17 z x y ph hbopab1 hbel ch x ax-17 hbbi (e. (cv z) (<,> A B)) y ax-17 z x y ph hbopab2 hbel ch y ax-17 hbbi (cv x) A (cv y) B opeq12 ({<,>|} x y ph) eleq1d x y ph opabid syl5bbr opelopabg.1 opelopabg.2 sylan9bb bitr3d 19.23ai 19.23ai syl)) thm (brabg ((x y) (A x) (A y) (B x) (B y) (ch x) (ch y)) ((opelopabg.1 (-> (= (cv x) A) (<-> ph ps))) (opelopabg.2 (-> (= (cv y) B) (<-> ps ch))) (brabg.5 (= R ({<,>|} x y ph)))) (-> (/\ (e. A C) (e. B D)) (<-> (br A R B) ch)) (opelopabg.1 opelopabg.2 C D opelopabg A R B df-br brabg.5 (<,> A B) eleq2i bitr syl5bb)) thm (opelopab ((x y) (A x) (A y) (B x) (B y) (ch x) (ch y)) ((opelopab.1 (e. A (V))) (opelopab.2 (e. B (V))) (opelopab.3 (-> (= (cv x) A) (<-> ph ps))) (opelopab.4 (-> (= (cv y) B) (<-> ps ch)))) (<-> (e. (<,> A B) ({<,>|} x y ph)) ch) (opelopab.1 opelopab.2 opelopab.3 opelopab.4 (V) (V) opelopabg mp2an)) thm (brab ((x y) (A x) (A y) (B x) (B y) (ch x) (ch y)) ((opelopab.1 (e. A (V))) (opelopab.2 (e. B (V))) (opelopab.3 (-> (= (cv x) A) (<-> ph ps))) (opelopab.4 (-> (= (cv y) B) (<-> ps ch))) (brab.5 (= R ({<,>|} x y ph)))) (<-> (br A R B) ch) (opelopab.1 opelopab.2 opelopab.3 opelopab.4 brab.5 (V) (V) brabg mp2an)) thm (ssopab2 ((ph z) (ps z) (x y) (x z) (y z)) () (<-> (C_ ({<,>|} x y ph) ({<,>|} x y ps)) (A. x (A. y (-> ph ps)))) (z x y ph hbopab1 z x y ps hbopab1 hbss z x y ph hbopab2 z x y ps hbopab2 hbss (cv x) (cv y) opex z isseti (cv z) x y ph copsexg (cv z) x y ps copsexg imbi12d z (E. x (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ph))) (E. x (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ps))) ss2ab z (-> (E. x (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ph))) (E. x (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ps)))) ax-4 sylbi syl5bir x y ph z df-opab x y ps z df-opab sseq12i syl5ib z 19.23aiv ax-mp 19.21ai 19.21ai x (A. y (-> ph ps)) hba1 y (-> ph ps) hba1 y (-> ph ps) ax-4 (= (cv z) (<,> (cv x) (cv y))) anim2d 19.22d x a4s 19.22d z 19.21aiv z (E. x (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ph))) (E. x (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ps))) ss2ab sylibr x y ph z df-opab x y ps z df-opab 3sstr4g impbi)) thm (ssopab2i ((x y)) ((ssopab2i.1 (-> ph ps))) (C_ ({<,>|} x y ph) ({<,>|} x y ps)) (x y ph ps ssopab2 ssopab2i.1 y ax-gen mpgbir)) thm (pwin ((A x) (B x)) () (= (P~ (i^i A B)) (i^i (P~ A) (P~ B))) ((cv x) A B ssin x visset A elpw x visset B elpw anbi12i x visset (i^i A B) elpw 3bitr4 ineqri eqcomi)) thm (pwunss ((A x) (B x)) () (C_ (u. (P~ A) (P~ B)) (P~ (u. A B))) ((cv x) A B ssun (cv x) (P~ A) (P~ B) elun x visset A elpw x visset B elpw orbi12i bitr x visset (u. A B) elpw 3imtr4 ssriv)) thm (pwssun ((A x) (A y) (x y) (B x) (B y)) () (<-> (\/ (C_ A B) (C_ B A)) (C_ (P~ (u. A B)) (u. (P~ A) (P~ B)))) ((C_ A B) (C_ B A) orcom B A ssequn2 (u. A B) A pweq (P~ (u. A B)) (P~ A) eqimss syl sylbi A B ssequn1 (u. A B) B pweq (P~ (u. A B)) (P~ B) eqimss syl sylbi orim12i sylbi (P~ (u. A B)) (P~ A) (P~ B) ssun syl (P~ (u. A B)) (u. (P~ A) (P~ B)) ({,} (cv x) (cv y)) ssel ({} (cv x)) A ({} (cv y)) B unss12 x visset A snss y visset B snss syl2anb x y zfpair (u. A B) elpw (cv x) (cv y) df-pr (u. A B) sseq1i bitr2 sylib syl5 exp3a com23 imp31 ({,} (cv x) (cv y)) (P~ A) (P~ B) elun sylib x y zfpair A elpw x visset y visset A prss bitr4 pm3.27bd x y zfpair B elpw x visset y visset B prss bitr4 pm3.26bd orim12i syl ord ex com23 imp ssrdv exp31 (e. (cv y) A) (C_ A B) bi2.15 syl6ib com23 imp ssrdv ex orrd impbi)) thm (pwun () () (<-> (\/ (C_ A B) (C_ B A)) (= (P~ (u. A B)) (u. (P~ A) (P~ B)))) (A B pwunss (C_ (P~ (u. A B)) (u. (P~ A) (P~ B))) biantru A B pwssun (P~ (u. A B)) (u. (P~ A) (P~ B)) eqss 3bitr4)) thm (epelc ((x y) (A x) (A y) (B x) (B y)) ((epelc.1 (e. A (V))) (epelc.2 (e. B (V)))) (<-> (br A (E) B) (e. A B)) (epelc.1 epelc.2 (cv x) A (cv y) eleq1 (cv y) B A eleq2 x y df-eprel brab)) thm (epel () () (<-> (br (cv x) (E) (cv y)) (e. (cv x) (cv y))) (x visset y visset epelc)) thm (ideqg ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A C) (e. B D)) (<-> (br A (I) B) (= A B))) ((cv x) A (cv y) eqeq1 (cv y) B A eqeq2 x y df-id C D brabg)) thm (ideq () ((ideq.1 (e. A (V))) (ideq.2 (e. B (V)))) (<-> (br A (I) B) (= A B)) (ideq.1 ideq.2 A (V) B (V) ideqg mp2an)) thm (poss ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (C_ A B) (-> (Po R B) (Po R A))) (A B (cv x) ssel A B (cv y) ssel A B (cv z) ssel 3anim123d (/\ (-. (br (cv x) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) imim1d z 19.20dv y 19.20dv x 19.20dv x B y B z B (/\ (-. (br (cv x) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) r3al x A y A z A (/\ (-. (br (cv x) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) r3al 3imtr4g R B x y z df-po R A x y z df-po 3imtr4g)) thm (poeq1 ((x y) (x z) (R x) (y z) (R y) (R z) (S x) (S y) (S z) (A x) (A y) (A z)) () (-> (= R S) (<-> (Po R A) (Po S A))) (R S (cv x) (cv x) breq negbid R S (cv x) (cv y) breq R S (cv y) (cv z) breq anbi12d R S (cv x) (cv z) breq imbi12d anbi12d z A ralbidv x A y A 2ralbidv R A x y z df-po S A x y z df-po 3bitr4g)) thm (poeq2 () () (-> (= A B) (<-> (Po R A) (Po R B))) (A B R poss B A R poss anim12i A B eqss (Po R B) (Po R A) bi 3imtr4 bicomd)) thm (pocl ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) () (-> (Po R A) (-> (/\/\ (e. B A) (e. C A) (e. D A)) (/\ (-. (br B R B)) (-> (/\ (br B R C) (br C R D)) (br B R D))))) ((= (cv x) B) id (= (cv x) B) id R breq12d negbid (cv x) B R (cv y) breq1 (br (cv y) R (cv z)) anbi1d (cv x) B R (cv z) breq1 imbi12d anbi12d (Po R A) imbi2d (cv y) C B R breq2 (cv y) C R (cv z) breq1 anbi12d (br B R (cv z)) imbi1d (-. (br B R B)) anbi2d (Po R A) imbi2d (cv z) D C R breq2 (br B R C) anbi2d (cv z) D B R breq2 imbi12d (-. (br B R B)) anbi2d (Po R A) imbi2d R A x y z df-po x A y A z A (/\ (-. (br (cv x) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) r3al bitr biimp 19.21bbi 19.21bi com12 vtocl3ga com12)) thm (poirr () () (-> (/\ (Po R A) (e. B A)) (-. (br B R B))) (R A B B B pocl imp pm3.26d (e. B A) (e. B A) (e. B A) df-3an (e. B A) (e. B A) anabs1 (e. B A) anidm 3bitrr sylan2b)) thm (potr () () (-> (/\ (Po R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-> (/\ (br B R C) (br C R D)) (br B R D))) (R A B C D pocl imp pm3.27d)) thm (po2nr () () (-> (/\ (Po R A) (/\ (e. B A) (e. C A))) (-. (/\ (br B R C) (br C R B)))) (R A B poirr (e. C A) adantrr R A B C B potr (e. B A) (e. C A) (e. B A) df-3an sylan2br exp44 com34 pm2.43d imp32 mtod)) thm (po3nr () () (-> (/\ (Po R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-. (/\/\ (br B R C) (br C R D) (br D R B)))) (R A B D po2nr (e. C A) 3adantr2 R A B C D potr (br D R B) anim1d (br B R C) (br C R D) (br D R B) df-3an syl5ib mtod)) thm (po0 ((x y) (x z) (R x) (y z) (R y) (R z)) () (Po R ({/})) (R ({/}) x y z df-po (cv x) noel (A.e. y ({/}) (A.e. z ({/}) (/\ (-. (br (cv x) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))))) pm2.21i mprgbir)) thm (sopo ((x y) (R x) (R y) (A x) (A y)) () (-> (Or R A) (Po R A)) (R A x y df-so pm3.26bd)) thm (soss ((x y) (R x) (R y) (A x) (A y) (B x) (B y)) () (-> (C_ A B) (-> (Or R B) (Or R A))) (A B R poss A B (cv x) ssel A B (cv y) ssel anim12d (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x))) imim1d y 19.20dv x 19.20dv x B y B (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x))) r2al x A y A (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x))) r2al 3imtr4g anim12d R B x y df-so R A x y df-so 3imtr4g)) thm (soeq1 ((x y) (R x) (R y) (S x) (S y) (A x) (A y)) () (-> (= R S) (<-> (Or R A) (Or S A))) (R S A poeq1 R S (cv x) (cv y) breq (= R S) (= (cv x) (cv y)) pm4.2i R S (cv y) (cv x) breq 3orbi123d x A y A 2ralbidv anbi12d R A x y df-so S A x y df-so 3bitr4g)) thm (soeq2 () () (-> (= A B) (<-> (Or R A) (Or R B))) (A B R soss B A R soss anim12i A B eqss (Or R B) (Or R A) bi 3imtr4 bicomd)) thm (sonr () () (-> (/\ (Or R A) (e. B A)) (-. (br B R B))) (R A B poirr R A sopo sylan)) thm (sotr () () (-> (/\ (Or R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-> (/\ (br B R C) (br C R D)) (br B R D))) (R A B C D potr R A sopo sylan)) thm (solin ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (R x) (R y)) () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (\/\/ (br B R C) (= B C) (br C R B))) ((cv x) B R (cv y) breq1 (cv x) B (cv y) eqeq1 (cv x) B (cv y) R breq2 3orbi123d (Or R A) imbi2d (cv y) C B R breq2 (cv y) C B eqeq2 (cv y) C R B breq1 3orbi123d (Or R A) imbi2d R A x y df-so x A y A (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x))) ra42 (Po R A) adantl sylbi com12 vtocl2ga impcom)) thm (so2nr () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (-. (/\ (br B R C) (br C R B)))) (R A B C po2nr R A sopo sylan)) thm (so3nr () () (-> (/\ (Or R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-. (/\/\ (br B R C) (br C R D) (br D R B)))) (R A B C D po3nr R A sopo sylan)) thm (sotric () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (<-> (br B R C) (-. (\/ (= B C) (br C R B))))) (B C B R breq2 negbid R A B sonr syl5bi com12 (e. C A) adantrr R A B C so2nr (br B R C) (br C R B) imnan sylibr con2d jaod R A B C solin (br B R C) (= B C) (br C R B) 3orass (br B R C) (\/ (= B C) (br C R B)) df-or bitr sylib impbid con2bid)) thm (sotrieq () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (<-> (= B C) (-. (\/ (br B R C) (br C R B))))) (B C B R breq2 negbid R A B sonr syl5bi B C C R breq2 negbid R A C sonr syl5bir anim12d com12 anandis R A B C sotric con2bid biimpar ord con1d ex imp3a impbid (br B R C) (br C R B) ioran syl6bbr)) thm (sotrieq2 () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (<-> (= B C) (/\ (-. (br B R C)) (-. (br C R B))))) (R A B C sotrieq (br B R C) (br C R B) ioran syl6bb)) thm (itlso ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z)) ((itlso.1 (-> (e. (cv x) A) (-. (br (cv x) R (cv x))))) (itlso.2 (-> (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A)) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))) (itlso.3 (-> (/\ (e. (cv x) A) (e. (cv y) A)) (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x)))))) (Or R A) (R A x y df-so itlso.1 (e. (cv y) A) (e. (cv z) A) 3ad2ant1 itlso.2 jca rgen3 R A x y z df-po mpbir itlso.3 rgen2 mpbir2an)) thm (so ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z)) ((so.1 (-> (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A)) (/\ (<-> (br (cv x) R (cv y)) (-. (\/ (= (cv x) (cv y)) (br (cv y) R (cv x))))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))))) (Or R A) ((cv x) eqid (= (cv x) (cv x)) (br (cv x) R (cv x)) orc ax-mp (cv y) (cv x) A eleq1 (e. (cv x) A) anbi2d (cv y) (cv x) (cv x) eqeq2 (cv y) (cv x) R (cv x) breq1 orbi12d (cv y) (cv x) (cv x) R breq2 negbid bibi12d imbi12d (e. (cv x) A) (e. (cv y) A) (e. (cv y) A) 3anass (e. (cv y) A) anidm (e. (cv x) A) anbi2i bitr2 (cv z) (cv y) A eleq1 (e. (cv x) A) (e. (cv y) A) 3anbi3d (<-> (br (cv x) R (cv y)) (-. (\/ (= (cv x) (cv y)) (br (cv y) R (cv x))))) imbi1d so.1 pm3.26d chvarv sylbi con2bid chvarv mpbii anidms so.1 pm3.27d (e. (cv x) A) (e. (cv y) A) (e. (cv y) A) 3anass (e. (cv y) A) anidm (e. (cv x) A) anbi2i bitr2 (cv z) (cv y) A eleq1 (e. (cv x) A) (e. (cv y) A) 3anbi3d (<-> (br (cv x) R (cv y)) (-. (\/ (= (cv x) (cv y)) (br (cv y) R (cv x))))) imbi1d so.1 pm3.26d chvarv sylbi con2bid biimprd (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x)) 3orass (br (cv x) R (cv y)) (\/ (= (cv x) (cv y)) (br (cv y) R (cv x))) df-or bitr sylibr itlso)) thm (so0 ((x y) (R x) (R y)) () (Or R ({/})) (R ({/}) x y df-so R po0 mpbiran (cv x) noel (A.e. y ({/}) (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x)))) pm2.21i mprgbir)) thm (axun ((w x) (w y) (w z) (x y) (x z) (y z)) () (E. x (A. y (-> (E. x (/\ (e. (cv y) (cv x)) (e. (cv x) (cv z)))) (e. (cv y) (cv x))))) (x y w z ax-un w x y elequ2 w x z elequ1 anbi12d cbvexv (e. (cv y) (cv x)) imbi1i y albii x exbii mpbi)) thm (axun2 ((w x) (w y) (w z) (x y) (x z) (y z)) () (E. y (A. z (<-> (e. (cv z) (cv y)) (E. w (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))))))) (y z w x ax-un bm1.3ii)) thm (uniex2 ((x y) (x z) (y z)) () (E. y (= (cv y) (U. (cv x)))) (y z x axun (cv z) (cv x) y eluni (e. (cv z) (cv y)) imbi1i z albii y exbii mpbir bm1.3ii (cv y) (U. (cv x)) z dfcleq y exbii mpbir)) thm (uniex ((x y) (A x) (A y)) ((uniex.1 (e. A (V)))) (e. (U. A) (V)) (uniex.1 (cv x) A unieq (V) eleq1d y x uniex2 issetri vtocl)) thm (uniexg ((A x)) () (-> (e. A B) (e. (U. A) (V))) ((cv x) A unieq (V) eleq1d x visset uniex B vtoclg)) thm (unex () ((unex.1 (e. A (V))) (unex.2 (e. B (V)))) (e. (u. A B) (V)) (unex.1 unex.2 unipr A B prex uniex eqeltrr)) thm (unexb ((x y) (A x) (A y) (B x) (B y)) () (<-> (/\ (e. A (V)) (e. B (V))) (e. (u. A B) (V))) ((cv x) A (cv y) uneq1 (V) eleq1d (cv y) B A uneq2 (V) eleq1d x visset y visset unex (V) (V) vtocl2g A B ssun1 A (u. A B) (V) ssexg mpan B A ssun2 B (u. A B) (V) ssexg mpan jca impbi)) thm (unexg () () (-> (/\ (e. A C) (e. B D)) (e. (u. A B) (V))) (A B unexb biimp A C elisset B D elisset syl2an)) thm (unisn2 () () (e. (U. ({} A)) ({,} ({/}) A)) (A (V) unisng A eqid (-. (= A ({/}))) a1i orri A (V) ({/}) A elprg mpbiri eqeltrd A snprc biimp unieqd uni0 0ex A pri1 eqeltr syl6eqel pm2.61i)) thm (difex2 () () (-> (e. B C) (<-> (e. A (V)) (e. (\ A B) (V)))) (A (V) B difexg (e. B C) a1i B C elisset (e. (\ A B) (V)) anim1i ancoms B (\ A B) unexb sylib B A undif2 syl5eqelr A B ssun2 A (u. B A) (V) ssexg mpan syl expcom impbid)) thm (tpex () () (e. ({,,} A B C) (V)) (A B C df-tp A B prex C snex unex eqeltr)) thm (opeluu () () (-> (e. (<,> (cv x) (cv y)) A) (/\ (e. (cv x) (U. (U. A))) (e. (cv y) (U. (U. A))))) ((cv x) (cv y) opi2 ({,} (cv x) (cv y)) (<,> (cv x) (cv y)) A elunii mpan x visset (cv y) pri1 (cv x) ({,} (cv x) (cv y)) (U. A) elunii mpan syl (cv x) (cv y) opi2 ({,} (cv x) (cv y)) (<,> (cv x) (cv y)) A elunii mpan y visset (cv x) pri2 (cv y) ({,} (cv x) (cv y)) (U. A) elunii mpan syl jca)) thm (euuni ((x y) (x z) (y z) (ph y) (ph z)) () (-> (E! x ph) (<-> ph (= (U. ({|} x ph)) (cv x)))) (x ph euabex ({|} x ph) (V) uniexg syl (U. ({|} x ph)) y eueq (cv y) (U. ({|} x ph)) eqcom y eubii z x ph hbab1 hbuni (e. (cv z) (cv y)) x ax-17 hbeq (= (U. ({|} x ph)) (cv x)) y ax-17 (cv y) (cv x) (U. ({|} x ph)) eqeq2 cbveu 3bitr sylib x ph eusn x visset snid ({|} x ph) ({} (cv x)) (cv x) eleq2 mpbiri x ph abid sylib ({|} x ph) ({} (cv x)) unieq x visset unisn syl6eq jca x 19.22i sylbi x ph (= (U. ({|} x ph)) (cv x)) eupickb 3exp imp3a mp2and)) thm (reuuni1 () () (-> (/\ (e. (cv x) A) (E!e. x A ph)) (<-> ph (= (U. ({e.|} x A ph)) (cv x)))) (x (/\ (e. (cv x) A) ph) euuni biimpd exp3a impcom x (/\ (e. (cv x) A) ph) euuni (e. (cv x) A) ph pm3.27 syl6bir (e. (cv x) A) adantl impbid x A ph df-rab unieqi (cv x) eqeq1i syl6bbr x A ph df-reu sylan2b)) thm (reuuni2f ((ph y) (x y) (A x) (A y) (B y)) ((reuuni2f.1 (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (reuuni2f.2 (-> (e. B A) (-> ps (A. x ps)))) (reuuni2f.3 (-> (= (cv x) B) (<-> ph ps)))) (-> (/\ (e. B A) (E!e. x A ph)) (<-> ps (= (U. ({e.|} x A ph)) B))) (reuuni2f.1 reuuni2f.1 (e. (cv y) A) x ax-17 hbel reuuni2f.1 (e. (cv y) A) x ax-17 hbel x A ph hbreu1 (e. B A) a1i reuuni2f.1 (e. (cv y) A) x ax-17 hbel reuuni2f.2 y x A ph hbrab1 hbuni reuuni2f.1 hbeq (e. B A) a1i hbbid hbimd hbim1 (cv x) B A eleq1 reuuni2f.3 (cv x) B (U. ({e.|} x A ph)) eqeq2 bibi12d (E!e. x A ph) imbi2d imbi12d x A ph reuuni1 ex A vtoclgf pm2.43i imp)) thm (reuuni2 ((ph y) (x y) (A x) (A y) (B x) (B y) (ps x)) ((reuuni2.1 (-> (= (cv x) B) (<-> ph ps)))) (-> (/\ (e. B A) (E!e. x A ph)) (<-> ps (= (U. ({e.|} x A ph)) B))) ((e. (cv y) B) x ax-17 ps x ax-17 (e. B A) a1i reuuni2.1 reuuni2f)) thm (reucl ((x y) (A x) (A y) (ph y)) () (-> (E!e. x A ph) (e. (U. ({e.|} x A ph)) A)) (x (/\ (e. (cv x) A) ph) eusn y x (/\ (e. (cv x) A) ph) hbab1 hbuni (e. (cv y) A) x ax-17 hbel ({|} x (/\ (e. (cv x) A) ph)) ({} (cv x)) unieq x visset unisn syl6req x visset snid ({|} x (/\ (e. (cv x) A) ph)) ({} (cv x)) (cv x) eleq2 mpbiri x (/\ (e. (cv x) A) ph) abid sylib pm3.26d eqeltrrd 19.23ai sylbi x A ph df-reu x A ph df-rab unieqi A eleq1i 3imtr4)) thm (reuuni3 ((x y) (A x) (A y) (ph y) (ps x) (ch x)) ((reuuni3.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (reuuni3.2 (-> (= (cv x) (U. ({e.|} y A ps))) (<-> ph ch)))) (-> (E!e. x A ph) ch) (x A ph reucl reuuni3.1 A cbvrabv unieqi syl5eqelr reuuni3.1 A cbvrabv unieqi reuuni3.2 A reuuni2 mpbiri mpancom)) thm (reuuni4 ((x y) (A x) (A y) (ph y)) () (-> (E!e. x A ph) ([/] (U. ({e.|} x A ph)) x ph)) (x A ph reucl x A ph reurex x A ph hbreu1 y x A ph hbrab1 hbuni (V) ph hbsbc1 x A ph reuuni1 x (U. ({e.|} x A ph)) ph sbceq1a eqcoms syl6bi ibd expcom (e. (U. ({e.|} x A ph)) (V)) a1i com4l r19.23ad mpd (U. ({e.|} x A ph)) A elisset syl5 mpd)) thm (reucl2 ((x y) (A x) (A y) (ph y)) () (-> (E!e. x A ph) (e. (U. ({e.|} x A ph)) ({e.|} x A ph))) (x A ph reucl x A ph reuuni4 jca (e. (cv y) A) x ax-17 (U. ({e.|} x A ph)) ph elrabsf sylibr)) thm (reuuniss ((x y) (A x) (A y) (B x) (B y) (ph y)) () (-> (/\/\ (C_ A B) (E.e. x A ph) (E!e. x B ph)) (= (U. ({e.|} x A ph)) (U. ({e.|} x B ph)))) (A B x ph reuss x A ph reuuni4 syl y x A ph hbrab1 hbuni y x A ph hbrab1 hbuni B ph hbsbc1g x (U. ({e.|} x A ph)) ph sbceq1a reuuni2f A B x ph reuss x A ph reucl syl A B (U. ({e.|} x A ph)) ssel (E.e. x A ph) (E!e. x B ph) 3ad2ant1 mpd (C_ A B) (E.e. x A ph) (E!e. x B ph) 3simp3 sylanc mpbid eqcomd)) thm (mouniss ((A x) (B x)) () (-> (/\/\ (C_ A B) (E.e. x A ph) (E* x (/\ (e. (cv x) B) ph))) (= (U. ({e.|} x A ph)) (U. ({e.|} x B ph)))) (A B (cv x) ssel ph anim1d r19.22dv2 imp (E* x (/\ (e. (cv x) B) ph)) anim1i x B ph reu5 sylibr A B x ph reuuniss 3expa syldan 3impa)) thm (reuuniss2 ((x y) (A x) (A y) (B x) (B y) (ph y) (ps y)) () (-> (/\ (/\ (C_ A B) (A.e. x A (-> ph ps))) (/\ (E.e. x A ph) (E!e. x B ps))) (= (U. ({e.|} x A ph)) (U. ({e.|} x B ps)))) (x A ph reuuni4 x A ph reucl y x A ph hbrab1 hbuni A (-> ph ps) ra4sbcf (U. ({e.|} x A ph)) A x ph ps sbcimg sylibd syl mpid A B x ph ps reuss2 (C_ A B) (A.e. x A (-> ph ps)) pm3.27 (/\ (E.e. x A ph) (E!e. x B ps)) adantr sylc y x A ph hbrab1 hbuni y x A ph hbrab1 hbuni B ps hbsbc1g x (U. ({e.|} x A ph)) ps sbceq1a reuuni2f A B x ph ps reuss2 x A ph reucl syl A B (U. ({e.|} x A ph)) ssel (A.e. x A (-> ph ps)) (/\ (E.e. x A ph) (E!e. x B ps)) ad2antrr mpd (E.e. x A ph) (E!e. x B ps) pm3.27 (/\ (C_ A B) (A.e. x A (-> ph ps))) adantl sylanc mpbid eqcomd)) thm (reusn ((y z) (A y) (A z) (ph y) (ph z) (x y) (x z)) () (<-> (E!e. x A ph) (E. y (= ({e.|} x A ph) ({} (cv y))))) (x A ph df-reu x (/\ (e. (cv x) A) ph) eusn x A ph df-rab ({} (cv x)) eqeq1i x exbii (= ({e.|} x A ph) ({} (cv x))) y ax-17 z x A ph hbrab1 (e. (cv z) ({} (cv y))) x ax-17 hbeq (cv x) (cv y) sneq ({e.|} x A ph) eqeq2d cbvex bitr3 3bitr)) thm (reusni ((A y) (B y) (ph y) (x y)) ((reusni.1 (e. B (V)))) (-> (= ({e.|} x A ph) ({} B)) (E!e. x A ph)) (reusni.1 (cv y) B sneq ({e.|} x A ph) eqeq2d cla4ev x A ph y reusn sylibr)) thm (reuunisn ((A y) (ph y) (x y)) () (-> (E!e. x A ph) (= ({e.|} x A ph) ({} (U. ({e.|} x A ph))))) (x A ph y reusn ({e.|} x A ph) ({} (cv y)) unieq y visset unisn syl6eq sneqd ({e.|} x A ph) ({} (cv y)) ({} (U. ({e.|} x A ph))) eqtr3t mpdan y 19.23aiv sylbi)) thm (ralxfr ((ps x) (ph y) (A x) (x y) (B x) (B y)) ((ralxfr.1 (-> (e. (cv y) B) (e. A B))) (ralxfr.2 (-> (e. (cv x) B) (E.e. y B (= (cv x) A)))) (ralxfr.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (A.e. x B ph) (A.e. y B ps)) (ralxfr.1 ralxfr.3 B rcla4v syl com12 r19.21aiv y B ps hbra1 ph y ax-17 y B ps ra4 ralxfr.3 biimprcd syl6 r19.23ad ralxfr.2 syl5 r19.21aiv impbi)) thm (rexxfr ((ps x) (ph y) (A x) (x y) (B x) (B y)) ((ralxfr.1 (-> (e. (cv y) B) (e. A B))) (ralxfr.2 (-> (e. (cv x) B) (E.e. y B (= (cv x) A)))) (ralxfr.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (E.e. x B ph) (E.e. y B ps)) (ralxfr.1 ralxfr.2 ralxfr.3 negbid ralxfr negbii x B ph dfrex2 y B ps dfrex2 3bitr4)) thm (rabxfr ((A x) (B z) (C z) (x y) (x z) (D x) (y z) (D y) (D z) (ph y) (ph z) (ps x) (ps z)) ((rabxfr.1 (-> (e. (cv z) B) (A. y (e. (cv z) B)))) (rabxfr.2 (-> (e. (cv z) C) (A. y (e. (cv z) C)))) (rabxfr.3 (-> (e. (cv y) D) (e. A D))) (rabxfr.4 (-> (= (cv x) A) (<-> ph ps))) (rabxfr.5 (-> (= (cv y) B) (= A C)))) (-> (e. B D) (<-> (e. C ({e.|} x D ph)) (e. B ({e.|} y D ps)))) (rabxfr.3 (e. (cv y) D) (e. A D) ibibr mpbi ps anbi1d rabxfr.4 D elrab y D ps rabid 3bitr4g rabbii B eleq2i rabxfr.1 (e. (cv z) D) y ax-17 rabxfr.2 (e. (cv z) ({e.|} x D ph)) y ax-17 hbel rabxfr.5 ({e.|} x D ph) eleq1d elrabf rabxfr.1 (e. (cv z) D) y ax-17 rabxfr.1 z y D ps hbrab1 hbel (cv y) B ({e.|} y D ps) eleq1 elrabf 3bitr3 (e. B D) (e. C ({e.|} x D ph)) (e. B ({e.|} y D ps)) pm5.32 mpbir)) thm (reuxfr2 ((ph x) (A x) (x y) (B x) (B y)) ((reuxfr2.1 (-> (e. (cv y) B) (e. A B))) (reuxfr2.2 (-> (e. (cv x) B) (E* y (/\ (e. (cv y) B) (= (cv x) A)))))) (<-> (E!e. x B (E.e. y B (/\ (= (cv x) A) ph))) (E!e. y B ph)) (x B y (/\ (= (cv x) A) ph) 2reuswap reuxfr2.2 y (/\ (e. (cv y) B) (= (cv x) A)) ph moan syl ph (/\ (e. (cv y) B) (= (cv x) A)) ancom (e. (cv y) B) (= (cv x) A) ph anass bitr y mobii sylib mprg y B x (/\ (= (cv x) A) ph) 2reuswap x A moeq (/\ (e. (cv x) B) ph) moani (/\ (e. (cv x) B) ph) (= (cv x) A) ancom (= (cv x) A) (e. (cv x) B) ph an12 bitr x mobii mpbi (e. (cv y) B) a1i mprg impbi reuxfr2.1 (= (cv x) A) ph pm4.2i B ceqsrexv syl reubiia bitr)) thm (reuxfr ((ps x) (ph y) (A x) (x y) (B x) (B y)) ((reuxfr.1 (-> (e. (cv y) B) (e. A B))) (reuxfr.2 (-> (e. (cv x) B) (E!e. y B (= (cv x) A)))) (reuxfr.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (E!e. x B ph) (E!e. y B ps)) (reuxfr.2 y B (= (cv x) A) reurex syl ph biantrurd y B (= (cv x) A) ph r19.41v reuxfr.3 pm5.32i y B rexbii bitr3 syl6bb reubiia reuxfr.1 reuxfr.2 y B (= (cv x) A) df-reu y (/\ (e. (cv y) B) (= (cv x) A)) eumo sylbi syl ps reuxfr2 bitr)) thm (reuhyp ((A x) (B y) (C y) (x y)) ((reuhyp.1 (-> (e. (cv x) C) (e. B C))) (reuhyp.2 (-> (/\ (e. (cv x) C) (e. (cv y) C)) (<-> (= (cv x) A) (= (cv y) B))))) (-> (e. (cv x) C) (E!e. y C (= (cv x) A))) (reuhyp.1 B C elisset syl B y eueq sylib (cv y) B C eleq1 reuhyp.1 syl5bir com12 pm4.71rd reuhyp.2 ex pm5.32d bitr4d y eubidv mpbid y C (= (cv x) A) df-reu sylibr)) thm (reuunixfr ((B x) (x z) (C x) (C z) (x y) (A x) (y z) (A y) (A z) (ph y) (ph z) (ps x) (ps z)) ((reuunixfr.1 (-> (e. (cv z) C) (A. y (e. (cv z) C)))) (reuunixfr.2 (-> (e. (cv y) A) (e. B A))) (reuunixfr.3 (-> (e. (U. ({e.|} y A ps)) A) (e. C A))) (reuunixfr.4 (-> (= (cv x) B) (<-> ph ps))) (reuunixfr.5 (-> (= (cv y) (U. ({e.|} y A ps))) (= B C))) (reuunixfr.6 (-> (e. (cv x) A) (E!e. y A (= (cv x) B))))) (-> (E!e. x A ph) (= (U. ({e.|} x A ph)) C)) (reuunixfr.2 reuunixfr.6 reuunixfr.4 reuxfr y A ps reucl2 y A ps reucl z y A ps hbrab1 hbuni reuunixfr.1 reuunixfr.2 reuunixfr.4 reuunixfr.5 rabxfr syl mpbird sylbi (e. (cv z) C) x ax-17 (e. (cv z) C) x ax-17 z x A ph hbrab1 hbel (e. C A) a1i (cv x) C ({e.|} x A ph) eleq1 reuuni2f reuunixfr.2 reuunixfr.6 reuunixfr.4 reuxfr y A ps reucl reuunixfr.3 syl sylbi x A ph rabid baibr reubiia biimp sylanc mpbid x A ph rabid baib rabbii unieqi syl5eqr)) thm (uniexb () () (<-> (e. A (V)) (e. (U. A) (V))) (A (V) uniexg A pwuni A (P~ (U. A)) (V) ssexg (U. A) (V) pwexg sylan2 mpan impbi)) thm (pwexb () () (<-> (e. A (V)) (e. (P~ A) (V))) ((P~ A) uniexb A unipw (V) eleq1i bitr2)) thm (univ () () (= (U. (V)) (V)) (pwv unieqi (V) unipw eqtr3)) thm (op1stb () ((op1stb.1 (e. A (V)))) (= (|^| (|^| (<,> A B))) A) (A B df-op inteqi A snex A B prex intpr A B snsspr ({} A) ({,} A B) df-ss mpbi 3eqtr inteqi op1stb.1 intsn eqtr)) thm (iunpw ((x y) (A x) (A y)) ((iunpw.1 (e. A (V)))) (<-> (E.e. x A (= (cv x) (U. A))) (= (P~ (U. A)) (U_ x A (P~ (cv x))))) ((cv x) (U. A) (cv y) sseq2 biimprcd x A r19.22sdv com12 x A (cv y) (cv x) ssiun A x uniiun syl6ssr (E.e. x A (= (cv x) (U. A))) a1i impbid y visset (U. A) elpw (cv y) x A (P~ (cv x)) eliun (cv x) y df-pw abeq2i x A rexbii bitr 3bitr4g eqrdv (U. A) ssid (P~ (U. A)) (U_ x A (P~ (cv x))) (U. A) eleq2 iunpw.1 uniex (U. A) elpw syl5bbr mpbii (U. A) x A (P~ (cv x)) eliun sylib (cv x) A elssuni iunpw.1 uniex (cv x) elpw biimp anim12i (cv x) (U. A) eqss sylibr ex r19.22i syl impbi)) thm (supeq1 ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (= B C) (= (sup B A R) (sup C A R))) (B C y (-. (br (cv x) R (cv y))) raleq1 B C z (br (cv y) R (cv z)) rexeq1 (br (cv y) R (cv x)) imbi2d y A ralbidv anbi12d x A rabbisdv unieqd B A R x y z df-sup C A R x y z df-sup 3eqtr4g)) thm (supmo ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (R x) (R y) (R z) (R w) (B x) (B y) (B z) (B w)) ((supmo.1 (Or R A))) (E* x (/\ (e. (cv x) A) (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z)))))))) ((cv x) (cv w) A eleq1 (cv x) (cv w) R (cv y) breq1 negbid y B ralbidv (cv x) (cv w) (cv y) R breq2 (E.e. z B (br (cv y) R (cv z))) imbi1d y A ralbidv anbi12d anbi12d mo4 (cv y) (cv x) R (cv w) breq1 (cv y) (cv x) R (cv z) breq1 z B rexbidv imbi12d A rcla4va con3d imp (cv y) (cv z) (cv x) R breq2 negbid B cbvralv z B (br (cv x) R (cv z)) ralnex bitr sylan2b an1rs (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))) adantlrr (A.e. y B (-. (br (cv w) R (cv y)))) adantrl (e. (cv w) A) adantrl (cv y) (cv w) R (cv x) breq1 (cv y) (cv w) R (cv z) breq1 z B rexbidv imbi12d A rcla4cva con3d imp (cv y) (cv z) (cv w) R breq2 negbid B cbvralv z B (br (cv w) R (cv z)) ralnex bitr sylan2b anasss (A.e. y A (-> (br (cv y) R (cv w)) (E.e. z B (br (cv y) R (cv z))))) adantrrr (A.e. y B (-. (br (cv x) R (cv y)))) adantll (e. (cv x) A) adantll jca supmo.1 R A (cv x) (cv w) sotrieq2 mpan (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z)))))) (/\ (A.e. y B (-. (br (cv w) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv w)) (E.e. z B (br (cv y) R (cv z)))))) ad2ant2r mpbird w ax-gen mpgbir)) thm (supex ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((supmo.1 (Or R A))) (e. (sup B A R) (V)) (B A R x y z df-sup x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z)))))) df-rab unieqi eqtr supmo.1 x y B z supmo x (/\ (e. (cv x) A) (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) moabex ax-mp uniex eqeltr)) thm (supeu ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((supmo.1 (Or R A))) (-> (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (E!e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z)))))))) (supmo.1 x y B z supmo (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) jctr x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z)))))) reu5 sylibr)) thm (supcl ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((supmo.1 (Or R A))) (-> (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (e. (sup B A R) A)) (supmo.1 x y B z supeu x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z)))))) reucl syl B A R x y z df-sup syl5eqel)) thm (supub ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (R x) (R y) (R z) (R w) (B x) (B y) (B z) (B w) (C w)) ((supmo.1 (Or R A))) (-> (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (-> (e. C B) (-. (br (sup B A R) R C)))) ((cv w) C (sup B A R) R breq2 negbid (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) imbi2d (cv y) (cv w) (sup B A R) R breq2 negbid B rcla4v B A R x y z df-sup eqcomi (cv x) (sup B A R) R (cv y) breq1 negbid y B ralbidv (cv x) (sup B A R) (cv y) R breq2 (E.e. z B (br (cv y) R (cv z))) imbi1d y A ralbidv anbi12d A reuuni2 supmo.1 x y B z supcl supmo.1 x y B z supeu sylanc mpbiri pm3.26d syl5 vtoclga com12)) thm (suplub ((C z) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (R x) (R y) (R z) (R w) (B x) (B y) (B z) (B w) (C w)) ((supmo.1 (Or R A))) (-> (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (-> (/\ (e. C A) (br C R (sup B A R))) (E.e. z B (br C R (cv z))))) ((cv w) C R (sup B A R) breq1 (cv w) C R (cv z) breq1 z B rexbidv imbi12d (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) imbi2d (cv y) (cv w) R (sup B A R) breq1 (cv y) (cv w) R (cv z) breq1 z B rexbidv imbi12d A rcla4v B A R x y z df-sup eqcomi (cv x) (sup B A R) R (cv y) breq1 negbid y B ralbidv (cv x) (sup B A R) (cv y) R breq2 (E.e. z B (br (cv y) R (cv z))) imbi1d y A ralbidv anbi12d A reuuni2 supmo.1 x y B z supcl supmo.1 x y B z supeu sylanc mpbiri pm3.27d syl5 vtoclga com12 imp3a)) thm (supnub ((C z) (x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((supmo.1 (Or R A))) (-> (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (-> (/\ (e. C A) (A.e. z B (-. (br C R (cv z))))) (-. (br C R (sup B A R))))) (supmo.1 x y B z C suplub exp3a imp z B (br C R (cv z)) dfrex2 syl6ib con2d ex imp3a)) thm (supeui ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((sup.1 (Or R A)) (sup.2 (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))))) (E!e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (sup.2 sup.1 x y B z supeu ax-mp)) thm (supcli ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((sup.1 (Or R A)) (sup.2 (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))))) (e. (sup B A R) A) (sup.2 sup.1 x y B z supcl ax-mp)) thm (supubi ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((sup.1 (Or R A)) (sup.2 (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))))) (-> (e. C B) (-. (br (sup B A R) R C))) (sup.2 sup.1 x y B z C supub ax-mp)) thm (suplubi ((C z) (x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((sup.1 (Or R A)) (sup.2 (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))))) (-> (/\ (e. C A) (br C R (sup B A R))) (E.e. z B (br C R (cv z)))) (sup.2 sup.1 x y B z C suplub ax-mp)) thm (supnubi ((C z) (x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((sup.1 (Or R A)) (sup.2 (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))))) (-> (/\ (e. C A) (A.e. z B (-. (br C R (cv z))))) (-. (br C R (sup B A R)))) (sup.2 sup.1 x y B z C supnub ax-mp)) thm (supsn ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (R x) (R y) (R z)) ((supsn.1 (Or R A))) (-> (e. B A) (= (sup ({} B) A R) B)) ((cv y) B elsni (cv y) B B R breq2 negbid supsn.1 R A B sonr mpan syl5bir syl com12 r19.21aiv (cv z) B (cv y) R breq2 ({} B) rcla4ev B A snidg sylan ex (e. (cv y) A) a1d r19.21aiv jca (cv y) B elsni (cv y) B B R breq2 negbid supsn.1 R A B sonr mpan syl5bir syl com12 r19.21aiv (cv z) B (cv y) R breq2 ({} B) rcla4ev B A snidg sylan ex (e. (cv y) A) a1d r19.21aiv jca (cv x) B R (cv y) breq1 negbid y ({} B) ralbidv (cv x) B (cv y) R breq2 (E.e. z ({} B) (br (cv y) R (cv z))) imbi1d y A ralbidv anbi12d A rcla4ev mpdan supsn.1 x y ({} B) z supmo jctir x A (/\ (A.e. y ({} B) (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z ({} B) (br (cv y) R (cv z)))))) reu5 sylibr (cv x) B R (cv y) breq1 negbid y ({} B) ralbidv (cv x) B (cv y) R breq2 (E.e. z ({} B) (br (cv y) R (cv z))) imbi1d y A ralbidv anbi12d A reuuni2 mpdan mpbid ({} B) A R x y z df-sup syl5eq)) thm (fri ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (e. B C) (Fr R A)) (/\ (C_ B A) (-. (= B ({/}))))) (E.e. x B (A.e. y B (-. (br (cv y) R (cv x)))))) ((cv z) B A sseq1 (cv z) B ({/}) eqeq1 negbid anbi12d (cv z) B y (-. (br (cv y) R (cv x))) raleq1 x rexeqd imbi12d C cla4gv R A z x y df-fr syl5ib imp31)) thm (dffr2 ((x y) (x z) (w x) (v x) (R x) (y z) (w y) (v y) (R y) (w z) (v z) (R z) (v w) (R w) (R v) (A x) (A w) (A v)) () (<-> (Fr R A) (A. x (-> (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (E.e. y (cv x) (= (i^i (cv x) ({|} z (br (cv z) R (cv y)))) ({/})))))) (R A x v w df-fr (cv x) ({|} z (br (cv z) R (cv y))) w disj w visset (cv z) (cv w) R (cv y) breq1 elab negbii w (cv x) ralbii bitr y (cv x) rexbii (cv y) (cv v) (cv w) R breq2 negbid w (cv x) ralbidv (cv x) cbvrexv bitr (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) imbi2i x albii bitr4)) thm (frc ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) ((frc.1 (e. B (V)))) (-> (Fr R A) (-> (/\ (C_ B A) (-. (= B ({/})))) (E.e. x B (= (i^i B ({|} y (br (cv y) R (cv x)))) ({/}))))) (R A z x y dffr2 frc.1 (cv z) B A sseq1 (cv z) B ({/}) eqeq1 negbid anbi12d (cv z) B ({|} y (br (cv y) R (cv x))) ineq1 ({/}) eqeq1d x rexeqd imbi12d cla4v sylbi)) thm (frss ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (C_ A B) (-> (Fr R B) (Fr R A))) ((cv x) A B sstr2 com12 (-. (= (cv x) ({/}))) anim1d (E.e. y (cv x) (= (i^i (cv x) ({|} z (br (cv z) R (cv y)))) ({/}))) imim1d x 19.20dv R B x y z dffr2 R A x y z dffr2 3imtr4g)) thm (freq1 ((x y) (x z) (R x) (y z) (R y) (R z) (S x) (S y) (S z) (A x) (A y) (A z)) () (-> (= R S) (<-> (Fr R A) (Fr S A))) (R S (cv z) (cv y) breq negbid y (cv x) z (cv x) rexralbidv (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) imbi2d x albidv R A x y z df-fr S A x y z df-fr 3bitr4g)) thm (freq2 () () (-> (= A B) (<-> (Fr R A) (Fr R B))) (A B R frss B A R frss anim12i A B eqss (Fr R B) (Fr R A) bi 3imtr4 bicomd)) thm (frirr ((x y) (x z) (R x) (y z) (R y) (R z) (A y) (A z)) () (-> (/\ (Fr R A) (e. (cv x) A)) (-. (br (cv x) R (cv x)))) (x visset snnz (cv x) snex R A y z frc exp3a x visset A snss syl5ib mpii y (cv x) elsn (cv y) (cv x) (cv z) R breq2 z abbidv ({} (cv x)) ineq2d ({/}) eqeq1d x visset (cv z) (cv x) R (cv x) breq1 elab biimpr x visset snid jctil (cv x) ({} (cv x)) ({|} z (br (cv z) R (cv x))) elin sylibr (cv x) (i^i ({} (cv x)) ({|} z (br (cv z) R (cv x)))) n0i syl con2i syl6bi sylbi r19.23aiv syl6 imp)) thm (fr2nr ((w z) (R z) (R w) (x z) (w x) (y z) (w y) (A z) (A w)) () (-> (/\ (Fr R A) (/\ (e. (cv x) A) (e. (cv y) A))) (-. (/\ (br (cv x) R (cv y)) (br (cv y) R (cv x))))) (y visset (cv x) prnz y x zfpair R A w z frc (cv w) (cv y) (cv z) R breq2 z abbidv ({,} (cv y) (cv x)) ineq2d ({/}) eqeq1d negbid x R y z brab1 x visset (cv y) pri2 (cv x) ({,} (cv y) (cv x)) ({|} z (br (cv z) R (cv y))) inelcm mpan sylbi syl5bir com12 (cv w) (cv x) (cv z) R breq2 z abbidv ({,} (cv y) (cv x)) ineq2d ({/}) eqeq1d negbid y R x z brab1 y visset (cv x) pri1 (cv y) ({,} (cv y) (cv x)) ({|} z (br (cv z) R (cv x))) inelcm mpan sylbi syl5bir com12 jaao w visset (cv y) (cv x) elpr syl5ib con3i expi r19.23aiv syl6 mpan2i y visset x visset A prss syl5ib ancomsd imp)) thm (fr3nr ((v w) (w x) (v x) (w y) (v y) (w z) (v z) (R w) (R v) (A w) (A v)) () (-> (/\ (Fr R A) (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A))) (-. (/\/\ (br (cv x) R (cv y)) (br (cv y) R (cv z)) (br (cv z) R (cv x))))) (y visset (cv z) (cv x) tpnz (cv y) (cv z) (cv x) tpex R A v w frc (= (cv v) (cv y)) (-. (= (i^i ({,,} (cv y) (cv z) (cv x)) ({|} w (br (cv w) R (cv v)))) ({/}))) (= (cv v) (cv z)) (= (cv v) (cv x)) 3jao (cv v) (cv y) (cv w) R breq2 w abbidv ({,,} (cv y) (cv z) (cv x)) ineq2d ({/}) eqeq1d negbid x R y w brab1 x visset (cv y) (cv z) tpi3 (cv x) ({,,} (cv y) (cv z) (cv x)) ({|} w (br (cv w) R (cv y))) inelcm mpan sylbi syl5bir com12 (cv v) (cv z) (cv w) R breq2 w abbidv ({,,} (cv y) (cv z) (cv x)) ineq2d ({/}) eqeq1d negbid y R z w brab1 y visset (cv z) (cv x) tpi1 (cv y) ({,,} (cv y) (cv z) (cv x)) ({|} w (br (cv w) R (cv z))) inelcm mpan sylbi syl5bir com12 (cv v) (cv x) (cv w) R breq2 w abbidv ({,,} (cv y) (cv z) (cv x)) ineq2d ({/}) eqeq1d negbid z R x w brab1 z visset (cv y) (cv x) tpi2 (cv z) ({,,} (cv y) (cv z) (cv x)) ({|} w (br (cv w) R (cv x))) inelcm mpan sylbi syl5bir com12 syl3an v visset (cv y) (cv z) (cv x) eltp syl5ib con3i expi r19.23aiv syl6 mpan2i y visset z visset x visset A tpss syl5ib (e. (cv x) A) (e. (cv y) A) (e. (cv z) A) 3anrot syl5ib imp)) thm (fr0 ((x y) (x z) (R x) (y z) (R y) (R z)) () (Fr R ({/})) (R ({/}) x y z dffr2 (cv x) ss0 (C_ (cv x) ({/})) (= (cv x) ({/})) iman mpbi (E.e. y (cv x) (= (i^i (cv x) ({|} z (br (cv z) R (cv y)))) ({/}))) pm2.21i mpgbir)) thm (efrirr ((A x)) () (-> (Fr (E) A) (-. (e. A A))) ((cv x) A (E) freq2 (cv x) A (cv x) eleq1 (cv x) A A eleq2 bitrd negbid imbi12d (E) (cv x) x frirr ex x x epel negbii syl6ib pm2.01d A vtoclg com12 pm2.01d)) thm (efrn2lp () () (-> (/\ (Fr (E) A) (/\ (e. (cv x) A) (e. (cv y) A))) (-. (/\ (e. (cv x) (cv y)) (e. (cv y) (cv x))))) ((E) A x y fr2nr x y epel y x epel anbi12i negbii sylib)) thm (epne3 () () (-> (/\ (Fr (E) A) (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A))) (-. (/\/\ (e. (cv x) (cv y)) (e. (cv y) (cv z)) (e. (cv z) (cv x))))) ((E) A x y z fr3nr x y epel y z epel z x epel 3anbi123i negbii sylib)) thm (tz7.2 () () (-> (/\ (/\ (Tr A) (Fr (E) A)) (e. B A)) (/\ (C_ B A) (-. (= B A)))) (A B trss B A A eleq1 negbid A efrirr syl5bir com12 con2d anim12i (e. B A) (C_ B A) (-. (= B A)) jcab sylibr imp)) thm (dfepfr ((x y) (x z) (y z) (A x) (A z)) () (<-> (Fr (E) A) (A. x (-> (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (E.e. y (cv x) (= (i^i (cv x) (cv y)) ({/})))))) ((E) A x y z dffr2 z y epel z abbii z (cv y) abid2 eqtr (cv x) ineq2i ({/}) eqeq1i y (cv x) rexbii (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) imbi2i x albii bitr)) thm (epfrc ((x y) (A x) (A y) (B x) (B y)) ((epfrc.1 (e. B (V)))) (-> (/\ (Fr (E) A) (/\ (C_ B A) (-. (= B ({/}))))) (E.e. x B (= (i^i B (cv x)) ({/})))) (epfrc.1 (E) A x y frc imp y x epel y abbii y (cv x) abid2 eqtr2 B ineq2i ({/}) eqeq1i x B rexbii sylibr)) thm (dfwe2 ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z)) () (<-> (We R A) (/\ (Fr R A) (A.e. x A (A.e. y A (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x))))))) (R A df-we R A (cv x) (cv y) solin ex x y 19.21aivv x A y A (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x))) r2al sylibr (Fr R A) anim2i R A x y fr2nr (e. (cv z) A) 3adantr3 (cv x) (cv z) (cv y) R breq2 biimprd (br (cv x) R (cv y)) anim2d impcom nsyl (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (= (cv x) (cv z)) imnan sylibr R A x y z fr3nr (br (cv x) R (cv y)) (br (cv y) R (cv z)) (br (cv z) R (cv x)) df-3an negbii sylib (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv z) R (cv x)) imnan sylibr jcad (= (cv x) (cv z)) (br (cv z) R (cv x)) ioran syl6ibr (br (cv x) R (cv z)) imim1d (br (cv x) R (cv z)) (= (cv x) (cv z)) (br (cv z) R (cv x)) 3orass (br (cv x) R (cv z)) (\/ (= (cv x) (cv z)) (br (cv z) R (cv x))) orcom (\/ (= (cv x) (cv z)) (br (cv z) R (cv x))) (br (cv x) R (cv z)) df-or 3bitr syl5ib R A x frirr (e. (cv y) A) adantrr (e. (cv z) A) 3adantr3 jctild ex a2d z 19.20dv y 19.20dv x 19.20dv x A y A z A (\/\/ (br (cv x) R (cv z)) (= (cv x) (cv z)) (br (cv z) R (cv x))) r3al x A y A z A (/\ (-. (br (cv x) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) r3al 3imtr4g y A (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x))) ralidm (cv y) (cv z) (cv x) R breq2 (cv y) (cv z) (cv x) eqeq2 (cv y) (cv z) R (cv x) breq1 3orbi123d A cbvralv y A ralbii bitr3 x A ralbii R A x y z df-po 3imtr4g ancrd R A x y df-so syl6ibr imdistani impbi bitr)) thm (wess () () (-> (C_ A B) (-> (We R B) (We R A))) (A B R frss A B R soss anim12d R B df-we R A df-we 3imtr4g)) thm (weeq1 () () (-> (= R S) (<-> (We R A) (We S A))) (R S A freq1 R S A soeq1 anbi12d R A df-we S A df-we 3bitr4g)) thm (weeq2 () () (-> (= A B) (<-> (We R A) (We R B))) (A B R freq2 A B R soeq2 anbi12d R A df-we R B df-we 3bitr4g)) thm (wefr () () (-> (We R A) (Fr R A)) (R A df-we pm3.26bd)) thm (weso () () (-> (We R A) (Or R A)) (R A df-we pm3.27bd)) thm (wecmpep () () (-> (/\ (We (E) A) (/\ (e. (cv x) A) (e. (cv y) A))) (\/\/ (e. (cv x) (cv y)) (= (cv x) (cv y)) (e. (cv y) (cv x)))) ((E) A (cv x) (cv y) solin x y epel (= (cv x) (cv y)) pm4.2 y x epel 3orbi123i sylib (E) A weso sylan)) thm (wetrep () () (-> (/\ (We (E) A) (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A))) (-> (/\ (e. (cv x) (cv y)) (e. (cv y) (cv z))) (e. (cv x) (cv z)))) ((E) A (cv x) (cv y) (cv z) sotr (E) A weso sylan x y epel y z epel anbi12i x z epel 3imtr3g)) thm (wefrc ((x y) (x z) (y z) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (We (E) A) (/\ (C_ B A) (-. (= B ({/}))))) (E.e. x B (= (i^i B (cv x)) ({/})))) (B A (E) wess (cv x) (cv y) B ineq2 ({/}) eqeq1d B rcla4ev ex (We (E) B) adantl B (cv y) inss1 y visset B inex2 B x epfrc (E) B wefr sylan exp32 mpi (cv x) B (cv y) elin (= (i^i (i^i B (cv y)) (cv x)) ({/})) anbi1i (e. (cv x) B) (e. (cv x) (cv y)) (= (i^i (i^i B (cv y)) (cv x)) ({/})) anass bitr rexbii2 syl6ib (e. (cv y) B) adantr B z x y wetrep exp3a (e. (cv y) B) (e. (cv z) B) (e. (cv x) B) df-3an (e. (cv y) B) (e. (cv z) B) (e. (cv x) B) 3anrot bitr3 sylan2b exp44 imp com34 imp3a (cv z) B (cv x) elin syl5ib imp4a com23 r19.21adv (i^i B (cv x)) (cv y) z dfss3 syl6ibr (i^i B (cv x)) (cv y) dfss B (cv x) (cv y) in23 (i^i B (cv x)) eqeq2i bitr biimp ({/}) eqeq1d biimprd syl6 exp3a imp4a r19.22dv syld pm2.61d ex y 19.23adv B y n0 syl5ib syl6com imp32)) thm (we0 () () (We R ({/})) (R ({/}) df-we R fr0 R so0 mpbir2an)) thm (wereu ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) ((wereu.1 (e. B (V)))) (-> (/\ (We R A) (/\ (C_ B A) (-. (= B ({/}))))) (E!e. x B (A.e. y B (-. (br (cv y) R (cv x)))))) (wereu.1 B (V) R A x y fri mpanl1 R A wefr sylan R B (cv x) (cv z) solin R B weso sylan (br (cv x) R (cv z)) (= (cv x) (cv z)) (br (cv z) R (cv x)) df-3or (br (cv x) R (cv z)) (= (cv x) (cv z)) (br (cv z) R (cv x)) or23 (\/ (br (cv x) R (cv z)) (br (cv z) R (cv x))) (= (cv x) (cv z)) df-or 3bitr sylib (br (cv x) R (cv z)) (br (cv z) R (cv x)) ioran syl5ibr B A R wess impcom sylan (cv y) (cv x) R (cv z) breq1 negbid B rcla4v (cv y) (cv z) R (cv x) breq1 negbid B rcla4v im2anan9 ancomsd imp syl5 exp4b pm2.43d (-. (= B ({/}))) adantrr x z 19.21aivv x B z B (-> (/\ (A.e. y B (-. (br (cv y) R (cv x)))) (A.e. y B (-. (br (cv y) R (cv z))))) (= (cv x) (cv z))) r2al sylibr jca (cv x) (cv z) (cv y) R breq2 negbid y B ralbidv B reu4 sylibr)) thm (ordeq () () (-> (= A B) (<-> (Ord A) (Ord B))) (A B treq A B (E) weeq2 anbi12d A df-ord B df-ord 3bitr4g)) thm (elong ((A x)) () (-> (e. A B) (<-> (e. A (On)) (Ord A))) ((cv x) A ordeq x df-on B elab2g)) thm (elon () ((elon.1 (e. A (V)))) (<-> (e. A (On)) (Ord A)) (elon.1 A (V) elong ax-mp)) thm (eloni () () (-> (e. A (On)) (Ord A)) (A (On) elong ibi)) thm (elon2 () () (<-> (e. A (On)) (/\ (Ord A) (e. A (V)))) (A eloni A (On) elisset jca A (V) elong biimparc impbi)) thm (limeq () () (-> (= A B) (<-> (Lim A) (Lim B))) (A B ordeq A B ({/}) eqeq1 negbid A B unieq A eqeq2d A B (U. B) eqeq1 bitrd 3anbi123d A df-lim B df-lim 3bitr4g)) thm (ordwe () () (-> (Ord A) (We (E) A)) (A df-ord pm3.27bd)) thm (ordtr () () (-> (Ord A) (Tr A)) (A df-ord pm3.26bd)) thm (ordfr () () (-> (Ord A) (Fr (E) A)) (A ordwe (E) A wefr syl)) thm (ordelss () () (-> (/\ (Ord A) (e. B A)) (C_ B A)) (A B trss imp A ordtr sylan)) thm (trssord () () (-> (/\/\ (Tr A) (C_ A B) (Ord B)) (Ord A)) (A B (E) wess imp B ordwe sylan2 (Tr A) anim2i (Tr A) (C_ A B) (Ord B) 3anass A df-ord 3imtr4)) thm (ordeirr () () (-> (Ord A) (-. (e. A A))) (A ordfr A efrirr syl)) thm (nordeq () () (-> (/\ (Ord A) (e. B A)) (-. (= A B))) (A B A eleq1 negbid A ordeirr syl5bi com12 con2d imp)) thm (ordn2lp () () (-> (Ord A) (-. (/\ (e. A B) (e. B A)))) (A ordeirr A ordtr A A B trel syl mtod)) thm (tz7.5 ((A x) (B x)) () (-> (/\ (Ord A) (/\ (C_ B A) (-. (= B ({/}))))) (E.e. x B (= (i^i B (cv x)) ({/})))) (A B x wefrc A ordwe sylan)) thm (ordelord ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (Ord A) (e. B A)) (Ord B)) ((cv x) B A eleq1 (Ord A) anbi2d (cv x) B ordeq imbi12d A z y x wetrep A ordwe sylan (Ord A) (e. (cv x) A) pm3.26 (/\ (e. (cv z) (cv y)) (e. (cv y) (cv x))) adantr A ordtr A (cv z) (cv y) (cv x) trel3 syl (e. (cv x) A) (e. (cv z) (cv y)) (e. (cv y) (cv x)) 3anrot (e. (cv x) A) (e. (cv z) (cv y)) (e. (cv y) (cv x)) 3anass bitr3 syl5ibr exp3a imp31 A ordtr A (cv y) (cv x) trel syl exp3a com23 imp31 (e. (cv z) (cv y)) adantrl (Ord A) (e. (cv x) A) pm3.27 (/\ (e. (cv z) (cv y)) (e. (cv y) (cv x))) adantr 3jca sylanc ex pm2.43d z y 19.21aivv (cv x) z y dftr2 sylibr A ordtr A (cv x) trss syl (cv x) A (E) wess A ordwe syl5com syld imp jca (cv x) df-ord sylibr A vtoclg anabsi7)) thm (ordelon () () (-> (/\ (Ord A) (e. B A)) (e. B (On))) (A B ordelord B A elong (Ord A) adantl mpbird)) thm (onelon () () (-> (/\ (e. A (On)) (e. B A)) (e. B (On))) (A B ordelon A eloni sylan)) thm (tz7.7 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (Ord A) (Tr B)) (<-> (e. B A) (/\ (C_ B A) (-. (= B A))))) (A B tz7.2 A ordtr A ordfr jca sylan ex (Tr B) adantr A ordtr A (cv x) trss (cv x) A B eldifi syl5 A B (cv x) difin0ss com12 syl6 syl (Tr B) (C_ B A) ad2antrr imp32 (cv y) (cv x) B eleq1 biimpcd (cv x) A B eldifn nsyli imp (C_ B A) adantll (/\ (Ord A) (Tr B)) adantl B (cv x) (cv y) trel exp3a com23 imp (cv x) A B eldifn nsyli ex (C_ B A) adantld imp32 (Ord A) adantll A y x wecmpep A ordwe B A (cv y) ssel2 (cv x) A B eldifi anim12i syl2an (Tr B) adantlr ecase23d exp44 com34 imp31 ssrdv (= (i^i (\ A B) (cv x)) ({/})) adantrr eqssd (cv x) A B eldifi (/\ (/\ (Ord A) (Tr B)) (C_ B A)) (= (i^i (\ A B) (cv x)) ({/})) ad2antrl eqeltrrd exp32 r19.23adv A B difss A (\ A B) x tz7.5 mpanr1 syl5 exp4b com23 (Tr B) adantrd pm2.43i B A pssdifn0 syl7 exp4a pm2.43d imp3a impbid)) thm (ordelssne () () (-> (/\ (Ord A) (Ord B)) (<-> (e. A B) (/\ (C_ A B) (-. (= A B))))) (B A tz7.7 A ordtr sylan2 ancoms)) thm (ordelpss () () (-> (/\ (Ord A) (Ord B)) (<-> (e. A B) (C: A B))) (A B ordelssne A B dfpss2 syl6bbr)) thm (ordsseleq () () (-> (/\ (Ord A) (Ord B)) (<-> (C_ A B) (\/ (e. A B) (= A B)))) (A B ordelssne biimprd exp3a (e. A B) (= A B) orcom (= A B) (e. A B) df-or bitr syl6ibr A B ordelssne (C_ A B) (-. (= A B)) pm3.26 syl6bi A B eqimss jctir (e. A B) (= A B) (C_ A B) jaob sylibr impbid)) thm (ordin () () (-> (/\ (Ord A) (Ord B)) (Ord (i^i A B))) (A B inss2 (i^i A B) B trssord mp3an2 A B trin A ordtr B ordtr syl2an sylan anabss3)) thm (onin () () (-> (/\ (e. A (On)) (e. B (On))) (e. (i^i A B) (On))) (A B ordin A eloni B eloni syl2an (e. A (On)) (e. B (On)) pm3.26 A (On) B inex1g (i^i A B) (V) elong 3syl mpbird)) thm (ordtri3or () () (-> (/\ (Ord A) (Ord B)) (\/\/ (e. A B) (= A B) (e. B A))) (A B ordin (i^i A B) ordeirr syl (i^i A B) A B elin A B incom B eleq1i (e. (i^i A B) A) anbi2i bitr negbii (e. (i^i A B) A) (e. (i^i B A) B) ianor bitr sylib A B inss1 (i^i A B) A ordsseleq mpbii A B ordin sylan anabss1 ord A B df-ss syl6ibr B A inss1 (i^i B A) B ordsseleq mpbii B A ordin sylan anabss4 ord B A df-ss syl6ibr orim12d mpd A B ordsseleq B A ordsseleq ancoms orbi12d mpbid (e. A B) (= A B) (e. B A) df-3or (= A B) oridm A B eqcom (= A B) orbi2i bitr3 (\/ (e. A B) (e. B A)) orbi2i (e. A B) (= A B) (e. B A) or23 (e. A B) (= A B) (e. B A) (= B A) or4 3bitr4 bitr sylibr)) thm (ordtri1 () () (-> (/\ (Ord A) (Ord B)) (<-> (C_ A B) (-. (e. B A)))) (A B ordsseleq A B ordn2lp (e. A B) (e. B A) imnan sylibr A B B eleq2 negbid B ordeirr syl5bir com12 jaao A B ordtri3or (e. A B) (= A B) (e. B A) df-3or (\/ (e. A B) (= A B)) (e. B A) orcom (e. B A) (\/ (e. A B) (= A B)) df-or 3bitr sylib impbid bitrd)) thm (ontri1 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (C_ A B) (-. (e. B A)))) (A B ordtri1 A eloni B eloni syl2an)) thm (ordtri2 () () (-> (/\ (Ord A) (Ord B)) (<-> (e. A B) (-. (\/ (= A B) (e. B A))))) (B A ordsseleq B A eqcom (e. B A) orbi2i (e. B A) (= A B) orcom bitr syl6bb B A ordtri1 bitr3d ancoms con2bid)) thm (ordtri3 () () (-> (/\ (Ord A) (Ord B)) (<-> (= A B) (-. (\/ (e. A B) (e. B A))))) (A B A eleq2 negbid A ordeirr syl5bi A B B eleq2 negbid B ordeirr syl5bir anim12d (e. A B) (e. B A) ioran syl6ibr com12 A B ordtri3or (e. A B) (= A B) (e. B A) df-3or (e. A B) (= A B) (e. B A) or23 (\/ (e. A B) (e. B A)) (= A B) df-or 3bitr sylib impbid)) thm (ordtri4 () () (-> (/\ (Ord A) (Ord B)) (<-> (= A B) (/\ (C_ A B) (-. (e. A B))))) (A B ordtri3 A B ordtri1 (-. (e. A B)) anbi1d (-. (e. B A)) (-. (e. A B)) ancom (e. A B) (e. B A) ioran bitr4 syl6bb bitr4d)) thm (orddisj () () (-> (Ord A) (= (i^i A ({} A)) ({/}))) (A ordeirr A A disjsn sylibr)) thm (onfr ((x y) (x z) (y z)) () (Fr (E) (On)) ((On) x z dfepfr (cv z) (cv y) (cv x) ineq2 ({/}) eqeq1d (cv x) rcla4ev expcom (C_ (cv x) (On)) a1d (cv x) (On) (cv y) ssel y visset elon syl6ib (cv x) (cv y) inss2 x visset (cv y) inex1 (cv y) z epfrc ex mpani (Tr (cv y)) z ax-17 z (cv x) (= (i^i (cv x) (cv z)) ({/})) hbre1 (cv x) (cv y) inss1 (cv z) sseli (cv y) (cv z) trss (cv x) (cv y) inss2 (cv z) sseli syl5 (cv z) (cv y) sseqin2 (i^i (cv y) (cv z)) (cv z) (cv x) ineq2 (cv x) (cv y) (cv z) inass syl5req sylbi ({/}) eqeq1d biimprcd sylan9 imp anim12i exp32 pm2.43b ex com23 z (cv x) (= (i^i (cv x) (cv z)) ({/})) ra4e syl8 r19.23ad sylan9 (cv y) ordfr (cv y) ordtr sylanc syl6 com3r pm2.61i y 19.23adv (cv x) y n0 syl5ib imp mpgbir)) thm (ordon ((x y)) () (Ord (On)) ((On) df-ord (On) x dftr3 (cv x) (cv y) ordelord x visset elon sylanb ex y visset elon syl6ibr ssrdv mprgbir (E) (On) x y dfwe2 onfr (cv x) (cv y) ordtri3or x y epel (= (cv x) (cv y)) pm4.2 y x epel 3orbi123i sylibr (cv x) eloni (cv y) eloni syl2an rgen2 mpbir2an mpbir2an)) thm (epweon () () (We (E) (On)) (ordon (On) ordwe ax-mp)) thm (onprc () () (-. (e. (On) (V))) (ordon (On) ordeirr ax-mp ordon (On) (V) elong mpbiri mto)) thm (ordeleqon () () (<-> (Ord A) (\/ (e. A (On)) (= A (On)))) (onprc (On) A elisset mto ordon A (On) ordtri3or mpan2 (e. A (On)) (= A (On)) (e. (On) A) df-3or sylib ord mt3i A eloni ordon A (On) ordeq mpbiri jaoi impbi)) thm (ordsson () () (-> (Ord A) (C_ A (On))) (ordon A (On) ordelssne mpan2 (C_ A (On)) (-. (= A (On))) pm3.26 syl6bi A ordeleqon biimp ord A (On) eqimss syl6 pm2.61d)) thm (onsst () () (-> (e. A (On)) (C_ A (On))) (A eloni A ordsson syl)) thm (ssorduni ((x y) (A x) (A y)) () (-> (C_ A (On)) (Ord (U. A))) (ordon (U. A) (On) trssord 3exp mpii A (On) (cv y) ssel (cv y) eloni (cv y) ordtr (cv y) (cv x) trss 3syl syl6 (e. (cv y) A) (e. (cv x) (cv y)) (C_ (cv x) (cv y)) anc2r syl (cv x) (cv y) A ssuni syl8 r19.23adv (cv x) A y eluni2 syl5ib r19.21aiv (U. A) x dftr3 sylibr A (On) (cv y) ssel (cv y) eloni (cv y) (cv x) ordelord ex x visset elon syl6ibr syl syl6 r19.23adv (cv x) A y eluni2 syl5ib ssrdv sylc)) thm (ssonunit () () (-> (e. A B) (-> (C_ A (On)) (e. (U. A) (On)))) (A B uniexg (U. A) (V) elong syl A ssorduni syl5bir)) thm (ssonuni () ((ssonuni.1 (e. A (V)))) (-> (C_ A (On)) (e. (U. A) (On))) (ssonuni.1 A (V) ssonunit ax-mp)) thm (onuni () () (-> (e. A (On)) (e. (U. A) (On))) (A onsst A (On) ssonunit mpd)) thm (orduni () () (-> (Ord A) (Ord (U. A))) (A ordsson A ssorduni syl)) thm (onelpsst () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. A B) (/\ (C_ A B) (-. (= A B))))) (A B ordelssne A eloni B eloni syl2an)) thm (onsseleq () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (C_ A B) (\/ (e. A B) (= A B)))) (A B ordsseleq A eloni B eloni syl2an)) thm (onelsst () () (-> (e. A (On)) (-> (e. B A) (C_ B A))) (A eloni A ordtr A B trss 3syl)) thm (ordtr1 () () (-> (Ord C) (-> (/\ (e. A B) (e. B C)) (e. A C))) (C ordtr C A B trel syl)) thm (ordtr2 () () (-> (/\ (Ord A) (Ord C)) (-> (/\ (C_ A B) (e. B C)) (e. A C))) (A B ordsseleq biimpd C A B ordtr1 exp3a B C A eleq1a com12 (Ord C) a1i jaod com23 imp syl9 ex C B ordelord syl5 pm2.43d exp3a imp com23 imp3a)) thm (ontr1 () () (-> (e. C (On)) (-> (/\ (e. A B) (e. B C)) (e. A C))) (C eloni C A B ordtr1 syl)) thm (ontr2 () () (-> (/\ (e. A (On)) (e. C (On))) (-> (/\ (C_ A B) (e. B C)) (e. A C))) (A C B ordtr2 A eloni C eloni syl2an)) thm (ordunidif ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (Ord A) (e. B A)) (= (U. (\ A B)) (U. A))) (A B ordelon B (cv x) onelsst syl A B ordelon B A B eldif biimpr ex B eloni B ordeirr syl syl5 (Ord A) adantl mpd jctild (e. (cv x) A) adantr (cv y) B (cv x) sseq2 (\ A B) rcla4ev syl6 (cv x) A B eldif biimpr (cv x) ssid jctir ex (cv y) (cv x) (cv x) sseq2 (\ A B) rcla4ev syl6 (/\ (Ord A) (e. B A)) adantl pm2.61d r19.21aiva x A y B unidif syl)) thm (onint ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (C_ A (On)) (-. (= A ({/})))) (e. (|^| A) A)) (A (On) (cv z) ssel (cv x) (cv z) ontri1 (cv x) (cv z) (cv y) ssel syl6bir ex sylan9 com4r imp31 r19.20dva A (cv x) z disj y visset A z elint2 3imtr4g A (On) (cv x) ssel imdistani sylan2 exp32 com4l imp32 ssrdv (cv x) A intss1 (C_ A (On)) (= (i^i A (cv x)) ({/})) ad2antrl eqssd A eleq1d biimpd exp32 com34 pm2.43d r19.23adv ordon (On) A x tz7.5 mpan syl5 anabsi5)) thm (onint0 () () (-> (C_ A (On)) (<-> (= (|^| A) ({/})) (e. ({/}) A))) (A onint 0ex (|^| A) ({/}) (V) eleq1 mpbiri A intex sylibr sylan2 (|^| A) ({/}) A eleq1 (C_ A (On)) adantl mpbid ex A int0el (C_ A (On)) a1i impbid)) thm (onssmin ((x y) (A x) (A y)) () (-> (/\ (C_ A (On)) (-. (= A ({/})))) (E.e. x A (A.e. y A (C_ (cv x) (cv y))))) (A onint (cv y) A intss1 rgen jctir (cv x) (|^| A) (cv y) sseq1 y A ralbidv A rcla4ev syl)) thm (onminsb ((x y) (ph y)) ((onminsb.1 (-> ps (A. x ps))) (onminsb.2 (-> (= (cv x) (|^| ({e.|} x (On) ph))) (<-> ph ps)))) (-> (E.e. x (On) ph) ps) (x (On) ph rabn0 x (On) ph ssrab2 ({e.|} x (On) ph) onint mpan sylbir y x (On) ph hbrab1 hbint (e. (cv y) (On)) x ax-17 onminsb.1 onminsb.2 elrabf pm3.27bd syl)) thm (onminesb ((x y)) () (-> (E.e. x (On) ph) ([/] (|^| ({e.|} x (On) ph)) x ph)) (x (On) ph rabn0 x (On) ph ssrab2 ({e.|} x (On) ph) onint mpan sylbir (e. (cv y) (On)) x ax-17 (|^| ({e.|} x (On) ph)) ph elrabsf pm3.27bd syl)) thm (onintss ((ps x) (A x)) ((onintss.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A (On)) (-> ps (C_ (|^| ({e.|} x (On) ph)) A))) (onintss.1 (On) elrab A ({e.|} x (On) ph) intss1 sylbir ex)) thm (oninton () () (-> (/\ (C_ A (On)) (-. (= A ({/})))) (e. (|^| A) (On))) (A onint ex A (On) (|^| A) ssel syld imp)) thm (onintrab () () (<-> (e. (|^| ({e.|} x (On) ph)) (V)) (e. (|^| ({e.|} x (On) ph)) (On))) (({e.|} x (On) ph) intex x (On) ph ssrab2 ({e.|} x (On) ph) oninton mpan sylbir (|^| ({e.|} x (On) ph)) (On) elisset impbi)) thm (onintrab2 () () (<-> (E.e. x (On) ph) (e. (|^| ({e.|} x (On) ph)) (On))) (x (On) ph intexrab x ph onintrab bitr)) thm (onnmin () () (-> (/\ (C_ A (On)) (e. B A)) (-. (e. B (|^| A)))) (B A intss1 (C_ A (On)) adantl (|^| A) B ontri1 A oninton B A n0i sylan2 A (On) B ssel2 sylanc mpbid)) thm (onnminsb ((A x) (ps x)) ((onnminsb.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A (On)) (-> (e. A (|^| ({e.|} x (On) ph))) (-. ps))) (onnminsb.1 (On) elrab x (On) ph ssrab2 ({e.|} x (On) ph) A onnmin mpan sylbir ex con2d)) thm (oneqmini ((A x) (B x)) () (-> (C_ B (On)) (-> (/\ (e. A B) (A.e. x A (-. (e. (cv x) B)))) (= A (|^| B)))) (B (On) A ssel B (On) (cv x) ssel anim12d A (cv x) ontri1 syl6 exp3a imp pm5.74d (e. (cv x) B) (e. (cv x) A) bi2.03 syl6bb ralbidv2 A B x ssint syl5bb biimprd ex imp3a A B intss1 (C_ B (On)) a1i (A.e. x A (-. (e. (cv x) B))) adantrd jcad A (|^| B) eqss syl6ibr)) thm (oneqmin ((A x) (B x)) () (-> (/\ (C_ B (On)) (-. (= B ({/})))) (<-> (= A (|^| B)) (/\ (e. A B) (A.e. x A (-. (e. (cv x) B)))))) (A (|^| B) B eleq1 B onint syl5bir com12 A (|^| B) (cv x) eleq2 biimpd B (cv x) onnmin ex con2d syl9r r19.21adv (-. (= B ({/}))) adantr jcad B A x oneqmini (-. (= B ({/}))) adantr impbid)) thm (bm2.5ii ((x y) (A x) (A y)) ((bm2.5ii.1 (e. A (V)))) (-> (C_ A (On)) (= (U. A) (|^| ({e.|} x (On) (A.e. y A (C_ (cv y) (cv x))))))) (bm2.5ii.1 ssonuni (U. A) (On) x intmin A (cv x) y unissb (e. (cv x) (On)) a1i rabbii inteqi syl5reqr syl)) thm (onminex ((x y) (ph y) (ps x)) ((onminex.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (-> (E.e. x (On) ph) (E.e. x (On) (/\ ph (A.e. y (cv x) (-. ps))))) (y x (/\ (e. (cv x) (On)) ph) hbab1 hbint y x (/\ (e. (cv x) (On)) ph) hbab1 hbint y x (/\ (e. (cv x) (On)) ph) hbab1 hbel y x (/\ (e. (cv x) (On)) ph) hbab1 hbint (-. ps) x ax-17 hbim y hbal hban (cv x) (|^| ({|} x (/\ (e. (cv x) (On)) ph))) ({|} x (/\ (e. (cv x) (On)) ph)) eleq1 (cv x) (|^| ({|} x (/\ (e. (cv x) (On)) ph))) (cv y) eleq2 (-. ps) imbi1d y albidv anbi12d ({|} x (/\ (e. (cv x) (On)) ph)) cla4egf anabsi5 x (On) ph ssab2 ({|} x (/\ (e. (cv x) (On)) ph)) onint mpan x (On) ph ssab2 ({|} x (/\ (e. (cv x) (On)) ph)) oninton mpan (|^| ({|} x (/\ (e. (cv x) (On)) ph))) (cv y) onelon ex syl y visset (cv x) (cv y) (On) eleq1 onminex.1 anbi12d elab x (On) ph ssab2 ({|} x (/\ (e. (cv x) (On)) ph)) (cv y) onnmin mpan sylbir ex con2d syli y 19.21aiv sylanc x (/\ (e. (cv x) (On)) ph) abn0 x (/\ (e. (cv x) (On)) ph) abid bicomi y (cv x) (-. ps) df-ral anbi12i (e. (cv x) (On)) ph (A.e. y (cv x) (-. ps)) anass bitr3 x exbii 3imtr3 x (On) ph df-rex x (On) (/\ ph (A.e. y (cv x) (-. ps))) df-rex 3imtr4)) thm (ord0 () () (Ord ({/})) (({/}) df-ord tr0 (E) we0 mpbir2an)) thm (0elon () () (e. ({/}) (On)) (ord0 0ex elon mpbir)) thm (ord0eln0 () () (-> (Ord A) (<-> (e. ({/}) A) (-. (= A ({/}))))) (({/}) A n0i (Ord A) a1i ord0 A noel A ({/}) ordtri2 con2bid mpbiri mpan2 ord impbid)) thm (on0eln0 () () (-> (e. A (On)) (<-> (e. ({/}) A) (-. (= A ({/}))))) (A eloni A ord0eln0 syl)) thm (dflim2 () () (<-> (Lim A) (/\/\ (Ord A) (e. ({/}) A) (= A (U. A)))) (A df-lim A ord0eln0 (= A (U. A)) anbi1d pm5.32i (Ord A) (e. ({/}) A) (= A (U. A)) 3anass (Ord A) (-. (= A ({/}))) (= A (U. A)) 3anass 3bitr4 bitr4)) thm (inton () () (= (|^| (On)) ({/})) (0elon (On) int0el ax-mp)) thm (nlim0 () () (-. (Lim ({/}))) (({/}) eqid ({/}) df-lim (Ord ({/})) (-. (= ({/}) ({/}))) (= ({/}) (U. ({/}))) 3simp2 sylbi mt2)) thm (limord () () (-> (Lim A) (Ord A)) (A df-lim (Ord A) (-. 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B (suc A))))) (A eloni A B ordnbtwn syl)) thm (ordsuc () () (<-> (Ord A) (Ord (suc A))) (A (V) elong A suceloni (suc A) eloni syl syl6bir (suc A) A ordelord ex A (V) sucidg syl5com impbid A sucprc eqcomd A (suc A) ordeq syl pm2.61i)) thm (ordpwsuc ((A x)) () (-> (Ord A) (= (i^i (P~ A) (On)) (suc A))) ((cv x) A ordsssuc expcom pm5.32d (e. (cv x) (On)) (e. (cv x) (suc A)) pm3.27 (Ord A) a1i A ordsuc (suc A) (cv x) ordelon ex sylbi ancrd impbid bitrd (cv x) (P~ A) (On) elin x visset A elpw (e. (cv x) (On)) anbi1i (C_ (cv x) A) (e. (cv x) (On)) ancom 3bitr syl5bb eqrdv)) thm (onpwsuc () () (-> (e. A (On)) (= (i^i (P~ A) (On)) (suc A))) (A eloni A ordpwsuc syl)) thm (sucelon () () (<-> (e. A (On)) (e. (suc A) (On))) (A ordsuc A sucexb anbi12i A elon2 (suc A) elon2 3bitr4)) thm (ordsucss () () (-> (Ord B) (-> (e. A B) (C_ (suc A) B))) (A B ordnbtwn (e. A B) (e. B (suc A)) imnan sylibr (Ord B) adantr (suc A) B ordtri1 A ordsuc sylanb sylibrd B A ordelord sylan exp31 pm2.43b pm2.43b)) thm (sucssel () () (-> (e. A C) (-> (C_ (suc A) B) (e. A B))) ((suc A) B A ssel A C sucidg syl5com)) thm (ordelsuc () () (-> (/\ (e. A C) (Ord B)) (<-> (e. A B) (C_ (suc A) B))) (B A ordsucss (e. A C) adantl A C B sucssel (Ord B) adantr impbid)) thm (onsucmin ((A x)) () (-> (e. A (On)) (= (suc A) (|^| ({e.|} x (On) (e. A (cv x)))))) (A (On) (cv x) ordelsuc (cv x) eloni sylan2 ex rabbidv inteqd A sucelon (suc A) (On) x intmin sylbi eqtr2d)) thm (ordsucelsuc () () (-> (Ord B) (<-> (e. A B) (e. (suc A) (suc B)))) ((suc A) B ordsseleq A ordsuc sylanb (e. A (V)) adantl B A ordsucss (e. 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A (V)) (Ord A) ad2antll A (V) B sucssel (/\ (Ord A) (Ord B)) adantr impbid A sucexb (suc A) (V) B elsucg sylbi (/\ (Ord A) (Ord B)) adantr 3bitr4d ex A B elisset (suc A) (suc B) elisset A sucexb sylibr pm5.21ni (/\ (Ord A) (Ord B)) a1d pm2.61i biimprd (suc B) (suc A) ordelord A ordsuc sylibr B ordsuc sylanb sylan exp31 pm2.43a pm2.43d impbid)) thm (ordsucsssuc () () (-> (/\ (Ord A) (Ord B)) (<-> (C_ A B) (C_ (suc A) (suc B)))) (A B ordsucelsuc negbid (Ord B) adantr A B ordtri1 (suc A) (suc B) ordtri1 A ordsuc B ordsuc syl2anb 3bitr4d)) thm (orddif () () (-> (Ord A) (= A (\ (suc A) ({} A)))) (A orddisj A ({} A) disj3 A df-suc ({} A) difeq1i A ({} A) difun2 eqtr A eqeq2i bitr4 sylib)) thm (orduniss () () (-> (Ord A) (C_ (U. A) A)) (A ordtr A df-tr sylib)) thm (ordtri2or () () (-> (/\ (Ord A) (Ord B)) (\/ (e. A B) (C_ B A))) (A B ordtri3or (e. A B) (= A B) (e. B A) 3orass sylib B A ordsseleq ancoms (e. B A) (= B A) orcom B A eqcom (e. B A) orbi1i bitr syl6bb (e. A B) orbi2d mpbird)) thm (ordtri2or2 () () (-> (/\ (Ord A) (Ord B)) (\/ (C_ A B) (C_ B A))) (A B ordtri2or B A ordelss ex (C_ B A) orim1d (Ord A) adantl mpd)) thm (ordssun () () (-> (/\ (Ord B) (Ord C)) (<-> (C_ A (u. B C)) (\/ (C_ A B) (C_ A C)))) (B C ordtri2or2 B C ssequn1 (u. B C) C A sseq2 sylbi (C_ A C) (C_ A B) olc syl6bi C B ssequn2 (u. B C) B A sseq2 sylbi (C_ A B) (C_ A C) orc syl6bi jaoi syl A B C ssun (/\ (Ord B) (Ord C)) a1i impbid)) thm (ordequn () () (-> (/\ (Ord B) (Ord C)) (-> (= A (u. B C)) (\/ (= A B) (= A C)))) (B C ordtri2or2 B C ssequn1 (u. B C) C A eqeq2 sylbi (= A C) (= A B) olc syl6bi C B ssequn2 (u. B C) B A eqeq2 sylbi (= A B) (= A C) orc syl6bi jaoi syl)) thm (ordun () () (-> (/\ (Ord A) (Ord B)) (Ord (u. A B))) ((u. A B) eqid A B (u. A B) ordequn mpi (u. A B) A ordeq biimprcd (u. A B) B ordeq biimprcd jaao mpd)) thm (ordsucun ((A x) (B x)) () (-> (/\ (Ord A) (Ord B)) (= (suc (u. A B)) (u. (suc A) (suc B)))) (A B (cv x) ordssun (e. (cv x) (On)) adantl (cv x) (u. A B) ordsssuc A B ordun sylan2 (cv x) A ordsssuc (Ord B) adantrr (cv x) B ordsssuc (Ord A) adantrl orbi12d 3bitr3d (cv x) (suc A) (suc B) elun syl6bbr expcom pm5.32d A B ordun (u. A B) ordsuc (suc (u. A B)) (cv x) ordelon ex sylbi syl pm4.71rd (suc A) (suc B) ordun (u. (suc A) (suc B)) (cv x) ordelon ex syl A ordsuc B ordsuc syl2anb pm4.71rd 3bitr4d eqrdv)) thm (ordunisssuc ((A x) (B x)) () (-> (/\ (C_ A (On)) (Ord B)) (<-> (C_ (U. A) B) (C_ A (suc B)))) ((cv x) B ordsssuc A (On) (cv x) ssel2 sylan an1rs ralbidva A B x unissb A (suc B) x dfss3 3bitr4g)) thm (ordunel () () (-> (/\/\ (Ord A) (e. B A) (e. C A)) (e. (u. B C) A)) (A B ordsucss A C ordsucss anim12d 3impib B C ordsucun A B ordelord (e. C A) 3adant3 A C ordelord (e. B A) 3adant2 sylanc A sseq1d biimprd (u. B C) (V) A ordelsuc B A C A unexg (Ord A) 3adant1 (Ord A) (e. B A) (e. C A) 3simp1 sylanc sylibrd (suc B) A (suc C) unss syl5ib mpd)) thm (onsucuni () () (-> (C_ A (On)) (C_ A (suc (U. A)))) (A ssorduni (U. A) ssid A (U. A) ordunisssuc mpbii mpdan)) thm (ordsucuni () () (-> (Ord A) (C_ A (suc (U. A)))) (A ordsson A onsucuni syl)) thm (orduniorsuc () () (-> (Ord A) (\/ (= A (U. A)) (= A (suc (U. A))))) (A orduniss A orduni (U. A) A ordelssne mpancom biimprd mpand A (U. A) ordsucss syld A ordsucuni jctild A (U. A) eqcom negbii A (suc (U. A)) eqss 3imtr4g orrd)) thm (unon ((x y)) () (= (U. (On)) (On)) ((cv x) (On) y eluni2 (cv y) (cv x) onelon ex r19.23aiv sylbi (cv x) suceloni x visset sucid jctil (cv x) (suc (cv x)) (On) elunii syl impbi eqriv)) thm (ordunisuc ((A x)) () (-> (Ord A) (= (U. (suc A)) A)) (A ordeleqon (cv x) A suceq unieqd (= (cv x) A) id eqeq12d (cv x) eloni (cv x) ordtr syl x visset unisuc sylib vtoclga onprc A (On) (V) eleq1 mtbiri A sucprc syl unieqd unon A (On) unieq (= A (On)) id eqeq12d mpbiri eqtrd jaoi sylbi)) thm (orduniss2 ((A x)) () (-> (Ord A) (= (U. ({e.|} x (On) (C_ (cv x) A))) A)) (A ordpwsuc x (On) (C_ (cv x) A) df-rab x (e. (cv x) (On)) (C_ (cv x) A) inab ({|} x (e. (cv x) (On))) ({|} x (C_ (cv x) A)) incom eqtr3 A x df-pw eqcomi x (On) abid2 ineq12i 3eqtr syl5eq unieqd A ordunisuc eqtrd)) thm (onsucuni2 () () (-> (/\ (e. A (On)) (= A (suc B))) (= (suc (U. A)) A)) (A (suc B) (On) eleq1 B sucelon syl6bbr biimpac B eloni B ordeirr syl (suc B) B B eleq2 B (On) sucidg syl5bi com12 mtod B eloni B ordunisuc syl (suc B) eqeq2d mtbird (= A (suc B)) adantl (= A (suc B)) id A (suc B) unieq eqeq12d (e. B (On)) adantr mtbird ex (e. A (On)) adantl mpd A eloni A orduniorsuc ord syl (= A (suc B)) adantr mpd eqcomd)) thm (0elsuc () () (-> (Ord A) (e. ({/}) (suc A))) (A ordsuc A nsuceq0 (suc A) ord0eln0 mpbiri sylbi)) thm (suc11 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (= (suc A) (suc B)) (= A B))) (A eloni A B ordn2lp (e. A B) (e. B A) ianor sylib syl (e. B (On)) adantr A (On) (suc B) sucssel (suc A) (suc B) eqimss syl5 A B elsuci ord com12 syl9 B (On) (suc A) sucssel (suc A) (suc B) eqimss2 syl5 B A elsuci ord com12 B A eqcom syl6ib syl9 jaao mpd A B suceq (/\ (e. A (On)) (e. B (On))) a1i impbid)) thm (limon () () (Lim (On)) (ordon onne0 unon eqcomi 3pm3.2i (On) df-lim mpbir)) thm (onord () ((on.1 (e. A (On)))) (Ord A) (on.1 A eloni ax-mp)) thm (ontrc () ((on.1 (e. A (On)))) (Tr A) (on.1 onord A ordtr ax-mp)) thm (oneirr () ((on.1 (e. A (On)))) (-. (e. A A)) (on.1 onord A ordeirr ax-mp)) thm (onel () ((on.1 (e. A (On)))) (-> (e. B A) (e. B (On))) (on.1 A B onelon mpan)) thm (onss () ((on.1 (e. A (On)))) (C_ A (On)) (on.1 A onsst ax-mp)) thm (onelss () ((on.1 (e. A (On)))) (-> (e. B A) (C_ B A)) (on.1 A B onelsst ax-mp)) thm (onssneli () ((on.1 (e. A (On)))) (-> (C_ A B) (-. (e. B A))) (on.1 B onel B eloni B ordeirr 3syl A B B ssel com12 mtod con2i)) thm (onssneli2 () ((on.1 (e. A (On)))) (-> (C_ B A) (-. (e. A B))) (on.1 oneirr B A A ssel mtoi)) thm (onelin () ((on.1 (e. A (On)))) (-> (e. B A) (= B (i^i B A))) (on.1 B onelss B A dfss sylib)) thm (onelun () ((on.1 (e. A (On)))) (-> (e. B A) (= (u. A B) A)) (on.1 B onelss B A ssequn2 sylib)) thm (onsuc () ((on.1 (e. A (On)))) (e. (suc A) (On)) (on.1 A suceloni ax-mp)) thm (onunisuc () ((on.1 (e. A (On)))) (= (U. (suc A)) A) (on.1 ontrc on.1 elisseti unisuc mpbi)) thm (onuniorsuc () ((on.1 (e. A (On)))) (\/ (= A (U. A)) (= A (suc (U. A)))) (on.1 onord A orduniorsuc ax-mp)) thm (onuninsuc ((A x)) ((on.1 (e. A (On)))) (<-> (= A (U. A)) (-. (E.e. x (On) (= A (suc (cv x)))))) (on.1 oneirr (= A (U. A)) id (cv x) df-suc A eqeq2i A (u. (cv x) ({} (cv x))) unieq sylbi (cv x) ({} (cv x)) uniun x visset unisn (U. (cv x)) uneq2i eqtr syl6eq on.1 A (suc (cv x)) (On) eleq1 mpbii ordon (On) ordtr ax-mp (On) (cv x) trsuc mpan (cv x) eloni (cv x) ordtr syl (cv x) df-tr sylib 3syl (U. (cv x)) (cv x) ssequn1 sylib eqtrd sylan9eqr x visset sucid A (suc (cv x)) (cv x) eleq2 mpbiri (= A (U. A)) adantr eqeltrd mto (= A (suc (cv x))) (= A (U. A)) imnan mpbir (e. (cv x) (On)) a1i r19.23aiv on.1 onuniorsuc ori on.1 A onuni ax-mp jctil (cv x) (U. A) suceq A eqeq2d (On) rcla4ev syl impbi con2bii)) thm (onssel () ((on.1 (e. A (On))) (on.2 (e. B (On)))) (<-> (C_ A B) (\/ (e. A B) (= A B))) (on.1 on.2 A B onsseleq mp2an)) thm (onun () ((on.1 (e. A (On))) (on.2 (e. B (On)))) (e. (u. A B) (On)) (on.2 onord on.1 onord B A ordtri2or mp2an on.1 B onelun on.1 syl6eqel A B ssequn1 on.2 (u. A B) B (On) eleq1 mpbiri sylbi jaoi ax-mp)) thm (onsucss () ((on.1 (e. A (On))) (on.2 (e. B (On)))) (<-> (e. A B) (C_ (suc A) B)) (on.1 on.2 onord A (On) B ordelsuc mp2an)) thm (nlimsucg () () (-> (e. A B) (-. (Lim (suc A)))) (A ordsuc (suc A) ordeirr sylbi (e. A B) adantl (e. A B) (= (U. (suc A)) A) pm3.27 A B sucidg (= (U. (suc A)) A) adantr eqeltrd (= (suc A) (U. (suc A))) adantr (suc A) (U. (suc A)) (suc A) eleq1 (/\ (e. A B) (= (U. (suc A)) A)) adantl mpbird ex A ordunisuc sylan2 mtod ex (suc A) limuni (e. A B) a1i nsyld (Ord (suc A)) (-. (= (suc A) ({/}))) (= (suc A) (U. (suc A))) 3simp1 (suc A) df-lim A ordsuc 3imtr4 con3i pm2.61d1)) thm (unizlim () () (-> (Ord A) (<-> (= A (U. A)) (\/ (= A ({/})) (Lim A)))) (A df-lim biimpr 3exp com23 imp orrd ex uni0 eqcomi (= A ({/})) id A ({/}) unieq eqeq12d mpbiri A limuni jaoi (Ord A) a1i impbid)) thm (orduninsuc ((A x)) () (-> (Ord A) (<-> (= A (U. A)) (-. (E.e. x (On) (= A (suc (cv x))))))) (A ordeleqon (= A (if (e. A (On)) A ({/}))) id A (if (e. A (On)) A ({/})) unieq eqeq12d A (if (e. A (On)) A ({/})) (suc (cv x)) eqeq1 x (On) rexbidv negbid bibi12d 0elon A elimel x onuninsuc dedth unon eqcomi onprc x visset sucex (On) (suc (cv x)) (V) eleq1 mpbiri mto (e. (cv x) (On)) a1i nrex 2th (= A (On)) id A (On) unieq eqeq12d A (On) (suc (cv x)) eqeq1 x (On) rexbidv negbid bibi12d mpbiri jaoi sylbi)) thm (ordunisuc2 ((A x)) () (-> (Ord A) (<-> (= A (U. A)) (A.e. x A (e. (suc (cv x)) A)))) (A x orduninsuc A (suc (cv x)) ordtri3 (cv x) suceloni (suc (cv x)) eloni syl sylan2 con2bid (cv x) A onnbtwn (e. (cv x) A) (e. A (suc (cv x))) imnan sylibr con2d (e. (cv x) A) (e. (suc (cv x)) A) pm2.21 syl6 (Ord A) adantl (e. (suc (cv x)) A) (e. (cv x) A) ax-1 (/\ (Ord A) (e. (cv x) (On))) a1i jaod (cv x) A ordtri2or (cv x) eloni sylan ancoms (e. (cv x) A) (C_ A (cv x)) orcom sylib (-> (e. (cv x) A) (e. (suc (cv x)) A)) adantr A (cv x) ordsssuc2 biimpd (-> (e. (cv x) A) (e. (suc (cv x)) A)) adantr (/\ (Ord A) (e. (cv x) (On))) (-> (e. (cv x) A) (e. (suc (cv x)) A)) pm3.27 orim12d mpd ex impbid bitr3d ex pm5.74d (e. (cv x) (On)) (e. (cv x) A) pm3.27 (Ord A) a1i A (cv x) ordelon ex ancrd impbid (e. (suc (cv x)) A) imbi1d (e. (cv x) (On)) (e. (cv x) A) (e. (suc (cv x)) A) impexp syl5bbr bitrd ralbidv2 x (On) (= A (suc (cv x))) ralnex syl5bbr bitrd)) thm (ordzsl ((A x)) () (<-> (Ord A) (\/\/ (= A ({/})) (E.e. x (On) (= A (suc (cv x)))) (Lim A))) (A x orduninsuc biimprd A unizlim sylibd orrd (= A ({/})) (E.e. x (On) (= A (suc (cv x)))) (Lim A) 3orass (= A ({/})) (E.e. x (On) (= A (suc (cv x)))) (Lim A) or12 bitr sylibr ord0 A ({/}) ordeq mpbiri A (suc (cv x)) (On) eleq1 (cv x) suceloni syl5bir A eloni syl6com r19.23aiv A limord 3jaoi impbi)) thm (onzsl ((A x)) () (<-> (e. A (On)) (\/\/ (= A ({/})) (E.e. x (On) (= A (suc (cv x)))) (/\ (e. A (V)) (Lim A)))) (A (On) elisset pm4.71ri A (V) elong pm5.32i (e. A (V)) (\/ (= A ({/})) (E.e. x (On) (= A (suc (cv x))))) (Lim A) andi 0ex A ({/}) (V) eleq1 mpbiri x visset sucex A (suc (cv x)) (V) eleq1 mpbiri (e. (cv x) (On)) a1i r19.23aiv jaoi pm4.71ri (/\ (e. A (V)) (Lim A)) orbi1i bitr4 A x ordzsl (= A ({/})) (E.e. x (On) (= A (suc (cv x)))) (Lim A) df-3or bitr (e. A (V)) anbi2i (= A ({/})) (E.e. x (On) (= A (suc (cv x)))) (/\ (e. A (V)) (Lim A)) df-3or 3bitr4 3bitr)) thm (dflim3 ((A x)) () (<-> (Lim A) (/\ (Ord A) (-. (\/ (= A ({/})) (E.e. x (On) (= A (suc (cv x)))))))) (A limord nlim0 A ({/}) limeq mtbiri con2i A limuni A limord A x orduninsuc syl mpbid jca (= A ({/})) (E.e. x (On) (= A (suc (cv x)))) ioran sylibr jca A x ordzsl biimp (= A ({/})) (E.e. x (On) (= A (suc (cv x)))) (Lim A) df-3or sylib orcanai impbi)) thm (dflim4 ((A x)) () (<-> (Lim A) (/\/\ (Ord A) (e. ({/}) A) (A.e. x A (e. (suc (cv x)) A)))) (A dflim2 A x ordunisuc2 (e. ({/}) A) anbi2d pm5.32i (Ord A) (e. ({/}) A) (= A (U. A)) 3anass (Ord A) (e. ({/}) A) (A.e. x A (e. (suc (cv x)) A)) 3anass 3bitr4 bitr)) thm (limsuc ((A x) (B x)) () (-> (Lim A) (<-> (e. B A) (e. (suc B) A))) (A x dflim4 (cv x) B suceq A eleq1d A rcla4cv (Ord A) (e. ({/}) A) 3ad2ant3 sylbi A limord A ordtr A B trsuc ex 3syl impbid)) thm (limsssuc ((A x) (B x)) () (-> (Lim A) (<-> (C_ A B) (C_ A (suc B)))) (B sssucid A B (suc B) sstr2 mpi (Lim A) a1i (cv x) B A eleq1 biimpcd A B limsuc biimpa A (suc B) ordtri1 A limord (e. B A) adantr A B ordelord A limord sylan B ordsuc sylib sylanc con2bid mpbid ex sylan9r con2d ex com23 imp31 A (suc B) (cv x) ssel2 x visset B elsuc sylib ord con1d (Lim A) adantll mpd ex ssrdv ex impbid)) thm (nlimon ((x y)) () (= ({e.|} x (On) (\/ (= (cv x) ({/})) (E.e. y (On) (= (cv x) (suc (cv y)))))) ({e.|} x (On) (-. (Lim (cv x))))) ((cv x) eloni (cv x) y dflim3 baib con2bid syl rabbii)) thm (limuni3 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (-. (= A ({/}))) (A.e. x A (Lim (cv x)))) (Lim (U. A))) ((cv x) (cv z) limeq A rcla4v z visset (cv z) (V) limelon mpan syl6com ssrdv A ssorduni syl (-. (= A ({/}))) adantl A z n0 (cv x) (cv z) limeq A rcla4v ({/}) (cv z) A elunii expcom (cv z) 0ellim syl5 syld z 19.23aiv sylbi imp (cv x) (cv z) limeq A rcla4cv (cv z) (cv y) limsuc (e. (cv z) A) anbi1d (suc (cv y)) (cv z) A elunii syl6bi exp3a com3r sylcom r19.23adv (cv y) A z eluni2 syl5ib r19.21aiv (-. (= A ({/}))) adantl 3jca (U. A) y dflim4 sylibr)) thm (on0eqelt () () (-> (e. A (On)) (\/ (= A ({/})) (e. ({/}) A))) (A 0ss 0elon ({/}) A onsseleq mpan mpbii ({/}) A eqcom (e. ({/}) A) orbi2i (e. ({/}) A) (= A ({/})) orcom bitr sylib)) thm (snsn0non () () (-. (e. ({} ({} ({/}))) (On))) (0ex snnz 0ex ({} ({/})) elsnc ({/}) ({} ({/})) eqcom bitr mtbir 0ex snid ({} ({/})) ({} ({} ({/}))) ({/}) ssel mpi mto p0ex snid ({} ({} ({/}))) ({} ({/})) onelsst mpi mto)) thm (tfi ((A x)) () (-> (/\ (C_ A (On)) (A.e. x (On) (-> (C_ (cv x) A) (e. (cv x) A)))) (= A (On))) ((cv x) (On) A eldifn (-> (e. (cv x) (On)) (-> (C_ (cv x) A) (e. (cv x) A))) adantl (On) A (cv x) difin0ss (cv x) onsst syl5com (e. (cv x) A) imim1d a2i (cv x) (On) A eldifi syl5 imp mtod ex r19.20i2 x (\ (On) A) (= (i^i (\ (On) A) (cv x)) ({/})) ralnex sylib (On) A ssdif0 negbii (On) A difss ordon (On) (\ (On) A) x tz7.5 mpan mpan sylbi nsyl2 (C_ A (On)) anim2i A (On) eqss sylibr)) thm (tfis ((ph y) (y z) (ph z) (x y) (x z)) ((tfis.1 (-> (e. (cv x) (On)) (-> (A.e. y (cv x) ([/] (cv y) x ph)) ph)))) (-> (e. (cv x) (On)) ph) (x (On) ph ssrab2 (e. (cv z) (On)) x ax-17 (e. (cv y) (cv z)) x ax-17 y x ph hbs1 hbral z x ph hbs1 hbim hbim (cv x) (cv z) (On) eleq1 (cv x) (cv z) y ([/] (cv y) x ph) raleq1 x z ph sbequ12 imbi12d imbi12d tfis.1 chvar (cv z) ({e.|} x (On) ph) y dfss3 (e. (cv z) (On)) x ax-17 (cv y) ph elrabsf pm3.27bd y (cv z) r19.20si sylbi syl5 anc2li (e. (cv z) (On)) x ax-17 (cv z) ph elrabsf syl6ibr rgen ({e.|} x (On) ph) z tfi mp2an eqcomi rabeq2i pm3.27bd)) thm (tfis2f ((ph y) (x y)) ((tfis2f.1 (-> ps (A. x ps))) (tfis2f.2 (-> (= (cv x) (cv y)) (<-> ph ps))) (tfis2f.3 (-> (e. (cv x) (On)) (-> (A.e. y (cv x) ps) ph)))) (-> (e. (cv x) (On)) ph) (tfis2f.3 tfis2f.1 tfis2f.2 sbie y (cv x) ralbii syl5ib tfis)) thm (tfis2 ((ps x) (ph y) (x y)) ((tfis2.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (tfis2.2 (-> (e. (cv x) (On)) (-> (A.e. y (cv x) ps) ph)))) (-> (e. (cv x) (On)) ph) (ps x ax-17 tfis2.1 tfis2.2 tfis2f)) thm (tfis3 ((ps x) (ph y) (ch x) (A x) (x y)) ((tfis3.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (tfis3.2 (-> (= (cv x) A) (<-> ph ch))) (tfis3.3 (-> (e. (cv x) (On)) (-> (A.e. y (cv x) ps) ph)))) (-> (e. A (On)) ch) (tfis3.2 tfis3.1 tfis3.3 tfis2 vtoclga)) thm (dfom2 ((x y) (x z) (y z)) () (= (om) ({e.|} x (On) (C_ (suc (cv x)) ({e.|} y (On) (-. (Lim (cv y))))))) (x visset elon (A. z (-> (Lim (cv z)) (e. (cv x) (cv z)))) anbi1i (cv z) (cv x) onsssuc (cv z) (cv x) ontri1 bitr3d ancoms (cv y) (cv z) limeq negbid (On) elrab (/\ (e. (cv x) (On)) (e. (cv z) (On))) a1i imbi12d ex pm5.74d z visset (cv z) (V) limelon mpan pm4.71ri (e. (cv x) (cv z)) imbi1i (e. (cv z) (On)) (Lim (cv z)) (e. (cv x) (cv z)) impexp (e. (cv z) (On)) (-. (Lim (cv z))) ibar (-. (e. (cv x) (cv z))) imbi2d (Lim (cv z)) (e. (cv x) (cv z)) pm4.1 syl5bb pm5.74i 3bitr syl6rbbr (e. (cv z) (On)) (e. (cv z) (suc (cv x))) pm3.27 (e. (cv x) (On)) a1i (cv x) suceloni (suc (cv x)) (cv z) onelon ex syl ancrd impbid (e. (cv z) ({e.|} y (On) (-. (Lim (cv y))))) imbi1d (e. (cv z) (On)) (e. (cv z) (suc (cv x))) (e. (cv z) ({e.|} y (On) (-. (Lim (cv y))))) impexp syl5bbr bitrd z albidv (suc (cv x)) ({e.|} y (On) (-. (Lim (cv y)))) z dfss2 syl6bbr pm5.32i bitr3 x abbii x z df-om x (On) (C_ (suc (cv x)) ({e.|} y (On) (-. (Lim (cv y))))) df-rab 3eqtr4)) thm (elom ((A x) (A y) (x y)) ((elom.1 (e. A (V)))) (<-> (e. A (om)) (/\ (Ord A) (A. x (-> (Lim (cv x)) (e. A (cv x)))))) (elom.1 (cv y) A ordeq (cv y) A (cv x) eleq1 (Lim (cv x)) imbi2d x albidv anbi12d y x df-om elab2)) thm (elomg ((A x) (A y) (x y)) () (-> (e. A B) (<-> (e. A (om)) (/\ (Ord A) (A. x (-> (Lim (cv x)) (e. A (cv x))))))) ((cv y) A (om) eleq1 (cv y) A ordeq (cv y) A (cv x) eleq1 (Lim (cv x)) imbi2d x albidv anbi12d y visset x elom B vtoclbg)) thm (omsson ((x y)) () (C_ (om) (On)) (x visset y elom pm3.26bd x visset elon sylibr ssriv)) thm (limomss ((x y) (A x) (A y)) () (-> (Lim A) (C_ (om) A)) (A limord A ordeleqon (cv y) A limeq (cv y) A (cv x) eleq2 imbi12d (On) cla4gv x visset y elom pm3.27bd syl5 com23 imp ssrdv ex omsson A (On) (om) sseq2 mpbiri (Lim A) a1d jaoi sylbi mpcom)) thm (nnont () () (-> (e. A (om)) (e. A (On))) (omsson A sseli)) thm (nnon () ((nnon.1 (e. A (om)))) (e. A (On)) (nnon.1 A nnont ax-mp)) thm (nnord () () (-> (e. A (om)) (Ord A)) (A nnont A eloni syl)) thm (ordom ((x y) (x z) (y z)) () (Ord (om)) ((om) y x dftr2 (cv x) (cv y) ordelord (cv x) nnord sylan ancoms (cv z) (cv y) (cv x) trel exp3a com12 (cv z) limord (cv z) ordtr syl syl5 a2d z 19.20dv x visset z elom pm3.27bd syl5 imp jca y visset z elom sylibr x ax-gen mpgbir omsson ordon (om) (On) trssord mp3an)) thm (elnn () () (-> (/\ (e. A B) (e. B (om))) (e. A (om))) (ordom (om) ordtr ax-mp (om) A B trel ax-mp)) thm (omon () () (\/ (e. (om) (On)) (= (om) (On))) (ordom (om) ordeleqon mpbi)) thm (nnlim ((A x)) () (-> (e. A (om)) (-. (Lim A))) (A nnord A ordeirr syl A (om) x elomg ibi pm3.27d (cv x) A limeq (cv x) A A eleq2 imbi12d (om) cla4gv mpd mtod)) thm (omssnlim ((x y)) () (C_ (om) ({e.|} x (On) (-. (Lim (cv x))))) ((cv y) nnont (cv y) nnlim jca (cv x) (cv y) limeq negbid (On) elrab sylibr ssriv)) thm (limom () () (Lim (om)) (ordom (om) ordeleqon ordom (om) ordeirr ax-mp (om) (On) x elomg ordom (cv x) (om) ordtri1 (-. (Lim (om))) adantr (cv x) (om) ordsseleq biimpd (cv x) nnlim (-. (Lim (om))) a1i (cv x) (om) limeq biimpd con3d com12 jaod sylan9 sylbird a3d mpanl2 (cv x) limord sylan ex pm2.43b x 19.21aiv ordom jctil syl5bir mt3i limon (om) (On) limeq mpbiri jaoi sylbi ax-mp)) thm (peano2b () () (<-> (e. A (om)) (e. (suc A) (om))) (limom (om) A limsuc ax-mp)) thm (nnsuc ((A x)) () (-> (/\ (e. A (om)) (-. (= A ({/})))) (E.e. x (om) (= A (suc (cv x))))) (A nnlim (-. (= A ({/}))) adantr A x orduninsuc (-. (= A ({/}))) adantr A df-lim biimpr 3exp imp sylbird A nnord sylan mt3d A (suc (cv x)) (om) eleq1 biimpcd (cv x) peano2b syl6ibr ancrd (e. (cv x) (On)) adantld x 19.22dv x (On) (= A (suc (cv x))) df-rex x (om) (= A (suc (cv x))) df-rex 3imtr4g (-. (= A ({/}))) adantr mpd)) thm (peano1 () () (e. ({/}) (om)) (limom (om) 0ellim ax-mp)) thm (peano2 () () (-> (e. A (om)) (e. (suc A) (om))) (A peano2b biimp)) thm (peano3 () () (-> (e. A (om)) (-. (= (suc A) ({/})))) (A nsuceq0 (e. A (om)) a1i)) thm (peano4 () () (-> (/\ (e. A (om)) (e. B (om))) (<-> (= (suc A) (suc B)) (= A B))) (A B suc11 A nnont B nnont syl2an)) thm (peano5 ((x y) (A x) (A y)) () (-> (/\ (e. ({/}) A) (A.e. x (om) (-> (e. (cv x) A) (e. (suc (cv x)) A)))) (C_ (om) A)) ((cv y) (om) A eldifn (/\ (e. ({/}) A) (A.e. x (om) (-> (e. (cv x) A) (e. (suc (cv x)) A)))) adantl (cv y) x nnsuc (cv y) (om) A eldifi (e. ({/}) A) adantl (cv y) ({/}) (\ (om) A) eleq1 biimpcd con3d ({/}) A (om) elndif syl5com imp sylanc (A.e. x (om) (-> (e. (cv x) A) (e. (suc (cv x)) A))) adantlr (= (i^i (\ (om) A) (cv y)) ({/})) adantr x (om) (-> (e. (cv x) A) (e. (suc (cv x)) A)) hbra1 (/\ (e. (cv y) (\ (om) A)) (= (i^i (\ (om) A) (cv y)) ({/}))) x ax-17 hban (e. (cv y) A) x ax-17 x (om) (-> (e. (cv x) A) (e. (suc (cv x)) A)) ra4 x visset sucid (cv y) (suc (cv x)) (cv x) eleq2 mpbiri (cv y) (suc (cv x)) (om) eleq1 (cv x) peano2b syl6bbr (cv x) (om) A neldif (cv x) (cv y) (\ (om) A) minel sylan2 exp32 syl6bi mpid (cv y) (om) A eldifi syl5 imp3a (suc (cv x)) A (cv y) eleq1a com12 imim12d com13 sylan9 r19.23ad exp32 (e. ({/}) A) a1i imp41 mpd ex mtod nrexdv (om) A difss ordom (om) (\ (om) A) y tz7.5 mpan mpan nsyl2 (om) A ssdif0 sylibr)) thm (nn0suc ((A x)) () (-> (e. A (om)) (\/ (= A ({/})) (E.e. x (om) (= A (suc (cv x)))))) (A x nnsuc ex orrd)) thm (find ((A x)) ((find.1 (/\/\ (C_ A (om)) (e. ({/}) A) (A.e. x A (e. (suc (cv x)) A))))) (= A (om)) (find.1 3simp1i find.1 (e. (suc (cv x)) A) (e. (cv x) (om)) ax-1 x A r19.20si x A (om) (e. (suc (cv x)) A) ralcom3 sylib (e. ({/}) A) anim2i (C_ A (om)) anim2i (C_ A (om)) (e. ({/}) A) (A.e. x A (e. (suc (cv x)) A)) 3anass (C_ A (om)) (e. ({/}) A) (A.e. x (om) (-> (e. (cv x) A) (e. (suc (cv x)) A))) 3anass 3imtr4 ax-mp A x peano5 (C_ A (om)) 3adant1 ax-mp eqssi)) thm (finds ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((finds.1 (-> (= (cv x) ({/})) (<-> ph ps))) (finds.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (finds.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (finds.4 (-> (= (cv x) A) (<-> ph ta))) (finds.5 ps) (finds.6 (-> (e. (cv y) (om)) (-> ch th)))) (-> (e. A (om)) ta) (finds.5 0ex finds.1 elab mpbir finds.6 y visset finds.2 elab y visset sucex finds.3 elab 3imtr4g rgen ({|} x ph) y peano5 mp2an A sseli finds.4 (om) elabg mpbid)) thm (findsg ((A x) (x y) (B x) (B y) (ps x) (ch x) (th x) (ta x) (ph y)) ((findsg.1 (-> (= (cv x) B) (<-> ph ps))) (findsg.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (findsg.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (findsg.4 (-> (= (cv x) A) (<-> ph ta))) (findsg.5 (-> (e. B (om)) ps)) (findsg.6 (-> (/\ (/\ (e. (cv y) (om)) (e. B (om))) (C_ B (cv y))) (-> ch th)))) (-> (/\ (/\ (e. A (om)) (e. B (om))) (C_ B A)) ta) ((cv x) ({/}) B sseq2 (= B ({/})) adantl B ({/}) (cv x) eqeq2 findsg.1 syl6bir imp imbi12d (cv x) ({/}) B sseq2 ph imbi1d B ss0 con3i (<-> ph ps) pm2.21d pm5.74d sylan9bbr pm2.61ian (e. B (om)) imbi2d (cv x) (cv y) B sseq2 findsg.2 imbi12d (e. B (om)) imbi2d (cv x) (suc (cv y)) B sseq2 findsg.3 imbi12d (e. B (om)) imbi2d (cv x) A B sseq2 findsg.4 imbi12d (e. B (om)) imbi2d findsg.5 (C_ B ({/})) a1d y visset sucex B x eqvinc findsg.1 findsg.5 syl5bir findsg.3 biimpd sylan9r x 19.23aiv sylbi eqcoms (C_ B (suc (cv y))) imim2i (-> (C_ B (cv y)) ch) a1d com4r (e. (cv y) (om)) adantl B (cv y) onsssuc B (suc (cv y)) onelpsst (cv y) suceloni sylan2 bitrd B nnont (cv y) nnont syl2an ancoms findsg.6 ex th (C_ B (suc (cv y))) ax-1 syl8 a2d com23 sylbird (C_ B (suc (cv y))) (= B (suc (cv y))) annim syl5ibr pm2.61d ex a2d finds imp31)) thm (finds2 ((x y) (ta x) (ta y) (ps x) (ch x) (th x) (ph y)) ((finds2.1 (-> (= (cv x) ({/})) (<-> ph ps))) (finds2.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (finds2.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (finds2.4 (-> ta ps)) (finds2.5 (-> (e. (cv y) (om)) (-> ta (-> ch th))))) (-> (e. (cv x) (om)) (-> ta ph)) (finds2.4 0ex finds2.1 ta imbi2d elab mpbir finds2.5 a2d y visset finds2.2 ta imbi2d elab y visset sucex finds2.3 ta imbi2d elab 3imtr4g rgen ({|} x (-> ta ph)) y peano5 mp2an (cv x) sseli x (-> ta ph) abid sylib)) thm (finds1 ((x y) (ps x) (ch x) (th x) (ph y)) ((finds1.1 (-> (= (cv x) ({/})) (<-> ph ps))) (finds1.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (finds1.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (finds1.4 ps) (finds1.5 (-> (e. (cv y) (om)) (-> ch th)))) (-> (e. (cv x) (om)) ph) (({/}) eqid finds1.1 finds1.2 finds1.3 finds1.4 (= ({/}) ({/})) a1i finds1.5 (= ({/}) ({/})) a1d finds2 mpi)) thm (findes ((x y) (x z) (y z) (ph y) (ph z)) ((findes.1 ([/] ({/}) x ph)) (findes.2 (-> (e. (cv x) (om)) (-> ph ([/] (suc (cv x)) x ph))))) (-> (e. (cv x) (om)) ph) ((cv z) ({/}) x ph dfsbcq z y x ph sbequ (cv z) (suc (cv y)) x ph dfsbcq z x ph sbequ12r findes.1 (e. (cv y) (om)) x ax-17 y x ph hbs1 y visset sucex x ph hbsbc1v hbim hbim (cv x) (cv y) (om) eleq1 x y ph sbequ12 (cv x) (cv y) suceq (suc (cv x)) (suc (cv y)) x ph dfsbcq syl imbi12d imbi12d findes.2 chvar finds)) thm (tfinds ((x y) (x z) (y z) (A x) (ps x) (ch x) (ch z) (th x) (ta x) (ph y) (ph z)) ((tfinds.1 (-> (= (cv x) ({/})) (<-> ph ps))) (tfinds.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (tfinds.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (tfinds.4 (-> (= (cv x) A) (<-> ph ta))) (tfinds.5 ps) (tfinds.6 (-> (e. (cv y) (On)) (-> ch th))) (tfinds.7 (-> (Lim (cv x)) (-> (A.e. y (cv x) ch) ph)))) (-> (e. A (On)) ta) (tfinds.2 tfinds.4 (cv x) eloni (cv x) df-lim biimpr 3exp (= (cv x) (U. (cv x))) (Lim (cv x)) con3 syl6 com23 (cv x) y orduninsuc biimprd con1d syl6d (= (cv x) ({/})) (E.e. y (On) (= (cv x) (suc (cv y)))) df-or syl6ibr syl tfinds.5 tfinds.1 mpbiri (A.e. y (cv x) ch) a1d (e. (cv x) (On)) a1d (e. (cv x) (On)) y ax-17 y (cv x) ch hbra1 ph y ax-17 hbim hbim (cv x) (suc (cv y)) z ([/] (cv z) x ph) raleq1 y z x ph sbequ ch x ax-17 tfinds.2 sbie syl5bbr (cv x) cbvralv ph z ax-17 z x ph hbs1 x z ph sbequ12 (suc (cv y)) cbvral 3bitr4g biimpd tfinds.6 y visset sucid tfinds.2 (suc (cv y)) rcla4v ax-mp syl5 sylan9r tfinds.3 (e. (cv y) (On)) adantl sylibrd (e. (cv x) (On)) a1d ex r19.23ai jaoi syl6 pm2.43a tfinds.7 pm2.61d2 tfis3)) thm (tfindsg ((A x) (x y) (B x) (B y) (ps x) (ch x) (th x) (ta x) (ph y)) ((tfindsg.1 (-> (= (cv x) B) (<-> ph ps))) (tfindsg.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (tfindsg.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (tfindsg.4 (-> (= (cv x) A) (<-> ph ta))) (tfindsg.5 (-> (e. B (On)) ps)) (tfindsg.6 (-> (/\ (/\ (e. (cv y) (On)) (e. B (On))) (C_ B (cv y))) (-> ch th))) (tfindsg.7 (-> (/\ (/\ (Lim (cv x)) (e. B (On))) (C_ B (cv x))) (-> (A.e. y (cv x) (-> (C_ B (cv y)) ch)) ph)))) (-> (/\ (/\ (e. A (On)) (e. B (On))) (C_ B A)) ta) ((cv x) ({/}) B sseq2 (= B ({/})) adantl B ({/}) (cv x) eqeq2 tfindsg.1 syl6bir imp imbi12d (cv x) ({/}) B sseq2 ph imbi1d B ss0 con3i (<-> ph ps) pm2.21d pm5.74d sylan9bbr pm2.61ian (e. B (On)) imbi2d (cv x) (cv y) B sseq2 tfindsg.2 imbi12d (e. B (On)) imbi2d (cv x) (suc (cv y)) B sseq2 tfindsg.3 imbi12d (e. B (On)) imbi2d (cv x) A B sseq2 tfindsg.4 imbi12d (e. B (On)) imbi2d tfindsg.5 (C_ B ({/})) a1d y visset sucex B x eqvinc tfindsg.1 tfindsg.5 syl5bir tfindsg.3 biimpd sylan9r x 19.23aiv sylbi eqcoms (C_ B (suc (cv y))) imim2i (-> (C_ B (cv y)) ch) a1d com4r (e. (cv y) (On)) adantl B (cv y) onsssuc B (suc (cv y)) onelpsst (cv y) suceloni sylan2 bitrd ancoms tfindsg.6 ex th (C_ B (suc (cv y))) ax-1 syl8 a2d com23 sylbird (C_ B (suc (cv y))) (= B (suc (cv y))) annim syl5ibr pm2.61d ex a2d (e. B (On)) (-> (C_ B (cv y)) ch) pm2.27 y (cv x) r19.20sdv (Lim (cv x)) (C_ B (cv x)) ad2antlr tfindsg.7 syld exp31 com3l com4t tfinds imp31)) thm (tfindsg2 ((A x) (x y) (B x) (B y) (ps x) (ch x) (th x) (ta x) (ph y)) ((tfindsg2.1 (-> (= (cv x) (suc B)) (<-> ph ps))) (tfindsg2.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (tfindsg2.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (tfindsg2.4 (-> (= (cv x) A) (<-> ph ta))) (tfindsg2.5 (-> (e. B (On)) ps)) (tfindsg2.6 (-> (/\ (e. (cv y) (On)) (e. B (cv y))) (-> ch th))) (tfindsg2.7 (-> (/\ (Lim (cv x)) (e. B (cv x))) (-> (A.e. y (cv x) (-> (e. B (cv y)) ch)) ph)))) (-> (/\ (e. A (On)) (e. B A)) ta) (A B onelon B sucelon sylib A eloni A B ordsucss syl imp tfindsg2.1 tfindsg2.2 tfindsg2.3 tfindsg2.4 B sucelon tfindsg2.5 sylbir B (On) (cv y) ordelsuc (cv y) eloni sylan2 ancoms tfindsg2.6 ex (e. B (On)) adantr sylbird B sucelon sylan2br imp tfindsg2.7 ex (e. B (On)) adantr B (On) (cv x) ordelsuc (cv x) eloni sylan2 B (On) (cv y) ordelsuc (cv x) (cv y) onelon (cv y) eloni syl sylan2 anassrs ch imbi1d ralbidva ph imbi1d imbi12d x visset (cv x) (V) limelon mpan sylan2 ancoms mpbid B sucelon sylan2br imp tfindsg exp31 imp3a (e. B A) adantr mp2and)) thm (tfindes ((x y) (x z) (y z) (ph y) (ph z)) ((tfindes.1 ([/] ({/}) x ph)) (tfindes.2 (-> (e. (cv x) (On)) (-> ph ([/] (suc (cv x)) x ph)))) (tfindes.3 (-> (Lim (cv y)) (-> (A.e. x (cv y) ph) ([/] (cv y) x ph))))) (-> (e. (cv x) (On)) ph) ((cv y) ({/}) x ph dfsbcq y z x ph sbequ (cv y) (suc (cv z)) x ph dfsbcq y x ph sbequ12r tfindes.1 (e. (cv z) (On)) x ax-17 z x ph hbs1 z visset sucex x ph hbsbc1v hbim hbim (cv x) (cv z) (On) eleq1 x z ph sbequ12 (cv x) (cv z) suceq (suc (cv x)) (suc (cv z)) x ph dfsbcq syl imbi12d imbi12d tfindes.2 chvar tfindes.3 ph z ax-17 z x ph hbs1 x z ph sbequ12 (cv y) cbvral syl5ibr tfinds)) thm (tfinds2 ((x y) (ta x) (ta y) (ps x) (ch x) (th x) (ph y)) ((tfinds2.1 (-> (= (cv x) ({/})) (<-> ph ps))) (tfinds2.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (tfinds2.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (tfinds2.4 (-> ta ps)) (tfinds2.5 (-> (e. (cv y) (On)) (-> ta (-> ch th)))) (tfinds2.6 (-> (Lim (cv x)) (-> ta (-> (A.e. y (cv x) ch) ph))))) (-> (e. (cv x) (On)) (-> ta ph)) (tfinds2.4 0ex tfinds2.1 ta imbi2d sbcie mpbir tfinds2.5 a2d x y sbimi x visset (cv x) (V) y (On) sbcel1gv ax-mp x y (-> ta ch) (-> ta th) sbim 3imtr3 x visset tfinds2.2 bicomd equcoms ta imbi2d sbcie y visset sucex tfinds2.3 ta imbi2d sbcie x y sbbii (cv x) (cv y) suceq (-> ta ph) sbcco2 bitr3 3imtr3g tfinds2.6 a2d y (cv x) ta ch r19.21v syl5ib y x sbimi (Lim (cv y)) x ax-17 (cv x) (cv y) limeq sbie y x (A.e. y (cv x) (-> ta ch)) (-> ta ph) sbim 3imtr3 tfinds2.2 bicomd equcoms ta imbi2d sbralie syl5ibr tfindes)) thm (tfinds3 ((ph y) (A x) (ps x) (ch x) (th x) (ta x) (x y) (et x) (et y)) ((tfinds3.1 (-> (= (cv x) ({/})) (<-> ph ps))) (tfinds3.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (tfinds3.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (tfinds3.4 (-> (= (cv x) A) (<-> ph ta))) (tfinds3.5 (-> et ps)) (tfinds3.6 (-> (e. (cv y) (On)) (-> et (-> ch th)))) (tfinds3.7 (-> (Lim (cv x)) (-> et (-> (A.e. y (cv x) ch) ph))))) (-> (e. A (On)) (-> et ta)) (tfinds3.1 et imbi2d tfinds3.2 et imbi2d tfinds3.3 et imbi2d tfinds3.4 et imbi2d tfinds3.5 tfinds3.6 a2d tfinds3.7 a2d y (cv x) et ch r19.21v syl5ib tfinds)) thm (ssnlim ((A x)) () (-> (/\ (Ord A) (C_ A ({e.|} x (On) (-. (Lim (cv x)))))) (C_ A (om))) (limom A ({e.|} x (On) (-. (Lim (cv x)))) (om) ssel (cv x) (om) limeq negbid (On) elrab pm3.27bd syl6 mt2i (Ord A) adantl ordom A (om) ordtri1 mpan2 (C_ A ({e.|} x (On) (-. (Lim (cv x))))) adantr mpbird)) var (set h) var (set j) var (set k) var (set m) var (set n) var (class H) var (class J) var (class K) var (class L) var (class M) var (class N) var (class W) var (class X) var (class Y) var (class Z) thm (xpeq1 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (= A B) (= (X. A C) (X. B C))) (A B (cv x) eleq2 (e. (cv y) C) anbi1d x y opabbidv A C x y df-xp B C x y df-xp 3eqtr4g)) thm (xpeq2 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (= A B) (= (X. C A) (X. C B))) (A B (cv y) eleq2 (e. (cv x) C) anbi2d x y opabbidv C A x y df-xp C B x y df-xp 3eqtr4g)) thm (elxp ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (<-> (e. A (X. B C)) (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) (/\ (e. (cv x) B) (e. (cv y) C)))))) (B C x y df-xp A eleq2i A x y (/\ (e. (cv x) B) (e. (cv y) C)) elopab bitr)) thm (elxp2 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (<-> (e. A (X. B C)) (E.e. x B (E.e. y C (= A (<,> (cv x) (cv y)))))) (y C (/\ (e. (cv x) B) (= A (<,> (cv x) (cv y)))) df-rex y C (e. (cv x) B) (= A (<,> (cv x) (cv y))) r19.42v (e. (cv x) B) (e. (cv y) C) (= A (<,> (cv x) (cv y))) anass (= A (<,> (cv x) (cv y))) (/\ (e. (cv x) B) (e. (cv y) C)) ancom (e. (cv y) C) (e. (cv x) B) (= A (<,> (cv x) (cv y))) an12 3bitr4r y exbii 3bitr3 x exbii x B (E.e. y C (= A (<,> (cv x) (cv y)))) df-rex A B C x y elxp 3bitr4r)) thm (hbxp ((y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (x y) (x z) (w x)) ((hbxp.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbxp.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (X. A B)) (A. x (e. (cv y) (X. A B)))) ((= (cv y) (<,> (cv z) (cv w))) x ax-17 (e. (cv y) (cv z)) x ax-17 hbxp.1 hbel (e. (cv y) (cv w)) x ax-17 hbxp.2 hbel hban hban w hbex z hbex (cv y) A B z w elxp (cv y) A B z w elxp x albii 3imtr4)) thm (opelxpex ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) () (-> (e. (<,> A B) (X. C D)) (e. A (V))) ((<,> A B) C D x y elxp x visset (<,> A B) (<,> (cv x) (cv y)) ({/}) eleq2 A B opprc1b (cv x) (cv y) opprc1b 3bitr4g con4bid mpbiri (/\ (e. (cv x) C) (e. (cv y) D)) adantr x y 19.23aivv sylbi)) thm (brrelex () () (-> (/\ (Rel R) (br A R B)) (e. A (V))) (R df-rel R (X. (V) (V)) (<,> A B) ssel A R B df-br syl5ib A B (V) (V) opelxpex syl6 sylbi imp)) thm (brrelexi () ((brrelexi.1 (Rel R))) (-> (br A R B) (e. A (V))) (brrelexi.1 R A B brrelex mpan)) thm (nprrel () ((nprrel.1 (Rel R)) (nprrel.2 (-. (e. A (V))))) (-. (br A R B)) (nprrel.2 nprrel.1 A B brrelexi mto)) thm (fconstopab ((x y) (A x) (A y) (B x) (B y)) () (= (X. A ({} B)) ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B)))) (A ({} B) x y df-xp y B elsn (e. (cv x) A) anbi2i x y opabbii eqtr)) thm (vtoclr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((vtoclr.1 (Rel R)) (vtoclr.2 (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))) (-> (e. C D) (-> (/\ (br A R B) (br B R C)) (br A R C))) (C D elisset (cv x) A R (cv y) breq1 (br (cv y) R C) anbi1d (cv x) A R C breq1 imbi12d (e. C (V)) imbi2d (cv y) B A R breq2 (cv y) B R C breq1 anbi12d (br A R C) imbi1d (e. C (V)) imbi2d (cv z) C (cv y) R breq2 (br (cv x) R (cv y)) anbi2d (cv z) C (cv x) R breq2 imbi12d vtoclr.2 (V) vtoclg (V) (V) vtocl2g vtoclr.1 A B brrelexi vtoclr.1 B C brrelexi syl2an pm2.43b syl)) thm (vtoclrbr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((vtoclr.1 (Rel R)) (vtoclr.2 (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) (vtoclrbr.3 (br (cv x) R (cv x)))) (-> (/\ (br A R B) (br B R C)) (br A R C)) (vtoclr.1 vtoclr.2 C (V) A B vtoclr C A R brprc (cv x) A R (cv x) breq1 (cv x) A A R breq2 bitrd vtoclrbr.3 (V) vtoclg syl5bir vtoclr.1 A B brrelexi syl5 (br B R C) adantrd pm2.61i)) thm (vtoclibr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((vtoclr.1 (Rel R)) (vtoclr.2 (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) (vtoclibr.3 (-. (br (cv x) R (cv x))))) (-> (/\ (br A R B) (br B R C)) (br A R C)) ((cv x) B R (cv x) breq1 (cv x) B B R breq2 bitrd negbid vtoclibr.3 (V) vtoclg vtoclr.1 B B brrelexi con3i pm2.61i C B R brprc mtbiri a3i vtoclr.1 vtoclr.2 C (V) A B vtoclr syl anabsi7)) thm (opelxp ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((opelxp.1 (e. B (V)))) (<-> (e. (<,> A B) (X. C D)) (/\ (e. A C) (e. B D))) (A B C D opelxpex A C elisset (e. B D) adantr (cv z) A B opeq1 (X. C D) eleq1d (cv z) A C eleq1 (e. B D) anbi1d (<,> (cv z) B) (<,> (cv x) (cv y)) eqcom x visset y visset opelxp.1 (cv z) opth bitr (/\ (e. (cv x) C) (e. (cv y) D)) anbi1i (= (cv x) (cv z)) (= (cv y) B) (e. (cv x) C) (e. (cv y) D) an4 bitr y exbii y (/\ (= (cv x) (cv z)) (e. (cv x) C)) (/\ (= (cv y) B) (e. (cv y) D)) 19.42v bitr x exbii x (/\ (= (cv x) (cv z)) (e. (cv x) C)) (E. y (/\ (= (cv y) B) (e. (cv y) D))) 19.41v bitr (<,> (cv z) B) C D x y elxp (cv z) C x df-clel B D y df-clel anbi12i 3bitr4 (V) vtoclbg pm5.21nii)) thm (brxp () ((opelxp.1 (e. B (V)))) (<-> (br A (X. C D) B) (/\ (e. A C) (e. B D))) (A (X. C D) B df-br opelxp.1 A C D opelxp bitr)) thm (opelxpg ((A x) (B x) (C x) (D x)) () (-> (e. B R) (<-> (e. (<,> A B) (X. C D)) (/\ (e. A C) (e. B D)))) ((cv x) B A opeq2 (X. C D) eleq1d (cv x) B D eleq1 (e. A C) anbi2d x visset A C D opelxp R vtoclbg)) thm (opelxpi () () (-> (/\ (e. A C) (e. B D)) (e. (<,> A B) (X. C D))) (B D A C D opelxpg biimprd anabsi7)) thm (ralxp ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (ph y) (ph z) (ps x)) ((ralxp.1 (-> (= (cv x) (<,> (cv y) (cv z))) (<-> ph ps)))) (<-> (A.e. x (X. A B) ph) (A.e. y A (A.e. z B ps))) (ralxp.1 (X. A B) rcla4cv z visset (cv y) A B opelxp syl5ibr r19.21aivv (cv x) A B y z elxp (= (cv x) (<,> (cv y) (cv z))) (/\ (e. (cv y) A) (e. (cv z) B)) pm3.26 y z 19.22i2 sylbi y A (A.e. z B ps) hbra1 (-> (e. (cv x) (X. A B)) ph) y ax-17 hbim (e. (cv y) A) z ax-17 z B ps hbra1 hbral (-> (e. (cv x) (X. A B)) ph) z ax-17 hbim (cv x) (<,> (cv y) (cv z)) (X. A B) eleq1 z visset (cv y) A B opelxp syl6bb ralxp.1 imbi12d y A z B ps ra42 syl5bir 19.23ai 19.23ai syl pm2.43b r19.21aiv impbi)) thm (rexxp ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (ph y) (ph z) (ps x)) ((ralxp.1 (-> (= (cv x) (<,> (cv y) (cv z))) (<-> ph ps)))) (<-> (E.e. x (X. A B) ph) (E.e. y A (E.e. z B ps))) (ralxp.1 negbid A B ralxp z B ps ralnex y A ralbii bitr negbii x (X. A B) ph dfrex2 y A (E.e. z B ps) dfrex2 3bitr4)) thm (ralxpf ((u v) (u w) (u x) (u y) (A u) (v w) (v x) (v y) (A v) (w x) (w y) (A w) (x y) (A x) (A y) (u z) (B u) (v z) (B v) (w z) (B w) (x z) (B x) (y z) (B y) (B z) (ph u) (ph v) (ph w) (ps u) (ps v) (ps w)) ((ralxpf.1 (-> ph (A. y ph))) (ralxpf.2 (-> ph (A. z ph))) (ralxpf.3 (-> ps (A. x ps))) (ralxpf.4 (-> (= (cv x) (<,> (cv y) (cv z))) (<-> ph ps)))) (<-> (A.e. x (X. A B) ph) (A.e. y A (A.e. z B ps))) (x (X. A B) ph w cbvsbcv z B ([/] (cv u) y ps) v cbvsbcv u A ralbii (A.e. z B ps) u ax-17 (e. (cv z) B) y ax-17 u y ps hbs1 hbral y u ps sbequ12 z B ralbidv A cbvral u visset v visset (cv w) y z eqvinop w visset (e. (cv x) (cv w)) y ax-17 ralxpf.1 (V) x hbsbcg ax-mp v visset (e. (cv x) (cv v)) y ax-17 u y ps hbs1 (V) z hbsbcg ax-mp hbbi w visset (e. (cv x) (cv w)) z ax-17 ralxpf.2 (V) x hbsbcg ax-mp v z ([/] (cv u) y ps) hbs1 hbbi w visset (<,> (cv y) (cv z)) x eqvinc w x ph hbs1 ralxpf.3 hbbi x w ph sbequ12 bicomd ralxpf.4 sylan9bb 19.23ai sylbi y visset z visset v visset (cv u) opth y u ps sbequ12 z v ([/] (cv u) y ps) sbequ12 sylan9bb sylbi sylan9bb 19.23ai 19.23ai sylbi A B ralxp 3bitr4r bitr)) thm (opthprc ((A x) (B x) (C x) (D x)) () (<-> (= (u. (X. A ({} ({/}))) (X. B ({} ({} ({/}))))) (u. (X. C ({} ({/}))) (X. D ({} ({} ({/})))))) (/\ (= A C) (= B D))) ((u. (X. A ({} ({/}))) (X. B ({} ({} ({/}))))) (u. (X. C ({} ({/}))) (X. D ({} ({} ({/}))))) (<,> (cv x) ({/})) eleq2 0ex (cv x) A ({} ({/})) opelxp 0ex snid mpbiran2 0ex (cv x) B ({} ({} ({/}))) opelxp 0nep0 0ex ({} ({/})) elsnc mtbir (e. (cv x) B) bianfi bitr4 orbi12i (<,> (cv x) ({/})) (X. A ({} ({/}))) (X. B ({} ({} ({/})))) elun 0nep0 0ex ({} ({/})) elsnc mtbir (e. (cv x) A) biorfi 3bitr4r 0ex (cv x) C ({} ({/})) opelxp 0ex snid mpbiran2 0ex (cv x) D ({} ({} ({/}))) opelxp 0nep0 0ex ({} ({/})) elsnc mtbir (e. (cv x) D) bianfi bitr4 orbi12i (<,> (cv x) ({/})) (X. C ({} ({/}))) (X. D ({} ({} ({/})))) elun 0nep0 0ex ({} ({/})) elsnc mtbir (e. (cv x) C) biorfi 3bitr4r 3bitr4g eqrdv (u. (X. A ({} ({/}))) (X. B ({} ({} ({/}))))) (u. (X. C ({} ({/}))) (X. D ({} ({} ({/}))))) (<,> (cv x) ({} ({/}))) eleq2 p0ex (cv x) A ({} ({/})) opelxp 0nep0 p0ex ({/}) elsnc ({} ({/})) ({/}) eqcom bitr mtbir (e. (cv x) A) bianfi bitr4 p0ex (cv x) B ({} ({} ({/}))) opelxp p0ex snid mpbiran2 orbi12i (<,> (cv x) ({} ({/}))) (X. A ({} ({/}))) (X. B ({} ({} ({/})))) elun 0nep0 p0ex ({/}) elsnc ({} ({/})) ({/}) eqcom bitr mtbir (e. ({} ({/})) ({} ({/}))) (e. (cv x) B) biorf ax-mp 3bitr4r p0ex (cv x) C ({} ({/})) opelxp 0nep0 p0ex ({/}) elsnc ({} ({/})) ({/}) eqcom bitr mtbir (e. (cv x) C) bianfi bitr4 p0ex (cv x) D ({} ({} ({/}))) opelxp p0ex snid mpbiran2 orbi12i (<,> (cv x) ({} ({/}))) (X. C ({} ({/}))) (X. D ({} ({} ({/})))) elun 0nep0 p0ex ({/}) elsnc ({} ({/})) ({/}) eqcom bitr mtbir (e. ({} ({/})) ({} ({/}))) (e. (cv x) D) biorf ax-mp 3bitr4r 3bitr4g eqrdv jca (X. A ({} ({/}))) (X. C ({} ({/}))) (X. B ({} ({} ({/})))) (X. D ({} ({} ({/})))) uneq12 A C ({} ({/})) xpeq1 B D ({} ({} ({/}))) xpeq1 syl2an impbi)) thm (brelg () ((brelg.1 (C_ R (X. C D)))) (-> (e. B S) (-> (br A R B) (/\ (e. A C) (e. B D)))) (B S A C D opelxpg brelg.1 (<,> A B) sseli syl5bi A R B df-br syl5ib)) thm (brel () ((brel.1 (e. B (V))) (brel.2 (C_ R (X. C D)))) (-> (br A R B) (/\ (e. A C) (e. B D))) (brel.1 brel.2 B (V) A brelg ax-mp)) thm (elxp3 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (<-> (e. A (X. B C)) (E. x (E. y (/\ (= (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) (X. B C)))))) (A B C x y elxp (<,> (cv x) (cv y)) A eqcom y visset (cv x) B C opelxp anbi12i x y 2exbii bitr4)) thm (xpundi ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (= (X. A (u. B C)) (u. (X. A B) (X. A C))) ((cv y) B C elun (e. (cv x) A) anbi2i (e. (cv x) A) (e. (cv y) B) (e. (cv y) C) andi bitr x y opabbii x y (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv x) A) (e. (cv y) C)) unopab eqtr4 A (u. B C) x y df-xp A B x y df-xp A C x y df-xp uneq12i 3eqtr4)) thm (xpundir ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (= (X. (u. A B) C) (u. (X. A C) (X. B C))) ((cv x) A B elun (e. (cv y) C) anbi1i (e. (cv x) A) (e. (cv x) B) (e. (cv y) C) andir bitr x y opabbii x y (/\ (e. (cv x) A) (e. (cv y) C)) (/\ (e. (cv x) B) (e. (cv y) C)) unopab eqtr4 (u. A B) C x y df-xp A C x y df-xp B C x y df-xp uneq12i 3eqtr4)) thm (xpun () () (= (X. (u. A B) (u. C D)) (u. (u. (X. A C) (X. A D)) (u. (X. B C) (X. B D)))) ((u. A B) C D xpundi A B C xpundir A B D xpundir uneq12i (X. A C) (X. B C) (X. A D) (X. B D) un4 3eqtr)) thm (elvv ((x y) (A x) (A y)) () (<-> (e. A (X. (V) (V))) (E. x (E. y (= A (<,> (cv x) (cv y)))))) (A (V) (V) x y elxp x visset y visset pm3.2i (= A (<,> (cv x) (cv y))) biantru x y 2exbii bitr4)) thm (elvvuni ((x y) (A x) (A y)) () (-> (e. A (X. (V) (V))) (e. (U. A) A)) (A x y elvv (cv x) (cv y) uniop (cv x) (cv y) opi2 eqeltr A (<,> (cv x) (cv y)) unieq (= A (<,> (cv x) (cv y))) id eleq12d mpbiri x y 19.23aivv sylbi)) thm (xpss ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (C_ (X. A B) (X. (V) (V))) ((= (cv z) (<,> (cv x) (cv y))) (/\ (e. (cv x) A) (e. (cv y) B)) pm3.26 x y 19.22i2 (cv z) A B x y elxp (cv z) x y elvv 3imtr4 ssriv)) thm (brinxp () () (-> (/\ (e. A C) (e. B D)) (<-> (br A R B) (br A (i^i R (X. C D)) B))) ((/\ (e. A C) (e. B D)) (br A R B) ibar A C B D opelxpi (/\ (e. A C) (e. B D)) (e. (<,> A B) (X. C D)) ibib mpbi A R B df-br (/\ (e. A C) (e. B D)) a1i anbi12d A (i^i R (X. C D)) B df-br (<,> A B) R (X. C D) elin (e. (<,> A B) R) (e. (<,> A B) (X. C D)) ancom 3bitr syl6bbr bitrd)) thm (weinxp ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z)) () (<-> (We R A) (We (i^i R (X. A A)) A)) ((cv z) A (cv x) ssel (cv z) A (cv y) ssel anim12d (-. (= (cv z) ({/}))) adantr (cv y) A (cv x) A R brinxp ancoms syl6 exp3a imp31 negbid ralbidva rexbidva pm5.74i z albii R A z x y df-fr (i^i R (X. A A)) A z x y df-fr 3bitr4 (cv x) A (cv y) A R brinxp (/\ (e. (cv x) A) (e. (cv y) A)) (= (cv x) (cv y)) pm4.2i (cv y) A (cv x) A R brinxp ancoms 3orbi123d pm5.74i x y 2albii x A y A (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x))) r2al x A y A (\/\/ (br (cv x) (i^i R (X. A A)) (cv y)) (= (cv x) (cv y)) (br (cv y) (i^i R (X. A A)) (cv x))) r2al 3bitr4 anbi12i R A x y dfwe2 (i^i R (X. A A)) A x y dfwe2 3bitr4)) thm (opabssxp ((x y) (A x) (A y) (B x) (B y)) () (C_ ({<,>|} x y (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph)) (X. A B)) ((/\ (e. (cv x) A) (e. (cv y) B)) ph pm3.26 x y ssopab2i A B x y df-xp sseqtr4)) thm (optocl ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (ps x) (ps y)) ((optocl.1 (= D (X. B C))) (optocl.2 (-> (= (<,> (cv x) (cv y)) A) (<-> ph ps))) (optocl.3 (-> (/\ (e. (cv x) B) (e. (cv y) C)) ph))) (-> (e. A D) ps) (optocl.1 A eleq2i A B C x y elxp3 optocl.2 y visset (cv x) B C opelxp optocl.3 sylbi syl5bi imp x y 19.23aivv sylbi sylbi)) thm (2optocl ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w) (ps x) (ps y) (ch z) (ch w) (R z) (R w)) ((2optocl.1 (= R (X. C D))) (2optocl.2 (-> (= (<,> (cv x) (cv y)) A) (<-> ph ps))) (2optocl.3 (-> (= (<,> (cv z) (cv w)) B) (<-> ps ch))) (2optocl.4 (-> (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (/\ (e. (cv z) C) (e. (cv w) D))) ph))) (-> (/\ (e. A R) (e. B R)) ch) (2optocl.1 2optocl.3 (e. A R) imbi2d 2optocl.1 2optocl.2 (/\ (e. (cv z) C) (e. (cv w) D)) imbi2d 2optocl.4 ex optocl com12 optocl impcom)) thm (3optocl ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B z) (B w) (B v) (B u) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (F x) (F y) (F z) (F w) (F v) (F u) (R z) (R w) (R v) (R u) (ps x) (ps y) (ch z) (ch w) (th v) (th u)) ((3optocl.1 (= R (X. D F))) (3optocl.2 (-> (= (<,> (cv x) (cv y)) A) (<-> ph ps))) (3optocl.3 (-> (= (<,> (cv z) (cv w)) B) (<-> ps ch))) (3optocl.4 (-> (= (<,> (cv v) (cv u)) C) (<-> ch th))) (3optocl.5 (-> (/\/\ (/\ (e. (cv x) D) (e. (cv y) F)) (/\ (e. (cv z) D) (e. (cv w) F)) (/\ (e. (cv v) D) (e. (cv u) F))) ph))) (-> (/\/\ (e. A R) (e. B R) (e. C R)) th) (3optocl.1 3optocl.4 (/\ (e. A R) (e. B R)) imbi2d 3optocl.1 3optocl.2 (/\ (e. (cv v) D) (e. (cv u) F)) imbi2d 3optocl.3 (/\ (e. (cv v) D) (e. (cv u) F)) imbi2d 3optocl.5 3expa ex 2optocl com12 optocl impcom 3impa)) thm (opbrop ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (S x) (S y) (S z) (S w) (S v) (S u) (ph x) (ph y) (ps z) (ps w) (ps v) (ps u)) ((opbrop.1 (-> (/\ (/\ (= (cv z) A) (= (cv w) B)) (/\ (= (cv v) C) (= (cv u) D))) (<-> ph ps))) (opbrop.2 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) ph)))))))))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (<-> (br (<,> A B) R (<,> C D)) ps)) (opbrop.1 S S copsex4g (/\ (e. (<,> A B) (X. S S)) (e. (<,> C D) (X. S S))) anbi2d A B opex C D opex (cv x) (<,> A B) (X. S S) eleq1 (e. (cv y) (X. S S)) anbi1d (cv x) (<,> A B) (<,> (cv z) (cv w)) eqeq1 (= (cv y) (<,> (cv v) (cv u))) anbi1d ph anbi1d z w v u 4exbidv anbi12d (cv y) (<,> C D) (X. S S) eleq1 (e. (<,> A B) (X. S S)) anbi2d (cv y) (<,> C D) (<,> (cv v) (cv u)) eqeq1 (= (<,> A B) (<,> (cv z) (cv w))) anbi2d ph anbi1d z w v u 4exbidv anbi12d opbrop.2 brab syl5bb A S B S opelxpi C S D S opelxpi anim12i ps biantrurd bitr4d)) thm (xp0r ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (X. ({/}) A) ({/})) ((cv z) ({/}) A x y elxp (cv x) noel (e. (cv x) ({/})) (e. (cv y) A) pm3.26 (= (cv z) (<,> (cv x) (cv y))) adantl mto y nex x nex (cv z) noel 2false bitr eqriv)) thm (0nelxp ((x y) (A x) (A y) (B x) (B y)) () (-. (e. ({/}) (X. A B))) (({} (cv x)) noel (cv x) (cv y) opi1 ({/}) (<,> (cv x) (cv y)) ({} (cv x)) eleq2 mpbiri mto (/\ (e. (cv x) A) (e. (cv y) B)) intnanr y nex x nex ({/}) A B x y elxp mtbir)) thm (onxpdisj ((x y) (x z) (y z)) () (= (i^i (On) (X. (V) (V))) ({/})) ((On) (X. (V) (V)) x disj (cv x) on0eqelt (V) (V) 0nelxp (cv x) ({/}) (X. (V) (V)) eleq1 mtbiri (cv x) y z elvv y visset (cv y) (cv z) opprc1b con1bii mpbir (cv x) (<,> (cv y) (cv z)) ({/}) eleq2 mtbiri y z 19.23aivv sylbi con2i jaoi syl mprgbir)) thm (releq () () (-> (= A B) (<-> (Rel A) (Rel B))) (A B (X. (V) (V)) sseq1 A df-rel B df-rel 3bitr4g)) thm (hbrel ((A y) (x y)) ((hbrel.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (Rel A) (A. x (Rel A))) (hbrel.1 (e. (cv y) (X. (V) (V))) x ax-17 hbss A df-rel A df-rel x albii 3imtr4)) thm (relss () () (-> (C_ A B) (-> (Rel B) (Rel A))) (A B (X. (V) (V)) sstr2 B df-rel A df-rel 3imtr4g)) thm (ssrel ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (Rel A) (<-> (C_ A B) (A. x (A. y (-> (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B)))))) (A B (<,> (cv x) (cv y)) ssel (Rel A) a1i y 19.21adv x 19.21adv A df-rel A (X. (V) (V)) (cv z) ssel sylbi (cv z) x y elvv syl6ib (-> (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B)) id (= (cv z) (<,> (cv x) (cv y))) anim2d (cv z) (<,> (cv x) (cv y)) B eleq1 biimpar syl6 (cv z) (<,> (cv x) (cv y)) A eleq1 pm5.32i syl5ib exp3a y 19.20i y (= (cv z) (<,> (cv x) (cv y))) (-> (e. (cv z) A) (e. (cv z) B)) 19.23v sylib x 19.20i x (E. y (= (cv z) (<,> (cv x) (cv y)))) (-> (e. (cv z) A) (e. (cv z) B)) 19.23v sylib syl9 (e. (cv z) A) (e. (cv z) B) pm2.43 syl6 z 19.21adv A B z dfss2 syl6ibr impbid)) thm (relssi ((x y) (A x) (A y) (B x) (B y)) ((relssi.1 (Rel A)) (relssi.2 (-> (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B)))) (C_ A B) (relssi.1 A B x y ssrel ax-mp relssi.2 y ax-gen mpgbir)) thm (relssdv ((x y) (A x) (A y) (B x) (B y) (ph x) (ph y)) ((relssdv.1 (-> ph (Rel A))) (relssdv.2 (-> ph (-> (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B))))) (-> ph (C_ A B)) (relssdv.2 x y 19.21aivv relssdv.1 A B x y ssrel syl mpbird)) thm (cleqrel ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (Rel A) (Rel B)) (<-> (= A B) (A. x (A. y (<-> (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B)))))) (A B x y ssrel B A x y ssrel bi2anan9 A B eqss x y (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B) 2albi 3bitr4g)) thm (cleqreli ((x y) (A x) (A y) (B x) (B y)) ((cleqreli.1 (Rel A)) (cleqreli.2 (Rel B)) (cleqreli.3 (<-> (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B)))) (= A B) (cleqreli.1 cleqreli.2 A B x y cleqrel mp2an cleqreli.3 y ax-gen mpgbir)) thm (elrel ((x y) (A x) (A y)) () (-> (/\ (Rel R) (e. A R)) (E. x (E. y (= A (<,> (cv x) (cv y)))))) (R df-rel biimp A sseld imp A x y elvv sylib)) thm (relsn () ((relsn.1 (e. A (V)))) (Rel ({} (<,> A B))) (relsn.1 A (V) B (V) opelxpi mpan B A opprc2 relsn.1 A (V) (V) opelxp relsn.1 relsn.1 mpbir2an syl6eqel pm2.61i (<,> A B) (X. (V) (V)) snssi ax-mp ({} (<,> A B)) df-rel mpbir)) thm (relxp () () (Rel (X. A B)) (A B xpss (X. A B) df-rel mpbir)) thm (ssxp ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) () (-> (/\ (C_ A B) (C_ C D)) (C_ (X. A C) (X. B D))) (A C relxp (/\ (C_ A B) (C_ C D)) a1i (e. (cv x) A) (e. (cv x) B) (e. (cv y) C) (e. (cv y) D) prth y visset (cv x) A C opelxp y visset (cv x) B D opelxp 3imtr4g A B (cv x) ssel C D (cv y) ssel syl2an relssdv)) thm (xpsspw ((x y) (A x) (A y) (B x) (B y)) () (C_ (X. A B) (P~ (P~ (u. A B)))) (A B relxp y visset (cv x) A B opelxp (cv x) A snssi ({} (cv x)) A B ssun3 syl (cv x) snex (u. A B) elpw sylibr (e. (cv y) B) adantr (cv x) A snssi ({} (cv x)) A B ssun3 syl (cv y) B snssi ({} (cv y)) B A ssun4 syl anim12i ({} (cv x)) (u. A B) ({} (cv y)) unss sylib (cv x) (cv y) df-pr syl5ss x y zfpair (u. A B) elpw sylibr jca ({} (cv x)) ({,} (cv x) (cv y)) prex (P~ (u. A B)) elpw (cv x) (cv y) df-op (P~ (P~ (u. A B))) eleq1i (cv x) snex x y zfpair (P~ (u. A B)) prss 3bitr4r sylib sylbi relssi)) thm (unixpss () () (C_ (U. (U. (X. A B))) (u. A B)) (A B xpsspw (X. A B) (P~ (P~ (u. A B))) uniss ax-mp (P~ (u. A B)) unipw sseqtr (U. (X. A B)) (P~ (u. A B)) uniss ax-mp (u. A B) unipw sseqtr)) thm (xpexg () () (-> (/\ (e. A C) (e. B D)) (e. (X. A B) (V))) (A C B D unexg (u. A B) (V) pwexg syl (P~ (u. A B)) (V) pwexg A B xpsspw (X. A B) (P~ (P~ (u. A B))) (V) ssexg mpan 3syl)) thm (xpex () ((xpex.1 (e. A (V))) (xpex.2 (e. B (V)))) (e. (X. A B) (V)) (xpex.1 xpex.2 A (V) B (V) xpexg mp2an)) thm (relun () () (<-> (Rel (u. A B)) (/\ (Rel A) (Rel B))) (A (X. (V) (V)) B unss A df-rel B df-rel anbi12i (u. A B) df-rel 3bitr4r)) thm (relin1 () () (-> (Rel A) (Rel (i^i A B))) (A B inss1 (i^i A B) A relss ax-mp)) thm (relin2 () () (-> (Rel B) (Rel (i^i A B))) (A B inss2 (i^i A B) B relss ax-mp)) thm (reldif () () (-> (Rel A) (Rel (\ A B))) (A B difss (\ A B) A relss ax-mp)) thm (reluni ((x y) (A x) (A y)) () (<-> (Rel (U. A)) (A.e. x A (Rel (cv x)))) (x A (e. (cv y) (cv x)) (e. (cv y) (X. (V) (V))) r19.23v (cv y) A x eluni2 (e. (cv y) (X. (V) (V))) imbi1i bitr4 y albii (cv x) df-rel (cv x) (X. (V) (V)) y dfss2 bitr x A ralbii x A y (-> (e. (cv y) (cv x)) (e. (cv y) (X. (V) (V)))) ralcom4 bitr (U. A) df-rel (U. A) (X. (V) (V)) y dfss2 bitr 3bitr4r)) thm (relopab ((x y)) () (Rel ({<,>|} x y ph)) (x visset y visset pm3.2i ph a1i x y ssopab2i (V) (V) x y df-xp sseqtr4 ({<,>|} x y ph) df-rel mpbir)) thm (inopab ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w) (ps z) (ps w)) () (= (i^i ({<,>|} x y ph) ({<,>|} x y ps)) ({<,>|} x y (/\ ph ps))) (x y ph relopab ({<,>|} x y ph) ({<,>|} x y ps) relin1 ax-mp x y (/\ ph ps) relopab w y ([/] (cv z) x ph) ([/] (cv z) x ps) sban z x ph ps sban w y sbbii z w x y ph opabsb z w x y ps opabsb anbi12i 3bitr4r (<,> (cv z) (cv w)) ({<,>|} x y ph) ({<,>|} x y ps) elin z w x y (/\ ph ps) opabsb 3bitr4 cleqreli)) thm (inxp ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) () (= (i^i (X. A B) (X. C D)) (X. (i^i A C) (i^i B D))) (A B relxp (X. A B) (X. C D) relin1 ax-mp (i^i A C) (i^i B D) relxp (e. (cv x) A) (e. (cv y) B) (e. (cv x) C) (e. (cv y) D) an4 y visset (cv x) A B opelxp y visset (cv x) C D opelxp anbi12i (cv x) A C elin (cv y) B D elin anbi12i 3bitr4 (<,> (cv x) (cv y)) (X. A B) (X. C D) elin y visset (cv x) (i^i A C) (i^i B D) opelxp 3bitr4 cleqreli)) thm (xpindi () () (= (X. A (i^i B C)) (i^i (X. A B) (X. A C))) (A B A C inxp A inidm (i^i A A) A (i^i B C) xpeq1 ax-mp eqtr2)) thm (xpindir () () (= (X. (i^i A B) C) (i^i (X. A C) (X. B C))) (A C B C inxp C inidm (i^i C C) C (i^i A B) xpeq2 ax-mp eqtr2)) thm (rel0 () () (Rel ({/})) ((X. (V) (V)) 0ss ({/}) df-rel mpbir)) thm (reli ((x y)) () (Rel (I)) (x y (= (cv x) (cv y)) relopab x y df-id (I) ({<,>|} x y (= (cv x) (cv y))) releq ax-mp mpbir)) thm (rele ((x y)) () (Rel (E)) (x y (e. (cv x) (cv y)) relopab x y df-eprel (E) ({<,>|} x y (e. (cv x) (cv y))) releq ax-mp mpbir)) thm (coeq1 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (= A B) (= (o. A C) (o. B C))) (A B (cv z) (cv y) breq (br (cv x) C (cv z)) anbi2d z exbidv x y opabbidv A C x y z df-co B C x y z df-co 3eqtr4g)) thm (coeq2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (= A B) (= (o. C A) (o. C B))) (A B (cv x) (cv z) breq (br (cv z) C (cv y)) anbi1d z exbidv x y opabbidv C A x y z df-co C B x y z df-co 3eqtr4g)) thm (coeq1i () ((coeq1i.1 (= A B))) (= (o. A C) (o. B C)) (coeq1i.1 A B C coeq1 ax-mp)) thm (coeq2i () ((coeq1i.1 (= A B))) (= (o. C A) (o. C B)) (coeq1i.1 A B C coeq2 ax-mp)) thm (coeq1d () ((coeq1d.1 (-> ph (= A B)))) (-> ph (= (o. A C) (o. B C))) (coeq1d.1 A B C coeq1 syl)) thm (coeq2d () ((coeq1d.1 (-> ph (= A B)))) (-> ph (= (o. C A) (o. C B))) (coeq1d.1 A B C coeq2 syl)) thm (hbco ((y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B y) (B z) (B w) (B v) (x y) (x z) (w x) (v x)) ((hbco.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbco.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (o. A B)) (A. x (e. (cv y) (o. A B)))) ((e. (cv y) (cv z)) x ax-17 hbco.2 (e. (cv y) (cv v)) x ax-17 hbbr (e. (cv y) (cv v)) x ax-17 hbco.1 (e. (cv y) (cv w)) x ax-17 hbbr hban v hbex y z w hbopab A B z w v df-co (cv y) eleq2i A B z w v df-co (cv y) eleq2i x albii 3imtr4)) thm (opelco ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((opelco.1 (e. A (V))) (opelco.2 (e. B (V)))) (<-> (e. (<,> A B) (o. C D)) (E. x (/\ (br A D (cv x)) (br (cv x) C B)))) (C D y z x df-co (<,> A B) eleq2i opelco.1 opelco.2 (cv y) A D (cv x) breq1 (br (cv x) C (cv z)) anbi1d x exbidv (cv z) B (cv x) C breq2 (br A D (cv x)) anbi2d x exbidv opelopab bitr)) thm (brco ((A x) (B x) (C x) (D x)) ((opelco.1 (e. A (V))) (opelco.2 (e. B (V)))) (<-> (br A (o. C D) B) (E. x (/\ (br A D (cv x)) (br (cv x) C B)))) (A (o. C D) B df-br opelco.1 opelco.2 C D x opelco bitr)) thm (opelcog ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) () (-> (/\ (e. A R) (e. B S)) (<-> (e. (<,> A B) (o. C D)) (E. x (/\ (e. (<,> A (cv x)) D) (e. (<,> (cv x) B) C))))) ((cv y) A (cv z) opeq1 (o. C D) eleq1d (cv y) A D (cv x) breq1 (br (cv x) C (cv z)) anbi1d x exbidv bibi12d (cv z) B A opeq2 (o. C D) eleq1d (cv z) B (cv x) C breq2 (br A D (cv x)) anbi2d x exbidv bibi12d y visset z visset C D x opelco R S vtocl2g A D (cv x) df-br (cv x) C B df-br anbi12i x exbii syl6bb)) thm (cnvss ((x y) (A x) (A y) (B x) (B y)) () (-> (C_ A B) (C_ (`' A) (`' B))) (A B (<,> (cv y) (cv x)) ssel (cv y) A (cv x) df-br (cv y) B (cv x) df-br 3imtr4g x y 19.21aivv x y (br (cv y) A (cv x)) (br (cv y) B (cv x)) ssopab2 sylibr A x y df-cnv B x y df-cnv 3sstr4g)) thm (cnveq () () (-> (= A B) (= (`' A) (`' B))) (A B cnvss B A cnvss anim12i A B eqss (`' A) (`' B) eqss 3imtr4)) thm (elcnv ((x y) (A x) (A y) (R x) (R y)) () (<-> (e. A (`' R)) (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) (br (cv y) R (cv x)))))) (R x y df-cnv A eleq2i A x y (br (cv y) R (cv x)) elopab bitr)) thm (elcnv2 ((x y) (A x) (A y) (R x) (R y)) () (<-> (e. A (`' R)) (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) (e. (<,> (cv y) (cv x)) R))))) (A R x y elcnv (cv y) R (cv x) df-br (= A (<,> (cv x) (cv y))) anbi2i x y 2exbii bitr)) thm (hbcnv ((y z) (w y) (A y) (w z) (A z) (A w) (x y) (x z) (w x)) ((hbcnv.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (`' A)) (A. x (e. (cv y) (`' A)))) ((= (cv y) (<,> (cv z) (cv w))) x ax-17 (e. (cv y) (cv w)) x ax-17 hbcnv.1 (e. (cv y) (cv z)) x ax-17 hbbr hban w hbex z hbex (cv y) A z w elcnv (cv y) A z w elcnv x albii 3imtr4)) thm (opelcnvg ((x y) (A x) (A y) (B x) (B y) (R x) (R y)) () (-> (/\ (e. A C) (e. B D)) (<-> (e. (<,> A B) (`' R)) (e. (<,> B A) R))) ((cv x) A (cv y) R breq2 (cv y) B R A breq1 C D opelopabg R x y df-cnv (<,> A B) eleq2i syl5bb B R A df-br syl6bb)) thm (brcnvg () () (-> (/\ (e. A C) (e. B D)) (<-> (br A (`' R) B) (br B R A))) (A C B D R opelcnvg A (`' R) B df-br B R A df-br 3bitr4g)) thm (opelcnv () ((opelcnv.1 (e. A (V))) (opelcnv.2 (e. B (V)))) (<-> (e. (<,> A B) (`' R)) (e. (<,> B A) R)) (opelcnv.1 opelcnv.2 A (V) B (V) R opelcnvg mp2an)) thm (brcnv () ((opelcnv.1 (e. A (V))) (opelcnv.2 (e. B (V)))) (<-> (br A (`' R) B) (br B R A)) (opelcnv.1 opelcnv.2 A (V) B (V) R brcnvg mp2an)) thm (cnvco ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (= (`' (o. A B)) (o. (`' B) (`' A))) ((cv x) (o. A B) (cv y) df-br x visset y visset A B z opelco (br (cv x) B (cv z)) (br (cv z) A (cv y)) ancom y visset z visset A brcnv z visset x visset B brcnv anbi12i bitr4 z exbii 3bitr y x opabbii (o. A B) y x df-cnv (`' B) (`' A) y x z df-co 3eqtr4)) thm (cnvuni ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (= (`' (U. A)) (U_ x A (`' (cv x)))) ((cv y) (U. A) z w elcnv2 (<,> (cv w) (cv z)) A x eluni2 (= (cv y) (<,> (cv z) (cv w))) anbi2i x A (= (cv y) (<,> (cv z) (cv w))) (e. (<,> (cv w) (cv z)) (cv x)) r19.42v bitr4 z w 2exbii (cv y) (cv x) z w elcnv2 x A rexbii x A z (E. w (/\ (= (cv y) (<,> (cv z) (cv w))) (e. (<,> (cv w) (cv z)) (cv x)))) rexcom4 x A w (/\ (= (cv y) (<,> (cv z) (cv w))) (e. (<,> (cv w) (cv z)) (cv x))) rexcom4 z exbii 3bitrr 3bitr (cv y) x A (`' (cv x)) eliun bitr4 eqriv)) thm (dfdm3 ((x y) (A x) (A y)) () (= (dom A) ({|} x (E. y (e. (<,> (cv x) (cv y)) A)))) (A x y df-dm (cv x) A (cv y) df-br y exbii x abbii eqtr)) thm (dfrn2 ((x y) (A x) (A y)) () (= (ran A) ({|} y (E. x (br (cv x) A (cv y))))) (A df-rn (`' A) y x df-dm y visset x visset A brcnv x exbii y abbii 3eqtr)) thm (dfrn3 ((x y) (A x) (A y)) () (= (ran A) ({|} y (E. x (e. (<,> (cv x) (cv y)) A)))) (A y x dfrn2 (cv x) A (cv y) df-br x exbii y abbii eqtr)) thm (dfdm4 ((x y) (A x) (A y)) () (= (dom A) (ran (`' A))) (y visset x visset A brcnv y exbii x abbii (`' A) x y dfrn2 A x y df-dm 3eqtr4r)) thm (dfdmf ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A z) (A w) (A v)) ((dfdmf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (dfdmf.2 (-> (e. (cv z) A) (A. y (e. (cv z) A))))) (= (dom A) ({|} x (E. y (e. (<,> (cv x) (cv y)) A)))) (A w v dfdm3 (e. (cv z) (<,> (cv w) (cv v))) y ax-17 dfdmf.2 hbel (e. (<,> (cv w) (cv y)) A) v ax-17 (cv v) (cv y) (cv w) opeq2 A eleq1d cbvex w abbii (e. (cv z) (<,> (cv w) (cv y))) x ax-17 dfdmf.1 hbel y hbex (E. y (e. (<,> (cv x) (cv y)) A)) w ax-17 (cv w) (cv x) (cv y) opeq1 A eleq1d y exbidv cbvab 3eqtr)) thm (eldm ((x y) (A x) (A y) (B x) (B y)) ((eldm.1 (e. A (V)))) (<-> (e. A (dom B)) (E. y (br A B (cv y)))) (eldm.1 (cv x) A B (cv y) breq1 y exbidv B x y df-dm elab2)) thm (eldm2 ((A y) (B y)) ((eldm.1 (e. A (V)))) (<-> (e. A (dom B)) (E. y (e. (<,> A (cv y)) B))) (eldm.1 B y eldm A B (cv y) df-br y exbii bitr)) thm (eldm2g ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A C) (<-> (e. A (dom B)) (E. y (e. (<,> A (cv y)) B)))) ((cv x) A (dom B) eleq1 (cv x) A (cv y) opeq1 B eleq1d y exbidv x visset B y eldm2 C vtoclbg)) thm (dmss ((x y) (A x) (A y) (B x) (B y)) () (-> (C_ A B) (C_ (dom A) (dom B))) (A B (<,> (cv x) (cv y)) ssel y 19.22dv x visset A y eldm2 x visset B y eldm2 3imtr4g ssrdv)) thm (dmeq () () (-> (= A B) (= (dom A) (dom B))) (A B dmss B A dmss anim12i A B eqss (dom A) (dom B) eqss 3imtr4)) thm (dmeqi () ((dmeqi.1 (= A B))) (= (dom A) (dom B)) (dmeqi.1 A B dmeq ax-mp)) thm (dmeqd () ((dmeqd.1 (-> ph (= A B)))) (-> ph (= (dom A) (dom B))) (dmeqd.1 A B dmeq syl)) thm (opeldm ((A y) (B y) (C y)) ((opeldm.1 (e. A (V)))) (-> (e. (<,> A B) C) (e. A (dom C))) ((cv y) B A opeq2 C eleq1d (V) cla4egv opeldm.1 C y eldm2 syl6ibr B A opprc2 C eleq1d opeldm.1 (cv y) A A opeq2 C eleq1d cla4ev opeldm.1 C y eldm2 sylibr syl6bi pm2.61i)) thm (breldm () ((breldm.1 (e. A (V)))) (-> (br A R B) (e. A (dom R))) (A R B df-br breldm.1 B R opeldm sylbi)) thm (dmun ((x y) (A x) (A y) (B x) (B y)) () (= (dom (u. A B)) (u. (dom A) (dom B))) ((<,> (cv x) (cv y)) A B elun y exbii y (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B) 19.43 bitr x visset (u. A B) y eldm2 (cv x) (dom A) (dom B) elun x visset A y eldm2 x visset B y eldm2 orbi12i bitr 3bitr4 eqriv)) thm (dmin ((x y) (A x) (A y) (B x) (B y)) () (C_ (dom (i^i A B)) (i^i (dom A) (dom B))) (y (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B) 19.40 x visset (i^i A B) y eldm2 (<,> (cv x) (cv y)) A B elin y exbii bitr (cv x) (dom A) (dom B) elin x visset A y eldm2 x visset B y eldm2 anbi12i bitr 3imtr4 ssriv)) thm (dmuni ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (dom (U. A)) (U_ x A (dom (cv x)))) ((<,> (cv y) (cv z)) A x eluni z exbii z x (/\ (e. (<,> (cv y) (cv z)) (cv x)) (e. (cv x) A)) excom (E. z (e. (<,> (cv y) (cv z)) (cv x))) (e. (cv x) A) ancom z (e. (<,> (cv y) (cv z)) (cv x)) (e. (cv x) A) 19.41v y visset (cv x) z eldm2 (e. (cv x) A) anbi2i 3bitr4 x exbii 3bitr x A (e. (cv y) (dom (cv x))) df-rex bitr4 y visset (U. A) z eldm2 (cv y) x A (dom (cv x)) eliun 3bitr4 eqriv)) thm (dmopab ((x y) (x z) (y z) (ph z)) () (= (dom ({<,>|} x y ph)) ({|} x (E. y ph))) (z x y ph hbopab1 z x y ph hbopab2 dfdmf x y ph opabid y exbii x abbii eqtr)) thm (dmopabss ((x y) (A x) (A y)) () (C_ (dom ({<,>|} x y (/\ (e. (cv x) A) ph))) A) (x y (/\ (e. (cv x) A) ph) dmopab y (e. (cv x) A) ph 19.42v x abbii x A (E. y ph) ssab2 eqsstr eqsstr)) thm (dmopab3 ((x y) (A x) (A y)) () (<-> (A.e. x A (E. y ph)) (= (dom ({<,>|} x y (/\ (e. (cv x) A) ph))) A)) (x A (E. y ph) df-ral (e. (cv x) A) (E. y ph) pm4.71 x albii x y (/\ (e. (cv x) A) ph) dmopab y (e. (cv x) A) ph 19.42v x abbii eqtr A eqeq1i A ({|} x (/\ (e. (cv x) A) (E. y ph))) eqcom A x (/\ (e. (cv x) A) (E. y ph)) abeq2 3bitr2r 3bitr)) thm (dm0 ((x y)) () (= (dom ({/})) ({/})) ((<,> (cv x) (cv y)) noel y nex (cv x) eqid (= (cv x) (cv x)) negb ax-mp 2false x abbii ({/}) x y dfdm3 x dfnul2 3eqtr4)) thm (dmsn0 ((x y)) () (= (dom ({} ({/}))) ({/})) ((cv x) (cv y) opnz (cv x) (cv y) opex ({/}) elsnc mtbir y nex (cv x) eqid (= (cv x) (cv x)) negb ax-mp 2false x abbii ({} ({/})) x y dfdm3 x dfnul2 3eqtr4)) thm (dmsnsn0 ((x y)) () (= (dom ({} ({} ({/})))) ({/})) (y visset (-. (e. (cv x) (V))) a1i orri (e. (cv x) (V)) (e. (cv y) (V)) oran mpbi (cv x) (cv y) opprc3 mtbi (cv x) (cv y) opex ({} ({/})) elsnc mtbir y nex (cv x) eqid (= (cv x) (cv x)) negb ax-mp 2false x abbii ({} ({} ({/}))) x y dfdm3 x dfnul2 3eqtr4)) thm (dmi ((x y)) () (= (dom (I)) (V)) (y x a9e x visset y visset ideq (cv x) (cv y) eqcom bitr y exbii mpbir (cv x) eqid 2th x abbii (I) x y df-dm x df-v 3eqtr4)) thm (dmv () () (= (dom (V)) (V)) ((dom (V)) ssv dmi (I) ssv (I) (V) dmss ax-mp eqsstr3 eqssi)) thm (dmsnop ((x y) (A x) (A y) (B x) (B y)) () (= (dom ({} (<,> A B))) ({} A)) (x visset y visset B (V) A opthg (cv x) (cv y) opex (<,> A B) elsnc syl5bb y exbidv y (= (cv x) A) (= (cv y) B) 19.42v syl6bb B y isset (E. y (= (cv y) B)) (= (cv x) A) iba sylbi bitr4d x abbidv ({} (<,> A B)) x y dfdm3 A x df-sn 3eqtr4g B A opprc2 (<,> A B) (<,> A A) sneq ({} (<,> A B)) ({} (<,> A A)) dmeq 3syl x visset y visset A (V) A opthg (cv x) (cv y) opex (<,> A A) elsnc syl5bb y exbidv y (= (cv x) A) (= (cv y) A) 19.42v syl6bb A y isset (E. y (= (cv y) A)) (= (cv x) A) iba sylbi bitr4d x abbidv ({} (<,> A A)) x y dfdm3 A x df-sn 3eqtr4g (-. (e. A (V))) anidm A A opprc3 bitr3 (<,> A A) ({} ({/})) sneq dmeqd sylbi dmsnsn0 syl6eq A snprc biimp eqtr4d pm2.61i syl6eq pm2.61i)) thm (dmsnsnsn () () (= (dom ({} ({} ({} A)))) ({} A)) (A dfsn2 ({} A) ({,} A A) ({} A) preq2 ax-mp ({} A) dfsn2 A A df-op 3eqtr4r sneqi dmeqi A A dmsnop eqtr3)) thm (dm0rn0 ((x y) (A x) (A y)) () (<-> (= (dom A) ({/})) (= (ran A) ({/}))) (x y (br (cv x) A (cv y)) excom negbii x (E. y (br (cv x) A (cv y))) alnex y (E. x (br (cv x) A (cv y))) alnex 3bitr4 (cv x) noel (E. y (br (cv x) A (cv y))) nbn x albii (cv y) noel (E. x (br (cv x) A (cv y))) nbn y albii 3bitr3 x (E. y (br (cv x) A (cv y))) ({/}) abeq1 y (E. x (br (cv x) A (cv y))) ({/}) abeq1 3bitr4 A x y df-dm ({/}) eqeq1i A y x dfrn2 ({/}) eqeq1i 3bitr4)) thm (reldm0 ((x y) (A x) (A y)) () (-> (Rel A) (<-> (= A ({/})) (= (dom A) ({/})))) (rel0 A ({/}) x y cleqrel mpan2 (dom A) x eq0 x visset A y eldm2 negbii y (e. (<,> (cv x) (cv y)) A) alnex (<,> (cv x) (cv y)) noel (e. (<,> (cv x) (cv y)) A) nbn y albii 3bitr2 x albii bitr2 syl6bb)) thm (dmexg ((x y) (A x) (A y)) () (-> (e. A B) (e. (dom A) (V))) (A B uniexg (U. A) (V) uniexg A x y dfdm3 x y A opeluu pm3.26d y 19.23aiv abssi eqsstr (dom A) (U. (U. A)) (V) ssexg mpan 3syl)) thm (dmxp ((x y) (A x) (A y) (B x) (B y)) () (-> (-. (= B ({/}))) (= (dom (X. A B)) A)) (B y n0 biimp (e. (cv x) A) a1d r19.21aiv x A y (e. (cv y) B) dmopab3 A B x y df-xp dmeqi A eqeq1i bitr4 sylib)) thm (dmxpid () () (= (dom (X. A A)) A) (A ({/}) A xpeq1 A xp0r syl6eq dmeqd dm0 syl6eq (= A ({/})) id eqtr4d A A dmxp pm2.61i)) thm (elreldm ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (Rel A) (e. B A)) (e. (|^| (|^| B)) (dom A))) (A df-rel A (X. (V) (V)) B ssel sylbi B x y elvv syl6ib B (<,> (cv x) (cv y)) A eleq1 x visset (cv y) A opeldm syl6bi B (<,> (cv x) (cv y)) inteq inteqd x visset (cv y) op1stb syl6eq (dom A) eleq1d sylibrd x y 19.23aivv syli imp)) thm (rneq () () (-> (= A B) (= (ran A) (ran B))) (A B cnveq dmeqd A df-rn B df-rn 3eqtr4g)) thm (rneqi () ((rneqi.1 (= A B))) (= (ran A) (ran B)) (rneqi.1 A B rneq ax-mp)) thm (rneqd () ((rneqd.1 (-> ph (= A B)))) (-> ph (= (ran A) (ran B))) (rneqd.1 A B rneq syl)) thm (rnss () () (-> (C_ A B) (C_ (ran A) (ran B))) (A B cnvss (`' A) (`' B) dmss syl A df-rn B df-rn 3sstr4g)) thm (brelrn () ((brelrn.1 (e. A (V))) (brelrn.2 (e. B (V)))) (-> (br A C B) (e. B (ran C))) (brelrn.2 brelrn.1 C brcnv brelrn.2 (`' C) A breldm sylbir C df-rn syl6eleqr)) thm (opelrn () ((brelrn.1 (e. A (V))) (brelrn.2 (e. B (V)))) (-> (e. (<,> A B) C) (e. B (ran C))) (A C B df-br brelrn.1 brelrn.2 C brelrn sylbir)) thm (dfrnf ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A z) (A w) (A v)) ((dfrnf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (dfrnf.2 (-> (e. (cv z) A) (A. y (e. (cv z) A))))) (= (ran A) ({|} y (E. x (e. (<,> (cv x) (cv y)) A)))) (A w v dfrn3 (e. (cv z) (<,> (cv v) (cv w))) x ax-17 dfrnf.1 hbel (e. (<,> (cv x) (cv w)) A) v ax-17 (cv v) (cv x) (cv w) opeq1 A eleq1d cbvex w abbii (e. (cv z) (<,> (cv x) (cv w))) y ax-17 dfrnf.2 hbel x hbex (E. x (e. (<,> (cv x) (cv y)) A)) w ax-17 (cv w) (cv y) (cv x) opeq2 A eleq1d x exbidv cbvab 3eqtr)) thm (elrn2 ((x y) (A x) (A y) (B x) (B y)) ((elrn.1 (e. A (V)))) (<-> (e. A (ran B)) (E. x (e. (<,> (cv x) A) B))) (elrn.1 (cv y) A (cv x) opeq2 B eleq1d x exbidv B y x dfrn3 elab2)) thm (elrn ((A x) (B x)) ((elrn.1 (e. A (V)))) (<-> (e. A (ran B)) (E. x (br (cv x) B A))) (elrn.1 B x elrn2 (cv x) B A df-br x exbii bitr4)) thm (hbrn ((x y) (x z) (w x) (y z) (w y) (w z) (A y) (A z) (A w)) ((hbrn.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (ran A)) (A. x (e. (cv y) (ran A)))) ((e. (cv w) (<,> (cv z) (cv y))) x ax-17 (e. (cv y) (cv z)) x ax-17 hbrn.1 hbel hbel z hbex y visset A z elrn2 y visset A z elrn2 x albii 3imtr4)) thm (hbdm ((x y) (A y)) ((hbdm.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (dom A)) (A. x (e. (cv y) (dom A)))) (hbdm.1 hbcnv hbrn A dfdm4 (cv y) eleq2i A dfdm4 (cv y) eleq2i x albii 3imtr4)) thm (rnopab ((x y) (x z) (y z) (ph z)) () (= (ran ({<,>|} x y ph)) ({|} y (E. x ph))) (z x y ph hbopab1 z x y ph hbopab2 dfrnf x y ph opabid x exbii y abbii eqtr)) thm (rnopab2 ((x y)) () (= (ran ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B)))) ({|} y (E.e. x A (= (cv y) B)))) (x y (/\ (e. (cv x) A) (= (cv y) B)) rnopab x A (= (cv y) B) df-rex y abbii eqtr4)) thm (rn0 () () (= (ran ({/})) ({/})) (dm0 ({/}) dm0rn0 mpbi)) thm (relrn0 () () (-> (Rel A) (<-> (= A ({/})) (= (ran A) ({/})))) (A reldm0 A dm0rn0 syl6bb)) thm (rnexg ((x y) (A x) (A y)) () (-> (e. A B) (e. (ran A) (V))) (A B uniexg (U. A) (V) uniexg A y x dfrn3 x y A opeluu pm3.27d x 19.23aiv abssi eqsstr (ran A) (U. (U. A)) (V) ssexg mpan 3syl)) thm (dmco ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (C_ (dom (o. A B)) (dom B)) ((cv x) (o. A B) (cv y) df-br x visset y visset A B z opelco bitr y exbii y z (/\ (br (cv x) B (cv z)) (br (cv z) A (cv y))) excom bitr y (br (cv x) B (cv z)) (br (cv z) A (cv y)) 19.42v pm3.26bd z 19.22i sylbi x ss2abi (o. A B) x y df-dm B x z df-dm 3sstr4)) thm (rnco () () (C_ (ran (o. A B)) (ran A)) ((`' B) (`' A) dmco (o. A B) df-rn A B cnvco dmeqi eqtr A df-rn 3sstr4)) thm (dmcosseq ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (C_ (ran B) (dom A)) (= (dom (o. A B)) (dom B))) (x (br (cv x) B (cv z)) hbe1 (E. y (br (cv z) A (cv y))) x ax-17 hbim z hbal z (-> (E. x (br (cv x) B (cv z))) (E. y (br (cv z) A (cv y)))) hba1 (br (cv x) B (cv z)) x 19.8a (E. y (br (cv z) A (cv y))) imim1i ancld z a4s 19.22d 19.21ai (br (cv x) B (cv z)) (E. y (br (cv z) A (cv y))) pm3.26 z 19.22i x ax-gen jctil x (E. z (/\ (br (cv x) B (cv z)) (E. y (br (cv z) A (cv y))))) (E. z (br (cv x) B (cv z))) albi sylibr z visset B x elrn z visset A y eldm imbi12i z albii x visset (o. A B) y eldm2 x visset y visset A B z opelco y exbii y z (/\ (br (cv x) B (cv z)) (br (cv z) A (cv y))) excom y (br (cv x) B (cv z)) (br (cv z) A (cv y)) 19.42v z exbii bitr 3bitr x visset B z eldm bibi12i x albii 3imtr4 (ran B) (dom A) z dfss2 (dom (o. A B)) (dom B) x dfcleq 3imtr4)) thm (dmcoeq () () (-> (= (dom A) (ran B)) (= (dom (o. A B)) (dom B))) ((dom A) (ran B) eqimss2 B A dmcosseq syl)) thm (rncoeq () () (-> (= (dom A) (ran B)) (= (ran (o. A B)) (ran A))) ((`' B) (`' A) dmcoeq (dom A) (ran B) eqcom B df-rn A dfdm4 eqeq12i bitr (o. A B) df-rn A B cnvco dmeqi eqtr A df-rn eqeq12i 3imtr4)) thm (reseq1 () () (-> (= A B) (= (|` A C) (|` B C))) (A B (X. C (V)) ineq1 A C df-res B C df-res 3eqtr4g)) thm (reseq2 () () (-> (= A B) (= (|` C A) (|` C B))) (A B (V) xpeq1 C ineq2d C A df-res C B df-res 3eqtr4g)) thm (hbres ((A y) (B y) (x y)) ((hbres.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbres.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (|` A B)) (A. x (e. (cv y) (|` A B)))) (hbres.1 hbres.2 (e. (cv y) (V)) x ax-17 hbxp hbin A B df-res (cv y) eleq2i A B df-res (cv y) eleq2i x albii 3imtr4)) thm (res0 () () (= (|` A ({/})) ({/})) (A ({/}) df-res (V) xp0r A ineq2i A in0 3eqtr)) thm (opelres () ((opelres.1 (e. B (V)))) (<-> (e. (<,> A B) (|` C D)) (/\ (e. (<,> A B) C) (e. A D))) (C D df-res (<,> A B) eleq2i (<,> A B) C (X. D (V)) elin opelres.1 A D (V) opelxp opelres.1 mpbiran2 (e. (<,> A B) C) anbi2i 3bitr)) thm (opelresg ((A y) (B y) (C y) (D y)) () (-> (e. B R) (<-> (e. (<,> A B) (|` C D)) (/\ (e. (<,> A B) C) (e. A D)))) ((cv y) B A opeq2 (|` C D) eleq1d (cv y) B A opeq2 C eleq1d (e. A D) anbi1d y visset A C D opelres R vtoclbg)) thm (opres () ((opres.1 (e. B (V)))) (-> (e. A D) (<-> (e. (<,> A B) (|` C D)) (e. (<,> A B) C))) (opres.1 A C D opelres pm3.26bd (e. A D) a1i opres.1 A C D opelres biimpr expcom impbid)) thm (resieq ((A x) (B x) (C x)) () (-> (/\ (e. B A) (e. C A)) (<-> (br B (|` (I) A) C) (= B C))) ((cv x) C B (|` (I) A) breq2 (cv x) C B eqeq2 bibi12d (e. B A) imbi2d x visset B A (I) opres x visset B A (cv x) (V) ideqg mpan2 B (I) (cv x) df-br syl5bbr bitrd B (|` (I) A) (cv x) df-br syl5bb A vtoclg impcom)) thm (resundi () () (= (|` A (u. B C)) (u. (|` A B) (|` A C))) (B C (V) xpundir A ineq2i A (X. B (V)) (X. C (V)) indi eqtr A (u. B C) df-res A B df-res A C df-res uneq12i 3eqtr4)) thm (resundir () () (= (|` (u. A B) C) (u. (|` A C) (|` B C))) (A B (X. C (V)) indir (u. A B) C df-res A C df-res B C df-res uneq12i 3eqtr4)) thm (dmres ((x y) (A x) (A y) (B x) (B y)) () (= (dom (|` A B)) (i^i B (dom A))) (y visset (cv x) A B opelres y exbii x visset (|` A B) y eldm2 x visset A y eldm2 (e. (cv x) B) anbi1i y (e. (<,> (cv x) (cv y)) A) (e. (cv x) B) 19.41v bitr4 3bitr4r ineqri (dom A) B incom eqtr3)) thm (ssdmres () () (<-> (C_ A (dom B)) (= (dom (|` B A)) A)) (A (dom B) df-ss B A dmres A eqeq1i bitr4)) thm (dmresexg () () (-> (e. B C) (e. (dom (|` A B)) (V))) (B C (dom A) inex1g A B dmres syl5eqel)) thm (resss () () (C_ (|` A B) A) (A B df-res A (X. B (V)) inss1 eqsstr)) thm (rescom () () (= (|` (|` A B) C) (|` (|` A C) B)) (A (X. B (V)) (X. C (V)) in23 A B df-res (X. C (V)) ineq1i A C df-res (X. B (V)) ineq1i 3eqtr4 (|` A B) C df-res (|` A C) B df-res 3eqtr4)) thm (ssres () () (-> (C_ A B) (C_ (|` A C) (|` B C))) (A B (X. C (V)) ssrin A C df-res B C df-res 3sstr4g)) thm (ssres2 () () (-> (C_ A B) (C_ (|` C A) (|` C B))) ((V) ssid A B (V) (V) ssxp mpan2 (X. A (V)) (X. B (V)) C sslin syl C A df-res C B df-res 3sstr4g)) thm (relres () () (Rel (|` A B)) (A B df-res A (X. B (V)) inss2 B (V) xpss sstri eqsstr (|` A B) df-rel mpbir)) thm (resabs1 () () (-> (C_ B C) (= (|` (|` A C) B) (|` A B))) (B C sseqin2 (i^i C B) B (V) xpeq1 sylbi C B (V) xpindir syl5reqr A ineq2d A (X. C (V)) (X. B (V)) inass syl6reqr (|` A C) B df-res A C df-res (X. B (V)) ineq1i eqtr A B df-res 3eqtr4g)) thm (resabs2 () () (-> (C_ B C) (= (|` (|` A B) C) (|` A B))) ((V) ssid B C (V) (V) ssxp (X. B (V)) (X. C (V)) dfss sylib mpan2 A ineq2d A (X. B (V)) (X. C (V)) inass syl6reqr (|` A B) C df-res A B df-res (X. C (V)) ineq1i eqtr A B df-res 3eqtr4g)) thm (residm () () (= (|` (|` A B) B) (|` A B)) (B ssid B B A resabs2 ax-mp)) thm (resima () () (= (" (|` A B) B) (" A B)) (A B residm rneqi (|` A B) B df-ima A B df-ima 3eqtr4)) thm (relssres ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (Rel A) (C_ (dom A) B)) (= (|` A B) A)) ((Rel A) (C_ (dom A) B) pm3.26 (dom A) B (cv x) ssel x visset (cv y) A opeldm syl5 ancld y visset (cv x) A B opelres syl6ibr (Rel A) adantl relssdv A B resss jctil (|` A B) A eqss sylibr)) thm (resexg () () (-> (e. A C) (e. (|` A B) (V))) (A C (X. B (V)) inex1g A B df-res syl5eqel)) thm (resopab ((x y) (A x) (A y)) () (= (|` ({<,>|} x y ph) A) ({<,>|} x y (/\ (e. (cv x) A) ph))) (({<,>|} x y ph) A df-res A (V) x y df-xp y visset (e. (cv x) A) biantru x y opabbii eqtr4 ({<,>|} x y ph) ineq2i ({<,>|} x y ph) ({<,>|} x y (e. (cv x) A)) incom eqtr x y (e. (cv x) A) ph inopab 3eqtr)) thm (iss ((x y) (A x) (A y)) () (<-> (C_ A (I)) (= A (|` (I) (dom A)))) (A (I) (<,> (cv x) (cv y)) ssel (cv x) (I) (cv y) df-br x visset y visset ideq bitr3 syl6ib pm4.71rd A (I) (<,> (cv x) (cv y)) ssel (cv x) (I) (cv y) df-br x visset y visset ideq bitr3 syl6ib pm4.71rd (cv x) (cv y) eqcom (e. (<,> (cv x) (cv y)) A) anbi1i syl6bb y exbidv x visset (cv y) (cv x) (cv x) opeq2 A eleq1d ceqsexv syl6bb x visset A y eldm2 syl5bb (= (cv x) (cv y)) anbi2d (cv x) (cv y) (cv x) opeq2 A eleq1d pm5.32i syl6bb bitr4d y visset (cv x) (I) (dom A) opelres (cv x) (I) (cv y) df-br x visset y visset ideq bitr3 (e. (cv x) (dom A)) anbi1i bitr2 syl6bb x y 19.21aivv reli A (I) relss mpi (I) (dom A) relres A (|` (I) (dom A)) x y cleqrel mpan2 syl mpbird (I) (dom A) resss A (|` (I) (dom A)) (I) sseq1 mpbiri impbi)) thm (dmresi () () (= (dom (|` (I) A)) A) (A ssv dmi sseqtr4 A (I) ssdmres mpbi)) thm (resid () () (-> (Rel A) (= (|` A (V)) A)) ((dom A) ssv A (V) relssres mpan2)) thm (imaeq1 () () (-> (= A B) (= (" A C) (" B C))) (A B C reseq1 rneqd A C df-ima B C df-ima 3eqtr4g)) thm (imaeq2 () () (-> (= A B) (= (" C A) (" C B))) (A B C reseq2 rneqd C A df-ima C B df-ima 3eqtr4g)) thm (dfima2 ((x y) (A x) (A y) (B x) (B y)) () (= (" A B) ({|} y (E.e. x B (br (cv x) A (cv y))))) (A B df-ima (|` A B) y x dfrn3 A B df-res (<,> (cv x) (cv y)) eleq2i (<,> (cv x) (cv y)) A (X. B (V)) elin (e. (cv x) B) (br (cv x) A (cv y)) ancom (cv x) A (cv y) df-br y visset (e. (cv x) B) biantru y visset (cv x) B (V) opelxp bitr4 anbi12i bitr2 3bitr x exbii x B (br (cv x) A (cv y)) df-rex bitr4 y abbii 3eqtr)) thm (dfima3 ((x y) (A x) (A y) (B x) (B y)) () (= (" A B) ({|} y (E. x (/\ (e. (cv x) B) (e. (<,> (cv x) (cv y)) A))))) (A B y x dfima2 x B (br (cv x) A (cv y)) df-rex (cv x) A (cv y) df-br (e. (cv x) B) anbi2i x exbii bitr y abbii eqtr)) thm (elima ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) ((elima.1 (e. A (V)))) (<-> (e. A (" B C)) (E.e. x C (br (cv x) B A))) (elima.1 (cv y) A (cv x) B breq2 x C rexbidv B C y x dfima2 elab2)) thm (elima2 ((A x) (B x) (C x)) ((elima.1 (e. A (V)))) (<-> (e. A (" B C)) (E. x (/\ (e. (cv x) C) (br (cv x) B A)))) (elima.1 B C x elima x C (br (cv x) B A) df-rex bitr)) thm (elima3 ((A x) (B x) (C x)) ((elima.1 (e. A (V)))) (<-> (e. A (" B C)) (E. x (/\ (e. (cv x) C) (e. (<,> (cv x) A) B)))) (elima.1 B C x elima2 (cv x) B A df-br (e. (cv x) C) anbi2i x exbii bitr)) thm (hbima ((y z) (A y) (A z) (B y) (B z) (x y) (x z) (w x) (w y) (w z)) ((hbima.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbima.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (" A B)) (A. x (e. (cv y) (" A B)))) ((e. (cv w) (cv z)) x ax-17 hbima.2 hbel (e. (cv w) (cv z)) x ax-17 (e. (cv w) (cv y)) x ax-17 hbop hbima.1 hbel hban z hbex y visset A B z elima3 y visset A B z elima3 x albii 3imtr4)) thm (imadmrn ((x y) (A x) (A y)) () (= (" A (dom A)) (ran A)) (x visset (cv y) A opeldm pm4.71i (e. (<,> (cv x) (cv y)) A) (e. (cv x) (dom A)) ancom bitr2 x exbii y abbii A (dom A) y x dfima3 A y x dfrn3 3eqtr4)) thm (imassrn ((x y) (A x) (A y) (B x) (B y)) () (C_ (" A B) (ran A)) ((e. (cv x) B) (e. (<,> (cv x) (cv y)) A) pm3.27 x 19.22i y ss2abi A B y x dfima3 A y x dfrn3 3sstr4)) thm (imaexg () () (-> (e. A C) (e. (" A B) (V))) (A C rnexg A B imassrn (" A B) (ran A) (V) ssexg mpan syl)) thm (imai ((x y) (A x) (A y)) () (= (" (I) A) A) ((I) A y x dfima3 (cv x) (I) (cv y) df-br x visset y visset ideq bitr3 (e. (cv x) A) anbi2i (e. (cv x) A) (= (cv x) (cv y)) ancom bitr x exbii y visset (cv x) (cv y) A eleq1 ceqsexv bitr y abbii y A abid2 3eqtr)) thm (rnresi () () (= (ran (|` (I) A)) A) ((I) A df-ima A imai eqtr3)) thm (ima0 () () (= (" A ({/})) ({/})) (A ({/}) df-ima A res0 rneqi rn0 3eqtr)) thm (imasng ((x y) (A y) (A x) (R y) (R x)) () (-> (e. A B) (= (" R ({} A)) ({|} y (br A R (cv y))))) (A B elisset (cv x) A R (cv y) breq1 (V) ceqsexgv x ({} A) (br (cv x) R (cv y)) df-rex x A elsn (br (cv x) R (cv y)) anbi1i x exbii bitr syl5bb y abbidv R ({} A) y x dfima2 syl5eq syl)) thm (relimasn ((A y) (R y)) () (-> (Rel R) (= (" R ({} A)) ({|} y (br A R (cv y))))) (A snprc ({} A) ({/}) R imaeq2 sylbi R ima0 syl6eq (Rel R) adantl R A (cv y) brrelex ex con3d imp y nexdv y (br A R (cv y)) abn0 con1bii sylib eqtr4d ex A (V) R y imasng pm2.61d2)) thm (elimasn ((A x) (B x) (C x)) ((elimasn.1 (e. B (V))) (elimasn.2 (e. C (V)))) (<-> (e. C (" A ({} B))) (e. (<,> B C) A)) (elimasn.2 A ({} B) x elima3 x B elsn (e. (<,> (cv x) C) A) anbi1i x exbii elimasn.1 (cv x) B C opeq1 A eleq1d ceqsexv 3bitr)) thm (args ((x y) (F x) (F y)) () (= ({|} x (E. y (= (" F ({} (cv x))) ({} (cv y))))) ({|} x (E! y (br (cv x) F (cv y))))) (x visset (cv x) (V) F y imasng ax-mp ({} (cv y)) eqeq1i y exbii y (br (cv x) F (cv y)) eusn bitr4 x abbii)) thm (eliniseg ((A x) (B x) (C x)) ((eliniseg.1 (e. C (V)))) (-> (e. B D) (<-> (e. C (" (`' A) ({} B))) (br C A B))) ((cv x) B sneq ({} (cv x)) ({} B) (`' A) imaeq2 syl C eleq2d (cv x) B C A breq2 bibi12d x visset eliniseg.1 (`' A) elimasn (cv x) (`' A) C df-br x visset eliniseg.1 A brcnv 3bitr2 D vtoclg)) thm (iniseg ((A x) (B x)) () (-> (e. B C) (= (" (`' A) ({} B)) ({|} x (br (cv x) A B)))) (B C elisset x visset B (V) A eliniseg abbi2dv syl)) thm (dffr3 ((x y) (x z) (R x) (y z) (R y) (R z) (A x)) () (<-> (Fr R A) (A. x (-> (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (E.e. y (cv x) (= (i^i (cv x) (" (`' R) ({} (cv y)))) ({/})))))) (R A x y z dffr2 y visset (cv y) (V) R z iniseg ax-mp (cv x) ineq2i ({/}) eqeq1i y (cv x) rexbii (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) imbi2i x albii bitr4)) thm (imass1 () () (-> (C_ A B) (C_ (" A C) (" B C))) (A B C ssres (|` A C) (|` B C) rnss syl A C df-ima B C df-ima 3sstr4g)) thm (imass2 () () (-> (C_ A B) (C_ (" C A) (" C B))) (A B C ssres2 (|` C A) (|` C B) rnss syl C A df-ima C B df-ima 3sstr4g)) thm (ndmima () () (-> (-. (e. A (dom B))) (= (" B ({} A)) ({/}))) ((dom B) A disjsn biimpr B ({} A) dmres ({} A) (dom B) incom eqtr syl5eq (|` B ({} A)) dm0rn0 sylib B ({} A) df-ima syl5eq)) thm (relcnv ((x y) (A x) (A y)) () (Rel (`' A)) (x y (br (cv y) A (cv x)) relopab A x y df-cnv (`' A) ({<,>|} x y (br (cv y) A (cv x))) releq ax-mp mpbir)) thm (cotr ((x y) (x z) (R x) (y z) (R y) (R z)) () (<-> (C_ (o. R R) R) (A. x (A. y (A. z (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))))) (({<,>|} x z (E. y (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))))) R (<,> (cv x) (cv z)) ssel (cv x) R (cv z) df-br syl6ibr x z (E. y (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z)))) opabid syl5ibr R R x z y df-co R sseq1i y (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)) 19.23v 3imtr4 z 19.21aiv y z (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))) alcom sylibr x 19.21aiv x z (E. y (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z)))) (br (cv x) R (cv z)) ssopab2 y z (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))) alcom y (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)) 19.23v z albii bitr x albii bitr4 x z R opabss ({<,>|} x z (E. y (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))))) ({<,>|} x z (br (cv x) R (cv z))) R sstr2 mpi sylbir R R x z y df-co syl5ss impbi)) thm (cnvsym ((x y) (R x) (R y)) () (<-> (C_ (`' R) R) (A. x (A. y (-> (br (cv x) R (cv y)) (br (cv y) R (cv x)))))) (R y x df-cnv R sseq1i ({<,>|} y x (br (cv x) R (cv y))) R (<,> (cv y) (cv x)) ssel (cv y) R (cv x) df-br syl6ibr y x (br (cv x) R (cv y)) opabid syl5ibr sylbi x y 19.21aivv y x (br (cv x) R (cv y)) (br (cv y) R (cv x)) ssopab2 y x (-> (br (cv x) R (cv y)) (br (cv y) R (cv x))) alcom bitr y x R opabss ({<,>|} y x (br (cv x) R (cv y))) ({<,>|} y x (br (cv y) R (cv x))) R sstr2 mpi sylbir R y x df-cnv syl5ss impbi)) thm (intasym ((x y) (R x) (R y)) () (<-> (C_ (i^i R (`' R)) (I)) (A. x (A. y (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv x))) (= (cv x) (cv y)))))) (R (`' R) inss2 R relcnv (i^i R (`' R)) (`' R) relss mp2 (i^i R (`' R)) (I) x y ssrel ax-mp (cv x) R (cv y) df-br x visset y visset R brcnv (cv x) (`' R) (cv y) df-br bitr3 anbi12i (<,> (cv x) (cv y)) R (`' R) elin bitr4 x visset y visset ideq (cv x) (I) (cv y) df-br bitr3 imbi12i x y 2albii bitr4)) thm (intirr ((x y) (R x) (R y)) () (<-> (= (i^i R (I)) ({/})) (A. x (-. (br (cv x) R (cv x))))) (R (I) inss2 reli (i^i R (I)) (I) relss mp2 rel0 (i^i R (I)) ({/}) x y cleqrel mp2an (cv x) R (cv x) df-br x visset (cv y) (cv x) (cv x) opeq2 R eleq1d ceqsexv bitr4 (<,> (cv x) (cv y)) noel (e. (<,> (cv x) (cv y)) (i^i R (I))) nbn con1bii x visset y visset ideq (cv x) (I) (cv y) df-br (cv x) (cv y) eqcom 3bitr3r (e. (<,> (cv x) (cv y)) R) anbi2i (= (cv y) (cv x)) (e. (<,> (cv x) (cv y)) R) ancom (<,> (cv x) (cv y)) R (I) elin 3bitr4r bitr2 y exbii y (<-> (e. (<,> (cv x) (cv y)) (i^i R (I))) (e. (<,> (cv x) (cv y)) ({/}))) exnal 3bitr con2bii x albii bitr)) thm (soirri () ((soi.1 (e. A (V))) (soi.2 (Or R S)) (soi.3 (C_ R (X. S S)))) (-. (br A R A)) (soi.2 R S A sonr mpan (e. A S) adantl soi.1 soi.3 A brel con3i pm2.61i)) thm (sotri () ((soi.1 (e. A (V))) (soi.2 (Or R S)) (soi.3 (C_ R (X. S S))) (sotri.4 (e. B (V))) (sotri.5 (e. C (V)))) (-> (/\ (br A R B) (br B R C)) (br A R C)) ((/\/\ (e. A S) (e. B S) (e. C S)) id 3exp (e. B S) a1dd imp43 sotri.4 soi.3 A brel sotri.5 soi.3 B brel syl2an soi.2 R S A B C sotr mpan mpcom)) thm (son2lpi () ((soi.1 (e. A (V))) (soi.2 (Or R S)) (soi.3 (C_ R (X. S S))) (son2lpi.4 (e. B (V)))) (-. (/\ (br A R B) (br B R A))) (soi.1 soi.2 soi.3 soirri soi.1 soi.2 soi.3 son2lpi.4 soi.1 sotri mto)) thm (cnvopab ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (= (`' ({<,>|} x y ph)) ({<,>|} y x ph)) (({<,>|} x y ph) relcnv y x ph relopab w visset z visset ({<,>|} x y ph) opelcnv (e. (cv y) (<,> (cv z) (cv w))) x ax-17 z x y ph hbopab1 hbel (e. (cv y) (<,> (cv w) (cv z))) x ax-17 z y x ph hbopab2 hbel hbbi (cv x) (cv z) (cv w) opeq1 ({<,>|} x y ph) eleq1d (cv x) (cv z) (cv w) opeq2 ({<,>|} y x ph) eleq1d bibi12d (e. (cv z) (<,> (cv x) (cv w))) y ax-17 z x y ph hbopab2 hbel (e. (cv z) (<,> (cv w) (cv x))) y ax-17 z y x ph hbopab1 hbel hbbi (cv y) (cv w) (cv x) opeq2 ({<,>|} x y ph) eleq1d (cv y) (cv w) (cv x) opeq1 ({<,>|} y x ph) eleq1d bibi12d x y ph opabid y x ph opabid bitr4 chvar chvar bitr cleqreli)) thm (cnv0 ((x y)) () (= (`' ({/})) ({/})) (({/}) relcnv rel0 x visset y visset ({/}) opelcnv (<,> (cv x) (cv y)) noel (<,> (cv y) (cv x)) noel 2false bitr4 cleqreli)) thm (cnvi ((x y)) () (= (`' (I)) (I)) ((I) relcnv reli (cv x) (cv y) eqcom (cv x) (I) (cv y) df-br x visset y visset ideq bitr3 x visset y visset (I) brcnv (cv x) (`' (I)) (cv y) df-br y visset x visset ideq 3bitr3 3bitr4r cleqreli)) thm (op1sta () ((op1sta.1 (e. A (V)))) (= (U. (dom ({} (<,> A B)))) A) (A B dmsnop unieqi op1sta.1 unisn eqtr)) thm (cnvsn ((x y) (A x) (A y) (B x) (B y)) ((cnvsn.1 (e. A (V))) (cnvsn.2 (e. B (V)))) (= (`' ({} (<,> A B))) ({} (<,> B A))) (({} (<,> A B)) relcnv cnvsn.2 A relsn (= (cv y) B) (= (cv x) A) ancom (cv y) (cv x) opex (<,> B A) elsnc y visset x visset cnvsn.1 B opth bitr y visset x visset ({} (<,> A B)) opelcnv (cv x) (cv y) opex (<,> A B) elsnc x visset y visset cnvsn.2 A opth 3bitr 3bitr4r cleqreli)) thm (rnsnop () ((cnvsn.1 (e. A (V))) (cnvsn.2 (e. B (V)))) (= (ran ({} (<,> A B))) ({} B)) (({} (<,> A B)) df-rn cnvsn.1 cnvsn.2 cnvsn dmeqi B A dmsnop 3eqtr)) thm (op2ndb () ((cnvsn.1 (e. A (V))) (cnvsn.2 (e. B (V)))) (= (|^| (|^| (|^| (`' ({} (<,> A B)))))) B) (cnvsn.1 cnvsn.2 cnvsn inteqi B A opex intsn eqtr inteqi inteqi cnvsn.2 A op1stb eqtr)) thm (op2nda () ((cnvsn.1 (e. A (V))) (cnvsn.2 (e. B (V)))) (= (U. (ran ({} (<,> A B)))) B) (({} (<,> A B)) df-rn cnvsn.1 cnvsn.2 cnvsn dmeqi eqtr unieqi cnvsn.2 A op1sta eqtr)) thm (elxp4 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (<-> (e. A (X. B C)) (/\ (= A (<,> (U. (dom ({} A))) (U. (ran ({} A))))) (/\ (e. (U. (dom ({} A))) B) (e. (U. (ran ({} A))) C)))) (A B C x y elxp A (<,> (cv x) (cv y)) sneq rneqd unieqd x visset y visset op2nda syl6req pm4.71ri (/\ (e. (cv x) B) (e. (cv y) C)) anbi1i (= (cv y) (U. (ran ({} A)))) (= A (<,> (cv x) (cv y))) (/\ (e. (cv x) B) (e. (cv y) C)) anass bitr y exbii A snex ({} A) (V) rnexg ax-mp uniex (cv y) (U. (ran ({} A))) (cv x) opeq2 A eqeq2d (cv y) (U. (ran ({} A))) C eleq1 (e. (cv x) B) anbi2d anbi12d ceqsexv bitr A (<,> (cv x) (U. (ran ({} A)))) sneq dmeqd unieqd x visset (U. (ran ({} A))) op1sta syl6req pm4.71ri (/\ (e. (cv x) B) (e. (U. (ran ({} A))) C)) anbi1i (= (cv x) (U. (dom ({} A)))) (= A (<,> (cv x) (U. (ran ({} A))))) (/\ (e. (cv x) B) (e. (U. (ran ({} A))) C)) anass 3bitr x exbii A snex ({} A) (V) dmexg ax-mp uniex (cv x) (U. (dom ({} A))) (U. (ran ({} A))) opeq1 A eqeq2d (cv x) (U. (dom ({} A))) B eleq1 (e. (U. (ran ({} A))) C) anbi1d anbi12d ceqsexv 3bitr)) thm (elxp5 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (<-> (e. A (X. B C)) (/\ (= A (<,> (|^| (|^| A)) (U. (ran ({} A))))) (/\ (e. (|^| (|^| A)) B) (e. (U. (ran ({} A))) C)))) (A B C x y elxp A (<,> (cv x) (cv y)) sneq rneqd unieqd x visset y visset op2nda syl6req pm4.71ri (/\ (e. (cv x) B) (e. (cv y) C)) anbi1i (= (cv y) (U. (ran ({} A)))) (= A (<,> (cv x) (cv y))) (/\ (e. (cv x) B) (e. (cv y) C)) anass bitr y exbii A snex ({} A) (V) rnexg ax-mp uniex (cv y) (U. (ran ({} A))) (cv x) opeq2 A eqeq2d (cv y) (U. (ran ({} A))) C eleq1 (e. (cv x) B) anbi2d anbi12d ceqsexv bitr A (<,> (cv x) (U. (ran ({} A)))) inteq inteqd x visset (U. (ran ({} A))) op1stb syl6req pm4.71ri (/\ (e. (cv x) B) (e. (U. (ran ({} A))) C)) anbi1i (= (cv x) (|^| (|^| A))) (= A (<,> (cv x) (U. (ran ({} A))))) (/\ (e. (cv x) B) (e. (U. (ran ({} A))) C)) anass 3bitr x exbii x visset (cv x) (|^| (|^| A)) (V) eleq1 mpbii (/\ (= A (<,> (cv x) (U. (ran ({} A))))) (/\ (e. (cv x) B) (e. (U. (ran ({} A))) C))) adantr x 19.23aiv (|^| (|^| A)) B elisset (= A (<,> (|^| (|^| A)) (U. (ran ({} A))))) (e. (U. (ran ({} A))) C) ad2antrl (cv x) (|^| (|^| A)) (U. (ran ({} A))) opeq1 A eqeq2d (cv x) (|^| (|^| A)) B eleq1 (e. (U. (ran ({} A))) C) anbi1d anbi12d (V) ceqsexgv pm5.21nii 3bitr)) thm (cnvun ((x y) (A x) (A y) (B x) (B y)) () (= (`' (u. A B)) (u. (`' A) (`' B))) ((u. A B) relcnv (`' A) (`' B) relun A relcnv B relcnv mpbir2an (<,> (cv y) (cv x)) A B elun x visset y visset A opelcnv x visset y visset B opelcnv orbi12i bitr4 x visset y visset (u. A B) opelcnv (<,> (cv x) (cv y)) (`' A) (`' B) elun 3bitr4 cleqreli)) thm (cnvin ((x y) (A x) (A y) (B x) (B y)) () (= (`' (i^i A B)) (i^i (`' A) (`' B))) ((i^i A B) relcnv A relcnv (`' A) (`' B) relin1 ax-mp (<,> (cv y) (cv x)) A B elin x visset y visset A opelcnv x visset y visset B opelcnv anbi12i bitr4 x visset y visset (i^i A B) opelcnv (<,> (cv x) (cv y)) (`' A) (`' B) elin 3bitr4 cleqreli)) thm (rnun () () (= (ran (u. A B)) (u. (ran A) (ran B))) (A B cnvun dmeqi (`' A) (`' B) dmun eqtr (u. A B) df-rn A df-rn B df-rn uneq12i 3eqtr4)) thm (rnin () () (C_ (ran (i^i A B)) (i^i (ran A) (ran B))) (A B cnvin dmeqi (`' A) (`' B) dmin eqsstr (i^i A B) df-rn A df-rn B df-rn ineq12i 3sstr4)) thm (rnuni ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (ran (U. A)) (U_ x A (ran (cv x)))) ((<,> (cv y) (cv z)) A x eluni y exbii y x (/\ (e. (<,> (cv y) (cv z)) (cv x)) (e. (cv x) A)) excom (E. y (e. (<,> (cv y) (cv z)) (cv x))) (e. (cv x) A) ancom y (e. (<,> (cv y) (cv z)) (cv x)) (e. (cv x) A) 19.41v z visset (cv x) y elrn2 (e. (cv x) A) anbi2i 3bitr4 x exbii 3bitr x A (e. (cv z) (ran (cv x))) df-rex bitr4 z visset (U. A) y elrn2 (cv z) x A (ran (cv x)) eliun 3bitr4 eqriv)) thm (imaun () () (= (" A (u. B C)) (u. (" A B) (" A C))) (A B C resundi rneqi (|` A B) (|` A C) rnun eqtr A (u. B C) df-ima A B df-ima A C df-ima uneq12i 3eqtr4)) thm (dminss ((x y) (A x) (A y) (R x) (R y)) () (C_ (i^i (dom R) A) (" (`' R) (" R A))) ((/\ (e. (cv x) A) (br (cv x) R (cv y))) x 19.8a ancoms y visset R A x elima2 sylibr (br (cv x) R (cv y)) (e. (cv x) A) pm3.26 y visset x visset R brcnv sylibr jca y 19.22i x visset R y eldm (e. (cv x) A) anbi1i (cv x) (dom R) A elin y (br (cv x) R (cv y)) (e. (cv x) A) 19.41v 3bitr4 x visset (`' R) (" R A) y elima2 3imtr4 ssriv)) thm (imainss ((x y) (A x) (A y) (B x) (B y) (R x) (R y)) () (C_ (i^i (" R A) B) (" R (i^i A (" (`' R) B)))) ((/\ (e. (cv y) B) (br (cv y) (`' R) (cv x))) y 19.8a y visset x visset R brcnv sylan2br ancoms (e. (cv x) A) anim2i (br (cv x) R (cv y)) (e. (cv y) B) pm3.26 (e. (cv x) A) adantl jca anassrs (cv x) A (" (`' R) B) elin x visset (`' R) B y elima2 (e. (cv x) A) anbi2i bitr (br (cv x) R (cv y)) anbi1i sylibr x 19.22i y visset R A x elima2 (e. (cv y) B) anbi1i (cv y) (" R A) B elin x (/\ (e. (cv x) A) (br (cv x) R (cv y))) (e. (cv y) B) 19.41v 3bitr4 y visset R (i^i A (" (`' R) B)) x elima2 3imtr4 ssriv)) thm (cnvxp ((x y) (A x) (A y) (B x) (B y)) () (= (`' (X. A B)) (X. B A)) ((X. A B) relcnv B A relxp x visset y visset (X. A B) opelcnv (e. (cv y) A) (e. (cv x) B) ancom x visset (cv y) A B opelxp y visset (cv x) B A opelxp 3bitr4 bitr cleqreli)) thm (xp0 () () (= (X. A ({/})) ({/})) (A xp0r (X. ({/}) A) ({/}) cnveq ax-mp ({/}) A cnvxp cnv0 3eqtr3)) thm (xpnzOLD ((A x) (A y) (A z) (x y) (x z) (y z) (B x) (B y) (B z)) () (-> (/\ (-. (= A ({/}))) (-. (= B ({/})))) (-. (= (X. A B) ({/})))) (A x n0 B y n0 anbi12i x y (e. (cv x) A) (e. (cv y) B) eeanv bitr4 (cv x) (cv y) opex (cv z) (<,> (cv x) (cv y)) (X. A B) eleq1 y visset (cv x) A B opelxp syl6bb cla4ev (X. A B) z n0 sylibr x y 19.23aivv sylbi)) thm (xpnz ((A x) (A y) (A z) (x y) (x z) (y z) (B x) (B y) (B z)) () (<-> (/\ (-. (= A ({/}))) (-. (= B ({/})))) (-. (= (X. A B) ({/})))) (A x n0 B y n0 anbi12i x y (e. (cv x) A) (e. (cv y) B) eeanv bitr4 (cv x) (cv y) opex (cv z) (<,> (cv x) (cv y)) (X. A B) eleq1 y visset (cv x) A B opelxp syl6bb cla4ev (X. A B) z n0 sylibr x y 19.23aivv sylbi A ({/}) B xpeq1 B xp0r syl6eq con3i B ({/}) A xpeq2 A xp0 syl6eq con3i jca impbi)) thm (xpeq0 () () (<-> (= (X. A B) ({/})) (\/ (= A ({/})) (= B ({/})))) (A B xpnz (= A ({/})) (= B ({/})) pm4.56 bitr3 con4bii)) thm (xpdisj1 () () (-> (= (i^i A B) ({/})) (= (i^i (X. A C) (X. B D)) ({/}))) ((i^i A B) ({/}) (i^i C D) xpeq1 (i^i C D) xp0r syl6eq A C B D inxp syl5eq)) thm (xpdisj2 () () (-> (= (i^i A B) ({/})) (= (i^i (X. C A) (X. D B)) ({/}))) ((i^i A B) ({/}) (i^i C D) xpeq2 (i^i C D) xp0 syl6eq C A D B inxp syl5eq)) thm (xpsndisj () () (-> (-. (= B D)) (= (i^i (X. A ({} B)) (X. C ({} D))) ({/}))) (B D disjsn2 ({} B) ({} D) A C xpdisj2 syl)) thm (resdisj () () (-> (= (i^i A B) ({/})) (= (|` (|` C A) B) ({/}))) (A B (V) (V) xpdisj1 C ineq2d C in0 syl6eq (|` C A) B df-res C A df-res (X. B (V)) ineq1i C (X. A (V)) (X. B (V)) inass 3eqtr syl5eq)) thm (rnxp () () (-> (-. (= A ({/}))) (= (ran (X. A B)) B)) (A B dmxp (X. A B) df-rn A B cnvxp dmeqi eqtr syl5eq)) thm (rnxpss () () (C_ (ran (X. A B)) B) (B 0ss A ({/}) B xpeq1 B xp0r syl6eq rneqd rn0 syl6eq B sseq1d mpbiri A B rnxp (ran (X. A B)) B eqimss syl pm2.61i)) thm (xpexr () () (-> (e. (X. A B) C) (\/ (e. A (V)) (e. B (V)))) (0ex A ({/}) (V) eleq1 mpbiri (e. B (V)) pm2.21nd (e. (X. A B) C) a1d A B rnxp (V) eleq1d (X. A B) C rnexg syl5bi (-. (e. A (V))) a1dd pm2.61i orrd)) thm (ssrnres ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (<-> (C_ B (ran (|` C A))) (= (ran (i^i C (X. A B))) B)) ((ran (i^i C (X. A B))) B eqss C (X. A B) inss2 (i^i C (X. A B)) (X. A B) rnss ax-mp A B rnxpss sstri (C_ B (ran (i^i C (X. A B)))) biantrur A ssid B ssv A A B (V) ssxp mp2an (X. A B) (X. A (V)) C sslin ax-mp C A df-res sseqtr4 (i^i C (X. A B)) (|` C A) rnss ax-mp B (ran (i^i C (X. A B))) (ran (|` C A)) sstr mpan2 B (ran (|` C A)) (cv y) ssel y visset (|` C A) x elrn2 syl6ib ancrd y visset (i^i C (X. A B)) x elrn2 (<,> (cv x) (cv y)) C (X. A B) elin y visset (cv x) A B opelxp (e. (<,> (cv x) (cv y)) C) anbi2i y visset (cv x) C A opelres (e. (cv y) B) anbi1i (e. (<,> (cv x) (cv y)) C) (e. (cv x) A) (e. (cv y) B) anass bitr2 3bitr x exbii x (e. (<,> (cv x) (cv y)) (|` C A)) (e. (cv y) B) 19.41v 3bitr syl6ibr ssrdv impbi 3bitr2r)) thm (rninxp ((x y) (A x) (A y) (B y) (C x) (C y)) () (<-> (= (ran (i^i C (X. A B))) B) (A.e. y B (E.e. x A (br (cv x) C (cv y))))) (B (ran (|` C A)) y dfss3 B C A ssrnres (e. (<,> (cv x) (cv y)) C) (e. (cv x) A) ancom y visset (cv x) C A opelres (cv x) C (cv y) df-br (e. (cv x) A) anbi2i 3bitr4 x exbii y visset (|` C A) x elrn2 x A (br (cv x) C (cv y)) df-rex 3bitr4 y B ralbii 3bitr3)) thm (dminxp ((A x) (x y) (B x) (B y) (C x) (C y)) () (<-> (= (dom (i^i C (X. A B))) A) (A.e. x A (E.e. y B (br (cv x) C (cv y))))) ((i^i C (X. A B)) dfdm4 C (X. A B) cnvin A B cnvxp (`' C) ineq2i eqtr rneqi eqtr A eqeq1i (`' C) B A x y rninxp y visset x visset C brcnv y B rexbii x A ralbii 3bitr)) thm (relco ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (Rel (o. A B)) (x y (E. z (/\ (br (cv x) B (cv z)) (br (cv z) A (cv y)))) relopab A B x y z df-co (o. A B) ({<,>|} x y (E. z (/\ (br (cv x) B (cv z)) (br (cv z) A (cv y))))) releq ax-mp mpbir)) thm (cores ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (C_ (ran B) C) (= (o. (|` A C) B) (o. A B))) ((br (cv x) B (cv z)) (e. (cv z) C) pm3.26 (br (cv z) A (cv y)) anim1i (C_ (ran B) C) a1i (ran B) C (cv z) ssel x visset z visset B brelrn syl5 ancld (br (cv z) A (cv y)) anim1d impbid y visset (cv z) A C opelres (cv z) (|` A C) (cv y) df-br (cv z) A (cv y) df-br (e. (cv z) C) anbi1i 3bitr4 (br (cv z) A (cv y)) (e. (cv z) C) ancom bitr (br (cv x) B (cv z)) anbi2i (br (cv x) B (cv z)) (e. (cv z) C) (br (cv z) A (cv y)) anass bitr4 syl5bb z exbidv x visset y visset (|` A C) B z opelco x visset y visset A B z opelco 3bitr4g x y 19.21aivv (|` A C) B relco A B relco (o. (|` A C) B) (o. A B) x y cleqrel mp2an sylibr)) thm (dfrel2 ((x y) (R x) (R y)) () (<-> (Rel R) (= (`' (`' R)) R)) ((`' R) relcnv x visset y visset (`' R) opelcnv y visset x visset R opelcnv bitr x y gen2 (`' (`' R)) R x y cleqrel mpbiri mpan (`' R) relcnv (`' (`' R)) R releq mpbii impbi)) thm (cnvcnv () () (= (`' (`' A)) (i^i A (X. (V) (V)))) (A (X. (V) (V)) cnvin (`' (i^i A (X. (V) (V)))) (i^i (`' A) (`' (X. (V) (V)))) cnveq ax-mp A (X. (V) (V)) inss2 (i^i A (X. (V) (V))) df-rel mpbir (i^i A (X. (V) (V))) dfrel2 mpbi (`' A) (`' (X. (V) (V))) cnvin (`' A) relcnv (`' (`' A)) df-rel mpbi (V) (V) relxp (X. (V) (V)) dfrel2 mpbi sseqtr4 (`' (`' A)) (`' (`' (X. (V) (V)))) dfss mpbi eqtr4 3eqtr3r)) thm (cnvcnvss () () (C_ (`' (`' A)) A) (A cnvcnv A (X. (V) (V)) inss1 eqsstr)) thm (co02 ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (o. A ({/})) ({/})) (A ({/}) relco rel0 (<,> (cv x) (cv z)) noel (cv x) ({/}) (cv z) df-br mtbir (br (cv z) A (cv y)) intnanr z nex x visset y visset A ({/}) z opelco mtbir (<,> (cv x) (cv y)) noel 2false cleqreli)) thm (co01 () () (= (o. ({/}) A) ({/})) (cnv0 (`' A) coeq2i (`' A) co02 eqtr2 cnv0 ({/}) A cnvco 3eqtr4 (`' ({/})) (`' (o. ({/}) A)) cnveq ax-mp rel0 ({/}) dfrel2 mpbi ({/}) A relco (o. ({/}) A) dfrel2 mpbi 3eqtr3r)) thm (coi1 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (Rel A) (= (o. A (I)) A)) (A (I) relco x visset y visset A (I) z opelco x visset z visset ideq (cv x) (cv z) eqcom bitr (br (cv z) A (cv y)) anbi1i z exbii x visset (cv z) (cv x) A (cv y) breq1 ceqsexv 3bitr (cv x) A (cv y) df-br bitr x y gen2 (o. A (I)) A x y cleqrel mpbiri mpan)) thm (coi2 () () (-> (Rel A) (= (o. (I) A) A)) (A dfrel2 cnvi (`' (`' A)) A (`' (I)) coeq2 (`' (I)) (I) A coeq1 sylan9eq mpan2 sylbi (`' A) (I) cnvco A relcnv (`' A) coi1 ax-mp (o. (`' A) (I)) (`' A) cnveq ax-mp eqtr3 syl5reqr A dfrel2 biimp eqtrd)) thm (coass ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) () (= (o. (o. A B) C) (o. A (o. B C))) ((o. A B) C relco A (o. B C) relco z w (/\ (br (cv x) C (cv z)) (/\ (br (cv z) B (cv w)) (br (cv w) A (cv y)))) excom (br (cv x) C (cv z)) (br (cv z) B (cv w)) (br (cv w) A (cv y)) anass w z 2exbii bitr4 (cv z) (o. A B) (cv y) df-br z visset y visset A B w opelco bitr (br (cv x) C (cv z)) anbi2i z exbii x visset y visset (o. A B) C z opelco w (br (cv x) C (cv z)) (/\ (br (cv z) B (cv w)) (br (cv w) A (cv y))) 19.42v z exbii 3bitr4 (cv x) (o. B C) (cv w) df-br x visset w visset B C z opelco bitr (br (cv w) A (cv y)) anbi1i w exbii x visset y visset A (o. B C) w opelco z (/\ (br (cv x) C (cv z)) (br (cv z) B (cv w))) (br (cv w) A (cv y)) 19.41v w exbii 3bitr4 3bitr4 cleqreli)) thm (relssdr ((x y) (A x) (A y)) () (-> (Rel A) (C_ A (X. (dom A) (ran A)))) ((Rel A) id (e. (<,> (cv x) (cv y)) A) y 19.8a (e. (<,> (cv x) (cv y)) A) x 19.8a jca y visset (cv x) (dom A) (ran A) opelxp x visset A y eldm2 y visset A x elrn2 anbi12i bitr sylibr (Rel A) a1i relssdv)) thm (unielrel ((x y) (A x) (A y) (R x) (R y)) () (-> (/\ (Rel R) (e. A R)) (e. (U. A) (U. R))) ((cv x) (cv y) opi2 (<,> (cv x) (cv y)) R elssuni ({,} (cv x) (cv y)) sseld mpi (= A (<,> (cv x) (cv y))) a1i A (<,> (cv x) (cv y)) R eleq1 A (<,> (cv x) (cv y)) unieq (cv x) (cv y) uniop syl6eq (U. R) eleq1d 3imtr4d x y 19.23aivv R A x y elrel (Rel R) (e. A R) pm3.27 sylc)) thm (relfld ((x y) (R x) (R y)) () (-> (Rel R) (= (U. (U. R)) (u. (dom R) (ran R)))) (R relssdr R (X. (dom R) (ran R)) uniss (U. R) (U. (X. (dom R) (ran R))) uniss 3syl (dom R) (ran R) unixpss (Rel R) a1i sstrd x visset (cv y) pri1 R (<,> (cv x) (cv y)) unielrel (cv x) (cv y) uniop syl5eqelr ({,} (cv x) (cv y)) (U. R) elssuni syl (cv x) sseld mpi ex y 19.23adv x visset R y eldm2 syl5ib ssrdv y visset (cv x) pri2 R (<,> (cv x) (cv y)) unielrel (cv x) (cv y) uniop syl5eqelr ({,} (cv x) (cv y)) (U. R) elssuni syl (cv y) sseld mpi ex x 19.23adv y visset R x elrn2 syl5ib ssrdv jca (dom R) (U. (U. R)) (ran R) unss sylib eqssd)) thm (unixp () () (-> (=/= (X. A B) ({/})) (= (U. (U. (X. A B))) (u. A B))) ((X. A B) ({/}) df-ne A B xpeq0 negbii (= A ({/})) (= B ({/})) ioran 3bitr B A dmxp (-. (= A ({/}))) adantl A B rnxp (-. (= B ({/}))) adantr uneq12d sylbi A B relxp (X. A B) relfld ax-mp syl5eq)) thm (unixp0 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (<-> (= (X. A B) ({/})) (= (U. (X. A B)) ({/}))) ((X. A B) ({/}) unieq uni0 syl6eq (X. A B) z n0 (cv z) A B x y elxp3 (<,> (cv x) (cv y)) (X. A B) snssi ({} (<,> (cv x) (cv y))) (X. A B) uniss (cv x) (cv y) opex unisn syl5ssr (cv x) (cv y) opnz (U. (X. A B)) ({/}) (<,> (cv x) (cv y)) sseq2 biimpd (<,> (cv x) (cv y)) ss0 syl6com mtoi 3syl (= (<,> (cv x) (cv y)) (cv z)) adantl x y 19.23aivv sylbi z 19.23aiv sylbi a3i impbi)) thm (cnvexg () () (-> (e. A B) (e. (`' A) (V))) (A relcnv (`' A) relssdr ax-mp (`' A) (X. (dom (`' A)) (ran (`' A))) (V) ssexg (dom (`' A)) (V) (ran (`' A)) (V) xpexg A B rnexg A df-rn syl5eqelr A B dmexg A dfdm4 syl5eqelr sylanc sylan2 mpan)) thm (cnvex () ((cnvex.1 (e. A (V)))) (e. (`' A) (V)) (cnvex.1 A (V) cnvexg ax-mp)) thm (cnvpo ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z)) () (<-> (Po R A) (Po (`' R) A)) (z A (-. (br (cv x) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))) r19.26 x A ralbii x A (A.e. z A (-. (br (cv x) R (cv x)))) (A.e. z A (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) r19.26 x A (-. (br (cv x) R (cv x))) ralidm (A.e. x A (-. (br (cv x) R (cv x)))) (A.e. x A (A.e. z A (-. (br (cv x) R (cv x))))) pm5.1 A x (-. (br (cv x) R (cv x))) rzal A x (A.e. z A (-. (br (cv x) R (cv x)))) rzal sylanc A (-. (br (cv x) R (cv x))) z r19.3rzvOLD x A ralbidv pm2.61i bitr2 (A.e. x A (A.e. z A (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))) anbi1i 3bitr x A (A.e. x A (-. (br (cv x) R (cv x)))) (A.e. z A (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) r19.26 bitr4 z A (-. (br (cv z) (`' R) (cv z))) (-> (/\ (br (cv z) (`' R) (cv y)) (br (cv y) (`' R) (cv x))) (br (cv z) (`' R) (cv x))) r19.26 (= (cv z) (cv x)) id (= (cv z) (cv x)) id R breq12d z visset z visset R brcnv syl5bb negbid A cbvralv z visset y visset R brcnv y visset x visset R brcnv anbi12i (br (cv y) R (cv z)) (br (cv x) R (cv y)) ancom bitr z visset x visset R brcnv imbi12i z A ralbii anbi12i bitr2 x A ralbii x A z A (/\ (-. (br (cv z) (`' R) (cv z))) (-> (/\ (br (cv z) (`' R) (cv y)) (br (cv y) (`' R) (cv x))) (br (cv z) (`' R) (cv x)))) ralcom 3bitr y A ralbii x A y A (A.e. z A (/\ (-. (br (cv x) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))) ralcom z A y A (A.e. x A (/\ (-. (br (cv z) (`' R) (cv z))) (-> (/\ (br (cv z) (`' R) (cv y)) (br (cv y) (`' R) (cv x))) (br (cv z) (`' R) (cv x))))) ralcom 3bitr4 R A x y z df-po (`' R) A z y x df-po 3bitr4)) thm (cnvso ((x y) (A x) (A y) (R x) (R y)) () (<-> (Or R A) (Or (`' R) A)) (R A cnvpo x A y A (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x))) ralcom y visset x visset R brcnv (cv y) (cv x) eqcom x visset y visset R brcnv 3orbi123i y A x A 2ralbii bitr4 anbi12i R A x y df-so (`' R) A y x df-so 3bitr4)) thm (coexg () () (-> (/\ (e. A C) (e. B D)) (e. (o. A B) (V))) (A B relco (o. A B) relssdr A B dmco A B rnco (dom (o. A B)) (dom B) (ran (o. A B)) (ran A) ssxp mp2an (o. A B) (X. (dom (o. A B)) (ran (o. A B))) (X. (dom B) (ran A)) sstr2 mpi syl ax-mp (o. A B) (X. (dom B) (ran A)) (V) ssexg (dom B) (V) (ran A) (V) xpexg B D dmexg A C rnexg syl2an ancoms sylan2 mpan)) thm (coex () ((coex.1 (e. A (V))) (coex.2 (e. B (V)))) (e. (o. A B) (V)) (coex.1 coex.2 A (V) B (V) coexg mp2an)) thm (dmco2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (= (dom (o. A B)) (" (`' B) (dom A))) (x visset y visset A B z opelco y exbii y z (/\ (br (cv x) B (cv z)) (br (cv z) A (cv y))) excom y (br (cv x) B (cv z)) (br (cv z) A (cv y)) 19.42v z visset x visset B opelcnv (cv x) B (cv z) df-br bitr4 A z y df-dm abeq2i anbi12i (e. (<,> (cv z) (cv x)) (`' B)) (e. (cv z) (dom A)) ancom 3bitr2 z exbii 3bitr x visset (o. A B) y eldm2 x visset (`' B) (dom A) z elima3 3bitr4 eqriv)) thm (dffun2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (Fun A) (/\ (Rel A) (A. x (A. y (A. z (-> (/\ (br (cv x) A (cv y)) (br (cv x) A (cv z))) (= (cv y) (cv z)))))))) (A df-fun y z df-id (o. A (`' A)) sseq2i A (`' A) y z x df-co ({<,>|} y z (= (cv y) (cv z))) sseq1i y z (E. x (/\ (br (cv y) (`' A) (cv x)) (br (cv x) A (cv z)))) (= (cv y) (cv z)) ssopab2 3bitr y visset x visset A brcnv (br (cv x) A (cv z)) anbi1i x exbii (= (cv y) (cv z)) imbi1i x (/\ (br (cv x) A (cv y)) (br (cv x) A (cv z))) (= (cv y) (cv z)) 19.23v bitr4 z albii z x (-> (/\ (br (cv x) A (cv y)) (br (cv x) A (cv z))) (= (cv y) (cv z))) alcom bitr y albii y x (A. z (-> (/\ (br (cv x) A (cv y)) (br (cv x) A (cv z))) (= (cv y) (cv z)))) alcom 3bitr (Rel A) anbi2i bitr)) thm (dffun3 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (Fun A) (/\ (Rel A) (A. x (E. z (A. y (-> (br (cv x) A (cv y)) (= (cv y) (cv z)))))))) (A x y z dffun2 (cv y) (cv z) (cv x) A breq2 mo4 (br (cv x) A (cv y)) z ax-17 y mo2 bitr3 x albii (Rel A) anbi2i bitr)) thm (dffun4 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (Fun A) (/\ (Rel A) (A. x (A. y (A. z (-> (/\ (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv z)) A)) (= (cv y) (cv z)))))))) (A x y z dffun2 (cv x) A (cv y) df-br (cv x) A (cv z) df-br anbi12i (= (cv y) (cv z)) imbi1i z albii x y 2albii (Rel A) anbi2i bitr)) thm (dffun5 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (Fun A) (/\ (Rel A) (A. x (E. z (A. y (-> (e. (<,> (cv x) (cv y)) A) (= (cv y) (cv z)))))))) (A x z y dffun3 (cv x) A (cv y) df-br (= (cv y) (cv z)) imbi1i y albii z exbii x albii (Rel A) anbi2i bitr)) thm (dffunmof ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A z) (A w) (A v) (A u)) ((dffunmof.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (dffunmof.2 (-> (e. (cv z) A) (A. y (e. (cv z) A))))) (<-> (Fun A) (/\ (Rel A) (A. x (E* y (br (cv x) A (cv y)))))) (A w u v dffun3 (e. (cv z) (cv w)) y ax-17 dffunmof.2 (e. (cv z) (cv v)) y ax-17 hbbr (br (cv w) A (cv y)) v ax-17 (cv v) (cv y) (cv w) A breq2 cbvmo w albii (br (cv w) A (cv v)) u ax-17 v mo2 w albii (e. (cv z) (cv w)) x ax-17 dffunmof.1 (e. (cv z) (cv y)) x ax-17 hbbr y hbmo (E* y (br (cv x) A (cv y))) w ax-17 (= (cv w) (cv x)) y ax-17 (cv w) (cv x) A (cv y) breq1 mobid cbval 3bitr3r (Rel A) anbi2i bitr4)) thm (dffunmo ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (Fun A) (/\ (Rel A) (A. x (E* y (br (cv x) A (cv y)))))) ((e. (cv z) A) x ax-17 (e. (cv z) A) y ax-17 dffunmof)) thm (funmo ((x y) (A x) (A y)) () (-> (Fun A) (E* y (br (cv x) A (cv y)))) (A x y dffunmo pm3.27bd 19.21bi)) thm (funrel () () (-> (Fun A) (Rel A)) (A df-fun pm3.26bd)) thm (funss ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (C_ A B) (-> (Fun B) (Fun A))) (A B relss B funrel syl5 A B (<,> (cv x) (cv y)) ssel (= (cv y) (cv z)) imim1d y 19.20dv z 19.22dv x 19.20dv B x z y dffun5 pm3.27bd syl5 jcad A x z y dffun5 syl6ibr)) thm (funeq () () (-> (= A B) (<-> (Fun A) (Fun B))) (B A funss A B funss anim12i ancoms A B eqss (Fun A) (Fun B) bi 3imtr4)) thm (hbfun ((F y) (x y)) ((hbfun.1 (-> (e. (cv y) F) (A. x (e. (cv y) F))))) (-> (Fun F) (A. x (Fun F))) (hbfun.1 hbrel hbfun.1 hbfun.1 hbcnv hbco (e. (cv y) (I)) x ax-17 hbss hban F df-fun F df-fun x albii 3imtr4)) thm (funeu ((x y) (x z) (F x) (y z) (F y) (F z)) () (-> (/\ (Fun F) (br (cv x) F (cv y))) (E! y (br (cv x) F (cv y)))) ((br (cv x) F (cv y)) y 19.8a F x z y dffun3 pm3.27bd 19.21bi anim12i (br (cv x) F (cv y)) z ax-17 y eu3 sylibr ancoms)) thm (funeu2 ((x y) (F x) (F y)) () (-> (/\ (Fun F) (e. (<,> (cv x) (cv y)) F)) (E! y (e. (<,> (cv x) (cv y)) F))) (F x y funeu (cv x) F (cv y) df-br (Fun F) anbi2i (cv x) F (cv y) df-br y eubii 3imtr3)) thm (dffun6 ((x y) (A x) (A y)) () (<-> (Fun A) (/\ (Rel A) (A.e. x (dom A) (E* y (br (cv x) A (cv y)))))) (A x y dffunmo y (br (cv x) A (cv y)) moabs x visset A y eldm (E* y (br (cv x) A (cv y))) imbi1i bitr4 x albii x (dom A) (E* y (br (cv x) A (cv y))) df-ral bitr4 (Rel A) anbi2i bitr)) thm (dffun7 ((x y) (A x) (A y)) () (<-> (Fun A) (/\ (Rel A) (A.e. x (dom A) (E! y (br (cv x) A (cv y)))))) (A funrel (Fun A) y ax-17 y (e. (<,> (cv x) (cv y)) A) hbeu1 A x y funeu2 ex 19.23ad x visset A y eldm2 (cv x) A (cv y) df-br y eubii 3imtr4g r19.21aiv jca y (br (cv x) A (cv y)) eumo x (dom A) r19.20si (Rel A) anim2i A x y dffun6 sylibr impbi)) thm (funfn () () (<-> (Fun A) (Fn A (dom A))) ((dom A) eqid (Fun A) biantru A (dom A) df-fn bitr4)) thm (funsn ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((funsn.1 (e. A (V))) (funsn.2 (e. B (V)))) (Fun ({} (<,> A B))) (({} (<,> A B)) x y z dffun4 funsn.1 B relsn (cv y) B (cv z) eqtr3t (cv x) (cv y) opex (<,> A B) elsnc y visset funsn.2 (cv x) A opth2 sylbi (cv x) (cv z) opex (<,> A B) elsnc z visset funsn.2 (cv x) A opth2 sylbi syl2an z ax-gen x y gen2 mpbir2an)) thm (fun0 () () (Fun ({/})) (({} (<,> ({/}) ({/}))) 0ss 0ex 0ex funsn ({/}) ({} (<,> ({/}) ({/}))) funss mp2)) thm (funi () () (Fun (I)) ((I) df-fun reli (I) relcnv (`' (I)) coi2 ax-mp cnvi eqtr (I) ssid eqsstr mpbir2an)) thm (nfunv () () (-. 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(cv v) (ran ({<,>|} x y (/\ (e. (cv x) (cv z)) ph)))) w ax-17 (A.e. x (cv z) (E.e. y (ran ({<,>|} x y (/\ (e. (cv x) (cv z)) ph))) ph)) w ax-17 w x y (/\ (e. (cv x) (cv z)) ph) hbopab1 hbrn hbeleq (e. (cv v) (cv w)) y ax-17 v x y (/\ (e. (cv x) (cv z)) ph) hbopab2 hbrn ph rexeq1f (cv z) ralbid (V) cla4egf ({<,>|} x y (/\ (e. (cv x) (cv z)) ph)) (V) funrnex y ph euex x (cv z) r19.20si (cv z) x (E. y ph) rabid2 sylibr y (e. (cv x) (cv z)) ph 19.42v x abbii x y (/\ (e. (cv x) (cv z)) ph) dmopab x (cv z) (E. y ph) df-rab 3eqtr4 syl6reqr z visset syl6eqel y ph eumo (e. (cv x) (cv z)) imim2i y (e. (cv x) (cv z)) ph moanimv sylibr x 19.20i x (cv z) (E! y ph) df-ral x y (/\ (e. (cv x) (cv z)) ph) funopab 3imtr4 sylc x (cv z) (E! y ph) hbra1 y ph euex x (cv z) r19.20si (cv z) x (E. y ph) rabid2 sylibr y (e. (cv x) (cv z)) ph 19.42v x abbii x y (/\ (e. (cv x) (cv z)) ph) dmopab x (cv z) (E. y ph) df-rab 3eqtr4 syl6reqr (cv x) eleq2d x y (/\ (e. (cv x) (cv z)) ph) opabid x visset y visset ({<,>|} x y (/\ (e. (cv x) (cv z)) ph)) opelrn sylbir ex impac y 19.22i x y (/\ (e. (cv x) (cv z)) ph) dmopab abeq2i y (ran ({<,>|} x y (/\ (e. (cv x) (cv z)) ph))) ph df-rex 3imtr4 syl6bir r19.21ai sylc)) thm (fnopabg ((x y) (x z) (A x) (y z) (A y) (A z) (F z) (w x) (w y) (w z)) ((fnopabg.1 (= F ({<,>|} x y (/\ (e. (cv x) A) ph))))) (<-> (A.e. x A (E! y ph)) (Fn F A)) (x A (E! y ph) hbra1 x A (E! y ph) ra4 y ph eumo (e. (cv x) A) imim2i y (e. (cv x) A) ph moanimv sylibr syl 19.21ai x y (/\ (e. (cv x) A) ph) funopab sylibr y ph euex x A r19.20si x A y ph dmopab3 sylib jca ({<,>|} x y (/\ (e. (cv x) A) ph)) A df-fn sylibr fnopabg.1 F ({<,>|} x y (/\ (e. (cv x) A) ph)) A fneq1 ax-mp sylibr z x y (/\ (e. (cv x) A) ph) hbopab1 fnopabg.1 (cv z) eleq2i fnopabg.1 (cv z) eleq2i x albii 3imtr4 (e. (cv z) A) x ax-17 hbfn F A (cv x) z fneu2 (e. (cv w) (<,> (cv x) (cv z))) y ax-17 z x y (/\ (e. (cv x) A) ph) hbopab2 fnopabg.1 (cv z) eleq2i fnopabg.1 (cv z) eleq2i y albii 3imtr4 hbel (e. (<,> (cv x) (cv y)) F) z ax-17 (cv z) (cv y) (cv x) opeq2 F eleq1d cbveu sylib fnopabg.1 (<,> (cv x) (cv y)) eleq2i x y (/\ (e. (cv x) A) ph) opabid bitr y eubii y (e. (cv x) A) ph euanv bitr pm3.27bd syl ex r19.21ai impbi)) thm (fnopab2g ((x y) (A x) (A y) (B y)) ((fnopab2g.1 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B)))))) (<-> (A.e. x A (e. B (V))) (Fn F A)) (B y eueq x A ralbii fnopab2g.1 fnopabg bitr)) thm (fnopab ((x y) (A x) (A y)) ((fnopab.1 (-> (e. (cv x) A) (E! y ph))) (fnopab.2 (= F ({<,>|} x y (/\ (e. (cv x) A) ph))))) (Fn F A) (fnopab.1 rgen fnopab.2 fnopabg mpbi)) thm (fnopab2 ((x y) (A x) (A y) (B y)) ((fnopab2.1 (e. B (V))) (fnopab2.2 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B)))))) (Fn F A) (fnopab2.1 y eueq1 (e. (cv x) A) a1i fnopab2.2 fnopab)) thm (dmopab2 ((x y) (A x) (A y) (B y)) ((fnopab2.1 (e. B (V))) (fnopab2.2 (= F ({<,>|} x y (/\ (e. 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F G) (u. A C) (u. B D))) (F A G C fnun ex (`' F) B (`' G) D fnun F G cnvun (`' (u. F G)) (u. (`' F) (`' G)) (u. B D) fneq1 ax-mp sylibr ex im2anan9 an4s F A B f1o4 G C D f1o4 syl2anb (u. F G) (u. A C) (u. B D) f1o4 syl6ibr imp)) thm (f1oco () () (-> (/\ (:-1-1-onto-> F B C) (:-1-1-onto-> G A B)) (:-1-1-onto-> (o. F G) A C)) (F B C G A f1co F B C G A foco anim12i an4s F B C df-f1o G A B df-f1o anbi12i (o. F G) A C df-f1o 3imtr4)) thm (f1ococnv2 () () (-> (:-1-1-onto-> F A B) (= (o. F (`' F)) (|` (I) B))) (F A B f1of F A B ffun F df-fun pm3.27bd 3syl (o. F (`' F)) iss sylib F A B f1of F A B fdm syl F A B f1ocnv (`' F) B A f1ofo (`' F) B A forn 3syl eqtr4d F (`' F) dmcoeq syl F A B f1ocnv (`' F) B A f1of (`' F) B A fdm 3syl eqtrd (dom (o. F (`' F))) B (I) reseq2 syl eqtrd)) thm (f1ococnv1 () () (-> (:-1-1-onto-> F A B) (= (o. (`' F) F) (|` (I) A))) (F A B f1orel F dfrel2 sylib (`' F) coeq2d F A B f1ocnv (`' F) B A f1ococnv2 syl eqtr3d)) thm (f1dmex () () (-> (/\ (:-1-1-> F A B) (e. B C)) (e. A (V))) ((ran F) B C ssexg F A B f1f F A B frn syl sylan ex (ran F) (V) (`' F) A fornex F A B f1f1orn F A (ran F) f1ocnv (`' F) (ran F) A f1ofo 3syl syl5com syld imp)) thm (ffoss ((F x) (A x) (B x)) ((f11o.1 (e. F (V)))) (<-> (:--> F A B) (E. x (/\ (:-onto-> F A (cv x)) (C_ (cv x) B)))) (F A B df-f F A fnforn (C_ (ran F) B) anbi1i bitr f11o.1 F (V) rnexg ax-mp (cv x) (ran F) F A foeq3 (cv x) (ran F) B sseq1 anbi12d cla4ev sylbi F A (cv x) B fss F A (cv x) fof sylan x 19.23aiv impbi)) thm (f11o ((F x) (A x) (B x)) ((f11o.1 (e. F (V)))) (<-> (:-1-1-> F A B) (E. x (/\ (:-1-1-onto-> F A (cv x)) (C_ (cv x) B)))) (f11o.1 A B x ffoss (Fun (`' F)) anbi1i F A B df-f1 F A (cv x) f1o3 (C_ (cv x) B) anbi1i (:-onto-> F A (cv x)) (Fun (`' F)) (C_ (cv x) B) an23 bitr x exbii x (/\ (:-onto-> F A (cv x)) (C_ (cv x) B)) (Fun (`' F)) 19.41v bitr 3bitr4)) thm (f10 () () (:-1-1-> ({/}) ({/}) A) (({/}) ({/}) A df-f1 A f0 fun0 cnv0 (`' ({/})) ({/}) funeq ax-mp mpbir mpbir2an)) thm (f1o00 () () (<-> (:-1-1-onto-> F ({/}) A) (/\ (= F ({/})) (= A ({/})))) (F ({/}) A f1o4 F fn0 biimp (Fn (`' F) A) adantr F fn0 F ({/}) cnveq cnv0 syl6eq sylbi (`' F) ({/}) A fneq1 syl biimpa ({/}) A fndm syl dm0 syl5reqr jca F fn0 biimpr (= A ({/})) adantr ({/}) eqid ({/}) fn0 mpbir F ({/}) cnveq cnv0 syl6eq (`' F) ({/}) A fneq1 syl A ({/}) ({/}) fneq2 sylan9bb mpbiri jca impbi bitr)) thm (fo00 () () (<-> (:-onto-> F ({/}) A) (/\ (= F ({/})) (= A ({/})))) (F ({/}) A fof F ({/}) A ffn F fn0 A f10 F ({/}) ({/}) A f1eq1 mpbiri sylbi 3syl ancri F ({/}) A df-f1o sylibr F ({/}) A f1ofo impbi F A f1o00 bitr)) thm (f1o0 () () (:-1-1-onto-> ({/}) ({/}) ({/})) (({/}) ({/}) f1o00 ({/}) eqid ({/}) eqid mpbir2an)) thm (f1oi () () (:-1-1-onto-> (|` (I) A) A A) ((|` (I) A) A A f1o3 (|` (I) A) A A df-fo A fnresi A rnresi mpbir2an funi cnvi (`' (I)) (I) funeq ax-mp mpbir (I) A funres11 ax-mp mpbir2an)) thm (f1ovi () () (:-1-1-onto-> (I) (V) (V)) ((V) f1oi reli (I) resid ax-mp (|` (I) (V)) (I) (V) (V) f1oeq1 ax-mp mpbi)) thm (f1osn () ((f1osn.1 (e. 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A B) (= (` (|` F B) A) (` F A))) (A B snssi ({} A) B F resabs1 (|` (|` F B) ({} A)) (|` F ({} A)) rneq 3syl (|` F B) ({} A) df-ima F ({} A) df-ima 3eqtr4g ({} (cv x)) eqeq1d x abbidv unieqd (|` F B) A x df-fv F A x df-fv 3eqtr4g)) thm (funssfv () () (-> (/\ (/\ (Fun F) (C_ G F)) (e. A (dom G))) (= (` F A) (` G A))) (F G funssres A fveq1d eqcomd A (dom G) F fvres sylan9eq eqcomd)) thm (tz6.12-1 ((y z) (F y) (F z) (A y) (A z)) ((tz6.12.1 (e. A (V)))) (-> (/\ (br A F (cv y)) (E! y (br A F (cv y)))) (= (` F A) (cv y))) (tz6.12.1 F z y fv3 abeq2i y (e. (cv z) (cv y)) (br A F (cv y)) exancom (E! y (br A F (cv y))) anbi1i (E. y (/\ (br A F (cv y)) (e. (cv z) (cv y)))) (E! y (br A F (cv y))) ancom bitr y (br A F (cv y)) (e. (cv z) (cv y)) eupick sylbi sylbi com12 (E! y (br A F (cv y))) adantr (/\ (e. (cv z) (cv y)) (br A F (cv y))) y 19.8a (E! y (br A F (cv y))) anim1i anasss tz6.12.1 F z y fv3 abeq2i sylibr expcom impbid eqrdv)) thm (tz6.12 ((F y) (A y)) ((tz6.12.1 (e. A (V)))) (-> (/\ (e. (<,> A (cv y)) F) (E! y (e. (<,> A (cv y)) F))) (= (` F A) (cv y))) (tz6.12.1 F y tz6.12-1 A F (cv y) df-br A F (cv y) df-br y eubii syl2anbr)) thm (tz6.12f ((x y) (x z) (w x) (y z) (w y) (w z) (F z) (F w)) ((tz6.12f.1 (-> (e. (cv w) F) (A. y (e. (cv w) F))))) (-> (/\ (e. (<,> (cv x) (cv y)) F) (E! y (e. (<,> (cv x) (cv y)) F))) (= (` F (cv x)) (cv y))) ((-> (/\ (e. (<,> (cv x) (cv y)) F) (E! y (e. (<,> (cv x) (cv y)) F))) (= (` F (cv x)) (cv y))) z ax-17 (cv z) (cv y) (cv x) opeq2 F eleq1d (e. (cv w) (<,> (cv x) (cv z))) y ax-17 tz6.12f.1 hbel (e. (<,> (cv x) (cv y)) F) z ax-17 (cv z) (cv y) (cv x) opeq2 F eleq1d cbveu (= (cv z) (cv y)) a1i anbi12d (cv z) (cv y) (` F (cv x)) eqeq2 imbi12d x visset z F tz6.12 chvar)) thm (tz6.12-2 ((x y) (x z) (A x) (y z) (A y) (A z) (F x) (F y) (F z)) () (-> (-. (E! y (br A F (cv y)))) (= (` F A) ({/}))) ((-. (E! y (br A F (cv y)))) z ax-17 ({|} x (E! y (br A F (cv y)))) z eq0 z visset (= (cv x) (cv z)) (E! y (br A F (cv y))) pm4.2i elab negbii z albii bitr2 sylib (` F A) sseq2d (cv z) A F fveq2 (cv z) A F (cv y) breq1 y eubidv x abbidv sseq12d z visset F x y fv3 (E. y (/\ (e. (cv x) (cv y)) (br (cv z) F (cv y)))) (E! y (br (cv z) F (cv y))) pm3.27 x ss2abi eqsstr (V) vtoclg syl5bi (` F A) ss0 syl6com A F fvprc (-. (E! y (br A F (cv y)))) a1d pm2.61i)) thm (tz6.12c ((F y) (A y)) ((tz6.12c.1 (e. A (V)))) (-> (E! y (br A F (cv y))) (<-> (= (` F A) (cv y)) (br A F (cv y)))) (y (br A F (cv y)) euex y (br A F (cv y)) hbeu1 (br A F (` F A)) y ax-17 hbim tz6.12c.1 F y tz6.12-1 expcom (` F A) (cv y) A F breq2 biimprd syli com12 19.23ai mpcom (` F A) (cv y) A F breq2 biimpcd syl tz6.12c.1 F y tz6.12-1 expcom impbid)) thm (tz6.12i ((F y) (A y) (B y)) ((tz6.12i.1 (e. A (V)))) (-> (-. (= B ({/}))) (-> (= (` F A) B) (br A F B))) (F A fvex (` F A) B (V) eleq1 mpbii (cv y) B (` F A) eqeq2 (cv y) B ({/}) eqeq1 negbid (cv y) B A F breq2 imbi12d imbi12d (` F A) (cv y) ({/}) eqeq1 negbid tz6.12i.1 y F tz6.12c y A F tz6.12-2 nsyl4 biimpd syl6bir pm2.43a (V) vtoclg mpcom com12)) thm (csbfv12g ((y z) (A y) (A z) (B y) (B z) (C y) (C z) (F y) (F z) (x y) (x z)) () (-> (e. A C) (= ([_/]_ A x (` F B)) (` ([_/]_ A x F) ([_/]_ A x B)))) ((e. A C) y ax-17 (e. (cv z) A) y ax-17 C ([_/]_ (cv y) x F) hbcsb1g (e. (cv z) A) y ax-17 C ([_/]_ (cv y) x B) hbcsb1g hbfvd x y a9e y visset (e. (cv z) (cv y)) x ax-17 (` F B) hbcsb1 y visset (e. (cv z) (cv y)) x ax-17 F hbcsb1 y visset (e. (cv z) (cv y)) x ax-17 B hbcsb1 hbfv hbeq x (cv y) F csbeq1a B fveq1d x (cv y) (` F B) csbeq1a x (cv y) B csbeq1a ([_/]_ (cv y) x F) fveq2d 3eqtr3d 19.23ai ax-mp (= (cv y) A) a1i y A ([_/]_ (cv y) x F) csbeq1a ([_/]_ (cv y) x B) fveq1d y A ([_/]_ (cv y) x B) csbeq1a ([_/]_ A y ([_/]_ (cv y) x F)) fveq2d 3eqtrd csbiegf A C y x (` F B) csbcog A C y x F csbcog ([_/]_ A y ([_/]_ (cv y) x B)) fveq1d A C y x B csbcog ([_/]_ A x F) fveq2d eqtrd 3eqtr3d)) thm (csbfv2g ((A y) (C y) (x y) (F x) (F y)) () (-> (e. A C) (= ([_/]_ A x (` F B)) (` F ([_/]_ A x B)))) (A C x F B csbfv12g (e. (cv y) F) x ax-17 A C csbconstgf ([_/]_ A x B) fveq1d eqtrd)) thm (csbfvg ((F x)) () (-> (e. A C) (= ([_/]_ A x (` F (cv x))) (` F A))) (A C x F (cv x) csbfv2g A C x csbvarg F fveq2d eqtrd)) thm (ndmfv ((x y) (A x) (A y) (F x) (F y)) () (-> (-. (e. A (dom F))) (= (` F A) ({/}))) ((cv x) A (dom F) eleq1 (cv x) A F (cv y) breq1 y exbidv x visset F y eldm (V) vtoclbg y (br A F (cv y)) euex syl5bir con3d y A F tz6.12-2 syl6 A F fvprc (-. (e. A (dom F))) a1d pm2.61i)) thm (ndmfvrcl () ((ndmfvrcl.1 (= (dom F) S)) (ndmfvrcl.2 (-. (e. ({/}) S)))) (-> (e. (` F A) S) (e. A S)) (ndmfvrcl.2 A F ndmfv S eleq1d mtbiri a3i ndmfvrcl.1 syl6eleq)) thm (nfvres () () (-> (-. (e. A B)) (= (` (|` F B) A) ({/}))) (F B dmres A eleq2i A B (dom F) elin bitr pm3.26bd con3i A (|` F B) ndmfv syl)) thm (fveqres () () (-> (= (` F A) (` G A)) (= (` (|` F B) A) (` (|` G B) A))) (A B F fvres A B G fvres eqeq12d biimprd A B F nfvres A B G nfvres eqtr4d (= (` F A) (` G A)) a1d pm2.61i)) thm (funbrfv ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) ((funbrfv.1 (e. B (V)))) (-> (Fun F) (-> (br A F B) (= (` F A) B))) (F A B brrelex F funrel sylan funbrfv.1 (cv x) A F (cv y) breq1 (Fun F) anbi2d (cv x) A F fveq2 (cv y) eqeq1d imbi12d (cv y) B A F breq2 (Fun F) anbi2d (cv y) B (` F A) eqeq2 imbi12d x visset F y tz6.12-1 F x y funeu sylan2 anabss7 (V) (V) vtocl2g mpan2 mpcom ex)) thm (funopfv () ((funopfv.1 (e. B (V)))) (-> (Fun F) (-> (e. (<,> A B) F) (= (` F A) B))) (funopfv.1 F A funbrfv A F B df-br syl5ibr)) thm (funopfvg ((A x) (B x) (F x)) () (-> (/\ (e. B C) (Fun F)) (-> (e. (<,> A B) F) (= (` F A) B))) ((cv x) B A opeq2 F eleq1d (cv x) B (` F A) eqeq2 imbi12d (Fun F) imbi2d x visset F A funopfv C vtoclg imp)) thm (fnbrfvb ((x y) (F x) (F y) (A x) (B x) (B y) (C x)) ((fnfvbr.1 (e. C (V)))) (-> (/\ (Fn F A) (e. B A)) (<-> (= (` F B) C) (br B F C))) (fnfvbr.1 (cv x) C (` F B) eqeq2 (cv x) C B F breq2 bibi12d (/\ (Fn F A) (e. B A)) imbi2d F A B x fneu (cv y) B F (cv x) breq1 x eubidv (cv y) B F fveq2 (cv x) eqeq1d (cv y) B F (cv x) breq1 bibi12d imbi12d y visset x F tz6.12c A vtoclg (Fn F A) adantl mpd vtocl)) thm (fnopfvb () ((fnfvbr.1 (e. C (V)))) (-> (/\ (Fn F A) (e. B A)) (<-> (= (` F B) C) (e. (<,> B C) F))) (fnfvbr.1 F A B fnbrfvb B F C df-br syl6bb)) thm (fnbrfvbg ((A x) (B x) (C x) (F x)) () (-> (/\/\ (Fn F A) (e. B A) (e. C D)) (<-> (= (` F B) C) (br B F C))) ((cv x) C (` F B) eqeq2 (cv x) C B F breq2 bibi12d (/\ (Fn F A) (e. B A)) imbi2d x visset F A B fnbrfvb D vtoclg 3impib 3coml)) thm (funbrfvb () ((funbrfvb.1 (e. B (V)))) (-> (/\ (Fun F) (e. A (dom F))) (<-> (= (` F A) B) (br A F B))) (funbrfvb.1 F (dom F) A fnbrfvb F (dom F) df-fn (dom F) eqid mpbiran2 sylanbr)) thm (funopfvb () ((funbrfvb.1 (e. B (V)))) (-> (/\ (Fun F) (e. A (dom F))) (<-> (= (` F A) B) (e. (<,> A B) F))) (funbrfvb.1 F A funbrfvb A F B df-br syl6bb)) thm (fnopabfv ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (F y) (F z) (F w)) () (<-> (Fn F A) (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) (` F (cv x))))))) (z visset w visset F A fnop ex pm4.71rd w visset F A (cv z) fnopfvb (cv w) (` F (cv z)) eqcom syl5bb ex pm5.32d bitr4d z visset w visset (cv x) (cv z) A eleq1 (cv x) (cv z) F fveq2 (cv y) eqeq2d anbi12d (cv y) (cv w) (` F (cv z)) eqeq1 (e. (cv z) A) anbi2d opelopab syl6bbr z w 19.21aivv F A fnrel x y (/\ (e. (cv x) A) (= (cv y) (` F (cv x)))) relopab jctir F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) (` F (cv x))))) z w cleqrel syl mpbird F (cv x) fvex ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) (` F (cv x))))) eqid fnopab2 F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) (` F (cv x))))) A fneq1 mpbiri impbi)) thm (fvelima ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (-> (/\ (Fun F) (e. A (" F B))) (E.e. x B (= (` F (cv x)) A))) ((cv y) A (" F B) eleq1 (cv y) A (` F (cv x)) eqeq2 x B rexbidv imbi12d (Fun F) imbi2d y visset F (cv x) funbrfv x B r19.22sdv y visset F B x elima syl5ib (" F B) vtoclg pm2.43b imp)) thm (fnrnfv ((x y) (A x) (A y) (F x) (F y)) () (-> (Fn F A) (= (ran F) ({|} y (E.e. x A (= (cv y) (` F (cv x))))))) (F A fndm (cv x) eleq2d x visset (cv y) F opeldm syl5bi pm4.71rd y visset F A (cv x) fnopfvb (` F (cv x)) (cv y) eqcom syl5rbbr ex pm5.32d bitrd x exbidv x A (= (cv y) (` F (cv x))) df-rex syl6bbr y abbidv F y x dfrn3 syl5eq)) thm (fniinfv ((x y) (A x) (A y) (F x) (F y)) () (-> (Fn F A) (= (|^|_ x A (` F (cv x))) (|^| (ran F)))) (F A y x fnrnfv inteqd F (cv x) fvex x A y dfiin2 syl6reqr)) thm (fnsnfv ((A y) (B y) (F y)) () (-> (/\ (Fn F A) (e. B A)) (= ({} (` F B)) (" F ({} B)))) (y visset F A B fnbrfvb (cv y) (` F B) eqcom syl5bb y abbidv (` F B) y df-sn (/\ (Fn F A) (e. B A)) a1i F A fnrel F B y relimasn syl (e. B A) adantr 3eqtr4d)) thm (funfv () () (-> (Fun F) (= (` F A) (U. (" F ({} A))))) (F (dom F) A fnsnfv F (dom F) df-fn (dom F) eqid mpbiran2 sylanbr unieqd F A fvex unisn syl5eqr ex A F ndmfv A F ndmima unieqd uni0 syl6eq eqtr4d pm2.61d1)) thm (funfv2 ((A y) (F y)) () (-> (Fun F) (= (` F A) (U. ({|} y (br A F (cv y)))))) (F A funfv F funrel F A y relimasn syl unieqd eqtrd)) thm (funfv2f ((w z) (A w) (A z) (F w) (F z) (w y) (y z)) ((funfv2f.1 (-> (e. (cv z) A) (A. y (e. (cv z) A)))) (funfv2f.2 (-> (e. (cv z) F) (A. y (e. (cv z) F))))) (-> (Fun F) (= (` F A) (U. ({|} y (br A F (cv y)))))) (F A w funfv2 funfv2f.1 funfv2f.2 (e. (cv z) (cv w)) y ax-17 hbbr (br A F (cv y)) w ax-17 (cv w) (cv y) A F breq2 cbvab unieqi syl6eq)) thm (dmfco ((y z) (A y) (A z) (F y) (F z) (G y) (G z)) () (-> (/\ (Fun G) (e. A (dom G))) (<-> (e. A (dom (o. F G))) (e. (` G A) (dom F)))) (A (dom G) (o. F G) y eldm2g y visset A (dom G) (cv y) (V) F G z opelcog mpan2 y exbidv bitrd (Fun G) adantl z visset G A funopfvb (cv z) (` G A) eqcom syl5bb (e. (<,> (cv z) (cv y)) F) anbi1d z exbidv G A fvex (cv z) (` G A) (cv y) opeq1 F eleq1d ceqsexv syl5bbr y exbidv G A fvex F y eldm2 syl5bb bitr4d)) thm (fvco ((A z) (F z) (G z)) () (-> (/\/\ (Fun F) (Fun G) (e. A (dom G))) (= (` (o. F G) A) (` F (` G A)))) (G A F dmfco (Fun F) anbi2d F (` G A) fvex A (dom G) (` F (` G A)) (V) F G z opelcog mpan2 (Fun G) adantl z visset G A funopfvb (cv z) (` G A) eqcom syl5bb (e. (<,> (cv z) (` F (` G A))) F) anbi1d z exbidv G A fvex (cv z) (` G A) (` F (` G A)) opeq1 F eleq1d ceqsexv syl5bbr bitr4d (` F (` G A)) eqid F (` G A) fvex F (` G A) funopfvb mpbii syl5bir sylbid exp4b com3r 3imp1 F (` G A) fvex (o. F G) A funopfvb F G funco sylan (e. A (dom G)) 3adantl3 mpbird ex A (o. F G) ndmfv (/\ (Fun G) (e. A (dom G))) adantl G A F dmfco negbid (` G A) F ndmfv syl6bi imp eqtr4d ex (Fun F) 3adant1 pm2.61d)) thm (fvco2 () () (-> (/\/\ (Fun F) (Fn G A) (e. C A)) (= (` (o. F G) C) (` F (` G C)))) (F G C fvco 3exp com3l imp A funfni ex com3r 3imp)) thm (fvco3 () () (-> (/\/\ (Fun F) (:--> G A B) (e. C A)) (= (` (o. F G) C) (` F (` G C)))) (F G A C fvco2 G A B ffn syl3an2)) thm (fvopab3 ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (ch x) (ch y)) ((fvopab3.1 (e. B (V))) (fvopab3.2 (-> (= (cv x) A) (<-> ph ps))) (fvopab3.3 (-> (= (cv y) B) (<-> ps ch))) (fvopab3.4 (-> (e. (cv x) C) (E! y ph))) (fvopab3.5 (= F ({<,>|} x y (/\ (e. (cv x) C) ph))))) (-> (e. A C) (<-> (= (` F A) B) ch)) (fvopab3.1 (cv x) A C eleq1 fvopab3.2 anbi12d fvopab3.3 (e. A C) anbi2d C (V) opelopabg mpan2 fvopab3.4 fvopab3.5 fnopab fvopab3.1 F C A fnopfvb mpan fvopab3.5 (<,> A B) eleq2i syl6bb (e. A C) ch ibar 3bitr4d)) thm (fvopab3ig ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (ch x) (ch y)) ((fvopab3ig.1 (-> (= (cv x) A) (<-> ph ps))) (fvopab3ig.2 (-> (= (cv y) B) (<-> ps ch))) (fvopab3ig.3 (-> (e. (cv x) C) (E* y ph))) (fvopab3ig.4 (= F ({<,>|} x y (/\ (e. (cv x) C) ph))))) (-> (/\ (e. A C) (e. B D)) (-> ch (= (` F A) B))) ((cv x) A C eleq1 fvopab3ig.1 anbi12d fvopab3ig.2 (e. A C) anbi2d C D opelopabg biimpar exp43 pm2.43a imp x y (/\ (e. (cv x) C) ph) funopab fvopab3ig.3 y (e. (cv x) C) ph moanimv mpbir mpgbir B D ({<,>|} x y (/\ (e. (cv x) C) ph)) A funopfvg mpan2 (e. A C) adantl syld fvopab3ig.4 A fveq1i B eqeq1i syl6ibr)) thm (fvopab4g ((x y) (A x) (A y) (B y) (C x) (C y) (D x) (D y)) ((fvopab4g.1 (-> (= (cv x) A) (= B C))) (fvopab4g.2 (= F ({<,>|} x y (/\ (e. (cv x) D) (= (cv y) B)))))) (-> (/\ (e. A D) (e. C R)) (= (` F A) C)) (C eqid fvopab4g.1 (cv y) eqeq2d (cv y) C C eqeq1 y B moeq (e. (cv x) D) a1i fvopab4g.2 R fvopab3ig mpi)) thm (fvopab4 ((x y) (A x) (A y) (B y) (C x) (C y) (D x) (D y)) ((fvopab4g.1 (-> (= (cv x) A) (= B C))) (fvopab4g.2 (= F ({<,>|} x y (/\ (e. (cv x) D) (= (cv y) B))))) (fvopab4.3 (e. C (V)))) (-> (e. A D) (= (` F A) C)) (fvopab4.3 fvopab4g.1 fvopab4g.2 (V) fvopab4g mpan2)) thm (fvopab4gf ((v w) (v z) (A v) (w z) (A w) (A z) (v y) (B v) (w y) (B w) (B y) (C z) (C w) (C v) (v x) (D v) (w x) (D w) (x y) (D x) (D y) (x z)) ((fvopab4gf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (fvopab4gf.2 (-> (e. (cv z) C) (A. x (e. (cv z) C)))) (fvopab4gf.3 (-> (= (cv x) A) (= B C))) (fvopab4gf.4 (= F ({<,>|} x y (/\ (e. (cv x) D) (= (cv y) B)))))) (-> (/\ (e. A D) (e. C R)) (= (` F A) C)) ((e. (cv z) (cv w)) x ax-17 fvopab4gf.1 w visset eqvincf w visset (e. (cv v) (cv w)) x ax-17 B hbcsb1 fvopab4gf.2 hbeq x (cv w) B csbeq1a eqcomd fvopab4gf.3 sylan9eq 19.23ai sylbi ({<,>|} w v (/\ (e. (cv w) D) (= (cv v) ([_/]_ (cv w) x B)))) eqid R fvopab4g fvopab4gf.4 A fveq1i (/\ (e. (cv x) D) (= (cv y) B)) w ax-17 (/\ (e. (cv x) D) (= (cv y) B)) v ax-17 (e. (cv w) D) x ax-17 w visset (e. (cv v) (cv w)) x ax-17 B hbcsb1 hbeleq hban (/\ (e. (cv w) D) (= (cv v) ([_/]_ (cv w) x B))) y ax-17 (cv x) (cv w) D eleq1 (= (cv y) (cv v)) adantr (= (cv y) (cv v)) id x (cv w) B csbeq1a eqeqan12rd anbi12d cbvopab A fveq1i eqtr syl5eq)) thm (fvopab4sf ((A z) (y z) (B y) (B z) (x y) (C x) (C y) (x z)) ((fvopab4sf.1 (e. A (V))) (fvopab4sf.2 (e. B (V))) (fvopab4sf.3 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (fvopab4sf.4 (= F ({<,>|} x y (/\ (e. (cv x) C) (= (cv y) B)))))) (-> (e. A C) (= (` F A) ([_/]_ A x B))) (fvopab4sf.1 fvopab4sf.2 x ax-gen A (V) x B (V) csbexg mp2an fvopab4sf.3 fvopab4sf.1 fvopab4sf.3 B hbcsb1 x A B csbeq1a fvopab4sf.4 (V) fvopab4gf mpan2)) thm (fvopab4s ((x z) (A x) (A z) (y z) (B y) (B z) (x y) (C x) (C y)) ((fvopab4s.1 (e. A (V))) (fvopab4s.2 (e. B (V))) (fvopab4s.3 (= F ({<,>|} x y (/\ (e. (cv x) C) (= (cv y) B)))))) (-> (e. A C) (= (` F A) ([_/]_ A x B))) (fvopab4s.1 fvopab4s.2 (e. (cv z) A) x ax-17 fvopab4s.3 fvopab4sf)) thm (fvopabg ((x y) (A x) (A y) (B y) (C x) (C y)) ((fvopabg.1 (-> (= (cv x) A) (= B C)))) (-> (/\ (e. A D) (e. C R)) (= (` ({<,>|} x y (= (cv y) B)) A) C)) (fvopabg.1 x visset (= (cv y) B) biantrur x y opabbii R fvopab4g A D elisset sylan)) thm (fvopabn ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((fvopabg.1 (-> (= (cv x) A) (= B C)))) (-> (-. (e. C (V))) (= (` ({<,>|} x y (= (cv y) B)) A) ({/}))) (z visset snnz (cv z) A (cv w) opeq1 ({<,>|} x y (= (cv y) B)) eleq1d (V) ceqsexgv z A elsn (e. (<,> (cv z) (cv w)) ({<,>|} x y (= (cv y) B))) anbi1i z exbii syl5bb w visset fvopabg.1 (cv y) eqeq2d (cv y) (cv w) C eqeq1 (V) (V) opelopabg mpan2 bitrd w abbidv w visset (cv w) C (V) eleq1 mpbii w 19.23aiv con3i w (= (cv w) C) abn0 con1bii sylib sylan9eq ({<,>|} x y (= (cv y) B)) ({} A) w z dfima3 syl5eq ({} (cv z)) eqeq1d ({/}) ({} (cv z)) eqcom syl6bb mtbiri z 19.21aiv z (= (" ({<,>|} x y (= (cv y) B)) ({} A)) ({} (cv z))) alnex z (= (" ({<,>|} x y (= (cv y) B)) ({} A)) ({} (cv z))) abn0 con1bii bitr sylib unieqd ({<,>|} x y (= (cv y) B)) A z df-fv syl5eq uni0 syl6eq ex A ({<,>|} x y (= (cv y) B)) fvprc (-. (e. C (V))) a1d pm2.61i)) thm (fvopabgf ((w z) (v z) (A z) (v w) (A w) (A v) (w y) (v y) (u y) (B y) (u w) (B w) (u v) (B v) (B u) (C z) (C w) (C v) (x y) (w x) (v x) (u x) (x z)) ((fvopabgf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (fvopabgf.2 (-> (e. (cv z) C) (A. x (e. (cv z) C)))) (fvopabgf.3 (-> (= (cv x) A) (= B C)))) (-> (/\ (e. A D) (e. C R)) (= (` ({<,>|} x y (= (cv y) B)) A) C)) ((e. (cv z) (cv w)) x ax-17 fvopabgf.1 w visset eqvincf w x (e. (cv u) B) hbs1 v u hbab fvopabgf.2 hbeq x w B u sbab eqcomd fvopabgf.3 sylan9eq 19.23ai sylbi D R v fvopabg (= (cv y) B) w ax-17 (= (cv y) B) v ax-17 w x (e. (cv u) B) hbs1 v u hbab hbeleq (= (cv v) ({|} u ([/] (cv w) x (e. (cv u) B)))) y ax-17 (= (cv y) (cv v)) id x w B u sbab eqeqan12rd cbvopab A fveq1i syl5eq)) thm (fvopabnf ((w z) (v z) (A z) (v w) (A w) (A v) (w y) (v y) (u y) (B y) (u w) (B w) (u v) (B v) (B u) (C z) (C w) (C v) (x y) (w x) (v x) (u x) (x z)) ((fvopabgf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (fvopabgf.2 (-> (e. (cv z) C) (A. x (e. (cv z) C)))) (fvopabgf.3 (-> (= (cv x) A) (= B C)))) (-> (-. (e. C (V))) (= (` ({<,>|} x y (= (cv y) B)) A) ({/}))) ((e. (cv z) (cv w)) x ax-17 fvopabgf.1 w visset eqvincf w x (e. (cv u) B) hbs1 v u hbab fvopabgf.2 hbeq x w B u sbab eqcomd fvopabgf.3 sylan9eq 19.23ai sylbi v fvopabn (= (cv y) B) w ax-17 (= (cv y) B) v ax-17 w x (e. (cv u) B) hbs1 v u hbab hbeleq (= (cv v) ({|} u ([/] (cv w) x (e. (cv u) B)))) y ax-17 (= (cv y) (cv v)) id x w B u sbab eqeqan12rd cbvopab A fveq1i syl5eq)) thm (fvopabf ((A z) (B y) (C z) (x y) (x z)) ((fvopabf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (fvopabf.2 (-> (e. (cv z) C) (A. x (e. (cv z) C)))) (fvopabf.3 (e. A (V))) (fvopabf.4 (e. C (V))) (fvopabf.5 (-> (= (cv x) A) (= B C)))) (= (` ({<,>|} x y (= (cv y) B)) A) C) (fvopabf.3 fvopabf.4 fvopabf.1 fvopabf.2 fvopabf.5 (V) (V) y fvopabgf mp2an)) thm (fvopab ((x z) (A x) (A z) (B y) (C x) (C z) (x y) (y z)) ((fvopab.1 (e. A (V))) (fvopab.2 (e. C (V))) (fvopab.3 (-> (= (cv x) A) (= B C)))) (= (` ({<,>|} x y (= (cv y) B)) A) C) ((e. (cv z) A) x ax-17 (e. (cv z) C) x ax-17 fvopab.1 fvopab.2 fvopab.3 y fvopabf)) thm (fvopab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) () (-> (/\ (e. (cv x) A) (e. B C)) (= (` ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B))) (cv x)) B)) (B C y elex (e. (cv x) A) y ax-17 z x y (/\ (e. (cv x) A) (= (cv y) B)) hbopab2 (e. (cv z) (cv x)) y ax-17 hbfv (e. (cv z) B) y ax-17 hbeq hbim y visset (cv y) B (V) eleq1 mpbii z x y (/\ (e. (cv x) A) (= (cv y) B)) hbopab2 x tz6.12f x y (/\ (e. (cv x) A) (= (cv y) B)) opabid sylanbr y (e. (cv x) A) (= (cv y) B) euanv x y (/\ (e. (cv x) A) (= (cv y) B)) opabid y eubii B y eueq (e. (cv x) A) anbi2i 3bitr4r sylan2b exp43 pm2.43a mpdi com12 (cv y) B (` ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B))) (cv x)) eqeq2 sylibd 19.23ai syl impcom)) thm (fvopabs ((x z) (A x) (A z) (y z) (B y) (B z) (x y)) ((fvopabs.1 (e. A (V))) (fvopabs.2 (e. B (V)))) (= (` ({<,>|} x y (= (cv y) B)) A) ([_/]_ A x B)) ((e. (cv z) A) x ax-17 fvopabs.1 (e. (cv z) A) x ax-17 B hbcsb1 fvopabs.1 fvopabs.1 fvopabs.2 x ax-gen A (V) x B (V) csbexg mp2an x A B csbeq1a y fvopabf)) thm (fvopab5 ((x y) (x z) (A x) (y z) (A y) (A z) (F z) (ph z) (ps x)) ((fvopab5.1 (= F ({<,>|} x y ph))) (fvopab5.2 (-> (= (cv x) A) (<-> ph ps)))) (-> (/\ (Fun F) (e. A B)) (= (` F A) (U. ({|} y ps)))) ((e. (cv z) A) y ax-17 z x y ph hbopab2 fvopab5.1 (cv z) eleq2i fvopab5.1 (cv z) eleq2i y albii 3imtr4 funfv2f A B x elex (e. (cv z) (<,> A (cv y))) x ax-17 z x y ph hbopab1 hbel ps x ax-17 hbbi (cv x) A (cv y) opeq1 ({<,>|} x y ph) eleq1d fvopab5.2 x y ph opabid syl5bb bitr3d 19.23ai A F (cv y) df-br fvopab5.1 (<,> A (cv y)) eleq2i bitr syl5bb y abbidv syl unieqd sylan9eq)) thm (fvsn () ((fvsn.1 (e. A (V))) (fvsn.2 (e. B (V)))) (= (` ({} (<,> A B)) A) B) (fvsn.1 fvsn.2 funsn A B opex snid fvsn.2 ({} (<,> A B)) A funopfv mp2)) thm (eqfnfv ((x y) (A x) (A y) (B x) (B y) (F x) (F y) (G x) (G y)) () (-> (/\ (Fn F A) (Fn G B)) (<-> (= F G) (/\ (= A B) (A.e. x A (= (` F (cv x)) (` G (cv x))))))) ((dom F) A (dom G) B eqeq12 F G dmeq syl5bi F A fndm G B fndm syl2an F G (cv x) fveq1 (e. (cv x) A) a1d r19.21aiv (/\ (Fn F A) (Fn G B)) a1i jcad y visset F A (cv x) fnopfvb (Fn G A) adantlr y visset G A (cv x) fnopfvb (Fn F A) adantll bibi12d (` F (cv x)) (` G (cv x)) (cv y) eqeq1 syl5bi ex a2d com3r F A fndm (cv x) eleq2d x visset (cv y) F opeldm syl5bi con3d (Fn G A) adantr G A fndm (cv x) eleq2d x visset (cv y) G opeldm syl5bi con3d (Fn F A) adantl jcad (e. (<,> (cv x) (cv y)) F) (e. (<,> (cv x) (cv y)) G) pm5.21 (-> (e. (cv x) A) (= (` F (cv x)) (` G (cv x)))) a1d syl6com pm2.61i y 19.21adv x 19.20dv x A (= (` F (cv x)) (` G (cv x))) df-ral syl5ib F G x y cleqrel F A fnrel G A fnrel syl2an sylibrd A B G fneq2 biimparc sylan2 exp32 imp4b impbid)) thm (eqfnfvf ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (F y) (F z) (G y) (G z)) ((eqfnfvf.1 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (eqfnfvf.2 (-> (e. (cv y) G) (A. x (e. (cv y) G))))) (-> (/\ (Fn F A) (Fn G B)) (<-> (= F G) (/\ (= A B) (A.e. x A (= (` F (cv x)) (` G (cv x))))))) (F A G B z eqfnfv eqfnfvf.1 (e. (cv y) (cv z)) x ax-17 hbfv eqfnfvf.2 (e. (cv y) (cv z)) x ax-17 hbfv hbeq (= (` F (cv x)) (` G (cv x))) z ax-17 (cv z) (cv x) F fveq2 (cv z) (cv x) G fveq2 eqeq12d A cbvral (= A B) anbi2i syl6bb)) thm (fvreseq ((B x) (F x) (G x)) () (-> (/\ (/\ (Fn F A) (Fn G A)) (C_ B A)) (<-> (= (|` F B) (|` G B)) (A.e. x B (= (` F (cv x)) (` G (cv x)))))) (F A B fnssres G A B fnssres anim12i anandirs (|` F B) B (|` G B) B x eqfnfv (cv x) B F fvres (cv x) B G fvres eqeq12d ralbiia B eqid (A.e. x B (= (` (|` F B) (cv x)) (` (|` G B) (cv x)))) biantrur bitr3 syl6bbr syl)) thm (fvelrn ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) () (-> (Fn F A) (<-> (e. B (ran F)) (E.e. x A (= (` F (cv x)) B)))) ((cv y) B (ran F) eleq1 (cv y) B (` F (cv x)) eqeq2 x A rexbidv bibi12d (Fn F A) imbi2d x visset y visset F A fnbr x visset F y tz6.12-1 ex F x y funeu F A fnfun sylan syl5 anabsi7 jca ex y visset F A (cv x) fnbrfvb biimpd ex imp3a impbid x exbidv y visset F x elrn x A (= (` F (cv x)) (cv y)) df-rex 3bitr4g (V) vtoclg B (ran F) elisset F (cv x) fvex (` F (cv x)) B (V) eleq1 mpbii (e. (cv x) A) a1i r19.23aiv pm5.21ni (Fn F A) a1d pm2.61i)) thm (elrnopab ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (F z) (w x) (w z) (C w) (C x) (C z) (F w)) ((elrnopab.1 (e. B (V))) (elrnopab.2 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B)))))) (<-> (e. C (ran F)) (E.e. x A (= C B))) (elrnopab.1 y eueq1 (e. (cv x) A) a1i elrnopab.2 fnopab F A C z fvelrn w x y (/\ (e. (cv x) A) (= (cv y) B)) hbopab1 elrnopab.2 (cv w) eleq2i elrnopab.2 (cv w) eleq2i x albii 3imtr4 (e. (cv w) (cv z)) x ax-17 hbfv (e. (cv w) C) x ax-17 hbeq (= (` F (cv x)) C) z ax-17 (cv z) (cv x) F fveq2 C eqeq1d A cbvrex elrnopab.1 x A B (V) y fvopab2 mpan2 elrnopab.2 (cv x) fveq1i syl5eq C eqeq1d B C eqcom syl6bb rexbiia bitr syl6bb ax-mp)) thm (chfnrn ((x y) (A x) (A y) (F x) (F y)) () (-> (/\ (Fn F A) (A.e. x A (e. (` F (cv x)) (cv x)))) (C_ (ran F) (U. A))) (F A (cv y) x fvelrn biimpd x A (e. (` F (cv x)) (cv x)) hbra1 x A (e. (` F (cv x)) (cv x)) ra4 (` F (cv x)) (cv y) (cv x) eleq1 biimpcd syl6 r19.22d sylan9 (cv y) A x eluni2 syl6ibr ssrdv)) thm (funfvop ((F x) (A x)) () (-> (/\ (Fun F) (e. A (dom F))) (e. (<,> A (` F A)) F)) (F A fvex x isseti x visset F A funopfvb (` F A) (cv x) A opeq2 F eleq1d biimprcd syl6bi pm2.43d (cv x) (` F A) eqcom syl5ib x 19.23adv mpi)) thm (fnopfv () () (-> (/\ (Fn F A) (e. B A)) (e. (<,> B (` F B)) F)) (F B funfvop A funfni)) thm (fvrn ((x y) (F x) (F y) (A x)) () (-> (/\ (Fun F) (e. A (dom F))) (e. (` F A) (ran F))) ((cv x) A (dom F) eleq1 (Fun F) anbi2d (cv x) A F fveq2 (ran F) eleq1d imbi12d F (cv x) funfvop x visset (cv y) (cv x) (` F (cv x)) opeq1 F eleq1d cla4ev syl F (cv x) fvex F y elrn2 sylibr (dom F) vtoclg anabsi7)) thm (fnfvrn () () (-> (/\ (Fn F A) (e. B A)) (e. (` F B) (ran F))) (F B fvrn A funfni)) thm (ffvrn () () (-> (/\ (:--> F A B) (e. C A)) (e. (` F C) B)) (F A C fnfvrn F A B ffn sylan F A B frn (` F C) sseld (e. C A) adantr mpd)) thm (ffvrni () ((ffvrni.1 (:--> F A B))) (-> (e. C A) (e. (` F C) B)) (ffvrni.1 F A B C ffvrn mpan)) thm (fopab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C y) (C z) (F z)) ((fopab2.1 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C)))))) (<-> (A.e. x A (e. C B)) (:--> F A B)) (C B elisset C y eueq sylib x A r19.20si fopab2.1 fnopabg sylib x A (e. C B) hbra1 (e. (cv y) B) x ax-17 x A (e. C B) ra4 C B (cv y) eleq1a (e. (cv x) A) imim2i imp3a syl 19.23ad x y (/\ (e. (cv x) A) (= (cv y) C)) rnopab abeq2i syl5ib y 19.21aiv z x y (/\ (e. (cv x) A) (= (cv y) C)) hbopab2 hbrn (e. (cv z) B) y ax-17 dfss2f sylibr fopab2.1 rneqi syl5ss jca F A B df-f sylibr F A B fdm x A y (= (cv y) C) dmopab3 C y isset x A ralbii fopab2.1 dmeqi A eqeq1i 3bitr4r sylib z x y (/\ (e. (cv x) A) (= (cv y) C)) hbopab1 (e. (cv z) A) x ax-17 (e. (cv z) B) x ax-17 hbf fopab2.1 F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) A B feq1 ax-mp fopab2.1 F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) A B feq1 ax-mp x albii 3imtr4 F A B (cv x) ffvrn (e. C (V)) adantr x A C (V) y fvopab2 fopab2.1 (cv x) fveq1i syl5eq B eleq1d (:--> F A B) adantll mpbid ex r19.20da mpd impbi)) thm (rnssopab ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C y) (C z) (F z)) ((fopab2.1 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))))) (rnssopab.2 (e. C (V)))) (<-> (A.e. x A (e. C B)) (C_ (ran F) B)) (fopab2.1 B fopab2 F A B frn sylbi z x y (/\ (e. (cv x) A) (= (cv y) C)) hbopab1 fopab2.1 (cv z) eleq2i fopab2.1 (cv z) eleq2i x albii 3imtr4 hbrn (e. (cv z) B) x ax-17 hbss (ran F) B C ssel rnssopab.2 x A C (V) y fvopab2 mpan2 fopab2.1 (cv x) fveq1i syl5eq rnssopab.2 fopab2.1 fnopab2 F A (cv x) fnfvrn mpan eqeltrrd syl5 r19.21ai impbi)) thm (fopab ((x y) (A x) (A y) (B x) (B y) (C y)) ((fopab2.1 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))))) (fopab.2 (-> (e. (cv x) A) (e. C B)))) (:--> F A B) (fopab.2 rgen fopab2.1 B fopab2 mpbi)) thm (fopabco ((x y) (x z) (A x) (y z) (A y) (A z) (w x) (w y) (w z) (B w) (B x) (B y) (B z) (C w) (C y) (C z) (D w) (D y) (D z) (F w) (F z) (G w) (G z)) ((fopabco.1 (e. C (V))) (fopabco.2 (e. D (V))) (fopabco.3 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))))) (fopabco.4 (= G ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) D)))))) (-> (C_ (ran G) A) (= (o. F G) ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) ([_/]_ D x C)))))) (z visset fopabco.2 fopabco.4 fvopab4s (C_ (ran G) A) adantl fopabco.2 fopabco.4 fnopab2 G B (cv z) fnfvrn mpan (C_ (ran G) A) adantl (ran G) A (` G (cv z)) ssel (e. (cv z) B) adantr mpd eqeltrrd z visset fopabco.2 x ax-gen (cv z) (V) x D (V) csbexg mp2an fopabco.1 z visset (e. (cv w) (cv z)) x ax-17 D hbcsb1 fopabco.3 fvopab4sf syl fopabco.2 fopabco.4 dmopab2 (cv z) eleq2i fopabco.1 fopabco.3 fnopab2 F A fnfun ax-mp fopabco.2 fopabco.4 fnopab2 G B fnfun ax-mp F G (cv z) fvco mp3an12 sylbir z visset fopabco.2 fopabco.4 fvopab4s F fveq2d eqtrd (C_ (ran G) A) adantl z visset fopabco.2 fopabco.1 x ax-gen D (V) x C (V) csbexg mp2an ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) ([_/]_ D x C)))) eqid fvopab4s z visset fopabco.2 x ax-gen (cv z) (V) x D (V) C csbnest1g mp2an syl6eq (C_ (ran G) A) adantl 3eqtr4d ex r19.21aiv B eqid jctil fopabco.1 fopabco.3 fnopab2 fopabco.2 fopabco.4 fnopab2 F A G B fnco mp3an12 fopabco.2 fopabco.1 x ax-gen D (V) x C (V) csbexg mp2an ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) ([_/]_ D x C)))) eqid fnopab2 (e. (cv w) (o. F G)) z ax-17 (e. (cv w) ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) ([_/]_ D x C))))) z ax-17 B B eqfnfvf mpan2 syl mpbird)) thm (ffnfv ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (<-> (:--> F A B) (/\ (Fn F A) (A.e. x A (e. (` F (cv x)) B)))) (F A B ffn F A B (cv x) ffvrn r19.21aiva jca (Fn F A) (A.e. x A (e. (` F (cv x)) B)) pm3.26 F A (cv y) x fvelrn biimpd x A (e. (` F (cv x)) B) hbra1 (e. (cv y) B) x ax-17 x A (e. (` F (cv x)) B) ra4 (` F (cv x)) (cv y) B eleq1 biimpcd syl6 r19.23ad sylan9 r19.21aiv (ran F) B y dfss3 sylibr jca F A B df-f sylibr impbi)) thm (ffnfvf ((y z) (A y) (A z) (B y) (B z) (F y) (F z) (x y) (x z)) ((ffnfvf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (ffnfvf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (ffnfvf.3 (-> (e. (cv y) F) (A. x (e. (cv y) F))))) (<-> (:--> F A B) (/\ (Fn F A) (A.e. x A (e. (` F (cv x)) B)))) (F A B z ffnfv (e. (cv y) A) z ax-17 ffnfvf.1 ffnfvf.3 (e. (cv y) (cv z)) x ax-17 hbfv ffnfvf.2 hbel (e. (` F (cv x)) B) z ax-17 (cv z) (cv x) F fveq2 B eleq1d cbvralf (Fn F A) anbi2i bitr)) thm (fnfvrnss ((A x) (B x) (F x)) () (-> (/\ (Fn F A) (A.e. x A (e. (` F (cv x)) B))) (C_ (ran F) B)) (F A B x ffnfv F A B frn sylbir)) thm (fopabfv ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (<-> (:--> F A B) (/\ (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) (` F (cv x)))))) (A.e. x A (e. (` F (cv x)) B)))) (F A B x ffnfv F A x y fnopabfv (A.e. x A (e. (` F (cv x)) B)) anbi1i bitr)) thm (fsn ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) ((fsn.1 (e. A (V))) (fsn.2 (e. B (V)))) (<-> (:--> F ({} A) ({} B)) (= F ({} (<,> A B)))) (y visset F ({} A) ({} B) (cv x) opelf x A elsn y B elsn anbi12i sylib ex (cv x) A (cv y) B opeq12 F eleq1d fsn.1 snid F ({} A) ({} B) A y feu mpan2 fsn.2 y eueq1 (e. (<,> A B) F) biantru y (e. (<,> A B) F) (= (cv y) B) euanv (cv y) B A opeq2 F eleq1d pm5.32i y B elsn (e. (<,> A (cv y)) F) anbi1i (e. (<,> A B) F) (= (cv y) B) ancom 3bitr4r y eubii y ({} B) (e. (<,> A (cv y)) F) df-reu bitr4 3bitr2 sylibr syl5bir com12 impbid (cv x) (cv y) opex (<,> A B) elsnc x visset y visset fsn.2 A opth bitr2 syl6bb x y 19.21aivv F ({} A) ({} B) frel fsn.1 B relsn jctir F ({} (<,> A B)) x y cleqrel syl mpbird fsn.1 fsn.2 f1osn F ({} (<,> A B)) ({} A) ({} B) f1oeq1 mpbiri F ({} A) ({} B) f1of syl impbi)) thm (fsn2 () ((fsn2.1 (e. A (V)))) (<-> (:--> F ({} A) B) (/\ (e. (` F A) B) (= F ({} (<,> A (` F A)))))) (fsn2.1 snid F ({} A) B A ffvrn mpan2 F ({} A) B ffn F ({} A) fnfrn biimp F ({} A) fndm (dom F) ({} A) F imaeq2 syl F imadmrn syl5eqr fsn2.1 snid F ({} A) A fnsnfv mpan2 eqtr4d (ran F) ({} (` F A)) F ({} A) feq3 syl mpbid syl jca F ({} A) ({} (` F A)) B fss ancoms (` F A) B snssi sylan impbi fsn2.1 F A fvex F fsn (e. (` F A) B) anbi2i bitr)) thm (fnressn ((A x) (B x) (F x)) () (-> (/\ (Fn F A) (e. B A)) (= (|` F ({} B)) ({} (<,> B (` F B))))) ((cv x) B sneq ({} (cv x)) ({} B) F reseq2 syl (cv x) B F fveq2 (cv x) B (` F (cv x)) (` F B) opeq12 mpdan sneqd eqeq12d (Fn F A) imbi2d F A ({} (cv x)) fnssres x visset A snss sylan2b (|` F ({} (cv x))) ({} (cv x)) fnf x visset (|` F ({} (cv x))) (V) fsn2 (|` F ({} (cv x))) (cv x) fvex (= (|` F ({} (cv x))) ({} (<,> (cv x) (` (|` F ({} (cv x))) (cv x))))) biantrur x visset snid (cv x) ({} (cv x)) F fvres ax-mp (` (|` F ({} (cv x))) (cv x)) (` F (cv x)) (cv x) opeq2 ax-mp sneqi (|` F ({} (cv x))) eqeq2i bitr3 3bitr sylib expcom vtoclga impcom)) thm (fressnfv ((A x) (B x) (C x) (F x)) () (-> (/\ (Fn F A) (e. B A)) (<-> (:--> (|` F ({} B)) ({} B) C) (e. (` F B) C))) ((cv x) B sneq ({} (cv x)) ({} B) F reseq2 (|` F ({} (cv x))) (|` F ({} B)) ({} (cv x)) C feq1 syl ({} (cv x)) ({} B) (|` F ({} B)) C feq2 bitrd syl (cv x) B F fveq2 C eleq1d bibi12d (Fn F A) imbi2d F A (cv x) fnressn x visset snid (cv x) ({} (cv x)) F fvres ax-mp (` (|` F ({} (cv x))) (cv x)) (` F (cv x)) (cv x) opeq2 ax-mp sneqi (|` F ({} (cv x))) eqeq2i (= (|` F ({} (cv x))) ({} (<,> (cv x) (` (|` F ({} (cv x))) (cv x))))) (e. (` (|` F ({} (cv x))) (cv x)) C) iba x visset snid (cv x) ({} (cv x)) F fvres ax-mp C eleq1i syl5rbbr x visset (|` F ({} (cv x))) C fsn2 syl5bb sylbir syl expcom vtoclga impcom)) thm (fvconst () () (-> (/\ (:--> F A ({} B)) (e. C A)) (= (` F C) B)) (F A ({} B) C ffvrn (` F C) B elsni syl)) thm (fopabsn ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((fopabsn.1 (e. A (V))) (fopabsn.2 (e. B (V)))) (= ({} (<,> A B)) ({<,>|} x y (/\ (e. (cv x) ({} A)) (= (cv y) B)))) (fopabsn.1 fopabsn.2 f1osn ({} (<,> A B)) ({} A) ({} B) f1of ax-mp ({} (<,> A B)) ({} A) ({} B) ffn ax-mp fopabsn.2 ({<,>|} x y (/\ (e. (cv x) ({} A)) (= (cv y) B))) eqid fnopab2 (e. (cv z) ({} (<,> A B))) x ax-17 z x y (/\ (e. (cv x) ({} A)) (= (cv y) B)) hbopab1 ({} A) ({} A) eqfnfvf mp2an ({} A) eqid x A elsn (cv x) A ({} (<,> A B)) fveq2 fopabsn.1 fopabsn.2 fvsn syl6eq x A elsn fopabsn.2 x ({} A) B (V) y fvopab2 mpan2 sylbir eqtr4d sylbi rgen mpbir2an)) thm (fvi ((A x)) () (-> (e. A B) (= (` (I) A) A)) ((cv x) A (I) fveq2 (= (cv x) A) id eqeq12d (I) (V) df-fn funi dmi mpbir2an x visset (cv x) eqid x visset x visset ideq mpbir (cv x) (I) (cv x) df-br mpbi x visset (I) (V) (cv x) fnopfvb mpbiri mp2an B vtoclg)) thm (fvresi () () (-> (e. B A) (= (` (|` (I) A) B) B)) (B A (I) fvres B A fvi eqtrd)) thm (fconst2 ((A x) (B x) (F x)) ((fconst2.1 (e. B (V)))) (<-> (:--> F A ({} B)) (= F (X. A ({} B)))) (F A B (cv x) fvconst fconst2.1 A fconst (X. A ({} B)) A B (cv x) fvconst mpan (:--> F A ({} B)) adantl eqtr4d r19.21aiva A eqid jctil F A ({} B) ffn fconst2.1 A fconst (X. A ({} B)) A ({} B) ffn ax-mp F A (X. A ({} B)) A x eqfnfv mpan2 syl mpbird fconst2.1 A fconst F (X. A ({} B)) A ({} B) feq1 mpbiri impbi)) thm (fvconst2 () ((fconst2.1 (e. B (V)))) (-> (e. C A) (= (` (X. A ({} B)) C) B)) (fconst2.1 A fconst (X. A ({} B)) A B C fvconst mpan)) thm (fvconst2g ((A x) (B x) (C x)) () (-> (/\ (e. B D) (e. C A)) (= (` (X. A ({} B)) C) B)) ((cv x) B sneq ({} (cv x)) ({} B) A xpeq2 syl C fveq1d (= (cv x) B) id eqeq12d (e. C A) imbi2d x visset C A fvconst2 D vtoclg imp)) thm (fconstfv ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (F x) (F y) (F z)) () (<-> (:--> F A ({} B)) (/\ (Fn F A) (A.e. x A (= (` F (cv x)) B)))) (F A ({} B) ffn F A B (cv x) fvconst r19.21aiva jca A ({/}) F fneq2 F fn0 syl6bb ({} B) f0 F ({/}) ({/}) ({} B) feq1 mpbiri syl6bi A ({/}) F ({} B) feq2 sylibrd (A.e. x A (= (` F (cv x)) B)) adantrd F A (cv y) z fvelrn (cv x) (cv z) F fveq2 B eqeq1d A rcla4cva (cv y) eqeq1d rexbidva A (= B (cv y)) z r19.9rzvOLD bicomd sylan9bbr sylan9bbr y B elsn (cv y) B eqcom bitr2 syl6bb eqrdv an1rs exp31 imdistand F A ({} B) df-fo F A ({} B) fof sylbir syl6 pm2.61i impbi)) thm (funfvima () () (-> (/\ (Fun F) (e. B (dom F))) (-> (e. B A) (e. (` F B) (" F A)))) (B A F fvres (ran (|` F A)) eleq1d F A df-ima (` F B) eleq2i syl6rbbr (|` F A) B fvrn F A funres sylan syl5bir com12 ex F A dmres B eleq2i B A (dom F) elin bitr syl5ibr exp3a com12 imp3a pm2.43b)) thm (funfvima2 () () (-> (/\ (Fun F) (C_ A (dom F))) (-> (e. B A) (e. (` F B) (" F A)))) (F B A funfvima ex com23 a2d A (dom F) B ssel syl5 imp)) thm (funfvima3 ((A x) (F x) (G x)) () (-> (/\ (Fun F) (C_ F G)) (-> (e. A (dom F)) (e. (` F A) (" G ({} A))))) (F G (<,> A (` F A)) ssel F A funfvop syl5 imp (cv x) A sneq ({} (cv x)) ({} A) G imaeq2 syl (` F A) eleq2d (cv x) A (` F A) opeq1 G eleq1d x visset F A fvex G elimasn (dom F) vtoclbg (C_ F G) (Fun F) ad2antll mpbird exp32 impcom)) thm (fvclss ((x y) (F x) (F y)) () (C_ ({|} y (E. x (= (cv y) (` F (cv x))))) (u. (ran F) ({} ({/})))) (x visset (cv y) F tz6.12i (cv y) (` F (cv x)) eqcom syl5ib x 19.22dv y visset F x elrn syl6ibr com12 con1d y ({/}) elsn syl6ibr orrd y ss2abi (ran F) ({} ({/})) y df-un sseqtr4)) thm (fvclex ((x y) (F x) (F y)) ((fvclex.1 (e. F (V)))) (e. ({|} y (E. x (= (cv y) (` F (cv x))))) (V)) (fvclex.1 F (V) rnexg ax-mp p0ex unex y x F fvclss ssexi)) thm (fvresex ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A x) (A y)) ((fvresex.1 (e. A (V)))) (e. ({|} y (E. x (= (cv y) (` (|` F A) (cv x))))) (V)) (x visset F (cv x) fvex (cv z) (cv x) F fveq2 w fvopab ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x) F A fveqres ax-mp (cv y) eqeq2i x exbii y abbii z w (` F (cv z)) funopabeq fvresex.1 ({<,>|} z w (= (cv w) (` F (cv z)))) A (V) resfunexg mp2an y x fvclex eqeltrr)) thm (abrexexlem1 ((x y) (F x) (F y) (A x) (A y)) ((abrexexlem1.1 (e. A (V)))) (e. ({|} y (E.e. x A (= (cv y) (` F (cv x))))) (V)) (abrexexlem1.1 y x F fvresex x A (= (cv y) (` F (cv x))) df-rex (cv x) A F fvres (cv y) eqeq2d biimpar x 19.22i sylbi y ss2abi ssexi)) thm (abrexexlem2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w)) ((abrexexlem2.1 (e. A (V))) (abrexexlem2.2 (e. B (V)))) (e. ({|} y (E.e. x A (= (cv y) B))) (V)) (x visset (= (cv y) B) biantrur x y opabbii (cv x) fveq1i x visset abrexexlem2.2 x (V) B (V) y fvopab2 mp2an eqtr (cv y) eqeq2i x A rexbii (= (cv y) (` ({<,>|} x y (= (cv y) B)) (cv x))) z ax-17 (e. (cv w) (cv y)) x ax-17 w x y (= (cv y) B) hbopab1 (e. (cv w) (cv z)) x ax-17 hbfv hbeq (cv x) (cv z) ({<,>|} x y (= (cv y) B)) fveq2 (cv y) eqeq2d A cbvrex bitr3 y abbii (E.e. z A (= (cv y) (` ({<,>|} x y (= (cv y) B)) (cv z)))) w ax-17 (e. (cv z) A) y ax-17 (e. (cv x) (cv w)) y ax-17 w x y (= (cv y) B) hbopab2 (e. (cv w) (cv z)) y ax-17 hbfv hbeq hbrex (cv y) (cv w) (` ({<,>|} x y (= (cv y) B)) (cv z)) eqeq1 z A rexbidv cbvab eqtr abrexexlem2.1 w z ({<,>|} x y (= (cv y) B)) abrexexlem1 eqeltr)) thm (abrexex ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) ((abrexex.1 (e. A (V)))) (e. ({|} y (E.e. x A (= (cv y) B))) (V)) (abrexex.1 z B class2set y x abrexexlem2 y visset (cv y) B (V) eleq1 mpbii (e. B (V)) (e. (cv z) B) ax-1 r19.21aiv B z (e. B (V)) rabid2 sylibr (cv y) eqeq2d syl ibi x A r19.22si y ss2abi ssexi)) thm (abrexexg ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) () (-> (e. A C) (e. ({|} y (E.e. x A (= (cv y) B))) (V))) ((cv z) A x (= (cv y) B) rexeq1 y abbidv (V) eleq1d z visset y x B abrexex C vtoclg)) thm (iunexg ((x y) (A x) (A y) (B y)) () (-> (/\ (e. A C) (A.e. x A (e. B D))) (e. (U_ x A B) (V))) (x A B D y dfiun2g (e. A C) adantl A C y x B abrexexg ({|} y (E.e. x A (= (cv y) B))) (V) uniexg syl (A.e. x A (e. B D)) adantr eqeltrd)) thm (iunex ((A x)) ((iunex.1 (e. A (V))) (iunex.2 (e. B (V)))) (e. (U_ x A B) (V)) (iunex.1 iunex.2 (e. (cv x) A) a1i rgen A (V) x B (V) iunexg mp2an)) thm (imaiun ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (= (" A (U. B)) (U_ x B (" A (cv x)))) ((cv z) B x eluni (e. (<,> (cv z) (cv y)) A) anbi1i z exbii x (/\ (e. (cv z) (cv x)) (e. (cv x) B)) (e. (<,> (cv z) (cv y)) A) 19.41v (e. (cv z) (cv x)) (e. (cv x) B) (e. (<,> (cv z) (cv y)) A) anass (e. (cv z) (cv x)) (e. (cv x) B) (e. (<,> (cv z) (cv y)) A) an12 bitr x exbii bitr3 z exbii z x (/\ (e. (cv x) B) (/\ (e. (cv z) (cv x)) (e. (<,> (cv z) (cv y)) A))) excom x z (e. (cv x) B) (/\ (e. (cv z) (cv x)) (e. (<,> (cv z) (cv y)) A)) exdistr 3bitr y visset A (cv x) z elima3 x B rexbii x B (E. z (/\ (e. (cv z) (cv x)) (e. (<,> (cv z) (cv y)) A))) df-rex bitr2 3bitr y visset A (U. B) z elima3 (cv y) x B (" A (cv x)) eliun 3bitr4 eqriv)) thm (fniunfv ((x y) (A x) (A y) (F x) (F y)) () (-> (Fn F A) (= (U_ x A (` F (cv x))) (U. (ran F)))) (F A y x fnrnfv unieqd F (cv x) fvex x A y dfiun2 syl6reqr)) thm (funiunfv ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (F w) (F x) (F y) (F z)) () (-> (Fun F) (= (U_ x A (` F (cv x))) (U. (" F A)))) (F (cv y) fvex ({<,>|} y z (/\ (e. (cv y) A) (= (cv z) (` F (cv y))))) eqid fnopab2 ({<,>|} y z (/\ (e. (cv y) A) (= (cv z) (` F (cv y))))) A x fniunfv ax-mp (Fun F) a1i (cv y) (cv x) F fveq2 ({<,>|} y z (/\ (e. (cv y) A) (= (cv z) (` F (cv y))))) eqid F (cv x) fvex fvopab4 iuneq2i syl5eqr z visset F (cv y) funbrfvb biimpd (` F (cv y)) (cv z) ({/}) eqeq1 (cv y) F ndmfv syl5bi con1d impcom (cv w) (cv z) n0i sylan sylan2 anassrs ex pm2.43d z visset F (cv y) funbrfv (e. (cv w) (cv z)) adantr impbid (cv z) (` F (cv y)) eqcom syl5bb y A rexbidv ex pm5.32d z exbidv (cv w) (" F A) z eluni z visset F A y elima (e. (cv w) (cv z)) anbi2i z exbii bitr2 syl6bb (cv w) z (E.e. y A (= (cv z) (` F (cv y)))) eluniab syl5bb eqrdv y z A (` F (cv y)) rnopab2 unieqi syl5eq eqtrd)) thm (eluniima ((A x) (B x) (F x)) () (-> (Fun F) (<-> (e. B (U. (" F A))) (E.e. x A (e. B (` F (cv x)))))) (F x A funiunfv B eleq2d B x A (` F (cv x)) eliun syl5rbbr)) thm (elunirn ((x y) (A x) (A y) (F x) (F y)) () (-> (Fun F) (<-> (e. A (U. (ran F))) (E.e. x (dom F) (e. A (` F (cv x)))))) (F funfn F (dom F) (cv y) x fvelrn sylbi (e. A (cv y)) anbi2d x (dom F) (e. A (cv y)) (= (` F (cv x)) (cv y)) r19.42v syl6bbr (` F (cv x)) (cv y) A eleq2 biimparc x (dom F) r19.22si syl6bi y 19.23adv F (cv x) fvrn (e. A (` F (cv x))) a1d ancld F (cv x) fvex (cv y) (` F (cv x)) A eleq2 (cv y) (` F (cv x)) (ran F) eleq1 anbi12d cla4ev syl6 ex r19.23adv impbid A (ran F) y eluni syl5bb)) thm (funiunfvf ((x y) (x z) (A x) (y z) (A y) (A z) (F y) (F z)) ((funiunfvf.1 (-> (e. (cv y) F) (A. x (e. (cv y) F))))) (-> (Fun F) (= (U_ x A (` F (cv x))) (U. (" F A)))) (F z A funiunfv funiunfvf.1 (e. (cv y) (cv z)) x ax-17 hbfv (e. (cv y) (` F (cv x))) z ax-17 (cv z) (cv x) F fveq2 A cbviun syl5eqr)) thm (abrexex2 ((x y) (x z) (A x) (y z) (A y) (A z) (ph z)) ((abrexex2.1 (e. A (V))) (abrexex2.2 (e. ({|} y ph) (V)))) (e. ({|} y (E.e. x A ph)) (V)) ((E.e. x A ph) z ax-17 (e. (cv x) A) y ax-17 z y ph hbs1 hbrex y z ph sbequ12 x A rexbidv cbvab z y ph df-clab x A rexbii z abbii eqtr4 x A ({|} y ph) z df-iun abrexex2.1 abrexex2.2 x iunex eqeltrr eqeltr)) thm (abexssex ((x y) (A x) (A y)) ((abrexex2.1 (e. A (V))) (abrexex2.2 (e. ({|} y ph) (V)))) (e. ({|} y (E. x (/\ (C_ (cv x) A) ph))) (V)) (x (P~ A) ph df-rex x visset A elpw ph anbi1i x exbii bitr y abbii abrexex2.1 pwex abrexex2.2 x abrexex2 eqeltrr)) thm (f1fv ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (F x) (F y) (F z)) () (<-> (:-1-1-> F A B) (/\ (:--> F A B) (A.e. x A (A.e. y A (-> (= (` F (cv x)) (` F (cv y))) (= (cv x) (cv y))))))) (F A B z x f11 F A B ffn F A fndm (cv x) eleq2d x visset F (cv z) breldm syl5bi F A fndm (cv y) eleq2d y visset F (cv z) breldm syl5bi anim12d pm4.71rd z visset F A (cv x) fnbrfvb (cv z) (` F (cv x)) eqcom syl5bb z visset F A (cv y) fnbrfvb (cv z) (` F (cv y)) eqcom syl5bb bi2anan9 anandis ex pm5.32d bitr4d (= (cv x) (cv y)) imbi1d (/\ (e. (cv x) A) (e. (cv y) A)) (/\ (= (cv z) (` F (cv x))) (= (cv z) (` F (cv y)))) (= (cv x) (cv y)) impexp syl6bb z albidv z (/\ (e. (cv x) A) (e. (cv y) A)) (-> (/\ (= (cv z) (` F (cv x))) (= (cv z) (` F (cv y)))) (= (cv x) (cv y))) 19.21v z (/\ (= (cv z) (` F (cv x))) (= (cv z) (` F (cv y)))) (= (cv x) (cv y)) 19.23v F (cv x) fvex (` F (cv y)) z eqvinc (= (cv x) (cv y)) imbi1i bitr4 (/\ (e. (cv x) A) (e. (cv y) A)) imbi2i bitr syl6bb x y 2albidv (cv x) (cv y) F (cv z) breq1 mo4 z albii z x (A. y (-> (/\ (br (cv x) F (cv z)) (br (cv y) F (cv z))) (= (cv x) (cv y)))) alcom z y (-> (/\ (br (cv x) F (cv z)) (br (cv y) F (cv z))) (= (cv x) (cv y))) alcom x albii 3bitr x A y A (-> (= (` F (cv x)) (` F (cv y))) (= (cv x) (cv y))) r2al 3bitr4g syl pm5.32i bitr)) thm (f1fvf ((x y) (w x) (v x) (A x) (w y) (v y) (A y) (v w) (A w) (A v) (B x) (B y) (B w) (B v) (w z) (v z) (F z) (F w) (F v) (x z) (y z)) ((f1fvf.1 (-> (e. (cv z) F) (A. x (e. (cv z) F)))) (f1fvf.2 (-> (e. (cv z) F) (A. y (e. (cv z) F))))) (<-> (:-1-1-> F A B) (/\ (:--> F A B) (A.e. x A (A.e. y A (-> (= (` F (cv x)) (` F (cv y))) (= (cv x) (cv y))))))) (F A B w v f1fv f1fvf.2 (e. (cv z) (cv w)) y ax-17 hbfv f1fvf.2 (e. (cv z) (cv v)) y ax-17 hbfv hbeq (= (cv w) (cv v)) y ax-17 hbim (= (` F (cv w)) (` F (cv y))) v ax-17 (= (cv w) (cv y)) v ax-17 hbim (cv v) (cv y) F fveq2 (` F (cv w)) eqeq2d (cv v) (cv y) (cv w) eqeq2 imbi12d A cbvral w A ralbii (e. (cv y) A) x ax-17 f1fvf.1 (e. (cv z) (cv w)) x ax-17 hbfv f1fvf.1 (e. (cv z) (cv y)) x ax-17 hbfv hbeq (= (cv w) (cv y)) x ax-17 hbim hbral (A.e. y A (-> (= (` F (cv x)) (` F (cv y))) (= (cv x) (cv y)))) w ax-17 (cv w) (cv x) F fveq2 (` F (cv y)) eqeq1d (cv w) (cv x) (cv y) eqeq1 imbi12d y A ralbidv A cbvral bitr (:--> F A B) anbi2i bitr)) thm (f1fveq ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y) (F x) (F y)) () (-> (/\ (:-1-1-> F A B) (/\ (e. C A) (e. D A))) (<-> (= (` F C) (` F D)) (= C D))) ((cv x) C F fveq2 (` F (cv y)) eqeq1d (cv x) C (cv y) eqeq1 imbi12d (:-1-1-> F A B) imbi2d (cv y) D F fveq2 (` F C) eqeq2d (cv y) D C eqeq2 imbi12d (:-1-1-> F A B) imbi2d F A B x y f1fv pm3.27bd x A y A (-> (= (` F (cv x)) (` F (cv y))) (= (cv x) (cv y))) ra42 syl com12 vtocl2ga impcom C D F fveq2 (/\ (:-1-1-> F A B) (/\ (e. C A) (e. D A))) a1i impbid)) thm (f1ocnvfv1 () () (-> (/\ (:-1-1-onto-> F A B) (e. C A)) (= (` (`' F) (` F C)) C)) (F A B f1ococnv1 C fveq1d (e. C A) adantr (`' F) F A B C fvco3 3expa F A B f1ocnv (`' F) B A f1ofun syl F A B f1of jca sylan C A fvresi (:-1-1-onto-> F A B) adantl 3eqtr3d)) thm (f1ocnvfv2 () () (-> (/\ (:-1-1-onto-> F A B) (e. C B)) (= (` F (` (`' F) C)) C)) (F A B f1ococnv2 C fveq1d (e. C B) adantr F (`' F) B A C fvco3 3expa F A B f1ofun F A B f1ocnv (`' F) B A f1of syl jca sylan C B fvresi (:-1-1-onto-> F A B) adantl 3eqtr3d)) thm (f1ocnvfv () () (-> (/\ (:-1-1-onto-> F A B) (e. C A)) (-> (= (` F C) D) (= (` (`' F) D) C))) (F A B C f1ocnvfv1 (` (`' F) D) eqeq2d D (` F C) (`' F) fveq2 eqcoms syl5bi)) thm (f1ocnvfvb () () (-> (/\/\ (:-1-1-onto-> F A B) (e. C A) (e. D B)) (<-> (= (` F C) D) (= (` (`' F) D) C))) (F A B C D f1ocnvfv (e. D B) 3adant3 F A B D f1ocnvfv2 (` F C) eqeq2d C (` (`' F) D) F fveq2 eqcoms syl5bi (e. C A) 3adant2 impbid)) thm (f1ofveu ((A x) (B x) (C x) (F x)) () (-> (/\ (:-1-1-onto-> F A B) (e. C B)) (E!e. x A (= (` F (cv x)) C))) ((`' F) B A C x feu F A B f1ocnv (`' F) B A f1of syl sylan F A B (cv x) C f1ocnvfvb 3com23 x visset (`' F) B C fnopfvb (e. (cv x) A) 3adant3 F A B f1o4 pm3.27bd syl3an1 bitrd 3expa reubidva mpbird)) thm (f1ocnvdm () () (-> (/\ (:-1-1-onto-> F A B) (e. C B)) (e. (` (`' F) C) A)) ((`' F) B A C ffvrn F A B f1ocnv (`' F) B A f1of syl sylan)) thm (cbvfo ((x y) (A x) (A y) (B x) (B y) (F x) (F y) (ph y) (ps x)) ((cbvfo.1 (-> (= (` F (cv x)) (cv y)) (<-> ph ps)))) (-> (:-onto-> F A B) (<-> (A.e. x A ph) (A.e. y B ps))) (F A B fof F A B ffun x visset F (cv y) breldm (Fun F) a1i y visset F (cv x) funbrfv jcad x 19.22dv y visset F x elrn syl5ib x (-> (e. (cv x) (dom F)) ph) hba1 ps x ax-17 cbvfo.1 biimpcd (e. (cv x) (dom F)) imim2i imp3a x a4s 19.23ad syl9 y 19.21adv x visset y visset F brelrn (Fun F) a1i y visset F (cv x) funbrfv jcad y 19.22dv x visset F y eldm syl5ib y (-> (e. (cv y) (ran F)) ps) hba1 ph y ax-17 cbvfo.1 biimprcd (e. (cv y) (ran F)) imim2i imp3a y a4s 19.23ad syl9 x 19.21adv impbid x (dom F) ph df-ral y (ran F) ps df-ral 3bitr4g 3syl F A B fof F A B fdm (dom F) A x ph raleq1 3syl F A B forn y ps raleq1d 3bitr3d)) thm (cbvexfo ((x y) (A x) (A y) (B x) (B y) (F x) (F y) (ph y) (ps x)) ((cbvfo.1 (-> (= (` F (cv x)) (cv y)) (<-> ph ps)))) (-> (:-onto-> F A B) (<-> (E.e. x A ph) (E.e. y B ps))) (cbvfo.1 negbid A B cbvfo negbid x A ph dfrex2 y B ps dfrex2 3bitr4g)) thm (isoeq1 ((x y) (A x) (A y) (B x) (B y) (H x) (H y) (G x) (G y) (R x) (R y) (S x) (S y)) () (-> (= H G) (<-> (Isom H R S A B) (Isom G R S A B))) (H G A B f1oeq1 H G (cv x) fveq1 H G (cv y) fveq1 S breq12d (br (cv x) R (cv y)) bibi2d x A y A 2ralbidv anbi12d H R S A B x y df-iso G R S A B x y df-iso 3bitr4g)) thm (isoeq2 ((x y) (A x) (A y) (B x) (B y) (H x) (H y) (R x) (R y) (S x) (S y) (T x) (T y)) () (-> (= R T) (<-> (Isom H R S A B) (Isom H T S A B))) (R T (cv x) (cv y) breq (br (` H (cv x)) S (` H (cv y))) bibi1d x A y A 2ralbidv (:-1-1-onto-> H A B) anbi2d H R S A B x y df-iso H T S A B x y df-iso 3bitr4g)) thm (isoeq3 ((x y) (A x) (A y) (B x) (B y) (H x) (H y) (R x) (R y) (S x) (S y) (T x) (T y)) () (-> (= S T) (<-> (Isom H R S A B) (Isom H R T A B))) (S T (` H (cv x)) (` H (cv y)) breq (br (cv x) R (cv y)) bibi2d x A y A 2ralbidv (:-1-1-onto-> H A B) anbi2d H R S A B x y df-iso H R T A B x y df-iso 3bitr4g)) thm (isoeq4 ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (H x) (H y) (R x) (R y) (S x) (S y)) () (-> (= A C) (<-> (Isom H R S A B) (Isom H R S C B))) (A C H B f1oeq2 A C y (<-> (br (cv x) R (cv y)) (br (` H (cv x)) S (` H (cv y)))) raleq1 x raleqd anbi12d H R S A B x y df-iso H R S C B x y df-iso 3bitr4g)) thm (isoeq5 ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (H x) (H y) (R x) (R y) (S x) (S y)) () (-> (= B C) (<-> (Isom H R S A B) (Isom H R S A C))) (B C H A f1oeq3 (A.e. x A (A.e. y A (<-> (br (cv x) R (cv y)) (br (` H (cv x)) S (` H (cv y)))))) anbi1d H R S A B x y df-iso H R S A C x y df-iso 3bitr4g)) thm (hbiso ((y z) (w y) (H y) (w z) (H z) (H w) (R y) (R z) (R w) (S y) (S z) (S w) (A y) (A z) (A w) (B y) (B z) (B w) (x y) (x z) (w x)) ((hbiso.1 (-> (e. (cv y) H) (A. x (e. (cv y) H)))) (hbiso.2 (-> (e. (cv y) R) (A. x (e. (cv y) R)))) (hbiso.3 (-> (e. (cv y) S) (A. x (e. (cv y) S)))) (hbiso.4 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbiso.5 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (Isom H R S A B) (A. x (Isom H R S A B))) (hbiso.1 hbiso.4 hbiso.5 hbf1o (e. (cv y) (cv z)) x ax-17 hbiso.4 hbel (e. (cv y) (cv w)) x ax-17 hbiso.4 hbel (e. (cv y) (cv z)) x ax-17 hbiso.2 (e. (cv y) (cv w)) x ax-17 hbbr hbiso.1 (e. (cv y) (cv z)) x ax-17 hbfv hbiso.3 hbiso.1 (e. (cv y) (cv w)) x ax-17 hbfv hbbr hbbi hbral hbral hban H R S A B z w df-iso H R S A B z w df-iso x albii 3imtr4)) thm (isof1o ((x y) (A x) (A y) (B x) (B y) (R x) (R y) (S x) (S y) (H x) (H y)) () (-> (Isom H R S A B) (:-1-1-onto-> H A B)) (H R S A B x y df-iso pm3.26bd)) thm (isorel ((x y) (A x) (A y) (B x) (B y) (R x) (R y) (S x) (S y) (H x) (H y) (C x) (C y) (D x) (D y)) () (-> (/\ (Isom H R S A B) (/\ (e. C A) (e. D A))) (<-> (br C R D) (br (` H C) S (` H D)))) ((cv x) C R (cv y) breq1 (cv x) C H fveq2 S (` H (cv y)) breq1d bibi12d (cv y) D C R breq2 (cv y) D H fveq2 (` H C) S breq2d bibi12d A A rcla42v H R S A B x y df-iso pm3.27bd syl5com imp)) thm (isoid ((x y) (A x) (A y) (R x) (R y)) () (Isom (|` (I) A) R R A A) ((|` (I) A) R R A A x y df-iso A f1oi (cv x) A fvresi (cv y) A fvresi R breqan12d bicomd rgen2 mpbir2an)) thm (isocnv ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (R x) (R y) (R z) (R w) (S x) (S y) (S z) (S w) (H x) (H y) (H z) (H w)) () (-> (Isom H R S A B) (Isom (`' H) S R B A)) (H A B f1ocnv (A.e. x A (A.e. y A (<-> (br (cv x) R (cv y)) (br (` H (cv x)) S (` H (cv y)))))) adantr H A B (cv z) f1ocnvfv2 (e. (cv w) B) adantrr H A B (cv w) f1ocnvfv2 (e. (cv z) B) adantrl S breq12d (A.e. x A (A.e. y A (<-> (br (cv x) R (cv y)) (br (` H (cv x)) S (` H (cv y)))))) adantlr (cv x) (` (`' H) (cv z)) R (cv y) breq1 (cv x) (` (`' H) (cv z)) H fveq2 S (` H (cv y)) breq1d bibi12d (br (` (`' H) (cv z)) R (cv y)) (br (` H (` (`' H) (cv z))) S (` H (cv y))) bicom syl6bb (cv y) (` (`' H) (cv w)) H fveq2 (` H (` (`' H) (cv z))) S breq2d (cv y) (` (`' H) (cv w)) (` (`' H) (cv z)) R breq2 bibi12d A A rcla42v imp (`' H) B A (cv z) ffvrn (`' H) B A (cv w) ffvrn anim12i anandis sylan an1rs H A B f1ocnv (`' H) B A f1of syl sylanl1 bitr3d exp32 r19.21adv r19.21aiv jca H R S A B x y df-iso (`' H) S R B A z w df-iso 3imtr4)) thm (isotr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (R x) (R y) (R z) (R w) (R v) (R u) (S x) (S y) (S z) (S w) (S v) (S u) (T x) (T y) (T z) (T w) (T v) (T u) (G x) (G y) (G z) (G w) (G v) (G u) (H x) (H y) (H z) (H w) (H v) (H u)) () (-> (/\ (Isom H R S A B) (Isom G S T B C)) (Isom (o. G H) R T A C)) ((:-1-1-onto-> G B C) (A.e. v B (A.e. u B (<-> (br (cv v) S (cv u)) (br (` G (cv v)) T (` G (cv u)))))) pm3.26 (:-1-1-onto-> H A B) (A.e. z A (A.e. w A (<-> (br (cv z) R (cv w)) (br (` H (cv z)) S (` H (cv w)))))) pm3.26 anim12i ancoms G B C H A f1oco syl H A B f1of H A B (cv x) ffvrn ex H A B (cv y) ffvrn ex anim12d syl (A.e. z A (A.e. w A (<-> (br (cv z) R (cv w)) (br (` H (cv z)) S (` H (cv w)))))) adantr (cv v) (` H (cv x)) S (cv u) breq1 (cv v) (` H (cv x)) G fveq2 T (` G (cv u)) breq1d bibi12d (cv u) (` H (cv y)) (` H (cv x)) S breq2 (cv u) (` H (cv y)) G fveq2 (` G (` H (cv x))) T breq2d bibi12d B B rcla42v com12 (:-1-1-onto-> G B C) adantl sylan9 imp (cv z) (cv x) R (cv w) breq1 (cv z) (cv x) H fveq2 S (` H (cv w)) breq1d bibi12d (cv w) (cv y) (cv x) R breq2 (cv w) (cv y) H fveq2 (` H (cv x)) S breq2d bibi12d A A rcla42v impcom (:-1-1-onto-> H A B) adantll (/\ (:-1-1-onto-> G B C) (A.e. v B (A.e. u B (<-> (br (cv v) S (cv u)) (br (` G (cv v)) T (` G (cv u))))))) adantlr G H A B (cv x) fvco3 3expa G H A B (cv y) fvco3 3expa T breqan12d anandis G B C f1ofun H A B f1of anim12i sylan (:-1-1-onto-> G B C) (A.e. v B (A.e. u B (<-> (br (cv v) S (cv u)) (br (` G (cv v)) T (` G (cv u)))))) pm3.26 (:-1-1-onto-> H A B) (A.e. z A (A.e. w A (<-> (br (cv z) R (cv w)) (br (` H (cv z)) S (` H (cv w)))))) pm3.26 anim12i ancoms sylan 3bitr4d exp32 r19.21adv r19.21aiv jca H R S A B z w df-iso G S T B C v u df-iso anbi12i (o. G H) R T A C x y df-iso 3imtr4)) thm (isotrALT ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (R x) (R y) (R z) (R w) (S x) (S y) (S z) (S w) (T x) (T y) (T z) (T w) (G x) (G y) (G z) (G w) (H x) (H y) (H z) (H w)) () (-> (/\ (Isom H R S A B) (Isom G S T B C)) (Isom (o. G H) R T A C)) ((:-1-1-onto-> G B C) (A.e. z B (A.e. w B (<-> (br (cv z) S (cv w)) (br (` G (cv z)) T (` G (cv w)))))) pm3.26 (:-1-1-onto-> H A B) (A.e. x A (A.e. y A (<-> (br (cv x) R (cv y)) (br (` H (cv x)) S (` H (cv y)))))) pm3.26 anim12i ancoms G B C H A f1oco syl (:-1-1-onto-> H A B) x ax-17 x A (A.e. y A (<-> (br (cv x) R (cv y)) (br (` H (cv x)) S (` H (cv y))))) hbra1 hban (/\ (:-1-1-onto-> G B C) (A.e. z B (A.e. w B (<-> (br (cv z) S (cv w)) (br (` G (cv z)) T (` G (cv w))))))) x ax-17 hban (:-1-1-onto-> H A B) y ax-17 (e. (cv x) A) y ax-17 y A (<-> (br (cv x) R (cv y)) (br (` H (cv x)) S (` H (cv y)))) hbra1 hbral hban (/\ (:-1-1-onto-> G B C) (A.e. z B (A.e. w B (<-> (br (cv z) S (cv w)) (br (` G (cv z)) T (` G (cv w))))))) y ax-17 hban (e. (cv x) A) y ax-17 H A B f1of H A B (cv x) ffvrn ex H A B (cv y) ffvrn ex anim12d syl (A.e. x A (A.e. y A (<-> (br (cv x) R (cv y)) (br (` H (cv x)) S (` H (cv y)))))) adantr (cv z) (` H (cv x)) S (cv w) breq1 (cv z) (` H (cv x)) G fveq2 T (` G (cv w)) breq1d bibi12d (cv w) (` H (cv y)) (` H (cv x)) S breq2 (cv w) (` H (cv y)) G fveq2 (` G (` H (cv x))) T breq2d bibi12d B B rcla42v com12 (:-1-1-onto-> G B C) adantl sylan9 imp x A y A (<-> (br (cv x) R (cv y)) (br (` H (cv x)) S (` H (cv y)))) ra42 imp (:-1-1-onto-> H A B) adantll (/\ (:-1-1-onto-> G B C) (A.e. z B (A.e. w B (<-> (br (cv z) S (cv w)) (br (` G (cv z)) T (` G (cv w))))))) adantlr G H A B (cv x) fvco3 3expa G H A B (cv y) fvco3 3expa T breqan12d anandis G B C f1ofun H A B f1of anim12i sylan (:-1-1-onto-> G B C) (A.e. z B (A.e. w B (<-> (br (cv z) S (cv w)) (br (` G (cv z)) T (` G (cv w)))))) pm3.26 (:-1-1-onto-> H A B) (A.e. x A (A.e. y A (<-> (br (cv x) R (cv y)) (br (` H (cv x)) S (` H (cv y)))))) pm3.26 anim12i ancoms sylan 3bitr4d exp32 r19.21ad r19.21ai jca H R S A B x y df-iso G S T B C z w df-iso anbi12i (o. G H) R T A C x y df-iso 3imtr4)) thm (isomin ((x y) (A x) (A y) (B x) (B y) (R x) (R y) (S x) (S y) (H x) (H y) (C x) (C y) (D x) (D y)) () (-> (/\ (Isom H R S A B) (/\ (C_ C A) (e. D A))) (<-> (= (i^i C (" (`' R) ({} D))) ({/})) (= (i^i (" H C) (" (`' S) ({} (` H D)))) ({/})))) (C A (cv x) ssel H R S A B isof1o H A B f1ofn y visset H A (cv x) fnbrfvb ex 3syl syl9r imp31 rexbidva y visset H C x elima syl6rbbr H D fvex y visset (` H D) (V) S eliniseg ax-mp (/\ (Isom H R S A B) (C_ C A)) a1i anbi12d (cv y) (" H C) (" (`' S) ({} (` H D))) elin x C (= (` H (cv x)) (cv y)) (br (cv y) S (` H D)) r19.41v 3bitr4g (e. D A) adantrr C A (cv x) ssel x visset D A R eliniseg (Isom H R S A B) (e. (cv x) A) ad2antll H R S A B (cv x) D isorel bitrd (` H (cv x)) (cv y) S (` H D) breq1 biimpar syl5bir exp32 syl9r com34 imp32 r19.22dv sylbid (cv x) C (" (`' R) ({} D)) elin x exbii (i^i C (" (`' R) ({} D))) x n0 x C (e. (cv x) (" (`' R) ({} D))) df-rex 3bitr4 syl6ibr y 19.23adv (i^i (" H C) (" (`' S) ({} (` H D)))) y n0 syl5ib a3d C A (cv x) ssel com12 H (cv x) C funfvima A funfni ex com13 syld com13 imp (e. D A) adantrr H R S A B isof1o H A B f1ofn syl sylan (e. (cv x) (" (`' R) ({} D))) adantrd C A (cv x) ssel x visset D A R eliniseg (Isom H R S A B) (e. (cv x) A) ad2antll H R S A B (cv x) D isorel biimpd H D fvex H (cv x) fvex (` H D) (V) S eliniseg ax-mp syl6ibr sylbid exp32 syl9r com34 imp32 imp3a jcad (cv x) C (" (`' R) ({} D)) elin (` H (cv x)) (" H C) (" (`' S) ({} (` H D))) elin 3imtr4g (` H (cv x)) (i^i (" H C) (" (`' S) ({} (` H D)))) n0i syl6 x 19.23adv (i^i C (" (`' R) ({} D))) x n0 syl5ib a3d impbid)) thm (isoini ((x y) (A x) (A y) (B x) (B y) (R x) (R y) (S x) (S y) (H x) (H y) (D x) (D y)) () (-> (/\ (Isom H R S A B) (e. D A)) (= (" H (i^i A (" (`' R) ({} D)))) (i^i B (" (`' S) ({} (` H D)))))) (H R S A B isof1o H A B f1ofo H A B forn (cv y) eleq2d 3syl H R S A B isof1o H A B f1ofn H A (cv y) x fvelrn 3syl bitr3d H D fvex y visset (` H D) (V) S eliniseg ax-mp (Isom H R S A B) a1i anbi12d (e. D A) adantr x visset D A R eliniseg (e. (cv x) A) anbi2d (cv x) A (" (`' R) ({} D)) elin syl5bb (br (cv x) H (cv y)) anbi1d (e. (cv x) A) (br (cv x) R D) (br (cv x) H (cv y)) anass syl6bb (Isom H R S A B) adantl H R S A B (cv x) D isorel y visset H A (cv x) fnbrfvb bicomd H R S A B isof1o H A B f1ofn syl sylan (e. D A) adantrr anbi12d (br (` H (cv x)) S (` H D)) (= (` H (cv x)) (cv y)) ancom (` H (cv x)) (cv y) S (` H D) breq1 pm5.32i bitr syl6bb exp32 com23 imp pm5.32d bitrd rexbidv2 x A (= (` H (cv x)) (cv y)) (br (cv y) S (` H D)) r19.41v syl6bb bitr4d (cv y) B (" (`' S) ({} (` H D))) elin syl5bb abbi2dv H (i^i A (" (`' R) ({} D))) y x dfima2 syl6reqr)) thm (isofrlem ((x y) (w x) (A x) (w y) (A y) (A w) (x z) (B x) (y z) (B y) (w z) (B z) (B w) (R x) (R y) (R w) (S x) (S y) (S z) (S w) (H x) (H y) (H z) (H w)) () (-> (Isom H R S A B) (-> (Fr S B) (Fr R A))) (H R S A B isof1o H A B f1ofun x visset H funimaex (cv z) (" H (cv x)) B sseq1 (cv z) (" H (cv x)) ({/}) eqeq1 negbid anbi12d (cv z) (" H (cv x)) (" (`' S) ({} (cv w))) ineq1 ({/}) eqeq1d w rexeqd imbi12d (V) cla4gv syl 3syl S B z w dffr3 syl5ib H R S A B isof1o H A B f1ofn (cv x) A (cv y) ssel H (cv y) (cv x) funfvima A funfni (` H (cv y)) (" H (cv x)) n0i syl6 ex sylan9r pm2.43d y 19.23adv (cv x) y n0 syl5ib ex imp3a syl H A B f1ofo H (cv x) imassrn H A B forn (" H (cv x)) sseq2d mpbii syl jctild syl syl5d H (cv w) (cv x) y fvelima H R S A B isof1o (C_ (cv x) A) adantr H A B f1ofun syl (e. (cv w) (" H (cv x))) (= (i^i (" H (cv x)) (" (`' S) ({} (cv w)))) ({/})) pm3.26 syl2an H R S A B (cv x) (cv y) isomin (cv x) A (cv y) ssel imdistani sylan2 (` H (cv y)) (cv w) sneq ({} (` H (cv y))) ({} (cv w)) (`' S) imaeq2 syl (" H (cv x)) ineq2d ({/}) eqeq1d sylan9bb (e. (cv w) (" H (cv x))) (= (i^i (" H (cv x)) (" (`' S) ({} (cv w)))) ({/})) pm3.27 syl5bir exp42 imp com3l com4t imp r19.22dv mpd exp32 r19.23adv ex (-. (= (cv x) ({/}))) adantrd a2d syld x 19.21adv R A x y dffr3 syl6ibr)) thm (isofr () () (-> (Isom H R S A B) (<-> (Fr R A) (Fr S B))) (H R S A B isocnv (`' H) S R B A isofrlem syl H R S A B isofrlem impbid)) thm (isowe ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (R x) (R y) (S x) (S y) (S z) (S w) (H x) (H y) (H z) (H w)) () (-> (Isom H R S A B) (<-> (We R A) (We S B))) (H R S A B isofr H R S A B (cv x) (cv y) isorel H A B (cv x) (cv y) f1fveq H R S A B isof1o H A B f1of1 syl sylan bicomd H R S A B (cv y) (cv x) isorel ancom2s 3orbi123d 2ralbidva H R S A B isof1o H A B f1ofo (` H (cv y)) (cv w) (` H (cv x)) S breq2 (` H (cv y)) (cv w) (` H (cv x)) eqeq2 (` H (cv y)) (cv w) S (` H (cv x)) breq1 3orbi123d A B cbvfo x A ralbidv (` H (cv x)) (cv z) S (cv w) breq1 (` H (cv x)) (cv z) (cv w) eqeq1 (` H (cv x)) (cv z) (cv w) S breq2 3orbi123d w B ralbidv A B cbvfo bitrd 3syl bitrd anbi12d R A x y dfwe2 S B z w dfwe2 3bitr4g)) thm (f1oiso ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B v) (B u) (H x) (H y) (H z) (H w) (H v) (H u) (R x) (R y) (R z) (R w) (R v) (R u) (S v) (S u)) () (-> (/\ (:-1-1-onto-> H A B) (= S ({<,>|} z w (E.e. x A (E.e. y A (/\ (/\ (= (cv z) (` H (cv x))) (= (cv w) (` H (cv y)))) (br (cv x) R (cv y)))))))) (Isom H R S A B)) ((:-1-1-onto-> H A B) (= S ({<,>|} z w (E.e. x A (E.e. y A (/\ (/\ (= (cv z) (` H (cv x))) (= (cv w) (` H (cv y)))) (br (cv x) R (cv y))))))) pm3.26 S ({<,>|} z w (E.e. x A (E.e. y A (/\ (/\ (= (cv z) (` H (cv x))) (= (cv w) (` H (cv y)))) (br (cv x) R (cv y)))))) (<,> (` H (cv v)) (` H (cv u))) eleq2 H A B (cv v) (cv x) f1fveq (cv v) (cv x) eqcom syl6bb anassrs (/\ (= (` H (cv u)) (` H (cv y))) (br (cv x) R (cv y))) anbi1d (= (` H (cv v)) (` H (cv x))) (= (` H (cv u)) (` H (cv y))) (br (cv x) R (cv y)) anass syl5bb y A rexbidv y A (= (cv x) (cv v)) (/\ (= (` H (cv u)) (` H (cv y))) (br (cv x) R (cv y))) r19.42v syl6bb rexbidva (cv x) (cv v) R (cv y) breq1 (= (` H (cv u)) (` H (cv y))) anbi2d y A rexbidv A ceqsrexv (:-1-1-> H A B) adantl bitrd H A B (cv u) (cv y) f1fveq (cv u) (cv y) eqcom syl6bb anassrs (br (cv v) R (cv y)) anbi1d rexbidva (cv y) (cv u) (cv v) R breq2 A ceqsrexv (:-1-1-> H A B) adantl bitrd sylan9bb anandis H (cv v) fvex H (cv u) fvex (cv z) (` H (cv v)) (` H (cv x)) eqeq1 (= (cv w) (` H (cv y))) anbi1d (br (cv x) R (cv y)) anbi1d x A y A 2rexbidv (cv w) (` H (cv u)) (` H (cv y)) eqeq1 (= (` H (cv v)) (` H (cv x))) anbi2d (br (cv x) R (cv y)) anbi1d x A y A 2rexbidv opelopab syl5bb sylan9bbr an1rs (` H (cv v)) S (` H (cv u)) df-br syl5rbb exp32 r19.21adv r19.21aiv H A B f1of1 sylan jca H R S A B v u df-iso sylibr)) thm (f1owe ((w z) (R z) (R w) (x y) (x z) (w x) (S x) (y z) (w y) (S y) (S z) (S w) (A z) (A w) (B z) (B w) (F x) (F y) (F z) (F w)) ((f1owe.1 (= R ({<,>|} x y (br (` F (cv x)) S (` F (cv y))))))) (-> (:-1-1-onto-> F A B) (-> (We S B) (We R A))) ((cv x) (cv z) F fveq2 S (` F (cv y)) breq1d (cv y) (cv w) F fveq2 (` F (cv z)) S breq2d f1owe.1 A A brabg rgen2 F R S A B z w df-iso F R S A B isowe sylbir mpan2 biimprd)) thm (f1oweALT ((w z) (v z) (u z) (f z) (R z) (v w) (u w) (f w) (R w) (u v) (f v) (R v) (f u) (R u) (R f) (x y) (x z) (w x) (v x) (u x) (f x) (S x) (y z) (w y) (v y) (u y) (f y) (S y) (S z) (S w) (S v) (S u) (S f) (A z) (A w) (A v) (A u) (A f) (B z) (B w) (B v) (B u) (B f) (F x) (F y) (F z) (F w) (F v) (F u) (F f)) ((f1oweALT.1 (= R ({<,>|} x y (br (` F (cv x)) S (` F (cv y))))))) (-> (:-1-1-onto-> F A B) (-> (We S B) (We R A))) (F A B f1ofo F A B df-fo (ran F) B S freq2 biimprd F A df-fn z visset F funimaex (cv w) (" F (cv z)) (ran F) sseq1 (cv w) (" F (cv z)) ({/}) eqeq1 negbid anbi12d (cv w) (" F (cv z)) f (-. (br (cv f) S (cv u))) raleq1 u rexeqd imbi12d (V) cla4gv F (cv z) (cv w) funfvima2 (` F (cv w)) (" F (cv z)) n0i syl6 w 19.23adv (cv z) w n0 syl5ib imp F (cv z) imassrn jctil syl7 syl S (ran F) w u f df-fr syl5ib com23 exp3a anabsi5 imp3a F (cv z) fores (` (|` F (cv z)) (cv v)) (cv f) S (` (|` F (cv z)) (cv w)) breq1 negbid (cv z) (" F (cv z)) cbvfo w (cv z) rexbidv (` (|` F (cv z)) (cv w)) (cv u) (cv f) S breq2 negbid f (" F (cv z)) ralbidv (cv z) (" F (cv z)) cbvexfo bitrd (cv v) (cv z) F fvres (cv w) (cv z) F fvres S breqan12rd v visset w visset (cv x) (cv v) F fveq2 S (` F (cv y)) breq1d (cv y) (cv w) F fveq2 (` F (cv v)) S breq2d f1oweALT.1 brab syl6rbbr negbid ralbidva rexbiia syl5bb syl sylibrd exp4b com34 com23 imp4a z 19.21adv R (dom F) z w v df-fr syl6ibr (dom F) A R freq2 biimpd sylan9 sylbi sylan9r sylbi syl F A B df-f1o w visset v visset (cv x) (cv w) F fveq2 S (` F (cv y)) breq1d (cv y) (cv v) F fveq2 (` F (cv w)) S breq2d f1oweALT.1 brab (/\ (:-1-1-> F A B) (/\ (e. (cv w) A) (e. (cv v) A))) a1i F A B (cv w) (cv v) f1fveq bicomd v visset w visset (cv x) (cv v) F fveq2 S (` F (cv y)) breq1d (cv y) (cv w) F fveq2 (` F (cv v)) S breq2d f1oweALT.1 brab (/\ (:-1-1-> F A B) (/\ (e. (cv w) A) (e. (cv v) A))) a1i 3orbi123d 2ralbidva (` F (cv w)) (cv u) S (` F (cv v)) breq1 (` F (cv w)) (cv u) (` F (cv v)) eqeq1 (` F (cv w)) (cv u) (` F (cv v)) S breq2 3orbi123d v A ralbidv A B cbvfo (` F (cv v)) (cv f) (cv u) S breq2 (` F (cv v)) (cv f) (cv u) eqeq2 (` F (cv v)) (cv f) S (cv u) breq1 3orbi123d A B cbvfo u B ralbidv bitrd sylan9bb sylbi biimprd anim12d S B u f dfwe2 R A w v dfwe2 3imtr4g)) thm (canth ((x y) (A x) (A y) (F x) (F y)) ((canth.1 (e. A (V)))) (-. (:-onto-> F A (P~ A))) (F A (P~ A) forn F A (P~ A) fof (= (cv x) (cv y)) id (cv x) (cv y) F fveq2 eleq12d negbid A elrab baibr (e. (cv y) (` F (cv y))) (e. (cv y) ({e.|} x A (-. (e. (cv x) (` F (cv x)))))) nbbn (` F (cv y)) ({e.|} x A (-. (e. (cv x) (` F (cv x))))) (cv y) eleq2 con3i sylbi syl rgen F A (P~ A) ffn F A ({e.|} x A (-. (e. (cv x) (` F (cv x))))) y fvelrn biimpd syl con3d y A (= (` F (cv y)) ({e.|} x A (-. (e. (cv x) (` F (cv x)))))) ralnex syl5ib mpi x A (-. (e. (cv x) (` F (cv x)))) ssrab2 canth.1 x (-. (e. (cv x) (` F (cv x)))) rabex A elpw mpbir (ran F) (P~ A) ({e.|} x A (-. (e. (cv x) (` F (cv x))))) eleq2 mpbiri con3i 3syl pm2.65i)) thm (ncanth () () (:-onto-> (I) (V) (P~ (V))) (f1ovi pwv (P~ (V)) (V) (I) (V) f1oeq3 ax-mp mpbir (I) (V) (P~ (V)) f1ofo ax-mp)) thm (iunon ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) ((iunon.1 (e. A (V))) (iunon.2 (e. B (V)))) (-> (A.e. x A (e. B (On))) (e. (U_ x A B) (On))) (x A (e. B (On)) hbra1 (e. (cv y) (On)) x ax-17 x A (e. B (On)) ra4 B (On) (cv y) eleq1a syl6 r19.23ad y (E.e. x A (= (cv y) B)) abid syl5ib y 19.21aiv z y (E.e. x A (= (cv y) B)) hbab1 (e. (cv z) (On)) y ax-17 dfss2f sylibr iunon.1 y x B abrexex ssonuni syl iunon.2 x A y dfiun2 syl5eqel)) thm (iinon ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) ((iinon.1 (e. B (V)))) (-> (/\ (A.e. x A (e. B (On))) (-. (= A ({/})))) (e. (|^|_ x A B) (On))) (({|} y (E.e. x A (= (cv y) B))) oninton x A (= (cv y) B) df-rex y exbii x y (/\ (e. (cv x) A) (= (cv y) B)) excom y (e. (cv x) A) (= (cv y) B) 19.42v iinon.1 y isseti mpbiran2 x exbii 3bitr2r A x n0 y (E.e. x A (= (cv y) B)) abn0 3bitr4 sylan2b x A (e. B (On)) hbra1 (e. (cv y) (On)) x ax-17 x A (e. B (On)) ra4 B (On) (cv y) eleq1a syl6 r19.23ad y (E.e. x A (= (cv y) B)) abid syl5ib y 19.21aiv z y (E.e. x A (= (cv y) B)) hbab1 (e. (cv z) (On)) y ax-17 dfss2f sylibr sylan iinon.1 x A y dfiin2 syl5eqel)) thm (tfrlem1 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (w x) (F x) (w y) (F y) (w z) (F z) (F w) (G x) (G y) (G z) (G w)) () (-> (e. A (On)) (-> (/\ (Fn F A) (Fn G A)) (-> (A.e. x A (/\ (= (` F (cv x)) (` B (|` F (cv x)))) (= (` G (cv x)) (` B (|` G (cv x)))))) (A.e. x A (= (` F (cv x)) (` G (cv x))))))) (A ssid (cv y) A A sseq1 (cv y) A x (/\ (= (` F (cv x)) (` B (|` F (cv x)))) (= (` G (cv x)) (` B (|` G (cv x))))) raleq1 (cv y) A x (= (` F (cv x)) (` G (cv x))) raleq1 imbi12d imbi12d (/\ (Fn F A) (Fn G A)) imbi2d (cv y) (cv z) A sseq1 (/\ (Fn F A) (Fn G A)) anbi2d (cv y) (cv z) x (/\ (= (` F (cv x)) (` B (|` F (cv x)))) (= (` G (cv x)) (` B (|` G (cv x))))) raleq1 anbi12d (cv y) (cv z) x (= (` F (cv x)) (` G (cv x))) raleq1 imbi12d (cv y) (cv z) onelsst (cv z) (cv y) A sstr2 (/\ (Fn F A) (Fn G A)) anim2d (C_ (cv z) (cv y)) x ax-17 x (cv y) (/\ (= (` F (cv x)) (` B (|` F (cv x)))) (= (` G (cv x)) (` B (|` G (cv x))))) hbra1 (cv z) (cv y) (cv x) ssel x (cv y) (/\ (= (` F (cv x)) (` B (|` F (cv x)))) (= (` G (cv x)) (` B (|` G (cv x))))) ra4 syl9 r19.21ad anim12d syl6 com23 r19.21adv (cv y) (cv x) onelsst (cv x) (cv y) A sstr2 (` F (cv x)) (` B (|` F (cv x))) (` G (cv x)) (` B (|` G (cv x))) eqeq12 F A G (cv x) w fvreseq biimpar B fveq2d syl5bir exp4c com4l syl9 syl6 imp4a com23 imp31 r19.20dva x (cv y) (/\ (= (` F (cv x)) (` B (|` F (cv x)))) (= (` G (cv x)) (` B (|` G (cv x))))) (= (` F (cv x)) (` G (cv x))) r19.20 syl6 x (cv z) (= (` F (cv x)) (` G (cv x))) hbra1 (A.e. w (cv x) (= (` F (cv w)) (` G (cv w)))) z ax-17 (cv z) (cv x) w (= (` F (cv w)) (` G (cv w))) raleq1 (cv x) (cv w) F fveq2 (cv x) (cv w) G fveq2 eqeq12d (cv z) cbvralv syl5bb (cv y) cbvral syl5ib ex com23 imp4a imim12d (/\ (/\ (/\ (Fn F A) (Fn G A)) (C_ (cv y) A)) (A.e. x (cv y) (/\ (= (` F (cv x)) (` B (|` F (cv x)))) (= (` G (cv x)) (` B (|` G (cv x))))))) (A.e. x (cv y) (= (` F (cv x)) (` G (cv x)))) pm2.43 syl6 z (cv y) (/\ (/\ (/\ (Fn F A) (Fn G A)) (C_ (cv z) A)) (A.e. x (cv z) (/\ (= (` F (cv x)) (` B (|` F (cv x)))) (= (` G (cv x)) (` B (|` G (cv x))))))) (A.e. x (cv z) (= (` F (cv x)) (` G (cv x)))) r19.20 syl5 tfis2 exp4c vtoclga mpii)) thm (tfrlem2 ((A w) (B w) (F w) (G w) (w x) (w y)) () (-> (/\ (Fn F A) (Fn G A)) (-> (/\ (e. (<,> (cv x) (cv y)) F) (e. (<,> (cv x) (cv z)) G)) (-> (e. A (On)) (-> (A. w (-> (e. A (On)) (-> (e. (cv w) A) (/\ (= (` F (cv w)) (` B (|` F (cv w)))) (= (` G (cv w)) (` B (|` G (cv w)))))))) (= (cv y) (cv z)))))) (A F G w B tfrlem1 com12 imp3a (e. (<,> (cv x) (cv y)) F) adantr x visset y visset F A fnop (Fn G A) adantlr (cv w) (cv x) F fveq2 (cv w) (cv x) G fveq2 eqeq12d A rcla4v syl syld imp (e. (<,> (cv x) (cv z)) G) adantlrr w A (/\ (= (` F (cv w)) (` B (|` F (cv w)))) (= (` G (cv w)) (` B (|` G (cv w))))) df-ral (e. A (On)) anbi2i sylan2br y visset F (cv x) funopfv imp z visset G (cv x) funopfv imp anim12i an4s F A fnfun G A fnfun anim12i sylan (` F (cv x)) (cv y) (` G (cv x)) (cv z) eqeq12 syl (/\ (e. A (On)) (A. w (-> (e. (cv w) A) (/\ (= (` F (cv w)) (` B (|` F (cv w)))) (= (` G (cv w)) (` B (|` G (cv w)))))))) adantr mpbid (e. A (On)) (-> (e. (cv w) A) (/\ (= (` F (cv w)) (` B (|` F (cv w)))) (= (` G (cv w)) (` B (|` G (cv w)))))) abai w albii w (e. A (On)) (-> (e. (cv w) A) (/\ (= (` F (cv w)) (` B (|` F (cv w)))) (= (` G (cv w)) (` B (|` G (cv w)))))) 19.28v w (e. A (On)) (-> (e. A (On)) (-> (e. (cv w) A) (/\ (= (` F (cv w)) (` B (|` F (cv w)))) (= (` G (cv w)) (` B (|` G (cv w))))))) 19.28v 3bitr3r sylan2b exp43)) thm (tfrlem3 ((x y) (f x) (g x) (f y) (g y) (f g) (x z) (y z) (g z) (G f) (G g) (G x) (G z)) ((tfrlem3.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))))) (= A ({|} g (E.e. z (On) (/\ (Fn (cv g) (cv z)) (A.e. y (cv z) (= (` (cv g) (cv y)) (` G (|` (cv g) (cv y))))))))) (tfrlem3.1 g visset (cv f) (cv g) (cv x) fneq1 (cv f) (cv g) (cv y) fveq1 (cv f) (cv g) (cv y) reseq1 G fveq2d eqeq12d y (cv x) ralbidv anbi12d x (On) rexbidv elab (cv x) (cv z) (cv g) fneq2 (cv x) (cv z) y (= (` (cv g) (cv y)) (` G (|` (cv g) (cv y)))) raleq1 anbi12d (On) cbvrexv bitr abbi2i eqtr)) thm (tfrlem4 ((x y) (f x) (g x) (A x) (f y) (g y) (A y) (f g) (A f) (A g) (x z) (F x) (y z) (F y) (f z) (F z) (F f) (G x) (G y) (g z) (G z) (G f) (G g)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (-> (e. (cv g) A) (Fun (cv g))) (tfrlem.1 g z tfrlem3 abeq2i (cv g) (cv z) fnfun (A.e. y (cv z) (= (` (cv g) (cv y)) (` G (|` (cv g) (cv y))))) adantr (e. (cv z) (On)) a1i r19.23aiv sylbi)) thm (tfrlem5 ((x y) (f x) (g x) (h x) (A x) (f y) (g y) (h y) (A y) (f g) (f h) (A f) (g h) (A g) (A h) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (w z) (v z) (u z) (f z) (F z) (v w) (u w) (f w) (F w) (u v) (f v) (F v) (f u) (F u) (F f) (G x) (G y) (g z) (h z) (G z) (g w) (h w) (G w) (G f) (G g) (G h) (g v) (h v) (g u) (h u)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (-> (/\ (e. (cv g) A) (e. (cv h) A)) (-> (/\ (e. (<,> (cv x) (cv u)) (cv g)) (e. (<,> (cv x) (cv v)) (cv h))) (= (cv u) (cv v)))) (tfrlem.1 g z tfrlem3 abeq2i tfrlem.1 h w tfrlem3 abeq2i anbi12i z (On) w (On) (/\ (Fn (cv g) (cv z)) (A.e. y (cv z) (= (` (cv g) (cv y)) (` G (|` (cv g) (cv y)))))) (/\ (Fn (cv h) (cv w)) (A.e. y (cv w) (= (` (cv h) (cv y)) (` G (|` (cv h) (cv y)))))) reeanv bitr4 (cv g) (cv z) (cv h) (cv w) x u v 2elresin (|` (cv g) (i^i (cv z) (cv w))) (i^i (cv z) (cv w)) (|` (cv h) (i^i (cv z) (cv w))) x u v y G tfrlem2 (cv g) (cv z) (cv w) fnresin1 (cv h) (cv w) (cv z) fnresin2 syl2an sylbid com24 com3r imp32 (cv z) (cv w) onin y (cv z) (= (` (cv g) (cv y)) (` G (|` (cv g) (cv y)))) (cv w) (= (` (cv h) (cv y)) (` G (|` (cv h) (cv y)))) r19.26m (e. (cv y) (cv z)) (= (` (cv g) (cv y)) (` G (|` (cv g) (cv y)))) (e. (cv y) (cv w)) (= (` (cv h) (cv y)) (` G (|` (cv h) (cv y)))) prth (e. (i^i (cv z) (cv w)) (On)) (e. (cv y) (i^i (cv z) (cv w))) pm3.27 (cv y) (cv z) (cv w) elin sylib syl5 (i^i (cv z) (cv w)) (cv y) onelsst impac (cv y) (i^i (cv z) (cv w)) (cv g) fvres (cv y) (i^i (cv z) (cv w)) (cv g) resabs1 G fveq2d eqeqan12rd (cv y) (i^i (cv z) (cv w)) (cv h) fvres (cv y) (i^i (cv z) (cv w)) (cv h) resabs1 G fveq2d eqeqan12rd anbi12d syl bicomd mpbidi exp3a y 19.20i sylbir (/\ (Fn (cv g) (cv z)) (Fn (cv h) (cv w))) anim2i an4s syl2an ex r19.23aivv sylbi)) thm (tfrlem6 ((x y) (f x) (g x) (A x) (f y) (g y) (A y) (f g) (A f) (A g) (F x) (F y) (F f) (G x) (G y) (G f) (G g)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (Rel F) (tfrlem.2 A g reluni tfrlem.1 tfrlem.2 g tfrlem4 (cv g) funrel syl mprgbir F (U. A) releq mpbiri ax-mp)) thm (tfrlem7 ((x y) (f x) (g x) (h x) (A x) (f y) (g y) (h y) (A y) (f g) (f h) (A f) (g h) (A g) (A h) (v x) (u x) (F x) (v y) (u y) (F y) (u v) (f v) (F v) (f u) (F u) (F f) (G x) (G y) (G f) (G g) (G h) (g v) (h v) (g u) (h u)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (Fun F) (F x u v dffun4 tfrlem.1 tfrlem.2 tfrlem6 tfrlem.2 (<,> (cv x) (cv u)) eleq2i (<,> (cv x) (cv u)) A g eluni bitr tfrlem.2 (<,> (cv x) (cv v)) eleq2i (<,> (cv x) (cv v)) A h eluni bitr anbi12i g h (/\ (e. (<,> (cv x) (cv u)) (cv g)) (e. (cv g) A)) (/\ (e. (<,> (cv x) (cv v)) (cv h)) (e. (cv h) A)) eeanv bitr4 (e. (<,> (cv x) (cv u)) (cv g)) (e. (cv g) A) (e. (<,> (cv x) (cv v)) (cv h)) (e. (cv h) A) an4 (/\ (e. (<,> (cv x) (cv u)) (cv g)) (e. (<,> (cv x) (cv v)) (cv h))) (/\ (e. (cv g) A) (e. (cv h) A)) ancom bitr tfrlem.1 tfrlem.2 g h u v tfrlem5 imp sylbi g h 19.23aivv sylbi v ax-gen x u gen2 mpbir2an)) thm (tfrlem8 ((x y) (f x) (A x) (f y) (A y) (A f) (x z) (w x) (v x) (F x) (y z) (w y) (v y) (F y) (w z) (v z) (f z) (F z) (v w) (f w) (F w) (f v) (F v) (F f) (G x) (G y) (G z) (G w) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (Ord (dom F)) ((dom F) v w dftr2 w visset F z eldm2 tfrlem.2 tfrlem.1 unieqi eqtr (<,> (cv w) (cv z)) eleq2i (<,> (cv w) (cv z)) f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))) eluniab bitr z exbii bitr x (On) (e. (<,> (cv w) (cv z)) (cv f)) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))) r19.42v w visset z visset (cv f) (cv x) fnop ancoms (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))) adantrr (e. (cv x) (On)) adantl x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))) ra4e tfrlem.1 abeq2i (cv f) A elssuni tfrlem.2 (cv f) sseq2i (cv f) F dmss sylbir syl sylbir syl (cv f) (cv x) fndm (dom F) sseq1d (e. (cv x) (On)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))) ad2antrl mpbid (e. (<,> (cv w) (cv z)) (cv f)) adantrl jca ex r19.22i sylbir z f 19.23aivv sylbi (e. (cv v) (cv w)) anim2i x (On) (e. (cv v) (cv w)) (/\ (e. (cv w) (cv x)) (C_ (cv x) (dom F))) r19.42v sylibr (cv x) (cv v) (cv w) ontr1 (cv x) (dom F) (cv v) ssel sylan9r exp4b com4l imp4d r19.23aiv syl w ax-gen mpgbir w visset F z eldm2 tfrlem.2 tfrlem.1 unieqi eqtr (<,> (cv w) (cv z)) eleq2i (<,> (cv w) (cv z)) f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))) eluniab bitr (cv x) (cv w) onelon w visset z visset (cv f) (cv x) fnop sylan2 exp32 com12 (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))) adantr com13 r19.23adv imp f 19.23aiv sylbi z 19.23aiv sylbi ssriv ordon (dom F) (On) trssord mp3an)) thm (tfrlem9 ((x y) (f x) (A x) (f y) (A y) (A f) (x z) (F x) (y z) (F y) (f z) (F z) (F f) (G x) (G y) (G z) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (-> (e. (cv y) (dom F)) (= (` F (cv y)) (` G (|` F (cv y))))) (y visset F z eldm2 tfrlem.2 tfrlem.1 unieqi eqtr (<,> (cv y) (cv z)) eleq2i (<,> (cv y) (cv z)) f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))) eluniab bitr y visset z visset (cv f) (cv x) fnop x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))) ra4e tfrlem.1 abeq2i (cv f) A elssuni tfrlem.2 syl6ssr sylbir syl y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))) ra4 com12 (cv f) (cv x) fndm (cv y) eleq2d tfrlem.1 tfrlem.2 tfrlem7 F (cv f) (cv y) funssfv (/\ (Fn (cv f) (cv x)) (e. (cv x) (On))) adantrl F (cv f) (cv y) fun2ssres G fveq2d (cv f) (cv x) fndm (On) eleq1d (dom (cv f)) (cv y) onelsst syl6bir imp31 sylan2 eqeq12d mpanl1 biimprd ex com3l exp31 com34 com24 sylbird com3l syld com24 imp4d mpdi syl exp4d ex com4r pm2.43i com3l imp4a r19.23adv imp f 19.23aiv sylbi z 19.23aiv sylbi)) thm (tfrlem10 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A))) (tfrlem.3 (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F))))))))) (-> (e. (dom F) (On)) (Fn C (suc (dom F)))) ((cv x) (dom F) (` G (|` F (dom F))) opeq1 (<,> (cv x) (` G (|` F (dom F)))) (<,> (dom F) (` G (|` F (dom F)))) sneq ({} (<,> (cv x) (` G (|` F (dom F))))) ({} (<,> (dom F) (` G (|` F (dom F))))) funeq 3syl x visset G (|` F (dom F)) fvex funsn (On) vtoclg tfrlem.1 tfrlem.2 tfrlem7 F ({} (<,> (dom F) (` G (|` F (dom F))))) funun tfrlem.3 C (u. F ({} (<,> (dom F) (` G (|` F (dom F)))))) funeq ax-mp sylibr (cv x) (dom F) (` G (|` F (dom F))) opeq1 sneqd dmeqd (cv x) (dom F) sneq eqeq12d (cv x) (` G (|` F (dom F))) dmsnop (On) vtoclg (cv x) eleq2d (cv x) (dom F) (dom F) eleq1 negbid (dom F) eloni (dom F) ordeirr syl syl5bir com12 (cv x) (dom F) elsni syl5 sylbid con2d r19.21aiv (dom F) (dom ({} (<,> (dom F) (` G (|` F (dom F)))))) x disj sylibr sylan2 ex mpan mpcom (cv x) (dom F) (` G (|` F (dom F))) opeq1 (<,> (cv x) (` G (|` F (dom F)))) (<,> (dom F) (` G (|` F (dom F)))) sneq ({} (<,> (cv x) (` G (|` F (dom F))))) ({} (<,> (dom F) (` G (|` F (dom F))))) dmeq 3syl (cv x) (dom F) sneq eqeq12d (cv x) (` G (|` F (dom F))) dmsnop (On) vtoclg (dom F) uneq2d tfrlem.3 dmeqi F ({} (<,> (dom F) (` G (|` F (dom F))))) dmun eqtr (dom F) df-suc 3eqtr4g jca C (suc (dom F)) df-fn sylibr)) thm (tfrlem11 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A))) (tfrlem.3 (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F))))))))) (-> (e. (dom F) (On)) (-> (e. (cv y) (suc (dom F))) (= (` C (cv y)) (` G (|` C (cv y)))))) (F ({} (<,> (dom F) (` G (|` F (dom F))))) ssun1 tfrlem.3 sseqtr4 C F (cv y) funssfv (e. (dom F) (On)) adantrl C F (cv y) fun2ssres G fveq2d (dom F) (cv y) onelsst imp sylan2 eqeq12d tfrlem.1 tfrlem.2 tfrlem9 syl5bir mpanl2 tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 C (suc (dom F)) fnfun syl sylan exp32 pm2.43i pm2.43d (cv y) (dom F) (` G (|` C (cv y))) opeq1 (e. (dom F) (On)) adantl F ({} (<,> (dom F) (` G (|` F (dom F))))) ssun1 tfrlem.3 sseqtr4 C F (cv y) fun2ssres mpanl2 tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 C (suc (dom F)) fnfun syl (cv y) (dom F) eqimss syl2an (cv y) (dom F) F reseq2 (e. (dom F) (On)) adantl eqtrd (|` C (cv y)) (|` F (dom F)) G fveq2 (` G (|` C (cv y))) (` G (|` F (dom F))) (dom F) opeq2 3syl eqtrd sneqd (cv y) (` G (|` C (cv y))) opex snid syl5eleq (<,> (cv y) (` G (|` C (cv y)))) ({} (<,> (dom F) (` G (|` F (dom F))))) F elun2 syl tfrlem.3 syl6eleqr G (|` C (cv y)) fvex C (suc (dom F)) (cv y) fnopfvb tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 y visset (dom F) eqelsuc syl2an mpbird ex jaod (cv y) (dom F) elsuci syl5)) thm (tfrlem12 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A))) (tfrlem.3 (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F))))))))) (-> (e. (dom F) (On)) (e. C A)) ((cv x) (suc (dom F)) C fneq2 (cv x) (suc (dom F)) y (= (` C (cv y)) (` G (|` C (cv y)))) raleq1 anbi12d (On) rcla4ev (dom F) suceloni tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 tfrlem.1 tfrlem.2 tfrlem.3 tfrlem11 r19.21aiv jca sylanc C (suc (dom F)) (On) fnex tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 (dom F) suceloni sylanc (cv f) C (cv x) fneq1 (cv f) C (cv y) fveq1 (cv f) C (cv y) reseq1 G fveq2d eqeq12d y (cv x) ralbidv anbi12d x (On) rexbidv tfrlem.1 (V) elab2g syl mpbird)) thm (tfrlem13 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A))) (tfrlem.3 (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F))))))))) (= (dom F) (On)) (tfrlem.1 tfrlem.2 tfrlem8 (dom F) ordeirr C A elssuni tfrlem.2 syl6ssr C F dmss (dom C) (dom F) (dom F) ssel 3syl tfrlem.1 tfrlem.2 tfrlem.3 tfrlem12 (dom F) (On) sucidg tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 C (suc (dom F)) fndm syl eleqtrrd sylc nsyl ax-mp tfrlem.1 tfrlem.2 tfrlem8 (dom F) ordeleqon mpbi (e. (dom F) (On)) (= (dom F) (On)) orel1 mp2)) thm (tfr1 ((x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((tfr.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfr.2 (= F (U. A)))) (Fn F (On)) (F (On) df-fn tfr.1 tfr.2 tfrlem7 tfr.1 tfr.2 (u. F ({} (<,> (dom F) (` G (|` F (dom F)))))) eqid tfrlem13 mpbir2an)) thm (tfr2 ((x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f) (y z)) ((tfr.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfr.2 (= F (U. A)))) (-> (e. (cv z) (On)) (= (` F (cv z)) (` G (|` F (cv z))))) ((cv y) (cv z) F fveq2 (cv y) (cv z) F reseq2 G fveq2d eqeq12d tfr.1 tfr.2 (u. F ({} (<,> (dom F) (` G (|` F (dom F)))))) eqid tfrlem13 (cv y) eleq2i tfr.1 tfr.2 tfrlem9 sylbir vtoclga)) thm (tfr3 ((x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f) (B x) (B y)) ((tfr.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfr.2 (= F (U. A)))) (-> (/\ (Fn B (On)) (A.e. x (On) (= (` B (cv x)) (` G (|` B (cv x)))))) (= B F)) ((Fn B (On)) x ax-17 x (On) (= (` B (cv x)) (` G (|` B (cv x)))) hbra1 hban (Fn B (On)) x ax-17 x (On) (= (` B (cv x)) (` G (|` B (cv x)))) hbra1 hban (= (` B (cv y)) (` F (cv y))) x ax-17 hbim (cv x) (cv y) B fveq2 (cv x) (cv y) F fveq2 eqeq12d (/\ (Fn B (On)) (A.e. x (On) (= (` B (cv x)) (` G (|` B (cv x)))))) imbi2d x (On) (= (` B (cv x)) (` G (|` B (cv x)))) ra4 tfr.1 tfr.2 tfr1 B (On) F (cv x) y fvreseq mpanl2 (|` B (cv x)) (|` F (cv x)) G fveq2 syl6bir (cv x) onsst sylan2 ancoms imp (/\ (-> (e. (cv x) (On)) (= (` B (cv x)) (` G (|` B (cv x))))) (e. (cv x) (On))) adantr tfr.1 tfr.2 x tfr2 (-> (e. (cv x) (On)) (= (` B (cv x)) (` G (|` B (cv x))))) jctr (e. (cv x) (On)) (= (` B (cv x)) (` G (|` B (cv x)))) (= (` F (cv x)) (` G (|` F (cv x)))) jcab sylibr (` B (cv x)) (` G (|` B (cv x))) (` F (cv x)) (` G (|` F (cv x))) eqeq12 syl6 imp (/\ (/\ (e. (cv x) (On)) (Fn B (On))) (A.e. y (cv x) (= (` B (cv y)) (` F (cv y))))) adantl mpbird exp43 com4t exp4a pm2.43d syl com3l imp3a a2d y (cv x) (/\ (Fn B (On)) (A.e. x (On) (= (` B (cv x)) (` G (|` B (cv x)))))) (= (` B (cv y)) (` F (cv y))) r19.21v syl5ib tfis2f com12 r19.21ai (On) eqid tfr.1 tfr.2 tfr1 B (On) F (On) x eqfnfv mpan2 biimpar mpanr1 syldan)) thm (tz7.44lem1 ((x y) (A x) (A y) (G x) (H y)) ((tz7.44lem1.1 (= G ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x)))))))))) (Fun G) (x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x)))))) funopab H (` (cv x) (U. (dom (cv x)))) fvex x visset (cv x) (V) rnexg (ran (cv x)) (V) uniexg syl ax-mp nlim0 dm0 (dom ({/})) ({/}) limeq ax-mp mtbir (cv x) ({/}) dmeq (dom (cv x)) (dom ({/})) limeq syl biimpa mto y A moeq3 mpgbir tz7.44lem1.1 G ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))))) funeq ax-mp mpbir)) thm (tz7.44-1 ((x y) (A x) (A y) (F x) (G x) (H y)) ((tz7.44.1 (= G ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))))))) (tz7.44.2 (Fn F (On))) (tz7.44.3 (-> (e. (cv x) (On)) (= (` F (cv x)) (` G (|` F (cv x)))))) (tz7.44.4 (e. A (V)))) (= (` F ({/})) A) (0elon (cv x) ({/}) F fveq2 (cv x) ({/}) F reseq2 G fveq2d eqeq12d tz7.44.3 vtoclga ax-mp F res0 G fveq2i tz7.44.1 tz7.44lem1 (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))) 3mix1 x y ssopab2i tz7.44.1 sseqtr4 0ex tz7.44.4 (cv x) ({/}) ({/}) eqeq1 (= (cv y) A) anbi1d (cv y) A A eqeq1 (= ({/}) ({/})) anbi2d opelopab ({/}) eqid A eqid mpbir2an sselii tz7.44.4 G ({/}) funopfv mp2 3eqtr)) thm (tz7.44-2 ((x y) (B x) (B y) (F y) (H x) (x y) (A x) (A y) (F x) (G x) (H y)) ((tz7.44.1 (= G ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))))))) (tz7.44.2 (Fn F (On))) (tz7.44.3 (-> (e. (cv x) (On)) (= (` F (cv x)) (` G (|` F (cv x)))))) (tz7.44.5 (e. B (On)))) (= (` F (suc B)) (` H (` F B))) (tz7.44.5 onsuc (cv x) (suc B) F fveq2 (cv x) (suc B) F reseq2 G fveq2d eqeq12d tz7.44.3 vtoclga ax-mp tz7.44.1 tz7.44lem1 (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))) 3mix2 x y ssopab2i tz7.44.1 sseqtr4 tz7.44.2 F (On) fnfun ax-mp tz7.44.5 onsuc F (suc B) (On) resfunexg mp2an H (` F B) fvex (cv x) (|` F (suc B)) ({/}) eqeq1 (cv x) (|` F (suc B)) dmeq (dom (cv x)) (dom (|` F (suc B))) limeq syl orbi12d negbid (cv x) (|` F (suc B)) dmeq (dom (cv x)) (dom (|` F (suc B))) unieq (U. (dom (cv x))) (U. (dom (|` F (suc B)))) (cv x) fveq2 3syl (cv x) (|` F (suc B)) (U. (dom (|` F (suc B)))) fveq1 eqtrd H fveq2d (cv y) eqeq2d anbi12d (cv y) (` H (` F B)) (` H (` (|` F (suc B)) (U. (dom (|` F (suc B)))))) eqeq1 (-. (\/ (= (|` F (suc B)) ({/})) (Lim (dom (|` F (suc B)))))) anbi2d opelopab B nsuceq0 tz7.44.2 F (On) fndm ax-mp (suc B) ineq2i F (suc B) dmres tz7.44.5 onsuc onss (suc B) (On) dfss mpbi 3eqtr4 ({/}) eqeq1i mtbir (|` F (suc B)) ({/}) dmeq dm0 syl6eq mto tz7.44.5 elisseti B (V) nlimsucg ax-mp tz7.44.2 F (On) fndm ax-mp (suc B) ineq2i F (suc B) dmres tz7.44.5 onsuc onss (suc B) (On) dfss mpbi 3eqtr4 (dom (|` F (suc B))) (suc B) limeq ax-mp mtbir pm3.2ni tz7.44.5 elisseti sucid B (suc B) F fvres ax-mp tz7.44.2 F (On) fndm ax-mp (suc B) ineq2i F (suc B) dmres tz7.44.5 onsuc onss (suc B) (On) dfss mpbi 3eqtr4 unieqi tz7.44.5 onunisuc eqtr2 (|` F (suc B)) fveq2i eqtr3 H fveq2i mpbir2an sselii H (` F B) fvex G (|` F (suc B)) funopfv mp2 eqtr)) thm (tz7.44-3 ((x y) (B x) (B y) (F y) (H x) (x y) (A x) (A y) (F x) (G x) (H y)) ((tz7.44.1 (= G ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))))))) (tz7.44.2 (Fn F (On))) (tz7.44.3 (-> (e. (cv x) (On)) (= (` F (cv x)) (` G (|` F (cv x)))))) (tz7.44.5 (e. B (On)))) (-> (Lim B) (= (` F B) (U. (" F B)))) (tz7.44.2 F (On) fndm ax-mp B ineq2i F B dmres tz7.44.5 onss B (On) dfss mpbi 3eqtr4 (dom (|` F B)) B limeq ax-mp biimpr F B df-ima unieqi jctir tz7.44.2 F (On) fnfun ax-mp tz7.44.5 F B (On) resfunexg mp2an tz7.44.2 F (On) fnfun ax-mp tz7.44.5 elisseti F funimaex ax-mp uniex (cv x) (|` F B) dmeq (dom (cv x)) (dom (|` F B)) limeq syl (cv x) (|` F B) rneq unieqd (cv y) eqeq2d anbi12d (cv y) (U. (" F B)) (U. (ran (|` F B))) eqeq1 (Lim (dom (|` F B))) anbi2d opelopab sylibr (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))) (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) 3mix3 x y ssopab2i tz7.44.1 sseqtr4 (<,> (|` F B) (U. (" F B))) sseli tz7.44.1 tz7.44lem1 tz7.44.2 F (On) fnfun ax-mp tz7.44.5 elisseti F funimaex ax-mp uniex G (|` F B) funopfv ax-mp 3syl tz7.44.5 (cv x) B F fveq2 (cv x) B F reseq2 G fveq2d eqeq12d tz7.44.3 vtoclga ax-mp syl5eq)) thm (dfrdg2 ((x y) (x z) (f x) (g x) (F x) (y z) (f y) (g y) (F y) (f z) (g z) (F z) (f g) (F f) (F g) (A x) (A y) (A z) (A f) (A g)) () (= (rec F A) (U. ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` ({<,>|} g z (\/\/ (/\ (= (cv g) ({/})) (= (cv z) A)) (/\ (-. (\/ (= (cv g) ({/})) (Lim (dom (cv g))))) (= (cv z) (` F (` (cv g) (U. (dom (cv g))))))) (/\ (Lim (dom (cv g))) (= (cv z) (U. (ran (cv g))))))) (|` (cv f) (cv y)))))))))) (F A f x y g z df-rdg (cv z) (Lim (dom (cv g))) (U. (ran (cv g))) (` F (` (cv g) (U. (dom (cv g))))) eqif (-. (= (cv g) ({/}))) anbi2i (/\ (Lim (dom (cv g))) (= (cv z) (U. (ran (cv g))))) (/\ (-. (Lim (dom (cv g)))) (= (cv z) (` F (` (cv g) (U. (dom (cv g))))))) orcom (-. (= (cv g) ({/}))) anbi2i (-. (= (cv g) ({/}))) (/\ (-. (Lim (dom (cv g)))) (= (cv z) (` F (` (cv g) (U. (dom (cv g))))))) (/\ (Lim (dom (cv g))) (= (cv z) (U. (ran (cv g))))) andi (= (cv g) ({/})) (Lim (dom (cv g))) ioran (= (cv z) (` F (` (cv g) (U. (dom (cv g)))))) anbi1i (-. (= (cv g) ({/}))) (-. (Lim (dom (cv g)))) (= (cv z) (` F (` (cv g) (U. (dom (cv g)))))) anass bitr (cv g) ({/}) dmeq dm0 syl6eq nlim0 (dom (cv g)) ({/}) limeq mtbiri syl con2i pm4.71ri (= (cv z) (U. (ran (cv g)))) anbi1i (-. (= (cv g) ({/}))) (Lim (dom (cv g))) (= (cv z) (U. (ran (cv g)))) anass bitr orbi12i bitr4 3bitr (/\ (= (cv g) ({/})) (= (cv z) A)) orbi2i (cv z) (= (cv g) ({/})) A (if (Lim (dom (cv g))) (U. (ran (cv g))) (` F (` (cv g) (U. (dom (cv g)))))) eqif (/\ (= (cv g) ({/})) (= (cv z) A)) (/\ (-. (\/ (= (cv g) ({/})) (Lim (dom (cv g))))) (= (cv z) (` F (` (cv g) (U. (dom (cv g))))))) (/\ (Lim (dom (cv g))) (= (cv z) (U. (ran (cv g))))) 3orass 3bitr4 g z opabbii (|` (cv f) (cv y)) fveq1i (` (cv f) (cv y)) eqeq2i y (cv x) ralbii (Fn (cv f) (cv x)) anbi2i x (On) rexbii f abbii unieqi eqtr)) thm (rdgeq1 ((x y) (x z) (f x) (g x) (F x) (y z) (f y) (g y) (F y) (f z) (g z) (F z) (f g) (F f) (F g) (G x) (G y) (G z) (G f) (G g) (A x) (A y) (A z) (A f) (A g)) () (-> (= F G) (= (rec F A) (rec G A))) (F G (` (cv g) (U. (dom (cv g)))) fveq1 (Lim (dom (cv g))) (U. (ran (cv g))) ifeq2d (= (cv g) ({/})) A ifeq2d (cv z) eqeq2d g z opabbidv (|` (cv f) (cv y)) fveq1d (` (cv f) (cv y)) eqeq2d y (cv x) ralbidv (Fn (cv f) (cv x)) anbi2d x (On) rexbidv f abbidv unieqd F A f x y g z df-rdg G A f x y g z df-rdg 3eqtr4g)) thm (rdgeq2 ((x y) (x z) (f x) (g x) (F x) (y z) (f y) (g y) (F y) (f z) (g z) (F z) (f g) (F f) (F g) (A x) (A y) (A z) (A f) (A g) (B x) (B y) (B z) (B f) (B g)) () (-> (= A B) (= (rec F A) (rec F B))) (A B (= (cv g) ({/})) (if (Lim (dom (cv g))) (U. (ran (cv g))) (` F (` (cv g) (U. (dom (cv g)))))) ifeq1 (cv z) eqeq2d g z opabbidv (|` (cv f) (cv y)) fveq1d (` (cv f) (cv y)) eqeq2d y (cv x) ralbidv (Fn (cv f) (cv x)) anbi2d x (On) rexbidv f abbidv unieqd F A f x y g z df-rdg F B f x y g z df-rdg 3eqtr4g)) thm (hbrdg ((y z) (w y) (v y) (f y) (g y) (F y) (w z) (v z) (f z) (g z) (F z) (v w) (f w) (g w) (F w) (f v) (g v) (F v) (f g) (F f) (F g) (A y) (A z) (A w) (A v) (A f) (A g) (x y) (x z) (w x) (v x) (f x) (g x)) ((hbrdg.1 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (hbrdg.2 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (rec F A)) (A. x (e. (cv y) (rec F A)))) ((e. (cv w) (On)) x ax-17 (Fn (cv f) (cv w)) x ax-17 (e. (cv v) (cv w)) x ax-17 (e. (cv y) (` (cv f) (cv v))) x ax-17 (e. (cv y) (cv z)) x ax-17 (= (cv g) ({/})) x ax-17 hbrdg.2 (Lim (dom (cv g))) x ax-17 (e. (cv y) (U. (ran (cv g)))) x ax-17 hbrdg.1 (e. (cv y) (` (cv g) (U. (dom (cv g))))) x ax-17 hbfv hbif hbif hbeq y g z hbopab (e. (cv y) (|` (cv f) (cv v))) x ax-17 hbfv hbeq hbral hban hbrex y f hbab hbuni F A f w v g z df-rdg (cv y) eleq2i F A f w v g z df-rdg (cv y) eleq2i x albii 3imtr4)) thm (rdglem1 ((x y) (f x) (g x) (f y) (g y) (f g) (x z) (y z) (g z) (G f) (G g) (G x) (G z) (w y) (G y) (w z) (g w) (G w)) () (= ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))) ({|} g (E.e. z (On) (/\ (Fn (cv g) (cv z)) (A.e. w (cv z) (= (` (cv g) (cv w)) (` G (|` (cv g) (cv w))))))))) (({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))) eqid g z tfrlem3 (cv y) (cv w) (cv g) fveq2 (cv y) (cv w) (cv g) reseq2 G fveq2d eqeq12d (cv z) cbvralv (Fn (cv g) (cv z)) anbi2i z (On) rexbii g abbii eqtr)) thm (rdglem2 ((x y) (x z) (w x) (y z) (w y) (w z) (A x) (A z) (A w) (H x) (H z) (H w)) () (= ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))))) ({<,>|} z y (\/\/ (/\ (= (cv z) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv z) ({/})) (Lim (dom (cv z))))) (= (cv y) (` H (` (cv z) (U. (dom (cv z))))))) (/\ (Lim (dom (cv z))) (= (cv y) (U. (ran (cv z)))))))) ((cv x) (cv z) (cv y) opeq1 (cv w) eqeq2d (cv x) (cv z) ({/}) eqeq1 (= (cv y) A) anbi1d (cv x) (cv z) ({/}) eqeq1 (cv x) (cv z) dmeq (dom (cv x)) (dom (cv z)) limeq syl orbi12d negbid (cv x) (cv z) dmeq (dom (cv x)) (dom (cv z)) unieq (U. (dom (cv x))) (U. (dom (cv z))) (cv x) fveq2 3syl (cv x) (cv z) (U. (dom (cv z))) fveq1 eqtrd H fveq2d (cv y) eqeq2d anbi12d (cv x) (cv z) dmeq (dom (cv x)) (dom (cv z)) limeq syl (cv x) (cv z) rneq unieqd (cv y) eqeq2d anbi12d 3orbi123d anbi12d y exbidv cbvexv w abbii x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x)))))) w df-opab z y (\/\/ (/\ (= (cv z) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv z) ({/})) (Lim (dom (cv z))))) (= (cv y) (` H (` (cv z) (U. (dom (cv z))))))) (/\ (Lim (dom (cv z))) (= (cv y) (U. (ran (cv z)))))) w df-opab 3eqtr4)) thm (rdgfnon ((x y) (x z) (f x) (g x) (u x) (v x) (w x) (F x) (y z) (f y) (g y) (u y) (v y) (w y) (F y) (f z) (g z) (u z) (v z) (w z) (F z) (f g) (f u) (f v) (f w) (F f) (g u) (g v) (g w) (F g) (u v) (u w) (F u) (v w) (F v) (F w) (A x) (A y) (A z) (A f) (A g) (A u) (A v) (A w)) () (Fn (rec F A) (On)) (w u v ({<,>|} g z (= (cv z) (if (= (cv g) ({/})) A (if (Lim (dom (cv g))) (U. (ran (cv g))) (` F (` (cv g) (U. (dom (cv g))))))))) f x y rdglem1 F A w u v g z df-rdg tfr1)) thm (rdgval ((x y) (x z) (f x) (g x) (u x) (v x) (w x) (F x) (y z) (f y) (g y) (u y) (v y) (w y) (F y) (f z) (g z) (u z) (v z) (w z) (F z) (f g) (f u) (f v) (f w) (F f) (g u) (g v) (g w) (F g) (u v) (u w) (F u) (v w) (F v) (F w) (A x) (A y) (A z) (A f) (A g) (A u) (A v) (A w)) () (-> (e. (cv g) (On)) (= (` (rec F A) (cv g)) (` ({<,>|} w z (\/\/ (/\ (= (cv w) ({/})) (= (cv z) A)) (/\ (-. (\/ (= (cv w) ({/})) (Lim (dom (cv w))))) (= (cv z) (` F (` (cv w) (U. (dom (cv w))))))) (/\ (Lim (dom (cv w))) (= (cv z) (U. (ran (cv w))))))) (|` (rec F A) (cv g))))) (w u v ({<,>|} g z (\/\/ (/\ (= (cv g) ({/})) (= (cv z) A)) (/\ (-. (\/ (= (cv g) ({/})) (Lim (dom (cv g))))) (= (cv z) (` F (` (cv g) (U. (dom (cv g))))))) (/\ (Lim (dom (cv g))) (= (cv z) (U. (ran (cv g))))))) f x y rdglem1 F A w u v g z dfrdg2 g tfr2 g z A F w rdglem2 (|` (rec F A) (cv g)) fveq1i syl6eq)) thm (rdg0 ((g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((rdg.1 (e. A (V)))) (= (` (rec F A) ({/})) A) (w z A F g rdglem2 F A rdgfnon g F A w z rdgval rdg.1 tz7.44-1)) thm (rdgsuc ((g z) (B z) (B g) (g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((rdg.2 (e. B (On)))) (= (` (rec F A) (suc B)) (` F (` (rec F A) B))) (w z A F g rdglem2 F A rdgfnon g F A w z rdgval rdg.2 tz7.44-2)) thm (rdglim ((g z) (B z) (B g) (g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((rdg.2 (e. B (On)))) (-> (Lim B) (= (` (rec F A) B) (U. (" (rec F A) B)))) (w z A F g rdglem2 F A rdgfnon g F A w z rdgval rdg.2 tz7.44-3)) thm (rdg0t ((A x) (F x)) () (-> (e. A C) (= (` (rec F A) ({/})) A)) ((cv x) A F rdgeq2 ({/}) fveq1d (= (cv x) A) id eqeq12d x visset F rdg0 C vtoclg)) thm (rdgsuct () () (-> (e. B (On)) (= (` (rec F A) (suc B)) (` F (` (rec F A) B)))) (B (if (e. B (On)) B ({/})) suceq (rec F A) fveq2d B (if (e. B (On)) B ({/})) (rec F A) fveq2 F fveq2d eqeq12d 0elon B elimel F A rdgsuc dedth)) thm (rdgsucopab ((D z) (y z) (C y) (C z) (A z) (B z) (x y) (x z)) ((rdgsucopab.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (rdgsucopab.2 (-> (e. (cv z) B) (A. x (e. (cv z) B)))) (rdgsucopab.3 (-> (e. (cv z) D) (A. x (e. (cv z) D)))) (rdgsucopab.4 (= F (rec ({<,>|} x y (= (cv y) C)) A))) (rdgsucopab.5 (-> (= (cv x) (` F B)) (= C D)))) (-> (/\ (e. B (On)) (e. D R)) (= (` F (suc B)) D)) (B ({<,>|} x y (= (cv y) C)) A rdgsuct rdgsucopab.4 (suc B) fveq1i syl5eq (rec ({<,>|} x y (= (cv y) C)) A) B fvex z x y (= (cv y) C) hbopab1 rdgsucopab.1 hbrdg rdgsucopab.2 hbfv rdgsucopab.3 rdgsucopab.4 B fveq1i (cv x) eqeq2i rdgsucopab.5 sylbir (V) R y fvopabgf mpan sylan9eq)) thm (rdgsucopabn ((D z) (y z) (C y) (C z) (A z) (B z) (x y) (x z)) ((rdgsucopab.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (rdgsucopab.2 (-> (e. (cv z) B) (A. x (e. (cv z) B)))) (rdgsucopab.3 (-> (e. (cv z) D) (A. x (e. (cv z) D)))) (rdgsucopab.4 (= F (rec ({<,>|} x y (= (cv y) C)) A))) (rdgsucopab.5 (-> (= (cv x) (` F B)) (= C D)))) (-> (-. (e. D (V))) (= (` F (suc B)) ({/}))) (B ({<,>|} x y (= (cv y) C)) A rdgsuct rdgsucopab.4 (suc B) fveq1i syl5eq z x y (= (cv y) C) hbopab1 rdgsucopab.1 hbrdg rdgsucopab.2 hbfv rdgsucopab.3 rdgsucopab.4 B fveq1i (cv x) eqeq2i rdgsucopab.5 sylbir y fvopabnf sylan9eq ex B sucelon rdgsucopab.4 dmeqi ({<,>|} x y (= (cv y) C)) A rdgfnon (rec ({<,>|} x y (= (cv y) C)) A) (On) fndm ax-mp eqtr (suc B) eleq2i bitr4 negbii (suc B) F ndmfv sylbi (-. (e. D (V))) a1d pm2.61i)) thm (rdglimt () () (-> (/\ (e. B C) (Lim B)) (= (` (rec F A) B) (U. (" (rec F A) B)))) (B (if (e. B (On)) B ({/})) limeq B (if (e. B (On)) B ({/})) (rec F A) fveq2 B (if (e. B (On)) B ({/})) (rec F A) imaeq2 unieqd eqeq12d imbi12d 0elon B elimel F A rdglim dedth imp B C limelon sylan anabss3)) thm (rdglim2 ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (-> (/\ (e. B C) (Lim B)) (= (` (rec F A) B) (U. ({|} y (E.e. x B (= (cv y) (` (rec F A) (cv x)))))))) (B C F A rdglimt B limord B (cv x) ordelord ex x visset elon syl6ibr syl F A rdgfnon y visset (rec F A) (On) (cv x) fnopfvb mpan (cv y) (` (rec F A) (cv x)) eqcom syl5bb syl6 pm5.32d x exbidv x B (= (cv y) (` (rec F A) (cv x))) df-rex syl5rbb y abbidv (rec F A) B y x dfima3 syl5eq unieqd (e. B C) adantl eqtrd)) thm (rdglim2a ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (-> (/\ (e. B C) (Lim B)) (= (` (rec F A) B) (U_ x B (` (rec F A) (cv x))))) (B C F A y x rdglim2 (rec F A) (cv x) fvex x B y dfiun2 syl6eqr)) thm (frfnom () () (Fn (|` (rec F A) (om)) (om)) ((|` (rec F A) (om)) (om) df-fn F A rdgfnon (rec F A) (On) fnfun ax-mp (rec F A) (om) funres ax-mp F A rdgfnon (rec F A) (On) fndm ax-mp (om) ineq2i (rec F A) (om) dmres omsson (om) (On) dfss mpbi 3eqtr4 mpbir2an)) thm (fr0t () () (-> (e. A B) (= (` (|` (rec F A) (om)) ({/})) A)) (A B F rdg0t peano1 ({/}) (om) (rec F A) fvres ax-mp syl5eq)) thm (frsuct () () (-> (e. B (om)) (= (` (|` (rec F A) (om)) (suc B)) (` F (` (|` (rec F A) (om)) B)))) (B nnont B F A rdgsuct syl B peano2b (suc B) (om) (rec F A) fvres sylbi B (om) (rec F A) fvres F fveq2d 3eqtr4d)) thm (frsucopab ((D z) (y z) (C y) (C z) (A z) (B z) (x y) (x z)) ((frsucopab.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (frsucopab.2 (-> (e. (cv z) B) (A. x (e. (cv z) B)))) (frsucopab.3 (-> (e. (cv z) D) (A. x (e. (cv z) D)))) (frsucopab.4 (= F (|` (rec ({<,>|} x y (= (cv y) C)) A) (om)))) (frsucopab.5 (-> (= (cv x) (` F B)) (= C D)))) (-> (/\ (e. B (om)) (e. D R)) (= (` F (suc B)) D)) (B ({<,>|} x y (= (cv y) C)) A frsuct frsucopab.4 (suc B) fveq1i syl5eq (|` (rec ({<,>|} x y (= (cv y) C)) A) (om)) B fvex z x y (= (cv y) C) hbopab1 frsucopab.1 hbrdg (e. (cv z) (om)) x ax-17 hbres frsucopab.2 hbfv frsucopab.3 frsucopab.4 B fveq1i (cv x) eqeq2i frsucopab.5 sylbir (V) R y fvopabgf mpan sylan9eq)) thm (tz7.48lem ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A x) (A y) (A z) (A w)) ((tz7.48.1 (Fn F (On)))) (-> (/\ (C_ A (On)) (A.e. x A (A.e. y (cv x) (-. (= (` F (cv x)) (` F (cv y))))))) (Fun (`' (|` F A)))) ((cv x) A F fvres (cv y) A F fvres eqeqan12d (C_ A (On)) (/\ (-> (e. (cv x) (cv y)) (-. (= (` F (cv y)) (` F (cv x))))) (-> (e. (cv y) (cv x)) (-. (= (` F (cv x)) (` F (cv y)))))) ad2antrl A (On) (cv x) ssel A (On) (cv y) ssel anim12d (e. (cv x) (cv y)) (-. (= (` F (cv y)) (` F (cv x)))) (e. (cv y) (cv x)) (-. (= (` F (cv x)) (` F (cv y)))) pm3.48 (-. (= (` F (cv x)) (` F (cv y)))) oridm (` F (cv x)) (` F (cv y)) eqcom negbii (-. (= (` F (cv x)) (` F (cv y)))) orbi1i bitr3 syl6ibr con2d (cv x) (cv y) ordtri3 biimprd (cv x) eloni (cv y) eloni syl2an syl9r syl6 imp32 sylbid exp32 a2d y 19.20dv x 19.20dv x A y A (/\ (-> (e. (cv x) (cv y)) (-. (= (` F (cv y)) (` F (cv x))))) (-> (e. (cv y) (cv x)) (-. (= (` F (cv x)) (` F (cv y)))))) r2al x A y A (-> (= (` (|` F A) (cv x)) (` (|` F A) (cv y))) (= (cv x) (cv y))) r2al 3imtr4g (e. (cv x) A) (e. (cv y) A) pm3.26 (e. (cv y) (cv x)) anim1i (-. (= (` F (cv x)) (` F (cv y)))) imim1i exp3a x y 19.20i2 x A y (cv x) (-. (= (` F (cv x)) (` F (cv y)))) r2al x A y A (-> (e. (cv y) (cv x)) (-. (= (` F (cv x)) (` F (cv y))))) r2al 3imtr4 (cv y) (cv w) (cv x) eleq1 (cv y) (cv w) F fveq2 (` F (cv x)) eqeq2d negbid imbi12d A cbvralv x A ralbii (cv x) (cv z) (cv w) eleq2 (cv x) (cv z) F fveq2 (` F (cv w)) eqeq1d negbid imbi12d w A ralbidv A cbvralv (cv w) (cv x) (cv z) eleq1 (cv w) (cv x) F fveq2 (` F (cv z)) eqeq2d negbid imbi12d A cbvralv z A ralbii (cv z) (cv y) (cv x) eleq2 (cv z) (cv y) F fveq2 (` F (cv x)) eqeq1d negbid imbi12d x A ralbidv A cbvralv bitr 3bitr y A x (-> (e. (cv x) (cv y)) (-. (= (` F (cv y)) (` F (cv x))))) ralcom2 sylbi ancri x A y A (-> (e. (cv x) (cv y)) (-. (= (` F (cv y)) (` F (cv x))))) (-> (e. (cv y) (cv x)) (-. (= (` F (cv x)) (` F (cv y))))) r19.26-2 sylibr syl syl5 imdistani (|` F A) A (V) x y f1fv (|` F A) A (V) df-f1 bitr3 pm3.27bd (|` F A) A fnf sylanb tz7.48.1 F (On) A fnssres mpan sylan syl)) thm (tz7.48-1 ((x y) (F x) (F y) (A x) (A y)) ((tz7.48.1 (Fn F (On)))) (-> (A.e. x (On) (e. (` F (cv x)) (\ A (" F (cv x))))) (C_ (ran F) A)) (x (On) (e. (` F (cv x)) (\ A (" F (cv x)))) hbra1 (e. (cv y) A) x ax-17 x (On) (e. (` F (cv x)) (\ A (" F (cv x)))) ra4 (` F (cv x)) (cv y) A eleq1 (` F (cv x)) A (" F (cv x)) eldifi syl5bi com12 (e. (cv x) (On)) imim2i imp3a syl 19.23ad y visset F x elrn2 x visset (cv y) F opeldm tz7.48.1 F (On) fndm ax-mp syl6eleq ancri tz7.48.1 y visset F (On) (cv x) fnopfvb mpan pm5.32i sylibr x 19.22i sylbi syl5 ssrdv)) thm (tz7.48-2 ((x y) (F x) (F y) (A x) (A y)) ((tz7.48.1 (Fn F (On)))) (-> (A.e. x (On) (e. (` F (cv x)) (\ A (" F (cv x))))) (Fun (`' F))) ((cv x) (cv y) onelon ancoms F (cv x) dmres (cv y) eleq2i (cv y) (cv x) (dom F) elin bitr tz7.48.1 F (On) fnfun ax-mp F (cv x) funres ax-mp (|` F (cv x)) (cv y) fvrn mpan sylbir (cv y) (cv x) F fvres (ran (|` F (cv x))) eleq1d F (cv x) df-ima (` F (cv y)) eleq2i syl6rbbr (e. (cv y) (dom F)) adantr mpbird (` F (cv y)) (" F (cv x)) (` F (cv x)) eleq1a (` F (cv x)) A (" F (cv x)) eldifn nsyli syl tz7.48.1 F (On) fndm ax-mp (cv y) eleq2i sylan2br syldan ex imp3a com12 r19.21aiv ex r19.20i (On) ssid tz7.48.1 (On) x y tz7.48lem mpan syl tz7.48.1 F (On) fnrel ax-mp tz7.48.1 F (On) fndm ax-mp (On) ssid eqsstr F (On) relssres mp2an (|` F (On)) F cnveq ax-mp (`' (|` F (On))) (`' F) funeq ax-mp sylib)) thm (tz7.48-3 ((F x) (A x)) ((tz7.48.1 (Fn F (On)))) (-> (A.e. x (On) (e. (` F (cv x)) (\ A (" F (cv x))))) (-. (e. A (V)))) (onprc tz7.48.1 F (On) fndm ax-mp (V) eleq1i mtbir tz7.48.1 x A tz7.48-2 (`' F) (V) funrnex com12 F df-rn (V) eleq1i F dfdm4 (V) eleq1i 3imtr4g syl mtoi tz7.48.1 x A tz7.48-1 (ran F) A (V) ssexg ex syl mtod)) thm (tz7.49 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((tz7.48.1 (Fn F (On))) (tz7.49.2 (e. A (V)))) (-> (A.e. x (On) (-> (-. (= (\ A (" F (cv x))) ({/}))) (e. (` F (cv x)) (\ A (" F (cv x)))))) (E.e. x (On) (/\/\ (A.e. y (cv x) (-. (= (\ A (" F (cv y))) ({/})))) (= (" F (cv x)) A) (Fun (`' (|` F (cv x))))))) (tz7.49.2 tz7.48.1 x A tz7.48-3 mt2 x (On) (-. (= (\ A (" F (cv x))) ({/}))) (e. (` F (cv x)) (\ A (" F (cv x)))) r19.20 mtoi x (On) (= (\ A (" F (cv x))) ({/})) dfrex2 sylibr (cv x) (cv y) F imaeq2 A difeq2d ({/}) eqeq1d onminex syl x (On) (-> (-. (= (\ A (" F (cv x))) ({/}))) (e. (` F (cv x)) (\ A (" F (cv x))))) hbra1 (A.e. x (On) (-> (-. (= (\ A (" F (cv x))) ({/}))) (e. (` F (cv x)) (\ A (" F (cv x)))))) (A.e. y (cv x) (-. (= (\ A (" F (cv y))) ({/})))) pm3.27 (e. (cv x) (On)) (= (\ A (" F (cv x))) ({/})) ad2antrr (A.e. x (On) (-> (-. (= (\ A (" F (cv x))) ({/}))) (e. (` F (cv x)) (\ A (" F (cv x)))))) y ax-17 y (cv x) (-. (= (\ A (" F (cv y))) ({/}))) hbra1 hban (-> (e. (cv x) (On)) (e. (cv z) A)) y ax-17 y (cv x) (-. (= (\ A (" F (cv y))) ({/}))) ra4 (e. (cv x) (On)) adantld (cv x) (cv y) onelon (cv x) (cv y) F imaeq2 A difeq2d ({/}) eqeq1d negbid (cv x) (cv y) F fveq2 (cv x) (cv y) F imaeq2 A difeq2d eleq12d imbi12d (On) rcla4v com23 syl sylcom com3r imp exp3a com23 (` F (cv y)) (cv z) A eleq1 (` F (cv y)) A (" F (cv y)) eldifi syl5bi com12 syl8 com34 r19.23ad tz7.48.1 F (On) fnfun ax-mp F (cv z) (cv x) y fvelima mpan syl5 com23 imp ssrdv A (" F (cv x)) ssdif0 biimpr anim12i (" F (cv x)) A eqss sylibr y (cv x) (-. (= (\ A (" F (cv y))) ({/}))) hbra1 (A.e. x (On) (-> (-. (= (\ A (" F (cv x))) ({/}))) (e. (` F (cv x)) (\ A (" F (cv x)))))) y ax-17 hban (C_ (cv x) (On)) y ax-17 (cv x) (On) (cv y) ssel (cv y) onsst tz7.48.1 F (On) fndm ax-mp syl6ssr tz7.48.1 F (On) fnfun ax-mp F (cv y) (cv z) funfvima2 mpan syl syl6 y (cv x) (-. (= (\ A (" F (cv y))) ({/}))) ra4 com12 (C_ (cv x) (On)) a1i (cv x) (On) (cv y) ssel (cv x) (cv y) F imaeq2 A difeq2d ({/}) eqeq1d negbid (cv x) (cv y) F fveq2 (cv x) (cv y) F imaeq2 A difeq2d eleq12d imbi12d (On) rcla4v com23 syl6 syldd imp4a (` F (cv z)) (" F (cv y)) (` F (cv y)) eleq1a con3d (` F (cv y)) A (" F (cv y)) eldifn syl5com syl8 com34 syldd com4r imp4a z 19.21adv 19.21ad y (cv x) z (cv y) (-. (= (` F (cv y)) (` F (cv z)))) r2al syl6ibr ancld tz7.48.1 (cv x) y z tz7.48lem syl6 (cv x) onsst syl5 ancoms imp (= (\ A (" F (cv x))) ({/})) adantr 3jca exp41 com23 com34 imp4a r19.22d mpd)) thm (tz7.49c ((x y) (F x) (F y) (A x) (A y)) ((tz7.48.1 (Fn F (On))) (tz7.49.2 (e. A (V)))) (-> (A.e. x (On) (-> (-. (= (\ A (" F (cv x))) ({/}))) (e. (` F (cv x)) (\ A (" F (cv x)))))) (E.e. x (On) (:-1-1-onto-> (|` F (cv x)) (cv x) A))) (tz7.48.1 tz7.49.2 x y tz7.49 (cv x) onsst tz7.48.1 F (On) (cv x) fnssres mpan syl F (cv x) df-ima A eqeq1i biimp anim12i (Fun (`' (|` F (cv x)))) anim1i (|` F (cv x)) (cv x) A f1o2 (Fn (|` F (cv x)) (cv x)) (Fun (`' (|` F (cv x)))) (= (ran (|` F (cv x))) A) df-3an (Fn (|` F (cv x)) (cv x)) (Fun (`' (|` F (cv x)))) (= (ran (|` F (cv x))) A) an23 3bitr sylibr exp31 imp3a (A.e. y (cv x) (-. (= (\ A (" F (cv y))) ({/})))) (= (" F (cv x)) A) (Fun (`' (|` F (cv x)))) 3simpc syl5 r19.22i syl)) thm (abianfplem ((x y) (v x) (A x) (v y) (A y) (A v) (x z) (w x) (u x) (F x) (y z) (w y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (G y) (G v) (G u)) ((abianfp.1 (e. A (V))) (abianfp.2 (= G (rec ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x))))) (-> (e. (cv v) (On)) (-> (= (` F (cv x)) (cv x)) (= (` G (cv v)) (cv x)))) ((cv v) ({/}) G fveq2 (cv x) eqeq1d (cv v) (cv y) G fveq2 (cv x) eqeq1d (cv v) (suc (cv y)) G fveq2 (cv x) eqeq1d abianfp.2 ({/}) fveq1i x visset ({<,>|} z w (= (cv w) (` F (cv z)))) rdg0 eqtr (= (` F (cv x)) (cv x)) a1i F (` G (cv y)) fvex (e. (cv u) (cv x)) z ax-17 (e. (cv u) (cv y)) z ax-17 (e. (cv u) F) z ax-17 u z w (= (cv w) (` F (cv z))) hbopab1 (e. (cv u) (cv x)) z ax-17 hbrdg abianfp.2 (cv u) eleq2i abianfp.2 (cv u) eleq2i z albii 3imtr4 (e. (cv u) (cv y)) z ax-17 hbfv hbfv abianfp.2 (cv z) (` G (cv y)) F fveq2 (V) rdgsucopab mpan2 (` G (cv y)) (cv x) F fveq2 (= (` F (cv x)) (cv x)) id sylan9eqr sylan9eq exp32 v visset (cv v) (V) ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x) y rdglim2a mpan abianfp.2 (cv v) fveq1i abianfp.2 (cv y) fveq1i (e. (cv y) (cv v)) a1i iuneq2i 3eqtr4g (A.e. y (cv v) (= (` G (cv y)) (cv x))) adantr y (cv v) (` G (cv y)) (cv x) iuneq2 (cv v) df-lim (Ord (cv v)) (-. (= (cv v) ({/}))) (= (cv v) (U. (cv v))) 3simp2 sylbi (cv v) y (cv x) iunconst syl sylan9eqr eqtrd ex (= (` F (cv x)) (cv x)) a1d tfinds2)) thm (abianfp ((x y) (v x) (A x) (v y) (A y) (A v) (x z) (w x) (u x) (F x) (y z) (w y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (G y) (G v) (G u)) ((abianfp.1 (e. A (V))) (abianfp.2 (= G (rec ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x))))) (<-> (E.e. x A (= (` F (cv x)) (cv x))) (E.e. x A (A.e. v (On) (/\ (e. (` G (cv v)) A) (-> (-. (= (` F (` G (cv v))) (` G (cv v)))) (A.e. u (cv v) (-. (= (` G (cv v)) (` G (cv u)))))))))) (abianfp.1 abianfp.2 v abianfplem imp A eleq1d biimprd abianfp.1 abianfp.2 v abianfplem (` G (cv v)) (cv x) F fveq2 (= (` G (cv v)) (cv x)) id eqeq12d biimprcd sylcom imp (A.e. u (cv v) (-. (= (` G (cv v)) (` G (cv u))))) pm2.21nd jctird ex com13 r19.21adv r19.22i onprc v (On) (e. (` G (cv v)) A) (-> (-. (= (` F (` G (cv v))) (` G (cv v)))) (A.e. u (cv v) (-. (= (` G (cv v)) (` G (cv u)))))) r19.26 (cv y) (` G (cv v)) F fveq2 (= (cv y) (` G (cv v))) id eqeq12d negbid A rcla4cv (A.e. u (cv v) (-. (= (` G (cv v)) (` G (cv u))))) imim1d v (On) r19.20sdv v (On) (e. (` G (cv v)) A) (A.e. u (cv v) (-. (= (` G (cv v)) (` G (cv u))))) r19.20 syl6 imp com12 ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x) rdgfnon abianfp.2 G (rec ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x)) (On) fneq1 ax-mp mpbir G (On) A v ffnfv biimpr mpan (On) ssid ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x) rdgfnon abianfp.2 G (rec ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x)) (On) fneq1 ax-mp mpbir (On) v u tz7.48lem mpan ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x) rdgfnon abianfp.2 G (rec ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x)) (On) fneq1 ax-mp mpbir G (On) fnresdm ax-mp (|` G (On)) G cnveq ax-mp (`' (|` G (On))) (`' G) funeq ax-mp sylib anim12i G (On) A df-f1 sylibr abianfp.1 G (On) A (V) f1dmex mpan2 syl ex syld exp3a com23 imp sylbi mtoi (e. (cv x) A) a1i r19.23aiv (cv x) (cv y) F fveq2 (= (cv x) (cv y)) id eqeq12d A cbvrexv y A (= (` F (cv y)) (cv y)) dfrex2 bitr2 sylib impbi)) thm (opreq () () (-> (= F G) (= (opr A F B) (opr A G B))) (F G (<,> A B) fveq1 A F B df-opr A G B df-opr 3eqtr4g)) thm (opreq1 () () (-> (= A B) (= (opr A F C) (opr B F C))) (A B C opeq1 F fveq2d A F C df-opr B F C df-opr 3eqtr4g)) thm (opreq2 () () (-> (= A B) (= (opr C F A) (opr C F B))) (A B C opeq2 F fveq2d C F A df-opr C F B df-opr 3eqtr4g)) thm (opreq12 () () (-> (/\ (= A B) (= C D)) (= (opr A F C) (opr B F D))) (A B F C opreq1 C D B F opreq2 sylan9eq)) thm (opreq1i () ((opreq1i.1 (= A B))) (= (opr A F C) (opr B F C)) (opreq1i.1 A B F C opreq1 ax-mp)) thm (opreq2i () ((opreq1i.1 (= A B))) (= (opr C F A) (opr C F B)) (opreq1i.1 A B C F opreq2 ax-mp)) thm (opreq12i () ((opreq1i.1 (= A B)) (opreq12i.2 (= C D))) (= (opr A F C) (opr B F D)) (opreq1i.1 F C opreq1i opreq12i.2 B F opreq2i eqtr)) thm (opreq1d () ((opreq1d.1 (-> ph (= A B)))) (-> ph (= (opr A F C) (opr B F C))) (opreq1d.1 A B F C opreq1 syl)) thm (opreq2d () ((opreq1d.1 (-> ph (= A B)))) (-> ph (= (opr C F A) (opr C F B))) (opreq1d.1 A B C F opreq2 syl)) thm (opreqd () ((opreq1d.1 (-> ph (= A B)))) (-> ph (= (opr C A D) (opr C B D))) (opreq1d.1 A B C D opreq syl)) thm (opreq12d () ((opreq1d.1 (-> ph (= A B))) (opreq12d.2 (-> ph (= C D)))) (-> ph (= (opr A F C) (opr B F D))) (opreq1d.1 F C opreq1d opreq12d.2 B F opreq2d eqtrd)) thm (opreqan12d () ((opreq1d.1 (-> ph (= A B))) (opreqan12i.2 (-> ps (= C D)))) (-> (/\ ph ps) (= (opr A F C) (opr B F D))) (A B C D F opreq12 opreq1d.1 opreqan12i.2 syl2an)) thm (opreqan12rd () ((opreq1d.1 (-> ph (= A B))) (opreqan12i.2 (-> ps (= C D)))) (-> (/\ ps ph) (= (opr A F C) (opr B F D))) (opreq1d.1 opreqan12i.2 F opreqan12d ancoms)) thm (hbopr ((F y) (A y) (B y) (x y)) ((hbopr.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbopr.2 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (hbopr.3 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (opr A F B)) (A. x (e. (cv y) (opr A F B)))) (hbopr.2 hbopr.1 hbopr.3 hbop hbfv A F B df-opr (cv y) eleq2i A F B df-opr (cv y) eleq2i x albii 3imtr4)) thm (hboprd ((y z) (A y) (A z) (B y) (B z) (F y) (F z) (x y) (x z) (ph y)) ((hboprd.1 (-> ph (A. x ph))) (hboprd.2 (-> ph (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (hboprd.3 (-> ph (-> (e. (cv y) F) (A. x (e. (cv y) F))))) (hboprd.4 (-> ph (-> (e. (cv y) B) (A. x (e. (cv y) B)))))) (-> ph (-> (e. (cv y) (opr A F B)) (A. x (e. (cv y) (opr A F B))))) (x (e. (cv z) A) hba1 y z hbab x (e. (cv z) F) hba1 y z hbab x (e. (cv z) B) hba1 y z hbab hbopr ph a1i hboprd.2 y 19.21aiv y A x z abidhb syl hboprd.4 y 19.21aiv y B x z abidhb syl ({|} z (A. x (e. (cv z) F))) opreq12d hboprd.3 y 19.21aiv y F x z abidhb ({|} z (A. x (e. (cv z) F))) F A B opreq 3syl eqtrd (cv y) eleq2d hboprd.1 hboprd.2 y 19.21aiv y A x z abidhb syl hboprd.4 y 19.21aiv y B x z abidhb syl ({|} z (A. x (e. (cv z) F))) opreq12d hboprd.3 y 19.21aiv y F x z abidhb ({|} z (A. x (e. (cv z) F))) F A B opreq 3syl eqtrd (cv y) eleq2d albid 3imtr3d)) thm (oprex () () (e. (opr A F B) (V)) (A F B df-opr F (<,> A B) fvex eqeltr)) thm (oprprc1 () ((oprprc1.1 (Rel (dom F)))) (-> (-. (e. A (V))) (= (opr A F B) ({/}))) (A (dom F) B df-br oprprc1.1 A B brrelexi sylbir con3i (<,> A B) F ndmfv syl A F B df-opr syl5eq)) thm (oprprc2 () () (-> (-. (e. B (V))) (= (opr A F B) (opr A F A))) (B A opprc2 F fveq2d A F B df-opr A F A df-opr 3eqtr4g)) thm (csboprg ((y z) (A y) (A z) (B y) (B z) (C y) (C z) (D y) (D z) (F y) (F z) (x y) (x z)) () (-> (e. A D) (= ([_/]_ A x (opr B F C)) (opr ([_/]_ A x B) ([_/]_ A x F) ([_/]_ A x C)))) ((e. A D) y ax-17 (e. (cv z) A) y ax-17 D ([_/]_ (cv y) x B) hbcsb1g (e. (cv z) A) y ax-17 D ([_/]_ (cv y) x F) hbcsb1g (e. (cv z) A) y ax-17 D ([_/]_ (cv y) x C) hbcsb1g hboprd x y a9e y visset (e. (cv z) (cv y)) x ax-17 (opr B F C) hbcsb1 y visset (e. (cv z) (cv y)) x ax-17 B hbcsb1 y visset (e. (cv z) (cv y)) x ax-17 F hbcsb1 y visset (e. (cv z) (cv y)) x ax-17 C hbcsb1 hbopr hbeq x (cv y) B csbeq1a x (cv y) C csbeq1a F opreq12d x (cv y) (opr B F C) csbeq1a x (cv y) F csbeq1a ([_/]_ (cv y) x B) ([_/]_ (cv y) x C) opreqd 3eqtr3d 19.23ai ax-mp (= (cv y) A) a1i y A ([_/]_ (cv y) x B) csbeq1a y A ([_/]_ (cv y) x C) csbeq1a ([_/]_ (cv y) x F) opreq12d y A ([_/]_ (cv y) x F) csbeq1a ([_/]_ A y ([_/]_ (cv y) x B)) ([_/]_ A y ([_/]_ (cv y) x C)) opreqd 3eqtrd csbiegf A D y x (opr B F C) csbcog A D y x B csbcog A D y x C csbcog ([_/]_ A y ([_/]_ (cv y) x F)) opreq12d A D y x F csbcog ([_/]_ A x B) ([_/]_ A x C) opreqd eqtrd 3eqtr3d)) thm (csbopr12g ((A y) (C y) (D y) (x y) (F x) (F y)) () (-> (e. A D) (= ([_/]_ A x (opr B F C)) (opr ([_/]_ A x B) F ([_/]_ A x C)))) (A D x B F C csboprg (e. (cv y) F) x ax-17 A D csbconstgf ([_/]_ A x B) ([_/]_ A x C) opreqd eqtrd)) thm (csbopr1g ((A y) (x y) (C x) (C y) (D y) (F x) (F y)) () (-> (e. A D) (= ([_/]_ A x (opr B F C)) (opr ([_/]_ A x B) F C))) (A D x B F C csbopr12g (e. (cv y) C) x ax-17 A D csbconstgf ([_/]_ A x B) F opreq2d eqtrd)) thm (csbopr2g ((A y) (x y) (B x) (B y) (D y) (F x) (F y)) () (-> (e. A D) (= ([_/]_ A x (opr B F C)) (opr B F ([_/]_ A x C)))) (A D x B F C csbopr12g (e. (cv y) B) x ax-17 A D csbconstgf F ([_/]_ A x C) opreq1d eqtrd)) thm (dfoprab2 ((x z) (w x) (v x) (w z) (v z) (v w) (y z) (w y) (v y) (ph w) (ph v)) () (= ({<<,>,>|} x y z ph) ({<,>|} w z (E. x (E. y (/\ (= (cv w) (<,> (cv x) (cv y))) ph))))) (z w (E. x (E. y (/\ (= (cv v) (<,> (cv w) (cv z))) (/\ (= (cv w) (<,> (cv x) (cv y))) ph)))) excom z w x y (/\ (= (cv v) (<,> (cv w) (cv z))) (/\ (= (cv w) (<,> (cv x) (cv y))) ph)) exrot4 w (/\ (= (cv v) (<,> (<,> (cv x) (cv y)) (cv z))) ph) (= (cv w) (<,> (cv x) (cv y))) 19.42v (cv w) (<,> (cv x) (cv y)) (cv z) opeq1 (cv v) eqeq2d pm5.32ri ph anbi1i (= (cv v) (<,> (cv w) (cv z))) (= (cv w) (<,> (cv x) (cv y))) ph anass (= (cv v) (<,> (<,> (cv x) (cv y)) (cv z))) (= (cv w) (<,> (cv x) (cv y))) ph an23 3bitr3 w exbii (cv x) (cv y) opex w isseti (/\ (= (cv v) (<,> (<,> (cv x) (cv y)) (cv z))) ph) biantru 3bitr4 x y z 3exbi bitr x y (= (cv v) (<,> (cv w) (cv z))) (/\ (= (cv w) (<,> (cv x) (cv y))) ph) 19.42vv w z 2exbii 3bitr3 v abbii x y z ph v df-oprab w z (E. x (E. y (/\ (= (cv w) (<,> (cv x) (cv y))) ph))) v df-opab 3eqtr4)) thm (reloprab ((x z) (w x) (w z) (y z) (w y) (ph w)) () (Rel ({<<,>,>|} x y z ph)) (w z (E. x (E. y (/\ (= (cv w) (<,> (cv x) (cv y))) ph))) relopab x y z ph w dfoprab2 ({<<,>,>|} x y z ph) ({<,>|} w z (E. x (E. y (/\ (= (cv w) (<,> (cv x) (cv y))) ph)))) releq ax-mp mpbir)) thm (hboprab1 ((x y) (x z) (v x) (y z) (v y) (v z) (w x) (ph v)) () (-> (e. (cv w) ({<<,>,>|} x y z ph)) (A. x (e. (cv w) ({<<,>,>|} x y z ph)))) (x (E. y (E. z (/\ (= (cv v) (<,> (<,> (cv x) (cv y)) (cv z))) ph))) hbe1 w v hbab x y z ph v df-oprab (cv w) eleq2i x y z ph v df-oprab (cv w) eleq2i x albii 3imtr4)) thm (hboprab2 ((x y) (x z) (v x) (y z) (v y) (v z) (w y) (ph v)) () (-> (e. (cv w) ({<<,>,>|} x y z ph)) (A. y (e. (cv w) ({<<,>,>|} x y z ph)))) (y (E. z (/\ (= (cv v) (<,> (<,> (cv x) (cv y)) (cv z))) ph)) hbe1 x hbex w v hbab x y z ph v df-oprab (cv w) eleq2i x y z ph v df-oprab (cv w) eleq2i y albii 3imtr4)) thm (oprabbid ((x z) (w x) (w z) (y z) (w y) (ph w) (ps w) (ch w)) ((oprabbid.1 (-> ph (A. x ph))) (oprabbid.2 (-> ph (A. y ph))) (oprabbid.3 (-> ph (A. z ph))) (oprabbid.4 (-> ph (<-> ps ch)))) (-> ph (= ({<<,>,>|} x y z ps) ({<<,>,>|} x y z ch))) (ph w ax-17 oprabbid.3 oprabbid.1 oprabbid.2 oprabbid.4 (= (cv w) (<,> (cv x) (cv y))) anbi2d exbid exbid opabbid x y z ps w dfoprab2 x y z ch w dfoprab2 3eqtr4g)) thm (oprabbidv ((x z) (ph x) (ph z) (y z) (ph y)) ((oprabbidv.1 (-> ph (<-> ps ch)))) (-> ph (= ({<<,>,>|} x y z ps) ({<<,>,>|} x y z ch))) (ph x ax-17 ph y ax-17 ph z ax-17 oprabbidv.1 oprabbid)) thm (oprabbii ((x y) (x z) (w x) (y z) (w y) (w z) (ph w) (ps w)) ((oprabbii.1 (<-> ph ps))) (= ({<<,>,>|} x y z ph) ({<<,>,>|} x y z ps)) ((cv w) eqid oprabbii.1 (= (cv w) (cv w)) a1i x y z oprabbidv ax-mp)) thm (cbvoprab12 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph u) (ps u)) ((cbvoprab12.1 (-> ph (A. w ph))) (cbvoprab12.2 (-> ph (A. v ph))) (cbvoprab12.3 (-> ps (A. x ps))) (cbvoprab12.4 (-> ps (A. y ps))) (cbvoprab12.5 (-> (/\ (= (cv x) (cv w)) (= (cv y) (cv v))) (<-> ph ps)))) (= ({<<,>,>|} x y z ph) ({<<,>,>|} w v z ps)) ((= (cv u) (<,> (<,> (cv x) (cv y)) (cv z))) w ax-17 cbvoprab12.1 hban z hbex (= (cv u) (<,> (<,> (cv x) (cv y)) (cv z))) v ax-17 cbvoprab12.2 hban z hbex (= (cv u) (<,> (<,> (cv w) (cv v)) (cv z))) x ax-17 cbvoprab12.3 hban z hbex (= (cv u) (<,> (<,> (cv w) (cv v)) (cv z))) y ax-17 cbvoprab12.4 hban z hbex (cv x) (cv w) (cv y) (cv v) opeq12 (<,> (cv x) (cv y)) (<,> (cv w) (cv v)) (cv z) opeq1 syl (cv u) eqeq2d cbvoprab12.5 anbi12d z exbidv cbvex2 u abbii x y z ph u df-oprab w v z ps u df-oprab 3eqtr4)) thm (cbvoprab12v ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph w) (ph v) (ps x) (ps y)) ((cbvoprab12v.1 (-> (/\ (= (cv x) (cv w)) (= (cv y) (cv v))) (<-> ph ps)))) (= ({<<,>,>|} x y z ph) ({<<,>,>|} w v z ps)) (ph w ax-17 ph v ax-17 ps x ax-17 ps y ax-17 cbvoprab12v.1 z cbvoprab12)) thm (cbvoprab3v ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph w) (ph v) (ps z) (ps v)) ((cbvoprab3v.1 (-> (= (cv z) (cv w)) (<-> ph ps)))) (= ({<<,>,>|} x y z ph) ({<<,>,>|} x y w ps)) (cbvoprab3v.1 (= (cv v) (<,> (cv x) (cv y))) anbi2d x y 2exbidv v cbvopab2v x y z ph v dfoprab2 x y w ps v dfoprab2 3eqtr4)) thm (dmoprab ((x y) (x z) (w x) (y z) (w y) (w z) (ph w)) () (= (dom ({<<,>,>|} x y z ph)) ({<,>|} x y (E. z ph))) (x y z ph w dfoprab2 dmeqi w z (E. x (E. y (/\ (= (cv w) (<,> (cv x) (cv y))) ph))) dmopab z x y (/\ (= (cv w) (<,> (cv x) (cv y))) ph) exrot3 z (= (cv w) (<,> (cv x) (cv y))) ph 19.42v x y 2exbii bitr w abbii x y (E. z ph) w df-opab eqtr4 3eqtr)) thm (dmoprabss ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (C_ (dom ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph))) (X. A B)) (x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph) dmoprab z (/\ (e. (cv x) A) (e. (cv y) B)) ph 19.42v x y opabbii x y A B (E. z ph) opabssxp eqsstr eqsstr)) thm (rnoprab ((x y) (x z) (w x) (y z) (w y) (w z) (ph w)) () (= (ran ({<<,>,>|} x y z ph)) ({|} z (E. x (E. y ph)))) (x y z ph w dfoprab2 rneqi w z (E. x (E. y (/\ (= (cv w) (<,> (cv x) (cv y))) ph))) rnopab w x y (/\ (= (cv w) (<,> (cv x) (cv y))) ph) exrot3 w (= (cv w) (<,> (cv x) (cv y))) ph 19.41v (cv x) (cv y) opex w isseti mpbiran x y 2exbii bitr z abbii 3eqtr)) thm (reldmoprab ((x y) (x z) (y z)) () (Rel (dom ({<<,>,>|} x y z ph))) (x y (E. z ph) relopab x y z ph dmoprab (dom ({<<,>,>|} x y z ph)) ({<,>|} x y (E. z ph)) releq ax-mp mpbir)) thm (eloprabg ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (ph w) (ps x) (ch x) (ch y) (th x) (th y) (th z) (th w)) ((eloprabg.1 (-> (= (cv x) A) (<-> ph ps))) (eloprabg.2 (-> (= (cv y) B) (<-> ps ch))) (eloprabg.3 (-> (= (cv z) C) (<-> ch th)))) (-> (/\/\ (e. A D) (e. B R) (e. C S)) (<-> (e. (<,> (<,> A B) C) ({<<,>,>|} x y z ph)) th)) ((<,> A B) C opex (cv w) (<,> (<,> A B) C) (<,> (<,> (cv x) (cv y)) (cv z)) eqeq1 (<,> (<,> A B) C) (<,> (<,> (cv x) (cv y)) (cv z)) eqcom syl6bb x visset y visset z visset B (V) C (V) A otthg (e. A (V)) 3adant1 sylan9bbr ph anbi1d eloprabg.1 eloprabg.2 eloprabg.3 syl3an9b pm5.32i syl6bb x y z 3exbidv (cv w) (<,> (<,> A B) C) ({<<,>,>|} x y z ph) eleq1 x y z ph w df-oprab (cv w) eleq2i w (E. x (E. y (E. z (/\ (= (cv w) (<,> (<,> (cv x) (cv y)) (cv z))) ph)))) abid bitr2 syl5bb (/\/\ (e. A (V)) (e. B (V)) (e. C (V))) adantl A (V) x elex B (V) y elex C (V) z elex 3anim123i x y z (= (cv x) A) (= (cv y) B) (= (cv z) C) eeeanv sylibr th biantrurd x y z (/\/\ (= (cv x) A) (= (cv y) B) (= (cv z) C)) th 19.41vvv syl6rbbr (= (cv w) (<,> (<,> A B) C)) adantr 3bitr3d expcom vtocle A D elisset B R elisset C S elisset syl3an)) thm (ssoprab2i ((ph w) (ps w) (x y) (x z) (w x) (y z) (w y) (w z)) ((ssoprab2i.1 (-> ph ps))) (C_ ({<<,>,>|} x y z ph) ({<<,>,>|} x y z ps)) (ssoprab2i.1 (= (cv w) (<,> (cv x) (cv y))) anim2i x y 19.22i2 w z ssopab2i x y z ph w dfoprab2 x y z ps w dfoprab2 3sstr4)) thm (funoprab ((x y) (x z) (w x) (y z) (w y) (w z) (ph w)) ((funoprab.1 (E* z ph))) (Fun ({<<,>,>|} x y z ph)) (w z (E. x (E. y (/\ (= (cv w) (<,> (cv x) (cv y))) ph))) funopab funoprab.1 x y (cv w) mosubop mpgbir x y z ph w dfoprab2 ({<<,>,>|} x y z ph) ({<,>|} w z (E. x (E. y (/\ (= (cv w) (<,> (cv x) (cv y))) ph)))) funeq ax-mp mpbir)) thm (fnoprab ((x y) (x z) (y z) (ph z)) ((fnoprab.1 (-> ph (E! z ps)))) (Fn ({<<,>,>|} x y z (/\ ph ps)) ({<,>|} x y ph)) (({<<,>,>|} x y z (/\ ph ps)) ({<,>|} x y ph) df-fn fnoprab.1 z ps eumo syl z ph ps moanimv mpbir x y funoprab x y z (/\ ph ps) dmoprab ph ps pm3.26 z 19.23aiv fnoprab.1 z ps euex syl ancli z ph ps 19.42v sylibr impbi x y opabbii eqtr mpbir2an)) thm (fnoprab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((fnoprab2.1 (e. C (V))) (fnoprab2.2 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (Fn F (X. A B)) (fnoprab2.1 z eueq1 (/\ (e. (cv x) A) (e. (cv y) B)) a1i x y fnoprab fnoprab2.2 F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))) (X. A B) fneq1 ax-mp A B x y df-xp (X. A B) ({<,>|} x y (/\ (e. (cv x) A) (e. (cv y) B))) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))) fneq2 ax-mp bitr mpbir)) thm (dmoprab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((fnoprab2.1 (e. C (V))) (fnoprab2.2 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (= (dom F) (X. A B)) (fnoprab2.1 fnoprab2.2 fnoprab2 F (X. A B) fndm ax-mp)) thm (ffnoprval ((x y) (w x) (A x) (w y) (A y) (A w) (B x) (B y) (B w) (C x) (C y) (C w) (F x) (F y) (F w)) () (<-> (:--> F (X. A B) C) (/\ (Fn F (X. A B)) (A.e. x A (A.e. y B (e. (opr (cv x) F (cv y)) C))))) (F (X. A B) C w ffnfv (cv w) (<,> (cv x) (cv y)) F fveq2 (cv x) F (cv y) df-opr syl6eqr C eleq1d A B ralxp (Fn F (X. A B)) anbi2i bitr)) thm (eqfnoprval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z) (D z) (F x) (F y) (F z) (G x) (G y) (G z)) () (-> (/\ (Fn F (X. A B)) (Fn G (X. C D))) (<-> (= F G) (/\ (= (X. A B) (X. C D)) (A.e. x A (A.e. y B (= (opr (cv x) F (cv y)) (opr (cv x) G (cv y)))))))) (F (X. A B) G (X. C D) z eqfnfv (cv z) (<,> (cv x) (cv y)) F fveq2 (cv z) (<,> (cv x) (cv y)) G fveq2 eqeq12d (cv x) F (cv y) df-opr (cv x) G (cv y) df-opr eqeq12i syl6bbr A B ralxp (= (X. A B) (X. C D)) anbi2i syl6bb)) thm (fnoprval ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w)) () (<-> (Fn F (X. A B)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (opr (cv x) F (cv y))))))) (F (X. A B) w z fnopabfv (cv w) A B x y elxp (= (cv z) (` F (cv w))) anbi1i x y (/\ (= (cv w) (<,> (cv x) (cv y))) (/\ (e. (cv x) A) (e. (cv y) B))) (= (cv z) (` F (cv w))) 19.41vv (= (cv w) (<,> (cv x) (cv y))) (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (` F (cv w))) anass (cv w) (<,> (cv x) (cv y)) F fveq2 (cv x) F (cv y) df-opr syl6eqr (cv z) eqeq2d (/\ (e. (cv x) A) (e. (cv y) B)) anbi2d pm5.32i bitr x y 2exbii 3bitr2 w z opabbii x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (opr (cv x) F (cv y)))) w dfoprab2 eqtr4 F eqeq2i bitr)) thm (foprval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (F x) (F y) (F z)) () (<-> (:--> F (X. A B) C) (/\ (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (opr (cv x) F (cv y)))))) (A.e. x A (A.e. y B (e. (opr (cv x) F (cv y)) C))))) (F A B C x y ffnoprval F A B x y z fnoprval (A.e. x A (A.e. y B (e. (opr (cv x) F (cv y)) C))) anbi1i bitr)) thm (oprabex ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((oprabex.1 (e. A (V))) (oprabex.2 (e. B (V))) (oprabex.3 (-> (/\ (e. (cv x) A) (e. (cv y) B)) (E* z ph))) (oprabex.4 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph))))) (e. F (V)) (oprabex.4 oprabex.3 z (/\ (e. (cv x) A) (e. (cv y) B)) ph moanimv mpbir x y funoprab oprabex.1 oprabex.2 xpex x y z A B ph dmoprabss ssexi ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph)) (V) funex mp2an eqeltr)) thm (oprabex2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((oprabex2.1 (e. A (V))) (oprabex2.2 (e. B (V))) (oprabex2.3 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (e. F (V)) (oprabex2.1 oprabex2.2 z C moeq (/\ (e. (cv x) A) (e. (cv y) B)) a1i oprabex2.3 oprabex)) thm (oprabex3 ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (H x) (H y) (H z) (H w) (H v) (H u) (H f) (R x) (R y) (R z)) ((oprabex3.1 (e. H (V))) (oprabex3.2 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. H H)) (e. (cv y) (X. H H))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) R))))))))))) (e. F (V)) (oprabex3.1 oprabex3.1 xpex oprabex3.1 oprabex3.1 xpex z R moeq u f (cv y) mosubop w v (cv x) mosubop (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f))) (= (cv z) R) anass u f 2exbii u f (= (cv x) (<,> (cv w) (cv v))) (/\ (= (cv y) (<,> (cv u) (cv f))) (= (cv z) R)) 19.42vv bitr w v 2exbii z mobii mpbir (/\ (e. (cv x) (X. H H)) (e. (cv y) (X. H H))) a1i oprabex3.2 oprabex)) thm (oprabval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((oprabval.1 (e. C (V))) (oprabval.2 (-> (= (cv x) A) (<-> ph ps))) (oprabval.3 (-> (= (cv y) B) (<-> ps ch))) (oprabval.4 (-> (= (cv z) C) (<-> ch th))) (oprabval.5 (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E! z ph))) (oprabval.6 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph))))) (-> (/\ (e. A R) (e. B S)) (<-> (= (opr A F B) C) th)) ((cv x) A R eleq1 (e. (cv y) S) anbi1d (cv y) B S eleq1 (e. A R) anbi2d R S opelopabg ibir oprabval.5 x y fnoprab oprabval.1 ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph)) ({<,>|} x y (/\ (e. (cv x) R) (e. (cv y) S))) (<,> A B) fnopfvb mpan syl oprabval.1 (cv x) A R eleq1 (e. (cv y) S) anbi1d oprabval.2 anbi12d (cv y) B S eleq1 (e. A R) anbi2d oprabval.3 anbi12d oprabval.4 (/\ (e. A R) (e. B S)) anbi2d R S (V) eloprabg mp3an3 bitrd A F B df-opr oprabval.6 (<,> A B) fveq1i eqtr C eqeq1i syl5bb (/\ (e. A R) (e. B S)) th ibar bitr4d)) thm (oprabvalig ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((oprabvalig.1 (-> (= (cv x) A) (<-> ph ps))) (oprabvalig.2 (-> (= (cv y) B) (<-> ps ch))) (oprabvalig.3 (-> (= (cv z) C) (<-> ch th))) (oprabvalig.4 (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E* z ph))) (oprabvalig.5 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph))))) (-> (/\/\ (e. A R) (e. B S) (e. C D)) (-> th (= (opr A F B) C))) ((cv x) A R eleq1 (e. (cv y) S) anbi1d oprabvalig.1 anbi12d (cv y) B S eleq1 (e. A R) anbi2d oprabvalig.2 anbi12d oprabvalig.3 (/\ (e. A R) (e. B S)) anbi2d R S D eloprabg biimpar exp32 com12 (e. C D) 3adant3 pm2.43i oprabvalig.4 z (/\ (e. (cv x) R) (e. (cv y) S)) ph moanimv mpbir x y funoprab C D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph)) (<,> A B) funopfvg mpan2 (e. A R) (e. B S) 3ad2ant3 syld A F B df-opr oprabvalig.5 (<,> A B) fveq1i eqtr C eqeq1i syl6ibr)) thm (oprabvali ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((oprabvali.1 (e. C (V))) (oprabvali.2 (-> (= (cv x) A) (<-> ph ps))) (oprabvali.3 (-> (= (cv y) B) (<-> ps ch))) (oprabvali.4 (-> (= (cv z) C) (<-> ch th))) (oprabvali.5 (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E* z ph))) (oprabvali.6 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph))))) (-> (/\ (e. A R) (e. B S)) (-> th (= (opr A F B) C))) (oprabvali.1 oprabvali.2 oprabvali.3 oprabvali.4 oprabvali.5 oprabvali.6 (V) oprabvalig mp3an3)) thm (oprabval2gf ((u v) (u x) (u y) (u z) (A u) (v x) (v y) (v z) (A v) (x y) (x z) (A x) (y z) (A y) (A z) (B u) (B v) (B x) (B y) (B z) (C u) (C v) (C x) (C y) (C z) (u w) (G u) (G w) (D u) (D v) (D x) (D y) (D z) (R u) (v w) (R v) (w z) (R w) (R z) (S u) (S v) (S w) (S z) (w x) (w y)) ((oprabval2gf.1 (-> (e. (cv w) G) (A. x (e. (cv w) G)))) (oprabval2gf.2 (-> (e. (cv w) S) (A. y (e. (cv w) S)))) (oprabval2gf.3 (-> (= (cv x) A) (= R G))) (oprabval2gf.4 (-> (= (cv y) B) (= G S))) (oprabval2gf.5 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R)))))) (-> (/\/\ (e. A C) (e. B D) (e. S H)) (= (opr A F B) S)) (S eqid v visset (= (cv u) A) y ax-17 u visset A x eqvinc u visset (e. (cv w) (cv u)) x ax-17 R hbcsb1 oprabval2gf.1 hbeq x (cv u) R csbeq1a eqcomd oprabval2gf.3 sylan9eq 19.23ai sylbi (cv v) (V) csbeq2d mpan2 (cv z) eqeq2d v visset B y eqvinc v visset (e. (cv w) (cv v)) y ax-17 G hbcsb1 oprabval2gf.2 hbeq y (cv v) G csbeq1a eqcomd oprabval2gf.4 sylan9eq 19.23ai sylbi (cv z) eqeq2d (cv z) S S eqeq1 z ([_/]_ (cv v) y ([_/]_ (cv u) x R)) moeq (/\ (e. (cv u) C) (e. (cv v) D)) a1i ({<<,>,>|} u v z (/\ (/\ (e. (cv u) C) (e. (cv v) D)) (= (cv z) ([_/]_ (cv v) y ([_/]_ (cv u) x R))))) eqid H oprabvalig mpi oprabval2gf.5 (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R)) u ax-17 (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R)) v ax-17 (/\ (e. (cv u) C) (e. (cv v) D)) x ax-17 (e. (cv w) (cv z)) x ax-17 v visset (e. (cv w) (cv v)) x ax-17 u visset (e. (cv w) (cv u)) x ax-17 R hbcsb1 (V) y hbcsbg ax-mp hbeq hban (/\ (e. (cv u) C) (e. (cv v) D)) y ax-17 (e. (cv w) (cv z)) y ax-17 v visset (e. (cv w) (cv v)) y ax-17 ([_/]_ (cv u) x R) hbcsb1 hbeq hban (cv x) (cv u) C eleq1 (e. (cv y) D) anbi1d x (cv u) R csbeq1a (cv z) eqeq2d anbi12d (cv y) (cv v) D eleq1 (e. (cv u) C) anbi2d y (cv v) ([_/]_ (cv u) x R) csbeq1a (cv z) eqeq2d anbi12d sylan9bb z cbvoprab12 eqtr F ({<<,>,>|} u v z (/\ (/\ (e. (cv u) C) (e. (cv v) D)) (= (cv z) ([_/]_ (cv v) y ([_/]_ (cv u) x R))))) A B opreq ax-mp syl5eq)) thm (oprabval2g ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (w x) (G x) (G w) (w z) (R w) (R z) (w y) (S w) (S y) (S z)) ((oprabval2g.1 (-> (= (cv x) A) (= R G))) (oprabval2g.2 (-> (= (cv y) B) (= G S))) (oprabval2g.3 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R)))))) (-> (/\/\ (e. A C) (e. B D) (e. S H)) (= (opr A F B) S)) ((e. (cv w) G) x ax-17 (e. (cv w) S) y ax-17 oprabval2g.1 oprabval2g.2 oprabval2g.3 H oprabval2gf)) thm (oprabval2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (R z) (S y) (S z)) ((oprabval2.1 (e. S (V))) (oprabval2.2 (-> (= (cv x) A) (= R G))) (oprabval2.3 (-> (= (cv y) B) (= G S))) (oprabval2.4 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R)))))) (-> (/\ (e. A C) (e. B D)) (= (opr A F B) S)) (oprabval2.1 oprabval2.2 oprabval2.3 oprabval2.4 (V) oprabval2g mp3an3)) thm (oprabval5 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (G x) (R z) (S y) (S z)) ((oprabval5.1 (e. S (V))) (oprabval5.2 (-> (= (cv x) A) (= R G))) (oprabval5.3 (-> (= (cv y) B) (= G S))) (oprabval5.4 (= F ({<<,>,>|} x y z (= (cv z) R))))) (-> (/\ (e. A C) (e. B D)) (= (opr A F B) S)) (oprabval5.1 oprabval5.2 oprabval5.3 oprabval5.4 x visset y visset pm3.2i (= (cv z) R) biantrur x y z oprabbii eqtr oprabval2 A C elisset B D elisset syl2an)) thm (oprabval3 ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f) (H x) (H y) (H z) (H w) (H v) (H u) (H f) (R x) (R y) (R z) (S x) (S y) (S z) (S w) (S v) (S u) (S f)) ((oprabval3.1 (e. S (V))) (oprabval3.2 (-> (/\ (/\ (= (cv w) A) (= (cv v) B)) (/\ (= (cv u) C) (= (cv f) D))) (= R S))) (oprabval3.3 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. H H)) (e. (cv y) (X. H H))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) R))))))))))) (-> (/\ (/\ (e. A H) (e. B H)) (/\ (e. C H) (e. D H))) (= (opr (<,> A B) F (<,> C D)) S)) (oprabval3.1 (cv x) (<,> A B) (<,> (cv w) (cv v)) eqeq1 (= (cv y) (<,> (cv u) (cv f))) anbi1d (= (cv z) R) anbi1d w v u f 4exbidv (cv y) (<,> C D) (<,> (cv u) (cv f)) eqeq1 (= (<,> A B) (<,> (cv w) (cv v))) anbi2d (= (cv z) R) anbi1d w v u f 4exbidv (cv z) S R eqeq1 (/\ (= (<,> A B) (<,> (cv w) (cv v))) (= (<,> C D) (<,> (cv u) (cv f)))) anbi2d w v u f 4exbidv z R moeq u f (cv y) mosubop w v (cv x) mosubop (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f))) (= (cv z) R) anass u f 2exbii u f (= (cv x) (<,> (cv w) (cv v))) (/\ (= (cv y) (<,> (cv u) (cv f))) (= (cv z) R)) 19.42vv bitr w v 2exbii z mobii mpbir (/\ (e. (cv x) (X. H H)) (e. (cv y) (X. H H))) a1i oprabval3.3 oprabvali A H B H opelxpi C H D H opelxpi anim12i S eqid oprabval3.2 S eqeq2d H H copsex4g mpbiri sylc)) thm (oprabval4g ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C z) (C w) (C v) (C u) (C f) (D w) (D v)) ((oprabval4g.1 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (-> (/\/\ (e. (cv x) A) (e. (cv y) B) (e. C D)) (= (opr (cv x) F (cv y)) C)) (x w ({|} u ([/] (cv v) y (e. (cv u) C))) f sbab eqcomd equcoms y v C u sbab eqcomd equcoms oprabval4g.1 (/\ (e. (cv w) A) (e. (cv v) B)) x ax-17 w x (e. (cv f) ({|} u ([/] (cv v) y (e. (cv u) C)))) hbs1 z f hbab hbeleq hban (/\ (e. (cv w) A) (e. (cv v) B)) y ax-17 v y (e. (cv u) C) hbs1 f u hbab w x hbsb z f hbab hbeleq hban (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)) w ax-17 (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)) v ax-17 (cv w) (cv x) A eleq1 (cv v) (cv y) B eleq1 bi2anan9 x w ({|} u ([/] (cv v) y (e. (cv u) C))) f sbab eqcomd equcoms y v C u sbab eqcomd equcoms sylan9eq (cv z) eqeq2d anbi12d z cbvoprab12 eqtr4 (V) oprabval2g C D elisset syl3an3)) thm (foprab2OLD ((u v) (u w) (u x) (u y) (u z) (A u) (v w) (v x) (v y) (v z) (A v) (w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (B u) (B v) (B w) (B x) (B y) (B z) (t u) (t v) (t w) (t z) (C t) (C u) (C v) (C w) (C z) (t x) (t y) (D t) (D u) (D v) (D w) (D x) (D y) (F u) (F v) (F w)) ((foprab2OLD.1 (e. C (V))) (foprab2OLD.2 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (<-> (A.e. x A (A.e. y B (e. C D))) (:--> F (X. A B) D)) ((e. ([_/]_ (cv w) x C) D) v ax-17 v visset (e. (cv t) (cv v)) y ax-17 ([_/]_ (cv w) x C) hbcsb1 (e. (cv t) D) y ax-17 hbel y (cv v) ([_/]_ (cv w) x C) csbeq1a D eleq1d B cbvral w A ralbii (A.e. y B (e. C D)) w ax-17 (e. (cv y) B) x ax-17 w visset (e. (cv t) (cv w)) x ax-17 C hbcsb1 (e. (cv t) D) x ax-17 hbel hbral x (cv w) C csbeq1a D eleq1d y B ralbidv A cbvral v visset w visset foprab2OLD.1 x ax-gen (cv w) (V) x C (V) csbexg mp2an y ax-gen (cv v) (V) y ([_/]_ (cv w) x C) (V) csbexg mp2an w visset (e. (cv t) (cv w)) x ax-17 C hbcsb1 v visset (e. (cv t) (cv v)) y ax-17 ([_/]_ (cv w) x C) hbcsb1 x (cv w) C csbeq1a y (cv v) ([_/]_ (cv w) x C) csbeq1a foprab2OLD.2 (V) oprabval2gf mp3an3 D eleq1d ralbidva ralbiia 3bitr4 F A B D w v u foprval (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)) w ax-17 (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)) v ax-17 (/\ (e. (cv w) A) (e. (cv v) B)) x ax-17 (e. (cv t) (cv z)) x ax-17 v visset (e. (cv t) (cv v)) x ax-17 w visset (e. (cv t) (cv w)) x ax-17 C hbcsb1 (V) y hbcsbg ax-mp hbeq hban (/\ (e. (cv w) A) (e. (cv v) B)) y ax-17 (e. (cv t) (cv z)) y ax-17 v visset (e. (cv t) (cv v)) y ax-17 ([_/]_ (cv w) x C) hbcsb1 hbeq hban (cv x) (cv w) A eleq1 (e. (cv y) B) anbi1d x (cv w) C csbeq1a (cv z) eqeq2d anbi12d (cv y) (cv v) B eleq1 (e. (cv w) A) anbi2d y (cv v) ([_/]_ (cv w) x C) csbeq1a (cv z) eqeq2d anbi12d sylan9bb z cbvoprab12 (cv z) (cv u) ([_/]_ (cv v) y ([_/]_ (cv w) x C)) eqeq1 (/\ (e. (cv w) A) (e. (cv v) B)) anbi2d w v cbvoprab3v eqtr foprab2OLD.2 v visset w visset foprab2OLD.1 x ax-gen (cv w) (V) x C (V) csbexg mp2an y ax-gen (cv v) (V) y ([_/]_ (cv w) x C) (V) csbexg mp2an w visset (e. (cv t) (cv w)) x ax-17 C hbcsb1 v visset (e. (cv t) (cv v)) y ax-17 ([_/]_ (cv w) x C) hbcsb1 x (cv w) C csbeq1a y (cv v) ([_/]_ (cv w) x C) csbeq1a foprab2OLD.2 (V) oprabval2gf mp3an3 (cv u) eqeq2d pm5.32i w v u oprabbii 3eqtr4 mpbiran bitr4)) thm (foprrn () () (-> (/\/\ (:--> F (X. R S) C) (e. A R) (e. B S)) (e. (opr A F B) C)) (F (X. R S) C (<,> A B) ffvrn A F B df-opr syl5eqel A R B S opelxpi sylan2 3impb)) thm (elrnoprab ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C z) (D x) (D y) (D w) (D v) (D u) (F w) (F v) (F u)) ((elrnoprab.1 (e. C (V))) (elrnoprab.2 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (<-> (e. D (ran F)) (E.e. x A (E.e. y B (= D C)))) (elrnoprab.1 elrnoprab.2 fnoprab2 F (X. A B) D w fvelrn (cv w) (<,> (cv v) (cv u)) F fveq2 D eqeq1d A B rexxp (e. (cv u) B) x ax-17 w x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)) hboprab1 elrnoprab.2 (cv w) eleq2i elrnoprab.2 (cv w) eleq2i x albii 3imtr4 (e. (cv w) (<,> (cv v) (cv u))) x ax-17 hbfv (e. (cv w) D) x ax-17 hbeq hbrex (E.e. u B (= (` F (<,> (cv x) (cv u))) D)) v ax-17 (cv v) (cv x) (cv u) opeq1 F fveq2d D eqeq1d u B rexbidv A cbvrex w x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)) hboprab2 elrnoprab.2 (cv w) eleq2i elrnoprab.2 (cv w) eleq2i y albii 3imtr4 (e. (cv w) (<,> (cv x) (cv u))) y ax-17 hbfv (e. (cv w) D) y ax-17 hbeq (= (` F (<,> (cv x) (cv y))) D) u ax-17 (cv u) (cv y) (cv x) opeq2 F fveq2d D eqeq1d B cbvrex x A rexbii 3bitr (cv x) F (cv y) df-opr D eqeq1i x A y B 2rexbii elrnoprab.1 elrnoprab.2 (V) oprabval4g mp3an3 D eqeq1d C D eqcom syl6bb 2rexbiia 3bitr2 syl6bb ax-mp)) thm (oprvalex ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A x) (A y) (A z) (A w) (B x) (B y) (B z) (B w)) ((oprvalex.1 (e. A (V))) (oprvalex.2 (e. B (V)))) (e. ({|} z (E.e. x A (E.e. y B (= (cv z) (opr (cv x) F (cv y)))))) (V)) ((cv w) (<,> (cv x) (cv y)) F fveq2 (cv x) F (cv y) df-opr syl6eqr (cv z) eqeq2d A B rexxp z abbii oprvalex.1 oprvalex.2 xpex z w (` F (cv w)) abrexex eqeltrr)) thm (oprssdm ((x y) (S x) (S y) (F x) (F y)) ((oprssdm.1 (-. (e. ({/}) S))) (oprssdm.2 (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S)))) (C_ (X. S S) (dom F)) (S S relxp y visset (cv x) S S opelxp (<,> (cv x) (cv y)) F ndmfv (cv x) F (cv y) df-opr ({/}) eqeq1i oprssdm.1 (opr (cv x) F (cv y)) ({/}) S eleq1 mtbiri oprssdm.2 nsyl sylbir syl a3i sylbi relssi)) thm (ndmoprg () ((ndmoprg.1 (= (dom F) (X. S S)))) (-> (/\ (e. B C) (-. (/\ (e. A S) (e. B S)))) (= (opr A F B) ({/}))) (B C A S S opelxpg ndmoprg.1 (<,> A B) eleq2i syl5bb negbid (<,> A B) F ndmfv A F B df-opr syl5eq syl6bir imp)) thm (ndmoprcl ((A x) (B x) (F x) (S x)) ((ndmoprcl.1 (= (dom F) (X. S S))) (ndmoprcl.2 (-> (/\ (e. A S) (e. (cv x) S)) (e. (opr A F (cv x)) S))) (ndmoprcl.3 (e. ({/}) S))) (e. (opr A F B) S) (B A F oprprc2 S eleq1d ndmoprcl.1 A (V) A ndmoprg ndmoprcl.3 syl6eqel ex (cv x) A A F opreq2 S eleq1d (e. A S) imbi2d ndmoprcl.2 expcom vtoclga imp pm2.61d2 syl5bir com12 ndmoprcl.1 B (V) A ndmoprg ndmoprcl.3 syl6eqel ex (cv x) B A F opreq2 S eleq1d (e. A S) imbi2d ndmoprcl.2 expcom vtoclga impcom pm2.61d2 pm2.61d2 S S relxp ndmoprcl.1 (dom F) (X. S S) releq ax-mp mpbir A B oprprc1 ndmoprcl.3 syl6eqel pm2.61i)) thm (ndmopr () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S)))) (-> (-. (/\ (e. A S) (e. B S))) (= (opr A F B) ({/}))) (ndmopr.1 ndmopr.2 B (V) A ndmoprg mpan)) thm (ndmoprrcl () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S))) (ndmoprrcl.3 (-. (e. ({/}) S)))) (-> (e. (opr A F B) S) (/\ (e. A S) (e. B S))) (ndmoprrcl.3 ndmopr.1 ndmopr.2 A ndmopr S eleq1d mtbiri a3i)) thm (ndmoprcom () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S))) (ndmopr.3 (e. A (V)))) (-> (-. (/\ (e. A S) (e. B S))) (= (opr A F B) (opr B F A))) (ndmopr.1 ndmopr.2 A ndmopr (e. A S) (e. B S) ancom negbii ndmopr.3 ndmopr.2 B ndmopr sylbi eqtr4d)) thm (ndmoprass () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S))) (ndmopr.4 (e. C (V))) (ndmopr.5 (-. (e. ({/}) S)))) (-> (-. (/\/\ (e. A S) (e. B S) (e. C S))) (= (opr (opr A F B) F C) (opr A F (opr B F C)))) (ndmopr.1 ndmopr.2 ndmopr.5 A ndmoprrcl (e. C S) anim1i (e. A S) (e. B S) (e. C S) df-3an sylibr con3i ndmopr.4 ndmopr.2 (opr A F B) ndmopr syl ndmopr.4 ndmopr.2 ndmopr.5 B ndmoprrcl (e. A S) anim2i (e. A S) (e. B S) (e. C S) 3anass sylibr con3i B F C oprex ndmopr.2 A ndmopr syl eqtr4d)) thm (ndmoprdistr () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S))) (ndmopr.4 (e. C (V))) (ndmopr.5 (-. (e. ({/}) S))) (ndmopr.6 (= (dom G) (X. S S)))) (-> (-. (/\/\ (e. A S) (e. B S) (e. C S))) (= (opr A G (opr B F C)) (opr (opr A G B) F (opr A G C)))) (ndmopr.4 ndmopr.2 ndmopr.5 B ndmoprrcl (e. A S) anim2i (e. A S) (e. B S) (e. C S) 3anass sylibr con3i B F C oprex ndmopr.6 A ndmopr syl ndmopr.1 ndmopr.6 ndmopr.5 A ndmoprrcl ndmopr.4 ndmopr.6 ndmopr.5 A ndmoprrcl anim12i (e. A S) (e. B S) (e. C S) 3anass (e. A S) (e. B S) (e. C S) anandi bitr sylibr con3i A G C oprex ndmopr.2 (opr A G B) ndmopr syl eqtr4d)) thm (ndmord () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S))) (ndmord.3 (e. A (V))) (ndmord.4 (C_ R (X. S S))) (ndmord.5 (-. (e. ({/}) S))) (ndmord.6 (-> (/\/\ (e. A S) (e. B S) (e. C S)) (<-> (br A R B) (br (opr C F A) R (opr C F B)))))) (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B)))) (ndmord.6 3exp imp ndmopr.1 ndmord.4 A brel C F B oprex ndmord.4 (opr C F A) brel ndmord.3 ndmopr.2 ndmord.5 C ndmoprrcl pm3.27d ndmopr.1 ndmopr.2 ndmord.5 C ndmoprrcl pm3.27d anim12i syl pm5.21ni (e. C S) a1d pm2.61i)) thm (ndmordi () ((ndmordi.3 (e. A (V))) (ndmordi.2 (= (dom F) (X. S S))) (ndmordi.4 (C_ R (X. S S))) (ndmordi.5 (-. (e. ({/}) S))) (ndmordi.6 (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B)))))) (-> (br (opr C F A) R (opr C F B)) (br A R B)) (C F B oprex ndmordi.4 (opr C F A) brel pm3.26d ndmordi.3 ndmordi.2 ndmordi.5 C ndmoprrcl pm3.26d syl ndmordi.6 biimprd mpcom)) thm (caoprcl ((x y) (F x) (F y) (S x) (S y) (A x) (A y) (B x) (B y)) ((caoprcl.1 (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S)))) (-> (/\ (e. A S) (e. B S)) (e. (opr A F B) S)) ((cv x) A F (cv y) opreq1 S eleq1d (cv y) B A F opreq2 S eleq1d caoprcl.1 vtocl2ga)) thm (caoprcom ((x y) (F x) (F y) (A x) (A y) (B x) (B y)) ((caoprcom.1 (e. A (V))) (caoprcom.2 (e. B (V))) (caoprcom.3 (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))))) (= (opr A F B) (opr B F A)) (caoprcom.1 caoprcom.2 (cv x) A F (cv y) opreq1 (cv x) A (cv y) F opreq2 eqeq12d (cv y) B A F opreq2 (cv y) B F A opreq1 eqeq12d sylan9bb caoprcom.3 vtocl2)) thm (caoprass ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caoprass.1 (e. A (V))) (caoprass.2 (e. B (V))) (caoprass.3 (e. C (V))) (caoprass.4 (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr (opr A F B) F C) (opr A F (opr B F C))) (caoprass.1 caoprass.2 caoprass.3 (cv x) A F (cv y) opreq1 F (cv z) opreq1d (cv x) A F (opr (cv y) F (cv z)) opreq1 eqeq12d (cv y) B A F opreq2 F (cv z) opreq1d (cv y) B F (cv z) opreq1 A F opreq2d eqeq12d (cv z) C (opr A F B) F opreq2 (cv z) C B F opreq2 A F opreq2d eqeq12d syl3an9b caoprass.4 vtocl3)) thm (caoprcan ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caoprcan.1 (e. C (V))) (caoprcan.2 (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z)))))) (-> (/\ (e. A S) (e. B S)) (-> (= (opr A F B) (opr A F C)) (= B C))) ((cv x) A F (cv y) opreq1 (cv x) A F C opreq1 eqeq12d (= (cv y) C) imbi1d (cv y) B A F opreq2 (opr A F C) eqeq1d (cv y) B C eqeq1 imbi12d caoprcan.1 (cv z) C (cv x) F opreq2 (opr (cv x) F (cv y)) eqeq2d (cv z) C (cv y) eqeq2 imbi12d (/\ (e. (cv x) S) (e. (cv y) S)) imbi2d caoprcan.2 vtocl vtocl2ga)) thm (caoprord ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((caoprord.1 (e. A (V))) (caoprord.2 (e. B (V))) (caoprord.3 (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y))))))) (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B)))) ((cv z) C F A opreq1 (cv z) C F B opreq1 R breq12d (br A R B) bibi2d caoprord.1 caoprord.2 (cv x) A R (cv y) breq1 (cv x) A (cv z) F opreq2 R (opr (cv z) F (cv y)) breq1d bibi12d (cv y) B A R breq2 (cv y) B (cv z) F opreq2 (opr (cv z) F A) R breq2d bibi12d sylan9bb (e. (cv z) S) imbi2d caoprord.3 vtocl2 vtoclga)) thm (caoprord2 ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((caoprord.1 (e. A (V))) (caoprord.2 (e. B (V))) (caoprord.3 (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y)))))) (caoprord2.3 (e. C (V))) (caoprord2.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))))) (-> (e. C S) (<-> (br A R B) (br (opr A F C) R (opr B F C)))) (caoprord.1 caoprord.2 caoprord.3 C caoprord caoprord2.3 caoprord.1 caoprord2.com caoprcom caoprord2.3 caoprord.2 caoprord2.com caoprcom R breq12i syl6bb)) thm (caoprord3 ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (R x) (R y) (R z)) ((caoprord.1 (e. A (V))) (caoprord.2 (e. B (V))) (caoprord.3 (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y)))))) (caoprord2.3 (e. C (V))) (caoprord2.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caoprord3.4 (e. D (V)))) (-> (/\ (/\ (e. B S) (e. C S)) (= (opr A F B) (opr C F D))) (<-> (br A R C) (br D R B))) (caoprord.1 caoprord2.3 caoprord.3 caoprord.2 caoprord2.com caoprord2 (e. C S) adantr (opr A F B) (opr C F D) R (opr C F B) breq1 sylan9bb caoprord3.4 caoprord.2 caoprord.3 C caoprord (e. B S) (= (opr A F B) (opr C F D)) ad2antlr bitr4d)) thm (caoprdistr ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (G x) (G y) (G z)) ((caoprdistr.1 (e. A (V))) (caoprdistr.2 (e. B (V))) (caoprdistr.3 (e. C (V))) (caoprdistr.4 (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z)))))) (= (opr A G (opr B F C)) (opr (opr A G B) F (opr A G C))) (caoprdistr.1 caoprdistr.2 caoprdistr.3 (cv x) A G (opr (cv y) F (cv z)) opreq1 (cv x) A G (cv y) opreq1 (cv x) A G (cv z) opreq1 F opreq12d eqeq12d (cv y) B F (cv z) opreq1 A G opreq2d (cv y) B A G opreq2 F (opr A G (cv z)) opreq1d eqeq12d (cv z) C B F opreq2 A G opreq2d (cv z) C A G opreq2 (opr A G B) F opreq2d eqeq12d syl3an9b caoprdistr.4 vtocl3)) thm (caopr32 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr (opr A F B) F C) (opr (opr A F C) F B)) (caopr.2 caopr.3 caopr.com caoprcom A F opreq2i caopr.1 caopr.2 caopr.3 caopr.ass caoprass caopr.1 caopr.3 caopr.2 caopr.ass caoprass 3eqtr4)) thm (caopr12 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr A F (opr B F C)) (opr B F (opr A F C))) (caopr.1 caopr.2 caopr.com caoprcom F C opreq1i caopr.1 caopr.2 caopr.3 caopr.ass caoprass caopr.2 caopr.1 caopr.3 caopr.ass caoprass 3eqtr3)) thm (caopr31 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr (opr A F B) F C) (opr (opr C F B) F A)) (caopr.1 caopr.3 caopr.2 caopr.ass caoprass caopr.1 caopr.3 caopr.2 caopr.com caopr.ass caopr12 eqtr caopr.1 caopr.2 caopr.3 caopr.com caopr.ass caopr32 caopr.3 caopr.1 caopr.2 caopr.com caopr.ass caopr32 caopr.3 caopr.1 caopr.2 caopr.ass caoprass eqtr3 3eqtr4)) thm (caopr13 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr A F (opr B F C)) (opr C F (opr B F A))) (caopr.1 caopr.2 caopr.3 caopr.com caopr.ass caopr31 caopr.1 caopr.2 caopr.3 caopr.ass caoprass caopr.3 caopr.2 caopr.1 caopr.ass caoprass 3eqtr3)) thm (caopr4 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (caopr.4 (e. D (V)))) (= (opr (opr A F B) F (opr C F D)) (opr (opr A F C) F (opr B F D))) (caopr.2 caopr.3 caopr.4 caopr.com caopr.ass caopr12 A F opreq2i caopr.1 caopr.2 C F D oprex caopr.ass caoprass caopr.1 caopr.3 B F D oprex caopr.ass caoprass 3eqtr4)) thm (caopr411 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (caopr.4 (e. D (V)))) (= (opr (opr A F B) F (opr C F D)) (opr (opr C F B) F (opr A F D))) (caopr.1 caopr.2 caopr.3 caopr.com caopr.ass caopr31 F D opreq1i A F B oprex caopr.3 caopr.4 caopr.ass caoprass C F B oprex caopr.1 caopr.4 caopr.ass caoprass 3eqtr3)) thm (caopr42 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (caopr.4 (e. D (V)))) (= (opr (opr A F B) F (opr C F D)) (opr (opr A F C) F (opr D F B))) (caopr.1 caopr.2 caopr.3 caopr.com caopr.ass caopr.4 caopr4 caopr.2 caopr.4 caopr.com caoprcom (opr A F C) F opreq2i eqtr)) thm (caoprdistrr ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (G x) (G y) (G z)) ((caoprd.1 (e. A (V))) (caoprd.2 (e. B (V))) (caoprd.3 (e. C (V))) (caoprd.com (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x)))) (caoprd.distr (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z)))))) (= (opr (opr A F B) G C) (opr (opr A G C) F (opr B G C))) (caoprd.3 caoprd.1 caoprd.2 caoprd.distr caoprdistr caoprd.3 A F B oprex caoprd.com caoprcom caoprd.3 caoprd.1 caoprd.com caoprcom caoprd.3 caoprd.2 caoprd.com caoprcom F opreq12i 3eqtr3)) thm (caoprdilem ((x y) (x z) (H x) (y z) (H y) (H z) (x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (G y) (G z)) ((caoprd.1 (e. A (V))) (caoprd.2 (e. B (V))) (caoprd.3 (e. C (V))) (caoprd.com (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x)))) (caoprd.distr (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z))))) (caoprdl.4 (e. D (V))) (caoprdl.5 (e. H (V))) (caoprdl.ass (= (opr (opr (cv x) G (cv y)) G (cv z)) (opr (cv x) G (opr (cv y) G (cv z)))))) (= (opr (opr (opr A G C) F (opr B G D)) G H) (opr (opr A G (opr C G H)) F (opr B G (opr D G H)))) (A G C oprex B G D oprex caoprdl.5 caoprd.com caoprd.distr caoprdistrr caoprd.1 caoprd.3 caoprdl.5 caoprdl.ass caoprass caoprd.2 caoprdl.4 caoprdl.5 caoprdl.ass caoprass F opreq12i eqtr)) thm (caoprlem2 ((x y) (x z) (H x) (y z) (H y) (H z) (R x) (R y) (R z) (x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (G y) (G z) (R x) (R y) (R z)) ((caoprd.1 (e. A (V))) (caoprd.2 (e. B (V))) (caoprd.3 (e. C (V))) (caoprd.com (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x)))) (caoprd.distr (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z))))) (caoprdl.4 (e. D (V))) (caoprdl.5 (e. H (V))) (caoprdl.ass (= (opr (opr (cv x) G (cv y)) G (cv z)) (opr (cv x) G (opr (cv y) G (cv z))))) (caoprdl2.6 (e. R (V))) (caoprdl2.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caoprdl2.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr (opr (opr (opr A G C) F (opr B G D)) G H) F (opr (opr (opr A G D) F (opr B G C)) G R)) (opr (opr A G (opr (opr C G H) F (opr D G R))) F (opr B G (opr (opr C G R) F (opr D G H))))) (A G (opr C G H) oprex B G (opr D G H) oprex A G (opr D G R) oprex caoprdl2.com caoprdl2.ass B G (opr C G R) oprex caopr42 caoprd.1 caoprd.2 caoprd.3 caoprd.com caoprd.distr caoprdl.4 caoprdl.5 caoprdl.ass caoprdilem caoprd.1 caoprd.2 caoprdl.4 caoprd.com caoprd.distr caoprd.3 caoprdl2.6 caoprdl.ass caoprdilem F opreq12i caoprd.1 C G H oprex D G R oprex caoprd.distr caoprdistr caoprd.2 C G R oprex D G H oprex caoprd.distr caoprdistr F opreq12i 3eqtr4)) thm (caoprmo ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (S w) (S v) (A w) (A v) (B w) (B v) (F w) (F v) (x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z)) ((caoprmo.1 (e. A (V))) (caoprmo.2 (e. B S)) (caoprmo.dom (= (dom F) (X. S S))) (caoprmo.3 (-. (e. ({/}) S))) (caoprmo.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caoprmo.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (caoprmo.id (-> (e. (cv x) S) (= (opr (cv x) F B) (cv x))))) (E* w (= (opr A F (cv w)) B)) ((cv w) (cv v) S eleq1 (cv w) (cv v) A F opreq2 B eqeq1d anbi12d mo4 (opr A F (cv v)) B (cv w) F opreq2 (cv x) (cv w) F B opreq1 (= (cv x) (cv w)) id eqeq12d caoprmo.id vtoclga sylan9eqr caoprmo.1 w visset v visset caoprmo.ass caoprass caoprmo.1 w visset v visset caoprmo.com caoprmo.ass caopr12 eqtr syl5eq (e. (cv v) S) adantrl (= (opr A F (cv w)) B) adantlr (opr A F (cv w)) B F (cv v) opreq1 (cv x) (cv v) F B opreq1 (= (cv x) (cv v)) id eqeq12d caoprmo.id vtoclga caoprmo.2 elisseti v visset caoprmo.com caoprcom syl5eq sylan9eq (= (opr A F (cv v)) B) adantrr (e. (cv w) S) adantll eqtr3d v ax-gen mpgbir caoprmo.2 (opr A F (cv w)) B S eleq1 mpbiri w visset caoprmo.dom caoprmo.3 A ndmoprrcl pm3.27d syl ancri w immoi ax-mp)) thm (1stval ((x y) (A x) (A y)) () (= (` (1st) A) (U. (dom ({} A)))) (A snex ({} A) (V) dmexg ax-mp uniex (cv x) A sneq dmeqd unieqd (V) (V) y fvopabg x y df-1st A fveq1i syl5eq mpan2 A (1st) fvprc A snprc biimp dmeqd dm0 syl6eq unieqd uni0 syl6eq eqtr4d pm2.61i)) thm (2ndval ((x y) (A x) (A y)) () (= (` (2nd) A) (U. (ran ({} A)))) (A snex ({} A) (V) rnexg ax-mp uniex (cv x) A sneq rneqd unieqd (V) (V) y fvopabg x y df-2nd A fveq1i syl5eq mpan2 A (2nd) fvprc A snprc biimp rneqd rn0 syl6eq unieqd uni0 syl6eq eqtr4d pm2.61i)) thm (op1st () ((op1st.1 (e. A (V)))) (= (` (1st) (<,> A B)) A) ((<,> A B) 1stval op1st.1 B op1sta eqtr)) thm (op2nd () ((op1st.1 (e. A (V))) (op2n.2 (e. B (V)))) (= (` (2nd) (<,> A B)) B) ((<,> A B) 2ndval op1st.1 op2n.2 op2nda eqtr)) thm (op1stg ((A x) (B x)) () (-> (e. A C) (= (` (1st) (<,> A B)) A)) ((cv x) A B opeq1 (1st) fveq2d (= (cv x) A) id eqeq12d x visset B op1st C vtoclg)) thm (op2ndg ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A C) (e. B D)) (= (` (2nd) (<,> A B)) B)) ((cv x) A (cv y) opeq1 (2nd) fveq2d (cv y) eqeq1d (cv y) B A opeq2 (2nd) fveq2d (= (cv y) B) id eqeq12d x visset y visset op2nd C D vtocl2g)) thm (1stval2 ((x y) (A x) (A y)) () (-> (e. A (X. (V) (V))) (= (` (1st) A) (|^| (|^| A)))) (A x y elvv x visset (cv y) op1st x visset (cv y) op1stb eqtr4 (= A (<,> (cv x) (cv y))) a1i A (<,> (cv x) (cv y)) (1st) fveq2 A (<,> (cv x) (cv y)) inteq inteqd 3eqtr4d x y 19.23aivv sylbi)) thm (2ndval2 ((x y) (A x) (A y)) () (-> (e. A (X. (V) (V))) (= (` (2nd) A) (|^| (|^| (|^| (`' ({} A))))))) (A x y elvv x visset y visset op2nd x visset y visset op2ndb eqtr4 (= A (<,> (cv x) (cv y))) a1i A (<,> (cv x) (cv y)) (2nd) fveq2 A (<,> (cv x) (cv y)) sneq ({} A) ({} (<,> (cv x) (cv y))) cnveq syl inteqd inteqd inteqd 3eqtr4d x y 19.23aivv sylbi)) thm (fo1st ((x y)) () (:-onto-> (1st) (V) (V)) (({<,>|} x y (= (cv y) (U. (dom ({} (cv x)))))) (V) (V) df-fo (cv x) snex ({} (cv x)) (V) dmexg ax-mp uniex x visset (= (cv y) (U. (dom ({} (cv x))))) biantrur x y opabbii fnopab2 y visset (cv y) op1sta eqcomi (cv y) (cv y) opex (cv x) (<,> (cv y) (cv y)) sneq dmeqd unieqd (cv y) eqeq2d cla4ev ax-mp y equid 2th y abbii x y (= (cv y) (U. (dom ({} (cv x))))) rnopab y df-v 3eqtr4 mpbir2an x y df-1st (1st) ({<,>|} x y (= (cv y) (U. (dom ({} (cv x)))))) (V) (V) foeq1 ax-mp mpbir)) thm (fo2nd ((x y)) () (:-onto-> (2nd) (V) (V)) (({<,>|} x y (= (cv y) (U. (ran ({} (cv x)))))) (V) (V) df-fo (cv x) snex ({} (cv x)) (V) rnexg ax-mp uniex x visset (= (cv y) (U. (ran ({} (cv x))))) biantrur x y opabbii fnopab2 y visset y visset op2nda eqcomi (cv y) (cv y) opex (cv x) (<,> (cv y) (cv y)) sneq rneqd unieqd (cv y) eqeq2d cla4ev ax-mp y equid 2th y abbii x y (= (cv y) (U. (ran ({} (cv x))))) rnopab y df-v 3eqtr4 mpbir2an x y df-2nd (2nd) ({<,>|} x y (= (cv y) (U. (ran ({} (cv x)))))) (V) (V) foeq1 ax-mp mpbir)) thm (f1stres ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (:--> (|` (1st) (X. A B)) (X. A B) A) (y visset (cv z) op1sta A eleq1i biimpr (e. (cv z) B) adantr rgen2a (cv x) (<,> (cv y) (cv z)) sneq dmeqd unieqd A eleq1d A B ralxp mpbir x y df-1st (1st) ({<,>|} x y (= (cv y) (U. (dom ({} (cv x)))))) (X. A B) reseq1 ax-mp x y (= (cv y) (U. (dom ({} (cv x))))) (X. A B) resopab eqtr A fopab2 mpbi)) thm (1st2val ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (= (` ({<<,>,>|} x y z (= (cv z) (cv x))) A) (` (1st) A)) (w visset (cv v) op1st w visset v visset w visset (= (cv x) (cv w)) id (cv w) eqid (= (cv y) (cv v)) a1i ({<<,>,>|} x y z (= (cv z) (cv x))) eqid (V) (V) oprabval5 mp2an (cv w) ({<<,>,>|} x y z (= (cv z) (cv x))) (cv v) df-opr 3eqtr2r (<,> (cv w) (cv v)) A ({<<,>,>|} x y z (= (cv z) (cv x))) fveq2 (<,> (cv w) (cv v)) A (1st) fveq2 eqeq12d mpbii w v 19.23aivv x visset y visset pm3.2i z x a9e 2th x y opabbii (V) (V) x y df-xp x y z (= (cv z) (cv x)) dmoprab 3eqtr4r A eleq2i A w v elvv A (<,> (cv w) (cv v)) eqcom w v 2exbii 3bitr negbii A ({<<,>,>|} x y z (= (cv z) (cv x))) ndmfv sylbir (dom ({} A)) w n0 w visset ({} A) v eldm2 (cv w) (cv v) opex A elsnc v exbii bitr w exbii bitr biimp con1i unieqd uni0 syl6eq A 1stval syl5eq eqtr4d pm2.61i)) thm (2nd2val ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (= (` ({<<,>,>|} x y z (= (cv z) (cv y))) A) (` (2nd) A)) (w visset v visset op2nd w visset v visset v visset y equid (= (cv x) (cv w)) a1i (= (cv y) (cv v)) id ({<<,>,>|} x y z (= (cv z) (cv y))) eqid (V) (V) oprabval5 mp2an (cv w) ({<<,>,>|} x y z (= (cv z) (cv y))) (cv v) df-opr 3eqtr2r (<,> (cv w) (cv v)) A ({<<,>,>|} x y z (= (cv z) (cv y))) fveq2 (<,> (cv w) (cv v)) A (2nd) fveq2 eqeq12d mpbii w v 19.23aivv x visset y visset pm3.2i z y a9e 2th x y opabbii (V) (V) x y df-xp x y z (= (cv z) (cv y)) dmoprab 3eqtr4r A eleq2i A w v elvv A (<,> (cv w) (cv v)) eqcom w v 2exbii 3bitr negbii A ({<<,>,>|} x y z (= (cv z) (cv y))) ndmfv sylbir (ran ({} A)) v n0 v visset ({} A) w elrn2 (cv w) (cv v) opex A elsnc w exbii bitr v exbii v w (= (<,> (cv w) (cv v)) A) excom 3bitr biimp con1i unieqd uni0 syl6eq A 2ndval syl5eq eqtr4d pm2.61i)) thm (df1st2 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (u x) (u y) (u z) (u w) (u v)) () (= ({<<,>,>|} x y z (= (cv z) (cv x))) (|` (1st) (X. (V) (V)))) (x visset x visset y visset pm3.2i (= (cv z) (cv x)) biantrur x y z oprabbii fnoprab2 fo1st (1st) (V) (V) fof ax-mp (1st) (V) (V) ffn ax-mp (X. (V) (V)) ssv (1st) (V) (X. (V) (V)) fnssres mp2an ({<<,>,>|} x y z (= (cv z) (cv x))) (X. (V) (V)) (|` (1st) (X. (V) (V))) (X. (V) (V)) u eqfnfv mp2an (X. (V) (V)) eqid x y z (<,> (cv w) (cv v)) 1st2val v visset (cv w) (V) (V) opelxp w visset v visset mpbir2an (<,> (cv w) (cv v)) (X. (V) (V)) (1st) fvres ax-mp eqtr4 (/\ (e. (cv w) (V)) (e. (cv v) (V))) a1i rgen2 (cv u) (<,> (cv w) (cv v)) ({<<,>,>|} x y z (= (cv z) (cv x))) fveq2 (cv u) (<,> (cv w) (cv v)) (|` (1st) (X. (V) (V))) fveq2 eqeq12d (V) (V) ralxp mpbir mpbir2an)) thm (elxp6 () () (<-> (e. A (X. B C)) (/\ (= A (<,> (` (1st) A) (` (2nd) A))) (/\ (e. (` (1st) A) B) (e. (` (2nd) A) C)))) (A B C elxp4 A 1stval A 2ndval (` (1st) A) (U. (dom ({} A))) (` (2nd) A) (U. (ran ({} A))) opeq12 mp2an A eqeq2i A 1stval B eleq1i A 2ndval C eleq1i anbi12i anbi12i bitr4)) thm (elxp7 () () (<-> (e. A (X. B C)) (/\ (e. A (X. (V) (V))) (/\ (e. (` (1st) A) B) (e. (` (2nd) A) C)))) (A B C elxp6 A (V) (V) elxp6 (1st) A fvex (2nd) A fvex pm3.2i mpbiran2 (/\ (e. (` (1st) A) B) (e. (` (2nd) A) C)) anbi1i bitr4)) thm (1st2nd () () (-> (/\ (Rel B) (e. A B)) (= A (<,> (` (1st) A) (` (2nd) A)))) (B (X. (V) (V)) A ssel2 B df-rel sylanb A (V) (V) elxp6 pm3.26bd syl)) thm (elopabi ((x y) (A x) (A y) (ch x) (ch y)) ((elopabi.1 (-> (= (cv x) (` (1st) A)) (<-> ph ps))) (elopabi.2 (-> (= (cv y) (` (2nd) A)) (<-> ps ch)))) (-> (e. A ({<,>|} x y ph)) ch) (x y ph relopab ({<,>|} x y ph) A 1st2nd mpan A (<,> (` (1st) A) (` (2nd) A)) ({<,>|} x y ph) eleq1 (1st) A fvex (2nd) A fvex elopabi.1 elopabi.2 opelopab syl6bb biimpcd mpd)) thm (csbopeq1a () () (-> (= (<,> (cv x) (cv y)) A) (= B ([_/]_ (` (1st) A) x ([_/]_ (` (2nd) A) y B)))) (y (` (2nd) A) B csbeq1a x (` (1st) A) ([_/]_ (` (2nd) A) y B) csbeq1a sylan9eq (<,> (cv x) (cv y)) A (2nd) fveq2 x visset y visset op2nd syl5eqr (<,> (cv x) (cv y)) A (1st) fveq2 x visset (cv y) op1st syl5eqr sylanc)) thm (dfoprab3 ((ph w) (x y) (x z) (w x) (y z) (w y) (w z)) () (= ({<<,>,>|} x y z ph) ({<,>|} w z (/\ (e. (cv w) (X. (V) (V))) ([/] (` (1st) (cv w)) x ([/] (` (2nd) (cv w)) y ph))))) (x y z ph w dfoprab2 (1st) (cv w) fvex x ([/] (` (2nd) (cv w)) y ph) hbsbc1v (E. y (= (cv w) (<,> (cv x) (cv y)))) 19.41 (cv w) (<,> (cv x) (cv y)) (2nd) fveq2 x visset y visset op2nd syl6req y (` (2nd) (cv w)) ph sbceq1a syl (cv w) (<,> (cv x) (cv y)) (1st) fveq2 x visset (cv y) op1st syl6req x (` (1st) (cv w)) ([/] (` (2nd) (cv w)) y ph) sbceq1a syl bitrd pm5.32i y exbii (1st) (cv w) fvex (e. (cv x) (` (1st) (cv w))) y ax-17 (2nd) (cv w) fvex y ph hbsbc1v (V) x hbsbcg ax-mp (= (cv w) (<,> (cv x) (cv y))) 19.41 bitr x exbii (cv w) x y elvv ([/] (` (1st) (cv w)) x ([/] (` (2nd) (cv w)) y ph)) anbi1i 3bitr4 w z opabbii eqtr)) thm (foprab2 ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (B w) (B x) (B y) (B z) (C w) (C z) (D w) (D x) (D y) (D z)) ((foprab2.1 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (<-> (A.e. x A (A.e. y B (e. C D))) (:--> F (X. A B) D)) ((1st) (cv w) fvex (e. (cv z) (` (1st) (cv w))) x ax-17 ([_/]_ (` (2nd) (cv w)) y C) hbcsb1 (e. (cv z) D) x ax-17 hbel (1st) (cv w) fvex (e. (cv z) (` (1st) (cv w))) y ax-17 (2nd) (cv w) fvex (e. (cv z) (` (2nd) (cv w))) y ax-17 C hbcsb1 (V) x hbcsbg ax-mp (e. (cv z) D) y ax-17 hbel (e. C D) w ax-17 x y (cv w) C csbopeq1a eqcoms eqcomd D eleq1d A B ralxpf bicomi foprab2.1 x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)) w dfoprab3 (cv w) A B elxp7 (= (cv z) ([_/]_ (` (1st) (cv w)) x ([_/]_ (` (2nd) (cv w)) y C))) anbi1i (e. (cv w) (X. (V) (V))) (/\ (e. (` (1st) (cv w)) A) (e. (` (2nd) (cv w)) B)) (= (cv z) ([_/]_ (` (1st) (cv w)) x ([_/]_ (` (2nd) (cv w)) y C))) anass bitr (1st) (cv w) fvex (2nd) (cv w) fvex (` (2nd) (cv w)) (V) y (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C) sbcang ax-mp (2nd) (cv w) fvex (` (2nd) (cv w)) (V) y (e. (cv x) A) (e. (cv y) B) sbcang ax-mp (2nd) (cv w) fvex (e. (cv x) A) y ax-17 (` (2nd) (cv w)) (V) sbcgf ax-mp (2nd) (cv w) fvex (` (2nd) (cv w)) (V) y B sbcel1gv ax-mp anbi12i bitr (2nd) (cv w) fvex (` (2nd) (cv w)) (V) y (cv z) C sbceq2dig ax-mp anbi12i bitr (` (1st) (cv w)) (V) x sbcbii ax-mp (1st) (cv w) fvex (` (1st) (cv w)) (V) x (/\ (e. (cv x) A) (e. (` (2nd) (cv w)) B)) (= (cv z) ([_/]_ (` (2nd) (cv w)) y C)) sbcang ax-mp (1st) (cv w) fvex (` (1st) (cv w)) (V) x (e. (cv x) A) (e. (` (2nd) (cv w)) B) sbcang ax-mp (1st) (cv w) fvex (` (1st) (cv w)) (V) x A sbcel1gv ax-mp (1st) (cv w) fvex (e. (` (2nd) (cv w)) B) x ax-17 (` (1st) (cv w)) (V) sbcgf ax-mp anbi12i bitr (1st) (cv w) fvex (` (1st) (cv w)) (V) x (cv z) ([_/]_ (` (2nd) (cv w)) y C) sbceq2dig ax-mp anbi12i bitr bitr (e. (cv w) (X. (V) (V))) anbi2i bicomi bitr w z opabbii eqcomi eqtr eqtr D fopab2 bitr)) thm (1on () () (e. (1o) (On)) (df-1o 0elon onsuc eqeltr)) thm (2on () () (e. (2o) (On)) (1on (1o) sucelon df-2o (On) eleq1i bitr4 mpbi)) thm (df1o2 () () (= (1o) ({} ({/}))) (df-1o suc0 eqtr)) thm (df2o2 () () (= (2o) ({,} ({/}) ({} ({/})))) (df-2o (1o) df-suc df1o2 df1o2 sneqi uneq12i ({/}) ({} ({/})) df-pr eqtr4 3eqtr)) thm (1ne0 () () (-. (= (1o) ({/}))) (0nep0 df1o2 ({/}) eqeq1i ({} ({/})) ({/}) eqcom bitr mtbir)) thm (ordgt0ge1 () () (-> (Ord A) (<-> (e. ({/}) A) (C_ (1o) A))) (0elon ({/}) (On) A ordelsuc mpan df-1o A sseq1i syl6bbr)) thm (ordge1n0 () () (-> (Ord A) (<-> (C_ (1o) A) (-. (= A ({/}))))) (A ordgt0ge1 A ord0eln0 bitr3d)) thm (el1o () () (<-> (e. A (1o)) (= A ({/}))) (df1o2 A eleq2i 0ex A elsnc2 bitr)) thm (0lt1o () () (e. ({/}) (1o)) (({/}) eqid ({/}) el1o mpbir)) thm (fnoa ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (Fn (+o) (X. (On) (On))) ((rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv x)) (cv y) fvex x y z w v df-oadd fnoprab2)) thm (fnom ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (Fn (.o) (X. (On) (On))) ((rec ({<,>|} w v (= (cv v) (opr (cv w) (+o) (cv x)))) ({/})) (cv y) fvex x y z w v df-omul fnoprab2)) thm (oav ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A z) (A w) (A v) (B z) (B w) (B v)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (+o) B) (` (rec ({<,>|} x y (= (cv y) (suc (cv x)))) A) B))) ((rec ({<,>|} x y (= (cv y) (suc (cv x)))) A) B fvex (cv w) A ({<,>|} x y (= (cv y) (suc (cv x)))) rdgeq2 (cv v) fveq1d (cv v) B (rec ({<,>|} x y (= (cv y) (suc (cv x)))) A) fveq2 w v z x y df-oadd oprabval2)) thm (omv ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B z) (B w) (B v)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (.o) B) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+o) A))) ({/})) B))) ((rec ({<,>|} x y (= (cv y) (opr (cv x) (+o) A))) ({/})) B fvex (cv w) A (cv x) (+o) opreq2 (cv y) eqeq2d x y opabbidv ({<,>|} x y (= (cv y) (opr (cv x) (+o) (cv w)))) ({<,>|} x y (= (cv y) (opr (cv x) (+o) A))) ({/}) rdgeq1 (rec ({<,>|} x y (= (cv y) (opr (cv x) (+o) (cv w)))) ({/})) (rec ({<,>|} x y (= (cv y) (opr (cv x) (+o) A))) ({/})) (cv v) fveq1 3syl (cv v) B (rec ({<,>|} x y (= (cv y) (opr (cv x) (+o) A))) ({/})) fveq2 w v z x y df-omul oprabval2)) thm (oe0lem () ((oe0lem.1 (-> (/\ ph (= A ({/}))) ps)) (oe0lem.2 (-> (/\ (/\ (e. A (On)) ph) (e. ({/}) A)) ps))) (-> (/\ (e. A (On)) ph) ps) (oe0lem.1 ex (e. A (On)) adantl A on0eln0 ph adantr oe0lem.2 ex sylbird pm2.61d)) thm (oev ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B z) (B w) (B v)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (^o) B) (if (= A ({/})) (\ (1o) B) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B)))) (1on (1o) (On) B difexg ax-mp (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B fvex (= A ({/})) ifex (cv w) A ({/}) eqeq1 (\ (1o) (cv v)) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) (cv w)))) (1o)) (cv v)) ifbid (cv w) A (cv x) (.o) opreq2 (cv y) eqeq2d x y opabbidv ({<,>|} x y (= (cv y) (opr (cv x) (.o) (cv w)))) ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o) rdgeq1 (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) (cv w)))) (1o)) (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) (cv v) fveq1 3syl (= A ({/})) (\ (1o) (cv v)) ifeq2d eqtrd (cv v) B (1o) difeq2 (= A ({/})) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) (cv v)) ifeq1d (cv v) B (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) fveq2 (= A ({/})) (\ (1o) B) ifeq2d eqtrd w v z x y df-oexp oprabval2)) thm (oevn0 ((x y) (A x) (A y)) () (-> (/\ (/\ (e. A (On)) (e. B (On))) (e. ({/}) A)) (= (opr A (^o) B) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B))) (A on0eln0 (e. B (On)) adantr A B x y oev (= A ({/})) (\ (1o) B) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B) iffalse sylan9eq ex sylbid imp)) thm (oa0 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr A (+o) ({/})) A)) (0elon A ({/}) x y oav mpan2 A (On) ({<,>|} x y (= (cv y) (suc (cv x)))) rdg0t eqtrd)) thm (om0 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr A (.o) ({/})) ({/}))) (0elon A ({/}) x y omv mpan2 0ex ({<,>|} x y (= (cv y) (opr (cv x) (+o) A))) rdg0 syl6eq)) thm (om0x () () (= (opr A (.o) ({/})) ({/})) (A om0 (e. ({/}) (On)) adantr 0ex fnom (.o) (X. (On) (On)) fndm ax-mp A ndmopr pm2.61i)) thm (oe0m ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr ({/}) (^o) A) (\ (1o) A))) (0elon ({/}) A x y oev mpan ({/}) eqid (= ({/}) ({/})) (\ (1o) A) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) ({/})))) (1o)) A) iftrue ax-mp syl6eq)) thm (oe0m0 () () (= (opr ({/}) (^o) ({/})) (1o)) (0elon ({/}) oe0m (1o) dif0 syl6eq ax-mp)) thm (oe0m1 () () (-> (e. A (On)) (<-> (e. ({/}) A) (= (opr ({/}) (^o) A) ({/})))) (A eloni A ordgt0ge1 syl A oe0m ({/}) eqeq1d (1o) A ssdif0 syl6rbbr bitrd)) thm (oe0 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr A (^o) ({/})) (1o))) (A ({/}) (^o) ({/}) opreq1 oe0m0 syl6eq (e. A (On)) adantl 0elon A ({/}) x y oevn0 mpanl2 1on elisseti ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) rdg0 syl6eq (e. A (On)) adantll oe0lem anidms)) thm (oev2 ((x y) (A x) (A y)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (^o) B) (i^i (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B) (u. (\ (V) (|^| A)) (|^| B))))) (A ({/}) B ({/}) (^o) opreq12 oe0m0 syl6eq B ({/}) (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) fveq2 1on elisseti ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) rdg0 syl6eq B ({/}) inteq int0 syl6eq ineq12d (1o) inv (= A ({/})) a1i sylan9eqr eqtr4d A ({/}) (^o) B opreq1 B oe0m1 biimpa sylan9eqr an1rs B int0el (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B) ineq2d (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B) in0 syl6eq (/\ (e. B (On)) (= A ({/}))) adantl eqtr4d oe0lem A ({/}) inteq int0 syl6eq (V) difeq2d (V) difid syl6eq (|^| B) uneq2d (|^| B) (\ (V) (|^| A)) uncom (|^| B) un0 3eqtr3g (e. B (On)) adantl (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B) ineq2d eqtr4d A B x y oevn0 A int0el (V) difeq2d (V) dif0 syl6eq (|^| B) uneq2d (|^| B) (\ (V) (|^| A)) uncom (|^| B) unv 3eqtr3g (/\ (e. A (On)) (e. B (On))) adantl (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B) ineq2d (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B) inv syl6req eqtrd oe0lem)) thm (oasuc ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (+o) (suc B)) (suc (opr A (+o) B)))) (B ({<,>|} x y (= (cv y) (suc (cv x)))) A rdgsuct (e. A (On)) adantl A (suc B) x y oav B suceloni sylan2 A B x y oav ({<,>|} x y (= (cv y) (suc (cv x)))) fveq2d A (+o) B oprex A (+o) B oprex sucex (cv x) (opr A (+o) B) suceq y fvopab syl5eqr 3eqtr4d)) thm (oa1suc () () (-> (e. A (On)) (= (opr A (+o) (1o)) (suc A))) (0elon A ({/}) oasuc mpan2 df-1o A (+o) opreq2i syl5eq A oa0 (opr A (+o) ({/})) A suceq syl eqtrd)) thm (omsuc ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (.o) (suc B)) (opr (opr A (.o) B) (+o) A))) (B ({<,>|} x y (= (cv y) (opr (cv x) (+o) A))) ({/}) rdgsuct (e. A (On)) adantl A (suc B) x y omv B suceloni sylan2 A B x y omv ({<,>|} x y (= (cv y) (opr (cv x) (+o) A))) fveq2d A (.o) B oprex (opr A (.o) B) (+o) A oprex (cv x) (opr A (.o) B) (+o) A opreq1 y fvopab syl5eqr 3eqtr4d)) thm (oesuc ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (^o) (suc B)) (opr (opr A (^o) B) (.o) A))) (A ({/}) (^o) (suc B) opreq1 (suc B) oe0m1 biimpa B suceloni B eloni B 0elsuc syl sylanc sylan9eqr A ({/}) (^o) B opreq1 (= A ({/})) id (.o) opreq12d B ({/}) ({/}) (^o) opreq2 oe0m0 1on eqeltr syl6eqel (e. B (On)) adantl B oe0m1 biimpa 0elon syl6eqel (e. B (On)) adantll oe0lem anidms (opr ({/}) (^o) B) om0 syl sylan9eqr eqtr4d B ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o) rdgsuct (e. A (On)) (e. ({/}) A) ad2antlr A (suc B) x y oevn0 B suceloni sylanl2 A B x y oevn0 ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) fveq2d A (^o) B oprex (opr A (^o) B) (.o) A oprex (cv x) (opr A (^o) B) (.o) A opreq1 y fvopab syl5eqr 3eqtr4d oe0lem)) thm (oalim ((x y) (x z) (A x) (y z) (A y) (A z) (B x)) () (-> (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (= (opr A (+o) B) (U_ x B (opr A (+o) (cv x))))) (B (On) ({<,>|} y z (= (cv z) (suc (cv y)))) A x rdglim2a (e. A (On)) adantl A B y z oav A (cv x) y z oav B (cv x) onelon sylan2 anassrs iuneq2dv eqeq12d (Lim B) adantrr mpbird B C limelon (e. B C) (Lim B) pm3.27 jca sylan2)) thm (omlim ((x y) (x z) (A x) (y z) (A y) (A z) (B x)) () (-> (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (= (opr A (.o) B) (U_ x B (opr A (.o) (cv x))))) (B (On) ({<,>|} y z (= (cv z) (opr (cv y) (+o) A))) ({/}) x rdglim2a (e. A (On)) adantl A B y z omv A (cv x) y z omv B (cv x) onelon sylan2 anassrs iuneq2dv eqeq12d (Lim B) adantrr mpbird B C limelon (e. B C) (Lim B) pm3.27 jca sylan2)) thm (oelim ((x y) (x z) (A x) (y z) (A y) (A z) (B x)) () (-> (/\ (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (e. ({/}) A)) (= (opr A (^o) B) (U_ x B (opr A (^o) (cv x))))) (B (On) ({<,>|} y z (= (cv z) (opr (cv y) (.o) A))) (1o) x rdglim2a (e. A (On)) (e. ({/}) A) ad2antlr A B y z oevn0 A (cv x) y z oevn0 B (cv x) onelon sylanl2 exp42 com34 imp41 iuneq2dv eqeq12d (Lim B) adantlrr mpbird B C limelon (e. B C) (Lim B) pm3.27 jca sylanl2)) thm (oacl ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (On)) (e. B (On))) (e. (opr A (+o) B) (On))) ((cv x) ({/}) A (+o) opreq2 (On) eleq1d (cv x) (cv y) A (+o) opreq2 (On) eleq1d (cv x) (suc (cv y)) A (+o) opreq2 (On) eleq1d (cv x) B A (+o) opreq2 (On) eleq1d A oa0 (On) eleq1d ibir A (cv y) oasuc (On) eleq1d (opr A (+o) (cv y)) suceloni syl5bir expcom x visset A (cv x) (V) y oalim mpanr1 (On) eleq1d x visset A (+o) (cv y) oprex y iunon syl5bir expcom tfinds3 impcom)) thm (omcl ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (On)) (e. B (On))) (e. (opr A (.o) B) (On))) ((cv x) ({/}) A (.o) opreq2 (On) eleq1d (cv x) (cv y) A (.o) opreq2 (On) eleq1d (cv x) (suc (cv y)) A (.o) opreq2 (On) eleq1d (cv x) B A (.o) opreq2 (On) eleq1d A om0 0elon syl6eqel A (cv y) omsuc (On) eleq1d (opr A (.o) (cv y)) A oacl syl5bir exp4b com24 pm2.43i com3r x visset A (cv x) (V) y omlim mpanr1 ancoms (On) eleq1d x visset A (.o) (cv y) oprex y iunon syl5bir ex tfinds3 impcom)) thm (oecl ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (On)) (e. B (On))) (e. (opr A (^o) B) (On))) (A ({/}) (^o) B opreq1 (On) eleq1d B ({/}) ({/}) (^o) opreq2 oe0m0 1on eqeltr syl6eqel (e. B (On)) adantl B oe0m1 biimpa 0elon syl6eqel (e. B (On)) adantll oe0lem anidms syl5bir impcom (cv x) ({/}) A (^o) opreq2 (On) eleq1d (cv x) (cv y) A (^o) opreq2 (On) eleq1d (cv x) (suc (cv y)) A (^o) opreq2 (On) eleq1d (cv x) B A (^o) opreq2 (On) eleq1d A oe0 1on syl6eqel (e. ({/}) A) adantr A (cv y) oesuc (On) eleq1d (opr A (^o) (cv y)) A omcl syl5bir exp4b com24 pm2.43i com3r (e. ({/}) A) adantrd x visset A (cv x) (V) y oelim mpanlr1 anasss an1s (On) eleq1d x visset A (^o) (cv y) oprex y iunon syl5bir ex tfinds3 exp3a com12 imp31 oe0lem)) thm (oa0r ((x y) (A x)) () (-> (e. A (On)) (= (opr ({/}) (+o) A) A)) ((cv x) ({/}) ({/}) (+o) opreq2 (= (cv x) ({/})) id eqeq12d (cv x) (cv y) ({/}) (+o) opreq2 (= (cv x) (cv y)) id eqeq12d (cv x) (suc (cv y)) ({/}) (+o) opreq2 (= (cv x) (suc (cv y))) id eqeq12d (cv x) A ({/}) (+o) opreq2 (= (cv x) A) id eqeq12d 0elon ({/}) oa0 ax-mp 0elon ({/}) (cv y) oasuc mpan (opr ({/}) (+o) (cv y)) (cv y) suceq sylan9eq ex x visset 0elon ({/}) (cv x) (V) y oalim mpan mpan (cv x) limuni eqeq12d y (cv x) (opr ({/}) (+o) (cv y)) (cv y) iuneq2 (cv x) y uniiun syl6eqr syl5bir tfinds)) thm (om0r ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr ({/}) (.o) A) ({/}))) ((cv x) ({/}) ({/}) (.o) opreq2 ({/}) eqeq1d (cv x) (cv y) ({/}) (.o) opreq2 ({/}) eqeq1d (cv x) (suc (cv y)) ({/}) (.o) opreq2 ({/}) eqeq1d (cv x) A ({/}) (.o) opreq2 ({/}) eqeq1d 0elon ({/}) om0 ax-mp 0elon ({/}) (cv y) omsuc mpan 0elon ({/}) oa0 ax-mp eqcomi (e. (cv y) (On)) a1i eqeq12d (opr ({/}) (.o) (cv y)) ({/}) (+o) ({/}) opreq1 syl5bir x visset 0elon ({/}) (cv x) (V) y omlim mpan mpan ({/}) eqeq1d y (cv x) (opr ({/}) (.o) (cv y)) ({/}) iuneq2 y (cv x) iun0 syl6eq syl5bir tfinds)) thm (o1p1e2 () () (= (opr (1o) (+o) (1o)) (2o)) (1on (1o) oa1suc ax-mp df-2o eqtr4)) thm (om1 () () (-> (e. A (On)) (= (opr A (.o) (1o)) A)) (0elon A ({/}) omsuc mpan2 df-1o A (.o) opreq2i syl5eq A om0 (+o) A opreq1d A oa0r 3eqtrd)) thm (om1r ((x y) (A x)) () (-> (e. A (On)) (= (opr (1o) (.o) A) A)) ((cv x) ({/}) (1o) (.o) opreq2 (= (cv x) ({/})) id eqeq12d (cv x) (cv y) (1o) (.o) opreq2 (= (cv x) (cv y)) id eqeq12d (cv x) (suc (cv y)) (1o) (.o) opreq2 (= (cv x) (suc (cv y))) id eqeq12d (cv x) A (1o) (.o) opreq2 (= (cv x) A) id eqeq12d 1on (1o) om0 ax-mp 1on (1o) (cv y) omsuc mpan (opr (1o) (.o) (cv y)) (cv y) (+o) (1o) opreq1 sylan9eq (cv y) oa1suc (= (opr (1o) (.o) (cv y)) (cv y)) adantr eqtrd ex x visset 1on (1o) (cv x) (V) y omlim mpan mpan (cv x) limuni eqeq12d y (cv x) (opr (1o) (.o) (cv y)) (cv y) iuneq2 (cv x) y uniiun syl6eqr syl5bir tfinds)) thm (oe1 () () (-> (e. A (On)) (= (opr A (^o) (1o)) A)) (0elon A ({/}) oesuc mpan2 df-1o A (^o) opreq2i syl5eq A oe0 (.o) A opreq1d A om1r 3eqtrd)) thm (oe1m ((x y) (A x)) () (-> (e. A (On)) (= (opr (1o) (^o) A) (1o))) ((cv x) ({/}) (1o) (^o) opreq2 (1o) eqeq1d (cv x) (cv y) (1o) (^o) opreq2 (1o) eqeq1d (cv x) (suc (cv y)) (1o) (^o) opreq2 (1o) eqeq1d (cv x) A (1o) (^o) opreq2 (1o) eqeq1d 1on (1o) oe0 ax-mp 1on (1o) (cv y) oesuc mpan (opr (1o) (^o) (cv y)) (1o) (.o) (1o) opreq1 1on (1o) om1 ax-mp syl6eq sylan9eq ex x visset 1on 0lt1o (1o) (cv x) (V) y oelim mpan2 mpan mpan (1o) eqeq1d (cv x) 0ellim ({/}) (cv x) n0i (cv x) y (1o) iunconst 3syl (U_ y (cv x) (opr (1o) (^o) (cv y))) eqeq2d bitr4d y (cv x) (opr (1o) (^o) (cv y)) (1o) iuneq2 syl5bir tfinds)) thm (oaordi ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (e. B (On)) (e. C (On))) (-> (e. A B) (e. (opr C (+o) A) (opr C (+o) B)))) (B A onelon (e. C (On)) adantll B eloni B A ordsucss syl (e. C (On)) (e. A (On)) ad2antlr (cv x) (suc A) C (+o) opreq2 (opr C (+o) (suc A)) sseq2d (e. C (On)) imbi2d (cv x) (cv y) C (+o) opreq2 (opr C (+o) (suc A)) sseq2d (e. C (On)) imbi2d (cv x) (suc (cv y)) C (+o) opreq2 (opr C (+o) (suc A)) sseq2d (e. C (On)) imbi2d (cv x) B C (+o) opreq2 (opr C (+o) (suc A)) sseq2d (e. C (On)) imbi2d (opr C (+o) (suc A)) ssid (e. C (On)) a1i (e. (suc A) (On)) a1i C (cv y) oasuc ancoms (opr C (+o) (suc A)) sseq2d (opr C (+o) (cv y)) sssucid (opr C (+o) (suc A)) (opr C (+o) (cv y)) (suc (opr C (+o) (cv y))) sstr2 mpi syl5bir ex (e. (suc A) (On)) (C_ (suc A) (cv y)) ad2antrr a2d A sucelon A (On) (cv x) sucssel sylbir (cv x) A limsuc biimpd sylan9r imp (cv y) (suc A) C (+o) opreq2 (cv x) ssiun2s syl (e. C (On)) adantr x visset C (cv x) (V) y oalim mpanr1 ancoms (e. (suc A) (On)) adantlr (C_ (suc A) (cv x)) adantlr sseqtr4d ex (A.e. y (cv x) (-> (C_ (suc A) (cv y)) (-> (e. C (On)) (C_ (opr C (+o) (suc A)) (opr C (+o) (cv y)))))) a1d tfindsg exp31 A sucelon syl5ib com4r imp31 syld C A oasuc (opr C (+o) B) sseq1d C (+o) A oprex (opr C (+o) A) (V) (opr C (+o) B) sucssel ax-mp syl6bi (e. B (On)) adantlr syld imp an1rs mpdan ex ancoms)) thm (oaord () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (<-> (e. A B) (e. (opr C (+o) A) (opr C (+o) B)))) (B C A oaordi (e. A (On)) 3adant1 A B C (+o) opreq2 (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) a1i A C B oaordi (e. B (On)) 3adant2 orim12d con3d (e. A (On)) (e. B (On)) (e. C (On)) df-3an (/\ (e. A (On)) (e. B (On))) (e. C (On)) ancom (e. C (On)) (e. A (On)) (e. B (On)) anandi 3bitr C A oacl (opr C (+o) A) eloni syl C B oacl (opr C (+o) B) eloni syl anim12i sylbi (opr C (+o) A) (opr C (+o) B) ordtri2 syl (e. A (On)) (e. B (On)) (e. C (On)) 3simpa A eloni B eloni anim12i A B ordtri2 3syl 3imtr4d impbid)) thm (oacan () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (<-> (= (opr A (+o) B) (opr A (+o) C)) (= B C))) (B C A oaord 3comr C B A oaord 3com13 orbi12d negbid B C ordtri3 B eloni C eloni syl2an (e. A (On)) 3adant1 (opr A (+o) B) (opr A (+o) C) ordtri3 A B oacl (opr A (+o) B) eloni syl A C oacl (opr A (+o) C) eloni syl syl2an 3impdi 3bitr4rd)) thm (oaword () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (<-> (C_ A B) (C_ (opr C (+o) A) (opr C (+o) B)))) (A B C oaord C A B oacan 3coml bicomd orbi12d A B onsseleq (e. C (On)) 3adant3 C A oacl C B oacl anim12i anandis ancoms 3impa (opr C (+o) A) (opr C (+o) B) onsseleq syl 3bitr4d)) thm (oawordri ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (-> (C_ A B) (C_ (opr A (+o) C) (opr B (+o) C)))) ((cv x) ({/}) A (+o) opreq2 (cv x) ({/}) B (+o) opreq2 sseq12d (cv x) (cv y) A (+o) opreq2 (cv x) (cv y) B (+o) opreq2 sseq12d (cv x) (suc (cv y)) A (+o) opreq2 (cv x) (suc (cv y)) B (+o) opreq2 sseq12d (cv x) C A (+o) opreq2 (cv x) C B (+o) opreq2 sseq12d A oa0 (e. B (On)) adantr B oa0 (e. A (On)) adantl sseq12d biimpar (opr A (+o) (cv y)) (opr B (+o) (cv y)) ordsucsssuc A (cv y) oacl (opr A (+o) (cv y)) eloni syl B (cv y) oacl (opr B (+o) (cv y)) eloni syl syl2an anandirs A (cv y) oasuc (e. B (On)) adantlr B (cv y) oasuc (e. A (On)) adantll sseq12d bitr4d biimpd expcom (C_ A B) adantrd x visset A (cv x) (V) y oalim (e. B (On)) adantlr B (cv x) (V) y oalim (e. A (On)) adantll sseq12d y (cv x) (opr A (+o) (cv y)) (opr B (+o) (cv y)) ss2iun syl5bir mpanr1 expcom (C_ A B) adantrd tfinds3 exp4c com3l 3imp)) thm (oaord1 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. ({/}) B) (e. A (opr A (+o) B)))) (0elon ({/}) B A oaord mp3an1 A oa0 (e. B (On)) adantl (opr A (+o) B) eleq1d bitrd ancoms)) thm (oaword1 () () (-> (/\ (e. A (On)) (e. B (On))) (C_ A (opr A (+o) B))) (A oa0 (e. B (On)) adantr B 0ss 0elon ({/}) B A oaword 3com13 mp3an3 mpbii eqsstr3d)) thm (oaword2 () () (-> (/\ (e. A (On)) (e. B (On))) (C_ A (opr B (+o) A))) (B 0ss 0elon ({/}) B A oawordri mp3an1 A oa0r (e. B (On)) adantl (opr B (+o) A) sseq1d sylibd mpi ancoms)) thm (oawordeulem ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (S x) (S z)) ((oawordeulem.1 (e. A (On))) (oawordeulem.2 (e. B (On))) (oawordeulem.3 (= S ({e.|} y (On) (C_ B (opr A (+o) (cv y))))))) (-> (C_ A B) (E!e. x (On) (= (opr A (+o) (cv x)) B))) (oawordeulem.3 y (On) (C_ B (opr A (+o) (cv y))) ssrab2 eqsstr oawordeulem.3 B eleq2i (cv y) B A (+o) opreq2 B sseq2d (On) elrab bitr oawordeulem.2 oawordeulem.2 oawordeulem.1 B A oaword2 mp2an mpbir2an B S n0i ax-mp S oninton mp2an (|^| S) z onzsl mpbi (|^| S) ({/}) A (+o) opreq2 oawordeulem.1 A oa0 ax-mp syl6eq B sseq1d biimprd (|^| S) (suc (cv z)) A (+o) opreq2 oawordeulem.1 A (cv z) oasuc mpan sylan9eqr z visset sucid (|^| S) (suc (cv z)) (cv z) eleq2 mpbiri oawordeulem.3 y (On) (C_ B (opr A (+o) (cv y))) ssrab2 eqsstr oawordeulem.3 B eleq2i (cv y) B A (+o) opreq2 B sseq2d (On) elrab bitr oawordeulem.2 oawordeulem.2 oawordeulem.1 B A oaword2 mp2an mpbir2an B S n0i ax-mp S oninton mp2an (cv z) onel (cv y) (cv z) A (+o) opreq2 B sseq2d onnminsb oawordeulem.3 inteqi (cv z) eleq2i syl5ib oawordeulem.1 A (cv z) oacl mpan oawordeulem.2 B (opr A (+o) (cv z)) ontri1 mpan syl con2bid sylibrd mpcom oawordeulem.2 onord B (opr A (+o) (cv z)) ordsucss ax-mp 3syl (e. (cv z) (On)) adantl eqsstrd ex r19.23aiv (C_ A B) a1d z (|^| S) (opr A (+o) (cv z)) B iunss oawordeulem.3 y (On) (C_ B (opr A (+o) (cv y))) ssrab2 eqsstr oawordeulem.3 B eleq2i (cv y) B A (+o) opreq2 B sseq2d (On) elrab bitr oawordeulem.2 oawordeulem.2 oawordeulem.1 B A oaword2 mp2an mpbir2an B S n0i ax-mp S oninton mp2an (cv z) onel (cv y) (cv z) A (+o) opreq2 B sseq2d onnminsb oawordeulem.3 inteqi (cv z) eleq2i syl5ib oawordeulem.1 A (cv z) oacl mpan oawordeulem.2 B (opr A (+o) (cv z)) ontri1 mpan syl con2bid sylibrd mpcom oawordeulem.2 (opr A (+o) (cv z)) onelss syl mprgbir oawordeulem.1 A (|^| S) (V) z oalim mpan B sseq1d mpbiri (C_ A B) a1d 3jaoi ax-mp oawordeulem.2 oawordeulem.2 oawordeulem.1 B A oaword2 mp2an (cv y) B A (+o) opreq2 B sseq2d (On) rcla4ev mp2an (e. (cv z) B) y ax-17 (e. (cv z) A) y ax-17 (e. (cv z) (+o)) y ax-17 z y (On) (C_ B (opr A (+o) (cv y))) hbrab1 hbint hbopr hbss (cv y) (|^| ({e.|} y (On) (C_ B (opr A (+o) (cv y))))) A (+o) opreq2 B sseq2d onminsb ax-mp oawordeulem.3 inteqi A (+o) opreq2i sseqtr4 jctir (opr A (+o) (|^| S)) B eqss sylibr oawordeulem.3 y (On) (C_ B (opr A (+o) (cv y))) ssrab2 eqsstr oawordeulem.3 B eleq2i (cv y) B A (+o) opreq2 B sseq2d (On) elrab bitr oawordeulem.2 oawordeulem.2 oawordeulem.1 B A oaword2 mp2an mpbir2an B S n0i ax-mp S oninton mp2an jctil (cv x) (|^| S) A (+o) opreq2 B eqeq1d (On) rcla4ev syl oawordeulem.1 A (cv x) (cv y) oacan mp3an1 (opr A (+o) (cv x)) B (opr A (+o) (cv y)) eqtr3t syl5bi rgen2 jctir (cv x) (cv y) A (+o) opreq2 B eqeq1d (On) reu4 sylibr)) thm (oawordeu ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (e. A (On)) (e. B (On))) (C_ A B)) (E!e. x (On) (= (opr A (+o) (cv x)) B))) (A (if (e. A (On)) A ({/})) B sseq1 A (if (e. A (On)) A ({/})) (+o) (cv x) opreq1 B eqeq1d x (On) reubidv imbi12d B (if (e. B (On)) B ({/})) (if (e. A (On)) A ({/})) sseq2 B (if (e. B (On)) B ({/})) (opr (if (e. A (On)) A ({/})) (+o) (cv x)) eqeq2 x (On) reubidv imbi12d 0elon A elimel 0elon B elimel ({e.|} y (On) (C_ (if (e. B (On)) B ({/})) (opr (if (e. A (On)) A ({/})) (+o) (cv y)))) eqid x oawordeulem dedth2h imp)) thm (oawordexr ((A x) (B x)) () (-> (/\ (e. A (On)) (E.e. x (On) (= (opr A (+o) (cv x)) B))) (C_ A B)) (A (cv x) oaword1 (opr A (+o) (cv x)) B A sseq2 biimpcd syl ex r19.23adv imp)) thm (oawordex ((A x) (B x)) () (-> (/\ (e. A (On)) (e. B (On))) (<-> (C_ A B) (E.e. x (On) (= (opr A (+o) (cv x)) B)))) (A B x oawordeu ex x (On) (= (opr A (+o) (cv x)) B) reurex syl6 A x B oawordexr ex (e. B (On)) adantr impbid)) thm (oaordex ((A x) (B x)) () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. A B) (E.e. x (On) (/\ (e. ({/}) (cv x)) (= (opr A (+o) (cv x)) B))))) (B A onelsst (e. A (On)) adantl A B x oawordex sylibd A (cv x) oaord1 (opr A (+o) (cv x)) B A eleq2 sylan9bb biimprcd exp4c com12 imp4b (e. (cv x) (On)) (= (opr A (+o) (cv x)) B) pm3.27 (/\ (e. A (On)) (e. A B)) a1i jcad exp3a r19.22dv ex (e. B (On)) adantr mpdd A (cv x) oaord1 (opr A (+o) (cv x)) B A eleq2 sylan9bb biimpd exp31 com34 imp4a r19.23adv (e. B (On)) adantr impbid)) thm (oa00 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (= (opr A (+o) B) ({/})) (/\ (= A ({/})) (= B ({/}))))) (A on0eln0 (e. B (On)) adantr A B oaword1 ({/}) sseld sylbird ({/}) (opr A (+o) B) n0i syl6 a3d B on0eln0 (e. A (On)) adantl 0elon ({/}) B A oaord mp3an1 ancoms bitr3d (opr A (+o) ({/})) (opr A (+o) B) n0i syl6bi a3d jcad A ({/}) B ({/}) (+o) opreq12 0elon ({/}) oa0 ax-mp syl6eq (/\ (e. A (On)) (e. B (On))) a1i impbid)) thm (oalimcl ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (Lim (opr A (+o) B))) (A B oacl (opr A (+o) B) eloni syl B C limelon sylan2 B 0ellim ({/}) B n0i syl (e. A (On)) (e. B C) ad2antll A B oa00 (= A ({/})) (= B ({/})) pm3.27 syl6bi con3d B C limelon sylan2 mpd (opr A (+o) B) (suc (cv y)) (U_ x B (opr A (+o) (cv x))) eqeq1 A B C x oalim syl5bi imp y visset sucid syl5eleq (cv y) x B (opr A (+o) (cv x)) eliun sylib B (cv x) onelon B C limelon sylan (cv x) B onnbtwn (e. (cv x) B) (e. B (suc (cv x))) imnan sylibr com12 (/\ (e. B C) (Lim B)) adantl mpd (e. A (On)) (e. (cv y) (opr A (+o) (cv x))) ad2antrl A (cv x) oacl (opr A (+o) (cv x)) eloni (opr A (+o) (cv x)) (cv y) ordsucelsuc 3syl A (cv x) oasuc (suc (cv y)) eleq2d bitr4d B (cv x) onelon B C limelon sylan sylan2 (opr A (+o) B) (suc (cv y)) (opr A (+o) (suc (cv x))) eleq1 bicomd sylan9bbr B (suc (cv x)) A oaord 3expa B C limelon (e. (cv x) B) adantr B (cv x) onelon B C limelon sylan (cv x) sucelon sylib jca sylan ancoms (= (opr A (+o) B) (suc (cv y))) adantl bitr4d biimpd exp32 com4l imp32 mtod exp44 imp r19.23adv (= (opr A (+o) B) (suc (cv y))) adantl mpd expcom pm2.01d (e. (cv y) (On)) adantr nrexdv jca (= (opr A (+o) B) ({/})) (E.e. y (On) (= (opr A (+o) B) (suc (cv y)))) ioran sylibr jca (opr A (+o) B) y dflim3 sylibr)) thm (oaass ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x)) () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (= (opr (opr A (+o) B) (+o) C) (opr A (+o) (opr B (+o) C)))) ((cv x) ({/}) (opr A (+o) B) (+o) opreq2 (cv x) ({/}) B (+o) opreq2 A (+o) opreq2d eqeq12d (cv x) (cv y) (opr A (+o) B) (+o) opreq2 (cv x) (cv y) B (+o) opreq2 A (+o) opreq2d eqeq12d (cv x) (suc (cv y)) (opr A (+o) B) (+o) opreq2 (cv x) (suc (cv y)) B (+o) opreq2 A (+o) opreq2d eqeq12d (cv x) C (opr A (+o) B) (+o) opreq2 (cv x) C B (+o) opreq2 A (+o) opreq2d eqeq12d A B oacl (opr A (+o) B) oa0 syl B oa0 A (+o) opreq2d (e. A (On)) adantl eqtr4d (opr A (+o) B) (cv y) oasuc A B oacl sylan B (cv y) oasuc A (+o) opreq2d (e. A (On)) adantl A (opr B (+o) (cv y)) oasuc B (cv y) oacl sylan2 eqtrd anassrs eqeq12d (opr (opr A (+o) B) (+o) (cv y)) (opr A (+o) (opr B (+o) (cv y))) suceq syl5bir expcom y (cv x) (opr (opr A (+o) B) (+o) (cv y)) (opr A (+o) (opr B (+o) (cv y))) iuneq2 (/\ (Lim (cv x)) (/\ (e. A (On)) (e. B (On)))) adantl x visset (opr A (+o) B) (cv x) (V) y oalim mpanr1 A B oacl sylan ancoms (A.e. y (cv x) (= (opr (opr A (+o) B) (+o) (cv y)) (opr A (+o) (opr B (+o) (cv y))))) adantr B (+o) (cv x) oprex A (opr B (+o) (cv x)) (V) z oalim mpanr1 x visset B (cv x) (V) oalimcl mpanr1 ancoms sylan2 x visset (cv x) (V) limelon mpan B (cv x) oacl ancoms (opr B (+o) (cv x)) (cv z) onelon ex syl (e. A (On)) adantld (Lim (cv x)) adantl (cv z) B ordtri2or ancoms B eloni (cv z) eloni syl2an (e. (cv x) (On)) (e. (cv z) (opr B (+o) (cv x))) ad2ant2l (Lim (cv x)) adantl (cv y) ({/}) B (+o) opreq2 (cv z) sseq2d (cv x) rcla4ev (cv x) 0ellim B (cv z) onelsst B oa0 (cv z) sseq2d sylibrd imp syl2an exp32 imp (e. (cv x) (On)) adantrl (/\ (e. (cv z) (opr B (+o) (cv x))) (e. (cv z) (On))) adantrr B (cv z) y oawordex (e. (cv x) (On)) (e. (cv z) (opr B (+o) (cv x))) ad2ant2l (cv y) (cv x) B oaord 3expb (opr B (+o) (cv y)) (cv z) (opr B (+o) (cv x)) eleq1 sylan9bb an1rs biimpar (opr B (+o) (cv y)) (cv z) eqimss2 (e. (cv y) (On)) adantl (/\ (e. (cv x) (On)) (e. B (On))) (e. (cv z) (opr B (+o) (cv x))) ad2antrr jca anasss expcom r19.22dv2 (e. (cv z) (On)) adantrr sylbid (Lim (cv x)) adantl jaod mpd exp45 imp (e. A (On)) adantld imp32 (cv z) (opr B (+o) (cv y)) A oaword (/\ (e. A (On)) (e. (cv z) (opr B (+o) (cv x)))) (e. (cv z) (On)) pm3.27 (/\ (Lim (cv x)) (/\ (e. (cv x) (On)) (e. B (On)))) (e. (cv y) (cv x)) ad2antlr B (cv y) oacl (cv x) (cv y) onelon sylan2 exp32 com12 imp31 (Lim (cv x)) adantll (/\ (/\ (e. A (On)) (e. (cv z) (opr B (+o) (cv x)))) (e. (cv z) (On))) adantlr (e. A (On)) (e. (cv z) (opr B (+o) (cv x))) pm3.26 (e. (cv z) (On)) adantr (/\ (Lim (cv x)) (/\ (e. (cv x) (On)) (e. B (On)))) (e. (cv y) (cv x)) ad2antlr syl3anc rexbidva mpbid exp32 mpdd exp32 mpd exp4a imp31 r19.21aiv z (opr B (+o) (cv x)) y (cv x) (opr A (+o) (cv z)) (opr A (+o) (opr B (+o) (cv y))) iunss2 syl ancoms (cv x) B (cv y) oaordi (e. (cv w) (opr A (+o) (opr B (+o) (cv y)))) anim1d (cv z) (opr B (+o) (cv y)) A (+o) opreq2 (cv w) eleq2d (opr B (+o) (cv x)) rcla4ev syl6 exp3a r19.23adv (cv w) y (cv x) (opr A (+o) (opr B (+o) (cv y))) eliun (cv w) z (opr B (+o) (cv x)) (opr A (+o) (cv z)) eliun 3imtr4g ssrdv x visset (cv x) (V) limelon mpan sylan (e. A (On)) adantl eqssd eqtrd an1s (A.e. y (cv x) (= (opr (opr A (+o) B) (+o) (cv y)) (opr A (+o) (opr B (+o) (cv y))))) adantr 3eqtr4d exp31 tfinds3 com12 3impia)) thm (oarec ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (+o) B) (u. A ({|} x (E.e. y B (= (cv x) (opr A (+o) (cv y)))))))) ((cv z) ({/}) A (+o) opreq2 (cv z) ({/}) y (= (cv x) (opr A (+o) (cv y))) rexeq1 x abbidv A uneq2d eqeq12d (cv z) (cv w) A (+o) opreq2 (cv z) (cv w) y (= (cv x) (opr A (+o) (cv y))) rexeq1 x abbidv A uneq2d eqeq12d (cv z) (suc (cv w)) A (+o) opreq2 (cv z) (suc (cv w)) y (= (cv x) (opr A (+o) (cv y))) rexeq1 x abbidv A uneq2d eqeq12d (cv z) B A (+o) opreq2 (cv z) B y (= (cv x) (opr A (+o) (cv y))) rexeq1 x abbidv A uneq2d eqeq12d A oa0 y (= (cv x) (opr A (+o) (cv y))) rex0 x nex x (E.e. y ({/}) (= (cv x) (opr A (+o) (cv y)))) abn0 con1bii mpbi A uneq2i A un0 eqtr2 syl6eq A (cv w) oasuc (opr A (+o) (cv w)) x df-sn (opr A (+o) (cv w)) (u. A ({|} x (E.e. y (cv w) (= (cv x) (opr A (+o) (cv y)))))) ({} (opr A (+o) (cv w))) ({|} x (= (cv x) (opr A (+o) (cv w)))) uneq12 mpan2 (opr A (+o) (cv w)) df-suc y visset (cv w) elsuc (= (cv x) (opr A (+o) (cv y))) anbi1i (e. (cv y) (cv w)) (= (cv y) (cv w)) (= (cv x) (opr A (+o) (cv y))) andir bitr y exbii y (/\ (e. (cv y) (cv w)) (= (cv x) (opr A (+o) (cv y)))) (/\ (= (cv y) (cv w)) (= (cv x) (opr A (+o) (cv y)))) 19.43 bitr y (suc (cv w)) (= (cv x) (opr A (+o) (cv y))) df-rex y (cv w) (= (cv x) (opr A (+o) (cv y))) df-rex w visset (cv y) (cv w) A (+o) opreq2 (cv x) eqeq2d ceqsexv bicomi orbi12i 3bitr4 x abbii x (E.e. y (cv w) (= (cv x) (opr A (+o) (cv y)))) (= (cv x) (opr A (+o) (cv w))) unab eqtr4 A uneq2i A ({|} x (E.e. y (cv w) (= (cv x) (opr A (+o) (cv y))))) ({|} x (= (cv x) (opr A (+o) (cv w)))) unass eqtr4 3eqtr4g sylan9eq exp31 com12 z visset A (cv z) (V) w oalim mpanr1 ancoms (A.e. w (cv z) (= (opr A (+o) (cv w)) (u. A ({|} x (E.e. y (cv w) (= (cv x) (opr A (+o) (cv y)))))))) adantr w (cv z) (opr A (+o) (cv w)) (u. A ({|} x (E.e. y (cv w) (= (cv x) (opr A (+o) (cv y)))))) iuneq2 (/\ (Lim (cv z)) (e. A (On))) adantl (cv z) 0ellim ({/}) (cv z) n0i (cv z) w A iunconst 3syl (cv z) limord (cv z) (cv y) (cv w) ordtr1 syl exp3a com23 imp (= (cv x) (opr A (+o) (cv y))) anim1d r19.22dv2 ex r19.23adv (Lim (cv z)) y ax-17 (e. (cv w) (cv z)) y ax-17 y (cv w) (= (cv x) (opr A (+o) (cv y))) hbre1 hbrex (cv z) (cv y) limsuc biimpd (cv y) (cv z) sucidg (Lim (cv z)) a1i jcad (= (cv x) (opr A (+o) (cv y))) anim1d (cv w) (suc (cv y)) (cv y) eleq2 (= (cv x) (opr A (+o) (cv y))) anbi1d (cv z) rcla4ev y (cv w) (= (cv x) (opr A (+o) (cv y))) ra4e w (cv z) r19.22si syl anassrs syl6 exp3a r19.23ad impbid x abbidv w (cv z) x (E.e. y (cv w) (= (cv x) (opr A (+o) (cv y)))) iunab syl5eq uneq12d w (cv z) A ({|} x (E.e. y (cv w) (= (cv x) (opr A (+o) (cv y))))) iunun syl5eq (e. A (On)) (A.e. w (cv z) (= (opr A (+o) (cv w)) (u. A ({|} x (E.e. y (cv w) (= (cv x) (opr A (+o) (cv y)))))))) ad2antrr 3eqtrd exp31 tfinds3 impcom)) thm (omordi ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (/\ (e. B (On)) (e. C (On))) (e. ({/}) C)) (-> (e. A B) (e. (opr C (.o) A) (opr C (.o) B)))) (B A onelon ex (cv x) ({/}) A eleq2 (cv x) ({/}) C (.o) opreq2 (opr C (.o) A) eleq2d imbi12d (cv x) (cv y) A eleq2 (cv x) (cv y) C (.o) opreq2 (opr C (.o) A) eleq2d imbi12d (cv x) (suc (cv y)) A eleq2 (cv x) (suc (cv y)) C (.o) opreq2 (opr C (.o) A) eleq2d imbi12d (cv x) B A eleq2 (cv x) B C (.o) opreq2 (opr C (.o) A) eleq2d imbi12d A noel (e. (opr C (.o) A) (opr C (.o) ({/}))) pm2.21i (/\ (/\ (e. A (On)) (e. C (On))) (e. ({/}) C)) a1i (opr C (.o) (cv y)) C oaword1 (opr C (.o) A) sseld (e. A (cv y)) imim2d imp (e. ({/}) C) adantrl A (cv y) C (.o) opreq2 (opr (opr C (.o) (cv y)) (+o) C) eleq1d (opr C (.o) (cv y)) C oaord1 biimpa syl5bir com12 (-> (e. A (cv y)) (e. (opr C (.o) A) (opr C (.o) (cv y)))) adantrr jaod C (cv y) omcl (e. C (On)) (e. (cv y) (On)) pm3.26 jca sylan A (cv y) elsuci syl5 C (cv y) omsuc (opr C (.o) A) eleq2d (/\ (e. ({/}) C) (-> (e. A (cv y)) (e. (opr C (.o) A) (opr C (.o) (cv y))))) adantr sylibrd exp43 com12 (e. A (On)) adantld imp3a (cv x) A limsuc biimpa (cv y) (suc A) C (.o) opreq2 (cv x) ssiun2s syl (e. C (On)) adantll x visset C (cv x) (V) y omlim mpanr1 (e. A (cv x)) adantr sseqtr4d (/\ (e. C (On)) (Lim (cv x))) id (e. A (On)) (e. ({/}) C) ad2ant2r sylan (opr C (.o) A) C oaord1 C A omcl sylan anabss1 biimpa C A omsuc (e. ({/}) C) adantr eleqtrrd (Lim (cv x)) adantrl (e. A (cv x)) adantr sseldd ex exp43 com13 imp4c (A.e. y (cv x) (-> (e. A (cv y)) (e. (opr C (.o) A) (opr C (.o) (cv y))))) a1dd tfinds3 com23 exp4a exp4a mpdd com34 com24 imp31)) thm (omord2 () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) C)) (<-> (e. A B) (e. (opr C (.o) A) (opr C (.o) B)))) (B C A omordi (e. A (On)) 3adantl1 A B C (.o) opreq2 (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) C)) a1i A C B omordi (e. B (On)) 3adantl2 orim12d con3d (opr C (.o) A) (opr C (.o) B) ordtri2 (opr C (.o) A) eloni (opr C (.o) B) eloni syl2an C A omcl C B omcl syl2an anandis ancoms 3impa (e. ({/}) C) adantr A B ordtri2 A eloni B eloni syl2an (e. C (On)) 3adant3 (e. ({/}) C) adantr 3imtr4d impbid)) thm (omord () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (<-> (/\ (e. A B) (e. ({/}) C)) (e. (opr C (.o) A) (opr C (.o) B)))) (A B C omord2 ex pm5.32rd (e. (opr C (.o) A) (opr C (.o) B)) (e. ({/}) C) pm3.26 (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) a1i C ({/}) (.o) B opreq1 ({/}) eqeq1d B om0r syl5bir com12 (opr C (.o) A) (opr C (.o) B) n0i nsyli (e. C (On)) adantr C on0eln0 (e. B (On)) adantl sylibrd (e. A (On)) 3adant1 ancld impbid bitrd)) thm (omcan () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) A)) (<-> (= (opr A (.o) B) (opr A (.o) C)) (= B C))) (C A B omordi ex ancoms (e. B (On)) 3adant2 imp B A C omordi ex ancoms (e. C (On)) 3adant3 imp orim12d con3d (opr A (.o) B) (opr A (.o) C) ordtri3 A B omcl (opr A (.o) B) eloni syl A C omcl (opr A (.o) C) eloni syl syl2an 3impdi (e. ({/}) A) adantr B C ordtri3 B eloni C eloni syl2an (e. A (On)) 3adant1 (e. ({/}) A) adantr 3imtr4d B C A (.o) opreq2 (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) A)) a1i impbid)) thm (omword () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) C)) (<-> (C_ A B) (C_ (opr C (.o) A) (opr C (.o) B)))) (A B C omord2 C A B omcan (e. C (On)) (e. A (On)) (e. B (On)) 3anrot sylanbr bicomd orbi12d A B onsseleq (e. C (On)) 3adant3 (e. ({/}) C) adantr C A omcl C B omcl anim12i anandis ancoms 3impa (opr C (.o) A) (opr C (.o) B) onsseleq syl (e. ({/}) C) adantr 3bitr4d)) thm (omwordi () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (-> (C_ A B) (C_ (opr C (.o) A) (opr C (.o) B)))) (A B C omword biimpd ex C eloni C ord0eln0 con2bid syl (e. A (On)) (e. B (On)) 3ad2ant3 C ({/}) (.o) A opreq1 C ({/}) (.o) B opreq1 sseq12d ({/}) ssid A om0r (e. B (On)) adantr B om0r (e. A (On)) adantl sseq12d mpbiri syl5bir com12 (e. C (On)) 3adant3 sylbird (C_ A B) a1dd pm2.61d)) thm (omwordri ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (-> (C_ A B) (C_ (opr A (.o) C) (opr B (.o) C)))) ((cv x) ({/}) A (.o) opreq2 (cv x) ({/}) B (.o) opreq2 sseq12d (cv x) (cv y) A (.o) opreq2 (cv x) (cv y) B (.o) opreq2 sseq12d (cv x) (suc (cv y)) A (.o) opreq2 (cv x) (suc (cv y)) B (.o) opreq2 sseq12d (cv x) C A (.o) opreq2 (cv x) C B (.o) opreq2 sseq12d (opr B (.o) ({/})) 0ss A om0 (opr B (.o) ({/})) sseq1d mpbiri (e. B (On)) (C_ A B) ad2antrr (opr A (.o) (cv y)) (opr B (.o) (cv y)) A oawordri A (cv y) omcl (e. B (On)) 3adant2 B (cv y) omcl (e. A (On)) 3adant1 (e. A (On)) (e. B (On)) (e. (cv y) (On)) 3simp1 syl3anc imp (C_ A B) adantrl A B (opr B (.o) (cv y)) oaword (e. A (On)) (e. B (On)) (e. (cv y) (On)) 3simp1 (e. A (On)) (e. B (On)) (e. (cv y) (On)) 3simp2 B (cv y) omcl (e. A (On)) 3adant1 syl3anc biimpa (C_ (opr A (.o) (cv y)) (opr B (.o) (cv y))) adantrr sstrd A (cv y) omsuc (e. B (On)) 3adant2 (/\ (C_ A B) (C_ (opr A (.o) (cv y)) (opr B (.o) (cv y)))) adantr B (cv y) omsuc (e. A (On)) 3adant1 (/\ (C_ A B) (C_ (opr A (.o) (cv y)) (opr B (.o) (cv y)))) adantr 3sstr4d exp32 3exp com3r imp4c x visset A (cv x) (V) y omlim (/\ (e. B (On)) (/\ (e. (cv x) (V)) (Lim (cv x)))) adantr B (cv x) (V) y omlim (/\ (e. A (On)) (/\ (e. (cv x) (V)) (Lim (cv x)))) adantl sseq12d y (cv x) (opr A (.o) (cv y)) (opr B (.o) (cv y)) ss2iun syl5bir anandirs mpanr1 expcom (C_ A B) adantrd tfinds3 exp3a 3impib 3coml)) thm (omword1 () () (-> (/\ (/\ (e. A (On)) (e. B (On))) (e. ({/}) B)) (C_ A (opr A (.o) B))) (B eloni B ordgt0ge1 syl (e. A (On)) adantl 1on (1o) B A omwordi mp3an1 ancoms A om1 (e. B (On)) adantr (opr A (.o) B) sseq1d sylibd sylbid imp)) thm (omword2 () () (-> (/\ (/\ (e. A (On)) (e. B (On))) (e. ({/}) B)) (C_ A (opr B (.o) A))) (A om1r (e. B (On)) (e. ({/}) B) ad2antrr B ordgt0ge1 biimpa B eloni sylan (e. A (On)) adantll 1on (1o) B A omwordri mp3an1 ancoms (e. ({/}) B) adantr mpd eqsstr3d)) thm (om00 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (= (opr A (.o) B) ({/})) (\/ (= A ({/})) (= B ({/}))))) (A eloni A ordge1n0 syl biimprd (e. B (On)) adantr B on0eln0 (e. A (On)) adantl A B omword1 ex sylbird anim12d (1o) A (opr A (.o) B) sstr syl6 (= A ({/})) (= B ({/})) ioran syl5ib A B omcl (opr A (.o) B) eloni (opr A (.o) B) ordge1n0 3syl sylibd a3d A ({/}) (.o) B opreq1 B om0r sylan9eqr ex (e. A (On)) adantl B ({/}) A (.o) opreq2 A om0 sylan9eqr ex (e. B (On)) adantr jaod impbid)) thm (om00el () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. ({/}) (opr A (.o) B)) (/\ (e. ({/}) A) (e. ({/}) B)))) (A B om00 negbid A B omcl (opr A (.o) B) on0eln0 syl A on0eln0 B on0eln0 bi2anan9 (= A ({/})) (= B ({/})) ioran syl6bbr 3bitr4d)) thm (omordlim ((A x) (B x) (C x)) () (-> (/\ (/\ (e. A (On)) (/\ (e. B D) (Lim B))) (e. C (opr A (.o) B))) (E.e. x B (e. C (opr A (.o) (cv x))))) (A B D x omlim C eleq2d C x B (opr A (.o) (cv x)) eliun syl6bb biimpa)) thm (omlimcl ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (e. ({/}) A)) (Lim (opr A (.o) B))) (A B omcl (opr A (.o) B) eloni syl B C limelon sylan2 (e. ({/}) A) adantr ({/}) A n0i B 0ellim ({/}) B n0i syl anim12i ancoms (e. B C) adantll (e. A (On)) adantll A B om00 negbid (= A ({/})) (= B ({/})) ioran syl6bb B C limelon sylan2 (e. ({/}) A) adantr mpbird (opr A (.o) B) (suc (cv y)) (U_ x B (opr A (.o) (cv x))) eqeq1 biimpac A B C x omlim sylan y visset sucid syl5eleq (cv y) x B (opr A (.o) (cv x)) eliun sylib (e. ({/}) A) adantlr B (cv x) onelon B C limelon sylan (cv x) B onnbtwn (e. (cv x) B) (e. B (suc (cv x))) imnan sylibr com12 (/\ (e. B C) (Lim B)) adantl mpd (e. A (On)) adantll (e. ({/}) A) adantlr (e. (cv y) (opr A (.o) (cv x))) adantr A (cv x) omcl (opr A (.o) (cv x)) eloni (opr A (.o) (cv x)) (cv y) ordsucelsuc syl (opr A (.o) (cv x)) oa1suc (suc (cv y)) eleq2d bitr4d syl (e. ({/}) A) adantr A eloni A ordgt0ge1 syl (e. (cv x) (On)) adantr A (cv x) omcl 1on (1o) A (opr A (.o) (cv x)) oaword mp3an1 syldan bitrd biimpa A (cv x) omsuc (e. ({/}) A) adantr sseqtr4d (suc (cv y)) sseld sylbid (opr A (.o) B) (suc (cv y)) (opr A (.o) (suc (cv x))) eleq1 biimprd syl9 com23 (e. B (On)) adantlrl B (suc (cv x)) A omord (e. B (suc (cv x))) (e. ({/}) A) pm3.26 syl6bir (cv x) sucelon syl3an2b 3comr 3expb (e. ({/}) A) adantr syl6d (e. B (On)) (e. (cv x) B) pm3.26 B (cv x) onelon jca B C limelon sylan (e. A (On)) anim2i anassrs sylan an1rs imp mtod exp31 r19.23adv (= (opr A (.o) B) (suc (cv y))) adantr mpd ex pm2.01d (e. (cv y) (On)) adantr nrexdv jca (= (opr A (.o) B) ({/})) (E.e. y (On) (= (opr A (.o) B) (suc (cv y)))) ioran sylibr jca (opr A (.o) B) y dflim3 sylibr)) thm (odi ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B x) (B y) (B z) (B w) (B v) (C x) (C y)) () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (= (opr A (.o) (opr B (+o) C)) (opr (opr A (.o) B) (+o) (opr A (.o) C)))) ((cv x) ({/}) B (+o) opreq2 A (.o) opreq2d (cv x) ({/}) A (.o) opreq2 (opr A (.o) B) (+o) opreq2d eqeq12d (cv x) (cv y) B (+o) opreq2 A (.o) opreq2d (cv x) (cv y) A (.o) opreq2 (opr A (.o) B) (+o) opreq2d eqeq12d (cv x) (suc (cv y)) B (+o) opreq2 A (.o) opreq2d (cv x) (suc (cv y)) A (.o) opreq2 (opr A (.o) B) (+o) opreq2d eqeq12d (cv x) C B (+o) opreq2 A (.o) opreq2d (cv x) C A (.o) opreq2 (opr A (.o) B) (+o) opreq2d eqeq12d A B omcl (opr A (.o) B) oa0 syl A om0 (e. B (On)) adantr (opr A (.o) B) (+o) opreq2d B oa0 (e. A (On)) adantl A (.o) opreq2d 3eqtr4rd B (cv y) oasuc (e. A (On)) 3adant1 A (.o) opreq2d A (opr B (+o) (cv y)) omsuc B (cv y) oacl sylan2 3impb eqtrd A (cv y) omsuc (e. B (On)) 3adant2 (opr A (.o) B) (+o) opreq2d (opr A (.o) B) (opr A (.o) (cv y)) A oaass A B omcl syl3an1 A (cv y) omcl syl3an2 3exp exp4b pm2.43a com4r pm2.43i 3imp eqtr4d eqeq12d (opr A (.o) (opr B (+o) (cv y))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv y))) (+o) A opreq1 syl5bir 3exp com3r imp3a A ({/}) (.o) (opr B (+o) (cv x)) opreq1 A ({/}) (.o) B opreq1 A ({/}) (.o) (cv x) opreq1 (+o) opreq12d eqeq12d B (cv x) oacl (opr B (+o) (cv x)) om0r syl B om0r (cv x) om0r (+o) opreqan12d 0elon ({/}) oa0 ax-mp syl6req eqtrd x visset (cv x) (V) limelon mpan sylan2 ancoms syl5bir exp3a com3r imp (A.e. y (cv x) (= (opr A (.o) (opr B (+o) (cv y))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv y))))) a1dd B (cv z) ontri1 B (cv z) v oawordex bitr3d (e. (cv x) (On)) (e. B (On)) pm3.27 (e. (cv z) (opr B (+o) (cv x))) adantr (opr B (+o) (cv x)) (cv z) onelon B (cv x) oacl ancoms sylan sylanc (cv v) (cv x) B oaord 3expb (opr B (+o) (cv v)) (cv z) (opr B (+o) (cv x)) eleq1 sylan9bb (= (opr B (+o) (cv v)) (cv z)) (e. (cv v) (cv x)) iba (/\ (e. (cv v) (On)) (/\ (e. (cv x) (On)) (e. B (On)))) adantl bitr3d an1rs biimpcd exp4c com4r imp r19.22dv sylbid orrd x visset (cv x) (V) limelon mpan sylanl1 (e. A (On)) adantlrl (/\ (e. ({/}) A) (A.e. y (cv x) (= (opr A (.o) (opr B (+o) (cv y))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv y)))))) adantlr A (cv x) om00el biimprd (cv x) 0ellim sylan2i x visset (cv x) (V) limelon mpan sylan2 exp4b com4r pm2.43a imp31 (e. (cv z) B) a1d (e. B (On)) adantlrr B A (cv z) omordi ancom1s A B omcl (opr A (.o) B) (opr A (.o) (cv z)) onelsst (opr A (.o) B) oa0 (opr A (.o) (cv z)) sseq2d sylibrd syl (e. ({/}) A) adantr syld (Lim (cv x)) adantll jcad (cv w) ({/}) (opr A (.o) B) (+o) opreq2 (opr A (.o) (cv z)) sseq2d (opr A (.o) (cv x)) rcla4ev syl6 (A.e. y (cv x) (= (opr A (.o) (opr B (+o) (cv y))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv y))))) adantrr (cv x) A (cv v) omordi x visset (cv x) (V) limelon mpan sylanl1 (= (opr B (+o) (cv v)) (cv z)) adantrd (A.e. y (cv x) (= (opr A (.o) (opr B (+o) (cv y))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv y))))) adantrr (cv y) (cv v) B (+o) opreq2 A (.o) opreq2d (cv y) (cv v) A (.o) opreq2 (opr A (.o) B) (+o) opreq2d eqeq12d (cv x) rcla4cv (opr A (.o) (opr B (+o) (cv v))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv v))) (opr A (.o) (cv z)) eqeq1 (opr B (+o) (cv v)) (cv z) A (.o) opreq2 syl5bi (opr (opr A (.o) B) (+o) (opr A (.o) (cv v))) (opr A (.o) (cv z)) eqimss2 syl6 (e. (cv v) (cv x)) imim2i imp3a syl (/\ (Lim (cv x)) (e. A (On))) (e. ({/}) A) ad2antll jcad (cv w) (opr A (.o) (cv v)) (opr A (.o) B) (+o) opreq2 (opr A (.o) (cv z)) sseq2d (opr A (.o) (cv x)) rcla4ev syl6 (e. (cv v) (On)) a1d r19.23adv (e. B (On)) adantlrr jaod (e. (cv z) (opr B (+o) (cv x))) adantr mpd r19.21aiva z (opr B (+o) (cv x)) w (opr A (.o) (cv x)) (opr A (.o) (cv z)) (opr (opr A (.o) B) (+o) (cv w)) iunss2 syl x visset A (cv x) (V) (cv w) v omordlim ex mpanr1 ancoms imp (e. B (On)) adantlrr (A.e. y (cv x) (= (opr A (.o) (opr B (+o) (cv y))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv y))))) adantlr (cv x) B (cv v) oaordi x visset (cv x) (V) limelon mpan sylan imp (e. A (On)) adantlrl (e. (cv w) (opr A (.o) (cv v))) a1d (A.e. y (cv x) (= (opr A (.o) (opr B (+o) (cv y))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv y))))) adantlr (opr A (.o) (cv v)) (opr A (.o) B) (cv w) oaordi A (cv v) omcl ancoms (e. B (On)) adantrr A B omcl (e. (cv v) (On)) adantl sylanc (cv x) (cv v) ordelon (cv x) limord sylan sylan an1rs (A.e. y (cv x) (= (opr A (.o) (opr B (+o) (cv y))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv y))))) adantlr (cv y) (cv v) B (+o) opreq2 A (.o) opreq2d (cv y) (cv v) A (.o) opreq2 (opr A (.o) B) (+o) opreq2d eqeq12d (cv x) rcla4cva (opr (opr A (.o) B) (+o) (cv w)) eleq2d (/\ (Lim (cv x)) (/\ (e. A (On)) (e. B (On)))) adantll sylibrd A (opr B (+o) (cv v)) omcl B (cv v) oacl ancoms sylan2 an1s (cv x) (cv v) ordelon (cv x) limord sylan sylan an1rs (opr A (.o) (opr B (+o) (cv v))) (opr (opr A (.o) B) (+o) (cv w)) onelsst syl (A.e. y (cv x) (= (opr A (.o) (opr B (+o) (cv y))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv y))))) adantlr syld jcad (cv z) (opr B (+o) (cv v)) A (.o) opreq2 (opr (opr A (.o) B) (+o) (cv w)) sseq2d (opr B (+o) (cv x)) rcla4ev syl6 ex r19.23adv (e. (cv w) (opr A (.o) (cv x))) adantr mpd r19.21aiva w (opr A (.o) (cv x)) z (opr B (+o) (cv x)) (opr (opr A (.o) B) (+o) (cv w)) (opr A (.o) (cv z)) iunss2 syl (e. ({/}) A) adantrl eqssd x visset B (cv x) (V) oalimcl mpanr1 ancoms (e. A (On)) anim2i an1s B (+o) (cv x) oprex A (opr B (+o) (cv x)) (V) z omlim mpanr1 syl (/\ (e. ({/}) A) (A.e. y (cv x) (= (opr A (.o) (opr B (+o) (cv y))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv y)))))) adantr (opr A (.o) B) (opr A (.o) (cv x)) (V) w oalim A B omcl (Lim (cv x)) (e. ({/}) A) ad2antlr A (cv x) (V) omlimcl x visset (Lim (cv x)) jctl (e. A (On)) anim2i ancoms sylan (e. B (On)) adantlrr A (.o) (cv x) oprex jctil sylanc (A.e. y (cv x) (= (opr A (.o) (opr B (+o) (cv y))) (opr (opr A (.o) B) (+o) (opr A (.o) (cv y))))) adantrr 3eqtr4d exp43 com3l imp oe0lem com12 tfinds3 exp3a com3l 3imp)) thm (omass ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x)) () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (= (opr (opr A (.o) B) (.o) C) (opr A (.o) (opr B (.o) C)))) ((cv x) ({/}) (opr A (.o) B) (.o) opreq2 (cv x) ({/}) B (.o) opreq2 A (.o) opreq2d eqeq12d (cv x) (cv y) (opr A (.o) B) (.o) opreq2 (cv x) (cv y) B (.o) opreq2 A (.o) opreq2d eqeq12d (cv x) (suc (cv y)) (opr A (.o) B) (.o) opreq2 (cv x) (suc (cv y)) B (.o) opreq2 A (.o) opreq2d eqeq12d (cv x) C (opr A (.o) B) (.o) opreq2 (cv x) C B (.o) opreq2 A (.o) opreq2d eqeq12d A B omcl (opr A (.o) B) om0 syl B om0 A (.o) opreq2d A om0 sylan9eqr eqtr4d (opr A (.o) B) (cv y) omsuc A B omcl sylan 3impa B (cv y) omsuc (e. A (On)) 3adant1 A (.o) opreq2d A (opr B (.o) (cv y)) B odi B (cv y) omcl syl3an2 3exp exp3a com34 pm2.43d 3imp eqtrd eqeq12d (opr (opr A (.o) B) (.o) (cv y)) (opr A (.o) (opr B (.o) (cv y))) (+o) (opr A (.o) B) opreq1 syl5bir 3exp com3r imp3a x visset (opr A (.o) B) (cv x) (V) y omlim mpanr1 A B omcl ancoms sylan an1rs (e. ({/}) B) (A.e. y (cv x) (= (opr (opr A (.o) B) (.o) (cv y)) (opr A (.o) (opr B (.o) (cv y))))) ad2antrr y (cv x) (opr (opr A (.o) B) (.o) (cv y)) (opr A (.o) (opr B (.o) (cv y))) iuneq2 (cv x) B (cv y) omordi x visset (cv x) (V) limelon mpan (e. B (On)) anim1i ancoms sylan (opr A (.o) (opr B (.o) (cv y))) ssid (cv z) (opr B (.o) (cv y)) A (.o) opreq2 (opr A (.o) (opr B (.o) (cv y))) sseq2d (opr B (.o) (cv x)) rcla4ev mpan2 syl6 r19.21aiv y (cv x) z (opr B (.o) (cv x)) (opr A (.o) (opr B (.o) (cv y))) (opr A (.o) (cv z)) iunss2 syl (e. A (On)) adantlr (opr B (.o) (cv x)) (cv z) onelon B (cv x) omcl x visset (cv x) (V) limelon mpan sylan2 sylan (e. A (On)) adantlr x visset B (cv x) (V) (cv z) y omordlim ex mpanr1 (e. (cv z) (On)) (e. A (On)) ad2antlr (opr B (.o) (cv y)) (cv z) onelsst (e. (cv z) (On)) (e. A (On)) 3ad2ant2 (cv z) (opr B (.o) (cv y)) A omwordi syld 3exp B (cv y) omcl (cv x) (cv y) onelon x visset (cv x) (V) limelon mpan sylan sylan2 syl5 exp4d imp32 com23 imp r19.22dv syld exp31 imp4c mpcom r19.21aiva z (opr B (.o) (cv x)) y (cv x) (opr A (.o) (cv z)) (opr A (.o) (opr B (.o) (cv y))) iunss2 syl (e. ({/}) B) adantr eqssd B (.o) (cv x) oprex A (opr B (.o) (cv x)) (V) z omlim mpanr1 x visset B (cv x) (V) omlimcl mpanlr1 sylan2 ancoms an1rs eqtr4d sylan9eqr eqtrd exp31 B eloni B ord0eln0 con2bid syl (Lim (cv x)) (e. A (On)) ad2antrr B ({/}) A (.o) opreq2 A om0 sylan9eqr (.o) (cv x) opreq1d (cv x) om0r sylan9eqr anassrs B ({/}) (.o) (cv x) opreq1 (cv x) om0r sylan9eqr A (.o) opreq2d A om0 sylan9eq an1rs eqtr4d ex x visset (cv x) (V) limelon mpan sylan (e. B (On)) adantll sylbird (A.e. y (cv x) (= (opr (opr A (.o) B) (.o) (cv y)) (opr A (.o) (opr B (.o) (cv y))))) a1dd pm2.61d exp31 com3l imp3a tfinds3 exp3a com3l 3imp)) thm (oneo () () (-> (/\/\ (e. A (On)) (e. B (On)) (= C (opr (2o) (.o) A))) (-. (= (suc C) (opr (2o) (.o) B)))) (A B onnbtwn (e. B (On)) (= C (opr (2o) (.o) A)) 3ad2ant1 C (opr (2o) (.o) A) suceq (opr (2o) (.o) B) eqeq1d (e. A (On)) (e. B (On)) 3ad2ant3 2on A B (2o) omord mp3an3 (e. A B) (e. ({/}) (2o)) pm3.26 syl6bir (2o) (.o) A oprex sucid (suc (opr (2o) (.o) A)) (opr (2o) (.o) B) (opr (2o) (.o) A) eleq2 mpbii syl5 (/\ (e. A (On)) (e. B (On))) (= (suc (opr (2o) (.o) A)) (opr (2o) (.o) B)) pm3.27 2on (2o) A omcl mpan (opr (2o) (.o) A) oa1suc syl 1on elisseti sucid df-2o eleqtrr 2on (2o) A omcl mpan 1on 2on (1o) (2o) (opr (2o) (.o) A) oaord mp3an12 syl mpbii 2on (2o) A omsuc mpan eleqtrrd eqeltrrd (e. B (On)) (= (suc (opr (2o) (.o) A)) (opr (2o) (.o) B)) ad2antrr eqeltrrd 2on B (suc A) (2o) omord mp3an3 A suceloni sylan2 ancoms (= (suc (opr (2o) (.o) A)) (opr (2o) (.o) B)) adantr mpbird pm3.26d ex jcad (= C (opr (2o) (.o) A)) 3adant3 sylbid mtod)) thm (oen0 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (e. A (On)) (e. B (On))) (e. ({/}) A)) (e. ({/}) (opr A (^o) B))) ((cv x) ({/}) A (^o) opreq2 ({/}) eleq2d (cv x) (cv y) A (^o) opreq2 ({/}) eleq2d (cv x) (suc (cv y)) A (^o) opreq2 ({/}) eleq2d (cv x) B A (^o) opreq2 ({/}) eleq2d A oe0 0lt1o syl5eleqr (e. ({/}) A) adantr A (opr A (^o) (cv y)) ({/}) omordi (opr A (^o) (cv y)) om0 (opr (opr A (^o) (cv y)) (.o) A) eleq1d (e. A (On)) (e. ({/}) (opr A (^o) (cv y))) ad2antlr sylibd (e. A (On)) (e. (cv y) (On)) pm3.26 A (cv y) oecl jca sylan A (cv y) oesuc ({/}) eleq2d (e. ({/}) (opr A (^o) (cv y))) adantr sylibrd exp31 com12 com34 imp3a (cv x) 0ellim A oe0 (opr A (^o) ({/})) (1o) eqimss2 syl anim12i (cv y) ({/}) A (^o) opreq2 (1o) sseq2d (cv x) rcla4ev y (cv x) (1o) (opr A (^o) (cv y)) ssiun 3syl (e. ({/}) A) adantrr x visset A (cv x) (V) y oelim mpanlr1 anasss an1s sseqtr4d A (cv x) oecl ancoms x visset (cv x) (V) limelon mpan sylan (opr A (^o) (cv x)) eloni (opr A (^o) (cv x)) ordgt0ge1 3syl (e. ({/}) A) adantrr mpbird ex (A.e. y (cv x) (e. ({/}) (opr A (^o) (cv y)))) a1dd tfinds3 exp3a com12 imp31)) thm (oeordi ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (/\ (e. B (On)) (e. C (On))) (e. (1o) C)) (-> (e. A B) (e. (opr C (^o) A) (opr C (^o) B)))) ((cv x) (suc A) C (^o) opreq2 (opr C (^o) A) eleq2d (/\ (e. C (On)) (e. (1o) C)) imbi2d (cv x) (cv y) C (^o) opreq2 (opr C (^o) A) eleq2d (/\ (e. C (On)) (e. (1o) C)) imbi2d (cv x) (suc (cv y)) C (^o) opreq2 (opr C (^o) A) eleq2d (/\ (e. C (On)) (e. (1o) C)) imbi2d (cv x) B C (^o) opreq2 (opr C (^o) A) eleq2d (/\ (e. C (On)) (e. (1o) C)) imbi2d 0lt1o C ({/}) (1o) ontr1 mpani (e. A (On)) adantr C A oen0 ex syld (e. C (On)) (e. A (On)) pm3.26 C A oecl jca jctild C (opr C (^o) A) (1o) omordi syli C A oecl (opr C (^o) A) om1 syl (opr (opr C (^o) A) (.o) C) eleq1d C A oesuc (opr C (^o) A) eleq2d bitr4d sylibd expcom imp3a 0lt1o C ({/}) (1o) ontr1 mpani (e. (cv y) (On)) adantr C (cv y) oen0 ex syld (e. C (On)) (e. (cv y) (On)) pm3.26 C (cv y) oecl jca jctild C (opr C (^o) (cv y)) (1o) omordi syli C (cv y) oecl (opr C (^o) (cv y)) om1 syl (opr (opr C (^o) (cv y)) (.o) C) eleq1d C (cv y) oesuc (opr C (^o) (cv y)) eleq2d bitr4d sylibd C (suc (cv y)) oecl (cv y) sucelon sylan2b (opr C (^o) (suc (cv y))) (opr C (^o) A) (opr C (^o) (cv y)) ontr1 syl exp3a com23 syld expcom imp3a (e. A (cv y)) adantr a2d (cv x) A limsuc biimpa (suc A) (cv x) elisset A sucexb A (V) sucidg sylbir syl (cv y) (suc A) A eleq2 (cv y) (suc A) C (^o) opreq2 (opr C (^o) A) eleq2d imbi12d (cv x) rcla4v mpid anc2li (cv y) (suc A) C (^o) opreq2 (opr C (^o) A) eleq2d (cv x) rcla4ev (opr C (^o) A) y (cv x) (opr C (^o) (cv y)) eliun sylibr syl6 syl (/\ (e. C (On)) (e. (1o) C)) adantr 0lt1o C ({/}) (1o) ontr1 mpani (Lim (cv x)) adantr x visset C (cv x) (V) y oelim mpanlr1 ex syld imp anasss an1s (e. A (cv x)) adantlr (opr C (^o) A) eleq2d sylibrd ex a2d (e. A (cv y)) (/\ (e. C (On)) (e. (1o) C)) (e. (opr C (^o) A) (opr C (^o) (cv y))) bi2.04 y (cv x) ralbii y (cv x) (/\ (e. C (On)) (e. (1o) C)) (-> (e. A (cv y)) (e. (opr C (^o) A) (opr C (^o) (cv y)))) r19.21v bitr syl5ib tfindsg2 exp4b com34 com24 imp31)) thm (oeord () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. (1o) C)) (<-> (e. A B) (e. (opr C (^o) A) (opr C (^o) B)))) (B C A oeordi (e. A (On)) 3adantl1 A B C (^o) opreq2 (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. (1o) C)) a1i A C B oeordi (e. B (On)) 3adantl2 orim12d con3d (opr C (^o) A) (opr C (^o) B) ordtri2 (opr C (^o) A) eloni (opr C (^o) B) eloni syl2an C A oecl C B oecl syl2an anandis ancoms 3impa (e. (1o) C) adantr A B ordtri2 A eloni B eloni syl2an (e. C (On)) 3adant3 (e. (1o) C) adantr 3imtr4d impbid)) thm (oecan () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. (1o) A)) (<-> (= (opr A (^o) B) (opr A (^o) C)) (= B C))) (C A B oeordi ex ancoms (e. B (On)) 3adant2 imp B A C oeordi ex ancoms (e. C (On)) 3adant3 imp orim12d con3d (opr A (^o) B) (opr A (^o) C) ordtri3 A B oecl (opr A (^o) B) eloni syl A C oecl (opr A (^o) C) eloni syl syl2an 3impdi (e. (1o) A) adantr B C ordtri3 B eloni C eloni syl2an (e. A (On)) 3adant1 (e. (1o) A) adantr 3imtr4d B C A (^o) opreq2 (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. (1o) A)) a1i impbid)) thm (oeword () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. (1o) C)) (<-> (C_ A B) (C_ (opr C (^o) A) (opr C (^o) B)))) (A B C oeord C A B oecan (e. C (On)) (e. A (On)) (e. B (On)) 3anrot sylanbr bicomd orbi12d A B onsseleq (e. C (On)) 3adant3 (e. (1o) C) adantr C A oecl C B oecl anim12i anandis ancoms 3impa (opr C (^o) A) (opr C (^o) B) onsseleq syl (e. (1o) C) adantr 3bitr4d)) thm (oewordi () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) C)) (-> (C_ A B) (C_ (opr C (^o) A) (opr C (^o) B)))) (C eloni C ordgt0ge1 syl 1on (1o) C onsseleq mpan bitrd (e. A (On)) (e. B (On)) 3ad2ant3 A B C oeword biimpd ex (1o) C (^o) A opreq1 (1o) C (^o) B opreq1 sseq12d A oe1m (e. B (On)) adantr B oe1m (e. A (On)) adantl eqtr4d (opr (1o) (^o) A) (opr (1o) (^o) B) eqimss syl syl5bi com12 (e. C (On)) 3adant3 (C_ A B) a1dd jaod sylbid imp)) thm (oewordri ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (e. B (On)) (e. C (On))) (-> (e. A B) (C_ (opr A (^o) C) (opr B (^o) C)))) ((cv x) ({/}) A (^o) opreq2 (cv x) ({/}) B (^o) opreq2 sseq12d (cv x) (cv y) A (^o) opreq2 (cv x) (cv y) B (^o) opreq2 sseq12d (cv x) (suc (cv y)) A (^o) opreq2 (cv x) (suc (cv y)) B (^o) opreq2 sseq12d (cv x) C A (^o) opreq2 (cv x) C B (^o) opreq2 sseq12d B A onelon A oe0 syl B oe0 (e. A B) adantr eqtr4d (opr A (^o) ({/})) (opr B (^o) ({/})) eqimss syl (opr A (^o) (cv y)) (opr B (^o) (cv y)) A omwordri A (cv y) oecl (e. B (On)) 3adant2 B (cv y) oecl (e. A (On)) 3adant1 (e. A (On)) (e. B (On)) (e. (cv y) (On)) 3simp1 syl3anc imp (C_ A B) adantrl A B (opr B (^o) (cv y)) omwordi (e. A (On)) (e. B (On)) (e. (cv y) (On)) 3simp1 (e. A (On)) (e. B (On)) (e. (cv y) (On)) 3simp2 B (cv y) oecl (e. A (On)) 3adant1 syl3anc imp (C_ (opr A (^o) (cv y)) (opr B (^o) (cv y))) adantrr sstrd A (cv y) oesuc (e. B (On)) 3adant2 (/\ (C_ A B) (C_ (opr A (^o) (cv y)) (opr B (^o) (cv y)))) adantr B (cv y) oesuc (e. A (On)) 3adant1 (/\ (C_ A B) (C_ (opr A (^o) (cv y)) (opr B (^o) (cv y)))) adantr 3sstr4d exp32 3exp com3r imp4c B A onelon (e. B (On)) (e. A B) pm3.26 jca B A onelsst imp jca syl5 A ({/}) (^o) (cv x) opreq1 (opr B (^o) (cv x)) sseq1d (cv x) oe0m1 biimpa x visset (cv x) (V) limelon mpan (cv x) 0ellim sylanc (opr B (^o) (cv x)) 0ss (Lim (cv x)) a1i eqsstrd syl5bir (/\ (e. B (On)) (e. A B)) adantl (A.e. y (cv x) (C_ (opr A (^o) (cv y)) (opr B (^o) (cv y)))) a1dd x visset A (cv x) (V) y oelim mpanlr1 an1rs (/\ (e. B (On)) (e. A B)) adantllr x visset B (cv x) (V) y oelim mpanlr1 (e. B (On)) (e. A B) pm3.26 (Lim (cv x)) anim1i B on0eln0 A B n0i syl5bir imp (Lim (cv x)) adantr sylanc (e. ({/}) A) adantlr (e. A (On)) adantlll sseq12d y (cv x) (opr A (^o) (cv y)) (opr B (^o) (cv y)) ss2iun syl5bir ex oe0lem com12 B A onelon ancri syl5 tfinds3 exp3a impcom)) thm (oeworde ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (e. A (On)) (e. B (On))) (e. (1o) A)) (C_ B (opr A (^o) B))) ((= (cv x) ({/})) id (cv x) ({/}) A (^o) opreq2 sseq12d (= (cv x) (cv y)) id (cv x) (cv y) A (^o) opreq2 sseq12d (= (cv x) (suc (cv y))) id (cv x) (suc (cv y)) A (^o) opreq2 sseq12d (= (cv x) B) id (cv x) B A (^o) opreq2 sseq12d (opr A (^o) ({/})) 0ss (/\ (e. A (On)) (e. (1o) A)) a1i (cv y) (opr A (^o) (cv y)) ordsucsssuc (cv y) eloni (e. A (On)) adantr A (cv y) oecl ancoms (opr A (^o) (cv y)) eloni syl sylanc (e. (1o) A) adantrr (suc (cv y)) (suc (opr A (^o) (cv y))) (opr A (^o) (suc (cv y))) sstr2 y visset sucid (suc (cv y)) A (cv y) oeordi mpi anasss A (suc (cv y)) oecl (opr A (^o) (suc (cv y))) eloni syl ancoms (opr A (^o) (suc (cv y))) (opr A (^o) (cv y)) ordsucss syl (e. (1o) A) adantrr mpd (cv y) sucelon sylanb syl5com sylbid ex (cv x) limuni (cv x) y uniiun syl6eq (/\ (e. A (On)) (e. ({/}) A)) adantr x visset A (cv x) (V) y oelim mpanlr1 anasss an1s sseq12d y (cv x) (cv y) (opr A (^o) (cv y)) ss2iun syl5bir ex A on0eln0 (1o) A n0i syl5bir imdistani syl5 tfinds3 exp3a com12 imp31)) thm (oeordsuc () () (-> (/\ (e. B (On)) (e. C (On))) (-> (e. A B) (e. (opr A (^o) (suc C)) (opr B (^o) (suc C))))) (B A onelon ex (e. C (On)) adantr B C A oewordri (e. A (On)) 3adant1 (opr A (^o) C) (opr B (^o) C) A omwordri A C oecl (e. B (On)) 3adant2 B C oecl (e. A (On)) 3adant1 (e. A (On)) (e. B (On)) (e. C (On)) 3simp1 syl3anc syld A C oesuc (e. B (On)) 3adant2 (opr (opr B (^o) C) (.o) A) sseq1d sylibrd B on0eln0 A B n0i syl5bir (e. C (On)) adantr B C oen0 ex syld B (opr B (^o) C) A omordi (e. B (On)) (e. C (On)) pm3.26 B C oecl jca sylan ex com23 mpdd (e. A (On)) 3adant1 B C oesuc (e. A (On)) 3adant1 (opr (opr B (^o) C) (.o) A) eleq2d sylibrd jcad 3expa (opr A (^o) (suc C)) (opr B (^o) (suc C)) (opr (opr B (^o) C) (.o) A) ontr2 A (suc C) oecl B (suc C) oecl syl2an anandirs C sucelon sylan2b syld exp31 com4l imp mpdd)) thm (oelim2 ((x y) (A x) (A y) (B x) (B y) (C y)) () (-> (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (= (opr A (^o) B) (U_ x (\ B (1o)) (opr A (^o) (cv x))))) (A ({/}) (^o) B opreq1 A ({/}) (^o) (cv x) opreq1 (e. (cv x) (\ B (1o))) adantr iuneq2dv eqeq12d B oe0m1 biimpa B C limelon B 0ellim (e. B C) adantl sylanc B (cv x) ordelon B limord sylan (cv x) on0eln0 (cv x) el1o negbii syl6bbr (cv x) oe0m1 biimpd sylbird syl ex imp32 (cv x) B (1o) eldif sylan2b iuneq2dv B 0ellim B ({/}) limsuc mpbid df-1o syl5eqel 1on oneirr jctir (1o) B (1o) eldif sylibr (1o) (\ B (1o)) n0i (\ B (1o)) x ({/}) iunconst 3syl eqtrd (e. B C) adantl eqtr4d syl5bir impcom A B C y oelim B (cv y) limsuc biimpa (cv y) nsuceq0 (suc (cv y)) el1o mtbir jctir (suc (cv y)) B (1o) eldif sylibr ex (e. A (On)) (e. ({/}) A) ad2antlr (cv y) sssucid (cv y) (suc (cv y)) A oewordi (/\/\ (e. (cv y) (On)) (e. (suc (cv y)) (On)) (e. A (On))) id 3expa ancoms B (cv y) ordelon B limord sylan (cv y) suceloni ancli syl sylan2 anassrs sylan an1rs mpi ex jcad (cv x) (suc (cv y)) A (^o) opreq2 (opr A (^o) (cv y)) sseq2d (\ B (1o)) rcla4ev syl6 r19.21aiv y B x (\ B (1o)) (opr A (^o) (cv y)) (opr A (^o) (cv x)) iunss2 syl B (1o) difss (\ B (1o)) B x (opr A (^o) (cv x)) iunss1 ax-mp (cv x) (cv y) A (^o) opreq2 B cbviunv sseqtr (/\ (/\ (e. A (On)) (Lim B)) (e. ({/}) A)) a1i eqssd (e. B C) adantlrl eqtrd oe0lem)) thm (nna0 () () (-> (e. A (om)) (= (opr A (+o) ({/})) A)) (A nnont A oa0 syl)) thm (nnm0 () () (-> (e. A (om)) (= (opr A (.o) ({/})) ({/}))) (A nnont A om0 syl)) thm (nnasuc () () (-> (/\ (e. A (om)) (e. B (om))) (= (opr A (+o) (suc B)) (suc (opr A (+o) B)))) (A B oasuc A nnont B nnont syl2an)) thm (nnmsuc () () (-> (/\ (e. A (om)) (e. B (om))) (= (opr A (.o) (suc B)) (opr (opr A (.o) B) (+o) A))) (A B omsuc A nnont B nnont syl2an)) thm (nna0r () () (-> (e. A (om)) (= (opr ({/}) (+o) A) A)) (A nnont A oa0r syl)) thm (nnm0r () () (-> (e. A (om)) (= (opr ({/}) (.o) A) ({/}))) (A nnont A om0r syl)) thm (nnacl ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (om)) (e. B (om))) (e. (opr A (+o) B) (om))) ((cv x) ({/}) A (+o) opreq2 (om) eleq1d (e. A (om)) imbi2d (cv x) (cv y) A (+o) opreq2 (om) eleq1d (e. A (om)) imbi2d (cv x) (suc (cv y)) A (+o) opreq2 (om) eleq1d (e. A (om)) imbi2d (cv x) B A (+o) opreq2 (om) eleq1d (e. A (om)) imbi2d A nna0 (om) eleq1d ibir A (cv y) nnasuc (om) eleq1d (opr A (+o) (cv y)) peano2 syl5bir expcom a2d finds impcom)) thm (nnmcl ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (om)) (e. B (om))) (e. (opr A (.o) B) (om))) ((cv x) ({/}) A (.o) opreq2 (om) eleq1d (e. A (om)) imbi2d (cv x) (cv y) A (.o) opreq2 (om) eleq1d (e. A (om)) imbi2d (cv x) (suc (cv y)) A (.o) opreq2 (om) eleq1d (e. A (om)) imbi2d (cv x) B A (.o) opreq2 (om) eleq1d (e. A (om)) imbi2d A nnm0 peano1 syl6eqel A (cv y) nnmsuc (om) eleq1d (opr A (.o) (cv y)) A nnacl syl5bir exp4b com24 pm2.43i com3r a2d finds impcom)) thm (nnarcl () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. (opr A (+o) B) (om)) (/\ (e. A (om)) (e. B (om))))) (A B oaword1 A eloni ordom A (om) (opr A (+o) B) ordtr2 mpan2 syl exp3a (e. B (On)) adantr mpd B A oaword2 ancoms B eloni ordom B (om) (opr A (+o) B) ordtr2 mpan2 syl exp3a (e. A (On)) adantl mpd jcad A B nnacl (/\ (e. A (On)) (e. B (On))) a1i impbid)) thm (nnacom ((A x) (x y) (x z) (B x) (y z) (B y) (B z)) () (-> (/\ (e. A (om)) (e. B (om))) (= (opr A (+o) B) (opr B (+o) A))) ((cv x) ({/}) (+o) B opreq1 (cv x) ({/}) B (+o) opreq2 eqeq12d (e. B (om)) imbi2d (cv x) (cv y) (+o) B opreq1 (cv x) (cv y) B (+o) opreq2 eqeq12d (e. B (om)) imbi2d (cv x) (suc (cv y)) (+o) B opreq1 (cv x) (suc (cv y)) B (+o) opreq2 eqeq12d (e. B (om)) imbi2d (cv x) A (+o) B opreq1 (cv x) A B (+o) opreq2 eqeq12d (e. B (om)) imbi2d B nna0r B nna0 eqtr4d (cv x) ({/}) (suc (cv y)) (+o) opreq2 (cv x) ({/}) (cv y) (+o) opreq2 (opr (cv y) (+o) (cv x)) (opr (cv y) (+o) ({/})) suceq syl eqeq12d (e. (cv y) (om)) imbi2d (cv x) (cv z) (suc (cv y)) (+o) opreq2 (cv x) (cv z) (cv y) (+o) opreq2 (opr (cv y) (+o) (cv x)) (opr (cv y) (+o) (cv z)) suceq syl eqeq12d (e. (cv y) (om)) imbi2d (cv x) (suc (cv z)) (suc (cv y)) (+o) opreq2 (cv x) (suc (cv z)) (cv y) (+o) opreq2 (opr (cv y) (+o) (cv x)) (opr (cv y) (+o) (suc (cv z))) suceq syl eqeq12d (e. (cv y) (om)) imbi2d (cv x) B (suc (cv y)) (+o) opreq2 (cv x) B (cv y) (+o) opreq2 (opr (cv y) (+o) (cv x)) (opr (cv y) (+o) B) suceq syl eqeq12d (e. (cv y) (om)) imbi2d (cv y) peano2b (suc (cv y)) nna0 sylbi (cv y) nna0 (opr (cv y) (+o) ({/})) (cv y) suceq syl eqtr4d (suc (cv y)) (cv z) oasuc (cv y) nnont (cv y) suceloni syl (cv z) nnont syl2an (cv y) nnont (cv z) nnont anim12i (cv y) (cv z) oasuc (opr (cv y) (+o) (suc (cv z))) (suc (opr (cv y) (+o) (cv z))) suceq 3syl eqeq12d (opr (suc (cv y)) (+o) (cv z)) (suc (opr (cv y) (+o) (cv z))) suceq syl5bir expcom a2d finds imp B (cv y) nnasuc eqeq12d (opr (cv y) (+o) B) (opr B (+o) (cv y)) suceq syl5bir expcom a2d finds imp)) thm (nnaordi () () (-> (/\ (e. B (om)) (e. C (om))) (-> (e. A B) (e. (opr C (+o) A) (opr C (+o) B)))) (B C A oaordi B nnont C nnont syl2an)) thm (nnaord () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (<-> (e. A B) (e. (opr C (+o) A) (opr C (+o) B)))) (A B C oaord A nnont B nnont C nnont syl3an)) thm (nnaordr () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (<-> (e. A B) (e. (opr A (+o) C) (opr B (+o) C)))) (A B C nnaord C A nnacom ancoms (e. B (om)) 3adant2 C B nnacom ancoms (e. A (om)) 3adant1 eleq12d bitrd)) thm (nnaass () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (= (opr (opr A (+o) B) (+o) C) (opr A (+o) (opr B (+o) C)))) (A B C oaass A nnont B nnont C nnont syl3an)) thm (nndi () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (= (opr A (.o) (opr B (+o) C)) (opr (opr A (.o) B) (+o) (opr A (.o) C)))) (A B C odi A nnont B nnont C nnont syl3an)) thm (nnmass () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (= (opr (opr A (.o) B) (.o) C) (opr A (.o) (opr B (.o) C)))) (A B C omass A nnont B nnont C nnont syl3an)) thm (nnmsucr ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (om)) (e. B (om))) (= (opr (suc A) (.o) B) (opr (opr A (.o) B) (+o) B))) ((cv x) ({/}) (suc A) (.o) opreq2 (cv x) ({/}) A (.o) opreq2 (= (cv x) ({/})) id (+o) opreq12d eqeq12d (e. A (om)) imbi2d (cv x) (cv y) (suc A) (.o) opreq2 (cv x) (cv y) A (.o) opreq2 (= (cv x) (cv y)) id (+o) opreq12d eqeq12d (e. A (om)) imbi2d (cv x) (suc (cv y)) (suc A) (.o) opreq2 (cv x) (suc (cv y)) A (.o) opreq2 (= (cv x) (suc (cv y))) id (+o) opreq12d eqeq12d (e. A (om)) imbi2d (cv x) B (suc A) (.o) opreq2 (cv x) B A (.o) opreq2 (= (cv x) B) id (+o) opreq12d eqeq12d (e. A (om)) imbi2d A peano2b (suc A) nnm0 sylbi A nnm0 (+o) ({/}) opreq1d peano1 ({/}) nna0 ax-mp syl6eq eqtr4d (suc A) (cv y) nnmsuc A peano2b sylanb A (cv y) nnmsuc (+o) (suc (cv y)) opreq1d A (cv y) nnacom (opr A (+o) (cv y)) (opr (cv y) (+o) A) suceq syl A (cv y) nnasuc (cv y) A nnasuc ancoms 3eqtr4d (opr A (.o) (cv y)) (+o) opreq2d (opr A (.o) (cv y)) A (suc (cv y)) nnaass (cv y) peano2b syl3an3b A (cv y) nnmcl syl3an1 3expb anidms (opr A (.o) (cv y)) (cv y) (suc A) nnaass A peano2b syl3an3b A (cv y) nnmcl syl3an1 3expb an42s anidms 3eqtr4d eqtrd eqeq12d (opr (suc A) (.o) (cv y)) (opr (opr A (.o) (cv y)) (+o) (cv y)) (+o) (suc A) opreq1 syl5bir expcom a2d finds impcom)) thm (nnmcom ((A x) (x y) (B x) (B y)) () (-> (/\ (e. A (om)) (e. B (om))) (= (opr A (.o) B) (opr B (.o) A))) ((cv x) ({/}) (.o) B opreq1 (cv x) ({/}) B (.o) opreq2 eqeq12d (e. B (om)) imbi2d (cv x) (cv y) (.o) B opreq1 (cv x) (cv y) B (.o) opreq2 eqeq12d (e. B (om)) imbi2d (cv x) (suc (cv y)) (.o) B opreq1 (cv x) (suc (cv y)) B (.o) opreq2 eqeq12d (e. B (om)) imbi2d (cv x) A (.o) B opreq1 (cv x) A B (.o) opreq2 eqeq12d (e. B (om)) imbi2d B nnm0r B nnm0 eqtr4d (cv y) B nnmsucr B (cv y) nnmsuc ancoms eqeq12d (opr (cv y) (.o) B) (opr B (.o) (cv y)) (+o) B opreq1 syl5bir ex a2d finds imp)) thm (nnacan () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (<-> (= (opr A (+o) B) (opr A (+o) C)) (= B C))) (A B C oacan A nnont B nnont C nnont syl3an)) thm (nnaword () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (<-> (C_ A B) (C_ (opr C (+o) A) (opr C (+o) B)))) (A B C oaword A nnont B nnont C nnont syl3an)) thm (nnaword1 () () (-> (/\ (e. A (om)) (e. B (om))) (C_ A (opr A (+o) B))) (A B oaword1 A nnont B nnont syl2an)) thm (nnaword2 () () (-> (/\ (e. A (om)) (e. B (om))) (C_ A (opr B (+o) A))) (A B nnaword1 A B nnacom sseqtrd)) thm (nnmordi () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (-> (/\ (e. A B) (e. ({/}) C)) (e. (opr C (.o) A) (opr C (.o) B)))) (B C A omordi ex B nnont C nnont syl2an com23 imp3a (e. A (om)) 3adant1)) thm (nnmord () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (<-> (/\ (e. A B) (e. ({/}) C)) (e. (opr C (.o) A) (opr C (.o) B)))) (A B C omord A nnont B nnont C nnont syl3an)) thm (nnmcan () () (-> (/\ (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (e. ({/}) A)) (<-> (= (opr A (.o) B) (opr A (.o) C)) (= B C))) (A B C omcan A nnont B nnont C nnont 3anim123i sylan)) thm (nnaordex ((A x) (B x)) () (-> (/\ (e. A (om)) (e. B (om))) (<-> (e. A B) (E.e. x (om) (/\ (e. ({/}) (cv x)) (= (opr A (+o) (cv x)) B))))) (A B x oaordex B nnont sylan2 (opr A (+o) (cv x)) B (om) eleq1 bicomd A (cv x) nnarcl sylan9bbr (e. A (om)) (e. (cv x) (om)) pm3.27 syl6bi exp31 com23 (e. ({/}) (cv x)) adantld com24 imp4b (e. (cv x) (On)) (/\ (e. ({/}) (cv x)) (= (opr A (+o) (cv x)) B)) pm3.27 (/\ (e. A (On)) (e. B (om))) a1i jcad (cv x) nnont (/\ (e. ({/}) (cv x)) (= (opr A (+o) (cv x)) B)) anim1i (/\ (e. A (On)) (e. B (om))) a1i impbid rexbidv2 bitrd A nnont sylan)) thm (nnawordex ((A x) (B x)) () (-> (/\ (e. A (om)) (e. B (om))) (<-> (C_ A B) (E.e. x (om) (= (opr A (+o) (cv x)) B)))) (A B x oawordex B nnont sylan2 (opr A (+o) (cv x)) B (om) eleq1 bicomd A (cv x) nnarcl sylan9bbr (e. A (om)) (e. (cv x) (om)) pm3.27 syl6bi exp31 com23 com24 imp4b (e. (cv x) (On)) (= (opr A (+o) (cv x)) B) pm3.27 (/\ (e. A (On)) (e. B (om))) a1i jcad (cv x) nnont (= (opr A (+o) (cv x)) B) anim1i (/\ (e. A (On)) (e. B (om))) a1i impbid rexbidv2 bitrd A nnont sylan)) thm (oaabslem ((A x)) () (-> (/\ (e. (om) (On)) (e. A (om))) (= (opr A (+o) (om)) (om))) (A (om) (On) x oalim A nnont limom (e. (om) (On)) jctr syl2an A (cv x) nnacl ordom (om) (opr A (+o) (cv x)) ordelss mpan syl r19.21aiva x (om) (opr A (+o) (cv x)) (om) iunss sylibr (e. (om) (On)) adantr eqsstrd ancoms (om) A oaword2 A nnont sylan2 eqssd)) thm (oaabs ((x y) (A x) (A y) (B x)) () (-> (/\ (/\ (e. A (om)) (e. B (On))) (C_ (om) B)) (= (opr A (+o) B) B)) ((om) B (On) ssexg ex ordom (om) (V) elong mpbiri syl6com imp (cv x) (om) A (+o) opreq2 (= (cv x) (om)) id eqeq12d (e. A (om)) imbi2d (cv x) (cv y) A (+o) opreq2 (= (cv x) (cv y)) id eqeq12d (e. A (om)) imbi2d (cv x) (suc (cv y)) A (+o) opreq2 (= (cv x) (suc (cv y))) id eqeq12d (e. A (om)) imbi2d (cv x) B A (+o) opreq2 (= (cv x) B) id eqeq12d (e. A (om)) imbi2d A oaabslem ex A (cv y) oasuc A nnont sylan (opr A (+o) (cv y)) (cv y) suceq sylan9eq exp31 com12 (e. (om) (On)) (C_ (om) (cv y)) ad2antrr a2d A (om) (On) y oalim A nnont limom (e. (om) (On)) jctr syl2an A oaabslem ancoms limom (om) limuni ax-mp syl6eq eqtr3d (/\ (/\ (Lim (cv x)) (C_ (om) (cv x))) (A.e. y (cv x) (-> (C_ (om) (cv y)) (= (opr A (+o) (cv y)) (cv y))))) adantl (cv x) (cv y) ordelon (cv x) limord sylan (cv y) eloni ordom (om) (cv y) ordtri1 mpan 3syl ex pm5.32d (cv y) (cv x) (om) eldif syl6bbr (= (opr A (+o) (cv y)) (cv y)) imbi1d (e. (cv y) (cv x)) (C_ (om) (cv y)) (= (opr A (+o) (cv y)) (cv y)) impexp syl5bbr ralbidv2 y (\ (cv x) (om)) (opr A (+o) (cv y)) (cv y) iuneq2 (\ (cv x) (om)) y uniiun syl6eqr syl6bi imp (C_ (om) (cv x)) adantlr (/\ (e. A (om)) (e. (om) (On))) adantr uneq12d A (cv x) (V) y oalim A nnont x visset (Lim (cv x)) jctl syl2an ancoms (om) (cv x) ssequn1 biimp (om) (cv x) undif2 syl5eq (u. (om) (\ (cv x) (om))) (cv x) y (opr A (+o) (cv y)) iuneq1 syl y (om) (\ (cv x) (om)) (opr A (+o) (cv y)) iunxun syl5reqr sylan9eq an1rs (A.e. y (cv x) (-> (C_ (om) (cv y)) (= (opr A (+o) (cv y)) (cv y)))) (e. (om) (On)) ad2ant2r (cv x) limuni (om) (cv x) ssequn1 biimp (om) (cv x) undif2 syl5eq unieqd (om) (\ (cv x) (om)) uniun syl5reqr sylan9eq (A.e. y (cv x) (-> (C_ (om) (cv y)) (= (opr A (+o) (cv y)) (cv y)))) (/\ (e. A (om)) (e. (om) (On))) ad2antrr 3eqtr4d exp43 com23 a2d (C_ (om) (cv y)) (e. A (om)) (= (opr A (+o) (cv y)) (cv y)) bi2.04 y (cv x) ralbii y (cv x) (e. A (om)) (-> (C_ (om) (cv y)) (= (opr A (+o) (cv y)) (cv y))) r19.21v bitr syl5ib com4r impcom an1rs tfindsg an1rs mpdan ex com3r imp31)) thm (1onn () () (e. (1o) (om)) (df-1o peano1 ({/}) peano2 ax-mp eqeltr)) thm (2onn () () (e. (2o) (om)) (df-2o 1onn (1o) peano2 ax-mp eqeltr)) thm (nneob ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (om)) (<-> (E.e. x (om) (= A (opr (2o) (.o) (cv x)))) (-. (E.e. x (om) (= (suc A) (opr (2o) (.o) (cv x))))))) ((cv x) (cv y) (2o) (.o) opreq2 A eqeq2d (om) cbvrexv (cv y) (cv x) A oneo (cv y) nnont (cv x) nnont (= A (opr (2o) (.o) (cv y))) id syl3an 3com23 3expa nrexdv ex r19.23aiv sylbi (e. A (om)) a1i (cv y) ({/}) suceq (opr (2o) (.o) (cv x)) eqeq1d x (om) rexbidv negbid (cv y) ({/}) (opr (2o) (.o) (cv x)) eqeq1 x (om) rexbidv imbi12d (cv y) (cv z) suceq (opr (2o) (.o) (cv x)) eqeq1d x (om) rexbidv negbid (cv y) (cv z) (opr (2o) (.o) (cv x)) eqeq1 x (om) rexbidv imbi12d (cv y) (suc (cv z)) suceq (opr (2o) (.o) (cv x)) eqeq1d x (om) rexbidv negbid (cv y) (suc (cv z)) (opr (2o) (.o) (cv x)) eqeq1 x (om) rexbidv imbi12d (cv y) A suceq (opr (2o) (.o) (cv x)) eqeq1d x (om) rexbidv negbid (cv y) A (opr (2o) (.o) (cv x)) eqeq1 x (om) rexbidv imbi12d peano1 ({/}) eqid (cv x) ({/}) (2o) (.o) opreq2 2on (2o) om0 ax-mp syl6eq ({/}) eqeq2d (om) rcla4ev mp2an (-. (E.e. x (om) (= (suc ({/})) (opr (2o) (.o) (cv x))))) a1i (cv z) nnont 1on 1on (cv z) (1o) (1o) oaass mp3an23 o1p1e2 (cv z) (+o) opreq2i syl6req (cv z) oa1suc (+o) (1o) opreq1d (cv z) suceloni (suc (cv z)) oa1suc syl 3eqtrd syl eqcomd (e. (cv y) (om)) (= (cv z) (opr (2o) (.o) (cv y))) ad2antrr (cv z) (opr (2o) (.o) (cv y)) (+o) (2o) opreq1 2onn 1onn (2o) (cv y) (1o) nndi mp3an13 2on (2o) om1 ax-mp (opr (2o) (.o) (cv y)) (+o) opreq2i syl6req sylan9eqr (e. (cv z) (om)) adantll eqtrd ex 1onn (cv y) (1o) nnacl mpan2 (e. (cv z) (om)) adantl jctild (cv x) (opr (cv y) (+o) (1o)) (2o) (.o) opreq2 (suc (suc (cv z))) eqeq2d (om) rcla4ev syl6 ex r19.23adv (cv x) (cv y) (2o) (.o) opreq2 (cv z) eqeq2d (om) cbvrexv syl5ib con3d (E.e. x (om) (= (suc (cv z)) (opr (2o) (.o) (cv x)))) (E.e. x (om) (= (cv z) (opr (2o) (.o) (cv x)))) con1 syl9 finds impbid)) thm (omsmolem ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (F y) (F z) (F w)) () (-> (e. (cv y) (om)) (-> (/\ (/\ (C_ A (On)) (:--> F (om) A)) (A.e. x (om) (e. (` F (cv x)) (` F (suc (cv x)))))) (-> (e. (cv z) (cv y)) (e. (` F (cv z)) (` F (cv y)))))) ((cv y) ({/}) (cv z) eleq2 (cv y) ({/}) F fveq2 (` F (cv z)) eleq2d imbi12d (cv y) (cv w) (cv z) eleq2 (cv y) (cv w) F fveq2 (` F (cv z)) eleq2d imbi12d (cv y) (suc (cv w)) (cv z) eleq2 (cv y) (suc (cv w)) F fveq2 (` F (cv z)) eleq2d imbi12d (cv z) noel (e. (` F (cv z)) (` F ({/}))) pm2.21i (/\ (/\ (C_ A (On)) (:--> F (om) A)) (A.e. x (om) (e. (` F (cv x)) (` F (suc (cv x)))))) a1i (cv x) (cv w) F fveq2 (cv x) (cv w) suceq F fveq2d eleq12d (om) rcla4cva (/\ (C_ A (On)) (:--> F (om) A)) adantll A (On) (` F (suc (cv w))) ssel F (om) A (suc (cv w)) ffvrn (cv w) peano2b sylan2b syl5 (` F (suc (cv w))) (` F (cv z)) (` F (cv w)) ontr1 exp3a com23 syl6 exp3a imp31 (A.e. x (om) (e. (` F (cv x)) (` F (suc (cv x))))) adantlr mpd (e. (cv z) (cv w)) imim2d imp (cv z) (cv w) F fveq2 (` F (suc (cv w))) eleq1d (cv x) (cv w) F fveq2 (cv x) (cv w) suceq F fveq2d eleq12d (om) rcla4cva syl5bir com12 (/\ (C_ A (On)) (:--> F (om) A)) adantll (-> (e. (cv z) (cv w)) (e. (` F (cv z)) (` F (cv w)))) adantr jaod z visset (cv w) elsuc syl5ib exp31 com12 finds2)) thm (omsmo ((x y) (x z) (A x) (y z) (A y) (A z) (F x) (F y) (F z)) () (-> (/\ (/\ (C_ A (On)) (:--> F (om) A)) (A.e. x (om) (e. (` F (cv x)) (` F (suc (cv x)))))) (:-1-1-> F (om) A)) ((C_ A (On)) (:--> F (om) A) pm3.27 (A.e. x (om) (e. (` F (cv x)) (` F (suc (cv x))))) adantr z A F x y omsmolem (e. (cv y) (om)) adantl imp y A F x z omsmolem (e. (cv z) (om)) adantr imp orim12d ancoms con3d A (On) (` F (cv y)) ssel F (om) A (cv y) ffvrn syl5 exp3a imp (` F (cv y)) eloni syl6 A (On) (` F (cv z)) ssel F (om) A (cv z) ffvrn syl5 exp3a imp (` F (cv z)) eloni syl6 anim12d imp (` F (cv y)) (` F (cv z)) ordtri3 syl (A.e. x (om) (e. (` F (cv x)) (` F (suc (cv x))))) adantlr (cv y) (cv z) ordtri3 (cv y) nnord (cv z) nnord syl2an (/\ (/\ (C_ A (On)) (:--> F (om) A)) (A.e. x (om) (e. (` F (cv x)) (` F (suc (cv x)))))) adantl 3imtr4d exp32 r19.21adv r19.21aiv jca F (om) A y z f1fv sylibr)) thm (er2 ((x y) (x z) (R x) (y z) (R y) (R z)) () (<-> (Er R) (A. x (A. y (A. z (/\ (-> (br (cv x) R (cv y)) (br (cv y) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))))))) (R df-er R x y cnvsym R x y z cotr anbi12i (`' R) R (o. R R) unss z (-> (br (cv x) R (cv y)) (br (cv y) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))) 19.28v y albii y (-> (br (cv x) R (cv y)) (br (cv y) R (cv x))) (A. z (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) 19.26 bitr x albii x (A. y (-> (br (cv x) R (cv y)) (br (cv y) R (cv x)))) (A. y (A. z (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))) 19.26 bitr2 3bitr3 bitr)) thm (ec2 ((A y) (x y) (A x) (R y) (R x)) ((ec2.1 (e. A (V)))) (= ([] A R) ({|} y (br A R (cv y)))) (R ({} A) y x dfima3 A R df-ec ec2.1 (cv x) A (cv y) opeq1 R eleq1d ceqsexv x A elsn (e. (<,> (cv x) (cv y)) R) anbi1i x exbii A R (cv y) df-br 3bitr4r y abbii 3eqtr4)) thm (ecexg () () (-> (e. R B) (e. ([] A R) (V))) (R B ({} A) imaexg A R df-ec syl5eqel)) thm (ereq () () (-> (= R S) (<-> (Er R) (Er S))) (R S cnveq R S R coeq1 R S S coeq2 eqtrd uneq12d R sseq1d R S (u. (`' S) (o. S S)) sseq2 bitrd R df-er S df-er 3bitr4g)) thm (ster ((x y) (x z) (R x) (y z) (R y) (R z)) ((ster.1 (-> (br (cv x) R (cv y)) (br (cv y) R (cv x)))) (ster.2 (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))) (Er R) (R x y z er2 ster.1 ster.2 pm3.2i y z gen2 mpgbir)) thm (ider ((x y) (x z) (y z)) () (Er (I)) (x y equcomi x visset y visset ideq y visset x visset ideq 3imtr4 (cv x) (cv y) (cv z) eqtrt x visset y visset ideq y visset z visset ideq anbi12i x visset z visset ideq 3imtr4 ster)) thm (ersym ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) ((ersym.1 (e. A (V))) (ersym.2 (e. B (V))) (ersym.3 (Er R))) (-> (br A R B) (br B R A)) (ersym.1 ersym.2 (cv x) A (cv y) B R breq12 (cv y) B (cv x) A R breq12 ancoms imbi12d ersym.3 R x y z er2 mpbi a4i a4i a4i pm3.26i vtocl2)) thm (ersymb () ((ersymb.1 (e. A (V))) (ersymb.2 (e. B (V))) (ersymb.3 (Er R))) (<-> (br A R B) (br B R A)) (ersymb.1 ersymb.2 ersymb.3 ersym ersymb.2 ersymb.1 ersymb.3 ersym impbi)) thm (ertr ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((ertr.1 (e. A (V))) (ertr.2 (e. B (V))) (ertr.3 (e. C (V))) (ertr.4 (Er R))) (-> (/\ (br A R B) (br B R C)) (br A R C)) (ertr.1 ertr.2 ertr.3 (cv x) A R (cv y) breq1 (br (cv y) R (cv z)) anbi1d (cv x) A R (cv z) breq1 imbi12d (cv y) B A R breq2 (cv y) B R (cv z) breq1 anbi12d (br A R (cv z)) imbi1d (cv z) C B R breq2 (br A R B) anbi2d (cv z) C A R breq2 imbi12d syl3an9b ertr.4 R x y z er2 mpbi a4i a4i a4i pm3.27i vtocl3)) thm (erref ((x y) (A x) (A y) (R x) (R y)) ((erref.1 (Er R))) (-> (e. A (u. (dom R) (ran R))) (br A R A)) ((cv x) A R (cv x) breq1 (cv x) A A R breq2 bitrd (cv x) (dom R) (ran R) elun x visset R y eldm x visset y visset x visset erref.1 ertr x visset y visset erref.1 ersymb sylan2b anidms y 19.23aiv sylbi x visset R y elrn x visset y visset x visset erref.1 ertr y visset x visset erref.1 ersymb sylanb anidms y 19.23aiv sylbi jaoi sylbi vtoclga)) thm (erdmrn ((x y) (R x) (R y)) ((erdmrn.1 (Er R))) (= (dom R) (ran R)) (x visset y visset erdmrn.1 ersymb y exbii x visset R y eldm x visset R y elrn 3bitr4 eqriv)) thm (eceq1 () () (-> (= A B) (= ([] C A) ([] C B))) (A B ({} C) imaeq1 C A df-ec C B df-ec 3eqtr4g)) thm (eceq2 () () (-> (= A B) (= ([] A C) ([] B C))) (A B sneq ({} A) ({} B) C imaeq2 syl A C df-ec B C df-ec 3eqtr4g)) thm (elec ((A x) (B x) (R x)) ((elec.1 (e. A (V))) (elec.2 (e. B (V)))) (<-> (e. A ([] B R)) (br B R A)) (elec.1 (cv x) A B R breq2 elec.2 R x ec2 elab2)) thm (ecdmn0 ((R x) (A x)) ((ecdmn0.1 (e. A (V)))) (<-> (e. A (dom R)) (-. (= ([] A R) ({/})))) (x visset ecdmn0.1 R elec x exbii ([] A R) x n0 ecdmn0.1 R x eldm 3bitr4r)) thm (erthi ((A x) (B x) (R x)) ((erthi.1 (e. A (V))) (erthi.2 (e. B (V))) (erthi.3 (Er R))) (-> (br A R B) (= ([] A R) ([] B R))) (erthi.1 erthi.2 erthi.3 ersymb erthi.2 erthi.1 x visset erthi.3 ertr ex sylbi erthi.1 erthi.2 x visset erthi.3 ertr ex impbid x visset erthi.1 R elec x visset erthi.2 R elec 3bitr4g eqrdv)) thm (erth ((A x) (B x) (R x)) ((erth.1 (e. B (V))) (erth.2 (Er R))) (-> (e. A (u. (dom R) (ran R))) (<-> (= ([] A R) ([] B R)) (br A R B))) ((cv x) A R eceq2 ([] B R) eqeq1d (cv x) A R B breq1 bibi12d ([] (cv x) R) ([] B R) (cv x) eleq2 x visset x visset R elec x visset erth.1 R elec 3bitr3g erth.1 x visset erth.2 ersym syl6bi erth.2 (cv x) erref syl5com x visset erth.1 erth.2 erthi (e. (cv x) (u. (dom R) (ran R))) a1i impbid vtoclga)) thm (erthdm () ((erthdm.1 (e. B (V))) (erthdm.2 (Er R))) (-> (e. A (dom R)) (<-> (= ([] A R) ([] B R)) (br A R B))) (A (dom R) (ran R) elun1 erthdm.1 erthdm.2 A erth syl)) thm (erthdmr () ((erthdmr.1 (e. A (V))) (erthdmr.2 (e. B (V))) (erthdmr.3 (Er R))) (-> (e. B (dom R)) (<-> (= ([] A R) ([] B R)) (br A R B))) (erthdmr.1 erthdmr.3 B erthdm ([] A R) ([] B R) eqcom erthdmr.1 erthdmr.2 erthdmr.3 ersymb 3bitr4g)) thm (ereldm () ((ereldm.1 (e. A (V))) (ereldm.2 (e. B (V))) (ereldm.3 (Er R)) (ereldm.4 (= (dom R) D))) (-> (= ([] A R) ([] B R)) (<-> (e. A D) (e. B D))) (ereldm.2 ereldm.3 A erthdm biimpcd ereldm.1 ereldm.2 ereldm.3 ersymb ereldm.2 R A breldm sylbi syl6 ereldm.1 ereldm.2 ereldm.3 erthdmr biimpcd ereldm.1 R B breldm syl6 impbid ereldm.4 A eleq2i ereldm.4 B eleq2i 3bitr3g)) thm (erdisj ((A x) (B x) (R x)) ((erdisj.1 (e. A (V))) (erdisj.2 (e. B (V))) (erdisj.3 (Er R))) (\/ (= ([] A R) ([] B R)) (= (i^i ([] A R) ([] B R)) ({/}))) (x visset erdisj.1 R elec erdisj.1 x visset erdisj.2 erdisj.3 ertr ex erdisj.1 erdisj.2 erdisj.3 erthi syl6 x visset erdisj.2 R elec erdisj.2 x visset erdisj.3 ersymb bitr syl5ib sylbi con3d com12 x 19.21aiv ([] A R) ([] B R) x disj1 sylibr orri)) thm (ecidsn () () (= ([] A (I)) ({} A)) (A (I) df-ec ({} A) imai eqtr)) thm (qseq1 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (= A B) (= (/. A C) (/. B C))) (A B x (= (cv y) ([] (cv x) C)) rexeq1 y abbidv A C y x df-qs B C y x df-qs 3eqtr4g)) thm (qseq2 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (= A B) (= (/. C A) (/. C B))) (A B (cv x) eceq1 (cv y) eqeq2d x C rexbidv y abbidv C A y x df-qs C B y x df-qs 3eqtr4g)) thm (elqs ((x y) (A x) (A y) (B x) (B y) (R x) (R y)) ((elqs.1 (e. B (V)))) (<-> (e. B (/. A R)) (E. x (/\ (e. (cv x) A) (= B ([] (cv x) R))))) (elqs.1 (cv y) B ([] (cv x) R) eqeq1 x A rexbidv A R y x df-qs elab2 x A (= B ([] (cv x) R)) df-rex bitr)) thm (elqsi ((x y) (A x) (A y) (B x) (B y) (R x) (R y)) () (-> (e. B (/. A R)) (E. x (/\ (e. (cv x) A) (= B ([] (cv x) R))))) ((cv y) B ([] (cv x) R) eqeq1 (e. (cv x) A) anbi2d x exbidv y visset A R x elqs biimp vtoclga)) thm (ecelqsi ((x y) (A x) (A y) (B x) (B y) (R x) (R y)) ((ecelqsi.1 (e. R (V)))) (-> (e. B A) (e. ([] B R) (/. A R))) ((cv y) B R eceq2 (/. A R) eleq1d x y a9e ([] (cv y) R) eqid (cv x) (cv y) A eleq1 (cv x) (cv y) R eceq2 ([] (cv y) R) eqeq2d anbi12d biimprcd mpan2 x 19.22dv mpi ecelqsi.1 R (V) (cv y) ecexg ax-mp A R x elqs sylibr vtoclga)) thm (ecopqsi () ((ecopqsi.1 (e. R (V))) (ecopqsi.2 (= S (/. (X. A A) R)))) (-> (/\ (e. B A) (e. C A)) (e. ([] (<,> B C) R) S)) (B A C A opelxpi ecopqsi.1 (<,> B C) (X. A A) ecelqsi ecopqsi.2 syl6eleqr syl)) thm (qsex ((x y) (A x) (A y) (R x) (R y)) ((qsex.1 (e. A (V)))) (e. (/. A R) (V)) (x A (= (cv y) ([] (cv x) R)) df-rex y abbii A R y x df-qs x y (/\ (e. (cv x) A) (= (cv y) ([] (cv x) R))) rnopab 3eqtr4 qsex.1 x y ([] (cv x) R) funopabex2 ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) ([] (cv x) R)))) (V) rnexg ax-mp eqeltr)) thm (snec ((x y) (A x) (A y) (R x) (R y)) ((snec.1 (e. A (V)))) (= ({} ([] A R)) (/. ({} A) R)) (x ({} A) (= (cv y) ([] (cv x) R)) df-rex x A elsn (= (cv y) ([] (cv x) R)) anbi1i x exbii snec.1 (cv x) A R eceq2 (cv y) eqeq2d ceqsexv 3bitrr y abbii ([] A R) y df-sn ({} A) R y x df-qs 3eqtr4)) thm (ecqs () ((ecqs.1 (e. A (V))) (ecqs.2 (e. R (V)))) (= ([] A R) (U. (/. ({} A) R))) (ecqs.2 R (V) A ecexg ax-mp unisn ecqs.1 R snec unieqi eqtr3)) thm (0nelqs ((R x) (A x)) ((0nelqs.1 (Er R)) (0nelqs.2 (= (dom R) A))) (-. (e. ({/}) (/. A R))) (x visset R ecdmn0 0nelqs.2 (cv x) eleq2i ([] (cv x) R) ({/}) eqcom negbii 3bitr3 biimp (e. (cv x) A) (= ({/}) ([] (cv x) R)) imnan mpbi x nex ({/}) A R x elqsi mto)) thm (ecelqsdm () ((ecelqsdm.1 (e. B (V))) (ecelqsdm.2 (Er R)) (ecelqsdm.3 (= (dom R) A))) (-> (e. ([] B R) (/. A R)) (e. B A)) (ecelqsdm.2 ecelqsdm.3 0nelqs ecelqsdm.1 R ecdmn0 ecelqsdm.3 B eleq2i bitr3 con1bii ([] B R) ({/}) (/. A R) eleq1 sylbi mtbiri a3i)) thm (ecid ((A y)) ((ecid.1 (e. A (V)))) (= ([] A (`' (E))) A) (ecid.1 (`' (E)) y ec2 ecid.1 y visset (E) brcnv y visset ecid.1 epelc bitr y abbii y A abid2 3eqtr)) thm (qsid ((x y) (A x) (A y)) ((qsid.1 (e. A (V)))) (= (/. A (`' (E))) A) (A (`' (E)) y x df-qs x visset ecid (cv y) eqeq2i (cv y) (cv x) eqcom bitr x A rexbii (cv y) A x risset bitr4 y abbii y A abid2 3eqtr)) thm (ectocl ((A x) (B x) (R x) (ps x)) ((ectocl.1 (= S (/. B R))) (ectocl.2 (-> (= ([] (cv x) R) A) (<-> ph ps))) (ectocl.3 (-> (e. (cv x) B) ph))) (-> (e. A S) ps) (ectocl.1 A eleq2i A B R x elqsi ectocl.2 eqcoms ectocl.3 syl5bi impcom x 19.23aiv syl sylbi)) thm (ecoptocl ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (ps x) (ps y) (ps z)) ((ecoptocl.1 (= S (/. (X. B C) R))) (ecoptocl.2 (-> (= ([] (<,> (cv x) (cv y)) R) A) (<-> ph ps))) (ecoptocl.3 (-> (/\ (e. (cv x) B) (e. (cv y) C)) ph))) (-> (e. A S) ps) (ecoptocl.1 A eleq2i A (X. B C) R z elqsi (X. B C) eqid (<,> (cv x) (cv y)) (cv z) R eceq2 A eqeq2d ps imbi1d ecoptocl.2 eqcoms ecoptocl.3 syl5bi com12 optocl imp z 19.23aiv syl sylbi)) thm (2ecoptocl ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w) (S z) (S w) (R x) (R y) (R z) (R w) (ps x) (ps y) (ch z) (ch w)) ((2ecoptocl.1 (= S (/. (X. C D) R))) (2ecoptocl.2 (-> (= ([] (<,> (cv x) (cv y)) R) A) (<-> ph ps))) (2ecoptocl.3 (-> (= ([] (<,> (cv z) (cv w)) R) B) (<-> ps ch))) (2ecoptocl.4 (-> (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (/\ (e. (cv z) C) (e. (cv w) D))) ph))) (-> (/\ (e. A S) (e. B S)) ch) (2ecoptocl.1 2ecoptocl.3 (e. A S) imbi2d 2ecoptocl.1 2ecoptocl.2 (/\ (e. (cv z) C) (e. (cv w) D)) imbi2d 2ecoptocl.4 ex ecoptocl com12 ecoptocl impcom)) thm (3ecoptocl ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B z) (B w) (B v) (B u) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (S z) (S w) (S v) (S u) (R x) (R y) (R z) (R w) (R v) (R u) (ps x) (ps y) (ch z) (ch w) (th v) (th u)) ((3ecoptocl.1 (= S (/. (X. D D) R))) (3ecoptocl.2 (-> (= ([] (<,> (cv x) (cv y)) R) A) (<-> ph ps))) (3ecoptocl.3 (-> (= ([] (<,> (cv z) (cv w)) R) B) (<-> ps ch))) (3ecoptocl.4 (-> (= ([] (<,> (cv v) (cv u)) R) C) (<-> ch th))) (3ecoptocl.5 (-> (/\/\ (/\ (e. (cv x) D) (e. (cv y) D)) (/\ (e. (cv z) D) (e. (cv w) D)) (/\ (e. (cv v) D) (e. (cv u) D))) ph))) (-> (/\/\ (e. A S) (e. B S) (e. C S)) th) (3ecoptocl.1 3ecoptocl.3 (e. A S) imbi2d 3ecoptocl.4 (e. A S) imbi2d 3ecoptocl.1 3ecoptocl.2 (/\ (/\ (e. (cv z) D) (e. (cv w) D)) (/\ (e. (cv v) D) (e. (cv u) D))) imbi2d 3ecoptocl.5 3exp imp3a ecoptocl com12 2ecoptocl com12 3impib)) thm (brecop ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (S x) (S y) (S z) (S w) (S v) (S u) (H x) (H y) (G z) (G w) (G v) (G u) (ph x) (ph y) (ps z) (ps w) (ps v) (ps u)) ((brecop.1 (e. S (V))) (brecop.2 (Er S)) (brecop.3 (= (dom S) (X. G G))) (brecop.4 (= H (/. (X. G G) S))) (brecop.5 (= R ({<,>|} x y (/\ (/\ (e. (cv x) H) (e. (cv y) H)) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) ([] (<,> (cv z) (cv w)) S)) (= (cv y) ([] (<,> (cv v) (cv u)) S))) ph))))))))) (brecop.6 (-> (/\ (/\ (/\ (e. (cv z) G) (e. (cv w) G)) (/\ (e. A G) (e. B G))) (/\ (/\ (e. (cv v) G) (e. (cv u) G)) (/\ (e. C G) (e. D G)))) (-> (/\ (= ([] (<,> (cv z) (cv w)) S) ([] (<,> A B) S)) (= ([] (<,> (cv v) (cv u)) S) ([] (<,> C D) S))) (<-> ph ps))))) (-> (/\ (/\ (e. A G) (e. B G)) (/\ (e. C G) (e. D G))) (<-> (br ([] (<,> A B) S) R ([] (<,> C D) S)) ps)) ((cv x) ([] (<,> A B) S) H eleq1 (e. (cv y) H) anbi1d (cv x) ([] (<,> A B) S) ([] (<,> (cv z) (cv w)) S) eqeq1 (= (cv y) ([] (<,> (cv v) (cv u)) S)) anbi1d ph anbi1d z w v u 4exbidv anbi12d (cv y) ([] (<,> C D) S) H eleq1 (e. ([] (<,> A B) S) H) anbi2d (cv y) ([] (<,> C D) S) ([] (<,> (cv v) (cv u)) S) eqeq1 (= ([] (<,> A B) S) ([] (<,> (cv z) (cv w)) S)) anbi2d ph anbi1d z w v u 4exbidv anbi12d H H opelopabg (/\ (e. ([] (<,> A B) S) H) (e. ([] (<,> C D) S) H)) (E. z (E. w (E. v (E. u (/\ (/\ (= ([] (<,> A B) S) ([] (<,> (cv z) (cv w)) S)) (= ([] (<,> C D) S) ([] (<,> (cv v) (cv u)) S))) ph))))) ibar bitr4d ([] (<,> A B) S) R ([] (<,> C D) S) df-br brecop.5 (<,> ([] (<,> A B) S) ([] (<,> C D) S)) eleq2i bitr syl5bb brecop.1 brecop.4 A B ecopqsi brecop.1 brecop.4 C D ecopqsi syl2an (cv z) A (cv w) B opeq12 (<,> (cv z) (cv w)) (<,> A B) S eceq2 syl (cv v) C (cv u) D opeq12 (<,> (cv v) (cv u)) (<,> C D) S eceq2 syl anim12i (cv z) (cv w) opex A B opex brecop.2 brecop.3 ereldm w visset (cv z) G G opelxp syl5bbr A G B G opelxpi syl5bir (cv v) (cv u) opex C D opex brecop.2 brecop.3 ereldm u visset (cv v) G G opelxp syl5bbr C G D G opelxpi syl5bir im2anan9 brecop.6 an4s ex com13 mpdd pm5.74d G G cgsex4g ([] (<,> A B) S) ([] (<,> (cv z) (cv w)) S) eqcom ([] (<,> C D) S) ([] (<,> (cv v) (cv u)) S) eqcom anbi12i (/\ (/\ (e. A G) (e. B G)) (/\ (e. C G) (e. D G))) a1i (/\ (/\ (e. A G) (e. B G)) (/\ (e. C G) (e. D G))) ph biimt anbi12d z w v u 4exbidv (/\ (/\ (e. A G) (e. B G)) (/\ (e. C G) (e. D G))) ps biimt 3bitr4d bitrd)) thm (brecop2 () ((brecop2.1 (e. S (V))) (brecop2.2 (e. B (V))) (brecop2.3 (e. C (V))) (brecop2.4 (e. D (V))) (brecop2.5 (Er S)) (brecop2.6 (= (dom S) (X. G G))) (brecop2.7 (= H (/. (X. G G) S))) (brecop2.8 (C_ R (X. H H))) (brecop2.9 (C_ Q (X. G G))) (brecop2.10 (-. (e. ({/}) G))) (brecop2.11 (= (dom F) (X. G G))) (brecop2.12 (-> (/\ (/\ (e. A G) (e. B G)) (/\ (e. C G) (e. D G))) (<-> (br ([] (<,> A B) S) R ([] (<,> C D) S)) (br (opr A F D) Q (opr B F C)))))) (<-> (br ([] (<,> A B) S) R ([] (<,> C D) S)) (br (opr A F D) Q (opr B F C))) (brecop2.1 S (V) (<,> C D) ecexg ax-mp brecop2.8 ([] (<,> A B) S) brel brecop2.7 ([] (<,> A B) S) eleq2i A B opex brecop2.5 brecop2.6 ecelqsdm sylbi brecop2.2 A G G opelxp sylib brecop2.7 ([] (<,> C D) S) eleq2i C D opex brecop2.5 brecop2.6 ecelqsdm sylbi brecop2.4 C G G opelxp sylib anim12i syl B F C oprex brecop2.9 (opr A F D) brel brecop2.4 brecop2.11 brecop2.10 A ndmoprrcl brecop2.3 brecop2.11 brecop2.10 B ndmoprrcl anim12i syl (e. A G) (e. D G) (e. B G) (e. C G) an42 sylib brecop2.12 pm5.21nii)) var (set s) var (set r) thm (ecopopreq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (x y) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (S x) (S y) (S z) (S w) (S v) (S u)) ((ecopopr.1 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v))))))))))))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (<-> (br (<,> A B) R (<,> C D)) (= (opr A F D) (opr B F C)))) ((cv z) A (cv u) D F opreq12 (cv w) B (cv v) C F opreq12 eqeqan12d an42s ecopopr.1 opbrop)) thm (ecopoprdm ((f g) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (F y) (w z) (v z) (u z) (f z) (g z) (F z) (v w) (u w) (f w) (g w) (F w) (u v) (f v) (g v) (F v) (f u) (g u) (F u) (F f) (F g) (R f) (R g) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g)) ((ecopopr.1 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))))))) (ecopopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))))) (= (dom R) (X. S S)) (ecopopr.1 x y (X. S S) (X. S S) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))) opabssxp eqsstr R (X. (X. S S) (X. S S)) dmss ax-mp (X. S S) dmxpid sseqtr S S relxp g visset (cv f) S S opelxp f visset g visset ecopopr.com caoprcom ecopopr.1 (cv f) (cv g) (cv f) (cv g) ecopopreq anidms mpbiri (<,> (cv f) (cv g)) R (<,> (cv f) (cv g)) df-br sylib (cv f) (cv g) opex (<,> (cv f) (cv g)) R opeldm syl sylbi relssi eqssi)) thm (ecopoprsym ((f g) (f h) (f t) (A f) (g h) (g t) (A g) (h t) (A h) (A t) (B f) (B g) (B h) (B t) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (F y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (F z) (v w) (u w) (f w) (g w) (h w) (t w) (F w) (u v) (f v) (g v) (h v) (t v) (F v) (f u) (g u) (h u) (t u) (F u) (F f) (F g) (F h) (F t) (R f) (R g) (R h) (R t) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g) (S h) (S t)) ((ecopopr.1 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))))))) (ecopopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (ecopopr.2 (e. B (V)))) (-> (br A R B) (br B R A)) (ecopopr.2 ecopopr.1 x y (X. S S) (X. S S) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))) opabssxp eqsstr A brel (X. S S) eqid (<,> (cv f) (cv g)) A R (<,> (cv h) (cv t)) breq1 (<,> (cv f) (cv g)) A (<,> (cv h) (cv t)) R breq2 bibi12d (<,> (cv h) (cv t)) B A R breq2 (<,> (cv h) (cv t)) B R A breq1 bibi12d ecopopr.1 (cv f) (cv g) (cv h) (cv t) ecopopreq f visset t visset ecopopr.com caoprcom g visset h visset ecopopr.com caoprcom eqeq12i (opr (cv t) F (cv f)) (opr (cv h) F (cv g)) eqcom bitr syl6bb ecopopr.1 (cv h) (cv t) (cv f) (cv g) ecopopreq ancoms bitr4d 2optocl syl ibi)) thm (ecopoprtrn ((f g) (f h) (f t) (f s) (f r) (A f) (g h) (g t) (g s) (g r) (A g) (h t) (h s) (h r) (A h) (s t) (r t) (A t) (r s) (A s) (A r) (B f) (B g) (B h) (B t) (B s) (B r) (C f) (C g) (C h) (C t) (C s) (C r) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (s x) (r x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (s y) (r y) (F y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (s z) (r z) (F z) (v w) (u w) (f w) (g w) (h w) (t w) (s w) (r w) (F w) (u v) (f v) (g v) (h v) (t v) (s v) (r v) (F v) (f u) (g u) (h u) (t u) (s u) (r u) (F u) (F f) (F g) (F h) (F t) (F s) (F r) (R f) (R g) (R h) (R t) (R s) (R r) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g) (S h) (S t) (S s) (S r)) ((ecopopr.1 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))))))) (ecopopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (ecopopr.cl (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S))) (ecopopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (ecopopr.can (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z))))) (ecopopr.3 (e. B (V))) (ecopopr.4 (e. C (V)))) (-> (/\ (br A R B) (br B R C)) (br A R C)) (ecopopr.3 ecopopr.1 x y (X. S S) (X. S S) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))) opabssxp eqsstr A brel pm3.26d ecopopr.4 ecopopr.1 x y (X. S S) (X. S S) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))) opabssxp eqsstr B brel anim12i (e. A (X. S S)) (e. B (X. S S)) (e. C (X. S S)) 3anass sylibr (X. S S) eqid (<,> (cv f) (cv g)) A R (<,> (cv h) (cv t)) breq1 (br (<,> (cv h) (cv t)) R (<,> (cv s) (cv r))) anbi1d (<,> (cv f) (cv g)) A R (<,> (cv s) (cv r)) breq1 imbi12d (<,> (cv h) (cv t)) B A R breq2 (<,> (cv h) (cv t)) B R (<,> (cv s) (cv r)) breq1 anbi12d (br A R (<,> (cv s) (cv r))) imbi1d (<,> (cv s) (cv r)) C B R breq2 (br A R B) anbi2d (<,> (cv s) (cv r)) C A R breq2 imbi12d ecopopr.1 (cv f) (cv g) (cv h) (cv t) ecopopreq (/\ (e. (cv s) S) (e. (cv r) S)) 3adant3 ecopopr.1 (cv h) (cv t) (cv s) (cv r) ecopopreq (/\ (e. (cv f) S) (e. (cv g) S)) 3adant1 anbi12d (opr (cv f) F (cv t)) (opr (cv g) F (cv h)) (opr (cv h) F (cv r)) (opr (cv t) F (cv s)) F opreq12 h visset t visset f visset ecopopr.com ecopopr.ass r visset caopr411 g visset t visset h visset ecopopr.com ecopopr.ass s visset caopr411 g visset t visset h visset ecopopr.com ecopopr.ass s visset caopr4 eqtr3 3eqtr4g syl6bi (cv g) F (cv s) oprex ecopopr.can (opr (cv h) F (cv t)) (opr (cv f) F (cv r)) caoprcan ecopopr.cl (cv h) (cv t) caoprcl ecopopr.cl (cv f) (cv r) caoprcl syl2an 3impb 3com12 (e. (cv s) S) 3adant3l (e. (cv g) S) 3adant1r syld ecopopr.1 (cv f) (cv g) (cv s) (cv r) ecopopreq (/\ (e. (cv h) S) (e. (cv t) S)) 3adant2 sylibrd 3optocl mpcom)) thm (ecopoprer ((f g) (f h) (g h) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (F y) (w z) (v z) (u z) (f z) (g z) (h z) (F z) (v w) (u w) (f w) (g w) (h w) (F w) (u v) (f v) (g v) (h v) (F v) (f u) (g u) (h u) (F u) (F f) (F g) (F h) (R f) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g) (S h)) ((ecopopr.1 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))))))) (ecopopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (ecopopr.cl (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S))) (ecopopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (ecopopr.can (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z)))))) (Er R) (ecopopr.1 ecopopr.com g visset (cv f) ecopoprsym ecopopr.1 ecopopr.com ecopopr.cl ecopopr.ass ecopopr.can g visset h visset (cv f) ecopoprtrn ster)) thm (eceqopreq ((x y) (F x) (F y) (S x) (S y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) ((eceqopreq.2 (e. B (V))) (eceqopreq.3 (e. C (V))) (eceqopreq.4 (e. D (V))) (eceqopreq.5 (Er R)) (eceqopreq.6 (= (dom R) (X. S S))) (eceqopreq.7 (= (dom F) (X. S S))) (eceqopreq.8 (-. (e. ({/}) S))) (eceqopreq.9 (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S))) (eceqopreq.10 (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (<-> (br (<,> A B) R (<,> C D)) (= (opr A F D) (opr B F C)))))) (-> (/\ (e. A S) (e. C S)) (<-> (= ([] (<,> A B) R) ([] (<,> C D) R)) (= (opr A F D) (opr B F C)))) ((/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S)) pm3.26 A S B S opelxpi eceqopreq.6 syl6eleqr C D opex eceqopreq.5 (<,> A B) erthdm 3syl eceqopreq.10 bitrd exp43 3imp ([] (<,> A B) R) ([] (<,> C D) R) ({/}) eqeq1 biimprcd con3d eceqopreq.6 (<,> C D) eleq2i C D opex R ecdmn0 eceqopreq.4 C S S opelxp 3bitr3 pm3.27bd nsyl4 eceqopreq.6 (<,> A B) eleq2i A B opex R ecdmn0 eceqopreq.2 A S S opelxp 3bitr3 syl5ibr com12 (e. C S) 3adant3 (opr A F D) (opr B F C) S eleq1 eceqopreq.9 B C caoprcl syl5bir eceqopreq.4 eceqopreq.7 eceqopreq.8 A ndmoprrcl pm3.27d syl6com con3d (e. A S) 3adant1 jcad (= ([] (<,> A B) R) ([] (<,> C D) R)) (= (opr A F D) (opr B F C)) pm5.21 syl6 pm2.61d 3exp com23 imp ([] (<,> A B) R) ([] (<,> C D) R) ({/}) eqeq1 biimpcd con3d eceqopreq.6 (<,> A B) eleq2i A B opex R ecdmn0 eceqopreq.2 A S S opelxp 3bitr3 pm3.27bd nsyl4 eceqopreq.6 (<,> C D) eleq2i C D opex R ecdmn0 eceqopreq.4 C S S opelxp 3bitr3 syl5ibr com12 (e. A S) 3adant1 (opr A F D) (opr B F C) S eleq1 eceqopreq.9 A D caoprcl syl5bi eceqopreq.3 eceqopreq.7 eceqopreq.8 B ndmoprrcl pm3.26d syl6com con3d (e. C S) 3adant2 jcad (= ([] (<,> A B) R) ([] (<,> C D) R)) (= (opr A F D) (opr B F C)) pm5.21 syl6 3exp imp ([] (<,> C D) R) ({/}) ([] (<,> A B) R) eqeq2 eceqopreq.6 (<,> C D) eleq2i C D opex R ecdmn0 eceqopreq.4 C S S opelxp 3bitr3 pm3.27bd nsyl4 eceqopreq.6 (<,> A B) eleq2i A B opex R ecdmn0 eceqopreq.2 A S S opelxp 3bitr3 pm3.27bd con1i syl5bir (e. B S) (e. C S) pm3.26 con3i eceqopreq.3 eceqopreq.7 B ndmopr (opr B F C) ({/}) (opr A F D) eqeq2 (e. A S) (e. D S) pm3.27 con3i eceqopreq.4 eceqopreq.7 A ndmopr syl syl5bir 3syl com12 jcad (= ([] (<,> A B) R) ([] (<,> C D) R)) (= (opr A F D) (opr B F C)) pm5.1 syl6 pm2.61d1 pm2.61d)) thm (th3qlem1 ((x y) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (R x) (R y) (R z) (R w) (R v) (R u) (S x) (S y) (S z) (S w) (S v) (S u) (A x) (A y) (A z) (A w) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) ((th3qlem1.1 (Er R)) (th3qlem1.2 (= (dom R) S)) (th3qlem1.3 (-> (/\ (/\ (e. (cv y) S) (e. (cv w) S)) (/\ (e. (cv z) S) (e. (cv v) S))) (-> (/\ (br (cv y) R (cv w)) (br (cv z) R (cv v))) (br (opr (cv y) F (cv z)) R (opr (cv w) F (cv v))))))) (-> (/\ (e. A (/. S R)) (e. B (/. S R))) (E* x (E. y (E. z (/\ (/\ (= A ([] (cv y) R)) (= B ([] (cv z) R))) (= (cv x) ([] (opr (cv y) F (cv z)) R))))))) ((= A ([] (cv y) R)) (= B ([] (cv z) R)) (= A ([] (cv w) R)) (= B ([] (cv v) R)) an4 (/\ (/\ (e. A (/. S R)) (e. A (/. S R))) (/\ (e. B (/. S R)) (e. B (/. S R)))) anbi2i (/\ (e. A (/. S R)) (e. A (/. S R))) (/\ (e. B (/. S R)) (e. B (/. S R))) (/\ (= A ([] (cv y) R)) (= A ([] (cv w) R))) (/\ (= B ([] (cv z) R)) (= B ([] (cv v) R))) an4 (e. A (/. S R)) (e. A (/. S R)) (= A ([] (cv y) R)) (= A ([] (cv w) R)) an4 A ([] (cv y) R) (/. S R) eleq1 pm5.32ri A ([] (cv w) R) (/. S R) eleq1 pm5.32ri anbi12i bitr (e. B (/. S R)) (e. B (/. S R)) (= B ([] (cv z) R)) (= B ([] (cv v) R)) an4 B ([] (cv z) R) (/. S R) eleq1 pm5.32ri B ([] (cv v) R) (/. S R) eleq1 pm5.32ri anbi12i bitr anbi12i 3bitr (e. (cv y) S) (e. (cv w) S) pm3.26 th3qlem1.2 syl6eleqr w visset th3qlem1.1 (cv y) erthdm syl (e. (cv z) S) (e. (cv v) S) pm3.26 th3qlem1.2 syl6eleqr v visset th3qlem1.1 (cv z) erthdm syl bi2anan9 th3qlem1.3 sylbid (cv x) ([] (opr (cv y) F (cv z)) R) (cv u) ([] (opr (cv w) F (cv v)) R) eqeq12 (cv y) F (cv z) oprex (cv w) F (cv v) oprex th3qlem1.1 erthi syl5bir syl9 com23 imp y visset th3qlem1.1 th3qlem1.2 ecelqsdm w visset th3qlem1.1 th3qlem1.2 ecelqsdm anim12i z visset th3qlem1.1 th3qlem1.2 ecelqsdm v visset th3qlem1.1 th3qlem1.2 ecelqsdm anim12i anim12i A ([] (cv y) R) ([] (cv w) R) eqtr2t B ([] (cv z) R) ([] (cv v) R) eqtr2t anim12i syl2an an4s (e. ([] (cv y) R) (/. S R)) (= A ([] (cv y) R)) (e. ([] (cv w) R) (/. S R)) (= A ([] (cv w) R)) an4 (e. ([] (cv z) R) (/. S R)) (= B ([] (cv z) R)) (e. ([] (cv v) R) (/. S R)) (= B ([] (cv v) R)) an4 syl2anb sylbi (e. A (/. S R)) anidm (e. B (/. S R)) anidm anbi12i sylanbr ex imp3a (/\ (= A ([] (cv y) R)) (= B ([] (cv z) R))) (= (cv x) ([] (opr (cv y) F (cv z)) R)) (/\ (= A ([] (cv w) R)) (= B ([] (cv v) R))) (= (cv u) ([] (opr (cv w) F (cv v)) R)) an4 syl5ib w v 19.23advv y z 19.23advv y z w v (/\ (/\ (= A ([] (cv y) R)) (= B ([] (cv z) R))) (= (cv x) ([] (opr (cv y) F (cv z)) R))) (/\ (/\ (= A ([] (cv w) R)) (= B ([] (cv v) R))) (= (cv u) ([] (opr (cv w) F (cv v)) R))) ee4anv syl5ibr x u 19.21aivv (cv x) (cv u) ([] (opr (cv y) F (cv z)) R) eqeq1 (/\ (= A ([] (cv y) R)) (= B ([] (cv z) R))) anbi2d y z 2exbidv (cv y) (cv w) R eceq2 A eqeq2d (cv z) (cv v) R eceq2 B eqeq2d bi2anan9 (cv y) (cv w) (cv z) (cv v) F opreq12 (opr (cv y) F (cv z)) (opr (cv w) F (cv v)) R eceq2 syl (cv u) eqeq2d anbi12d cbvex2v syl6bb mo4 sylibr)) thm (th3qlem2 ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (A x) (A y) (A z) (A w) (A v) (A u) (A t) (A s) (A f) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h)) ((th3q.1 (e. R (V))) (th3q.2 (Er R)) (th3q.3 (= (dom R) (X. S S))) (th3q.4 (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h)))))))) (-> (/\ (e. A (/. (X. S S) R)) (e. B (/. (X. S S) R))) (E* z (E. w (E. v (E. u (E. t (/\ (/\ (= A ([] (<,> (cv w) (cv v)) R)) (= B ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R))))))))) (th3q.2 th3q.3 (X. S S) eqid (<,> (cv w) (cv v)) (cv s) R (<,> (cv u) (cv t)) breq1 (br (cv x) R (cv y)) anbi1d (<,> (cv w) (cv v)) (cv s) F (cv x) opreq1 R (opr (<,> (cv u) (cv t)) F (cv y)) breq1d imbi12d (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) imbi2d (<,> (cv u) (cv t)) (cv f) (cv s) R breq2 (br (cv x) R (cv y)) anbi1d (<,> (cv u) (cv t)) (cv f) F (cv y) opreq1 (opr (cv s) F (cv x)) R breq2d imbi12d (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) imbi2d (X. S S) eqid (<,> (cv s) (cv f)) (cv x) R (<,> (cv g) (cv h)) breq1 (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) anbi2d (<,> (cv s) (cv f)) (cv x) (<,> (cv w) (cv v)) F opreq2 R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h))) breq1d imbi12d (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) imbi2d (<,> (cv g) (cv h)) (cv y) (cv x) R breq2 (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) anbi2d (<,> (cv g) (cv h)) (cv y) (<,> (cv u) (cv t)) F opreq2 (opr (<,> (cv w) (cv v)) F (cv x)) R breq2d imbi12d (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) imbi2d th3q.4 expcom 2optocl com12 2optocl imp A B z th3qlem1 (cv w) (cv v) opex (cv u) (cv t) opex (cv s) (<,> (cv w) (cv v)) R eceq2 A eqeq2d (cv x) (<,> (cv u) (cv t)) R eceq2 B eqeq2d bi2anan9 (cv s) (<,> (cv w) (cv v)) (cv x) (<,> (cv u) (cv t)) F opreq12 (opr (cv s) F (cv x)) (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R eceq2 syl (cv z) eqeq2d anbi12d cla4e2v u t 19.23aivv w v 19.23aivv z immoi syl)) thm (th3qcor ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h) (G x) (G y) (G z) (G w) (G v) (G u) (G t) (G s) (G f) (G g) (G h)) ((th3q.1 (e. R (V))) (th3q.2 (Er R)) (th3q.3 (= (dom R) (X. S S))) (th3q.4 (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h))))))) (th3q.5 (= G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (/. (X. S S) R)) (e. (cv y) (/. (X. S S) R))) (E. w (E. v (E. u (E. t (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) R)) (= (cv y) ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R)))))))))))) (Fun G) (th3q.1 th3q.2 th3q.3 th3q.4 (cv x) (cv y) z th3qlem2 z (/\ (e. (cv x) (/. (X. S S) R)) (e. (cv y) (/. (X. S S) R))) (E. w (E. v (E. u (E. t (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) R)) (= (cv y) ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R))))))) moanimv mpbir x y funoprab th3q.5 G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (/. (X. S S) R)) (e. (cv y) (/. (X. S S) R))) (E. w (E. v (E. u (E. t (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) R)) (= (cv y) ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R))))))))) funeq ax-mp mpbir)) thm (th3q ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (A x) (A y) (A z) (A w) (A v) (A u) (A t) (A s) (A f) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h) (G x) (G y) (G z) (G w) (G v) (G u) (G t) (G s) (G f) (G g) (G h)) ((th3q.1 (e. R (V))) (th3q.2 (Er R)) (th3q.3 (= (dom R) (X. S S))) (th3q.4 (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h))))))) (th3q.5 (= G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (/. (X. S S) R)) (e. (cv y) (/. (X. S S) R))) (E. w (E. v (E. u (E. t (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) R)) (= (cv y) ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R)))))))))))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (= (opr ([] (<,> A B) R) G ([] (<,> C D) R)) ([] (opr (<,> A B) F (<,> C D)) R))) (th3q.1 R (V) (opr (<,> A B) F (<,> C D)) ecexg ax-mp (cv x) ([] (<,> A B) R) ([] (<,> (cv w) (cv v)) R) eqeq1 (= (cv y) ([] (<,> (cv u) (cv t)) R)) anbi1d (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R)) anbi1d w v u t 4exbidv (cv y) ([] (<,> C D) R) ([] (<,> (cv u) (cv t)) R) eqeq1 (= ([] (<,> A B) R) ([] (<,> (cv w) (cv v)) R)) anbi2d (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R)) anbi1d w v u t 4exbidv (cv z) ([] (opr (<,> A B) F (<,> C D)) R) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R) eqeq1 (/\ (= ([] (<,> A B) R) ([] (<,> (cv w) (cv v)) R)) (= ([] (<,> C D) R) ([] (<,> (cv u) (cv t)) R))) anbi2d w v u t 4exbidv th3q.1 th3q.2 th3q.3 th3q.4 (cv x) (cv y) z th3qlem2 th3q.5 oprabvali A S B S opelxpi th3q.1 (<,> A B) (X. S S) ecelqsi syl C S D S opelxpi th3q.1 (<,> C D) (X. S S) ecelqsi syl anim12i ([] (<,> A B) R) eqid ([] (<,> C D) R) eqid pm3.2i ([] (opr (<,> A B) F (<,> C D)) R) eqid (cv w) A (cv v) B opeq12 (<,> (cv w) (cv v)) (<,> A B) R eceq2 ([] (<,> A B) R) eqeq2d (= ([] (<,> C D) R) ([] (<,> C D) R)) anbi1d (<,> (cv w) (cv v)) (<,> A B) F (<,> C D) opreq1 (opr (<,> (cv w) (cv v)) F (<,> C D)) (opr (<,> A B) F (<,> C D)) R eceq2 syl ([] (opr (<,> A B) F (<,> C D)) R) eqeq2d anbi12d syl S S cla4e2gv (cv u) C (cv t) D opeq12 (<,> (cv u) (cv t)) (<,> C D) R eceq2 ([] (<,> C D) R) eqeq2d (= ([] (<,> A B) R) ([] (<,> (cv w) (cv v)) R)) anbi2d (<,> (cv u) (cv t)) (<,> C D) (<,> (cv w) (cv v)) F opreq2 (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) (opr (<,> (cv w) (cv v)) F (<,> C D)) R eceq2 syl ([] (opr (<,> A B) F (<,> C D)) R) eqeq2d anbi12d syl S S cla4e2gv w v 19.22dvv sylan9 mp2ani sylc)) var (set a) var (set b) var (set c) var (set d) thm (oprec ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d) (F x) (F y) (F z) (F t) (F s) (F g) (F h) (F a) (F b) (F c) (F d) (G x) (G y) (G z) (G t) (G s) (G g) (G h) (G a) (G b) (G c) (G d) (H x) (H y) (H z) (H w) (H v) (H u) (H f) (J x) (J y) (J z) (K x) (K y) (K z) (K w) (K v) (K u) (K f) (L x) (L y) (L z) (L w) (L v) (L u) (L f) (Q x) (Q y) (Q z) (Q a) (Q b) (Q c) (Q d) (R x) (R y) (R z) (R t) (R s) (R g) (R h) (R a) (R b) (R c) (R d) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (S a) (S b) (S c) (S d) (ph x) (ph y) (ps z) (ps w) (ps v) (ps u) (ch z) (ch w) (ch v) (ch u)) ((oprec.1 (e. H (V))) (oprec.2 (e. K (V))) (oprec.3 (e. L (V))) (oprec.4 (e. R (V))) (oprec.5 (Er R)) (oprec.6 (= (dom R) (X. S S))) (oprec.7 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) ph))))))))) (oprec.8 (-> (/\ (/\ (= (cv z) (cv a)) (= (cv w) (cv b))) (/\ (= (cv v) (cv c)) (= (cv u) (cv d)))) (<-> ph ps))) (oprec.9 (-> (/\ (/\ (= (cv z) (cv g)) (= (cv w) (cv h))) (/\ (= (cv v) (cv t)) (= (cv u) (cv s)))) (<-> ph ch))) (oprec.10 (= G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) J)))))))))) (oprec.11 (-> (/\ (/\ (= (cv w) (cv a)) (= (cv v) (cv b))) (/\ (= (cv u) (cv g)) (= (cv f) (cv h)))) (= J K))) (oprec.12 (-> (/\ (/\ (= (cv w) (cv c)) (= (cv v) (cv d))) (/\ (= (cv u) (cv t)) (= (cv f) (cv s)))) (= J L))) (oprec.13 (-> (/\ (/\ (= (cv w) A) (= (cv v) B)) (/\ (= (cv u) C) (= (cv f) D))) (= J H))) (oprec.14 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) Q) (e. (cv y) Q)) (E. a (E. b (E. c (E. d (/\ (/\ (= (cv x) ([] (<,> (cv a) (cv b)) R)) (= (cv y) ([] (<,> (cv c) (cv d)) R))) (= (cv z) ([] (opr (<,> (cv a) (cv b)) G (<,> (cv c) (cv d))) R))))))))))) (oprec.15 (= Q (/. (X. S S) R))) (oprec.16 (-> (/\ (/\ (/\ (e. (cv a) S) (e. (cv b) S)) (/\ (e. (cv c) S) (e. (cv d) S))) (/\ (/\ (e. (cv g) S) (e. (cv h) S)) (/\ (e. (cv t) S) (e. (cv s) S)))) (-> (/\ ps ch) (br K R L))))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (= (opr ([] (<,> A B) R) F ([] (<,> C D) R)) ([] H R))) (oprec.4 oprec.5 oprec.6 oprec.16 oprec.8 oprec.7 opbrop oprec.9 oprec.7 opbrop bi2anan9 oprec.2 oprec.11 oprec.10 oprabval3 oprec.3 oprec.12 oprec.10 oprabval3 R breqan12d an4s 3imtr4d oprec.14 oprec.15 (cv x) eleq2i oprec.15 (cv y) eleq2i anbi12i (E. a (E. b (E. c (E. d (/\ (/\ (= (cv x) ([] (<,> (cv a) (cv b)) R)) (= (cv y) ([] (<,> (cv c) (cv d)) R))) (= (cv z) ([] (opr (<,> (cv a) (cv b)) G (<,> (cv c) (cv d))) R))))))) anbi1i x y z oprabbii eqtr A B C D th3q oprec.1 oprec.13 oprec.10 oprabval3 (opr (<,> A B) G (<,> C D)) H R eceq2 syl eqtrd)) thm (ecoprcom ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w) (R x) (R y) (R z) (R w) (S x) (S y) (S z) (S w) (C z) (C w)) ((ecoprcom.1 (= C (/. (X. S S) R))) (ecoprcom.2 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (= (opr ([] (<,> (cv x) (cv y)) R) F ([] (<,> (cv z) (cv w)) R)) ([] (<,> D G) R)))) (ecoprcom.3 (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv x) S) (e. (cv y) S))) (= (opr ([] (<,> (cv z) (cv w)) R) F ([] (<,> (cv x) (cv y)) R)) ([] (<,> H J) R)))) (ecoprcom.4 (= D H)) (ecoprcom.5 (= G J))) (-> (/\ (e. A C) (e. B C)) (= (opr A F B) (opr B F A))) (ecoprcom.1 ([] (<,> (cv x) (cv y)) R) A F ([] (<,> (cv z) (cv w)) R) opreq1 ([] (<,> (cv x) (cv y)) R) A ([] (<,> (cv z) (cv w)) R) F opreq2 eqeq12d ([] (<,> (cv z) (cv w)) R) B A F opreq2 ([] (<,> (cv z) (cv w)) R) B F A opreq1 eqeq12d ecoprcom.2 ecoprcom.4 ecoprcom.5 D H G J opeq12 (<,> D G) (<,> H J) R eceq2 syl mp2an syl6eq ecoprcom.3 ancoms eqtr4d 2ecoptocl)) thm (ecoprass ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (F x) (F y) (F z) (F w) (F v) (F u) (R x) (R y) (R z) (R w) (R v) (R u) (S x) (S y) (S z) (S w) (S v) (S u) (D z) (D w) (D v) (D u)) ((ecoprass.1 (= D (/. (X. S S) R))) (ecoprass.2 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (= (opr ([] (<,> (cv x) (cv y)) R) F ([] (<,> (cv z) (cv w)) R)) ([] (<,> G H) R)))) (ecoprass.3 (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (= (opr ([] (<,> (cv z) (cv w)) R) F ([] (<,> (cv v) (cv u)) R)) ([] (<,> N Q) R)))) (ecoprass.4 (-> (/\ (/\ (e. G S) (e. H S)) (/\ (e. (cv v) S) (e. (cv u) S))) (= (opr ([] (<,> G H) R) F ([] (<,> (cv v) (cv u)) R)) ([] (<,> J K) R)))) (ecoprass.5 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. N S) (e. Q S))) (= (opr ([] (<,> (cv x) (cv y)) R) F ([] (<,> N Q) R)) ([] (<,> L M) R)))) (ecoprass.6 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (/\ (e. G S) (e. H S)))) (ecoprass.7 (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (/\ (e. N S) (e. Q S)))) (ecoprass.8 (= J L)) (ecoprass.9 (= K M))) (-> (/\/\ (e. A D) (e. B D) (e. C D)) (= (opr (opr A F B) F C) (opr A F (opr B F C)))) (ecoprass.1 ([] (<,> (cv x) (cv y)) R) A F ([] (<,> (cv z) (cv w)) R) opreq1 F ([] (<,> (cv v) (cv u)) R) opreq1d ([] (<,> (cv x) (cv y)) R) A F (opr ([] (<,> (cv z) (cv w)) R) F ([] (<,> (cv v) (cv u)) R)) opreq1 eqeq12d ([] (<,> (cv z) (cv w)) R) B A F opreq2 F ([] (<,> (cv v) (cv u)) R) opreq1d ([] (<,> (cv z) (cv w)) R) B F ([] (<,> (cv v) (cv u)) R) opreq1 A F opreq2d eqeq12d ([] (<,> (cv v) (cv u)) R) C (opr A F B) F opreq2 ([] (<,> (cv v) (cv u)) R) C B F opreq2 A F opreq2d eqeq12d ecoprass.2 F ([] (<,> (cv v) (cv u)) R) opreq1d (/\ (e. (cv v) S) (e. (cv u) S)) adantr ecoprass.4 ecoprass.6 sylan eqtrd 3impa ecoprass.8 ecoprass.9 J L K M opeq12 (<,> J K) (<,> L M) R eceq2 syl mp2an syl6eq ecoprass.3 ([] (<,> (cv x) (cv y)) R) F opreq2d (/\ (e. (cv x) S) (e. (cv y) S)) adantl ecoprass.5 ecoprass.7 sylan2 eqtrd 3impb eqtr4d 3ecoptocl)) thm (ecoprdi ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (F x) (F y) (F z) (F w) (F v) (F u) (R x) (R y) (R z) (R w) (R v) (R u) (S x) (S y) (S z) (S w) (S v) (S u) (G x) (G y) (G z) (G w) (G v) (G u) (D z) (D w) (D v) (D u)) ((ecoprdi.1 (= D (/. (X. S S) R))) (ecoprdi.2 (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (= (opr ([] (<,> (cv z) (cv w)) R) F ([] (<,> (cv v) (cv u)) R)) ([] (<,> M N) R)))) (ecoprdi.3 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. M S) (e. N S))) (= (opr ([] (<,> (cv x) (cv y)) R) G ([] (<,> M N) R)) ([] (<,> H J) R)))) (ecoprdi.4 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (= (opr ([] (<,> (cv x) (cv y)) R) G ([] (<,> (cv z) (cv w)) R)) ([] (<,> W X) R)))) (ecoprdi.5 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (= (opr ([] (<,> (cv x) (cv y)) R) G ([] (<,> (cv v) (cv u)) R)) ([] (<,> Y Z) R)))) (ecoprdi.6 (-> (/\ (/\ (e. W S) (e. X S)) (/\ (e. Y S) (e. Z S))) (= (opr ([] (<,> W X) R) F ([] (<,> Y Z) R)) ([] (<,> K L) R)))) (ecoprdi.7 (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (/\ (e. M S) (e. N S)))) (ecoprdi.8 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (/\ (e. W S) (e. X S)))) (ecoprdi.9 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (/\ (e. Y S) (e. Z S)))) (ecoprdi.10 (= H K)) (ecoprdi.11 (= J L))) (-> (/\/\ (e. A D) (e. B D) (e. C D)) (= (opr A G (opr B F C)) (opr (opr A G B) F (opr A G C)))) (ecoprdi.1 ([] (<,> (cv x) (cv y)) R) A G (opr ([] (<,> (cv z) (cv w)) R) F ([] (<,> (cv v) (cv u)) R)) opreq1 ([] (<,> (cv x) (cv y)) R) A G ([] (<,> (cv z) (cv w)) R) opreq1 ([] (<,> (cv x) (cv y)) R) A G ([] (<,> (cv v) (cv u)) R) opreq1 F opreq12d eqeq12d ([] (<,> (cv z) (cv w)) R) B F ([] (<,> (cv v) (cv u)) R) opreq1 A G opreq2d ([] (<,> (cv z) (cv w)) R) B A G opreq2 F (opr A G ([] (<,> (cv v) (cv u)) R)) opreq1d eqeq12d ([] (<,> (cv v) (cv u)) R) C B F opreq2 A G opreq2d ([] (<,> (cv v) (cv u)) R) C A G opreq2 (opr A G B) F opreq2d eqeq12d ecoprdi.2 ([] (<,> (cv x) (cv y)) R) G opreq2d (/\ (e. (cv x) S) (e. (cv y) S)) adantl ecoprdi.3 ecoprdi.7 sylan2 eqtrd 3impb ecoprdi.10 ecoprdi.11 H K J L opeq12 (<,> H J) (<,> K L) R eceq2 syl mp2an syl6eq ecoprdi.4 ecoprdi.5 F opreqan12d ecoprdi.6 ecoprdi.8 ecoprdi.9 syl2an eqtrd 3impdi eqtr4d 3ecoptocl)) thm (mapprc ((A f) (B f)) () (-> (-. (e. A (V))) (= ({|} f (:--> (cv f) A B)) ({/}))) (f (:--> (cv f) A B) abn0 (cv f) A B fdm f visset (cv f) (V) dmexg ax-mp syl6eqelr f 19.23aiv sylbi con1i)) thm (mapex ((A f) (B f)) () (-> (/\ (e. A C) (e. B D)) (e. ({|} f (:--> (cv f) A B)) (V))) ((cv f) A B fssxp f ss2abi (X. A B) f df-pw sseqtr4 ({|} f (:--> (cv f) A B)) (P~ (X. A B)) (V) ssexg A C B D xpexg (X. A B) (V) pwexg syl sylan2 mpan)) thm (fnmap ((f x) (f y) (f z) (x y) (x z) (y z)) () (Fn (^m) (X. (V) (V))) (y visset x visset (cv y) (V) (cv x) (V) f mapex mp2an x y z f df-map x visset y visset pm3.2i (= (cv z) ({|} f (:--> (cv f) (cv y) (cv x)))) biantrur x y z oprabbii eqtr fnoprab2)) thm (mapvalg ((x y) (x z) (f x) (A x) (y z) (f y) (A y) (f z) (A z) (A f) (B x) (B y) (B z) (B f)) () (-> (/\ (e. A C) (e. B D)) (= (opr A (^m) B) ({|} f (:--> (cv f) B A)))) (B D A C f mapex ancoms (cv x) A (cv f) (cv y) feq3 f abbidv (cv y) B (cv f) A feq2 f abbidv x y z f df-map x visset y visset pm3.2i (= (cv z) ({|} f (:--> (cv f) (cv y) (cv x)))) biantrur x y z oprabbii eqtr (V) oprabval2g 3exp imp A C elisset B D elisset syl2an mpd)) thm (mapval ((A f) (B f)) ((mapval.1 (e. A (V))) (mapval.2 (e. B (V)))) (= (opr A (^m) B) ({|} f (:--> (cv f) B A))) (mapval.1 mapval.2 A (V) B (V) f mapvalg mp2an)) thm (elmap ((A f) (B f) (C f)) ((elmap.1 (e. A (V))) (elmap.2 (e. B (V)))) (<-> (e. C (opr A (^m) B)) (:--> C B A)) (elmap.1 elmap.2 f mapval C eleq2i elmap.2 C B A (V) fex mpan2 (cv f) C B A feq1 elab3 bitr)) thm (elmapg ((x y) (A x) (A y) (B y) (C x) (C y)) () (-> (/\ (e. A R) (e. B S)) (<-> (e. C (opr A (^m) B)) (:--> C B A))) ((cv x) A (^m) (cv y) opreq1 C eleq2d (cv x) A C (cv y) feq3 bibi12d (cv y) B A (^m) opreq2 C eleq2d (cv y) B C A feq2 bibi12d x visset y visset C elmap R S vtocl2g)) thm (map0e ((A f)) ((map0e.1 (e. A (V)))) (= (opr A (^m) ({/})) (1o)) ((cv f) fn0 (C_ (ran (cv f)) A) anbi1i (cv f) ({/}) A df-f A 0ss (cv f) ({/}) rneq rn0 syl6eq A sseq1d mpbiri pm4.71i 3bitr4 f abbii map0e.1 0ex f mapval ({/}) f df-sn 3eqtr4 df1o2 eqtr4)) thm (map0b ((A f)) ((map0e.1 (e. A (V)))) (-> (-. (= A ({/}))) (= (opr ({/}) (^m) A) ({/}))) (f (:--> (cv f) A ({/})) abn0 (cv f) A ({/}) fdm (cv f) A ({/}) frn (ran (cv f)) ss0 syl (cv f) dm0rn0 sylibr eqtr3d f 19.23aiv sylbi con1i 0ex map0e.1 f mapval syl5eq)) thm (map0 ((f x) (A f) (A x) (B f) (B x)) ((map0.1 (e. A (V))) (map0.2 (e. B (V)))) (<-> (= (opr A (^m) B) ({/})) (/\ (= A ({/})) (-. (= B ({/}))))) (map0.1 map0.2 f mapval ({/}) eqeq1i (cv x) A snssi x visset B fconst (X. B ({} (cv x))) B ({} (cv x)) A fss mpan map0.2 (cv x) snex xpex (cv f) (X. B ({} (cv x))) B A feq1 cla4ev 3syl x 19.23aiv A x n0 f (:--> (cv f) B A) abn0 3imtr4 a3i sylbi 0nep0 map0.1 map0e ({/}) eqeq1i df1o2 ({/}) eqeq1i ({} ({/})) ({/}) eqcom 3bitr mtbir B ({/}) A (^m) opreq2 ({/}) eqeq1d mtbiri con2i jca A ({/}) (^m) B opreq1 map0.2 map0b sylan9eq impbi)) thm (mapsn ((f y) (A f) (A y) (B f) (B y)) ((map0.1 (e. A (V))) (map0.2 (e. B (V)))) (= (opr A (^m) ({} B)) ({|} f (E.e. y A (= (cv f) ({} (<,> B (cv y))))))) (map0.1 B snex (cv f) elmap map0.2 snid (cv f) ({} B) B y fneu (cv f) ({} B) A ffn sylan mpan2 (cv f) ({} B) A frel (cv f) B y relimasn syl (cv f) ({} B) A fdm (dom (cv f)) ({} B) (cv f) imaeq2 syl (cv f) imadmrn syl5reqr eqtr3d ({} (cv y)) eqeq1d y exbidv y (br B (cv f) (cv y)) eusn syl5bb mpbid (cv f) ({} B) A frn (cv y) sseld y visset snid (ran (cv f)) ({} (cv y)) (cv y) eleq2 mpbiri syl5 (ran (cv f)) ({} (cv y)) (cv f) ({} B) feq3 (cv f) ({} B) A ffn (cv f) ({} B) fnforn sylib (cv f) ({} B) (ran (cv f)) fof syl syl5bi com12 map0.2 y visset (cv f) fsn syl6ib jcad y 19.22dv mpd y A (= (cv f) ({} (<,> B (cv y)))) df-rex sylibr (cv f) ({} B) ({} (cv y)) A fss map0.2 y visset f1osn (cv f) ({} (<,> B (cv y))) ({} B) ({} (cv y)) f1oeq1 mpbiri (cv f) ({} B) ({} (cv y)) f1of syl (cv y) A snssi syl2an expcom r19.23aiv impbi bitr abbi2i)) thm (mapss ((A f) (B f) (C f)) ((mapss.1 (e. B (V))) (mapss.2 (e. C (V)))) (-> (C_ A B) (C_ (opr A (^m) C) (opr B (^m) C))) ((cv f) C A B fss expcom f 19.21aiv f (:--> (cv f) C A) (:--> (cv f) C B) ss2ab sylibr mapss.1 A ssex mapss.2 A (V) C (V) f mapvalg mpan2 syl mapss.1 mapss.2 f mapval (C_ A B) a1i 3sstr4d)) thm (elixp2 ((A f) (B f) (f x) (F f) (F x)) () (<-> (e. F (X_ x A B)) (/\/\ (e. F (V)) (Fn F A) (A.e. x A (e. (` F (cv x)) B)))) ((cv f) F A fneq1 (cv f) F (cv x) fveq1 B eleq1d x A ralbidv anbi12d x A B f df-ixp (V) elab2g pm5.32i F (X_ x A B) elisset pm4.71ri (e. F (V)) (Fn F A) (A.e. x A (e. (` F (cv x)) B)) 3anass 3bitr4)) thm (elixp ((F x)) ((elixp.1 (e. F (V)))) (<-> (e. F (X_ x A B)) (/\ (Fn F A) (A.e. x A (e. (` F (cv x)) B)))) ((e. F (V)) (Fn F A) (A.e. x A (e. (` F (cv x)) B)) 3anass F x A B elixp2 elixp.1 (/\ (Fn F A) (A.e. x A (e. (` F (cv x)) B))) biantrur 3bitr4)) thm (ss2ixp ((A f) (B f) (C f) (f x)) () (-> (A.e. x A (C_ B C)) (C_ (X_ x A B) (X_ x A C))) (B C (` (cv f) (cv x)) ssel x A r19.20si x A (e. (` (cv f) (cv x)) B) (e. (` (cv f) (cv x)) C) r19.20 syl (Fn (cv f) A) anim2d f visset x A B elixp f visset x A C elixp 3imtr4g ssrdv)) thm (ixpf ((x y) (A x) (A y) (B y) (F x) (F y)) () (-> (e. F (X_ x A B)) (:--> F A (U_ x A B))) (F x A B elixp2 x A B ssiun2 (` F (cv x)) sseld r19.20i (Fn F A) anim2i (e. (cv y) A) x ax-17 y x A B hbiu1 (e. (cv y) F) x ax-17 ffnfvf sylibr (e. F (V)) 3adant1 sylbi)) thm (0elixp () () (e. ({/}) (X_ x ({/}) A)) (0ex x ({/}) A elixp ({/}) eqid ({/}) fn0 mpbir x (e. (` ({/}) (cv x)) A) ral0 mpbir2an)) thm (ixp0 ((A f) (B f) (f x)) () (-> (E.e. x A (= B ({/}))) (= (X_ x A B) ({/}))) ((` (cv f) (cv x)) B n0i con2i x A r19.22si x A (e. (` (cv f) (cv x)) B) rexnal sylib (Fn (cv f) A) intnand (cv f) noel jctir (/\ (Fn (cv f) A) (A.e. x A (e. (` (cv f) (cv x)) B))) (e. (cv f) ({/})) pm5.21 syl f visset x A B elixp syl5bb eqrdv)) thm (mapixp ((B x) (f x) (A f) (A x) (B f)) ((mapixp.1 (e. A (V))) (mapixp.2 (e. B (V)))) (= (opr B (^m) A) (X_ x A B)) ((cv f) A B x ffnfv mapixp.2 mapixp.1 (cv f) elmap f visset x A B elixp 3bitr4 eqriv)) thm (relen ((x y) (f x) (f y)) () (Rel (~~)) (x y (E. f (:-1-1-onto-> (cv f) (cv x) (cv y))) relopab x y f df-en (~~) ({<,>|} x y (E. f (:-1-1-onto-> (cv f) (cv x) (cv y)))) releq ax-mp mpbir)) thm (reldom ((x y) (f x) (f y)) () (Rel (~<_)) (x y (E. f (:-1-1-> (cv f) (cv x) (cv y))) relopab x y f df-dom (~<_) ({<,>|} x y (E. f (:-1-1-> (cv f) (cv x) (cv y)))) releq ax-mp mpbir)) thm (relsdom () () (Rel (~<)) (reldom (~<_) (~~) reldif df-sdom (~<) (\ (~<_) (~~)) releq ax-mp sylibr ax-mp)) thm (breng ((f x) (f y) (A f) (x y) (A x) (A y) (B f) (B x) (B y) (C y)) () (-> (e. B C) (<-> (br A (~~) B) (E. f (:-1-1-onto-> (cv f) A B)))) ((cv x) A (cv f) (cv y) f1oeq2 f exbidv (cv y) B (cv f) A f1oeq3 f exbidv x y f df-en (V) C brabg ex relen A B brrelexi (cv f) A B f1ofn (cv f) A fndm f visset (cv f) (V) dmexg ax-mp syl6eqelr syl f 19.23aiv pm5.21ni (e. B C) a1d pm2.61i)) thm (brdomg ((f x) (f y) (A f) (x y) (A x) (A y) (B f) (B x) (B y) (C y)) () (-> (e. B C) (<-> (br A (~<_) B) (E. f (:-1-1-> (cv f) A B)))) ((cv x) A (cv f) (cv y) f1eq2 f exbidv (cv y) B (cv f) A f1eq3 f exbidv x y f df-dom (V) C brabg ex reldom A B brrelexi (cv f) A B f1f (cv f) A B fdm f visset (cv f) (V) dmexg ax-mp syl6eqelr syl f 19.23aiv pm5.21ni (e. B C) a1d pm2.61i)) thm (bren ((A f) (B f)) ((bren.1 (e. B (V)))) (<-> (br A (~~) B) (E. f (:-1-1-onto-> (cv f) A B))) (bren.1 B (V) A f breng ax-mp)) thm (bren3 ((f x) (A f) (A x) (B f) (B x)) ((bren.1 (e. B (V)))) (<-> (br A (~~) B) (E. x (/\ (/\ (= (dom (i^i (cv x) (X. A B))) A) (= (ran (i^i (cv x) (X. A B))) B)) (/\ (Fun (`' (`' (cv x)))) (Fun (`' (cv x))))))) (bren.1 A f bren (cv f) (cv x) A B f1oeq1 cbvexv (cv x) A B f1of (cv x) A B fssxp (cv x) (X. A B) df-ss sylib dmeqd (cv x) A B fdm eqtrd syl (cv x) A B f1of (cv x) A B fssxp (cv x) (X. A B) df-ss sylib syl rneqd (cv x) A B f1o5 pm3.27bd eqtrd jca (cv x) A B f1ofun (cv x) funcnvcnv syl (cv x) A B f1o3 pm3.27bd jca jca x 19.22i sylbi (cv x) (X. A B) inss1 (i^i (cv x) (X. A B)) (cv x) cnvss ax-mp (`' (i^i (cv x) (X. A B))) (`' (cv x)) cnvss ax-mp (`' (`' (i^i (cv x) (X. A B)))) (`' (`' (cv x))) funss ax-mp A B relxp (X. A B) (cv x) relin2 ax-mp jctil (i^i (cv x) (X. A B)) svrelfun sylibr (= (dom (i^i (cv x) (X. A B))) A) anim1i (i^i (cv x) (X. A B)) A df-fn sylibr ancoms (= (ran (i^i (cv x) (X. A B))) B) (Fun (`' (cv x))) ad2ant2r (cv x) (X. A B) inss1 (i^i (cv x) (X. A B)) (cv x) cnvss ax-mp (`' (i^i (cv x) (X. A B))) (`' (cv x)) funss ax-mp (/\ (= (dom (i^i (cv x) (X. A B))) A) (= (ran (i^i (cv x) (X. A B))) B)) (Fun (`' (`' (cv x)))) ad2antll (= (dom (i^i (cv x) (X. A B))) A) (= (ran (i^i (cv x) (X. A B))) B) pm3.27 (/\ (Fun (`' (`' (cv x)))) (Fun (`' (cv x)))) adantr 3jca (i^i (cv x) (X. A B)) A B f1o2 sylibr x visset (X. A B) inex1 (cv f) (i^i (cv x) (X. A B)) A B f1oeq1 cla4ev syl x 19.23aiv impbi bitr)) thm (brdom ((A f) (B f)) ((bren.1 (e. B (V)))) (<-> (br A (~<_) B) (E. f (:-1-1-> (cv f) A B))) (bren.1 B (V) A f brdomg ax-mp)) thm (domen ((f x) (A f) (A x) (B f) (B x)) ((bren.1 (e. B (V)))) (<-> (br A (~<_) B) (E. x (/\ (br A (~~) (cv x)) (C_ (cv x) B)))) (f visset A B x f11o f exbii f x (/\ (:-1-1-onto-> (cv f) A (cv x)) (C_ (cv x) B)) excom bitr bren.1 A f brdom x visset A f bren (C_ (cv x) B) anbi1i f (:-1-1-onto-> (cv f) A (cv x)) (C_ (cv x) B) 19.41v bitr4 x exbii 3bitr4)) thm (domeng ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B C) (<-> (br A (~<_) B) (E. x (/\ (br A (~~) (cv x)) (C_ (cv x) B))))) ((cv y) B A (~<_) breq2 (cv y) B (cv x) sseq2 (br A (~~) (cv x)) anbi2d x exbidv bibi12d y visset A x domen C vtoclg)) thm (brsdom () () (<-> (br A (~<) B) (/\ (br A (~<_) B) (-. 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A C) (-> (:-1-1-onto-> F A B) (br A (~~) B))) (F A C fnex expcom F A B f1ofn syl5 (cv f) F A B f1oeq1 (V) cla4egv syli B A (~~) brprc A C enrefg syl5bir (E. f (:-1-1-onto-> (cv f) A B)) a1d com3r B (V) A f breng biimprd pm2.61d2 syld)) thm (f1domg ((A f) (B f) (F f)) () (-> (e. A C) (-> (:-1-1-> F A B) (br A (~<_) B))) (F A B C fex F A B f1f sylan expcom (cv f) F A B f1eq1 (V) cla4egv syli B A (~<_) brprc A C domrefg syl5bir (E. f (:-1-1-> (cv f) A B)) a1d com3r B (V) A f brdomg biimprd pm2.61d2 syld)) thm (f1dom2g () () (-> (e. B C) (-> (:-1-1-> F A B) (br A (~<_) B))) (F A B C f1dmex expcom A (V) F B f1domg syli)) thm (f1oen () ((f1oen.1 (e. A (V)))) (-> (:-1-1-onto-> F A B) (br A (~~) B)) (f1oen.1 A (V) F B f1oeng ax-mp)) thm (f1dom () ((f1oen.1 (e. A (V)))) (-> (:-1-1-> F A B) (br A (~<_) B)) (f1oen.1 A (V) F B f1domg ax-mp)) thm (en2d ((x y) (A x) (A y) (B x) (B y) (C y) (D x) (ph x) (ph y)) ((en2d.1 (-> ph (e. A (V)))) (en2d.2 (-> ph (-> (e. (cv x) A) (e. C (V))))) (en2d.3 (-> ph (-> (e. (cv y) B) (e. D (V))))) (en2d.4 (-> ph (<-> (/\ (e. (cv x) A) (= (cv y) C)) (/\ (e. (cv y) B) (= (cv x) D)))))) (-> ph (br A (~~) B)) (A (V) ({<,>|} x y (/\ (e. (cv y) B) (= (cv x) D))) B f1oeng en2d.1 en2d.2 C y eueq syl6ib r19.21aiv ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) eqid fnopabg sylib en2d.4 x y opabbidv ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) ({<,>|} x y (/\ (e. (cv y) B) (= (cv x) D))) A fneq1 syl mpbid en2d.3 D x eueq syl6ib r19.21aiv x y (/\ (e. (cv y) B) (= (cv x) D)) cnvopab fnopabg sylib jca ({<,>|} x y (/\ (e. (cv y) B) (= (cv x) D))) A B f1o4 sylibr sylc)) thm (en3d ((x y) (A x) (A y) (B x) (B y) (C y) (D x) (ph x) (ph y)) ((en3d.1 (-> ph (e. A (V)))) (en3d.2 (-> ph (-> (e. (cv x) A) (e. C B)))) (en3d.3 (-> ph (-> (e. (cv y) B) (e. D A)))) (en3d.4 (-> ph (-> (/\ (e. (cv x) A) (e. (cv y) B)) (<-> (= (cv x) D) (= (cv y) C)))))) (-> ph (br A (~~) B)) (en3d.1 en3d.2 C B elisset syl6 en3d.3 D A elisset syl6 en3d.2 C B (cv y) eleq1a syl6 imp32 en3d.4 imp biimpar exp42 com34 imp32 jcai ex en3d.3 D A (cv x) eleq1a syl6 imp32 en3d.4 imp biimpa exp42 com23 com34 imp32 jcai ex impbid en2d)) thm (en2 ((x y) (A x) (A y) (B x) (B y) (C y) (D x)) ((en2.1 (e. A (V))) (en2.2 (-> (e. (cv x) A) (e. C (V)))) (en2.3 (-> (e. (cv y) B) (e. D (V)))) (en2.4 (<-> (/\ (e. (cv x) A) (= (cv y) C)) (/\ (e. (cv y) B) (= (cv x) D))))) (br A (~~) B) (A eqid en2.1 (= A A) a1i en2.2 (= A A) a1i en2.3 (= A A) a1i en2.4 (= A A) a1i en2d ax-mp)) thm (en3 ((x y) (A x) (A y) (B x) (B y) (C y) (D x)) ((en3.1 (e. A (V))) (en3.2 (-> (e. (cv x) A) (e. C B))) (en3.3 (-> (e. (cv y) B) (e. D A))) (en3.4 (-> (/\ (e. (cv x) A) (e. (cv y) B)) (<-> (= (cv x) D) (= (cv y) C))))) (br A (~~) B) (A eqid en3.1 (= A A) a1i en3.2 (= A A) a1i en3.3 (= A A) a1i en3.4 (= A A) a1i en3d ax-mp)) thm (dom2d ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C y) (C z) (D x) (D z) (ph x) (ph y)) ((dom2d.1 (-> ph (-> (e. (cv x) A) (e. C B)))) (dom2d.2 (-> ph (-> (/\ (e. (cv x) A) (e. (cv y) A)) (<-> (= C D) (= (cv x) (cv y))))))) (-> ph (-> (e. A R) (br A (~<_) B))) (A R ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) B f1domg dom2d.1 r19.21aiv ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) eqid B fopab2 sylib dom2d.1 imp x A C B y fvopab2 ph adantll mpdan (e. (cv y) A) adantrr (/\ ph (e. (cv y) A)) x ax-17 z x y (/\ (e. (cv x) A) (= (cv y) C)) hbopab1 (e. (cv z) (cv y)) x ax-17 hbfv (e. (cv z) D) x ax-17 hbeq hbim (cv x) (cv y) A eleq1 ph anbi2d (= (` ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) (cv x)) C) imbi1d (cv x) (cv y) A eleq1 (e. (cv y) A) anbi1d (e. (cv y) A) anidm syl6bb ph anbi2d (cv x) (cv y) ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) fveq2 (/\ ph (/\ (e. (cv x) A) (e. (cv y) A))) adantr dom2d.2 imp biimparc eqeq12d ex sylbird pm5.74d bitrd dom2d.1 imp x A C B y fvopab2 ph adantll mpdan chvar (e. (cv x) A) adantrl eqeq12d dom2d.2 imp biimpd sylbid ex r19.21aivv jca z x y (/\ (e. (cv x) A) (= (cv y) C)) hbopab1 z x y (/\ (e. (cv x) A) (= (cv y) C)) hbopab2 A B f1fvf sylibr syl5com)) thm (dom2 ((x y) (A x) (A y) (B x) (B y) (C y) (D x)) ((dom2.1 (-> (e. (cv x) A) (e. C B))) (dom2.2 (-> (/\ (e. (cv x) A) (e. (cv y) A)) (<-> (= C D) (= (cv x) (cv y)))))) (-> (e. A R) (br A (~<_) B)) (A eqid dom2.1 (= A A) a1i dom2.2 (= A A) a1i R dom2d ax-mp)) thm (idssen ((x y)) () (C_ (I) (~~)) (reli x visset y visset ideq (cv x) f1oi (cv x) (cv y) (|` (I) (cv x)) (cv x) f1oeq3 mpbii x visset (|` (I) (cv x)) (cv y) f1oen syl sylbi (cv x) (I) (cv y) df-br (cv x) (~~) (cv y) df-br 3imtr3 relssi)) thm (dmen () () (= (dom (~~)) (V)) (idssen (I) (~~) dmss dmi syl5ssr ax-mp (dom (~~)) vss mpbi)) thm (ssdomg () () (-> (e. A C) (-> (C_ A B) (br A (~<_) B))) (A C (|` (I) A) B f1domg A f1oi (|` (I) A) A A f1o3 mpbi pm3.26i (|` (I) A) A A fof ax-mp (|` (I) A) A A B fss mpan funi cnvi (`' (I)) (I) funeq ax-mp mpbir (I) A funres11 ax-mp jctir (|` (I) A) A B df-f1 sylibr syl5)) thm (ssdom2g () () (-> (e. B C) (-> (C_ A B) (br A (~<_) B))) (A B C ssexg expcom A (V) B ssdomg syli)) thm (ener ((f g) (f x) (f y) (f z) (g x) (g y) (g z) (x y) (x z) (y z)) () (Er (~~)) (y visset (cv x) f bren (cv f) (cv x) (cv y) f1ocnv y visset (`' (cv f)) (cv x) f1oen syl f 19.23aiv sylbi g f (:-1-1-onto-> (cv g) (cv x) (cv y)) (:-1-1-onto-> (cv f) (cv y) (cv z)) eeanv (cv f) (cv y) (cv z) (cv g) (cv x) f1oco ancoms x visset (o. (cv f) (cv g)) (cv z) f1oen syl g f 19.23aivv sylbir y visset (cv x) g bren z visset (cv y) f bren syl2anb ster)) thm (sdomirr () () (-. (br A (~<) A)) (A A sdomnen A (V) enrefg nsyl3 relsdom A A brrelexi con3i pm2.61i)) thm (sdomex () () (-> (br A (~<) B) (/\ (e. A (V)) (e. B (V)))) (relsdom A B brrelexi A sdomirr B A (~<) brprc mtbiri a3i jca)) thm (ensymg ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B C) (-> (br A (~~) B) (br B (~~) A))) (relen A B brrelexi (cv x) A (~~) (cv y) breq1 (cv x) A (cv y) (~~) breq2 imbi12d (cv y) B A (~~) breq2 (cv y) B (~~) A breq1 imbi12d x visset y visset ener ersym (V) C vtocl2g ex syl pm2.43b)) thm (ensym () ((ensym.1 (e. B (V)))) (-> (br A (~~) B) (br B (~~) A)) (ensym.1 B (V) A ensymg ax-mp)) thm (ensymi () ((ensym.1 (e. B (V))) (ensymi.2 (br A (~~) B))) (br B (~~) A) (ensymi.2 ensym.1 A ensym ax-mp)) thm (entrt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (/\ (br A (~~) B) (br B (~~) C)) (br A (~~) C)) (relen x visset y visset z visset ener ertr x visset enref A B C vtoclrbr)) thm (domtr ((x y) (x z) (f x) (g x) (A x) (y z) (f y) (g y) (A y) (f z) (g z) (A z) (f g) (A f) (A g) (B x) (B y) (B z) (B f) (B g) (C x) (C y) (C z) (C f) (C g)) () (-> (/\ (br A (~<_) B) (br B (~<_) C)) (br A (~<_) C)) (reldom g f (:-1-1-> (cv g) (cv x) (cv y)) (:-1-1-> (cv f) (cv y) (cv z)) eeanv (cv f) (cv y) (cv z) (cv g) (cv x) f1co ancoms x visset (o. (cv f) (cv g)) (cv z) f1dom syl g f 19.23aivv sylbir y visset (cv x) g brdom z visset (cv y) f brdom syl2anb x visset (cv x) (V) domrefg ax-mp A B C vtoclrbr)) thm (entr () ((entr.1 (br A (~~) B)) (entr.2 (br B (~~) C))) (br A (~~) C) (entr.1 entr.2 A B C entrt mp2an)) thm (entr2 () ((entr2.1 (e. C (V))) (entr2.2 (br A (~~) B)) (entr2.3 (br B (~~) C))) (br C (~~) A) (entr2.1 entr2.2 entr2.3 entr ensymi)) thm (entr3 () ((entr3.1 (e. B (V))) (entr3.2 (br A (~~) B)) (entr3.3 (br A (~~) C))) (br B (~~) C) (entr3.1 entr3.2 ensymi entr3.3 entr)) thm (entr4 () ((entr4.1 (e. B (V))) (entr4.2 (br A (~~) B)) (entr4.3 (br C (~~) B))) (br A (~~) C) (entr4.2 entr4.1 entr4.3 ensymi entr)) thm (endomtr () () (-> (/\ (br A (~~) B) (br B (~<_) C)) (br A (~<_) C)) (A B C domtr A B endom sylan)) thm (domentr () () (-> (/\ (br A (~<_) B) (br B (~~) C)) (br A (~<_) C)) (A B C domtr B C endom sylan2)) thm (f1imaen () ((f1imaen.1 (e. C (V)))) (-> (/\ (:-1-1-> F A B) (C_ C A)) (br (" F C) (~~) C)) (F A B C f1ores f1imaen.1 (|` F C) (" F C) f1oen (|` F C) C (" F C) f1ofo f1imaen.1 C (V) (|` F C) (" F C) fornex ax-mp (" F C) (V) C ensymg 3syl mpd syl)) thm (en0 ((A f)) () (<-> (br A (~~) ({/})) (= A ({/}))) (0ex A f bren (cv f) A ({/}) f1ocnv (`' (cv f)) A f1o00 pm3.27bd syl f 19.23aiv sylbi 0ex enref A ({/}) (~~) ({/}) breq1 mpbiri impbi)) thm (ensn1 ((A f)) ((ensn1.1 (e. A (V)))) (br ({} A) (~~) (1o)) (ensn1.1 0ex f1osn (<,> A ({/})) snex (cv f) ({} (<,> A ({/}))) ({} A) ({} ({/})) f1oeq1 cla4ev ax-mp p0ex ({} A) f bren mpbir df1o2 breqtrr)) thm (ensn1g ((A x) (B x)) () (-> (e. A B) (br ({} A) (~~) (1o))) ((cv x) A sneq (~~) (1o) breq1d x visset ensn1 B vtoclg)) thm (en1 ((f x) (A x) (A f)) () (<-> (br A (~~) (1o)) (E. x (= A ({} (cv x))))) (df1o2 A (~~) breq2i p0ex A f bren bitr (cv f) A ({} ({/})) f1ocnv (`' (cv f)) ({} ({/})) A f1ofo (`' (cv f)) ({} ({/})) A forn syl (`' (cv f)) ({} ({/})) A f1of 0ex (`' (cv f)) A fsn2 pm3.27bd syl rneqd 0ex (`' (cv f)) ({/}) fvex rnsnop syl6eq eqtr3d (`' (cv f)) ({/}) fvex (cv x) (` (`' (cv f)) ({/})) sneq A eqeq2d cla4ev 3syl f 19.23aiv sylbi x visset ensn1 A ({} (cv x)) (~~) (1o) breq1 mpbiri x 19.23aiv impbi)) thm (2dom ((x y) (f x) (A x) (f y) (A y) (A f)) ((2dom.1 (e. A (V)))) (-> (br (2o) (~<_) A) (E.e. x A (E.e. y A (-. (= (cv x) (cv y)))))) (df2o2 (~<_) A breq1i 2dom.1 ({,} ({/}) ({} ({/}))) f brdom bitr (cv x) (` (cv f) ({/})) (cv y) eqeq1 negbid (cv y) (` (cv f) ({} ({/}))) (` (cv f) ({/})) eqeq2 negbid A A rcla42ev (cv f) ({,} ({/}) ({} ({/}))) A f1f 0ex ({} ({/})) pri1 (cv f) ({,} ({/}) ({} ({/}))) A ({/}) ffvrn mpan2 p0ex ({/}) pri2 (cv f) ({,} ({/}) ({} ({/}))) A ({} ({/})) ffvrn mpan2 jca syl 0nep0 0ex ({} ({/})) pri1 p0ex ({/}) pri2 pm3.2i (cv f) ({,} ({/}) ({} ({/}))) A ({/}) ({} ({/})) f1fveq mpan2 mtbiri sylanc f 19.23aiv sylbi)) thm (fundmen ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w)) ((fundmen.1 (e. F (V)))) (-> (Fun F) (br (dom F) (~~) F)) (fundmen.1 F (V) dmexg ax-mp (Fun F) a1i F (cv x) funfvop ex F funrel F (cv y) elreldm ex syl F (X. (V) (V)) (cv y) ssel2 F funrel F df-rel sylib sylan (cv y) z w elvv sylib (cv x) (|^| (|^| (cv y))) (cv z) eqeq1 (cv y) (<,> (cv z) (cv w)) inteq inteqd z visset (cv w) op1stb syl6eq syl5bir (cv x) (cv z) (cv w) opeq1 syl6 imp (<,> (cv x) (cv w)) (<,> (cv z) (cv w)) (cv y) eqeq2 biimprcd (= (cv x) (|^| (|^| (cv y)))) adantl mpd ancoms (/\ (Fun F) (e. (cv y) F)) adantl (cv x) (|^| (|^| (cv y))) (cv z) eqeq1 (cv y) (<,> (cv z) (cv w)) inteq inteqd z visset (cv w) op1stb syl6eq syl5bir (cv x) (cv z) (cv w) opeq1 syl6 imp (<,> (cv x) (cv w)) (<,> (cv z) (cv w)) (cv y) eqeq2 biimprcd (= (cv x) (|^| (|^| (cv y)))) adantl mpd F eleq1d (Fun F) adantl w visset F (cv x) funopfv (/\ (= (cv x) (|^| (|^| (cv y)))) (= (cv y) (<,> (cv z) (cv w)))) adantr sylbid exp32 com24 imp43 (` F (cv x)) (cv w) (cv x) opeq2 syl eqtr4d exp32 z w 19.23advv mpd (e. (cv x) (dom F)) adantrl (cv y) (<,> (cv x) (` F (cv x))) inteq inteqd x visset (` F (cv x)) op1stb syl6req (/\ (Fun F) (/\ (e. (cv x) (dom F)) (e. (cv y) F))) a1i impbid ex en3d)) thm (mapsnen ((y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w)) ((mapsnen.1 (e. A (V))) (mapsnen.2 (e. B (V)))) (br (opr A (^m) ({} B)) (~~) A) (A (^m) ({} B) oprex (cv z) B fvex (e. (cv z) (opr A (^m) ({} B))) a1i (<,> B (cv w)) snex (e. (cv w) A) a1i mapsnen.1 mapsnen.2 z y mapsn abeq2i (= (cv w) (` (cv z) B)) anbi1i y A (= (cv z) ({} (<,> B (cv y)))) (= (cv w) (` (cv z) B)) r19.41v y A (/\ (= (cv z) ({} (<,> B (cv y)))) (= (cv w) (` (cv z) B))) df-rex 3bitr2 (cv z) ({} (<,> B (cv y))) B fveq1 mapsnen.2 y visset fvsn syl6eq (cv w) eqeq2d w y equcom syl6bb pm5.32i (e. (cv y) A) anbi2i (e. (cv y) A) (= (cv z) ({} (<,> B (cv y)))) (= (cv y) (cv w)) anass (/\ (e. (cv y) A) (= (cv z) ({} (<,> B (cv y))))) (= (cv y) (cv w)) ancom 3bitr2 y exbii w visset (cv y) (cv w) A eleq1 (cv y) (cv w) B opeq2 sneqd (cv z) eqeq2d anbi12d ceqsexv 3bitr en2)) thm (map1 ((x y) (A x) (A y)) ((map1.1 (e. A (V)))) (br (opr (1o) (^m) A) (~~) (1o)) ((1o) (^m) A oprex 0ex (e. (cv x) (opr (1o) (^m) A)) a1i map1.1 p0ex xpex (e. (cv y) (1o)) a1i (e. (cv y) (1o)) (= (cv x) (X. A ({} ({/})))) ancom df1o2 (^m) A opreq1i (cv x) eleq2i p0ex map1.1 (cv x) elmap 0ex (cv x) A fconst2 3bitrr (cv y) el1o anbi12i bitr2 en2)) thm (en2sn () () (-> (/\ (e. A C) (e. B D)) (br ({} A) (~~) ({} B))) (({} A) (1o) ({} B) entrt A C ensn1g B D ensn1g 1on (1o) (On) ({} B) ensymg ax-mp syl syl2an)) thm (snfi ((A x)) () (E.e. x (om) (br ({} A) (~~) (cv x))) (A (V) ensn1g 1onn (cv x) (1o) ({} A) (~~) breq2 (om) rcla4ev mpan syl A snprc ({} A) en0 peano1 (cv x) ({/}) ({} A) (~~) breq2 (om) rcla4ev mpan sylbir sylbi pm2.61i)) thm (unen ((f g) (f h) (A f) (g h) (A g) (A h) (B f) (B g) (B h) (C f) (C g) (C h) (D f) (D g) (D h)) () (-> (/\ (/\ (br A (~~) B) (br C (~~) D)) (/\ (= (i^i A C) ({/})) (= (i^i B D) ({/})))) (br (u. A C) (~~) (u. B D))) (B D unexb B (V) A f breng D (V) C g breng bi2anan9 sylbir (u. B D) (V) (u. A C) h breng (cv f) A B (cv g) C D f1oun f visset g visset unex (cv h) (u. (cv f) (cv g)) (u. A C) (u. B D) f1oeq1 cla4ev syl syl5bir exp3a f g 19.23advv f g (:-1-1-onto-> (cv f) A B) (:-1-1-onto-> (cv g) C D) eeanv syl5ibr sylbid imp3a (u. B D) (u. A C) (~~) brprc relen A B brrelexi relen C D brrelexi anim12i A C unexb sylib (u. A C) (V) enrefg syl syl5bir (/\ (= (i^i A C) ({/})) (= (i^i B D) ({/}))) adantrd pm2.61i)) thm (xpsnen ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((xpsnen.1 (e. A (V))) (xpsnen.2 (e. B (V)))) (br (X. A ({} B)) (~~) A) (xpsnen.1 B snex xpex (cv y) A ({} B) x z elxp (cv y) (<,> (cv x) (cv z)) inteq inteqd x visset (cv z) op1stb syl6eq x visset syl6eqel (/\ (e. (cv x) A) (e. (cv z) ({} B))) adantr x z 19.23aivv sylbi (cv x) B opex (e. (cv x) A) a1i x visset (cv x) (|^| (|^| (cv y))) (V) eleq1 mpbii (cv x) (|^| (|^| (cv y))) B opeq1 (cv y) eqeq2d (cv x) (|^| (|^| (cv y))) A eleq1 anbi12d (V) ceqsexgv (cv y) A ({} B) x z elxp (/\ (= (cv y) (<,> (cv x) (cv z))) (e. (cv x) A)) (e. (cv z) ({} B)) ancom (= (cv y) (<,> (cv x) (cv z))) (e. (cv x) A) (e. (cv z) ({} B)) anass z B elsn (/\ (= (cv y) (<,> (cv x) (cv z))) (e. (cv x) A)) anbi1i 3bitr3 z exbii xpsnen.2 (cv z) B (cv x) opeq2 (cv y) eqeq2d (e. (cv x) A) anbi1d ceqsexv (cv y) (<,> (cv x) B) inteq inteqd x visset B op1stb syl6req pm4.71ri (e. (cv x) A) anbi1i (= (cv x) (|^| (|^| (cv y)))) (= (cv y) (<,> (cv x) B)) (e. (cv x) A) anass bitr 3bitr x exbii bitr syl5bb syl pm5.32ri (cv y) (<,> (cv x) B) inteq inteqd x visset B op1stb syl6req (e. (cv x) A) adantr pm4.71i (cv x) (|^| (|^| (cv y))) B opeq1 (cv y) eqeq2d (cv x) (|^| (|^| (cv y))) A eleq1 anbi12d pm5.32ri bitr2 (= (cv y) (<,> (cv x) B)) (e. (cv x) A) ancom 3bitr en2)) thm (xpsneng ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A C) (e. B D)) (br (X. A ({} B)) (~~) A)) ((cv x) A ({} (cv y)) xpeq1 (= (cv x) A) id (~~) breq12d (cv y) B sneq ({} (cv y)) ({} B) A xpeq2 syl (~~) A breq1d x visset y visset xpsnen C D vtocl2g)) thm (endisj ((x y) (A x) (A y) (B x) (B y)) ((endisj.1 (e. A (V))) (endisj.2 (e. B (V)))) (E. x (E. y (/\ (/\ (br (cv x) (~~) A) (br (cv y) (~~) B)) (= (i^i (cv x) (cv y)) ({/}))))) (endisj.1 0ex xpsnen endisj.2 p0ex xpsnen pm3.2i 0nep0 ({/}) ({} ({/})) A B xpsndisj ax-mp endisj.1 p0ex xpex endisj.2 ({} ({/})) snex xpex (cv x) (X. A ({} ({/}))) (~~) A breq1 (cv y) (X. B ({} ({} ({/})))) (~~) B breq1 bi2anan9 (cv x) (X. A ({} ({/}))) (cv y) (X. B ({} ({} ({/})))) ineq12 ({/}) eqeq1d anbi12d cla4e2v mp2an)) thm (undom ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) ((undom.1 (e. B (V))) (undom.2 (e. C (V))) (undom.3 (e. D (V)))) (-> (/\ (/\ (br A (~<_) B) (br C (~<_) D)) (= (i^i B D) ({/}))) (br (u. A C) (~<_) (u. B D))) ((u. A C) (u. (cv x) (cv y)) (u. B D) endomtr A (cv x) (\ C A) (cv y) unen A C undif2 syl5eqbrr (i^i B D) ({/}) (i^i (cv x) (cv y)) sseq2 (i^i (cv x) (cv y)) ss0b syl6bb (cv x) B (cv y) D ss2in syl5bi imp A C difdisj jctil sylan2 anassrs an1rs (cv x) B (cv y) D unss12 undom.1 undom.3 unex (u. B D) (V) (u. (cv x) (cv y)) ssdom2g ax-mp syl (/\ (br A (~~) (cv x)) (br (\ C A) (~~) (cv y))) (= (i^i B D) ({/})) ad2antlr sylanc ex an4s ex x 19.23aiv y 19.23adv imp undom.1 A x domen undom.3 (\ C A) y domen syl2anb undom.2 C A difss C (V) (\ C A) ssdom2g mp2 (\ C A) C D domtr mpan sylan2 imp)) thm (xpcomen ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((xpcomen.1 (e. A (V))) (xpcomen.2 (e. B (V)))) (br (X. A B) (~~) (X. B A)) (xpcomen.1 xpcomen.2 xpex (cv x) snex cnvex uniex (e. (cv x) (X. A B)) a1i (cv y) snex cnvex uniex (e. (cv y) (X. B A)) a1i (cv x) (<,> (cv z) (cv w)) sneq ({} (cv x)) ({} (<,> (cv z) (cv w))) cnveq syl z visset w visset cnvsn syl6eq unieqd (cv w) (cv z) opex unisn syl6req (cv y) (<,> (cv w) (cv z)) sneq ({} (cv y)) ({} (<,> (cv w) (cv z))) cnveq syl w visset z visset cnvsn syl6eq unieqd (cv z) (cv w) opex unisn syl6req eq2tr (e. (cv z) A) (e. (cv w) B) ancom anbi12i (= (cv x) (<,> (cv z) (cv w))) (/\ (e. (cv z) A) (e. (cv w) B)) (= (cv y) (U. (`' ({} (cv x))))) an23 (= (cv y) (<,> (cv w) (cv z))) (/\ (e. (cv w) B) (e. (cv z) A)) (= (cv x) (U. (`' ({} (cv y))))) an23 3bitr4 z w 2exbii z w (/\ (= (cv x) (<,> (cv z) (cv w))) (/\ (e. (cv z) A) (e. (cv w) B))) (= (cv y) (U. (`' ({} (cv x))))) 19.41vv z w (/\ (= (cv y) (<,> (cv w) (cv z))) (/\ (e. (cv w) B) (e. (cv z) A))) (= (cv x) (U. (`' ({} (cv y))))) 19.41vv 3bitr3 (cv x) A B z w elxp (= (cv y) (U. (`' ({} (cv x))))) anbi1i (cv y) B A w z elxp w z (/\ (= (cv y) (<,> (cv w) (cv z))) (/\ (e. (cv w) B) (e. (cv z) A))) excom bitr (= (cv x) (U. (`' ({} (cv y))))) anbi1i 3bitr4 en2)) thm (xpcomeng ((x y) (A x) (A y) (B y)) () (-> (/\ (e. A C) (e. B D)) (br (X. A B) (~~) (X. B A))) ((cv x) A (cv y) xpeq1 (cv x) A (cv y) xpeq2 (~~) breq12d (cv y) B A xpeq2 (cv y) B A xpeq1 (~~) breq12d x visset y visset xpcomen C D vtocl2g)) thm (xpassen ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((xpassen.1 (e. A (V))) (xpassen.2 (e. B (V))) (xpassen.3 (e. C (V)))) (br (X. (X. A B) C) (~~) (X. A (X. B C))) (xpassen.1 xpassen.2 xpex xpassen.3 xpex (U. (dom ({} (U. (dom ({} (cv x))))))) (<,> (U. (ran ({} (U. (dom ({} (cv x))))))) (U. (ran ({} (cv x))))) opex (e. (cv x) (X. (X. A B) C)) a1i (<,> (U. (dom ({} (cv y)))) (U. (dom ({} (U. (ran ({} (cv y)))))))) (U. (ran ({} (U. (ran ({} (cv y))))))) opex (e. (cv y) (X. A (X. B C))) a1i (cv z) (U. (dom ({} (U. (dom ({} (cv x))))))) (<,> (cv w) (cv v)) (<,> (U. (ran ({} (U. (dom ({} (cv x))))))) (U. (ran ({} (cv x))))) opeq12 (cv x) (<,> (<,> (cv z) (cv w)) (cv v)) sneq dmeqd unieqd sneqd dmeqd unieqd (cv z) (cv w) opex (cv v) op1sta sneqi dmeqi unieqi z visset (cv w) op1sta eqtr syl6req (cv w) (U. (ran ({} (U. (dom ({} (cv x))))))) (cv v) (U. (ran ({} (cv x)))) opeq12 (cv x) (<,> (<,> (cv z) (cv w)) (cv v)) sneq dmeqd unieqd sneqd rneqd unieqd (cv z) (cv w) opex (cv v) op1sta sneqi rneqi unieqi z visset w visset op2nda eqtr syl6req (cv x) (<,> (<,> (cv z) (cv w)) (cv v)) sneq rneqd unieqd (cv z) (cv w) opex v visset op2nda syl6req sylanc sylanc (<,> (cv z) (cv w)) (<,> (U. (dom ({} (cv y)))) (U. (dom ({} (U. (ran ({} (cv y)))))))) (cv v) (U. (ran ({} (U. (ran ({} (cv y))))))) opeq12 (cv z) (U. (dom ({} (cv y)))) (cv w) (U. (dom ({} (U. (ran ({} (cv y))))))) opeq12 (cv y) (<,> (cv z) (<,> (cv w) (cv v))) sneq dmeqd unieqd z visset (<,> (cv w) (cv v)) op1sta syl6req (cv y) (<,> (cv z) (<,> (cv w) (cv v))) sneq rneqd unieqd sneqd dmeqd unieqd z visset (cv w) (cv v) opex op2nda sneqi dmeqi unieqi w visset (cv v) op1sta eqtr syl6req sylanc (cv y) (<,> (cv z) (<,> (cv w) (cv v))) sneq rneqd unieqd sneqd rneqd unieqd z visset (cv w) (cv v) opex op2nda sneqi rneqi unieqi w visset v visset op2nda eqtr syl6req sylanc eq2tr (e. (cv z) A) (e. (cv w) B) (e. (cv v) C) anass anbi12i (= (cv x) (<,> (<,> (cv z) (cv w)) (cv v))) (/\ (/\ (e. (cv z) A) (e. (cv w) B)) (e. (cv v) C)) (= (cv y) (<,> (U. (dom ({} (U. (dom ({} (cv x))))))) (<,> (U. (ran ({} (U. (dom ({} (cv x))))))) (U. (ran ({} (cv x))))))) an23 (= (cv y) (<,> (cv z) (<,> (cv w) (cv v)))) (/\ (e. (cv z) A) (/\ (e. (cv w) B) (e. (cv v) C))) (= (cv x) (<,> (<,> (U. (dom ({} (cv y)))) (U. (dom ({} (U. (ran ({} (cv y)))))))) (U. (ran ({} (U. (ran ({} (cv y))))))))) an23 3bitr4 v exbii v (/\ (= (cv x) (<,> (<,> (cv z) (cv w)) (cv v))) (/\ (/\ (e. (cv z) A) (e. (cv w) B)) (e. (cv v) C))) (= (cv y) (<,> (U. (dom ({} (U. (dom ({} (cv x))))))) (<,> (U. (ran ({} (U. (dom ({} (cv x))))))) (U. (ran ({} (cv x))))))) 19.41v v (/\ (= (cv y) (<,> (cv z) (<,> (cv w) (cv v)))) (/\ (e. (cv z) A) (/\ (e. (cv w) B) (e. (cv v) C)))) (= (cv x) (<,> (<,> (U. (dom ({} (cv y)))) (U. (dom ({} (U. (ran ({} (cv y)))))))) (U. (ran ({} (U. (ran ({} (cv y))))))))) 19.41v 3bitr3 z w 2exbii z w (E. v (/\ (= (cv x) (<,> (<,> (cv z) (cv w)) (cv v))) (/\ (/\ (e. (cv z) A) (e. (cv w) B)) (e. (cv v) C)))) (= (cv y) (<,> (U. (dom ({} (U. (dom ({} (cv x))))))) (<,> (U. (ran ({} (U. (dom ({} (cv x))))))) (U. (ran ({} (cv x))))))) 19.41vv z w (E. v (/\ (= (cv y) (<,> (cv z) (<,> (cv w) (cv v)))) (/\ (e. (cv z) A) (/\ (e. (cv w) B) (e. (cv v) C))))) (= (cv x) (<,> (<,> (U. (dom ({} (cv y)))) (U. (dom ({} (U. (ran ({} (cv y)))))))) (U. (ran ({} (U. (ran ({} (cv y))))))))) 19.41vv 3bitr3 (cv x) (X. A B) C u v elxp u v (/\ (= (cv x) (<,> (cv u) (cv v))) (/\ (e. (cv u) (X. A B)) (e. (cv v) C))) excom (cv u) A B z w elxp (/\ (= (cv x) (<,> (cv u) (cv v))) (e. (cv v) C)) anbi1i (= (cv x) (<,> (cv u) (cv v))) (e. (cv u) (X. A B)) (e. (cv v) C) an12 z w (/\ (= (cv u) (<,> (cv z) (cv w))) (/\ (e. (cv z) A) (e. (cv w) B))) (/\ (= (cv x) (<,> (cv u) (cv v))) (e. (cv v) C)) 19.41vv 3bitr4 v u 2exbii v u z w (/\ (/\ (= (cv u) (<,> (cv z) (cv w))) (/\ (e. (cv z) A) (e. (cv w) B))) (/\ (= (cv x) (<,> (cv u) (cv v))) (e. (cv v) C))) exrot4 (= (cv u) (<,> (cv z) (cv w))) (/\ (e. (cv z) A) (e. (cv w) B)) (/\ (= (cv x) (<,> (cv u) (cv v))) (e. (cv v) C)) anass u exbii (cv z) (cv w) opex (cv u) (<,> (cv z) (cv w)) (cv v) opeq1 (cv x) eqeq2d (e. (cv v) C) anbi1d (/\ (e. (cv z) A) (e. (cv w) B)) anbi2d ceqsexv (/\ (e. (cv z) A) (e. (cv w) B)) (= (cv x) (<,> (<,> (cv z) (cv w)) (cv v))) (e. (cv v) C) an12 3bitr z w v 3exbi 3bitr 3bitr (= (cv y) (<,> (U. (dom ({} (U. (dom ({} (cv x))))))) (<,> (U. (ran ({} (U. (dom ({} (cv x))))))) (U. (ran ({} (cv x))))))) anbi1i (cv y) A (X. B C) z u elxp (cv u) B C w v elxp (/\ (= (cv y) (<,> (cv z) (cv u))) (e. (cv z) A)) anbi2i (= (cv y) (<,> (cv z) (cv u))) (e. (cv z) A) (e. (cv u) (X. B C)) anass w v (/\ (= (cv y) (<,> (cv z) (cv u))) (e. (cv z) A)) (/\ (= (cv u) (<,> (cv w) (cv v))) (/\ (e. (cv w) B) (e. (cv v) C))) 19.42vv (/\ (= (cv y) (<,> (cv z) (cv u))) (e. (cv z) A)) (= (cv u) (<,> (cv w) (cv v))) (/\ (e. (cv w) B) (e. (cv v) C)) an12 (= (cv y) (<,> (cv z) (cv u))) (e. (cv z) A) (/\ (e. (cv w) B) (e. (cv v) C)) anass (= (cv u) (<,> (cv w) (cv v))) anbi2i bitr w v 2exbii bitr3 3bitr3 u exbii u w v (/\ (= (cv u) (<,> (cv w) (cv v))) (/\ (= (cv y) (<,> (cv z) (cv u))) (/\ (e. (cv z) A) (/\ (e. (cv w) B) (e. (cv v) C))))) exrot3 (cv w) (cv v) opex (cv u) (<,> (cv w) (cv v)) (cv z) opeq2 (cv y) eqeq2d (/\ (e. (cv z) A) (/\ (e. (cv w) B) (e. (cv v) C))) anbi1d ceqsexv w v 2exbii 3bitr z exbii bitr (= (cv x) (<,> (<,> (U. (dom ({} (cv y)))) (U. (dom ({} (U. (ran ({} (cv y)))))))) (U. (ran ({} (U. (ran ({} (cv y))))))))) anbi1i 3bitr4 en2)) thm (xpdom2 ((A t) (t x) (t y) (t z) (t w) (t v) (t u) (f t) (B t) (x y) (x z) (w x) (v x) (u x) (f x) (B x) (y z) (w y) (v y) (u y) (f y) (B y) (w z) (v z) (u z) (f z) (B z) (v w) (u w) (f w) (B w) (u v) (f v) (B v) (f u) (B u) (B f) (C t) (C x) (C y) (C z) (C w) (C v) (C u) (C f)) ((xpdom.1 (e. B (V))) (xpdom.2 (e. C (V)))) (-> (br A (~<_) B) (br (X. C A) (~<_) (X. C B))) (reldom A B brrelexi (cv t) A (~<_) B breq1 (cv t) A C xpeq2 (~<_) (X. C B) breq1d imbi12d xpdom.1 (cv t) f brdom xpdom.2 t visset xpex (cv f) (cv t) B f1f (cv f) (cv t) B (U. (ran ({} (cv x)))) ffvrn ex syl (e. (U. (dom ({} (cv x)))) C) anim2d (= (cv x) (<,> (U. (dom ({} (cv x)))) (U. (ran ({} (cv x)))))) adantld (cv x) C (cv t) elxp4 (cv f) (U. (ran ({} (cv x)))) fvex (U. (dom ({} (cv x)))) C B opelxp 3imtr4g (cv f) (cv t) B (cv w) (cv u) f1fveq ancoms (= (cv z) (cv v)) anbi2d z visset (cv f) (cv w) fvex (cv f) (cv u) fvex (cv v) opth syl5bb ex (e. (cv z) C) (e. (cv v) C) ad2ant2l imp (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) adantlr (cv x) (<,> (cv z) (cv w)) sneq dmeqd unieqd z visset (cv w) op1sta syl6eq (cv x) (<,> (cv z) (cv w)) sneq rneqd unieqd z visset w visset op2nda syl6eq (cv f) fveq2d jca (cv x) snex ({} (cv x)) (V) dmexg ax-mp uniex (cv f) (U. (ran ({} (cv x)))) fvex (cv f) (cv w) fvex (cv z) opth sylibr (cv y) (<,> (cv v) (cv u)) sneq dmeqd unieqd v visset (cv u) op1sta syl6eq (cv y) (<,> (cv v) (cv u)) sneq rneqd unieqd v visset u visset op2nda syl6eq (cv f) fveq2d jca (cv y) snex ({} (cv y)) (V) dmexg ax-mp uniex (cv f) (U. (ran ({} (cv y)))) fvex (cv f) (cv u) fvex (cv v) opth sylibr eqeqan12d (/\ (/\ (e. (cv z) C) (e. (cv w) (cv t))) (/\ (e. (cv v) C) (e. (cv u) (cv t)))) (:-1-1-> (cv f) (cv t) B) ad2antlr (cv x) (<,> (cv z) (cv w)) (cv y) (<,> (cv v) (cv u)) eqeq12 z visset w visset u visset (cv v) opth syl6bb (/\ (/\ (e. (cv z) C) (e. (cv w) (cv t))) (/\ (e. (cv v) C) (e. (cv u) (cv t)))) (:-1-1-> (cv f) (cv t) B) ad2antlr 3bitr4d ex exp43 com23 r19.23aivv r19.23advv imp (cv x) C (cv t) z w elxp2 (cv y) C (cv t) v u elxp2 syl2anb com12 (V) dom2d mpi f 19.23aiv sylbi (V) vtoclg mpcom)) thm (xpdom1 () ((xpdom.1 (e. B (V))) (xpdom.2 (e. C (V)))) (-> (br A (~<_) B) (br (X. A C) (~<_) (X. B C))) ((X. A C) (X. C A) (X. C B) endomtr reldom A B brrelexi xpdom.2 A (V) C (V) xpcomeng mpan2 syl xpdom.1 xpdom.2 A xpdom2 sylanc xpdom.2 xpdom.1 xpcomen (X. A C) (X. C B) (X. B C) domentr mpan2 syl)) thm (xpdom1g ((y z) (A y) (A z) (B y) (B z) (C z)) () (-> (/\/\ (e. B R) (e. C S) (br A (~<_) B)) (br (X. A C) (~<_) (X. B C))) ((cv y) B A (~<_) breq2 (cv y) B (cv z) xpeq1 (X. A (cv z)) (~<_) breq2d imbi12d (cv z) C A xpeq2 (cv z) C B xpeq2 (~<_) breq12d (br A (~<_) B) imbi2d y visset z visset A xpdom1 R S vtocl2g 3impia)) thm (xpdom3 ((A x) (B x)) ((xpdom3.1 (e. A (V)))) (-> (-. (= B ({/}))) (br A (~<_) (X. A B))) (B x n0 x visset B snss A ssid A A ({} (cv x)) B ssxp mpan xpdom3.1 (cv x) snex xpex (X. A ({} (cv x))) (V) (X. A B) ssdomg ax-mp xpdom3.1 xpdom3.1 x visset xpsnen ensymi A (X. A ({} (cv x))) (X. A B) endomtr mpan 3syl sylbi x 19.23aiv sylbi)) thm (pw2en ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u)) ((pw2en.1 (e. A (V)))) (br (P~ A) (~~) (opr (2o) (^m) A)) (pw2en.1 pwex pw2en.1 z w ({|} v (/\ (= (cv v) ({/})) (e. (cv z) (cv x)))) funopabex2 (e. (cv x) (P~ A)) a1i y visset cnvex (`' (cv y)) (V) ({} ({} ({/}))) imaexg ax-mp (e. (cv y) (opr (2o) (^m) A)) a1i (cv x) A sseqin2 biimp (" (`' (cv y)) ({} ({} ({/})))) eqeq1d (cv y) ({<,>|} z w (/\ (e. (cv z) A) (= (cv w) ({|} v (/\ (= (cv v) ({/})) (e. (cv z) (cv x))))))) (<,> (cv u) ({} ({/}))) eleq2 p0ex u visset (`' (cv y)) elimasn p0ex u visset (cv y) opelcnv bitr syl5bb v (= (cv v) ({/})) (/\ (= (cv v) ({/})) (e. (cv u) (cv x))) eq2ab ({/}) v df-sn ({|} v (/\ (= (cv v) ({/})) (e. (cv u) (cv x)))) eqeq1i (e. (cv u) (cv x)) (= (cv v) ({/})) iba v 19.21aiv 0ex (cv v) ({/}) ({/}) eqeq1 (cv v) ({/}) ({/}) eqeq1 (e. (cv u) (cv x)) anbi1d bibi12d cla4v ({/}) eqid (e. (cv u) (cv x)) a1bi (= ({/}) ({/})) (e. (cv u) (cv x)) pm4.71 bitr sylibr impbi 3bitr4r (e. (cv u) A) anbi2i (cv u) A (cv x) elin u visset p0ex (cv z) (cv u) A eleq1 (cv z) (cv u) (cv x) eleq1 (= (cv v) ({/})) anbi2d v abbidv (cv w) eqeq2d anbi12d (cv w) ({} ({/})) ({|} v (/\ (= (cv v) ({/})) (e. (cv u) (cv x)))) eqeq1 (e. (cv u) A) anbi2d opelopab 3bitr4 syl6rbbr eqrdv syl5bi com12 (cv x) (" (`' (cv y)) ({} ({} ({/})))) A sseq1 (`' (cv y)) ({} ({} ({/}))) imassrn ({/}) v df-sn p0ex eqeltrr (= (cv v) ({/})) (e. (cv z) (cv x)) pm3.26 v ss2abi ssexi ({<,>|} z w (/\ (e. (cv z) A) (= (cv w) ({|} v (/\ (= (cv v) ({/})) (e. (cv z) (cv x))))))) eqid fnopab2 (cv y) ({<,>|} z w (/\ (e. (cv z) A) (= (cv w) ({|} v (/\ (= (cv v) ({/})) (e. (cv z) (cv x))))))) A fneq1 mpbiri (cv y) A fndm syl (cv y) dfdm4 syl5eqr (" (`' (cv y)) ({} ({} ({/})))) sseq2d mpbii syl5bir com12 impbid x visset A elpw syl5bb pm5.32ri (= (cv x) (" (`' (cv y)) ({} ({} ({/}))))) (= (cv y) ({<,>|} z w (/\ (e. (cv z) A) (= (cv w) ({|} v (/\ (= (cv v) ({/})) (e. (cv z) (cv x)))))))) ancom ({<,>|} z w (/\ (e. (cv z) A) (= (cv w) ({|} v (/\ (= (cv v) ({/})) (e. (cv z) (cv x))))))) eqid (e. (cv z) (cv x)) (= (cv v) ({/})) iba v ({/}) elsn syl5bb abbi2dv p0ex ({/}) pri2 df2o2 eleqtrr syl6eqelr v (/\ (= (cv v) ({/})) (e. (cv z) (cv x))) abn0 (= (cv v) ({/})) (e. (cv z) (cv x)) pm3.27 v 19.23aiv sylbi con1i 0ex ({} ({/})) pri1 df2o2 eleqtrr syl6eqel pm2.61i (e. (cv z) A) a1i fopab (cv y) ({<,>|} z w (/\ (e. (cv z) A) (= (cv w) ({|} v (/\ (= (cv v) ({/})) (e. (cv z) (cv x))))))) A (2o) feq1 mpbiri (= (cv x) (" (`' (cv y)) ({} ({} ({/}))))) a1i (cv x) (" (`' (cv y)) ({} ({} ({/})))) (cv z) eleq2 p0ex z visset (`' (cv y)) elimasn p0ex z visset (cv y) opelcnv bitr syl6bb p0ex (cv y) A (cv z) fnopfvb (cv y) A (2o) ffn sylan bicomd sylan9bb biimpa (e. (cv z) (cv x)) (= (cv v) ({/})) iba v ({/}) elsn syl5bb abbi2dv (/\ (= (cv x) (" (`' (cv y)) ({} ({} ({/}))))) (/\ (:--> (cv y) A (2o)) (e. (cv z) A))) adantl eqtrd (cv y) A (2o) (cv z) ffvrn df2o2 (` (cv y) (cv z)) eleq2i (cv y) (cv z) fvex ({/}) ({} ({/})) elpr bitr sylib ord (= (cv x) (" (`' (cv y)) ({} ({} ({/}))))) adantl (cv x) (" (`' (cv y)) ({} ({} ({/})))) (cv z) eleq2 p0ex z visset (`' (cv y)) elimasn p0ex z visset (cv y) opelcnv bitr syl6bb p0ex (cv y) A (cv z) fnopfvb (cv y) A (2o) ffn sylan bicomd sylan9bb sylibrd con1d imp v (/\ (= (cv v) ({/})) (e. (cv z) (cv x))) abn0 (= (cv v) ({/})) (e. (cv z) (cv x)) pm3.27 v 19.23aiv sylbi con1i (/\ (= (cv x) (" (`' (cv y)) ({} ({} ({/}))))) (/\ (:--> (cv y) A (2o)) (e. (cv z) A))) adantl eqtr4d pm2.61dan ({/}) v df-sn p0ex eqeltrr (= (cv v) ({/})) (e. (cv z) (cv x)) pm3.26 v ss2abi ssexi z A ({|} v (/\ (= (cv v) ({/})) (e. (cv z) (cv x)))) (V) w fvopab2 mpan2 (= (cv x) (" (`' (cv y)) ({} ({} ({/}))))) (:--> (cv y) A (2o)) ad2antll eqtr4d exp32 imp r19.21aiv (cv y) A (2o) ffn ({/}) v df-sn p0ex eqeltrr (= (cv v) ({/})) (e. (cv z) (cv x)) pm3.26 v ss2abi ssexi ({<,>|} z w (/\ (e. (cv z) A) (= (cv w) ({|} v (/\ (= (cv v) ({/})) (e. (cv z) (cv x))))))) eqid fnopab2 (e. (cv u) (cv y)) z ax-17 u z w (/\ (e. (cv z) A) (= (cv w) ({|} v (/\ (= (cv v) ({/})) (e. (cv z) (cv x)))))) hbopab1 A A eqfnfvf mpan2 A eqid (A.e. z A (= (` (cv y) (cv z)) (` ({<,>|} z w (/\ (e. (cv z) A) (= (cv w) ({|} v (/\ (= (cv v) ({/})) (e. (cv z) (cv x))))))) (cv z)))) biantrur syl6bbr syl (= (cv x) (" (`' (cv y)) ({} ({} ({/}))))) adantl mpbird ex impbid 2on elisseti pw2en.1 (cv y) elmap syl6bbr pm5.32ri 3bitr en2)) thm (sbthlem1 ((A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))))) (C_ (U. D) (\ A (" (cv g) (\ B (" (cv f) (U. D)))))) (D (\ A (" (cv g) (\ B (" (cv f) (U. D))))) x unissb sbthlem.2 abeq2i (cv x) A (" (cv g) (\ B (" (cv f) (cv x)))) ssconb biimprd ex A (cv x) difss (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)) A sstr2 mpi syl5 pm2.43d imp sylbi (cv x) D elssuni (cv x) (U. D) (cv f) imass2 (" (cv f) (cv x)) (" (cv f) (U. D)) B sscon 3syl (\ B (" (cv f) (U. D))) (\ B (" (cv f) (cv x))) (cv g) imass2 (" (cv g) (\ B (" (cv f) (U. D)))) (" (cv g) (\ B (" (cv f) (cv x)))) A sscon 3syl sstrd mprgbir)) thm (sbthlem2 ((A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))))) (-> (C_ (ran (cv g)) A) (C_ (\ A (" (cv g) (\ B (" (cv f) (U. D))))) (U. D))) (sbthlem.1 sbthlem.2 sbthlem1 (U. D) (\ A (" (cv g) (\ B (" (cv f) (U. D))))) (cv f) imass2 ax-mp (" (cv f) (U. D)) (" (cv f) (\ A (" (cv g) (\ B (" (cv f) (U. D)))))) B sscon ax-mp (\ B (" (cv f) (\ A (" (cv g) (\ B (" (cv f) (U. D))))))) (\ B (" (cv f) (U. D))) (cv g) imass2 ax-mp (" (cv g) (\ B (" (cv f) (\ A (" (cv g) (\ B (" (cv f) (U. D)))))))) (" (cv g) (\ B (" (cv f) (U. D)))) A sscon ax-mp (cv g) (\ B (" (cv f) (\ A (" (cv g) (\ B (" (cv f) (U. D))))))) imassrn (" (cv g) (\ B (" (cv f) (\ A (" (cv g) (\ B (" (cv f) (U. D)))))))) (ran (cv g)) A sstr2 ax-mp A (" (cv g) (\ B (" (cv f) (U. D)))) difss (" (cv g) (\ B (" (cv f) (\ A (" (cv g) (\ B (" (cv f) (U. D)))))))) A (\ A (" (cv g) (\ B (" (cv f) (U. D))))) ssconb mpan2 syl mpbiri A (" (cv g) (\ B (" (cv f) (U. D)))) difss jctil sbthlem.1 A (" (cv g) (\ B (" (cv f) (U. D)))) difss ssexi (cv x) (\ A (" (cv g) (\ B (" (cv f) (U. D))))) A sseq1 (cv x) (\ A (" (cv g) (\ B (" (cv f) (U. D))))) A difeq2 (" (cv g) (\ B (" (cv f) (cv x)))) sseq2d (cv x) (\ A (" (cv g) (\ B (" (cv f) (U. D))))) (cv f) imaeq2 B difeq2d (\ B (" (cv f) (cv x))) (\ B (" (cv f) (\ A (" (cv g) (\ B (" (cv f) (U. D))))))) (cv g) imaeq2 (" (cv g) (\ B (" (cv f) (cv x)))) (" (cv g) (\ B (" (cv f) (\ A (" (cv g) (\ B (" (cv f) (U. D)))))))) (\ A (\ A (" (cv g) (\ B (" (cv f) (U. D)))))) sseq1 3syl bitrd anbi12d elab sylibr sbthlem.2 syl6eleqr (\ A (" (cv g) (\ B (" (cv f) (U. D))))) D elssuni syl)) thm (sbthlem3 ((A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))))) (-> (C_ (ran (cv g)) A) (= (" (cv g) (\ B (" (cv f) (U. D)))) (\ A (U. D)))) (sbthlem.1 sbthlem.2 sbthlem2 sbthlem.1 sbthlem.2 sbthlem1 jctil (U. D) (\ A (" (cv g) (\ B (" (cv f) (U. D))))) eqss sylibr A difeq2d (cv g) (\ B (" (cv f) (U. D))) imassrn (" (cv g) (\ B (" (cv f) (U. D)))) (ran (cv g)) A sstr2 ax-mp (" (cv g) (\ B (" (cv f) (U. D)))) A dfss4 sylib eqtr2d)) thm (sbthlem4 ((A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))))) (-> (/\ (/\ (= (dom (cv g)) B) (C_ (ran (cv g)) A)) (Fun (`' (cv g)))) (= (" (`' (cv g)) (\ A (U. D))) (\ B (" (cv f) (U. D))))) (B (" (cv f) (U. D)) difss (dom (cv g)) B (\ B (" (cv f) (U. D))) sseq2 mpbiri (\ B (" (cv f) (U. D))) (cv g) ssdmres sylib (|` (cv g) (\ B (" (cv f) (U. D)))) dfdm4 syl5reqr (cv g) (\ B (" (cv f) (U. D))) funcnvres sbthlem.1 sbthlem.2 sbthlem3 (" (cv g) (\ B (" (cv f) (U. D)))) (\ A (U. D)) (`' (cv g)) reseq2 syl sylan9eqr rneqd sylan9eq anassrs (`' (cv g)) (\ A (U. D)) df-ima syl6reqr)) thm (sbthlem5 ((H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))))))) (-> (/\ (= (dom (cv f)) A) (C_ (ran (cv g)) A)) (= (dom H) A)) (sbthlem.1 sbthlem.2 sbthlem1 A (" (cv g) (\ B (" (cv f) (U. D)))) difss sstri (dom (cv f)) A (U. D) sseq2 mpbiri (U. D) (dom (cv f)) dfss sylib (\ A (U. D)) uneq1d (cv g) (\ B (" (cv f) (U. D))) imassrn sbthlem.1 sbthlem.2 sbthlem3 (ran (cv g)) sseq1d mpbii (\ A (U. D)) (ran (cv g)) dfss sylib (i^i (U. D) (dom (cv f))) uneq2d sylan9eq sbthlem.3 dmeqi (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))) dmun (cv f) (U. D) dmres (`' (cv g)) (\ A (U. D)) dmres (cv g) df-rn eqcomi (\ A (U. D)) ineq2i eqtr uneq12i 3eqtr syl6reqr sbthlem.1 sbthlem.2 sbthlem1 A (" (cv g) (\ B (" (cv f) (U. D)))) difss sstri (U. D) A ssundif mpbi syl6eq)) thm (sbthlem6 ((H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))))))) (-> (/\ (C_ (ran (cv f)) B) (/\ (/\ (= (dom (cv g)) B) (C_ (ran (cv g)) A)) (Fun (`' (cv g))))) (= (ran H) B)) (sbthlem.1 sbthlem.2 sbthlem4 (`' (cv g)) (\ A (U. D)) df-ima syl5reqr (" (cv f) (U. D)) uneq2d (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))) rnun sbthlem.3 rneqi (cv f) (U. D) df-ima (ran (|` (`' (cv g)) (\ A (U. D)))) uneq1i 3eqtr4 syl6reqr (cv f) (U. D) imassrn (" (cv f) (U. D)) (ran (cv f)) B sstr2 ax-mp (" (cv f) (U. D)) B ssundif sylib sylan9eqr)) thm (sbthlem7 ((H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))))))) (-> (/\ (Fun (cv f)) (Fun (`' (cv g)))) (Fun H)) ((cv f) (U. D) dmres (U. D) (dom (cv f)) inss1 eqsstr (dom (|` (cv f) (U. D))) (U. D) (dom (|` (`' (cv g)) (\ A (U. D)))) ssrin ax-mp (`' (cv g)) (\ A (U. D)) dmres (\ A (U. D)) (dom (`' (cv g))) inss1 eqsstr (dom (|` (`' (cv g)) (\ A (U. D)))) (\ A (U. D)) (U. D) sslin ax-mp sstri (U. D) A difdisj sseqtr (i^i (dom (|` (cv f) (U. D))) (dom (|` (`' (cv g)) (\ A (U. D))))) ss0 ax-mp (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))) funun mpan2 (cv f) (U. D) funres (`' (cv g)) (\ A (U. D)) funres syl2an sbthlem.3 H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D)))) funeq ax-mp sylibr)) thm (sbthlem8 ((H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))))))) (-> (/\ (Fun (`' (cv f))) (/\ (/\ (/\ (Fun (cv g)) (= (dom (cv g)) B)) (C_ (ran (cv g)) A)) (Fun (`' (cv g))))) (Fun (`' H))) ((`' (|` (cv f) (U. D))) (`' (|` (`' (cv g)) (\ A (U. D)))) funun (cv f) (U. D) funres11 (cv g) funcnvcnv (`' (cv g)) (\ A (U. D)) funres11 syl (= (dom (cv g)) B) adantr (C_ (ran (cv g)) A) (Fun (`' (cv g))) ad2antrr anim12i sbthlem.1 sbthlem.2 sbthlem4 (`' (cv g)) (\ A (U. D)) df-ima (|` (`' (cv g)) (\ A (U. D))) df-rn eqtr syl5eqr (cv f) (U. D) df-ima (|` (cv f) (U. D)) df-rn eqtr2 jctil (dom (`' (|` (cv f) (U. D)))) (" (cv f) (U. D)) (dom (`' (|` (`' (cv g)) (\ A (U. D))))) (\ B (" (cv f) (U. D))) ineq12 syl (" (cv f) (U. D)) B difdisj syl6eq (Fun (cv g)) adantlll (Fun (`' (cv f))) adantl sylanc sbthlem.3 H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D)))) cnveq ax-mp (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))) cnvun eqtr (`' H) (u. (`' (|` (cv f) (U. D))) (`' (|` (`' (cv g)) (\ A (U. D))))) funeq ax-mp sylibr)) thm (sbthlem9 ((H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))))))) (-> (/\ (:-1-1-> (cv f) A B) (:-1-1-> (cv g) B A)) (:-1-1-onto-> H A B)) (sbthlem.1 sbthlem.2 sbthlem.3 sbthlem7 sbthlem.1 sbthlem.2 sbthlem.3 sbthlem5 (/\ (Fun (cv g)) (= (dom (cv g)) B)) adantrl anim12i an42s (C_ (ran (cv f)) B) adantlr (Fun (`' (cv f))) adantlr sbthlem.1 sbthlem.2 sbthlem.3 sbthlem8 (/\ (/\ (Fun (cv f)) (= (dom (cv f)) A)) (C_ (ran (cv f)) B)) adantll sbthlem.1 sbthlem.2 sbthlem.3 sbthlem6 H df-rn syl5eqr (Fun (cv g)) (= (dom (cv g)) B) pm3.27 (C_ (ran (cv g)) A) anim1i sylanr1 (/\ (Fun (cv f)) (= (dom (cv f)) A)) adantll (Fun (`' (cv f))) adantlr jca jca (cv f) A B df-f1 (cv f) A B df-f (cv f) A df-fn (C_ (ran (cv f)) B) anbi1i bitr (Fun (`' (cv f))) anbi1i bitr (cv g) B A df-f1 (cv g) B A df-f (cv g) B df-fn (C_ (ran (cv g)) A) anbi1i bitr (Fun (`' (cv g))) anbi1i bitr anbi12i H A B f1o4 H A df-fn (`' H) B df-fn anbi12i bitr 3imtr4)) thm (sbthlem10 ((f g) (A f) (A g) (B f) (B g) (H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D)))))) (sbthlem.4 (e. B (V)))) (-> (/\ (br A (~<_) B) (br B (~<_) A)) (br A (~~) B)) (sbthlem.4 A f brdom sbthlem.1 B g brdom anbi12i f g (:-1-1-> (cv f) A B) (:-1-1-> (cv g) B A) eeanv bitr4 sbthlem.1 sbthlem.2 sbthlem.3 sbthlem9 sbthlem.1 H B f1oen syl f g 19.23aivv sylbi)) thm (sbth ((x y) (x z) (w x) (f x) (g x) (A x) (y z) (w y) (f y) (g y) (A y) (w z) (f z) (g z) (A z) (f w) (g w) (A w) (f g) (A f) (A g) (B x) (B y) (B z) (B w) (B f) (B g)) () (-> (/\ (br A (~<_) B) (br B (~<_) A)) (br A (~~) B)) ((cv z) A (~<_) (cv w) breq1 (cv z) A (cv w) (~<_) breq2 anbi12d (cv z) A (~~) (cv w) breq1 imbi12d (cv w) B A (~<_) breq2 (cv w) B (~<_) A breq1 anbi12d (cv w) B A (~~) breq2 imbi12d z visset (cv y) (cv x) (cv z) sseq1 (cv y) (cv x) (cv f) imaeq2 (cv w) difeq2d (\ (cv w) (" (cv f) (cv y))) (\ (cv w) (" (cv f) (cv x))) (cv g) imaeq2 (" (cv g) (\ (cv w) (" (cv f) (cv y)))) (" (cv g) (\ (cv w) (" (cv f) (cv x)))) (\ (cv z) (cv y)) sseq1 3syl (cv y) (cv x) (cv z) difeq2 (" (cv g) (\ (cv w) (" (cv f) (cv x)))) sseq2d bitrd anbi12d cbvabv (u. (|` (cv f) (U. ({|} y (/\ (C_ (cv y) (cv z)) (C_ (" (cv g) (\ (cv w) (" (cv f) (cv y)))) (\ (cv z) (cv y))))))) (|` (`' (cv g)) (\ (cv z) (U. ({|} y (/\ (C_ (cv y) (cv z)) (C_ (" (cv g) (\ (cv w) (" (cv f) (cv y)))) (\ (cv z) (cv y))))))))) eqid w visset sbthlem10 (V) (V) vtocl2g reldom A B brrelexi reldom B A brrelexi syl2an pm2.43i)) thm (sbthbg () () (-> (e. B C) (<-> (/\ (br A (~<_) B) (br B (~<_) A)) (br A (~~) B))) (A B sbth (e. B C) a1i A B endom (e. B C) a1i B C A ensymg B A endom syl6 jcad impbid)) thm (sbthcl ((x y)) () (= (~~) (i^i (~<_) (`' (~<_)))) (relen reldom (~<_) (`' (~<_)) relin1 ax-mp y visset (cv y) (V) (cv x) sbthbg ax-mp (cv x) (~<_) (cv y) df-br (cv y) (~<_) (cv x) df-br x visset y visset (~<_) opelcnv bitr4 anbi12i (<,> (cv x) (cv y)) (~<_) (`' (~<_)) elin bitr4 (cv x) (~~) (cv y) df-br 3bitr3r cleqreli)) thm (dfsdom2 () () (= (~<) (\ (~<_) (`' (~<_)))) (df-sdom sbthcl (~<_) difeq2i (~<_) (`' (~<_)) difin 3eqtr)) thm (brsdom2 () ((brsdom2.1 (e. A (V))) (brsdom2.2 (e. B (V)))) (<-> (br A (~<) B) (/\ (br A (~<_) B) (-. (br B (~<_) A)))) (dfsdom2 (<,> A B) eleq2i A (~<) B df-br A (~<_) B df-br B (~<_) A df-br brsdom2.1 brsdom2.2 (~<_) opelcnv bitr4 negbii anbi12i (<,> A B) (~<_) (`' (~<_)) eldif bitr4 3bitr4)) thm (sdomnsym () () (-> (br A (~<) B) (-. (br B (~<) A))) (A B sdomnen A B sbth A B sdomdom B A sdomdom syl2an ex mtod)) thm (domnsym () () (-> (br A (~<_) B) (-. (br B (~<) A))) (A B brdom2 A B sdomnsym B (V) A ensymg con3d relsdom B A brrelexi B A sdomnen sylc con2i jaoi sylbi)) thm (0dom () () (br ({/}) (~<_) A) (0ex A 0ss ({/}) (V) A ssdomg mp2)) thm (dom0 () () (<-> (br A (~<_) ({/})) (= A ({/}))) (A 0dom (br A (~<_) ({/})) biantru 0ex ({/}) (V) A sbthbg ax-mp A en0 3bitr)) thm (0sdomg () () (-> (e. A B) (<-> (br ({/}) (~<) A) (-. (= A ({/}))))) (A B ({/}) ensymg 0ex A ensym (e. A B) a1i impbid A en0 syl6bb negbid ({/}) A brsdom A 0dom mpbiran syl5bb)) thm (0sdom () ((0sdom.1 (e. A (V)))) (<-> (br ({/}) (~<) A) (-. (= A ({/})))) (0sdom.1 A (V) 0sdomg ax-mp)) thm (sdom0 () () (-. (br A (~<) ({/}))) (A 0dom ({/}) A domnsym ax-mp)) thm (sdomdomtr () () (-> (e. C D) (-> (/\ (br A (~<) B) (br B (~<_) C)) (br A (~<) C))) (A B sdomnen (e. C D) (br B (~<_) C) ad2antrl A B C domtr A B sdomdom sylan A C brdom2 (br A (~<) C) (br A (~~) C) df-or bitr sylib (e. C D) adantl A B C domtr A B sdomdom sylan A C brdom2 (br A (~<) C) (br A (~~) C) df-or bitr sylib C D A ensymg C A endom syl6 sylan9r A B sdomdom (e. C D) (br B (~<_) C) ad2antrl jctird C A B domtr syl6 (br A (~<) B) (br B (~<_) C) pm3.27 (e. C D) adantl jctird C B sbth syl6 jcad A C B entrt syl6 mt3d ex)) thm (sdomentr () () (-> (e. C D) (-> (/\ (br A (~<) B) (br B (~~) C)) (br A (~<) C))) (C D A B sdomdomtr B C endom sylan2i)) thm (ensdomtr () () (-> (/\ (br A (~~) B) (br B (~<) C)) (br A (~<) C)) (A B C endomtr ex (e. B (V)) adantl B (V) A ensymg imp B A C entrt ex syl con3d anim12d B C brsdom A C brsdom 3imtr4g ex imp3a relsdom B C brrelexi con3i (br A (~<) C) pm2.21d (br A (~~) B) adantld pm2.61i)) thm (sdomtr () () (-> (/\ (br A (~<) B) (br B (~<) C)) (br A (~<) C)) (B C sdomex pm3.27d C (V) A B sdomdomtr B C sdomdom sylan2i syl anabsi7)) thm (domsdomtr () () (-> (/\ (br A (~<_) B) (br B (~<) C)) (br A (~<) C)) (A B brdom2 A B C sdomtr ex relsdom B C brrelexi A B C endomtr ex (e. B (V)) adantl B (V) A ensymg B A C entrt ex syl6 imp con3d anim12d B C brsdom A C brsdom 3imtr4g ex syl pm2.43b jaoi sylbi imp)) thm (enen1 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br A (~~) C) (br B (~~) C))) (B D A ensymg imp B A C entrt ex syl A B C entrt (e. B D) adantll ex impbid)) thm (enen2 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br C (~~) A) (br C (~~) B))) (C A B entrt expcom (e. B D) adantl C B A entrt ex B D A ensymg imp syl5com impbid)) thm (domen1 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br A (~<_) C) (br B (~<_) C))) (B D A ensymg imp B A C endomtr ex syl A B C endomtr ex (e. B D) adantl impbid)) thm (domen2 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br C (~<_) A) (br C (~<_) B))) (C A B domentr expcom (e. B D) adantl C B A domentr ex B D A ensymg imp syl5com impbid)) thm (sdomen1 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br A (~<) C) (br B (~<) C))) (B D A ensymg imp B A C ensdomtr ex syl A B C ensdomtr (e. B D) adantll ex impbid)) thm (sdomen2 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br C (~<) A) (br C (~<) B))) (B D C A sdomentr exp3a com23 imp (e. A (V)) adantll B D A ensymg A (V) C B sdomentr exp3a com23 syl9r imp31 impbid exp31 imp3a relen A B brrelexi con3i (<-> (br C (~<) A) (br C (~<) B)) pm2.21d (e. B D) adantld pm2.61i)) thm (fodomr ((f g) (f z) (A f) (g z) (A g) (A z) (B f) (B g) (B z)) () (-> (/\/\ (e. A C) (br ({/}) (~<) B) (br B (~<_) A)) (E. f (:-onto-> (cv f) A B))) (A (V) (ran (cv g)) difexg (cv z) snex (\ A (ran (cv g))) (V) ({} (cv z)) (V) xpexg mpan2 syl g visset cnvex jctil (`' (cv g)) (X. (\ A (ran (cv g))) ({} (cv z))) unexb sylib (cv f) (u. (`' (cv g)) (X. (\ A (ran (cv g))) ({} (cv z)))) A B foeq1 (V) cla4egv syl (cv g) B A df-f1 pm3.27bd z visset (\ A (ran (cv g))) fconst (X. (\ A (ran (cv g))) ({} (cv z))) (\ A (ran (cv g))) ({} (cv z)) ffun ax-mp jctir (cv g) df-rn eqcomi z visset snnz ({} (cv z)) (\ A (ran (cv g))) dmxp ax-mp ineq12i (ran (cv g)) A difdisj eqtr (`' (cv g)) (X. (\ A (ran (cv g))) ({} (cv z))) funun mpan2 syl (e. (cv z) B) adantl (cv g) B A f1f (cv g) B A frn syl (ran (cv g)) A ssundif sylib (`' (cv g)) (X. (\ A (ran (cv g))) ({} (cv z))) dmun (cv g) df-rn (dom (X. (\ A (ran (cv g))) ({} (cv z)))) uneq1i z visset snnz ({} (cv z)) (\ A (ran (cv g))) dmxp ax-mp (ran (cv g)) uneq2i 3eqtr2 syl5eq (e. (cv z) B) adantl jca (u. (`' (cv g)) (X. (\ A (ran (cv g))) ({} (cv z)))) A df-fn sylibr (cv g) B A f1f (cv g) B A fdm syl (cv g) dfdm4 syl5eqr (ran (X. (\ A (ran (cv g))) ({} (cv z)))) uneq1d B 0ss (\ A (ran (cv g))) ({/}) ({} (cv z)) xpeq1 ({} (cv z)) xp0r syl6eq rneqd rn0 syl6eq B sseq1d mpbiri (e. (cv z) B) a1d (\ A (ran (cv g))) ({} (cv z)) rnxp (e. (cv z) B) adantr (cv z) B snssi (-. (= (\ A (ran (cv g))) ({/}))) adantl eqsstrd ex pm2.61i (ran (X. (\ A (ran (cv g))) ({} (cv z)))) B ssequn2 sylib sylan9eqr (`' (cv g)) (X. (\ A (ran (cv g))) ({} (cv z))) rnun syl5eq jca (u. (`' (cv g)) (X. (\ A (ran (cv g))) ({} (cv z)))) A B df-fo sylibr syl5 exp3a imp g 19.23adv ex z 19.23adv 3imp A C elisset (br ({/}) (~<) B) (br B (~<_) A) 3ad2ant1 reldom B A brrelexi B (V) 0sdomg B z n0 syl6bb syl biimpac (e. A C) 3adant1 A C B g brdomg biimpa (br ({/}) (~<) B) 3adant2 syl3anc)) thm (canth2 ((x y) (f x) (A x) (f y) (A y) (A f)) ((canth2.1 (e. A (V)))) (br A (~<) (P~ A)) (A (P~ A) brsdom canth2.1 x visset A snelpw biimp x visset (cv y) sneqr (cv x) (cv y) sneq impbi (/\ (e. (cv x) A) (e. (cv y) A)) a1i (V) dom2 ax-mp canth2.1 (cv f) canth (cv f) A (P~ A) f1ofo mto f nex canth2.1 pwex A f bren mtbir mpbir2an)) thm (canth2g ((A x)) () (-> (e. A B) (br A (~<) (P~ A))) ((cv x) A pweq (cv x) A (P~ (cv x)) (P~ A) (~<) breq12 mpdan x visset canth2 B vtoclg)) thm (xpen () ((xpen.1 (e. A (V))) (xpen.2 (e. B (V))) (xpen.3 (e. C (V))) (xpen.4 (e. D (V)))) (-> (/\ (br A (~~) B) (br C (~~) D)) (br (X. A C) (~~) (X. B D))) ((X. A C) (X. B C) (X. B D) entrt xpen.2 xpen.3 A xpdom2 xpen.1 xpen.3 B xpdom2 anim12i xpen.2 B (V) A sbthbg ax-mp xpen.3 xpen.2 xpex (X. C B) (V) (X. C A) sbthbg ax-mp 3imtr3 xpen.2 xpen.3 xpex xpen.3 xpen.2 xpcomen (X. B C) (V) (X. C B) (X. C A) enen2 mp2an xpen.1 xpen.3 xpex xpen.3 xpen.1 xpcomen (X. A C) (V) (X. C A) (X. B C) enen1 mp2an bitr sylib xpen.4 xpen.2 C xpdom2 xpen.3 xpen.2 D xpdom2 anim12i xpen.4 D (V) C sbthbg ax-mp xpen.2 xpen.4 xpex (X. B D) (V) (X. B C) sbthbg ax-mp 3imtr3 syl2an)) thm (mapenlem1 ((f g) (f x) (f y) (f z) (f v) (A f) (g x) (g y) (g z) (g v) (A g) (x y) (x z) (v x) (A x) (y z) (v y) (A y) (v z) (A z) (A v) (B f) (B g) (B x) (B y) (B z) (B v) (C f) (C g) (C x) (C y) (C z) (C v) (D f) (D g) (D x) (D y) (D z) (D v) (H z) (H v)) ((mapenlem.1 (e. A (V))) (mapenlem.2 (e. B (V))) (mapenlem.3 (e. C (V))) (mapenlem.4 (e. D (V))) (mapenlem.5 (= H ({<,>|} x y (/\ (e. (cv x) (opr A (^m) C)) (= (cv y) (o. (o. (cv f) (cv x)) (`' (cv g))))))))) (-> (/\ (/\ (/\ (:-1-1-onto-> (cv f) A B) (:-1-1-onto-> (cv g) C D)) (:--> (cv z) C A)) (e. (cv v) C)) (= (` (` H (cv z)) (` (cv g) (cv v))) (` (cv f) (` (cv z) (cv v))))) (mapenlem.1 mapenlem.3 (cv z) elmap (cv x) (cv z) (cv f) coeq2 (`' (cv g)) coeq1d mapenlem.5 f visset z visset coex g visset cnvex coex fvopab4 sylbir (` (cv g) (cv v)) fveq1d (/\ (:-1-1-onto-> (cv f) A B) (:-1-1-onto-> (cv g) C D)) (e. (cv v) C) ad2antlr (cv g) C D f1ococnv1 (o. (cv f) (cv z)) coeq2d (o. (cv f) (cv z)) C B fcoi1 sylan9eqr (cv f) A B (cv z) C fco (cv f) A B f1of sylan sylan an1rs (o. (cv f) (cv z)) (`' (cv g)) (cv g) coass syl5eq (cv v) fveq1d (e. (cv v) C) adantr (o. (o. (cv f) (cv z)) (`' (cv g))) (cv g) C D (cv v) fvco3 3expa (o. (cv f) (cv z)) (`' (cv g)) funco (cv f) (cv z) funco (cv f) A B f1ofun (cv z) C A ffun syl2an (cv g) C D f1o3 pm3.27bd syl2an (cv g) C D f1of anim12i anabss3 an1rs sylan (cv f) (cv z) C A (cv v) fvco3 3expb (cv f) A B f1ofun sylan (:-1-1-onto-> (cv g) C D) adantlr anassrs 3eqtr3d eqtrd)) thm (mapenlem2 ((f g) (f x) (f y) (f z) (f w) (f v) (A f) (g x) (g y) (g z) (g w) (g v) (A g) (x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B f) (B g) (B x) (B y) (B z) (B w) (B v) (C f) (C g) (C x) (C y) (C z) (C w) (C v) (D f) (D g) (D x) (D y) (D z) (D w) (D v) (H z) (H w) (H v)) ((mapenlem.1 (e. A (V))) (mapenlem.2 (e. B (V))) (mapenlem.3 (e. C (V))) (mapenlem.4 (e. D (V))) (mapenlem.5 (= H ({<,>|} x y (/\ (e. (cv x) (opr A (^m) C)) (= (cv y) (o. (o. (cv f) (cv x)) (`' (cv g))))))))) (-> (/\ (:-1-1-onto-> (cv f) A B) (:-1-1-onto-> (cv g) C D)) (:-1-1-onto-> H (opr A (^m) C) (opr B (^m) D))) ((o. (cv f) (cv x)) C B (`' (cv g)) D fco (cv f) A B (cv x) C fco (cv f) A B f1of sylan (cv g) C D f1ocnv (`' (cv g)) D C f1of syl syl2an exp31 com23 imp mapenlem.1 mapenlem.3 (cv x) elmap mapenlem.2 mapenlem.4 (o. (o. (cv f) (cv x)) (`' (cv g))) elmap 3imtr4g r19.21aiv mapenlem.5 (opr B (^m) D) fopab2 sylib (` H (cv z)) (` H (cv w)) (` (cv g) (cv v)) fveq1 (/\ (:--> (cv z) C A) (:--> (cv w) C A)) adantl (/\ (:-1-1-onto-> (cv f) A B) (:-1-1-onto-> (cv g) C D)) (e. (cv v) C) ad2antlr mapenlem.1 mapenlem.2 mapenlem.3 mapenlem.4 mapenlem.5 z v mapenlem1 (= (` H (cv z)) (` H (cv w))) adantrl exp43 (:--> (cv w) C A) adantrd imp42 mapenlem.1 mapenlem.2 mapenlem.3 mapenlem.4 mapenlem.5 w v mapenlem1 (= (` H (cv z)) (` H (cv w))) adantrl exp43 (:--> (cv z) C A) adantld imp42 3eqtr3d (cv f) A B (` (cv z) (cv v)) (` (cv w) (cv v)) f1fveq (cv f) A B f1of1 (cv z) C A (cv v) ffvrn (:--> (cv w) C A) adantlr (cv w) C A (cv v) ffvrn (:--> (cv z) C A) adantll jca (= (` H (cv z)) (` H (cv w))) adantlr syl2an (:-1-1-onto-> (cv g) C D) adantlr anassrs mpbid r19.21aiva (cv z) C (cv w) C v eqfnfv C eqid (A.e. v C (= (` (cv z) (cv v)) (` (cv w) (cv v)))) biantrur syl6bbr (cv z) C A ffn (cv w) C A ffn syl2an (/\ (:-1-1-onto-> (cv f) A B) (:-1-1-onto-> (cv g) C D)) (= (` H (cv z)) (` H (cv w))) ad2antrl mpbird exp32 mapenlem.1 mapenlem.3 (cv z) elmap mapenlem.1 mapenlem.3 (cv w) elmap anbi12i syl5ib exp3a r19.21adv r19.21aiv jca H (opr A (^m) C) (opr B (^m) D) z w f1fv sylibr (o. (cv f) (cv x)) C B (`' (cv g)) D fco (cv f) A B (cv x) C fco (cv f) A B f1of sylan (cv g) C D f1ocnv (`' (cv g)) D C f1of syl syl2an exp31 com23 imp mapenlem.1 mapenlem.3 (cv x) elmap mapenlem.2 mapenlem.4 (o. (o. (cv f) (cv x)) (`' (cv g))) elmap 3imtr4g r19.21aiv mapenlem.5 (opr B (^m) D) fopab2 sylib H (opr A (^m) C) (opr B (^m) D) frn syl (o. (`' (cv f)) (cv z)) D A (cv g) C fco (`' (cv f)) B A (cv z) D fco (cv f) A B f1ocnv (`' (cv f)) B A f1of syl sylan (cv g) C D f1of syl2an an1rs mapenlem.1 mapenlem.3 (o. (o. (`' (cv f)) (cv z)) (cv g)) elmap sylibr (o. (`' (cv f)) (cv z)) D A (cv g) C fco (`' (cv f)) B A (cv z) D fco (cv f) A B f1ocnv (`' (cv f)) B A f1of syl sylan (cv g) C D f1of syl2an an1rs mapenlem.1 mapenlem.3 (o. (o. (`' (cv f)) (cv z)) (cv g)) elmap sylibr (cv x) (o. (o. (`' (cv f)) (cv z)) (cv g)) (cv f) coeq2 (`' (cv g)) coeq1d mapenlem.5 f visset f visset cnvex z visset coex g visset coex coex g visset cnvex coex fvopab4 syl (cv f) A B f1ococnv2 (cv z) coeq1d (cv z) D B fcoi2 sylan9eq (o. (cv g) (`' (cv g))) coeq1d (:-1-1-onto-> (cv g) C D) adantlr (cv g) C D f1ococnv2 (cv z) coeq2d (cv z) D B fcoi1 sylan9eq (:-1-1-onto-> (cv f) A B) adantll eqtrd (cv f) (o. (o. (`' (cv f)) (cv z)) (cv g)) (`' (cv g)) coass (o. (`' (cv f)) (cv z)) (cv g) (`' (cv g)) coass (cv f) coeq2i (cv f) (`' (cv f)) (cv z) coass (o. (cv g) (`' (cv g))) coeq1i (cv f) (o. (`' (cv f)) (cv z)) (o. (cv g) (`' (cv g))) coass eqtr2 3eqtr syl5eq eqtrd jca ex mapenlem.2 mapenlem.4 (cv z) elmap syl5ib (cv w) (o. (o. (`' (cv f)) (cv z)) (cv g)) H fveq2 (cv z) eqeq1d (opr A (^m) C) rcla4ev syl6 (o. (cv f) (cv x)) C B (`' (cv g)) D fco (cv f) A B (cv x) C fco (cv f) A B f1of sylan (cv g) C D f1ocnv (`' (cv g)) D C f1of syl syl2an exp31 com23 imp mapenlem.1 mapenlem.3 (cv x) elmap mapenlem.2 mapenlem.4 (o. (o. (cv f) (cv x)) (`' (cv g))) elmap 3imtr4g r19.21aiv mapenlem.5 (opr B (^m) D) fopab2 sylib H (opr A (^m) C) (opr B (^m) D) ffn H (opr A (^m) C) (cv z) w fvelrn 3syl sylibrd ssrdv eqssd jca H (opr A (^m) C) (opr B (^m) D) f1o5 sylibr)) thm (mapen ((f g) (f x) (f y) (A f) (g x) (g y) (A g) (x y) (A x) (A y) (B f) (B g) (B x) (B y) (C f) (C g) (C x) (C y) (D f) (D g) (D x) (D y)) ((mapen.1 (e. A (V))) (mapen.2 (e. B (V))) (mapen.3 (e. C (V))) (mapen.4 (e. D (V)))) (-> (/\ (br A (~~) B) (br C (~~) D)) (br (opr A (^m) C) (~~) (opr B (^m) D))) (mapen.1 mapen.2 mapen.3 mapen.4 ({<,>|} x y (/\ (e. (cv x) (opr A (^m) C)) (= (cv y) (o. (o. (cv f) (cv x)) (`' (cv g)))))) eqid mapenlem2 A (^m) C oprex ({<,>|} x y (/\ (e. (cv x) (opr A (^m) C)) (= (cv y) (o. (o. (cv f) (cv x)) (`' (cv g)))))) (opr B (^m) D) f1oen syl ex f 19.23aiv g 19.23adv imp mapen.2 A f bren mapen.4 C g bren syl2anb)) thm (mapdom1 ((A x) (B x) (C x)) ((mapdom1.1 (e. A (V))) (mapdom1.2 (e. B (V))) (mapdom1.3 (e. C (V)))) (-> (br A (~<_) B) (br (opr A (^m) C) (~<_) (opr B (^m) C))) (mapdom1.2 A x domen (opr A (^m) C) (opr (cv x) (^m) C) (opr B (^m) C) endomtr mapdom1.3 enref mapdom1.1 x visset mapdom1.3 mapdom1.3 mapen mpan2 mapdom1.2 mapdom1.3 (cv x) mapss (cv x) (^m) C oprex (opr (cv x) (^m) C) (V) (opr B (^m) C) ssdomg ax-mp syl syl2an x 19.23aiv sylbi)) thm (mapdom2lem ((x z) (w x) (A x) (w z) (A z) (A w) (B x) (B z) (B w) (C x) (C z) (C w)) ((mapdom1.1 (e. A (V))) (mapdom1.2 (e. B (V))) (mapdom1.3 (e. C (V)))) (-> (e. (cv x) (opr C (^m) (cv z))) (= (i^i (cv x) (X. (\ B (cv z)) ({} (cv w)))) ({/}))) (mapdom1.3 z visset (cv x) elmap (cv x) (cv z) C fdm sylbi w visset (\ B (cv z)) fconst (X. (\ B (cv z)) ({} (cv w))) (\ B (cv z)) ({} (cv w)) fdm ax-mp (e. (cv x) (opr C (^m) (cv z))) a1i ineq12d (cv z) B difdisj syl6eq (cv x) (X. (\ B (cv z)) ({} (cv w))) dmin (i^i (dom (cv x)) (dom (X. (\ B (cv z)) ({} (cv w))))) ({/}) (dom (i^i (cv x) (X. (\ B (cv z)) ({} (cv w))))) sseq2 mpbii (dom (i^i (cv x) (X. (\ B (cv z)) ({} (cv w))))) ss0 syl syl (\ B (cv z)) ({} (cv w)) relxp (X. (\ B (cv z)) ({} (cv w))) (cv x) relin1 ax-mp (X. (\ B (cv z)) ({} (cv w))) (cv x) incom (i^i (X. (\ B (cv z)) ({} (cv w))) (cv x)) (i^i (cv x) (X. (\ B (cv z)) ({} (cv w)))) releq ax-mp mpbi (i^i (cv x) (X. (\ B (cv z)) ({} (cv w)))) reldm0 ax-mp sylibr)) thm (mapdom2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((mapdom1.1 (e. A (V))) (mapdom1.2 (e. B (V))) (mapdom1.3 (e. C (V)))) (-> (/\ (br A (~<_) B) (-. (/\ (= A ({/})) (= C ({/}))))) (br (opr C (^m) A) (~<_) (opr C (^m) B))) (C ({/}) (^m) A opreq1 mapdom1.1 map0b sylan9eqr (opr C (^m) B) 0dom syl6eqbr (br A (~<_) B) a1i mapdom1.2 A z domen (opr C (^m) A) (opr C (^m) (cv z)) (opr C (^m) B) endomtr mapdom1.3 enref mapdom1.3 mapdom1.3 mapdom1.1 z visset mapen mpan C (^m) (cv z) oprex (cv z) B ssundif (u. (cv z) (\ B (cv z))) B (u. (cv x) (X. (\ B (cv z)) ({} (cv w)))) (u. C ({} (cv w))) feq2 sylbi (cv w) C snssi ({} (cv w)) C ssequn2 sylib (u. C ({} (cv w))) C (u. (cv x) (X. (\ B (cv z)) ({} (cv w)))) B feq3 syl sylan9bb w visset (\ B (cv z)) fconst (cv z) B difdisj (cv x) (cv z) C (X. (\ B (cv z)) ({} (cv w))) (\ B (cv z)) ({} (cv w)) fun mpan2 mpan2 syl5bi mapdom1.3 z visset (cv x) elmap mapdom1.3 mapdom1.2 (u. (cv x) (X. (\ B (cv z)) ({} (cv w)))) elmap 3imtr4g mapdom1.1 mapdom1.2 mapdom1.3 x z w mapdom2lem mapdom1.1 mapdom1.2 mapdom1.3 y z w mapdom2lem eqcomd sylan9eq (= (u. (cv x) (X. (\ B (cv z)) ({} (cv w)))) (u. (cv y) (X. (\ B (cv z)) ({} (cv w))))) biantrud (cv x) (X. (\ B (cv z)) ({} (cv w))) (cv y) unineq syl6bb (/\ (C_ (cv z) B) (e. (cv w) C)) a1i (V) dom2d mpi syl2an anassrs ex z 19.23aiv sylbi w 19.23adv C w n0 syl5ib jaod imp (= C ({/})) exmid (\/ (-. (= A ({/}))) (-. (= C ({/})))) biantru (= A ({/})) (= C ({/})) ianor (-. (= A ({/}))) (= C ({/})) (-. (= C ({/}))) ordir 3bitr4 sylan2b)) thm (mapxpen ((x y) (x z) (w x) (v x) (f x) (g x) (A x) (y z) (w y) (v y) (f y) (g y) (A y) (w z) (v z) (f z) (g z) (A z) (v w) (f w) (g w) (A w) (f v) (g v) (A v) (f g) (A f) (A g) (B x) (B y) (B z) (B w) (B v) (B f) (B g) (C x) (C y) (C z) (C w) (C v) (C f) (C g)) ((mapxpen.1 (e. A (V))) (mapxpen.2 (e. B (V))) (mapxpen.3 (e. C (V)))) (br (opr (opr A (^m) B) (^m) C) (~~) (opr A (^m) (X. B C))) ((opr A (^m) B) (^m) C oprex mapxpen.2 mapxpen.3 ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z))))) eqid oprabex2 (e. (cv x) (opr (opr A (^m) B) (^m) C)) a1i mapxpen.3 w f ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w))))) funopabex2 (e. (cv y) (opr A (^m) (X. B C))) a1i (` (cv x) (cv w)) (cv z) fvex ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z))))) eqid fnoprab2 (cv y) ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z))))) (X. B C) fneq1 mpbiri (e. (cv x) (opr (opr A (^m) B) (^m) C)) adantl (e. (cv x) (opr (opr A (^m) B) (^m) C)) z ax-17 y z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z)))) hboprab1 hbeleq hban (e. (cv x) (opr (opr A (^m) B) (^m) C)) w ax-17 y z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z)))) hboprab2 hbeleq hban (e. (cv z) B) w ax-17 (` (cv x) (cv w)) B A (cv z) ffvrn mapxpen.1 mapxpen.2 (` (cv x) (cv w)) elmap sylanb (cv x) C (opr A (^m) B) (cv w) ffvrn A (^m) B oprex mapxpen.3 (cv x) elmap sylanb sylan an1rs anasss (= (cv y) ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z)))))) adantlr (cv y) ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z))))) (cv z) (cv w) opreq (` (cv x) (cv w)) (cv z) fvex ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z))))) eqid (V) oprabval4g mp3an3 sylan9eq A eleq1d (e. (cv x) (opr (opr A (^m) B) (^m) C)) adantll mpbird exp32 r19.21ad r19.21ai jca mapxpen.1 mapxpen.2 mapxpen.3 xpex (cv y) elmap (cv y) B C A z w ffnoprval bitr sylibr A (^m) B oprex mapxpen.3 (cv x) elmap (cv x) C (opr A (^m) B) w f fopabfv pm3.26bd sylbi (= (cv y) ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z)))))) adantr (e. (cv x) (opr (opr A (^m) B) (^m) C)) w ax-17 y z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z)))) hboprab2 hbeleq hban (/\ (e. (cv x) (opr (opr A (^m) B) (^m) C)) (= (cv y) ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z))))))) f ax-17 (cv x) C (opr A (^m) B) (cv w) ffvrn A (^m) B oprex mapxpen.3 (cv x) elmap sylanb (= (cv y) ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z)))))) adantlr y z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z)))) hboprab1 hbeleq (e. (cv w) C) z ax-17 hban (/\ (= (cv y) ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z)))))) (e. (cv w) C)) g ax-17 (cv y) ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z))))) (cv z) (cv w) opreq (` (cv x) (cv w)) (cv z) fvex ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z))))) eqid (V) oprabval4g mp3an3 sylan9eq anassrs an1rs (cv g) eqeq2d ex pm5.32d opabbid (` (cv x) (cv w)) eqeq2d mapxpen.1 mapxpen.2 (` (cv x) (cv w)) elmap (` (cv x) (cv w)) B A z g fopabfv pm3.26bd sylbi syl5bir (e. (cv x) (opr (opr A (^m) B) (^m) C)) adantll mpd (cv f) eqeq2d ex pm5.32d opabbid eqtrd jca (e. (cv w) C) z ax-17 f z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w)))) hbopab1 hbeleq hban x w f hbopab hbeleq x w f (/\ (e. (cv w) C) (= (cv f) ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w))))))) hbopab1 hbeleq (= (cv x) ({<,>|} w f (/\ (e. (cv w) C) (= (cv f) ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w))))))))) v ax-17 (cv x) ({<,>|} w f (/\ (e. (cv w) C) (= (cv f) ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w)))))))) (cv w) fveq1 mapxpen.2 z g (opr (cv z) (cv y) (cv w)) funopabex2 w C ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w))))) (V) f fvopab2 mpan2 sylan9eq (cv z) fveq1d (cv z) (cv y) (cv w) oprex z B (opr (cv z) (cv y) (cv w)) (V) g fvopab2 mpan2 sylan9eq an1rs anasss (cv v) eqeq2d ex pm5.32d oprabbid (cv y) eqeq2d (cv y) B C z w v fnoprval syl6bbr (A.e. z B (A.e. w C (e. (opr (cv z) (cv y) (cv w)) A))) anbi1d mapxpen.1 mapxpen.2 mapxpen.3 xpex (cv y) elmap (cv y) B C A z w ffnoprval bitr syl6bbr biimparc (= (cv y) ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z)))))) (A.e. z B (A.e. w C (e. (opr (cv z) (cv y) (cv w)) A))) ancom sylib mapxpen.2 z g (opr (cv z) (cv y) (cv w)) funopabex2 ({<,>|} w f (/\ (e. (cv w) C) (= (cv f) ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w)))))))) eqid fnopab2 (cv x) ({<,>|} w f (/\ (e. (cv w) C) (= (cv f) ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w)))))))) C fneq1 mpbiri (A.e. w C (e. (` (cv x) (cv w)) (opr A (^m) B))) biantrurd A (^m) B oprex mapxpen.3 (cv x) elmap (cv x) C (opr A (^m) B) w ffnfv bitr2 syl6bb x w f (/\ (e. (cv w) C) (= (cv f) ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w))))))) hbopab1 hbeleq (e. (cv w) C) z ax-17 f z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w)))) hbopab1 hbeleq hban x w f hbopab hbeleq (e. (cv w) C) z ax-17 hban (cv x) ({<,>|} w f (/\ (e. (cv w) C) (= (cv f) ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w)))))))) (cv w) fveq1 mapxpen.2 z g (opr (cv z) (cv y) (cv w)) funopabex2 w C ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w))))) (V) f fvopab2 mpan2 sylan9eq (cv z) fveq1d (cv z) (cv y) (cv w) oprex z B (opr (cv z) (cv y) (cv w)) (V) g fvopab2 mpan2 sylan9eq A eleq1d ralbida (Fn (` (cv x) (cv w)) B) anbi2d mapxpen.1 mapxpen.2 (` (cv x) (cv w)) elmap (` (cv x) (cv w)) B A z ffnfv bitr syl5bb (cv x) ({<,>|} w f (/\ (e. (cv w) C) (= (cv f) ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w)))))))) (cv w) fveq1 mapxpen.2 z g (opr (cv z) (cv y) (cv w)) funopabex2 w C ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w))))) (V) f fvopab2 mpan2 sylan9eq (cv z) (cv y) (cv w) oprex ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w))))) eqid fnopab2 (` (cv x) (cv w)) ({<,>|} z g (/\ (e. (cv z) B) (= (cv g) (opr (cv z) (cv y) (cv w))))) B fneq1 mpbiri syl (A.e. z B (e. (opr (cv z) (cv y) (cv w)) A)) biantrurd bitr4d ralbida bitr3d w C z B (e. (opr (cv z) (cv y) (cv w)) A) ralcom syl6bb (e. (cv y) (opr A (^m) (X. B C))) adantl (= (cv y) ({<<,>,>|} z w v (/\ (/\ (e. (cv z) B) (e. (cv w) C)) (= (cv v) (` (` (cv x) (cv w)) (cv z)))))) anbi1d mpbird impbi en2)) thm (xpmapenlem1 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V))) (xpmapenlem.4 (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))))) (xpmapenlem.5 (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))))) (xpmapenlem.6 (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))))))) (/\ (-> (= (cv y) (<,> D R)) (A. z (= (cv y) (<,> D R)))) (-> (= (cv y) (<,> D R)) (A. w (= (cv y) (<,> D R))))) (y z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z))))))) hbopab1 xpmapenlem.4 (cv y) eleq2i xpmapenlem.4 (cv y) eleq2i z albii 3imtr4 y z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z))))))) hbopab1 xpmapenlem.5 (cv y) eleq2i xpmapenlem.5 (cv y) eleq2i z albii 3imtr4 hbop hbeleq y z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z))))))) hbopab2 xpmapenlem.4 (cv y) eleq2i xpmapenlem.4 (cv y) eleq2i w albii 3imtr4 y z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z))))))) hbopab2 xpmapenlem.5 (cv y) eleq2i xpmapenlem.5 (cv y) eleq2i w albii 3imtr4 hbop hbeleq pm3.2i)) thm (xpmapenlem2 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V))) (xpmapenlem.4 (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))))) (xpmapenlem.5 (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))))) (xpmapenlem.6 (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))))))) (-> (/\ (= (cv y) (<,> D R)) (e. (cv z) C)) (/\ (= (` (U. (dom ({} (cv y)))) (cv z)) (U. (dom ({} (` (cv x) (cv z)))))) (= (` (U. (ran ({} (cv y)))) (cv z)) (U. (ran ({} (` (cv x) (cv z)))))))) ((cv y) (<,> D R) sneq xpmapenlem.4 D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))) R opeq1 ax-mp sneqi syl6eq dmeqd unieqd xpmapen.3 z w (U. (dom ({} (` (cv x) (cv z))))) funopabex2 R op1sta syl6eq (cv z) fveq1d (` (cv x) (cv z)) snex ({} (` (cv x) (cv z))) (V) dmexg ax-mp uniex z C (U. (dom ({} (` (cv x) (cv z))))) (V) w fvopab2 mpan2 sylan9eq (cv y) (<,> D R) sneq xpmapenlem.5 R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))) D opeq2 ax-mp sneqi syl6eq rneqd unieqd xpmapenlem.4 xpmapen.3 z w (U. (dom ({} (` (cv x) (cv z))))) funopabex2 eqeltr xpmapen.3 z w (U. (ran ({} (` (cv x) (cv z))))) funopabex2 op2nda syl6eq (cv z) fveq1d (` (cv x) (cv z)) snex ({} (` (cv x) (cv z))) (V) rnexg ax-mp uniex z C (U. (ran ({} (` (cv x) (cv z))))) (V) w fvopab2 mpan2 sylan9eq jca)) thm (xpmapenlem3 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V))) (xpmapenlem.4 (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))))) (xpmapenlem.5 (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))))) (xpmapenlem.6 (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))))))) (-> (/\ (:--> (cv x) C (X. A B)) (= (cv y) (<,> D R))) (= (cv x) S)) ((cv x) C (X. A B) ffn (cv x) C z w fnopabfv sylib (= (cv y) (<,> D R)) adantr (:--> (cv x) C (X. A B)) z ax-17 xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.26i hban (:--> (cv x) C (X. A B)) w ax-17 xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.27i hban (cv x) C (X. A B) (cv z) ffvrn (` (cv x) (cv z)) A B elxp4 pm3.26bd syl (= (cv y) (<,> D R)) adantlr xpmapen.1 xpmapen.1 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem2 (` (U. (dom ({} (cv y)))) (cv z)) (U. (dom ({} (` (cv x) (cv z))))) (` (U. (ran ({} (cv y)))) (cv z)) (U. (ran ({} (` (cv x) (cv z))))) opeq12 syl (:--> (cv x) C (X. A B)) adantll eqtr4d (cv w) eqeq2d ex pm5.32d opabbid xpmapenlem.6 syl6eqr eqtrd)) thm (xpmapenlem4 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V))) (xpmapenlem.4 (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))))) (xpmapenlem.5 (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))))) (xpmapenlem.6 (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))))))) (-> (/\ (/\ (= (cv y) (<,> (U. (dom ({} (cv y)))) (U. (ran ({} (cv y)))))) (/\ (:--> (U. (dom ({} (cv y)))) C A) (:--> (U. (ran ({} (cv y)))) C B))) (= (cv x) S)) (/\ (:--> (cv x) C (X. A B)) (= (cv y) (<,> D R)))) (xpmapenlem.6 (cv x) S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)))))) eqtrt (cv x) ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)))))) C (X. A B) feq1 syl mpan2 (U. (ran ({} (cv y)))) (cv z) fvex (` (U. (dom ({} (cv y)))) (cv z)) A B opelxp z C ralbii ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)))))) eqid (X. A B) fopab2 z C (e. (` (U. (dom ({} (cv y)))) (cv z)) A) (e. (` (U. (ran ({} (cv y)))) (cv z)) B) r19.26 3bitr3r syl6rbbr biimpac (U. (dom ({} (cv y)))) C A (cv z) ffvrn r19.21aiva (U. (ran ({} (cv y)))) C B (cv z) ffvrn r19.21aiva anim12i sylan (= (cv y) (<,> (U. (dom ({} (cv y)))) (U. (ran ({} (cv y)))))) adantll (cv y) (<,> (U. (dom ({} (cv y)))) (U. (ran ({} (cv y))))) (<,> D R) eqeq1 (U. (dom ({} (cv y)))) C A z w fopabfv pm3.26bd x z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))) hbopab1 xpmapenlem.6 (cv x) eleq2i xpmapenlem.6 (cv x) eleq2i z albii 3imtr4 hbeleq x z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))) hbopab2 xpmapenlem.6 (cv x) eleq2i xpmapenlem.6 (cv x) eleq2i w albii 3imtr4 hbeleq xpmapenlem.6 (cv x) S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)))))) eqtrt mpan2 (cv z) fveq1d (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)) opex z C (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))) (V) w fvopab2 mpan2 sylan9eq sneqd dmeqd unieqd (U. (dom ({} (cv y)))) (cv z) fvex (` (U. (ran ({} (cv y)))) (cv z)) op1sta syl6eq (cv w) eqeq2d ex pm5.32d opabbid xpmapenlem.4 syl5req sylan9eq (:--> (U. (ran ({} (cv y)))) C B) adantlr (U. (ran ({} (cv y)))) C B z w fopabfv pm3.26bd x z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))) hbopab1 xpmapenlem.6 (cv x) eleq2i xpmapenlem.6 (cv x) eleq2i z albii 3imtr4 hbeleq x z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))) hbopab2 xpmapenlem.6 (cv x) eleq2i xpmapenlem.6 (cv x) eleq2i w albii 3imtr4 hbeleq xpmapenlem.6 (cv x) S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)))))) eqtrt mpan2 (cv z) fveq1d (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)) opex z C (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))) (V) w fvopab2 mpan2 sylan9eq sneqd rneqd unieqd (U. (dom ({} (cv y)))) (cv z) fvex (U. (ran ({} (cv y)))) (cv z) fvex op2nda syl6eq (cv w) eqeq2d ex pm5.32d opabbid xpmapenlem.5 syl5req sylan9eq (:--> (U. (dom ({} (cv y)))) C A) adantll jca (cv y) snex ({} (cv y)) (V) dmexg ax-mp uniex (cv y) snex ({} (cv y)) (V) rnexg ax-mp uniex xpmapenlem.5 xpmapen.3 z w (U. (ran ({} (` (cv x) (cv z))))) funopabex2 eqeltr D opth sylibr syl5bir exp3a imp31 jca)) thm (xpmapenlem5 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V))) (xpmapenlem.4 (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))))) (xpmapenlem.5 (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))))) (xpmapenlem.6 (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))))))) (br (opr (X. A B) (^m) C) (~~) (X. (opr A (^m) C) (opr B (^m) C))) ((X. A B) (^m) C oprex D R opex (e. (cv x) (opr (X. A B) (^m) C)) a1i xpmapenlem.6 xpmapen.3 z w (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))) funopabex2 eqeltr (e. (cv y) (X. (opr A (^m) C) (opr B (^m) C))) a1i xpmapenlem.5 xpmapen.3 z w (U. (ran ({} (` (cv x) (cv z))))) funopabex2 eqeltr D (V) (V) opelxp xpmapenlem.4 xpmapen.3 z w (U. (dom ({} (` (cv x) (cv z))))) funopabex2 eqeltr xpmapenlem.5 xpmapen.3 z w (U. (ran ({} (` (cv x) (cv z))))) funopabex2 eqeltr mpbir2an (cv y) (<,> D R) (X. (V) (V)) eleq1 mpbiri (:--> (cv x) C (X. A B)) adantl (cv y) (V) (V) elxp4 pm3.26bd syl (cv y) (<,> D R) sneq dmeqd unieqd xpmapenlem.4 xpmapen.3 z w (U. (dom ({} (` (cv x) (cv z))))) funopabex2 eqeltr R op1sta syl6eq xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.26i xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.27i xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem2 pm3.26d (cv w) eqeq2d ex pm5.32d opabbid xpmapenlem.4 syl6eqr eqtr4d (cv y) (<,> D R) sneq rneqd unieqd xpmapenlem.4 xpmapen.3 z w (U. (dom ({} (` (cv x) (cv z))))) funopabex2 eqeltr xpmapenlem.5 xpmapen.3 z w (U. (ran ({} (` (cv x) (cv z))))) funopabex2 eqeltr op2nda syl6eq xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.26i xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.27i xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem2 pm3.27d (cv w) eqeq2d ex pm5.32d opabbid xpmapenlem.5 syl6eqr eqtr4d jca (:--> (cv x) C (X. A B)) adantl (cv x) C (X. A B) (cv z) ffvrn r19.21aiva (= (cv y) (<,> D R)) adantr (:--> (cv x) C (X. A B)) z ax-17 xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.26i hban xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem3 (cv z) fveq1d (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)) opex z C (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))) (V) w fvopab2 mpan2 xpmapenlem.6 (cv z) fveq1i syl5eq sylan9eq (X. A B) eleq1d (U. (ran ({} (cv y)))) (cv z) fvex (` (U. (dom ({} (cv y)))) (cv z)) A B opelxp syl6bb ralbida z C (e. (` (U. (dom ({} (cv y)))) (cv z)) A) (e. (` (U. (ran ({} (cv y)))) (cv z)) B) r19.26 syl6bb mpbid jca (= (U. (dom ({} (cv y)))) ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (` (U. (dom ({} (cv y)))) (cv z)))))) (= (U. (ran ({} (cv y)))) ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (` (U. (ran ({} (cv y)))) (cv z)))))) (A.e. z C (e. (` (U. (dom ({} (cv y)))) (cv z)) A)) (A.e. z C (e. (` (U. (ran ({} (cv y)))) (cv z)) B)) an4 (U. (dom ({} (cv y)))) C A z w fopabfv (U. (ran ({} (cv y)))) C B z w fopabfv anbi12i bitr4 sylib jca xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem3 jca xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem4 impbi xpmapen.1 xpmapen.2 xpex xpmapen.3 (cv x) elmap (= (cv y) (<,> D R)) anbi1i (cv y) (opr A (^m) C) (opr B (^m) C) elxp4 xpmapen.1 xpmapen.3 (U. (dom ({} (cv y)))) elmap xpmapen.2 xpmapen.3 (U. (ran ({} (cv y)))) elmap anbi12i (= (cv y) (<,> (U. (dom ({} (cv y)))) (U. (ran ({} (cv y)))))) anbi2i bitr (= (cv x) S) anbi1i 3bitr4 en2)) thm (xpmapen ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V)))) (br (opr (X. A B) (^m) C) (~~) (X. (opr A (^m) C) (opr B (^m) C))) (xpmapen.1 xpmapen.2 xpmapen.3 ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))) eqid ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))) eqid ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)))))) eqid xpmapenlem5)) thm (mapunen ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) ((mapunen.1 (e. A (V))) (mapunen.2 (e. B (V))) (mapunen.3 (e. C (V)))) (-> (= (i^i A B) ({/})) (br (opr C (^m) (u. A B)) (~~) (X. (opr C (^m) A) (opr C (^m) B)))) (C (^m) (u. A B) oprex (= (i^i A B) ({/})) a1i A B ssun1 (cv x) (u. A B) C A fssres mpan2 mapunen.3 mapunen.1 (|` (cv x) A) elmap sylibr B A ssun2 (cv x) (u. A B) C B fssres mpan2 mapunen.3 mapunen.2 (|` (cv x) B) elmap sylibr jca mapunen.3 mapunen.1 mapunen.2 unex (cv x) elmap x visset (cv x) (V) B resexg ax-mp (|` (cv x) A) (opr C (^m) A) (opr C (^m) B) opelxp 3imtr4 (= (i^i A B) ({/})) a1i (|^| (|^| (cv y))) A C (U. (ran ({} (cv y)))) B C fun mapunen.3 mapunen.1 mapunen.2 unex (u. (|^| (|^| (cv y))) (U. (ran ({} (cv y))))) elmap C unidm (u. C C) C (u. (|^| (|^| (cv y))) (U. (ran ({} (cv y))))) (u. A B) feq3 ax-mp bitr4 sylibr (cv y) (opr C (^m) A) (opr C (^m) B) elxp5 pm3.27bd mapunen.3 mapunen.1 (|^| (|^| (cv y))) elmap mapunen.3 mapunen.2 (U. (ran ({} (cv y)))) elmap anbi12i sylib sylan expcom (cv y) (opr C (^m) A) (opr C (^m) B) elxp5 (|` (cv x) A) (|^| (|^| (cv y))) (|` (cv x) B) (U. (ran ({} (cv y)))) opeq12 (cv x) (u. (|^| (|^| (cv y))) (U. (ran ({} (cv y))))) A reseq1 (|^| (|^| (cv y))) (U. (ran ({} (cv y)))) A resundir syl6eq (|^| (|^| (cv y))) A fnresdm (|` (U. (ran ({} (cv y)))) A) uneq1d (U. (ran ({} (cv y)))) B A fnresdisj biimpa A B incom ({/}) eqeq1i sylan2b (|^| (|^| (cv y))) uneq2d (|^| (|^| (cv y))) un0 syl6eq sylan9eq anassrs sylan9eqr (cv x) (u. (|^| (|^| (cv y))) (U. (ran ({} (cv y))))) B reseq1 (|^| (|^| (cv y))) (U. (ran ({} (cv y)))) B resundir syl6eq (U. (ran ({} (cv y)))) B fnresdm (|` (|^| (|^| (cv y))) B) uneq2d (|^| (|^| (cv y))) A B fnresdisj biimpa (U. (ran ({} (cv y)))) uneq1d ({/}) (U. (ran ({} (cv y)))) uncom (U. (ran ({} (cv y)))) un0 eqtr syl6eq sylan9eqr an1rs sylan9eqr sylanc (cv y) eqeq2d biimparc exp44 imp mapunen.3 mapunen.1 (|^| (|^| (cv y))) elmap (|^| (|^| (cv y))) A C ffn sylbi mapunen.3 mapunen.2 (U. (ran ({} (cv y)))) elmap (U. (ran ({} (cv y)))) B C ffn sylbi anim12i sylan2 sylbi impcom (e. (cv x) (opr C (^m) (u. A B))) adantrl mapunen.3 mapunen.1 mapunen.2 unex (cv x) elmap (cv x) (u. A B) C ffn sylbi (cv x) (u. A B) fnresdm syl eqcomd (cv y) (<,> (|` (cv x) A) (|` (cv x) B)) inteq inteqd x visset (cv x) (V) A resexg ax-mp (|` (cv x) B) op1stb syl6eq (cv y) (<,> (|` (cv x) A) (|` (cv x) B)) sneq rneqd unieqd x visset (cv x) (V) A resexg ax-mp x visset (cv x) (V) B resexg ax-mp op2nda syl6eq uneq12d (cv x) A B resundi syl6reqr sylan9eq ex (= (i^i A B) ({/})) (e. (cv y) (X. (opr C (^m) A) (opr C (^m) B))) ad2antrl impbid ex en3d)) thm (pwen () ((pwen.1 (e. A (V))) (pwen.2 (e. B (V)))) (-> (br A (~~) B) (br (P~ A) (~~) (P~ B))) (2on (2o) (On) enrefg ax-mp 2on elisseti 2on elisseti pwen.1 pwen.2 mapen mpan (2o) (^m) A oprex pwen.1 pw2en (opr (2o) (^m) A) (V) (P~ A) (P~ B) enen1 mp2an (2o) (^m) B oprex pwen.2 pw2en (opr (2o) (^m) B) (V) (P~ B) (opr (2o) (^m) A) enen2 mp2an bitr2 sylib)) thm (ssenen ((x y) (x z) (f x) (A x) (y z) (f y) (A y) (f z) (A z) (A f) (B x) (B y) (B z) (B f) (C x) (C y) (C z) (C f)) ((ssenen.1 (e. A (V))) (ssenen.2 (e. B (V))) (ssenen.3 (e. C (V)))) (-> (br A (~~) B) (br ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) C))) (~~) ({|} x (/\ (C_ (cv x) B) (br (cv x) (~~) C))))) (ssenen.2 A f bren ssenen.1 pwex ({|} x (br (cv x) (~~) C)) inex1 (:-1-1-onto-> (cv f) A B) a1i (" (cv f) (cv y)) (cv y) C entrt y visset (cv f) A B f1imaen (cv f) A B f1of1 sylan sylan exp31 imp3a (cv f) A B f1ofo (cv f) (cv y) imassrn (cv f) A B forn (" (cv f) (cv y)) sseq2d mpbii syl jctild (cv y) (P~ A) ({|} x (br (cv x) (~~) C)) elin y visset A elpw y visset (cv x) (cv y) (~~) C breq1 elab anbi12i bitr (" (cv f) (cv y)) (P~ B) ({|} x (br (cv x) (~~) C)) elin f visset (cv f) (V) (cv y) imaexg ax-mp B elpw f visset (cv f) (V) (cv y) imaexg ax-mp (cv x) (" (cv f) (cv y)) (~~) C breq1 elab anbi12i bitr 3imtr4g (cv f) A B f1ocnv (" (`' (cv f)) (cv z)) (cv z) C entrt z visset (`' (cv f)) B A f1imaen (`' (cv f)) B A f1of1 sylan sylan exp31 imp3a (`' (cv f)) B A f1ofo (`' (cv f)) (cv z) imassrn (`' (cv f)) B A forn (" (`' (cv f)) (cv z)) sseq2d mpbii syl jctild syl (cv z) (P~ B) ({|} x (br (cv x) (~~) C)) elin z visset B elpw z visset (cv x) (cv z) (~~) C breq1 elab anbi12i bitr (" (`' (cv f)) (cv z)) (P~ A) ({|} x (br (cv x) (~~) C)) elin f visset cnvex (`' (cv f)) (V) (cv z) imaexg ax-mp A elpw f visset cnvex (`' (cv f)) (V) (cv z) imaexg ax-mp (cv x) (" (`' (cv f)) (cv z)) (~~) C breq1 elab anbi12i bitr 3imtr4g (cv y) (" (`' (cv f)) (cv z)) (cv f) imaeq2 (cv f) A B f1orel (cv f) dfrel2 sylib (`' (`' (cv f))) (cv f) (" (`' (cv f)) (cv z)) imaeq1 syl (C_ (cv z) B) adantr (`' (cv f)) B A (cv z) f1imacnv (cv f) A B f1ocnv (`' (cv f)) B A f1of1 syl sylan eqtr3d sylan9eqr eqcomd ex (cv z) (P~ B) ({|} x (br (cv x) (~~) C)) elin (e. (cv z) (P~ B)) (e. (cv z) ({|} x (br (cv x) (~~) C))) pm3.26 z visset B elpw sylib sylbi sylan2 (e. (cv y) (i^i (P~ A) ({|} x (br (cv x) (~~) C)))) adantrl (cv z) (" (cv f) (cv y)) (`' (cv f)) imaeq2 (cv f) A B (cv y) f1imacnv (cv f) A B f1of1 sylan sylan9eqr eqcomd ex (cv y) (P~ A) ({|} x (br (cv x) (~~) C)) elin (e. (cv y) (P~ A)) (e. (cv y) ({|} x (br (cv x) (~~) C))) pm3.26 y visset A elpw sylib sylbi sylan2 (e. (cv z) (i^i (P~ B) ({|} x (br (cv x) (~~) C)))) adantrr impbid ex en3d f 19.23aiv sylbi A x df-pw ({|} x (br (cv x) (~~) C)) ineq1i x (C_ (cv x) A) (br (cv x) (~~) C) inab eqtr B x df-pw ({|} x (br (cv x) (~~) C)) ineq1i x (C_ (cv x) B) (br (cv x) (~~) C) inab eqtr 3brtr3g)) thm (limenpsi ((x y) (A x) (A y)) ((limenpsi.1 (Lim A))) (-> (e. A B) (br A (~~) (\ A ({} ({/}))))) (A (\ A ({} ({/}))) sbth limenpsi.1 A (cv x) limsuc ax-mp biimp (cv x) nsuceq0 x visset sucex ({/}) elsnc mtbir jctir (suc (cv x)) A ({} ({/})) eldif sylibr (cv x) (cv y) suc11 limenpsi.1 A limord ax-mp A (cv x) ordelon mpan limenpsi.1 A limord ax-mp A (cv y) ordelon mpan syl2an B dom2 A ({} ({/})) difss A B (\ A ({} ({/}))) ssdom2g mpi sylanc)) thm (limensuci () ((limensuci.1 (Lim A))) (-> (e. A B) (br A (~~) (suc A))) ((\ A ({} ({/}))) ({} ({/})) incom ({} ({/})) A difdisj eqtr limensuci.1 A limord ax-mp A ordeirr ax-mp A A disjsn mpbir pm3.2i (\ A ({} ({/}))) A ({} ({/})) ({} A) unen mpan2 (\ A ({} ({/}))) (V) A ensymg A B ({} ({/})) difexg limensuci.1 B limenpsi sylc 0ex ({/}) (V) A B en2sn mpan sylanc limensuci.1 A 0ellim ax-mp 0ex A snss mpbi ({} ({/})) A ssundif mpbi ({} ({/})) (\ A ({} ({/}))) uncom eqtr3 A df-suc 3brtr4g)) thm (limensuc () () (-> (/\ (e. A B) (Lim A)) (br A (~~) (suc A))) (A (if (Lim A) A (On)) B eleq1 (= A (if (Lim A) A (On))) id A (if (Lim A) A (On)) suceq (~~) breq12d imbi12d A (if (Lim A) A (On)) limeq (On) (if (Lim A) A (On)) limeq limon elimhyp B limensuci dedth impcom)) thm (phplem1 () () (-> (/\ (e. A (om)) (e. B A)) (= (u. ({} A) (\ A ({} B))) (\ (suc A) ({} B)))) (A B nordeq A B disjsn2 syl A nnord sylan ({} A) ({} B) A undif4 A df-suc A ({} A) uncom eqtr ({} B) difeq1i syl6eqr syl)) thm (phplem2 () ((phplem2.1 (e. A (V))) (phplem2.2 (e. B (V)))) (-> (/\ (e. A (om)) (e. B A)) (br A (~~) (\ (suc A) ({} B)))) (A ({} B) difss (\ A ({} B)) A ({} A) ssrin ax-mp A nnord A orddisj syl (i^i (\ A ({} B)) ({} A)) sseq2d mpbii (i^i (\ A ({} B)) ({} A)) ss0 syl ({} A) (\ A ({} B)) incom syl5eq ({} B) A difdisj jctil phplem2.2 phplem2.1 f1osn B snex ({} (<,> B A)) ({} A) f1oen ax-mp phplem2.1 A ({} B) difss ssexi enref pm3.2i jctil ({} B) ({} A) (\ A ({} B)) (\ A ({} B)) unen syl (e. B A) adantr B A difsnid ({} B) (\ A ({} B)) uncom syl5eq (e. A (om)) adantl A B phplem1 3brtr3d)) thm (phplem3 () ((phplem2.1 (e. A (V))) (phplem2.2 (e. B (V)))) (-> (/\ (e. A (om)) (e. B (suc A))) (br A (~~) (\ (suc A) ({} B)))) (phplem2.1 phplem2.2 phplem2 A nnord A orddif syl A B sneq (suc A) difeq2d eqcoms sylan9eq phplem2.1 enref syl5breq jaodan B A elsuci sylan2)) thm (phplem4 ((A f) (B f)) ((phplem2.1 (e. A (V))) (phplem2.2 (e. B (V)))) (-> (/\ (e. A (om)) (e. B (om))) (-> (br (suc A) (~~) (suc B)) (br A (~~) B))) (A (\ (suc B) ({} (` (cv f) A))) B entrt (cv f) (suc A) (suc B) f1of1 A sssucid jctir (cv f) (suc A) (suc B) A f1ores phplem2.1 (|` (cv f) A) (" (cv f) A) f1oen 3syl (e. A (om)) adantl A nnord A orddif A (\ (suc A) ({} A)) (cv f) imaeq2 3syl (cv f) (suc A) (suc B) f1ofn phplem2.1 sucid (cv f) (suc A) A fnsnfv mpan2 ({} (` (cv f) A)) (" (cv f) ({} A)) (" (cv f) (suc A)) difeq2 3syl (cv f) imadmrn eqcomi (:-1-1-onto-> (cv f) (suc A) (suc B)) a1i (cv f) (suc A) (suc B) f1ofo (cv f) (suc A) (suc B) forn syl (cv f) (suc A) (suc B) f1ofn (cv f) (suc A) fndm (dom (cv f)) (suc A) (cv f) imaeq2 3syl 3eqtr3d ({} (` (cv f) A)) difeq1d (cv f) (suc A) (suc B) f1o3 pm3.27bd (cv f) (suc A) ({} A) imadif syl 3eqtr4rd sylan9eq breqtrd phplem2.2 (cv f) A fvex phplem3 (cv f) (suc A) (suc B) f1ofn phplem2.1 sucid (cv f) (suc A) A fnfvrn mpan2 syl (cv f) (suc A) (suc B) f1ofo (cv f) (suc A) (suc B) forn (` (cv f) A) eleq2d syl mpbid sylan2 phplem2.2 sucex (suc B) ({} (` (cv f) A)) difss ssexi B ensym syl syl2an anandirs ex f 19.23adv phplem2.2 sucex (suc A) f bren syl5ib)) thm (nneneq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~~) B) (= A B))) ((cv z) B A (~~) breq2 (cv z) B A eqeq2 imbi12d (om) rcla4v (cv x) ({/}) (~~) (cv z) breq1 (cv x) ({/}) (cv z) eqeq1 imbi12d z (om) ralbidv (cv x) (cv y) (~~) (cv z) breq1 (cv x) (cv y) (cv z) eqeq1 imbi12d z (om) ralbidv (cv x) (suc (cv y)) (~~) (cv z) breq1 (cv x) (suc (cv y)) (cv z) eqeq1 imbi12d z (om) ralbidv (cv x) A (~~) (cv z) breq1 (cv x) A (cv z) eqeq1 imbi12d z (om) ralbidv z visset ({/}) ensym (cv z) en0 (cv z) ({/}) eqcom bitr sylib (e. (cv z) (om)) a1i rgen (suc (cv y)) en0 (cv w) ({/}) (suc (cv y)) (~~) breq2 (cv w) ({/}) (suc (cv y)) eqeq2 bibi12d mpbiri biimpd (/\ (e. (cv y) (om)) (A.e. z (om) (-> (br (cv y) (~~) (cv z)) (= (cv y) (cv z))))) a1i (e. (cv y) (om)) z ax-17 z (om) (-> (br (cv y) (~~) (cv z)) (= (cv y) (cv z))) hbra1 hban (-> (br (suc (cv y)) (~~) (cv w)) (= (suc (cv y)) (cv w))) z ax-17 y visset z visset phplem4 (= (cv y) (cv z)) imim1d ex a2d z (om) (-> (br (cv y) (~~) (cv z)) (= (cv y) (cv z))) ra4 syl5 imp (cv y) (cv z) suceq syl8 (cv w) (suc (cv z)) (suc (cv y)) (~~) breq2 (cv w) (suc (cv z)) (suc (cv y)) eqeq2 imbi12d biimprcd syl6 r19.23ad jaod ex (cv w) z nn0suc syl7 r19.21adv (cv w) (cv z) (suc (cv y)) (~~) breq2 (cv w) (cv z) (suc (cv y)) eqeq2 imbi12d (om) cbvralv syl6ib finds syl5 impcom A (om) B eqeng (e. B (om)) adantr impbid)) thm (php ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (om)) (C: B A)) (-. (br A (~~) B))) (A x nn0suc orcanai B 0ss ({/}) B A sspsstr mpan A 0pss sylib sylan2 A (suc (cv x)) B psseq2 A (suc (cv x)) (~~) B breq1 negbid imbi12d B (suc (cv x)) y pssnel B (\ (suc (cv x)) ({} (cv y))) (cv x) domentr B (cv y) disjsn B ({} (cv y)) disj3 bitr3 B (\ B ({} (cv y))) (\ (suc (cv x)) ({} (cv y))) sseq1 sylbi B (suc (cv x)) ({} (cv y)) ssdif syl5bir x visset sucex (suc (cv x)) ({} (cv y)) difss ssexi (\ (suc (cv x)) ({} (cv y))) (V) B ssdom2g ax-mp syl6 B (suc (cv x)) pssss syl5 imp x visset y visset phplem3 x visset sucex (suc (cv x)) ({} (cv y)) difss ssexi (cv x) ensym syl syl2an exp43 com4r imp y 19.23aiv mpcom (suc (cv x)) B (cv x) endomtr x visset sucex (cv x) sssucid (suc (cv x)) (V) (cv x) ssdom2g mp2 jctir (suc (cv x)) (cv x) sbth syl expcom (cv x) peano2b (suc (cv x)) nnord sylbi x visset sucid (suc (cv x)) (cv x) nordeq mpan2 syl (suc (cv x)) (cv x) nneneq (cv x) peano2b sylanb anidms mtbird nsyli syli com12 syl5bir com12 r19.23aiv syl ex pm2.43d imp)) thm (php2 ((A x) (B x)) () (-> (/\ (e. A (om)) (C: B A)) (br B (~<) A)) ((cv x) A (om) eleq1 (cv x) A B psseq2 anbi12d (cv x) A B (~<) breq2 imbi12d B (cv x) pssss x visset (cv x) (V) B ssdom2g ax-mp syl (e. (cv x) (om)) adantl (cv x) B php x visset B ensym nsyl jca B (cv x) brsdom sylibr (om) vtoclg anabsi5)) thm (php3 ((A x) (x y) (x z) (f x) (A y) (A z) (A f) (y z) (f y) (f z) (B y) (B z) (B f)) () (-> (/\ (E.e. x (om) (br A (~~) (cv x))) (C: B A)) (br B (~<) A)) (A (V) B ssdom2g imp relen A (cv z) brrelexi B A pssss syl2an (e. (cv z) (om)) adantll (cv z) (" (cv f) B) php B A pssss B A (cv f) imass2 syl (:-1-1-onto-> (cv f) A (cv z)) adantl (cv f) (\ A B) (cv y) funfvima2 (cv f) A (cv z) f1ofun A B difss (cv f) A (cv z) f1ofn (cv f) A fndm (dom (cv f)) A (\ A B) sseq2 3syl mpbiri sylanc (cv f) A (cv z) f1o3 pm3.27bd (cv f) A B imadif syl (` (cv f) (cv y)) eleq2d sylibd (` (cv f) (cv y)) (\ (" (cv f) A) (" (cv f) B)) n0i syl6 (cv y) A B eldif syl5ibr y 19.23adv imp B A y pssnel sylan2 (" (cv f) A) (" (cv f) B) ssdif0 negbii sylibr jca (" (cv f) B) (" (cv f) A) dfpss3 sylibr (cv f) imadmrn (:-1-1-onto-> (cv f) A (cv z)) a1i (cv f) A (cv z) f1ofn (cv f) A fndm (dom (cv f)) A (cv f) imaeq2 3syl (cv f) A (cv z) f1ofo (cv f) A (cv z) forn syl 3eqtr3d (" (cv f) B) psseq2d (C: B A) adantr mpbid sylan2 (cv f) A (cv z) B f1ores (cv f) A (cv z) f1of1 B A pssss syl2an f visset (cv f) (V) B resexg ax-mp (cv y) (|` (cv f) B) B (" (cv f) B) f1oeq1 cla4ev f visset (cv f) (V) B imaexg ax-mp B y bren sylibr (cv z) B (" (cv f) B) entrt expcom 3syl (e. (cv z) (om)) adantl mtod exp32 f 19.23adv z visset A f bren syl5ib imp31 B A (cv z) entrt ex z visset B ensym syl6com (e. (cv z) (om)) (C: B A) ad2antlr mtod jca B A brsdom sylibr exp31 r19.23aiv imp (cv x) (cv z) A (~~) breq2 (om) cbvrexv sylanb)) thm (php4 () () (-> (e. A (om)) (br A (~<) (suc A))) (A (om) sucidg A nnord A ordsuc biimp ancli A (suc A) ordelpss 3syl mpbid (suc A) A php2 A peano2b sylanb mpdan)) thm (php5 () () (-> (e. A (om)) (-. (br A (~~) (suc A)))) (A php4 A (suc A) sdomnen syl)) thm (onomeneq () () (-> (/\ (e. A (On)) (e. B (om))) (<-> (br A (~~) B) (= A B))) (A B nneneq biimpa B php5 (br A (~~) B) adantr B (om) A (suc B) enen1 mtbird (e. A (On)) adantll A B (suc B) endomtr B sssucid B (om) (suc B) ssdomg mpi sylan2 ancoms (C_ (om) A) a1d (e. A (On)) adantll (om) A B ssel com12 (e. A (On)) adantr B (om) A ordelsuc A eloni sylan2 sylibd A (On) (suc B) ssdom2g (e. B (om)) adantl syld ancoms (br A (~~) B) adantr jcad A (suc B) sbth syl6 mtod A eloni ordom (om) A ordtri1 mpan syl con2bid (e. B (om)) (br A (~~) B) ad2antrr mpbird (e. A (On)) (e. B (om)) pm3.27 (br A (~~) B) adantr jca (/\ (e. A (On)) (e. B (om))) (br A (~~) B) pm3.27 sylanc ex A (On) B eqeng (e. B (om)) adantr impbid)) thm (onfin ((A x)) () (-> (e. A (On)) (<-> (E.e. x (om) (br A (~~) (cv x))) (e. A (om)))) (A (cv x) onomeneq (cv x) (om) A eleq1a (e. A (On)) adantl sylbid ex r19.23adv A (om) enrefg (cv x) A A (~~) breq2 (om) rcla4ev mpdan (e. A (On)) a1i impbid)) thm (nndomo () () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~<_) B) (C_ A B))) (A B php2 ex A B domnsym nsyli (e. B (om)) adantr A B ordtri1 B A ordelpss ancoms negbid bitrd A nnord B nnord syl2an sylibrd A (om) B ssdomg (e. B (om)) adantr impbid)) thm (nnsdomo () () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~<) B) (C: A B))) (A B nndomo A B nneneq negbid anbi12d A B brsdom A B dfpss2 3bitr4g)) thm (omsucdom () () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~<) B) (br (suc A) (~<_) B))) (A B ordelpss A nnord sylan A (om) B ordelsuc bitr3d B nnord sylan2 A B nnsdomo (suc A) B nndomo A peano2b sylanb 3bitr4d)) thm (sucdomi () () (-> (/\ (e. A (om)) (e. B C)) (-> (br (suc A) (~<_) B) (br A (~<) B))) (B C A (suc A) sdomdomtr exp3a impcom A php4 sylan)) thm (0sdom1dom ((A x)) ((0sdom1dom.1 (e. A (V)))) (<-> (br ({/}) (~<) A) (br (1o) (~<_) A)) (0sdom1dom.1 0sdom A x n0 bitr (cv x) A snssi 0sdom1dom.1 A (V) ({} (cv x)) ssdom2g ax-mp 1on elisseti x visset ensn1 ensymi (1o) ({} (cv x)) A endomtr mpan 3syl x 19.23aiv sylbi df-1o (~<_) A breq1i peano1 0sdom1dom.1 ({/}) A (V) sucdomi mp2an sylbi impbi)) thm (finsucdom ((A x) (B x)) () (-> (/\ (e. 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(cv y) (om)) w ax-17 w (-> (C: (cv w) (cv y)) (E.e. x (cv y) (br (cv w) (~~) (cv x)))) hba1 z y w elequ1 biimpcd con3d (C: (cv w) (suc (cv y))) adantl (cv w) (suc (cv y)) pssss (cv z) sseld (cv z) (cv y) elsuci ord con1d syl6 imp syld ex com23 imp ssrdv (-. (= (cv w) (cv y))) anim1i (cv w) (cv y) dfpss2 sylibr (cv x) (cv y) elelsuc (br (cv w) (~~) (cv x)) anim1i r19.22i2 imim12i exp4c w a4s (e. (cv y) (om)) adantl com4t (cv y) nnord (cv y) orddif syl (\ (cv w) ({} (cv y))) sseq2d (cv w) (suc (cv y)) ({} (cv y)) ssdif syl5bir (cv w) (suc (cv y)) pssss syl5 (\ (cv w) ({} (cv y))) (cv y) (cv z) eleq2 (cv z) (cv w) ({} (cv y)) eldifi syl6bir (/\ (e. (cv y) (cv w)) (e. (cv z) (suc (cv y)))) adantl (cv z) (cv y) elsuci ord (cv y) (cv w) (cv z) eleq1a sylan9r (= (\ (cv w) ({} (cv y))) (cv y)) adantr pm2.61d ex con3d ex imp3a z 19.23adv (cv w) (suc (cv y)) z pssnel syl5 im2anan9r (C: (cv w) (suc (cv y))) anidm syl5ibr (\ (cv w) ({} (cv y))) (cv y) dfpss2 syl6ibr (cv w) (cv z) (cv y) psseq1 (cv w) (cv z) (~~) (cv x) breq1 x (cv y) rexbidv imbi12d cbvalv w visset (cv w) ({} (cv y)) difss ssexi (cv z) (\ (cv w) ({} (cv y))) (cv y) psseq1 (cv z) (\ (cv w) ({} (cv y))) (~~) (cv x) breq1 x (cv y) rexbidv imbi12d cla4v sylbi sylan9 (cv y) nnord (cv y) (cv x) ordsucelsuc biimpd syl (e. (cv y) (cv w)) adantl (br (\ (cv w) ({} (cv y))) (~~) (cv x)) adantrd (cv y) (cv w) difsnid eqcomd (cv x) df-suc (e. (cv y) (cv w)) a1i (~~) breq12d (\ (cv w) ({} (cv y))) (cv x) ({} (cv y)) ({} (cv x)) unen y visset x visset f1osn (cv y) snex ({} (<,> (cv y) (cv x))) ({} (cv x)) f1oen ax-mp (br (\ (cv w) ({} (cv y))) (~~) (cv x)) jctr (cv x) nnord (cv x) orddisj syl ({} (cv y)) (\ (cv w) ({} (cv y))) incom ({} (cv y)) (cv w) difdisj eqtr3 jctil syl2an syl5bir (cv x) (cv y) elnn sylan2i exp4d com24 imp4b jcad (cv z) (suc (cv x)) (cv w) (~~) breq2 (suc (cv y)) rcla4ev syl6 x 19.23adv x (cv y) (br (\ (cv w) ({} (cv y))) (~~) (cv x)) df-rex (cv x) (cv z) (cv w) (~~) breq2 (suc (cv y)) cbvrexv 3imtr4g (A. w (-> (C: (cv w) (cv y)) (E.e. x (cv y) (br (cv w) (~~) (cv x))))) adantr syld exp31 imp3a w visset (cv y) eqelsuc w visset enref jctir (cv x) (cv w) (cv w) (~~) breq2 (suc (cv y)) rcla4ev syl (C: (cv w) (suc (cv y))) a1d (/\ (e. (cv y) (om)) (A. w (-> (C: (cv w) (cv y)) (E.e. x (cv y) (br (cv w) (~~) (cv x)))))) a1d pm2.61ii ex 19.21ad finds syl5 com3l imp mpd)) thm (ssnn ((A x) (B x)) () (-> (/\ (e. A (om)) (C_ B A)) (E.e. x (om) (br B (~~) (cv x)))) (A B x pssnn (cv x) A elnn expcom (br B (~~) (cv x)) anim1d x 19.22dv x A (br B (~~) (cv x)) df-rex x (om) (br B (~~) (cv x)) df-rex 3imtr4g (C: B A) adantr mpd B A (om) eleq1 biimparc B (om) enrefg ancli (cv x) B B (~~) breq2 (om) rcla4ev 3syl jaodan B A sspss sylan2b)) thm (ssfi ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (E.e. x (om) (br A (~~) (cv x))) (C_ B A)) (E.e. x (om) (br B (~~) (cv x)))) ((cv x) (om) A z breng (cv x) (" (cv z) B) y ssnn (cv z) A (cv x) f1ofo (cv z) B imassrn (cv z) A (cv x) forn (" (cv z) B) sseq2d mpbii syl sylan2 (C_ B A) adantrr B (" (cv z) B) (cv y) entrt z visset (cv z) (V) B resexg ax-mp (cv x) (|` (cv z) B) B (" (cv z) B) f1oeq1 cla4ev z visset (cv z) (V) B imaexg ax-mp B x bren sylibr sylan (cv z) A (cv x) B f1ores (cv z) A (cv x) f1of1 sylan sylan ex y (om) r19.22sdv (e. (cv x) (om)) adantl mpd exp32 z 19.23adv sylbid r19.23aiv imp (cv y) (cv x) B (~~) breq2 (om) cbvrexv sylib)) thm (domfi ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (E.e. x (om) (br A (~~) (cv x))) (br B (~<_) A)) (E.e. x (om) (br B (~~) (cv x)))) (relen A (cv x) brrelexi (e. (cv x) (om)) a1i r19.23aiv A (V) B y domeng syl y visset (cv y) (V) B (cv x) enen1 mpan x (om) rexbidv x A (cv y) ssfi syl5bir exp3a com12 imp3a y 19.23adv sylbid imp)) thm (unblem1 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (C_ B (om)) (A.e. x (om) (E.e. y B (e. (cv x) (cv y))))) (e. A B)) (e. (|^| (\ B (suc A))) B)) (omsson B (om) (On) sstr mpan2 B (suc A) difss (\ B (suc A)) B (On) sstr mpan syl (A.e. x (om) (E.e. y B (e. (cv x) (cv y)))) (e. A B) ad2antrr B (om) (cv y) ssel (cv y) nnord (cv y) (suc A) ordn2lp (e. (cv y) (suc A)) (e. (suc A) (cv y)) imnan sylibr con2d syl syl6 imdistand (cv y) B (suc A) eldif (cv y) (\ B (suc A)) n0i sylbir syl6 exp3a r19.23adv (cv x) (suc A) (cv y) eleq1 y B rexbidv (om) rcla4cva syl5 B (om) A ssel A peano2b syl6ib sylan2d exp3a imp31 jca (\ B (suc A)) onint (|^| (\ B (suc A))) B (suc A) eldifi 3syl)) thm (unblem2 ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (F z) (F w) (F v) (F u)) ((unblem.1 (-> (e. (cv w) F) (A. x (e. (cv w) F)))) (unblem.2 (= F (|` (rec ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) (|^| A)) (om))))) (-> (/\ (C_ A (om)) (A.e. w (om) (E.e. v A (e. (cv w) (cv v))))) (-> (e. (cv z) (om)) (e. (` F (cv z)) A))) ((cv z) ({/}) F fveq2 A eleq1d (cv z) (cv u) F fveq2 A eleq1d (cv z) (suc (cv u)) F fveq2 A eleq1d A onint omsson A (om) (On) sstr mpan2 peano1 (cv w) ({/}) (cv v) eleq1 v A rexbidv (om) rcla4v ax-mp v A (e. ({/}) (cv v)) df-rex sylib (e. (cv v) A) (e. ({/}) (cv v)) pm3.26 v 19.22i syl A v n0 sylibr syl2an (|^| A) A ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) fr0t unblem.2 ({/}) fveq1i syl5req A eleq1d ibi syl (e. (cv w) (|^| A)) x ax-17 (e. (cv w) (cv u)) x ax-17 (e. (cv w) A) x ax-17 unblem.1 (e. (cv w) (cv u)) x ax-17 hbfv hbsuc hbdif hbint unblem.2 (cv x) (` F (cv u)) suceq A difeq2d inteqd A frsucopab eqcomd A eleq1d ex ibd A w v (` F (cv u)) unblem1 syl5 exp3a finds2 com12)) thm (unblem3 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F z) (F w) (F v)) ((unblem.1 (-> (e. (cv w) F) (A. x (e. (cv w) F)))) (unblem.2 (= F (|` (rec ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) (|^| A)) (om))))) (-> (/\ (C_ A (om)) (A.e. w (om) (E.e. v A (e. (cv w) (cv v))))) (-> (e. (cv z) (om)) (e. (` F (cv z)) (` F (suc (cv z)))))) (unblem.1 unblem.2 v z unblem2 imp omsson A (om) (On) sstr mpan2 A (On) (` F (cv z)) ssel anc2li syl (A.e. w (om) (E.e. v A (e. (cv w) (cv v)))) (e. (cv z) (om)) ad2antrr mpd A (` F (cv z)) onmindif syl unblem.1 unblem.2 v z unblem2 A w v (` F (cv z)) unblem1 ex syld (e. (cv w) (|^| A)) x ax-17 (e. (cv w) (cv z)) x ax-17 (e. (cv w) A) x ax-17 unblem.1 (e. (cv w) (cv z)) x ax-17 hbfv hbsuc hbdif hbint unblem.2 (cv x) (` F (cv z)) suceq A difeq2d inteqd A frsucopab ex sylcom imp eleqtrrd ex)) thm (unblem4 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F z) (F w) (F v)) ((unblem.1 (-> (e. (cv w) F) (A. x (e. (cv w) F)))) (unblem.2 (= F (|` (rec ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) (|^| A)) (om))))) (-> (/\ (C_ A (om)) (A.e. w (om) (E.e. v A (e. (cv w) (cv v))))) (:-1-1-> F (om) A)) (A F z omsmo omsson A (om) (On) sstr mpan2 (A.e. w (om) (E.e. v A (e. (cv w) (cv v)))) adantr unblem.1 unblem.2 v z unblem2 r19.21aiv ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) (|^| A) frfnom unblem.2 F (|` (rec ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) (|^| A)) (om)) (om) fneq1 ax-mp mpbir F (om) A z ffnfv biimpr mpan syl jca unblem.1 unblem.2 v z unblem3 r19.21aiv sylanc)) thm (unbnn ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) ((unbnn.1 (e. A (V)))) (-> (/\ (C_ A (om)) (A.e. x (om) (E.e. y A (e. (cv x) (cv y))))) (br A (~~) (om))) (A (om) sbth unbnn.1 A (V) (om) ssdomg ax-mp (A.e. x (om) (E.e. y A (e. (cv x) (cv y)))) adantr x z w (= (cv w) (|^| (\ A (suc (cv z))))) hbopab1 (e. (cv x) (|^| A)) z ax-17 hbrdg (e. (cv x) (om)) z ax-17 hbres (|` (rec ({<,>|} z w (= (cv w) (|^| (\ A (suc (cv z)))))) (|^| A)) (om)) eqid y unblem4 unbnn.1 A (V) (|` (rec ({<,>|} z w (= (cv w) (|^| (\ A (suc (cv z)))))) (|^| A)) (om)) (om) f1dom2g ax-mp syl sylanc)) thm (unbnn2 ((x y) (x z) (A x) (y z) (A y) (A z)) ((unbnn.1 (e. A (V)))) (-> (/\ (C_ A (om)) (A.e. x (om) (E.e. y A (C_ (cv x) (cv y))))) (br A (~~) (om))) (unbnn.1 z y unbnn (cv x) (suc (cv z)) (cv y) sseq1 y A rexbidv (om) rcla4v z visset (cv z) (V) (cv y) sucssel ax-mp y A r19.22si syl6com (cv z) peano2 syl5 r19.21aiv sylan2)) thm (isfinite2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (br A (~<) (om)) (E.e. x (om) (br A (~~) (cv x)))) (A (om) sdomex pm3.27d (om) (V) A y domeng A (om) sdomdom syl5bi y visset z w unbnn ex (cv y) (om) sdomnen nsyli (cv y) A (om) ensdomtr y visset A ensym sylan syl5 (cv w) (cv z) ordtri1 (cv y) (On) (cv w) ssel2 w visset elon sylib sylan an1rs ralbidva (cv y) (cv z) w unissb w (cv y) (e. (cv z) (cv w)) ralnex bicomi 3bitr4g (cv y) (cv z) ordunisssuc bitr3d omsson (cv y) (om) (On) sstr mpan2 (cv z) nnord syl2an (suc (cv z)) (cv y) x ssnn (cv z) peano2b sylanb ex (C_ (cv y) (om)) adantl sylbid ex r19.23adv z (om) (E.e. w (cv y) (e. (cv z) (cv w))) rexnal syl5ibr syld imp A (cv y) (cv x) entrt ex x (om) r19.22sdv (C_ (cv y) (om)) (br A (~<) (om)) ad2antrl mpd exp32 com13 imp3a y 19.23adv sylcom mpcom)) thm (unfilem1 ((x y) (A x) (A y) (B x) (B y)) ((unfilem1.1 (e. A (om))) (unfilem1.2 (e. B (om))) (unfilem1.3 (= F ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x)))))))) (= (ran F) (\ (opr A (+o) B) A)) (x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x)))) rnopab unfilem1.3 rneqi (cv y) (opr A (+o) B) A eldif unfilem1.1 unfilem1.2 A B nnacl mp2an (cv y) (opr A (+o) B) elnn mpan2 unfilem1.1 A (cv y) ordtri1 A nnord (cv y) nnord syl2an A (cv y) x nnawordex bitr3d mpan x (om) (= (opr A (+o) (cv x)) (cv y)) df-rex syl6bb syl unfilem1.2 unfilem1.1 (cv x) B A nnaord mp3an23 (opr A (+o) (cv x)) (cv y) (opr A (+o) B) eleq1 sylan9bb biimprcd (opr A (+o) (cv x)) (cv y) eqcom biimp (e. (cv x) (om)) adantl (e. (cv y) (opr A (+o) B)) a1i jcad x 19.22dv sylbid imp (cv y) (opr A (+o) (cv x)) (opr A (+o) B) eleq1 (cv y) (opr A (+o) (cv x)) A eleq1 negbid anbi12d biimparc unfilem1.2 (cv x) B elnn mpan2 unfilem1.2 unfilem1.1 (cv x) B A nnaord mp3an23 syl ibi unfilem1.2 (cv x) B elnn mpan2 unfilem1.1 A (cv x) nnaword1 A (cv x) nnacl (opr A (+o) (cv x)) nnord unfilem1.1 A nnord ax-mp A (opr A (+o) (cv x)) ordtri1 mpan 3syl mpbid mpan syl jca sylan x 19.23aiv impbi bitr abbi2i 3eqtr4)) thm (unfilem2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F z) (F w)) ((unfilem1.1 (e. A (om))) (unfilem1.2 (e. B (om))) (unfilem1.3 (= F ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x)))))))) (:-1-1-onto-> F B (\ (opr A (+o) B) A)) (F B (\ (opr A (+o) B) A) df-f1o F B (\ (opr A (+o) B) A) z w f1fv F B (\ (opr A (+o) B) A) df-fo A (+o) (cv x) oprex unfilem1.3 fnopab2 unfilem1.1 unfilem1.2 unfilem1.3 unfilem1 mpbir2an F B (\ (opr A (+o) B) A) fof ax-mp (cv x) (cv z) A (+o) opreq2 unfilem1.3 A (+o) (cv z) oprex fvopab4 (cv x) (cv w) A (+o) opreq2 unfilem1.3 A (+o) (cv w) oprex fvopab4 eqeqan12d unfilem1.1 A (cv z) (cv w) nnacan mp3an1 unfilem1.2 (cv z) B elnn mpan2 unfilem1.2 (cv w) B elnn mpan2 syl2an bitrd biimpd rgen2 mpbir2an F B (\ (opr A (+o) B) A) df-fo A (+o) (cv x) oprex unfilem1.3 fnopab2 unfilem1.1 unfilem1.2 unfilem1.3 unfilem1 mpbir2an mpbir2an)) thm (unfilem3 ((x y) (f x) (A x) (f y) (A y) (A f) (B x) (B y) (B f)) () (-> (/\ (e. A (om)) (e. B (om))) (br B (~~) (\ (opr A (+o) B) A))) ((cv f) ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x))))) B (\ (opr A (+o) B) A) f1oeq1 (V) cla4egv A (if (e. A (om)) A ({/})) (+o) (cv x) opreq1 (cv y) eqeq2d (e. (cv x) B) anbi2d x y opabbidv ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x))))) ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) B (\ (opr A (+o) B) A) f1oeq1 syl A (if (e. A (om)) A ({/})) (+o) B opreq1 A difeq1d A (if (e. A (om)) A ({/})) (opr (if (e. A (om)) A ({/})) (+o) B) difeq2 eqtrd (\ (opr A (+o) B) A) (\ (opr (if (e. A (om)) A ({/})) (+o) B) (if (e. A (om)) A ({/}))) ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) B f1oeq3 syl bitrd B (if (e. B (om)) B ({/})) (cv x) eleq2 (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))) anbi1d x y opabbidv ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) ({<,>|} x y (/\ (e. (cv x) (if (e. B (om)) B ({/}))) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) B (\ (opr (if (e. A (om)) A ({/})) (+o) B) (if (e. A (om)) A ({/}))) f1oeq1 syl B (if (e. B (om)) B ({/})) ({<,>|} x y (/\ (e. (cv x) (if (e. B (om)) B ({/}))) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) (\ (opr (if (e. A (om)) A ({/})) (+o) B) (if (e. A (om)) A ({/}))) f1oeq2 B (if (e. B (om)) B ({/})) (if (e. A (om)) A ({/})) (+o) opreq2 (if (e. A (om)) A ({/})) difeq1d (\ (opr (if (e. A (om)) A ({/})) (+o) B) (if (e. A (om)) A ({/}))) (\ (opr (if (e. A (om)) A ({/})) (+o) (if (e. B (om)) B ({/}))) (if (e. A (om)) A ({/}))) ({<,>|} x y (/\ (e. (cv x) (if (e. B (om)) B ({/}))) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) (if (e. B (om)) B ({/})) f1oeq3 syl 3bitrd peano1 A elimel peano1 B elimel ({<,>|} x y (/\ (e. (cv x) (if (e. B (om)) B ({/}))) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) eqid unfilem2 dedth2h ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x))))) B (\ (opr A (+o) B) A) f1ofn syl ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x))))) B (om) fnex expcom (e. A (om)) adantl mpd A (if (e. A (om)) A ({/})) (+o) (cv x) opreq1 (cv y) eqeq2d (e. (cv x) B) anbi2d x y opabbidv ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x))))) ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) B (\ (opr A (+o) B) A) f1oeq1 syl A (if (e. A (om)) A ({/})) (+o) B opreq1 A difeq1d A (if (e. A (om)) A ({/})) (opr (if (e. A (om)) A ({/})) (+o) B) difeq2 eqtrd (\ (opr A (+o) B) A) (\ (opr (if (e. A (om)) A ({/})) (+o) B) (if (e. A (om)) A ({/}))) ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) B f1oeq3 syl bitrd B (if (e. B (om)) B ({/})) (cv x) eleq2 (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))) anbi1d x y opabbidv ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) ({<,>|} x y (/\ (e. (cv x) (if (e. B (om)) B ({/}))) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) B (\ (opr (if (e. A (om)) A ({/})) (+o) B) (if (e. A (om)) A ({/}))) f1oeq1 syl B (if (e. B (om)) B ({/})) ({<,>|} x y (/\ (e. (cv x) (if (e. B (om)) B ({/}))) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) (\ (opr (if (e. A (om)) A ({/})) (+o) B) (if (e. A (om)) A ({/}))) f1oeq2 B (if (e. B (om)) B ({/})) (if (e. A (om)) A ({/})) (+o) opreq2 (if (e. A (om)) A ({/})) difeq1d (\ (opr (if (e. A (om)) A ({/})) (+o) B) (if (e. A (om)) A ({/}))) (\ (opr (if (e. A (om)) A ({/})) (+o) (if (e. B (om)) B ({/}))) (if (e. A (om)) A ({/}))) ({<,>|} x y (/\ (e. (cv x) (if (e. B (om)) B ({/}))) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) (if (e. B (om)) B ({/})) f1oeq3 syl 3bitrd peano1 A elimel peano1 B elimel ({<,>|} x y (/\ (e. (cv x) (if (e. B (om)) B ({/}))) (= (cv y) (opr (if (e. A (om)) A ({/})) (+o) (cv x))))) eqid unfilem2 dedth2h sylc A (+o) B oprex (opr A (+o) B) (V) A difexg ax-mp B f bren sylibr)) thm (fin2inf () () (-> (br A (~<) (om)) (e. (om) (V))) (A (om) sdomex pm3.27d)) thm (unfi ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (E.e. x (om) (br A (~~) (cv x))) (E.e. x (om) (br B (~~) (cv x)))) (E.e. x (om) (br (u. A B) (~~) (cv x)))) (x (om) y (om) (br A (~~) (cv x)) (br (\ B A) (~~) (cv y)) reeanv A B undif2 (/\ (e. (cv x) (om)) (e. (cv y) (om))) a1i (cv x) (cv y) nnaword1 (cv x) (opr (cv x) (+o) (cv y)) ssundif sylib (~~) breq12d A B difdisj (cv x) (opr (cv x) (+o) (cv y)) difdisj pm3.2i A (cv x) (\ B A) (\ (opr (cv x) (+o) (cv y)) (cv x)) unen mpan2 syl5bi (cv x) (cv y) unfilem3 (\ B A) (cv y) (\ (opr (cv x) (+o) (cv y)) (cv x)) entrt expcom syl sylan2d (cv x) (cv y) nnacl (cv z) (opr (cv x) (+o) (cv y)) (u. A B) (~~) breq2 (om) rcla4ev ex syl syld r19.23aivv sylbir (cv x) (cv y) B (~~) breq2 (om) cbvrexv B A difss y B (\ B A) ssfi mpan2 sylbi sylan2 (cv z) (cv x) (u. A B) (~~) breq2 (om) cbvrexv sylib)) thm (unfi2 ((A x) (B x)) () (-> (/\ (br A (~<) (om)) (br B (~<) (om))) (br (u. A B) (~<) (om))) (x A B unfi A x isfinite2 B x isfinite2 syl2an x (u. A B) isfinite1 syl A fin2inf (om) (V) (u. A B) ensymg syl con3d (br B (~<) (om)) adantr (br (u. A B) (~<_) (om)) anim2d mpd (u. A B) (om) brsdom sylibr)) thm (infcntss ((A x)) ((infcntss.1 (e. A (V)))) (-> (br (om) (~<_) A) (E. x (/\ (C_ (cv x) A) (br (cv x) (~~) (om))))) (infcntss.1 (om) x domen x visset (om) ensym (C_ (cv x) A) anim2i ancoms x 19.22i sylbi)) thm (prfi ((A x) (B x)) () (E.e. x (om) (br ({,} A B) (~~) (cv x))) (x A snfi x B snfi x ({} A) ({} B) unfi mp2an A B df-pr (~~) (cv x) breq1i x (om) rexbii mpbir)) thm (unifi ((f k) (f m) (f n) (f x) (f y) (f z) (A f) (k m) (k n) (k x) (k y) (k z) (A k) (m n) (m x) (m y) (m z) (A m) (n x) (n y) (n z) (A n) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (E.e. n (om) (br A (~~) (cv n))) (A.e. x A (E.e. n (om) (br (cv x) (~~) (cv n))))) (E.e. n (om) (br (U. A) (~~) (cv n)))) ((cv n) (cv m) A (~~) breq2 (om) cbvrexv (cv m) ({/}) (cv y) (~~) breq2 (A.e. x (cv y) (E.e. n (om) (br (cv x) (~~) (cv n)))) anbi1d (E.e. n (om) (br (U. (cv y)) (~~) (cv n))) imbi1d y albidv (cv m) (cv k) (cv y) (~~) breq2 (A.e. x (cv y) (E.e. n (om) (br (cv x) (~~) (cv n)))) anbi1d (E.e. n (om) (br (U. (cv y)) (~~) (cv n))) imbi1d y albidv (cv m) (suc (cv k)) (cv y) (~~) breq2 (A.e. x (cv y) (E.e. n (om) (br (cv x) (~~) (cv n)))) anbi1d (E.e. n (om) (br (U. (cv y)) (~~) (cv n))) imbi1d y albidv (cv y) en0 (cv y) ({/}) unieq uni0 0ex enref eqbrtr syl6eqbr peano1 (cv n) ({/}) (U. (cv y)) (~~) breq2 (om) rcla4ev mpan syl sylbi (A.e. x (cv y) (E.e. n (om) (br (cv x) (~~) (cv n)))) adantr y ax-gen (cv y) (cv z) (~~) (cv k) breq1 (cv y) (cv z) x (E.e. n (om) (br (cv x) (~~) (cv n))) raleq1 anbi12d (cv y) (cv z) unieq (~~) (cv n) breq1d n (om) rexbidv imbi12d cbvalv k visset sucex (cv y) ensym y visset (suc (cv k)) f bren sylib n (U. (" (cv f) (cv k))) (` (cv f) (cv k)) unfi f visset (cv f) (V) (cv k) imaexg ax-mp (cv z) (" (cv f) (cv k)) (~~) (cv k) breq1 (cv z) (" (cv f) (cv k)) x (E.e. n (om) (br (cv x) (~~) (cv n))) raleq1 anbi12d (cv z) (" (cv f) (cv k)) unieq (~~) (cv n) breq1d n (om) rexbidv imbi12d cla4v imp (cv f) (suc (cv k)) (cv y) f1of1 (cv k) sssucid (cv f) (suc (cv k)) (cv y) (cv k) f1ores mpan2 k visset (|` (cv f) (cv k)) (" (cv f) (cv k)) f1oen 3syl f visset (cv f) (V) (cv k) imaexg ax-mp (cv k) ensym syl (A.e. x (cv y) (E.e. n (om) (br (cv x) (~~) (cv n)))) adantr (" (cv f) (cv k)) ({} (` (cv f) (cv k))) ssun1 (:-1-1-onto-> (cv f) (suc (cv k)) (cv y)) a1i (cv f) (suc (cv k)) (cv y) f1ofn k visset sucid (cv f) (suc (cv k)) (cv k) fnsnfv mpan2 syl (" (cv f) (cv k)) uneq2d (cv k) df-suc (suc (cv k)) (u. (cv k) ({} (cv k))) (cv f) imaeq2 ax-mp (cv f) (cv k) ({} (cv k)) imaun eqtr2 syl6eq (cv f) (suc (cv k)) (cv y) f1ofo (cv f) (suc (cv k)) (cv y) foima syl eqtrd sseqtrd (" (cv f) (cv k)) (cv y) x (E.e. n (om) (br (cv x) (~~) (cv n))) ssralv syl imp jca sylan2 an1s (cv x) (` (cv f) (cv k)) (~~) (cv n) breq1 n (om) rexbidv (cv y) rcla4va (cv f) (suc (cv k)) (cv y) f1of k visset sucid (cv f) (suc (cv k)) (cv y) (cv k) ffvrn mpan2 syl sylan (A. z (-> (/\ (br (cv z) (~~) (cv k)) (A.e. x (cv z) (E.e. n (om) (br (cv x) (~~) (cv n))))) (E.e. n (om) (br (U. (cv z)) (~~) (cv n))))) adantrl sylanc (cv f) (suc (cv k)) (cv y) f1ofn k visset sucid (cv f) (suc (cv k)) (cv k) fnsnfv mpan2 syl (" (cv f) (cv k)) uneq2d (cv k) df-suc (suc (cv k)) (u. (cv k) ({} (cv k))) (cv f) imaeq2 ax-mp (cv f) (cv k) ({} (cv k)) imaun eqtr2 syl6eq (cv f) (suc (cv k)) (cv y) f1ofo (cv f) (suc (cv k)) (cv y) foima syl eqtrd unieqd (" (cv f) (cv k)) ({} (` (cv f) (cv k))) uniun (cv f) (cv k) fvex unisn (U. (" (cv f) (cv k))) uneq2i eqtr2 syl5eq (~~) (cv n) breq1d n (om) rexbidv (/\ (A. z (-> (/\ (br (cv z) (~~) (cv k)) (A.e. x (cv z) (E.e. n (om) (br (cv x) (~~) (cv n))))) (E.e. n (om) (br (U. (cv z)) (~~) (cv n))))) (A.e. x (cv y) (E.e. n (om) (br (cv x) (~~) (cv n))))) adantr mpbid exp32 f 19.23aiv syl com12 imp3a y 19.21aiv sylbi (e. (cv k) (om)) a1i finds1 relen A (cv m) brrelexi (cv y) A (~~) (cv m) breq1 (cv y) A x (E.e. n (om) (br (cv x) (~~) (cv n))) raleq1 anbi12d (cv y) A unieq (~~) (cv n) breq1d n (om) rexbidv imbi12d (V) cla4gv exp4a syl pm2.43b syl r19.23aiv sylbi imp)) thm (unifi2 ((n x) (A n) (A x)) () (-> (/\ (br A (~<) (om)) (A.e. x A (br (cv x) (~<) (om)))) (br (U. A) (~<) (om))) (n A x unifi A n isfinite2 (cv x) n isfinite2 x A r19.20si syl2an n (U. A) isfinite1 syl A fin2inf (om) (V) (U. A) ensymg syl con3d (A.e. x A (br (cv x) (~<) (om))) adantr (br (U. A) (~<_) (om)) anim2d mpd (U. A) (om) brsdom sylibr)) thm (fiint ((x y) (x z) (w x) (v x) (f x) (A x) (y z) (w y) (v y) (f y) (A y) (w z) (v z) (f z) (A z) (v w) (f w) (A w) (f v) (A v) (A f)) () (<-> (A.e. x A (A.e. y A (e. (i^i (cv x) (cv y)) A))) (A. x (-> (/\/\ (C_ (cv x) A) (-. (= (cv x) ({/}))) (E.e. y (om) (br (cv x) (~~) (cv y)))) (e. (|^| (cv x)) A)))) ((cv y) ({/}) (~~) (cv x) breq1 (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) anbi2d (e. (|^| (cv x)) A) imbi1d x albidv (cv y) (cv v) (~~) (cv x) breq1 (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) anbi2d (e. (|^| (cv x)) A) imbi1d x albidv (cv y) (suc (cv v)) (~~) (cv x) breq1 (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) anbi2d (e. (|^| (cv x)) A) imbi1d x albidv x visset ({/}) ensym (cv x) en0 sylib (-. (= (cv x) ({/}))) anim1i ancoms (C_ (cv x) A) adantll (= (cv x) ({/})) pm3.24 (e. (|^| (cv x)) A) pm2.21i syl x ax-gen (A.e. z A (A.e. w A (e. (i^i (cv z) (cv w)) A))) a1i (A.e. z A (A.e. w A (e. (i^i (cv z) (cv w)) A))) x ax-17 x (-> (/\ (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (br (cv v) (~~) (cv x))) (e. (|^| (cv x)) A)) hba1 (cv x) A (` (cv f) (cv v)) ssel (cv f) (suc (cv v)) (cv x) f1ofo (cv f) (suc (cv v)) (cv x) fof v visset sucid (cv f) (suc (cv v)) (cv x) (cv v) ffvrn mpan2 3syl syl5 imp (/\ (A. x (-> (/\ (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (br (cv v) (~~) (cv x))) (e. (|^| (cv x)) A))) (A.e. z A (A.e. w A (e. (i^i (cv z) (cv w)) A)))) adantr (cv f) (cv v) imassrn (cv f) (suc (cv v)) (cv x) f1o2 (Fn (cv f) (suc (cv v))) (Fun (`' (cv f))) (= (ran (cv f)) (cv x)) 3simp3 sylbi (" (cv f) (cv v)) sseq2d mpbii (" (cv f) (cv v)) (cv x) A sstr2 syl (-. (= (" (cv f) (cv v)) ({/}))) anim1d (cv f) (suc (cv v)) (cv x) f1of1 (cv v) sssucid (cv f) (suc (cv v)) (cv x) (cv v) f1ores mpan2 v visset (|` (cv f) (cv v)) (" (cv f) (cv v)) f1oen 3syl jctird f visset (cv f) (V) (cv v) imaexg ax-mp (cv x) (" (cv f) (cv v)) A sseq1 (cv x) (" (cv f) (cv v)) ({/}) eqeq1 negbid anbi12d (cv x) (" (cv f) (cv v)) (cv v) (~~) breq2 anbi12d (cv x) (" (cv f) (cv v)) inteq A eleq1d imbi12d cla4v sylan9 (cv z) (|^| (" (cv f) (cv v))) (cv w) ineq1 A eleq1d (cv w) (` (cv f) (cv v)) (|^| (" (cv f) (cv v))) ineq2 A eleq1d A A rcla42v ex syl6 com4r exp4a exp3a com14 imp43 (" (cv f) (cv v)) ({/}) inteq int0 syl6eq (` (cv f) (cv v)) ineq1d (` (cv f) (cv v)) ssv (` (cv f) (cv v)) (V) sseqin2 mpbi syl6eq A eleq1d biimprd pm2.61d2 mpd (cv f) (suc (cv v)) (cv x) f1ofn v visset sucid (cv f) (suc (cv v)) (cv v) fnsnfv mpan2 syl (" (cv f) (cv v)) uneq2d (cv v) df-suc (suc (cv v)) (u. (cv v) ({} (cv v))) (cv f) imaeq2 ax-mp (cv f) (cv v) ({} (cv v)) imaun eqtr2 syl6eq (cv f) (suc (cv v)) (cv x) f1ofo (cv f) (suc (cv v)) (cv x) foima syl eqtrd inteqd (cv f) (cv v) fvex (" (cv f) (cv v)) intunsn syl5eqr A eleq1d (C_ (cv x) A) (/\ (A. x (-> (/\ (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (br (cv v) (~~) (cv x))) (e. (|^| (cv x)) A))) (A.e. z A (A.e. w A (e. (i^i (cv z) (cv w)) A)))) ad2antlr mpbid exp43 f 19.23adv x visset (suc (cv v)) f bren syl5ib imp (-. (= (cv x) ({/}))) adantlr com13 19.21ad (e. (cv v) (om)) a1i finds2 x (-> (/\ (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (br (cv y) (~~) (cv x))) (e. (|^| (cv x)) A)) ax-4 syl6 exp4a com24 y visset (cv x) ensym syl5 r19.23aiv com13 imp3a x 19.21aiv z w zfpair (cv x) ({,} (cv z) (cv w)) A sseq1 (cv x) ({,} (cv z) (cv w)) ({/}) eqeq1 negbid anbi12d (cv x) ({,} (cv z) (cv w)) (~~) (cv y) breq1 y (om) rexbidv anbi12d (cv x) ({,} (cv z) (cv w)) inteq A eleq1d imbi12d cla4v z visset w visset A prss z visset (cv w) prnz (C_ ({,} (cv z) (cv w)) A) biantru y (cv z) (cv w) prfi (/\ (C_ ({,} (cv z) (cv w)) A) (-. (= ({,} (cv z) (cv w)) ({/})))) biantru 3bitrr z visset w visset intpr A eleq1i 3imtr3g exp3a r19.21adv r19.21aiv impbi (cv x) (cv z) (cv y) ineq1 A eleq1d y A ralbidv (cv y) (cv w) (cv z) ineq2 A eleq1d A cbvralv syl6bb A cbvralv (C_ (cv x) A) (-. (= (cv x) ({/}))) (E.e. y (om) (br (cv x) (~~) (cv y))) df-3an (e. (|^| (cv x)) A) imbi1i x albii 3bitr4)) thm (abfii1 ((x y) (x z) (y z)) () (= (|^| ({|} x (/\ (C_ A (cv x)) (A.e. y (cv x) (A.e. z (cv x) (e. (i^i (cv y) (cv z)) (cv x))))))) (|^| ({|} x (/\ (C_ A (cv x)) (A. y (-> (/\/\ (C_ (cv y) (cv x)) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x)))))))) (y (cv x) z fiint (C_ A (cv x)) anbi2i x abbii inteqi)) thm (abfii2 ((x y) (x z) (A x) (y z) (A y) (A z)) ((abfii2.1 (e. A (V)))) (= ({|} x (E. y (/\/\ (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) (= (cv x) (|^| (cv y)))))) (|^| ({|} x (A. y (-> (/\/\ (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x))))))) (abfii2.1 uniex (|^| (cv y)) inex2 abfii2.1 (i^i (|^| (cv y)) (U. A)) x df-sn (i^i (|^| (cv y)) (U. A)) snex eqeltrr y abexssex (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z))) 3simp1 (= (cv x) (i^i (|^| (cv y)) (U. A))) anim1i y 19.22i x ss2abi ssexi intab (cv y) A intssuni2 (|^| (cv y)) (U. A) dfss sylib (E.e. z (om) (br (cv y) (~~) (cv z))) 3adant3 (cv x) eleq1d pm5.74i y albii x abbii inteqi (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) (= (cv x) (|^| (cv y))) df-3an x visset (cv x) (|^| (cv y)) (V) eleq1 mpbii (cv y) intex sylibr pm4.71ri (/\ (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z)))) anbi2i (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) (-. (= (cv y) ({/}))) (= (cv x) (|^| (cv y))) an4 (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z))) df-3an (= (cv x) (|^| (cv y))) anbi1i (cv y) A intssuni2 (|^| (cv y)) (U. A) dfss sylib (cv x) eqeq2d (E.e. z (om) (br (cv y) (~~) (cv z))) 3adant3 pm5.32i (/\ (C_ (cv y) A) (-. (= (cv y) ({/})))) (E.e. z (om) (br (cv y) (~~) (cv z))) (= (cv x) (|^| (cv y))) anass 3bitr3r bitr 3bitr y exbii x abbii 3eqtr4r)) thm (abfii3 ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z)) ((abfii2.1 (e. A (V)))) (= (|^| ({|} x (/\ (C_ A (cv x)) (A. y (-> (/\/\ (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x))))))) (|^| ({|} x (A. y (-> (/\/\ (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x))))))) ((cv w) A snssi z (cv w) snfi (cv w) eqid (cv w) snex (cv y) ({} (cv w)) A sseq1 (cv y) ({} (cv w)) (~~) (cv z) breq1 z (om) rexbidv (cv y) ({} (cv w)) inteq w visset intsn syl6eq (cv w) eqeq2d 3anbi123d cla4ev mp3an23 syl w visset (cv x) (cv w) (|^| (cv y)) eqeq1 (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) 3anbi3d y exbidv elab sylibr ssriv abfii2.1 x y z abfii2 sseqtr A x (A. y (-> (/\/\ (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x)))) intmin4 ax-mp)) thm (abfii4 ((f t) (f u) (f v) (f w) (f x) (f y) (f z) (A f) (t u) (t v) (t w) (t x) (t y) (t z) (A t) (u v) (u w) (u x) (u y) (u z) (A u) (v w) (v x) (v y) (v z) (A v) (w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z)) ((abfii2.1 (e. A (V)))) (= (|^| ({|} x (/\ (C_ A (cv x)) (A. y (-> (/\/\ (C_ (cv y) (cv x)) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x))))))) (|^| ({|} x (/\ (C_ A (cv x)) (A. y (-> (/\/\ (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x)))))))) (abfii2.1 (|^| (cv v)) w df-sn (|^| (cv v)) snex eqeltrr v abexssex (C_ (cv v) A) (E.e. u (om) (br (cv v) (~~) (cv u))) (= (cv w) (|^| (cv v))) 3simpb v 19.22i w ss2abi ssexi (cv x) ({|} w (E. v (/\/\ (C_ (cv v) A) (E.e. u (om) (br (cv v) (~~) (cv u))) (= (cv w) (|^| (cv v)))))) A sseq2 (cv x) ({|} w (E. v (/\/\ (C_ (cv v) A) (E.e. u (om) (br (cv v) (~~) (cv u))) (= (cv w) (|^| (cv v)))))) (cv y) sseq2 (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z))) 3anbi1d (cv x) ({|} w (E. v (/\/\ (C_ (cv v) A) (E.e. u (om) (br (cv v) (~~) (cv u))) (= (cv w) (|^| (cv v)))))) (|^| (cv y)) eleq2 imbi12d y albidv anbi12d A w (A. v (-> (/\/\ (C_ (cv v) A) (-. (= (cv v) ({/}))) (E.e. u (om) (br (cv v) (~~) (cv u)))) (e. (|^| (cv v)) (cv w)))) ssmin abfii2.1 w v u abfii3 abfii2.1 w v u abfii2 eqtr4 sseqtr t f (/\/\ (C_ (cv t) A) (E.e. u (om) (br (cv t) (~~) (cv u))) (= (cv y) (|^| (cv t)))) (/\/\ (C_ (cv f) A) (E.e. u (om) (br (cv f) (~~) (cv u))) (= (cv z) (|^| (cv f)))) eeanv (C_ (cv t) A) (E.e. u (om) (br (cv t) (~~) (cv u))) (= (cv y) (|^| (cv t))) (C_ (cv f) A) (E.e. u (om) (br (cv f) (~~) (cv u))) (= (cv z) (|^| (cv f))) an6 t visset f visset unex (cv v) (u. (cv t) (cv f)) A sseq1 (cv v) (u. (cv t) (cv f)) (~~) (cv u) breq1 u (om) rexbidv (cv v) (u. (cv t) (cv f)) inteq (i^i (cv y) (cv z)) eqeq2d 3anbi123d cla4ev (cv t) A (cv f) unss biimp u (cv t) (cv f) unfi (cv y) (|^| (cv t)) (cv z) (|^| (cv f)) ineq12 (cv t) (cv f) intun syl6eqr syl3an sylbi t f 19.23aivv sylbir y visset (cv z) inex1 (cv w) (i^i (cv y) (cv z)) (|^| (cv v)) eqeq1 (C_ (cv v) A) (E.e. u (om) (br (cv v) (~~) (cv u))) 3anbi3d v exbidv elab sylibr y visset (cv w) (cv y) (|^| (cv v)) eqeq1 (C_ (cv v) A) (E.e. u (om) (br (cv v) (~~) (cv u))) 3anbi3d v exbidv elab (cv v) (cv t) A sseq1 (cv v) (cv t) (~~) (cv u) breq1 u (om) rexbidv (cv v) (cv t) inteq (cv y) eqeq2d 3anbi123d cbvexv bitr z visset (cv w) (cv z) (|^| (cv v)) eqeq1 (C_ (cv v) A) (E.e. u (om) (br (cv v) (~~) (cv u))) 3anbi3d v exbidv elab (cv v) (cv f) A sseq1 (cv v) (cv f) (~~) (cv u) breq1 u (om) rexbidv (cv v) (cv f) inteq (cv z) eqeq2d 3anbi123d cbvexv bitr syl2anb rgen2 y ({|} w (E. v (/\/\ (C_ (cv v) A) (E.e. u (om) (br (cv v) (~~) (cv u))) (= (cv w) (|^| (cv v)))))) z fiint mpbi pm3.2i (V) intmin3 ax-mp (cv w) (cv x) (|^| (cv v)) eqeq1 (C_ (cv v) A) (E.e. u (om) (br (cv v) (~~) (cv u))) 3anbi3d v exbidv (cv v) (cv y) A sseq1 (cv v) (cv y) (~~) (cv z) breq1 z (om) rexbidv (cv u) (cv z) (cv v) (~~) breq2 (om) cbvrexv syl5bb (cv v) (cv y) inteq (cv x) eqeq2d 3anbi123d cbvexv syl6bb cbvabv abfii2.1 x y z abfii2 eqtr sseqtr (cv y) A (cv x) sstr2 com12 (C_ A (cv x)) (-. (= (cv y) ({/}))) idd (C_ A (cv x)) (E.e. z (om) (br (cv y) (~~) (cv z))) idd 3anim123d (e. (|^| (cv y)) (cv x)) imim1d y 19.20dv imp x ss2abi ({|} x (/\ (C_ A (cv x)) (A. y (-> (/\/\ (C_ (cv y) (cv x)) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x)))))) ({|} x (A. y (-> (/\/\ (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x))))) intss ax-mp eqssi abfii2.1 x y z abfii3 eqtr4)) thm (abfii5 ((x y) (x z) (A x) (y z) (A y) (A z)) ((abfii2.1 (e. A (V)))) (= (|^| ({|} x (/\ (C_ A (cv x)) (A.e. y (cv x) (A.e. z (cv x) (e. (i^i (cv y) (cv z)) (cv x))))))) ({|} x (E. y (/\/\ (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) (= (cv x) (|^| (cv y))))))) (abfii2.1 x y z abfii4 abfii2.1 x y z abfii3 eqtr x A y z abfii1 abfii2.1 x y z abfii2 3eqtr4)) thm (fodomfi ((f g) (f m) (f n) (f k) (f y) (f x) (A f) (g m) (g n) (g k) (g y) (g x) (A g) (m n) (k m) (m y) (m x) (A m) (k n) (n y) (n x) (A n) (k y) (k x) (A k) (x y) (A y) (A x) (B f) (B g) (B m) (B x) (F f) (F g) (F m)) () (-> (/\ (E.e. n (om) (br A (~~) (cv n))) (:-onto-> F A B)) (br B (~<_) A)) (B (cv m) A domentr F A B (V) fex ancoms relen A (cv m) brrelexi F A B fof syl2an (e. (cv m) (om)) adantl g visset F (V) (cv g) (V) coexg mpan2 (cv f) (o. F (cv g)) (cv m) B foeq1 (V) cla4egv syl m visset (cv m) (V) (cv f) B fornex ax-mp f 19.23aiv (cv x) B (cv f) (cv m) foeq3 f exbidv (cv x) B (~<_) (cv m) breq1 imbi12d (V) cla4gv (cv m) ({/}) (cv f) (cv x) foeq2 f exbidv (cv m) ({/}) (cv x) (~<_) breq2 imbi12d x albidv (cv m) (cv k) (cv f) (cv x) foeq2 f exbidv (cv m) (cv k) (cv x) (~<_) breq2 imbi12d x albidv (cv m) (suc (cv k)) (cv f) (cv x) foeq2 f exbidv (cv m) (suc (cv k)) (cv x) (~<_) breq2 imbi12d x albidv (cv f) (cv x) fo00 pm3.27bd ({/}) 0dom syl6eqbr f 19.23aiv x ax-gen (cv f) (suc (cv k)) (cv x) fof (cv f) (suc (cv k)) (cv x) ffn k visset sucid (cv f) (suc (cv k)) (cv k) fnsnfv mpan2 3syl (" (cv f) (cv k)) uneq2d (cv f) (suc (cv k)) (cv x) foima (cv k) df-suc (suc (cv k)) (u. (cv k) ({} (cv k))) (cv f) imaeq2 ax-mp (cv f) (cv k) ({} (cv k)) imaun eqtr2 syl5eq eqtrd (/\ (e. (cv k) (om)) (A. y (-> (E. g (:-onto-> (cv g) (cv k) (cv y))) (br (cv y) (~<_) (cv k))))) adantl k visset (` (cv f) (cv k)) snex (cv k) snex (" (cv f) (cv k)) undom f visset (cv f) (V) (cv k) imaexg ax-mp (cv y) (" (cv f) (cv k)) (cv g) (cv k) foeq3 g exbidv (cv y) (" (cv f) (cv k)) (~<_) (cv k) breq1 imbi12d cla4v imp (cv f) (cv k) fores (cv f) (suc (cv k)) (cv x) fof (cv f) (suc (cv k)) (cv x) ffun syl (cv f) (suc (cv k)) (cv x) fof (cv k) sssucid (cv f) (suc (cv k)) (cv x) fdm (cv k) sseq2d mpbiri syl sylanc f visset (cv f) (V) (cv k) resexg ax-mp (cv g) (|` (cv f) (cv k)) (cv k) (" (cv f) (cv k)) foeq1 cla4ev syl sylan2 (e. (cv k) (om)) adantll (cv f) (cv k) fvex k visset (` (cv f) (cv k)) (V) (cv k) (V) en2sn mp2an ({} (` (cv f) (cv k))) ({} (cv k)) endom ax-mp jctir (cv k) nnord (cv k) orddisj syl (A. y (-> (E. g (:-onto-> (cv g) (cv k) (cv y))) (br (cv y) (~<_) (cv k)))) (:-onto-> (cv f) (suc (cv k)) (cv x)) ad2antrr sylanc eqbrtrrd (cv k) df-suc syl6breqr ex f 19.23adv ex x 19.21adv (cv x) (cv y) (cv f) (cv k) foeq3 f exbidv (cv f) (cv g) (cv k) (cv y) foeq1 cbvexv syl6bb (cv x) (cv y) (~<_) (cv k) breq1 imbi12d cbvalv syl5ib finds1 syl5 syl pm2.43b sylan9 g 19.23adv g (:-onto-> F A B) (:-onto-> (cv g) (cv m) A) 19.42v F A B (cv g) (cv m) foco g 19.22i sylbir m visset A f bren (cv f) A (cv m) f1ocnv (`' (cv f)) (cv m) A f1ofo f visset cnvex (cv g) (`' (cv f)) (cv m) A foeq1 cla4ev 3syl f 19.23aiv sylbi sylan2 ancoms syl5 imp anasss mpancom m visset A ensym (e. (cv m) (om)) (:-onto-> F A B) ad2antrl sylanc exp32 r19.23aiv imp (cv n) (cv m) A (~~) breq2 (om) cbvrexv sylanb)) thm (fodomfib ((f n) (A f) (A n) (B f)) () (-> (E.e. n (om) (br A (~~) (cv n))) (<-> (/\ (-. (= A ({/}))) (E. f (:-onto-> (cv f) A B))) (/\ (br ({/}) (~<) B) (br B (~<_) A)))) ((cv f) A B fof (cv f) A B fdm syl ({/}) eqeq1d (cv f) A B forn ({/}) eqeq1d (cv f) dm0rn0 syl5bb bitr3d negbid biimpac (e. A (V)) adantll A (V) (cv f) B fornex imp B (V) 0sdomg syl (-. (= A ({/}))) adantlr mpbird ex relen A (cv n) brrelexi (e. (cv n) (om)) a1i r19.23aiv sylan n A (cv f) B fodomfi ex (-. (= A ({/}))) adantr jcad f 19.23adv ex imp3a relen A (cv n) brrelexi (e. (cv n) (om)) a1i r19.23aiv A (V) ({/}) B sdomdomtr A (V) 0sdomg sylibd A (V) B f fodomr 3exp imp3a jcad syl impbid)) thm (iunfi ((w x) (w y) (w z) (n w) (A w) (x y) (x z) (n x) (A x) (y z) (n y) (A y) (n z) (A z) (A n) (B n) (B w) (B y) (B z)) () (-> (/\ (E.e. n (om) (br A (~~) (cv n))) (A.e. x A (E.e. n (om) (br B (~~) (cv n))))) (E.e. n (om) (br (U_ x A B) (~~) (cv n)))) (n ({|} y (E.e. x A (= (cv y) B))) z unifi w y (E.e. x A (= (cv y) B)) hbab1 (e. (cv w) ({|} y (E.e. x A (= (cv y) B)))) z ax-17 (E.e. n (om) (br (cv y) (~~) (cv n))) z ax-17 (E.e. n (om) (br (cv z) (~~) (cv n))) y ax-17 (cv y) (cv z) (~~) (cv n) breq1 n (om) rexbidv cbvralf sylan2b n A ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B))) (ran ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B)))) fodomfi ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B))) eqid fnopab2g ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B))) A fnforn bitr sylan2b x y A B rnopab2 syl5eqbrr n A ({|} y (E.e. x A (= (cv y) B))) domfi syldan relen B (cv n) brrelexi (e. (cv n) (om)) a1i r19.23aiv x A r19.20si sylan2 x A (E.e. n (om) (br B (~~) (cv n))) (= (cv y) B) r19.29 (cv y) B (~~) (cv n) breq1 n (om) rexbidv biimparc (e. (cv x) A) a1i r19.23aiv syl ex (E.e. n (om) (br A (~~) (cv n))) adantl y (E.e. x A (= (cv y) B)) abid syl5ib r19.21aiv sylanc (e. (cv x) A) n ax-17 n (om) (br B (~~) (cv n)) hbre1 hbral relen B (cv n) brrelexi (e. (cv n) (om)) a1i r19.23aiv x A r19.20si x A B (V) y dfiun2g syl (~~) (cv n) breq1d (om) rexbid (E.e. n (om) (br A (~~) (cv n))) adantl mpbird)) thm (axreg ((x y) (x z) (y z)) () (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y)))))))) (x y z ax-reg 19.23bi)) thm (zfregcl ((x y) (A x) (x z) (A y) (y z) (A z)) ((zfregcl.1 (e. A (V)))) (-> (E. x (e. (cv x) A)) (E.e. x A (A.e. y (cv x) (-. (e. (cv y) A))))) (zfregcl.1 (cv z) A (cv x) eleq2 x exbidv (cv z) A (cv y) eleq2 negbid y (cv x) ralbidv x rexeqd imbi12d x (cv z) (A.e. y (cv x) (-. (e. (cv y) (cv z)))) hbre1 x z y axreg y (cv x) (-. (e. (cv y) (cv z))) df-ral x (cv z) rexbii x (cv z) (A. y (-> (e. (cv y) (cv x)) (-. (e. (cv y) (cv z))))) df-rex bitr2 sylib 19.23ai vtocl)) thm (zfreg ((A x) (x y) (A y)) ((zfreg.1 (e. A (V)))) (-> (-. (= A ({/}))) (E.e. x A (= (i^i (cv x) A) ({/})))) (zfreg.1 x y zfregcl A x n0 (cv x) A y disj x A rexbii 3imtr4)) thm (zfreg2 ((A x) (x y) (A y)) ((zfreg2.1 (e. A (V)))) (-> (-. (= A ({/}))) (E.e. x A (= (i^i A (cv x)) ({/})))) (zfreg2.1 x y zfregcl A x n0 A (cv x) incom ({/}) eqeq1i (cv x) A y disj bitr x A rexbii 3imtr4)) thm (eirrv ((x y) (x z) (y z)) () (-. (e. (cv x) (cv x))) ((cv y) (cv x) ({} (cv x)) eleq1 x visset snid a4w1 (cv x) snex y z zfregcl ax-mp x y x ax-14 equcoms com12 y (cv x) elsn syl5ib x visset snid (cv z) (cv x) ({} (cv x)) eleq1 negbid (cv y) rcla4cv mt2i nsyli con2d r19.21aiv y ({} (cv x)) (A.e. z (cv y) (-. (e. (cv z) ({} (cv x))))) ralnex sylib mt2)) thm (eirr ((A x)) () (-. (e. A A)) ((cv x) A (cv x) eleq2 (cv x) A A eleq1 bitrd negbid x eirrv A vtoclg (e. A A) pm2.01 ax-mp)) thm (sucprcreg ((A x)) () (<-> (-. (e. A (V))) (= (suc A) A)) (A sucprc A eirr (e. A A) x ax-17 (cv x) A A eleq1 (V) ceqsalg mtbiri A ssid A df-suc A eqeq1i (u. A ({} A)) A A sseq1 sylbi mpbiri (cv x) sseld (cv x) A ({} A) elun syl5ibr (e. (cv x) ({} A)) (e. (cv x) A) olc syl5 x A elsn syl5ibr x 19.21aiv nsyl3 impbi)) thm (zfregfr ((x y) (A x) (A y)) () (Fr (E) A) (A x y dfepfr x visset y zfreg2 (C_ (cv x) A) adantl mpgbir)) thm (en2lp ((A x) (A y) (x y) (B x) (B y)) () (-. (/\ (e. A B) (e. B A))) ((cv x) A (cv y) eleq1 (cv x) A (cv y) eleq2 anbi12d negbid (cv y) B A eleq2 (cv y) B A eleq1 anbi12d negbid (V) zfregfr x visset y visset pm3.2i (V) x y efrn2lp mp2an (V) (V) vtocl2g A B elisset B A elisset anim12i con3i pm2.61i)) thm (preleq () ((preleq.1 (e. A (V))) (preleq.2 (e. B (V))) (preleq.3 (e. C (V))) (preleq.4 (e. D (V)))) (-> (/\ (/\ (e. A B) (e. C D)) (= ({,} A B) ({,} C D))) (/\ (= A C) (= B D))) (preleq.1 preleq.2 preleq.3 preleq.4 preq12b biimp ord D C en2lp A D B C eleq12 (e. C D) anbi1d mtbiri syl6 a3d impcom)) thm (opthreg () ((preleq.1 (e. A (V))) (preleq.2 (e. B (V))) (preleq.3 (e. C (V))) (preleq.4 (e. D (V)))) (-> (= ({,} A ({,} A B)) ({,} C ({,} C D))) (/\ (= A C) (= B D))) (preleq.1 B pri1 preleq.3 D pri1 preleq.1 A B prex preleq.3 C D prex preleq mpanl12 A C B preq1 ({,} C D) eqeq1d preleq.2 preleq.4 C prer2 syl6bi imdistani syl)) thm (suc11reg () () (<-> (= (suc A) (suc B)) (= A B)) (A B en2lp (e. A B) (e. B A) ianor mpbi (suc A) (suc B) A eleq2 A (V) sucidg syl5bi com12 A (V) B elsucg sylibd imp ord ex com23 (suc A) (suc B) B eleq2 B (V) sucidg syl5bir com12 B (V) A elsucg sylibd imp ord B A eqcom syl6ib ex com23 jaao mpi (suc A) (V) (suc B) nelneq A sucexb B sucexb negbii syl2anb (= A B) pm2.21d (suc B) (V) (suc A) nelneq B sucexb A sucexb negbii syl2anb ancoms (= A B) pm2.21d (suc A) (suc B) eqcom syl5ib A sucprc B sucprc eqeqan12d biimpd 4cases A B suceq impbi)) thm (inf0 ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) ((inf0.1 (e. (om) (V)))) (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))) (y visset (cv y) (V) ({<,>|} w v (= (cv v) (suc (cv w)))) fr0t ax-mp ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y) frfnom peano1 (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (om) ({/}) fnfvrn mp2an eqeltrr ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y) frfnom (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (om) (cv u) f fvelrn ax-mp (` (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (cv f)) (cv u) (` (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (suc (cv f))) eleq1 (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (cv f) fvex sucex (e. (cv z) (cv y)) w ax-17 (e. (cv z) (cv f)) w ax-17 z w v (= (cv v) (suc (cv w))) hbopab1 (e. (cv z) (cv y)) w ax-17 hbrdg (e. (cv z) (om)) w ax-17 hbres (e. (cv z) (cv f)) w ax-17 hbfv hbsuc (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) eqid (cv w) (` (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (cv f)) suceq (V) frsucopab mpan2 (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (cv f) fvex sucid syl5eleqr syl5bi (cv f) peano2b ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y) frfnom (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (om) (suc (cv f)) fnfvrn mpan sylbi (= (` (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (cv f)) (cv u)) a1i jcad (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (suc (cv f)) fvex (cv z) (` (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (suc (cv f))) (cv u) eleq2 (cv z) (` (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (suc (cv f))) (ran (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om))) eleq1 anbi12d cla4ev syl6com r19.23aiv sylbi u ax-gen ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y) frfnom (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (om) fndm ax-mp inf0.1 eqeltr ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y) frfnom (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (om) fnfun ax-mp (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om)) (V) funrnex mp2 (cv x) (ran (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om))) (cv y) eleq2 (cv x) (ran (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om))) (cv u) eleq2 (cv x) (ran (|` (rec ({<,>|} w v (= (cv v) (suc (cv w)))) (cv y)) (om))) (cv z) eleq2 (e. (cv u) (cv z)) anbi2d z exbidv imbi12d u albidv anbi12d cla4ev mp2an (cv y) (cv u) (cv x) eleq1 (cv y) (cv u) (cv z) eleq1 (e. (cv z) (cv x)) anbi1d z exbidv imbi12d cbvalv (e. (cv y) (cv x)) anbi2i x exbii mpbir)) thm (inf1 ((x y) (x z) (y z)) ((inf1.1 (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))))) (E. x (/\ (-. (= (cv x) ({/}))) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))) (inf1.1 (cv y) (cv x) n0i (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))) anim1i x 19.22i ax-mp)) thm (inf2 ((x y) (x z) (y z)) ((inf1.1 (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))))) (E. x (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x))))) (inf1.1 inf1 (cv x) (U. (cv x)) y dfss2 (cv y) (cv x) z eluni (e. (cv y) (cv x)) imbi2i y albii bitr (-. (= (cv x) ({/}))) anbi2i x exbii mpbir)) thm (inf3lema ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (G v) (G u) (G f) (F v) (F u) (F f) (A v) (A u) (A f) (B v) (B u) (B f)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (<-> (e. A (` G B)) (/\ (e. A (cv x)) (C_ (i^i A (cv x)) B))) (inf3lem.1 (= (cv z) (cv u)) id (cv y) (cv v) (i^i (cv w) (cv x)) sseq2 w (cv x) rabbisdv (cv w) (cv f) (cv x) ineq1 (cv v) sseq1d (cv x) cbvrabv syl6eq eqeqan12rd cbvopabv eqtr B fveq1i inf3lem.4 x visset f (C_ (i^i (cv f) (cv x)) B) rabex (cv v) B (i^i (cv f) (cv x)) sseq2 f (cv x) rabbisdv u fvopab eqtr A eleq2i (cv f) A (cv x) ineq1 B sseq1d (cv x) elrab bitr)) thm (inf3lemb ((x y) (x z) (w x) (y z) (w y) (w z)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (= (` F ({/})) ({/})) (inf3lem.2 ({/}) fveq1i 0ex ({/}) (V) G fr0t ax-mp eqtr)) thm (inf3lemc ((x y) (x z) (w x) (y z) (w y) (w z)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (e. A (om)) (= (` F (suc A)) (` G (` F A)))) (A G ({/}) frsuct inf3lem.2 (suc A) fveq1i inf3lem.2 A fveq1i G fveq2i 3eqtr4g)) thm (inf3lemd ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (e. A (om)) (C_ (` F A) (cv x))) (A v nnsuc A (suc (cv v)) F fveq2 (cv x) sseq1d inf3lem.1 inf3lem.2 v visset inf3lem.4 inf3lemc (cv u) eleq2d inf3lem.1 inf3lem.2 u visset F (cv v) fvex inf3lema pm3.26bd syl6bi ssrdv syl5bir com12 r19.23aiv syl ex (cv x) 0ss A ({/}) F fveq2 inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lemb syl6eq (cv x) sseq1d mpbiri pm2.61d2)) thm (inf3lem1 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (e. A (om)) (C_ (` F A) (` F (suc A)))) ((cv v) ({/}) F fveq2 (cv v) ({/}) suceq F fveq2d sseq12d (cv v) (cv u) F fveq2 (cv v) (cv u) suceq F fveq2d sseq12d (cv v) (suc (cv u)) F fveq2 (cv v) (suc (cv u)) suceq F fveq2d sseq12d (cv v) A F fveq2 (cv v) A suceq F fveq2d sseq12d inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.3 inf3lemb (` F (suc ({/}))) 0ss eqsstr inf3lem.1 inf3lem.2 u visset u visset inf3lemc (cv v) eleq2d inf3lem.1 inf3lem.2 v visset F (cv u) fvex inf3lema syl6bb (cv u) peano2b inf3lem.1 inf3lem.2 u visset sucex u visset inf3lemc sylbi (cv v) eleq2d inf3lem.1 inf3lem.2 v visset F (suc (cv u)) fvex inf3lema syl6bb imbi12d (i^i (cv v) (cv x)) (` F (cv u)) (` F (suc (cv u))) sstr2 com12 (e. (cv v) (cv x)) anim2d syl5bir imp ssrdv ex finds)) thm (inf3lem2 ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (G v) (G u) (G f) (F v) (F u) (F f) (A v) (A u) (A f) (B v) (B u) (B f)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (e. A (om)) (-. (= (` F A) (cv x))))) ((cv v) ({/}) F fveq2 (cv x) eqeq1d negbid (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) imbi2d (cv v) (cv u) F fveq2 (cv x) eqeq1d negbid (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) imbi2d (cv v) (suc (cv u)) F fveq2 (cv x) eqeq1d negbid (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) imbi2d (cv v) A F fveq2 (cv x) eqeq1d negbid (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) imbi2d inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lemb (cv x) eqeq1i ({/}) (cv x) eqcom bitr biimp con3i (C_ (cv x) (U. (cv x))) adantr (cv x) (U. (cv x)) (cv v) ssel (cv v) (cv x) f eluni syl6ib inf3lem.1 inf3lem.2 u visset inf3lem.4 inf3lemc (cv f) eleq2d inf3lem.1 inf3lem.2 f visset F (cv u) fvex inf3lema pm3.27bd (cv v) sseld (cv v) (cv f) (cv x) elin syl5ibr syl6bi (` F (suc (cv u))) (cv x) (cv f) eleq2 biimparc syl5 com23 exp4a exp3a com34 imp3a f 19.23adv sylan9r pm2.43d (= (` F (suc (cv u))) (cv x)) (e. (cv v) (` F (cv u))) con3 syl6 imp3a v 19.23adv (` F (cv u)) (cv x) dfpss2 (` F (cv u)) (cv x) v pssnel sylbir syl5 inf3lem.1 inf3lem.2 u visset inf3lem.4 inf3lemd sylani exp4b pm2.43a (-. (= (cv x) ({/}))) adantld a2d finds com12)) thm (inf3lem3 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G v) (F v) (A v) (B v)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (e. A (om)) (-. (= (` F A) (` F (suc A)))))) (inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lem2 com12 inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lemd jctild (` F A) (cv x) pssdifn0 syl6 inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lemc (cv v) eleq2d (cv v) (cv x) (` F A) eldifi (cv v) (cv x) (` F A) inssdif0 biimpr anim12i inf3lem.1 inf3lem.2 v visset F A fvex inf3lema sylibr syl5bir (cv v) (cv x) (` F A) eldifn (= (i^i (cv v) (\ (cv x) (` F A))) ({/})) adantr (e. A (om)) a1i jcad (` F A) (` F (suc A)) (cv v) eleq2 biimprd (e. (cv v) (` F (suc A))) (e. (cv v) (` F A)) iman sylib con2i syl6 exp3a r19.23adv x visset (cv x) (` F A) difss ssexi v zfreg syl5 syld com12)) thm (inf3lem4 ((x y) (x z) (w x) (y z) (w y) (w z)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (e. A (om)) (C: (` F A) (` F (suc A))))) (inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lem1 (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) a1i inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lem3 jcad (` F A) (` F (suc A)) dfpss2 syl6ibr)) thm (inf3lem5 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (/\ (e. A (om)) (e. B A)) (C: (` F B) (` F A)))) (B A elnn ancoms A nnord A B ordsucss syl (e. B (om)) adantr (cv v) (suc B) F fveq2 (` F B) psseq2d (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) imbi2d (cv v) (cv u) F fveq2 (` F B) psseq2d (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) imbi2d (cv v) (suc (cv u)) F fveq2 (` F B) psseq2d (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) imbi2d (cv v) A F fveq2 (` F B) psseq2d (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) imbi2d B peano2b inf3lem.1 inf3lem.2 inf3lem.4 inf3lem.4 inf3lem4 com12 sylbir inf3lem.1 inf3lem.2 u visset inf3lem.4 inf3lem4 (` F B) (` F (cv u)) (` F (suc (cv u))) psstr expcom syl6com a2d (e. (suc B) (om)) (C_ (suc B) (cv u)) ad2antrr findsg ex B peano2b sylan2b syld ex com23 imp mpd com12)) thm (inf3lem6 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (:-1-1-> F (om) (P~ (cv x)))) (inf3lem.1 inf3lem.2 u visset v visset inf3lem5 (` F (cv v)) (` F (cv u)) dfpss2 pm3.27bd syl6 exp3a imp (e. (cv v) (om)) adantrl inf3lem.1 inf3lem.2 v visset u visset inf3lem5 (` F (cv u)) (` F (cv v)) dfpss2 pm3.27bd (` F (cv u)) (` F (cv v)) eqcom negbii sylib syl6 exp3a imp (e. (cv u) (om)) adantrr jaod con2d (cv v) (cv u) ordtri3 (cv v) nnord (cv u) nnord syl2an (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) adantl sylibrd ex v u 19.21aivv v (om) u (om) (-> (= (` F (cv v)) (` F (cv u))) (= (cv v) (cv u))) r2al sylibr inf3lem.2 G ({/}) frfnom F (|` (rec G ({/})) (om)) (om) fneq1 mpbiri ax-mp F (om) (cv u) v fvelrn (` F (cv v)) (cv u) (P~ (cv x)) eleq1 inf3lem.1 inf3lem.2 v visset inf3lem.4 inf3lemd F (cv v) fvex (cv x) elpw sylibr syl5bi com12 r19.23aiv syl6bi ssrdv ancli ax-mp F (om) (P~ (cv x)) df-f mpbir jctil F (om) (P~ (cv x)) v u f1fv sylibr)) thm (inf3lem7 ((x y) (x z) (w x) (y z) (w y) (w z)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (e. (om) (V))) (inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lem6 F (om) (P~ (cv x)) f1f F (om) (P~ (cv x)) fdm syl F dfdm4 syl5eqr (`' F) (V) funrnex F (om) (P~ (cv x)) f1f F (om) (P~ (cv x)) frn x visset pwex (ran F) ssex 3syl F df-rn syl5eqelr F (om) (P~ (cv x)) df-f1 pm3.27bd sylc eqeltrrd syl)) thm (inf3 ((x y) (x z) (w x) (y z) (w y) (w z)) ((inf3.1 (E. x (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x))))))) (e. (om) (V)) (inf3.1 ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))) eqid (|` (rec ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))) ({/})) (om)) eqid x visset x visset inf3lem7 x 19.23aiv ax-mp)) thm (infeq5 ((x y) (x z) (w x) (y z) (w y) (w z)) () (<-> (E. x (C: (cv x) (U. (cv x)))) (e. (om) (V))) ((cv x) (U. (cv x)) dfpss2 pm3.27bd (cv x) ({/}) unieq uni0 syl6req (cv x) ({/}) (U. (cv x)) eqtrt mpdan nsyl (cv x) (U. (cv x)) pssss jca x 19.22i ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))) eqid (|` (rec ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))) ({/})) (om)) eqid x visset x visset inf3lem7 x 19.23aiv syl (om) (V) ({} ({/})) difexg 0ex snid (om) ({} ({/})) disj4 (om) ({} ({/})) disj3 bitr3 peano1 (om) (\ (om) ({} ({/}))) ({/}) eleq2 mpbii ({/}) (om) ({} ({/})) eldif sylib pm3.27d sylbi a3i ax-mp (om) unidif0 limom (om) limuni ax-mp eqtr4 (\ (om) ({} ({/}))) psseq2i mpbir (cv x) (\ (om) ({} ({/}))) (U. (cv x)) psseq1 (cv x) (\ (om) ({} ({/}))) unieq (\ (om) ({} ({/}))) psseq2d bitrd (V) cla4egv mpi syl impbi)) thm (axinf ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))) (x y w z ax-inf w y x elequ1 w y z elequ1 (e. (cv z) (cv x)) anbi1d z exbidv imbi12d cbvalv (e. (cv y) (cv x)) anbi2i x exbii mpbi)) thm (inf4 ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (/\ (E. y (/\ (e. (cv y) (cv x)) (A. z (-. (e. (cv z) (cv y)))))) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv z) (cv x)) (A. w (<-> (e. (cv w) (cv z)) (\/ (e. (cv w) (cv y)) (= (cv w) (cv y))))))))))) (peano1 (cv y) peano2 y ax-gen x y z axinf inf2 inf3 (cv x) (om) ({/}) eleq2 (cv x) (om) (cv y) eleq2 (cv x) (om) (suc (cv y)) eleq2 imbi12d y albidv anbi12d cla4ev mp2an (cv x) y z 0el y (cv x) (A. z (-. (e. (cv z) (cv y)))) df-rex bitr (cv y) (cv x) z w sucel z (cv x) (A. w (<-> (e. (cv w) (cv z)) (\/ (e. (cv w) (cv y)) (= (cv w) (cv y))))) df-rex bitr (e. (cv y) (cv x)) imbi2i y albii anbi12i x exbii mpbi)) thm (zfinf ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (/\ (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x))))) (x y z w inf4 (cv x) y z 0el y (cv x) (A. z (-. (e. (cv z) (cv y)))) df-rex bitr (cv y) (cv x) z w sucel z (cv x) (A. w (<-> (e. (cv w) (cv z)) (\/ (e. (cv w) (cv y)) (= (cv w) (cv y))))) df-rex bitr y (cv x) ralbii y (cv x) (E. z (/\ (e. (cv z) (cv x)) (A. w (<-> (e. (cv w) (cv z)) (\/ (e. (cv w) (cv y)) (= (cv w) (cv y))))))) df-ral bitr anbi12i x exbii mpbir)) thm (omex ((x y)) () (e. (om) (V)) (x y zfinf (cv x) y peano5 (-> (e. (cv y) (cv x)) (e. (suc (cv y)) (cv x))) (e. (cv y) (om)) ax-1 r19.20i2 sylan2 x 19.22i ax-mp x visset (om) ssex x 19.23aiv ax-mp)) thm (inf5 () () (E. x (C: (cv x) (U. (cv x)))) (omex x infeq5 mpbir)) thm (omelon () () (e. (om) (On)) (omex onprc (om) (On) (V) eleq1 mtbiri mt2 omon ori mt3)) thm (dfom3 ((x y) (x z) (y z)) () (= (om) (|^| ({|} x (/\ (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x))))))) (0ex x (/\ (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x)))) elintab (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x))) pm3.26 mpgbir (cv y) (cv z) suceq (cv x) eleq1d (cv x) rcla4cv (e. ({/}) (cv x)) adantl a2i x 19.20i z visset x (/\ (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x)))) elintab z visset sucex x (/\ (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x)))) elintab 3imtr4 (e. (cv z) (om)) a1i rgen (|^| ({|} x (/\ (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x)))))) z peano5 mp2an peano1 (cv y) peano2 rgen z visset x (/\ (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x)))) elintab omex (cv x) (om) ({/}) eleq2 (cv x) (om) (suc (cv y)) eleq2 y raleqd anbi12d (cv x) (om) (cv z) eleq2 imbi12d cla4v sylbi mp2ani ssriv eqssi)) thm (elom3 ((A x)) () (<-> (e. A (om)) (A. x (-> (Lim (cv x)) (e. A (cv x))))) (A (om) x elomg ibi pm3.27d limom omex (cv x) (om) limeq (cv x) (om) A eleq2 imbi12d cla4v mpi impbi)) thm (dfom4 ((x y)) () (= (om) ({|} x (A. y (-> (Lim (cv y)) (e. (cv x) (cv y)))))) ((cv x) y elom3 abbi2i)) thm (oancom () () (=/= (opr (1o) (+o) (om)) (opr (om) (+o) (1o))) (omelon 1onn (1o) oaabslem mp2an omex sucid omelon (om) oa1suc ax-mp eleqtrr eqeltr 1on omelon (1o) (om) oacl mp2an omelon 1on (om) (1o) oacl mp2an (opr (1o) (+o) (om)) (opr (om) (+o) (1o)) onelpsst mp2an pm3.27bd ax-mp (opr (1o) (+o) (om)) (opr (om) (+o) (1o)) df-ne mpbir)) thm (isfinite ((A x)) () (<-> (br A (~<) (om)) (E.e. x (om) (br A (~~) (cv x)))) (A x isfinite2 x A isfinite1 omex A ensym con3i (br A (~<_) (om)) anim2i A (om) brsdom sylibr syl impbi)) thm (nnsdom () () (-> (e. A (om)) (br A (~<) (om))) (A omsdomnn omex A ensym con3i (br A (~<_) (om)) anim2i syl A (om) brsdom sylibr)) thm (omenps () () (br (om) (~~) (\ (om) ({} ({/})))) (omex limom (V) limenpsi ax-mp)) thm (omensuc () () (br (om) (~~) (suc (om))) (omex limom (V) limensuci ax-mp)) thm (infensuc ((x y) (A x) (A y)) () (-> (/\ (e. A (On)) (C_ (om) A)) (br A (~~) (suc A))) (omelon (= (cv x) (om)) id (cv x) (om) suceq (~~) breq12d (= (cv x) (cv y)) id (cv x) (cv y) suceq (~~) breq12d (= (cv x) (suc (cv y))) id (cv x) (suc (cv y)) suceq (~~) breq12d (= (cv x) A) id (cv x) A suceq (~~) breq12d omensuc (e. (om) (On)) a1i y visset y visset sucex (cv y) (V) (suc (cv y)) (V) en2sn mp2an (cv y) (suc (cv y)) ({} (cv y)) ({} (suc (cv y))) unen (cv y) df-suc (suc (cv y)) df-suc 3brtr4g ex (cv y) eloni (cv y) ordeirr syl (cv y) (cv y) disjsn sylibr (suc (cv y)) eloni (suc (cv y)) ordeirr syl (cv y) sucelon (suc (cv y)) (suc (cv y)) disjsn 3imtr4 jca syl5 mpan2 com12 (e. (om) (On)) (C_ (om) (cv y)) ad2antrr x visset (cv x) (V) limensuc mpan (e. (om) (On)) (C_ (om) (cv x)) ad2antrr (A.e. y (cv x) (-> (C_ (om) (cv y)) (br (cv y) (~~) (suc (cv y))))) a1d tfindsg mpanl2)) thm (unbnnt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (C_ A (om)) (A.e. x (om) (E.e. y A (e. (cv x) (cv y))))) (br A (~~) (om))) (omex A ssex (cv z) A (om) sseq1 (cv z) A y (e. (cv x) (cv y)) rexeq1 x (om) ralbidv anbi12d (cv z) A (~~) (om) breq1 imbi12d z visset x y unbnn (V) vtoclg exp3a mpcom imp)) thm (noinfep ((x y) (x z) (F x) (y z) (F y) (F z)) () (-> (Fn F (om)) (E.e. x (om) (-. (e. (` F (suc (cv x))) (` F (cv x)))))) ((ran F) zfregfr (ran F) ssid (ran F) (V) (E) (ran F) y z fri mpanr1 mpanl2 F (V) funrnex F (om) fndm omex syl6eqel F (om) fnfun sylc F (om) fndm peano1 (dom F) (om) ({/}) eleq2 mpbiri ({/}) (dom F) n0i syl F dm0rn0 negbii sylib syl sylanc F (om) (cv y) x fvelrn (A.e. z (ran F) (-. (br (cv z) (E) (cv y)))) adantr (` F (cv x)) (cv y) (` F (suc (cv x))) eleq2 negbid (cv z) (` F (suc (cv x))) (cv y) eleq1 z y epel syl5bb negbid (ran F) rcla4va syl5bir F (om) (suc (cv x)) fnfvrn (A.e. z (ran F) (-. (br (cv z) (E) (cv y)))) adantlr (Fn F (om)) (A.e. z (ran F) (-. (br (cv z) (E) (cv y)))) pm3.27 (e. (suc (cv x)) (om)) adantr jca syl5 (cv x) peano2 sylan2i com12 r19.22dva sylbid ex com23 r19.23adv mpd)) thm (trcl ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A y) (A z) (A v) (F y) (F v) (F u)) ((trcl.1 (e. A (V))) (trcl.2 (= F (|` (rec ({<,>|} z w (= (cv w) (u. (cv z) (U. (cv z))))) A) (om)))) (trcl.3 (= C (U_ y (om) (` F (cv y)))))) (/\/\ (C_ A C) (Tr C) (A. x (-> (/\ (C_ A (cv x)) (Tr (cv x))) (C_ C (cv x))))) (peano1 trcl.2 ({/}) fveq1i trcl.1 A (V) ({<,>|} z w (= (cv w) (u. (cv z) (U. (cv z))))) fr0t ax-mp eqtr2 A (` F ({/})) eqimss ax-mp (cv y) ({/}) F fveq2 A sseq2d (om) rcla4ev mp2an y (om) A (` F (cv y)) ssiun ax-mp trcl.3 sseqtr4 (U_ y (om) (` F (cv y))) v u dftr2 (cv u) y (om) (` F (cv y)) eliun (e. (cv v) (cv u)) anbi2i y (om) (e. (cv v) (cv u)) (e. (cv u) (` F (cv y))) r19.42v bitr4 (U. (` F (cv y))) (` F (cv y)) ssun2 F (cv y) fvex F (cv y) fvex uniex unex (e. (cv v) A) z ax-17 (e. (cv v) (cv y)) z ax-17 v z w (= (cv w) (u. (cv z) (U. (cv z)))) hbopab1 (e. (cv v) A) z ax-17 hbrdg (e. (cv v) (om)) z ax-17 hbres trcl.2 (cv v) eleq2i trcl.2 (cv v) eleq2i z albii 3imtr4 (e. (cv v) (cv y)) z ax-17 hbfv v z w (= (cv w) (u. (cv z) (U. (cv z)))) hbopab1 (e. (cv v) A) z ax-17 hbrdg (e. (cv v) (om)) z ax-17 hbres trcl.2 (cv v) eleq2i trcl.2 (cv v) eleq2i z albii 3imtr4 (e. (cv v) (cv y)) z ax-17 hbfv hbuni hbun trcl.2 (cv z) (` F (cv y)) unieq (cv z) (` F (cv y)) (U. (cv z)) (U. (` F (cv y))) uneq12 mpdan (V) frsucopab mpan2 (U. (` F (cv y))) sseq2d mpbiri (cv v) sseld (cv v) (cv u) (` F (cv y)) elunii syl5 r19.22i sylbi (cv y) peano2 (cv u) (suc (cv y)) F fveq2 (cv v) eleq2d (om) rcla4ev ex syl r19.23aiv (cv y) (cv u) F fveq2 (cv v) eleq2d (om) cbvrexv sylibr (cv v) y (om) (` F (cv y)) eliun sylibr syl u ax-gen mpgbir trcl.3 C (U_ y (om) (` F (cv y))) treq ax-mp mpbir (cv v) ({/}) F fveq2 (cv x) sseq1d (cv v) (cv y) F fveq2 (cv x) sseq1d (cv v) (suc (cv y)) F fveq2 (cv x) sseq1d trcl.2 ({/}) fveq1i trcl.1 A (V) ({<,>|} z w (= (cv w) (u. (cv z) (U. (cv z))))) fr0t ax-mp eqtr (cv x) sseq1i biimpr (Tr (cv x)) adantr (` F (cv y)) (cv x) uniss (U. (` F (cv y))) (U. (cv x)) (cv x) sstr2 (cv x) df-tr syl5ib syl anc2li (` F (cv y)) (cv x) (U. (` F (cv y))) unss syl6ib F (cv y) fvex F (cv y) fvex uniex unex (e. (cv v) A) z ax-17 (e. (cv v) (cv y)) z ax-17 v z w (= (cv w) (u. (cv z) (U. (cv z)))) hbopab1 (e. (cv v) A) z ax-17 hbrdg (e. (cv v) (om)) z ax-17 hbres trcl.2 (cv v) eleq2i trcl.2 (cv v) eleq2i z albii 3imtr4 (e. (cv v) (cv y)) z ax-17 hbfv v z w (= (cv w) (u. (cv z) (U. (cv z)))) hbopab1 (e. (cv v) A) z ax-17 hbrdg (e. (cv v) (om)) z ax-17 hbres trcl.2 (cv v) eleq2i trcl.2 (cv v) eleq2i z albii 3imtr4 (e. (cv v) (cv y)) z ax-17 hbfv hbuni hbun trcl.2 (cv z) (` F (cv y)) unieq (cv z) (` F (cv y)) (U. (cv z)) (U. (` F (cv y))) uneq12 mpdan (V) frsucopab mpan2 (cv x) sseq1d biimprd syl9r com23 (C_ A (cv x)) adantld finds2 com12 r19.21aiv trcl.3 (cv y) (cv v) F fveq2 (om) cbviunv eqtr (cv x) sseq1i v (om) (` F (cv v)) (cv x) iunss bitr sylibr x ax-gen 3pm3.2i)) thm (tz9.1 ((A x) (A y) (A z) (A w) (A v) (x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) ((tz9.1.1 (e. A (V)))) (E. x (/\/\ (C_ A (cv x)) (Tr (cv x)) (A. y (-> (/\ (C_ A (cv y)) (Tr (cv y))) (C_ (cv x) (cv y)))))) (tz9.1.1 (|` (rec ({<,>|} w v (= (cv v) (u. (cv w) (U. (cv w))))) A) (om)) eqid (U_ z (om) (` (|` (rec ({<,>|} w v (= (cv v) (u. (cv w) (U. (cv w))))) A) (om)) (cv z))) eqid y trcl omex (|` (rec ({<,>|} w v (= (cv v) (u. (cv w) (U. (cv w))))) A) (om)) (cv z) fvex z iunex (cv x) (U_ z (om) (` (|` (rec ({<,>|} w v (= (cv v) (u. (cv w) (U. (cv w))))) A) (om)) (cv z))) A sseq2 (cv x) (U_ z (om) (` (|` (rec ({<,>|} w v (= (cv v) (u. (cv w) (U. (cv w))))) A) (om)) (cv z))) treq (cv x) (U_ z (om) (` (|` (rec ({<,>|} w v (= (cv v) (u. (cv w) (U. (cv w))))) A) (om)) (cv z))) (cv y) sseq1 (/\ (C_ A (cv y)) (Tr (cv y))) imbi2d y albidv 3anbi123d cla4ev ax-mp)) thm (zfregs ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (-. (= A ({/}))) (E.e. x A (= (i^i (cv x) A) ({/})))) (A z n0 (cv z) snex y w tz9.1 (cv y) (cv w) (cv x) trel (cv y) A (cv x) inass A (cv x) incom (cv y) ineq2i eqtr (cv w) eleq2i (cv w) (cv y) (i^i (cv x) A) elin bitr2 (cv w) (i^i (i^i (cv y) A) (cv x)) n0i sylbi ex syl6 exp3a com34 imp3a (cv x) A inss1 (cv w) sseli ancri syl5 w 19.23adv (i^i (cv x) A) w n0 syl5ib com23 imp a3d (e. (cv x) A) anim2d ex imp3a (cv x) (cv y) A elin (= (i^i (i^i (cv y) A) (cv x)) ({/})) anbi1i (e. (cv x) (cv y)) (e. (cv x) A) (= (i^i (i^i (cv y) A) (cv x)) ({/})) anass bitr syl5ib r19.22dv2 y visset A inex1 x zfreg2 syl5 (cv z) A snssi (C_ ({} (cv z)) (cv y)) anim2i ({} (cv z)) (cv y) A ssin z visset (i^i (cv y) A) snss bitr4 sylib (cv z) (i^i (cv y) A) n0i syl syl5 exp3a impcom (A. w (-> (/\ (C_ ({} (cv z)) (cv w)) (Tr (cv w))) (C_ (cv y) (cv w)))) 3adant3 y 19.23aiv ax-mp z 19.23aiv sylbi)) thm (setind ((x y) (A x) (A y)) () (-> (A. x (-> (C_ (cv x) A) (e. (cv x) A))) (= A (V))) ((cv x) (cv y) A sseq1 (cv x) (cv y) A eleq1 imbi12d a4b1 (cv y) A ssindif0 syl5ibr (cv y) (V) A eldifn nsyli imp nrexdv (\ (V) A) y zfregs nsyl2 A vdif0 sylibr)) thm (setind2 ((A x)) () (-> (C_ (P~ A) A) (= A (V))) ((P~ A) A x dfss2 x visset A elpw (e. (cv x) A) imbi1i x albii bitr x A setind sylbi)) thm (r1fnon ((x y)) () (Fn (R1) (On)) (({<,>|} x y (= (cv y) (P~ (cv x)))) ({/}) rdgfnon x y df-r1 (R1) (rec ({<,>|} x y (= (cv y) (P~ (cv x)))) ({/})) (On) fneq1 ax-mp mpbir)) thm (r10 ((x y)) () (= (` (R1) ({/})) ({/})) (x y df-r1 ({/}) fveq1i 0ex ({<,>|} x y (= (cv y) (P~ (cv x)))) rdg0 eqtr)) thm (r1suc ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (On)) (= (` (R1) (suc A)) (P~ (` (R1) A)))) ((R1) A fvex pwex (e. (cv z) ({/})) x ax-17 (e. (cv z) A) x ax-17 (e. (cv z) (P~ (` (R1) A))) x ax-17 x y df-r1 (cv x) (` (R1) A) pweq (V) rdgsucopab mpan2)) thm (r1lim ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (e. A B) (Lim A)) (= (` (R1) A) (U_ x A (` (R1) (cv x))))) (A B ({<,>|} y z (= (cv z) (P~ (cv y)))) ({/}) x rdglim2a y z df-r1 A fveq1i y z df-r1 (cv x) fveq1i (e. (cv x) A) a1i iuneq2i 3eqtr4g)) thm (r1tr ((x y) (x z) (A x) (y z) (A y) (A z)) () (Tr (` (R1) A)) ((cv x) ({/}) (R1) fveq2 (` (R1) (cv x)) (` (R1) ({/})) treq syl (cv x) (cv y) (R1) fveq2 (` (R1) (cv x)) (` (R1) (cv y)) treq syl (cv x) (suc (cv y)) (R1) fveq2 (` (R1) (cv x)) (` (R1) (suc (cv y))) treq syl (cv x) A (R1) fveq2 (` (R1) (cv x)) (` (R1) A) treq syl tr0 r10 (` (R1) ({/})) ({/}) treq ax-mp mpbir (cv y) r1suc (cv x) eleq2d x visset (` (R1) (cv y)) elpw syl6bb (Tr (` (R1) (cv y))) adantr (cv x) (` (R1) (cv y)) (cv z) ssel (` (R1) (cv y)) dftr4 (` (R1) (cv y)) (P~ (` (R1) (cv y))) (cv z) ssel sylbi (cv y) r1suc (cv z) eleq2d biimprd sylan9r sylan9r ssrdv ex sylbid r19.21aiv (` (R1) (suc (cv y))) x dftr3 sylibr ex x visset (cv x) (V) y r1lim mpan (cv z) eleq2d (cv z) y (cv x) (` (R1) (cv y)) eliun biimp syl6bi y (cv x) (Tr (` (R1) (cv y))) hbra1 y (cv x) (Tr (` (R1) (cv y))) ra4 (` (R1) (cv y)) (cv z) trss syl6 r19.22d sylan9 x visset (cv x) (V) y r1lim mpan (cv z) sseq2d y (cv x) (cv z) (` (R1) (cv y)) ssiun syl5bir (A.e. y (cv x) (Tr (` (R1) (cv y)))) adantr syld r19.21aiv (` (R1) (cv x)) z dftr3 sylibr ex tfinds tr0 r1fnon (R1) (On) fndm ax-mp A eleq2i negbii A (R1) ndmfv sylbir (` (R1) A) ({/}) treq syl mpbiri pm2.61i)) thm (r1ord ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B (On)) (-> (e. A B) (e. (` (R1) A) (` (R1) B)))) ((cv x) (suc A) A eleq2 (cv x) (suc A) (R1) fveq2 (` (R1) A) eleq2d imbi12d (cv x) (cv y) A eleq2 (cv x) (cv y) (R1) fveq2 (` (R1) A) eleq2d imbi12d (cv x) (suc (cv y)) A eleq2 (cv x) (suc (cv y)) (R1) fveq2 (` (R1) A) eleq2d imbi12d (cv x) B A eleq2 (cv x) B (R1) fveq2 (` (R1) A) eleq2d imbi12d (suc A) A onelon A r1suc (R1) A fvex pwid syl5eleqr syl ex (cv y) r1suc (R1) (cv y) fvex pwid syl5eleqr (suc (cv y)) r1tr (` (R1) (suc (cv y))) (` (R1) (cv y)) trss ax-mp syl (` (R1) A) sseld (e. A (cv y)) imim2d (suc A) (On) elisset A sucexb sylibr A (V) (cv y) sucssel syl imp syl7 (e. A (suc (cv y))) a1d com24 exp3a imp31 (cv y) (suc A) (R1) fveq2 (` (R1) A) eleq2d (cv x) rcla4ev (cv x) A limsuc biimpa (cv x) A onelon (cv x) limord x visset elon sylibr sylan A r1suc (R1) A fvex pwid syl5eleqr syl sylanc (` (R1) A) y (cv x) (` (R1) (cv y)) eliun sylibr x visset (cv x) (V) y r1lim mpan (` (R1) A) eleq2d (e. A (cv x)) adantr mpbird ex (e. (suc A) (On)) (C_ (suc A) (cv x)) ad2antrr (A.e. y (cv x) (-> (C_ (suc A) (cv y)) (-> (e. A (cv y)) (e. (` (R1) A) (` (R1) (cv y)))))) a1d tfindsg (e. B (On)) (e. A B) pm3.26 B A onelon A suceloni syl jca B eloni B A ordsucss syl imp sylanc ex pm2.43d)) thm (r1ord2 () () (-> (e. B (On)) (-> (e. A B) (C_ (` (R1) A) (` (R1) B)))) (B A r1ord B r1tr (` (R1) B) (` (R1) A) trss ax-mp syl6)) thm (r1ord3 () () (-> (/\ (e. A (On)) (e. B (On))) (-> (C_ A B) (C_ (` (R1) A) (` (R1) B)))) (A B onsseleq B A r1ord2 (e. A (On)) adantl A B (R1) fveq2 (` (R1) A) (` (R1) B) eqimss syl (/\ (e. A (On)) (e. B (On))) a1i jaod sylbid)) thm (r1val1 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (` (R1) A) (U_ x A (P~ (` (R1) (cv x)))))) (A x onzsl (U_ x A (P~ (` (R1) (cv x)))) 0ss A ({/}) (R1) fveq2 r10 syl6eq (U_ x A (P~ (` (R1) (cv x)))) sseq1d mpbiri (e. (cv y) (` (R1) A)) x ax-17 y x A (P~ (` (R1) (cv x))) hbiu1 hbss A (suc (cv x)) (R1) fveq2 (cv x) r1suc sylan9eqr x visset sucid A (suc (cv x)) (cv x) eleq2 mpbiri x A (P~ (` (R1) (cv x))) ssiun2 syl (e. (cv x) (On)) adantl eqsstrd ex r19.23ai A (V) x r1lim A (cv x) ordelon A limord sylan (cv x) sucelon x visset sucid (suc (cv x)) (cv x) r1ord2 mpi sylbi (cv x) r1suc sseqtrd syl r19.21aiva x A (` (R1) (cv x)) (P~ (` (R1) (cv x))) ss2iun syl (e. A (V)) adantl eqsstrd 3jaoi sylbi A (cv x) onelon (cv x) r1suc syl (suc (cv x)) A r1ord3 A (cv x) onelon (cv x) sucelon sylib (e. A (On)) (e. (cv x) A) pm3.26 jca A eloni A (cv x) ordsucss syl imp sylc eqsstr3d r19.21aiva x A (P~ (` (R1) (cv x))) (` (R1) A) iunss sylibr eqssd)) thm (tz9.12lem1 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F x) (F y)) ((tz9.12lem.1 (e. A (V))) (tz9.12lem.2 (= F ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v)))))))))) (C_ (" F A) (On)) (y visset F A x elima3 tz9.12lem.2 (<,> (cv x) (cv y)) eleq2i x visset y visset (cv z) (cv x) (` (R1) (cv v)) eleq1 v (On) rabbisdv inteqd (cv w) eqeq2d (cv w) (cv y) (|^| ({e.|} v (On) (e. (cv x) (` (R1) (cv v))))) eqeq1 opelopab bitr (= (cv y) (|^| ({e.|} v (On) (e. (cv x) (` (R1) (cv v)))))) y 19.8a (|^| ({e.|} v (On) (e. (cv x) (` (R1) (cv v))))) y isset sylibr ({e.|} v (On) (e. (cv x) (` (R1) (cv v)))) intex (cv y) (|^| ({e.|} v (On) (e. (cv x) (` (R1) (cv v))))) (On) eleq1 v (On) (e. (cv x) (` (R1) (cv v))) ssrab2 ({e.|} v (On) (e. (cv x) (` (R1) (cv v)))) oninton mpan syl5bir com12 sylbir mpcom sylbi (e. (cv x) A) adantl x 19.23aiv sylbi ssriv)) thm (tz9.12lem2 ((w z) (v z) (A z) (v w) (A w) (A v)) ((tz9.12lem.1 (e. A (V))) (tz9.12lem.2 (= F ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v)))))))))) (e. (suc (U. (" F A))) (On)) (tz9.12lem.1 tz9.12lem.2 tz9.12lem1 z w (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v))))) funopabeq tz9.12lem.2 F ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v))))))) funeq ax-mp mpbir tz9.12lem.1 F funimaex ax-mp ssonuni ax-mp onsuc)) thm (tz9.12lem3 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F x) (F y)) ((tz9.12lem.1 (e. A (V))) (tz9.12lem.2 (= F ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v)))))))))) (-> (A.e. x A (E.e. y (On) (e. (cv x) (` (R1) (cv y))))) (e. A (` (R1) (suc (suc (U. (" F A))))))) ((cv v) (cv y) (R1) fveq2 (cv x) eleq2d (On) rcla4ev v (On) (e. (cv x) (` (R1) (cv v))) rabn0 sylibr tz9.12lem.2 (<,> (cv x) (cv y)) eleq2i x visset y visset (cv z) (cv x) (` (R1) (cv v)) eleq1 v (On) rabbisdv inteqd (cv w) eqeq2d (cv w) (cv y) (|^| ({e.|} v (On) (e. (cv x) (` (R1) (cv v))))) eqeq1 opelopab bitr y exbii x visset F y eldm2 (|^| ({e.|} v (On) (e. (cv x) (` (R1) (cv v))))) y isset 3bitr4 ({e.|} v (On) (e. (cv x) (` (R1) (cv v)))) intex bitr4 sylibr z w (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v))))) funopabeq tz9.12lem.2 F ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v))))))) funeq ax-mp mpbir F (cv x) A funfvima mpan syl tz9.12lem.1 tz9.12lem.2 tz9.12lem1 (" F A) onsucuni ax-mp (` F (cv x)) sseli tz9.12lem.1 tz9.12lem.2 tz9.12lem2 (suc (U. (" F A))) (` F (cv x)) r1ord2 ax-mp syl syl6 imp (cv v) (cv y) (R1) fveq2 (cv x) eleq2d (On) rcla4ev v (On) (e. (cv x) (` (R1) (cv v))) rabn0 sylibr ({e.|} v (On) (e. (cv x) (` (R1) (cv v)))) intex x visset (cv z) (cv x) (` (R1) (cv v)) eleq1 v (On) rabbisdv inteqd (V) (V) w fvopabg mpan tz9.12lem.2 (cv x) fveq1i syl5eq sylbi v (On) (e. (cv x) (` (R1) (cv v))) ssrab2 ({e.|} v (On) (e. (cv x) (` (R1) (cv v)))) onint mpan eqeltrd w v (On) (e. (cv z) (` (R1) (cv v))) hbrab1 hbint hbeleq y z w hbopab tz9.12lem.2 (cv y) eleq2i tz9.12lem.2 (cv y) eleq2i v albii 3imtr4 (e. (cv y) (cv x)) v ax-17 hbfv (e. (cv y) (On)) v ax-17 (e. (cv y) (cv x)) v ax-17 (e. (cv y) (R1)) v ax-17 w v (On) (e. (cv z) (` (R1) (cv v))) hbrab1 hbint hbeleq y z w hbopab tz9.12lem.2 (cv y) eleq2i tz9.12lem.2 (cv y) eleq2i v albii 3imtr4 (e. (cv y) (cv x)) v ax-17 hbfv hbfv hbel (cv v) (` F (cv x)) (R1) fveq2 (cv x) eleq2d elrabf pm3.27bd 3syl (e. (cv x) A) adantr sseldd exp31 com3r r19.23adv r19.20i tz9.12lem.1 tz9.12lem.2 tz9.12lem2 (suc (U. (" F A))) r1suc ax-mp A eleq2i tz9.12lem.1 (` (R1) (suc (U. (" F A)))) elpw A (` (R1) (suc (U. (" F A)))) x dfss3 3bitr sylibr)) thm (tz9.12 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) ((tz9.12.1 (e. A (V)))) (-> (A.e. x A (E.e. y (On) (e. (cv x) (` (R1) (cv y))))) (E.e. y (On) (e. A (` (R1) (cv y))))) (tz9.12.1 ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v))))))) eqid x y tz9.12lem3 tz9.12.1 ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v))))))) eqid tz9.12lem2 onsuc (cv y) (suc (suc (U. (" ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v))))))) A)))) (R1) fveq2 A eleq2d (On) rcla4ev mpan syl)) thm (tz9.13 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) ((tz9.13.1 (e. A (V)))) (E.e. x (On) (e. A (` (R1) (cv x)))) (tz9.13.1 z ({|} y (E.e. x (On) (e. (cv y) (` (R1) (cv x))))) setind (cv z) ({|} y (E.e. x (On) (e. (cv y) (` (R1) (cv x))))) (cv w) ssel w visset (cv y) (cv w) (` (R1) (cv x)) eleq1 x (On) rexbidv elab syl6ib r19.21aiv z visset w x tz9.12 syl z visset (cv y) (cv z) (` (R1) (cv x)) eleq1 x (On) rexbidv elab sylibr mpg eleqtrr tz9.13.1 (cv y) A (` (R1) (cv x)) eleq1 x (On) rexbidv elab mpbi)) thm (tz9.13g ((x y) (A x) (A y)) () (-> (e. A B) (E.e. x (On) (e. A (` (R1) (cv x))))) ((= (cv y) A) x ax-17 (cv y) A (` (R1) (cv x)) eleq1 (On) rexbid y visset x tz9.13 B vtoclg)) thm (rankwflem ((A x)) () (-> (e. A B) (E.e. x (On) (e. A (` (R1) (suc (cv x)))))) (A B x tz9.13g (cv x) suceloni x visset sucid (suc (cv x)) (cv x) r1ord2 mpi syl A sseld r19.22i syl)) thm (jech9.3 ((x y)) () (= (U_ x (On) (` (R1) (cv x))) (V)) (y visset x tz9.13 (cv y) x (On) (` (R1) (cv x)) eliun mpbir y visset 2th eqriv)) thm (unir1 () () (= (U. (" (R1) (On))) (V)) (r1fnon (R1) (On) fndm ax-mp (dom (R1)) (On) (R1) imaeq2 ax-mp (R1) imadmrn eqtr3 unieqi r1fnon (R1) (On) x fniunfv ax-mp x jech9.3 3eqtr2)) thm (rankval ((x y) (x z) (A x) (y z) (A y) (A z)) ((rankval.1 (e. A (V)))) (= (` (rank) A) (|^| ({e.|} x (On) (e. A (` (R1) (suc (cv x))))))) (y z x df-rank A fveq1i rankval.1 rankval.1 A (V) x rankwflem ax-mp x (On) (e. A (` (R1) (suc (cv x)))) rabn0 mpbir ({e.|} x (On) (e. A (` (R1) (suc (cv x))))) intex mpbi (cv y) A (` (R1) (suc (cv x))) eleq1 x (On) rabbisdv inteqd z fvopab eqtr)) thm (rankvalg ((x y) (A x) (A y)) () (-> (e. A B) (= (` (rank) A) (|^| ({e.|} x (On) (e. A (` (R1) (suc (cv x)))))))) ((cv y) A (rank) fveq2 (cv y) A (` (R1) (suc (cv x))) eleq1 x (On) rabbisdv inteqd eqeq12d y visset x rankval B vtoclg)) thm (rankval2 ((A x)) () (-> (e. A B) (= (` (rank) A) (|^| ({e.|} x (On) (C_ A (` (R1) (cv x))))))) (A B x rankvalg (cv x) r1suc A eleq2d (R1) (cv x) fvex (` (R1) (cv x)) (V) A elpw2g ax-mp syl6bb rabbii inteqi syl6eq)) thm (rankon ((A x)) () (e. (` (rank) A) (On)) (A (V) x rankvalg A (V) x rankwflem x (On) (e. A (` (R1) (suc (cv x)))) rabn0 sylibr x (On) (e. A (` (R1) (suc (cv x)))) ssrab2 ({e.|} x (On) (e. A (` (R1) (suc (cv x))))) oninton mpan syl eqeltrd A (rank) fvprc 0elon syl6eqel pm2.61i)) thm (rankid ((x y) (A x) (A y)) ((rankid.1 (e. A (V)))) (e. A (` (R1) (suc (` (rank) A)))) (rankid.1 A (V) x rankwflem ax-mp (e. (cv y) A) x ax-17 (e. (cv y) (R1)) x ax-17 y x (On) (e. A (` (R1) (suc (cv x)))) hbrab1 hbint hbsuc hbfv hbel (cv x) (|^| ({e.|} x (On) (e. A (` (R1) (suc (cv x)))))) suceq (R1) fveq2d A eleq2d onminsb ax-mp rankid.1 x rankval (` (rank) A) (|^| ({e.|} x (On) (e. A (` (R1) (suc (cv x)))))) suceq ax-mp (R1) fveq2i eleqtrr)) thm (rankr1lem () ((rankr1lem.1 (e. A (V)))) (-> (e. B (On)) (-> (-. (e. A (` (R1) B))) (C_ B (` (rank) A)))) (A rankon onsuc (suc (` (rank) A)) B ontri1 mpan rankr1lem.1 rankid A rankon onsuc (suc (` (rank) A)) B r1ord3 mpan (` (R1) (suc (` (rank) A))) (` (R1) B) A ssel syl6 mpii sylbird con1d A rankon B (` (rank) A) onsssuc mpan2 sylibrd)) thm (rankr1 ((x y) (A x) (A y) (B x) (B y)) ((rankr1.1 (e. A (V)))) (<-> (= B (` (rank) A)) (/\ (-. (e. A (` (R1) B))) (e. A (` (R1) (suc B))))) (A rankon B (` (rank) A) (On) eleq1 mpbiri B eloni B x ordzsl A noel B ({/}) (R1) fveq2 r10 syl6eq A eleq2d mtbiri (= B (` (rank) A)) a1d x visset sucid B (suc (cv x)) (cv x) eleq2 mpbiri (e. (cv x) (On)) adantl B (cv x) ontri1 con2bid (suc (cv x)) (On) B eleq1a imp (cv x) suceloni sylan (e. (cv x) (On)) (= B (suc (cv x))) pm3.26 sylanc mpbid (= B (` (rank) A)) adantr rankr1.1 y rankval B eqeq2i biimp (cv x) sseq1d (cv y) (cv x) suceq (R1) fveq2d A eleq2d onintss imp syl5bir B (suc (cv x)) (R1) fveq2 A eleq2d biimpa sylan2i exp4d com3l imp31 mtod exp31 r19.23aiv B (` (rank) A) (cv x) eleq2 A rankon (cv x) onel syl6bi anc2li (cv x) suceloni x visset sucid (suc (cv x)) (cv x) r1ord2 mpi syl A sseld (cv y) (cv x) suceq (R1) fveq2d A eleq2d onintss rankr1.1 y rankval (cv x) sseq1i syl6ibr syld B (` (rank) A) (cv x) sseq1 biimprd sylan9r B (cv x) ontri1 A rankon B (` (rank) A) (On) eleq1 mpbiri sylan sylibd syl6 com3l (= B (` (rank) A)) (e. (cv x) B) con2 syl6 pm2.43a r19.23aiv con2i (Lim B) adantr B (V) x r1lim (rank) A fvex B (` (rank) A) (V) eleq1 mpbiri sylan A eleq2d A x B (` (R1) (cv x)) eliun syl6bb mtbird expcom 3jaoi sylbi syl mpcom rankr1.1 y rankval y (On) (e. A (` (R1) (suc (cv y)))) ssrab2 rankr1.1 A (V) y rankwflem ax-mp y (On) (e. A (` (R1) (suc (cv y)))) rabn0 mpbir ({e.|} y (On) (e. A (` (R1) (suc (cv y))))) onint mp2an eqeltr B (` (rank) A) ({e.|} y (On) (e. A (` (R1) (suc (cv y))))) eleq1 mpbiri (cv y) B suceq (R1) fveq2d A eleq2d (On) elrab pm3.27bd syl jca rankr1.1 B rankr1lem com12 A noel (suc B) (R1) ndmfv A eleq2d mtbiri a3i r1fnon (R1) (On) fndm ax-mp (suc B) eleq2i B sucelon bitr4 sylib syl5 imp A noel (suc B) (R1) ndmfv A eleq2d mtbiri a3i r1fnon (R1) (On) fndm ax-mp (suc B) eleq2i B sucelon bitr4 sylib (cv y) B suceq (R1) fveq2d A eleq2d onintss mpcom rankr1.1 y rankval syl5ss (-. (e. A (` (R1) B))) adantl eqssd impbi)) thm (rankr1g ((A x) (B x)) () (-> (e. A C) (<-> (= B (` (rank) A)) (/\ (-. (e. A (` (R1) B))) (e. A (` (R1) (suc B)))))) ((cv x) A (rank) fveq2 B eqeq2d (cv x) A (` (R1) B) eleq1 negbid (cv x) A (` (R1) (suc B)) eleq1 anbi12d bibi12d x visset B rankr1 C vtoclg)) thm (ssrankr1 () ((ssrankr1.1 (e. A (V)))) (-> (e. B (On)) (<-> (C_ B (` (rank) A)) (-. (e. A (` (R1) B))))) ((` (rank) A) eqid ssrankr1.1 (` (rank) A) rankr1 mpbi pm3.26i A rankon B (` (rank) A) r1ord3 mpan2 imp A sseld mtoi ex ssrankr1.1 B rankr1lem impbid)) thm (rankr1a () ((rankr1a.1 (e. A (V)))) (-> (e. B (On)) (<-> (e. A (` (R1) B)) (e. (` (rank) A) B))) (rankr1a.1 B ssrankr1 A rankon B (` (rank) A) ontri1 mpan2 bitr3d con4bid)) thm (r1val2 ((A x)) () (-> (e. A (On)) (= (` (R1) A) ({|} x (e. (` (rank) (cv x)) A)))) (x visset A rankr1a abbi2dv)) thm (r1val3 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (` (R1) A) (U_ x A (P~ ({|} y (e. (` (rank) (cv y)) (cv x))))))) (A x r1val1 A (cv x) onelon (cv x) y r1val2 (` (R1) (cv x)) ({|} y (e. (` (rank) (cv y)) (cv x))) pweq 3syl r19.21aiva x A (P~ (` (R1) (cv x))) (P~ ({|} y (e. (` (rank) (cv y)) (cv x)))) iuneq2 syl eqtrd)) thm (rankel () ((rankel.1 (e. B (V)))) (-> (e. A B) (e. (` (rank) A) (` (rank) B))) ((` (rank) A) eqid A B (` (rank) A) rankr1g mpbii pm3.26d A rankon (` (rank) A) r1suc ax-mp B eleq2i rankel.1 (` (R1) (` (rank) A)) elpw bitr B (` (R1) (` (rank) A)) A ssel sylbi com12 mtod B rankon A rankon (` (rank) B) (` (rank) A) ontri1 mp2an B rankon onord A rankon onord (` (rank) B) (` (rank) A) ordsucsssuc mp2an rankel.1 rankid B rankon onsuc A rankon onsuc (suc (` (rank) B)) (suc (` (rank) A)) r1ord3 mp2an B sseld mpi sylbi sylbir nsyl2)) thm (rankval3 ((x y) (A x) (A y)) ((rankval3.1 (e. A (V)))) (= (` (rank) A) (|^| ({e.|} x (On) (A.e. y A (e. (` (rank) (cv y)) (cv x)))))) (rankval3.1 x rankval y visset rankid (cv x) eloni (cv x) (` (rank) (cv y)) ordsucss syl (cv y) rankon onsuc (suc (` (rank) (cv y))) (cv x) r1ord3 mpan syld (` (R1) (suc (` (rank) (cv y)))) (` (R1) (cv x)) (cv y) ssel syl6 mpii y A r19.20sdv (cv x) r1suc A eleq2d rankval3.1 (` (R1) (cv x)) elpw A (` (R1) (cv x)) y dfss3 bitr syl6bb sylibrd ss2rabi ({e.|} x (On) (A.e. y A (e. (` (rank) (cv y)) (cv x)))) ({e.|} x (On) (e. A (` (R1) (suc (cv x))))) intss ax-mp eqsstr A rankon rankval3.1 (cv y) rankel rgen (cv x) (` (rank) A) (` (rank) (cv y)) eleq2 y A ralbidv onintss mp2 eqssi)) thm (bndrank ((x y) (A x) (A y)) () (-> (E.e. x (On) (A.e. y A (C_ (` (rank) (cv y)) (cv x)))) (e. A (V))) (y visset rankid (cv x) eloni (cv y) rankon onord (` (rank) (cv y)) (cv x) ordsucsssuc mpan syl (cv x) suceloni (cv y) rankon onsuc (suc (` (rank) (cv y))) (suc (cv x)) r1ord3 mpan syl sylbid (` (R1) (suc (` (rank) (cv y)))) (` (R1) (suc (cv x))) (cv y) ssel syl6 mpii y A r19.20sdv A (` (R1) (suc (cv x))) y dfss3 (R1) (suc (cv x)) fvex A ssex sylbir syl6 r19.23aiv)) thm (unbndrank ((x y) (A x) (A y)) () (-> (-. (e. A (V))) (A.e. x (On) (E.e. y A (e. (cv x) (` (rank) (cv y)))))) ((cv y) rankon (` (rank) (cv y)) (cv x) ontri1 mpan y A ralbidv y A (e. (cv x) (` (rank) (cv y))) ralnex syl6bb rexbiia x (On) (E.e. y A (e. (cv x) (` (rank) (cv y)))) rexnal bitr x y A bndrank sylbir con1i)) thm (rankpw () ((rankpw.1 (e. A (V)))) (= (` (rank) (P~ A)) (suc (` (rank) A))) (rankpw.1 pwex (suc (` (rank) A)) rankr1 (` (rank) A) eqid rankpw.1 (` (rank) A) rankr1 pm3.26bd ax-mp rankpw.1 pwid (P~ A) (` (R1) (` (rank) A)) A ssel mpi mto rankpw.1 pwex (` (R1) (` (rank) A)) elpw mtbir A rankon (` (rank) A) r1suc ax-mp (P~ A) eleq2i mtbir rankpw.1 rankid A rankon (` (rank) A) r1suc ax-mp eleqtr rankpw.1 (` (R1) (` (rank) A)) elpw mpbi A (` (R1) (` (rank) A)) sspwb mpbi A rankon (` (rank) A) r1suc ax-mp sseqtr4 rankpw.1 pwex (` (R1) (suc (` (rank) A))) elpw mpbir A rankon onsuc (suc (` (rank) A)) r1suc ax-mp eleqtrr mpbir2an eqcomi)) thm (ranklim ((A x) (B x)) () (-> (Lim B) (<-> (e. (` (rank) A) B) (e. (` (rank) (P~ A)) B))) (B (` (rank) A) limsuc (e. A (V)) adantl (cv x) A pweq (rank) fveq2d (cv x) A (rank) fveq2 (` (rank) (cv x)) (` (rank) A) suceq syl eqeq12d x visset rankpw (V) vtoclg B eleq1d (Lim B) adantr bitr4d A (rank) fvprc A pwexb negbii (P~ A) (rank) fvprc sylbi eqtr4d B eleq1d (Lim B) adantr pm2.61ian)) thm (r1pw ((A x) (B x)) () (-> (e. B (On)) (<-> (e. A (` (R1) B)) (e. (P~ A) (` (R1) (suc B))))) ((cv x) A (` (R1) B) eleq1 (cv x) A pweq (` (R1) (suc B)) eleq1d bibi12d (e. B (On)) imbi2d x visset B rankr1a B eloni B (` (rank) (cv x)) ordsucelsuc syl bitrd x visset rankpw (suc B) eleq1i syl6bbr B suceloni x visset pwex (suc B) rankr1a syl bitr4d (V) vtoclg A (` (R1) B) elisset (P~ A) (` (R1) (suc B)) elisset A pwexb sylibr pm5.21ni (e. B (On)) a1d pm2.61i)) thm (r1pwcl ((A x) (A y) (x y) (B x) (B y)) () (-> (Lim B) (<-> (e. A (` (R1) B)) (e. (P~ A) (` (R1) B)))) (B (V) x r1lim A eleq2d A x B (` (R1) (cv x)) eliun syl6bb B (cv x) onelon B (V) limelon sylan (cv x) A r1pw syl rexbidva B (cv x) limsuc (e. (P~ A) (` (R1) (suc (cv x)))) anbi1d x visset sucex (cv y) (suc (cv x)) B eleq1 (cv y) (suc (cv x)) (R1) fveq2 (P~ A) eleq2d anbi12d cla4ev syl6bi x 19.23adv x B (e. (P~ A) (` (R1) (suc (cv x)))) df-rex y B (e. (P~ A) (` (R1) (cv y))) df-rex 3imtr4g (cv x) (cv y) (R1) fveq2 (P~ A) eleq2d B cbvrexv syl6ibr (e. B (V)) adantl B (cv x) onelon B (V) limelon sylan ex (cv x) sssucid (cv x) (suc (cv x)) r1ord3 mpi (cv x) sucelon sylan2b anidms (P~ A) sseld syl6 r19.22dv impbid 3bitrd B (V) x r1lim (P~ A) eleq2d (P~ A) x B (` (R1) (cv x)) eliun syl6bb bitr4d ex A (` (R1) B) n0i B (R1) fvprc nsyl2 (P~ A) (` (R1) B) n0i B (R1) fvprc nsyl2 pm5.21ni (Lim B) a1d pm2.61i)) thm (rankss () ((rankss.1 (e. B (V)))) (-> (C_ A B) (C_ (` (rank) A) (` (rank) B))) (rankss.1 pwex A rankel rankss.1 B (V) A elpw2g ax-mp rankss.1 rankpw (` (rank) A) eleq2i A rankon B rankon (` (rank) A) (` (rank) B) onsssuc mp2an bitr4 3imtr3)) thm (ranksn ((x y) (A x) (A y)) ((ranksn.1 (e. A (V)))) (= (` (rank) ({} A)) (suc (` (rank) A))) (y ({} A) (e. (` (rank) (cv y)) (cv x)) df-ral y A elsn (e. (` (rank) (cv y)) (cv x)) imbi1i y albii ranksn.1 (cv y) A (rank) fveq2 (cv x) eleq1d ceqsalv 3bitr (e. (cv x) (On)) a1i rabbii inteqi A snex x y rankval3 A rankon (` (rank) A) x onsucmin ax-mp 3eqtr4)) thm (rankuni2 ((x y) (x z) (A x) (y z) (A y) (A z)) ((ranksn.1 (e. A (V)))) (= (` (rank) (U. A)) (U_ x A (` (rank) (cv x)))) (ranksn.1 uniex z y rankval3 (cv z) (U_ x A (` (rank) (cv x))) (` (rank) (cv y)) eleq2 y (U. A) ralbidv (On) elrab ranksn.1 (rank) (cv x) fvex x iunon (cv x) rankon (e. (cv x) A) a1i mprg (cv y) A x eluni2 (e. (cv z) (` (rank) (cv y))) x ax-17 z x A (` (rank) (cv x)) hbiu1 hbel x A (` (rank) (cv x)) ssiun2 (` (rank) (cv y)) sseld x visset (cv y) rankel syl5 r19.23ai sylbi rgen mpbir2an (U_ x A (` (rank) (cv x))) ({e.|} z (On) (A.e. y (U. A) (e. (` (rank) (cv y)) (cv z)))) intss1 ax-mp eqsstr x A (` (rank) (cv x)) (` (rank) (U. A)) iunss (cv x) A elssuni ranksn.1 uniex (cv x) rankss syl mprgbir eqssi)) thm (rankun ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((rankun.1 (e. A (V))) (rankun.2 (e. B (V)))) (= (` (rank) (u. A B)) (u. (` (rank) A) (` (rank) B))) (rankun.1 rankun.2 unex y z rankval3 (cv x) eleq2i x visset y (On) (A.e. z (u. A B) (e. (` (rank) (cv z)) (cv y))) elintrab bitr (cv z) A B elun rankun.1 (cv z) rankel (` (rank) (cv z)) (` (rank) A) (` (rank) B) elun1 syl rankun.2 (cv z) rankel (` (rank) (cv z)) (` (rank) B) (` (rank) A) elun2 syl jaoi sylbi rgen A rankon B rankon onun (cv y) (u. (` (rank) A) (` (rank) B)) (` (rank) (cv z)) eleq2 z (u. A B) ralbidv (cv y) (u. (` (rank) A) (` (rank) B)) (cv x) eleq2 imbi12d (On) rcla4v ax-mp mpi sylbi ssriv A B ssun1 rankun.1 rankun.2 unex A rankss ax-mp B A ssun2 rankun.1 rankun.2 unex B rankss ax-mp unssi eqssi)) thm (rankpr () ((rankun.1 (e. A (V))) (rankun.2 (e. B (V)))) (= (` (rank) ({,} A B)) (suc (u. (` (rank) A) (` (rank) B)))) (A B df-pr (rank) fveq2i A snex B snex rankun rankun.1 ranksn rankun.2 ranksn uneq12i 3eqtr A rankon onord B rankon onord (` (rank) A) (` (rank) B) ordsucun mp2an eqtr4)) thm (rankop () ((rankun.1 (e. A (V))) (rankun.2 (e. B (V)))) (= (` (rank) (<,> A B)) (suc (suc (u. (` (rank) A) (` (rank) B))))) (A B df-op (rank) fveq2i A snex A B prex rankpr A B snsspr ({} A) ({,} A B) ssequn1 mpbi (rank) fveq2i A snex A B prex rankun rankun.1 rankun.2 rankpr 3eqtr3 (u. (` (rank) ({} A)) (` (rank) ({,} A B))) (suc (u. (` (rank) A) (` (rank) B))) suceq ax-mp 3eqtr)) thm (r1rankid () () (-> (e. A B) (C_ A (` (R1) (` (rank) A)))) ((` (rank) A) eqid A B (` (rank) A) rankr1g mpbii pm3.27d A rankon (` (rank) A) r1suc ax-mp syl6eleq A B (` (R1) (` (rank) A)) elpwg mpbid)) thm (rankonid ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (e. A (On)) (= (` (rank) A) A)) ((cv x) (cv y) (rank) fveq2 (= (cv x) (cv y)) id eqeq12d (cv x) A (rank) fveq2 (= (cv x) A) id eqeq12d (` (rank) (cv y)) (cv y) (cv z) eleq1 y (cv x) r19.20si y (cv x) (e. (` (rank) (cv y)) (cv z)) (e. (cv y) (cv z)) r19.15 syl (cv x) (cv z) y dfss3 syl6bbr z (On) rabbisdv inteqd x visset z y rankval3 syl5eq (cv x) (On) z intmin sylan9eqr ex tfis3 A rankon (` (rank) A) A (On) eleq1 mpbii impbi)) thm (rankeq0 ((A x)) ((rankeq0.1 (e. A (V)))) (<-> (= A ({/})) (= (` (rank) A) ({/}))) (A ({/}) (rank) fveq2 0elon ({/}) rankonid mpbi syl6eq rankeq0.1 A (V) x rankval2 ax-mp ({/}) eqeq1i x (On) (C_ A (` (R1) (cv x))) ssrab2 ({e.|} x (On) (C_ A (` (R1) (cv x)))) onint0 ax-mp (cv x) ({/}) (R1) fveq2 A sseq2d (On) elrab 0elon (C_ A (` (R1) ({/}))) biantrur r10 A sseq2i bitr3 3bitr A ss0 sylbi sylbi impbi)) thm (rankr1id ((x y) (A x) (A y)) () (<-> (e. A (On)) (= (` (rank) (` (R1) A)) A)) ((cv x) A (R1) fveq2 (rank) fveq2d (= (cv x) A) id eqeq12d (cv x) (cv y) r1ord3 ss2rabdv ({e.|} y (On) (C_ (cv x) (cv y))) ({e.|} y (On) (C_ (` (R1) (cv x)) (` (R1) (cv y)))) intss syl (cv x) (On) y intmin sseqtrd (R1) (cv x) fvex (` (R1) (cv x)) (V) y rankval2 ax-mp syl5ss (cv x) rankonid x visset (cv x) (V) r1rankid ax-mp (` (rank) (cv x)) (cv x) (R1) fveq2 (cv x) sseq2d mpbii (R1) (cv x) fvex (cv x) rankss syl (` (rank) (cv x)) (cv x) (` (rank) (` (R1) (cv x))) sseq1 mpbid sylbi eqssd vtoclga (` (R1) A) rankon (` (rank) (` (R1) A)) A (On) eleq1 mpbii impbi)) thm (rankuni ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (` (rank) (U. A)) (U. (` (rank) A))) ((cv x) A unieq (rank) fveq2d (cv x) A (rank) fveq2 unieqd eqeq12d x visset z rankuni2 (rank) (cv z) fvex z (cv x) y dfiun2 eqtr z (cv x) (= (cv y) (` (rank) (cv z))) df-rex x visset (cv z) rankel (= (cv y) (` (rank) (cv z))) anim1i z 19.22i z (e. (cv y) (` (rank) (cv x))) (= (cv y) (` (rank) (cv z))) 19.42v (cv y) (` (rank) (cv z)) (` (rank) (cv x)) eleq1 pm5.32ri z exbii (e. (cv y) (` (rank) (cv x))) (E. z (= (cv y) (` (rank) (cv z)))) pm3.26 (cv x) rankon (cv y) onel (cv y) rankr1id sylib eqcomd (R1) (cv y) fvex (cv z) (` (R1) (cv y)) (rank) fveq2 (cv y) eqeq2d cla4ev syl ancli impbi 3bitr3 sylib sylbi abssi ({|} y (E.e. z (cv x) (= (cv y) (` (rank) (cv z))))) (` (rank) (cv x)) uniss ax-mp eqsstr (cv x) pwuni x visset uniex pwex (cv x) rankss ax-mp x visset uniex rankpw sseqtr (` (rank) (cv x)) (suc (` (rank) (U. (cv x)))) uniss ax-mp (U. (cv x)) rankon onunisuc sseqtr eqssi (V) vtoclg A uniexb negbii (U. A) (rank) fvprc sylbi uni0 syl6eqr A (rank) fvprc unieqd eqtr4d pm2.61i)) thm (rankr1b () ((rankr1b.1 (e. A (V)))) (-> (e. B (On)) (<-> (C_ A (` (R1) B)) (C_ (` (rank) A) B))) (B rankr1id biimp (` (rank) A) sseq2d (R1) B fvex A rankss syl5bi A rankon (` (rank) A) B r1ord3 mpan rankr1b.1 A (V) r1rankid ax-mp A (` (R1) (` (rank) A)) (` (R1) B) sstr mpan syl6 impbid)) thm (rankuniOLD ((x y) (A x) (A y)) ((rankr1b.1 (e. A (V)))) (= (` (rank) (U. A)) (U. (` (rank) A))) (rankr1b.1 y rankuni2 (rank) (cv y) fvex y A x dfiun2 eqtr y A (= (cv x) (` (rank) (cv y))) df-rex rankr1b.1 (cv y) rankel (= (cv x) (` (rank) (cv y))) anim1i y 19.22i y (e. (cv x) (` (rank) A)) (= (cv x) (` (rank) (cv y))) 19.42v (cv x) (` (rank) (cv y)) (` (rank) A) eleq1 pm5.32ri y exbii (e. (cv x) (` (rank) A)) (E. y (= (cv x) (` (rank) (cv y)))) pm3.26 A rankon (cv x) onel (cv x) rankr1id sylib eqcomd (R1) (cv x) fvex (cv y) (` (R1) (cv x)) (rank) fveq2 (cv x) eqeq2d cla4ev syl ancli impbi 3bitr3 sylib sylbi abssi ({|} x (E.e. y A (= (cv x) (` (rank) (cv y))))) (` (rank) A) uniss ax-mp eqsstr A pwuni rankr1b.1 uniex pwex A rankss ax-mp rankr1b.1 uniex rankpw sseqtr (` (rank) A) (suc (` (rank) (U. A))) uniss ax-mp (U. A) rankon onunisuc sseqtr eqssi)) thm (rankuniss () ((rankr1b.1 (e. A (V)))) (C_ (` (rank) (U. A)) (` (rank) A)) (A rankuni A rankon onord (` (rank) A) orduniss ax-mp eqsstr)) thm (rankval4 ((x y) (A x) (A y)) ((rankr1b.1 (e. A (V)))) (= (` (rank) A) (U_ x A (suc (` (rank) (cv x))))) ((e. (cv y) A) x ax-17 (e. (cv y) (R1)) x ax-17 y x A (suc (` (rank) (cv x))) hbiu1 hbfv dfss2f x visset rankid x A (suc (` (rank) (cv x))) ssiun2 (cv x) rankon onsuc rankr1b.1 (rank) (cv x) fvex sucex x iunon (cv x) rankon onsuc (e. (cv x) A) a1i mprg (suc (` (rank) (cv x))) (U_ x A (suc (` (rank) (cv x)))) r1ord3 mp2an syl (cv x) sseld mpi mpgbir (R1) (U_ x A (suc (` (rank) (cv x)))) fvex A rankss ax-mp rankr1b.1 (rank) (cv x) fvex sucex x iunon (cv x) rankon onsuc (e. (cv x) A) a1i mprg (U_ x A (suc (` (rank) (cv x)))) (cv y) r1ord3 mpan ss2rabi ({e.|} y (On) (C_ (U_ x A (suc (` (rank) (cv x)))) (cv y))) ({e.|} y (On) (C_ (` (R1) (U_ x A (suc (` (rank) (cv x))))) (` (R1) (cv y)))) intss ax-mp (R1) (U_ x A (suc (` (rank) (cv x)))) fvex (` (R1) (U_ x A (suc (` (rank) (cv x))))) (V) y rankval2 ax-mp rankr1b.1 (rank) (cv x) fvex sucex x iunon (cv x) rankon onsuc (e. (cv x) A) a1i mprg (U_ x A (suc (` (rank) (cv x)))) (On) y intmin ax-mp eqcomi 3sstr4 sstri x A (suc (` (rank) (cv x))) (` (rank) A) iunss rankr1b.1 (cv x) rankel (cv x) rankon A rankon onsucss sylib mprgbir eqssi)) thm (rankbnd ((A x) (B x)) ((rankr1b.1 (e. A (V)))) (<-> (A.e. x A (C_ (suc (` (rank) (cv x))) B)) (C_ (` (rank) A) B)) (rankr1b.1 x rankval4 B sseq1i x A (suc (` (rank) (cv x))) B iunss bitr2)) thm (rankelun () ((rankelun.1 (e. A (V))) (rankelun.2 (e. B (V))) (rankelun.3 (e. C (V))) (rankelun.4 (e. D (V)))) (-> (/\ (e. (` (rank) A) (` (rank) C)) (e. (` (rank) B) (` (rank) D))) (e. (` (rank) (u. A B)) (` (rank) (u. C D)))) (A (` (R1) (` (rank) C)) (` (R1) (` (rank) D)) elun1 B (` (R1) (` (rank) D)) (` (R1) (` (rank) C)) elun2 anim12i rankelun.1 rankelun.2 (u. (` (R1) (` (rank) C)) (` (R1) (` (rank) D))) prss sylib (R1) (` (rank) C) fvex (R1) (` (rank) D) fvex unex ({,} A B) rankss syl rankelun.1 rankelun.2 rankpr rankelun.1 rankelun.2 rankun (` (rank) (u. A B)) (u. (` (rank) A) (` (rank) B)) suceq ax-mp eqtr4 syl5ssr (rank) (u. A B) fvex (` (rank) (u. A B)) (V) (` (rank) (u. (` (R1) (` (rank) C)) (` (R1) (` (rank) D)))) sucssel ax-mp syl C rankon rankelun.1 (` (rank) C) rankr1a ax-mp D rankon rankelun.2 (` (rank) D) rankr1a ax-mp syl2anbr C rankon (` (rank) C) rankr1id mpbi D rankon (` (rank) D) rankr1id mpbi uneq12i (R1) (` (rank) C) fvex (R1) (` (rank) D) fvex rankun rankelun.3 rankelun.4 rankun 3eqtr4 syl6eleq)) thm (rankxpl () ((rankxpl.1 (e. A (V))) (rankxpl.2 (e. B (V)))) (-> (=/= (X. A B) ({/})) (C_ (` (rank) (u. A B)) (` (rank) (X. A B)))) (A B unixp (rank) fveq2d rankxpl.1 rankxpl.2 xpex uniex rankuniss rankxpl.1 rankxpl.2 xpex rankuniss sstri (=/= (X. A B) ({/})) a1i eqsstr3d)) thm (rankxpu () ((rankxpl.1 (e. A (V))) (rankxpl.2 (e. B (V)))) (C_ (` (rank) (X. A B)) (suc (suc (` (rank) (u. A B))))) (A B xpsspw rankxpl.1 rankxpl.2 unex pwex pwex (X. A B) rankss ax-mp rankxpl.1 rankxpl.2 unex pwex rankpw rankxpl.1 rankxpl.2 unex rankpw (` (rank) (P~ (u. A B))) (suc (` (rank) (u. A B))) suceq ax-mp eqtr sseqtr)) thm (rankxplim ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((rankxplim.1 (e. A (V))) (rankxplim.2 (e. B (V)))) (-> (/\ (Lim (` (rank) (u. A B))) (=/= (X. A B) ({/}))) (= (` (rank) (X. A B)) (` (rank) (u. A B)))) (x visset y visset rankxplim.1 rankxplim.2 rankelun rankxplim.1 (cv x) rankel rankxplim.2 (cv y) rankel syl2an (Lim (` (rank) (u. A B))) adantl (` (rank) (u. A B)) (u. (cv x) (cv y)) ranklim (` (rank) (u. A B)) (P~ (u. (cv x) (cv y))) ranklim bitrd (/\ (e. (cv x) A) (e. (cv y) B)) adantr mpbid (<,> (cv x) (cv y)) pwuni (cv x) (cv y) uniop (U. (<,> (cv x) (cv y))) ({,} (cv x) (cv y)) pweq ax-mp sseqtr ({,} (cv x) (cv y)) pwuni x visset y visset unipr (U. ({,} (cv x) (cv y))) (u. (cv x) (cv y)) pweq ax-mp sseqtr ({,} (cv x) (cv y)) (P~ (u. (cv x) (cv y))) sspwb mpbi sstri x visset y visset unex pwex pwex (<,> (cv x) (cv y)) rankss ax-mp (<,> (cv x) (cv y)) rankon (u. A B) rankon (` (rank) (<,> (cv x) (cv y))) (` (rank) (u. A B)) (` (rank) (P~ (P~ (u. (cv x) (cv y))))) ontr2 mp2an mpan syl (<,> (cv x) (cv y)) rankon (u. A B) rankon onsucss sylib ex r19.21aivv (cv z) (<,> (cv x) (cv y)) (rank) fveq2 (` (rank) (cv z)) (` (rank) (<,> (cv x) (cv y))) suceq syl (` (rank) (u. A B)) sseq1d A B ralxp rankxplim.1 rankxplim.2 xpex z (` (rank) (u. A B)) rankbnd bitr3 sylib (=/= (X. A B) ({/})) adantr rankxplim.1 rankxplim.2 rankxpl (Lim (` (rank) (u. A B))) adantl eqssd)) thm (rankxplim2 () ((rankxplim.1 (e. A (V))) (rankxplim.2 (e. B (V)))) (-> (Lim (` (rank) (X. A B))) (Lim (` (rank) (u. A B)))) ((` (rank) (X. A B)) 0ellim ({/}) (` (rank) (X. A B)) n0i (X. A B) ({/}) df-ne rankxplim.1 rankxplim.2 xpex rankeq0 negbii bitr2 biimp 3syl A B unixp (rank) fveq2d (U. (X. A B)) rankuni (X. A B) rankuni unieqi eqtr2 syl5eq (U. (U. (` (rank) (X. A B)))) (` (rank) (u. A B)) limeq syl (` (rank) (X. A B)) limuni2 (U. (` (rank) (X. A B))) limuni2 syl syl5bi mpcom)) thm (rankxplim3 ((x y) (A x) (A y) (B x) (B y)) ((rankxplim.1 (e. A (V))) (rankxplim.2 (e. B (V)))) (<-> (Lim (` (rank) (X. A B))) (Lim (U. (` (rank) (X. A B))))) ((` (rank) (X. A B)) limuni2 (u. A B) rankon onord (` (rank) (u. A B)) x ordzsl mpbi 3ori A B un00 (= B ({/})) (= A ({/})) olc (= A ({/})) adantl sylbir A B xpeq0 sylibr con3i rankxplim.1 rankxplim.2 xpex rankeq0 negbii rankxplim.1 rankxplim.2 unex rankeq0 negbii 3imtr3 sylan ex (X. A B) rankon onord (` (rank) (X. A B)) y ordzsl mpbi 3ori ex orim12d (E.e. x (On) (= (` (rank) (u. A B)) (suc (cv x)))) (E.e. y (On) (= (` (rank) (X. A B)) (suc (cv y)))) ianor syl5ib imp (Lim (` (rank) (u. A B))) (-. (= (` (rank) (X. A B)) ({/}))) pm3.26 rankxplim.1 rankxplim.2 rankxplim (X. A B) ({/}) df-ne rankxplim.1 rankxplim.2 xpex rankeq0 negbii bitr sylan2br (` (rank) (X. A B)) (` (rank) (u. A B)) limeq syl mpbird expcom (-. (= (` (rank) (X. A B)) ({/}))) (Lim (` (rank) (X. A B))) idd jaod (-. (/\ (E.e. x (On) (= (` (rank) (u. A B)) (suc (cv x)))) (E.e. y (On) (= (` (rank) (X. A B)) (suc (cv y)))))) adantr mpd (U. (` (rank) (X. A B))) 0ellim ({/}) (U. (` (rank) (X. A B))) n0i (` (rank) (X. A B)) ({/}) unieq uni0 syl6eq con3i 3syl (u. A B) rankon onsuc onsuc elisseti sucid (u. A B) rankon onsuc onsuc onsuc (u. A B) rankon onsuc onsuc (suc (suc (suc (` (rank) (u. A B))))) (suc (suc (` (rank) (u. A B)))) ontri1 mp2an con2bii mpbi rankxplim.1 rankxplim.2 rankxpu (suc (suc (suc (` (rank) (u. A B))))) (` (rank) (X. A B)) (suc (suc (` (rank) (u. A B)))) sstr mpan2 mto (= (` (rank) (u. A B)) (suc (cv x))) (= (` (rank) (X. A B)) (suc (cv y))) pm3.26 (Lim (U. (` (rank) (X. A B)))) adantl (Lim (U. (` (rank) (X. A B)))) (= (` (rank) (u. A B)) (suc (cv x))) pm3.27 (U. (` (rank) (X. A B))) 0ellim ({/}) (U. (` (rank) (X. A B))) n0i (` (rank) (X. A B)) ({/}) unieq uni0 syl6eq con3i 3syl (X. A B) ({/}) df-ne rankxplim.1 rankxplim.2 xpex rankeq0 negbii bitr2 sylib A B unixp syl (rank) fveq2d (U. (X. A B)) rankuni (X. A B) rankuni unieqi eqtr syl5reqr (` (rank) (u. A B)) (U. (U. (` (rank) (X. A B)))) eqimss syl (= (` (rank) (u. A B)) (suc (cv x))) adantr eqsstr3d (= (` (rank) (X. A B)) (suc (cv y))) adantrr (U. (` (rank) (X. A B))) limuni (/\ (= (` (rank) (u. A B)) (suc (cv x))) (= (` (rank) (X. A B)) (suc (cv y)))) adantr sseqtr4d x visset (X. A B) rankon onord (` (rank) (X. A B)) orduni ax-mp (cv x) (V) (U. (` (rank) (X. A B))) ordelsuc mp2an sylibr (U. (` (rank) (X. A B))) (cv x) limsuc (/\ (= (` (rank) (u. A B)) (suc (cv x))) (= (` (rank) (X. A B)) (suc (cv y)))) adantr mpbid eqeltrd (U. (` (rank) (X. A B))) (` (rank) (u. A B)) limsuc (/\ (= (` (rank) (u. A B)) (suc (cv x))) (= (` (rank) (X. A B)) (suc (cv y)))) adantr mpbid (X. A B) rankon onord (` (rank) (X. A B)) orduni ax-mp (U. (` (rank) (X. A B))) (suc (` (rank) (u. A B))) ordsucelsuc ax-mp sylib (X. A B) rankon (` (rank) (X. A B)) (cv y) onsucuni2 mpan (Lim (U. (` (rank) (X. A B)))) (= (` (rank) (u. A B)) (suc (cv x))) ad2antll eleqtrd (u. A B) rankon onsuc onsuc elisseti (X. A B) rankon onord (suc (suc (` (rank) (u. A B)))) (V) (` (rank) (X. A B)) ordelsuc mp2an sylib ex (/\ (e. (cv x) (On)) (e. (cv y) (On))) a1d r19.23advv x (On) y (On) (= (` (rank) (u. A B)) (suc (cv x))) (= (` (rank) (X. A B)) (suc (cv y))) reeanv syl5ibr mtoi sylanc impbi)) thm (rankxpsuc ((A x) (B x)) ((rankxplim.1 (e. A (V))) (rankxplim.2 (e. B (V)))) (-> (/\ (= (` (rank) (u. A B)) (suc C)) (=/= (X. A B) ({/}))) (= (` (rank) (X. A B)) (suc (suc (` (rank) (u. A B)))))) (A B unixp (rank) fveq2d (U. (X. A B)) rankuni (X. A B) rankuni unieqi eqtr syl5reqr (u. A B) rankon (X. A B) rankon (` (rank) (X. A B)) onuni ax-mp (U. (` (rank) (X. A B))) onuni ax-mp (` (rank) (u. A B)) (U. (U. (` (rank) (X. A B)))) suc11 mp2an sylibr (= (` (rank) (u. A B)) (suc C)) adantl (rank) (u. A B) fvex (` (rank) (u. A B)) (suc C) (V) eleq1 mpbii C sucexb sylibr C (V) nlimsucg syl (` (rank) (u. A B)) (suc C) limeq mtbird rankxplim.1 rankxplim.2 rankxplim2 con3i (X. A B) ({/}) df-ne rankxplim.1 rankxplim.2 xpex rankeq0 negbii bitr (X. A B) rankon onord (` (rank) (X. A B)) x ordzsl mpbi (= (` (rank) (X. A B)) ({/})) (E.e. x (On) (= (` (rank) (X. A B)) (suc (cv x)))) (Lim (` (rank) (X. A B))) 3orass mpbi ori sylbi ord con1d com12 3syl x visset (cv x) (V) nlimsucg ax-mp (` (rank) (X. A B)) (suc (cv x)) limeq mtbiri (e. (cv x) (On)) a1i r19.23aiv rankxplim.1 rankxplim.2 rankxplim3 negbii sylib syl6com (X. A B) ({/}) df-ne A B unixp0 rankxplim.1 rankxplim.2 xpex uniex rankeq0 (X. A B) rankuni ({/}) eqeq1i 3bitr negbii bitr (X. A B) rankon (` (rank) (X. A B)) onuni ax-mp onord (U. (` (rank) (X. A B))) x ordzsl mpbi (= (U. (` (rank) (X. A B))) ({/})) (E.e. x (On) (= (U. (` (rank) (X. A B))) (suc (cv x)))) (Lim (U. (` (rank) (X. A B)))) 3orass mpbi ori sylbi ord con1d syld impcom (X. A B) rankon (` (rank) (X. A B)) onuni ax-mp (U. (` (rank) (X. A B))) (cv x) onsucuni2 mpan (e. (cv x) (On)) a1i r19.23aiv syl eqtrd (u. A B) rankon onsuc (X. A B) rankon (` (rank) (X. A B)) onuni ax-mp (suc (` (rank) (u. A B))) (U. (` (rank) (X. A B))) suc11 mp2an sylibr (rank) (u. A B) fvex (` (rank) (u. A B)) (suc C) (V) eleq1 mpbii C sucexb sylibr C (V) nlimsucg syl (` (rank) (u. A B)) (suc C) limeq mtbird rankxplim.1 rankxplim.2 rankxplim2 con3i (X. A B) ({/}) df-ne rankxplim.1 rankxplim.2 xpex rankeq0 negbii bitr (X. A B) rankon onord (` (rank) (X. A B)) x ordzsl mpbi (= (` (rank) (X. A B)) ({/})) (E.e. x (On) (= (` (rank) (X. A B)) (suc (cv x)))) (Lim (` (rank) (X. A B))) 3orass mpbi ori sylbi ord con1d com12 3syl imp (X. A B) rankon (` (rank) (X. A B)) (cv x) onsucuni2 mpan (e. (cv x) (On)) a1i r19.23aiv syl eqtr2d)) thm (scottex ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (e. ({e.|} x A (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y))))) (V)) (0ex A ({/}) (V) eleq1 mpbiri A (V) x (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y)))) rabexg syl A y n0 y A (C_ (` (rank) (cv x)) (` (rank) (cv y))) hbra1 (e. (cv z) A) y ax-17 x hbrab (e. (cv z) (V)) y ax-17 hbel y A (C_ (` (rank) (cv x)) (` (rank) (cv y))) ra4 com12 (e. (cv x) A) a1d r19.21aiv x A (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y)))) (C_ (` (rank) (cv x)) (` (rank) (cv y))) ss2rab sylibr (cv y) rankon (cv x) (cv w) (rank) fveq2 (` (rank) (cv y)) sseq1d A elrab pm3.27bd rgen (cv z) (` (rank) (cv y)) (` (rank) (cv w)) sseq2 w ({e.|} x A (C_ (` (rank) (cv x)) (` (rank) (cv y)))) ralbidv (On) rcla4ev mp2an z w ({e.|} x A (C_ (` (rank) (cv x)) (` (rank) (cv y)))) bndrank ax-mp ({e.|} x A (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y))))) ssex syl 19.23ai sylbi pm2.61i)) thm (scott0 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (= A ({/})) (= ({e.|} x A (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y))))) ({/}))) (A ({/}) x (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y)))) rabeq x (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y)))) rab0 syl6eq A x n0 x A (= (` (rank) (cv x)) (` (rank) (cv x))) hbre1 (` (rank) (cv x)) eqid x A (= (` (rank) (cv x)) (` (rank) (cv x))) ra4e mpan2 19.23ai sylbi (rank) (cv x) fvex (cv y) (` (rank) (cv x)) (` (rank) (cv x)) eqeq1 (e. (cv x) A) anbi2d cla4ev x 19.22i y x (/\ (e. (cv x) A) (= (cv y) (` (rank) (cv x)))) excom sylibr x A (= (` (rank) (cv x)) (` (rank) (cv x))) df-rex x A (= (cv y) (` (rank) (cv x))) df-rex y exbii 3imtr4 syl y (E.e. x A (= (cv y) (` (rank) (cv x)))) abn0 sylibr z y (E.e. x A (= (cv y) (` (rank) (cv x)))) hbab1 (e. (cv z) (On)) y ax-17 dfss2f y (E.e. x A (= (cv y) (` (rank) (cv x)))) abid (cv x) rankon (cv y) (` (rank) (cv x)) (On) eleq1 mpbiri (e. (cv x) A) a1i r19.23aiv sylbi mpgbir ({|} y (E.e. x A (= (cv y) (` (rank) (cv x))))) onint mpan (rank) (cv x) fvex x A y dfiin2 syl5eqel syl y x A (` (rank) (cv x)) hbii1 hbeleq (cv y) (|^|_ x A (` (rank) (cv x))) (` (rank) (cv x)) eqeq1 A rexbid ({|} y (E.e. x A (= (cv y) (` (rank) (cv x))))) elabg ibi (|^|_ x A (` (rank) (cv x))) (` (rank) (cv x)) (` (rank) (cv y)) sseq1 (` (rank) (cv y)) ssid (cv x) (cv y) (rank) fveq2 (` (rank) (cv y)) sseq1d A rcla4ev mpan2 x A (` (rank) (cv x)) (` (rank) (cv y)) iinss syl syl5bi r19.21aiv x A r19.22si 3syl x A (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y)))) rabn0 sylibr a3i impbi)) thm (scottexs ((x y) (x z) (y z) (ph y) (ph z)) () (e. ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) (V)) ((e. (cv y) ({|} x ph)) z ax-17 y x ph hbab1 y x ph hbab1 (C_ (` (rank) (cv z)) (` (rank) (cv y))) x ax-17 hbral (A.e. y ({|} x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))) z ax-17 (cv z) (cv x) (rank) fveq2 (` (rank) (cv y)) sseq1d y ({|} x ph) ralbidv cbvrab x ({|} x ph) (A.e. y ({|} x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))) df-rab x ph abid y ({|} x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))) df-ral y x ph df-clab (C_ (` (rank) (cv x)) (` (rank) (cv y))) imbi1i y albii bitr anbi12i x abbii 3eqtr z ({|} x ph) y scottex eqeltrr)) thm (scott0s ((x y) (x z) (y z) (ph y) (ph z)) () (<-> (E. x ph) (-. (= ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) ({/})))) (x ph abn0 ({|} x ph) z y scott0 (e. (cv y) ({|} x ph)) z ax-17 y x ph hbab1 y x ph hbab1 (C_ (` (rank) (cv z)) (` (rank) (cv y))) x ax-17 hbral (A.e. y ({|} x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))) z ax-17 (cv z) (cv x) (rank) fveq2 (` (rank) (cv y)) sseq1d y ({|} x ph) ralbidv cbvrab x ({|} x ph) (A.e. y ({|} x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))) df-rab x ph abid y ({|} x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))) df-ral y x ph df-clab (C_ (` (rank) (cv x)) (` (rank) (cv y))) imbi1i y albii bitr anbi12i x abbii 3eqtr ({/}) eqeq1i bitr negbii bitr3)) thm (cplem1 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C w) (D w)) ((cplem1.1 (= C ({e.|} y B (A.e. z B (C_ (` (rank) (cv y)) (` (rank) (cv z))))))) (cplem1.2 (= D (U_ x A C)))) (A.e. x A (-> (-. (= B ({/}))) (-. (= (i^i B D) ({/}))))) (cplem1.1 y B (A.e. z B (C_ (` (rank) (cv y)) (` (rank) (cv z)))) ssrab2 eqsstr (cv w) sseli (e. (cv x) A) a1i x A C ssiun2 cplem1.2 syl6ssr (cv w) sseld jcad (cv w) B D inelcm syl6 w 19.23adv B y z scott0 cplem1.1 ({/}) eqeq1i bitr4 negbii C w n0 bitr syl5ib rgen)) thm (cplem2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w)) ((cplem2.1 (e. A (V)))) (E. y (A.e. x A (-> (-. (= B ({/}))) (-. (= (i^i B (cv y)) ({/})))))) (({e.|} z B (A.e. w B (C_ (` (rank) (cv z)) (` (rank) (cv w))))) eqid (U_ x A ({e.|} z B (A.e. w B (C_ (` (rank) (cv z)) (` (rank) (cv w)))))) eqid cplem1 cplem2.1 z B w scottex x iunex y x A ({e.|} z B (A.e. w B (C_ (` (rank) (cv z)) (` (rank) (cv w))))) hbiu1 hbeleq (cv y) (U_ x A ({e.|} z B (A.e. w B (C_ (` (rank) (cv z)) (` (rank) (cv w)))))) B ineq2 ({/}) eqeq1d negbid (-. (= B ({/}))) imbi2d A ralbid cla4ev ax-mp)) thm (cp ((ph z) (ph w) (w z) (x y) (x z) (w x) (y z) (w y)) () (E. w (A.e. x (cv z) (-> (E. y ph) (E.e. y (cv w) ph)))) (z visset w x ({|} y ph) cplem2 y ph abn0 (cv y) ({|} y ph) (cv w) elin y ph abid (e. (cv y) (cv w)) anbi1i ph (e. (cv y) (cv w)) ancom 3bitr y exbii z y ph hbab1 (e. (cv z) (cv w)) y ax-17 hbin n0f y (cv w) ph df-rex 3bitr4 imbi12i x (cv z) ralbii w exbii mpbi)) thm (bnd ((ph z) (ph w) (w z) (x y) (x z) (w x) (y z) (w y)) () (-> (A.e. x (cv z) (E. y ph)) (E. w (A.e. x (cv z) (E.e. y (cv w) ph)))) (w x z y ph cp x (cv z) (E. y ph) (E.e. y (cv w) ph) r19.20 w 19.22i ax-mp w (A.e. x (cv z) (E. y ph)) (A.e. x (cv z) (E.e. y (cv w) ph)) 19.37v mpbi)) thm (bnd2 ((ph z) (ph w) (ph v) (w z) (v z) (v w) (x z) (w x) (v x) (A x) (A z) (A w) (A v) (x y) (B x) (y z) (w y) (v y) (B y) (B z) (B w) (B v)) ((bnd2.1 (e. A (V)))) (-> (A.e. x A (E.e. y B ph)) (E. z (/\ (C_ (cv z) B) (A.e. x A (E.e. y (cv z) ph))))) (y B ph df-rex x A ralbii bnd2.1 (cv v) A x (E. y (/\ (e. (cv y) B) ph)) raleq1 (cv v) A x (E.e. y (cv w) (/\ (e. (cv y) B) ph)) raleq1 w exbidv imbi12d x v y (/\ (e. (cv y) B) ph) w bnd vtocl sylbi w visset B inex1 (cv w) B inss2 (cv z) (i^i (cv w) B) B sseq1 mpbiri (A.e. x A (E.e. y (cv z) ph)) biantrurd (cv z) (i^i (cv w) B) y ph rexeq1 (cv y) (cv w) B elin ph anbi1i (e. (cv y) (cv w)) (e. (cv y) B) ph anass bitr rexbii2 syl6bb x A ralbidv bitr3d cla4ev w 19.23aiv syl)) thm (kardex ((x y) (x z) (A x) (y z) (A y) (A z)) ((kardex.1 (e. A (V)))) (e. ({|} x (/\ (br (cv x) (~~) A) (A. y (-> (br (cv y) (~~) A) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) (V)) (x ({|} z (br (cv z) (~~) A)) (A.e. y ({|} z (br (cv z) (~~) A)) (C_ (` (rank) (cv x)) (` (rank) (cv y)))) df-rab x visset (cv z) (cv x) (~~) A breq1 elab y ({|} z (br (cv z) (~~) A)) (C_ (` (rank) (cv x)) (` (rank) (cv y))) df-ral y visset (cv z) (cv y) (~~) A breq1 elab (C_ (` (rank) (cv x)) (` (rank) (cv y))) imbi1i y albii bitr anbi12i x abbii eqtr x ({|} z (br (cv z) (~~) A)) y scottex eqeltrr)) thm (karden ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C z) (D z)) ((karden.1 (e. A (V))) (karden.2 (e. B (V))) (karden.3 (= C ({|} x (/\ (br (cv x) (~~) A) (A. y (-> (br (cv y) (~~) A) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))))) (karden.4 (= D ({|} x (/\ (br (cv x) (~~) B) (A. y (-> (br (cv y) (~~) B) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))))))) (<-> (= C D) (br A (~~) B)) (karden.1 enref karden.1 (cv w) A (~~) A breq1 cla4ev ax-mp w (br (cv w) (~~) A) abn0 mpbir ({|} w (br (cv w) (~~) A)) z y scott0 mtbi z ({|} w (br (cv w) (~~) A)) (A.e. y ({|} w (br (cv w) (~~) A)) (C_ (` (rank) (cv z)) (` (rank) (cv y)))) rabn0 mpbi (br (cv z) (~~) A) (A. y (-> (br (cv y) (~~) A) (C_ (` (rank) (cv z)) (` (rank) (cv y))))) pm3.26 (= C D) a1i karden.3 karden.4 eqeq12i x (/\ (br (cv x) (~~) A) (A. y (-> (br (cv y) (~~) A) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))) (/\ (br (cv x) (~~) B) (A. y (-> (br (cv y) (~~) B) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))) eq2ab bitr (cv x) (cv z) (~~) A breq1 (cv x) (cv z) (rank) fveq2 (` (rank) (cv y)) sseq1d (br (cv y) (~~) A) imbi2d y albidv anbi12d (cv x) (cv z) (~~) B breq1 (cv x) (cv z) (rank) fveq2 (` (rank) (cv y)) sseq1d (br (cv y) (~~) B) imbi2d y albidv anbi12d bibi12d a4b1 sylbi (br (cv z) (~~) B) (A. y (-> (br (cv y) (~~) B) (C_ (` (rank) (cv z)) (` (rank) (cv y))))) pm3.26 syl6bi jcad A (cv z) B entrt karden.1 (cv z) ensym sylan syl6 z visset (cv w) (cv z) (~~) A breq1 elab y ({|} w (br (cv w) (~~) A)) (C_ (` (rank) (cv z)) (` (rank) (cv y))) df-ral y visset (cv w) (cv y) (~~) A breq1 elab (C_ (` (rank) (cv z)) (` (rank) (cv y))) imbi1i y albii bitr anbi12i syl5ib exp3a r19.23adv mpi karden.2 B (V) A (cv x) enen2 mpan karden.2 B (V) A (cv y) enen2 mpan (C_ (` (rank) (cv x)) (` (rank) (cv y))) imbi1d y albidv anbi12d x abbidv karden.3 karden.4 3eqtr4g impbi)) thm (htalem ((x y) (A x) (A y) (R x) (R y)) ((htalem.1 (e. A (V))) (htalem.2 (= B (U. ({e.|} x A (A.e. y A (-. (br (cv y) R (cv x))))))))) (-> (/\ (We R A) (-. (= A ({/})))) (e. B A)) (A ssid htalem.1 R A x y wereu mpanr1 x A (A.e. y A (-. (br (cv y) R (cv x)))) reucl syl htalem.2 syl5eqel)) thm (hta ((v z) (A z) (A v) (x y) (R x) (x z) (v x) (R y) (y z) (v y) (R z) (R v) (ph y) (ph z) (ph v)) ((hta.1 (= A ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))))) (hta.2 (= B (U. ({e.|} x A (A.e. y A (-. (br (cv y) R (cv x))))))))) (-> (We R A) (-> ph ([/] B x ph))) (hta.1 A ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) R weeq2 ax-mp x ph y scottexs hta.2 hta.1 ph y ax-17 y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))) hba1 hban z x hbab hta.1 (cv z) eleq2i hta.1 (cv z) eleq2i y albii 3imtr4 ph y ax-17 y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))) hba1 hban z x hbab (-. (br (cv y) R (cv x))) raleq1f ax-mp (e. (cv x) A) a1i rabbii hta.1 z x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))) hbab1 hta.1 (cv z) eleq2i hta.1 (cv z) eleq2i x albii 3imtr4 z x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))) hbab1 (A.e. y ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) (-. (br (cv y) R (cv x)))) rabeqf ax-mp v x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))) hbab1 (e. (cv v) ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))))) z ax-17 (A.e. y ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) (-. (br (cv y) R (cv x)))) z ax-17 y x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))) hbab1 (-. (br (cv y) R (cv z))) x ax-17 hbral (cv x) (cv z) (cv y) R breq2 negbid y ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) ralbidv cbvrab ph y ax-17 y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))) hba1 hban z x hbab (e. (cv z) ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))))) v ax-17 (-. (br (cv y) R (cv z))) v ax-17 (-. (br (cv v) R (cv z))) y ax-17 (cv y) (cv v) R (cv z) breq1 negbid cbvralf (e. (cv z) ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))))) a1i rabbii eqtr 3eqtr unieqi eqtr htalem ex sylbi ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))) pm3.26 x ss2abi B sseli hta.2 hta.1 x ph y scottexs eqeltr (e. (cv z) (cv v)) x ax-17 z x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))) hbab1 hta.1 (cv z) eleq2i hta.1 (cv z) eleq2i x albii 3imtr4 hbel (e. (cv v) A) z ax-17 (A.e. y A (-. (br (cv y) R (cv x)))) z ax-17 (e. (cv z) (cv y)) x ax-17 z x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))) hbab1 hta.1 (cv z) eleq2i hta.1 (cv z) eleq2i x albii 3imtr4 hbel (-. (br (cv y) R (cv z))) x ax-17 hbral (cv x) (cv z) (cv y) R breq2 negbid y A ralbidv cbvrab z A (A.e. y A (-. (br (cv y) R (cv z)))) ssrab2 eqsstr ssexi uniex eqeltr x ph elabs sylib syl6 ph x 19.8a x ph y scott0s sylib syl5)) thm (aceq1 ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (y z) (w y) (v y) (u y) (t y) (h y) (w z) (v z) (u z) (t z) (h z) (v w) (u w) (t w) (h w) (u v) (t v) (h v) (t u) (h u) (h t)) () (<-> (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. x (A. z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x)))))))))) ((= (cv w) (cv t)) (E!e. v (cv h) (E.e. u (cv y) (/\ (e. (cv h) (cv u)) (e. (cv v) (cv u))))) pm4.2i (cv h) cbvralv (cv v) (cv z) (cv u) eleq1 (e. (cv h) (cv u)) anbi2d u (cv y) rexbidv (cv h) cbvreuv t (cv h) ralbii bitr h (cv x) ralbii (cv z) (cv h) (cv u) eleq1 (e. (cv v) (cv u)) anbi1d u (cv y) rexbidv v reueqd w raleqd (cv x) cbvralv (cv w) (cv h) (cv u) eleq1 (e. (cv z) (cv u)) anbi1d u (cv y) rexbidv z reueqd t raleqd (cv x) cbvralv 3bitr4 y exbii z (e. (cv w) (cv x)) (-> (e. (cv z) (cv w)) (E. x (A. z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x)))))) 19.21v (e. (cv z) (cv w)) (e. (cv w) (cv x)) (E. x (A. z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x))))) impexp (e. (cv z) (cv w)) (e. (cv w) (cv x)) (E. x (A. z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x))))) bi2.04 bitr z albii z (/\ (e. (cv z) (cv w)) (E.e. u (cv y) (/\ (e. (cv w) (cv u)) (e. (cv z) (cv u))))) x df-eu z (cv w) (E.e. u (cv y) (/\ (e. (cv w) (cv u)) (e. (cv z) (cv u)))) df-reu x (e. (cv z) (cv w)) (/\ (e. (cv x) (cv y)) (/\ (e. (cv w) (cv x)) (e. (cv z) (cv x)))) 19.42v (e. (cv z) (cv w)) (e. (cv x) (cv y)) (e. (cv w) (cv x)) (e. (cv z) (cv x)) an42 (e. (cv z) (cv w)) (e. (cv x) (cv y)) (/\ (e. (cv w) (cv x)) (e. (cv z) (cv x))) anass bitr3 x exbii u (cv y) (/\ (e. (cv w) (cv u)) (e. (cv z) (cv u))) df-rex (cv u) (cv x) (cv y) eleq1 (cv u) (cv x) (cv w) eleq2 (cv u) (cv x) (cv z) eleq2 anbi12d anbi12d cbvexv bitr (e. (cv z) (cv w)) anbi2i 3bitr4 (= (cv z) (cv x)) bibi1i z albii x exbii 3bitr4 (e. (cv t) (cv w)) imbi2i t albii t (cv w) (E!e. z (cv w) (E.e. u (cv y) (/\ (e. (cv w) (cv u)) (e. (cv z) (cv u))))) df-ral (-> (e. (cv z) (cv w)) (E. x (A. z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x)))))) t ax-17 (e. (cv t) (cv w)) z ax-17 z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x))) hba1 x hbex hbim (cv z) (cv t) (cv w) eleq1 (E. x (A. z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x))))) imbi1d cbval 3bitr4 (e. (cv w) (cv x)) imbi2i 3bitr4 w albii z w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. x (A. z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x)))))) alcom w (cv x) (A.e. t (cv w) (E!e. z (cv w) (E.e. u (cv y) (/\ (e. (cv w) (cv u)) (e. (cv z) (cv u)))))) df-ral 3bitr4r y exbii bitr)) thm (aceq0 ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (<-> (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. v (A. u (<-> (E. t (/\ (/\ (e. (cv u) (cv w)) (e. (cv w) (cv t))) (/\ (e. (cv u) (cv t)) (e. (cv t) (cv y))))) (= (cv u) (cv v)))))))))) (y z x w v u aceq1 v x u equequ2 (E. t (/\ (/\ (e. (cv u) (cv w)) (e. (cv w) (cv t))) (/\ (e. (cv u) (cv t)) (e. (cv t) (cv y))))) bibi2d t x w elequ2 (e. (cv u) (cv w)) anbi2d t x u elequ2 t x y elequ1 anbi12d anbi12d cbvexv (= (cv u) (cv x)) bibi1i syl6bb u albidv u z w elequ1 (e. (cv w) (cv x)) anbi1d u z x elequ1 (e. (cv x) (cv y)) anbi1d anbi12d x exbidv u z x equequ1 bibi12d cbvalv syl6bb cbvexv (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) imbi2i z w 2albii y exbii bitr4)) thm (aceq2 ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (<-> (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v))))))))) (t (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u))))) df-ral t (e. (cv t) (cv z)) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u))))) 19.23v bitr (= (cv w) (cv t)) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u))))) pm4.2i (cv z) cbvralv (cv z) t n0 (cv v) (cv u) (cv z) eleq2 (cv v) (cv u) (cv w) eleq2 anbi12d (cv y) cbvrexv w (cv z) reubii (cv w) (cv v) (cv u) eleq1 (e. (cv z) (cv u)) anbi2d u (cv y) rexbidv (cv z) cbvreuv bitr imbi12i 3bitr4 z (cv x) ralbii y exbii)) thm (aceq3lem ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (g x) (f x) (y z) (w y) (v y) (u y) (t y) (h y) (g y) (f y) (w z) (v z) (u z) (t z) (h z) (g z) (f z) (v w) (u w) (t w) (h w) (g w) (f w) (u v) (t v) (h v) (g v) (f v) (t u) (h u) (g u) (f u) (h t) (g t) (f t) (g h) (f h) (f g) (F t) (F h) (F g)) ((aceq3lem.1 (= F ({<,>|} w v (/\ (e. (cv w) (dom (cv y))) (= (cv v) (` (cv f) ({|} u (br (cv w) (cv y) (cv u)))))))))) (-> (A. x (E. f (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z)))))) (E. f (/\ (C_ (cv f) (cv y)) (Fn (cv f) (dom (cv y)))))) (y visset (cv y) (V) rnexg ax-mp pwex (cv x) (P~ (ran (cv y))) z (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z))) raleq1 f exbidv cla4v w v (/\ (e. (cv w) (dom (cv y))) (= (cv v) (` (cv f) ({|} u (br (cv w) (cv y) (cv u)))))) relopab aceq3lem.1 F ({<,>|} w v (/\ (e. (cv w) (dom (cv y))) (= (cv v) (` (cv f) ({|} u (br (cv w) (cv y) (cv u))))))) releq ax-mp mpbir (A.e. z (P~ (ran (cv y))) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z)))) a1i aceq3lem.1 (<,> (cv t) (cv h)) eleq2i t visset h visset (cv w) (cv t) (dom (cv y)) eleq1 (cv w) (cv t) (cv y) (cv u) breq1 u abbidv (cv f) fveq2d (cv v) eqeq2d anbi12d (cv v) (cv h) (` (cv f) ({|} u (br (cv t) (cv y) (cv u)))) eqeq1 (e. (cv t) (dom (cv y))) anbi2d opelopab bitr pm3.26bd (br (cv t) (cv y) (cv u)) t 19.8a u ss2abi (cv y) u t dfrn2 sseqtr4 y visset (cv y) (V) rnexg ax-mp (ran (cv y)) (V) ({|} u (br (cv t) (cv y) (cv u))) elpw2g ax-mp mpbir (cv z) ({|} u (br (cv t) (cv y) (cv u))) ({/}) eqeq1 negbid (cv z) ({|} u (br (cv t) (cv y) (cv u))) (cv f) fveq2 (= (cv z) ({|} u (br (cv t) (cv y) (cv u)))) id eleq12d imbi12d (P~ (ran (cv y))) rcla4v ax-mp t visset (cv y) u eldm u (br (cv t) (cv y) (cv u)) abn0 bitr4 syl5ib com12 (cv w) (cv t) (cv y) (cv u) breq1 u abbidv (cv f) fveq2d aceq3lem.1 (cv f) ({|} u (br (cv t) (cv y) (cv u))) fvex fvopab4 ({|} u (br (cv t) (cv y) (cv u))) eleq1d F (cv t) fvex (cv h) (` F (cv t)) (cv t) (cv y) breq2 (cv u) (cv h) (cv t) (cv y) breq2 cbvabv elab2 (cv t) (cv y) (` F (cv t)) df-br bitr2 syl5bb sylibrd syl (cv f) ({|} u (br (cv w) (cv y) (cv u))) fvex aceq3lem.1 fnopab2 F (dom (cv y)) fnfun ax-mp h visset F (cv t) funopfv ax-mp (` F (cv t)) (cv h) (cv t) opeq2 (cv y) eleq1d syl sylibd com12 relssdv (cv f) ({|} u (br (cv w) (cv y) (cv u))) fvex aceq3lem.1 fnopab2 jctir (cv f) ({|} u (br (cv w) (cv y) (cv u))) fvex aceq3lem.1 fnopab2 y visset (cv y) (V) dmexg ax-mp F (dom (cv y)) (V) fnex mp2an (cv g) F (cv y) sseq1 (cv g) F (dom (cv y)) fneq1 anbi12d cla4ev syl f 19.23aiv syl (cv g) (cv f) (cv y) sseq1 (cv g) (cv f) (dom (cv y)) fneq1 anbi12d cbvexv sylib)) thm (aceq3 ((f x) (f y) (f z) (f w) (f v) (f u) (x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. f (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z))))))) ((cv x) (cv y) (cv f) sseq2 (cv x) (cv y) dmeq (dom (cv x)) (dom (cv y)) (cv f) fneq2 syl anbi12d f exbidv cbvalv x visset x visset uniex xpex (e. (cv w) (cv x)) (e. (cv v) (cv w)) pm3.26 (cv v) (cv w) (cv x) elunii ancoms jca w v ssopab2i (cv x) (U. (cv x)) w v df-xp sseqtr4 ssexi (cv y) ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w)))) (cv f) sseq2 (cv y) ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w)))) dmeq (dom (cv y)) (dom ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w))))) (cv f) fneq2 syl anbi12d f exbidv cla4v (cv f) (dom ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w))))) fndm (dom (cv f)) (dom ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w))))) (cv z) eleq2 w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w))) dmopab (cv z) eleq2i v (e. (cv z) (cv x)) (e. (cv v) (cv z)) 19.42v z visset (cv w) (cv z) (cv x) eleq1 (cv w) (cv z) (cv v) eleq2 anbi12d v exbidv elab (cv z) v n0 (e. (cv z) (cv x)) anbi2i 3bitr4 bitr2 syl6rbbr syl (C_ (cv f) ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w))))) adantl (cv f) ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w)))) (cv z) funfvima3 ancoms (cv f) (dom ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w))))) fnfun sylan2 sylbid imp (e. (cv z) (cv x)) (e. (cv u) (cv z)) ibar abbi2dv z visset (cv z) (V) ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w)))) u imasng ax-mp z visset u visset (cv w) (cv z) (cv x) eleq1 (cv w) (cv z) (cv v) eleq2 anbi12d (cv v) (cv u) (cv z) eleq1 (e. (cv z) (cv x)) anbi2d ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w)))) eqid brab u abbii eqtr syl6reqr (` (cv f) (cv z)) eleq2d (/\ (C_ (cv f) ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w))))) (Fn (cv f) (dom ({<,>|} w v (/\ (e. (cv w) (cv x)) (e. (cv v) (cv w))))))) (-. (= (cv z) ({/}))) ad2antrl mpbid exp32 r19.21aiv f 19.22i syl x 19.21aiv ({<,>|} w v (/\ (e. (cv w) (dom (cv y))) (= (cv v) (` (cv f) ({|} u (br (cv w) (cv y) (cv u))))))) eqid x z aceq3lem y 19.21aiv impbi bitr)) thm (aceq4 ((f x) (f z) (f y) (f w) (f v) (x z) (x y) (w x) (v x) (y z) (w z) (v z) (w y) (v y) (v w)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. f (/\ (Fn (cv f) (cv x)) (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z)))))))) (x f z aceq3 (cv f) (cv y) (cv z) fveq1 (cv z) eleq1d (-. (= (cv z) ({/}))) imbi2d z (cv x) ralbidv cbvexv (cv w) (cv z) (cv y) fveq2 ({<,>|} w v (/\ (e. (cv w) (cv x)) (= (cv v) (` (cv y) (cv w))))) eqid (cv y) (cv z) fvex fvopab4 (cv z) eleq1d (-. (= (cv z) ({/}))) imbi2d ralbiia (Fn ({<,>|} w v (/\ (e. (cv w) (cv x)) (= (cv v) (` (cv y) (cv w))))) (cv x)) anbi2i (cv y) (cv w) fvex ({<,>|} w v (/\ (e. (cv w) (cv x)) (= (cv v) (` (cv y) (cv w))))) eqid fnopab2 mpbiran x visset w v (` (cv y) (cv w)) funopabex2 (cv f) ({<,>|} w v (/\ (e. (cv w) (cv x)) (= (cv v) (` (cv y) (cv w))))) (cv x) fneq1 (cv f) ({<,>|} w v (/\ (e. (cv w) (cv x)) (= (cv v) (` (cv y) (cv w))))) (cv z) fveq1 (cv z) eleq1d (-. (= (cv z) ({/}))) imbi2d z (cv x) ralbidv anbi12d cla4ev sylbir y 19.23aiv sylbi (Fn (cv f) (cv x)) (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z)))) pm3.27 f 19.22i impbi x albii bitr)) thm (aceq5lem1 ((v w) (v y) (g v) (t v) (w y) (g w) (t w) (g y) (t y) (g t)) () (<-> (E! v (e. (cv v) (i^i (X. ({} (cv w)) (cv w)) (cv y)))) (E! g (/\ (e. (cv g) (cv w)) (e. (<,> (cv w) (cv g)) (cv y))))) ((cv v) (X. ({} (cv w)) (cv w)) (cv y) elin (cv v) ({} (cv w)) (cv w) t g elxp t g (/\ (= (cv v) (<,> (cv t) (cv g))) (/\ (e. (cv t) ({} (cv w))) (e. (cv g) (cv w)))) excom bitr (e. (cv v) (cv y)) anbi1i g t (/\ (= (cv v) (<,> (cv t) (cv g))) (/\ (e. (cv t) ({} (cv w))) (e. (cv g) (cv w)))) (e. (cv v) (cv y)) 19.41vv (= (cv v) (<,> (cv t) (cv g))) (/\ (e. (cv t) ({} (cv w))) (e. (cv g) (cv w))) (e. (cv v) (cv y)) an23 (cv v) (<,> (cv t) (cv g)) (cv y) eleq1 pm5.32i t (cv w) elsn (e. (cv g) (cv w)) anbi1i anbi12i (= (cv v) (<,> (cv t) (cv g))) (e. (<,> (cv t) (cv g)) (cv y)) (= (cv t) (cv w)) (e. (cv g) (cv w)) an4 (= (cv v) (<,> (cv t) (cv g))) (= (cv t) (cv w)) ancom (e. (<,> (cv t) (cv g)) (cv y)) (e. (cv g) (cv w)) ancom anbi12i (= (cv t) (cv w)) (= (cv v) (<,> (cv t) (cv g))) (/\ (e. (cv g) (cv w)) (e. (<,> (cv t) (cv g)) (cv y))) anass 3bitr 3bitr t exbii w visset (cv t) (cv w) (cv g) opeq1 (cv v) eqeq2d (cv t) (cv w) (cv g) opeq1 (cv y) eleq1d (e. (cv g) (cv w)) anbi2d anbi12d ceqsexv bitr g exbii bitr3 3bitr v eubii v g (cv w) (/\ (e. (cv g) (cv w)) (e. (<,> (cv w) (cv g)) (cv y))) euop2 bitr)) thm (aceq5lem2 ((u w) (t w) (h w) (g w) (t u) (h u) (g u) (h t) (g t) (g h) (A w) (A g)) ((aceq5lem.1 (= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t))))))))) (<-> (e. (<,> (cv w) (cv g)) (U. A)) (/\ (e. (cv w) (cv h)) (e. (cv g) (cv w)))) (aceq5lem.1 unieqi (<,> (cv w) (cv g)) eleq2i (<,> (cv w) (cv g)) u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t))))) eluniab t (cv h) (/\ (e. (<,> (cv w) (cv g)) (cv u)) (-. (= (cv u) ({/})))) (= (cv u) (X. ({} (cv t)) (cv t))) r19.42v (e. (<,> (cv w) (cv g)) (cv u)) (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t)))) anass bitr2 u exbii t (cv h) u (/\ (/\ (e. (<,> (cv w) (cv g)) (cv u)) (-. (= (cv u) ({/})))) (= (cv u) (X. ({} (cv t)) (cv t)))) rexcom4 t (cv h) (E. u (/\ (/\ (e. (<,> (cv w) (cv g)) (cv u)) (-. (= (cv u) ({/})))) (= (cv u) (X. ({} (cv t)) (cv t))))) df-rex bitr3 3bitr (/\ (e. (<,> (cv w) (cv g)) (cv u)) (-. (= (cv u) ({/})))) (= (cv u) (X. ({} (cv t)) (cv t))) ancom (<,> (cv w) (cv g)) (cv u) n0i pm4.71i (= (cv u) (X. ({} (cv t)) (cv t))) anbi2i bitr4 u exbii (cv t) snex t visset xpex (cv u) (X. ({} (cv t)) (cv t)) (<,> (cv w) (cv g)) eleq2 ceqsexv bitr (e. (cv t) (cv h)) anbi2i g visset (cv w) ({} (cv t)) (cv t) opelxp w (cv t) elsn (cv w) (cv t) eqcom bitr (e. (cv g) (cv t)) anbi1i bitr (e. (cv t) (cv h)) anbi2i (e. (cv t) (cv h)) (= (cv t) (cv w)) (e. (cv g) (cv t)) an12 3bitr t exbii w visset (cv t) (cv w) (cv h) eleq1 (cv t) (cv w) (cv g) eleq2 anbi12d ceqsexv bitr 3bitr)) thm (aceq5lem3 ((u w) (t w) (h w) (t u) (h u) (h t) (A w)) ((aceq5lem.1 (= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t))))))))) (<-> (e. (X. ({} (cv w)) (cv w)) A) (/\ (-. (= (cv w) ({/}))) (e. (cv w) (cv h)))) ((cv w) snex w visset xpex (cv u) (X. ({} (cv w)) (cv w)) ({/}) eqeq1 negbid (cv u) (X. ({} (cv w)) (cv w)) (X. ({} (cv t)) (cv t)) eqeq1 t (cv h) rexbidv anbi12d elab aceq5lem.1 (X. ({} (cv w)) (cv w)) eleq2i (cv w) ({/}) ({} (cv w)) xpeq2 ({} (cv w)) xp0 syl6eq (X. ({} (cv w)) (cv w)) ({/}) rneq w visset snnz ({} (cv w)) (cv w) rnxp ax-mp rn0 3eqtr3g impbi negbii t (cv h) (= (X. ({} (cv w)) (cv w)) (X. ({} (cv t)) (cv t))) df-rex (X. ({} (cv w)) (cv w)) (X. ({} (cv t)) (cv t)) rneq w visset snnz ({} (cv w)) (cv w) rnxp ax-mp t visset snnz ({} (cv t)) (cv t) rnxp ax-mp 3eqtr3g (cv w) (cv t) sneq ({} (cv w)) ({} (cv t)) (cv w) xpeq1 syl (cv w) (cv t) ({} (cv t)) xpeq2 eqtrd impbi (cv w) (cv t) eqcom bitr (e. (cv t) (cv h)) anbi2i (e. (cv t) (cv h)) (= (cv t) (cv w)) ancom bitr t exbii w visset (cv t) (cv w) (cv h) eleq1 ceqsexv 3bitrr anbi12i 3bitr4)) thm (aceq5lem4 ((x z) (x y) (w x) (v x) (u x) (t x) (h x) (g x) (y z) (w z) (v z) (u z) (t z) (h z) (g z) (w y) (v y) (u y) (t y) (h y) (g y) (v w) (u w) (t w) (h w) (g w) (u v) (t v) (h v) (g v) (t u) (h u) (g u) (h t) (g t) (g h) (B z) (B w) (B g) (A x) (A y) (A z) (A w) (A g)) ((aceq5lem.1 (= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t)))))))) (aceq5lem.2 (= B (i^i (U. A) (cv y)))) (aceq5lem.3 (<-> ph (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))))))) (-> ph (E. y (A.e. z A (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (aceq5lem.1 (cv z) eleq2i z visset (cv u) (cv z) ({/}) eqeq1 negbid (cv u) (cv z) (X. ({} (cv t)) (cv t)) eqeq1 t (cv h) rexbidv anbi12d elab pm3.26bd sylbi rgen (cv z) (X. ({} (cv t)) (cv t)) (cv x) eleq2 (cv x) ({} (cv t)) (cv t) u v elxp u v (/\ (= (cv x) (<,> (cv u) (cv v))) (/\ (e. (cv u) ({} (cv t))) (e. (cv v) (cv t)))) excom bitr syl6bb (cv w) (X. ({} (cv g)) (cv g)) (cv x) eleq2 (cv x) ({} (cv g)) (cv g) u y elxp u y (/\ (= (cv x) (<,> (cv u) (cv y))) (/\ (e. (cv u) ({} (cv g))) (e. (cv y) (cv g)))) excom bitr syl6bb bi2anan9 v y (E. u (/\ (= (cv x) (<,> (cv u) (cv v))) (/\ (e. (cv u) ({} (cv t))) (e. (cv v) (cv t))))) (E. u (/\ (= (cv x) (<,> (cv u) (cv y))) (/\ (e. (cv u) ({} (cv g))) (e. (cv y) (cv g))))) eeanv syl6bbr (cv u) (cv t) (cv v) opeq1 (cv x) eqeq2d biimpac u (cv t) elsn sylan2b (e. (cv v) (cv t)) adantrr u 19.23aiv (cv u) (cv g) (cv y) opeq1 (cv x) eqeq2d biimpac u (cv g) elsn sylan2b (e. (cv y) (cv g)) adantrr u 19.23aiv anim12i (cv x) (<,> (cv t) (cv v)) (<,> (cv g) (cv y)) eqtr2t t visset v visset y visset (cv g) opth pm3.26bd 3syl v y 19.23aivv syl6bi (cv t) (cv g) sneq ({} (cv t)) ({} (cv g)) (cv t) xpeq1 syl (cv t) (cv g) ({} (cv g)) xpeq2 eqtrd syl6 (cv z) (X. ({} (cv t)) (cv t)) (cv w) (X. ({} (cv g)) (cv g)) eqeq12 sylibrd ex (e. (cv t) (cv h)) a1i r19.23aiv (e. (cv g) (cv h)) a1d r19.23adv imp z visset (cv u) (cv z) ({/}) eqeq1 negbid (cv u) (cv z) (X. ({} (cv t)) (cv t)) eqeq1 t (cv h) rexbidv anbi12d aceq5lem.1 elab2 pm3.27bd w visset (cv u) (cv w) ({/}) eqeq1 negbid (cv u) (cv w) (X. ({} (cv t)) (cv t)) eqeq1 t (cv h) rexbidv anbi12d aceq5lem.1 elab2 pm3.27bd (cv t) (cv g) sneq ({} (cv t)) ({} (cv g)) (cv t) xpeq1 syl (cv t) (cv g) ({} (cv g)) xpeq2 eqtrd (cv w) eqeq2d (cv h) cbvrexv sylib syl2an (e. (cv x) (cv z)) (e. (cv x) (cv w)) df-an syl5ibr con1d x 19.21adv (cv z) (cv w) x disj1 syl6ibr rgen2 aceq5lem.3 aceq5lem.1 h visset u t (X. ({} (cv t)) (cv t)) abrexex (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t)))) pm3.27 u ss2abi ssexi eqeltr (cv x) A z (-. (= (cv z) ({/}))) raleq1 (cv x) A w (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/}))) raleq1 z raleqd anbi12d (cv x) A z (E! v (e. (cv v) (i^i (cv z) (cv y)))) raleq1 y exbidv imbi12d cla4v sylbi mp2ani)) thm (aceq5lem5 ((f x) (f z) (f y) (f w) (f v) (f u) (f t) (f h) (f g) (x z) (x y) (w x) (v x) (u x) (t x) (h x) (g x) (y z) (w z) (v z) (u z) (t z) (h z) (g z) (w y) (v y) (u y) (t y) (h y) (g y) (v w) (u w) (t w) (h w) (g w) (u v) (t v) (h v) (g v) (t u) (h u) (g u) (h t) (g t) (g h) (B z) (B w) (B f) (B g) (A x) (A y) (A z) (A w) (A g)) ((aceq5lem.1 (= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t)))))))) (aceq5lem.2 (= B (i^i (U. A) (cv y)))) (aceq5lem.3 (<-> ph (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))))))) (-> ph (E. f (A.e. w (cv h) (-> (-. (= (cv w) ({/}))) (e. (` (cv f) (cv w)) (cv w)))))) (aceq5lem.1 aceq5lem.2 aceq5lem.3 aceq5lem4 (-. (= (cv w) ({/}))) (e. (cv w) (cv h)) pm3.27 (A.e. z A (E! v (e. (cv v) (i^i (cv z) (cv y))))) a1i (cv z) (X. ({} (cv w)) (cv w)) (cv y) ineq1 (cv v) eleq2d v eubidv A rcla4cv aceq5lem.1 w aceq5lem3 v w y g aceq5lem1 3imtr3g jcad aceq5lem.2 (<,> (cv w) (cv g)) eleq2i (<,> (cv w) (cv g)) (U. A) (cv y) elin aceq5lem.1 w g aceq5lem2 (e. (<,> (cv w) (cv g)) (cv y)) anbi1i (e. (cv w) (cv h)) (e. (cv g) (cv w)) (e. (<,> (cv w) (cv g)) (cv y)) anass bitr 3bitr g eubii g (e. (cv w) (cv h)) (/\ (e. (cv g) (cv w)) (e. (<,> (cv w) (cv g)) (cv y))) euanv bitr2 syl6ib g (e. (<,> (cv w) (cv g)) B) euex g (e. (<,> (cv w) (cv g)) B) hbeu1 (e. (` B (cv w)) (cv w)) g ax-17 hbim aceq5lem.2 (<,> (cv w) (cv g)) eleq2i (<,> (cv w) (cv g)) (U. A) (cv y) elin aceq5lem.1 w g aceq5lem2 (e. (<,> (cv w) (cv g)) (cv y)) anbi1i (e. (cv w) (cv h)) (e. (cv g) (cv w)) (e. (<,> (cv w) (cv g)) (cv y)) anass bitr 3bitr pm3.27bd pm3.26d w visset g B tz6.12 (cv w) eleq1d biimparc exp32 mpcom 19.23ai mpcom syl6 exp3a com23 r19.21aiv aceq5lem.2 y visset (U. A) inex2 eqeltr (cv f) B (cv w) fveq1 (cv w) eleq1d (-. (= (cv w) ({/}))) imbi2d w (cv h) ralbidv cla4ev syl y 19.23aiv syl)) thm (aceq5 ((f x) (f z) (f y) (f w) (f v) (f h) (f u) (f t) (x z) (x y) (w x) (v x) (h x) (u x) (t x) (y z) (w z) (v z) (h z) (u z) (t z) (w y) (v y) (h y) (u y) (t y) (v w) (h w) (u w) (t w) (h v) (u v) (t v) (h u) (h t) (t u)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))))) (x f w aceq4 (cv w) (cv t) (cv f) fveq2 (= (cv w) (cv t)) id eleq12d (cv w) (cv t) (cv z) eqeq2 negbid (cv w) (cv t) (cv z) ineq2 ({/}) eqeq1d imbi12d anbi12d (cv x) rcla4v (cv v) (` (cv f) (cv t)) (cv t) eleq1 negbid (cv z) rcla4v (cv z) (cv t) v disj syl5ib (-. (= (cv z) (cv t))) imim2d (e. (` (cv f) (cv t)) (cv t)) (= (cv z) (cv t)) pm4.1 syl6ibr com13 imp31 (` (cv f) (cv t)) (cv v) (cv z) eleq1 biimpar sylan2 (` (cv f) (cv t)) (cv v) (` (cv f) (cv z)) eqeq2 (` (cv f) (cv z)) (cv v) eqcom syl6bb (cv z) (cv t) (cv f) fveq2 syl5bi (/\ (e. (` (cv f) (cv t)) (cv t)) (-> (-. (= (cv z) (cv t))) (= (i^i (cv z) (cv t)) ({/})))) (e. (cv v) (cv z)) ad2antrl mpd exp32 syl6com com14 r19.23adv (cv f) (cv x) (cv v) t fvelrn biimpac syl5 exp3a com4t imp4b (cv v) (cv z) (ran (cv f)) elin syl5ib w (cv x) (e. (` (cv f) (cv w)) (cv w)) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/}))) r19.26 sylan2br anassrs (e. (cv z) (cv x)) adantlr (cv v) (` (cv f) (cv z)) (i^i (cv z) (ran (cv f))) eleq1 (cv w) (cv z) (cv f) fveq2 (= (cv w) (cv z)) id eleq12d (cv x) rcla4v (cv f) (cv x) (cv z) fnfvrn expcom anim12d (` (cv f) (cv z)) (cv z) (ran (cv f)) elin syl6ibr exp3a com13 imp31 syl5bir com12 (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))) adantr impbid ex v 19.21adv (cv f) (cv z) fvex (cv h) (` (cv f) (cv z)) (cv v) eqeq2 (e. (cv v) (i^i (cv z) (ran (cv f)))) bibi2d v albidv cla4ev v (e. (cv v) (i^i (cv z) (ran (cv f)))) h df-eu sylibr syl6 r19.20dva ex (cv z) (cv w) ({/}) eqeq1 negbid (cv x) cbvralv (A.e. w (cv x) (-> (-. (= (cv w) ({/}))) (e. (` (cv f) (cv w)) (cv w)))) anbi2i w (cv x) (-> (-. (= (cv w) ({/}))) (e. (` (cv f) (cv w)) (cv w))) (-. (= (cv w) ({/}))) r19.26 bitr4 (-. (= (cv w) ({/}))) (e. (` (cv f) (cv w)) (cv w)) pm3.35 ancoms w (cv x) r19.20si sylbi syl5 exp3a imp4b f visset (cv f) (V) rnexg ax-mp (cv y) (ran (cv f)) (cv z) ineq2 (cv v) eleq2d v eubidv z (cv x) ralbidv cla4ev syl6 f 19.23aiv x 19.20i sylbi ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t)))))) eqid (i^i (U. ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t))))))) (cv y)) eqid (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) pm4.2 f aceq5lem5 h 19.21aiv x f w aceq3 (cv x) (cv h) w (-> (-. (= (cv w) ({/}))) (e. (` (cv f) (cv w)) (cv w))) raleq1 f exbidv cbvalv bitr2 sylib impbi)) thm (aceq6a ((x z) (f x) (x y) (w x) (v x) (u x) (g x) (f z) (y z) (w z) (v z) (u z) (g z) (f y) (f w) (f v) (f u) (f g) (w y) (v y) (u y) (g y) (v w) (u w) (g w) (u v) (g v) (g u)) () (-> (A. x (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v))))))))) (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x))))))) ((cv u) (cv z) (cv w) eleq2 (cv u) (cv z) (cv v) eleq1 (e. (cv w) (cv v)) anbi1d v (cv y) rexbidv anbi12d w abbidv w (cv u) (E.e. v (cv y) (/\ (e. (cv u) (cv v)) (e. (cv w) (cv v)))) df-rab w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v)))) df-rab 3eqtr4g unieqd ({<,>|} u g (/\ (e. (cv u) (cv x)) (= (cv g) (U. ({e.|} w (cv u) (E.e. v (cv y) (/\ (e. (cv u) (cv v)) (e. (cv w) (cv v))))))))) eqid z visset w (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v)))) rabex uniex fvopab4 (cv z) eleq1d w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v)))) reucl syl5bir (-. (= (cv z) ({/}))) imim2d r19.20i x visset u g (U. ({e.|} w (cv u) (E.e. v (cv y) (/\ (e. (cv u) (cv v)) (e. (cv w) (cv v)))))) funopabex2 (cv f) ({<,>|} u g (/\ (e. (cv u) (cv x)) (= (cv g) (U. ({e.|} w (cv u) (E.e. v (cv y) (/\ (e. (cv u) (cv v)) (e. (cv w) (cv v))))))))) (cv z) fveq1 (cv z) eleq1d (-. (= (cv z) ({/}))) imbi2d z (cv x) ralbidv cla4ev syl y 19.23aiv x 19.20i x f z aceq3 sylibr)) thm (aceq6b ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (y z) (w y) (v y) (u y) (f y) (g y) (w z) (v z) (u z) (f z) (g z) (v w) (u w) (f w) (g w) (u v) (f v) (g v) (f u) (g u) (f g)) () (-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v)))))))))) (x f z aceq3 z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z))) hbra1 z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z))) ra4 (cv f) (cv z) fvex (cv w) (` (cv f) (cv z)) (cv z) eleq1 (cv w) (` (cv f) (cv z)) (cv v) eleq1 (e. (cv z) (cv v)) anbi2d v ({|} g (E.e. u (cv x) (/\ (-. (= (cv u) ({/}))) (= (cv g) ({,} (` (cv f) (cv u)) (cv u)))))) rexbidv anbi12d cla4ev (cv z) eqid (cv u) (cv z) ({/}) eqeq1 negbid (cv u) (cv z) (cv z) eqeq1 anbi12d (cv x) rcla4ev mpanr2 (cv u) (cv z) (cv f) fveq2 (` (cv f) (cv u)) (` (cv f) (cv z)) (cv u) preq1 syl (cv u) (cv z) (` (cv f) (cv z)) preq2 eqtr2d (-. (= (cv u) ({/}))) anim2i u (cv x) r19.22si syl (` (cv f) (cv z)) (cv z) prex (cv g) ({,} (` (cv f) (cv z)) (cv z)) ({,} (` (cv f) (cv u)) (cv u)) eqeq1 (-. (= (cv u) ({/}))) anbi2d u (cv x) rexbidv elab sylibr z visset (` (cv f) (cv z)) pri2 (cv f) (cv z) fvex (cv z) pri1 pm3.2i jctir (cv v) ({,} (` (cv f) (cv z)) (cv z)) (cv z) eleq2 (cv v) ({,} (` (cv f) (cv z)) (cv z)) (` (cv f) (cv z)) eleq2 anbi12d ({|} g (E.e. u (cv x) (/\ (-. (= (cv u) ({/}))) (= (cv g) ({,} (` (cv f) (cv u)) (cv u)))))) rcla4ev syl sylan2 ex syl8 imp3a pm2.43d v visset (cv g) (cv v) ({,} (` (cv f) (cv u)) (cv u)) eqeq1 (-. (= (cv u) ({/}))) anbi2d u (cv x) rexbidv elab (cv z) (cv u) ({/}) eqeq1 negbid (cv z) (cv u) (cv f) fveq2 (cv z) eleq1d (cv z) (cv u) (` (cv f) (cv u)) eleq2 bitrd imbi12d (cv x) rcla4cv w visset z visset (cv f) (cv u) fvex u visset prel12 (cv v) ({,} (` (cv f) (cv u)) (cv u)) (cv w) eleq2 (cv v) ({,} (` (cv f) (cv u)) (cv u)) (cv z) eleq2 anbi12d (e. (cv w) (cv v)) (e. (cv z) (cv v)) ancom syl5rbbr sylan9bbr w eirrv (cv w) (cv z) (cv w) eleq2 mtbii con2i sylan2 (e. (` (cv f) (cv u)) (cv u)) adantrr ex pm5.32d w visset z visset (cv f) (cv u) fvex u visset preleq syl6bir (cv z) (cv u) (cv f) fveq2 (cv w) eqeq2d biimparc syl6 exp4c com13 syl8 com4r imp imp4a com3l r19.23aiv sylbi exp3a com13 imp4b v 19.23adv v ({|} g (E.e. u (cv x) (/\ (-. (= (cv u) ({/}))) (= (cv g) ({,} (` (cv f) (cv u)) (cv u)))))) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v))) df-rex syl5ib ex imp3a w 19.21aiv w (/\ (e. (cv w) (cv z)) (E.e. v ({|} g (E.e. u (cv x) (/\ (-. (= (cv u) ({/}))) (= (cv g) ({,} (` (cv f) (cv u)) (cv u)))))) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v))))) (` (cv f) (cv z)) mo2icl syl jctird w (cv z) (E.e. v ({|} g (E.e. u (cv x) (/\ (-. (= (cv u) ({/}))) (= (cv g) ({,} (` (cv f) (cv u)) (cv u)))))) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v)))) df-reu w (/\ (e. (cv w) (cv z)) (E.e. v ({|} g (E.e. u (cv x) (/\ (-. (= (cv u) ({/}))) (= (cv g) ({,} (` (cv f) (cv u)) (cv u)))))) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v))))) eu5 bitr syl6ibr exp3a r19.21ai x visset g u ({,} (` (cv f) (cv u)) (cv u)) abrexex (-. (= (cv u) ({/}))) (= (cv g) ({,} (` (cv f) (cv u)) (cv u))) pm3.27 u (cv x) r19.22si g ss2abi ssexi (cv y) ({|} g (E.e. u (cv x) (/\ (-. (= (cv u) ({/}))) (= (cv g) ({,} (` (cv f) (cv u)) (cv u)))))) v (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v))) rexeq1 w (cv z) reubidv (-. (= (cv z) ({/}))) imbi2d z (cv x) ralbidv cla4ev syl f 19.23aiv x 19.20i sylbi)) thm (aceq7 ((x z) (f x) (x y) (w x) (v x) (u x) (f z) (y z) (w z) (v z) (u z) (f y) (f w) (f v) (f u) (w y) (v y) (u y) (v w) (u w) (u v)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))))) (x f y z w v aceq6b x y z w v f aceq6a impbi y z x w v u aceq2 x albii bitr4)) thm (axac ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (E. x (A. y (A. z (-> (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))) (x y z w v u t ax-ac v w u equequ2 (E. t (/\ (/\ (e. (cv u) (cv z)) (e. (cv z) (cv t))) (/\ (e. (cv u) (cv t)) (e. (cv t) (cv x))))) bibi2d t w z elequ2 (e. (cv u) (cv z)) anbi2d t w u elequ2 t w x elequ1 anbi12d anbi12d cbvexv (= (cv u) (cv w)) bibi1i syl6bb u albidv u y z elequ1 (e. (cv z) (cv w)) anbi1d u y w elequ1 (e. (cv w) (cv x)) anbi1d anbi12d w exbidv u y w equequ1 bibi12d cbvalv syl6bb cbvexv (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) imbi2i y z 2albii x exbii mpbi)) thm (ac2 ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (y z w x v u t ax-ac y z x w v u t aceq0 mpbir)) thm (ac3 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v)))))))) (y z x w v u ac2 y z x w v u aceq2 mpbi)) thm (ac7 ((f x) (f y) (f z) (f w) (f v) (f u) (x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x))))) (x f y z w v u aceq7 y z x w v u ac2 mpgbir a4i)) thm (ac7g ((f x) (R f) (R x)) () (-> (e. R A) (E. f (/\ (C_ (cv f) R) (Fn (cv f) (dom R))))) ((cv x) R (cv f) sseq2 (cv x) R dmeq (dom (cv x)) (dom R) (cv f) fneq2 syl anbi12d f exbidv f x ac7 A vtoclg)) thm (ac4 ((x z) (f x) (f z)) () (E. f (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z))))) (x f z aceq3 f x ac7 mpgbi a4i)) thm (ac4c ((x y) (f x) (A x) (f y) (A y) (A f)) ((ac4c.1 (e. A (V)))) (E. f (A.e. x A (-> (-. (= (cv x) ({/}))) (e. (` (cv f) (cv x)) (cv x))))) (ac4c.1 (cv y) A x (-> (-. (= (cv x) ({/}))) (e. (` (cv f) (cv x)) (cv x))) raleq1 f exbidv f x y ac4 vtocl)) thm (ac5 ((f x) (f y) (A f) (x y) (A x) (A y)) ((ac5.1 (e. A (V)))) (E. f (/\ (Fn (cv f) A) (A.e. x A (-> (-. (= (cv x) ({/}))) (e. (` (cv f) (cv x)) (cv x)))))) (ac5.1 (cv y) A (cv f) fneq2 (cv y) A x (-> (-. (= (cv x) ({/}))) (e. (` (cv f) (cv x)) (cv x))) raleq1 anbi12d f exbidv y f x aceq3 f x y ac4 mpgbir y f x aceq4 mpbi a4i vtocl)) thm (ac5b ((f x) (A x) (A f)) ((ac5b.1 (e. A (V)))) (-> (A.e. x A (-. (= (cv x) ({/})))) (E. f (/\ (:--> (cv f) A (U. A)) (A.e. x A (e. (` (cv f) (cv x)) (cv x)))))) (ac5b.1 f x ac5 f (A.e. x A (-. (= (cv x) ({/})))) (/\ (Fn (cv f) A) (A.e. x A (-> (-. (= (cv x) ({/}))) (e. (` (cv f) (cv x)) (cv x))))) 19.42v (cv f) A x chfnrn ex anc2li (cv f) A (U. A) df-f syl6ibr impac x A (-. (= (cv x) ({/}))) (-> (-. (= (cv x) ({/}))) (e. (` (cv f) (cv x)) (cv x))) r19.26 (-. (= (cv x) ({/}))) (e. (` (cv f) (cv x)) (cv x)) pm3.35 x A r19.20si sylbir sylan2 an1s f 19.22i sylbir mpan2)) thm (ac6lem ((f z) (g z) (C z) (f g) (C f) (C g) (H z) (H f) (H g) (f g) (f x) (f y) (f z) (A f) (g x) (g y) (g z) (A g) (x y) (x z) (A x) (y z) (A y) (A z) (B f) (B g) (B x) (B y) (B z) (ph z) (f ph)) ((ac6.1 (e. A (V))) (ac6.2 (e. B (V))) (ac6lem.4 (= C ({e.|} y B ph))) (ac6lem.5 (= H ({<,>|} x z (/\ (e. (cv x) A) (= (cv z) C)))))) (-> (A.e. x A (E.e. y B ph)) (E. f (/\ (:--> (cv f) A B) (A.e. x A (e. (` (cv f) (cv x)) C))))) (ac6lem.4 (cv z) eqeq2i biimp ({/}) eqeq1d negbid y B ph rabn0 syl6bb biimprcd x A r19.20si x A (= (cv z) C) (-. (= (cv z) ({/}))) r19.23v sylib z (E. x (/\ (e. (cv x) A) (= (cv z) C))) abid ac6lem.5 rneqi x z (/\ (e. (cv x) A) (= (cv z) C)) rnopab eqtr (cv z) eleq2i x A (= (cv z) C) df-rex 3bitr4 syl5ib r19.21aiv ac6lem.4 ac6.2 y ph rabex eqeltr ac6lem.5 fnopab2 ac6.1 H A (V) fnex mp2an H (V) rnexg ax-mp z g ac5b ac6lem.4 ac6.2 y ph rabex eqeltr ac6lem.5 fnopab2 H A fnfrn mpbi (cv g) (ran H) B H A fco mpan2 (A.e. z (ran H) (e. (` (cv g) (cv z)) (cv z))) adantr (Fun (cv g)) x ax-17 z x z (/\ (e. (cv x) A) (= (cv z) C)) hbopab1 ac6lem.5 (cv z) eleq2i ac6lem.5 (cv z) eleq2i x albii 3imtr4 hbrn (e. (` (cv g) (cv z)) (cv z)) x ax-17 hbral hban ac6lem.4 ac6.2 y ph rabex eqeltr (/\ (e. (cv x) A) (= (cv z) C)) x 19.8a ac6lem.5 rneqi x z (/\ (e. (cv x) A) (= (cv z) C)) rnopab eqtr abeq2i sylibr expcom (cv z) C (ran H) eleq1 sylibd vtocle (cv z) C (cv g) fveq2 (= (cv z) C) id eleq12d (ran H) rcla4v syl impcom (Fun (cv g)) adantll ac6lem.4 ac6.2 y ph rabex eqeltr ac6lem.5 fnopab2 H A fnfun ax-mp (cv g) H (cv x) fvco mp3an2 ac6lem.4 ac6.2 y ph rabex eqeltr ac6lem.5 dmopab2 (cv x) eleq2i sylan2br ac6lem.4 ac6.2 y ph rabex eqeltr x A C (V) z fvopab2 mpan2 ac6lem.5 (cv x) fveq1i syl5eq (cv g) fveq2d (Fun (cv g)) adantl eqtrd C eleq1d (A.e. z (ran H) (e. (` (cv g) (cv z)) (cv z))) adantlr mpbird ex r19.21ai (cv g) (ran H) B ffun sylan jca (ran H) B z unissb z (E. x (/\ (e. (cv x) A) (= (cv z) C))) abid ac6lem.5 rneqi x z (/\ (e. (cv x) A) (= (cv z) C)) rnopab eqtr (cv z) eleq2i x A (= (cv z) C) df-rex 3bitr4 ac6lem.4 y B ph ssrab2 eqsstr (cv z) C B sseq1 mpbiri (e. (cv x) A) a1i r19.23aiv sylbi mprgbir (cv g) (ran H) (U. (ran H)) B fss mpan2 sylan g visset ac6lem.4 ac6.2 y ph rabex eqeltr ac6lem.5 fnopab2 ac6.1 H A (V) fnex mp2an coex (cv f) (o. (cv g) H) A B feq1 (e. (cv f) (cv g)) x ax-17 f x z (/\ (e. (cv x) A) (= (cv z) C)) hbopab1 ac6lem.5 (cv f) eleq2i ac6lem.5 (cv f) eleq2i x albii 3imtr4 hbco hbeleq (cv f) (o. (cv g) H) (cv x) fveq1 C eleq1d A ralbid anbi12d cla4ev syl g 19.23aiv 3syl)) thm (ac6 ((f x) (f y) (f z) (f w) (A f) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B f) (B x) (B y) (B z) (B w) (ph z) (ph w) (f ph) (ps y)) ((ac6.1 (e. A (V))) (ac6.2 (e. B (V))) (ac6.3 (-> (= (cv y) (` (cv f) (cv x))) (<-> ph ps)))) (-> (A.e. x A (E.e. y B ph)) (E. f (/\ (:--> (cv f) A B) (A.e. x A ps)))) (ac6.1 ac6.2 ({e.|} y B ph) eqid (cv w) (cv z) ({e.|} y B ph) eqeq1 (e. (cv x) A) anbi2d x cbvopab2v f ac6lem ac6.3 B elrab pm3.27bd x A r19.20si (:--> (cv f) A B) anim2i f 19.22i syl)) thm (ac6s ((x y) (f x) (x z) (A x) (f y) (y z) (A y) (f z) (A f) (A z) (B x) (B y) (B f) (B z) (f ph) (ph z) (ps y) (ps z)) ((ac6s.1 (e. A (V))) (ac6s.2 (-> (= (cv y) (` (cv f) (cv x))) (<-> ph ps)))) (-> (A.e. x A (E.e. y B ph)) (E. f (/\ (:--> (cv f) A B) (A.e. x A ps)))) (ac6s.1 x y B ph z bnd2 ac6s.1 z visset ac6s.2 ac6 (C_ (cv z) B) anim2i z 19.22i (cv f) A (cv z) B fss expcom (A.e. x A ps) anim1d f 19.22dv imp z 19.23aiv syl syl)) thm (ac6s2 ((x y) (f x) (A x) (f y) (A y) (A f) (f ph) (ps y)) ((ac6s.1 (e. A (V))) (ac6s.2 (-> (= (cv y) (` (cv f) (cv x))) (<-> ph ps)))) (-> (A.e. x A (E. y ph)) (E. f (/\ (Fn (cv f) A) (A.e. x A ps)))) (y ph rexv x A ralbii ac6s.1 ac6s.2 (V) ac6s (cv f) A (V) ffn (A.e. x A ps) anim1i f 19.22i syl sylbir)) thm (ac6s3 ((x y) (f x) (A x) (f y) (A y) (A f) (f ph) (ps y)) ((ac6s.1 (e. A (V))) (ac6s.2 (-> (= (cv y) (` (cv f) (cv x))) (<-> ph ps)))) (-> (A.e. x A (E. y ph)) (E. f (A.e. x A ps))) (ac6s.1 ac6s.2 ac6s2 (Fn (cv f) A) (A.e. x A ps) pm3.27 f 19.22i syl)) thm (ac6s4 ((f x) (f y) (A f) (x y) (A x) (A y) (B f) (B y)) ((ac6s4.1 (e. A (V)))) (-> (A.e. x A (=/= B ({/}))) (E. f (/\ (Fn (cv f) A) (A.e. x A (e. (` (cv f) (cv x)) B))))) (B y ne0 x A ralbii ac6s4.1 (cv y) (` (cv f) (cv x)) B eleq1 ac6s2 sylbi)) thm (ac8 ((x z) (x y) (w x) (v x) (f x) (y z) (w z) (v z) (f z) (w y) (v y) (f y) (v w) (f w) (f v)) () (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (x f z w y v aceq5 f x ac7 mpgbi a4i)) thm (ac9s ((f x) (A f) (A x) (B f)) ((ac9.1 (e. A (V)))) (<-> (A.e. x A (=/= B ({/}))) (=/= (X_ x A B) ({/}))) (ac9.1 x B f ac6s4 (X_ x A B) f ne0 f visset x A B elixp f exbii bitr2 sylib x A B ixp0 con3i (X_ x A B) ({/}) df-ne B ({/}) df-ne x A ralbii x A (= B ({/})) ralnex bitr 3imtr4 impbi)) thm (kmlem1 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph x) (ph y) (ph v) (ps x) (ps v)) () (-> (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) ph))) (E. y (A.e. z (cv x) ps)))) (A. x (-> (A.e. z (cv x) (A.e. w (cv x) ph)) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) ps)))))) (v visset u (-. (= (cv u) ({/}))) rabex (cv x) ({e.|} u (cv v) (-. (= (cv u) ({/})))) z (-. (= (cv z) ({/}))) raleq1 (cv x) ({e.|} u (cv v) (-. (= (cv u) ({/})))) w ph raleq1 z raleqd anbi12d (cv x) ({e.|} u (cv v) (-. (= (cv u) ({/})))) z ps raleq1 y exbidv imbi12d cla4v v 19.21aiv u (cv v) (-. (= (cv u) ({/}))) ssrab2 (cv z) sseli u (cv v) (-. (= (cv u) ({/}))) ssrab2 (cv w) sseli ph imim1i r19.20i2 imim12i r19.20i2 (cv u) (cv z) ({/}) eqeq1 negbid (cv v) elrab pm3.27bd rgen jctil (cv u) (cv z) ({/}) eqeq1 negbid (cv v) elrab biimpr ps imim1i exp3a r19.20i2 y 19.22i imim12i v 19.20i syl (cv v) (cv x) w ph raleq1 z raleqd (cv v) (cv x) z (-> (-. (= (cv z) ({/}))) ps) raleq1 y exbidv imbi12d cbvalv sylib)) thm (kmlem2 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph y) (ph w) (ph u)) () (<-> (E. y (A.e. z (cv x) (-> ph (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (E. y (/\ (-. (e. (cv y) (cv x))) (A.e. z (cv x) (-> ph (E! v (e. (cv v) (i^i (cv z) (cv y))))))))) ((cv y) (cv w) (cv z) ineq2 (cv v) eleq2d v eubidv ph imbi2d z (cv x) ralbidv cbvexv x visset uniex (cv y) (U. (cv x)) (cv u) eleq2 negbid u exbidv y u nalset y u (e. (cv u) (cv y)) alexn mpbir a4i vtocl (cv z) (cv x) elssuni (cv u) sseld con3d (cv z) (cv u) disjsn syl6ibr impcom (i^i (cv z) (cv w)) uneq2d (i^i (cv z) (cv w)) un0 syl6eq (cv z) (cv w) ({} (cv u)) indi syl5req (cv v) eleq2d v eubidv ph imbi2d ralbidva u visset snid (-. (e. (cv u) (cv w))) a1i orri (cv u) (cv w) ({} (cv u)) elun mpbir (u. (cv w) ({} (cv u))) (cv x) elssuni (cv u) sseld mpi con3i (A.e. z (cv x) (-> ph (E! v (e. (cv v) (i^i (cv z) (u. (cv w) ({} (cv u)))))))) biantrurd bitrd w visset (cv u) snex unex (cv y) (u. (cv w) ({} (cv u))) (cv x) eleq1 negbid (cv y) (u. (cv w) ({} (cv u))) (cv z) ineq2 (cv v) eleq2d v eubidv ph imbi2d z (cv x) ralbidv anbi12d cla4ev syl6bi u 19.23aiv ax-mp w 19.23aiv sylbi (-. (e. (cv y) (cv x))) (A.e. z (cv x) (-> ph (E! v (e. (cv v) (i^i (cv z) (cv y)))))) pm3.27 y 19.22i impbi)) thm (kmlem3 ((x z) (w x) (v x) (w z) (v z) (v w)) () (<-> (-. (= (\ (cv z) (U. (\ (cv x) ({} (cv z))))) ({/}))) (E.e. v (cv z) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (-. (e. (cv v) (i^i (cv z) (cv w)))))))) ((cv z) (U. (\ (cv x) ({} (cv z)))) v dfdif2 v (cv z) dfnul3 ({e.|} v (cv z) (-. (e. (cv v) (U. (\ (cv x) ({} (cv z))))))) uneq2i ({e.|} v (cv z) (-. (e. (cv v) (U. (\ (cv x) ({} (cv z))))))) un0 v (cv z) (-. (e. (cv v) (U. (\ (cv x) ({} (cv z)))))) (-. (e. (cv v) (cv z))) unrab 3eqtr3 (e. (cv v) (U. (\ (cv x) ({} (cv z))))) (e. (cv v) (cv z)) ianor (cv v) (\ (cv x) ({} (cv z))) w eluni (e. (cv v) (cv z)) anbi1i w (cv x) (-. (-> (-. (= (cv z) (cv w))) (-. (e. (cv v) (i^i (cv z) (cv w)))))) df-rex (cv w) (cv x) ({} (cv z)) eldif w (cv z) elsn (cv w) (cv z) eqcom bitr negbii (e. (cv w) (cv x)) anbi2i bitr (/\ (e. (cv v) (cv w)) (e. (cv v) (cv z))) anbi2i (e. (cv v) (cv w)) (e. (cv v) (cv z)) ancom (/\ (e. (cv w) (cv x)) (-. (= (cv z) (cv w)))) anbi1i (/\ (e. (cv v) (cv z)) (e. (cv v) (cv w))) (/\ (e. (cv w) (cv x)) (-. (= (cv z) (cv w)))) ancom bitr (e. (cv w) (cv x)) (-. (= (cv z) (cv w))) (/\ (e. (cv v) (cv z)) (e. (cv v) (cv w))) anass 3bitr (e. (cv v) (cv w)) (e. (cv v) (cv z)) (e. (cv w) (\ (cv x) ({} (cv z)))) an23 (cv v) (cv z) (cv w) elin (-. (= (cv z) (cv w))) anbi2i (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))) df-an bitr3 (e. (cv w) (cv x)) anbi2i 3bitr3r w exbii w (/\ (e. (cv v) (cv w)) (e. (cv w) (\ (cv x) ({} (cv z))))) (e. (cv v) (cv z)) 19.41v 3bitr w (cv x) (-> (-. (= (cv z) (cv w))) (-. (e. (cv v) (i^i (cv z) (cv w))))) rexnal 3bitr2r con1bii bitr3 (e. (cv v) (cv z)) a1i rabbii 3eqtr ({/}) eqeq1i negbii v (cv z) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (-. (e. (cv v) (i^i (cv z) (cv w)))))) rabn0 bitr)) thm (kmlem4 ((v y) (x y) (v x) (y z) (v z) (w y) (v w)) () (-> (/\ (e. (cv w) (cv x)) (-. (= (cv z) (cv w)))) (= (i^i (\ (cv z) (U. (\ (cv x) ({} (cv z))))) (cv w)) ({/}))) (w visset (cv v) (cv w) (cv x) eleq1 (cv v) (cv w) (cv z) eqeq2 negbid anbi12d (cv v) (cv w) (cv y) eleq2 negbid imbi12d cla4v com12 (cv y) (cv z) (U. (\ (cv x) ({} (cv z)))) eldif (e. (cv y) (cv z)) (-. (e. (cv y) (U. (\ (cv x) ({} (cv z)))))) pm3.27 (cv y) (\ (cv x) ({} (cv z))) v eluni negbii v (/\ (e. (cv y) (cv v)) (e. (cv v) (\ (cv x) ({} (cv z))))) alnex (e. (cv y) (cv v)) (e. (cv v) (\ (cv x) ({} (cv z)))) ianor (cv v) (cv x) ({} (cv z)) eldif v (cv z) elsn (cv v) (cv z) eqcom bitr negbii (e. (cv v) (cv x)) anbi2i bitr negbii (e. (cv v) (cv x)) (= (cv z) (cv v)) iman bitr4 (e. (cv y) (cv v)) imbi2i (e. (cv y) (cv v)) (e. (cv v) (\ (cv x) ({} (cv z)))) pm4.62 (e. (cv y) (cv v)) (= (cv z) (cv v)) pm4.1 (e. (cv v) (cv x)) imbi2i (e. (cv y) (cv v)) (e. (cv v) (cv x)) (= (cv z) (cv v)) bi2.04 (e. (cv v) (cv x)) (-. (= (cv z) (cv v))) (-. (e. (cv y) (cv v))) impexp 3bitr4 3bitr3 bitr v albii 3bitr2 sylib sylbi syl5 r19.21aiv (\ (cv z) (U. (\ (cv x) ({} (cv z))))) (cv w) y disj sylibr)) thm (kmlem5 () () (-> (/\ (e. (cv w) (cv x)) (-. (= (cv z) (cv w)))) (= (i^i (\ (cv z) (U. (\ (cv x) ({} (cv z))))) (\ (cv w) (U. (\ (cv x) ({} (cv w)))))) ({/}))) ((cv w) (U. (\ (cv x) ({} (cv w)))) difss (\ (cv w) (U. (\ (cv x) ({} (cv w))))) (cv w) (\ (cv z) (U. (\ (cv x) ({} (cv z))))) sslin ax-mp w x z kmlem4 (i^i (\ (cv z) (U. (\ (cv x) ({} (cv z))))) (\ (cv w) (U. (\ (cv x) ({} (cv w)))))) sseq2d mpbii (i^i (\ (cv z) (U. (\ (cv x) ({} (cv z))))) (\ (cv w) (U. (\ (cv x) ({} (cv w)))))) ss0b sylib)) thm (kmlem6 ((x z) (w x) (v x) (w z) (v z) (v w) (ph v) (A v)) () (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> ph (= A ({/})))))) (A.e. z (cv x) (E.e. v (cv z) (A.e. w (cv x) (-> ph (-. (e. (cv v) A))))))) (z (cv x) (-. (= (cv z) ({/}))) (A.e. w (cv x) (-> ph (= A ({/})))) r19.26 v (e. (cv v) (cv z)) (A.e. w (cv x) (-> ph (-. (e. (cv v) A)))) 19.29r v (cv z) (A.e. w (cv x) (-> ph (-. (e. (cv v) A)))) df-rex sylibr (cv z) v n0 biimp (cv v) A n0i con2i ph imim2i w (cv x) r19.20si v 19.21aiv syl2an z (cv x) r19.20si sylbir)) thm (kmlem7 ((x z) (w x) (v x) (w z) (v z) (v w)) () (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (-. (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))))))) (z x w (-. (= (cv z) (cv w))) (i^i (cv z) (cv w)) v kmlem6 w (cv x) (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))) ralinexa v (cv z) rexbii v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))) rexnal bitr z (cv x) ralbii z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w)))))) ralnex bitr sylib)) thm (kmlem8 ((x z) (w x) (u x) (t x) (h x) (w z) (u z) (t z) (h z) (u w) (t w) (h w) (t u) (h u) (h t) (A z) (A w) (A h)) ((kmlem8.1 (= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t))))))))))) (A.e. z A (A.e. w A (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/}))))) (t (cv x) h (cv x) (= (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))) (= (cv w) (\ (cv h) (U. (\ (cv x) ({} (cv h)))))) reeanv (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t))))) (cv w) (\ (cv h) (U. (\ (cv x) ({} (cv h))))) ineq12 ({/}) eqeq1d h x t kmlem5 syl5bir exp3a (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t))))) (cv w) (\ (cv h) (U. (\ (cv x) ({} (cv h))))) eqeq12 (cv t) (cv h) (U. (\ (cv x) ({} (cv t)))) difeq1 (cv t) (cv h) sneq (cv x) difeq2d unieqd (cv h) difeq2d eqtrd syl5bir con3d syl5d com12 (e. (cv t) (cv x)) adantl r19.23aivv sylbir z visset (cv u) (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t))))) eqeq1 t (cv x) rexbidv kmlem8.1 elab2 w visset (cv u) (cv w) (\ (cv t) (U. (\ (cv x) ({} (cv t))))) eqeq1 t (cv x) rexbidv kmlem8.1 elab2 (cv t) (cv h) (U. (\ (cv x) ({} (cv t)))) difeq1 (cv t) (cv h) sneq (cv x) difeq2d unieqd (cv h) difeq2d eqtrd (cv w) eqeq2d (cv x) cbvrexv bitr syl2anb rgen2)) thm (kmlem9 ((h ph) (x y) (x z) (w x) (u x) (t x) (h x) (y z) (w y) (u y) (t y) (h y) (w z) (u z) (t z) (h z) (u w) (t w) (h w) (t u) (h u) (h t) (A y) (A z) (A w) (A h)) ((kmlem8.1 (= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t))))))))))) (-> (A. h (-> (A.e. z (cv h) (A.e. w (cv h) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/}))))) (E. y (A.e. z (cv h) ph)))) (E. y (A.e. z A ph))) (kmlem8.1 z w kmlem8 kmlem8.1 x visset u t (\ (cv t) (U. (\ (cv x) ({} (cv t))))) abrexex eqeltr (cv h) A w (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/}))) raleq1 z raleqd (cv h) A z ph raleq1 y exbidv imbi12d cla4v mpi)) thm (kmlem10 ((x z) (u x) (t x) (u z) (t z) (t u) (A z)) ((kmlem8.1 (= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t))))))))))) (-> (e. (cv z) (cv x)) (= (i^i (cv z) (U. A)) (\ (cv z) (U. (\ (cv x) ({} (cv z))))))) ((cv z) (cv x) snssi ({} (cv z)) (cv x) ssequn1 sylib ({} (cv z)) (cv x) undif2 syl5req (cv x) (u. ({} (cv z)) (\ (cv x) ({} (cv z)))) t (i^i (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))) iuneq1 syl z x t kmlem4 (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t))))) incom syl5eq ex (cv t) (cv x) ({} (cv z)) eldifn t (cv z) elsn negbii sylib syl5 r19.21aiv t (\ (cv x) ({} (cv z))) (i^i (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))) ({/}) iuneq2 syl t (\ (cv x) ({} (cv z))) iun0 syl6eq (i^i (cv z) (\ (cv z) (U. (\ (cv x) ({} (cv z)))))) uneq2d t ({} (cv z)) (\ (cv x) ({} (cv z))) (i^i (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))) iunxun z visset (cv t) (cv z) (U. (\ (cv x) ({} (cv t)))) difeq1 (cv t) (cv z) sneq (cv x) difeq2d unieqd (cv z) difeq2d eqtrd (cv z) ineq2d iunxsn (U_ t (\ (cv x) ({} (cv z))) (i^i (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t))))))) uneq1i eqtr syl5eq eqtrd (i^i (cv z) (\ (cv z) (U. (\ (cv x) ({} (cv z)))))) un0 (cv z) (U. (\ (cv x) ({} (cv z)))) indif eqtr syl6eq kmlem8.1 unieqi t visset (cv t) (V) (U. (\ (cv x) ({} (cv t)))) difexg ax-mp t (cv x) u dfiun2 eqtr4 (cv z) ineq2i t (cv x) (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t))))) iunin2 eqtr4 syl5eq)) thm (kmlem11 ((x y) (x z) (v x) (u x) (t x) (y z) (v y) (u y) (t y) (v z) (u z) (t z) (u v) (t v) (t u) (A y) (A z) (A v)) ((kmlem8.1 (= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t))))))))))) (-> (A.e. z (cv x) (-. (= (\ (cv z) (U. (\ (cv x) ({} (cv z))))) ({/})))) (-> (A.e. z A (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))))) (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (i^i (cv y) (U. A))))))))) ((cv t) (cv z) (U. (\ (cv x) ({} (cv t)))) difeq1 (cv t) (cv z) sneq (cv x) difeq2d unieqd (cv z) difeq2d eqtrd ({/}) eqeq1d negbid (cv x) cbvralv (cv t) (cv z) (U. (\ (cv x) ({} (cv t)))) difeq1 (cv t) (cv z) sneq (cv x) difeq2d unieqd (cv z) difeq2d eqtrd (cv y) ineq1d (cv v) eleq2d v eubidv (cv x) cbvralv imbi12i kmlem8.1 z kmlem10 (cv y) ineq1d (cv z) (cv y) (U. A) in12 (cv y) (i^i (cv z) (U. A)) incom eqtr syl5req (cv v) eleq2d v eubidv (E! v (e. (cv v) (i^i (cv z) (i^i (cv y) (U. A))))) (-. (= (cv z) ({/}))) ax-1 syl6bi r19.20i (A.e. z (cv x) (-. (= (\ (cv z) (U. (\ (cv x) ({} (cv z))))) ({/})))) imim2i sylbi com12 kmlem8.1 A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))))) z (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y))))) raleq1 ax-mp z ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))))) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y))))) df-ral z visset (cv u) (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t))))) eqeq1 t (cv x) rexbidv elab (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y))))) imbi1i t (cv x) (= (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y))))) r19.23v bitr4 z albii t (cv x) z (-> (= (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))))) ralcom4 t visset (cv t) (V) (U. (\ (cv x) ({} (cv t)))) difexg ax-mp (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t))))) ({/}) eqeq1 negbid (cv z) (\ (cv t) (U. (\ (cv x) ({} (cv t))))) (cv y) ineq1 (cv v) eleq2d v eubidv imbi12d ceqsalv t (cv x) ralbii 3bitr2 3bitr t (cv x) (-. (= (\ (cv t) (U. (\ (cv x) ({} (cv t))))) ({/}))) (E! v (e. (cv v) (i^i (\ (cv t) (U. (\ (cv x) ({} (cv t))))) (cv y)))) r19.20 sylbi syl5)) thm (kmlem12 ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (g x) (y z) (w y) (v y) (u y) (t y) (h y) (g y) (w z) (v z) (u z) (t z) (h z) (g z) (v w) (u w) (t w) (h w) (g w) (u v) (t v) (h v) (g v) (t u) (h u) (g u) (h t) (g t) (g h) (A y) (A z) (A w) (A v) (A h) (A g)) ((kmlem8.1 (= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t))))))))))) (<-> (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (A. x (-> (-. (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w)))))))) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))))) (x z w (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/}))) y (E! v (e. (cv v) (i^i (cv z) (cv y)))) kmlem1 (cv x) (cv h) w (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/}))) raleq1 z raleqd (cv x) (cv h) z (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y))))) raleq1 y exbidv imbi12d cbvalv kmlem8.1 h z w y (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y))))) kmlem9 (cv y) (cv g) (cv z) ineq2 (cv v) eleq2d v eubidv (-. (= (cv z) ({/}))) imbi2d z A ralbidv cbvexv kmlem8.1 z v g kmlem11 g visset (U. A) inex1 (cv y) (i^i (cv g) (U. A)) (cv z) ineq2 (cv v) eleq2d v eubidv (-. (= (cv z) ({/}))) imbi2d z (cv x) ralbidv cla4ev syl6 g 19.23adv com12 z x v w kmlem3 w (cv x) (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))) ralinexa v (cv z) rexbii v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))) rexnal 3bitr z (cv x) ralbii z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w)))))) ralnex bitr syl5ibr sylbi syl x 19.21aiv sylbi syl z x w v kmlem7 (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y))))))) imim1i (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))) biimt z (cv x) r19.20si z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y))))) r19.15 syl y exbidv (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/}))))) adantr pm5.74i sylibr x 19.20i impbi)) thm (kmlem13 ((x y) (x z) (v x) (y z) (v y) (v z)) () (<-> (-> (-. (E.e. z (cv x) (A.e. v (cv z) ph))) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (\/ (E.e. z (cv x) (A.e. v (cv z) ph)) (E. y (/\ (-. (e. (cv y) (cv x))) (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))))) (z (cv x) (A.e. v (cv z) ph) ralnex v (cv z) (-. ph) df-rex v (cv z) ph rexnal bitr3 (e. (cv v) (cv z)) (-. ph) pm3.26 v 19.22i (cv z) v n0 sylibr sylbir z (cv x) r19.20si sylbir (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))) biimt z (cv x) r19.20si z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y))))) r19.15 syl (-. (e. (cv y) (cv x))) anbi2d y exbidv y z x (-. (= (cv z) ({/}))) v kmlem2 syl6rbbr syl pm5.74i (E.e. z (cv x) (A.e. v (cv z) ph)) (E. y (/\ (-. (e. (cv y) (cv x))) (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))) pm4.64 bitr)) thm (kmlem14 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph u)) ((kmlem14.1 (<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v)))))) (kmlem14.2 (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))))) (kmlem14.3 (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (<-> (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))))) (E. y (A. z (E. v (A. u (/\ (e. (cv y) (cv x)) ph)))))) ((cv z) (cv y) (cv w) eqeq1 negbid (cv z) (cv y) (cv w) ineq1 (cv v) eleq2d anbi12d w (cv x) rexbidv v raleqd (cv x) cbvrexv y (cv x) (A.e. v (cv y) (E.e. w (cv x) (/\ (-. (= (cv y) (cv w))) (e. (cv v) (i^i (cv y) (cv w)))))) df-rex (cv v) (cv z) (i^i (cv y) (cv w)) eleq1 (-. (= (cv y) (cv w))) anbi2d w (cv x) rexbidv (cv y) cbvralv z (cv y) (E.e. w (cv x) (/\ (-. (= (cv y) (cv w))) (e. (cv z) (i^i (cv y) (cv w))))) df-ral bitr (e. (cv y) (cv x)) anbi2i z (e. (cv y) (cv x)) (-> (e. (cv z) (cv y)) (E.e. w (cv x) (/\ (-. (= (cv y) (cv w))) (e. (cv z) (i^i (cv y) (cv w)))))) 19.28v (cv w) (cv v) (cv y) eqeq2 negbid (cv w) (cv v) (cv y) ineq2 (cv z) eleq2d anbi12d (cv x) cbvrexv v (cv x) (/\ (-. (= (cv y) (cv v))) (e. (cv z) (i^i (cv y) (cv v)))) df-rex bitr (e. (cv z) (cv y)) imbi2i v (e. (cv z) (cv y)) (/\ (e. (cv v) (cv x)) (/\ (-. (= (cv y) (cv v))) (e. (cv z) (i^i (cv y) (cv v))))) 19.37v bitr4 (e. (cv y) (cv x)) anbi2i v (e. (cv y) (cv x)) (-> (e. (cv z) (cv y)) (/\ (e. (cv v) (cv x)) (/\ (-. (= (cv y) (cv v))) (e. (cv z) (i^i (cv y) (cv v)))))) 19.42v kmlem14.1 (cv z) (cv y) (cv v) elin baibr (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) anbi2d (e. (cv v) (cv x)) (-. (= (cv y) (cv v))) (e. (cv z) (i^i (cv y) (cv v))) anass syl6bb pm5.74i bitr (e. (cv y) (cv x)) anbi2i (/\ (e. (cv y) (cv x)) ph) u ax-17 19.3r bitr3 v exbii 3bitr2 z albii 3bitr2 y exbii 3bitr)) thm (kmlem15 ((x y) (x z) (v x) (u x) (y z) (v y) (u y) (v z) (u z) (u v) (ph u)) ((kmlem14.1 (<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v)))))) (kmlem14.2 (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))))) (kmlem14.3 (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (<-> (/\ (-. (e. (cv y) (cv x))) ch) (A. z (E. v (A. u (/\ (-. (e. (cv y) (cv x))) ps))))) (kmlem14.3 (e. (cv v) (i^i (cv z) (cv y))) u ax-17 v eu1 (cv v) (cv z) (cv y) elin (e. (cv u) (i^i (cv z) (cv y))) v ax-17 (cv v) (cv u) (i^i (cv z) (cv y)) eleq1 sbie (cv u) (cv z) (cv y) elin bitr (cv v) (cv u) eqcom imbi12i u albii anbi12i u (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))) 19.28v bitr4 v exbii bitr z (cv x) ralbii z (cv x) (E. v (A. u (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))))) df-ral kmlem14.2 u albii u (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))) 19.21v bitr v exbii v (e. (cv z) (cv x)) (A. u (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))) 19.37v bitr z albii bitr4 3bitr (-. (e. (cv y) (cv x))) anbi2i z (-. (e. (cv y) (cv x))) (E. v (A. u ps)) 19.28v u (-. (e. (cv y) (cv x))) ps 19.28v v exbii v (-. (e. (cv y) (cv x))) (A. u ps) 19.42v bitr2 z albii 3bitr2)) thm (kmlem16 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph u)) ((kmlem14.1 (<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v)))))) (kmlem14.2 (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))))) (kmlem14.3 (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (<-> (\/ (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))))) (E. y (/\ (-. (e. (cv y) (cv x))) ch))) (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) ph) (/\ (-. (e. (cv y) (cv x))) ps))))))) (kmlem14.1 kmlem14.2 kmlem14.3 w kmlem14 kmlem14.1 kmlem14.2 kmlem14.3 kmlem15 y exbii orbi12i y (A. z (E. v (A. u (/\ (e. (cv y) (cv x)) ph)))) (A. z (E. v (A. u (/\ (-. (e. (cv y) (cv x))) ps)))) 19.43 (e. (cv y) (cv x)) pm3.24 (e. (cv y) (cv x)) ph pm3.26 u a4s z v 19.23aivv (-. (e. (cv y) (cv x))) ps pm3.26 u a4s z v 19.23aivv anim12i mto z (E. v (A. u (/\ (e. (cv y) (cv x)) ph))) (E. v (A. u (/\ (-. (e. (cv y) (cv x))) ps))) 19.33b ax-mp (e. (cv y) (cv x)) pm3.24 (e. (cv y) (cv x)) ph pm3.26 u 19.23aiv (-. (e. (cv y) (cv x))) ps pm3.26 u 19.23aiv anim12i mto u (/\ (e. (cv y) (cv x)) ph) (/\ (-. (e. (cv y) (cv x))) ps) 19.33b ax-mp v exbii v (A. u (/\ (e. (cv y) (cv x)) ph)) (A. u (/\ (-. (e. (cv y) (cv x))) ps)) 19.43 bitr2 z albii bitr3 y exbii 3bitr2)) thm (aceqkm ((x y) (x z) (w x) (v x) (u x) (f x) (t x) (h x) (y z) (w y) (v y) (u y) (f y) (t y) (h y) (w z) (v z) (u z) (f z) (t z) (h z) (v w) (u w) (f w) (t w) (h w) (u v) (f v) (t v) (h v) (f u) (t u) (h u) (f t) (f h) (h t)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (/\ (-. (e. (cv y) (cv x))) (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))))))))))) (x f z w y v aceq5 ({|} t (E.e. h (cv x) (= (cv t) (\ (cv h) (U. (\ (cv x) ({} (cv h)))))))) eqid z w y v kmlem12 z x v (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))) y kmlem13 x albii bitr (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v)))) pm4.2 (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))) pm4.2 (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))) pm4.2 w kmlem16 x albii 3bitr)) thm (ackm ((x y) (x z) (v x) (u x) (f x) (y z) (v y) (u y) (f y) (v z) (u z) (f z) (u v) (f v) (f u)) () (A. x (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (/\ (-. (e. (cv y) (cv x))) (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))))))))))) (x f y z v u aceqkm f x ac7 mpgbi)) thm (numthlem ((x y) (f x) (g x) (A x) (f y) (g y) (A y) (f g) (A f) (A g) (B x) (B y) (B f) (F x) (F y) (F f) (G x) (G y) (G f)) ((numthlem.1 (e. A (V))) (numthlem.2 (= B ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (numthlem.3 (= F (U. B))) (numthlem.4 (= G ({<,>|} f y (= (cv y) (` (cv g) (\ A (ran (cv f))))))))) (E.e. x (On) (E. f (:-1-1-onto-> (cv f) (cv x) A))) (numthlem.1 pwex g y ac4c numthlem.2 numthlem.3 x tfr2 numthlem.2 numthlem.3 tfrlem7 x visset F (cv x) (V) resfunexg mp2an (cv g) (\ A (ran (|` F (cv x)))) fvex (cv f) (|` F (cv x)) rneq (ran (cv f)) (ran (|` F (cv x))) A difeq2 (\ A (ran (cv f))) (\ A (ran (|` F (cv x)))) (cv g) fveq2 3syl y fvopab numthlem.4 (|` F (cv x)) fveq1i F (cv x) df-ima A difeq2i (cv g) fveq2i 3eqtr4 syl6eq (\ A (" F (cv x))) eleq1d A (" F (cv x)) difss numthlem.1 A (" F (cv x)) difss ssexi A elpw mpbir (cv y) (\ A (" F (cv x))) ({/}) eqeq1 negbid (cv y) (\ A (" F (cv x))) (cv g) fveq2 (= (cv y) (\ A (" F (cv x)))) id eleq12d imbi12d (P~ A) rcla4v ax-mp imp syl5bir exp3a com12 r19.21aiv numthlem.2 numthlem.3 tfr1 numthlem.1 x tz7.49c numthlem.2 numthlem.3 tfrlem7 x visset F (cv x) (V) resfunexg mp2an (cv f) (|` F (cv x)) (cv x) A f1oeq1 cla4ev x (On) r19.22si 3syl g 19.23aiv ax-mp)) thm (numth ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h)) ((numth.1 (e. A (V)))) (E.e. x (On) (E. f (:-1-1-onto-> (cv f) (cv x) A))) (numth.1 g z w ({<,>|} v u (= (cv u) (` (cv h) (\ A (ran (cv v)))))) f x y rdglem1 (U. ({|} g (E.e. z (On) (/\ (Fn (cv g) (cv z)) (A.e. w (cv z) (= (` (cv g) (cv w)) (` ({<,>|} v u (= (cv u) (` (cv h) (\ A (ran (cv v)))))) (|` (cv g) (cv w))))))))) eqid (= (cv u) (cv y)) id (cv v) (cv f) rneq (ran (cv v)) (ran (cv f)) A difeq2 (\ A (ran (cv v))) (\ A (ran (cv f))) (cv h) fveq2 3syl eqeqan12rd cbvopabv numthlem)) thm (numth2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (E.e. x (On) (br (cv x) (~~) A)) ((cv y) A (cv x) (~~) breq2 x (On) rexbidv y visset x z numth y visset (cv x) z bren x (On) rexbii mpbir (V) vtoclg 0elon ({/}) (On) enrefg ax-mp A ({/}) (~~) brprc mpbiri 0elon (cv x) ({/}) (~~) A breq1 (On) rcla4ev mpan syl pm2.61i)) thm (numthcor ((x y) (A x) (A y)) () (-> (e. A B) (E.e. x (On) (br A (~<) (cv x)))) ((cv y) A (~<) (cv x) breq1 x (On) rexbidv x (P~ (cv y)) numth2 y visset pwex (cv x) ensym y visset canth2 x visset (cv x) (V) (cv y) (P~ (cv y)) sdomentr ax-mp mpan syl x (On) r19.22si ax-mp B vtoclg)) thm (weth ((x y) (x z) (w x) (f x) (A x) (y z) (w y) (f y) (A y) (w z) (f z) (A z) (f w) (A w) (A f)) ((weth.1 (e. A (V)))) (E. x (We (cv x) A)) (weth.1 y f numth (cv f) (cv y) A f1ocnv ({<,>|} z w (br (` (`' (cv f)) (cv z)) (E) (` (`' (cv f)) (cv w)))) eqid A (cv y) f1owe ({<,>|} z w (br (` (`' (cv f)) (cv z)) (E) (` (`' (cv f)) (cv w)))) A weinxp weth.1 weth.1 xpex ({<,>|} z w (br (` (`' (cv f)) (cv z)) (E) (` (`' (cv f)) (cv w)))) inex2 (cv x) (i^i ({<,>|} z w (br (` (`' (cv f)) (cv z)) (E) (` (`' (cv f)) (cv w)))) (X. A A)) A weeq1 cla4ev sylbi syl6 (cv y) eloni (cv y) ordwe syl syl5 syl f 19.23aiv com12 r19.23aiv ax-mp)) thm (zornlem1 ((w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D u) (D v) (D f) (D t) (R x) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))))) (-> (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/}))))) (e. (` F (cv x)) D)) (zornlem.2 zornlem.3 x tfr2 zornlem.6 (|` F (cv x)) fveq1i zornlem.2 zornlem.3 tfrlem7 x visset F (cv x) (V) resfunexg mp2an zornlem.1 zornlem.5 z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))) ssrab2 eqsstr ssexi v (A.e. u D (-. (br (cv u) (cv w) (cv v)))) rabex uniex (cv f) (|` F (cv x)) rneq F (cv x) df-ima syl6eqr (cv g) eleq2d (br (cv g) R (cv z)) imbi1d ralbidv2 z A rabbisdv zornlem.4 zornlem.5 3eqtr4g (cv v) eleq2d (cv f) (|` F (cv x)) rneq F (cv x) df-ima syl6eqr (cv g) eleq2d (br (cv g) R (cv z)) imbi1d ralbidv2 z A rabbisdv zornlem.4 zornlem.5 3eqtr4g (cv u) eleq2d (-. (br (cv u) (cv w) (cv v))) imbi1d ralbidv2 anbi12d v abbidv v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))) df-rab v D (A.e. u D (-. (br (cv u) (cv w) (cv v)))) df-rab 3eqtr4g unieqd t fvopab eqtr syl6eq D eleq1d zornlem.5 z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))) ssrab2 eqsstr zornlem.1 zornlem.5 z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))) ssrab2 eqsstr ssexi (cv w) A v u wereu mpanr1 v D (A.e. u D (-. (br (cv u) (cv w) (cv v)))) reucl syl syl5bir imp)) thm (zornlem2 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))))) (-> (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/}))))) (-> (e. (cv y) (cv x)) (br (` F (cv y)) R (` F (cv x))))) ((cv x) onsst zornlem.2 zornlem.3 tfr1 F (On) fndm ax-mp syl6ssr zornlem.2 zornlem.3 tfrlem7 F (cv x) (cv y) funfvima2 mpan syl (/\ (We (cv w) A) (-. (= D ({/})))) adantr zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem1 zornlem.5 (` F (cv x)) eleq2i (cv z) (` F (cv x)) (cv g) R breq2 g (" F (cv x)) ralbidv A elrab bitr pm3.27bd (cv g) (` F (cv y)) R (` F (cv x)) breq1 (" F (cv x)) rcla4cv 3syl syld)) thm (zornlem3 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))))) (-> (/\ (Po R A) (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/})))))) (-> (e. (cv y) (cv x)) (-. (= (` F (cv x)) (` F (cv y)))))) (zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 y zornlem2 (Po R A) adantl (` F (cv x)) (` F (cv y)) R (` F (cv x)) breq1 biimprcd R A (` F (cv x)) poirr nsyli com12 zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem1 zornlem.5 z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))) ssrab2 eqsstr (` F (cv x)) sseli syl sylan2 syld)) thm (zornlem4 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))))) (-> (/\ (Po R A) (We (cv w) A)) (E.e. x (On) (= D ({/})))) ((e. (ran F) (V)) pm3.24 (We (cv w) A) x ax-17 x (-> (e. (cv x) (On)) (-. (= D ({/})))) hba1 hban (e. (cv y) A) x ax-17 (` F (cv x)) (cv y) A eleq1 zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem1 zornlem.5 z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))) ssrab2 eqsstr (` F (cv x)) sseli syl syl5bi com12 exp32 com12 a2d x a4sd imp r19.23ad zornlem.2 zornlem.3 tfr1 F (On) (cv y) x fvelrn ax-mp syl5ib ssrdv zornlem.1 (ran F) ssex syl ex (Po R A) adantl zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 y zornlem3 exp45 com23 imp a2d imp4a y 19.21adv x 19.20dv x (On) y (cv x) (-. (= (` F (cv x)) (` F (cv y)))) r2al syl6ibr (On) ssid zornlem.2 zornlem.3 tfr1 (On) x y tz7.48lem mpan zornlem.2 zornlem.3 tfrlem6 zornlem.2 zornlem.3 tfr1 F (On) fndm ax-mp (On) ssid eqsstr F (On) relssres mp2an (|` F (On)) F cnveq ax-mp (`' (|` F (On))) (`' F) funeq ax-mp sylib syl6 onprc (`' F) (V) funrnex com12 F df-rn (V) eleq1i zornlem.2 zornlem.3 tfr1 F (On) fndm ax-mp F dfdm4 eqtr3 (V) eleq1i 3imtr4g mtoi syl6 jcad x (e. (cv x) (On)) (= D ({/})) alinexa syl5ibr mt3i x (On) (= D ({/})) df-rex sylibr)) thm (zornlem5 ((u x) (v x) (f x) (t x) (s x) (H x) (u v) (f u) (t u) (s u) (H u) (f v) (t v) (s v) (H v) (f t) (f s) (H f) (s t) (H t) (H s) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (s x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (s y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (s w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h s) (A h) (t z) (f t) (g t) (t u) (t v) (s t) (A t) (f z) (g z) (u z) (v z) (s z) (A z) (f g) (f u) (f v) (f s) (A f) (g u) (g v) (g s) (A g) (u v) (s u) (A u) (s v) (A v) (A s) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F s) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (zornlem.7 (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z))))))) (-> (/\ (/\ (We (cv w) A) (e. (cv x) (On))) (A.e. y (cv x) (-. (= H ({/}))))) (C_ (" F (cv x)) A)) ((/\ (We (cv w) A) (e. (cv x) (On))) y ax-17 y (cv x) (-. (= H ({/}))) hbra1 hban (e. (cv s) A) y ax-17 (` F (cv y)) (cv s) A eleq1 zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.7 zornlem.6 zornlem1 zornlem.7 z A (A.e. g (" F (cv y)) (br (cv g) R (cv z))) ssrab2 eqsstr (` F (cv y)) sseli syl syl5bi (cv x) (cv y) onelon sylani com12 exp43 com3r imp a2d y a4sd y (cv x) (-. (= H ({/}))) df-ral syl5ib imp r19.23ad zornlem.2 zornlem.3 tfrlem7 F (cv s) (cv x) y fvelima mpan syl5 ssrdv)) thm (zornlem6 ((u x) (v x) (f x) (t x) (s x) (r x) (a x) (b x) (H x) (u v) (f u) (t u) (s u) (r u) (a u) (b u) (H u) (f v) (t v) (s v) (r v) (a v) (b v) (H v) (f t) (f s) (f r) (a f) (b f) (H f) (s t) (r t) (a t) (b t) (H t) (r s) (a s) (b s) (H s) (a r) (b r) (H r) (a b) (H a) (H b) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (r x) (s x) (a x) (b x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (r y) (s y) (a y) (b y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (r w) (s w) (a w) (b w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h r) (h s) (a h) (b h) (A h) (t z) (f t) (g t) (t u) (t v) (r t) (s t) (a t) (b t) (A t) (f z) (g z) (u z) (v z) (r z) (s z) (a z) (b z) (A z) (f g) (f u) (f v) (f r) (f s) (a f) (b f) (A f) (g u) (g v) (g r) (g s) (a g) (b g) (A g) (u v) (r u) (s u) (a u) (b u) (A u) (r v) (s v) (a v) (b v) (A v) (r s) (a r) (b r) (A r) (a s) (b s) (A s) (a b) (A a) (A b) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F r) (F s) (F a) (F b) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (D a) (D b) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s) (R r) (R a) (R b)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (zornlem.7 (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z))))))) (-> (Po R A) (-> (/\ (/\ (We (cv w) A) (e. (cv x) (On))) (A.e. y (cv x) (-. (= H ({/}))))) (Or R (" F (cv x))))) (zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem.7 zornlem5 (" F (cv x)) A R poss syl com12 (cv x) (cv a) onelon (cv x) (cv b) onelon anim12i anandis ex zornlem.1 zornlem.2 zornlem.3 zornlem.4 ({e.|} z A (A.e. g (" F (cv b)) (br (cv g) R (cv z)))) eqid zornlem.6 a zornlem2 (e. (cv a) (On)) adantll (` F (cv a)) (cv r) (` F (cv b)) (cv s) R breq12 biimpcd syl6 com23 (-. (= ({e.|} z A (A.e. g (" F (cv a)) (br (cv g) R (cv z)))) ({/}))) adantrrl imp (` F (cv a)) (cv r) (` F (cv b)) (cv s) eqeq12 (cv a) (cv b) F fveq2 syl5bi (/\ (/\ (e. (cv a) (On)) (e. (cv b) (On))) (/\ (We (cv w) A) (/\ (-. (= ({e.|} z A (A.e. g (" F (cv a)) (br (cv g) R (cv z)))) ({/}))) (-. (= ({e.|} z A (A.e. g (" F (cv b)) (br (cv g) R (cv z)))) ({/})))))) adantl zornlem.1 zornlem.2 zornlem.3 zornlem.4 ({e.|} z A (A.e. g (" F (cv a)) (br (cv g) R (cv z)))) eqid zornlem.6 b zornlem2 (e. (cv b) (On)) adantlr (` F (cv b)) (cv s) (` F (cv a)) (cv r) R breq12 ancoms biimpcd syl6 com23 (-. (= ({e.|} z A (A.e. g (" F (cv b)) (br (cv g) R (cv z)))) ({/}))) adantrrr imp 3orim123d (cv a) (cv b) ordtri3or (cv a) eloni (cv b) eloni syl2an syl5 exp31 com4r pm2.43i syl6 exp4a com3r imp a2d zornlem.7 ({/}) eqeq1i negbii y (cv x) ralbii (cv y) (cv a) F imaeq2 g (br (cv g) R (cv z)) raleq1d z A rabbisdv ({/}) eqeq1d negbid (cv x) rcla4cv (cv y) (cv b) F imaeq2 g (br (cv g) R (cv z)) raleq1d z A rabbisdv ({/}) eqeq1d negbid (cv x) rcla4cv anim12d sylbi syl5 imp4b a b 19.23advv zornlem.2 zornlem.3 tfr1 F (On) fnfun ax-mp F (cv r) (cv x) a fvelima a (cv x) (= (` F (cv a)) (cv r)) df-rex sylib ex F (cv s) (cv x) b fvelima b (cv x) (= (` F (cv b)) (cv s)) df-rex sylib ex anim12d ax-mp (e. (cv a) (cv x)) (e. (cv b) (cv x)) (= (` F (cv a)) (cv r)) (= (` F (cv b)) (cv s)) an4 a b 2exbii a b (/\ (e. (cv a) (cv x)) (= (` F (cv a)) (cv r))) (/\ (e. (cv b) (cv x)) (= (` F (cv b)) (cv s))) eeanv bitr sylibr syl5 r s 19.21aivv r (" F (cv x)) s (" F (cv x)) (\/\/ (br (cv r) R (cv s)) (= (cv r) (cv s)) (br (cv s) R (cv r))) r2al sylibr (Po R A) a1i jcad R (" F (cv x)) r s df-so syl6ibr)) thm (zornlem7 ((u x) (v x) (f x) (t x) (s x) (r x) (a x) (b x) (H x) (u v) (f u) (t u) (s u) (r u) (a u) (b u) (H u) (f v) (t v) (s v) (r v) (a v) (b v) (H v) (f t) (f s) (f r) (a f) (b f) (H f) (s t) (r t) (a t) (b t) (H t) (r s) (a s) (b s) (H s) (a r) (b r) (H r) (a b) (H a) (H b) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (r x) (s x) (a x) (b x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (r y) (s y) (a y) (b y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (r w) (s w) (a w) (b w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h r) (h s) (a h) (b h) (A h) (t z) (f t) (g t) (t u) (t v) (r t) (s t) (a t) (b t) (A t) (f z) (g z) (u z) (v z) (r z) (s z) (a z) (b z) (A z) (f g) (f u) (f v) (f r) (f s) (a f) (b f) (A f) (g u) (g v) (g r) (g s) (a g) (b g) (A g) (u v) (r u) (s u) (a u) (b u) (A u) (r v) (s v) (a v) (b v) (A v) (r s) (a r) (b r) (A r) (a s) (b s) (A s) (a b) (A a) (A b) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F r) (F s) (F a) (F b) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (D a) (D b) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s) (R r) (R a) (R b)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (zornlem.7 (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z))))))) (-> (/\ (Po R A) (A. s (-> (/\ (C_ (cv s) A) (Or R (cv s))) (E.e. a A (A.e. r (cv s) (\/ (br (cv r) R (cv a)) (= (cv r) (cv a)))))))) (E.e. a A (A.e. b A (-. (br (cv a) R (cv b)))))) (zornlem.1 w weth zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem4 (cv x) (cv y) F imaeq2 g (br (cv g) R (cv z)) raleq1d z A rabbisdv zornlem.5 zornlem.7 3eqtr4g ({/}) eqeq1d onminex zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem.7 zornlem5 (Po R A) a1i zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem.7 zornlem6 jcad zornlem.2 zornlem.3 tfrlem7 x visset F funimaex ax-mp (cv s) (" F (cv x)) A sseq1 (cv s) (" F (cv x)) R soeq2 anbi12d (cv s) (" F (cv x)) r (\/ (br (cv r) R (cv a)) (= (cv r) (cv a))) raleq1 a A rexbidv imbi12d cla4v sylan9 (= D ({/})) adantld imp (cv b) noel zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem.7 zornlem5 (cv r) sseld R A (cv r) (cv a) (cv b) potr (e. (cv r) A) (e. (cv a) A) (e. (cv b) A) 3anass sylan2br exp3a com23 imp (cv r) (cv a) R (cv b) breq1 biimprcd (/\ (Po R A) (/\ (e. (cv r) A) (/\ (e. (cv a) A) (e. (cv b) A)))) adantl jaod exp42 sylan9r com24 com23 imp31 a2d r 19.20dv r (" F (cv x)) (\/ (br (cv r) R (cv a)) (= (cv r) (cv a))) df-ral r (" F (cv x)) (br (cv r) R (cv b)) df-ral 3imtr4g zornlem.5 ({/}) eqeq1i ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))) ({/}) (cv b) eleq2 sylbi (cv z) (cv b) (cv g) R breq2 g (" F (cv x)) ralbidv A elrab syl5bbr biimpd exp3a imp (cv r) (cv g) R (cv b) breq1 (" F (cv x)) cbvralv syl5ib sylan9r exp32 com34 imp31 mtoi exp42 exp4a com34 ex com4r pm2.43a imp3a com4l imp3a b 19.21adv b A (-. (br (cv a) R (cv b))) df-ral syl6ibr exp3a imdistand a 19.22dv a A (A.e. r (" F (cv x)) (\/ (br (cv r) R (cv a)) (= (cv r) (cv a)))) df-rex a A (A.e. b A (-. (br (cv a) R (cv b)))) df-rex 3imtr4g exp32 com12 (A. s (-> (/\ (C_ (cv s) A) (Or R (cv s))) (E.e. a A (A.e. r (cv s) (\/ (br (cv r) R (cv a)) (= (cv r) (cv a))))))) adantr imp32 mpd exp45 com23 exp3a imp imp4a com3l r19.23aiv 3syl (A. s (-> (/\ (C_ (cv s) A) (Or R (cv s))) (E.e. a A (A.e. r (cv s) (\/ (br (cv r) R (cv a)) (= (cv r) (cv a))))))) adantlr pm2.43i ex w 19.23adv mpi)) thm (zorn2lem ((x y) (x z) (y z) (w x) (w y)) () (<-> (br (cv z) ({<,>|} x y (C: (cv x) (cv y))) (cv w)) (C: (cv z) (cv w))) (z visset w visset (cv x) (cv z) (cv y) psseq1 (cv y) (cv w) (cv z) psseq2 ({<,>|} x y (C: (cv x) (cv y))) eqid brab)) var (set q) thm (zorn ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (s x) (r x) (q x) (a x) (b x) (c x) (d x) (j x) (k x) (m x) (n x) (R x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (s y) (r y) (q y) (a y) (b y) (c y) (d y) (j y) (k y) (m y) (n y) (R y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (s z) (r z) (q z) (a z) (b z) (c z) (d z) (j z) (k z) (m z) (n z) (R z) (v w) (u w) (f w) (g w) (h w) (t w) (s w) (r w) (q w) (a w) (b w) (c w) (d w) (j w) (k w) (m w) (n w) (R w) (u v) (f v) (g v) (h v) (t v) (s v) (r v) (q v) (a v) (b v) (c v) (d v) (j v) (k v) (m v) (n v) (R v) (f u) (g u) (h u) (t u) (s u) (r u) (q u) (a u) (b u) (c u) (d u) (j u) (k u) (m u) (n u) (R u) (f g) (f h) (f t) (f s) (f r) (f q) (a f) (b f) (c f) (d f) (f j) (f k) (f m) (f n) (R f) (g h) (g t) (g s) (g r) (g q) (a g) (b g) (c g) (d g) (g j) (g k) (g m) (g n) (R g) (h t) (h s) (h r) (h q) (a h) (b h) (c h) (d h) (h j) (h k) (h m) (h n) (R h) (s t) (r t) (q t) (a t) (b t) (c t) (d t) (j t) (k t) (m t) (n t) (R t) (r s) (q s) (a s) (b s) (c s) (d s) (j s) (k s) (m s) (n s) (R s) (q r) (a r) (b r) (c r) (d r) (j r) (k r) (m r) (n r) (R r) (a q) (b q) (c q) (d q) (j q) (k q) (m q) (n q) (R q) (a b) (a c) (a d) (a j) (a k) (a m) (a n) (R a) (b c) (b d) (b j) (b k) (b m) (b n) (R b) (c d) (c j) (c k) (c m) (c n) (R c) (d j) (d k) (d m) (d n) (R d) (j k) (j m) (j n) (R j) (k m) (k n) (R k) (m n) (R m) (R n) (A x) (A y) (A z) (A w) (A v) (A u) (A f) (A g) (A h) (A t) (A s) (A r) (A q) (A a) (A b) (A c) (A d) (A j) (A k) (A m) (A n)) ((zorn.1 (e. A (V)))) (-> (/\ (Po R A) (A. w (-> (/\ (C_ (cv w) A) (Or R (cv w))) (E.e. x A (A.e. z (cv w) (\/ (br (cv z) R (cv x)) (= (cv z) (cv x)))))))) (E.e. x A (A.e. y A (-. (br (cv x) R (cv y)))))) (zorn.1 a b c ({<,>|} h k (= (cv k) (U. ({e.|} m ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) (A.e. k ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) (-. (br (cv k) (cv q) (cv m)))))))) d f g rdglem1 (U. ({|} a (E.e. b (On) (/\ (Fn (cv a) (cv b)) (A.e. c (cv b) (= (` (cv a) (cv c)) (` ({<,>|} h k (= (cv k) (U. ({e.|} m ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) (A.e. k ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) (-. (br (cv k) (cv q) (cv m)))))))) (|` (cv a) (cv c))))))))) eqid (cv v) (cv r) (cv s) R breq2 s (ran (cv d)) ralbidv (cv q) (cv s) R (cv v) breq1 (ran (cv d)) cbvralv syl5bb A cbvrabv ({e.|} r A (A.e. s (" (U. ({|} a (E.e. b (On) (/\ (Fn (cv a) (cv b)) (A.e. c (cv b) (= (` (cv a) (cv c)) (` ({<,>|} h k (= (cv k) (U. ({e.|} m ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) (A.e. k ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) (-. (br (cv k) (cv q) (cv m)))))))) (|` (cv a) (cv c))))))))) (cv t)) (br (cv s) R (cv r)))) eqid (= (cv k) (cv g)) id (cv h) (cv d) rneq q (br (cv q) R (cv v)) raleq1d v A rabbisdv (cv n) eleq2d (cv h) (cv d) rneq q (br (cv q) R (cv v)) raleq1d v A rabbisdv ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) ({e.|} v A (A.e. q (ran (cv d)) (br (cv q) R (cv v)))) j (-. (br (cv j) (cv q) (cv n))) raleq1 (cv k) (cv j) (cv q) (cv n) breq1 negbid ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) cbvralv syl5bb syl anbi12d n abbidv (cv m) (cv n) ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) eleq1 (cv m) (cv n) (cv k) (cv q) breq2 negbid k ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) ralbidv anbi12d cbvabv syl5eq m ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) (A.e. k ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) (-. (br (cv k) (cv q) (cv m)))) df-rab n ({e.|} v A (A.e. q (ran (cv d)) (br (cv q) R (cv v)))) (A.e. j ({e.|} v A (A.e. q (ran (cv d)) (br (cv q) R (cv v)))) (-. (br (cv j) (cv q) (cv n)))) df-rab 3eqtr4g unieqd eqeqan12rd cbvopabv ({e.|} r A (A.e. s (" (U. ({|} a (E.e. b (On) (/\ (Fn (cv a) (cv b)) (A.e. c (cv b) (= (` (cv a) (cv c)) (` ({<,>|} h k (= (cv k) (U. ({e.|} m ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) (A.e. k ({e.|} v A (A.e. q (ran (cv h)) (br (cv q) R (cv v)))) (-. (br (cv k) (cv q) (cv m)))))))) (|` (cv a) (cv c))))))))) (cv u)) (br (cv s) R (cv r)))) eqid w x z y zornlem7)) thm (zorn2 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A x) (A y) (A z) (A w) (A v) (A u)) ((zorn.1 (e. A (V)))) (-> (A. z (-> (/\ (C_ (cv z) A) (A.e. x (cv z) (A.e. y (cv z) (\/ (C_ (cv x) (cv y)) (C_ (cv y) (cv x)))))) (e. (U. (cv z)) A))) (E.e. x A (A.e. y A (-. (C: (cv x) (cv y)))))) (({<,>|} w v (C: (cv w) (cv v))) (cv z) x y df-so pm3.27bd x w v y zorn2lem (= (cv x) (cv y)) pm4.2 y w v x zorn2lem 3orbi123i (cv x) (cv y) sspsstri bitr4 x (cv z) y (cv z) 2ralbii sylib (C_ (cv z) A) anim2i (U. (cv z)) A x risset (cv x) (U. (cv z)) eqimss2 (cv z) (cv x) u unissb sylib u w v x zorn2lem (= (cv u) (cv x)) orbi1i (cv u) (cv x) sspss bitr4 u (cv z) ralbii sylibr x A r19.22si sylbi imim12i z 19.20i (cv u) pssirr u w v u zorn2lem mtbir (cv u) (cv y) (cv x) psstr u w v x zorn2lem sylibr u w v y zorn2lem y w v x zorn2lem syl2anb pm3.2i (/\/\ (e. (cv u) A) (e. (cv y) A) (e. (cv x) A)) a1i rgen3 ({<,>|} w v (C: (cv w) (cv v))) A u y x df-po mpbir zorn.1 ({<,>|} w v (C: (cv w) (cv v))) z x u y zorn mpan syl x w v y zorn2lem negbii y A ralbii x A rexbii sylib)) thm (fodom ((F f) (A f) (B f)) ((fodom.1 (e. A (V)))) (-> (:-onto-> F A B) (br B (~<_) A)) (F A B fof fodom.1 F A B (V) fex mpan2 syl F (V) cnvexg (`' F) (V) f ac7g 3syl F A B forn F df-rn syl5eqr (dom (`' F)) B (cv f) fneq2 syl B (ran (cv f)) A domtr (cv f) B fnfrn biimp (:-onto-> F A B) (C_ (cv f) (`' F)) ad2antlr (`' (cv f)) F funss impcom F A B fof F A B ffun syl (cv f) (`' F) cnvss F cnvcnvss (`' (cv f)) (`' (`' F)) F sstr mpan2 syl syl2an (Fn (cv f) B) adantlr jca (cv f) B (ran (cv f)) df-f1 sylibr f visset (cv f) (V) rnexg ax-mp (ran (cv f)) (V) (cv f) B f1dom2g ax-mp syl (cv f) (`' F) rnss (:-onto-> F A B) adantl F A B fof F A B fdm syl F dfdm4 syl5eqr (C_ (cv f) (`' F)) adantr sseqtrd f visset (cv f) (V) rnexg ax-mp (ran (cv f)) (V) A ssdomg ax-mp syl (Fn (cv f) B) adantlr sylanc exp31 sylbid com23 imp3a f 19.23adv mpd)) thm (fodomg ((A x) (B x) (F x)) () (-> (e. A C) (-> (:-onto-> F A B) (br B (~<_) A))) ((cv x) A F B foeq2 (cv x) A B (~<_) breq2 imbi12d x visset F B fodom C vtoclg)) thm (fodomb ((A f) (B f)) ((fodomb.1 (e. A (V)))) (<-> (/\ (-. (= A ({/}))) (E. f (:-onto-> (cv f) A B))) (/\ (br ({/}) (~<) B) (br B (~<_) A))) ((cv f) A B fof (cv f) A B fdm syl ({/}) eqeq1d (cv f) A B forn ({/}) eqeq1d (cv f) dm0rn0 syl5bb bitr3d negbid biimpac fodomb.1 A (V) (cv f) B fornex ax-mp B (V) 0sdomg syl (-. (= A ({/}))) adantl mpbird ex fodomb.1 (cv f) B fodom (-. (= A ({/}))) a1i jcad f 19.23adv imp fodomb.1 A (V) ({/}) B sdomdomtr ax-mp fodomb.1 0sdom sylib fodomb.1 A (V) B f fodomr mp3an1 jca impbi)) thm (imadomg () () (-> (e. A B) (-> (Fun F) (br (" F A) (~<_) A))) ((dom (|` F A)) (V) (|` F A) (ran (|` F A)) fodomg F A B resfunexg (|` F A) (V) dmexg syl F A funres (|` F A) funforn sylib (e. A B) adantr sylc F A df-ima syl5eqbr expcom (" F A) (dom (|` F A)) A domtr F A dmres A (dom F) inss1 eqsstr A B (dom (|` F A)) ssdom2g mpi sylan2 expcom syld)) thm (fnrndomg () () (-> (e. A B) (-> (Fn F A) (br (ran F) (~<_) A))) (A B F (ran F) fodomg F A fnforn syl5ib)) thm (unidom ((f g) (f h) (f x) (f y) (f z) (A f) (g h) (g x) (g y) (g z) (A g) (h x) (h y) (h z) (A h) (x y) (x z) (A x) (y z) (A y) (A z) (B f) (B g) (B h) (B x) (B y) (B z)) ((unidom.1 (e. A (V))) (unidom.2 (e. B (V)))) (-> (A.e. x A (br (cv x) (~<_) B)) (br (U. A) (~<_) (X. A B))) (unidom.1 uniex (= (cv x) (cv y)) id (cv x) (cv y) (cv h) fveq2 eleq12d (cv x) (cv y) (cv h) fveq2 A eleq1d anbi12d (U. A) rcla4cv (e. (cv y) (` (cv h) (cv y))) (e. (` (cv h) (cv y)) A) pm3.27 syl6 (A.e. x A (:-1-1-> (` (cv f) (cv x)) (cv x) B)) adantl (= (cv x) (cv y)) id (cv x) (cv y) (cv h) fveq2 eleq12d (cv x) (cv y) (cv h) fveq2 A eleq1d anbi12d (U. A) rcla4cv (cv x) (` (cv h) (cv y)) (cv f) fveq2 (` (cv f) (cv x)) (` (cv f) (` (cv h) (cv y))) (cv x) B f1eq1 syl (cv x) (` (cv h) (cv y)) (` (cv f) (` (cv h) (cv y))) B f1eq2 bitrd A rcla4v (` (cv f) (` (cv h) (cv y))) (` (cv h) (cv y)) B f1f syl6 (` (cv f) (` (cv h) (cv y))) (` (cv h) (cv y)) B (cv y) ffvrn expcom sylan9r syl6 com3r imp jcad (` (cv h) (cv y)) A (` (` (cv f) (` (cv h) (cv y))) (cv y)) B opelxpi syl6 (` (cv h) (cv y)) (` (cv h) (cv z)) (cv f) fveq2 (cv z) fveq1d (` (` (cv f) (` (cv h) (cv y))) (cv y)) eqeq2d (/\ (/\ (A.e. x A (:-1-1-> (` (cv f) (cv x)) (cv x) B)) (A.e. x (U. A) (/\ (e. (cv x) (` (cv h) (cv x))) (e. (` (cv h) (cv x)) A)))) (/\ (e. (cv y) (U. A)) (e. (cv z) (U. A)))) adantl (` (cv f) (` (cv h) (cv y))) (` (cv h) (cv y)) B (cv y) (cv z) f1fveq biimpd (= (cv x) (cv y)) id (cv x) (cv y) (cv h) fveq2 eleq12d (cv x) (cv y) (cv h) fveq2 A eleq1d anbi12d (U. A) rcla4cv (cv x) (` (cv h) (cv y)) (cv f) fveq2 (` (cv f) (cv x)) (` (cv f) (` (cv h) (cv y))) (cv x) B f1eq1 syl (cv x) (` (cv h) (cv y)) (` (cv f) (` (cv h) (cv y))) B f1eq2 bitrd A rcla4v (e. (cv y) (` (cv h) (cv y))) adantl syl6 com3r imp31 (e. (cv z) (U. A)) adantrr (= (` (cv h) (cv y)) (` (cv h) (cv z))) adantr (= (cv x) (cv y)) id (cv x) (cv y) (cv h) fveq2 eleq12d (cv x) (cv y) (cv h) fveq2 A eleq1d anbi12d (U. A) rcla4cv (e. (cv y) (` (cv h) (cv y))) (e. (` (cv h) (cv y)) A) pm3.26 syl6 (= (` (cv h) (cv y)) (` (cv h) (cv z))) adantr (= (cv x) (cv z)) id (cv x) (cv z) (cv h) fveq2 eleq12d (cv x) (cv z) (cv h) fveq2 A eleq1d anbi12d (U. A) rcla4cv (e. (cv z) (` (cv h) (cv z))) (e. (` (cv h) (cv z)) A) pm3.26 syl6 (` (cv h) (cv y)) (` (cv h) (cv z)) (cv z) eleq2 biimprd sylan9 anim12d imp an1rs (A.e. x A (:-1-1-> (` (cv f) (cv x)) (cv x) B)) adantlll sylanc sylbird ex imp3a (cv h) (cv y) fvex (` (cv f) (` (cv h) (cv y))) (cv y) fvex (` (cv f) (` (cv h) (cv z))) (cv z) fvex (` (cv h) (cv z)) opth syl5ib (` (cv h) (cv y)) (` (cv h) (cv z)) (` (` (cv f) (` (cv h) (cv y))) (cv y)) (` (` (cv f) (` (cv h) (cv z))) (cv z)) opeq12 (cv y) (cv z) (cv h) fveq2 (cv y) (cv z) (cv h) fveq2 (cv f) fveq2d (cv y) fveq1d (cv y) (cv z) (` (cv f) (` (cv h) (cv z))) fveq2 eqtrd sylanc (/\ (/\ (A.e. x A (:-1-1-> (` (cv f) (cv x)) (cv x) B)) (A.e. x (U. A) (/\ (e. (cv x) (` (cv h) (cv x))) (e. (` (cv h) (cv x)) A)))) (/\ (e. (cv y) (U. A)) (e. (cv z) (U. A)))) a1i impbid ex (V) dom2d mpi ex h 19.23adv f 19.23aiv unidom.2 (cv x) g brdom x A ralbii unidom.1 (cv g) (` (cv f) (cv x)) (cv x) B f1eq1 ac6s3 sylbi unidom.1 uniex (cv g) (` (cv h) (cv x)) (cv x) eleq2 (cv g) (` (cv h) (cv x)) A eleq1 anbi12d ac6s3 (cv x) A g eluni biimp mprg (A.e. x A (br (cv x) (~<_) B)) a1i sylc)) thm (unidomg ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B z)) () (-> (/\/\ (e. A C) (e. B D) (A.e. x A (br (cv x) (~<_) B))) (br (U. A) (~<_) (X. A B))) ((cv y) A x (br (cv x) (~<_) (cv z)) raleq1 (cv y) A unieq (cv y) A (cv z) xpeq1 (~<_) breq12d imbi12d (cv z) B (cv x) (~<_) breq2 x A ralbidv (cv z) B A xpeq2 (U. A) (~<_) breq2d imbi12d y visset z visset x unidom C D vtocl2g 3impia)) thm (uniimadom ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) ((uniimadom.1 (e. A (V))) (uniimadom.2 (e. B (V)))) (-> (/\ (Fun F) (A.e. x A (br (` F (cv x)) (~<_) B))) (br (U. (" F A)) (~<_) (X. A B))) ((U. (" F A)) (X. (" F A) B) (X. A B) domtr uniimadom.2 (" F A) (V) B (V) y unidomg mp3an2 uniimadom.1 F funimaex (A.e. x A (br (` F (cv x)) (~<_) B)) adantr F (cv y) A x fvelima ex (` F (cv x)) (cv y) (~<_) B breq1 biimpd x A r19.22si x A (br (` F (cv x)) (~<_) B) (br (cv y) (~<_) B) r19.36av syl syl6 com23 imp r19.21aiv sylanc uniimadom.1 A (V) F imadomg ax-mp uniimadom.1 uniimadom.2 (" F A) xpdom1 syl (A.e. x A (br (` F (cv x)) (~<_) B)) adantr sylanc)) thm (uniimadomf ((x z) (A x) (A z) (x y) (B x) (y z) (B y) (B z) (F y) (F z)) ((uniimadomf.1 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (uniimadomf.2 (e. A (V))) (uniimadomf.3 (e. B (V)))) (-> (/\ (Fun F) (A.e. x A (br (` F (cv x)) (~<_) B))) (br (U. (" F A)) (~<_) (X. A B))) (uniimadomf.2 uniimadomf.3 F z uniimadom (br (` F (cv x)) (~<_) B) z ax-17 uniimadomf.1 (e. (cv y) (cv z)) x ax-17 hbfv (e. (cv y) (~<_)) x ax-17 (e. (cv y) B) x ax-17 hbbr (cv x) (cv z) F fveq2 (~<_) B breq1d A cbvral sylan2b)) thm (iundom ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B z) (C y) (C z)) ((iundom.1 (e. A (V))) (iundom.2 (e. B (V))) (iundom.3 (e. C (V)))) (-> (A.e. x A (br C (~<_) B)) (br (U_ x A C) (~<_) (X. A B))) (iundom.3 x A C (V) y fvopab2 mpan2 (~<_) B breq1d ralbiia iundom.3 ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) eqid fnopab2 ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) A fnfun ax-mp z x y (/\ (e. (cv x) A) (= (cv y) C)) hbopab1 iundom.1 iundom.2 uniimadomf mpan iundom.3 x A C (V) y fvopab2 mpan2 iuneq2i iundom.3 ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) eqid fnopab2 ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))) A fnfun ax-mp z x y (/\ (e. (cv x) A) (= (cv y) C)) hbopab1 A funiunfvf ax-mp eqtr3 syl5eqbr sylbir)) thm (oncardval ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (On)) (= (` (card) A) (|^| ({e.|} x (On) (br (cv x) (~~) A))))) (A (On) enrefg (cv x) A (~~) A breq1 (On) rcla4ev mpdan x (On) (br (cv x) (~~) A) rabn0 sylibr x (On) (br (cv x) (~~) A) ssrab2 ({e.|} x (On) (br (cv x) (~~) A)) oninton mpan syl (cv y) A (cv x) (~~) breq2 x (On) rabbisdv inteqd (On) (On) z fvopabg mpdan y z x df-card A fveq1i syl5eq)) thm (oncardon ((A x)) () (-> (e. A (On)) (e. (` (card) A) (On))) (A x oncardval A x oncardval (card) A fvex syl6eqelr ({e.|} x (On) (br (cv x) (~~) A)) intex sylibr x (On) (br (cv x) (~~) A) ssrab2 ({e.|} x (On) (br (cv x) (~~) A)) oninton mpan syl eqeltrd)) thm (oncardid ((A x)) () (-> (e. A (On)) (br (` (card) A) (~~) A)) (A x oncardval A x oncardval (card) A fvex syl6eqelr ({e.|} x (On) (br (cv x) (~~) A)) intex sylibr x (On) (br (cv x) (~~) A) ssrab2 ({e.|} x (On) (br (cv x) (~~) A)) onint mpan syl eqeltrd (cv x) (` (card) A) (~~) A breq1 (On) elrab pm3.27bd syl)) thm (cardonle ((A x)) () (-> (e. A (On)) (C_ (` (card) A) A)) (A x oncardval A (On) enrefg (cv x) A (~~) A breq1 (On) elrab A ({e.|} x (On) (br (cv x) (~~) A)) intss1 sylbir mpdan eqsstrd)) thm (card0 () () (= (` (card) ({/})) ({/})) (0elon ({/}) cardonle ax-mp (` (card) ({/})) ss0b mpbi)) thm (cardnn () () (-> (e. A (om)) (= (` (card) A) A)) ((` (card) A) A sdomnen A nnont A oncardid syl nsyl3 A nnont A oncardon (` (card) A) A ordelpss (` (card) A) eloni A eloni syl2an mpancom syl A nnont A cardonle A oncardon (` (card) A) A onsseleq mpancom mpbid syl (` (card) A) A elnn expcom A (om) (` (card) A) eleq1a jaod mpd (` (card) A) A nnsdomo mpancom bitr4d mtbird A nnont A cardonle A oncardon (` (card) A) A onsseleq mpancom mpbid syl ord mpd)) thm (cardom () () (= (` (card) (om)) (om)) (omelon (om) oncardid ax-mp (` (card) (om)) nnsdom (` (card) (om)) (om) sdomnen syl mt2 omelon (om) cardonle ax-mp omelon (om) oncardon ax-mp omelon onssel mpbi ori ax-mp)) thm (cardval ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (` (card) A) (|^| ({e.|} x (On) (br (cv x) (~~) A)))) (x A numth2 x (On) (br (cv x) (~~) A) intexrab mpbi (cv y) A (cv x) (~~) breq2 x (On) rabbisdv inteqd (V) (V) z fvopabg mpan2 y z x df-card A fveq1i syl5eq A (card) fvprc x visset enref A (cv x) (~~) brprc mpbiri (e. (cv x) (On)) biantrud x abbidv x (On) (br (cv x) (~~) A) df-rab syl6reqr x (On) abid2 syl6eq inteqd inton syl6eq eqtr4d pm2.61i)) thm (cardon ((A x)) () (e. (` (card) A) (On)) (A x cardval x (On) (br (cv x) (~~) A) ssrab2 A x cardval (card) A fvex eqeltrr ({e.|} x (On) (br (cv x) (~~) A)) intex mpbir ({e.|} x (On) (br (cv x) (~~) A)) oninton mp2an eqeltr)) thm (cardid ((A x)) () (br (` (card) A) (~~) A) (A x cardval x (On) (br (cv x) (~~) A) ssrab2 A x cardval (card) A fvex eqeltrr ({e.|} x (On) (br (cv x) (~~) A)) intex mpbir ({e.|} x (On) (br (cv x) (~~) A)) onint mp2an eqeltr (cv x) (` (card) A) (~~) A breq1 (On) elrab pm3.27bd ax-mp)) thm (oncard ((x y) (A x) (A y)) () (<-> (E. x (= A (` (card) (cv x)))) (= A (` (card) A))) ((cv x) cardid A (` (card) (cv x)) (~~) (cv x) breq1 mpbiri A cardid (` (card) A) A (cv x) entrt mpan A cardon (cv y) (` (card) A) (~~) (cv x) breq1 onintss ax-mp (cv x) y cardval syl5ss 3syl A (` (card) (cv x)) (` (card) A) sseq1 mpbird (cv x) cardon A (` (card) (cv x)) (On) eleq1 mpbiri A cardonle syl eqssd x 19.23aiv (card) A fvex A (` (card) A) (V) eleq1 mpbiri (cv x) A (card) fveq2 A eqeq2d (V) cla4egv mpcom impbi)) thm (cardne ((A x) (B x)) () (-> (e. A (` (card) B)) (-. (br A (~~) B))) (B cardon A onel (cv x) A (~~) B breq1 onintss B x cardval A sseq1i syl6ibr B cardon (` (card) B) A ontri1 mpan sylibd con2d mpcom)) thm (carden () () (-> (/\ (e. A C) (e. B D)) (<-> (= (` (card) A) (` (card) B)) (br A (~~) B))) ((` (card) A) (` (card) B) A (~~) breq2 B cardid A (` (card) B) B entrt mpan2 syl6bi A cardid A C (` (card) A) ensymg mpi syl5com (e. B D) adantr B D A ensymg B cardid (` (card) B) B A entrt mpan (` (card) B) A cardne con2i A cardon B cardon (` (card) A) (` (card) B) ontri1 mp2an sylibr syl syl6 A cardid (` (card) A) A B entrt mpan (` (card) A) B cardne con2i B cardon A cardon (` (card) B) (` (card) A) ontri1 mp2an sylibr syl (e. B D) a1i jcad (` (card) A) (` (card) B) eqss syl6ibr (e. A C) adantl impbid)) thm (cardeq0 () () (-> (e. A B) (<-> (= (` (card) A) ({/})) (= A ({/})))) (0ex A B ({/}) (V) carden mpan2 card0 (` (card) A) eqeq2i A en0 3bitr3g)) thm (cardsn () () (-> (e. A B) (= (` (card) ({} A)) (1o))) (A B ensn1g A snex 1onn ({} A) (V) (1o) (om) carden mp2an sylibr 1onn (1o) cardnn ax-mp syl6eq)) thm (carddomi () () (-> (e. A C) (-> (C_ (` (card) A) (` (card) B)) (br A (~<_) B))) (A (` (card) A) (` (card) B) endomtr A cardid A C (` (card) A) ensymg mpi A cardon (` (card) A) (On) (` (card) B) ssdomg ax-mp syl2an B cardid A (` (card) B) B domentr mpan2 syl ex)) thm (carddom () () (-> (/\ (e. A C) (e. B D)) (<-> (C_ (` (card) A) (` (card) B)) (br A (~<_) B))) (A C B carddomi (e. B D) adantr B D A carddomi A cardon (` (card) B) onelss syl5 B A domnsym syl6 con2d A cardon B cardon (` (card) A) (` (card) B) ontri1 mp2an syl6ibr (e. A C) adantl A C B D carden (` (card) A) (` (card) B) eqimss syl6bir jaod A B brdom2 syl5ib impbid)) thm (cardsdom () () (-> (/\ (e. A C) (e. B D)) (<-> (e. (` (card) A) (` (card) B)) (br A (~<) B))) (A C B D carddom A C B D carden negbid anbi12d A cardon B cardon (` (card) A) (` (card) B) onelpsst mp2an A B brsdom 3bitr4g)) thm (domtri () () (-> (/\ (e. A C) (e. B D)) (<-> (br A (~<_) B) (-. (br B (~<) A)))) (A C B D carddom B D A C cardsdom ancoms negbid A cardon B cardon (` (card) A) (` (card) B) ontri1 mp2an syl5bb bitr3d)) thm (entri () () (-> (/\ (e. A C) (e. B D)) (\/\/ (br A (~<) B) (br A (~~) B) (br B (~<) A))) (A C B D domtri biimprd A B brdom2 syl6ib con1d orrd (br A (~<) B) (br A (~~) B) (br B (~<) A) df-3or sylibr)) thm (entri2 () () (-> (/\ (e. A C) (e. B D)) (\/ (br A (~<_) B) (br B (~<) A))) (A C B D entri A B brdom2 (br B (~<) A) orbi1i (br A (~<) B) (br A (~~) B) (br B (~<) A) df-3or bitr4 sylibr)) thm (entri3 () () (-> (/\ (e. A C) (e. B D)) (\/ (br A (~<_) B) (br B (~<_) A))) (A C B D entri2 B A sdomdom (br A (~<_) B) orim2i syl)) thm (sucdom ((A x) (B x)) () (-> (/\ (e. A (om)) (e. B C)) (<-> (br A (~<) B) (br (suc A) (~<_) B))) (omex (om) (V) B C entri2 mpan (e. A (om)) adantl B C A (om) sdomdomtr exp3a imp A nnsdom sylan2 ancoms A peano2b (suc A) nnsdom sylbi (suc A) (om) sdomdom (suc A) (om) B domtr ex 3syl (e. B C) adantr jcad (br A (~<) B) (br (suc A) (~<_) B) pm5.1 syl6 A x B finsucdom ex B x isfinite2 syl5 (e. B C) adantr jaod mpd)) thm (unxpdomlem ((x y) (x z) (w x) (v x) (u x) (t x) (f x) (A x) (y z) (w y) (v y) (u y) (t y) (f y) (A y) (w z) (v z) (u z) (t z) (f z) (A z) (v w) (u w) (t w) (f w) (A w) (u v) (t v) (f v) (A v) (t u) (f u) (A u) (f t) (A t) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B f)) ((unxpdomlem.1 (e. A (V))) (unxpdomlem.2 (e. B (V)))) (-> (/\ (br (1o) (~<) A) (br (1o) (~<) B)) (br (u. A B) (~<_) (X. A B))) ((cv x) (cv f) (cv w) eqeq1 (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv y) (cv v)) (cv v) (cv x)) ifbid (cv x) (cv f) (= (cv y) (cv v)) (cv v) ifeq2 (= (cv f) (cv w)) (if (= (cv y) (cv v)) (cv w) (cv y)) ifeq2d eqtrd (cv f) eqeq1d (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv v) (cv v)) (cv w) (cv v)) (if (= (cv y) (cv v)) (cv v) (cv f)) (if (= (cv v) (cv v)) (cv v) (cv f)) (= (cv f) (cv w)) ifeq12 (cv y) (cv v) (cv v) eqeq1 (cv w) (cv y) ifbid (cv y) (cv v) (= (cv v) (cv v)) (cv w) ifeq2 eqtrd (cv y) (cv v) (cv v) eqeq1 (cv v) (cv f) ifbid sylanc (cv f) eqeq1d A B rcla42ev (= (cv f) (cv w)) (if (= (cv v) (cv v)) (cv w) (cv v)) (if (= (cv v) (cv v)) (cv v) (cv f)) iftrue (cv v) eqid (= (cv v) (cv v)) (cv w) (cv v) iftrue ax-mp syl6eq (= (cv f) (cv w)) id eqtr4d sylan2 exp31 com3l (e. (cv t) B) (-. (= (cv t) (cv v))) ad2antlr (cv x) (cv f) (cv w) eqeq1 (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv y) (cv v)) (cv v) (cv x)) ifbid (cv x) (cv f) (= (cv y) (cv v)) (cv v) ifeq2 (= (cv f) (cv w)) (if (= (cv y) (cv v)) (cv w) (cv y)) ifeq2d eqtrd (cv f) eqeq1d (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv t) (cv v)) (cv w) (cv t)) (if (= (cv y) (cv v)) (cv v) (cv f)) (if (= (cv t) (cv v)) (cv v) (cv f)) (= (cv f) (cv w)) ifeq12 (cv y) (cv t) (cv v) eqeq1 (cv w) (cv y) ifbid (cv y) (cv t) (= (cv t) (cv v)) (cv w) ifeq2 eqtrd (cv y) (cv t) (cv v) eqeq1 (cv v) (cv f) ifbid sylanc (cv f) eqeq1d A B rcla42ev (= (cv f) (cv w)) (if (= (cv t) (cv v)) (cv w) (cv t)) (if (= (cv t) (cv v)) (cv v) (cv f)) iffalse (= (cv t) (cv v)) (cv v) (cv f) iffalse sylan9eqr sylan2 exp43 imp3a com3l (e. (cv v) B) adantlr pm2.61d (/\ (/\ (e. (cv u) A) (e. (cv w) A)) (-. (= (cv u) (cv w)))) adantl (cv x) (cv u) (cv w) eqeq1 (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv y) (cv v)) (cv v) (cv x)) ifbid (cv x) (cv u) (= (cv y) (cv v)) (cv v) ifeq2 (= (cv u) (cv w)) (if (= (cv y) (cv v)) (cv w) (cv y)) ifeq2d eqtrd (cv f) eqeq1d (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv f) (cv v)) (cv w) (cv f)) (if (= (cv y) (cv v)) (cv v) (cv u)) (if (= (cv f) (cv v)) (cv v) (cv u)) (= (cv u) (cv w)) ifeq12 (cv y) (cv f) (cv v) eqeq1 (cv w) (cv y) ifbid (cv y) (cv f) (= (cv f) (cv v)) (cv w) ifeq2 eqtrd (cv y) (cv f) (cv v) eqeq1 (cv v) (cv u) ifbid sylanc (cv f) eqeq1d A B rcla42ev (= (cv u) (cv w)) (if (= (cv f) (cv v)) (cv w) (cv f)) (if (= (cv f) (cv v)) (cv v) (cv u)) iffalse (= (cv f) (cv v)) (cv v) (cv u) iftrue f v equcomi eqtrd sylan9eqr sylan2 exp43 com24 imp (e. (cv w) A) adantlr (cv x) (cv w) (cv w) eqeq1 (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv y) (cv v)) (cv v) (cv x)) ifbid (cv x) (cv w) (= (cv y) (cv v)) (cv v) ifeq2 (= (cv w) (cv w)) (if (= (cv y) (cv v)) (cv w) (cv y)) ifeq2d eqtrd (cv f) eqeq1d (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv f) (cv v)) (cv w) (cv f)) (if (= (cv y) (cv v)) (cv v) (cv w)) (if (= (cv f) (cv v)) (cv v) (cv w)) (= (cv w) (cv w)) ifeq12 (cv y) (cv f) (cv v) eqeq1 (cv w) (cv y) ifbid (cv y) (cv f) (= (cv f) (cv v)) (cv w) ifeq2 eqtrd (cv y) (cv f) (cv v) eqeq1 (cv v) (cv w) ifbid sylanc (cv f) eqeq1d A B rcla42ev (= (cv f) (cv v)) (cv w) (cv f) iffalse (cv w) eqid (= (cv w) (cv w)) (if (= (cv f) (cv v)) (cv w) (cv f)) (if (= (cv f) (cv v)) (cv v) (cv w)) iftrue ax-mp syl5eq sylan2 exp31 com23 (e. (cv u) A) (-. (= (cv u) (cv w))) ad2antlr pm2.61d (/\ (/\ (e. (cv t) B) (e. (cv v) B)) (-. (= (cv t) (cv v)))) adantr jaod (cv f) A B elun syl5ib (if (= (cv x) (cv w)) (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv y) (cv v)) (cv v) (cv x))) (cv f) eqcom x A y B 2rexbii w visset y visset (= (cv y) (cv v)) ifex v visset x visset (= (cv y) (cv v)) ifex (= (cv x) (cv w)) ifex ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (if (= (cv x) (cv w)) (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv y) (cv v)) (cv v) (cv x)))))) eqid (cv f) elrnoprab bitr4 syl6ib ssrdv unxpdomlem.1 unxpdomlem.2 unex (u. A B) (V) (ran ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (if (= (cv x) (cv w)) (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv y) (cv v)) (cv v) (cv x))))))) ssdomg ax-mp unxpdomlem.1 unxpdomlem.2 xpex w visset y visset (= (cv y) (cv v)) ifex v visset x visset (= (cv y) (cv v)) ifex (= (cv x) (cv w)) ifex ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (if (= (cv x) (cv w)) (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv y) (cv v)) (cv v) (cv x)))))) eqid fnoprab2 (X. A B) (V) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (if (= (cv x) (cv w)) (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv y) (cv v)) (cv v) (cv x)))))) fnrndomg mp2 (u. A B) (ran ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (if (= (cv x) (cv w)) (if (= (cv y) (cv v)) (cv w) (cv y)) (if (= (cv y) (cv v)) (cv v) (cv x))))))) (X. A B) domtr mpan2 3syl exp43 r19.23aivv r19.23advv imp 1onn unxpdomlem.1 (1o) A (V) sucdom mp2an df-2o (~<_) A breq1i bitr4 unxpdomlem.1 u w 2dom sylbi 1onn unxpdomlem.2 (1o) B (V) sucdom mp2an df-2o (~<_) B breq1i bitr4 unxpdomlem.2 t v 2dom sylbi syl2an)) thm (unxpdom ((x y) (A x) (A y) (B y)) () (-> (/\ (br (1o) (~<) A) (br (1o) (~<) B)) (br (u. A B) (~<_) (X. A B))) ((1o) A sdomex pm3.27d (1o) B sdomex pm3.27d anim12i (cv x) A (1o) (~<) breq2 (br (1o) (~<) (cv y)) anbi1d (cv x) A (cv y) uneq1 (cv x) A (cv y) xpeq1 (~<_) breq12d imbi12d (cv y) B (1o) (~<) breq2 (br (1o) (~<) A) anbi2d (cv y) B A uneq2 (cv y) B A xpeq2 (~<_) breq12d imbi12d x visset y visset unxpdomlem (V) (V) vtocl2g mpcom)) thm (unxpdom2 () ((unxpdom2.1 (e. A (V))) (unxpdom2.2 (e. B (V)))) (-> (/\ (br (1o) (~<) A) (br B (~<_) A)) (br (u. A B) (~<_) (X. A A))) ((u. A B) (u. (X. A ({} (1o))) (X. A ({} ({/})))) (X. A A) domtr unxpdom2.1 unxpdom2.1 0ex xpsnen ensymi B A (X. A ({} ({/}))) domentr mpan2 unxpdom2.1 unxpdom2.1 1onn elisseti xpsnen ensymi A (X. A ({} (1o))) endom ax-mp 1ne0 (1o) ({/}) A A xpsndisj ax-mp unxpdom2.1 (1o) snex xpex unxpdom2.2 unxpdom2.1 p0ex xpex A undom mpan2 mpan syl (X. A ({} (1o))) (X. A ({} ({/}))) unxpdom unxpdom2.1 unxpdom2.1 1onn elisseti xpsnen ensymi unxpdom2.1 (1o) snex xpex (X. A ({} (1o))) (V) (1o) A sdomentr ax-mp mpan2 unxpdom2.1 unxpdom2.1 0ex xpsnen ensymi unxpdom2.1 p0ex xpex (X. A ({} ({/}))) (V) (1o) A sdomentr ax-mp mpan2 sylanc unxpdom2.1 1onn elisseti xpsnen unxpdom2.1 0ex xpsnen unxpdom2.1 (1o) snex xpex unxpdom2.1 unxpdom2.1 p0ex xpex unxpdom2.1 xpen mp2an (u. (X. A ({} (1o))) (X. A ({} ({/})))) (X. (X. A ({} (1o))) (X. A ({} ({/})))) (X. A A) domentr mpan2 syl syl2an ancoms)) thm (sucxpdom ((A x)) () (-> (br (1o) (~<) A) (br (suc A) (~<_) (X. A A))) ((1o) A sdomex pm3.27d (cv x) A (1o) (~<) breq2 (cv x) A suceq (cv x) A (cv x) xpeq1 (cv x) A A xpeq2 eqtrd (~<_) breq12d imbi12d x visset x visset 0ex xpsnen (cv x) (V) (X. (cv x) ({} ({/}))) ({} (cv x)) sdomen2 mp2an 1on elisseti x visset ensn1 (1o) (V) ({} (cv x)) (cv x) sdomen1 mp2an bitr ({} (cv x)) (X. (cv x) ({} ({/}))) sdomdom sylbir x visset (cv x) (V) domrefg ax-mp x visset x visset 1on elisseti xpsnen (cv x) (V) (X. (cv x) ({} (1o))) (cv x) domen2 mp2an mpbir 1ne0 (1o) ({/}) (cv x) (cv x) xpsndisj ax-mp x visset (1o) snex xpex (cv x) snex x visset p0ex xpex (cv x) undom mpan2 mpan syl (X. (cv x) ({} (1o))) (X. (cv x) ({} ({/}))) unxpdom x visset x visset 1on elisseti xpsnen (cv x) (V) (X. (cv x) ({} (1o))) (1o) sdomen2 mp2an x visset x visset 0ex xpsnen (cv x) (V) (X. (cv x) ({} ({/}))) (1o) sdomen2 mp2an sylancbr jca (u. (cv x) ({} (cv x))) (u. (X. (cv x) ({} (1o))) (X. (cv x) ({} ({/})))) (X. (X. (cv x) ({} (1o))) (X. (cv x) ({} ({/})))) domtr x visset 1on elisseti xpsnen x visset 0ex xpsnen x visset (1o) snex xpex x visset x visset p0ex xpex x visset xpen mp2an (u. (cv x) ({} (cv x))) (X. (X. (cv x) ({} (1o))) (X. (cv x) ({} ({/})))) (X. (cv x) (cv x)) domentr mpan2 3syl (cv x) df-suc syl5eqbr (V) vtoclg mpcom)) thm (sdomel () () (-> (/\ (e. A (On)) (e. B (On))) (-> (br A (~<) B) (e. A B))) (B (On) A ssdomg (e. A (On)) adantr B A ontri1 B (On) A (On) domtri 3imtr3d a3d ancoms)) thm (sdomsdomcard () () (<-> (br A (~<) B) (br A (~<) (` (card) B))) (A B sdomex pm3.27d A sdom0 B (card) fvprc A (~<) breq2d mtbiri a3i (card) B fvex (` (card) B) (V) A B sdomentr ax-mp ex B cardid B (V) (` (card) B) ensymg mpi syl5com B cardid B (V) A (` (card) B) sdomentr mpan2i impbid pm5.21nii)) thm (cardidm () () (= (` (card) (` (card) A)) (` (card) A)) (A cardid (card) A fvex (` (card) A) (V) A (V) carden mpan mpbiri A (card) fvprc (card) fveq2d card0 syl6eq A (card) fvprc eqtr4d pm2.61i)) thm (canth3 () () (-> (e. A B) (e. (` (card) A) (` (card) (P~ A)))) (A B canth2g A B pwexg A B (P~ A) (V) cardsdom mpdan mpbird)) thm (cardlim ((A x)) () (<-> (C_ (om) (` (card) A)) (Lim (` (card) A))) ((` (card) A) (suc (cv x)) (om) sseq2 biimpd (cv x) infensuc ex limom (om) (cv x) limsssuc ax-mp syl5ibr sylan9r (` (card) A) (suc (cv x)) (cv x) (~~) breq2 (e. (cv x) (On)) adantl sylibrd ex com3r imp x visset sucid (` (card) A) (suc (cv x)) (cv x) eleq2 mpbiri A cardidm syl6eleqr (cv x) (` (card) A) cardne syl (/\ (C_ (om) (` (card) A)) (e. (cv x) (On))) a1i pm2.65d nrexdv peano1 (om) (` (card) A) ({/}) ssel mpi ({/}) (` (card) A) n0i A cardon onord (` (card) A) x ordzsl mpbi (= (` (card) A) ({/})) (E.e. x (On) (= (` (card) A) (suc (cv x)))) (Lim (` (card) A)) 3orass mpbi ori 3syl ord mpd (` (card) A) limomss impbi)) thm (cardsdomel () () (-> (e. A (On)) (<-> (br A (~<) B) (e. A (` (card) B)))) (A (On) (` (card) B) ssdom2g B cardon (` (card) B) A ontri1 mpan B cardon (` (card) B) (On) A (On) domtri mpan 3imtr3d a3d B cardon A onelss B cardon (` (card) B) (On) A ssdom2g ax-mp syl B cardidm A eleq2i A (` (card) B) cardne sylbir jca A (` (card) B) brsdom sylibr (e. A (On)) a1i impbid A B sdomsdomcard syl5bb)) thm (iscard ((A x)) () (<-> (= (` (card) A) A) (/\ (e. A (On)) (A.e. x A (br (cv x) (~<) A)))) (A cardon (` (card) A) A (On) eleq1 mpbii pm4.71ri A cardonle (` (card) A) A eqss baibr syl A (cv x) onelon (cv x) A cardsdomel syl ralbidva A (` (card) A) x dfss3 syl6rbbr bitr3d pm5.32i bitr)) thm (iscard2 ((x y) (A x) (A y)) () (<-> (= (` (card) A) A) (/\ (e. A (On)) (A.e. x (On) (-> (br A (~~) (cv x)) (C_ A (cv x)))))) (A cardon (` (card) A) A (On) eleq1 mpbii pm4.71ri A cardonle (C_ A (` (card) A)) biantrurd (` (card) A) A eqss syl6rbbr A (On) (cv x) ensymg x visset A ensym (e. A (On)) a1i impbid (e. (cv x) (On)) anbi2d (cv y) (cv x) (~~) A breq1 (On) elrab syl5bb (C_ A (cv x)) imbi1d (e. (cv x) (On)) (br A (~~) (cv x)) (C_ A (cv x)) impexp syl6bb ralbidv2 A ({e.|} y (On) (br (cv y) (~~) A)) x ssint syl5bb A y cardval A sseq2i syl5bb bitrd pm5.32i bitr)) thm (cardval2 ((A x)) () (= (` (card) A) ({e.|} x (On) (br (cv x) (~<) A))) (x (On) (br (cv x) (~<) A) df-rab A cardon (cv x) onel pm4.71ri (cv x) A cardsdomel pm5.32i bitr4 x abbii x (` (card) A) abid2 3eqtr2r)) thm (ondomon ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A B) (e. ({e.|} x (On) (br (cv x) (~<_) A)) (On))) ((cv y) (cv z) A domtr (e. (cv y) (On)) anim2i anassrs (cv z) (cv y) onelon (cv z) (cv y) onelsst imp y visset (cv y) (V) (cv z) ssdomg ax-mp syl jca sylan exp31 com12 imp3a (cv x) (cv z) (~<_) A breq1 (On) elrab (cv x) (cv y) (~<_) A breq1 (On) elrab 3imtr4g imp y z gen2 ({e.|} x (On) (br (cv x) (~<_) A)) y z dftr2 mpbir x (On) (br (cv x) (~<_) A) ssrab2 ordon ({e.|} x (On) (br (cv x) (~<_) A)) (On) trssord mp3an A B elisset (cv x) A (P~ A) domsdomtr A (V) canth2g sylan2 expcom (e. (cv x) (On)) a1d r19.21aiv syl x (On) (br (cv x) (~<_) A) (br (cv x) (~<) (P~ A)) ss2rab sylibr (P~ A) x cardval2 (card) (P~ A) fvex eqeltrr ({e.|} x (On) (br (cv x) (~<_) A)) ssex ({e.|} x (On) (br (cv x) (~<_) A)) (V) elong 3syl mpbiri)) thm (ondomcard ((x y) (A x) (A y)) () (-> (e. A B) (= (` (card) ({e.|} x (On) (br (cv x) (~<_) A))) ({e.|} x (On) (br (cv x) (~<_) A)))) (A B elisset A (V) x ondomon (cv y) A ({e.|} x (On) (br (cv x) (~<_) A)) domsdomtr (cv x) (cv y) (~<_) A breq1 (On) elrab pm3.27bd A (V) x ondomon ({e.|} x (On) (br (cv x) (~<_) A)) eloni ({e.|} x (On) (br (cv x) (~<_) A)) ordeirr syl y x (On) (br (cv x) (~<_) A) hbrab1 (e. (cv y) (On)) x ax-17 y x (On) (br (cv x) (~<_) A) hbrab1 (e. (cv y) (~<_)) x ax-17 (e. (cv y) A) x ax-17 hbbr (cv x) ({e.|} x (On) (br (cv x) (~<_) A)) (~<_) A breq1 elrabf biimpr ex mtod syl A (V) x ondomon ({e.|} x (On) (br (cv x) (~<_) A)) (On) A (V) domtri con2bid mpancom mpbird syl2an (cv y) ({e.|} x (On) (br (cv x) (~<_) A)) sdomnen syl expcom con2d y visset ({e.|} x (On) (br (cv x) (~<_) A)) ensym syl5 (e. (cv y) (On)) adantr ({e.|} x (On) (br (cv x) (~<_) A)) (cv y) ontri1 A (V) x ondomon sylan sylibrd r19.21aiva jca ({e.|} x (On) (br (cv x) (~<_) A)) y iscard2 sylibr syl)) thm (carduni ((x y) (A x) (A y) (B y)) () (-> (e. A B) (-> (A.e. x A (= (` (card) (cv x)) (cv x))) (= (` (card) (U. A)) (U. A)))) (A B ssonunit (cv x) (cv y) (card) fveq2 (= (cv x) (cv y)) id eqeq12d A rcla4v (cv y) cardon (` (card) (cv y)) (cv y) (On) eleq1 mpbii syl6com ssrdv syl5 imp (U. A) cardonle syl (` (card) (U. A)) eirr (` (card) (U. A)) A y eluni (cv x) (cv y) (card) fveq2 (= (cv x) (cv y)) id eqeq12d A rcla4v A B uniexg y visset (cv y) (V) (U. A) (V) carddom mpan bicomd syl (` (card) (cv y)) (cv y) (` (card) (U. A)) sseq1 sylan9bb (cv y) A elssuni y visset (cv y) (V) (U. A) ssdomg ax-mp syl syl5bi (cv y) (` (card) (U. A)) (` (card) (U. A)) ssel syl6 ex com13 syld com4r imp y 19.23aiv sylbi com13 imp mtoi jca A B ssonunit (cv x) (cv y) (card) fveq2 (= (cv x) (cv y)) id eqeq12d A rcla4v (cv y) cardon (` (card) (cv y)) (cv y) (On) eleq1 mpbii syl6com ssrdv syl5 imp (U. A) eloni (U. A) cardon onord (` (card) (U. A)) (U. A) ordtri4 mpan 3syl mpbird ex)) thm (cardiun ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) () (-> (e. A C) (-> (A.e. x A (= (` (card) B) B)) (= (` (card) (U_ x A B)) (U_ x A B)))) (A C z x (` (card) B) abrexexg y visset (cv z) (cv y) (` (card) B) eqeq1 x A rexbidv elab (cv y) (` (card) B) (card) fveq2 B cardidm syl6eq (= (cv y) (` (card) B)) id eqtr4d (e. (cv x) A) a1i r19.23aiv sylbi rgen ({|} z (E.e. x A (= (cv z) (` (card) B)))) (V) y carduni mpi syl (card) B fvex x A z dfiun2 (card) fveq2i (card) B fvex x A z dfiun2 3eqtr4g (A.e. x A (= (` (card) B) B)) adantr x A (` (card) B) B iuneq2 (e. A C) adantl (card) fveq2d x A (` (card) B) B iuneq2 (e. A C) adantl 3eqtr3d ex)) thm (cardmin ((x y) (A x) (A y) (B y)) () (-> (e. A B) (= (` (card) (|^| ({e.|} x (On) (br A (~<) (cv x))))) (|^| ({e.|} x (On) (br A (~<) (cv x)))))) (A B x numthcor x (br A (~<) (cv x)) onintrab2 sylib A B x numthcor x (br A (~<) (cv x)) onintrab2 sylib (|^| ({e.|} x (On) (br A (~<) (cv x)))) (cv y) onelon ex syl (cv x) (cv y) A (~<) breq2 (On) elrab x (On) (br A (~<) (cv x)) ssrab2 ({e.|} x (On) (br A (~<) (cv x))) (cv y) onnmin mpan sylbir ex con2d syli y visset (cv y) (V) A B domtri mpan sylibrd A B x numthcor (e. (cv y) A) x ax-17 (e. (cv y) (~<)) x ax-17 y x (On) (br A (~<) (cv x)) hbrab1 hbint hbbr (cv x) (|^| ({e.|} x (On) (br A (~<) (cv x)))) A (~<) breq2 onminsb syl jctird (cv y) A (|^| ({e.|} x (On) (br A (~<) (cv x)))) domsdomtr syl6 r19.21aiv jca (|^| ({e.|} x (On) (br A (~<) (cv x)))) y iscard sylibr)) thm (cardprc ((x y)) () (-. (e. ({|} x (= (` (card) (cv x)) (cv x))) (V))) ((U. ({|} x (= (` (card) (cv x)) (cv x)))) (V) canth3 (card) (P~ (U. ({|} x (= (` (card) (cv x)) (cv x))))) fvex (P~ (U. ({|} x (= (` (card) (cv x)) (cv x))))) cardidm (card) (P~ (U. ({|} x (= (` (card) (cv x)) (cv x))))) fvex (e. (cv y) (card)) x ax-17 y x (= (` (card) (cv x)) (cv x)) hbab1 hbuni hbpw hbfv (e. (cv y) (card)) x ax-17 (e. (cv y) (card)) x ax-17 y x (= (` (card) (cv x)) (cv x)) hbab1 hbuni hbpw hbfv hbfv (e. (cv y) (card)) x ax-17 y x (= (` (card) (cv x)) (cv x)) hbab1 hbuni hbpw hbfv hbeq (cv x) (` (card) (P~ (U. ({|} x (= (` (card) (cv x)) (cv x)))))) (card) fveq2 (= (cv x) (` (card) (P~ (U. ({|} x (= (` (card) (cv x)) (cv x))))))) id eqeq12d (V) elabgf ax-mp mpbir (` (card) (P~ (U. ({|} x (= (` (card) (cv x)) (cv x)))))) ({|} x (= (` (card) (cv x)) (cv x))) elssuni ax-mp (` (card) (P~ (U. ({|} x (= (` (card) (cv x)) (cv x)))))) (V) (U. ({|} x (= (` (card) (cv x)) (cv x)))) ssdomg mp2 (card) (P~ (U. ({|} x (= (` (card) (cv x)) (cv x))))) fvex (` (card) (P~ (U. ({|} x (= (` (card) (cv x)) (cv x)))))) (V) (U. ({|} x (= (` (card) (cv x)) (cv x)))) (V) carddom mpan mpbiri (P~ (U. ({|} x (= (` (card) (cv x)) (cv x))))) cardidm syl5ssr (P~ (U. ({|} x (= (` (card) (cv x)) (cv x))))) cardon (U. ({|} x (= (` (card) (cv x)) (cv x)))) cardon (` (card) (P~ (U. ({|} x (= (` (card) (cv x)) (cv x)))))) (` (card) (U. ({|} x (= (` (card) (cv x)) (cv x))))) ontri1 mp2an sylib pm2.65i ({|} x (= (` (card) (cv x)) (cv x))) (V) uniexg mto)) thm (alephfnon ((x y) (x z) (y z)) () (Fn (aleph) (On)) (({<,>|} x y (= (cv y) (|^| ({e.|} z (On) (br (cv x) (~<) (cv z)))))) (om) rdgfnon x y z df-aleph (aleph) (rec ({<,>|} x y (= (cv y) (|^| ({e.|} z (On) (br (cv x) (~<) (cv z)))))) (om)) (On) fneq1 ax-mp mpbir)) thm (aleph0 ((x y) (x z) (y z)) () (= (` (aleph) ({/})) (om)) (x y z df-aleph ({/}) fveq1i omex ({<,>|} x y (= (cv y) (|^| ({e.|} z (On) (br (cv x) (~<) (cv z)))))) rdg0 eqtr)) thm (alephlim ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (/\ (e. A B) (Lim A)) (= (` (aleph) A) (U_ x A (` (aleph) (cv x))))) (A B ({<,>|} y z (= (cv z) (|^| ({e.|} w (On) (br (cv y) (~<) (cv w)))))) (om) x rdglim2a y z w df-aleph A fveq1i y z w df-aleph (cv x) fveq1i (e. (cv x) A) a1i iuneq2i 3eqtr4g)) thm (alephon ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (e. (` (aleph) A) (On)) ((cv x) ({/}) (aleph) fveq2 (On) eleq1d (cv x) (cv y) (aleph) fveq2 (On) eleq1d (cv x) (suc (cv y)) (aleph) fveq2 (On) eleq1d (cv x) A (aleph) fveq2 (On) eleq1d aleph0 omelon eqeltr (e. (cv w) (om)) z ax-17 (e. (cv w) (cv y)) z ax-17 (e. (cv w) (|^| ({e.|} x (On) (br (` (aleph) (cv y)) (~<) (cv x))))) z ax-17 z y x df-aleph (cv z) (` (aleph) (cv y)) (~<) (cv x) breq1 x (On) rabbisdv inteqd (V) rdgsucopab (On) eleq1d x (br (` (aleph) (cv y)) (~<) (cv x)) onintrab syl6rbbr ex ibd (e. (cv w) (om)) z ax-17 (e. (cv w) (cv y)) z ax-17 (e. (cv w) (|^| ({e.|} x (On) (br (` (aleph) (cv y)) (~<) (cv x))))) z ax-17 z y x df-aleph (cv z) (` (aleph) (cv y)) (~<) (cv x) breq1 x (On) rabbisdv inteqd rdgsucopabn 0elon syl6eqel pm2.61d1 (e. (` (aleph) (cv y)) (On)) a1d x visset (cv x) (V) y alephlim mpan (On) eleq1d x visset (aleph) (cv y) fvex y iunon syl5bir tfinds alephfnon (aleph) (On) fndm ax-mp A eleq2i negbii A (aleph) ndmfv sylbir 0elon syl6eqel pm2.61i)) thm (alephsuc ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (On)) (= (` (aleph) (suc A)) (|^| ({e.|} x (On) (br (` (aleph) A) (~<) (cv x)))))) ((aleph) A fvex (` (aleph) A) (V) x numthcor ax-mp x (On) (br (` (aleph) A) (~<) (cv x)) intexrab mpbi (e. (cv w) (om)) y ax-17 (e. (cv w) A) y ax-17 (e. (cv w) (|^| ({e.|} x (On) (br (` (aleph) A) (~<) (cv x))))) y ax-17 y z x df-aleph (cv y) (` (aleph) A) (~<) (cv x) breq1 x (On) rabbisdv inteqd (V) rdgsucopab mpan2)) thm (alephcard ((x y) (A x) (A y)) () (= (` (card) (` (aleph) A)) (` (aleph) A)) ((cv x) ({/}) (aleph) fveq2 (card) fveq2d (cv x) ({/}) (aleph) fveq2 eqeq12d (cv x) (cv y) (aleph) fveq2 (card) fveq2d (cv x) (cv y) (aleph) fveq2 eqeq12d (cv x) (suc (cv y)) (aleph) fveq2 (card) fveq2d (cv x) (suc (cv y)) (aleph) fveq2 eqeq12d (cv x) A (aleph) fveq2 (card) fveq2d (cv x) A (aleph) fveq2 eqeq12d cardom aleph0 (card) fveq2i aleph0 3eqtr4 (aleph) (cv y) fvex (` (aleph) (cv y)) (V) x cardmin ax-mp (e. (cv y) (On)) a1i (cv y) x alephsuc (card) fveq2d (cv y) x alephsuc 3eqtr4d (= (` (card) (` (aleph) (cv y))) (` (aleph) (cv y))) a1d x visset (cv x) (V) y (` (aleph) (cv y)) cardiun ax-mp (Lim (cv x)) adantl x visset (cv x) (V) y alephlim mpan (A.e. y (cv x) (= (` (card) (` (aleph) (cv y))) (` (aleph) (cv y)))) adantr (card) fveq2d x visset (cv x) (V) y alephlim mpan (A.e. y (cv x) (= (` (card) (` (aleph) (cv y))) (` (aleph) (cv y)))) adantr 3eqtr4d ex tfinds card0 alephfnon (aleph) (On) fndm ax-mp A eleq2i negbii A (aleph) ndmfv sylbir (card) fveq2d alephfnon (aleph) (On) fndm ax-mp A eleq2i negbii A (aleph) ndmfv sylbir eqeq12d mpbiri pm2.61i)) thm (alephnbtwn ((A x) (B x)) () (-> (= (` (card) B) B) (-. (/\ (e. (` (aleph) A) B) (e. B (` (aleph) (suc A)))))) ((` (card) B) B (` (aleph) A) eleq2 A alephon (` (aleph) A) B cardsdomel ax-mp syl5bb (e. A (On)) adantl A x alephsuc B eleq2d biimpd (cv x) B (` (aleph) A) (~<) breq2 onnminsb sylan9 con2d B cardon (` (card) B) B (On) eleq1 mpbii sylan2 sylbird (e. (` (aleph) A) B) (e. B (` (aleph) (suc A))) imnan sylib ex B (` (aleph) (suc A)) n0i alephfnon (aleph) (On) fndm ax-mp (suc A) eleq2i negbii (suc A) (aleph) ndmfv sylbir nsyl2 A sucelon sylibr (e. (` (aleph) A) B) adantl con3i (= (` (card) B) B) a1d pm2.61i)) thm (alephnbtwn2 () () (-. (/\ (br (` (aleph) A) (~<) B) (br B (~<) (` (aleph) (suc A))))) (B cardidm (` (card) B) A alephnbtwn ax-mp A alephon (` (aleph) A) B cardsdomel ax-mp (e. B (V)) a1i (suc A) alephon B (V) (` (aleph) (suc A)) (On) cardsdom mpan2 (suc A) alephcard (` (card) B) eleq2i syl5rbbr anbi12d mtbiri relsdom B (` (aleph) (suc A)) brrelexi (br (` (aleph) A) (~<) B) adantl con3i pm2.61i)) thm (alephsucpw () () (br (` (aleph) (suc A)) (~<_) (P~ (` (aleph) A))) ((aleph) A fvex canth2 A (P~ (` (aleph) A)) alephnbtwn2 (br (` (aleph) A) (~<) (P~ (` (aleph) A))) (br (P~ (` (aleph) A)) (~<) (` (aleph) (suc A))) imnan mpbir ax-mp (aleph) (suc A) fvex (aleph) A fvex pwex (` (aleph) (suc A)) (V) (P~ (` (aleph) A)) (V) domtri mp2an mpbir)) thm (aleph1 () () (br (` (aleph) (1o)) (~<_) (opr (2o) (^m) (` (aleph) ({/})))) (df-1o (aleph) fveq2i ({/}) alephsucpw (2o) (^m) (` (aleph) ({/})) oprex (aleph) ({/}) fvex pw2en (opr (2o) (^m) (` (aleph) ({/}))) (V) (P~ (` (aleph) ({/}))) (` (aleph) (suc ({/}))) domen2 mp2an mpbi eqbrtr)) thm (alephordlem1 ((x y) (A x) (A y)) () (-> (e. A (On)) (br (` (aleph) A) (~<) (` (aleph) (suc A)))) (A x alephsuc A alephon (` (aleph) A) (On) x numthcor ax-mp (e. (cv y) (` (aleph) A)) x ax-17 (e. (cv y) (~<)) x ax-17 y x (On) (br (` (aleph) A) (~<) (cv x)) hbrab1 hbint hbbr (cv x) (|^| ({e.|} x (On) (br (` (aleph) A) (~<) (cv x)))) (` (aleph) A) (~<) breq2 onminsb ax-mp syl5breqr)) thm (alephordlem2 ((A x) (B x)) () (-> (/\ (e. B (V)) (Lim B)) (-> (e. A B) (br (` (aleph) A) (~<_) (` (aleph) B)))) (B (V) x alephlim (` (aleph) A) sseq2d (cv x) A (aleph) fveq2 B ssiun2s syl5bir A alephon (` (aleph) A) (On) (` (aleph) B) ssdomg ax-mp syl6)) thm (alephordi ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B (On)) (-> (e. A B) (br (` (aleph) A) (~<) (` (aleph) B)))) ((cv x) ({/}) A eleq2 (cv x) ({/}) (aleph) fveq2 (` (aleph) A) (~<) breq2d imbi12d (cv x) (cv y) A eleq2 (cv x) (cv y) (aleph) fveq2 (` (aleph) A) (~<) breq2d imbi12d (cv x) (suc (cv y)) A eleq2 (cv x) (suc (cv y)) (aleph) fveq2 (` (aleph) A) (~<) breq2d imbi12d (cv x) B A eleq2 (cv x) B (aleph) fveq2 (` (aleph) A) (~<) breq2d imbi12d A noel (br (` (aleph) A) (~<) (` (aleph) ({/}))) pm2.21i (` (aleph) A) (` (aleph) (cv y)) (` (aleph) (suc (cv y))) sdomtr (cv y) alephordlem1 sylan2 expcom (e. A (cv y)) imim2d com23 A (cv y) (aleph) fveq2 (~<) (` (aleph) (suc (cv y))) breq1d (cv y) alephordlem1 syl5bir (-> (e. A (cv y)) (br (` (aleph) A) (~<) (` (aleph) (cv y)))) a1d com3r jaod y visset A elsuc2 syl5ib com23 x visset (cv x) (V) y alephlim mpan (` (aleph) A) sseq2d (cv y) A (aleph) fveq2 (cv x) ssiun2s syl5bir A alephon (` (aleph) A) (On) (` (aleph) (cv x)) ssdomg ax-mp syl6 (cv x) A limsuc x visset (cv x) (suc A) alephordlem2 mpan sylbid imp (` (aleph) (suc A)) (` (aleph) (cv x)) domnsym syl (aleph) (cv x) fvex (` (aleph) A) ensym (` (aleph) (cv x)) (` (aleph) A) (` (aleph) (suc A)) ensdomtr ex syl A alephordlem1 syl5 (cv x) A onelon x visset (cv x) (V) limelon mpan sylan syl5com mtod ex jcad (` (aleph) A) (` (aleph) (cv x)) brsdom syl6ibr (A.e. y (cv x) (-> (e. A (cv y)) (br (` (aleph) A) (~<) (` (aleph) (cv y))))) a1d tfinds)) thm (alephord () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. A B) (br (` (aleph) A) (~<) (` (aleph) B)))) (B A alephordi (e. A (On)) adantl A B alephordi con3d A alephon B alephon (` (aleph) A) (On) (` (aleph) B) (On) domtri mp2an syl5ib (e. B (On)) adantr A B ontri1 sylibrd A B (aleph) fveq2 A alephon (` (aleph) A) (On) (` (aleph) B) eqeng ax-mp syl con3i (/\ (e. A (On)) (e. B (On))) a1i anim12d A B onelpsst sylibrd (` (aleph) A) (` (aleph) B) brsdom syl5ib impbid)) thm (alephord2 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. A B) (e. (` (aleph) A) (` (aleph) B)))) (A B alephord A alephon (` (aleph) A) (` (aleph) B) cardsdomel ax-mp B alephcard (` (aleph) A) eleq2i bitr syl6bb)) thm (alephord2i () () (-> (e. B (On)) (-> (e. A B) (e. (` (aleph) A) (` (aleph) B)))) (B A onelon A B alephord2 biimpd ex imp3a mpcom ex)) thm (alephord3 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (C_ A B) (C_ (` (aleph) A) (` (aleph) B)))) (B A alephord2 ancoms negbid A B ontri1 A alephon B alephon (` (aleph) A) (` (aleph) B) ontri1 mp2an (/\ (e. A (On)) (e. B (On))) a1i 3bitr4d)) thm (aleph11 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (= (` (aleph) A) (` (aleph) B)) (= A B))) (A B alephord3 B A alephord3 ancoms anbi12d A B eqss (` (aleph) A) (` (aleph) B) eqss 3bitr4g bicomd)) thm (alephsucdom () () (-> (e. B (On)) (<-> (br A (~<_) (` (aleph) B)) (br A (~<) (` (aleph) (suc B))))) (A (` (aleph) B) (` (aleph) (suc B)) domsdomtr ex B alephordlem1 syl5com (e. A (V)) adantl (aleph) B fvex A (V) (` (aleph) B) (V) domtri mpan2 B A alephnbtwn2 (br (` (aleph) B) (~<) A) (br A (~<) (` (aleph) (suc B))) imnan mpbir con2i syl5bir (e. B (On)) adantr impbid ex reldom A (` (aleph) B) brrelexi relsdom A (` (aleph) (suc B)) brrelexi pm5.21ni (e. B (On)) a1d pm2.61i)) thm (alephsuc2 ((A x)) () (-> (e. A (On)) (= (` (aleph) (suc A)) ({e.|} x (On) (br (cv x) (~<_) (` (aleph) A))))) (A (cv x) alephsucdom x (On) rabbisdv (suc A) alephcard (` (aleph) (suc A)) x cardval2 eqtr3 syl6reqr)) thm (alephgeom () () (<-> (e. A (On)) (C_ (om) (` (aleph) A))) (A 0ss 0elon ({/}) A alephord3 mpan mpbii aleph0 syl5ssr peano1 ordom ord0 (om) ({/}) ordtri1 mp2an con2bii mpbi A (aleph) ndmfv (om) sseq2d mtbiri a3i alephfnon (aleph) (On) fndm ax-mp syl6eleq impbi)) thm (alephislim () () (<-> (e. A (On)) (Lim (` (aleph) A))) (A alephgeom (` (aleph) A) cardlim A alephcard (om) sseq2i A alephcard (` (card) (` (aleph) A)) (` (aleph) A) limeq ax-mp 3bitr3 bitr)) thm (alephle ((x y) (A x) (A y)) () (-> (e. A (On)) (C_ A (` (aleph) A))) ((= (cv x) (cv y)) id (cv x) (cv y) (aleph) fveq2 sseq12d (= (cv x) A) id (cv x) A (aleph) fveq2 sseq12d (cv x) (cv y) alephord2i imp (cv x) (cv y) onelon (cv x) alephon (cv y) (` (aleph) (cv x)) (` (aleph) (cv y)) ontr2 mpan2 syl mpan2d r19.20dva (cv x) alephon (cv x) (` (aleph) (cv x)) ontri1 mpan2 (` (aleph) (cv x)) eirr (cv y) (` (aleph) (cv x)) (` (aleph) (cv x)) eleq1 (cv x) rcla4cv mtoi syl5bir syld tfis3)) thm (cardaleph ((x y) (A x) (A y)) () (-> (/\ (C_ (om) A) (= (` (card) A) A)) (= A (` (aleph) (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x)))))))) (A cardon (` (card) A) A (On) eleq1 mpbii A alephle (cv x) A (aleph) fveq2 A sseq2d (On) rcla4ev mpdan x (C_ A (` (aleph) (cv x))) onintrab2 sylib (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) eloni (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) y ordzsl (= (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) ({/})) (E.e. y (On) (= (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) (suc (cv y)))) (Lim (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x)))))) 3orass bitr sylib 3syl (C_ (om) A) adantl A cardon (` (card) A) A (On) eleq1 mpbii A alephle (cv x) A (aleph) fveq2 A sseq2d (On) rcla4ev mpdan (e. (cv y) A) x ax-17 (e. (cv y) (aleph)) x ax-17 y x (On) (C_ A (` (aleph) (cv x))) hbrab1 hbint hbfv hbss (cv x) (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) (aleph) fveq2 A sseq2d onminsb 3syl (= (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) ({/})) a1i (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) ({/}) (aleph) fveq2 aleph0 syl6eq A sseq1d biimprd anim12d A (` (aleph) (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x)))))) eqss syl6ibr com12 ancoms (cv x) (cv y) (aleph) fveq2 A sseq2d onnminsb y visset sucid (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) (suc (cv y)) (cv y) eleq2 mpbiri syl5 imp (= (` (card) A) A) adantl (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) (suc (cv y)) (aleph) fveq2 (cv y) x alephsuc sylan9eqr A eleq2d biimpd A cardon (` (card) A) A (On) eleq1 mpbii (cv x) A (` (aleph) (cv y)) (~<) breq2 onnminsb (aleph) (cv y) fvex A (On) (` (aleph) (cv y)) (V) domtri mpan2 sylibrd (aleph) (cv y) fvex A (On) (` (aleph) (cv y)) (V) carddom mpan2 sylibrd syl (` (card) A) A (` (card) (` (aleph) (cv y))) sseq1 (cv y) alephcard A sseq2i syl6bb sylibd sylan9r mtod exp32 r19.23adv A cardon (` (card) A) A (On) eleq1 mpbii (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) (cv y) onelon A alephle (cv x) A (aleph) fveq2 A sseq2d (On) rcla4ev mpdan x (C_ A (` (aleph) (cv x))) onintrab2 sylib sylan (cv x) (cv y) (aleph) fveq2 A sseq2d onnminsb (e. A (On)) adantld mpcom (cv y) alephon A onelss nsyl nrexdv (Lim (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x)))))) adantr (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) (On) y alephlim A alephle (cv x) A (aleph) fveq2 A sseq2d (On) rcla4ev mpdan x (C_ A (` (aleph) (cv x))) onintrab2 sylib sylan A eleq2d A y (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) (` (aleph) (cv y)) eliun syl6bb mtbird ex syl jaod A cardon (` (card) A) A (On) eleq1 mpbii A alephle (cv x) A (aleph) fveq2 A sseq2d (On) rcla4ev mpdan (e. (cv y) A) x ax-17 (e. (cv y) (aleph)) x ax-17 y x (On) (C_ A (` (aleph) (cv x))) hbrab1 hbint hbfv hbss (cv x) (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) (aleph) fveq2 A sseq2d onminsb syl (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x))))) alephon A (` (aleph) (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x)))))) onsseleq mpan2 mpbid ord syl syld (C_ (om) A) adantl jaod mpd)) thm (cardalephex ((x y) (A x) (A y)) () (-> (C_ (om) A) (<-> (= (` (card) A) A) (E.e. x (On) (= A (` (aleph) (cv x)))))) ((cv x) (|^| ({e.|} y (On) (C_ A (` (aleph) (cv y))))) (aleph) fveq2 A eqeq2d (On) rcla4ev (C_ (om) A) (= (` (card) A) A) pm3.26 A y cardaleph (om) sseq2d (|^| ({e.|} y (On) (C_ A (` (aleph) (cv y))))) alephgeom syl6bbr mpbid A y cardaleph sylanc ex (cv x) alephcard A (` (aleph) (cv x)) (card) fveq2 (= A (` (aleph) (cv x))) id eqeq12d mpbiri (e. (cv x) (On)) a1i r19.23aiv (C_ (om) A) a1i impbid)) thm (isinfcard ((A x)) () (<-> (/\ (C_ (om) A) (= (` (card) A) A)) (e. A (ran (aleph)))) ((` (aleph) (cv x)) A eqcom x (On) rexbii alephfnon (aleph) (On) A x fvelrn ax-mp A x cardalephex pm5.32i A (` (aleph) (cv x)) (om) sseq2 (cv x) alephgeom biimp syl5bir com12 r19.23aiv pm4.71ri bitr4 3bitr4r)) thm (iscard3 () () (<-> (= (` (card) A) A) (e. A (u. (om) (ran (aleph))))) (A cardon (` (card) A) A (On) eleq1 mpbii A eloni ordom A (om) ordtri2or mpan2 3syl ord A isinfcard biimp expcom syld orrd A cardnn A isinfcard bicomi pm3.27bd jaoi impbi A (om) (ran (aleph)) elun bitr4)) thm (cardnum () () (= ({|} x (= (` (card) (cv x)) (cv x))) (u. (om) (ran (aleph)))) ((cv x) iscard3 bicomi abbi2i eqcomi)) thm (carduniima ((F x) (A x)) () (-> (e. A B) (-> (:--> F A (u. (om) (ran (aleph)))) (e. (U. (" F A)) (u. (om) (ran (aleph)))))) (F A B funimaexg F A (u. (om) (ran (aleph))) ffun sylan expcom (" F A) (V) x carduni F A (u. (om) (ran (aleph))) ffn F A fnima syl F A (u. (om) (ran (aleph))) frn eqsstrd (cv x) sseld (cv x) iscard3 syl6ibr r19.21aiv syl5 syli (U. (" F A)) iscard3 syl6ib)) thm (cardinfima ((F x) (A x)) () (-> (e. A B) (-> (/\ (:--> F A (u. (om) (ran (aleph)))) (E.e. x A (e. (` F (cv x)) (ran (aleph))))) (e. (U. (" F A)) (ran (aleph))))) (A B elisset (` F (cv x)) isinfcard bicomi pm3.26bd F A (cv x) fnfvrn ex F A fnima (` F (cv x)) eleq2d sylibrd (` F (cv x)) (" F A) elssuni syl6 imp F A (u. (om) (ran (aleph))) ffn sylan sylan9ssr anasss (e. A (V)) a1i A (V) F carduniima (U. (" F A)) iscard3 syl6ibr (/\ (e. (cv x) A) (e. (` F (cv x)) (ran (aleph)))) adantrd jcad (U. (" F A)) isinfcard syl6ib exp4d imp r19.23adv ex imp3a syl)) thm (alephiso ((x y) (x z) (y z)) () (Isom (aleph) (E) (E) (On) ({|} x (/\ (C_ (om) (cv x)) (= (` (card) (cv x)) (cv x))))) ((aleph) (E) (E) (On) ({|} x (/\ (C_ (om) (cv x)) (= (` (card) (cv x)) (cv x)))) y z df-iso (aleph) (On) ({|} x (/\ (C_ (om) (cv x)) (= (` (card) (cv x)) (cv x)))) df-f1o (aleph) (On) ({|} x (/\ (C_ (om) (cv x)) (= (` (card) (cv x)) (cv x)))) y z f1fv (aleph) (On) ({|} x (/\ (C_ (om) (cv x)) (= (` (card) (cv x)) (cv x)))) df-fo alephfnon (cv x) isinfcard bicomi abbi2i mpbir2an (aleph) (On) ({|} x (/\ (C_ (om) (cv x)) (= (` (card) (cv x)) (cv x)))) fof ax-mp (cv y) (cv z) aleph11 biimpd rgen2 mpbir2an (aleph) (On) ({|} x (/\ (C_ (om) (cv x)) (= (` (card) (cv x)) (cv x)))) df-fo alephfnon (cv x) isinfcard bicomi abbi2i mpbir2an mpbir2an (cv y) (cv z) alephord2 y z epel (aleph) (cv y) fvex (aleph) (cv z) fvex epelc 3bitr4g rgen2 mpbir2an)) thm (alephprc () () (-. (e. (ran (aleph)) (V))) (x cardprc x cardnum (V) eleq1i mtbi omex (om) (V) (ran (aleph)) (V) unexg mpan mto)) thm (alephsson () () (C_ (ran (aleph)) (On)) ((cv x) isinfcard (cv x) cardon (` (card) (cv x)) (cv x) (On) eleq1 mpbii (C_ (om) (cv x)) adantl sylbir ssriv)) thm (unialeph () () (= (U. (ran (aleph))) (On)) (alephprc (ran (aleph)) uniexb mtbi (U. (ran (aleph))) (On) elisset mto alephsson (ran (aleph)) ssorduni ax-mp (U. (ran (aleph))) ordeleqon mpbi ori ax-mp)) thm (alephfplem1 ((x y)) ((alephfplem.1 (= H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om))))) (e. (` H ({/})) (ran (aleph))) (alephfplem.1 ({/}) fveq1i (aleph) ({/}) fvex (` (aleph) ({/})) (V) ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) fr0t ax-mp eqtr alephfnon 0elon (aleph) (On) ({/}) fnfvrn mp2an eqeltr)) thm (alephfplem2 ((x y) (x z) (w x) (y z) (w y) (w z) (H z) (H w)) ((alephfplem.1 (= H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om))))) (-> (e. (cv w) (om)) (= (` H (suc (cv w))) (` (aleph) (` H (cv w))))) ((aleph) (` H (cv w)) fvex (e. (cv z) (` (aleph) ({/}))) x ax-17 (e. (cv z) (cv w)) x ax-17 (e. (cv z) (aleph)) x ax-17 z x y (= (cv y) (` (aleph) (cv x))) hbopab1 (e. (cv z) (` (aleph) ({/}))) x ax-17 hbrdg (e. (cv z) (om)) x ax-17 hbres alephfplem.1 (cv z) eleq2i alephfplem.1 (cv z) eleq2i x albii 3imtr4 (e. (cv z) (cv w)) x ax-17 hbfv hbfv alephfplem.1 (cv x) (` H (cv w)) (aleph) fveq2 (V) frsucopab mpan2)) thm (alephfplem3 ((x y) (w x) (v x) (w y) (v y) (v w) (H w) (H v)) ((alephfplem.1 (= H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om))))) (-> (e. (cv v) (om)) (e. (` H (cv v)) (ran (aleph)))) (y equid (cv v) ({/}) H fveq2 (ran (aleph)) eleq1d (cv v) (cv w) H fveq2 (ran (aleph)) eleq1d (cv v) (suc (cv w)) H fveq2 (ran (aleph)) eleq1d alephfplem.1 alephfplem1 (= (cv y) (cv y)) a1i alephfplem.1 w alephfplem2 (ran (aleph)) eleq1d alephsson (` H (cv w)) sseli alephfnon (aleph) (On) (` H (cv w)) fnfvrn mpan syl syl5bir (= (cv y) (cv y)) a1d finds2 mpi)) thm (alephfplem4 ((x y) (x z) (y z) (H z)) ((alephfplem.1 (= H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om))))) (e. (U. (" H (om))) (ran (aleph))) (H (om) (ran (aleph)) z ffnfv ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/})) frfnom alephfplem.1 H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om)) (om) fneq1 ax-mp mpbir alephfplem.1 z alephfplem3 rgen mpbir2an (ran (aleph)) (om) ssun2 H (om) (ran (aleph)) (u. (om) (ran (aleph))) fss mp2an peano1 alephfplem.1 alephfplem1 (cv z) ({/}) H fveq2 (ran (aleph)) eleq1d (om) rcla4ev mp2an omex (om) (V) H z cardinfima ax-mp mp2an)) thm (alephfp ((x y) (x z) (v x) (y z) (v y) (v z) (H z) (H v)) ((alephfplem.1 (= H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om))))) (= (` (aleph) (U. (" H (om)))) (U. (" H (om)))) (alephfplem.1 alephfplem4 (U. (" H (om))) isinfcard (U. (" H (om))) z cardalephex biimpa sylbir ax-mp (cv z) alephle (` (aleph) (cv z)) eirr ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/})) frfnom alephfplem.1 H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om)) (om) fneq1 ax-mp mpbir H (om) fnfun ax-mp H (cv z) (om) v eluniima ax-mp alephfplem.1 v alephfplem3 alephsson (` H (cv v)) sseli (` H (cv v)) (cv z) alephord2i 3syl alephfplem.1 v alephfplem2 (cv v) peano2 ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/})) frfnom alephfplem.1 H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om)) (om) fneq1 ax-mp mpbir H (om) (suc (cv v)) fnfvrn mpan ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/})) frfnom alephfplem.1 H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om)) (om) fneq1 ax-mp mpbir H (om) fnima ax-mp syl6eleqr syl eqeltrrd (` (aleph) (` H (cv v))) (" H (om)) elssuni syl (` (aleph) (cv z)) sseld syld r19.23aiv sylbi (U. (" H (om))) (` (aleph) (cv z)) (cv z) eleq2 (U. (" H (om))) (` (aleph) (cv z)) (` (aleph) (cv z)) eleq2 imbi12d mpbii mtoi anim12i (cv z) eloni (cv z) alephon onord (cv z) (` (aleph) (cv z)) ordtri4 mpan2 syl (= (U. (" H (om))) (` (aleph) (cv z))) adantr mpbird (U. (" H (om))) (` (aleph) (cv z)) (cv z) eqeq2 (e. (cv z) (On)) adantl mpbird eqcomd (aleph) fveq2d (U. (" H (om))) (` (aleph) (cv z)) (` (aleph) (U. (" H (om)))) eqeq2 (e. (cv z) (On)) adantl mpbird ex r19.23aiv ax-mp)) thm (alephfp2 ((x y) (x z) (y z)) () (E.e. x (On) (= (` (aleph) (cv x)) (cv x))) (alephsson (|` (rec ({<,>|} y z (= (cv z) (` (aleph) (cv y)))) (` (aleph) ({/}))) (om)) eqid alephfplem4 sselii (|` (rec ({<,>|} y z (= (cv z) (` (aleph) (cv y)))) (` (aleph) ({/}))) (om)) eqid alephfp (cv x) (U. (" (|` (rec ({<,>|} y z (= (cv z) (` (aleph) (cv y)))) (` (aleph) ({/}))) (om)) (om))) (aleph) fveq2 (= (cv x) (U. (" (|` (rec ({<,>|} y z (= (cv z) (` (aleph) (cv y)))) (` (aleph) ({/}))) (om)) (om)))) id eqeq12d (On) rcla4ev mp2an)) thm (alephval2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (e. A (On)) (e. ({/}) A)) (= (` (aleph) A) (|^| ({e.|} x (On) (A.e. y A (br (` (aleph) (cv y)) (~<) (cv x))))))) (x (On) (A.e. y A (br (` (aleph) (cv y)) (~<) (cv x))) ssrab2 ({e.|} x (On) (A.e. y A (br (` (aleph) (cv y)) (~<) (cv x)))) (` (aleph) A) z oneqmini ax-mp A (cv y) alephordi r19.21aiv A alephon jctil (cv x) (` (aleph) A) (` (aleph) (cv y)) (~<) breq2 y A ralbidv (On) elrab sylibr (e. ({/}) A) adantr omex z visset (om) (V) (cv z) (V) entri3 mp2an (cv x) A alephord ancoms (` (card) (cv z)) (` (aleph) (cv x)) (~<) (` (aleph) A) breq1 z visset (cv z) cardid (cv z) (V) (` (card) (cv z)) (` (aleph) A) sdomen1 mp2an syl5rbbr sylan9bb (cv y) (cv x) (aleph) fveq2 (~<) (cv z) breq1d A rcla4v (` (aleph) (cv x)) sdomirr (` (card) (cv z)) (` (aleph) (cv x)) (` (aleph) (cv x)) (~<) breq2 z visset (cv z) cardid (cv z) (V) (` (card) (cv z)) (` (aleph) (cv x)) sdomen2 mp2an syl5bbr mtbiri nsyli com12 (/\ (e. A (On)) (e. (cv x) (On))) adantl sylbird exp31 r19.23adv (cv z) cardidm (` (card) (cv z)) x cardalephex mpbii syl5 omex z visset (om) (V) (cv z) (V) carddom mp2an cardom (` (card) (cv z)) sseq1i bitr3 syl5ib (e. ({/}) A) adantr ({/}) A n0i A y (-. (br (` (aleph) (cv y)) (~<) (cv z))) r19.2zOLD (cv z) (om) (` (aleph) (cv y)) domtr (cv y) alephgeom omex (om) (V) (` (aleph) (cv y)) ssdomg ax-mp sylbi sylan2 (cv z) (` (aleph) (cv y)) domnsym syl A (cv y) onelon sylan2 exp32 imp r19.21aiv syl5 syl y A (br (` (aleph) (cv y)) (~<) (cv z)) rexnal syl6ib com12 ex imp3a (br (cv z) (~<) (` (aleph) A)) a1d com3r jaod mpi A alephon (cv z) onel (cv z) (` (aleph) A) cardsdomel A alephcard (cv z) eleq2i syl6rbb syl ibi syl5 (cv x) (cv z) (` (aleph) (cv y)) (~<) breq2 y A ralbidv (On) elrab pm3.27bd con3i syl6 r19.21aiv sylanc)) thm (alephval3 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (On)) (= (` (aleph) A) (|^| ({|} x (/\/\ (= (` (card) (cv x)) (cv x)) (C_ (om) (cv x)) (A.e. y A (-. (= (cv x) (` (aleph) (cv y)))))))))) ((cv x) cardon (` (card) (cv x)) (cv x) (On) eleq1 mpbii (C_ (om) (cv x)) (A.e. y A (-. (= (cv x) (` (aleph) (cv y))))) 3ad2ant1 abssi ({|} x (/\/\ (= (` (card) (cv x)) (cv x)) (C_ (om) (cv x)) (A.e. y A (-. (= (cv x) (` (aleph) (cv y))))))) (` (aleph) A) z oneqmini ax-mp A alephcard (e. A (On)) a1i A alephgeom biimp A (cv y) alephord2i (` (aleph) (cv y)) eirr (` (aleph) A) (` (aleph) (cv y)) (` (aleph) (cv y)) eleq2 mtbiri con2i syl6 r19.21aiv 3jca (aleph) A fvex (cv x) (` (aleph) A) (card) fveq2 (= (cv x) (` (aleph) A)) id eqeq12d (cv x) (` (aleph) A) (om) sseq2 (cv x) (` (aleph) A) (` (aleph) (cv y)) eqeq1 negbid y A ralbidv 3anbi123d elab sylibr (cv z) (` (aleph) (cv y)) (` (aleph) A) eleq1 (cv y) A alephord2 bicomd sylan9bbr biimpcd (/\ (e. (cv y) (On)) (e. A (On))) (= (cv z) (` (aleph) (cv y))) pm3.27 (e. (cv z) (` (aleph) A)) a1i jcad exp4c com3r imp4b y 19.22dv y (On) (= (cv z) (` (aleph) (cv y))) df-rex y A (= (cv z) (` (aleph) (cv y))) df-rex 3imtr4g (cv z) y cardalephex biimpac syl5 imp y A (= (cv z) (` (aleph) (cv y))) dfrex2 sylib (/\ (e. A (On)) (e. (cv z) (` (aleph) A))) (/\ (= (` (card) (cv z)) (cv z)) (C_ (om) (cv z))) (A.e. y A (-. (= (cv z) (` (aleph) (cv y))))) nan mpbir ex z visset (cv x) (cv z) (card) fveq2 (= (cv x) (cv z)) id eqeq12d (cv x) (cv z) (om) sseq2 (cv x) (cv z) (` (aleph) (cv y)) eqeq1 negbid y A ralbidv 3anbi123d elab (= (` (card) (cv z)) (cv z)) (C_ (om) (cv z)) (A.e. y A (-. (= (cv z) (` (aleph) (cv y))))) df-3an bitr negbii syl6ibr r19.21aiv sylanc)) thm (cflem ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A B) (E. x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))))))) (A ssid (cv z) ssid (cv w) (cv z) (cv z) sseq2 A rcla4ev mpan2 rgen (cv y) A A sseq1 (cv y) A w (C_ (cv z) (cv w)) rexeq1 z A ralbidv anbi12d B cla4egv mp2ani x (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))) 19.41v (card) (cv y) fvex x isseti mpbiran y exbii y x (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w)))))) excom bitr3 sylib)) thm (cfval ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u)) () (-> (e. A (On)) (= (` (cf) A) (|^| ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))))))))) (A (On) x y z w cflem x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))))) intexab sylib (cv v) A (cv y) sseq2 (cv v) A z (E.e. w (cv y) (C_ (cv z) (cv w))) raleq1 anbi12d (= (cv x) (` (card) (cv y))) anbi2d y exbidv x abbidv inteqd v u x y z w df-cf (V) fvopab4g mpdan)) thm (cffnon ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (Fn (cf) (On)) (v visset (cv v) (V) x y z w cflem ax-mp x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) (cv v)) (A.e. z (cv v) (E.e. w (cv y) (C_ (cv z) (cv w))))))) intexab mpbi v u x y z w df-cf fnopab2)) thm (cfub ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (C_ (` (cf) A) (|^| ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (C_ A (U. (cv y))))))))) (A x y z w cfval (cv y) A (cv w) ssel A (cv w) onelon ex sylan9r (cv w) (cv z) onelsst syl6 imdistand ancomsd w 19.22dv (cv z) (cv y) w eluni w (cv y) (C_ (cv z) (cv w)) df-rex 3imtr4g z A r19.20sdv A (U. (cv y)) z dfss3 syl5ib ex imdistand (= (cv x) (` (card) (cv y))) anim2d y 19.22dv x 19.21aiv x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (C_ A (U. (cv y)))))) (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))))) ss2ab sylibr ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (C_ A (U. (cv y))))))) ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w)))))))) intss syl eqsstrd (|^| ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (C_ A (U. (cv y)))))))) 0ss cffnon (cf) (On) fndm ax-mp A eleq2i negbii A (cf) ndmfv sylbir (|^| ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (C_ A (U. (cv y)))))))) sseq1d mpbiri pm2.61i)) thm (cflim ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (/\ (e. A B) (Lim A)) (= (` (cf) A) (|^| ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (= A (U. (cv y)))))))))) (A (cv v) limsuc biimpd (cv z) (suc (cv v)) (cv w) sseq1 w (cv y) rexbidv A rcla4v v visset (cv v) (V) (cv w) sucssel ax-mp w (cv y) r19.22si (cv v) (cv y) w eluni2 sylibr syl6com syl9 r19.21adv A (U. (cv y)) v dfss3 syl6ibr (C_ (cv y) A) adantr A limuni (U. (cv y)) sseq2d (cv y) A uniss syl5bir imp jctird A (U. (cv y)) eqss syl6ibr ex imdistand (= (cv x) (` (card) (cv y))) anim2d y 19.22dv x 19.21aiv x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))))) (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (= A (U. (cv y)))))) ss2ab sylibr ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w)))))))) ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (= A (U. (cv y))))))) intss syl (e. A (V)) adantl A (V) limelon A x y z w cfval syl sseqtr4d A x y cfub A (U. (cv y)) eqimss (C_ (cv y) A) anim2i (= (cv x) (` (card) (cv y))) anim2i y 19.22i x ss2abi ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (= A (U. (cv y))))))) ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (C_ A (U. (cv y))))))) intss ax-mp sstri jctil (` (cf) A) (|^| ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (= A (U. (cv y)))))))) eqss sylibr A B elisset sylan)) thm (cf0 ((x y)) () (= (` (cf) ({/})) ({/})) (({/}) x y cfub (U. (cv y)) 0ss (C_ (cv y) ({/})) biantru (cv y) ss0b bitr3 (= (cv x) (` (card) (cv y))) anbi2i (= (cv x) (` (card) (cv y))) (= (cv y) ({/})) ancom bitr y exbii 0ex (cv y) ({/}) (card) fveq2 (cv x) eqeq2d ceqsexv card0 (cv x) eqeq2i 3bitr x abbii ({/}) x df-sn eqtr4 inteqi 0ex intsn eqtr sseqtr (` (cf) ({/})) ss0b mpbi)) thm (cardcf ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (= (` (card) (` (cf) A)) (` (cf) A)) (A x y z w cfval A x y z w cfval (cf) A fvex syl6eqelr ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w)))))))) intex sylibr v visset (cv x) (cv v) (` (card) (cv y)) eqeq1 (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))) anbi1d y exbidv elab (cv v) (` (card) (cv y)) (card) fveq2 (cv y) cardidm syl6eq (cv v) (` (card) (cv y)) (` (card) (cv v)) eqeq2 mpbird (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))) adantr y 19.23aiv sylbi (cv v) cardon syl6eqelr ssriv ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w)))))))) onint mpan syl eqeltrd (cv v) (` (cf) A) (card) fveq2 (= (cv v) (` (cf) A)) id eqeq12d v visset (cv x) (cv v) (` (card) (cv y)) eqeq1 (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))) anbi1d y exbidv elab (cv v) (` (card) (cv y)) (card) fveq2 (cv y) cardidm syl6eq (cv v) (` (card) (cv y)) (` (card) (cv v)) eqeq2 mpbird (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))) adantr y 19.23aiv sylbi vtoclga syl cffnon (cf) (On) fndm ax-mp A eleq2i negbii A (cf) ndmfv sylbir card0 (` (cf) A) ({/}) (card) fveq2 (= (` (cf) A) ({/})) id eqeq12d mpbiri syl pm2.61i)) thm (cflecard ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (C_ (` (cf) A) (` (card) A)) (A x y z w cfval A ssid (cv z) ssid (cv w) (cv z) (cv z) sseq2 A rcla4ev mpan2 rgen pm3.2i (cv y) A (card) fveq2 (cv x) eqeq2d (cv y) A A sseq1 (cv y) A w (C_ (cv z) (cv w)) rexeq1 z A ralbidv anbi12d anbi12d (On) cla4egv mpan2i x 19.21aiv x (= (cv x) (` (card) A)) (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))))) ss2ab sylibr (` (card) A) x df-sn syl5ss ({} (` (card) A)) ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w)))))))) intss syl (card) A fvex intsn syl6ss eqsstrd (` (card) A) 0ss cffnon (cf) (On) fndm ax-mp A eleq2i negbii A (cf) ndmfv sylbir (` (card) A) sseq1d mpbiri pm2.61i)) thm (cfle () () (C_ (` (cf) A) A) (A cardonle A cflecard (` (cf) A) (` (card) A) A sstr2 ax-mp syl A 0ss cffnon (cf) (On) fndm ax-mp A eleq2i negbii A (cf) ndmfv sylbir A sseq1d mpbiri pm2.61i)) thm (cfeq0 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (e. A (On)) (<-> (= (` (cf) A) ({/})) (= A ({/})))) (A x y z w cfval ({/}) eqeq1d v visset (cv x) (cv v) (` (card) (cv y)) eqeq1 (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))) anbi1d y exbidv elab (cv v) (` (card) (cv y)) (card) fveq2 (cv y) cardidm syl6eq (cv v) (` (card) (cv y)) (` (card) (cv v)) eqeq2 mpbird (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))) adantr y 19.23aiv sylbi (cv v) cardon syl6eqelr ssriv ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w)))))))) onint0 ax-mp 0ex (cv x) ({/}) (` (card) (cv y)) eqeq1 (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))) anbi1d y exbidv elab (cv y) ({/}) A sseq1 (cv y) ({/}) w (C_ (cv z) (cv w)) rexeq1 z A ralbidv anbi12d biimpa ({/}) (` (card) (cv y)) eqcom y visset (cv y) (V) cardeq0 ax-mp bitr sylanb w (C_ (cv z) (cv w)) rex0 (e. (cv z) A) a1i rgen A z (-. (E.e. w ({/}) (C_ (cv z) (cv w)))) r19.2zOLD mpi z A (E.e. w ({/}) (C_ (cv z) (cv w))) rexnal sylib a3i (C_ ({/}) A) adantl syl y 19.23aiv sylbi sylbi syl6bi A ({/}) (cf) fveq2 cf0 syl6eq (e. A (On)) a1i impbid)) thm (cfsuc ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (e. A (On)) (= (` (cf) (suc A)) (1o))) (A sucelon (suc A) x y z w cfval sylbi A snex (cv y) ({} A) (card) fveq2 (1o) eqeq2d (cv y) ({} A) (suc A) sseq1 (cv y) ({} A) w (C_ (cv z) (cv w)) rexeq1 z (suc A) ralbidv anbi12d anbi12d cla4ev A (On) cardsn eqcomd A (cv z) onelsst (cv z) A eqimss (e. A (On)) a1i jaod (cv z) A elsuci syl5 A (On) snidg jctild (cv w) A (cv z) sseq2 ({} A) rcla4ev syl6 r19.21aiv ({} A) A ssun2 A df-suc sseqtr4 jctil sylanc 1on elisseti (cv x) (1o) (` (card) (cv y)) eqeq1 (/\ (C_ (cv y) (suc A)) (A.e. z (suc A) (E.e. w (cv y) (C_ (cv z) (cv w))))) anbi1d y exbidv elab sylibr (cv v) el1o ({/}) (` (card) (cv y)) eqcom y visset (cv y) (V) cardeq0 ax-mp bitr w (C_ (cv z) (cv w)) rex0 (e. (cv z) (suc A)) a1i nrex A nsuceq0 (suc A) z (E.e. w ({/}) (C_ (cv z) (cv w))) r19.2zOLD ax-mp mto (cv y) ({/}) w (C_ (cv z) (cv w)) rexeq1 z (suc A) ralbidv mtbiri sylbi (C_ (cv y) (suc A)) intnand (= ({/}) (` (card) (cv y))) (/\ (C_ (cv y) (suc A)) (A.e. z (suc A) (E.e. w (cv y) (C_ (cv z) (cv w))))) imnan mpbi y nex 0ex (cv x) ({/}) (` (card) (cv y)) eqeq1 (/\ (C_ (cv y) (suc A)) (A.e. z (suc A) (E.e. w (cv y) (C_ (cv z) (cv w))))) anbi1d y exbidv elab mtbir (cv v) ({/}) ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) (suc A)) (A.e. z (suc A) (E.e. w (cv y) (C_ (cv z) (cv w)))))))) eleq1 mtbiri sylbi rgen jctir (cv y) cardon (cv x) (` (card) (cv y)) (On) eleq1 mpbiri (/\ (C_ (cv y) (suc A)) (A.e. z (suc A) (E.e. w (cv y) (C_ (cv z) (cv w))))) adantr y 19.23aiv abssi ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) (suc A)) (A.e. z (suc A) (E.e. w (cv y) (C_ (cv z) (cv w)))))))) (1o) v oneqmini ax-mp syl eqtr4d)) thm (cfom ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (` (cf) (om)) (om)) ((om) cfle omex intsn (cv x) (` (card) (cv y)) (om) eqtrt y visset z w unbnn2 y visset omex (cv y) (V) (om) (V) carden mp2an sylibr cardom syl6eq sylan2 y 19.23aiv x ss2abi (om) x df-sn sseqtr4 ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) (om)) (A.e. z (om) (E.e. w (cv y) (C_ (cv z) (cv w)))))))) ({} (om)) intss ax-mp omelon (om) x y z w cfval ax-mp sseqtr4 eqsstr3 eqssi)) thm (cdavalt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A C) (e. B D)) (= (opr A (+c) B) (u. (X. A ({} ({/}))) (X. B ({} (1o)))))) (p0ex A (V) ({} ({/})) (V) xpexg mpan2 (1o) snex B (V) ({} (1o)) (V) xpexg mpan2 anim12i (X. A ({} ({/}))) (X. B ({} (1o))) unexb sylib (cv x) A ({} ({/})) xpeq1 (X. (cv y) ({} (1o))) uneq1d (cv y) B ({} (1o)) xpeq1 (X. A ({} ({/}))) uneq2d x y z df-cda x visset y visset pm3.2i (= (cv z) (u. (X. (cv x) ({} ({/}))) (X. (cv y) ({} (1o))))) biantrur x y z oprabbii eqtr (V) oprabval2g 3expa mpdan A C elisset B D elisset syl2an)) thm (cdaval () ((cdaval.1 (e. A (V))) (cdaval.2 (e. B (V)))) (= (opr A (+c) B) (u. (X. A ({} ({/}))) (X. B ({} (1o))))) (cdaval.1 cdaval.2 A (V) B (V) cdavalt mp2an)) thm (uncdadom () ((cdaval.1 (e. A (V))) (cdaval.2 (e. B (V)))) (br (u. A B) (~<_) (opr A (+c) B)) (cdaval.1 cdaval.1 0ex xpsnen ensymi A (X. A ({} ({/}))) endom ax-mp cdaval.2 cdaval.2 1on elisseti xpsnen ensymi B (X. B ({} (1o))) endom ax-mp pm3.2i 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) A B xpsndisj ax-mp cdaval.1 p0ex xpex cdaval.2 cdaval.2 (1o) snex xpex A undom mp2an cdaval.1 cdaval.2 cdaval breqtrr)) thm (cdaen () ((cdaen.1 (e. A (V))) (cdaen.2 (e. B (V))) (cdaen.3 (e. C (V))) (cdaen.4 (e. D (V)))) (-> (/\ (br A (~~) B) (br C (~~) D)) (br (opr A (+c) C) (~~) (opr B (+c) D))) (1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) A C xpsndisj ax-mp 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) B D xpsndisj ax-mp pm3.2i (X. A ({} ({/}))) (X. B ({} ({/}))) (X. C ({} (1o))) (X. D ({} (1o))) unen mpan2 cdaen.1 cdaen.1 0ex xpsnen A (V) (X. A ({} ({/}))) (X. B ({} ({/}))) enen1 mp2an cdaen.2 cdaen.2 0ex xpsnen B (V) (X. B ({} ({/}))) A enen2 mp2an bitr cdaen.3 cdaen.3 1on elisseti xpsnen C (V) (X. C ({} (1o))) (X. D ({} (1o))) enen1 mp2an cdaen.4 cdaen.4 1on elisseti xpsnen D (V) (X. D ({} (1o))) C enen2 mp2an bitr syl2anbr cdaen.1 cdaen.3 cdaval cdaen.2 cdaen.4 cdaval 3brtr4g)) thm (cda0en () ((cda0en.1 (e. A (V)))) (br (opr A (+c) ({/})) (~~) A) (cda0en.1 0ex cdaval ({} (1o)) xp0r (X. A ({} ({/}))) uneq2i (X. A ({} ({/}))) un0 3eqtr cda0en.1 0ex xpsnen eqbrtr)) thm (cda1en () ((cda0en.1 (e. A (V)))) (br (opr A (+c) (1o)) (~~) (suc (` (card) A))) (cda0en.1 cda0en.1 0ex xpsnen A cardid entr4 1on elisseti 1on elisseti 1on elisseti xpsnen (card) A fvex ensn1 entr4 pm3.2i 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) A (1o) xpsndisj ax-mp A cardon onord (` (card) A) orddisj ax-mp pm3.2i (X. A ({} ({/}))) (` (card) A) (X. (1o) ({} (1o))) ({} (` (card) A)) unen mp2an cda0en.1 1on elisseti cdaval (` (card) A) df-suc 3brtr4)) thm (xp1en () ((cda0en.1 (e. A (V)))) (br (X. A (1o)) (~~) A) (df1o2 (1o) ({} ({/})) A xpeq2 ax-mp cda0en.1 0ex xpsnen eqbrtr)) thm (xp2cda () ((cda0en.1 (e. A (V)))) (= (X. A (2o)) (opr A (+c) A)) (A ({} ({/})) ({} (1o)) xpundi ({/}) ({} ({/})) df-pr df2o2 df1o2 sneqi ({} ({/})) uneq2i 3eqtr4 (2o) (u. ({} ({/})) ({} (1o))) A xpeq2 ax-mp cda0en.1 cda0en.1 cdaval 3eqtr4)) thm (cdacomen () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V)))) (br (opr A (+c) B) (~~) (opr B (+c) A)) (1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) A B xpsndisj ax-mp 1ne0 (1o) ({/}) A B xpsndisj ax-mp cdacomen.1 cdacomen.1 0ex xpsnen cdacomen.1 1on elisseti xpsnen entr4 cdacomen.2 cdacomen.2 1on elisseti xpsnen cdacomen.2 0ex xpsnen entr4 (X. A ({} ({/}))) (X. A ({} (1o))) (X. B ({} (1o))) (X. B ({} ({/}))) unen mpanl12 mp2an (X. A ({} (1o))) (X. B ({} ({/}))) uncom breqtr cdacomen.1 cdacomen.2 cdaval cdacomen.2 cdacomen.1 cdaval 3brtr4)) thm (cdaassen () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V))) (cdaassen.3 (e. C (V)))) (br (opr (opr A (+c) B) (+c) C) (~~) (opr A (+c) (opr B (+c) C))) (cdacomen.1 p0ex xpex cdacomen.2 (1o) snex xpex unex p0ex xpex (X. A ({} ({/}))) (X. B ({} (1o))) ({} ({/})) xpundir (X. (u. (X. A ({} ({/}))) (X. B ({} (1o)))) ({} ({/}))) (V) (u. (X. (X. A ({} ({/}))) ({} ({/}))) (X. (X. B ({} (1o))) ({} ({/})))) eqeng mp2 cdaassen.3 (1o) snex xpex cdaassen.3 (1o) snex xpex 1on elisseti xpsnen ensymi pm3.2i 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) (u. (X. A ({} ({/}))) (X. B ({} (1o)))) C xpsndisj ax-mp 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) (X. A ({} ({/}))) (X. C ({} (1o))) xpsndisj ax-mp 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) (X. B ({} (1o))) (X. C ({} (1o))) xpsndisj ax-mp pm3.2i (X. (X. A ({} ({/}))) ({} ({/}))) (X. (X. C ({} (1o))) ({} (1o))) (X. (X. B ({} (1o))) ({} ({/}))) undisj1 mpbi pm3.2i (X. (u. (X. A ({} ({/}))) (X. B ({} (1o)))) ({} ({/}))) (u. (X. (X. A ({} ({/}))) ({} ({/}))) (X. (X. B ({} (1o))) ({} ({/})))) (X. C ({} (1o))) (X. (X. C ({} (1o))) ({} (1o))) unen mp2an cdacomen.1 p0ex xpex p0ex xpex cdacomen.2 p0ex xpex (1o) snex xpex cdaassen.3 (1o) snex xpex (1o) snex xpex unex unex cdacomen.1 p0ex xpex p0ex xpex enref cdacomen.2 (1o) snex p0ex xpassen cdacomen.2 p0ex (1o) snex xpex xpex cdacomen.2 enref (1o) snex p0ex xpcomen cdacomen.2 cdacomen.2 (1o) snex p0ex xpex p0ex (1o) snex xpex xpen mp2an cdacomen.2 p0ex (1o) snex xpassen entr4 entr pm3.2i 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) A B xpsndisj ax-mp (i^i (X. A ({} ({/}))) (X. B ({} (1o)))) ({/}) ({} ({/})) xpeq1 ax-mp (X. A ({} ({/}))) (X. B ({} (1o))) ({} ({/})) xpindir ({} ({/})) xp0r 3eqtr3 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) (X. A ({} ({/}))) (X. B ({} ({/}))) xpsndisj ax-mp pm3.2i (X. (X. A ({} ({/}))) ({} ({/}))) (X. (X. A ({} ({/}))) ({} ({/}))) (X. (X. B ({} (1o))) ({} ({/}))) (X. (X. B ({} ({/}))) ({} (1o))) unen mp2an cdaassen.3 (1o) snex xpex (1o) snex xpex enref pm3.2i 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) (X. A ({} ({/}))) (X. C ({} (1o))) xpsndisj ax-mp 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) (X. B ({} (1o))) (X. C ({} (1o))) xpsndisj ax-mp pm3.2i (X. (X. A ({} ({/}))) ({} ({/}))) (X. (X. C ({} (1o))) ({} (1o))) (X. (X. B ({} (1o))) ({} ({/}))) undisj1 mpbi 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) (X. A ({} ({/}))) (X. C ({} (1o))) xpsndisj ax-mp 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) B C xpsndisj ax-mp (i^i (X. B ({} ({/}))) (X. C ({} (1o)))) ({/}) ({} (1o)) xpeq1 ax-mp (X. B ({} ({/}))) (X. C ({} (1o))) ({} (1o)) xpindir ({} (1o)) xp0r 3eqtr3 pm3.2i (X. (X. A ({} ({/}))) ({} ({/}))) (X. (X. C ({} (1o))) ({} (1o))) (X. (X. B ({} ({/}))) ({} (1o))) undisj1 mpbi pm3.2i (u. (X. (X. A ({} ({/}))) ({} ({/}))) (X. (X. B ({} (1o))) ({} ({/})))) (u. (X. (X. A ({} ({/}))) ({} ({/}))) (X. (X. B ({} ({/}))) ({} (1o)))) (X. (X. C ({} (1o))) ({} (1o))) (X. (X. C ({} (1o))) ({} (1o))) unen mp2an (X. (X. A ({} ({/}))) ({} ({/}))) (X. (X. B ({} ({/}))) ({} (1o))) (X. (X. C ({} (1o))) ({} (1o))) unass breqtr cdacomen.1 p0ex xpex cdacomen.1 p0ex xpex 0ex xpsnen ensymi cdacomen.2 p0ex xpex cdaassen.3 (1o) snex xpex unex (1o) snex xpex (X. B ({} ({/}))) (X. C ({} (1o))) ({} (1o)) xpundir (X. (u. (X. B ({} ({/}))) (X. C ({} (1o)))) ({} (1o))) (V) (u. (X. (X. B ({} ({/}))) ({} (1o))) (X. (X. C ({} (1o))) ({} (1o)))) eqeng mp2 pm3.2i 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) A (u. (X. B ({} ({/}))) (X. C ({} (1o)))) xpsndisj ax-mp 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) (X. A ({} ({/}))) (X. B ({} ({/}))) xpsndisj ax-mp 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) (X. A ({} ({/}))) (X. C ({} (1o))) xpsndisj ax-mp pm3.2i (X. (X. A ({} ({/}))) ({} ({/}))) (X. (X. B ({} ({/}))) ({} (1o))) (X. (X. C ({} (1o))) ({} (1o))) undisj2 mpbi pm3.2i (X. A ({} ({/}))) (X. (X. A ({} ({/}))) ({} ({/}))) (X. (u. (X. B ({} ({/}))) (X. C ({} (1o)))) ({} (1o))) (u. (X. (X. B ({} ({/}))) ({} (1o))) (X. (X. C ({} (1o))) ({} (1o)))) unen mp2an entr4 entr cdacomen.1 cdacomen.2 cdaval (+c) C opreq1i cdacomen.1 p0ex xpex cdacomen.2 (1o) snex xpex unex cdaassen.3 cdaval eqtr cdacomen.2 cdaassen.3 cdaval A (+c) opreq2i cdacomen.1 cdacomen.2 p0ex xpex cdaassen.3 (1o) snex xpex unex cdaval eqtr 3brtr4)) thm (xpcdaen () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V))) (cdaassen.3 (e. C (V)))) (br (X. A (opr B (+c) C)) (~~) (opr (X. A B) (+c) (X. A C))) (cdacomen.1 cdacomen.2 p0ex xpex cdaassen.3 (1o) snex xpex unex xpex cdacomen.1 cdacomen.2 p0ex xpassen cdacomen.1 cdaassen.3 (1o) snex xpassen pm3.2i 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) (X. A B) (X. A C) xpsndisj ax-mp 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) B C xpsndisj ax-mp (i^i (X. B ({} ({/}))) (X. C ({} (1o)))) ({/}) A xpeq2 ax-mp A (X. B ({} ({/}))) (X. C ({} (1o))) xpindi A xp0 3eqtr3 pm3.2i (X. (X. A B) ({} ({/}))) (X. A (X. B ({} ({/})))) (X. (X. A C) ({} (1o))) (X. A (X. C ({} (1o)))) unen mp2an A (X. B ({} ({/}))) (X. C ({} (1o))) xpundi breqtrr ensymi cdacomen.2 cdaassen.3 cdaval (opr B (+c) C) (u. (X. B ({} ({/}))) (X. C ({} (1o)))) A xpeq2 ax-mp cdacomen.1 cdacomen.2 xpex cdacomen.1 cdaassen.3 xpex cdaval 3brtr4)) thm (mapcdaen () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V))) (cdaassen.3 (e. C (V)))) (br (opr A (^m) (opr B (+c) C)) (~~) (X. (opr A (^m) B) (opr A (^m) C))) (cdacomen.2 cdaassen.3 cdaval A (^m) opreq2i 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) B C xpsndisj ax-mp cdacomen.2 p0ex xpex cdaassen.3 (1o) snex xpex cdacomen.1 mapunen ax-mp cdacomen.1 enref cdacomen.2 0ex xpsnen cdacomen.1 cdacomen.1 cdacomen.2 p0ex xpex cdacomen.2 mapen mp2an cdacomen.1 enref cdaassen.3 1on elisseti xpsnen cdacomen.1 cdacomen.1 cdaassen.3 (1o) snex xpex cdaassen.3 mapen mp2an A (^m) (X. B ({} ({/}))) oprex A (^m) B oprex A (^m) (X. C ({} (1o))) oprex A (^m) C oprex xpen mp2an entr eqbrtr)) thm (cdadom1 () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V))) (cdaassen.3 (e. C (V)))) (-> (br A (~<_) B) (br (opr A (+c) C) (~<_) (opr B (+c) C))) (cdacomen.1 cdacomen.1 0ex xpsnen A (V) (X. A ({} ({/}))) (X. B ({} ({/}))) domen1 mp2an cdacomen.2 cdacomen.2 0ex xpsnen B (V) (X. B ({} ({/}))) A domen2 mp2an bitr cdaassen.3 (1o) snex xpex (X. C ({} (1o))) (V) domrefg ax-mp 1ne0 ({/}) (1o) eqcom mtbir ({/}) (1o) B C xpsndisj ax-mp cdacomen.2 p0ex xpex cdaassen.3 (1o) snex xpex cdaassen.3 (1o) snex xpex (X. A ({} ({/}))) undom mpan2 mpan2 sylbir cdacomen.1 cdaassen.3 cdaval cdacomen.2 cdaassen.3 cdaval 3brtr4g)) thm (cdadom2 () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V))) (cdaassen.3 (e. C (V)))) (-> (br A (~<_) B) (br (opr C (+c) A) (~<_) (opr C (+c) B))) (cdacomen.1 cdacomen.2 cdaassen.3 cdadom1 cdaassen.3 cdacomen.1 cdacomen (opr C (+c) A) (opr A (+c) C) (opr B (+c) C) endomtr mpan cdacomen.2 cdaassen.3 cdacomen (opr C (+c) A) (opr B (+c) C) (opr C (+c) B) domentr mpan2 3syl)) thm (cdadom3 () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V)))) (br A (~<_) (opr A (+c) B)) (cdacomen.1 A B ssun1 A (V) (u. A B) ssdomg mp2 cdacomen.1 cdacomen.2 uncdadom A (u. A B) (opr A (+c) B) domtr mp2an)) thm (cdafi () () (-> (/\ (br A (~<) (om)) (br B (~<) (om))) (br (opr A (+c) B) (~<) (om))) (A (V) B (V) cdavalt A (om) sdomex pm3.26d B (om) sdomex pm3.26d syl2an (X. A ({} ({/}))) (X. B ({} (1o))) unfi2 A (V) (X. A ({} ({/}))) (om) sdomen1 A (om) sdomex pm3.26d A (om) sdomex pm3.26d 0elon A (V) ({/}) (On) xpsneng mpan2 syl sylanc ibir B (V) (X. B ({} (1o))) (om) sdomen1 B (om) sdomex pm3.26d B (om) sdomex pm3.26d 1on B (V) (1o) (On) xpsneng mpan2 syl sylanc ibir syl2an eqbrtrd)) thm (cdainf () ((cdainf.1 (e. A (V)))) (<-> (br (om) (~<_) A) (br (om) (~<_) (opr A (+c) A))) (cdainf.1 cdainf.1 cdadom3 (om) A (opr A (+c) A) domtr mpan2 A A cdafi anidms con3i omex A (+c) A oprex (om) (V) (opr A (+c) A) (V) domtri mp2an omex cdainf.1 (om) (V) A (V) domtri mp2an 3imtr4 impbi)) thm (nd1 () () (-> (A. x (= (cv x) (cv y))) (-. (A. x (e. (cv y) (cv z))))) (z eirrv y (e. (cv y) (cv z)) z stdpc4 z eirrv (A. y (e. (cv z) (cv z))) pm2.21i y z z elequ1 sbie sylib mto x y (e. (cv y) (cv z)) ax-10 mtoi)) thm (nd2 () () (-> (A. x (= (cv x) (cv y))) (-. (A. x (e. (cv z) (cv y))))) (z eirrv y (e. (cv z) (cv y)) z stdpc4 z eirrv (A. y (e. (cv z) (cv z))) pm2.21i y z z elequ2 sbie sylib mto x y (e. (cv z) (cv y)) ax-10 mtoi)) thm (nd3 () () (-> (A. x (= (cv x) (cv y))) (-. (A. z (e. (cv x) (cv y))))) (x (= (cv x) (cv y)) ax-4 x eirrv x y x elequ2 mtbii z (e. (cv x) (cv y)) ax-4 con3i 3syl)) thm (nd4 () () (-> (A. x (= (cv x) (cv y))) (-. (A. z (e. (cv y) (cv x))))) (y x z nd3 alequcoms)) thm (nd5 ((x z)) () (-> (-. (A. y (= (cv y) (cv x)))) (-> (= (cv z) (cv y)) (A. x (= (cv z) (cv y))))) (x y z dveeq2 nalequcoms)) thm (axextnd ((w x) (w y) (w z)) () (E. x (-> (<-> (e. (cv x) (cv y)) (e. (cv x) (cv z))) (= (cv y) (cv z)))) (x y x hbnae x z x hbnae hban x y x hbnae x z x hbnae hban x y w dveel2 (-. (A. x (= (cv x) (cv z)))) adantr x z w dveel2 (-. (A. x (= (cv x) (cv y)))) adantl hbbid w x y elequ1 w x z elequ1 bibi12d (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) a1i cbvald w y z zfext2 syl6bir (= (cv y) (cv z)) x 19.8a syl6 ex x z a9e x y x hbae x y z ax-8 x a4s 19.22d mpi (A. x (<-> (e. (cv x) (cv y)) (e. (cv x) (cv z)))) a1d x y a9e x z x hbae x z y ax-8 z y equcomi syl6 x a4s 19.22d mpi (A. x (<-> (e. (cv x) (cv y)) (e. (cv x) (cv z)))) a1d pm2.61ii 19.35ri)) thm (axrepndlem1 ((x z) (w x) (w z) (x y) (w y) (ph w)) () (-> (-. (A. y (= (cv y) (cv z)))) (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv y)) (A. y ph)))))))) (x y w ([/] (cv w) z ph) axrep2 y z x hbnae y z y hbnae y z z hbnae y z z hbnae ph w ax-17 z hbsb3 (-. (A. y (= (cv y) (cv z)))) a1i y z w nd5 hbimd w z ph sbequ12r w z y equequ1 imbi12d (-. (A. y (= (cv y) (cv z)))) a1i cbvald exbid y z z hbnae y z z hbnae (e. (cv w) (cv x)) z ax-17 (-. (A. y (= (cv y) (cv z)))) a1i y z x hbnae z y x dveel2 nalequcoms ph w ax-17 z hbsb3 y hbal (-. (A. y (= (cv y) (cv z)))) a1i hband hbexd hbbid w z x elequ1 (-. (A. y (= (cv y) (cv z)))) adantl y z w dveeq2 imp y (= (cv w) (cv z)) hba1 w z ph sbequ12r y a4s albid syl (e. (cv x) (cv y)) anbi2d x exbidv bibi12d ex cbvald imbi12d exbid mpbii)) thm (axrepndlem2 ((w x) (w y) (w z) (ph w)) () (-> (/\ (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) (-. (A. y (= (cv y) (cv z))))) (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv y)) (A. y ph)))))))) (x y x hbnae x z x hbnae hban x y x hbnae x z x hbnae hban x y y hbnae x z y hbnae hban x y z hbnae x z z hbnae hban x y x hbnae x z x hbnae hban ph w ax-17 x hbsb3 (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) a1i x z y ax-12 impcom hbimd hbald hbexd x y z hbnae x z z hbnae hban x y x hbnae x z x hbnae hban x z w dveel1 (-. (A. x (= (cv x) (cv y)))) adantl (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) w ax-17 x y w dveel2 (-. (A. x (= (cv x) (cv z)))) adantr x y y hbnae x z y hbnae hban ph w ax-17 x hbsb3 (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) a1i hbald hband hbexd hbbid hbald hbimd x y w nd5 (-. (A. x (= (cv x) (cv z)))) adantr imdistani x y y hbnae x z y hbnae hban y (= (cv w) (cv x)) hba1 hban x z w nd5 imdistani x z z hbnae z (= (cv w) (cv x)) hba1 hban w x ph sbequ12r (= (cv z) (cv y)) imbi1d z a4s (-. (A. x (= (cv x) (cv z)))) adantl albid syl (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z)))) pm3.27 y (= (cv w) (cv x)) ax-4 syl2an exbid syl x z w nd5 (-. (A. x (= (cv x) (cv y)))) adantl imdistani x y z hbnae x z z hbnae hban z (= (cv w) (cv x)) hba1 hban w x z elequ2 (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) adantl x y x hbnae x z x hbnae hban x y w dveel2 (-. (A. x (= (cv x) (cv z)))) adantr x y y hbnae x z y hbnae hban ph w ax-17 x hbsb3 (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) a1i hbald hband w x y elequ1 (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) adantl y x w ph sbal2 nalequcoms bicomd w x (A. y ph) sbequ12r sylan9bb (-. (A. x (= (cv x) (cv z)))) adantlr anbi12d ex cbvexd (= (cv w) (cv x)) adantr bibi12d z (= (cv w) (cv x)) ax-4 sylan2 albid syl imbi12d ex cbvexd y z w ([/] (cv w) x ph) axrepndlem1 syl5bi imp)) thm (axrepnd () () (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (A. y (e. (cv z) (cv x))) (E. x (/\ (A. z (e. (cv x) (cv y))) (A. y ph))))))) (x y z ph axrepndlem2 x y x hbnae x z x hbnae hban y z x hbnae hban x y z hbnae x z z hbnae hban y z z hbnae hban y z x ax15 com12 nalequcoms imp (-. (A. x (= (cv x) (cv z)))) adantlr y (e. (cv z) (cv x)) ax-4 (/\ (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) (-. (A. y (= (cv y) (cv z))))) a1i impbid x y x hbnae x z x hbnae hban y z x hbnae hban z x y ax15 imp z x alequcom con3i z y alequcom con3i syl2an (-. (A. x (= (cv x) (cv y)))) adantll z (e. (cv x) (cv y)) ax-4 (/\ (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) (-. (A. y (= (cv y) (cv z))))) a1i impbid (A. y ph) anbi1d exbid bibi12d albid (E. y (A. z (-> ph (= (cv z) (cv y))))) imbi2d exbid mpbid exp31 x y z hbae (A. y (e. (cv z) (cv x))) (E. x (/\ (A. z (e. (cv x) (cv y))) (A. y ph))) pm5.21 y x z nd2 alequcoms x y x hbae x y z nd3 (A. y ph) intnanrd nexd sylanc 19.21ai (E. y (A. z (-> ph (= (cv z) (cv y))))) a1d (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (A. y (e. (cv z) (cv x))) (E. x (/\ (A. z (e. (cv x) (cv y))) (A. y ph)))))) x 19.8a syl x z z hbae (A. y (e. (cv z) (cv x))) (E. x (/\ (A. z (e. (cv x) (cv y))) (A. y ph))) pm5.21 x z y nd4 x z x hbae z x y nd1 alequcoms (A. y ph) intnanrd nexd sylanc 19.21ai (E. y (A. z (-> ph (= (cv z) (cv y))))) a1d (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (A. y (e. (cv z) (cv x))) (E. x (/\ (A. z (e. (cv x) (cv y))) (A. y ph)))))) x 19.8a syl y z z hbae (A. y (e. (cv z) (cv x))) (E. x (/\ (A. z (e. (cv x) (cv y))) (A. y ph))) pm5.21 y z x nd1 y z x hbae z y x nd2 alequcoms (A. y ph) intnanrd nexd sylanc 19.21ai (E. y (A. z (-> ph (= (cv z) (cv y))))) a1d (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (A. y (e. (cv z) (cv x))) (E. x (/\ (A. z (e. (cv x) (cv y))) (A. y ph)))))) x 19.8a syl pm2.61iii)) thm (axunndlem1 ((x y) (w x) (w y) (x z) (w z)) () (E. x (A. y (-> (E. x (/\ (e. (cv y) (cv x)) (e. (cv x) (cv z)))) (e. (cv y) (cv x))))) (y z x hbae (cv y) (cv x) en2lp y z x elequ2 (e. (cv y) (cv x)) anbi2d mtbii y a4s nexd (e. (cv y) (cv x)) pm2.21d a5i (A. y (-> (E. x (/\ (e. (cv y) (cv x)) (e. (cv x) (cv z)))) (e. (cv y) (cv x)))) x 19.8a syl x w z axun y z x hbnae y z y hbnae y z y hbnae y z x hbnae (e. (cv w) (cv x)) y ax-17 (-. (A. y (= (cv y) (cv z)))) a1i y z x dveel2 hband hbexd (e. (cv w) (cv x)) y ax-17 (-. (A. y (= (cv y) (cv z)))) a1i hbimd w y x elequ1 (e. (cv x) (cv z)) anbi1d x exbidv w y x elequ1 imbi12d (-. (A. y (= (cv y) (cv z)))) a1i cbvald exbid mpbii pm2.61i)) thm (axunnd ((w x) (w y) (w z)) () (E. x (A. y (-> (E. x (/\ (e. (cv y) (cv x)) (e. (cv x) (cv z)))) (e. (cv y) (cv x))))) (w y z axunndlem1 x y x hbnae x z x hbnae hban x y y hbnae x z y hbnae hban x y x hbnae x z x hbnae hban (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) w ax-17 x y w dveel1 (-. (A. x (= (cv x) (cv z)))) adantr x z w dveel2 (-. (A. x (= (cv x) (cv y)))) adantl hband hbexd x y w dveel1 (-. (A. x (= (cv x) (cv z)))) adantr hbimd hbald x y w nd5 (-. (A. x (= (cv x) (cv z)))) adantr imdistani x y y hbnae x z y hbnae hban y (= (cv w) (cv x)) hba1 hban x y x hbnae x z x hbnae hban x y w dveel1 (-. (A. x (= (cv x) (cv z)))) adantr x z w dveel2 (-. (A. x (= (cv x) (cv y)))) adantl hband w x y elequ2 w x z elequ1 anbi12d (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) a1i cbvexd (A. y (= (cv w) (cv x))) adantr w x y elequ2 y a4s (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) adantl imbi12d albid syl ex cbvexd mpbii ex x y y hbae x y x hbae y eirrv x y y elequ2 (e. (cv y) (cv x)) (e. (cv x) (cv z)) pm3.26 syl5bi mtoi x a4s nexd (e. (cv y) (cv x)) pm2.21d 19.21ai (A. y (-> (E. x (/\ (e. (cv y) (cv x)) (e. (cv x) (cv z)))) (e. (cv y) (cv x)))) x 19.8a syl x z y hbae x z x hbae z eirrv x z z elequ1 (e. (cv y) (cv x)) (e. (cv x) (cv z)) pm3.27 syl5bi mtoi x a4s nexd (e. (cv y) (cv x)) pm2.21d 19.21ai (A. y (-> (E. x (/\ (e. (cv y) (cv x)) (e. (cv x) (cv z)))) (e. (cv y) (cv x)))) x 19.8a syl pm2.61ii)) thm (axpowndlem1 () () (-> (A. x (= (cv x) (cv y))) (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x))))))) ((= (cv x) (cv y)) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x))))) pm2.24 x a4s)) thm (axpowndlem2 ((w x) (y z) (w z) (w y)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (-. (A. x (= (cv x) (cv z)))) (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x)))))))) (w y z axpow (e. (cv w) (cv y)) z 19.8a y (e. (cv w) (cv z)) ax-4 imim12i w 19.20i (e. (cv y) (cv w)) imim1i y 19.20i w 19.22i ax-mp (-. (= (cv w) (cv y))) a1i w ax-gen x y x hbnae x z x hbnae hban x y x hbnae x z x hbnae hban x y x hbnae x y w dveeq2 hbnd (-. (A. x (= (cv x) (cv z)))) adantr (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) w ax-17 x y y hbnae x z y hbnae hban x y x hbnae x z x hbnae hban x y w hbnae x z w hbnae hban x y x hbnae x z x hbnae hban x y z hbnae x z z hbnae hban x y w dveel2 (-. (A. x (= (cv x) (cv z)))) adantr hbexd x y y hbnae x z y hbnae hban x z w dveel2 (-. (A. x (= (cv x) (cv y)))) adantl hbald hbimd hbald x y w dveel1 (-. (A. x (= (cv x) (cv z)))) adantr hbimd hbald hbexd hbimd w x y equequ1 negbid (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) adantl x y x hbnae x z x hbnae hban x y y hbnae x z y hbnae hban x y x hbnae x z x hbnae hban (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) w ax-17 x y x hbnae x z x hbnae hban x y z hbnae x z z hbnae hban x y w dveel2 (-. (A. x (= (cv x) (cv z)))) adantr hbexd x y y hbnae x z y hbnae hban x z w dveel2 (-. (A. x (= (cv x) (cv y)))) adantl hbald hbimd hbald x y w dveel1 (-. (A. x (= (cv x) (cv z)))) adantr hbimd hbald x y w nd5 (-. (A. x (= (cv x) (cv z)))) adantr imdistani x y y hbnae x z y hbnae hban y (= (cv w) (cv x)) hba1 hban x y x hbnae x z x hbnae hban x y x hbnae x z x hbnae hban x y z hbnae x z z hbnae hban x y w dveel2 (-. (A. x (= (cv x) (cv z)))) adantr hbexd x y y hbnae x z y hbnae hban x z w dveel2 (-. (A. x (= (cv x) (cv y)))) adantl hbald hbimd x z w nd5 imdistani z (= (cv w) (cv x)) hba1 w x y elequ1 z a4s exbid (-. (A. x (= (cv x) (cv z)))) adantl syl (-. (A. x (= (cv x) (cv y)))) adantll x y w nd5 imdistani y (= (cv w) (cv x)) hba1 w x z elequ1 y a4s albid (-. (A. x (= (cv x) (cv y)))) adantl syl (-. (A. x (= (cv x) (cv z)))) adantlr imbi12d ex cbvald (A. y (= (cv w) (cv x))) adantr w x y elequ2 y a4s (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) adantl imbi12d albid syl ex cbvexd (= (cv w) (cv x)) adantr imbi12d ex cbvald mpbii 19.21bi ex)) thm (axpowndlem3 ((w x) (y z) (w y) (w z)) () (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x)))))) (x y z axpowndlem2 x y z axpowndlem1 x z x hbae x z y hbae x z y nd3 (A. y (e. (cv x) (cv z))) (E. z (e. (cv x) (cv y))) mtt syl z x (-. (e. (cv x) (cv y))) ax-10 alequcoms z (e. (cv x) (cv y)) alnex x (e. (cv x) (cv y)) alnex 3imtr3g sylbird x a4sd (e. (cv y) (cv x)) imim1d 19.20d 19.22d p0ex (cv x) ({} ({/})) (cv w) eleq2 (= (cv w) ({/})) imbi2d w albidv cla4ev 0ex snid (cv w) ({/}) ({} ({/})) eleq1 mpbiri mpg (cv w) x n0 con1bii (e. (cv w) (cv x)) imbi1i w albii x exbii mpbir x y x hbnae x y y hbnae x y y hbnae x y y hbnae x y x hbnae y x w dveel1 nalequcoms hbexd hbnd y x w dveel2 nalequcoms hbimd x y w dveeq2 imdistani x (= (cv w) (cv y)) hba1 w y x elequ2 x a4s exbid (-. (A. x (= (cv x) (cv y)))) adantl syl negbid w y x elequ1 (-. (A. x (= (cv x) (cv y)))) adantl imbi12d ex cbvald exbid mpbii syl5 (-. (= (cv x) (cv y))) a1dd x y z axpowndlem1 pm2.61d2 pm2.61ii)) thm (axpowndlem4 ((w x) (w y) (w z)) () (-> (-. (A. y (= (cv y) (cv x)))) (-> (-. (A. y (= (cv y) (cv z)))) (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x)))))))) (x w z axpowndlem3 w ax-gen y x y hbnae y z y hbnae hban y x y hbnae y z y hbnae hban y x y hbnae y x w dveeq1 hbnd (-. (A. y (= (cv y) (cv z)))) adantr y x x hbnae y z x hbnae hban (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv z))))) w ax-17 y x y hbnae y z y hbnae hban y x x hbnae y z x hbnae hban y x y hbnae y z y hbnae hban y x z hbnae y x w dveel1 hbexd (-. (A. y (= (cv y) (cv z)))) adantr (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv z))))) w ax-17 y x z ax15 imp hbald hbimd hbald y x w dveel2 (-. (A. y (= (cv y) (cv z)))) adantr hbimd hbald hbexd hbimd w y x equequ2 negbid (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv z))))) adantl y x x hbnae y z x hbnae hban y x y hbnae y z y hbnae hban y x y hbnae y z y hbnae hban y x x hbnae y z x hbnae hban y x y hbnae y z y hbnae hban y x z hbnae y x w dveel1 hbexd (-. (A. y (= (cv y) (cv z)))) adantr (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv z))))) w ax-17 y x z ax15 imp hbald hbimd hbald y x w dveel2 (-. (A. y (= (cv y) (cv z)))) adantr hbimd y x w nd5 (-. (A. y (= (cv y) (cv z)))) adantr imdistani y x x hbnae y z x hbnae hban x (= (cv w) (cv y)) hba1 hban y z w nd5 imdistani z (= (cv w) (cv y)) hba1 w y x elequ2 z a4s exbid (-. (A. y (= (cv y) (cv z)))) adantl syl x (= (cv w) (cv y)) ax-4 sylan2 (-. (A. y (= (cv y) (cv x)))) adantll y x y hbnae y z y hbnae hban y x z ax15 imp (= (cv w) (cv y)) (e. (cv x) (cv z)) pm4.2i (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv z))))) a1i cbvald (A. x (= (cv w) (cv y))) adantr imbi12d albid syl w y x elequ1 (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv z))))) adantl imbi12d ex cbvald exbid (= (cv w) (cv y)) adantr imbi12d ex cbvald mpbii 19.21bi ex)) thm (axpownd ((w x) (w y)) () (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x)))))) (y x z axpowndlem4 x y z axpowndlem1 alequcoms x y z axpowndlem1 (A. y (= (cv y) (cv z))) a1d x y y hbnae y z y hbae hban w x el x y y hbnae y x w dveel1 nalequcoms w y x elequ2 (-. (A. x (= (cv x) (cv y)))) a1i cbvexd mpbii (E. y (e. (cv x) (cv y))) x 19.8a syl x (E. y (e. (cv x) (cv y))) df-ex sylib (A. y (= (cv y) (cv z))) adantr (A. y (= (cv y) (cv z))) (-. (e. (cv x) (cv y))) pm4.2i dral1 y (e. (cv x) (cv y)) alnex z (e. (cv x) (cv y)) alnex 3bitr3g y z x nd2 (A. y (e. (cv x) (cv z))) (E. z (e. (cv x) (cv y))) mtt syl bitrd x dral2 (-. (A. x (= (cv x) (cv y)))) adantl mtbid (e. (cv y) (cv x)) pm2.21d 19.21ai (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x)))) x 19.8a syl (-. (= (cv x) (cv y))) a1d ex pm2.61i pm2.61ii)) thm (axregndlem1 () () (-> (A. x (= (cv x) (cv z))) (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y))))))))) (x z x hbae x z z hbae x eirrv x z x elequ1 mtbii x a4s (-. (e. (cv z) (cv y))) pm2.21d 19.21ai (e. (cv x) (cv y)) anim2i expcom 19.22d (e. (cv x) (cv y)) x 19.8a syl5)) thm (axregndlem2 ((w x) (y z) (w z) (w y)) () (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y)))))))) (w y z axreg w ax-gen x y x hbnae x z x hbnae hban x y x hbnae x z x hbnae hban x y w dveel2 (-. (A. x (= (cv x) (cv z)))) adantr (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) w ax-17 x y w dveel2 (-. (A. x (= (cv x) (cv z)))) adantr x y z hbnae x z z hbnae hban x y x hbnae x z x hbnae hban x z w dveel1 (-. (A. x (= (cv x) (cv y)))) adantl x y x hbnae x z x hbnae hban x z y ax15 impcom hbnd hbimd hbald hband hbexd hbimd w x y elequ1 (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) adantl x y x hbnae x z x hbnae hban x y w dveel2 (-. (A. x (= (cv x) (cv z)))) adantr x y z hbnae x z z hbnae hban x y x hbnae x z x hbnae hban x z w dveel1 (-. (A. x (= (cv x) (cv y)))) adantl x y x hbnae x z x hbnae hban x z y ax15 impcom hbnd hbimd hbald hband w x y elequ1 (-. (A. x (= (cv x) (cv z)))) adantl x z w nd5 imdistani x z z hbnae z (= (cv w) (cv x)) hba1 hban w x z elequ2 (-. (e. (cv z) (cv y))) imbi1d z a4s (-. (A. x (= (cv x) (cv z)))) adantl albid syl anbi12d ex (-. (A. x (= (cv x) (cv y)))) adantl cbvexd (= (cv w) (cv x)) adantr imbi12d ex cbvald mpbii 19.21bi ex x eirrv x y x elequ2 mtbii x a4s (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y))))))) pm2.21d x z y axregndlem1 pm2.61ii)) thm (axregnd ((w x) (w y) (w z)) () (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y)))))))) (z x x hbnae z y x hbnae hban z x z hbnae z y z hbnae hban z x z hbnae z y z hbnae hban z x w dveel2 (-. (A. z (= (cv z) (cv y)))) adantr z x z hbnae z y z hbnae hban z y w dveel2 (-. (A. z (= (cv z) (cv x)))) adantl hbnd hbimd w z x elequ1 w z y elequ1 negbid imbi12d (/\ (-. (A. z (= (cv z) (cv x)))) (-. (A. z (= (cv z) (cv y))))) a1i cbvald (e. (cv x) (cv y)) anbi2d exbid x y w axregndlem2 syl5bi ex x z y axregndlem1 alequcoms z y x hbae z eirrv z y z elequ2 mtbii z a4s (e. (cv z) (cv x)) a1d a5i (e. (cv x) (cv y)) anim2i expcom 19.22d (e. (cv x) (cv y)) x 19.8a syl5 pm2.61ii)) thm (axinfndlem1 ((w x) (y z) (w z) (w y)) () (-> (A. x (e. (cv y) (cv z))) (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x))))))))) (w y z axinf (A. w (e. (cv y) (cv z))) a1i x y x hbnae x z x hbnae hban x y z ax15 imp (= (cv w) (cv x)) (e. (cv y) (cv z)) pm4.2i (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) a1i cbvald x y x hbnae x z x hbnae hban x y w dveel1 (-. (A. x (= (cv x) (cv z)))) adantr x y y hbnae x z y hbnae hban x y x hbnae x z x hbnae hban x y w dveel1 (-. (A. x (= (cv x) (cv z)))) adantr x y z hbnae x z z hbnae hban x y z ax15 imp x z w dveel1 (-. (A. x (= (cv x) (cv y)))) adantl hband hbexd hbimd hbald hband w x y elequ2 (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) adantl x y w nd5 (-. (A. x (= (cv x) (cv z)))) adantr imdistani x z y hbnae y (= (cv w) (cv x)) hba1 hban w x y elequ2 (-. (A. x (= (cv x) (cv z)))) adantl x z w nd5 imdistani z (= (cv w) (cv x)) hba1 w x z elequ2 (e. (cv y) (cv z)) anbi2d z a4s exbid (-. (A. x (= (cv x) (cv z)))) adantl syl imbi12d y (= (cv w) (cv x)) ax-4 sylan2 albid (-. (A. x (= (cv x) (cv y)))) adantll syl anbi12d ex cbvexd imbi12d mpbii ex x y z nd1 (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))) pm2.21d x z y nd2 (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))) pm2.21d pm2.61ii)) thm (axinfnd ((w x) (w y) (w z)) () (E. x (-> (e. (cv y) (cv z)) (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x))))))))) (x w z axinfndlem1 w ax-gen y x y hbnae y z y hbnae hban y x y hbnae y z y hbnae hban y z x hbnae y z w dveel2 hbald (-. (A. y (= (cv y) (cv x)))) adantl y x x hbnae y z x hbnae hban y x w dveel2 (-. (A. y (= (cv y) (cv z)))) adantr (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv z))))) w ax-17 y x y hbnae y z y hbnae hban y x w dveel2 (-. (A. y (= (cv y) (cv z)))) adantr y x z hbnae y z z hbnae hban y z w dveel2 (-. (A. y (= (cv y) (cv x)))) adantl y z x ax15 impcom hband hbexd hbimd hbald hband hbexd hbimd y x w nd5 (-. (A. y (= (cv y) (cv z)))) adantr imdistani x (= (cv w) (cv y)) hba1 w y z elequ1 x a4s albid (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv z))))) adantl y x x hbnae y z x hbnae hban x (= (cv w) (cv y)) hba1 hban w y x elequ1 x a4s y x y hbnae y z y hbnae hban y x y hbnae y z y hbnae hban y x w dveel2 (-. (A. y (= (cv y) (cv z)))) adantr y x z hbnae y z z hbnae hban y z w dveel2 (-. (A. y (= (cv y) (cv x)))) adantl y z x ax15 impcom hband hbexd hbimd w y x elequ1 (-. (A. y (= (cv y) (cv z)))) adantl y z w nd5 imdistani z (= (cv w) (cv y)) hba1 w y z elequ1 (e. (cv z) (cv x)) anbi1d z a4s exbid (-. (A. y (= (cv y) (cv z)))) adantl syl imbi12d (-. (A. y (= (cv y) (cv x)))) adantll ex cbvald bi2anan9r exbid imbi12d syl ex cbvald mpbii 19.21bi ex x y z nd1 alequcoms (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))) pm2.21d y z x nd3 (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))) pm2.21d pm2.61ii 19.35ri)) thm (axacndlem1 () () (-> (A. x (= (cv x) (cv y))) (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))))) (x y y hbae x y z hbae x y z nd1 (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))) pm2.21d (e. (cv y) (cv z)) (e. (cv z) (cv w)) pm3.26 x 19.20i syl5 19.21ai 19.21ai (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))) x 19.8a syl)) thm (axacndlem2 () () (-> (A. x (= (cv x) (cv z))) (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))))) (x z y hbae x z z hbae x z w nd1 (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))) pm2.21d (e. (cv y) (cv z)) (e. (cv z) (cv w)) pm3.27 x 19.20i syl5 19.21ai 19.21ai (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))) x 19.8a syl)) thm (axacndlem3 () () (-> (A. y (= (cv y) (cv z))) (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))))) (y z z hbae y z x nd3 (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))) pm2.21d (e. (cv y) (cv z)) (e. (cv z) (cv w)) pm3.26 x 19.20i syl5 19.21ai a5i (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))) x 19.8a syl)) thm (axacndlem4 ((v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))) (v y z w axac v (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) ax-4 (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv v))))) (= (cv y) (cv w))))) imim1i y z 19.20i2 v 19.22i ax-mp x z x hbnae x y x hbnae x w x hbnae hban hban x z y hbnae x y y hbnae x w y hbnae hban hban x z z hbnae x y z hbnae x w z hbnae hban hban x z x hbnae x y x hbnae x w x hbnae hban hban (/\ (-. (A. x (= (cv x) (cv z)))) (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv w)))))) v ax-17 x y z ax15 impcom (-. (A. x (= (cv x) (cv w)))) adantrr x z w ax15 imp (-. (A. x (= (cv x) (cv y)))) adantrl hband hbald x z w hbnae x y w hbnae x w w hbnae hban hban x z y hbnae x y y hbnae x w y hbnae hban hban x z x hbnae x y x hbnae x w x hbnae hban hban x z w hbnae x y w hbnae x w w hbnae hban hban x y z ax15 impcom (-. (A. x (= (cv x) (cv w)))) adantrr x z w ax15 imp (-. (A. x (= (cv x) (cv y)))) adantrl hband x y w ax15 imp x w v dveel1 (-. (A. x (= (cv x) (cv y)))) adantl hband (-. (A. x (= (cv x) (cv z)))) adantl hband hbexd x y w ax-12 imp (-. (A. x (= (cv x) (cv z)))) adantl hbbid hbald hbexd hbimd hbald hbald x z v nd5 x y z hbnae x y v nd5 hbald sylan9 (-. (A. x (= (cv x) (cv w)))) adantrr x z y hbnae x y y hbnae x w y hbnae hban hban y (A. z (= (cv v) (cv x))) hba1 hban x z z hbnae x y z hbnae x w z hbnae hban hban z (= (cv v) (cv x)) hba1 y hbal hban x z x hbnae x y x hbnae x w x hbnae hban hban x y z ax15 impcom (-. (A. x (= (cv x) (cv w)))) adantrr x z w ax15 imp (-. (A. x (= (cv x) (cv y)))) adantrl hband (= (cv v) (cv x)) (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) pm4.2i (/\ (-. (A. x (= (cv x) (cv z)))) (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv w)))))) a1i cbvald (A. y (A. z (= (cv v) (cv x)))) adantr x w y hbnae x w v nd5 19.20d z (= (cv v) (cv x)) ax-4 y 19.20i syl5 (-. (A. x (= (cv x) (cv y)))) adantl imdistani w (= (cv v) (cv x)) hba1 y hbal y (A. w (= (cv v) (cv x))) hba1 w (= (cv v) (cv x)) hba1 v x w elequ2 (e. (cv y) (cv w)) anbi2d (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) anbi2d w a4s exbid (= (cv y) (cv w)) bibi1d y a4s albid exbid (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv w))))) adantl syl (-. (A. x (= (cv x) (cv z)))) adantll imbi12d albid albid ex syld cbvexd mpbii exp32 x z y w axacndlem2 x y z w axacndlem1 x w y hbae x w z hbae x w z nd2 (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))) pm2.21d (e. (cv y) (cv z)) (e. (cv z) (cv w)) pm3.27 x 19.20i syl5 19.21ai 19.21ai (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))) x 19.8a syl pm2.61iii)) thm (axacndlem5 ((v x) (v y) (w z) (v z) (v w)) () (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))) (x v z w axacndlem4 y z x hbnae y x x hbnae y w x hbnae hban hban y z y hbnae y x y hbnae y w y hbnae hban hban y z z hbnae y x z hbnae y w z hbnae hban hban y z y hbnae y x y hbnae y w y hbnae hban hban y z x hbnae y x x hbnae y w x hbnae hban hban y z v dveel2 (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv w))))) adantr y z w ax15 imp (-. (A. y (= (cv y) (cv x)))) adantrl hband hbald y z w hbnae y x w hbnae y w w hbnae hban hban (/\ (-. (A. y (= (cv y) (cv z)))) (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv w)))))) v ax-17 y z y hbnae y x y hbnae y w y hbnae hban hban y z w hbnae y x w hbnae y w w hbnae hban hban y z v dveel2 (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv w))))) adantr y z w ax15 imp (-. (A. y (= (cv y) (cv x)))) adantrl hband y w v dveel2 (-. (A. y (= (cv y) (cv z)))) (-. (A. y (= (cv y) (cv x)))) ad2antll y w x ax15 impcom (-. (A. y (= (cv y) (cv z)))) adantl hband hband hbexd y w v dveeq2 (-. (A. y (= (cv y) (cv z)))) (-. (A. y (= (cv y) (cv x)))) ad2antll hbbid hbald hbexd hbimd hbald y z v nd5 (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv w))))) adantr imdistani y z z hbnae y x z hbnae y w z hbnae hban hban z (= (cv v) (cv y)) hba1 hban y x v nd5 imdistani x (= (cv v) (cv y)) hba1 v y z elequ1 (e. (cv z) (cv w)) anbi1d x a4s albid (-. (A. y (= (cv y) (cv x)))) adantl syl z (= (cv v) (cv y)) ax-4 sylan2 (-. (A. y (= (cv y) (cv w)))) adantlr (-. (A. y (= (cv y) (cv z)))) adantll y z w hbnae y x w hbnae y w w hbnae hban hban y z y hbnae y x y hbnae y w y hbnae hban hban y z y hbnae y x y hbnae y w y hbnae hban hban y z w hbnae y x w hbnae y w w hbnae hban hban y z v dveel2 (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv w))))) adantr y z w ax15 imp (-. (A. y (= (cv y) (cv x)))) adantrl hband y w v dveel2 (-. (A. y (= (cv y) (cv z)))) (-. (A. y (= (cv y) (cv x)))) ad2antll y w x ax15 impcom (-. (A. y (= (cv y) (cv z)))) adantl hband hband hbexd y w v dveeq2 (-. (A. y (= (cv y) (cv z)))) (-. (A. y (= (cv y) (cv x)))) ad2antll hbbid y w v nd5 imdistani w (= (cv v) (cv y)) hba1 v y z elequ1 (e. (cv z) (cv w)) anbi1d v y w elequ1 (e. (cv w) (cv x)) anbi1d anbi12d w a4s exbid (-. (A. y (= (cv y) (cv w)))) adantl syl (-. (A. y (= (cv y) (cv x)))) adantll (-. (A. y (= (cv y) (cv z)))) adantll v y w equequ1 (/\ (-. (A. y (= (cv y) (cv z)))) (/\ (-. (A. y (= (cv y) (cv x)))) (-. (A. y (= (cv y) (cv w)))))) adantl bibi12d ex cbvald exbid (A. z (= (cv v) (cv y))) adantr imbi12d albid syl ex cbvald exbid mpbii exp32 y z x w axacndlem3 x y z w axacndlem1 alequcoms y w z hbae (cv y) (cv z) en2lp y w z elequ2 (e. (cv y) (cv z)) anbi2d mtbii y a4s (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))) pm2.21d x a4sd 19.21ai a5i (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))) x 19.8a syl pm2.61iii)) thm (axacnd ((v x) (v y) (v z) (v w)) () (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))) (x y v w axacndlem5 z x x hbnae z y x hbnae z w x hbnae hban hban z x y hbnae z y y hbnae z w y hbnae hban hban z x z hbnae z y z hbnae z w z hbnae hban hban z x z hbnae z y z hbnae z w z hbnae hban hban z x x hbnae z y x hbnae z w x hbnae hban hban z y v dveel1 (-. (A. z (= (cv z) (cv x)))) (-. (A. z (= (cv z) (cv w)))) ad2antrl z w v dveel2 (-. (A. z (= (cv z) (cv x)))) (-. (A. z (= (cv z) (cv y)))) ad2antll hband hbald z x w hbnae z y w hbnae z w w hbnae hban hban z x y hbnae z y y hbnae z w y hbnae hban hban z x z hbnae z y z hbnae z w z hbnae hban hban z x w hbnae z y w hbnae z w w hbnae hban hban z y v dveel1 (-. (A. z (= (cv z) (cv x)))) (-. (A. z (= (cv z) (cv w)))) ad2antrl z w v dveel2 (-. (A. z (= (cv z) (cv x)))) (-. (A. z (= (cv z) (cv y)))) ad2antll hband z y w ax15 imp (-. (A. z (= (cv z) (cv x)))) adantl z w x ax15 impcom (-. (A. z (= (cv z) (cv y)))) adantrl hband hband hbexd z y w ax-12 imp (-. (A. z (= (cv z) (cv x)))) adantl hbbid hbald hbexd hbimd z x v nd5 imdistani x (= (cv v) (cv z)) hba1 v z y elequ2 v z w elequ1 anbi12d x a4s albid (-. (A. z (= (cv z) (cv x)))) adantl syl (/\ (-. (A. z (= (cv z) (cv y)))) (-. (A. z (= (cv z) (cv w))))) adantlr z y v nd5 z w y hbnae z w v nd5 hbald sylan9 imdistani w (A. y (= (cv v) (cv z))) hba1 y (= (cv v) (cv z)) hba1 w hbal w (A. y (= (cv v) (cv z))) hba1 v z y elequ2 v z w elequ1 anbi12d y a4s w a4s (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))) anbi1d exbid (= (cv y) (cv w)) bibi1d albid exbid (/\ (-. (A. z (= (cv z) (cv y)))) (-. (A. z (= (cv z) (cv w))))) adantl syl (-. (A. z (= (cv z) (cv x)))) adantll imbi12d ex cbvald albid exbid mpbii exp32 x z y w axacndlem2 alequcoms y z x w axacndlem3 alequcoms z w y hbae z w x nd3 (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))) pm2.21d (e. (cv y) (cv z)) (e. (cv z) (cv w)) pm3.27 x 19.20i syl5 a5i 19.21ai (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))) x 19.8a syl pm2.61iii)) thm (zfcndext ((x y) (x z) (y z)) () (-> (A. z (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y)))) (= (cv x) (cv y))) (z x y axextnd 19.35i (= (cv x) (cv y)) z 19.9rv sylibr)) thm (zfcndrep ((x y) (x z) (w x) (y z) (w y) (w z)) () (-> (A. w (E. y (A. z (-> (A. y ph) (= (cv z) (cv y)))))) (E. y (A. z (<-> (e. (cv z) (cv y)) (E. w (/\ (e. (cv w) (cv x)) (A. y ph))))))) (y (A. z (-> (A. y ph) (= (cv z) (cv y)))) hbe1 (e. (cv z) (cv w)) y ax-17 (e. (cv w) (cv x)) y ax-17 y (A. y ph) hba1 hban w hbex hbbi z hbal hbim w hbex y x w elequ2 (A. y (A. y ph)) anbi1d w exbidv (e. (cv z) (cv w)) bibi2d z albidv (E. y (A. z (-> (A. y ph) (= (cv z) (cv y))))) imbi2d w exbidv w y z (A. y ph) axrepnd (e. (cv z) (cv w)) y ax-17 19.3r (e. (cv w) (cv y)) z ax-17 19.3r (A. y (A. y ph)) anbi1i w exbii bibi12i z albii (E. y (A. z (-> (A. y ph) (= (cv z) (cv y))))) imbi2i w exbii mpbir chvar 19.35i (e. (cv z) (cv y)) w ax-17 w (/\ (e. (cv w) (cv x)) (A. y ph)) hbe1 hbbi z hbal (e. (cv z) (cv w)) y ax-17 (e. (cv w) (cv x)) y ax-17 y (A. y ph) hba1 hban w hbex hbbi z hbal y w z elequ2 y ph hba1 19.3r (e. (cv w) (cv x)) anbi2i w exbii (= (cv y) (cv w)) a1i bibi12d z albidv cbvex sylibr)) thm (zfcndun ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (A. z (-> (E. w (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x)))) (e. (cv z) (cv y))))) (y z x axunnd w y z elequ2 w y x elequ1 anbi12d cbvexv (e. (cv z) (cv y)) imbi1i z albii y exbii mpbir)) thm (zfcndpow ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (A. z (-> (A. w (-> (e. (cv w) (cv z)) (e. (cv w) (cv x)))) (e. (cv z) (cv y))))) (y z dtru y (= (cv y) (cv z)) exnal mpbir y (A. z (-> (A. y (-> (E. x (e. (cv y) (cv z))) (A. z (e. (cv y) (cv x))))) (e. (cv z) (cv y)))) hbe1 y z x axpownd 19.23ai ax-mp (e. (cv y) (cv z)) x 19.9rv (e. (cv y) (cv x)) z ax-17 19.3r imbi12i y albii (e. (cv z) (cv y)) imbi1i z albii y exbii mpbir w y z elequ1 w y x elequ1 imbi12d cbvalv (e. (cv z) (cv y)) imbi1i z albii y exbii mpbir)) thm (zfcndreg ((x y) (x z) (y z)) () (-> (E. y (e. (cv y) (cv x))) (E. y (/\ (e. (cv y) (cv x)) (A. z (-> (e. (cv z) (cv y)) (-. (e. (cv z) (cv x)))))))) (y (/\ (e. (cv y) (cv x)) (A. z (-> (e. (cv z) (cv y)) (-. (e. (cv z) (cv x)))))) hbe1 y x z axregnd 19.23ai)) thm (zfcndinf ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv y)) (E. w (/\ (e. (cv z) (cv w)) (e. (cv w) (cv y)))))))) (w x el (e. (cv x) (cv y)) w ax-17 (e. (cv x) (cv y)) w ax-17 w (/\ (e. (cv x) (cv w)) (e. (cv w) (cv y))) hbe1 hbim x hbal hban y hbex (e. (cv x) (cv w)) y ax-17 y x w axinfnd 19.35i syl 19.23ai ax-mp z x y elequ1 z x w elequ1 (e. (cv w) (cv y)) anbi1d w exbidv imbi12d cbvalv (e. (cv x) (cv y)) anbi2i y exbii mpbir)) thm (zfcndac ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. v (A. u (<-> (E. t (/\ (/\ (e. (cv u) (cv w)) (e. (cv w) (cv t))) (/\ (e. (cv u) (cv t)) (e. (cv t) (cv y))))) (= (cv u) (cv v))))))))) (y z w x axacnd (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) y ax-17 19.3r (E. x (A. z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x))))) imbi1i z w 2albii y exbii mpbir v x u equequ2 (E. t (/\ (/\ (e. (cv u) (cv w)) (e. (cv w) (cv t))) (/\ (e. (cv u) (cv t)) (e. (cv t) (cv y))))) bibi2d t x w elequ2 (e. (cv u) (cv w)) anbi2d t x u elequ2 t x y elequ1 anbi12d anbi12d cbvexv (= (cv u) (cv x)) bibi1i syl6bb u albidv u z w elequ1 (e. (cv w) (cv x)) anbi1d u z x elequ1 (e. (cv x) (cv y)) anbi1d anbi12d x exbidv u z x equequ1 bibi12d cbvalv syl6bb cbvexv (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) imbi2i z w 2albii y exbii mpbir)) thm (grothlem ((v w) (u w) (h w) (g w) (f w) (u v) (h v) (g v) (f v) (h u) (g u) (f u) (g h) (f h) (f g)) () (<-> (br (cv u) (cv w) (cv v)) (E. g (/\ (e. (cv g) (cv w)) (A. h (<-> (e. (cv h) (cv g)) (\/ (A. f (<-> (e. (cv f) (cv h)) (= (cv f) (cv u)))) (A. f (<-> (e. (cv f) (cv h)) (\/ (= (cv f) (cv u)) (= (cv f) (cv v))))))))))) ((cv u) (cv w) (cv v) df-br (cv u) (cv v) df-op ({} (cv u)) ({,} (cv u) (cv v)) h dfpr2 eqtr (cv w) eleq1i h (\/ (= (cv h) ({} (cv u))) (= (cv h) ({,} (cv u) (cv v)))) (cv w) g clabel (cv h) ({} (cv u)) f dfcleq f (cv u) elsn (e. (cv f) (cv h)) bibi2i f albii bitr (cv h) ({,} (cv u) (cv v)) f dfcleq f visset (cv u) (cv v) elpr (e. (cv f) (cv h)) bibi2i f albii bitr orbi12i (e. (cv h) (cv g)) bibi2i h albii (e. (cv g) (cv w)) anbi2i g exbii bitr 3bitr)) thm (grothprim ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (g x) (f x) (k x) (y z) (w y) (v y) (u y) (t y) (h y) (g y) (f y) (k y) (w z) (v z) (u z) (t z) (h z) (g z) (f z) (k z) (v w) (u w) (t w) (h w) (g w) (f w) (k w) (u v) (t v) (h v) (g v) (f v) (k v) (t u) (h u) (g u) (f u) (k u) (h t) (g t) (f t) (k t) (g h) (f h) (h k) (f g) (g k) (f k)) () (E. y (/\ (e. (cv x) (cv y)) (/\ (A. z (-> (e. (cv z) (cv y)) (/\ (A. w (-> (A. v (-> (e. (cv v) (cv w)) (e. (cv v) (cv z)))) (e. (cv w) (cv y)))) (E. w (/\ (e. (cv w) (cv y)) (A. v (-> (A. u (-> (e. (cv u) (cv v)) (e. (cv u) (cv z)))) (e. (cv v) (cv w))))))))) (A. z (-> (A. w (-> (e. (cv w) (cv z)) (e. (cv w) (cv y)))) (\/ (E. w (/\ (/\ (A. v (-> (e. (cv v) (cv z)) (E. u (/\ (e. (cv u) (cv y)) (E. g (/\ (e. (cv g) (cv w)) (A. h (<-> (e. (cv h) (cv g)) (\/ (A. f (<-> (e. (cv f) (cv h)) (= (cv f) (cv v)))) (A. f (<-> (e. (cv f) (cv h)) (\/ (= (cv f) (cv v)) (= (cv f) (cv u)))))))))))))) (A. v (-> (e. (cv v) (cv y)) (E. u (/\ (e. (cv u) (cv z)) (E. g (/\ (e. (cv g) (cv w)) (A. h (<-> (e. (cv h) (cv g)) (\/ (A. f (<-> (e. (cv f) (cv h)) (= (cv f) (cv u)))) (A. f (<-> (e. (cv f) (cv h)) (\/ (= (cv f) (cv u)) (= (cv f) (cv v))))))))))))))) (A. v (A. u (A. t (A. k (-> (/\ (E. g (/\ (e. (cv g) (cv w)) (A. h (<-> (e. (cv h) (cv g)) (\/ (A. f (<-> (e. (cv f) (cv h)) (= (cv f) (cv v)))) (A. f (<-> (e. (cv f) (cv h)) (\/ (= (cv f) (cv v)) (= (cv f) (cv u)))))))))) (E. g (/\ (e. (cv g) (cv w)) (A. h (<-> (e. (cv h) (cv g)) (\/ (A. f (<-> (e. (cv f) (cv h)) (= (cv f) (cv t)))) (A. f (<-> (e. (cv f) (cv h)) (\/ (= (cv f) (cv t)) (= (cv f) (cv k))))))))))) (<-> (= (cv v) (cv t)) (= (cv u) (cv k)))))))))) (e. (cv z) (cv y)))))))) (y x z w v ax-groth (e. (cv x) (cv y)) (A.e. z (cv y) (/\ (A. w (-> (C_ (cv w) (cv z)) (e. (cv w) (cv y)))) (E.e. w (cv y) (A. v (-> (C_ (cv v) (cv z)) (e. (cv v) (cv w))))))) (A. z (-> (C_ (cv z) (cv y)) (\/ (br (cv z) (~~) (cv y)) (e. (cv z) (cv y))))) 3anass z (cv y) (/\ (A. w (-> (C_ (cv w) (cv z)) (e. (cv w) (cv y)))) (E.e. w (cv y) (A. v (-> (C_ (cv v) (cv z)) (e. (cv v) (cv w)))))) df-ral (cv w) (cv z) v dfss2 (e. (cv w) (cv y)) imbi1i w albii w (cv y) (A. v (-> (C_ (cv v) (cv z)) (e. (cv v) (cv w)))) df-rex (cv v) (cv z) u dfss2 (e. (cv v) (cv w)) imbi1i v albii (e. (cv w) (cv y)) anbi2i w exbii bitr anbi12i (e. (cv z) (cv y)) imbi2i z albii bitr (cv z) (cv y) w dfss2 y visset (cv z) w bren3 (cv w) (cv z) (cv y) v u dminxp v (cv z) (E.e. u (cv y) (br (cv v) (cv w) (cv u))) df-ral u (cv y) (br (cv v) (cv w) (cv u)) df-rex v w u g h f grothlem (e. (cv u) (cv y)) anbi2i u exbii bitr (e. (cv v) (cv z)) imbi2i v albii 3bitr (cv w) (cv z) (cv y) v u rninxp v (cv y) (E.e. u (cv z) (br (cv u) (cv w) (cv v))) df-ral u (cv z) (br (cv u) (cv w) (cv v)) df-rex u w v g h f grothlem (e. (cv u) (cv z)) anbi2i u exbii bitr (e. (cv v) (cv y)) imbi2i v albii 3bitr anbi12i (cv w) v u t k fun11 v w u g h f grothlem t w k g h f grothlem anbi12i (<-> (= (cv v) (cv t)) (= (cv u) (cv k))) imbi1i t k 2albii v u 2albii bitr anbi12i w exbii bitr (e. (cv z) (cv y)) orbi1i imbi12i z albii anbi12i (e. (cv x) (cv y)) anbi2i bitr y exbii mpbi)) thm (elni () () (<-> (e. A (N.)) (/\ (e. A (om)) (-. (= A ({/}))))) (df-ni A eleq2i A (om) ({} ({/})) eldif 0ex A elsnc2 negbii (e. A (om)) anbi2i 3bitr)) thm (elni2 () () (<-> (e. A (N.)) (/\ (e. A (om)) (e. ({/}) A))) (A elni A nnord A ord0eln0 syl pm5.32i bitr4)) thm (pinn () () (-> (e. A (N.)) (e. A (om))) (df-ni (om) ({} ({/})) difss eqsstr A sseli)) thm (pion () () (-> (e. A (N.)) (e. A (On))) (A pinn A nnont syl)) thm (piord () () (-> (e. A (N.)) (Ord A)) (A pinn A nnord syl)) thm (niex () () (e. (N.) (V)) (omex df-ni (om) ({} ({/})) difss eqsstr ssexi)) thm (0npi () () (-. (e. ({/}) (N.))) (({/}) eqid ({/}) elni pm3.27bd mt2)) thm (1pi () () (e. (1o) (N.)) ((1o) elni 1onn 1ne0 mpbir2an)) thm (addpiord () () (-> (/\ (e. A (N.)) (e. B (N.))) (= (opr A (+N) B) (opr A (+o) B))) (A (N.) B (N.) opelxpi (<,> A B) (X. (N.) (N.)) (+o) fvres A (+N) B df-opr df-pli (<,> A B) fveq1i eqtr A (+o) B df-opr 3eqtr4g syl)) thm (mulpiord () () (-> (/\ (e. A (N.)) (e. B (N.))) (= (opr A (.N) B) (opr A (.o) B))) (A (N.) B (N.) opelxpi (<,> A B) (X. (N.) (N.)) (.o) fvres A (.N) B df-opr df-mi (<,> A B) fveq1i eqtr A (.o) B df-opr 3eqtr4g syl)) thm (mulidpi () () (-> (e. A (N.)) (= (opr A (.N) (1o)) A)) (1pi A (1o) mulpiord mpan2 A pinn A nnont A om1 3syl eqtrd)) thm (ltpiord ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (N.)) (e. B (N.))) (<-> (br A ( (cv x) (cv y)) (X. (N.) (N.))) (e. (<,> (cv x) (cv y)) (E)) iba (cv x) (E) (cv y) df-br x y epel bitr3 syl5bbr (cv x) ( (cv x) (cv y)) eleq2i (<,> (cv x) (cv y)) (E) (X. (N.) (N.)) elin 3bitr syl6rbbr sylbir vtocl2ga)) thm (ltsopi ((x y) (x z) (y z)) () (Or ( (/\ (e. A (N.)) (e. B (N.))) (e. (opr A (+N) B) (N.))) (A B addpiord A B nnacl B pinn sylan2 B A ({/}) nnaordi (opr A (+o) ({/})) (opr A (+o) B) n0i syl6 expcom imp32 B elni2 sylan2b jca (opr A (+o) B) elni sylibr A pinn sylan eqeltrd)) thm (mulclpi () () (-> (/\ (e. A (N.)) (e. B (N.))) (e. (opr A (.N) B) (N.))) (A B mulpiord A B nnmcl A pinn B pinn syl2an peano1 ({/}) B A nnmordi mp3an1 imp an4s B elni2 A elni2 syl2anb ancoms (opr A (.o) ({/})) (opr A (.o) B) n0i syl jca (opr A (.o) B) elni sylibr eqeltrd)) thm (addcompi () ((addcompi.1 (e. A (V))) (addcompi.2 (e. B (V)))) (= (opr A (+N) B) (opr B (+N) A)) (A B nnacom A pinn B pinn syl2an A B addpiord B A addpiord ancoms 3eqtr4d addcompi.2 dmaddpi addcompi.1 ndmoprcom pm2.61i)) thm (addasspi () ((addasspi.1 (e. B (V))) (addasspi.2 (e. C (V)))) (= (opr (opr A (+N) B) (+N) C) (opr A (+N) (opr B (+N) C))) (A B C nnaass A pinn B pinn C pinn syl3an (opr A (+N) B) C addpiord A B addclpi sylan A B addpiord (+o) C opreq1d (e. C (N.)) adantr eqtrd 3impa A (opr B (+N) C) addpiord B C addclpi sylan2 B C addpiord A (+o) opreq2d (e. A (N.)) adantl eqtrd 3impb 3eqtr4d addasspi.1 dmaddpi addasspi.2 0npi A ndmoprass pm2.61i)) thm (mulcompi () ((mulcompi.1 (e. A (V))) (mulcompi.2 (e. B (V)))) (= (opr A (.N) B) (opr B (.N) A)) (A B nnmcom A pinn B pinn syl2an A B mulpiord B A mulpiord ancoms 3eqtr4d mulcompi.2 dmmulpi mulcompi.1 ndmoprcom pm2.61i)) thm (mulasspi () ((mulasspi.1 (e. B (V))) (mulasspi.2 (e. C (V)))) (= (opr (opr A (.N) B) (.N) C) (opr A (.N) (opr B (.N) C))) (A B C nnmass A pinn B pinn C pinn syl3an (opr A (.N) B) C mulpiord A B mulclpi sylan A B mulpiord (.o) C opreq1d (e. C (N.)) adantr eqtrd 3impa A (opr B (.N) C) mulpiord B C mulclpi sylan2 B C mulpiord A (.o) opreq2d (e. A (N.)) adantl eqtrd 3impb 3eqtr4d mulasspi.1 dmmulpi mulasspi.2 0npi A ndmoprass pm2.61i)) thm (distrpi () ((distrpi.1 (e. B (V))) (distrpi.2 (e. C (V)))) (= (opr A (.N) (opr B (+N) C)) (opr (opr A (.N) B) (+N) (opr A (.N) C))) (A B C nndi A pinn B pinn C pinn syl3an A (opr B (+N) C) mulpiord B C addclpi sylan2 B C addpiord A (.o) opreq2d (e. A (N.)) adantl eqtrd 3impb (opr A (.N) B) (opr A (.N) C) addpiord A B mulclpi A C mulclpi syl2an A B mulpiord A C mulpiord (+o) opreqan12d eqtrd 3impdi 3eqtr4d distrpi.1 dmaddpi distrpi.2 0npi dmmulpi A ndmoprdistr pm2.61i)) thm (mulcanpi () ((mulcanpi.1 (e. C (V)))) (-> (/\ (e. A (N.)) (e. B (N.))) (-> (= (opr A (.N) B) (opr A (.N) C)) (= B C))) (A B mulpiord (e. C (N.)) adantr A C mulpiord (e. B (N.)) adantlr eqeq12d A B C nnmcan biimpd A elni2 pm3.27bd sylan2 ex A pinn B pinn C pinn syl3an 3exp com4r pm2.43i imp31 sylbid (opr A (.N) B) (opr A (.N) C) (N.) eleq1 A B mulclpi syl5bi imp mulcanpi.1 dmmulpi 0npi A ndmoprrcl (e. A (N.)) (e. C (N.)) pm3.27 3syl sylan2 exp32 imp4b pm2.43i ex)) thm (addnidpi () ((addnidpi.1 (e. B (V)))) (-> (e. A (N.)) (-. (= (opr A (+N) B) A))) (B A ({/}) nnaordi A nna0 (opr A (+o) B) eleq1d (opr A (+o) B) A A eleq2 negbid A nnord A ordeirr syl syl5bir com12 con2d sylbid (e. B (om)) adantl syld expcom imp32 B elni2 sylan2b A pinn sylan A B addpiord A eqeq1d mtbird (e. A (N.)) a1d addnidpi.1 dmaddpi A ndmopr A eqeq1d 0npi ({/}) A (N.) eleq1 mtbii syl6bi con2d pm2.61i)) thm (ltexpi ((A x) (B x)) () (-> (/\ (e. A (N.)) (e. B (N.))) (<-> (br A ( (e. C (N.)) (<-> (br A ( (e. C (N.)) (<-> (br A ( (= (cv x) (1o)) (<-> ph ps))) (indpi.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (indpi.3 (-> (= (cv x) (opr (cv y) (+N) (1o))) (<-> ph th))) (indpi.4 (-> (= (cv x) A) (<-> ph ta))) (indpi.5 ps) (indpi.6 (-> (e. (cv y) (N.)) (-> ch th)))) (-> (e. A (N.)) ta) (A nlt1pi 1pi ltsopi ( (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (<-> (br (<,> A B) (~Q) (<,> C D)) (= (opr A (.N) D) (opr B (.N) C)))) (x y z w v u df-enq A B C D ecopopreq)) thm (dmenq ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (dom (~Q)) (X. (N.) (N.))) (x y z w v u df-enq x visset y visset mulcompi ecopoprdm)) thm (enqer ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (Er (~Q)) (x y z w v u df-enq x visset y visset mulcompi (cv x) (cv y) mulclpi y visset z visset (cv x) mulasspi z visset (cv x) (cv y) mulcanpi ecopoprer)) thm (enqeceq () () (-> (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (<-> (= ([] (<,> A B) (~Q)) ([] (<,> C D) (~Q))) (= (opr A (.N) D) (opr B (.N) C)))) ((/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.))) pm3.26 A (N.) B (N.) opelxpi dmenq syl6eleqr C D opex enqer (<,> A B) erthdm 3syl A B C D enqbreq bitrd)) thm (enqex ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (e. (~Q) (V)) (niex niex xpex niex niex xpex xpex x y z w v u df-enq x y (X. (N.) (N.)) (X. (N.) (N.)) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) (.N) (cv u)) (opr (cv w) (.N) (cv v)))))))) opabssxp eqsstr ssexi)) thm (nqex () () (e. (Q.) (V)) (df-nq niex niex xpex (~Q) qsex eqeltr)) thm (0npq () () (-. (e. ({/}) (Q.))) (enqer dmenq 0nelqs df-nq ({/}) eleq2i mtbir)) thm (ltrelpq ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (C_ ( (cv z) (cv w)) (~Q))) (= (cv y) ([] (<,> (cv v) (cv u)) (~Q)))) (br (opr (cv z) (.N) (cv u)) ( (/\ (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (/\ (/\ (e. F (N.)) (e. G (N.))) (/\ (e. R (N.)) (e. S (N.))))) (-> (/\ (= (opr A (.N) D) (opr B (.N) C)) (= (opr F (.N) S) (opr G (.N) R))) (br (<,> (opr (opr A (.N) G) (+N) (opr B (.N) F)) (opr B (.N) G)) (~Q) (<,> (opr (opr C (.N) S) (+N) (opr D (.N) R)) (opr D (.N) S))))) ((opr A (.N) G) (opr B (.N) F) addclpi A G mulclpi B F mulclpi syl2an an42s B G mulclpi (e. A (N.)) (e. F (N.)) ad2ant2l jca (opr C (.N) S) (opr D (.N) R) addclpi C S mulclpi D R mulclpi syl2an an42s D S mulclpi (e. C (N.)) (e. R (N.)) ad2ant2l jca anim12i an4s (opr (opr A (.N) G) (+N) (opr B (.N) F)) (opr B (.N) G) (opr (opr C (.N) S) (+N) (opr D (.N) R)) (opr D (.N) S) enqbreq syl (opr A (.N) D) (opr B (.N) C) (.N) (opr G (.N) S) opreq1 (opr F (.N) S) (opr G (.N) R) (opr B (.N) D) (.N) opreq2 (+N) opreqan12d A (.N) G oprex B (.N) F oprex D (.N) S oprex x visset y visset mulcompi y visset z visset (cv x) distrpi caoprdistrr cmpblnq.1 cmpblnq.6 cmpblnq.4 x visset y visset mulcompi y visset z visset (cv x) mulasspi cmpblnq.8 caopr4 cmpblnq.2 cmpblnq.5 cmpblnq.4 x visset y visset mulcompi y visset z visset (cv x) mulasspi cmpblnq.8 caopr4 (+N) opreq12i eqtr C (.N) S oprex D (.N) R oprex (opr B (.N) G) distrpi cmpblnq.2 cmpblnq.6 cmpblnq.3 x visset y visset mulcompi y visset z visset (cv x) mulasspi cmpblnq.8 caopr4 cmpblnq.2 cmpblnq.6 cmpblnq.4 x visset y visset mulcompi y visset z visset (cv x) mulasspi cmpblnq.7 caopr4 (+N) opreq12i eqtr 3eqtr4g syl5bir)) thm (mulcmpblnq ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (F x) (F y) (F z) (G x) (G y) (G z) (R x) (R y) (R z) (S x) (S y) (S z)) ((cmpblnq.1 (e. A (V))) (cmpblnq.2 (e. B (V))) (cmpblnq.3 (e. C (V))) (cmpblnq.4 (e. D (V))) (cmpblnq.5 (e. F (V))) (cmpblnq.6 (e. G (V))) (cmpblnq.7 (e. R (V))) (cmpblnq.8 (e. S (V)))) (-> (/\ (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (/\ (/\ (e. F (N.)) (e. G (N.))) (/\ (e. R (N.)) (e. S (N.))))) (-> (/\ (= (opr A (.N) D) (opr B (.N) C)) (= (opr F (.N) S) (opr G (.N) R))) (br (<,> (opr A (.N) F) (opr B (.N) G)) (~Q) (<,> (opr C (.N) R) (opr D (.N) S))))) (A F mulclpi B G mulclpi anim12i an4s C R mulclpi D S mulclpi anim12i an4s anim12i an4s (opr A (.N) F) (opr B (.N) G) (opr C (.N) R) (opr D (.N) S) enqbreq syl cmpblnq.1 cmpblnq.5 cmpblnq.4 x visset y visset mulcompi y visset z visset (cv x) mulasspi cmpblnq.8 caopr4 cmpblnq.2 cmpblnq.6 cmpblnq.3 x visset y visset mulcompi y visset z visset (cv x) mulasspi cmpblnq.7 caopr4 eqeq12i syl6bb (opr A (.N) D) (opr B (.N) C) (opr F (.N) S) (opr G (.N) R) (.N) opreq12 syl5bir)) thm (addpipq ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (= (opr ([] (<,> A B) (~Q)) (+Q) ([] (<,> C D) (~Q))) ([] (<,> (opr (opr A (.N) D) (+N) (opr B (.N) C)) (opr B (.N) D)) (~Q)))) ((opr (opr A (.N) D) (+N) (opr B (.N) C)) (opr B (.N) D) opex (opr (opr (cv a) (.N) (cv h)) (+N) (opr (cv b) (.N) (cv g))) (opr (cv b) (.N) (cv h)) opex (opr (opr (cv c) (.N) (cv s)) (+N) (opr (cv d) (.N) (cv t))) (opr (cv d) (.N) (cv s)) opex enqex enqer dmenq x y z w v u df-enq (cv z) (cv a) (cv u) (cv d) (.N) opreq12 (cv w) (cv b) (cv v) (cv c) (.N) opreq12 eqeqan12d an42s (cv z) (cv g) (cv u) (cv s) (.N) opreq12 (cv w) (cv h) (cv v) (cv t) (.N) opreq12 eqeqan12d an42s x y z w v u f df-plpq (opr (opr (cv w) (.N) (cv f)) (+N) (opr (cv v) (.N) (cv u))) (opr (opr (cv a) (.N) (cv h)) (+N) (opr (cv b) (.N) (cv g))) (opr (cv v) (.N) (cv f)) (opr (cv b) (.N) (cv h)) opeq12 (cv w) (cv a) (cv f) (cv h) (.N) opreq12 (cv v) (cv b) (cv u) (cv g) (.N) opreq12 (+N) opreqan12d an42s (cv v) (cv b) (cv f) (cv h) (.N) opreq12 (= (cv w) (cv a)) (= (cv u) (cv g)) ad2ant2l sylanc (opr (opr (cv w) (.N) (cv f)) (+N) (opr (cv v) (.N) (cv u))) (opr (opr (cv c) (.N) (cv s)) (+N) (opr (cv d) (.N) (cv t))) (opr (cv v) (.N) (cv f)) (opr (cv d) (.N) (cv s)) opeq12 (cv w) (cv c) (cv f) (cv s) (.N) opreq12 (cv v) (cv d) (cv u) (cv t) (.N) opreq12 (+N) opreqan12d an42s (cv v) (cv d) (cv f) (cv s) (.N) opreq12 (= (cv w) (cv c)) (= (cv u) (cv t)) ad2ant2l sylanc (opr (opr (cv w) (.N) (cv f)) (+N) (opr (cv v) (.N) (cv u))) (opr (opr A (.N) D) (+N) (opr B (.N) C)) (opr (cv v) (.N) (cv f)) (opr B (.N) D) opeq12 (cv w) A (cv f) D (.N) opreq12 (cv v) B (cv u) C (.N) opreq12 (+N) opreqan12d an42s (cv v) B (cv f) D (.N) opreq12 (= (cv w) A) (= (cv u) C) ad2ant2l sylanc x y z a b c d df-plq df-nq a visset b visset c visset d visset g visset h visset t visset s visset addcmpblnq oprec)) thm (mulpipq ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (= (opr ([] (<,> A B) (~Q)) (.Q) ([] (<,> C D) (~Q))) ([] (<,> (opr A (.N) C) (opr B (.N) D)) (~Q)))) ((opr A (.N) C) (opr B (.N) D) opex (opr (cv a) (.N) (cv g)) (opr (cv b) (.N) (cv h)) opex (opr (cv c) (.N) (cv t)) (opr (cv d) (.N) (cv s)) opex enqex enqer dmenq x y z w v u df-enq (cv z) (cv a) (cv u) (cv d) (.N) opreq12 (cv w) (cv b) (cv v) (cv c) (.N) opreq12 eqeqan12d an42s (cv z) (cv g) (cv u) (cv s) (.N) opreq12 (cv w) (cv h) (cv v) (cv t) (.N) opreq12 eqeqan12d an42s x y z w v u f df-mpq (opr (cv w) (.N) (cv u)) (opr (cv a) (.N) (cv g)) (opr (cv v) (.N) (cv f)) (opr (cv b) (.N) (cv h)) opeq12 (cv w) (cv a) (cv u) (cv g) (.N) opreq12 (cv v) (cv b) (cv f) (cv h) (.N) opreq12 syl2an an4s (opr (cv w) (.N) (cv u)) (opr (cv c) (.N) (cv t)) (opr (cv v) (.N) (cv f)) (opr (cv d) (.N) (cv s)) opeq12 (cv w) (cv c) (cv u) (cv t) (.N) opreq12 (cv v) (cv d) (cv f) (cv s) (.N) opreq12 syl2an an4s (opr (cv w) (.N) (cv u)) (opr A (.N) C) (opr (cv v) (.N) (cv f)) (opr B (.N) D) opeq12 (cv w) A (cv u) C (.N) opreq12 (cv v) B (cv f) D (.N) opreq12 syl2an an4s x y z a b c d df-mq df-nq a visset b visset c visset d visset g visset h visset t visset s visset mulcmpblnq oprec)) thm (ordpipq ((A x) (A y) (A z) (A w) (A v) (A u) (A f) (x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f)) ((ordpipq.1 (e. A (V))) (ordpipq.2 (e. B (V))) (ordpipq.3 (e. C (V))) (ordpipq.4 (e. D (V)))) (<-> (br ([] (<,> A B) (~Q)) ( C D) (~Q))) (br (opr A (.N) D) ( (1o) (1o)) (X. (N.) (N.)) ecelqsi ax-mp df-1q df-nq eleq12i mpbir)) thm (addclpq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (Q.)) (e. B (Q.))) (e. (opr A (+Q) B) (Q.))) (df-nq ([] (<,> (cv x) (cv y)) (~Q)) A (+Q) ([] (<,> (cv z) (cv w)) (~Q)) opreq1 (/. (X. (N.) (N.)) (~Q)) eleq1d ([] (<,> (cv z) (cv w)) (~Q)) B A (+Q) opreq2 (/. (X. (N.) (N.)) (~Q)) eleq1d (cv x) (cv y) (cv z) (cv w) addpipq (opr (cv x) (.N) (cv w)) (opr (cv y) (.N) (cv z)) addclpi (cv x) (cv w) mulclpi (cv y) (cv z) mulclpi syl2an an42s (cv y) (cv w) mulclpi (e. (cv x) (N.)) (e. (cv z) (N.)) ad2ant2l jca (opr (opr (cv x) (.N) (cv w)) (+N) (opr (cv y) (.N) (cv z))) (N.) (opr (cv y) (.N) (cv w)) (N.) opelxpi enqex (<,> (opr (opr (cv x) (.N) (cv w)) (+N) (opr (cv y) (.N) (cv z))) (opr (cv y) (.N) (cv w))) (X. (N.) (N.)) ecelqsi 3syl eqeltrd 2ecoptocl df-nq syl6eleqr)) thm (dmaddpq ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (+Q)) (X. (Q.) (Q.))) (x y z w v u f df-plq dmeqi x y z (Q.) (Q.) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) (~Q))) (= (cv y) ([] (<,> (cv u) (cv f)) (~Q)))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) (+pQ) (<,> (cv u) (cv f))) (~Q)))))))) dmoprabss eqsstr 0npq (cv x) (cv y) addclpq oprssdm eqssi)) thm (mulclpq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (Q.)) (e. B (Q.))) (e. (opr A (.Q) B) (Q.))) (df-nq ([] (<,> (cv x) (cv y)) (~Q)) A (.Q) ([] (<,> (cv z) (cv w)) (~Q)) opreq1 (/. (X. (N.) (N.)) (~Q)) eleq1d ([] (<,> (cv z) (cv w)) (~Q)) B A (.Q) opreq2 (/. (X. (N.) (N.)) (~Q)) eleq1d (cv x) (cv y) (cv z) (cv w) mulpipq (cv x) (cv z) mulclpi (cv y) (cv w) mulclpi anim12i an4s (opr (cv x) (.N) (cv z)) (N.) (opr (cv y) (.N) (cv w)) (N.) opelxpi enqex (<,> (opr (cv x) (.N) (cv z)) (opr (cv y) (.N) (cv w))) (X. (N.) (N.)) ecelqsi 3syl eqeltrd 2ecoptocl df-nq syl6eleqr)) thm (dmmulpq ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (.Q)) (X. (Q.) (Q.))) (x y z w v u f df-mq dmeqi x y z (Q.) (Q.) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) (~Q))) (= (cv y) ([] (<,> (cv u) (cv f)) (~Q)))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) (.pQ) (<,> (cv u) (cv f))) (~Q)))))))) dmoprabss eqsstr 0npq (cv x) (cv y) mulclpq oprssdm eqssi)) thm (addcompq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((addcompq.1 (e. A (V))) (addcompq.2 (e. B (V)))) (= (opr A (+Q) B) (opr B (+Q) A)) (df-nq (cv x) (cv y) (cv z) (cv w) addpipq (cv z) (cv w) (cv x) (cv y) addpipq x visset w visset mulcompi y visset z visset mulcompi (+N) opreq12i (cv w) (.N) (cv x) oprex (cv z) (.N) (cv y) oprex addcompi eqtr y visset w visset mulcompi A B ecoprcom addcompq.2 dmaddpq addcompq.1 ndmoprcom pm2.61i)) thm (addasspq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((addasspq.1 (e. B (V))) (addasspq.2 (e. C (V)))) (= (opr (opr A (+Q) B) (+Q) C) (opr A (+Q) (opr B (+Q) C))) (df-nq (cv x) (cv y) (cv z) (cv w) addpipq (cv z) (cv w) (cv v) (cv u) addpipq (opr (opr (cv x) (.N) (cv w)) (+N) (opr (cv y) (.N) (cv z))) (opr (cv y) (.N) (cv w)) (cv v) (cv u) addpipq (cv x) (cv y) (opr (opr (cv z) (.N) (cv u)) (+N) (opr (cv w) (.N) (cv v))) (opr (cv w) (.N) (cv u)) addpipq (opr (cv x) (.N) (cv w)) (opr (cv y) (.N) (cv z)) addclpi (cv x) (cv w) mulclpi (cv y) (cv z) mulclpi syl2an an42s (cv y) (cv w) mulclpi (e. (cv x) (N.)) (e. (cv z) (N.)) ad2ant2l jca (opr (cv z) (.N) (cv u)) (opr (cv w) (.N) (cv v)) addclpi (cv z) (cv u) mulclpi (cv w) (cv v) mulclpi syl2an an42s (cv w) (cv u) mulclpi (e. (cv z) (N.)) (e. (cv v) (N.)) ad2ant2l jca (cv y) (.N) (opr (cv z) (.N) (cv u)) oprex (cv y) (.N) (opr (cv w) (.N) (cv v)) oprex (opr (cv x) (.N) (opr (cv w) (.N) (cv u))) addasspi x visset y visset w visset f visset g visset mulcompi g visset h visset (cv f) distrpi z visset u visset g visset h visset (cv f) mulasspi caoprdilem w visset v visset (cv y) mulasspi (+N) opreq12i (cv z) (.N) (cv u) oprex (cv w) (.N) (cv v) oprex (cv y) distrpi (opr (cv x) (.N) (opr (cv w) (.N) (cv u))) (+N) opreq2i 3eqtr4 w visset u visset (cv y) mulasspi A B C ecoprass addasspq.1 dmaddpq addasspq.2 0npq A ndmoprass pm2.61i)) thm (mulcompq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((mulcompq.1 (e. A (V))) (mulcompq.2 (e. B (V)))) (= (opr A (.Q) B) (opr B (.Q) A)) (df-nq (cv x) (cv y) (cv z) (cv w) mulpipq (cv z) (cv w) (cv x) (cv y) mulpipq x visset z visset mulcompi y visset w visset mulcompi A B ecoprcom mulcompq.2 dmmulpq mulcompq.1 ndmoprcom pm2.61i)) thm (mulasspq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((mulasspq.1 (e. B (V))) (mulasspq.2 (e. C (V)))) (= (opr (opr A (.Q) B) (.Q) C) (opr A (.Q) (opr B (.Q) C))) (df-nq (cv x) (cv y) (cv z) (cv w) mulpipq (cv z) (cv w) (cv v) (cv u) mulpipq (opr (cv x) (.N) (cv z)) (opr (cv y) (.N) (cv w)) (cv v) (cv u) mulpipq (cv x) (cv y) (opr (cv z) (.N) (cv v)) (opr (cv w) (.N) (cv u)) mulpipq (cv x) (cv z) mulclpi (cv y) (cv w) mulclpi anim12i an4s (cv z) (cv v) mulclpi (cv w) (cv u) mulclpi anim12i an4s z visset v visset (cv x) mulasspi w visset u visset (cv y) mulasspi A B C ecoprass mulasspq.1 dmmulpq mulasspq.2 0npq A ndmoprass pm2.61i)) thm (distrpqlem ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((distrpqlem.1 (e. A (V))) (distrpqlem.2 (e. B (V))) (distrpqlem.3 (e. C (V)))) (-> (/\/\ (e. A (N.)) (e. B (N.)) (e. C (N.))) (= ([] (<,> (opr A (.N) B) (opr A (.N) C)) (~Q)) ([] (<,> B C) (~Q)))) (distrpqlem.1 distrpqlem.2 distrpqlem.3 x visset y visset mulcompi y visset z visset (cv x) mulasspi caopr32 A B mulclpi A C mulclpi anim12i (/\ (e. A (N.)) (e. A (N.))) (/\ (e. B (N.)) (e. C (N.))) pm3.27 an4s jca 3impdi (opr A (.N) B) (opr A (.N) C) B C enqbreq syl mpbiri (opr A (.N) B) (opr A (.N) C) opex B C opex enqer erthi syl)) thm (distrpq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((distrpq.1 (e. B (V))) (distrpq.2 (e. C (V)))) (= (opr A (.Q) (opr B (+Q) C)) (opr (opr A (.Q) B) (+Q) (opr A (.Q) C))) (df-nq (cv z) (cv w) (cv v) (cv u) addpipq (cv x) (cv y) (opr (opr (cv z) (.N) (cv u)) (+N) (opr (cv w) (.N) (cv v))) (opr (cv w) (.N) (cv u)) mulpipq (cv x) (opr (opr (cv z) (.N) (cv u)) (+N) (opr (cv w) (.N) (cv v))) mulclpi (e. (cv y) (N.)) (e. (opr (cv w) (.N) (cv u)) (N.)) pm3.26 (cv y) (opr (cv w) (.N) (cv u)) mulclpi jca anim12i (e. (opr (cv x) (.N) (opr (opr (cv z) (.N) (cv u)) (+N) (opr (cv w) (.N) (cv v)))) (N.)) (e. (cv y) (N.)) (e. (opr (cv y) (.N) (opr (cv w) (.N) (cv u))) (N.)) an12 (e. (cv y) (N.)) (e. (opr (cv x) (.N) (opr (opr (cv z) (.N) (cv u)) (+N) (opr (cv w) (.N) (cv v)))) (N.)) (e. (opr (cv y) (.N) (opr (cv w) (.N) (cv u))) (N.)) 3anass bitr4 sylib an4s y visset (cv x) (.N) (opr (opr (cv z) (.N) (cv u)) (+N) (opr (cv w) (.N) (cv v))) oprex (cv y) (.N) (opr (cv w) (.N) (cv u)) oprex distrpqlem syl eqtr4d (cv x) (cv y) (cv z) (cv w) mulpipq (cv x) (cv y) (cv v) (cv u) mulpipq (opr (cv x) (.N) (cv z)) (opr (cv y) (.N) (cv w)) (opr (cv x) (.N) (cv v)) (opr (cv y) (.N) (cv u)) addpipq (opr (cv z) (.N) (cv u)) (opr (cv w) (.N) (cv v)) addclpi (cv z) (cv u) mulclpi (cv w) (cv v) mulclpi syl2an an42s (cv w) (cv u) mulclpi (e. (cv z) (N.)) (e. (cv v) (N.)) ad2ant2l jca (cv x) (cv z) mulclpi (cv y) (cv w) mulclpi anim12i an4s (cv x) (cv v) mulclpi (cv y) (cv u) mulclpi anim12i an4s (cv z) (.N) (cv u) oprex (cv w) (.N) (cv v) oprex (opr (cv y) (.N) (cv x)) distrpi x visset (opr (cv z) (.N) (cv u)) (+N) (opr (cv w) (.N) (cv v)) oprex (cv y) mulasspi x visset y visset mulcompi (.N) (opr (cv z) (.N) (cv u)) opreq1i x visset y visset z visset f visset g visset mulcompi g visset h visset (cv f) mulasspi u visset caopr4 eqtr3 y visset x visset w visset f visset g visset mulcompi g visset h visset (cv f) mulasspi v visset caopr4 (+N) opreq12i 3eqtr3 y visset (cv w) (.N) (cv u) oprex (cv y) mulasspi y visset y visset w visset f visset g visset mulcompi g visset h visset (cv f) mulasspi u visset caopr4 eqtr3 A B C ecoprdi distrpq.1 dmaddpq distrpq.2 0npq dmmulpq A ndmoprdistr pm2.61i)) thm (1qec () ((1qec.1 (e. A (V)))) (-> (e. A (N.)) (= (1Q) ([] (<,> A A) (~Q)))) (1pi 1pi 1qec.1 1pi elisseti 1pi elisseti distrpqlem mp3an23 A mulidpi A mulidpi jca (opr A (.N) (1o)) A (opr A (.N) (1o)) A opeq12 (<,> (opr A (.N) (1o)) (opr A (.N) (1o))) (<,> A A) (~Q) eceq2 3syl eqtr3d df-1q syl5eq)) thm (mulidpq ((x y) (A x) (A y)) () (-> (e. A (Q.)) (= (opr A (.Q) (1Q)) A)) (df-nq ([] (<,> (cv x) (cv y)) (~Q)) A (.Q) (1Q) opreq1 (= ([] (<,> (cv x) (cv y)) (~Q)) A) id eqeq12d 1pi 1pi pm3.2i (cv x) (cv y) (1o) (1o) mulpipq mpan2 df-1q ([] (<,> (cv x) (cv y)) (~Q)) (.Q) opreq2i 1pi elisseti x visset mulcompi 1pi elisseti y visset mulcompi (opr (1o) (.N) (cv x)) (opr (cv x) (.N) (1o)) (opr (1o) (.N) (cv y)) (opr (cv y) (.N) (1o)) opeq12 mp2an (<,> (opr (1o) (.N) (cv x)) (opr (1o) (.N) (cv y))) (<,> (opr (cv x) (.N) (1o)) (opr (cv y) (.N) (1o))) (~Q) eceq2 ax-mp 3eqtr4g 1pi 1pi elisseti x visset y visset distrpqlem mp3an1 eqtrd ecoptocl)) thm (recmulpq ((x y) (A x) (A y) (B x) (B y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) ((recmulpq.1 (e. B (V)))) (-> (e. A (Q.)) (<-> (= (` (*Q) A) B) (= (opr A (.Q) B) (1Q)))) (recmulpq.1 (cv x) A (.Q) (cv y) opreq1 (1Q) eqeq1d (cv y) B A (.Q) opreq2 (1Q) eqeq1d df-nq ([] (<,> (cv z) (cv w)) (~Q)) (cv x) (.Q) (cv y) opreq1 (1Q) eqeq1d y exbidv (cv z) (cv w) (cv w) (cv z) mulpipq an42s anidms (cv z) (cv w) mulclpi (cv z) (.N) (cv w) oprex 1qec z visset w visset mulcompi (opr (cv z) (.N) (cv w)) (opr (cv w) (.N) (cv z)) (opr (cv z) (.N) (cv w)) opeq2 ax-mp (<,> (opr (cv z) (.N) (cv w)) (opr (cv z) (.N) (cv w))) (<,> (opr (cv z) (.N) (cv w)) (opr (cv w) (.N) (cv z))) (~Q) eceq2 ax-mp syl6eq syl eqtr4d enqex (~Q) (V) (<,> (cv w) (cv z)) ecexg ax-mp (cv y) ([] (<,> (cv w) (cv z)) (~Q)) ([] (<,> (cv z) (cv w)) (~Q)) (.Q) opreq2 (1Q) eqeq1d cla4ev syl ecoptocl y (= (opr (cv x) (.Q) (cv y)) (1Q)) eu5 x visset 1q dmmulpq 0npq z visset w visset mulcompq w visset v visset (cv z) mulasspq (cv z) mulidpq y caoprmo mpbiran2 sylibr x y df-rq fvopab3)) thm (recidpq () () (-> (e. A (Q.)) (= (opr A (.Q) (` (*Q) A)) (1Q))) ((` (*Q) A) eqid (*Q) A fvex A recmulpq mpbii)) thm (recclpq () () (-> (e. A (Q.)) (e. (` (*Q) A) (Q.))) (A recidpq 1q syl6eqel (*Q) A fvex dmmulpq 0npq A ndmoprrcl pm3.27d syl)) thm (recrecpq () ((recrecpq.1 (e. A (V)))) (-> (e. A (Q.)) (= (` (*Q) (` (*Q) A)) A)) (A recidpq (*Q) A fvex recrecpq.1 mulcompq syl5eq A recclpq recrecpq.1 (` (*Q) A) recmulpq syl mpbird)) thm (dmrecpq ((x y)) () (= (dom (*Q)) (Q.)) (x y df-rq dmeqi (cv x) recidpq (*Q) (cv x) fvex (cv y) (` (*Q) (cv x)) (cv x) (.Q) opreq2 (1Q) eqeq1d cla4ev syl rgen x (Q.) y (= (opr (cv x) (.Q) (cv y)) (1Q)) dmopab3 mpbi eqtr)) thm (ltsopq ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (w z) (v z) (u z) (f z) (g z) (h z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) () (Or ( (cv x) (cv y)) (~Q)) (cv f) ( (cv z) (cv w)) (~Q)) breq1 ([] (<,> (cv x) (cv y)) (~Q)) (cv f) ([] (<,> (cv z) (cv w)) (~Q)) eqeq1 ([] (<,> (cv x) (cv y)) (~Q)) (cv f) ([] (<,> (cv z) (cv w)) (~Q)) ( (cv x) (cv y)) (~Q)) (cv f) ( (cv z) (cv w)) (~Q)) breq1 (br ([] (<,> (cv z) (cv w)) (~Q)) ( (cv v) (cv u)) (~Q))) anbi1d ([] (<,> (cv x) (cv y)) (~Q)) (cv f) ( (cv v) (cv u)) (~Q)) breq1 imbi12d anbi12d ([] (<,> (cv z) (cv w)) (~Q)) (cv g) (cv f) ( (cv z) (cv w)) (~Q)) (cv g) (cv f) eqeq2 ([] (<,> (cv z) (cv w)) (~Q)) (cv g) ( (cv z) (cv w)) (~Q)) (cv g) (cv f) ( (cv z) (cv w)) (~Q)) (cv g) ( (cv v) (cv u)) (~Q)) breq1 anbi12d (br (cv f) ( (cv v) (cv u)) (~Q))) imbi1d anbi12d ([] (<,> (cv v) (cv u)) (~Q)) (cv h) (cv g) ( (cv v) (cv u)) (~Q)) (cv h) (cv f) ( (br (cv f) ( (e. C (Q.)) (<-> (br A ( (cv x) (cv y)) (~Q)) A ( (cv z) (cv w)) (~Q)) breq1 ([] (<,> (cv x) (cv y)) (~Q)) A ([] (<,> (cv v) (cv u)) (~Q)) (+Q) opreq2 ( (cv v) (cv u)) (~Q)) (+Q) ([] (<,> (cv z) (cv w)) (~Q))) breq1d bibi12d ([] (<,> (cv z) (cv w)) (~Q)) B A ( (cv z) (cv w)) (~Q)) B ([] (<,> (cv v) (cv u)) (~Q)) (+Q) opreq2 (opr ([] (<,> (cv v) (cv u)) (~Q)) (+Q) A) ( (cv v) (cv u)) (~Q)) C (+Q) A opreq1 ([] (<,> (cv v) (cv u)) (~Q)) C (+Q) B opreq1 ( (e. C (Q.)) (<-> (br A ( (cv x) (cv y)) (~Q)) A ( (cv z) (cv w)) (~Q)) breq1 ([] (<,> (cv x) (cv y)) (~Q)) A ([] (<,> (cv v) (cv u)) (~Q)) (.Q) opreq2 ( (cv v) (cv u)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q))) breq1d bibi12d ([] (<,> (cv z) (cv w)) (~Q)) B A ( (cv z) (cv w)) (~Q)) B ([] (<,> (cv v) (cv u)) (~Q)) (.Q) opreq2 (opr ([] (<,> (cv v) (cv u)) (~Q)) (.Q) A) ( (cv v) (cv u)) (~Q)) C (.Q) A opreq1 ([] (<,> (cv v) (cv u)) (~Q)) C (.Q) B opreq1 ( (/\ (e. A (Q.)) (e. B (Q.))) (br A ( (/\ (e. A (Q.)) (e. B (Q.))) (<-> (br A ( (cv y) (cv z)) (~Q)) A ( (cv w) (cv v)) (~Q)) breq1 ([] (<,> (cv y) (cv z)) (~Q)) A (+Q) (cv x) opreq1 ([] (<,> (cv w) (cv v)) (~Q)) eqeq1d x exbidv imbi12d ([] (<,> (cv w) (cv v)) (~Q)) B A ( (cv w) (cv v)) (~Q)) B (opr A (+Q) (cv x)) eqeq2 x exbidv imbi12d (cv y) (cv v) mulclpi (cv z) (cv w) mulclpi anim12i an42s (opr (cv y) (.N) (cv v)) (opr (cv z) (.N) (cv w)) u ltexpi syl (/\ (e. (cv y) (N.)) (e. (cv z) (N.))) (/\ (e. (cv w) (N.)) (e. (cv v) (N.))) pm3.26 (e. (cv u) (N.)) adantr (/\ (/\ (e. (cv y) (N.)) (e. (cv z) (N.))) (/\ (e. (cv w) (N.)) (e. (cv v) (N.)))) (e. (cv u) (N.)) pm3.27 (e. (cv y) (N.)) (e. (cv z) (N.)) pm3.27 (e. (cv w) (N.)) (e. (cv v) (N.)) pm3.27 anim12i (e. (cv u) (N.)) adantr (cv z) (cv v) mulclpi syl jca jca (= (opr (opr (cv y) (.N) (cv v)) (+N) (cv u)) (opr (cv z) (.N) (cv w))) adantrr (cv y) (cv z) (cv u) (opr (cv z) (.N) (cv v)) addpipq syl (e. (cv y) (N.)) (e. (cv z) (N.)) pm3.27 (/\ (e. (cv w) (N.)) (e. (cv v) (N.))) (e. (cv u) (N.)) ad2antrr (opr (cv y) (.N) (cv v)) (cv u) addclpi (cv y) (cv v) mulclpi (e. (cv y) (N.)) (e. (cv z) (N.)) pm3.26 (e. (cv w) (N.)) (e. (cv v) (N.)) pm3.27 syl2an sylan (e. (cv y) (N.)) (e. (cv z) (N.)) pm3.27 (e. (cv w) (N.)) (e. (cv v) (N.)) pm3.27 anim12i (e. (cv u) (N.)) adantr (cv z) (cv v) mulclpi syl 3jca (= (opr (opr (cv y) (.N) (cv v)) (+N) (cv u)) (opr (cv z) (.N) (cv w))) adantrr z visset (opr (cv y) (.N) (cv v)) (+N) (cv u) oprex (cv z) (.N) (cv v) oprex distrpqlem y visset z visset v visset x visset w visset mulcompi w visset u visset (cv x) mulasspi caopr12 (+N) (opr (cv z) (.N) (cv u)) opreq1i (cv y) (.N) (cv v) oprex u visset (cv z) distrpi eqtr4 (opr (opr (cv y) (.N) (opr (cv z) (.N) (cv v))) (+N) (opr (cv z) (.N) (cv u))) (opr (cv z) (.N) (opr (opr (cv y) (.N) (cv v)) (+N) (cv u))) (opr (cv z) (.N) (opr (cv z) (.N) (cv v))) opeq1 ax-mp (<,> (opr (opr (cv y) (.N) (opr (cv z) (.N) (cv v))) (+N) (opr (cv z) (.N) (cv u))) (opr (cv z) (.N) (opr (cv z) (.N) (cv v)))) (<,> (opr (cv z) (.N) (opr (opr (cv y) (.N) (cv v)) (+N) (cv u))) (opr (cv z) (.N) (opr (cv z) (.N) (cv v)))) (~Q) eceq2 ax-mp syl5eq syl (e. (cv z) (N.)) (e. (cv w) (N.)) (e. (cv v) (N.)) 3anass biimpr (e. (cv y) (N.)) adantll (= (opr (opr (cv y) (.N) (cv v)) (+N) (cv u)) (opr (cv z) (.N) (cv w))) anim1i (e. (cv u) (N.)) adantrl (opr (opr (cv y) (.N) (cv v)) (+N) (cv u)) (opr (cv z) (.N) (cv w)) (opr (cv z) (.N) (cv v)) opeq1 (<,> (opr (opr (cv y) (.N) (cv v)) (+N) (cv u)) (opr (cv z) (.N) (cv v))) (<,> (opr (cv z) (.N) (cv w)) (opr (cv z) (.N) (cv v))) (~Q) eceq2 syl z visset w visset v visset distrpqlem sylan9eqr syl 3eqtrd enqex (~Q) (V) (<,> (cv u) (opr (cv z) (.N) (cv v))) ecexg ax-mp (cv x) ([] (<,> (cv u) (opr (cv z) (.N) (cv v))) (~Q)) ([] (<,> (cv y) (cv z)) (~Q)) (+Q) opreq2 ([] (<,> (cv w) (cv v)) (~Q)) eqeq1d cla4ev syl ex u 19.23adv sylbid y visset z visset w visset v visset ordpipq syl5ib 2ecoptocl (opr A (+Q) (cv x)) B (Q.) eleq1 x visset dmaddpq 0npq A ndmoprrcl pm3.27d syl6bir ltexpq.1 x visset ltaddpq ex syl9 imp3a (opr A (+Q) (cv x)) B A ( (/\ (e. A (Q.)) (e. B (Q.))) (<-> (br A ( (e. A (Q.)) (E. x (= (opr (cv x) (+Q) (cv x)) A))) (df-nq ([] (<,> (cv y) (cv z)) (~Q)) A (opr (cv x) (+Q) (cv x)) eqeq2 x exbidv (cv y) (opr (cv z) (+N) (cv z)) (cv y) (opr (cv z) (+N) (cv z)) addpipq y visset y visset (opr (cv z) (+N) (cv z)) distrpi y visset (cv z) (+N) (cv z) oprex mulcompi (+N) (opr (opr (cv z) (+N) (cv z)) (.N) (cv y)) opreq1i eqtr4 (opr (opr (cv z) (+N) (cv z)) (.N) (opr (cv y) (+N) (cv y))) (opr (opr (cv y) (.N) (opr (cv z) (+N) (cv z))) (+N) (opr (opr (cv z) (+N) (cv z)) (.N) (cv y))) (opr (opr (cv z) (+N) (cv z)) (.N) (opr (cv z) (+N) (cv z))) opeq1 ax-mp (<,> (opr (opr (cv z) (+N) (cv z)) (.N) (opr (cv y) (+N) (cv y))) (opr (opr (cv z) (+N) (cv z)) (.N) (opr (cv z) (+N) (cv z)))) (<,> (opr (opr (cv y) (.N) (opr (cv z) (+N) (cv z))) (+N) (opr (opr (cv z) (+N) (cv z)) (.N) (cv y))) (opr (opr (cv z) (+N) (cv z)) (.N) (opr (cv z) (+N) (cv z)))) (~Q) eceq2 ax-mp syl6eqr (cv z) (cv z) addclpi anidms (e. (cv y) (N.)) anim2i (cv z) (cv z) addclpi anidms (e. (cv y) (N.)) anim2i sylanc (cv z) (+N) (cv z) oprex (cv y) (+N) (cv y) oprex (cv z) (+N) (cv z) oprex distrpqlem (cv z) (cv z) addclpi anidms (e. (cv y) (N.)) adantl (cv y) (cv y) addclpi anidms (e. (cv z) (N.)) adantr (cv z) (cv z) addclpi anidms (e. (cv y) (N.)) adantl syl3anc (cv y) mulidpi (cv y) mulidpi (+N) opreq12d y visset (1o) (+N) (1o) oprex mulcompi 1pi elisseti 1pi elisseti (cv y) distrpi eqtr3 syl5eq (cv z) mulidpi (cv z) mulidpi (+N) opreq12d z visset (1o) (+N) (1o) oprex mulcompi 1pi elisseti 1pi elisseti (cv z) distrpi eqtr3 syl5eq anim12i (opr (opr (1o) (+N) (1o)) (.N) (cv y)) (opr (cv y) (+N) (cv y)) (opr (opr (1o) (+N) (1o)) (.N) (cv z)) (opr (cv z) (+N) (cv z)) opeq12 (<,> (opr (opr (1o) (+N) (1o)) (.N) (cv y)) (opr (opr (1o) (+N) (1o)) (.N) (cv z))) (<,> (opr (cv y) (+N) (cv y)) (opr (cv z) (+N) (cv z))) (~Q) eceq2 3syl 1pi 1pi (1o) (1o) addclpi mp2an (1o) (+N) (1o) oprex y visset z visset distrpqlem mp3an1 eqtr3d 3eqtrd enqex (~Q) (V) (<,> (cv y) (opr (cv z) (+N) (cv z))) ecexg ax-mp (cv x) ([] (<,> (cv y) (opr (cv z) (+N) (cv z))) (~Q)) (cv x) ([] (<,> (cv y) (opr (cv z) (+N) (cv z))) (~Q)) (+Q) opreq12 anidms ([] (<,> (cv y) (cv z)) (~Q)) eqeq1d cla4ev syl ecoptocl)) thm (nsmallpq ((A x)) () (-> (e. A (Q.)) (E. x (br (cv x) ( (br A ( (br A ( (br (cv z) ( (e. A (P.)) (/\ (/\ (C: ({/}) A) (C: A (Q.))) (A.e. x A (/\ (A. y (-> (br (cv y) ( (br (cv y) ( (e. A (P.)) (-. (= A ({/})))) (A x y elnp (/\ (C: ({/}) A) (C: A (Q.))) (A.e. x A (/\ (A. y (-> (br (cv y) ( (e. A (P.)) (C: A (Q.))) (A x y elnp (/\ (C: ({/}) A) (C: A (Q.))) (A.e. x A (/\ (A. y (-> (br (cv y) ( (/\ (e. A (P.)) (e. B A)) (e. B (Q.))) (A prpssnq pssssd B sseld imp)) thm (0npr () () (-. (e. ({/}) (P.))) (({/}) eqid ({/}) prn0 mt2)) thm (prcdpq ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (e. A (P.)) (e. B A)) (-> (br C ( (/\ (/\ (e. A (P.)) (e. B A)) (e. C (Q.))) (-> (-. (e. C A)) (br B ( (/\ (e. A (P.)) (e. B A)) (E. x (/\ (e. (cv x) A) (br B ( (/\ (e. A (P.)) (e. B A)) (E. x (e. (opr B (+Q) (cv x)) A))) (A B y prnmax y visset ltrelpq B brel prnmadd.1 (cv y) x ltexpq biimpcd mpd (cv y) (opr B (+Q) (cv x)) eqcom x exbii sylibr (e. (cv y) A) anim1i x (= (cv y) (opr B (+Q) (cv x))) (e. (cv y) A) 19.41v sylibr ancoms y 19.22i y x (/\ (= (cv y) (opr B (+Q) (cv x))) (e. (cv y) A)) excomim 3syl (opr B (+Q) (cv x)) A y df-clel x exbii sylibr)) thm (ltrelpr ((x y)) () (C_ (,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))))) (-> (/\ (e. A (P.)) (e. B (P.))) (= (opr A F B) ({|} f (E. g (E. h (/\ (/\ (e. (cv g) A) (e. (cv h) B)) (= (cv f) (opr (cv g) G (cv h))))))))) ((cv f) A F (cv g) opreq1 (cv f) A y (E.e. z (cv g) (= (cv x) (opr (cv y) G (cv z)))) rexeq1 x abbidv eqeq12d (cv g) B A F opreq2 (cv g) B z (= (cv x) (opr (cv y) G (cv z))) rexeq1 y A rexbidv x abbidv eqeq12d f visset g visset x y z G oprvalex (cv w) (cv f) y (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))) rexeq1 x abbidv (cv v) (cv g) z (= (cv x) (opr (cv y) G (cv z))) rexeq1 y (cv f) rexbidv x abbidv genp.1 oprabval2 vtocl2ga (cv x) (cv f) (opr (cv g) G (cv h)) eqeq1 (/\ (e. (cv g) A) (e. (cv h) B)) anbi2d g h 2exbidv y A z B (= (cv x) (opr (cv y) G (cv z))) r2ex (cv y) (cv g) A eleq1 (cv z) (cv h) B eleq1 bi2anan9 (cv y) (cv g) (cv z) (cv h) G opreq12 (cv x) eqeq2d anbi12d cbvex2v bitr syl5bb cbvabv syl6eq)) thm (genpelv ((f g) (f h) (C f) (g h) (C g) (C h) (F h) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpelv.2 (e. C (V)))) (-> (/\ (e. A (P.)) (e. B (P.))) (<-> (e. C (opr A F B)) (E. f (E. g (/\ (/\ (e. (cv f) A) (e. (cv g) B)) (= C (opr (cv f) G (cv g)))))))) (genp.1 A B h f g genpv C eleq2d genpelv.2 (cv h) C (opr (cv f) G (cv g)) eqeq1 (/\ (e. (cv f) A) (e. (cv g) B)) anbi2d f g 2exbidv elab syl6bb)) thm (genpprecl ((f g) (f h) (C f) (g h) (C g) (C h) (D f) (D g) (D h) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))))) (-> (/\ (e. A (P.)) (e. B (P.))) (-> (/\ (e. C A) (e. D B)) (e. (opr C G D) (opr A F B)))) ((opr C G D) eqid genp.1 A B f g h genpv (opr C G D) eleq2d C G D oprex (cv f) (opr C G D) (opr (cv g) G (cv h)) eqeq1 (/\ (e. (cv g) A) (e. (cv h) B)) anbi2d g h 2exbidv elab syl6bb (cv g) C A eleq1 (cv h) D B eleq1 bi2anan9 (cv g) C (cv h) D G opreq12 (opr C G D) eqeq2d anbi12d A B cla4e2gv anabsi5 syl5bir mpan2i)) thm (genpdm ((x y) (x z) (y z) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (u v) (G v) (G u)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))))) (= (dom F) (X. (P.) (P.))) (w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))) dmoprab genp.1 dmeqi (P.) (P.) w v df-xp u (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))) 19.42v w visset v visset x y z G oprvalex u isseti mpbiran2 w v opabbii eqtr4 3eqtr4)) thm (genpn0 ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))))) (-> (/\ (e. A (P.)) (e. B (P.))) (C: ({/}) (opr A F B))) (A prn0 A g n0 sylib B prn0 B h n0 sylib anim12i f g h (/\ (/\ (e. (cv g) A) (e. (cv h) B)) (= (cv f) (opr (cv g) G (cv h)))) exrot3 f (/\ (e. (cv g) A) (e. (cv h) B)) (= (cv f) (opr (cv g) G (cv h))) 19.42v (cv g) G (cv h) oprex f isseti mpbiran2 g h 2exbii g h (e. (cv g) A) (e. (cv h) B) eeanv 3bitr sylibr genp.1 A B f g h genpv abeq2d f exbidv mpbird (opr A F B) 0pss (opr A F B) f n0 bitr sylibr)) thm (genpss ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpss.2 (-> (/\ (e. (cv g) (Q.)) (e. (cv h) (Q.))) (e. (opr (cv g) G (cv h)) (Q.))))) (-> (/\ (e. A (P.)) (e. B (P.))) (C_ (opr A F B) (Q.))) (genp.1 A B f g h genpv abeq2d A prpssnq pssssd (cv g) sseld B prpssnq pssssd (cv h) sseld im2anan9 genpss.2 syl6 (opr (cv g) G (cv h)) (Q.) (cv f) eleq1a syl6 imp3a g h 19.23advv sylbid ssrdv)) thm (genpnnp ((v w) (A w) (A v) (B w) (B v) (F w) (F v) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpnnp.2 (-> (/\ (e. (cv w) (Q.)) (e. (cv v) (Q.))) (e. (opr (cv w) G (cv v)) (Q.)))) (genpnnp.3 (-> (e. (cv z) (Q.)) (<-> (br (cv x) ( (/\ (e. A (P.)) (e. B (P.))) (-. (= (opr A F B) (Q.)))) (A prpssnq A (Q.) w pssnel syl B prpssnq B (Q.) v pssnel syl anim12i w v (/\ (e. (cv w) (Q.)) (-. (e. (cv w) A))) (/\ (e. (cv v) (Q.)) (-. (e. (cv v) B))) eeanv sylibr A (cv g) (cv w) prub B (cv h) (cv v) prub im2anan9 ltsopq (,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpcd.2 (-> (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) ( (/\ (e. A (P.)) (e. B (P.))) (-> (e. (cv f) (opr A F B)) (-> (br (cv x) (,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpnmax.2 (-> (e. (cv v) (Q.)) (<-> (br (cv z) ( (/\ (e. A (P.)) (e. B (P.))) (-> (e. (cv f) (opr A F B)) (E. x (/\ (e. (cv x) (opr A F B)) (br (cv f) (,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpcl.2 (-> (/\ (e. (cv x) (Q.)) (e. (cv y) (Q.))) (e. (opr (cv x) G (cv y)) (Q.)))) (genpcl.3 (-> (e. (cv h) (Q.)) (<-> (br (cv f) ( (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) ( (/\ (e. A (P.)) (e. B (P.))) (e. (opr A F B) (P.))) (genp.1 A B genpn0 genp.1 genpcl.2 (cv g) (cv h) caoprcl A B genpss genp.1 genpcl.2 (cv w) (cv v) caoprcl x visset y visset genpcl.3 (cv z) caoprord genpcl.4 A B genpnnp jca (opr A F B) (Q.) dfpss2 sylibr jca genp.1 genpcl.5 f genpcd x 19.21adv genp.1 z visset w visset genpcl.3 (cv v) caoprord z visset w visset genpcl.4 caoprcom A B f genpnmax x (opr A F B) (br (cv f) (,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpass.2 (e. B (V))) (genpass.3 (e. C (V))) (genpass.4 (= (dom F) (X. (P.) (P.)))) (genpass.5 (-> (/\ (e. (cv f) (P.)) (e. (cv g) (P.))) (e. (opr (cv f) F (cv g)) (P.)))) (genpass.6 (= (opr (opr (cv f) G (cv g)) G (cv h)) (opr (cv f) G (opr (cv g) G (cv h)))))) (= (opr (opr A F B) F C) (opr A F (opr B F C))) (genp.1 t visset B C g h genpelv (e. A (P.)) 3adant1 (/\ (e. (cv f) A) (= (cv x) (opr (cv f) G (cv t)))) anbi1d (e. (cv f) A) (e. (cv t) (opr B F C)) (= (cv x) (opr (cv f) G (cv t))) anass (e. (cv f) A) (e. (cv t) (opr B F C)) (= (cv x) (opr (cv f) G (cv t))) an12 bitr g h (/\ (/\ (e. (cv g) B) (e. (cv h) C)) (= (cv t) (opr (cv g) G (cv h)))) (/\ (e. (cv f) A) (= (cv x) (opr (cv f) G (cv t)))) 19.41vv 3bitr4g f t 2exbidv t g h (/\ (/\ (/\ (e. (cv g) B) (e. (cv h) C)) (= (cv t) (opr (cv g) G (cv h)))) (/\ (e. (cv f) A) (= (cv x) (opr (cv f) G (cv t))))) exrot3 (/\ (e. (cv g) B) (e. (cv h) C)) (= (cv t) (opr (cv g) G (cv h))) (e. (cv f) A) (= (cv x) (opr (cv f) G (cv t))) an4 (/\ (e. (cv g) B) (e. (cv h) C)) (e. (cv f) A) ancom (cv t) (opr (cv g) G (cv h)) (cv f) G opreq2 (cv x) eqeq2d pm5.32i anbi12i (/\ (e. (cv f) A) (/\ (e. (cv g) B) (e. (cv h) C))) (= (cv t) (opr (cv g) G (cv h))) (= (cv x) (opr (cv f) G (opr (cv g) G (cv h)))) an12 3bitr t exbii t (= (cv t) (opr (cv g) G (cv h))) (/\ (/\ (e. (cv f) A) (/\ (e. (cv g) B) (e. (cv h) C))) (= (cv x) (opr (cv f) G (opr (cv g) G (cv h))))) 19.41v (cv g) G (cv h) oprex t isseti mpbiran bitr g h 2exbii bitr f exbii syl6bb genp.1 x visset A (opr B F C) f t genpelv genpass.5 B C caoprcl sylan2 3impb genp.1 x visset (opr A F B) C t h genpelv genpass.5 A B caoprcl sylan 3impa genp.1 t visset A B f g genpelv (e. C (P.)) 3adant3 (/\ (e. (cv h) C) (= (cv x) (opr (cv t) G (cv h)))) anbi1d (e. (cv t) (opr A F B)) (e. (cv h) C) (= (cv x) (opr (cv t) G (cv h))) anass f g (/\ (/\ (e. (cv f) A) (e. (cv g) B)) (= (cv t) (opr (cv f) G (cv g)))) (/\ (e. (cv h) C) (= (cv x) (opr (cv t) G (cv h)))) 19.41vv 3bitr4g t h 2exbidv bitrd t h f g (/\ (/\ (/\ (e. (cv f) A) (e. (cv g) B)) (= (cv t) (opr (cv f) G (cv g)))) (/\ (e. (cv h) C) (= (cv x) (opr (cv t) G (cv h))))) exrot4 t h (/\ (/\ (/\ (e. (cv f) A) (e. (cv g) B)) (= (cv t) (opr (cv f) G (cv g)))) (/\ (e. (cv h) C) (= (cv x) (opr (cv t) G (cv h))))) excom (/\ (e. (cv f) A) (e. (cv g) B)) (= (cv t) (opr (cv f) G (cv g))) (e. (cv h) C) (= (cv x) (opr (cv t) G (cv h))) an4 (e. (cv f) A) (e. (cv g) B) (e. (cv h) C) anass (cv t) (opr (cv f) G (cv g)) G (cv h) opreq1 genpass.6 syl6eq (cv x) eqeq2d pm5.32i anbi12i (/\ (e. (cv f) A) (/\ (e. (cv g) B) (e. (cv h) C))) (= (cv t) (opr (cv f) G (cv g))) (= (cv x) (opr (cv f) G (opr (cv g) G (cv h)))) an12 3bitr t exbii t (= (cv t) (opr (cv f) G (cv g))) (/\ (/\ (e. (cv f) A) (/\ (e. (cv g) B) (e. (cv h) C))) (= (cv x) (opr (cv f) G (opr (cv g) G (cv h))))) 19.41v (cv f) G (cv g) oprex t isseti mpbiran bitr h exbii bitr f g 2exbii bitr syl6bb 3bitr4rd eqrdv genpass.2 genpass.4 genpass.3 0npr A ndmoprass pm2.61i)) thm (plpv ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u)) () (-> (/\ (e. A (P.)) (e. B (P.))) (= (opr A (+P.) B) ({|} x (E. y (E. z (/\ (/\ (e. (cv y) A) (e. (cv z) B)) (= (cv x) (opr (cv y) (+Q) (cv z))))))))) (w v u f g h df-plp A B x y z genpv)) thm (mpv ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u)) () (-> (/\ (e. A (P.)) (e. B (P.))) (= (opr A (.P.) B) ({|} x (E. y (E. z (/\ (/\ (e. (cv y) A) (e. (cv z) B)) (= (cv x) (opr (cv y) (.Q) (cv z))))))))) (w v u f g h df-mp A B x y z genpv)) thm (dmplp ((x y) (x z) (y z) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (dom (+P.)) (X. (P.) (P.))) (x y z w v u df-plp genpdm)) thm (dmmp ((x y) (x z) (y z) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (dom (.P.)) (X. (P.) (P.))) (x y z w v u df-mp genpdm)) thm (1pr ((x y)) () (e. (1P) (P.)) ((1P) x y elnp 1lt2pq 1q elisseti (1Q) (+Q) (1Q) oprex ltrpq (*Q) (1Q) fvex 1q elisseti mulcompq 1q (1Q) recclpq ax-mp (` (*Q) (1Q)) mulidpq ax-mp 1q (1Q) recidpq ax-mp 3eqtr3 syl6breq ax-mp (*Q) (opr (1Q) (+Q) (1Q)) fvex (cv x) (` (*Q) (opr (1Q) (+Q) (1Q))) ( (/\ (/\ (e. A (P.)) (e. (cv g) A)) (e. (cv x) (Q.))) (-> (br (cv x) ( (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) ( (/\ (e. A (P.)) (e. B (P.))) (e. (opr A (+P.) B) (P.))) (w v u x y z df-plp (cv x) (cv y) addclpq f visset g visset (cv h) ltapq x visset y visset addcompq A g B h x addclprlem2 genpcl)) thm (mulclprlem ((x y) (x z) (w x) (v x) (u x) (g x) (h x) (y z) (w y) (v y) (u y) (g y) (h y) (w z) (v z) (u z) (g z) (h z) (v w) (u w) (g w) (h w) (u v) (g v) (h v) (g u) (h u) (g h) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) ( (/\ (e. A (P.)) (e. B (P.))) (e. (opr A (.P.) B) (P.))) (w v u x y z df-mp (cv x) (cv y) mulclpq f visset g visset (cv h) ltmpq x visset y visset mulcompq A g B h x mulclprlem genpcl)) thm (addcompr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((addcompr.1 (e. A (V))) (addcompr.2 (e. B (V)))) (= (opr A (+P.) B) (opr B (+P.) A)) (A B x y z plpv B A x z y plpv (e. (cv z) B) (e. (cv y) A) ancom z visset y visset addcompq (cv x) eqeq2i anbi12i z y 2exbii z y (/\ (/\ (e. (cv y) A) (e. (cv z) B)) (= (cv x) (opr (cv y) (+Q) (cv z)))) excom bitr x abbii syl6eq ancoms eqtr4d addcompr.2 dmplp addcompr.1 ndmoprcom pm2.61i)) thm (addasspr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) ((addasspr.1 (e. B (V))) (addasspr.2 (e. C (V)))) (= (opr (opr A (+P.) B) (+P.) C) (opr A (+P.) (opr B (+P.) C))) (w v u x y z df-plp addasspr.1 addasspr.2 dmplp (cv f) (cv g) addclpr g visset h visset (cv f) addasspq A genpass)) thm (mulcompr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((mulcompr.1 (e. A (V))) (mulcompr.2 (e. B (V)))) (= (opr A (.P.) B) (opr B (.P.) A)) (A B x y z mpv B A x z y mpv (e. (cv z) B) (e. (cv y) A) ancom z visset y visset mulcompq (cv x) eqeq2i anbi12i z y 2exbii z y (/\ (/\ (e. (cv y) A) (e. (cv z) B)) (= (cv x) (opr (cv y) (.Q) (cv z)))) excom bitr x abbii syl6eq ancoms eqtr4d mulcompr.2 dmmp mulcompr.1 ndmoprcom pm2.61i)) thm (mulasspr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) ((mulasspr.1 (e. B (V))) (mulasspr.2 (e. C (V)))) (= (opr (opr A (.P.) B) (.P.) C) (opr A (.P.) (opr B (.P.) C))) (w v u x y z df-mp mulasspr.1 mulasspr.2 dmmp (cv f) (cv g) mulclpr g visset h visset (cv f) mulasspq A genpass)) thm (distrlem1pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) () (-> (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (C_ (opr A (.P.) (opr B (+P.) C)) (opr (opr A (.P.) B) (+P.) (opr A (.P.) C)))) (y z u f g h df-mp w visset A (opr B (+P.) C) x v genpelv B C addclpr sylan2 x w u f g h df-plp v visset B C y z genpelv (= (cv w) (opr (cv x) (.Q) (cv v))) anbi1d (e. (cv x) A) anbi2d (e. (cv x) A) (e. (cv v) (opr B (+P.) C)) (= (cv w) (opr (cv x) (.Q) (cv v))) anass y z (e. (cv x) A) (/\ (/\ (/\ (e. (cv y) B) (e. (cv z) C)) (= (cv v) (opr (cv y) (+Q) (cv z)))) (= (cv w) (opr (cv x) (.Q) (cv v)))) 19.42vv y z (/\ (/\ (e. (cv y) B) (e. (cv z) C)) (= (cv v) (opr (cv y) (+Q) (cv z)))) (= (cv w) (opr (cv x) (.Q) (cv v))) 19.41vv (e. (cv x) A) anbi2i bitr 3bitr4g (e. A (P.)) adantl x v 2exbidv bitrd x v y z (/\ (e. (cv x) A) (/\ (/\ (/\ (e. (cv y) B) (e. (cv z) C)) (= (cv v) (opr (cv y) (+Q) (cv z)))) (= (cv w) (opr (cv x) (.Q) (cv v))))) exrot4 (/\ (e. (cv y) B) (e. (cv z) C)) (= (cv v) (opr (cv y) (+Q) (cv z))) (= (cv w) (opr (cv x) (.Q) (cv v))) anass v exbii v (/\ (e. (cv y) B) (e. (cv z) C)) (/\ (= (cv v) (opr (cv y) (+Q) (cv z))) (= (cv w) (opr (cv x) (.Q) (cv v)))) 19.42v (cv y) (+Q) (cv z) oprex (cv v) (opr (cv y) (+Q) (cv z)) (cv x) (.Q) opreq2 (cv w) eqeq2d ceqsexv (/\ (e. (cv y) B) (e. (cv z) C)) anbi2i 3bitr (e. (cv x) A) anbi2i v (e. (cv x) A) (/\ (/\ (/\ (e. (cv y) B) (e. (cv z) C)) (= (cv v) (opr (cv y) (+Q) (cv z)))) (= (cv w) (opr (cv x) (.Q) (cv v)))) 19.42v (e. (cv x) A) (/\ (e. (cv y) B) (e. (cv z) C)) (= (cv w) (opr (cv x) (.Q) (opr (cv y) (+Q) (cv z)))) anass 3bitr4 y z x 3exbi bitr syl6bb w v u f g h df-mp A B (cv x) (cv y) genpprecl w v u f g h df-mp A C (cv x) (cv z) genpprecl im2anan9 w v u f g h df-plp (opr A (.P.) B) (opr A (.P.) C) (opr (cv x) (.Q) (cv y)) (opr (cv x) (.Q) (cv z)) genpprecl A B mulclpr A C mulclpr syl2an syld (e. (cv x) A) (e. (cv y) B) (e. (cv z) C) anandi y visset z visset (cv x) distrpq (opr (opr A (.P.) B) (+P.) (opr A (.P.) C)) eleq1i 3imtr4g anandis (opr (cv x) (.Q) (opr (cv y) (+Q) (cv z))) (opr (opr A (.P.) B) (+P.) (opr A (.P.) C)) (cv w) eleq1a syl6 imp3a z x 19.23advv y 19.23adv sylbid ssrdv)) thm (distrlem2pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) () (-> (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (-> (/\ (e. (cv x) A) (/\ (e. (cv y) B) (e. (cv z) C))) (e. (opr (opr (cv x) (.Q) (cv y)) (+Q) (opr (cv x) (.Q) (cv z))) (opr A (.P.) (opr B (+P.) C))))) (w v u f g h df-plp B C (cv y) (cv z) genpprecl (e. (cv x) A) anim2d (e. A (P.)) adantl w v u f g h df-mp A (opr B (+P.) C) (cv x) (opr (cv y) (+Q) (cv z)) genpprecl B C addclpr sylan2 syld y visset z visset (cv x) distrpq (opr A (.P.) (opr B (+P.) C)) eleq1i syl6ib)) thm (distrlem3pr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (/\ (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (/\ (e. (cv x) A) (/\ (e. (cv y) B) (e. (cv z) C)))) (/\ (e. (cv x) (Q.)) (/\ (e. (cv y) (Q.)) (e. (cv z) (Q.))))) ((e. A (P.)) (e. B (P.)) (e. C (P.)) (e. (cv x) A) (e. (cv y) B) (e. (cv z) C) an6 A (cv x) elprpq B (cv y) elprpq C (cv z) elprpq 3anim123i sylbi (e. A (P.)) (e. B (P.)) (e. C (P.)) 3anass (e. (cv x) A) (e. (cv y) B) (e. (cv z) C) 3anass anbi12i (e. (cv x) (Q.)) (e. (cv y) (Q.)) (e. (cv z) (Q.)) 3anass 3imtr3)) thm (distrlem4pr ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f)) () (-> (/\ (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (/\ (/\ (e. (cv x) A) (e. (cv f) A)) (/\ (e. (cv y) B) (e. (cv z) C)))) (e. (opr (opr (cv x) (.Q) (cv y)) (+Q) (opr (cv f) (.Q) (cv z))) (opr A (.P.) (opr B (+P.) C)))) (A B C f y z distrlem3pr x visset f visset w visset v visset (cv u) ltmpq y visset w visset v visset mulcompq caoprord2 (cv f) (cv z) mulclpq (cv x) (.Q) (cv y) oprex (cv f) (.Q) (cv y) oprex w visset v visset (cv u) ltapq (cv f) (.Q) (cv z) oprex w visset v visset addcompq caoprord2 syl sylan9bb an1s syl A B C f y z distrlem2pr A (opr B (+P.) C) mulclpr B C addclpr sylan2 (opr A (.P.) (opr B (+P.) C)) (opr (opr (cv f) (.Q) (cv y)) (+Q) (opr (cv f) (.Q) (cv z))) (opr (opr (cv x) (.Q) (cv y)) (+Q) (opr (cv f) (.Q) (cv z))) prcdpq ex syl syld imp sylbid (e. (cv x) A) adantrll A B C x y z distrlem3pr f visset x visset w visset v visset (cv u) ltmpq z visset w visset v visset mulcompq caoprord2 (cv x) (cv y) mulclpq (cv f) (.Q) (cv z) oprex (cv x) (.Q) (cv z) oprex (opr (cv x) (.Q) (cv y)) ltapq syl sylan9bbr anasss syl A B C x y z distrlem2pr A (opr B (+P.) C) mulclpr B C addclpr sylan2 (opr A (.P.) (opr B (+P.) C)) (opr (opr (cv x) (.Q) (cv y)) (+Q) (opr (cv x) (.Q) (cv z))) (opr (opr (cv x) (.Q) (cv y)) (+Q) (opr (cv f) (.Q) (cv z))) prcdpq ex syl syld imp sylbid (e. (cv f) A) adantrlr A (cv x) elprpq A (cv f) elprpq anim12i anandis ltsopq ( (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (C_ (opr (opr A (.P.) B) (+P.) (opr A (.P.) C)) (opr A (.P.) (opr B (+P.) C)))) (x y z f g h df-plp w visset (opr A (.P.) B) (opr A (.P.) C) v u genpelv A B mulclpr A C mulclpr syl2an w v u f g h df-mp v visset A B x y genpelv w v u x g h df-mp u visset A C f z genpelv bi2anan9 x y f z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv v) (opr (cv x) (.Q) (cv y)))) (/\ (/\ (e. (cv f) A) (e. (cv z) C)) (= (cv u) (opr (cv f) (.Q) (cv z)))) ee4anv syl6bbr (= (cv w) (opr (cv v) (+Q) (cv u))) anbi1d f z (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv v) (opr (cv x) (.Q) (cv y)))) (/\ (/\ (e. (cv f) A) (e. (cv z) C)) (= (cv u) (opr (cv f) (.Q) (cv z))))) (= (cv w) (opr (cv v) (+Q) (cv u))) 19.41vv x y 2exbii x y (E. f (E. z (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv v) (opr (cv x) (.Q) (cv y)))) (/\ (/\ (e. (cv f) A) (e. (cv z) C)) (= (cv u) (opr (cv f) (.Q) (cv z))))))) (= (cv w) (opr (cv v) (+Q) (cv u))) 19.41vv bitr syl6bbr v u 2exbidv v u x y (E. f (E. z (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv v) (opr (cv x) (.Q) (cv y)))) (/\ (/\ (e. (cv f) A) (e. (cv z) C)) (= (cv u) (opr (cv f) (.Q) (cv z))))) (= (cv w) (opr (cv v) (+Q) (cv u)))))) exrot4 v u f z (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv v) (opr (cv x) (.Q) (cv y)))) (/\ (/\ (e. (cv f) A) (e. (cv z) C)) (= (cv u) (opr (cv f) (.Q) (cv z))))) (= (cv w) (opr (cv v) (+Q) (cv u)))) exrot4 v u (/\ (/\ (e. (cv x) A) (e. (cv f) A)) (/\ (e. (cv y) B) (e. (cv z) C))) (/\ (/\ (= (cv v) (opr (cv x) (.Q) (cv y))) (= (cv u) (opr (cv f) (.Q) (cv z)))) (= (cv w) (opr (cv v) (+Q) (cv u)))) 19.42vv (/\ (/\ (e. (cv x) A) (e. (cv f) A)) (/\ (e. (cv y) B) (e. (cv z) C))) (/\ (= (cv v) (opr (cv x) (.Q) (cv y))) (= (cv u) (opr (cv f) (.Q) (cv z)))) (= (cv w) (opr (cv v) (+Q) (cv u))) anass (e. (cv x) A) (e. (cv f) A) (e. (cv y) B) (e. (cv z) C) an4 (/\ (= (cv v) (opr (cv x) (.Q) (cv y))) (= (cv u) (opr (cv f) (.Q) (cv z)))) anbi1i (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv f) A) (e. (cv z) C)) (= (cv v) (opr (cv x) (.Q) (cv y))) (= (cv u) (opr (cv f) (.Q) (cv z))) an4 bitr (= (cv w) (opr (cv v) (+Q) (cv u))) anbi1i bitr3 v u 2exbii (cv x) (.Q) (cv y) oprex (cv f) (.Q) (cv z) oprex (/\ (= (cv v) (opr (cv x) (.Q) (cv y))) (= (cv u) (opr (cv f) (.Q) (cv z)))) id (cv v) (opr (cv x) (.Q) (cv y)) (cv u) (opr (cv f) (.Q) (cv z)) (+Q) opreq12 (cv w) eqeq2d (V) (V) cgsex2g mp2an (/\ (/\ (e. (cv x) A) (e. (cv f) A)) (/\ (e. (cv y) B) (e. (cv z) C))) anbi2i 3bitr3 f z 2exbii bitr x y 2exbii bitr syl6bb bitrd anandis (cv w) (opr (opr (cv x) (.Q) (cv y)) (+Q) (opr (cv f) (.Q) (cv z))) (opr A (.P.) (opr B (+P.) C)) eleq1 A B C x f y z distrlem4pr syl5bir com12 ex imp3a f z 19.23advv x y 19.23advv sylbid ssrdv)) thm (distrpr () ((distrpr.1 (e. B (V))) (distrpr.2 (e. C (V)))) (= (opr A (.P.) (opr B (+P.) C)) (opr (opr A (.P.) B) (+P.) (opr A (.P.) C))) (A B C distrlem1pr A B C distrlem5pr eqssd 3impb distrpr.1 dmplp distrpr.2 0npr dmmp A ndmoprdistr pm2.61i)) thm (1idpr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (A y) (w z) (v z) (u z) (f z) (g z) (A z) (v w) (u w) (f w) (g w) (A w) (u v) (f v) (g v) (A v) (f u) (g u) (A u) (f g) (A f) (A g)) () (-> (e. A (P.)) (= (opr A (.P.) (1P)) A)) ((cv x) (opr (cv f) (.Q) (cv g)) ( (/\ (e. A (P.)) (e. B (P.))) (<-> (br A ( (/\ (e. A (P.)) (e. B (P.))) (\/\/ (C: A B) (= A B) (C: B A))) (B (cv y) (cv x) prub A (cv x) elprpq sylan2 A (cv x) (cv y) prcdpq (/\ (e. B (P.)) (e. (cv y) B)) adantl syld exp43 com3r imp imp4a com23 y 19.21adv x 19.23adv A B sspss negbii A B x nss bitr3 B A sspss B A y dfss2 bitr3 3imtr4g orrd (C: A B) (= A B) (C: B A) df-3or (C: A B) (= A B) (C: B A) or23 (C: A B) (C: B A) (= A B) orordir B A eqcom (C: B A) orbi2i (\/ (C: A B) (= A B)) orbi2i bitr4 3bitr sylibr)) thm (ltsopr ((x y) (x z) (y z)) () (Or ( (e. C (N.)) (-> (/\ (/\ (e. B (Q.)) (A. x (-> (e. (cv x) A) (e. (opr (cv x) (+Q) B) A)))) (e. (cv y) A)) (e. (opr (cv y) (+Q) (opr ([] (<,> C (1o)) (~Q)) (.Q) B)) A))) ((cv w) (1o) (1o) opeq1 (<,> (cv w) (1o)) (<,> (1o) (1o)) (~Q) eceq2 syl (.Q) B opreq1d (cv y) (+Q) opreq2d A eleq1d (/\ (/\ (e. B (Q.)) (A. x (-> (e. (cv x) A) (e. (opr (cv x) (+Q) B) A)))) (e. (cv y) A)) imbi2d (cv w) (cv z) (1o) opeq1 (<,> (cv w) (1o)) (<,> (cv z) (1o)) (~Q) eceq2 syl (.Q) B opreq1d (cv y) (+Q) opreq2d A eleq1d (/\ (/\ (e. B (Q.)) (A. x (-> (e. (cv x) A) (e. (opr (cv x) (+Q) B) A)))) (e. (cv y) A)) imbi2d (cv w) (opr (cv z) (+N) (1o)) (1o) opeq1 (<,> (cv w) (1o)) (<,> (opr (cv z) (+N) (1o)) (1o)) (~Q) eceq2 syl (.Q) B opreq1d (cv y) (+Q) opreq2d A eleq1d (/\ (/\ (e. B (Q.)) (A. x (-> (e. (cv x) A) (e. (opr (cv x) (+Q) B) A)))) (e. (cv y) A)) imbi2d (cv w) C (1o) opeq1 (<,> (cv w) (1o)) (<,> C (1o)) (~Q) eceq2 syl (.Q) B opreq1d (cv y) (+Q) opreq2d A eleq1d (/\ (/\ (e. B (Q.)) (A. x (-> (e. (cv x) A) (e. (opr (cv x) (+Q) B) A)))) (e. (cv y) A)) imbi2d (cv x) (cv y) A eleq1 (cv x) (cv y) (+Q) B opreq1 A eleq1d imbi12d a4b1 B mulidpq prlem934a.1 1q elisseti mulcompq syl5eqr df-1q (.Q) B opreq1i syl5eqr (cv y) (+Q) opreq2d A eleq1d biimprd sylan9r imp (cv y) (+Q) (opr ([] (<,> (cv z) (1o)) (~Q)) (.Q) B) oprex (cv x) (opr (cv y) (+Q) (opr ([] (<,> (cv z) (1o)) (~Q)) (.Q) B)) A eleq1 (cv x) (opr (cv y) (+Q) (opr ([] (<,> (cv z) (1o)) (~Q)) (.Q) B)) (+Q) B opreq1 A eleq1d imbi12d cla4v B mulidpq prlem934a.1 1q elisseti mulcompq syl5reqr (opr ([] (<,> (cv z) (1o)) (~Q)) (.Q) B) (+Q) opreq2d enqex (~Q) (V) (<,> (cv z) (1o)) ecexg ax-mp 1q elisseti prlem934a.1 v visset u visset mulcompq u visset f visset (cv v) distrpq caoprdistrr syl6eqr 1pi 1pi 1pi pm3.2i (cv z) (1o) (1o) (1o) addpipq mpan2 mpan2 df-1q ([] (<,> (cv z) (1o)) (~Q)) (+Q) opreq2i syl5eq (cv z) mulidpi 1pi (1o) mulidpi ax-mp (e. (cv z) (N.)) a1i (+N) opreq12d 1pi (1o) mulidpi ax-mp jctir (opr (opr (cv z) (.N) (1o)) (+N) (opr (1o) (.N) (1o))) (opr (cv z) (+N) (1o)) (opr (1o) (.N) (1o)) (1o) opeq12 (<,> (opr (opr (cv z) (.N) (1o)) (+N) (opr (1o) (.N) (1o))) (opr (1o) (.N) (1o))) (<,> (opr (cv z) (+N) (1o)) (1o)) (~Q) eceq2 3syl eqtrd (.Q) B opreq1d sylan9eqr (cv y) (+Q) opreq2d ([] (<,> (cv z) (1o)) (~Q)) (.Q) B oprex prlem934a.1 (cv y) addasspq syl5eq A eleq1d biimpd sylan9r exp31 imp3a (e. (cv y) A) adantrd a2d indpi)) thm (prlem934b ((w z) (v z) (u z) (v w) (u w) (u v) (f z) (f w) (f v) (f u) (g z) (h z) (g w) (h w) (g v) (h v) (g u) (h u) (f g) (f h) (g h)) () (-> (/\ (/\ (e. (cv u) (N.)) (e. (cv w) (N.))) (/\ (e. (cv v) (N.)) (e. (cv z) (N.)))) (\/ (= (opr ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q))) ([] (<,> (cv v) (cv u)) (~Q))) (br ([] (<,> (cv v) (cv u)) (~Q)) ( (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q)))))) ((cv u) (cv z) mulclpi (opr (cv u) (.N) (cv z)) nlt1pi 1pi ltsopi ( (opr (opr (cv w) (.N) (cv v)) (.N) (cv z)) (opr (1o) (.N) (cv w))) (<,> (opr (cv w) (.N) (opr (cv v) (.N) (cv z))) (opr (cv w) (.N) (1o))) (~Q) eceq2 ax-mp syl5eq eqtrd (e. (cv u) (N.)) adantll (opr ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q))) ([] (<,> (opr (cv v) (.N) (cv z)) (1o)) (~Q)) ([] (<,> (cv v) (cv u)) (~Q)) eqeq1 (opr ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q))) ([] (<,> (opr (cv v) (.N) (cv z)) (1o)) (~Q)) ([] (<,> (cv v) (cv u)) (~Q)) ( (/\ (e. A (P.)) (e. B (Q.))) (E. x (/\ (e. (cv x) A) (-. (e. (opr (cv x) (+Q) B) A))))) (A prpssnq A (Q.) dfpss3 sylib pm3.27d (e. B (Q.)) adantl df-nq ([] (<,> (cv v) (cv u)) (~Q)) (cv y) A eleq1 (A. x (-> (e. (cv x) A) (e. (opr (cv x) (+Q) ([] (<,> (cv z) (cv w)) (~Q))) A))) imbi2d (e. A (P.)) imbi2d ([] (<,> (cv z) (cv w)) (~Q)) B (cv x) (+Q) opreq2 A eleq1d (e. (cv x) A) imbi2d x albidv (e. (cv y) A) imbi1d (e. A (P.)) imbi2d (e. (cv u) (N.)) (e. (cv w) (N.)) pm3.27 (e. (cv u) (N.)) (e. (cv w) (N.)) pm3.27 jca (/\ (e. (cv v) (N.)) (e. (cv z) (N.))) anim1i (e. (cv w) (N.)) (e. (cv w) (N.)) (e. (cv v) (N.)) (e. (cv z) (N.)) an42 sylib (cv w) (cv v) mulclpi enqex (~Q) (V) (<,> (cv z) (cv w)) ecexg ax-mp (opr (cv w) (.N) (cv v)) x A y prlem934a exp4c enqex df-nq (cv z) (cv w) ecopqsi syl5 syl imp syl com23 (e. A (P.)) adantld A (opr (cv y) (+Q) (opr ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q)))) ([] (<,> (cv v) (cv u)) (~Q)) prcdpq ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q)) oprex y visset ltaddpq (e. (cv u) (N.)) (e. (cv w) (N.)) pm3.27 (e. (cv u) (N.)) (e. (cv w) (N.)) pm3.27 jca (/\ (e. (cv v) (N.)) (e. (cv z) (N.))) anim1i (e. (cv w) (N.)) (e. (cv w) (N.)) (e. (cv v) (N.)) (e. (cv z) (N.)) an42 sylib (cv w) (cv v) mulclpi 1pi enqex df-nq (opr (cv w) (.N) (cv v)) (1o) ecopqsi mpan2 syl enqex df-nq (cv z) (cv w) ecopqsi anim12i ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) ([] (<,> (cv z) (cv w)) (~Q)) mulclpq 3syl sylan u w v z prlem934b (opr ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q))) ([] (<,> (cv v) (cv u)) (~Q)) ( (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q))) (+Q) (cv y)) breq1 biimpd enqex (~Q) (V) (<,> (cv v) (cv u)) ecexg ax-mp ltsopq ltrelpq ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q)) oprex (opr ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q))) (+Q) (cv y) oprex sotri ex jaoi syl (e. (cv y) (Q.)) adantr mpd ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q)) oprex y visset addcompq syl6breq syl5 exp4b com24 A (cv y) elprpq syl5 anabsi5 com12 syldd y 19.23adv A prn0 A y n0 sylib ancli y (e. A (P.)) (e. (cv y) A) 19.42v sylibr syl5 ancoms an4s 2ecoptocl ex com4l imp y 19.21adv (Q.) A y dfss2 syl6ibr mtod x (e. (cv x) A) (e. (opr (cv x) (+Q) B) A) exanali sylibr ancoms)) thm (ltaddpr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (P.)) (e. B (P.))) (br A ( (e. C (P.)) (-> (= (opr A (+P.) B) C) (br A ( (e. B (P.)) (-> (C: A B) (-. (= C ({/}))))) (y visset B x prnmadd (-. (e. (cv y) A)) anim2i x (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B) 19.42v sylibr exp32 com3l imp3a y 19.22dv A B y pssnel syl5 ltexprlem.1 abeq2i x exbii C x n0 y x (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B)) excom 3bitr4 syl6ibr)) thm (ltexprlem2 ((x y) (A x) (A y) (B x) (B y) (C x)) ((ltexprlem.1 (= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B))))))) (-> (e. B (P.)) (C: C (Q.))) (C B (Q.) sspsstr B (opr (cv y) (+Q) (cv x)) elprpq x visset dmaddpq 0npq (cv y) ndmoprrcl syl B (opr (cv y) (+Q) (cv x)) (cv x) prcdpq x visset y visset ltaddpq ancoms x visset y visset addcompq syl6breq syl5 mpd ex (-. (e. (cv y) A)) adantld y 19.23adv ltexprlem.1 abeq2i syl5ib ssrdv B prpssnq sylanc)) thm (ltexprlem3 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C z)) ((ltexprlem.1 (= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B))))))) (-> (e. B (P.)) (-> (e. (cv x) C) (A. z (-> (br (cv z) ( (e. B (P.)) (-> (e. (cv x) C) (E. z (/\ (e. (cv z) C) (br (cv x) ( (/\ (e. B (P.)) (C: A B)) (e. C (P.))) (ltexprlem.1 ltexprlem1 C 0pss syl6ibr imp ltexprlem.1 ltexprlem2 (C: A B) adantr jca ltexprlem.1 z ltexprlem3 ltexprlem.1 z ltexprlem4 z C (br (cv x) ( (/\ (/\ (e. A (P.)) (e. B (P.))) (C: A B)) (C_ (opr A (+P.) C) B)) (z v u f g h df-plp z visset A C w x genpelv ltexprlem.1 ltexprlem5 sylan2 (cv z) (opr (cv w) (+Q) (cv x)) B eleq1 biimparc A (cv w) (cv y) prub B (opr (cv y) (+Q) (cv x)) elprpq x visset dmaddpq 0npq (cv y) ndmoprrcl pm3.26d syl sylan2 B (opr (cv y) (+Q) (cv x)) elprpq x visset dmaddpq 0npq (cv y) ndmoprrcl pm3.27d w visset y visset z visset v visset (cv u) ltapq x visset z visset v visset addcompq caoprord2 3syl B (opr (cv y) (+Q) (cv x)) (opr (cv w) (+Q) (cv x)) prcdpq sylbid (/\ (e. A (P.)) (e. (cv w) A)) adantl syld exp32 com34 imp4b y 19.23adv ltexprlem.1 abeq2i syl5ib exp31 com23 imp43 sylan exp31 imp3a w x 19.23advv (C: A B) adantrr sylbid ssrdv anassrs)) thm (ltexprlem7 ((x y) (x z) (w x) (v x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (f y) (g y) (h y) (A y) (w z) (v z) (f z) (g z) (h z) (A z) (v w) (f w) (g w) (h w) (A w) (f v) (g v) (h v) (A v) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (C x) (C z) (C w) (C v) (C f) (C g) (C h) (u x) (u y) (u z) (u w) (u v) (f u) (g u) (h u)) ((ltexprlem.1 (= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B))))))) (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (C: A B)) (C_ B (opr A (+P.) C))) (A C ltaddpr A C addclpr A (opr A (+P.) C) ltprord syldan mpbid pssssd (cv w) sseld (e. (cv w) B) a1d (e. B (P.)) a1d com4r exp3a w visset B v prnmadd B (opr (cv w) (+Q) (cv v)) elprpq v visset dmaddpq 0npq (cv w) ndmoprrcl syl A (cv v) z prlem934 A (cv z) (cv w) prub z visset (cv w) x ltexpq A (cv z) elprpq sylan sylibd ex (e. C (P.)) (-. (e. (opr (cv z) (+Q) (cv v)) A)) ad2ant2r z v u f g h df-plp A C (cv z) (cv x) genpprecl (cv z) (+Q) (cv v) oprex (cv y) (opr (cv z) (+Q) (cv v)) A eleq1 negbid (cv y) (opr (cv z) (+Q) (cv v)) (+Q) (cv x) opreq1 B eleq1d anbi12d cla4ev ltexprlem.1 abeq2i sylibr (opr (cv z) (+Q) (cv x)) (cv w) (+Q) (cv v) opreq1 z visset v visset x visset f visset g visset addcompq g visset h visset (cv f) addasspq caopr32 syl5eq B eleq1d biimpar sylan2 sylan2i exp4d imp42 (opr (cv z) (+Q) (cv x)) (cv w) (opr A (+P.) C) eleq1 (/\ (/\ (e. A (P.)) (e. C (P.))) (/\ (e. (cv z) A) (-. (e. (opr (cv z) (+Q) (cv v)) A)))) (e. (opr (cv w) (+Q) (cv v)) B) ad2antrl mpbid exp32 x 19.23adv syl6d exp31 com23 z 19.23adv (e. (cv v) (Q.)) adantr mpd com24 ex com14 imp4b com4r com12 (e. B (P.)) adantld mpcom ex v 19.23adv (e. (cv w) B) adantr mpd ex com4t ex pm2.61i ltexprlem.1 ltexprlem5 syl5 exp3a com34 pm2.43d imp31 ssrdv)) thm (ltexpri ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((ltexpri.1 (e. B (V)))) (-> (br A ( (e. C (P.)) (-> (br A ( (e. C (P.)) (<-> (br A ( (/\ (e. A (P.)) (e. B (P.))) (-> (= (opr A (+P.) B) (opr A (+P.) C)) (= B C))) ((opr A (+P.) B) (opr A (+P.) C) (P.) eleq1 addcanpr.2 dmplp 0npr A ndmoprrcl syl6bi A B addclpr syl5com addcanpr.1 addcanpr.2 A ltapr addcanpr.2 addcanpr.1 A ltapr orbi12d negbid (e. B (P.)) (/\ (e. A (P.)) (e. C (P.))) ad2antrr ltsopr ( (/\ (e. (cv x) (Q.)) (/\ (e. (cv z) (Q.)) (e. (cv y) (Q.)))) (<-> (br (opr (cv y) (+Q) (cv z)) ( (/\ (e. (opr (cv y) (.Q) B) A) ph) (e. (opr (cv y) (+Q) (cv z)) A))) (prlem936b.2 (-> (/\ (/\ (e. A (P.)) (e. (opr (cv y) (+Q) (cv z)) A)) (/\ (e. (cv x) (Q.)) (e. (cv z) (Q.)))) (-> ps ch))) (prlem936b.3 (-> (/\ (e. (cv x) (Q.)) (/\ (e. (cv z) (Q.)) (e. (cv y) (Q.)))) (<-> ch th))) (prlem936b.4 (-> (/\ (/\ (/\ (br (1Q) ( th ta))) (prlem936b.5 (-> (/\ (e. A (P.)) ta) (-> ps (-. (e. (opr (cv x) (.Q) B) A)))))) (-> (/\ (/\ (e. A (P.)) (e. (cv z) (Q.))) (/\ (/\ ph (e. (cv y) (Q.))) (/\ (br (1Q) ( (/\ (e. (cv x) A) ps) (/\ (e. (cv x) A) (-. (e. (opr (cv x) (.Q) B) A))))) (prlem936b.2 ex prlem936b.1 sylan2 com12 (br (1Q) ( (/\ (e. A (P.)) (br (1Q) ( (e. A (P.)) (C: ({/}) B)) (A prpssnq A (Q.) x pssnel (cv x) recclpq dmrecpq 0npq (cv x) ndmfvrcl impbi (-. (e. (` (*Q) (` (*Q) (cv x))) A)) anbi1i x visset recrecpq A eleq1d negbid pm5.32i bitr3 (*Q) (cv x) fvex (cv y) (` (*Q) (cv x)) (Q.) eleq1 (cv y) (` (*Q) (cv x)) (*Q) fveq2 A eleq1d negbid anbi12d cla4ev sylbir x 19.23aiv 3syl (cv y) x nsmallpq (-. (e. (` (*Q) (cv y)) A)) anim1i x (br (cv x) ( (e. A (P.)) (e. B (P.))) (reclempr.1 reclem1pr A prn0 A (cv z) elprpq z visset recrecpq A eleq1d (e. A (P.)) anbi2d syl (*Q) (cv z) fvex (cv x) (` (*Q) (cv z)) (*Q) fveq2 A eleq1d (e. A (P.)) anbi2d cla4ev syl6bir pm2.43i A (` (*Q) (cv x)) elprpq dmrecpq 0npq (cv x) ndmfvrcl syl A (` (*Q) (cv x)) (` (*Q) (cv y)) prcdpq x visset y visset ltrpq syl5 y 19.21aiv reclempr.1 abeq2i y (br (cv x) ( (e. A (P.)) (C_ (1P) (opr A (.P.) B))) ((*Q) (cv w) fvex A v prlem936 (` (*Q) (cv z)) (.Q) (cv w) oprex x isseti (*Q) (cv z) fvex (*Q) (cv v) fvex x visset y visset (cv u) ltmpq w visset x visset y visset mulcompq caoprord2 v visset z visset ltrpq syl5bi (e. (cv v) (Q.)) adantl (cv v) recidpq v visset (*Q) (cv v) fvex mulcompq syl5eqr (cv w) recidpq (.Q) opreqan12d (*Q) (cv v) fvex v visset w visset x visset y visset mulcompq y visset u visset (cv x) mulasspq (*Q) (cv w) fvex caopr4 1q (1Q) mulidpq ax-mp 3eqtr3g (` (*Q) (cv v)) (cv w) mulclpq (cv v) recclpq sylan (cv v) (.Q) (` (*Q) (cv w)) oprex (opr (` (*Q) (cv v)) (.Q) (cv w)) recmulpq syl mpbird A eleq1d negbid biimprd anim12d (cv x) (opr (` (*Q) (cv z)) (.Q) (cv w)) ( (e. A (P.)) (= (opr A (.P.) B) (1P))) (reclempr.1 reclem2pr y w v u f g df-mp w visset A B z x genpelv mpdan A (cv z) elprpq x visset y visset (cv z) ltmpq syl biimpd (e. (cv y) (Q.)) adantr A (cv z) (` (*Q) (cv y)) prub (cv y) recclpq sylan2 z visset (*Q) (cv y) fvex (cv y) ltmpq y visset z visset mulcompq (e. (cv y) (Q.)) a1i (cv y) recidpq ( (e. A (P.)) (E. x (/\ (e. (cv x) (P.)) (= (opr A (.P.) (cv x)) (1P))))) ((cv x) ({|} z (E. y (/\ (br (cv z) ( (/\ (/\ (C_ A (P.)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (P.)) (A. y (-> (e. (cv y) (P.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) A) (br (cv y) ( (C_ A (P.)) (/\ (-> (e. (cv y) A) (-. (br (U. A) ( (br (cv y) ( (/\ (/\ (C_ A (P.)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (P.)) (A. y (-> (e. (cv y) (P.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (P.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (e. (cv y) (P.)) (-> (e. (cv y) A) (br (cv y) ( (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (<-> (br (<,> A B) (~R) (<,> C D)) (= (opr A (+P.) D) (opr B (+P.) C)))) (x y z w v u df-enr A B C D ecopopreq)) thm (dmenr ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (dom (~R)) (X. (P.) (P.))) (x y z w v u df-enr x visset y visset addcompr ecopoprdm)) thm (enrer ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (Er (~R)) (x y z w v u df-enr x visset y visset addcompr (cv x) (cv y) addclpr y visset z visset (cv x) addasspr y visset z visset (cv x) addcanpr ecopoprer)) thm (enreceq () () (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (<-> (= ([] (<,> A B) (~R)) ([] (<,> C D) (~R))) (= (opr A (+P.) D) (opr B (+P.) C)))) ((/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.))) pm3.26 A (P.) B (P.) opelxpi dmenr syl6eleqr C D opex enrer (<,> A B) erthdm 3syl A B C D enrbreq bitrd)) thm (enrex ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (e. (~R) (V)) (npex npex xpex npex npex xpex xpex x y z w v u df-enr x y (X. (P.) (P.)) (X. (P.) (P.)) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) (+P.) (cv u)) (opr (cv w) (+P.) (cv v)))))))) opabssxp eqsstr ssexi)) thm (srex () () (e. (R.) (V)) (df-nr npex npex xpex (~R) qsex eqeltr)) thm (ltrelsr ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (C_ ( (cv z) (cv w)) (~R))) (= (cv y) ([] (<,> (cv v) (cv u)) (~R)))) (br (opr (cv z) (+P.) (cv u)) ( (/\ (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (/\ (/\ (e. F (P.)) (e. G (P.))) (/\ (e. R (P.)) (e. S (P.))))) (-> (/\ (= (opr A (+P.) D) (opr B (+P.) C)) (= (opr F (+P.) S) (opr G (+P.) R))) (br (<,> (opr A (+P.) F) (opr B (+P.) G)) (~R) (<,> (opr C (+P.) R) (opr D (+P.) S))))) (A F addclpr B G addclpr anim12i an4s C R addclpr D S addclpr anim12i an4s anim12i an4s (opr A (+P.) F) (opr B (+P.) G) (opr C (+P.) R) (opr D (+P.) S) enrbreq syl cmpblnr.1 cmpblnr.5 cmpblnr.4 x visset y visset addcompr y visset z visset (cv x) addasspr cmpblnr.8 caopr4 cmpblnr.2 cmpblnr.6 cmpblnr.3 x visset y visset addcompr y visset z visset (cv x) addasspr cmpblnr.7 caopr4 eqeq12i syl6bb (opr A (+P.) D) (opr B (+P.) C) (opr F (+P.) S) (opr G (+P.) R) (+P.) opreq12 syl5bir)) thm (mulcmpblnrlem ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (F x) (F y) (F z) (G x) (G y) (G z) (R x) (R y) (R z) (S x) (S y) (S z)) ((cmpblnr.1 (e. A (V))) (cmpblnr.2 (e. B (V))) (cmpblnr.3 (e. C (V))) (cmpblnr.4 (e. D (V))) (cmpblnr.5 (e. F (V))) (cmpblnr.6 (e. G (V))) (cmpblnr.7 (e. R (V))) (cmpblnr.8 (e. S (V)))) (-> (/\ (= (opr A (+P.) D) (opr B (+P.) C)) (= (opr F (+P.) S) (opr G (+P.) R))) (= (opr (opr D (.P.) F) (+P.) (opr (opr (opr A (.P.) F) (+P.) (opr B (.P.) G)) (+P.) (opr (opr C (.P.) S) (+P.) (opr D (.P.) R)))) (opr (opr D (.P.) F) (+P.) (opr (opr (opr A (.P.) G) (+P.) (opr B (.P.) F)) (+P.) (opr (opr C (.P.) R) (+P.) (opr D (.P.) S)))))) ((opr A (+P.) D) (opr B (+P.) C) (.P.) F opreq1 cmpblnr.1 cmpblnr.4 cmpblnr.5 x visset y visset mulcompr y visset z visset (cv x) distrpr caoprdistrr cmpblnr.2 cmpblnr.3 cmpblnr.5 x visset y visset mulcompr y visset z visset (cv x) distrpr caoprdistrr 3eqtr3g (+P.) (opr C (.P.) S) opreq1d (opr F (+P.) S) (opr G (+P.) R) C (.P.) opreq2 cmpblnr.5 cmpblnr.8 C distrpr cmpblnr.6 cmpblnr.7 C distrpr 3eqtr3g (opr B (.P.) F) (+P.) opreq2d C (.P.) F oprex C (.P.) S oprex (opr B (.P.) F) addasspr syl5eq sylan9eq A (.P.) F oprex D (.P.) F oprex C (.P.) S oprex x visset y visset addcompr y visset z visset (cv x) addasspr caopr32 B (.P.) F oprex C (.P.) G oprex C (.P.) R oprex x visset y visset addcompr y visset z visset (cv x) addasspr caopr12 3eqtr3g (opr (opr B (.P.) G) (+P.) (opr D (.P.) R)) (+P.) opreq2d (opr F (+P.) S) (opr G (+P.) R) D (.P.) opreq2 cmpblnr.5 cmpblnr.8 D distrpr cmpblnr.6 cmpblnr.7 D distrpr 3eqtr3g (opr A (.P.) G) (+P.) opreq2d D (.P.) G oprex D (.P.) R oprex (opr A (.P.) G) addasspr syl6eqr (opr A (+P.) D) (opr B (+P.) C) (.P.) G opreq1 cmpblnr.1 cmpblnr.4 cmpblnr.6 x visset y visset mulcompr y visset z visset (cv x) distrpr caoprdistrr cmpblnr.2 cmpblnr.3 cmpblnr.6 x visset y visset mulcompr y visset z visset (cv x) distrpr caoprdistrr 3eqtr3g (+P.) (opr D (.P.) R) opreq1d sylan9eqr A (.P.) G oprex D (.P.) F oprex D (.P.) S oprex x visset y visset addcompr y visset z visset (cv x) addasspr caopr12 B (.P.) G oprex C (.P.) G oprex D (.P.) R oprex x visset y visset addcompr y visset z visset (cv x) addasspr caopr32 3eqtr3g (+P.) (opr (opr B (.P.) F) (+P.) (opr C (.P.) R)) opreq1d C (.P.) G oprex (opr B (.P.) F) (+P.) (opr C (.P.) R) oprex (opr (opr B (.P.) G) (+P.) (opr D (.P.) R)) addasspr syl6eq eqtr4d (opr B (.P.) G) (+P.) (opr D (.P.) R) oprex (opr A (.P.) F) (+P.) (opr C (.P.) S) oprex D (.P.) F oprex x visset y visset addcompr y visset z visset (cv x) addasspr caopr13 (opr A (.P.) G) (+P.) (opr D (.P.) S) oprex (opr B (.P.) F) (+P.) (opr C (.P.) R) oprex (opr D (.P.) F) addasspr 3eqtr3g A (.P.) F oprex C (.P.) S oprex B (.P.) G oprex x visset y visset addcompr y visset z visset (cv x) addasspr D (.P.) R oprex caopr4 (opr D (.P.) F) (+P.) opreq2i A (.P.) G oprex D (.P.) S oprex B (.P.) F oprex x visset y visset addcompr y visset z visset (cv x) addasspr C (.P.) R oprex caopr42 (opr D (.P.) F) (+P.) opreq2i 3eqtr3g)) thm (mulcmpblnr () ((cmpblnr.1 (e. A (V))) (cmpblnr.2 (e. B (V))) (cmpblnr.3 (e. C (V))) (cmpblnr.4 (e. D (V))) (cmpblnr.5 (e. F (V))) (cmpblnr.6 (e. G (V))) (cmpblnr.7 (e. R (V))) (cmpblnr.8 (e. S (V)))) (-> (/\ (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (/\ (/\ (e. F (P.)) (e. G (P.))) (/\ (e. R (P.)) (e. S (P.))))) (-> (/\ (= (opr A (+P.) D) (opr B (+P.) C)) (= (opr F (+P.) S) (opr G (+P.) R))) (br (<,> (opr (opr A (.P.) F) (+P.) (opr B (.P.) G)) (opr (opr A (.P.) G) (+P.) (opr B (.P.) F))) (~R) (<,> (opr (opr C (.P.) R) (+P.) (opr D (.P.) S)) (opr (opr C (.P.) S) (+P.) (opr D (.P.) R)))))) (D F mulclpr (e. C (P.)) (e. D (P.)) pm3.27 (e. F (P.)) (e. G (P.)) pm3.26 syl2an (opr (opr A (.P.) F) (+P.) (opr B (.P.) G)) (opr (opr C (.P.) S) (+P.) (opr D (.P.) R)) addclpr (opr (opr A (.P.) F) (+P.) (opr B (.P.) G)) (+P.) (opr (opr C (.P.) S) (+P.) (opr D (.P.) R)) oprex (opr (opr A (.P.) G) (+P.) (opr B (.P.) F)) (+P.) (opr (opr C (.P.) R) (+P.) (opr D (.P.) S)) oprex (opr D (.P.) F) addcanpr cmpblnr.1 cmpblnr.2 cmpblnr.3 cmpblnr.4 cmpblnr.5 cmpblnr.6 cmpblnr.7 cmpblnr.8 mulcmpblnrlem syl5 expcom imp3a syl (e. (opr (opr C (.P.) R) (+P.) (opr D (.P.) S)) (P.)) adantrl (e. (opr (opr A (.P.) G) (+P.) (opr B (.P.) F)) (P.)) adantlr (opr (opr A (.P.) F) (+P.) (opr B (.P.) G)) (opr (opr A (.P.) G) (+P.) (opr B (.P.) F)) (opr (opr C (.P.) R) (+P.) (opr D (.P.) S)) (opr (opr C (.P.) S) (+P.) (opr D (.P.) R)) enrbreq sylibrd (e. A (P.)) (e. B (P.)) (e. F (P.)) (e. G (P.)) rnlem (opr A (.P.) F) (opr B (.P.) G) addclpr A F mulclpr B G mulclpr syl2an (opr A (.P.) G) (opr B (.P.) F) addclpr A G mulclpr B F mulclpr syl2an anim12i sylbi (e. C (P.)) (e. D (P.)) (e. R (P.)) (e. S (P.)) rnlem (opr C (.P.) R) (opr D (.P.) S) addclpr C R mulclpr D S mulclpr syl2an (opr C (.P.) S) (opr D (.P.) R) addclpr C S mulclpr D R mulclpr syl2an anim12i sylbi syl2an an42s an4s exp4b mpdi imp an42s)) thm (addsrpr ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (= (opr ([] (<,> A B) (~R)) (+R) ([] (<,> C D) (~R))) ([] (<,> (opr A (+P.) C) (opr B (+P.) D)) (~R)))) ((opr A (+P.) C) (opr B (+P.) D) opex (opr (cv a) (+P.) (cv g)) (opr (cv b) (+P.) (cv h)) opex (opr (cv c) (+P.) (cv t)) (opr (cv d) (+P.) (cv s)) opex enrex enrer dmenr x y z w v u df-enr (cv z) (cv a) (cv u) (cv d) (+P.) opreq12 (cv w) (cv b) (cv v) (cv c) (+P.) opreq12 eqeqan12d an42s (cv z) (cv g) (cv u) (cv s) (+P.) opreq12 (cv w) (cv h) (cv v) (cv t) (+P.) opreq12 eqeqan12d an42s x y z w v u f df-plpr (opr (cv w) (+P.) (cv u)) (opr (cv a) (+P.) (cv g)) (opr (cv v) (+P.) (cv f)) (opr (cv b) (+P.) (cv h)) opeq12 (cv w) (cv a) (cv u) (cv g) (+P.) opreq12 (cv v) (cv b) (cv f) (cv h) (+P.) opreq12 syl2an an4s (opr (cv w) (+P.) (cv u)) (opr (cv c) (+P.) (cv t)) (opr (cv v) (+P.) (cv f)) (opr (cv d) (+P.) (cv s)) opeq12 (cv w) (cv c) (cv u) (cv t) (+P.) opreq12 (cv v) (cv d) (cv f) (cv s) (+P.) opreq12 syl2an an4s (opr (cv w) (+P.) (cv u)) (opr A (+P.) C) (opr (cv v) (+P.) (cv f)) (opr B (+P.) D) opeq12 (cv w) A (cv u) C (+P.) opreq12 (cv v) B (cv f) D (+P.) opreq12 syl2an an4s x y z a b c d df-plr df-nr a visset b visset c visset d visset g visset h visset t visset s visset addcmpblnr oprec)) thm (mulsrpr ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (= (opr ([] (<,> A B) (~R)) (.R) ([] (<,> C D) (~R))) ([] (<,> (opr (opr A (.P.) C) (+P.) (opr B (.P.) D)) (opr (opr A (.P.) D) (+P.) (opr B (.P.) C))) (~R)))) ((opr (opr A (.P.) C) (+P.) (opr B (.P.) D)) (opr (opr A (.P.) D) (+P.) (opr B (.P.) C)) opex (opr (opr (cv a) (.P.) (cv g)) (+P.) (opr (cv b) (.P.) (cv h))) (opr (opr (cv a) (.P.) (cv h)) (+P.) (opr (cv b) (.P.) (cv g))) opex (opr (opr (cv c) (.P.) (cv t)) (+P.) (opr (cv d) (.P.) (cv s))) (opr (opr (cv c) (.P.) (cv s)) (+P.) (opr (cv d) (.P.) (cv t))) opex enrex enrer dmenr x y z w v u df-enr (cv z) (cv a) (cv u) (cv d) (+P.) opreq12 (cv w) (cv b) (cv v) (cv c) (+P.) opreq12 eqeqan12d an42s (cv z) (cv g) (cv u) (cv s) (+P.) opreq12 (cv w) (cv h) (cv v) (cv t) (+P.) opreq12 eqeqan12d an42s x y z w v u f df-mpr (opr (opr (cv w) (.P.) (cv u)) (+P.) (opr (cv v) (.P.) (cv f))) (opr (opr (cv a) (.P.) (cv g)) (+P.) (opr (cv b) (.P.) (cv h))) (opr (opr (cv w) (.P.) (cv f)) (+P.) (opr (cv v) (.P.) (cv u))) (opr (opr (cv a) (.P.) (cv h)) (+P.) (opr (cv b) (.P.) (cv g))) opeq12 (cv w) (cv a) (cv u) (cv g) (.P.) opreq12 (cv v) (cv b) (cv f) (cv h) (.P.) opreq12 (+P.) opreqan12d an4s (cv w) (cv a) (cv f) (cv h) (.P.) opreq12 (cv v) (cv b) (cv u) (cv g) (.P.) opreq12 (+P.) opreqan12d an42s sylanc (opr (opr (cv w) (.P.) (cv u)) (+P.) (opr (cv v) (.P.) (cv f))) (opr (opr (cv c) (.P.) (cv t)) (+P.) (opr (cv d) (.P.) (cv s))) (opr (opr (cv w) (.P.) (cv f)) (+P.) (opr (cv v) (.P.) (cv u))) (opr (opr (cv c) (.P.) (cv s)) (+P.) (opr (cv d) (.P.) (cv t))) opeq12 (cv w) (cv c) (cv u) (cv t) (.P.) opreq12 (cv v) (cv d) (cv f) (cv s) (.P.) opreq12 (+P.) opreqan12d an4s (cv w) (cv c) (cv f) (cv s) (.P.) opreq12 (cv v) (cv d) (cv u) (cv t) (.P.) opreq12 (+P.) opreqan12d an42s sylanc (opr (opr (cv w) (.P.) (cv u)) (+P.) (opr (cv v) (.P.) (cv f))) (opr (opr A (.P.) C) (+P.) (opr B (.P.) D)) (opr (opr (cv w) (.P.) (cv f)) (+P.) (opr (cv v) (.P.) (cv u))) (opr (opr A (.P.) D) (+P.) (opr B (.P.) C)) opeq12 (cv w) A (cv u) C (.P.) opreq12 (cv v) B (cv f) D (.P.) opreq12 (+P.) opreqan12d an4s (cv w) A (cv f) D (.P.) opreq12 (cv v) B (cv u) C (.P.) opreq12 (+P.) opreqan12d an42s sylanc x y z a b c d df-mr df-nr a visset b visset c visset d visset g visset h visset t visset s visset mulcmpblnr oprec)) thm (ltsrpr ((A x) (A y) (A z) (A w) (A v) (A u) (A f) (x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f)) ((ltsrpr.1 (e. A (V))) (ltsrpr.2 (e. B (V))) (ltsrpr.3 (e. C (V))) (ltsrpr.4 (e. D (V)))) (<-> (br ([] (<,> A B) (~R)) ( C D) (~R))) (br (opr A (+P.) D) ( (br (0R) ( A B) (~R))) (br B ( A B) (~R)) breq1i 1pr gt0srpr.2 gt0srpr.1 (1P) ltapr ax-mp 3bitr4)) thm (0nsr () () (-. (e. ({/}) (R.))) (enrer dmenr 0nelqs df-nr ({/}) eleq2i mtbir)) thm (0r () () (e. (0R) (R.)) (1pr 1pr (1P) (P.) (1P) (P.) opelxpi mp2an enrex (<,> (1P) (1P)) (X. (P.) (P.)) ecelqsi ax-mp df-0r df-nr eleq12i mpbir)) thm (1r () () (e. (1R) (R.)) (1pr 1pr (1P) (1P) addclpr mp2an 1pr (opr (1P) (+P.) (1P)) (P.) (1P) (P.) opelxpi mp2an enrex (<,> (opr (1P) (+P.) (1P)) (1P)) (X. (P.) (P.)) ecelqsi ax-mp df-1r df-nr eleq12i mpbir)) thm (m1r () () (e. (-1R) (R.)) (1pr 1pr 1pr (1P) (1P) addclpr mp2an (1P) (P.) (opr (1P) (+P.) (1P)) (P.) opelxpi mp2an enrex (<,> (1P) (opr (1P) (+P.) (1P))) (X. (P.) (P.)) ecelqsi ax-mp df-m1r df-nr eleq12i mpbir)) thm (addclsr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (R.)) (e. B (R.))) (e. (opr A (+R) B) (R.))) (df-nr ([] (<,> (cv x) (cv y)) (~R)) A (+R) ([] (<,> (cv z) (cv w)) (~R)) opreq1 (/. (X. (P.) (P.)) (~R)) eleq1d ([] (<,> (cv z) (cv w)) (~R)) B A (+R) opreq2 (/. (X. (P.) (P.)) (~R)) eleq1d (cv x) (cv y) (cv z) (cv w) addsrpr (cv x) (cv z) addclpr (cv y) (cv w) addclpr anim12i an4s (opr (cv x) (+P.) (cv z)) (P.) (opr (cv y) (+P.) (cv w)) (P.) opelxpi enrex (<,> (opr (cv x) (+P.) (cv z)) (opr (cv y) (+P.) (cv w))) (X. (P.) (P.)) ecelqsi 3syl eqeltrd 2ecoptocl df-nr syl6eleqr)) thm (mulclsr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (R.)) (e. B (R.))) (e. (opr A (.R) B) (R.))) (df-nr ([] (<,> (cv x) (cv y)) (~R)) A (.R) ([] (<,> (cv z) (cv w)) (~R)) opreq1 (/. (X. (P.) (P.)) (~R)) eleq1d ([] (<,> (cv z) (cv w)) (~R)) B A (.R) opreq2 (/. (X. (P.) (P.)) (~R)) eleq1d (cv x) (cv y) (cv z) (cv w) mulsrpr (opr (cv x) (.P.) (cv z)) (opr (cv y) (.P.) (cv w)) addclpr (cv x) (cv z) mulclpr (cv y) (cv w) mulclpr syl2an an4s (opr (cv x) (.P.) (cv w)) (opr (cv y) (.P.) (cv z)) addclpr (cv x) (cv w) mulclpr (cv y) (cv z) mulclpr syl2an an42s jca (opr (opr (cv x) (.P.) (cv z)) (+P.) (opr (cv y) (.P.) (cv w))) (P.) (opr (opr (cv x) (.P.) (cv w)) (+P.) (opr (cv y) (.P.) (cv z))) (P.) opelxpi enrex (<,> (opr (opr (cv x) (.P.) (cv z)) (+P.) (opr (cv y) (.P.) (cv w))) (opr (opr (cv x) (.P.) (cv w)) (+P.) (opr (cv y) (.P.) (cv z)))) (X. (P.) (P.)) ecelqsi 3syl eqeltrd 2ecoptocl df-nr syl6eleqr)) thm (dmaddsr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (+R)) (X. (R.) (R.))) (x y z w v u f df-plr dmeqi x y z (R.) (R.) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) (~R))) (= (cv y) ([] (<,> (cv u) (cv f)) (~R)))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) (+pR) (<,> (cv u) (cv f))) (~R)))))))) dmoprabss eqsstr 0nsr (cv x) (cv y) addclsr oprssdm eqssi)) thm (dmmulsr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (.R)) (X. (R.) (R.))) (x y z w v u f df-mr dmeqi x y z (R.) (R.) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) (~R))) (= (cv y) ([] (<,> (cv u) (cv f)) (~R)))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) (.pR) (<,> (cv u) (cv f))) (~R)))))))) dmoprabss eqsstr 0nsr (cv x) (cv y) mulclsr oprssdm eqssi)) thm (addcomsr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((addcomsr.1 (e. A (V))) (addcomsr.2 (e. B (V)))) (= (opr A (+R) B) (opr B (+R) A)) (df-nr (cv x) (cv y) (cv z) (cv w) addsrpr (cv z) (cv w) (cv x) (cv y) addsrpr x visset z visset addcompr y visset w visset addcompr A B ecoprcom addcomsr.2 dmaddsr addcomsr.1 ndmoprcom pm2.61i)) thm (addasssr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((addasssr.1 (e. B (V))) (addasssr.2 (e. C (V)))) (= (opr (opr A (+R) B) (+R) C) (opr A (+R) (opr B (+R) C))) (df-nr (cv x) (cv y) (cv z) (cv w) addsrpr (cv z) (cv w) (cv v) (cv u) addsrpr (opr (cv x) (+P.) (cv z)) (opr (cv y) (+P.) (cv w)) (cv v) (cv u) addsrpr (cv x) (cv y) (opr (cv z) (+P.) (cv v)) (opr (cv w) (+P.) (cv u)) addsrpr (cv x) (cv z) addclpr (cv y) (cv w) addclpr anim12i an4s (cv z) (cv v) addclpr (cv w) (cv u) addclpr anim12i an4s z visset v visset (cv x) addasspr w visset u visset (cv y) addasspr A B C ecoprass addasssr.1 dmaddsr addasssr.2 0nsr A ndmoprass pm2.61i)) thm (mulcomsr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((mulcomsr.1 (e. A (V))) (mulcomsr.2 (e. B (V)))) (= (opr A (.R) B) (opr B (.R) A)) (df-nr (cv x) (cv y) (cv z) (cv w) mulsrpr (cv z) (cv w) (cv x) (cv y) mulsrpr x visset z visset mulcompr y visset w visset mulcompr (+P.) opreq12i x visset w visset mulcompr y visset z visset mulcompr (+P.) opreq12i (cv w) (.P.) (cv x) oprex (cv z) (.P.) (cv y) oprex addcompr eqtr A B ecoprcom mulcomsr.2 dmmulsr mulcomsr.1 ndmoprcom pm2.61i)) thm (mulasssr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((mulasssr.1 (e. B (V))) (mulasssr.2 (e. C (V)))) (= (opr (opr A (.R) B) (.R) C) (opr A (.R) (opr B (.R) C))) (df-nr (cv x) (cv y) (cv z) (cv w) mulsrpr (cv z) (cv w) (cv v) (cv u) mulsrpr (opr (opr (cv x) (.P.) (cv z)) (+P.) (opr (cv y) (.P.) (cv w))) (opr (opr (cv x) (.P.) (cv w)) (+P.) (opr (cv y) (.P.) (cv z))) (cv v) (cv u) mulsrpr (cv x) (cv y) (opr (opr (cv z) (.P.) (cv v)) (+P.) (opr (cv w) (.P.) (cv u))) (opr (opr (cv z) (.P.) (cv u)) (+P.) (opr (cv w) (.P.) (cv v))) mulsrpr (opr (cv x) (.P.) (cv z)) (opr (cv y) (.P.) (cv w)) addclpr (cv x) (cv z) mulclpr (cv y) (cv w) mulclpr syl2an an4s (opr (cv x) (.P.) (cv w)) (opr (cv y) (.P.) (cv z)) addclpr (cv x) (cv w) mulclpr (cv y) (cv z) mulclpr syl2an an42s jca (opr (cv z) (.P.) (cv v)) (opr (cv w) (.P.) (cv u)) addclpr (cv z) (cv v) mulclpr (cv w) (cv u) mulclpr syl2an an4s (opr (cv z) (.P.) (cv u)) (opr (cv w) (.P.) (cv v)) addclpr (cv z) (cv u) mulclpr (cv w) (cv v) mulclpr syl2an an42s jca x visset y visset z visset f visset g visset mulcompr g visset h visset (cv f) distrpr w visset v visset g visset h visset (cv f) mulasspr u visset f visset g visset addcompr g visset h visset (cv f) addasspr caoprlem2 x visset y visset z visset f visset g visset mulcompr g visset h visset (cv f) distrpr w visset u visset g visset h visset (cv f) mulasspr v visset f visset g visset addcompr g visset h visset (cv f) addasspr caoprlem2 A B C ecoprass mulasssr.1 dmmulsr mulasssr.2 0nsr A ndmoprass pm2.61i)) thm (distrsr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((distrsr.1 (e. B (V))) (distrsr.2 (e. C (V)))) (= (opr A (.R) (opr B (+R) C)) (opr (opr A (.R) B) (+R) (opr A (.R) C))) (df-nr (cv z) (cv w) (cv v) (cv u) addsrpr (cv x) (cv y) (opr (cv z) (+P.) (cv v)) (opr (cv w) (+P.) (cv u)) mulsrpr (cv x) (cv y) (cv z) (cv w) mulsrpr (cv x) (cv y) (cv v) (cv u) mulsrpr (opr (opr (cv x) (.P.) (cv z)) (+P.) (opr (cv y) (.P.) (cv w))) (opr (opr (cv x) (.P.) (cv w)) (+P.) (opr (cv y) (.P.) (cv z))) (opr (opr (cv x) (.P.) (cv v)) (+P.) (opr (cv y) (.P.) (cv u))) (opr (opr (cv x) (.P.) (cv u)) (+P.) (opr (cv y) (.P.) (cv v))) addsrpr (cv z) (cv v) addclpr (cv w) (cv u) addclpr anim12i an4s (opr (cv x) (.P.) (cv z)) (opr (cv y) (.P.) (cv w)) addclpr (cv x) (cv z) mulclpr (cv y) (cv w) mulclpr syl2an an4s (opr (cv x) (.P.) (cv w)) (opr (cv y) (.P.) (cv z)) addclpr (cv x) (cv w) mulclpr (cv y) (cv z) mulclpr syl2an an42s jca (opr (cv x) (.P.) (cv v)) (opr (cv y) (.P.) (cv u)) addclpr (cv x) (cv v) mulclpr (cv y) (cv u) mulclpr syl2an an4s (opr (cv x) (.P.) (cv u)) (opr (cv y) (.P.) (cv v)) addclpr (cv x) (cv u) mulclpr (cv y) (cv v) mulclpr syl2an an42s jca z visset v visset (cv x) distrpr w visset u visset (cv y) distrpr (+P.) opreq12i (cv x) (.P.) (cv z) oprex (cv x) (.P.) (cv v) oprex (cv y) (.P.) (cv w) oprex f visset g visset addcompr g visset h visset (cv f) addasspr (cv y) (.P.) (cv u) oprex caopr4 eqtr w visset u visset (cv x) distrpr z visset v visset (cv y) distrpr (+P.) opreq12i (cv x) (.P.) (cv w) oprex (cv x) (.P.) (cv u) oprex (cv y) (.P.) (cv z) oprex f visset g visset addcompr g visset h visset (cv f) addasspr (cv y) (.P.) (cv v) oprex caopr4 eqtr A B C ecoprdi distrsr.1 dmaddsr distrsr.2 0nsr dmmulsr A ndmoprdistr pm2.61i)) thm (m1p1sr () () (= (opr (-1R) (+R) (1R)) (0R)) (1pr 1pr 1pr (1P) (1P) addclpr mp2an pm3.2i 1pr 1pr (1P) (1P) addclpr mp2an 1pr pm3.2i (1P) (opr (1P) (+P.) (1P)) (opr (1P) (+P.) (1P)) (1P) addsrpr mp2an 1pr elisseti 1pr elisseti (1P) addasspr (1P) (+P.) opreq2i 1pr 1pr pm3.2i 1pr 1pr 1pr (1P) (1P) addclpr mp2an (1P) (opr (1P) (+P.) (1P)) addclpr mp2an 1pr 1pr (1P) (1P) addclpr mp2an 1pr (opr (1P) (+P.) (1P)) (1P) addclpr mp2an pm3.2i (1P) (1P) (opr (1P) (+P.) (opr (1P) (+P.) (1P))) (opr (opr (1P) (+P.) (1P)) (+P.) (1P)) enreceq mp2an mpbir eqtr4 df-m1r df-1r (+R) opreq12i df-0r 3eqtr4)) thm (m1m1sr () () (= (opr (-1R) (.R) (-1R)) (1R)) (1pr 1pr 1pr (1P) (1P) addclpr mp2an pm3.2i 1pr 1pr 1pr (1P) (1P) addclpr mp2an pm3.2i (1P) (opr (1P) (+P.) (1P)) (1P) (opr (1P) (+P.) (1P)) mulsrpr mp2an 1pr elisseti (opr (1P) (.P.) (opr (1P) (+P.) (1P))) (+P.) (opr (opr (1P) (+P.) (1P)) (.P.) (1P)) oprex (1P) addasspr 1pr (1P) 1idpr ax-mp 1pr elisseti 1pr elisseti (opr (1P) (+P.) (1P)) distrpr 1pr elisseti (1P) (+P.) (1P) oprex mulcompr (+P.) (opr (opr (1P) (+P.) (1P)) (.P.) (1P)) opreq1i eqtr4 (+P.) opreq12i (1P) (+P.) opreq2i eqtr4 1pr 1pr (1P) (1P) addclpr mp2an 1pr pm3.2i 1pr 1pr (1P) (1P) mulclpr mp2an 1pr 1pr (1P) (1P) addclpr mp2an 1pr 1pr (1P) (1P) addclpr mp2an (opr (1P) (+P.) (1P)) (opr (1P) (+P.) (1P)) mulclpr mp2an (opr (1P) (.P.) (1P)) (opr (opr (1P) (+P.) (1P)) (.P.) (opr (1P) (+P.) (1P))) addclpr mp2an 1pr 1pr 1pr (1P) (1P) addclpr mp2an (1P) (opr (1P) (+P.) (1P)) mulclpr mp2an 1pr 1pr (1P) (1P) addclpr mp2an 1pr (opr (1P) (+P.) (1P)) (1P) mulclpr mp2an (opr (1P) (.P.) (opr (1P) (+P.) (1P))) (opr (opr (1P) (+P.) (1P)) (.P.) (1P)) addclpr mp2an pm3.2i (opr (1P) (+P.) (1P)) (1P) (opr (opr (1P) (.P.) (1P)) (+P.) (opr (opr (1P) (+P.) (1P)) (.P.) (opr (1P) (+P.) (1P)))) (opr (opr (1P) (.P.) (opr (1P) (+P.) (1P))) (+P.) (opr (opr (1P) (+P.) (1P)) (.P.) (1P))) enreceq mp2an mpbir eqtr4 df-m1r df-m1r (.R) opreq12i df-1r 3eqtr4)) thm (ltsosr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (w z) (v z) (u z) (f z) (g z) (h z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) () (Or ( (cv x) (cv y)) (~R)) (cv f) ( (cv z) (cv w)) (~R)) breq1 ([] (<,> (cv x) (cv y)) (~R)) (cv f) ([] (<,> (cv z) (cv w)) (~R)) eqeq1 ([] (<,> (cv x) (cv y)) (~R)) (cv f) ([] (<,> (cv z) (cv w)) (~R)) ( (cv x) (cv y)) (~R)) (cv f) ( (cv z) (cv w)) (~R)) breq1 (br ([] (<,> (cv z) (cv w)) (~R)) ( (cv v) (cv u)) (~R))) anbi1d ([] (<,> (cv x) (cv y)) (~R)) (cv f) ( (cv v) (cv u)) (~R)) breq1 imbi12d anbi12d ([] (<,> (cv z) (cv w)) (~R)) (cv g) (cv f) ( (cv z) (cv w)) (~R)) (cv g) (cv f) eqeq2 ([] (<,> (cv z) (cv w)) (~R)) (cv g) ( (cv z) (cv w)) (~R)) (cv g) (cv f) ( (cv z) (cv w)) (~R)) (cv g) ( (cv v) (cv u)) (~R)) breq1 anbi12d (br (cv f) ( (cv v) (cv u)) (~R))) imbi1d anbi12d ([] (<,> (cv v) (cv u)) (~R)) (cv h) (cv g) ( (cv v) (cv u)) (~R)) (cv h) (cv f) ( (br (cv f) ( (e. A (R.)) (= (opr A (+R) (0R)) A)) (df-nr ([] (<,> (cv x) (cv y)) (~R)) A (+R) (0R) opreq1 (= ([] (<,> (cv x) (cv y)) (~R)) A) id eqeq12d 1pr 1pr pm3.2i (cv x) (cv y) (1P) (1P) addsrpr mpan2 1pr (cv x) (1P) addclpr mpan2 1pr (cv y) (1P) addclpr mpan2 anim12i x visset y visset 1pr elisseti z visset w visset addcompr w visset v visset (cv z) addasspr caopr12 (cv x) (cv y) (opr (cv x) (+P.) (1P)) (opr (cv y) (+P.) (1P)) enreceq mpbiri mpdan eqtr4d df-0r ([] (<,> (cv x) (cv y)) (~R)) (+R) opreq2i syl5eq ecoptocl)) thm (1idsr ((x y) (A x) (A y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (-> (e. A (R.)) (= (opr A (.R) (1R)) A)) (df-nr ([] (<,> (cv x) (cv y)) (~R)) A (.R) (1R) opreq1 (= ([] (<,> (cv x) (cv y)) (~R)) A) id eqeq12d 1pr 1pr (1P) (1P) addclpr mp2an 1pr pm3.2i (cv x) (cv y) (opr (1P) (+P.) (1P)) (1P) mulsrpr mpan2 (cv x) 1idpr (+P.) (opr (cv x) (.P.) (1P)) opreq1d 1pr elisseti 1pr elisseti (cv x) distrpr syl5req (cv y) 1idpr (+P.) (opr (cv y) (.P.) (1P)) opreq1d 1pr elisseti 1pr elisseti (cv y) distrpr syl5eq (+P.) opreqan12d (cv x) (.P.) (1P) oprex (cv y) (.P.) (opr (1P) (+P.) (1P)) oprex (cv x) addasspr (cv x) (.P.) (opr (1P) (+P.) (1P)) oprex y visset (cv y) (.P.) (1P) oprex z visset w visset addcompr w visset v visset (cv z) addasspr caopr12 3eqtr3g (opr (cv x) (.P.) (opr (1P) (+P.) (1P))) (opr (cv y) (.P.) (1P)) addclpr 1pr 1pr (1P) (1P) addclpr mp2an (cv x) (opr (1P) (+P.) (1P)) mulclpr mpan2 1pr (cv y) (1P) mulclpr mpan2 syl2an (opr (cv x) (.P.) (1P)) (opr (cv y) (.P.) (opr (1P) (+P.) (1P))) addclpr 1pr (cv x) (1P) mulclpr mpan2 1pr 1pr (1P) (1P) addclpr mp2an (cv y) (opr (1P) (+P.) (1P)) mulclpr mpan2 syl2an anim12i (cv x) (cv y) (opr (opr (cv x) (.P.) (opr (1P) (+P.) (1P))) (+P.) (opr (cv y) (.P.) (1P))) (opr (opr (cv x) (.P.) (1P)) (+P.) (opr (cv y) (.P.) (opr (1P) (+P.) (1P)))) enreceq syldan anidms mpbird eqtr4d df-1r ([] (<,> (cv x) (cv y)) (~R)) (.R) opreq2i syl5eq ecoptocl)) thm (00sr ((x y) (A x) (A y)) () (-> (e. A (R.)) (= (opr A (.R) (0R)) (0R))) (df-nr ([] (<,> (cv x) (cv y)) (~R)) A (.R) (0R) opreq1 (0R) eqeq1d 1pr 1pr pm3.2i (cv x) (cv y) (1P) (1P) mulsrpr mpan2 1pr 1pr pm3.2i (opr (opr (opr (cv x) (.P.) (1P)) (+P.) (opr (cv y) (.P.) (1P))) (+P.) (1P)) eqid (opr (opr (cv x) (.P.) (1P)) (+P.) (opr (cv y) (.P.) (1P))) (opr (opr (cv x) (.P.) (1P)) (+P.) (opr (cv y) (.P.) (1P))) (1P) (1P) enreceq mpbiri (opr (cv x) (.P.) (1P)) (opr (cv y) (.P.) (1P)) addclpr 1pr (cv x) (1P) mulclpr mpan2 1pr (cv y) (1P) mulclpr mpan2 syl2an (opr (cv x) (.P.) (1P)) (opr (cv y) (.P.) (1P)) addclpr 1pr (cv x) (1P) mulclpr mpan2 1pr (cv y) (1P) mulclpr mpan2 syl2an anim12i sylan mpan2 anidms eqtrd df-0r ([] (<,> (cv x) (cv y)) (~R)) (.R) opreq2i df-0r 3eqtr4g ecoptocl)) thm (ltasr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (f y) (f z) (f w) (f v) (f u)) ((ltasr.1 (e. A (V))) (ltasr.2 (e. B (V)))) (-> (e. C (R.)) (<-> (br A ( (cv v) (cv u)) (~R)) C (+R) ([] (<,> (cv x) (cv y)) (~R)) opreq1 ([] (<,> (cv v) (cv u)) (~R)) C (+R) ([] (<,> (cv z) (cv w)) (~R)) opreq1 ( (cv x) (cv y)) (~R)) ( (cv z) (cv w)) (~R))) bibi2d ([] (<,> (cv x) (cv y)) (~R)) A ( (cv z) (cv w)) (~R)) breq1 ([] (<,> (cv x) (cv y)) (~R)) A C (+R) opreq2 ( (cv z) (cv w)) (~R))) breq1d bibi12d ([] (<,> (cv z) (cv w)) (~R)) B A ( (cv z) (cv w)) (~R)) B C (+R) opreq2 (opr C (+R) A) ( (e. A (R.)) (= (opr A (+R) (opr A (.R) (-1R))) (0R))) (A 1idsr (+R) (opr A (.R) (-1R)) opreq1d A 00sr m1r elisseti 1r elisseti A distrsr m1p1sr A (.R) opreq2i A (.R) (-1R) oprex A (.R) (1R) oprex addcomsr 3eqtr3 syl5eqr eqtr3d)) thm (negexsr ((A x)) () (-> (e. A (R.)) (E. x (/\ (e. (cv x) (R.)) (= (opr A (+R) (cv x)) (0R))))) (A (.R) (-1R) oprex (cv x) (opr A (.R) (-1R)) (R.) eleq1 (cv x) (opr A (.R) (-1R)) A (+R) opreq2 (0R) eqeq1d anbi12d cla4ev m1r A (-1R) mulclsr mpan2 A pn0sr sylanc)) thm (recexsrlem ((x y) (x z) (A x) (y z) (A y) (A z) (w x) (v x) (u x) (f x) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) ((recexsrlem.1 (e. A (V)))) (-> (br (0R) ( (cv y) (cv z)) (~R)) A (0R) ( (cv y) (cv z)) (~R)) A (.R) (cv x) opreq1 (1R) eqeq1d x exbidv imbi12d 1pr (cv v) (1P) addclpr mpan2 1pr jctir (/\ (e. (cv y) (P.)) (e. (cv z) (P.))) anim2i (/\ (= (opr (cv w) (.P.) (cv v)) (1P)) (= (opr (cv z) (+P.) (cv w)) (cv y))) adantr (cv y) (cv z) (opr (cv v) (+P.) (1P)) (1P) mulsrpr syl (opr (cv y) (.P.) (opr (cv v) (+P.) (1P))) (opr (cv z) (.P.) (1P)) addclpr (cv y) (opr (cv v) (+P.) (1P)) mulclpr 1pr (cv v) (1P) addclpr mpan2 sylan2 1pr (cv z) (1P) mulclpr mpan2 syl2an an1rs (opr (cv y) (.P.) (1P)) (opr (cv z) (.P.) (opr (cv v) (+P.) (1P))) addclpr 1pr (cv y) (1P) mulclpr mpan2 (cv z) (opr (cv v) (+P.) (1P)) mulclpr 1pr (cv v) (1P) addclpr mpan2 sylan2 syl2an anassrs jca 1pr 1pr (1P) (1P) addclpr mp2an 1pr pm3.2i jctir (opr (opr (cv y) (.P.) (opr (cv v) (+P.) (1P))) (+P.) (opr (cv z) (.P.) (1P))) (opr (opr (cv y) (.P.) (1P)) (+P.) (opr (cv z) (.P.) (opr (cv v) (+P.) (1P)))) (opr (1P) (+P.) (1P)) (1P) enreceq syl (opr (cv z) (+P.) (cv w)) (cv y) (.P.) (cv v) opreq1 eqcomd (opr (cv w) (.P.) (cv v)) (1P) (opr (cv z) (.P.) (cv v)) (+P.) opreq2 z visset w visset v visset u visset f visset mulcompr f visset x visset (cv u) distrpr caoprdistrr syl5eq sylan9eqr (+P.) (opr (opr (cv y) (.P.) (1P)) (+P.) (opr (cv z) (.P.) (1P))) opreq1d (cv z) (.P.) (cv v) oprex 1pr elisseti (opr (cv y) (.P.) (1P)) (+P.) (opr (cv z) (.P.) (1P)) oprex u visset f visset addcompr f visset x visset (cv u) addasspr caopr32 syl6eq (+P.) (1P) opreq1d 1pr elisseti 1pr elisseti (opr (opr (cv z) (.P.) (cv v)) (+P.) (opr (opr (cv y) (.P.) (1P)) (+P.) (opr (cv z) (.P.) (1P)))) addasspr syl6eq v visset 1pr elisseti (cv y) distrpr (+P.) (opr (cv z) (.P.) (1P)) opreq1i (cv y) (.P.) (1P) oprex (cv z) (.P.) (1P) oprex (opr (cv y) (.P.) (cv v)) addasspr eqtr (+P.) (1P) opreq1i v visset 1pr elisseti (cv z) distrpr (opr (cv y) (.P.) (1P)) (+P.) opreq2i (cv y) (.P.) (1P) oprex (cv z) (.P.) (cv v) oprex (cv z) (.P.) (1P) oprex u visset f visset addcompr f visset x visset (cv u) addasspr caopr12 eqtr (+P.) (opr (1P) (+P.) (1P)) opreq1i 3eqtr4g syl5bir imp eqtrd df-1r syl6eqr enrex (~R) (V) (<,> (opr (cv v) (+P.) (1P)) (1P)) ecexg ax-mp (cv x) ([] (<,> (opr (cv v) (+P.) (1P)) (1P)) (~R)) ([] (<,> (cv y) (cv z)) (~R)) (.R) opreq2 (1R) eqeq1d cla4ev syl exp43 imp3a v 19.23adv (cv w) v recexpr syl5 imp3a w 19.23adv y visset z visset gt0srpr y visset (cv z) w ltexpri sylbi syl5 ecoptocl mpcom 1r (opr A (.R) (cv x)) (1R) (R.) eleq1 mpbiri x visset dmmulsr 0nsr A ndmoprrcl syl pm3.27d ancri x 19.22i syl)) thm (addgt0sr () ((addgt0sr.1 (e. A (V))) (addgt0sr.2 (e. B (V)))) (-> (/\ (br (0R) ( (/\ (br (0R) ( (cv x) (cv y)) (~R)) A (0R) ( (cv z) (cv w)) (~R))) anbi1d ([] (<,> (cv x) (cv y)) (~R)) A (.R) ([] (<,> (cv z) (cv w)) (~R)) opreq1 (0R) ( (cv z) (cv w)) (~R)) B (0R) ( (cv z) (cv w)) (~R)) B A (.R) opreq2 (0R) ( (e. A (R.)) (-> (-. (= A (0R))) (br (0R) ( (e. A (R.)) (-> (-. (= A (0R))) (E. x (/\ (e. (cv x) (R.)) (= (opr A (.R) (cv x)) (1R)))))) (recexsr.1 sqgt0sr A (.R) (cv y) oprex (cv x) (opr A (.R) (cv y)) (R.) eleq1 (cv x) (opr A (.R) (cv y)) A (.R) opreq2 (1R) eqeq1d anbi12d cla4ev recexsr.1 y visset A mulasssr (1R) eqeq1i sylan2b A (cv y) mulclsr sylan exp31 imp3a y 19.23adv A (.R) A oprex y recexsrlem syl5 syld)) thm (ssgt0sr () ((ssgt0sr.1 (e. A (V))) (ssgt0sr.2 (e. B (V)))) (-> (/\ (e. A (R.)) (e. B (R.))) (-> (-. (/\ (= A (0R)) (= B (0R)))) (br (0R) ( (br (0R) ( (opr A (+P.) (1P)) (1P)) (~R))) (e. A (P.))) (df-0r ( (opr A (+P.) (1P)) (1P)) (~R)) breq1i 1pr elisseti 1pr elisseti A (+P.) (1P) oprex 1pr elisseti ltsrpr mappsrpr.1 1pr elisseti addcompr (1P) (+P.) opreq2i 1pr elisseti mappsrpr.1 (1P) addasspr eqtr4 (opr (1P) (+P.) (1P)) ( (br ([] (<,> (opr A (+P.) (1P)) (1P)) (~R)) ( (opr B (+P.) (1P)) (1P)) (~R))) (br A ( (br (0R) ( (opr (cv x) (+P.) (1P)) (1P)) (~R)) A)))) (map2psrpr.1 ltrelsr (0R) brel pm3.27d df-nr ([] (<,> (cv y) (cv z)) (~R)) A (0R) ( (cv y) (cv z)) (~R)) A ([] (<,> (opr (cv x) (+P.) (1P)) (1P)) (~R)) eqeq2 (e. (cv x) (P.)) anbi2d x exbidv imbi12d (opr (cv x) (+P.) (1P)) (1P) (cv y) (cv z) enreceq 1pr (cv x) (1P) addclpr mpan2 1pr jctir sylan 1pr elisseti z visset (cv x) addasspr x visset (1P) (+P.) (cv z) oprex addcompr eqtr (opr (1P) (+P.) (cv y)) eqeq1i syl6bb expcom pm5.32d x exbidv df-0r ( (cv y) (cv z)) (~R)) breq1i 1pr elisseti 1pr elisseti y visset z visset ltsrpr bitr (1P) (+P.) (cv y) oprex (opr (1P) (+P.) (cv z)) x ltexpri sylbi syl5bir ecoptocl mpcom ([] (<,> (opr (cv x) (+P.) (1P)) (1P)) (~R)) A (0R) ( (opr (cv w) (+P.) (1P)) (1P)) (~R)) A))))) (-> (/\ (A. x (-> (e. (cv x) A) (br (0R) ( (opr (cv w) (+P.) (1P)) (1P)) ecexg ax-mp (cv x) ([] (<,> (opr (cv w) (+P.) (1P)) (1P)) (~R)) A eleq1 (cv x) ([] (<,> (opr (cv w) (+P.) (1P)) (1P)) (~R)) (0R) ( (e. (cv x) A) (br (0R) ( (e. (cv x) A) (br (0R) ( (opr (cv w) (+P.) (1P)) (1P)) (~R)) (cv x) A eleq1 suppsr.1 abeq2i syl5bb biimprcd (cv w) B n0i syl6 (e. (cv w) (P.)) adantld w 19.23adv x visset w map2psrpr syl5ib syld 19.23ai com12 A x n0 syl5ib imp jca)) thm (suppsr ((w x) (v x) (u x) (A x) (v w) (u w) (A w) (u v) (A v) (A u) (x y) (x z) (B x) (y z) (w y) (v y) (u y) (B y) (w z) (v z) (u z) (B z) (B w) (B v) (B u)) ((suppsr.1 (= B ({|} w (e. ([] (<,> (opr (cv w) (+P.) (1P)) (1P)) (~R)) A))))) (-> (/\ (/\ (A. x (-> (e. (cv x) A) (br (0R) ( (br (0R) ( (e. (cv y) A) (br (cv y) ( (br (0R) ( (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (opr (cv w) (+P.) (1P)) (1P)) ecexg ax-mp ([] (<,> (opr (cv w) (+P.) (1P)) (1P)) (~R)) (cv x) ( (opr (cv w) (+P.) (1P)) (1P)) (~R)) (cv x) (cv y) ( (opr (cv v) (+P.) (1P)) (1P)) ecexg ax-mp ([] (<,> (opr (cv v) (+P.) (1P)) (1P)) (~R)) (cv y) A eleq1 v visset (cv w) (cv v) (+P.) (1P) opreq1 (opr (cv w) (+P.) (1P)) (opr (cv v) (+P.) (1P)) (1P) opeq1 (<,> (opr (cv w) (+P.) (1P)) (1P)) (<,> (opr (cv v) (+P.) (1P)) (1P)) (~R) eceq2 3syl A eleq1d suppsr.1 elab2 syl5bb ([] (<,> (opr (cv v) (+P.) (1P)) (1P)) (~R)) (cv y) ([] (<,> (opr (cv w) (+P.) (1P)) (1P)) (~R)) ( (opr (cv v) (+P.) (1P)) (1P)) (~R)) (cv y) ( (opr (cv w) (+P.) (1P)) (1P)) (~R)) breq1 v visset w visset ltpsrpr syl5bbr ([] (<,> (opr (cv v) (+P.) (1P)) (1P)) (~R)) (cv y) ( (opr (cv u) (+P.) (1P)) (1P)) ecexg ax-mp ([] (<,> (opr (cv u) (+P.) (1P)) (1P)) (~R)) (cv z) A eleq1 u visset (cv w) (cv u) (+P.) (1P) opreq1 (opr (cv w) (+P.) (1P)) (opr (cv u) (+P.) (1P)) (1P) opeq1 (<,> (opr (cv w) (+P.) (1P)) (1P)) (<,> (opr (cv u) (+P.) (1P)) (1P)) (~R) eceq2 3syl A eleq1d suppsr.1 elab2 syl5bb ([] (<,> (opr (cv u) (+P.) (1P)) (1P)) (~R)) (cv z) ([] (<,> (opr (cv v) (+P.) (1P)) (1P)) (~R)) ( (opr (cv u) (+P.) (1P)) (1P)) (~R)) (cv z) (0R) ( (opr (cv v) (+P.) (1P)) (1P)) (~R)) (cv y) (0R) ( (opr (cv w) (+P.) (1P)) (1P)) (~R)) (cv x) (0R) ( (opr (cv w) (+P.) (1P)) (1P)) ecexg ax-mp ([] (<,> (opr (cv w) (+P.) (1P)) (1P)) (~R)) (cv x) (cv y) ( (opr (cv v) (+P.) (1P)) (1P)) ecexg ax-mp ([] (<,> (opr (cv v) (+P.) (1P)) (1P)) (~R)) (cv y) A eleq1 v visset (cv w) (cv v) (+P.) (1P) opreq1 (opr (cv w) (+P.) (1P)) (opr (cv v) (+P.) (1P)) (1P) opeq1 (<,> (opr (cv w) (+P.) (1P)) (1P)) (<,> (opr (cv v) (+P.) (1P)) (1P)) (~R) eceq2 3syl A eleq1d suppsr.1 elab2 syl5bb ([] (<,> (opr (cv v) (+P.) (1P)) (1P)) (~R)) (cv y) ( (opr (cv w) (+P.) (1P)) (1P)) (~R)) breq1 v visset w visset ltpsrpr syl5bbr imbi12d ([] (<,> (opr (cv v) (+P.) (1P)) (1P)) (~R)) (cv y) (0R) ( (opr (cv w) (+P.) (1P)) (1P)) (~R)) (cv x) (0R) ( (/\ (/\ (A. x (-> (e. (cv x) A) (br (0R) ( (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (e. (cv x) A) (br (0R) ( (e. (cv y) A) (br (cv y) ( (e. (cv x) A) (br (0R) ( (opr (cv v) (+P.) (1P)) (1P)) (<,> (opr (cv w) (+P.) (1P)) (1P)) (~R) eceq2 3syl A eleq1d cbvabv x y z suppsr syl x (-> (e. (cv x) A) (br (0R) ( (e. (cv x) A) (br (0R) ( (e. (cv y) A) (-. (br (cv x) ( (e. (cv x) A) (br (0R) ( (br (0R) ( (br (cv y) ( (br (0R) ( (br (cv y) ( (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) ( (/\ (E. y (/\ (e. (cv y) A) (br (0R) ( (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (e. (cv y) (R.)) (/\ (-> (e. (cv y) B) (-. (br (cv x) ( (br (cv y) ( (e. (cv y) (R.)) (-> (e. (cv y) B) (-. (br (cv x) ( (/\ (e. (cv y) A) (br (0R) ( (e. (cv y) B) (-. (br (cv x) ( (br (cv y) ( (e. (cv y) (R.)) (-> (e. (cv y) B) (-. (br (cv x) ( (e. (cv y) (R.)) (-> (br (cv y) ( (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) ( (e. (opr C (+R) (opr A (+R) (-1R))) (R.)) (e. A (R.))) (A (+R) (-1R) oprex dmaddsr 0nsr C ndmoprrcl pm3.27d m1r elisseti dmaddsr 0nsr A ndmoprrcl pm3.26d syl m1r A (-1R) addclsr mpan2 supsrlem.1 C (opr A (+R) (-1R)) addclsr mpan syl impbi)) thm (supsrlem2 ((A x) (C x)) ((supsrlem.1 (e. C (R.)))) (<-> (e. A (R.)) (E. x (/\ (e. (cv x) (R.)) (= (opr C (+R) (opr (cv x) (+R) (-1R))) A)))) (m1r A (-1R) mulclsr mpan2 supsrlem.1 jctir (opr A (.R) (-1R)) C addclsr syl m1r (opr (opr A (.R) (-1R)) (+R) C) (-1R) addclsr mpan2 (opr (opr (opr A (.R) (-1R)) (+R) C) (+R) (-1R)) x negexsr 3syl A pn0sr (+R) (opr C (+R) (opr (cv x) (+R) (-1R))) opreq1d m1r (cv x) (-1R) addclsr mpan2 supsrlem.1 jctil C (opr (cv x) (+R) (-1R)) addclsr (opr C (+R) (opr (cv x) (+R) (-1R))) 0idsr 3syl C (+R) (opr (cv x) (+R) (-1R)) oprex 0r elisseti addcomsr syl5eqr sylan9eq A (.R) (-1R) oprex C (+R) (opr (cv x) (+R) (-1R)) oprex A addasssr syl5eqr A 0idsr (e. (cv x) (R.)) adantr eqeq12d m1r elisseti x visset (opr (opr A (.R) (-1R)) (+R) C) addasssr m1r elisseti x visset addcomsr (opr (opr A (.R) (-1R)) (+R) C) (+R) opreq2i supsrlem.1 elisseti (cv x) (+R) (-1R) oprex (opr A (.R) (-1R)) addasssr 3eqtr (0R) eqeq1i (opr (opr A (.R) (-1R)) (+R) (opr C (+R) (opr (cv x) (+R) (-1R)))) (0R) A (+R) opreq2 sylbi syl5bi ex imdistand x 19.22dv mpd (opr C (+R) (opr (cv x) (+R) (-1R))) A (R.) eleq1 m1r (cv x) (-1R) addclsr mpan2 supsrlem.1 jctil C (opr (cv x) (+R) (-1R)) addclsr syl syl5bi impcom x 19.23aiv impbi)) thm (supsrlem3 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((supsrlem.1 (e. C (R.))) (supsrlem3.2 (e. A (V))) (supsrlem3.3 (e. B (V)))) (<-> (br (opr C (+R) (opr A (+R) (-1R))) ( (e. D B) (e. (opr C (+R) (opr D (+R) (-1R))) A)) (supsrlem4.1 (cv x) D (+R) (-1R) opreq1 C (+R) opreq2d A eleq1d supsrlem.2 (cv w) (cv x) (+R) (-1R) opreq1 C (+R) opreq2d A eleq1d cbvabv eqtr elab2)) thm (supsrlem5 ((w y) (A y) (A w) (B y) (B w) (C y) (C w)) ((supsrlem.1 (e. C (R.))) (supsrlem.2 (= B ({|} w (e. (opr C (+R) (opr (cv w) (+R) (-1R))) A))))) (-> (e. C A) (E. y (/\ (e. (cv y) B) (br (0R) ( (/\ (e. C A) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (/\ (/\ (C_ A (R.)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (e. (<,> A B) (CC)) (/\ (e. A (R.)) (e. B (R.)))) (df-c (<,> A B) eleq2i opelcn.1 A (R.) (R.) opelxp bitr)) thm (opelreal () () (<-> (e. (<,> A (0R)) (RR)) (e. A (R.))) (df-r (<,> A (0R)) eleq2i 0r elisseti A (R.) ({} (0R)) opelxp 0r elisseti (0R) elsnc (e. A (R.)) anbi2i 3bitr (0R) eqid mpbiran2)) thm (elreal ((x y) (A x) (A y)) () (<-> (e. A (RR)) (E. x (/\ (e. (cv x) (R.)) (= (<,> (cv x) (0R)) A)))) (df-r A eleq2i A (R.) ({} (0R)) x y elxp (= A (<,> (cv x) (cv y))) (/\ (e. (cv x) (R.)) (e. (cv y) ({} (0R)))) ancom (e. (cv x) (R.)) (e. (cv y) ({} (0R))) (= A (<,> (cv x) (cv y))) anass y (0R) elsn A (<,> (cv x) (cv y)) eqcom anbi12i (cv y) (0R) (cv x) opeq2 A eqeq1d pm5.32i bitr (e. (cv x) (R.)) anbi2i (e. (cv x) (R.)) (= (cv y) (0R)) (= (<,> (cv x) (0R)) A) an12 3bitr bitr y exbii y (= (cv y) (0R)) (/\ (e. (cv x) (R.)) (= (<,> (cv x) (0R)) A)) 19.41v bitr 0r elisseti y isseti mpbiran x exbii 3bitr)) thm (0ncn () () (-. (e. ({/}) (CC))) ((R.) (R.) 0nelxp df-c ({/}) eleq2i mtbir)) thm (ltrelre ((x y) (x z) (w x) (y z) (w y) (w z)) () (C_ ( (cv z) (0R))) (= (cv y) (<,> (cv w) (0R)))) (br (cv z) ( (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr (<,> A B) (+) (<,> C D)) (<,> (opr A (+R) C) (opr B (+R) D)))) ((opr A (+R) C) (opr B (+R) D) opex (opr (cv w) (+R) (cv u)) (opr A (+R) (cv u)) (opr (cv v) (+R) (cv f)) (opr B (+R) (cv f)) opeq12 (cv w) A (+R) (cv u) opreq1 (cv v) B (+R) (cv f) opreq1 syl2an (opr A (+R) (cv u)) (opr A (+R) C) (opr B (+R) (cv f)) (opr B (+R) D) opeq12 (cv u) C A (+R) opreq2 (cv f) D B (+R) opreq2 syl2an sylan9eq x y z w v u f df-plus df-c (cv x) eleq2i df-c (cv y) eleq2i anbi12i (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) (<,> (opr (cv w) (+R) (cv u)) (opr (cv v) (+R) (cv f))))))))) anbi1i x y z oprabbii eqtr oprabval3)) thm (mulcnsr ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f)) () (-> (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr (<,> A B) (x.) (<,> C D)) (<,> (opr (opr A (.R) C) (+R) (opr (-1R) (.R) (opr B (.R) D))) (opr (opr B (.R) C) (+R) (opr A (.R) D))))) ((opr (opr A (.R) C) (+R) (opr (-1R) (.R) (opr B (.R) D))) (opr (opr B (.R) C) (+R) (opr A (.R) D)) opex (opr (opr (cv w) (.R) (cv u)) (+R) (opr (-1R) (.R) (opr (cv v) (.R) (cv f)))) (opr (opr A (.R) (cv u)) (+R) (opr (-1R) (.R) (opr B (.R) (cv f)))) (opr (opr (cv v) (.R) (cv u)) (+R) (opr (cv w) (.R) (cv f))) (opr (opr B (.R) (cv u)) (+R) (opr A (.R) (cv f))) opeq12 (cv w) A (.R) (cv u) opreq1 (cv v) B (.R) (cv f) opreq1 (-1R) (.R) opreq2d (+R) opreqan12d (cv v) B (.R) (cv u) opreq1 (cv w) A (.R) (cv f) opreq1 (+R) opreqan12rd sylanc (opr (opr A (.R) (cv u)) (+R) (opr (-1R) (.R) (opr B (.R) (cv f)))) (opr (opr A (.R) C) (+R) (opr (-1R) (.R) (opr B (.R) D))) (opr (opr B (.R) (cv u)) (+R) (opr A (.R) (cv f))) (opr (opr B (.R) C) (+R) (opr A (.R) D)) opeq12 (cv u) C A (.R) opreq2 (cv f) D B (.R) opreq2 (-1R) (.R) opreq2d (+R) opreqan12d (cv u) C B (.R) opreq2 (cv f) D A (.R) opreq2 (+R) opreqan12d sylanc sylan9eq x y z w v u f df-mul df-c (cv x) eleq2i df-c (cv y) eleq2i anbi12i (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) (<,> (opr (opr (cv w) (.R) (cv u)) (+R) (opr (-1R) (.R) (opr (cv v) (.R) (cv f)))) (opr (opr (cv v) (.R) (cv u)) (+R) (opr (cv w) (.R) (cv f)))))))))) anbi1i x y z oprabbii eqtr oprabval3)) thm (eqresr () ((eqresr.1 (e. A (V)))) (<-> (= (<,> A (0R)) (<,> B (0R))) (= A B)) (eqresr.1 0r elisseti 0r elisseti B opth (0R) eqid mpbiran2)) thm (addresr () () (-> (/\ (e. A (R.)) (e. B (R.))) (= (opr (<,> A (0R)) (+) (<,> B (0R))) (<,> (opr A (+R) B) (0R)))) (0r 0r pm3.2i A (0R) B (0R) addcnsr an4s mpan2 0r (0R) 0idsr ax-mp (opr (0R) (+R) (0R)) (0R) (opr A (+R) B) opeq2 ax-mp syl6eq)) thm (mulresr () ((mulresr.1 (e. B (V)))) (-> (/\ (e. A (R.)) (e. B (R.))) (= (opr (<,> A (0R)) (x.) (<,> B (0R))) (<,> (opr A (.R) B) (0R)))) (0r 0r pm3.2i A (0R) B (0R) mulcnsr an4s mpan2 (opr (opr A (.R) B) (+R) (opr (-1R) (.R) (opr (0R) (.R) (0R)))) (opr A (.R) B) (opr (opr (0R) (.R) B) (+R) (opr A (.R) (0R))) (0R) opeq12 A B mulclsr (opr A (.R) B) 0idsr syl 0r (0R) 00sr ax-mp (-1R) (.R) opreq2i m1r (-1R) 00sr ax-mp eqtr (opr A (.R) B) (+R) opreq2i syl5eq B 00sr 0r elisseti mulresr.1 mulcomsr syl5eq A 00sr (+R) opreqan12rd 0r (0R) 0idsr ax-mp syl6eq sylanc eqtrd)) thm (ltresr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((ltresr.1 (e. A (V))) (ltresr.2 (e. B (V)))) (<-> (br (<,> A (0R)) ( B (0R))) (br A ( A (0R)) brel A opelreal B opelreal anbi12i sylib ltresr.2 ltrelsr A brel A (0R) opex B (0R) opex (cv x) (<,> A (0R)) (RR) eleq1 (e. (cv y) (RR)) anbi1d (cv x) (<,> A (0R)) (<,> (cv z) (0R)) eqeq1 (= (cv y) (<,> (cv w) (0R))) anbi1d (br (cv z) ( B (0R)) (RR) eleq1 (e. (<,> A (0R)) (RR)) anbi2d (cv y) (<,> B (0R)) (<,> (cv w) (0R)) eqeq1 (= (<,> A (0R)) (<,> (cv z) (0R))) anbi2d (br (cv z) ( (cv w) (0R)) A))))) (-> (/\ (C_ A (RR)) (-. (= A ({/})))) (/\ (C_ B (R.)) (-. (= B ({/}))))) (A (RR) (<,> (cv w) (0R)) ssel supre.1 abeq2i syl5ib (cv w) opelreal syl6ib ssrdv (-. (= A ({/}))) adantr A (RR) (cv x) ssel com12 (<,> (cv w) (0R)) (cv x) A eleq1 supre.1 abeq2i syl5bb biimprcd (cv w) B n0i syl6 (e. (cv w) (R.)) adantld w 19.23adv (cv x) w elreal syl5ib syld x 19.23aiv com12 A x n0 syl5ib imp jca)) thm (supre ((w x) (v x) (u x) (A x) (v w) (u w) (A w) (u v) (A v) (A u) (x y) (x z) (B x) (y z) (w y) (v y) (u y) (B y) (w z) (v z) (u z) (B z) (B w) (B v) (B u)) ((supre.1 (= B ({|} w (e. (<,> (cv w) (0R)) A))))) (-> (/\ (/\ (C_ A (RR)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (RR)) (A. y (-> (e. (cv y) (RR)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (RR)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (cv w) (0R)) (cv x) ( (cv w) (0R)) (cv x) (cv y) ( (cv v) (0R)) (cv y) A eleq1 v visset (cv w) (cv v) (0R) opeq1 A eleq1d supre.1 elab2 syl5bb (<,> (cv v) (0R)) (cv y) (<,> (cv w) (0R)) ( (cv v) (0R)) (cv y) ( (cv w) (0R)) breq1 v visset w visset ltresr syl5bbr (<,> (cv v) (0R)) (cv y) ( (cv u) (0R)) (cv z) A eleq1 u visset (cv w) (cv u) (0R) opeq1 A eleq1d supre.1 elab2 syl5bb (<,> (cv u) (0R)) (cv z) (<,> (cv v) (0R)) ( (cv u) (0R)) (cv z) (RR) eleq1 (cv u) opelreal syl5bbr (cv z) u elreal gencbvex syl5bb imbi12d anbi12d (<,> (cv v) (0R)) (cv y) (RR) eleq1 (cv v) opelreal syl5bbr (cv y) v elreal gencbval syl5bb (<,> (cv w) (0R)) (cv x) (RR) eleq1 (cv w) opelreal syl5bbr (cv x) w elreal gencbvex sylib (cv w) (0R) opex (<,> (cv w) (0R)) (cv x) (cv y) ( (cv v) (0R)) (cv y) A eleq1 v visset (cv w) (cv v) (0R) opeq1 A eleq1d supre.1 elab2 syl5bb (<,> (cv v) (0R)) (cv y) ( (cv w) (0R)) breq1 v visset w visset ltresr syl5bbr imbi12d (<,> (cv v) (0R)) (cv y) (RR) eleq1 (cv v) opelreal syl5bbr (cv y) v elreal gencbval syl5bb (<,> (cv w) (0R)) (cv x) (RR) eleq1 (cv w) opelreal syl5bbr (cv x) w elreal gencbvex sylan2br supre.1 suprelem sylan)) thm (ltsor ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (/\ (Or ( (cv w) (0R)) (cv x) ( (cv v) (0R)) breq1 (<,> (cv w) (0R)) (cv x) (<,> (cv v) (0R)) eqeq1 (<,> (cv w) (0R)) (cv x) (<,> (cv v) (0R)) ( (cv w) (0R)) (cv x) ( (cv v) (0R)) breq1 (br (<,> (cv v) (0R)) ( (cv u) (0R))) anbi1d (<,> (cv w) (0R)) (cv x) ( (cv u) (0R)) breq1 imbi12d anbi12d (<,> (cv v) (0R)) (cv y) (cv x) ( (cv v) (0R)) (cv y) (cv x) eqeq2 (<,> (cv v) (0R)) (cv y) ( (cv v) (0R)) (cv y) (cv x) ( (cv v) (0R)) (cv y) ( (cv u) (0R)) breq1 anbi12d (br (cv x) ( (cv u) (0R))) imbi1d anbi12d (<,> (cv u) (0R)) (cv z) (cv y) ( (cv u) (0R)) (cv z) (cv x) ( (br (cv x) ( (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr ([] (<,> A B) (`' (E))) (+) ([] (<,> C D) (`' (E)))) ([] (<,> (opr A (+R) C) (opr B (+R) D)) (`' (E))))) (A B C D addcnsr A B opex ecid C D opex ecid (+) opreq12i (opr A (+R) C) (opr B (+R) D) opex ecid 3eqtr4g)) thm (mulcnsrec () () (-> (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr ([] (<,> A B) (`' (E))) (x.) ([] (<,> C D) (`' (E)))) ([] (<,> (opr (opr A (.R) C) (+R) (opr (-1R) (.R) (opr B (.R) D))) (opr (opr B (.R) C) (+R) (opr A (.R) D))) (`' (E))))) (A B C D mulcnsr A B opex ecid C D opex ecid (x.) opreq12i (opr (opr A (.R) C) (+R) (opr (-1R) (.R) (opr B (.R) D))) (opr (opr B (.R) C) (+R) (opr A (.R) D)) opex ecid 3eqtr4g)) thm (axaddopr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (:--> (+) (X. (CC) (CC)) (CC)) ((+) (CC) (CC) (CC) x y ffnoprval (+) (X. (CC) (CC)) df-fn z (<,> (opr (cv w) (+R) (cv u)) (opr (cv v) (+R) (cv f))) moeq u f (cv y) mosubop w v (cv x) mosubop (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f))) (= (cv z) (<,> (opr (cv w) (+R) (cv u)) (opr (cv v) (+R) (cv f)))) anass u f 2exbii u f (= (cv x) (<,> (cv w) (cv v))) (/\ (= (cv y) (<,> (cv u) (cv f))) (= (cv z) (<,> (opr (cv w) (+R) (cv u)) (opr (cv v) (+R) (cv f))))) 19.42vv bitr w v 2exbii z mobii mpbir (/\ (e. (cv x) (CC)) (e. (cv y) (CC))) moani x y funoprab x y z w v u f df-plus (+) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (CC))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) (<,> (opr (cv w) (+R) (cv u)) (opr (cv v) (+R) (cv f))))))))))) funeq ax-mp mpbir x y z w v u f df-plus dmeqi x y z (CC) (CC) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) (<,> (opr (cv w) (+R) (cv u)) (opr (cv v) (+R) (cv f))))))))) dmoprabss eqsstr 0ncn df-c (<,> (cv z) (cv w)) (cv x) (+) (<,> (cv v) (cv u)) opreq1 (X. (R.) (R.)) eleq1d (<,> (cv v) (cv u)) (cv y) (cv x) (+) opreq2 (X. (R.) (R.)) eleq1d (cv z) (cv w) (cv v) (cv u) addcnsr (cv z) (cv v) addclsr (cv w) (cv u) addclsr anim12i an4s (opr (cv z) (+R) (cv v)) (R.) (opr (cv w) (+R) (cv u)) (R.) opelxpi syl eqeltrd 2optocl df-c syl6eleqr oprssdm eqssi mpbir2an df-c (<,> (cv z) (cv w)) (cv x) (+) (<,> (cv v) (cv u)) opreq1 (X. (R.) (R.)) eleq1d (<,> (cv v) (cv u)) (cv y) (cv x) (+) opreq2 (X. (R.) (R.)) eleq1d (cv z) (cv w) (cv v) (cv u) addcnsr (cv z) (cv v) addclsr (cv w) (cv u) addclsr anim12i an4s (opr (cv z) (+R) (cv v)) (R.) (opr (cv w) (+R) (cv u)) (R.) opelxpi syl eqeltrd 2optocl df-c syl6eleqr rgen2 mpbir2an)) thm (axmulopr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (:--> (x.) (X. (CC) (CC)) (CC)) ((x.) (CC) (CC) (CC) x y ffnoprval (x.) (X. (CC) (CC)) df-fn z (<,> (opr (opr (cv w) (.R) (cv u)) (+R) (opr (-1R) (.R) (opr (cv v) (.R) (cv f)))) (opr (opr (cv v) (.R) (cv u)) (+R) (opr (cv w) (.R) (cv f)))) moeq u f (cv y) mosubop w v (cv x) mosubop (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f))) (= (cv z) (<,> (opr (opr (cv w) (.R) (cv u)) (+R) (opr (-1R) (.R) (opr (cv v) (.R) (cv f)))) (opr (opr (cv v) (.R) (cv u)) (+R) (opr (cv w) (.R) (cv f))))) anass u f 2exbii u f (= (cv x) (<,> (cv w) (cv v))) (/\ (= (cv y) (<,> (cv u) (cv f))) (= (cv z) (<,> (opr (opr (cv w) (.R) (cv u)) (+R) (opr (-1R) (.R) (opr (cv v) (.R) (cv f)))) (opr (opr (cv v) (.R) (cv u)) (+R) (opr (cv w) (.R) (cv f)))))) 19.42vv bitr w v 2exbii z mobii mpbir (/\ (e. (cv x) (CC)) (e. (cv y) (CC))) moani x y funoprab x y z w v u f df-mul (x.) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (CC))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) (<,> (opr (opr (cv w) (.R) (cv u)) (+R) (opr (-1R) (.R) (opr (cv v) (.R) (cv f)))) (opr (opr (cv v) (.R) (cv u)) (+R) (opr (cv w) (.R) (cv f)))))))))))) funeq ax-mp mpbir x y z w v u f df-mul dmeqi x y z (CC) (CC) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) (<,> (opr (opr (cv w) (.R) (cv u)) (+R) (opr (-1R) (.R) (opr (cv v) (.R) (cv f)))) (opr (opr (cv v) (.R) (cv u)) (+R) (opr (cv w) (.R) (cv f)))))))))) dmoprabss eqsstr 0ncn df-c (<,> (cv z) (cv w)) (cv x) (x.) (<,> (cv v) (cv u)) opreq1 (X. (R.) (R.)) eleq1d (<,> (cv v) (cv u)) (cv y) (cv x) (x.) opreq2 (X. (R.) (R.)) eleq1d (cv z) (cv w) (cv v) (cv u) mulcnsr (opr (opr (cv z) (.R) (cv v)) (+R) (opr (-1R) (.R) (opr (cv w) (.R) (cv u)))) (R.) (opr (opr (cv w) (.R) (cv v)) (+R) (opr (cv z) (.R) (cv u))) (R.) opelxpi (opr (cv z) (.R) (cv v)) (opr (-1R) (.R) (opr (cv w) (.R) (cv u))) addclsr (cv z) (cv v) mulclsr (cv w) (cv u) mulclsr m1r (-1R) (opr (cv w) (.R) (cv u)) mulclsr mpan syl syl2an an4s (opr (cv w) (.R) (cv v)) (opr (cv z) (.R) (cv u)) addclsr (cv w) (cv v) mulclsr (cv z) (cv u) mulclsr syl2an ancoms an42s sylanc eqeltrd 2optocl df-c syl6eleqr oprssdm eqssi mpbir2an df-c (<,> (cv z) (cv w)) (cv x) (x.) (<,> (cv v) (cv u)) opreq1 (X. (R.) (R.)) eleq1d (<,> (cv v) (cv u)) (cv y) (cv x) (x.) opreq2 (X. (R.) (R.)) eleq1d (cv z) (cv w) (cv v) (cv u) mulcnsr (opr (opr (cv z) (.R) (cv v)) (+R) (opr (-1R) (.R) (opr (cv w) (.R) (cv u)))) (R.) (opr (opr (cv w) (.R) (cv v)) (+R) (opr (cv z) (.R) (cv u))) (R.) opelxpi (opr (cv z) (.R) (cv v)) (opr (-1R) (.R) (opr (cv w) (.R) (cv u))) addclsr (cv z) (cv v) mulclsr (cv w) (cv u) mulclsr m1r (-1R) (opr (cv w) (.R) (cv u)) mulclsr mpan syl syl2an an4s (opr (cv w) (.R) (cv v)) (opr (cv z) (.R) (cv u)) addclsr (cv w) (cv v) mulclsr (cv z) (cv u) mulclsr syl2an ancoms an42s sylanc eqeltrd 2optocl df-c syl6eleqr rgen2 mpbir2an)) thm (axcnex () () (e. (CC) (V)) (df-c srex srex xpex eqeltr)) thm (axresscn () () (C_ (RR) (CC)) ((R.) ssid 0r (0R) (R.) snssi ax-mp (R.) (R.) ({} (0R)) (R.) ssxp mp2an df-r df-c 3sstr4)) thm (ax1re () () (e. (1) (RR)) (df-1 1r (1R) opelreal mpbir eqeltr)) thm (axicn () () (e. (i) (CC)) (df-i (CC) eleq1i 1r elisseti (0R) opelcn bitr 0r 1r mpbir2an)) thm (axaddcl ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CC)) (e. B (CC))) (e. (opr A (+) B) (CC))) (axaddopr (+) (CC) (CC) (CC) x y ffnoprval pm3.27bd ax-mp (cv x) A (+) (cv y) opreq1 (CC) eleq1d (cv y) B A (+) opreq2 (CC) eleq1d (CC) (CC) rcla42v mpi)) thm (axaddrcl ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (RR)) (e. B (RR))) (e. (opr A (+) B) (RR))) (A x elreal B y elreal (<,> (cv x) (0R)) A (+) (<,> (cv y) (0R)) opreq1 (RR) eleq1d (<,> (cv y) (0R)) B A (+) opreq2 (RR) eleq1d (cv x) (cv y) addresr (cv x) (cv y) addclsr (opr (cv x) (+R) (cv y)) opelreal sylibr eqeltrd 2gencl)) thm (axmulcl ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CC)) (e. B (CC))) (e. (opr A (x.) B) (CC))) (axmulopr (x.) (CC) (CC) (CC) x y ffnoprval pm3.27bd ax-mp (cv x) A (x.) (cv y) opreq1 (CC) eleq1d (cv y) B A (x.) opreq2 (CC) eleq1d (CC) (CC) rcla42v mpi)) thm (axmulrcl ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (RR)) (e. B (RR))) (e. (opr A (x.) B) (RR))) (A x elreal B y elreal (<,> (cv x) (0R)) A (x.) (<,> (cv y) (0R)) opreq1 (RR) eleq1d (<,> (cv y) (0R)) B A (x.) opreq2 (RR) eleq1d y visset (cv x) mulresr (cv x) (cv y) mulclsr (opr (cv x) (.R) (cv y)) opelreal sylibr eqeltrd 2gencl)) thm (axaddcom ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (+) B) (opr B (+) A))) (dfcnqs (cv x) (cv y) (cv z) (cv w) addcnsrec (cv z) (cv w) (cv x) (cv y) addcnsrec x visset z visset addcomsr y visset w visset addcomsr A B ecoprcom)) thm (axmulcom ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (x.) B) (opr B (x.) A))) (dfcnqs (cv x) (cv y) (cv z) (cv w) mulcnsrec (cv z) (cv w) (cv x) (cv y) mulcnsrec x visset z visset mulcomsr y visset w visset mulcomsr (-1R) (.R) opreq2i (+R) opreq12i y visset z visset mulcomsr x visset w visset mulcomsr (+R) opreq12i (cv z) (.R) (cv y) oprex (cv w) (.R) (cv x) oprex addcomsr eqtr A B ecoprcom)) thm (axaddass ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (+) C) (opr A (+) (opr B (+) C)))) (dfcnqs (cv x) (cv y) (cv z) (cv w) addcnsrec (cv z) (cv w) (cv v) (cv u) addcnsrec (opr (cv x) (+R) (cv z)) (opr (cv y) (+R) (cv w)) (cv v) (cv u) addcnsrec (cv x) (cv y) (opr (cv z) (+R) (cv v)) (opr (cv w) (+R) (cv u)) addcnsrec (cv x) (cv z) addclsr (cv y) (cv w) addclsr anim12i an4s (cv z) (cv v) addclsr (cv w) (cv u) addclsr anim12i an4s z visset v visset (cv x) addasssr w visset u visset (cv y) addasssr A B C ecoprass)) thm (axmulass ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (x.) B) (x.) C) (opr A (x.) (opr B (x.) C)))) (dfcnqs (cv x) (cv y) (cv z) (cv w) mulcnsrec (cv z) (cv w) (cv v) (cv u) mulcnsrec (opr (opr (cv x) (.R) (cv z)) (+R) (opr (-1R) (.R) (opr (cv y) (.R) (cv w)))) (opr (opr (cv y) (.R) (cv z)) (+R) (opr (cv x) (.R) (cv w))) (cv v) (cv u) mulcnsrec (cv x) (cv y) (opr (opr (cv z) (.R) (cv v)) (+R) (opr (-1R) (.R) (opr (cv w) (.R) (cv u)))) (opr (opr (cv w) (.R) (cv v)) (+R) (opr (cv z) (.R) (cv u))) mulcnsrec (opr (cv x) (.R) (cv z)) (opr (-1R) (.R) (opr (cv y) (.R) (cv w))) addclsr (cv x) (cv z) mulclsr (cv y) (cv w) mulclsr m1r (-1R) (opr (cv y) (.R) (cv w)) mulclsr mpan syl syl2an an4s (opr (cv y) (.R) (cv z)) (opr (cv x) (.R) (cv w)) addclsr (cv y) (cv z) mulclsr (cv x) (cv w) mulclsr syl2an ancoms an42s jca (opr (cv z) (.R) (cv v)) (opr (-1R) (.R) (opr (cv w) (.R) (cv u))) addclsr (cv z) (cv v) mulclsr (cv w) (cv u) mulclsr m1r (-1R) (opr (cv w) (.R) (cv u)) mulclsr mpan syl syl2an an4s (opr (cv w) (.R) (cv v)) (opr (cv z) (.R) (cv u)) addclsr (cv w) (cv v) mulclsr (cv z) (cv u) mulclsr syl2an ancoms an42s jca (cv x) (.R) (opr (cv z) (.R) (cv v)) oprex (cv x) (.R) (opr (-1R) (.R) (opr (cv w) (.R) (cv u))) oprex (-1R) (.R) (opr (cv y) (.R) (opr (cv w) (.R) (cv v))) oprex f visset g visset addcomsr g visset h visset (cv f) addasssr (-1R) (.R) (opr (cv y) (.R) (opr (cv z) (.R) (cv u))) oprex caopr42 (cv z) (.R) (cv v) oprex (-1R) (.R) (opr (cv w) (.R) (cv u)) oprex (cv x) distrsr (cv w) (.R) (cv v) oprex (cv z) (.R) (cv u) oprex (cv y) distrsr (-1R) (.R) opreq2i (cv y) (.R) (opr (cv w) (.R) (cv v)) oprex (cv y) (.R) (opr (cv z) (.R) (cv u)) oprex (-1R) distrsr eqtr (+R) opreq12i x visset m1r elisseti z visset f visset g visset mulcomsr g visset h visset (cv f) distrsr (cv y) (.R) (cv w) oprex v visset g visset h visset (cv f) mulasssr caoprdilem w visset v visset (cv y) mulasssr (-1R) (.R) opreq2i (opr (cv x) (.R) (opr (cv z) (.R) (cv v))) (+R) opreq2i eqtr y visset x visset z visset f visset g visset mulcomsr g visset h visset (cv f) distrsr w visset u visset g visset h visset (cv f) mulasssr caoprdilem (-1R) (.R) opreq2i (cv y) (.R) (opr (cv z) (.R) (cv u)) oprex (cv x) (.R) (opr (cv w) (.R) (cv u)) oprex (-1R) distrsr m1r elisseti x visset (cv w) (.R) (cv u) oprex f visset g visset mulcomsr g visset h visset (cv f) mulasssr caopr12 (opr (-1R) (.R) (opr (cv y) (.R) (opr (cv z) (.R) (cv u)))) (+R) opreq2i 3eqtr (+R) opreq12i 3eqtr4r (cv y) (.R) (opr (cv z) (.R) (cv v)) oprex (cv y) (.R) (opr (-1R) (.R) (opr (cv w) (.R) (cv u))) oprex (cv x) (.R) (opr (cv w) (.R) (cv v)) oprex f visset g visset addcomsr g visset h visset (cv f) addasssr (cv x) (.R) (opr (cv z) (.R) (cv u)) oprex caopr42 (cv z) (.R) (cv v) oprex (-1R) (.R) (opr (cv w) (.R) (cv u)) oprex (cv y) distrsr (cv w) (.R) (cv v) oprex (cv z) (.R) (cv u) oprex (cv x) distrsr (+R) opreq12i y visset x visset z visset f visset g visset mulcomsr g visset h visset (cv f) distrsr w visset v visset g visset h visset (cv f) mulasssr caoprdilem x visset m1r elisseti z visset f visset g visset mulcomsr g visset h visset (cv f) distrsr (cv y) (.R) (cv w) oprex u visset g visset h visset (cv f) mulasssr caoprdilem w visset u visset (cv y) mulasssr (-1R) (.R) opreq2i m1r elisseti y visset (cv w) (.R) (cv u) oprex f visset g visset mulcomsr g visset h visset (cv f) mulasssr caopr12 eqtr (opr (cv x) (.R) (opr (cv z) (.R) (cv u))) (+R) opreq2i eqtr (+R) opreq12i 3eqtr4r A B C ecoprass)) thm (axdistr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (x.) (opr B (+) C)) (opr (opr A (x.) B) (+) (opr A (x.) C)))) (dfcnqs (cv z) (cv w) (cv v) (cv u) addcnsrec (cv x) (cv y) (opr (cv z) (+R) (cv v)) (opr (cv w) (+R) (cv u)) mulcnsrec (cv x) (cv y) (cv z) (cv w) mulcnsrec (cv x) (cv y) (cv v) (cv u) mulcnsrec (opr (opr (cv x) (.R) (cv z)) (+R) (opr (-1R) (.R) (opr (cv y) (.R) (cv w)))) (opr (opr (cv y) (.R) (cv z)) (+R) (opr (cv x) (.R) (cv w))) (opr (opr (cv x) (.R) (cv v)) (+R) (opr (-1R) (.R) (opr (cv y) (.R) (cv u)))) (opr (opr (cv y) (.R) (cv v)) (+R) (opr (cv x) (.R) (cv u))) addcnsrec (cv z) (cv v) addclsr (cv w) (cv u) addclsr anim12i an4s (opr (cv x) (.R) (cv z)) (opr (-1R) (.R) (opr (cv y) (.R) (cv w))) addclsr (cv x) (cv z) mulclsr (cv y) (cv w) mulclsr m1r (-1R) (opr (cv y) (.R) (cv w)) mulclsr mpan syl syl2an an4s (opr (cv y) (.R) (cv z)) (opr (cv x) (.R) (cv w)) addclsr (cv y) (cv z) mulclsr (cv x) (cv w) mulclsr syl2an ancoms an42s jca (opr (cv x) (.R) (cv v)) (opr (-1R) (.R) (opr (cv y) (.R) (cv u))) addclsr (cv x) (cv v) mulclsr (cv y) (cv u) mulclsr m1r (-1R) (opr (cv y) (.R) (cv u)) mulclsr mpan syl syl2an an4s (opr (cv y) (.R) (cv v)) (opr (cv x) (.R) (cv u)) addclsr (cv y) (cv v) mulclsr (cv x) (cv u) mulclsr syl2an ancoms an42s jca z visset v visset (cv x) distrsr w visset u visset (cv y) distrsr (-1R) (.R) opreq2i (cv y) (.R) (cv w) oprex (cv y) (.R) (cv u) oprex (-1R) distrsr eqtr (+R) opreq12i (cv x) (.R) (cv z) oprex (cv x) (.R) (cv v) oprex (-1R) (.R) (opr (cv y) (.R) (cv w)) oprex f visset g visset addcomsr g visset h visset (cv f) addasssr (-1R) (.R) (opr (cv y) (.R) (cv u)) oprex caopr4 eqtr z visset v visset (cv y) distrsr w visset u visset (cv x) distrsr (+R) opreq12i (cv y) (.R) (cv z) oprex (cv y) (.R) (cv v) oprex (cv x) (.R) (cv w) oprex f visset g visset addcomsr g visset h visset (cv f) addasssr (cv x) (.R) (cv u) oprex caopr4 eqtr A B C ecoprdi)) thm (ax1ne0 () () (=/= (1) (0)) (1ne0sr 1r elisseti (0R) eqresr mtbir df-1 df-0 eqeq12i mtbir (1) (0) df-ne mpbir)) thm (ax0id ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (opr A (+) (0)) A)) (df-c (<,> (cv x) (cv y)) A (+) (0) opreq1 (= (<,> (cv x) (cv y)) A) id eqeq12d 0r 0r pm3.2i (cv x) (cv y) (0R) (0R) addcnsr mpan2 (opr (cv x) (+R) (0R)) (cv x) (opr (cv y) (+R) (0R)) (cv y) opeq12 (cv x) 0idsr (cv y) 0idsr syl2an eqtrd df-0 (<,> (cv x) (cv y)) (+) opreq2i syl5eq optocl)) thm (ax1id ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (opr A (x.) (1)) A)) (df-c (<,> (cv x) (cv y)) A (x.) (1) opreq1 (= (<,> (cv x) (cv y)) A) id eqeq12d 1r 0r pm3.2i (cv x) (cv y) (1R) (0R) mulcnsr mpan2 (opr (opr (cv x) (.R) (1R)) (+R) (opr (-1R) (.R) (opr (cv y) (.R) (0R)))) (cv x) (opr (opr (cv y) (.R) (1R)) (+R) (opr (cv x) (.R) (0R))) (cv y) opeq12 (cv y) 00sr (-1R) (.R) opreq2d m1r (-1R) 00sr ax-mp syl6eq (opr (cv x) (.R) (1R)) (+R) opreq2d (cv x) 1idsr (+R) (0R) opreq1d (cv x) 0idsr eqtrd sylan9eqr (cv x) 00sr (opr (cv y) (.R) (1R)) (+R) opreq2d (cv y) 1idsr (+R) (0R) opreq1d (cv y) 0idsr eqtrd sylan9eq sylanc eqtrd df-1 (<,> (cv x) (cv y)) (x.) opreq2i syl5eq optocl)) thm (axnegex ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (e. A (CC)) (E.e. x (CC) (= (opr A (+) (cv x)) (0)))) (df-c (<,> (cv y) (cv z)) A (+) (cv x) opreq1 (0) eqeq1d (e. (cv x) (CC)) anbi2d x exbidv (cv y) w negexsr (cv z) v negexsr anim12i w v (/\ (e. (cv w) (R.)) (= (opr (cv y) (+R) (cv w)) (0R))) (/\ (e. (cv v) (R.)) (= (opr (cv z) (+R) (cv v)) (0R))) eeanv sylibr (cv w) (cv v) opex (cv x) (<,> (cv w) (cv v)) (CC) eleq1 (cv x) (<,> (cv w) (cv v)) (<,> (cv y) (cv z)) (+) opreq2 (0) eqeq1d anbi12d cla4ev (cv w) (R.) (cv v) (R.) opelxpi df-c syl6eleqr (/\ (e. (cv y) (R.)) (e. (cv z) (R.))) (/\ (= (opr (cv y) (+R) (cv w)) (0R)) (= (opr (cv z) (+R) (cv v)) (0R))) ad2antlr (cv y) (cv z) (cv w) (cv v) addcnsr (opr (cv y) (+R) (cv w)) (0R) (opr (cv z) (+R) (cv v)) (0R) opeq12 df-0 syl6eqr sylan9eq sylanc exp31 imp3a (e. (cv w) (R.)) (= (opr (cv y) (+R) (cv w)) (0R)) (e. (cv v) (R.)) (= (opr (cv z) (+R) (cv v)) (0R)) an4 syl5ib w v 19.23advv mpd optocl x (CC) (= (opr A (+) (cv x)) (0)) df-rex sylibr)) thm (axrecex ((x y) (x z) (A x) (y z) (A y) (A z) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (-> (/\ (e. A (CC)) (=/= A (0))) (E.e. x (CC) (= (opr A (x.) (cv x)) (1)))) (df-c (<,> (cv y) (cv z)) A (0) neeq1 (<,> (cv y) (cv z)) A (x.) (cv x) opreq1 (1) eqeq1d x (CC) rexbidv imbi12d y visset z visset ssgt0sr (opr (cv y) (.R) (cv y)) (+R) (opr (cv z) (.R) (cv z)) oprex w recexsrlem syl6 (<,> (cv y) (cv z)) (0) df-ne df-0 (<,> (cv y) (cv z)) eqeq2i y visset z visset 0r elisseti (0R) opth bitr negbii bitr syl5ib (opr (cv y) (.R) (cv w)) (opr (-1R) (.R) (opr (cv z) (.R) (cv w))) opex (cv x) (<,> (opr (cv y) (.R) (cv w)) (opr (-1R) (.R) (opr (cv z) (.R) (cv w)))) (CC) eleq1 (cv x) (<,> (opr (cv y) (.R) (cv w)) (opr (-1R) (.R) (opr (cv z) (.R) (cv w)))) (<,> (cv y) (cv z)) (x.) opreq2 (1) eqeq1d anbi12d cla4ev (cv y) (cv w) mulclsr (cv z) (cv w) mulclsr m1r jctil (-1R) (opr (cv z) (.R) (cv w)) mulclsr syl anim12i (e. (cv y) (R.)) (e. (cv z) (R.)) (e. (cv w) (R.)) anandir (-1R) (.R) (opr (cv z) (.R) (cv w)) oprex (opr (cv y) (.R) (cv w)) opelcn 3imtr4 (= (opr (opr (opr (cv y) (.R) (cv y)) (+R) (opr (cv z) (.R) (cv z))) (.R) (cv w)) (1R)) adantrr (/\ (e. (cv y) (R.)) (e. (cv z) (R.))) (e. (cv w) (R.)) pm3.26 (cv y) (cv w) mulclsr (cv z) (cv w) mulclsr m1r jctil (-1R) (opr (cv z) (.R) (cv w)) mulclsr syl anim12i anandirs jca (= (opr (opr (opr (cv y) (.R) (cv y)) (+R) (opr (cv z) (.R) (cv z))) (.R) (cv w)) (1R)) adantrr (cv y) (cv z) (opr (cv y) (.R) (cv w)) (opr (-1R) (.R) (opr (cv z) (.R) (cv w))) mulcnsr syl (opr (opr (cv y) (.R) (opr (cv y) (.R) (cv w))) (+R) (opr (-1R) (.R) (opr (cv z) (.R) (opr (-1R) (.R) (opr (cv z) (.R) (cv w)))))) (1R) (opr (opr (cv z) (.R) (opr (cv y) (.R) (cv w))) (+R) (opr (cv y) (.R) (opr (-1R) (.R) (opr (cv z) (.R) (cv w))))) (0R) opeq12 (opr (opr (opr (cv y) (.R) (cv y)) (+R) (opr (cv z) (.R) (cv z))) (.R) (cv w)) (1R) (opr (opr (cv y) (.R) (opr (cv y) (.R) (cv w))) (+R) (opr (-1R) (.R) (opr (cv z) (.R) (opr (-1R) (.R) (opr (cv z) (.R) (cv w)))))) eqeq2 y visset w visset (cv y) mulasssr eqcomi (/\ (e. (cv z) (R.)) (e. (cv w) (R.))) a1i (e. (cv z) (R.)) (e. (cv w) (R.)) pm3.26 (cv z) (cv w) mulclsr jca (cv z) (opr (cv z) (.R) (cv w)) mulclsr (opr (cv z) (.R) (opr (cv z) (.R) (cv w))) 1idsr 3syl m1m1sr (.R) (opr (cv z) (.R) (opr (cv z) (.R) (cv w))) opreq1i z visset m1r elisseti (cv z) (.R) (cv w) oprex v visset u visset mulcomsr u visset x visset (cv v) mulasssr caopr12 (-1R) (.R) opreq2i m1r elisseti (cv z) (.R) (opr (cv z) (.R) (cv w)) oprex (-1R) mulasssr eqtr4 (cv z) (.R) (opr (cv z) (.R) (cv w)) oprex 1r elisseti mulcomsr 3eqtr4 z visset w visset (cv z) mulasssr 3eqtr4g (+R) opreq12d (cv y) (.R) (cv y) oprex (cv z) (.R) (cv z) oprex w visset v visset u visset mulcomsr u visset x visset (cv v) distrsr caoprdistrr syl6eqr syl5bi exp3a com3l imp32 (e. (cv y) (R.)) adantll (cv y) (opr (cv z) (.R) (cv w)) mulclsr (cv z) (cv w) mulclsr sylan2 anassrs (= (opr (opr (opr (cv y) (.R) (cv y)) (+R) (opr (cv z) (.R) (cv z))) (.R) (cv w)) (1R)) adantrr (opr (cv y) (.R) (opr (cv z) (.R) (cv w))) pn0sr z visset y visset w visset v visset u visset mulcomsr u visset x visset (cv v) mulasssr caopr12 y visset m1r elisseti (cv z) (.R) (cv w) oprex v visset u visset mulcomsr u visset x visset (cv v) mulasssr caopr12 m1r elisseti (cv y) (.R) (opr (cv z) (.R) (cv w)) oprex mulcomsr eqtr (+R) opreq12i syl5eq syl sylanc eqtrd df-1 syl6eqr sylanc ex w 19.23adv syld x (CC) (= (opr (<,> (cv y) (cv z)) (x.) (cv x)) (1)) df-rex syl6ibr optocl imp)) thm (axrnegex ((x y) (A x) (A y) (x z) (y z)) () (-> (e. A (RR)) (E.e. x (RR) (= (opr A (+) (cv x)) (0)))) (A y elreal (<,> (cv y) (0R)) A (+) (cv x) opreq1 (0) eqeq1d x (RR) rexbidv (cv y) z negexsr (cv y) (cv z) addresr (0) eqeq1d df-0 (<,> (opr (cv y) (+R) (cv z)) (0R)) eqeq2i (cv y) (+R) (cv z) oprex (0R) eqresr bitr syl6bb ex pm5.32d (cv z) (0R) opex (cv x) (<,> (cv z) (0R)) (RR) eleq1 (cv x) (<,> (cv z) (0R)) (<,> (cv y) (0R)) (+) opreq2 (0) eqeq1d anbi12d cla4ev (cv z) opelreal sylanbr x (RR) (= (opr (<,> (cv y) (0R)) (+) (cv x)) (0)) df-rex sylibr syl6bir z 19.23adv mpd gencl)) thm (axrrecex ((x y) (A x) (A y) (x z) (y z)) () (-> (/\ (e. A (RR)) (=/= A (0))) (E.e. x (RR) (= (opr A (x.) (cv x)) (1)))) (A y elreal (<,> (cv y) (0R)) A (0) neeq1 (<,> (cv y) (0R)) A (x.) (cv x) opreq1 (1) eqeq1d x (RR) rexbidv imbi12d y visset z recexsr z visset (cv y) mulresr (1) eqeq1d df-1 (<,> (opr (cv y) (.R) (cv z)) (0R)) eqeq2i (cv y) (.R) (cv z) oprex (1R) eqresr bitr syl6bb ex pm5.32d (cv z) opelreal (= (opr (<,> (cv y) (0R)) (x.) (<,> (cv z) (0R))) (1)) anbi1i syl5bb (cv z) (0R) opex (cv x) (<,> (cv z) (0R)) (RR) eleq1 (cv x) (<,> (cv z) (0R)) (<,> (cv y) (0R)) (x.) opreq2 (1) eqeq1d anbi12d cla4ev syl6bir z 19.23adv syld df-0 (<,> (cv y) (0R)) eqeq2i y visset (0R) eqresr bitr negbii syl5ib (<,> (cv y) (0R)) (0) df-ne x (RR) (= (opr (<,> (cv y) (0R)) (x.) (cv x)) (1)) df-rex 3imtr4g gencl imp)) thm (axi2m1 () () (= (opr (opr (i) (x.) (i)) (+) (1)) (0)) (0r 1r pm3.2i 0r 1r pm3.2i (0R) (1R) (0R) (1R) mulcnsr mp2an (opr (0R) (.R) (0R)) (+R) (opr (-1R) (.R) (opr (1R) (.R) (1R))) oprex (opr (1R) (.R) (0R)) (+R) (opr (0R) (.R) (1R)) oprex 0r elisseti (-1R) opth 0r (0R) 00sr ax-mp 1r (1R) 1idsr ax-mp (-1R) (.R) opreq2i m1r (-1R) 1idsr ax-mp eqtr (+R) opreq12i 0r elisseti m1r elisseti addcomsr m1r (-1R) 0idsr ax-mp 3eqtr 1r (1R) 00sr ax-mp 0r (0R) 1idsr ax-mp (+R) opreq12i 0r (0R) 0idsr ax-mp eqtr mpbir2an eqtr (+) (<,> (1R) (0R)) opreq1i m1r 1r (-1R) (1R) addresr mp2an m1p1sr (-1R) (+R) (1R) oprex (0R) eqresr mpbir 3eqtr df-i df-i (x.) opreq12i df-1 (+) opreq12i df-0 3eqtr4)) thm (pre-axlttri () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A ( (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\ (br A ( (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (br A ( (cv x) (0R)) A ( (cv y) (0R)) breq1 (<,> (cv x) (0R)) A (<,> (cv z) (0R)) (+) opreq2 ( (cv z) (0R)) (+) (<,> (cv y) (0R))) breq1d bibi12d (<,> (cv y) (0R)) B A ( (cv y) (0R)) B (<,> (cv z) (0R)) (+) opreq2 (opr (<,> (cv z) (0R)) (+) A) ( (cv z) (0R)) C (+) A opreq1 (<,> (cv z) (0R)) C (+) B opreq1 ( (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) ( (cv x) (0R)) A (0) ( (cv y) (0R))) anbi1d (<,> (cv x) (0R)) A (x.) (<,> (cv y) (0R)) opreq1 (0) ( (cv y) (0R)) B (0) ( (cv y) (0R)) B A (x.) opreq2 (0) ( (cv x) (0R)) breq1i 0r elisseti x visset ltresr bitr df-0 ( (cv y) (0R)) breq1i 0r elisseti y visset ltresr bitr syl2anb syl5bir 2gencl)) thm (axcnre ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (E.e. x (RR) (E.e. y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))))) (df-c (<,> (cv z) (cv w)) A (opr (cv x) (+) (opr (cv y) (x.) (i))) eqeq1 x (RR) y (RR) 2rexbidv (cv z) (0R) opex (cv w) (0R) opex (cv x) (<,> (cv z) (0R)) (RR) eleq1 (cv y) (<,> (cv w) (0R)) (RR) eleq1 bi2anan9 (cv x) (<,> (cv z) (0R)) (+) (opr (cv y) (x.) (i)) opreq1 (cv y) (<,> (cv w) (0R)) (x.) (i) opreq1 (<,> (cv z) (0R)) (+) opreq2d sylan9eq (<,> (cv z) (cv w)) eqeq2d anbi12d cla4e2v (cv z) opelreal (cv w) opelreal anbi12i biimpr 0r 0r 1r pm3.2i (cv w) (0R) (0R) (1R) mulcnsr mpan2 mpan2 (opr (opr (cv w) (.R) (0R)) (+R) (opr (-1R) (.R) (opr (0R) (.R) (1R)))) (0R) (opr (opr (0R) (.R) (0R)) (+R) (opr (cv w) (.R) (1R))) (cv w) opeq12 (cv w) 00sr (+R) (opr (-1R) (.R) (opr (0R) (.R) (1R))) opreq1d 0r (0R) 1idsr ax-mp (-1R) (.R) opreq2i m1r (-1R) 00sr ax-mp eqtr (0R) (+R) opreq2i 0r (0R) 0idsr ax-mp eqtr syl6eq (cv w) 1idsr (opr (0R) (.R) (0R)) (+R) opreq2d (cv w) 0idsr 0r (0R) 00sr ax-mp (+R) (cv w) opreq1i 0r elisseti w visset addcomsr eqtr syl5eq eqtrd sylanc eqtrd df-i (<,> (cv w) (0R)) (x.) opreq2i syl5eq (<,> (cv z) (0R)) (+) opreq2d (e. (cv z) (R.)) adantl 0r 0r (cv z) (0R) (0R) (cv w) addcnsr mpanl2 mpanr1 (opr (cv z) (+R) (0R)) (cv z) (opr (0R) (+R) (cv w)) (cv w) opeq12 (cv z) 0idsr (cv w) 0idsr 0r elisseti w visset addcomsr syl5eq syl2an 3eqtrrd sylanc x (RR) y (RR) (= (<,> (cv z) (cv w)) (opr (cv x) (+) (opr (cv y) (x.) (i)))) r2ex sylibr optocl)) thm (pre-axsup ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (/\/\ (C_ A (RR)) (=/= A ({/})) (E.e. x (RR) (A.e. y A (br (cv y) ( (br (cv y) ( (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (e. (cv y) (RR)) (-> (e. (cv y) A) (-. (br (cv x) ( (e. (cv y) (RR)) (-> (br (cv y) ( (br (cv y) ( (e. (cv y) A) (br (cv y) ( (e. A (RR)) (e. A (CC))) (axresscn A sseli)) thm (recn () ((recn.1 (e. A (RR)))) (e. A (CC)) (axresscn recn.1 sselii)) thm (recnd () ((recnd.1 (-> ph (e. A (RR))))) (-> ph (e. A (CC))) (recnd.1 A recnt syl)) thm (elimne0 () () (=/= (if (=/= A (0)) A (1)) (0)) (A (if (=/= A (0)) A (1)) (0) neeq1 (1) (if (=/= A (0)) A (1)) (0) neeq1 ax1ne0 elimhyp)) thm (addex () () (e. (+) (V)) (axaddopr axcnex axcnex xpex (+) (X. (CC) (CC)) (CC) (V) fex mp2an)) thm (mulex () () (e. (x.) (V)) (axmulopr axcnex axcnex xpex (x.) (X. (CC) (CC)) (CC) (V) fex mp2an)) thm (adddirt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (x.) C) (opr (opr A (x.) C) (+) (opr B (x.) C)))) (C A B axdistr 3coml (opr A (+) B) C axmulcom A B axaddcl sylan 3impa A C axmulcom (e. B (CC)) 3adant2 B C axmulcom (e. A (CC)) 3adant1 (+) opreq12d 3eqtr4d)) thm (addcl () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC)))) (e. (opr A (+) B) (CC)) (axi.1 axi.2 A B axaddcl mp2an)) thm (mulcl () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC)))) (e. (opr A (x.) B) (CC)) (axi.1 axi.2 A B axmulcl mp2an)) thm (addcom () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC)))) (= (opr A (+) B) (opr B (+) A)) (axi.1 axi.2 A B axaddcom mp2an)) thm (mulcom () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC)))) (= (opr A (x.) B) (opr B (x.) A)) (axi.1 axi.2 A B axmulcom mp2an)) thm (addass () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC))) (axi.3 (e. C (CC)))) (= (opr (opr A (+) B) (+) C) (opr A (+) (opr B (+) C))) (axi.1 axi.2 axi.3 A B C axaddass mp3an)) thm (mulass () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC))) (axi.3 (e. C (CC)))) (= (opr (opr A (x.) B) (x.) C) (opr A (x.) (opr B (x.) C))) (axi.1 axi.2 axi.3 A B C axmulass mp3an)) thm (adddi () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC))) (axi.3 (e. C (CC)))) (= (opr A (x.) (opr B (+) C)) (opr (opr A (x.) B) (+) (opr A (x.) C))) (axi.1 axi.2 axi.3 A B C axdistr mp3an)) thm (adddir () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC))) (axi.3 (e. C (CC)))) (= (opr (opr A (+) B) (x.) C) (opr (opr A (x.) C) (+) (opr B (x.) C))) (axi.1 axi.2 axi.3 A B C adddirt mp3an)) thm (1cn () () (e. (1) (CC)) (ax1re recn)) thm (0cn () () (e. (0) (CC)) (axi2m1 axicn axicn mulcl 1cn addcl eqeltrr)) thm (addid1 () ((addid1.1 (e. A (CC)))) (= (opr A (+) (0)) A) (addid1.1 A ax0id ax-mp)) thm (addid2 () ((addid1.1 (e. A (CC)))) (= (opr (0) (+) A) A) (0cn addid1.1 addcom addid1.1 addid1 eqtr)) thm (mulid1 () ((addid1.1 (e. A (CC)))) (= (opr A (x.) (1)) A) (addid1.1 A ax1id ax-mp)) thm (mulid2 () ((addid1.1 (e. A (CC)))) (= (opr (1) (x.) A) A) (1cn addid1.1 mulcom addid1.1 mulid1 eqtr)) thm (negex ((A x)) ((addid1.1 (e. A (CC)))) (E.e. x (CC) (= (opr A (+) (cv x)) (0))) (addid1.1 A x axnegex ax-mp)) thm (recex ((A x)) ((addid1.1 (e. A (CC))) (recex.2 (=/= A (0)))) (E.e. x (CC) (= (opr A (x.) (cv x)) (1))) (addid1.1 recex.2 A x axrecex mp2an)) thm (readdcl () ((axri.1 (e. A (RR))) (axri.2 (e. B (RR)))) (e. (opr A (+) B) (RR)) (axri.1 axri.2 A B axaddrcl mp2an)) thm (remulcl () ((axri.1 (e. A (RR))) (axri.2 (e. B (RR)))) (e. (opr A (x.) B) (RR)) (axri.1 axri.2 A B axmulrcl mp2an)) thm (addcan ((A x) (B x) (C x)) ((addcan.1 (e. A (CC))) (addcan.2 (e. B (CC))) (addcan.3 (e. C (CC)))) (<-> (= (opr A (+) B) (opr A (+) C)) (= B C)) (addcan.1 x negex addcan.1 addcan.2 (cv x) A B axaddass mp3an3 addcan.3 (cv x) A C axaddass mp3an3 eqeq12d mpan2 (opr A (+) B) (opr A (+) C) (cv x) (+) opreq2 syl5bir (= (opr A (+) (cv x)) (0)) adantr addcan.1 A (cv x) axaddcom mpan (0) eqeq1d (opr (cv x) (+) A) (0) (+) B opreq1 addcan.2 addid2 syl6eq (opr (cv x) (+) A) (0) (+) C opreq1 addcan.3 addid2 syl6eq eqeq12d syl6bi imp sylibd ex r19.23aiv ax-mp B C A (+) opreq2 impbi)) thm (addcan2 () ((addcan.1 (e. A (CC))) (addcan.2 (e. B (CC))) (addcan.3 (e. C (CC)))) (<-> (= (opr A (+) C) (opr B (+) C)) (= A B)) (addcan.1 addcan.3 addcom addcan.2 addcan.3 addcom eqeq12i addcan.3 addcan.1 addcan.2 addcan bitr)) thm (addcant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (+) B) (opr A (+) C)) (= B C))) (A (if (e. A (CC)) A (0)) (+) B opreq1 A (if (e. A (CC)) A (0)) (+) C opreq1 eqeq12d (= B C) bibi1d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (+) opreq2 (opr (if (e. A (CC)) A (0)) (+) C) eqeq1d B (if (e. B (CC)) B (0)) C eqeq1 bibi12d C (if (e. C (CC)) C (0)) (if (e. A (CC)) A (0)) (+) opreq2 (opr (if (e. A (CC)) A (0)) (+) (if (e. B (CC)) B (0))) eqeq2d C (if (e. C (CC)) C (0)) (if (e. B (CC)) B (0)) eqeq2 bibi12d 0cn A elimel 0cn B elimel 0cn C elimel addcan dedth3h)) thm (addcan2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (+) C) (opr B (+) C)) (= A B))) (C A axaddcom (e. B (CC)) 3adant3 C B axaddcom (e. A (CC)) 3adant2 eqeq12d C A B addcant bitr3d 3coml)) thm (add12t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (+) (opr B (+) C)) (opr B (+) (opr A (+) C)))) (A B axaddcom (+) C opreq1d (e. C (CC)) 3adant3 A B C axaddass B A C axaddass 3com12 3eqtr3d)) thm (add23t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (+) C) (opr (opr A (+) C) (+) B))) (B C axaddcom A (+) opreq2d (e. A (CC)) 3adant1 A B C axaddass A C B axaddass 3com23 3eqtr4d)) thm (add4t () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (= (opr (opr A (+) B) (+) (opr C (+) D)) (opr (opr A (+) C) (+) (opr B (+) D)))) (A B C add23t (+) D opreq1d 3expa (e. D (CC)) adantrr (opr A (+) B) C D axaddass 3expb A B axaddcl sylan (opr A (+) C) B D axaddass 3expb A C axaddcl sylan an4s 3eqtr3d)) thm (add42t () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (= (opr (opr A (+) B) (+) (opr C (+) D)) (opr (opr A (+) C) (+) (opr D (+) B)))) (A B C D add4t B D axaddcom (e. A (CC)) (e. C (CC)) ad2ant2l (opr A (+) C) (+) opreq2d eqtrd)) thm (add12 () ((add.1 (e. A (CC))) (add.2 (e. B (CC))) (add.3 (e. C (CC)))) (= (opr A (+) (opr B (+) C)) (opr B (+) (opr A (+) C))) (add.1 add.2 add.3 A B C add12t mp3an)) thm (add23 () ((add.1 (e. A (CC))) (add.2 (e. B (CC))) (add.3 (e. C (CC)))) (= (opr (opr A (+) B) (+) C) (opr (opr A (+) C) (+) B)) (add.1 add.2 add.3 A B C add23t mp3an)) thm (add4 () ((add4.1 (e. A (CC))) (add4.2 (e. B (CC))) (add4.3 (e. C (CC))) (add4.4 (e. D (CC)))) (= (opr (opr A (+) B) (+) (opr C (+) D)) (opr (opr A (+) C) (+) (opr B (+) D))) (add4.1 add4.2 pm3.2i add4.3 add4.4 pm3.2i A B C D add4t mp2an)) thm (add42 () ((add4.1 (e. A (CC))) (add4.2 (e. B (CC))) (add4.3 (e. C (CC))) (add4.4 (e. D (CC)))) (= (opr (opr A (+) B) (+) (opr C (+) D)) (opr (opr A (+) C) (+) (opr D (+) B))) (add4.1 add4.2 add4.3 add4.4 add4 add4.2 add4.4 addcom (opr A (+) C) (+) opreq2i eqtr)) thm (addid2t () () (-> (e. A (CC)) (= (opr (0) (+) A) A)) (0cn (0) A axaddcom mpan A ax0id eqtrd)) thm (peano2cn () () (-> (e. A (CC)) (e. (opr A (+) (1)) (CC))) (1cn A (1) axaddcl mpan2)) thm (peano2re () () (-> (e. A (RR)) (e. (opr A (+) (1)) (RR))) (ax1re A (1) axaddrcl mpan2)) thm (negeu ((x y) (A x) (A y) (B x) (B y)) ((negeu.1 (e. A (CC))) (negeu.2 (e. B (CC)))) (E!e. x (CC) (= (opr A (+) (cv x)) B)) ((cv x) (cv y) A (+) opreq2 B eqeq1d (CC) reu4 negeu.1 y negex negeu.2 (cv y) B axaddcl mpan2 (opr (cv y) (+) B) (CC) x risset sylib (cv x) (opr (cv y) (+) B) A (+) opreq2 negeu.1 negeu.2 A (cv y) B axaddass mp3an13 eqcomd sylan9eqr (opr A (+) (cv y)) (0) (+) B opreq1 negeu.2 addid2 syl6eq sylan9eqr exp32 impcom (e. (cv x) (CC)) a1d r19.21aiv ex x (CC) (= (cv x) (opr (cv y) (+) B)) (= (opr A (+) (cv x)) B) r19.22 syl6 mpid r19.23aiv ax-mp negeu.1 A (cv x) (cv y) addcant (opr A (+) (cv x)) B (opr A (+) (cv y)) eqtr3t syl5bi mp3an1 rgen2 mpbir2an)) thm (subval ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((subval.1 (e. A (CC))) (subval.2 (e. B (CC)))) (= (opr A (-) B) (U. ({e.|} x (CC) (= (opr B (+) (cv x)) A)))) (subval.1 subval.2 axcnex x (= (opr B (+) (cv x)) A) rabex uniex (cv y) A (opr (cv z) (+) (cv x)) eqeq2 x (CC) rabbisdv unieqd (cv z) B (+) (cv x) opreq1 A eqeq1d x (CC) rabbisdv unieqd y z w x df-sub oprabval2 mp2an)) thm (negeq () () (-> (= A B) (= (-u A) (-u B))) (A B (0) (-) opreq2 A df-neg B df-neg 3eqtr4g)) thm (negeqi () ((negeqi.1 (= A B))) (= (-u A) (-u B)) (negeqi.1 A B negeq ax-mp)) thm (negeqd () ((negeqd.1 (-> ph (= A B)))) (-> ph (= (-u A) (-u B))) (negeqd.1 A B negeq syl)) thm (hbneg ((A y) (x y)) ((hbneg.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (-u A)) (A. x (e. (cv y) (-u A)))) ((e. (cv y) (0)) x ax-17 (e. (cv y) (-)) x ax-17 hbneg.1 hbopr A df-neg (cv y) eleq2i A df-neg (cv y) eleq2i x albii 3imtr4)) thm (minusex () () (e. (-u A) (V)) (A df-neg (0) (-) A oprex eqeltr)) thm (subcl ((A x) (B x)) ((subcl.1 (e. A (CC))) (subcl.2 (e. B (CC)))) (e. (opr A (-) B) (CC)) (subcl.1 subcl.2 x subval subcl.2 subcl.1 x negeu x (CC) (= (opr B (+) (cv x)) A) reucl ax-mp eqeltr)) thm (subclt () () (-> (/\ (e. A (CC)) (e. B (CC))) (e. (opr A (-) B) (CC))) (A (if (e. A (CC)) A (0)) (-) B opreq1 (CC) eleq1d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (-) opreq2 (CC) eleq1d 0cn A elimel 0cn B elimel subcl dedth2h)) thm (negclt () () (-> (e. A (CC)) (e. (-u A) (CC))) (0cn (0) A subclt mpan A df-neg syl5eqel)) thm (negcl () ((negcl.1 (e. A (CC)))) (e. (-u A) (CC)) (negcl.1 A negclt ax-mp)) thm (subadd ((A x) (B x) (C x)) ((subadd.1 (e. A (CC))) (subadd.2 (e. B (CC))) (subadd.3 (e. C (CC)))) (<-> (= (opr A (-) B) C) (= (opr B (+) C) A)) (subadd.3 (cv x) C (opr A (-) B) eqeq2 (cv x) C B (+) opreq2 A eqeq1d bibi12d subadd.2 subadd.1 x negeu x (CC) (= (opr B (+) (cv x)) A) reuuni1 mpan2 subadd.1 subadd.2 x subval (cv x) eqeq1i syl6rbbr vtoclga ax-mp)) thm (subsub23 () ((subadd.1 (e. A (CC))) (subadd.2 (e. B (CC))) (subadd.3 (e. C (CC)))) (<-> (= (opr A (-) B) C) (= (opr A (-) C) B)) (subadd.2 subadd.3 addcom A eqeq1i subadd.1 subadd.2 subadd.3 subadd subadd.1 subadd.3 subadd.2 subadd 3bitr4)) thm (subaddt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (-) B) C) (= (opr B (+) C) A))) (A (if (e. A (CC)) A (0)) (-) B opreq1 C eqeq1d A (if (e. A (CC)) A (0)) (opr B (+) C) eqeq2 bibi12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (-) opreq2 C eqeq1d B (if (e. B (CC)) B (0)) (+) C opreq1 (if (e. A (CC)) A (0)) eqeq1d bibi12d C (if (e. C (CC)) C (0)) (opr (if (e. A (CC)) A (0)) (-) (if (e. B (CC)) B (0))) eqeq2 C (if (e. C (CC)) C (0)) (if (e. B (CC)) B (0)) (+) opreq2 (if (e. A (CC)) A (0)) eqeq1d bibi12d 0cn A elimel 0cn B elimel 0cn C elimel subadd dedth3h)) thm (pncan3t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (+) (opr B (-) A)) B)) ((opr B (-) A) eqid B A (opr B (-) A) subaddt (e. A (CC)) (e. B (CC)) pm3.27 (e. A (CC)) (e. B (CC)) pm3.26 B A subclt ancoms syl3anc mpbii)) thm (pncan3 () ((subaddeq.1 (e. A (CC))) (subaddeq.2 (e. B (CC)))) (= (opr A (+) (opr B (-) A)) B) (subaddeq.1 subaddeq.2 A B pncan3t mp2an)) thm (negidt () () (-> (e. A (CC)) (= (opr A (+) (-u A)) (0))) (0cn A (0) pncan3t mpan2 A df-neg A (+) opreq2i syl5eq)) thm (negid () ((negid.1 (e. A (CC)))) (= (opr A (+) (-u A)) (0)) (negid.1 A negidt ax-mp)) thm (negsub () ((negsub.1 (e. A (CC))) (negsub.2 (e. B (CC)))) (= (opr A (+) (-u B)) (opr A (-) B)) (negsub.2 negsub.1 negsub.2 negcl addcl addcom negsub.1 negsub.2 negcl negsub.2 addass negsub.2 negcl negsub.2 addcom negsub.2 negid eqtr A (+) opreq2i negsub.1 addid1 3eqtr eqtr negsub.1 negsub.2 negsub.1 negsub.2 negcl addcl subadd mpbir eqcomi)) thm (negsubt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (+) (-u B)) (opr A (-) B))) (A (if (e. A (CC)) A (0)) (+) (-u B) opreq1 A (if (e. A (CC)) A (0)) (-) B opreq1 eqeq12d B (if (e. B (CC)) B (0)) negeq (if (e. A (CC)) A (0)) (+) opreq2d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (-) opreq2 eqeq12d 0cn A elimel 0cn B elimel negsub dedth2h)) thm (addsubasst () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (-) C) (opr A (+) (opr B (-) C)))) (A B (-u C) axaddass C negclt syl3an3 (opr A (+) B) C negsubt A B axaddcl sylan 3impa B C negsubt (e. A (CC)) 3adant1 A (+) opreq2d 3eqtr3d)) thm (addsubt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (-) C) (opr (opr A (-) C) (+) B))) (A B axaddcom (-) C opreq1d (e. C (CC)) 3adant3 B A C addsubasst 3com12 B (opr A (-) C) axaddcom ex A C subclt syl5 exp3a com12 3imp 3eqtrd)) thm (addsub12t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (+) (opr B (-) C)) (opr B (+) (opr A (-) C)))) (A B axaddcom (-) C opreq1d (e. C (CC)) 3adant3 A B C addsubasst B A C addsubasst 3com12 3eqtr3d)) thm (addsubass () ((addsubass.1 (e. A (CC))) (addsubass.2 (e. B (CC))) (addsubass.3 (e. C (CC)))) (= (opr (opr A (+) B) (-) C) (opr A (+) (opr B (-) C))) (addsubass.1 addsubass.2 addsubass.3 A B C addsubasst mp3an)) thm (addsub () ((addsubass.1 (e. A (CC))) (addsubass.2 (e. B (CC))) (addsubass.3 (e. C (CC)))) (= (opr (opr A (+) B) (-) C) (opr (opr A (-) C) (+) B)) (addsubass.1 addsubass.2 addsubass.3 A B C addsubt mp3an)) thm (2addsubt () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (= (opr (opr (opr A (+) B) (+) C) (-) D) (opr (opr (opr A (+) C) (-) D) (+) B))) (A B C add23t 3expa (e. D (CC)) adantrr (-) D opreq1d (opr A (+) C) B D addsubt 3expb A C axaddcl sylan an4s eqtrd)) thm (negneg () ((negneg.1 (e. A (CC)))) (= (-u (-u A)) A) (negneg.1 negcl negid A (+) opreq2i negneg.1 negid (+) (-u (-u A)) opreq1i negneg.1 negneg.1 negcl negneg.1 negcl negcl addass negneg.1 negcl negcl addid2 3eqtr3 negneg.1 addid1 3eqtr3)) thm (subid () ((negneg.1 (e. A (CC)))) (= (opr A (-) A) (0)) (negneg.1 negneg.1 negsub negneg.1 negid eqtr3)) thm (subid1 () ((negneg.1 (e. A (CC)))) (= (opr A (-) (0)) A) (negneg.1 addid2 negneg.1 0cn negneg.1 subadd mpbir)) thm (negnegt () () (-> (e. A (CC)) (= (-u (-u A)) A)) (A (if (e. A (CC)) A (0)) negeq negeqd (= A (if (e. A (CC)) A (0))) id eqeq12d 0cn A elimel negneg dedth)) thm (subnegt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (-) (-u B)) (opr A (+) B))) (A (-u B) negsubt B negclt sylan2 B negnegt A (+) opreq2d (e. A (CC)) adantl eqtr3d)) thm (subidt () () (-> (e. A (CC)) (= (opr A (-) A) (0))) (A (if (e. A (CC)) A (0)) A (if (e. A (CC)) A (0)) (-) opreq12 anidms (0) eqeq1d 0cn A elimel subid dedth)) thm (subid1t () () (-> (e. A (CC)) (= (opr A (-) (0)) A)) (A (if (e. A (CC)) A (0)) (-) (0) opreq1 (= A (if (e. A (CC)) A (0))) id eqeq12d 0cn A elimel subid1 dedth)) thm (pncant () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr A (+) B) (-) B) A)) (A B B addsubasst 3expb anabsan2 B subidt A (+) opreq2d A ax0id sylan9eqr eqtrd)) thm (pncan2t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr A (+) B) (-) A) B)) (B A axaddcom (-) A opreq1d B A pncant eqtr3d ancoms)) thm (npcant () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr A (-) B) (+) B) A)) (A B B addsubt 3expb anabsan2 A B pncant eqtr3d)) thm (npncant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) B) (+) (opr B (-) C)) (opr A (-) C))) ((opr A (-) B) B C addsubasst A B subclt (e. C (CC)) 3adant3 (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp2 (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp3 syl3anc A B npcant (-) C opreq1d (e. C (CC)) 3adant3 eqtr3d)) thm (nppcant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr (opr A (-) B) (+) C) (+) B) (opr A (+) C))) ((opr A (-) B) C B add23t A B subclt (e. C (CC)) 3adant3 (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp3 (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp2 syl3anc A B npcant (+) C opreq1d (e. C (CC)) 3adant3 eqtrd)) thm (subneg () ((subneg.1 (e. A (CC))) (subneg.2 (e. B (CC)))) (= (opr A (-) (-u B)) (opr A (+) B)) (subneg.1 subneg.2 A B subnegt mp2an)) thm (subeq0 () ((subneg.1 (e. A (CC))) (subneg.2 (e. B (CC)))) (<-> (= (opr A (-) B) (0)) (= A B)) (subneg.1 subneg.2 negsub (0) eqeq1i (opr A (+) (-u B)) (0) (+) B opreq1 sylbir subneg.1 subneg.2 negcl subneg.2 add23 subneg.1 subneg.2 subneg.2 negcl addass subneg.2 negid A (+) opreq2i subneg.1 addid1 eqtr 3eqtr subneg.2 addid2 3eqtr3g A B (-) B opreq1 subneg.2 subid syl6eq impbi)) thm (neg11 () ((subneg.1 (e. A (CC))) (subneg.2 (e. B (CC)))) (<-> (= (-u A) (-u B)) (= A B)) (A df-neg B df-neg eqeq12i 0cn subneg.1 0cn subneg.2 subcl subadd B df-neg A (+) opreq2i subneg.1 subneg.2 negsub eqtr3 (0) eqeq1i subneg.1 subneg.2 subeq0 bitr 3bitr)) thm (negcon1 () ((subneg.1 (e. A (CC))) (subneg.2 (e. B (CC)))) (<-> (= (-u A) B) (= (-u B) A)) (subneg.1 negneg (-u B) eqeq1i subneg.1 negcl subneg.2 neg11 A (-u B) eqcom 3bitr3)) thm (negcon2 () ((subneg.1 (e. A (CC))) (subneg.2 (e. B (CC)))) (<-> (= A (-u B)) (= B (-u A))) (subneg.2 subneg.1 negcon1 A (-u B) eqcom B (-u A) eqcom 3bitr4)) thm (neg11t () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= (-u A) (-u B)) (= A B))) (A (if (e. A (CC)) A (0)) negeq (-u B) eqeq1d A (if (e. A (CC)) A (0)) B eqeq1 bibi12d B (if (e. B (CC)) B (0)) negeq (-u (if (e. A (CC)) A (0))) eqeq2d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) eqeq2 bibi12d 0cn A elimel 0cn B elimel neg11 dedth2h)) thm (negcon1t () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= (-u A) B) (= (-u B) A))) ((-u A) B neg11t A negclt sylan A negnegt (e. B (CC)) adantr (-u B) eqeq1d bitr3d A (-u B) eqcom syl6bb)) thm (negcon2t () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= A (-u B)) (= B (-u A)))) (A B negcon1t A (-u B) eqcom syl6rbbr (-u A) B eqcom syl6bb)) thm (subcant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (-) B) (opr A (-) C)) (= B C))) (A (-u B) (-u C) addcant C negclt syl3an3 B negclt syl3an2 A B negsubt (e. C (CC)) 3adant3 A C negsubt (e. B (CC)) 3adant2 eqeq12d B C neg11t (e. A (CC)) 3adant1 3bitr3d)) thm (subcan2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (-) C) (opr B (-) C)) (= A B))) (A C negsubt (e. B (CC)) 3adant2 B C negsubt (e. A (CC)) 3adant1 eqeq12d A B (-u C) addcan2t C negclt syl3an3 bitr3d)) thm (subcan () ((subcan.1 (e. A (CC))) (subcan.2 (e. B (CC))) (subcan.3 (e. C (CC)))) (<-> (= (opr A (-) B) (opr A (-) C)) (= B C)) (subcan.1 subcan.2 subcan.3 A B C subcant mp3an)) thm (subcan2 () ((subcan.1 (e. A (CC))) (subcan.2 (e. B (CC))) (subcan.3 (e. C (CC)))) (<-> (= (opr A (-) C) (opr B (-) C)) (= A B)) (subcan.1 subcan.2 subcan.3 A B C subcan2t mp3an)) thm (subeq0t () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= (opr A (-) B) (0)) (= A B))) (A (if (e. A (CC)) A (0)) (-) B opreq1 (0) eqeq1d A (if (e. A (CC)) A (0)) B eqeq1 bibi12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (-) opreq2 (0) eqeq1d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) eqeq2 bibi12d 0cn A elimel 0cn B elimel subeq0 dedth2h)) thm (neg0 () () (= (-u (0)) (0)) ((0) df-neg 0cn subid eqtr)) thm (renegcl ((A x)) ((renegcl.1 (e. A (RR)))) (e. (-u A) (RR)) (renegcl.1 A x axrnegex ax-mp x (RR) (= (opr A (+) (cv x)) (0)) df-rex mpbi (cv x) recnt 0cn renegcl.1 recn (0) A (cv x) subaddt mp3an12 syl A df-neg (cv x) eqeq1i syl5bb (cv x) (RR) (-u A) eleq1a sylbird imp x 19.23aiv ax-mp)) thm (renegclt () () (-> (e. A (RR)) (e. (-u A) (RR))) (A (if (e. A (RR)) A (1)) negeq (RR) eleq1d ax1re A elimel renegcl dedth)) thm (resubclt () () (-> (/\ (e. A (RR)) (e. B (RR))) (e. (opr A (-) B) (RR))) (A B negsubt A recnt B recnt syl2an A (-u B) axaddrcl B renegclt sylan2 eqeltrrd)) thm (resubcl () ((resubcl.1 (e. A (RR))) (resubcl.2 (e. B (RR)))) (e. (opr A (-) B) (RR)) (resubcl.1 resubcl.2 A B resubclt mp2an)) thm (0re () () (e. (0) (RR)) (1cn subid ax1re ax1re resubcl eqeltrr)) thm (mulid2t () () (-> (e. A (CC)) (= (opr (1) (x.) A) A)) (1cn (1) A axmulcom mpan A ax1id eqtrd)) thm (mul12t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (x.) (opr B (x.) C)) (opr B (x.) (opr A (x.) C)))) (A B axmulcom (x.) C opreq1d (e. C (CC)) 3adant3 A B C axmulass B A C axmulass 3com12 3eqtr3d)) thm (mul23t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (x.) B) (x.) C) (opr (opr A (x.) C) (x.) B))) (B C axmulcom A (x.) opreq2d (e. A (CC)) 3adant1 A B C axmulass A C B axmulass 3com23 3eqtr4d)) thm (mul4t () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (= (opr (opr A (x.) B) (x.) (opr C (x.) D)) (opr (opr A (x.) C) (x.) (opr B (x.) D)))) (A B C mul23t (x.) D opreq1d 3expa (e. D (CC)) adantrr (opr A (x.) B) C D axmulass 3expb A B axmulcl sylan (opr A (x.) C) B D axmulass 3expb A C axmulcl sylan an4s 3eqtr3d)) thm (muladdt () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (= (opr (opr A (+) B) (x.) (opr C (+) D)) (opr (opr (opr A (x.) C) (+) (opr D (x.) B)) (+) (opr (opr A (x.) D) (+) (opr C (x.) B))))) ((opr A (+) B) C D axdistr A B axaddcl (/\ (e. C (CC)) (e. D (CC))) adantr (e. C (CC)) (e. D (CC)) pm3.26 (/\ (e. A (CC)) (e. B (CC))) adantl (e. C (CC)) (e. D (CC)) pm3.27 (/\ (e. A (CC)) (e. B (CC))) adantl syl3anc A B C adddirt 3expa (e. D (CC)) adantrr A B D adddirt 3expa (e. C (CC)) adantrl (+) opreq12d (opr A (x.) C) (opr B (x.) C) (opr (opr A (x.) D) (+) (opr B (x.) D)) add23t A C axmulcl (e. B (CC)) (e. D (CC)) ad2ant2r B C axmulcl (e. D (CC)) adantrr (e. A (CC)) adantll (opr A (x.) D) (opr B (x.) D) axaddcl A D axmulcl B D axmulcl syl2an anandirs (e. C (CC)) adantrl syl3anc B D axmulcom (e. A (CC)) (e. C (CC)) ad2ant2l (opr (opr A (x.) C) (+) (opr A (x.) D)) (+) opreq2d (opr A (x.) C) (opr A (x.) D) (opr B (x.) D) axaddass A C axmulcl (e. B (CC)) (e. D (CC)) ad2ant2r A D axmulcl (e. C (CC)) adantrl (e. B (CC)) adantlr B D axmulcl (e. A (CC)) (e. C (CC)) ad2ant2l syl3anc (opr A (x.) C) (opr A (x.) D) (opr D (x.) B) add23t A C axmulcl (e. B (CC)) (e. D (CC)) ad2ant2r A D axmulcl (e. C (CC)) adantrl (e. B (CC)) adantlr D B axmulcl ancoms (e. A (CC)) (e. C (CC)) ad2ant2l syl3anc 3eqtr3d B C axmulcom (/\ (e. A (CC)) (e. D (CC))) adantl an42s (+) opreq12d (opr (opr A (x.) C) (+) (opr D (x.) B)) (opr A (x.) D) (opr C (x.) B) axaddass (opr A (x.) C) (opr D (x.) B) axaddcl A C axmulcl D B axmulcl ancoms syl2an an4s A D axmulcl (e. C (CC)) adantrl (e. B (CC)) adantlr C B axmulcl ancoms (/\ (e. A (CC)) (e. D (CC))) adantl an42s syl3anc 3eqtrd 3eqtrd)) thm (muladd11t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr (1) (+) A) (x.) (opr (1) (+) B)) (opr (opr (1) (+) A) (+) (opr B (+) (opr A (x.) B))))) (1cn (opr (1) (+) A) (1) B axdistr mp3an2 1cn (1) A axaddcl mpan sylan 1cn (1) A axaddcl mpan (opr (1) (+) A) ax1id syl (e. B (CC)) adantr 1cn (1) A B adddirt mp3an1 B mulid2t (e. A (CC)) adantl (+) (opr A (x.) B) opreq1d eqtrd (+) opreq12d eqtrd)) thm (mul12 () ((mul.1 (e. A (CC))) (mul.2 (e. B (CC))) (mul.3 (e. C (CC)))) (= (opr A (x.) (opr B (x.) C)) (opr B (x.) (opr A (x.) C))) (mul.1 mul.2 mulcom (x.) C opreq1i mul.1 mul.2 mul.3 mulass mul.2 mul.1 mul.3 mulass 3eqtr3)) thm (mul23 () ((mul.1 (e. A (CC))) (mul.2 (e. B (CC))) (mul.3 (e. C (CC)))) (= (opr (opr A (x.) B) (x.) C) (opr (opr A (x.) C) (x.) B)) (mul.1 mul.2 mul.3 A B C mul23t mp3an)) thm (mul4 () ((mul4.1 (e. A (CC))) (mul4.2 (e. B (CC))) (mul4.3 (e. C (CC))) (mul4.4 (e. D (CC)))) (= (opr (opr A (x.) B) (x.) (opr C (x.) D)) (opr (opr A (x.) C) (x.) (opr B (x.) D))) (mul4.1 mul4.2 pm3.2i mul4.3 mul4.4 pm3.2i A B C D mul4t mp2an)) thm (muladd () ((mul4.1 (e. A (CC))) (mul4.2 (e. B (CC))) (mul4.3 (e. C (CC))) (mul4.4 (e. D (CC)))) (= (opr (opr A (+) B) (x.) (opr C (+) D)) (opr (opr (opr A (x.) C) (+) (opr D (x.) B)) (+) (opr (opr A (x.) D) (+) (opr C (x.) B)))) (mul4.1 mul4.2 pm3.2i mul4.3 mul4.4 pm3.2i A B C D muladdt mp2an)) thm (subdit () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (x.) (opr B (-) C)) (opr (opr A (x.) B) (-) (opr A (x.) C)))) (A C (opr B (-) C) axdistr (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp1 (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp3 B C subclt (e. A (CC)) 3adant1 syl3anc C B pncan3t ancoms (e. A (CC)) 3adant1 A (x.) opreq2d eqtr3d (opr A (x.) B) (opr A (x.) C) (opr A (x.) (opr B (-) C)) subaddt A B axmulcl (e. C (CC)) 3adant3 A C axmulcl (e. B (CC)) 3adant2 A (opr B (-) C) axmulcl B C subclt sylan2 3impb syl3anc mpbird eqcomd)) thm (subdirt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) B) (x.) C) (opr (opr A (x.) C) (-) (opr B (x.) C)))) (C A B subdit 3coml (opr A (-) B) C axmulcom A B subclt sylan 3impa A C axmulcom (e. B (CC)) 3adant2 B C axmulcom (e. A (CC)) 3adant1 (-) opreq12d 3eqtr4d)) thm (subdi () ((subdi.1 (e. A (CC))) (subdi.2 (e. B (CC))) (subdi.3 (e. C (CC)))) (= (opr A (x.) (opr B (-) C)) (opr (opr A (x.) B) (-) (opr A (x.) C))) (subdi.1 subdi.2 subdi.3 A B C subdit mp3an)) thm (subdir () ((subdi.1 (e. A (CC))) (subdi.2 (e. B (CC))) (subdi.3 (e. C (CC)))) (= (opr (opr A (-) B) (x.) C) (opr (opr A (x.) C) (-) (opr B (x.) C))) (subdi.1 subdi.2 subdi.3 A B C subdirt mp3an)) thm (mul01 () ((mulzer.1 (e. A (CC)))) (= (opr A (x.) (0)) (0)) (mulzer.1 0cn 0cn subdi 0cn subid A (x.) opreq2i mulzer.1 0cn mulcl subid 3eqtr3)) thm (mul02 () ((mulzer.1 (e. A (CC)))) (= (opr (0) (x.) A) (0)) (0cn mulzer.1 mulcom mulzer.1 mul01 eqtr)) thm (1p1times () ((mulzer.1 (e. A (CC)))) (= (opr (opr (1) (+) (1)) (x.) A) (opr A (+) A)) (1cn 1cn mulzer.1 adddir mulzer.1 mulid2 mulzer.1 mulid2 (+) opreq12i eqtr)) thm (mul01t () () (-> (e. A (CC)) (= (opr A (x.) (0)) (0))) (A (if (e. A (CC)) A (0)) (x.) (0) opreq1 (0) eqeq1d 0cn A elimel mul01 dedth)) thm (mul02t () () (-> (e. A (CC)) (= (opr (0) (x.) A) (0))) (0cn (0) A axmulcom mpan A mul01t eqtrd)) thm (mulneg1 () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (opr (-u A) (x.) B) (-u (opr A (x.) B))) (mulneg.2 mul01 mulneg.2 mulneg.1 mulcom (-) opreq12i A df-neg (x.) B opreq1i 0cn mulneg.1 subcl mulneg.2 mulcom mulneg.2 0cn mulneg.1 subdi 3eqtr (opr A (x.) B) df-neg 3eqtr4)) thm (mulneg2 () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (opr A (x.) (-u B)) (-u (opr A (x.) B))) (mulneg.1 mulneg.2 negcl mulcom mulneg.2 mulneg.1 mulneg1 mulneg.2 mulneg.1 mulcom negeqi 3eqtr)) thm (mul2neg () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (opr (-u A) (x.) (-u B)) (opr A (x.) B)) (mulneg.1 mulneg.2 negcl mulneg1 mulneg.1 mulneg.2 negcl mulcom mulneg.2 mulneg.1 mulneg1 eqtr negeqi mulneg.2 mulneg.1 mulcl negneg 3eqtr mulneg.2 mulneg.1 mulcom eqtr)) thm (negdi () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (-u (opr A (+) B)) (opr (-u A) (+) (-u B))) (mulneg.1 mulneg.2 addcl mulid2 negeqi 1cn negcl mulneg.1 mulneg.2 adddi 1cn mulneg.1 mulneg.2 addcl mulneg1 1cn mulneg.1 mulneg1 mulneg.1 mulid2 negeqi eqtr 1cn mulneg.2 mulneg1 mulneg.2 mulid2 negeqi eqtr (+) opreq12i 3eqtr3 eqtr3)) thm (negsubdi () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (-u (opr A (-) B)) (opr (-u A) (+) B)) (mulneg.1 mulneg.2 negcl negdi mulneg.1 mulneg.2 negsub negeqi mulneg.2 negneg (-u A) (+) opreq2i 3eqtr3)) thm (negsubdi2 () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (-u (opr A (-) B)) (opr B (-) A)) (mulneg.1 mulneg.2 negsubdi mulneg.1 negcl mulneg.2 addcom mulneg.2 mulneg.1 negsub 3eqtr)) thm (mulneg1t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (-u A) (x.) B) (-u (opr A (x.) B)))) (A (if (e. A (CC)) A (0)) negeq (x.) B opreq1d A (if (e. A (CC)) A (0)) (x.) B opreq1 negeqd eqeq12d B (if (e. B (CC)) B (0)) (-u (if (e. A (CC)) A (0))) (x.) opreq2 B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (x.) opreq2 negeqd eqeq12d 0cn A elimel 0cn B elimel mulneg1 dedth2h)) thm (mulneg2t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (x.) (-u B)) (-u (opr A (x.) B)))) (B A mulneg1t ancoms A (-u B) axmulcom B negclt sylan2 A B axmulcom negeqd 3eqtr4d)) thm (mulneg12t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (-u A) (x.) B) (opr A (x.) (-u B)))) (A B mulneg1t A B mulneg2t eqtr4d)) thm (mul2negt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (-u A) (x.) (-u B)) (opr A (x.) B))) (A (if (e. A (CC)) A (0)) negeq (x.) (-u B) opreq1d A (if (e. A (CC)) A (0)) (x.) B opreq1 eqeq12d B (if (e. B (CC)) B (0)) negeq (-u (if (e. A (CC)) A (0))) (x.) opreq2d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (x.) opreq2 eqeq12d 0cn A elimel 0cn B elimel mul2neg dedth2h)) thm (negdit () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (-u (opr A (+) B)) (opr (-u A) (+) (-u B)))) (A (if (e. A (CC)) A (0)) (+) B opreq1 negeqd A (if (e. A (CC)) A (0)) negeq (+) (-u B) opreq1d eqeq12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (+) opreq2 negeqd B (if (e. B (CC)) B (0)) negeq (-u (if (e. A (CC)) A (0))) (+) opreq2d eqeq12d 0cn A elimel 0cn B elimel negdi dedth2h)) thm (negdi2t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (-u (opr A (+) B)) (opr (-u A) (-) B))) (A B negdit (-u A) B negsubt A negclt sylan eqtrd)) thm (negsubdit () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (-u (opr A (-) B)) (opr (-u A) (+) B))) (A (-u B) negdit B negclt sylan2 A B negsubt negeqd B negnegt (e. A (CC)) adantl (-u A) (+) opreq2d 3eqtr3d)) thm (negsubdi2t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (-u (opr A (-) B)) (opr B (-) A))) (A B negsubdit (-u A) B axaddcom A negclt sylan B A negsubt ancoms 3eqtrd)) thm (subsub2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (-) (opr B (-) C)) (opr A (+) (opr C (-) B)))) (A (opr B (-) C) negsubt B C subclt sylan2 3impb B C negsubdi2t A (+) opreq2d (e. A (CC)) 3adant1 eqtr3d)) thm (subsubt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (-) (opr B (-) C)) (opr (opr A (-) B) (+) C))) (A B C subsub2t A C B addsubasst A C B addsubt eqtr3d 3com23 eqtrd)) thm (subsub3t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (-) (opr B (-) C)) (opr (opr A (+) C) (-) B))) (A B C subsub2t A C B addsubasst 3com23 eqtr4d)) thm (subsub4t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) B) (-) C) (opr A (-) (opr B (+) C)))) (A B (-u C) subsubt C negclt syl3an3 B C subnegt (e. A (CC)) 3adant1 A (-) opreq2d (opr A (-) B) C negsubt A B subclt sylan 3impa 3eqtr3rd)) thm (sub23t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) B) (-) C) (opr (opr A (-) C) (-) B))) (B C axaddcom (e. A (CC)) 3adant1 A (-) opreq2d A B C subsub4t A C B subsub4t 3com23 3eqtr4d)) thm (nnncant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) (opr B (-) C)) (-) C) (opr A (-) B))) (A (opr B (-) C) C subsub4t (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp1 B C subclt (e. A (CC)) 3adant1 (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp3 syl3anc B C npcant A (-) opreq2d (e. A (CC)) 3adant1 eqtrd)) thm (nnncan1t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) B) (-) (opr A (-) C)) (opr C (-) B))) ((opr A (-) B) (opr A (-) C) negsubt (opr A (-) B) (-u (opr A (-) C)) axaddcom (opr A (-) C) negclt sylan2 eqtr3d A B subclt (e. C (CC)) 3adant3 A C subclt (e. B (CC)) 3adant2 sylanc A C negsubdi2t (e. B (CC)) 3adant2 (+) (opr A (-) B) opreq1d C A B npncant 3coml 3eqtrd)) thm (nnncan2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) C) (-) (opr B (-) C)) (opr A (-) B))) (A (opr B (-) C) C sub23t (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp1 B C subclt (e. A (CC)) 3adant1 (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp3 syl3anc A B C nnncant eqtr3d)) thm (nncant () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (-) (opr A (-) B)) B)) (0cn A (0) B nnncan1t mp3an2 A subid1t (e. B (CC)) adantr (-) (opr A (-) B) opreq1d B subid1t (e. A (CC)) adantl 3eqtr3d)) thm (nppcan2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) (opr B (+) C)) (+) C) (opr A (-) B))) (A (opr B (+) C) C subsubt (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp1 B C axaddcl (e. A (CC)) 3adant1 (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp3 syl3anc B C pncant (e. A (CC)) 3adant1 A (-) opreq2d eqtr3d)) thm (mulm1t () () (-> (e. A (CC)) (= (opr (-u (1)) (x.) A) (-u A))) (1cn (1) A mulneg1t mpan A mulid2t negeqd eqtrd)) thm (mulm1 () ((mulm1.1 (e. A (CC)))) (= (opr (-u (1)) (x.) A) (-u A)) (mulm1.1 A mulm1t ax-mp)) thm (sub4t () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (= (opr (opr A (+) B) (-) (opr C (+) D)) (opr (opr A (-) C) (+) (opr B (-) D)))) (C D negdit (/\ (e. A (CC)) (e. B (CC))) adantl (opr A (+) B) (+) opreq2d A B (-u C) (-u D) add4t C negclt D negclt anim12i sylan2 eqtrd (opr A (+) B) (opr C (+) D) negsubt A B axaddcl C D axaddcl syl2an A C negsubt (e. B (CC)) (e. D (CC)) ad2ant2r B D negsubt (e. A (CC)) (e. C (CC)) ad2ant2l (+) opreq12d 3eqtr3d)) thm (sub4 () ((sub4.1 (e. A (CC))) (sub4.2 (e. B (CC))) (sub4.3 (e. C (CC))) (sub4.4 (e. D (CC)))) (= (opr (opr A (+) B) (-) (opr C (+) D)) (opr (opr A (-) C) (+) (opr B (-) D))) (sub4.1 sub4.2 pm3.2i sub4.3 sub4.4 pm3.2i A B C D sub4t mp2an)) thm (mulsubt () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (= (opr (opr A (-) B) (x.) (opr C (-) D)) (opr (opr (opr A (x.) C) (+) (opr D (x.) B)) (-) (opr (opr A (x.) D) (+) (opr C (x.) B))))) (A B negsubt C D negsubt (x.) opreqan12d A (-u B) C (-u D) muladdt D negclt sylanr2 B negclt sylanl2 D B mul2negt ancoms (opr A (x.) C) (+) opreq2d (e. A (CC)) (e. C (CC)) ad2ant2l A D mulneg2t C B mulneg2t (+) opreqan12d (opr A (x.) D) (opr C (x.) B) negdit A D axmulcl C B axmulcl syl2an eqtr4d ancom2s an42s (+) opreq12d (opr (opr A (x.) C) (+) (opr D (x.) B)) (opr (opr A (x.) D) (+) (opr C (x.) B)) negsubt (opr A (x.) C) (opr D (x.) B) axaddcl A C axmulcl D B axmulcl ancoms syl2an an4s (opr A (x.) D) (opr C (x.) B) axaddcl A D axmulcl C B axmulcl ancoms syl2an an42s sylanc 3eqtrd eqtr3d)) thm (pnpcant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (-) (opr A (+) C)) (opr B (-) C))) (A B A C sub4t anandis A subidt (+) (opr B (-) C) opreq1d B C subclt (opr B (-) C) addid2t syl sylan9eq eqtrd 3impb)) thm (pnpcan2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) C) (-) (opr B (+) C)) (opr A (-) B))) (A C axaddcom (e. B (CC)) 3adant2 B C axaddcom (e. A (CC)) 3adant1 (-) opreq12d C A B pnpcant 3coml eqtrd)) thm (pnncant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (-) (opr A (-) C)) (opr B (+) C))) (A B (-u C) pnpcant C negclt syl3an3 A C negsubt (e. B (CC)) 3adant2 (opr A (+) B) (-) opreq2d B C subnegt (e. A (CC)) 3adant1 3eqtr3d)) thm (ppncant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (+) (opr C (-) B)) (opr A (+) C))) (A B axaddcom (e. C (CC)) 3adant3 (-) (opr B (-) C) opreq1d (opr A (+) B) B C subsub2t A B axaddcl (e. C (CC)) 3adant3 (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp2 (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp3 syl3anc B A C pnncant 3com12 3eqtr3d)) thm (pnncan () ((pnncan.1 (e. A (CC))) (pnncan.2 (e. B (CC))) (pnncan.3 (e. C (CC)))) (= (opr (opr A (+) B) (-) (opr A (-) C)) (opr B (+) C)) (pnncan.1 pnncan.2 pnncan.3 A B C pnncant mp3an)) thm (mulcan ((A x) (B x) (C x)) ((pnncan.1 (e. A (CC))) (pnncan.2 (e. B (CC))) (pnncan.3 (e. C (CC))) (mulcan.4 (=/= A (0)))) (<-> (= (opr A (x.) B) (opr A (x.) C)) (= B C)) (pnncan.1 mulcan.4 x recex pnncan.1 pnncan.2 (cv x) A B axmulass mp3an3 pnncan.3 (cv x) A C axmulass mp3an3 eqeq12d mpan2 (opr A (x.) B) (opr A (x.) C) (cv x) (x.) opreq2 syl5bir (= (opr A (x.) (cv x)) (1)) adantr pnncan.1 A (cv x) axmulcom mpan (1) eqeq1d (opr (cv x) (x.) A) (1) (x.) B opreq1 pnncan.2 mulid2 syl6eq (opr (cv x) (x.) A) (1) (x.) C opreq1 pnncan.3 mulid2 syl6eq eqeq12d syl6bi imp sylibd ex r19.23aiv ax-mp B C A (x.) opreq2 impbi)) thm (mulcant2 () ((mulcant2.1 (=/= A (0)))) (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (x.) B) (opr A (x.) C)) (= B C))) (A (if (e. A (CC)) A (1)) (x.) B opreq1 A (if (e. A (CC)) A (1)) (x.) C opreq1 eqeq12d (= B C) bibi1d B (if (e. B (CC)) B (1)) (if (e. A (CC)) A (1)) (x.) opreq2 (opr (if (e. A (CC)) A (1)) (x.) C) eqeq1d B (if (e. B (CC)) B (1)) C eqeq1 bibi12d C (if (e. C (CC)) C (1)) (if (e. A (CC)) A (1)) (x.) opreq2 (opr (if (e. A (CC)) A (1)) (x.) (if (e. B (CC)) B (1))) eqeq2d C (if (e. C (CC)) C (1)) (if (e. B (CC)) B (1)) eqeq2 bibi12d 1cn A elimel 1cn B elimel 1cn C elimel A (if (e. A (CC)) A (1)) (0) neeq1 (1) (if (e. A (CC)) A (1)) (0) neeq1 mulcant2.1 ax1ne0 keephyp mulcan dedth3h)) thm (mulcant () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= A (0))) (<-> (= (opr A (x.) B) (opr A (x.) C)) (= B C))) (A (if (=/= A (0)) A (1)) (CC) eleq1 (e. B (CC)) (e. C (CC)) 3anbi1d A (if (=/= A (0)) A (1)) (x.) B opreq1 A (if (=/= A (0)) A (1)) (x.) C opreq1 eqeq12d (= B C) bibi1d imbi12d A elimne0 B C mulcant2 dedth impcom)) thm (mulcan2t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (<-> (= (opr A (x.) C) (opr B (x.) C)) (= A B))) (A C axmulcom (e. B (CC)) 3adant2 B C axmulcom (e. A (CC)) 3adant1 eqeq12d (=/= C (0)) adantr C A B mulcant ex 3coml imp bitrd)) thm (mul0or () ((mul0or.1 (e. A (CC))) (mul0or.2 (e. B (CC)))) (<-> (= (opr A (x.) B) (0)) (\/ (= A (0)) (= B (0)))) (A (0) df-ne mul0or.1 mul0or.2 0cn 3pm3.2i A B (0) mulcant mpan mul0or.1 mul01 (opr A (x.) B) eqeq2i syl5bbr biimpd sylbir com12 orrd A (0) (x.) B opreq1 mul0or.2 mul02 syl6eq B (0) A (x.) opreq2 mul0or.1 mul01 syl6eq jaoi impbi)) thm (msq0 () ((mul0or.1 (e. A (CC)))) (<-> (= (opr A (x.) A) (0)) (= A (0))) (mul0or.1 mul0or.1 mul0or (= A (0)) oridm bitr)) thm (mul0ort () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= (opr A (x.) B) (0)) (\/ (= A (0)) (= B (0))))) (A (if (e. A (CC)) A (0)) (x.) B opreq1 (0) eqeq1d A (if (e. A (CC)) A (0)) (0) eqeq1 (= B (0)) orbi1d bibi12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (x.) opreq2 (0) eqeq1d B (if (e. B (CC)) B (0)) (0) eqeq1 (= (if (e. A (CC)) A (0)) (0)) orbi2d bibi12d 0cn A elimel 0cn B elimel mul0or dedth2h)) thm (muln0bt () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (/\ (=/= A (0)) (=/= B (0))) (=/= (opr A (x.) B) (0)))) (A B mul0ort negbid (= A (0)) (= B (0)) ioran syl6rbb A (0) df-ne B (0) df-ne anbi12i (opr A (x.) B) (0) df-ne 3bitr4g)) thm (muln0 () ((muln0.1 (e. A (CC))) (muln0.2 (e. B (CC))) (muln0.3 (=/= A (0))) (muln0.4 (=/= B (0)))) (=/= (opr A (x.) B) (0)) (muln0.1 muln0.2 muln0.3 muln0.4 pm3.2i A B muln0bt mpbii mp2an)) thm (receu ((x y) (A x) (A y) (B x) (B y)) ((receu.1 (e. A (CC))) (receu.2 (e. B (CC))) (receu.3 (=/= A (0)))) (E!e. x (CC) (= (opr A (x.) (cv x)) B)) ((cv x) (cv y) A (x.) opreq2 B eqeq1d (CC) reu4 receu.1 receu.3 y recex receu.2 (cv y) B axmulcl mpan2 (opr (cv y) (x.) B) (CC) x risset sylib (cv x) (opr (cv y) (x.) B) A (x.) opreq2 receu.1 receu.2 A (cv y) B axmulass mp3an13 eqcomd sylan9eqr (opr A (x.) (cv y)) (1) (x.) B opreq1 receu.2 mulid2 syl6eq sylan9eqr exp32 impcom (e. (cv x) (CC)) a1d r19.21aiv ex x (CC) (= (cv x) (opr (cv y) (x.) B)) (= (opr A (x.) (cv x)) B) r19.22 syl6 mpid r19.23aiv ax-mp receu.1 receu.3 (cv x) (cv y) mulcant2 (opr A (x.) (cv x)) B (opr A (x.) (cv y)) eqtr3t syl5bi mp3an1 rgen2 mpbir2an)) thm (divval ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((divval.1 (e. A (CC))) (divval.2 (e. B (CC))) (divval.3 (=/= B (0)))) (= (opr A (/) B) (U. ({e.|} x (CC) (= (opr B (x.) (cv x)) A)))) (divval.1 B (CC) ({} (0)) eldif divval.2 divval.3 B (0) df-ne mpbi divval.2 elisseti (0) elsnc mtbir mpbir2an axcnex x (= (opr B (x.) (cv x)) A) rabex uniex (cv y) A (opr (cv z) (x.) (cv x)) eqeq2 x (CC) rabbisdv unieqd (cv z) B (x.) (cv x) opreq1 A eqeq1d x (CC) rabbisdv unieqd y z w x df-div oprabval2 mp2an)) thm (divmul ((A x) (B x) (C x)) ((divmul.1 (e. A (CC))) (divmul.2 (e. B (CC))) (divmul.3 (e. C (CC))) (divmul.4 (=/= B (0)))) (<-> (= (opr A (/) B) C) (= (opr B (x.) C) A)) (divmul.3 (cv x) C (opr A (/) B) eqeq2 (cv x) C B (x.) opreq2 A eqeq1d bibi12d divmul.2 divmul.1 divmul.4 x receu x (CC) (= (opr B (x.) (cv x)) A) reuuni1 mpan2 divmul.1 divmul.2 divmul.4 x divval (cv x) eqeq1i syl6rbbr vtoclga ax-mp)) thm (divmulz () ((divmulz.1 (e. A (CC))) (divmulz.2 (e. B (CC))) (divmulz.3 (e. C (CC)))) (-> (=/= B (0)) (<-> (= (opr A (/) B) C) (= (opr B (x.) C) A))) (B (if (=/= B (0)) B (1)) A (/) opreq2 C eqeq1d B (if (=/= B (0)) B (1)) (x.) C opreq1 A eqeq1d bibi12d divmulz.1 divmulz.2 1cn (=/= B (0)) keepel divmulz.3 B elimne0 divmul dedth)) thm (divmult () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= B (0))) (<-> (= (opr A (/) B) C) (= (opr B (x.) C) A))) (A (if (e. A (CC)) A (0)) (/) B opreq1 C eqeq1d A (if (e. A (CC)) A (0)) (opr B (x.) C) eqeq2 bibi12d (=/= B (0)) imbi2d B (if (e. B (CC)) B (0)) (0) neeq1 B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (/) opreq2 C eqeq1d B (if (e. B (CC)) B (0)) (x.) C opreq1 (if (e. A (CC)) A (0)) eqeq1d bibi12d imbi12d C (if (e. C (CC)) C (0)) (opr (if (e. A (CC)) A (0)) (/) (if (e. B (CC)) B (0))) eqeq2 C (if (e. C (CC)) C (0)) (if (e. B (CC)) B (0)) (x.) opreq2 (if (e. A (CC)) A (0)) eqeq1d bibi12d (=/= (if (e. B (CC)) B (0)) (0)) imbi2d 0cn A elimel 0cn B elimel 0cn C elimel divmulz dedth3h imp)) thm (divmul2t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= B (0))) (<-> (= (opr A (/) B) C) (= A (opr B (x.) C)))) (A B C divmult (opr B (x.) C) A eqcom syl6bb)) thm (divmul3t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= B (0))) (<-> (= (opr A (/) B) C) (= A (opr C (x.) B)))) (A B C divmul2t B C axmulcom A eqeq2d (e. A (CC)) 3adant1 (=/= B (0)) adantr bitrd)) thm (divcl ((A x) (B x)) ((divcl.1 (e. A (CC))) (divcl.2 (e. B (CC))) (divcl.3 (=/= B (0)))) (e. (opr A (/) B) (CC)) (divcl.1 divcl.2 divcl.3 x divval divcl.2 divcl.1 divcl.3 x receu x (CC) (= (opr B (x.) (cv x)) A) reucl ax-mp eqeltr)) thm (divclz () ((divclz.1 (e. A (CC))) (divclz.2 (e. B (CC)))) (-> (=/= B (0)) (e. (opr A (/) B) (CC))) (B (if (=/= B (0)) B (1)) A (/) opreq2 (CC) eleq1d divclz.1 divclz.2 1cn (=/= B (0)) keepel B elimne0 divcl dedth)) thm (divclt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (e. (opr A (/) B) (CC))) (A (if (e. A (CC)) A (0)) (/) B opreq1 (CC) eleq1d (=/= B (0)) imbi2d B (if (e. B (CC)) B (0)) (0) neeq1 B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (/) opreq2 (CC) eleq1d imbi12d 0cn A elimel 0cn B elimel divclz dedth2h 3impia)) thm (reccl () ((reccl.1 (e. A (CC))) (reccl.2 (=/= A (0)))) (e. (opr (1) (/) A) (CC)) (1cn reccl.1 reccl.2 divcl)) thm (recclz () ((recclz.1 (e. A (CC)))) (-> (=/= A (0)) (e. (opr (1) (/) A) (CC))) (1cn recclz.1 divclz)) thm (recclt () () (-> (/\ (e. A (CC)) (=/= A (0))) (e. (opr (1) (/) A) (CC))) (1cn (1) A divclt mp3an1)) thm (divcan2 () ((divcan.1 (e. A (CC))) (divcan.2 (e. B (CC))) (divcan.3 (=/= A (0)))) (= (opr A (x.) (opr B (/) A)) B) ((opr B (/) A) eqid divcan.2 divcan.1 divcan.2 divcan.1 divcan.3 divcl divcan.3 divmul mpbi)) thm (divcan1 () ((divcan.1 (e. A (CC))) (divcan.2 (e. B (CC))) (divcan.3 (=/= A (0)))) (= (opr (opr B (/) A) (x.) A) B) (divcan.2 divcan.1 divcan.3 divcl divcan.1 mulcom divcan.1 divcan.2 divcan.3 divcan2 eqtr)) thm (divcan1z () ((divcan.1 (e. A (CC))) (divcan.2 (e. B (CC)))) (-> (=/= A (0)) (= (opr (opr B (/) A) (x.) A) B)) (A (if (=/= A (0)) A (1)) B (/) opreq2 (= A (if (=/= A (0)) A (1))) id (x.) opreq12d B eqeq1d divcan.1 1cn (=/= A (0)) keepel divcan.2 A elimne0 divcan1 dedth)) thm (divcan2z () ((divcan.1 (e. A (CC))) (divcan.2 (e. B (CC)))) (-> (=/= A (0)) (= (opr A (x.) (opr B (/) A)) B)) ((= A (if (=/= A (0)) A (1))) id A (if (=/= A (0)) A (1)) B (/) opreq2 (x.) opreq12d B eqeq1d divcan.1 1cn (=/= A (0)) keepel divcan.2 A elimne0 divcan2 dedth)) thm (divcan1t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= A (0))) (= (opr (opr B (/) A) (x.) A) B)) (A (if (e. A (CC)) A (0)) (0) neeq1 A (if (e. A (CC)) A (0)) B (/) opreq2 (= A (if (e. A (CC)) A (0))) id (x.) opreq12d B eqeq1d imbi12d B (if (e. B (CC)) B (0)) (/) (if (e. A (CC)) A (0)) opreq1 (x.) (if (e. A (CC)) A (0)) opreq1d (= B (if (e. B (CC)) B (0))) id eqeq12d (=/= (if (e. A (CC)) A (0)) (0)) imbi2d 0cn A elimel 0cn B elimel divcan1z dedth2h 3impia)) thm (divcan2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= A (0))) (= (opr A (x.) (opr B (/) A)) B)) (A (if (e. A (CC)) A (0)) (0) neeq1 (= A (if (e. A (CC)) A (0))) id A (if (e. A (CC)) A (0)) B (/) opreq2 (x.) opreq12d B eqeq1d imbi12d B (if (e. B (CC)) B (0)) (/) (if (e. A (CC)) A (0)) opreq1 (if (e. A (CC)) A (0)) (x.) opreq2d (= B (if (e. B (CC)) B (0))) id eqeq12d (=/= (if (e. A (CC)) A (0)) (0)) imbi2d 0cn A elimel 0cn B elimel divcan2z dedth2h 3impia)) thm (divne0bt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (<-> (=/= A (0)) (=/= (opr A (/) B) (0)))) (B mul01t A eqeq1d A (0) eqcom syl6rbbr (e. A (CC)) (=/= B (0)) 3ad2ant2 0cn A B (0) divmult mp3anl3 3impa bitr4d eqneqd)) thm (divne0 () ((divneq0.1 (e. A (CC))) (divneq0.2 (e. B (CC))) (divneq0.3 (=/= A (0))) (divneq0.4 (=/= B (0)))) (=/= (opr A (/) B) (0)) (divneq0.1 divneq0.2 divneq0.4 divneq0.3 A B divne0bt mpbii mp3an)) thm (recne0z () ((recneq0z.1 (e. A (CC)))) (-> (=/= A (0)) (=/= (opr (1) (/) A) (0))) (A (if (=/= A (0)) A (1)) (1) (/) opreq2 (0) neeq1d 1cn recneq0z.1 1cn (=/= A (0)) keepel ax1ne0 A elimne0 divne0 dedth)) thm (recne0t () () (-> (/\ (e. A (CC)) (=/= A (0))) (=/= (opr (1) (/) A) (0))) (A (if (e. A (CC)) A (0)) (0) neeq1 A (if (e. A (CC)) A (0)) (1) (/) opreq2 (0) neeq1d imbi12d 0cn A elimel recne0z dedth imp)) thm (recid () ((recid.1 (e. A (CC))) (recid.2 (=/= A (0)))) (= (opr A (x.) (opr (1) (/) A)) (1)) (recid.1 1cn recid.2 divcan2)) thm (recidz () ((recidz.1 (e. A (CC)))) (-> (=/= A (0)) (= (opr A (x.) (opr (1) (/) A)) (1))) (recidz.1 1cn divcan2z)) thm (recidt () () (-> (/\ (e. A (CC)) (=/= A (0))) (= (opr A (x.) (opr (1) (/) A)) (1))) (A (if (e. A (CC)) A (0)) (0) neeq1 (= A (if (e. A (CC)) A (0))) id A (if (e. A (CC)) A (0)) (1) (/) opreq2 (x.) opreq12d (1) eqeq1d imbi12d 0cn A elimel recidz dedth imp)) thm (recid2t () () (-> (/\ (e. A (CC)) (=/= A (0))) (= (opr (opr (1) (/) A) (x.) A) (1))) ((opr (1) (/) A) A axmulcom A recclt (e. A (CC)) (=/= A (0)) pm3.26 sylanc A recidt eqtrd)) thm (divrec () ((divrec.1 (e. A (CC))) (divrec.2 (e. B (CC))) (divrec.3 (=/= B (0)))) (= (opr A (/) B) (opr A (x.) (opr (1) (/) B))) (divrec.2 divrec.1 divrec.2 divrec.3 reccl mulcl mulcom divrec.1 divrec.2 divrec.3 reccl divrec.2 mulass divrec.2 1cn divrec.3 divcan1 A (x.) opreq2i divrec.1 mulid1 eqtr 3eqtr divrec.1 divrec.2 divrec.1 divrec.2 divrec.3 reccl mulcl divrec.3 divmul mpbir)) thm (divrecz () ((divrec.1 (e. A (CC))) (divrec.2 (e. B (CC)))) (-> (=/= B (0)) (= (opr A (/) B) (opr A (x.) (opr (1) (/) B)))) (B (if (=/= B (0)) B (1)) A (/) opreq2 B (if (=/= B (0)) B (1)) (1) (/) opreq2 A (x.) opreq2d eqeq12d divrec.1 divrec.2 1cn (=/= B (0)) keepel B elimne0 divrec dedth)) thm (divrect () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (= (opr A (/) B) (opr A (x.) (opr (1) (/) B)))) (A (if (e. A (CC)) A (0)) (/) B opreq1 A (if (e. A (CC)) A (0)) (x.) (opr (1) (/) B) opreq1 eqeq12d (=/= B (0)) imbi2d B (if (e. B (CC)) B (0)) (0) neeq1 B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (/) opreq2 B (if (e. B (CC)) B (0)) (1) (/) opreq2 (if (e. A (CC)) A (0)) (x.) opreq2d eqeq12d imbi12d 0cn A elimel 0cn B elimel divrecz dedth2h 3impia)) thm (divrec2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (= (opr A (/) B) (opr (opr (1) (/) B) (x.) A))) (A B divrect A (opr (1) (/) B) axmulcom (e. A (CC)) (e. B (CC)) (=/= B (0)) 3simp1 B recclt (e. A (CC)) 3adant1 sylanc eqtrd)) thm (divasst () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (= (opr (opr A (x.) B) (/) C) (opr A (x.) (opr B (/) C)))) ((e. A (CC)) id (e. B (CC)) id C recclt 3anim123i 3exp exp4a 3imp1 A B (opr (1) (/) C) axmulass syl (opr A (x.) B) C divrect 3expa A B axmulcl (e. C (CC)) anim1i 3impa sylan B C divrect 3expa (e. A (CC)) 3adantl1 A (x.) opreq2d 3eqtr4d)) thm (div23t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (= (opr (opr A (x.) B) (/) C) (opr (opr A (/) C) (x.) B))) (A B axmulcom (e. C (CC)) 3adant3 (=/= C (0)) adantr (/) C opreq1d B A C divasst ex 3com12 imp B (opr A (/) C) axmulcom (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp2 (=/= C (0)) adantr A C divclt 3expa (e. B (CC)) 3adantl2 sylanc 3eqtrd)) thm (div13t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= B (0))) (= (opr (opr A (/) B) (x.) C) (opr (opr C (/) B) (x.) A))) (A C axmulcom (/) B opreq1d (e. B (CC)) 3adant2 (=/= B (0)) adantr A C B div23t ex 3com23 imp C A B div23t ex 3coml imp 3eqtr3d)) thm (div12t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (= (opr A (x.) (opr B (/) C)) (opr B (x.) (opr A (/) C)))) (A (opr B (/) C) axmulcom (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp1 (=/= C (0)) adantr B C divclt 3expa (e. A (CC)) 3adantl1 sylanc B C A div13t ex 3comr imp (opr A (/) C) B axmulcom A C divclt 3expa (e. B (CC)) 3adantl2 (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp2 (=/= C (0)) adantr sylanc 3eqtrd)) thm (divassz () ((divass.1 (e. A (CC))) (divass.2 (e. B (CC))) (divass.3 (e. C (CC)))) (-> (=/= C (0)) (= (opr (opr A (x.) B) (/) C) (opr A (x.) (opr B (/) C)))) (divass.1 divass.2 divass.3 3pm3.2i A B C divasst mpan)) thm (divass () ((divass.1 (e. A (CC))) (divass.2 (e. B (CC))) (divass.3 (e. C (CC))) (divass.4 (=/= C (0)))) (= (opr (opr A (x.) B) (/) C) (opr A (x.) (opr B (/) C))) (divass.4 divass.1 divass.2 divass.3 divassz ax-mp)) thm (divdir () ((divass.1 (e. A (CC))) (divass.2 (e. B (CC))) (divass.3 (e. C (CC))) (divass.4 (=/= C (0)))) (= (opr (opr A (+) B) (/) C) (opr (opr A (/) C) (+) (opr B (/) C))) (divass.1 divass.2 divass.3 divass.4 reccl adddir divass.1 divass.2 addcl divass.3 divass.4 divrec divass.1 divass.3 divass.4 divrec divass.2 divass.3 divass.4 divrec (+) opreq12i 3eqtr4)) thm (div23 () ((divass.1 (e. A (CC))) (divass.2 (e. B (CC))) (divass.3 (e. C (CC))) (divass.4 (=/= C (0)))) (= (opr (opr A (x.) B) (/) C) (opr (opr A (/) C) (x.) B)) (divass.1 divass.2 mulcom (/) C opreq1i divass.2 divass.1 divass.3 divass.4 divass divass.2 divass.1 divass.3 divass.4 divcl mulcom 3eqtr)) thm (divdirz () ((divass.1 (e. A (CC))) (divass.2 (e. B (CC))) (divass.3 (e. C (CC)))) (-> (=/= C (0)) (= (opr (opr A (+) B) (/) C) (opr (opr A (/) C) (+) (opr B (/) C)))) (C (if (=/= C (0)) C (1)) (opr A (+) B) (/) opreq2 C (if (=/= C (0)) C (1)) A (/) opreq2 C (if (=/= C (0)) C (1)) B (/) opreq2 (+) opreq12d eqeq12d divass.1 divass.2 divass.3 1cn (=/= C (0)) keepel C elimne0 divdir dedth)) thm (divdirt () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (= (opr (opr A (+) B) (/) C) (opr (opr A (/) C) (+) (opr B (/) C)))) (A (if (e. A (CC)) A (0)) (+) B opreq1 (/) C opreq1d A (if (e. A (CC)) A (0)) (/) C opreq1 (+) (opr B (/) C) opreq1d eqeq12d (=/= C (0)) imbi2d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (+) opreq2 (/) C opreq1d B (if (e. B (CC)) B (0)) (/) C opreq1 (opr (if (e. A (CC)) A (0)) (/) C) (+) opreq2d eqeq12d (=/= C (0)) imbi2d C (if (e. C (CC)) C (0)) (0) neeq1 C (if (e. C (CC)) C (0)) (opr (if (e. A (CC)) A (0)) (+) (if (e. B (CC)) B (0))) (/) opreq2 C (if (e. C (CC)) C (0)) (if (e. A (CC)) A (0)) (/) opreq2 C (if (e. C (CC)) C (0)) (if (e. B (CC)) B (0)) (/) opreq2 (+) opreq12d eqeq12d imbi12d 0cn A elimel 0cn B elimel 0cn C elimel divdirz dedth3h imp)) thm (divcan3 () ((divcan3.1 (e. A (CC))) (divcan3.2 (e. B (CC))) (divcan3.3 (=/= A (0)))) (= (opr (opr A (x.) B) (/) A) B) (divcan3.1 divcan3.2 divcan3.1 divcan3.3 divass divcan3.1 divcan3.2 divcan3.3 divcan2 eqtr)) thm (divcan4 () ((divcan3.1 (e. A (CC))) (divcan3.2 (e. B (CC))) (divcan3.3 (=/= A (0)))) (= (opr (opr B (x.) A) (/) A) B) (divcan3.2 divcan3.1 mulcom (/) A opreq1i divcan3.1 divcan3.2 divcan3.3 divcan3 eqtr)) thm (divcan3z () ((divcan3.1 (e. A (CC))) (divcan3.2 (e. B (CC)))) (-> (=/= A (0)) (= (opr (opr A (x.) B) (/) A) B)) (A (if (=/= A (0)) A (1)) (x.) B opreq1 (= A (if (=/= A (0)) A (1))) id (/) opreq12d B eqeq1d divcan3.1 1cn (=/= A (0)) keepel divcan3.2 A elimne0 divcan3 dedth)) thm (divcan4z () ((divcan3.1 (e. A (CC))) (divcan3.2 (e. B (CC)))) (-> (=/= A (0)) (= (opr (opr B (x.) A) (/) A) B)) (divcan3.1 divcan3.2 divcan3z divcan3.2 divcan3.1 mulcom (/) A opreq1i syl5eq)) thm (divcan3t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= A (0))) (= (opr (opr A (x.) B) (/) A) B)) (A (if (e. A (CC)) A (0)) (0) neeq1 A (if (e. A (CC)) A (0)) (x.) B opreq1 (= A (if (e. A (CC)) A (0))) id (/) opreq12d B eqeq1d imbi12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (x.) opreq2 (/) (if (e. A (CC)) A (0)) opreq1d (= B (if (e. B (CC)) B (0))) id eqeq12d (=/= (if (e. A (CC)) A (0)) (0)) imbi2d 0cn A elimel 0cn B elimel divcan3z dedth2h 3impia)) thm (divcan4t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= A (0))) (= (opr (opr B (x.) A) (/) A) B)) (A B axmulcom (/) A opreq1d (=/= A (0)) 3adant3 A B divcan3t eqtr3d)) thm (div11 () ((div11.1 (e. A (CC))) (div11.2 (e. B (CC))) (div11.3 (e. C (CC))) (div11.4 (=/= C (0)))) (<-> (= (opr A (/) C) (opr B (/) C)) (= A B)) (div11.3 div11.1 div11.3 div11.4 divcl div11.2 div11.3 div11.4 divcl div11.4 mulcan div11.3 div11.1 div11.4 divcan2 div11.3 div11.2 div11.4 divcan2 eqeq12i bitr3)) thm (div11t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (/\ (e. C (CC)) (=/= C (0)))) (<-> (= (opr A (/) C) (opr B (/) C)) (= A B))) (A (if (e. A (CC)) A (1)) (/) C opreq1 (opr B (/) C) eqeq1d A (if (e. A (CC)) A (1)) B eqeq1 bibi12d B (if (e. B (CC)) B (1)) (/) C opreq1 (opr (if (e. A (CC)) A (1)) (/) C) eqeq2d B (if (e. B (CC)) B (1)) (if (e. A (CC)) A (1)) eqeq2 bibi12d C (if (/\ (e. C (CC)) (=/= C (0))) C (1)) (if (e. A (CC)) A (1)) (/) opreq2 C (if (/\ (e. C (CC)) (=/= C (0))) C (1)) (if (e. B (CC)) B (1)) (/) opreq2 eqeq12d (= (if (e. A (CC)) A (1)) (if (e. B (CC)) B (1))) bibi1d 1cn A elimel 1cn B elimel C (if (/\ (e. C (CC)) (=/= C (0))) C (1)) (CC) eleq1 C (if (/\ (e. C (CC)) (=/= C (0))) C (1)) (0) neeq1 anbi12d (1) (if (/\ (e. C (CC)) (=/= C (0))) C (1)) (CC) eleq1 (1) (if (/\ (e. C (CC)) (=/= C (0))) C (1)) (0) neeq1 anbi12d 1cn ax1ne0 pm3.2i elimhyp pm3.26i C (if (/\ (e. C (CC)) (=/= C (0))) C (1)) (CC) eleq1 C (if (/\ (e. C (CC)) (=/= C (0))) C (1)) (0) neeq1 anbi12d (1) (if (/\ (e. C (CC)) (=/= C (0))) C (1)) (CC) eleq1 (1) (if (/\ (e. C (CC)) (=/= C (0))) C (1)) (0) neeq1 anbi12d 1cn ax1ne0 pm3.2i elimhyp pm3.27i div11 dedth3h)) thm (dividt () () (-> (/\ (e. A (CC)) (=/= A (0))) (= (opr A (/) A) (1))) (A A divrect 3expa anabsan A recidt eqtrd)) thm (div0t () () (-> (/\ (e. A (CC)) (=/= A (0))) (= (opr (0) (/) A) (0))) (0cn (0) A divrect mp3an1 A recclt (opr (1) (/) A) mul02t syl eqtrd)) thm (diveq0t () () (-> (/\/\ (e. A (CC)) (e. C (CC)) (=/= C (0))) (<-> (= (opr A (/) C) (0)) (= A (0)))) (C div0t (opr A (/) C) eqeq2d (e. A (CC)) 3adant1 0cn A (0) C div11t mp3an2 3impb bitr3d)) thm (recrec () ((recrec.1 (e. A (CC))) (recrec.2 (=/= A (0)))) (= (opr (1) (/) (opr (1) (/) A)) A) (recrec.1 recrec.2 reccl 1cn recrec.1 ax1ne0 recrec.2 divne0 recid A (x.) opreq2i recrec.1 recrec.2 recid (x.) (opr (1) (/) (opr (1) (/) A)) opreq1i recrec.1 recrec.1 recrec.2 reccl recrec.1 recrec.2 reccl 1cn recrec.1 ax1ne0 recrec.2 divne0 reccl mulass recrec.1 recrec.2 reccl 1cn recrec.1 ax1ne0 recrec.2 divne0 reccl mulid2 3eqtr3 recrec.1 mulid1 3eqtr3)) thm (divid () ((recrec.1 (e. A (CC))) (recrec.2 (=/= A (0)))) (= (opr A (/) A) (1)) (recrec.1 recrec.1 recrec.2 divrec recrec.1 recrec.2 recid eqtr)) thm (div0 () ((recrec.1 (e. A (CC))) (recrec.2 (=/= A (0)))) (= (opr (0) (/) A) (0)) (recrec.1 recrec.2 A div0t mp2an)) thm (div1 () ((div1.1 (e. A (CC)))) (= (opr A (/) (1)) A) (div1.1 mulid2 div1.1 1cn div1.1 ax1ne0 divmul mpbir)) thm (div1t () () (-> (e. A (CC)) (= (opr A (/) (1)) A)) (A (if (e. A (CC)) A (1)) (/) (1) opreq1 (= A (if (e. A (CC)) A (1))) id eqeq12d 1cn A elimel div1 dedth)) thm (divnegt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (= (-u (opr A (/) B)) (opr (-u A) (/) B))) (A (opr (1) (/) B) mulneg1t B recclt sylan2 3impb (-u A) B divrect A negclt syl3an1 A B divrect negeqd 3eqtr4rd)) thm (divsubdirt () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (= (opr (opr A (-) B) (/) C) (opr (opr A (/) C) (-) (opr B (/) C)))) (A (-u B) C divdirt B negclt syl3anl2 A B negsubt (e. C (CC)) 3adant3 (/) C opreq1d (=/= C (0)) adantr B C divnegt 3expa (e. A (CC)) 3adantl1 (opr A (/) C) (+) opreq2d (opr A (/) C) (opr B (/) C) negsubt A C divclt 3expa (e. B (CC)) 3adantl2 B C divclt 3expa (e. A (CC)) 3adantl1 sylanc eqtr3d 3eqtr3d)) thm (recrect () () (-> (/\ (e. A (CC)) (=/= A (0))) (= (opr (1) (/) (opr (1) (/) A)) A)) (A (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (1) (/) opreq2 (1) (/) opreq2d (= A (if (/\ (e. A (CC)) (=/= A (0))) A (1))) id eqeq12d A (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (CC) eleq1 A (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (0) neeq1 anbi12d (1) (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (CC) eleq1 (1) (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (0) neeq1 anbi12d 1cn ax1ne0 pm3.2i elimhyp pm3.26i A (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (CC) eleq1 A (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (0) neeq1 anbi12d (1) (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (CC) eleq1 (1) (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (0) neeq1 anbi12d 1cn ax1ne0 pm3.2i elimhyp pm3.27i recrec dedth)) thm (rec11i () ((rec11.1 (e. A (CC))) (rec11.2 (e. B (CC))) (rec11i.3 (=/= A (0))) (rec11i.4 (=/= B (0)))) (<-> (= (opr (1) (/) A) (opr (1) (/) B)) (= A B)) ((opr (1) (/) A) (opr (1) (/) B) (opr A (x.) B) (x.) opreq2 rec11.1 rec11.2 rec11.1 rec11i.3 reccl mul23 rec11.1 rec11i.3 recid (x.) B opreq1i rec11.2 mulid2 3eqtr rec11.1 rec11.2 rec11.2 rec11i.4 reccl mulass rec11.2 rec11i.4 recid A (x.) opreq2i rec11.1 mulid1 3eqtr eqeq12i B A eqcom bitr sylib A B (1) (/) opreq2 impbi)) thm (rec11 () ((rec11.1 (e. A (CC))) (rec11.2 (e. B (CC)))) (-> (/\ (=/= A (0)) (=/= B (0))) (<-> (= (opr (1) (/) A) (opr (1) (/) B)) (= A B))) (A (if (=/= A (0)) A (1)) (1) (/) opreq2 (opr (1) (/) B) eqeq1d A (if (=/= A (0)) A (1)) B eqeq1 bibi12d B (if (=/= B (0)) B (1)) (1) (/) opreq2 (opr (1) (/) (if (=/= A (0)) A (1))) eqeq2d B (if (=/= B (0)) B (1)) (if (=/= A (0)) A (1)) eqeq2 bibi12d rec11.1 1cn (=/= A (0)) keepel rec11.2 1cn (=/= B (0)) keepel A elimne0 B elimne0 rec11i dedth2h)) thm (divmuldivt () () (-> (/\ (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (/\ (=/= B (0)) (=/= D (0)))) (= (opr (opr A (/) B) (x.) (opr C (/) D)) (opr (opr A (x.) C) (/) (opr B (x.) D)))) ((opr B (x.) D) (opr (opr A (/) B) (x.) (opr C (/) D)) divcan3t B D axmulcl (e. A (CC)) (e. C (CC)) ad2ant2l (/\ (=/= B (0)) (=/= D (0))) adantr (opr A (/) B) (opr C (/) D) axmulcl A B divclt 3expa C D divclt 3expa syl2an an4s B D muln0bt biimpd (e. A (CC)) (e. C (CC)) ad2ant2l imp syl3anc B (opr A (/) B) D (opr C (/) D) mul4t (e. A (CC)) (e. B (CC)) (=/= B (0)) 3simp2 A B divclt jca 3expa (e. C (CC)) (e. D (CC)) pm3.27 (=/= D (0)) adantr C D divclt 3expa jca syl2an B A divcan2t 3exp impcom imp D C divcan2t 3com12 3expa (x.) opreqan12d eqtr3d an4s (/) (opr B (x.) D) opreq1d eqtr3d)) thm (divcan5t () () (-> (/\/\ (e. A (CC)) (/\ (e. B (CC)) (=/= B (0))) (/\ (e. C (CC)) (=/= C (0)))) (= (opr (opr C (x.) A) (/) (opr C (x.) B)) (opr A (/) B))) (C dividt (x.) (opr A (/) B) opreq1d (e. A (CC)) (/\ (e. B (CC)) (=/= B (0))) 3ad2ant3 C C A B divmuldivt (e. C (CC)) (=/= C (0)) pm3.26 (e. C (CC)) (=/= C (0)) pm3.26 jca (e. B (CC)) (=/= B (0)) pm3.26 (e. A (CC)) anim2i anim12i 3impb 3coml (e. C (CC)) (=/= C (0)) pm3.27 (e. B (CC)) (=/= B (0)) pm3.27 anim12i ancoms (e. A (CC)) 3adant1 sylanc A B divclt 3expb (opr A (/) B) mulid2t syl (/\ (e. C (CC)) (=/= C (0))) 3adant3 3eqtr3d)) thm (divmul13t () () (-> (/\ (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (/\ (=/= B (0)) (=/= D (0)))) (= (opr (opr A (/) B) (x.) (opr C (/) D)) (opr (opr C (/) B) (x.) (opr A (/) D)))) (A C axmulcom (e. B (CC)) (e. D (CC)) ad2ant2r (/) (opr B (x.) D) opreq1d (/\ (=/= B (0)) (=/= D (0))) adantr A B C D divmuldivt C B A D divmuldivt (e. A (CC)) (e. B (CC)) ancom (/\ (e. C (CC)) (e. D (CC))) anbi1i (/\ (e. B (CC)) (e. A (CC))) (/\ (e. C (CC)) (e. D (CC))) ancom (e. C (CC)) (e. D (CC)) (e. B (CC)) (e. A (CC)) an42 3bitr sylanb 3eqtr4d)) thm (divmul24t () () (-> (/\ (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (/\ (=/= B (0)) (=/= D (0)))) (= (opr (opr A (/) B) (x.) (opr C (/) D)) (opr (opr A (/) D) (x.) (opr C (/) B)))) (B D axmulcom (e. A (CC)) (e. C (CC)) ad2ant2l (opr A (x.) C) (/) opreq2d (/\ (=/= B (0)) (=/= D (0))) adantr A B C D divmuldivt A D C B divmuldivt (e. A (CC)) (e. B (CC)) (e. C (CC)) (e. D (CC)) an42 (e. A (CC)) (e. C (CC)) (e. D (CC)) (e. B (CC)) an4 bitr (=/= B (0)) (=/= D (0)) ancom syl2anb 3eqtr4d)) thm (divadddivt () () (-> (/\ (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (/\ (=/= B (0)) (=/= D (0)))) (= (opr (opr A (/) B) (+) (opr C (/) D)) (opr (opr (opr A (x.) D) (+) (opr B (x.) C)) (/) (opr B (x.) D)))) (B D muln0bt (e. A (CC)) (e. C (CC)) ad2ant2l (opr A (x.) D) (opr B (x.) C) (opr B (x.) D) divdirt ex A D axmulcl (e. C (CC)) adantrl (e. B (CC)) adantlr B C axmulcl (e. D (CC)) adantrr (e. A (CC)) adantll B D axmulcl (e. A (CC)) (e. C (CC)) ad2ant2l syl3anc sylbid imp D dividt (e. C (CC)) (=/= B (0)) ad2ant2l (/\ (e. A (CC)) (e. B (CC))) adantll (opr A (/) B) (x.) opreq2d A B D D divmuldivt (e. C (CC)) (e. D (CC)) pm3.27 (e. C (CC)) (e. D (CC)) pm3.27 jca sylanl2 A B divclt 3expa (opr A (/) B) ax1id syl (/\ (e. C (CC)) (e. D (CC))) (=/= D (0)) ad2ant2r 3eqtr3d B dividt (=/= D (0)) adantrr (e. A (CC)) adantll (/\ (e. C (CC)) (e. D (CC))) adantlr (x.) (opr C (/) D) opreq1d B B C D divmuldivt (e. A (CC)) (e. B (CC)) pm3.27 (e. A (CC)) (e. B (CC)) pm3.27 jca sylanl1 C D divclt 3expa (opr C (/) D) mulid2t syl (/\ (e. A (CC)) (e. B (CC))) (=/= B (0)) ad2ant2l 3eqtr3d (+) opreq12d eqtr2d)) thm (divdivdivt () () (-> (/\ (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (/\/\ (=/= B (0)) (=/= C (0)) (=/= D (0)))) (= (opr (opr A (/) B) (/) (opr C (/) D)) (opr (opr A (x.) D) (/) (opr B (x.) C)))) ((opr D (/) C) (opr A (/) B) axmulcom D C divclt 3com12 3expa (=/= D (0)) adantrr (/\ (e. A (CC)) (e. B (CC))) (=/= B (0)) ad2ant2l A B divclt 3expa (/\ (e. C (CC)) (e. D (CC))) (/\ (=/= C (0)) (=/= D (0))) ad2ant2r sylanc A B D C divmuldivt (e. C (CC)) (e. D (CC)) pm3.22 (/\ (e. A (CC)) (e. B (CC))) anim2i (=/= C (0)) (=/= D (0)) pm3.26 (=/= B (0)) anim2i syl2an eqtrd (opr C (/) D) (x.) opreq2d (e. C (CC)) (e. D (CC)) pm3.22 ancli (=/= C (0)) (=/= D (0)) pm3.22 (=/= B (0)) adantl anim12i (/\ (e. A (CC)) (e. B (CC))) adantll C D D C divmuldivt syl C D axmulcom (/\ (e. A (CC)) (e. B (CC))) (/\ (=/= B (0)) (/\ (=/= C (0)) (=/= D (0)))) ad2antlr (/) (opr D (x.) C) opreq1d (opr D (x.) C) dividt D C axmulcl ancoms (/\ (e. A (CC)) (e. B (CC))) (/\ (=/= B (0)) (/\ (=/= C (0)) (=/= D (0)))) ad2antlr D C muln0bt (=/= C (0)) (=/= D (0)) ancom syl5bb ancoms biimpa (/\ (e. A (CC)) (e. B (CC))) (=/= B (0)) ad2ant2l sylanc 3eqtrd (x.) (opr A (/) B) opreq1d (opr C (/) D) (opr D (/) C) (opr A (/) B) axmulass C D divclt 3expa (=/= C (0)) adantrl (/\ (e. A (CC)) (e. B (CC))) (=/= B (0)) ad2ant2l D C divclt 3com12 3expa (=/= D (0)) adantrr (/\ (e. A (CC)) (e. B (CC))) (=/= B (0)) ad2ant2l A B divclt 3expa (/\ (e. C (CC)) (e. D (CC))) (/\ (=/= C (0)) (=/= D (0))) ad2ant2r syl3anc A B divclt 3expa (/\ (e. C (CC)) (e. D (CC))) (/\ (=/= C (0)) (=/= D (0))) ad2ant2r (opr A (/) B) mulid2t syl 3eqtr3d eqtr3d (opr A (/) B) (opr C (/) D) (opr (opr A (x.) D) (/) (opr B (x.) C)) divmult A B divclt 3expa (/\ (e. C (CC)) (e. D (CC))) (/\ (=/= C (0)) (=/= D (0))) ad2ant2r C D divclt 3expa (=/= C (0)) adantrl (/\ (e. A (CC)) (e. B (CC))) (=/= B (0)) ad2ant2l (opr A (x.) D) (opr B (x.) C) divclt A D axmulcl (e. C (CC)) adantrl (e. B (CC)) adantlr (/\ (=/= B (0)) (/\ (=/= C (0)) (=/= D (0)))) adantr B C axmulcl (e. D (CC)) adantrr (e. A (CC)) adantll (/\ (=/= B (0)) (/\ (=/= C (0)) (=/= D (0)))) adantr B C muln0bt biimpd (=/= C (0)) (=/= D (0)) pm3.26 sylan2i (e. D (CC)) adantrr (e. A (CC)) adantll imp syl3anc 3jca C D divne0bt biimpd 3exp com34 imp43 (/\ (e. A (CC)) (e. B (CC))) (=/= B (0)) ad2ant2l sylanc mpbird (=/= B (0)) (=/= C (0)) (=/= D (0)) 3anass sylan2b)) thm (divmuldiv () ((divmuldiv.1 (e. A (CC))) (divmuldiv.2 (e. B (CC))) (divmuldiv.3 (e. C (CC))) (divmuldiv.4 (e. D (CC))) (divmuldiv.5 (=/= B (0))) (divmuldiv.6 (=/= D (0)))) (= (opr (opr A (/) B) (x.) (opr C (/) D)) (opr (opr A (x.) C) (/) (opr B (x.) D))) (divmuldiv.1 divmuldiv.2 pm3.2i divmuldiv.3 divmuldiv.4 pm3.2i pm3.2i divmuldiv.5 divmuldiv.6 pm3.2i A B C D divmuldivt mp2an)) thm (divmul13 () ((divmuldiv.1 (e. A (CC))) (divmuldiv.2 (e. B (CC))) (divmuldiv.3 (e. C (CC))) (divmuldiv.4 (e. D (CC))) (divmuldiv.5 (=/= B (0))) (divmuldiv.6 (=/= D (0)))) (= (opr (opr A (/) B) (x.) (opr C (/) D)) (opr (opr C (/) B) (x.) (opr A (/) D))) (divmuldiv.3 divmuldiv.1 mulcom (/) (opr B (x.) D) opreq1i divmuldiv.3 divmuldiv.2 divmuldiv.1 divmuldiv.4 divmuldiv.5 divmuldiv.6 divmuldiv divmuldiv.1 divmuldiv.2 divmuldiv.3 divmuldiv.4 divmuldiv.5 divmuldiv.6 divmuldiv 3eqtr4r)) thm (divadddiv () ((divmuldiv.1 (e. A (CC))) (divmuldiv.2 (e. B (CC))) (divmuldiv.3 (e. C (CC))) (divmuldiv.4 (e. D (CC))) (divmuldiv.5 (=/= B (0))) (divmuldiv.6 (=/= D (0)))) (= (opr (opr A (/) B) (+) (opr C (/) D)) (opr (opr (opr A (x.) D) (+) (opr B (x.) C)) (/) (opr B (x.) D))) (divmuldiv.1 divmuldiv.2 pm3.2i divmuldiv.3 divmuldiv.4 pm3.2i pm3.2i divmuldiv.5 divmuldiv.6 pm3.2i A B C D divadddivt mp2an)) thm (divdivdiv () ((divmuldiv.1 (e. A (CC))) (divmuldiv.2 (e. B (CC))) (divmuldiv.3 (e. C (CC))) (divmuldiv.4 (e. D (CC))) (divmuldiv.5 (=/= B (0))) (divmuldiv.6 (=/= D (0))) (divdivdiv.7 (=/= C (0)))) (= (opr (opr A (/) B) (/) (opr C (/) D)) (opr (opr A (x.) D) (/) (opr B (x.) C))) (divmuldiv.1 divmuldiv.2 pm3.2i divmuldiv.3 divmuldiv.4 pm3.2i pm3.2i divmuldiv.5 divdivdiv.7 divmuldiv.6 3pm3.2i A B C D divdivdivt mp2an)) thm (recdivt () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (=/= A (0)) (=/= B (0)))) (= (opr (1) (/) (opr A (/) B)) (opr B (/) A))) (1cn 1cn pm3.2i (1) (1) A B divdivdivt ax1ne0 (/\ (=/= A (0)) (=/= B (0))) jctl (=/= (1) (0)) (=/= A (0)) (=/= B (0)) 3anass sylibr sylan2 mpanl1 1cn div1 (/) (opr A (/) B) opreq1i (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (=/= A (0)) (=/= B (0)))) a1i B mulid2t A mulid2t (/) opreqan12rd (/\ (=/= A (0)) (=/= B (0))) adantr 3eqtr3d)) thm (divdiv23t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (/\ (=/= B (0)) (=/= C (0)))) (= (opr (opr A (/) B) (/) C) (opr (opr A (/) C) (/) B))) (A (opr (1) (/) B) C div23t (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp1 (/\ (=/= B (0)) (=/= C (0))) adantr B recclt (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp2 sylan (=/= C (0)) adantrr (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp3 (/\ (=/= B (0)) (=/= C (0))) adantr 3jca (=/= B (0)) (=/= C (0)) pm3.27 (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) adantl sylanc A B divrect 3expa (e. C (CC)) 3adantl3 (=/= C (0)) adantrr (/) C opreq1d (opr A (/) C) B divrect A C divclt 3expa (e. B (CC)) 3adantl2 (=/= B (0)) adantrl (e. A (CC)) (e. B (CC)) (e. C (CC)) 3simp2 (/\ (=/= B (0)) (=/= C (0))) adantr (=/= B (0)) (=/= C (0)) pm3.26 (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) adantl syl3anc 3eqtr4d)) thm (divdiv23 () ((divdiv23.1 (e. A (CC))) (divdiv23.2 (e. B (CC))) (divdiv23.3 (e. C (CC))) (divdiv23.4 (=/= B (0))) (divdiv23.5 (=/= C (0)))) (= (opr (opr A (/) B) (/) C) (opr (opr A (/) C) (/) B)) (divdiv23.1 divdiv23.2 divdiv23.4 reccl divdiv23.3 divdiv23.5 div23 divdiv23.1 divdiv23.2 divdiv23.4 divrec (/) C opreq1i divdiv23.1 divdiv23.3 divdiv23.5 divcl divdiv23.2 divdiv23.4 divrec 3eqtr4)) thm (divdiv23z () ((divdiv23.1 (e. A (CC))) (divdiv23.2 (e. B (CC))) (divdiv23.3 (e. C (CC)))) (-> (/\ (=/= B (0)) (=/= C (0))) (= (opr (opr A (/) B) (/) C) (opr (opr A (/) C) (/) B))) (divdiv23.1 divdiv23.2 divdiv23.3 3pm3.2i A B C divdiv23t mpan)) thm (redivcl ((B x)) ((redivcl.1 (e. A (RR))) (redivcl.2 (e. B (RR))) (redivcl.3 (=/= B (0)))) (e. (opr A (/) B) (RR)) (redivcl.1 recn redivcl.2 recn redivcl.3 divrec redivcl.1 redivcl.2 redivcl.3 B x axrrecex mp2an x (RR) (= (opr B (x.) (cv x)) (1)) df-rex (cv x) recnt (cv x) (if (e. (cv x) (CC)) (cv x) (1)) (opr (1) (/) B) eqeq2 (cv x) (if (e. (cv x) (CC)) (cv x) (1)) B (x.) opreq2 (1) eqeq1d bibi12d 1cn redivcl.2 recn 1cn (cv x) elimel redivcl.3 divmul dedth syl (cv x) (RR) (opr (1) (/) B) eleq1a sylbird imp x 19.23aiv sylbi ax-mp remulcl eqeltr)) thm (redivclz () ((redivclz.1 (e. A (RR))) (redivclz.2 (e. B (RR)))) (-> (=/= B (0)) (e. (opr A (/) B) (RR))) (B (if (=/= B (0)) B (1)) A (/) opreq2 (RR) eleq1d redivclz.1 redivclz.2 ax1re (=/= B (0)) keepel B elimne0 redivcl dedth)) thm (redivclt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (=/= B (0))) (e. (opr A (/) B) (RR))) (A (if (e. A (RR)) A (0)) (/) B opreq1 (RR) eleq1d (=/= B (0)) imbi2d B (if (e. B (RR)) B (0)) (0) neeq1 B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (/) opreq2 (RR) eleq1d imbi12d 0re A elimel 0re B elimel redivclz dedth2h 3impia)) thm (rereccl () ((rereccl.1 (e. A (RR))) (rereccl.2 (=/= A (0)))) (e. (opr (1) (/) A) (RR)) (ax1re rereccl.1 rereccl.2 redivcl)) thm (rerecclz () ((rerecclz.1 (e. A (RR)))) (-> (=/= A (0)) (e. (opr (1) (/) A) (RR))) (ax1re rerecclz.1 redivclz)) thm (rerecclt () () (-> (/\ (e. A (RR)) (=/= A (0))) (e. (opr (1) (/) A) (RR))) (ax1re (1) A redivclt mp3an1)) thm (pnfxr () () (e. (+oo) (RR*)) (df-pnf axcnex eqeltr (-oo) pri1 (e. (+oo) ({,} (+oo) (-oo))) (e. (+oo) (RR)) olc ax-mp (+oo) (RR) ({,} (+oo) (-oo)) elun mpbir df-xr eleqtrr)) thm (mnfxr () () (e. (-oo) (RR*)) (df-mnf (CC) snex eqeltr (+oo) pri2 (e. (-oo) ({,} (+oo) (-oo))) (e. (-oo) (RR)) olc ax-mp (-oo) (RR) ({,} (+oo) (-oo)) elun mpbir df-xr eleqtrr)) thm (ltxrt () () (-> (/\ (e. A (RR*)) (e. B (RR*))) (<-> (br A (<) B) (\/ (\/ (/\ (/\ (e. A (RR)) (e. B (RR))) (br A ( A B) ( (-oo) (+oo)) elsnc syl5bb orbi12d (<,> A B) (i^i ( (-oo) (+oo))) elun syl5bb B (RR*) A (RR) ({} (+oo)) opelxpg B (RR*) (+oo) elsncg (e. A (RR)) anbi2d bitrd (e. A (RR*)) adantl B (RR*) A ({} (-oo)) (RR) opelxpg A (RR*) (-oo) elsncg (e. B (RR)) anbi1d sylan9bbr orbi12d (<,> A B) (X. (RR) ({} (+oo))) (X. ({} (-oo)) (RR)) elun syl5bb orbi12d A (<) B df-br df-ltxr (<,> A B) eleq2i (<,> A B) (u. (i^i ( (-oo) (+oo)))) (u. (X. (RR) ({} (+oo))) (X. ({} (-oo)) (RR))) elun 3bitr syl5bb)) thm (pnfnre () () (e/ (+oo) (RR)) ((CC) eirr df-pnf (CC) eleq1i mtbir (+oo) recnt mto (+oo) (RR) df-nel mpbir)) thm (minfnre () () (e/ (-oo) (RR)) (axcnex snid (CC) ({} (CC)) en2lp (e. (CC) ({} (CC))) (e. ({} (CC)) (CC)) imnan mpbir ax-mp df-mnf (CC) eleq1i mtbir (-oo) recnt mto (-oo) (RR) df-nel mpbir)) thm (ressxr () () (C_ (RR) (RR*)) ((RR) ({,} (+oo) (-oo)) ssun1 df-xr sseqtr4)) thm (rexrt () () (-> (e. A (RR)) (e. A (RR*))) (ressxr A sseli)) thm (ltxrltt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br A ( (/\ (e. A (RR*)) (e. B (RR*))) (<-> (br A (<_) B) (-. (br B (<) A)))) (A (RR*) B (RR*) opelxpi df-le (<,> A B) eleq2i (<,> A B) (X. (RR*) (RR*)) (`' (<)) eldif bitr baib syl A (<_) B df-br syl5bb A (RR*) B (RR*) (<) opelcnvg B (<) A df-br syl6rbbr negbid bitr4d)) thm (axlttri () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (-. (\/ (= A B) (br B (<) A))))) (A B pre-axlttri A B ltxrltt B A ltxrltt ancoms (= A B) orbi2d negbid 3bitr4d)) thm (axlttrn () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\ (br A (<) B) (br B (<) C)) (br A (<) C))) (A B C pre-axlttrn A B ltxrltt (e. C (RR)) 3adant3 B C ltxrltt (e. A (RR)) 3adant1 anbi12d A C ltxrltt (e. B (RR)) 3adant2 3imtr4d)) thm (axltadd () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (br A (<) B) (br (opr C (+) A) (<) (opr C (+) B)))) (A B C pre-axltadd A B ltxrltt (e. C (RR)) 3adant3 (opr C (+) A) (opr C (+) B) ltxrltt C A axaddrcl C B axaddrcl syl2an 3impdi 3coml 3imtr4d)) thm (axmulgt0 () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (br (0) (<) (opr A (x.) B)))) (A B pre-axmulgt0 0re (0) A ltxrltt mpan 0re (0) B ltxrltt mpan bi2anan9 A B axmulrcl 0re (0) (opr A (x.) B) ltxrltt mpan syl 3imtr4d)) thm (axsup ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (=/= A ({/})) (E.e. x (RR) (A.e. y A (br (cv y) (<) (cv x))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))))) (A x y z pre-axsup 3expa ex (cv y) (cv x) ltxrltt A (RR) (cv y) ssel2 sylan an1rs ralbidva rexbidva (=/= A ({/})) adantr (cv x) (cv y) ltxrltt ancoms A (RR) (cv y) ssel2 sylan an1rs negbid ralbidva (cv y) (cv x) ltxrltt ancoms (C_ A (RR)) adantll (cv y) (cv z) ltxrltt ancoms A (RR) (cv z) ssel2 sylan an1rs rexbidva (e. (cv x) (RR)) adantlr imbi12d ralbidva anbi12d rexbidva (=/= A ({/})) adantr 3imtr4d 3impia)) thm (lenltt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) B) (-. (br B (<) A)))) (A B xrlenltt A rexrt B rexrt syl2an)) thm (ltnlet () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (-. (br B (<_) A)))) (B A lenltt ancoms con2bid)) thm (ltso ((x y) (x z) (y z)) () (Or (<) (RR)) ((cv x) (cv y) axlttri (e. (cv z) (RR)) 3adant3 (cv x) (cv y) (cv z) axlttrn jca so)) thm (lttri2t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (-. (= A B)) (\/ (br A (<) B) (br B (<) A)))) (ltso (<) (RR) A B sotrieq mpan bicomd con1bid)) thm (lttri3t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (= A B) (/\ (-. (br A (<) B)) (-. (br B (<) A))))) (ltso (<) (RR) A B sotrieq2 mpan)) thm (ltnet () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (br A (<) B) (-. (= A B)))) (A B lttri2t (br A (<) B) (br B (<) A) orc syl5bir)) thm (letri3t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (= A B) (/\ (br A (<_) B) (br B (<_) A)))) (A B lttri3t (-. (br B (<) A)) (-. (br A (<) B)) ancom syl6bbr A B lenltt B A lenltt ancoms anbi12d bitr4d)) thm (leloet () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) B) (\/ (br A (<) B) (= A B)))) (A B lenltt B A axlttri ancoms con2bid B A eqcom (br A (<) B) orbi1i (= A B) (br A (<) B) orcom bitr syl5rbbr bitrd)) thm (eqleltt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (= A B) (/\ (br A (<_) B) (-. (br A (<) B))))) (A B letri3t B A lenltt ancoms (br A (<_) B) anbi2d bitrd)) thm (ltlet () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (br A (<) B) (br A (<_) B))) (A B leloet (br A (<) B) (= A B) orc syl5bir)) thm (leltnet () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) (<-> (br A (<) B) (-. (= A B)))) (A B lttri3t (-. (br A (<) B)) (-. (br B (<) A)) pm3.26 syl6bi (br A (<_) B) adantr A B leloet biimpa ord impbid con2bid 3impa)) thm (ltlent () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (/\ (br A (<_) B) (-. (= A B))))) (A B ltlet A B ltnet jcad A B leloet (br A (<) B) (-. (= A B)) ax-1 (= A B) (br A (<) B) pm2.24 jaoi syl6bi imp3a impbid)) thm (lelttrt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\ (br A (<_) B) (br B (<) C)) (br A (<) C))) (A B leloet (e. C (RR)) 3adant3 A B C axlttrn exp3a A B (<) C breq1 biimprd (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) a1i jaod sylbid imp3a)) thm (ltletrt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\ (br A (<) B) (br B (<_) C)) (br A (<) C))) (B C leloet (e. A (RR)) 3adant1 A B C axlttrn exp3a com23 B C A (<) breq2 biimpd (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) a1i jaod sylbid com23 imp3a)) thm (letrt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\ (br A (<_) B) (br B (<_) C)) (br A (<_) C))) (B C leloet (e. A (RR)) 3adant1 (br A (<_) B) adantr A B C lelttrt A C ltlet (e. B (RR)) 3adant2 syld exp3a imp B C A (<_) breq2 biimpcd (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) adantl jaod sylbid ex imp3a)) thm (letrd () ((letrd.1 (-> ph (e. A (RR)))) (letrd.2 (-> ph (e. B (RR)))) (letrd.3 (-> ph (e. C (RR)))) (letrd.4 (-> ph (br A (<_) B))) (letrd.5 (-> ph (br B (<_) C)))) (-> ph (br A (<_) C)) (letrd.4 letrd.5 A B C letrt letrd.1 letrd.2 letrd.3 syl3anc mp2and)) thm (lelttrd () ((letrd.1 (-> ph (e. A (RR)))) (letrd.2 (-> ph (e. B (RR)))) (letrd.3 (-> ph (e. C (RR)))) (lelttrd.4 (-> ph (br A (<_) B))) (lelttrd.5 (-> ph (br B (<) C)))) (-> ph (br A (<) C)) (lelttrd.4 lelttrd.5 A B C lelttrt letrd.1 letrd.2 letrd.3 syl3anc mp2and)) thm (ltletrd () ((letrd.1 (-> ph (e. A (RR)))) (letrd.2 (-> ph (e. B (RR)))) (letrd.3 (-> ph (e. C (RR)))) (ltletrd.4 (-> ph (br A (<) B))) (ltletrd.5 (-> ph (br B (<_) C)))) (-> ph (br A (<) C)) (ltletrd.4 ltletrd.5 A B C ltletrt letrd.1 letrd.2 letrd.3 syl3anc mp2and)) thm (lttrd () ((letrd.1 (-> ph (e. A (RR)))) (letrd.2 (-> ph (e. B (RR)))) (letrd.3 (-> ph (e. C (RR)))) (lttrd.4 (-> ph (br A (<) B))) (lttrd.5 (-> ph (br B (<) C)))) (-> ph (br A (<) C)) (lttrd.4 lttrd.5 A B C axlttrn letrd.1 letrd.2 letrd.3 syl3anc mp2and)) thm (ltnrt () () (-> (e. A (RR)) (-. (br A (<) A))) (ltso (<) (RR) A sonr mpan)) thm (leidt () () (-> (e. A (RR)) (br A (<_) A)) (A eqid (-. (br A (<) A)) a1i orri A A leloet mpbiri anidms)) thm (ltnsymt () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (br A (<) B) (-. (br B (<) A)))) (A B axlttri (= A B) (br B (<) A) pm2.46 syl6bi)) thm (ltnsym2t () () (-> (/\ (e. A (RR)) (e. B (RR))) (-. (/\ (br A (<) B) (br B (<) A)))) (A B ltnsymt (br A (<) B) (br B (<) A) imnan sylib)) thm (elxr () () (<-> (e. A (RR*)) (\/\/ (e. A (RR)) (= A (+oo)) (= A (-oo)))) (df-xr A eleq2i A (RR) ({,} (+oo) (-oo)) elun pnfxr elisseti mnfxr elisseti A elpr2 (e. A (RR)) orbi2i (e. A (RR)) (= A (+oo)) (= A (-oo)) 3orass bitr4 3bitr)) thm (pnfnemnf () () (=/= (+oo) (-oo)) ((CC) eirr axcnex snid (CC) ({} (CC)) (CC) eleq2 mpbiri mto df-pnf df-mnf eqeq12i mtbir (+oo) (-oo) df-ne mpbir)) thm (renepnft () () (-> (e. A (RR)) (=/= A (+oo))) (pnfnre (+oo) (RR) df-nel mpbi A (RR) (+oo) nelneq mpan2 A (+oo) df-ne sylibr)) thm (renemnft () () (-> (e. A (RR)) (=/= A (-oo))) (minfnre (-oo) (RR) df-nel mpbi A (RR) (-oo) nelneq mpan2 A (-oo) df-ne sylibr)) thm (renfdisj () () (= (i^i (RR) ({,} (+oo) (-oo))) ({/})) ((+oo) (-oo) df-pr (RR) ineq2i (+oo) eqid (+oo) renepnft (+oo) (+oo) df-ne sylib mt2 (RR) (+oo) disjsn mpbir (-oo) eqid (-oo) renemnft (-oo) (-oo) df-ne sylib mt2 (RR) (-oo) disjsn mpbir pm3.2i (RR) ({} (+oo)) ({} (-oo)) undisj2 mpbi eqtr)) thm (ssxr ((A x)) () (-> (C_ A (RR*)) (\/\/ (C_ A (RR)) (e. (+oo) A) (e. (-oo) A))) (A ({,} (+oo) (-oo)) (RR) disjssun df-xr (RR) ({,} (+oo) (-oo)) uncom eqtr A sseq2i syl5bb biimpcd A ({,} (+oo) (-oo)) x disj syl5ibr con3d x A (e. (cv x) ({,} (+oo) (-oo))) dfrex2 x visset (+oo) (-oo) elpr x A rexbii bitr3 x A (= (cv x) (+oo)) (= (cv x) (-oo)) r19.43 x A (= (cv x) (+oo)) df-rex x (e. (cv x) A) (= (cv x) (+oo)) exancom pnfxr elisseti (cv x) (+oo) A eleq1 ceqsexv 3bitr x A (= (cv x) (-oo)) df-rex x (e. (cv x) A) (= (cv x) (-oo)) exancom mnfxr elisseti (cv x) (-oo) A eleq1 ceqsexv 3bitr orbi12i 3bitr syl6ib orrd (C_ A (RR)) (e. (+oo) A) (e. (-oo) A) 3orass sylibr)) thm (xrltnrt () () (-> (e. A (RR*)) (-. (br A (<) A))) (A elxr A ltnrt pnfnre (+oo) (RR) df-nel mpbi (e. (+oo) (RR)) intnan (br (+oo) ( (e. A (RR)) (br A (<) (+oo))) ((+oo) eqid (e. A (RR)) jctr (/\ (e. A (RR)) (= (+oo) (+oo))) (/\ (= A (-oo)) (e. (+oo) (RR))) orc (\/ (/\ (e. A (RR)) (= (+oo) (+oo))) (/\ (= A (-oo)) (e. (+oo) (RR)))) (\/ (/\ (/\ (e. A (RR)) (e. (+oo) (RR))) (br A ( (e. A (RR)) (br (-oo) (<) A)) ((-oo) eqid (e. A (RR)) jctl (/\ (= (-oo) (-oo)) (e. A (RR))) (/\ (e. (-oo) (RR)) (= A (+oo))) olc (\/ (/\ (e. (-oo) (RR)) (= A (+oo))) (/\ (= (-oo) (-oo)) (e. A (RR)))) (\/ (/\ (/\ (e. (-oo) (RR)) (e. A (RR))) (br (-oo) ( (\/ (e. A (RR)) (= A (+oo))) (br (-oo) (<) A)) (A mnfltt mnfltpnf A (+oo) (-oo) (<) breq2 mpbiri jaoi)) thm (pnfnltt () () (-> (e. A (RR*)) (-. (br (+oo) (<) A))) (pnfnre (+oo) (RR) df-nel mpbi (e. A (RR)) intnanr (br (+oo) ( (e. A (RR*)) (-. (br A (<) (-oo)))) (minfnre (-oo) (RR) df-nel mpbi (e. A (RR)) intnan (br A ( (e. A (RR*)) (br A (<_) (+oo))) (A pnfnltt pnfxr A (+oo) xrlenltt mpan2 mpbird)) thm (mnflet () () (-> (e. A (RR*)) (br (-oo) (<_) A)) (A nltmnft mnfxr (-oo) A xrlenltt mpan mpbird)) thm (xrltnsymt () () (-> (/\ (e. A (RR*)) (e. B (RR*))) (-> (br A (<) B) (-. (br B (<) A)))) (A B ltnsymt A rexrt A pnfnltt syl (= B (+oo)) adantr B (+oo) (<) A breq1 (e. A (RR)) adantl mtbird (br A (<) B) a1d A rexrt A nltmnft syl (= B (-oo)) adantr B (-oo) A (<) breq2 (e. A (RR)) adantl mtbird (-. (br B (<) A)) pm2.21d 3jaodan B pnfnltt (= A (+oo)) adantl A (+oo) (<) B breq1 (e. B (RR*)) adantr mtbird (-. (br B (<) A)) pm2.21d B elxr sylan2br B rexrt B nltmnft syl (= A (-oo)) adantl A (-oo) B (<) breq2 (e. B (RR)) adantr mtbird (br A (<) B) a1d mnfxr (-oo) pnfnltt ax-mp B (+oo) A (-oo) (<) breq12 mtbiri ancoms (br A (<) B) a1d mnfxr (-oo) xrltnrt ax-mp A (-oo) B (-oo) (<) breq12 mtbiri (-. (br B (<) A)) pm2.21d 3jaodan 3jaoian A elxr B elxr syl2anb)) thm (xrlttrit () () (-> (/\ (e. A (RR*)) (e. B (RR*))) (<-> (br A (<) B) (-. (\/ (= A B) (br B (<) A))))) (A xrltnrt (= A B) adantr A B A (<) breq2 (e. A (RR*)) adantl mtbid ex (e. B (RR*)) adantr B A xrltnsymt ancoms jaod A B axlttri biimprd con1d A ltpnft (= B (+oo)) adantr B (+oo) A (<) breq2 (e. A (RR)) adantl mpbird (\/ (= A B) (br B (<) A)) pm2.21nd A mnfltt (= B (-oo)) adantr B (-oo) (<) A breq1 (e. A (RR)) adantl mpbird (br B (<) A) (= A B) olc syl (-. (br A (<) B)) a1d 3jaodan B ltpnft (= A (+oo)) adantl A (+oo) B (<) breq2 (e. B (RR)) adantr mpbird (br B (<) A) (= A B) olc syl (-. (br A (<) B)) a1d A (+oo) B eqtr3t (= A B) (br B (<) A) orc syl (-. (br A (<) B)) a1d mnfltpnf B (-oo) A (+oo) (<) breq12 mpbiri ancoms (br B (<) A) (= A B) olc syl (-. (br A (<) B)) a1d 3jaodan B mnfltt (= A (-oo)) adantl A (-oo) (<) B breq1 (e. B (RR)) adantr mpbird (\/ (= A B) (br B (<) A)) pm2.21nd mnfltpnf A (-oo) B (+oo) (<) breq12 mpbiri (\/ (= A B) (br B (<) A)) pm2.21nd A (-oo) B eqtr3t (= A B) (br B (<) A) orc syl (-. (br A (<) B)) a1d 3jaodan 3jaoian A elxr B elxr syl2anb impbid con2bid)) thm (xrlttrt () () (-> (/\/\ (e. A (RR*)) (e. B (RR*)) (e. C (RR*))) (-> (/\ (br A (<) B) (br B (<) C)) (br A (<) C))) (A B C axlttrn 3expa an1rs C rexrt C pnfnltt syl (= B (+oo)) adantr B (+oo) (<) C breq1 (e. C (RR)) adantl mtbird (br A (<) C) pm2.21d (e. A (RR)) adantll (br A (<) B) adantld A rexrt A nltmnft syl (= B (-oo)) adantr B (-oo) A (<) breq2 (e. A (RR)) adantl mtbird (br A (<) C) pm2.21d (e. C (RR)) adantlr (br B (<) C) adantrd 3jaodan B elxr sylan2b an1rs A ltpnft (= C (+oo)) adantr C (+oo) A (<) breq2 (e. A (RR)) adantl mpbird (e. B (RR*)) adantlr (/\ (br A (<) B) (br B (<) C)) a1d B nltmnft (= C (-oo)) adantr C (-oo) B (<) breq2 (e. B (RR*)) adantl mtbird (br A (<) C) pm2.21d (br A (<) B) adantld (e. A (RR)) adantll 3jaodan anasss B pnfnltt (= A (+oo)) adantl A (+oo) (<) B breq1 (e. B (RR*)) adantr mtbird (br A (<) C) pm2.21d (br B (<) C) adantrd (\/\/ (e. C (RR)) (= C (+oo)) (= C (-oo))) adantrr C mnfltt (= A (-oo)) adantl A (-oo) (<) C breq1 (e. C (RR)) adantr mpbird (/\ (br A (<) B) (br B (<) C)) a1d (e. B (RR*)) adantlr mnfltpnf A (-oo) C (+oo) (<) breq12 mpbiri (/\ (br A (<) B) (br B (<) C)) a1d (e. B (RR*)) adantlr B nltmnft (= C (-oo)) adantr C (-oo) B (<) breq2 (e. B (RR*)) adantl mtbird (br A (<) C) pm2.21d (br A (<) B) adantld (= A (-oo)) adantll 3jaodan anasss 3jaoian 3impb C elxr syl3an3b A elxr syl3an1b)) thm (xrltso ((x y) (x z) (y z)) () (Or (<) (RR*)) ((cv x) (cv y) xrlttrit (e. (cv z) (RR*)) 3adant3 (cv x) (cv y) (cv z) xrlttrt jca so)) thm (xrlttri3t () () (-> (/\ (e. A (RR*)) (e. B (RR*))) (<-> (= A B) (/\ (-. (br A (<) B)) (-. (br B (<) A))))) (xrltso (<) (RR*) A B sotrieq2 mpan)) thm (xrleloet () () (-> (/\ (e. A (RR*)) (e. B (RR*))) (<-> (br A (<_) B) (\/ (br A (<) B) (= A B)))) (A B xrlenltt B A xrlttrit ancoms con2bid B A eqcom (br A (<) B) orbi1i (= A B) (br A (<) B) orcom bitr syl5rbbr bitrd)) thm (xrleltnet () () (-> (/\/\ (e. A (RR*)) (e. B (RR*)) (br A (<_) B)) (<-> (br A (<) B) (-. (= A B)))) (A B xrlttri3t (-. (br A (<) B)) (-. (br B (<) A)) pm3.26 syl6bi (br A (<_) B) adantr A B xrleloet biimpa ord impbid con2bid 3impa)) thm (xrltlet () () (-> (/\ (e. A (RR*)) (e. B (RR*))) (-> (br A (<) B) (br A (<_) B))) (A B xrleloet (br A (<) B) (= A B) orc syl5bir)) thm (xrlelttrt () () (-> (/\/\ (e. A (RR*)) (e. B (RR*)) (e. C (RR*))) (-> (/\ (br A (<_) B) (br B (<) C)) (br A (<) C))) (A B xrleloet (e. C (RR*)) 3adant3 A B C xrlttrt exp3a A B (<) C breq1 biimprd (/\/\ (e. A (RR*)) (e. B (RR*)) (e. C (RR*))) a1i jaod sylbid imp3a)) thm (xrltletrt () () (-> (/\/\ (e. A (RR*)) (e. B (RR*)) (e. C (RR*))) (-> (/\ (br A (<) B) (br B (<_) C)) (br A (<) C))) (B C xrleloet (e. A (RR*)) 3adant1 A B C xrlttrt exp3a com23 B C A (<) breq2 biimpd (/\/\ (e. A (RR*)) (e. B (RR*)) (e. C (RR*))) a1i jaod sylbid com23 imp3a)) thm (xrletrt () () (-> (/\/\ (e. A (RR*)) (e. B (RR*)) (e. C (RR*))) (-> (/\ (br A (<_) B) (br B (<_) C)) (br A (<_) C))) (B C xrleloet (e. A (RR*)) 3adant1 (br A (<_) B) adantr A B C xrlelttrt A C xrltlet (e. B (RR*)) 3adant2 syld exp3a imp B C A (<_) breq2 biimpcd (/\/\ (e. A (RR*)) (e. B (RR*)) (e. C (RR*))) adantl jaod sylbid ex imp3a)) thm (xrltnet () () (-> (/\ (e. A (RR*)) (e. B (RR*))) (-> (br A (<) B) (-. (= A B)))) (xrltso (<) (RR*) A B sotrieq mpan con2bid (br A (<) B) (br B (<) A) orc syl5bi)) thm (nltpnftt () () (-> (e. A (RR*)) (<-> (= A (+oo)) (-. (br A (<) (+oo))))) (pnfxr (+oo) xrltnrt ax-mp A (+oo) (<) (+oo) breq1 mtbiri (e. A (RR*)) a1i A pnfget pnfxr A (+oo) xrleloet mpan2 mpbid ord impbid)) thm (ngtmnftt () () (-> (e. A (RR*)) (<-> (= A (-oo)) (-. (br (-oo) (<) A)))) (mnfxr (-oo) xrltnrt ax-mp A (-oo) (-oo) (<) breq2 mtbiri (e. A (RR*)) a1i A mnflet mnfxr (-oo) A xrleloet mpan mpbid ord (-oo) A eqcom syl6ib impbid)) thm (xrrebndt () () (-> (e. A (RR*)) (<-> (e. A (RR)) (/\ (br (-oo) (<) A) (br A (<) (+oo))))) (A mnfltt A ltpnft jca (e. A (RR*)) a1i A nltpnftt A ngtmnftt orbi12d (br (-oo) (<) A) (br A (<) (+oo)) ianor (-. (br (-oo) (<) A)) (-. (br A (<) (+oo))) orcom bitr2 syl6bb con2bid A elxr biimp (e. A (RR)) (= A (+oo)) (= A (-oo)) 3orass (e. A (RR)) (\/ (= A (+oo)) (= A (-oo))) orcom bitr sylib ord sylbid impbid)) thm (xrret () () (-> (/\ (/\ (e. A (RR*)) (e. B (RR))) (/\ (br (-oo) (<) A) (br A (<_) B))) (e. A (RR))) ((br (-oo) (<) A) (br A (<_) B) pm3.26 (/\ (e. A (RR*)) (e. B (RR))) adantl B ltpnft (e. A (RR*)) adantl pnfxr A B (+oo) xrlelttrt mp3an3 B rexrt sylan2 mpan2d imp (br (-oo) (<) A) adantrl jca A xrrebndt (e. B (RR)) (/\ (br (-oo) (<) A) (br A (<_) B)) ad2antrr mpbird)) thm (eqlet () () (-> (/\ (e. A (RR)) (= A B)) (br A (<_) B)) (A B A (<_) breq2 biimpac A leidt sylan)) thm (lttri2 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (-. (= A B)) (\/ (br A (<) B) (br B (<) A))) (lt.1 lt.2 A B lttri2t mp2an)) thm (lttri3 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (= A B) (/\ (-. (br A (<) B)) (-. (br B (<) A)))) (lt.1 lt.2 A B lttri3t mp2an)) thm (letri3 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (= A B) (/\ (br A (<_) B) (br B (<_) A))) (lt.1 lt.2 A B letri3t mp2an)) thm (leloe () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<_) B) (\/ (br A (<) B) (= A B))) (lt.1 lt.2 A B leloet mp2an)) thm (ltlen () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<) B) (/\ (br A (<_) B) (-. (= A B)))) (lt.1 lt.2 A B ltlent mp2an)) thm (ltnsym () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (br A (<) B) (-. (br B (<) A))) (lt.1 lt.2 A B ltnsymt mp2an)) thm (lenlt () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<_) B) (-. (br B (<) A))) (lt.1 lt.2 A B lenltt mp2an)) thm (ltnle () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<) B) (-. (br B (<_) A))) (lt.2 lt.1 lenlt con2bii)) thm (ltle () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (br A (<) B) (br A (<_) B)) (lt.1 lt.2 A B ltlet mp2an)) thm (ltlei () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (ltlei.1 (br A (<) B))) (br A (<_) B) (ltlei.1 lt.1 lt.2 ltle ax-mp)) thm (eqle () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (= A B) (br A (<_) B)) (lt.1 lt.2 letri3 pm3.26bd)) thm (ltne () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (br A (<) B) (-. (= A B))) (lt.1 lt.2 A B ltnet mp2an)) thm (letri () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (\/ (br A (<_) B) (br B (<_) A)) (lt.2 lt.1 ltnle lt.2 lt.1 ltle sylbir orri)) thm (lttr () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (-> (/\ (br A (<) B) (br B (<) C)) (br A (<) C)) (lt.1 lt.2 lt.3 A B C axlttrn mp3an)) thm (lelttr () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (-> (/\ (br A (<_) B) (br B (<) C)) (br A (<) C)) (lt.1 lt.2 lt.3 A B C lelttrt mp3an)) thm (ltletr () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (-> (/\ (br A (<) B) (br B (<_) C)) (br A (<) C)) (lt.1 lt.2 lt.3 A B C ltletrt mp3an)) thm (letr () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (-> (/\ (br A (<_) B) (br B (<_) C)) (br A (<_) C)) (lt.1 lt.2 lt.3 A B C letrt mp3an)) thm (le2tri3 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (/\/\ (br A (<_) B) (br B (<_) C) (br C (<_) A)) (/\/\ (= A B) (= B C) (= C A))) (lt.1 lt.2 letri3 biimpr lt.2 lt.3 lt.1 letr sylan2 3impb lt.2 lt.3 letri3 biimpr lt.3 lt.1 lt.2 letr sylan2 3impb 3comr lt.1 lt.3 letri3 biimpr eqcomd lt.1 lt.2 lt.3 letr sylan 3impa 3jca lt.1 lt.2 eqle lt.2 lt.3 eqle lt.3 lt.1 eqle 3anim123i impbi)) thm (ltadd2 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br A (<) B) (br (opr C (+) A) (<) (opr C (+) B))) (lt.1 lt.2 lt.3 A B C axltadd mp3an lt.3 lt.1 readdcl lt.3 lt.2 readdcl lt.3 renegcl (opr C (+) A) (opr C (+) B) (-u C) axltadd mp3an lt.3 renegcl recn lt.3 recn addcom lt.3 recn negid eqtr (+) A opreq1i lt.3 renegcl recn lt.3 recn lt.1 recn addass lt.1 recn addid2 3eqtr3 lt.3 renegcl recn lt.3 recn addcom lt.3 recn negid eqtr (+) B opreq1i lt.3 renegcl recn lt.3 recn lt.2 recn addass lt.2 recn addid2 3eqtr3 3brtr3g impbi)) thm (ltadd1 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br A (<) B) (br (opr A (+) C) (<) (opr B (+) C))) (lt.1 lt.2 lt.3 ltadd2 lt.3 recn lt.1 recn addcom lt.3 recn lt.2 recn addcom (<) breq12i bitr)) thm (leadd1 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br A (<_) B) (br (opr A (+) C) (<_) (opr B (+) C))) (lt.2 lt.1 lt.3 ltadd1 negbii lt.1 lt.2 lenlt lt.1 lt.3 readdcl lt.2 lt.3 readdcl lenlt 3bitr4)) thm (leadd2 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br A (<_) B) (br (opr C (+) A) (<_) (opr C (+) B))) (lt.1 lt.2 lt.3 leadd1 lt.1 recn lt.3 recn addcom lt.2 recn lt.3 recn addcom (<_) breq12i bitr)) thm (ltsubadd () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br (opr A (-) B) (<) C) (br A (<) (opr C (+) B))) (lt.1 lt.2 renegcl readdcl lt.3 lt.2 ltadd1 lt.1 recn lt.2 recn negsub (<) C breq1i lt.1 recn lt.2 renegcl recn lt.2 recn addass lt.2 renegcl recn lt.2 recn addcom lt.2 recn negid eqtr A (+) opreq2i lt.1 recn addid1 3eqtr (<) (opr C (+) B) breq1i 3bitr3)) thm (lesubadd () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br (opr A (-) B) (<_) C) (br A (<_) (opr C (+) B))) (lt.1 lt.2 lt.3 ltsubadd lt.1 recn lt.2 recn lt.3 recn subadd lt.2 recn lt.3 recn addcom A eqeq1i (opr C (+) B) A eqcom 3bitr orbi12i lt.1 lt.2 resubcl lt.3 leloe lt.1 lt.3 lt.2 readdcl leloe 3bitr4)) thm (lt2add () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR))) (lt.4 (e. D (RR)))) (-> (/\ (br A (<) C) (br B (<) D)) (br (opr A (+) B) (<) (opr C (+) D))) (lt.1 lt.2 readdcl lt.3 lt.2 readdcl lt.3 lt.4 readdcl lttr lt.1 lt.3 lt.2 ltadd1 lt.2 lt.4 lt.3 ltadd2 syl2anb)) thm (le2add () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR))) (lt.4 (e. D (RR)))) (-> (/\ (br A (<_) C) (br B (<_) D)) (br (opr A (+) B) (<_) (opr C (+) D))) (lt.1 lt.2 readdcl lt.3 lt.2 readdcl lt.3 lt.4 readdcl letr lt.1 lt.3 lt.2 leadd1 lt.2 lt.4 lt.3 leadd2 syl2anb)) thm (addgt0 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (br (0) (<) (opr A (+) B))) (0re lt.1 lt.1 lt.2 readdcl lttr 0re lt.2 lt.1 ltadd2 lt.1 recn addid1 (<) (opr A (+) B) breq1i bitr sylan2b)) thm (addge0 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (br (0) (<_) (opr A (+) B))) (lt.1 lt.2 addgt0 0re lt.1 lt.2 readdcl ltle syl (0) A (+) B opreq1 lt.2 recn addid2 syl5eqr (0) (<) breq2d biimpa 0re lt.1 lt.2 readdcl ltle syl (0) B A (+) opreq2 lt.1 recn addid1 syl5eqr (0) (<) breq2d biimpac 0re lt.1 lt.2 readdcl ltle syl (0) A (0) B (+) opreq12 0cn addid1 syl5eqr 0re lt.1 lt.2 readdcl eqle syl ccase 0re lt.1 leloe 0re lt.2 leloe syl2anb)) thm (addgegt0 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<) B)) (br (0) (<) (opr A (+) B))) (0re lt.1 leloe lt.1 lt.2 addgt0 ex (0) A (+) B opreq1 lt.2 recn addid2 syl5eqr (0) (<) breq2d biimpd jaoi sylbi imp)) thm (addgt0i () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (addgt0i.3 (br (0) (<) A)) (addgt0i.4 (br (0) (<) B))) (br (0) (<) (opr A (+) B)) (addgt0i.3 addgt0i.4 lt.1 lt.2 addgt0 mp2an)) thm (add20 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (opr A (+) B) (0)) (/\ (= A (0)) (= B (0))))) (lt.1 lt.2 readdcl 0re lttri3 pm3.27bd lt.1 lt.2 addgt0 nsyl3 (/\ (= A (0)) (= B (0))) pm2.21d (0) A (+) B opreq1 lt.2 recn addid2 syl5eqr (0) eqeq1d biimprd (0) A eqcom biimp jctild (0) B A (+) opreq2 lt.1 recn addid1 syl5eqr (0) eqeq1d biimprd (0) B eqcom biimp jctird ccase2 0re lt.1 leloe 0re lt.2 leloe syl2anb A (0) (+) B opreq1 B (0) (0) (+) opreq2 0cn addid1 syl6eq sylan9eq (/\ (br (0) (<_) A) (br (0) (<_) B)) a1i impbid)) thm (ltneg () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<) B) (br (-u B) (<) (-u A))) (lt.1 lt.2 lt.1 renegcl lt.2 renegcl readdcl ltadd1 lt.1 recn negid (+) (-u B) opreq1i lt.1 recn lt.1 renegcl recn lt.2 renegcl recn addass lt.2 renegcl recn addid2 3eqtr3 lt.2 recn lt.1 renegcl recn lt.2 renegcl recn add12 lt.2 recn negid (-u A) (+) opreq2i lt.1 renegcl recn addid1 3eqtr (<) breq12i bitr)) thm (leneg () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<_) B) (br (-u B) (<_) (-u A))) (lt.1 lt.2 ltneg lt.2 recn lt.1 recn neg11 B A eqcom bitr2 orbi12i lt.1 lt.2 leloe lt.2 renegcl lt.1 renegcl leloe 3bitr4)) thm (ltnegcon2 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<) (-u B)) (br B (<) (-u A))) (lt.1 lt.2 renegcl ltneg lt.2 recn negneg (<) (-u A) breq1i bitr)) thm (mulgt0 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (br (0) (<) (opr A (x.) B))) (lt.1 lt.2 A B axmulgt0 mp2an)) thm (mulge0 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (br (0) (<_) (opr A (x.) B))) (lt.1 lt.2 mulgt0 0re lt.1 lt.2 remulcl ltle syl (0) A (x.) B opreq1 lt.2 recn mul02 syl5eqr 0re lt.1 lt.2 remulcl eqle syl (0) B A (x.) opreq2 lt.1 recn mul01 syl5eqr 0re lt.1 lt.2 remulcl eqle syl ccase2 0re lt.1 leloe 0re lt.2 leloe syl2anb)) thm (mulgt0i () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (mulgt0i.3 (br (0) (<) A)) (mulgt0i.4 (br (0) (<) B))) (br (0) (<) (opr A (x.) B)) (mulgt0i.3 mulgt0i.4 lt.1 lt.2 mulgt0 mp2an)) thm (ltnr () ((lt.1 (e. A (RR)))) (-. (br A (<) A)) (lt.1 A ltnrt ax-mp)) thm (leid () ((lt.1 (e. A (RR)))) (br A (<_) A) (lt.1 A leidt ax-mp)) thm (gt0ne0 () ((lt.1 (e. A (RR)))) (-> (br (0) (<) A) (=/= A (0))) (0re lt.1 ltne (0) A df-ne sylibr A (0) necom sylibr)) thm (lesub0 () ((lt.1 (e. A (RR))) (lesub0.1 (e. B (RR)))) (<-> (/\ (br (0) (<_) A) (br B (<_) (opr B (-) A))) (= A (0))) (0re lt.1 renegcl lesub0.1 leadd2 lt.1 0re leneg neg0 (<_) (-u A) breq1i bitr2 lesub0.1 recn addid1 lesub0.1 recn lt.1 recn negsub (<_) breq12i 3bitr3r (br (0) (<_) A) anbi1i (br (0) (<_) A) (br B (<_) (opr B (-) A)) ancom lt.1 0re letri3 3bitr4)) thm (msqgt0 () ((lt.1 (e. A (RR)))) (-> (-. (= A (0))) (br (0) (<) (opr A (x.) A))) (lt.1 0re lttri2 lt.1 0re ltneg neg0 (<) (-u A) breq1i bitr lt.1 renegcl lt.1 renegcl mulgt0 anidms lt.1 recn lt.1 recn mul2neg syl6breq sylbi lt.1 lt.1 mulgt0 anidms jaoi sylbi)) thm (msqge0 () ((lt.1 (e. A (RR)))) (br (0) (<_) (opr A (x.) A)) (A (0) A (x.) opreq2 lt.1 recn mul01 syl6req (= (0) (opr A (x.) A)) (br (0) (<) (opr A (x.) A)) olc syl 0re lt.1 lt.1 remulcl leloe sylibr lt.1 msqgt0 0re lt.1 lt.1 remulcl ltle syl pm2.61i)) thm (gt0ne0i () ((gt0ne0i.1 (e. A (RR))) (gt0ne0i.2 (br (0) (<) A))) (=/= A (0)) (gt0ne0i.2 gt0ne0i.1 gt0ne0 ax-mp)) thm (gt0ne0t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (=/= A (0))) (A (if (e. A (RR)) A (0)) (0) (<) breq2 A (if (e. A (RR)) A (0)) (0) neeq1 imbi12d 0re A elimel gt0ne0 dedth imp)) thm (letrit () () (-> (/\ (e. A (RR)) (e. B (RR))) (\/ (br A (<_) B) (br B (<_) A))) (A (if (e. A (RR)) A (0)) (<_) B breq1 A (if (e. A (RR)) A (0)) B (<_) breq2 orbi12d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<_) breq2 B (if (e. B (RR)) B (0)) (<_) (if (e. A (RR)) A (0)) breq1 orbi12d 0re A elimel 0re B elimel letri dedth2h)) thm (lelttrit () () (-> (/\ (e. A (RR)) (e. B (RR))) (\/ (br A (<_) B) (br B (<) A))) (A B eqlet (br A (<_) B) (br B (<) A) orc syl ex (e. B (RR)) adantr A B lttri2t A B ltlet (br B (<) A) orim1d sylbid pm2.61d)) thm (ltadd1t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<) B) (br (opr A (+) C) (<) (opr B (+) C)))) (A (if (e. A (RR)) A (0)) (<) B breq1 A (if (e. A (RR)) A (0)) (+) C opreq1 (<) (opr B (+) C) breq1d bibi12d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<) breq2 B (if (e. B (RR)) B (0)) (+) C opreq1 (opr (if (e. A (RR)) A (0)) (+) C) (<) breq2d bibi12d C (if (e. C (RR)) C (0)) (if (e. A (RR)) A (0)) (+) opreq2 C (if (e. C (RR)) C (0)) (if (e. B (RR)) B (0)) (+) opreq2 (<) breq12d (br (if (e. A (RR)) A (0)) (<) (if (e. B (RR)) B (0))) bibi2d 0re A elimel 0re B elimel 0re C elimel ltadd1 dedth3h)) thm (ltadd2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<) B) (br (opr C (+) A) (<) (opr C (+) B)))) (A B C ltadd1t A C axaddcom A recnt C recnt syl2an (e. B (RR)) 3adant2 B C axaddcom B recnt C recnt syl2an (e. A (RR)) 3adant1 (<) breq12d bitrd)) thm (leadd1t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<_) B) (br (opr A (+) C) (<_) (opr B (+) C)))) (A (if (e. A (RR)) A (0)) (<_) B breq1 A (if (e. A (RR)) A (0)) (+) C opreq1 (<_) (opr B (+) C) breq1d bibi12d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<_) breq2 B (if (e. B (RR)) B (0)) (+) C opreq1 (opr (if (e. A (RR)) A (0)) (+) C) (<_) breq2d bibi12d C (if (e. C (RR)) C (0)) (if (e. A (RR)) A (0)) (+) opreq2 C (if (e. C (RR)) C (0)) (if (e. B (RR)) B (0)) (+) opreq2 (<_) breq12d (br (if (e. A (RR)) A (0)) (<_) (if (e. B (RR)) B (0))) bibi2d 0re A elimel 0re B elimel 0re C elimel leadd1 dedth3h)) thm (leadd2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<_) B) (br (opr C (+) A) (<_) (opr C (+) B)))) (A B C leadd1t A C axaddcom A recnt C recnt syl2an (e. B (RR)) 3adant2 B C axaddcom B recnt C recnt syl2an (e. A (RR)) 3adant1 (<_) breq12d bitrd)) thm (addge01t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<_) B) (br A (<_) (opr A (+) B)))) (0re (0) B A leadd2t mp3an1 ancoms A recnt A ax0id syl (e. B (RR)) adantr (<_) (opr A (+) B) breq1d bitrd)) thm (addge02t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<_) B) (br A (<_) (opr B (+) A)))) (A B addge01t A B axaddcom A recnt B recnt syl2an A (<_) breq2d bitrd)) thm (addge01tOLD () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (0) (<_) B)) (br A (<_) (opr A (+) B))) (A recnt A ax0id syl (e. B (RR)) (br (0) (<_) B) 3ad2ant1 0re (0) B A leadd2t mp3an1 ancoms biimp3a eqbrtrrd)) thm (subge0t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<_) (opr A (-) B)) (br B (<_) A))) (B A (-u B) leadd1t (e. A (RR)) (e. B (RR)) pm3.27 (e. A (RR)) (e. B (RR)) pm3.26 B renegclt (e. A (RR)) adantl syl3anc B negidt (e. A (CC)) adantl A B negsubt (<_) breq12d A recnt B recnt syl2an bitr2d)) thm (subge0 () ((subge0.1 (e. A (RR))) (subge0.2 (e. B (RR)))) (<-> (br (0) (<_) (opr A (-) B)) (br B (<_) A)) (subge0.1 subge0.2 A B subge0t mp2an)) thm (ltsubaddt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<) C) (br A (<) (opr C (+) B)))) (A (if (e. A (RR)) A (0)) (-) B opreq1 (<) C breq1d A (if (e. A (RR)) A (0)) (<) (opr C (+) B) breq1 bibi12d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (-) opreq2 (<) C breq1d B (if (e. B (RR)) B (0)) C (+) opreq2 (if (e. A (RR)) A (0)) (<) breq2d bibi12d C (if (e. C (RR)) C (0)) (opr (if (e. A (RR)) A (0)) (-) (if (e. B (RR)) B (0))) (<) breq2 C (if (e. C (RR)) C (0)) (+) (if (e. B (RR)) B (0)) opreq1 (if (e. A (RR)) A (0)) (<) breq2d bibi12d 0re A elimel 0re B elimel 0re C elimel ltsubadd dedth3h)) thm (ltsubadd2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<) C) (br A (<) (opr B (+) C)))) (A B C ltsubaddt B C axaddcom B recnt C recnt syl2an (e. A (RR)) 3adant1 A (<) breq2d bitr4d)) thm (lesubaddt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<_) C) (br A (<_) (opr C (+) B)))) (A (if (e. A (RR)) A (0)) (-) B opreq1 (<_) C breq1d A (if (e. A (RR)) A (0)) (<_) (opr C (+) B) breq1 bibi12d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (-) opreq2 (<_) C breq1d B (if (e. B (RR)) B (0)) C (+) opreq2 (if (e. A (RR)) A (0)) (<_) breq2d bibi12d C (if (e. C (RR)) C (0)) (opr (if (e. A (RR)) A (0)) (-) (if (e. B (RR)) B (0))) (<_) breq2 C (if (e. C (RR)) C (0)) (+) (if (e. B (RR)) B (0)) opreq1 (if (e. A (RR)) A (0)) (<_) breq2d bibi12d 0re A elimel 0re B elimel 0re C elimel lesubadd dedth3h)) thm (lesubadd2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<_) C) (br A (<_) (opr B (+) C)))) (A B C lesubaddt B C axaddcom B recnt C recnt syl2an (e. A (RR)) 3adant1 A (<_) breq2d bitr4d)) thm (ltaddsubt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (+) B) (<) C) (br A (<) (opr C (-) B)))) (C B A lesubaddt 3com13 (opr C (-) B) A lenltt C B resubclt sylan 3impa 3com13 C (opr A (+) B) lenltt A B axaddrcl sylan2 3impb 3coml 3bitr3rd con4bid)) thm (ltaddsub2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (+) B) (<) C) (br B (<) (opr C (-) A)))) (A B axaddcom A recnt B recnt syl2an (e. C (RR)) 3adant3 (<) C breq1d B A C ltaddsubt 3com12 bitrd)) thm (leaddsubt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (+) B) (<_) C) (br A (<_) (opr C (-) B)))) (C B A ltsubaddt 3com13 (opr C (-) B) A ltnlet C B resubclt sylan 3impa 3com13 C (opr A (+) B) ltnlet A B axaddrcl sylan2 3impb 3coml 3bitr3rd con4bid)) thm (leaddsub2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (+) B) (<_) C) (br B (<_) (opr C (-) A)))) (A B axaddcom A recnt B recnt syl2an (e. C (RR)) 3adant3 (<_) C breq1d B A C leaddsubt 3com12 bitrd)) thm (sublet () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<_) C) (br (opr A (-) C) (<_) B))) (A B C lesubaddt A C B lesubadd2t 3com23 bitr4d)) thm (lesubt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<_) (opr B (-) C)) (br C (<_) (opr B (-) A)))) (A C B leaddsubt 3com23 A C B leaddsub2t 3com23 bitr3d)) thm (ltsubadd2 () ((ltsubadd.1 (e. A (RR))) (ltsubadd.2 (e. B (RR))) (ltsubadd.3 (e. C (RR)))) (<-> (br (opr A (-) B) (<) C) (br A (<) (opr B (+) C))) (ltsubadd.1 ltsubadd.2 ltsubadd.3 A B C ltsubadd2t mp3an)) thm (lesubadd2 () ((ltsubadd.1 (e. A (RR))) (ltsubadd.2 (e. B (RR))) (ltsubadd.3 (e. C (RR)))) (<-> (br (opr A (-) B) (<_) C) (br A (<_) (opr B (+) C))) (ltsubadd.1 ltsubadd.2 ltsubadd.3 A B C lesubadd2t mp3an)) thm (ltaddsub () ((ltsubadd.1 (e. A (RR))) (ltsubadd.2 (e. B (RR))) (ltsubadd.3 (e. C (RR)))) (<-> (br (opr A (+) B) (<) C) (br A (<) (opr C (-) B))) (ltsubadd.1 ltsubadd.2 ltsubadd.3 A B C ltaddsubt mp3an)) thm (ltmullem () ((ltsubadd.1 (e. A (RR))) (ltsubadd.2 (e. B (RR))) (ltsubadd.3 (e. C (RR)))) (-> (br (0) (<) C) (-> (br A (<) B) (br (opr A (x.) C) (<) (opr B (x.) C)))) (ltsubadd.2 ltsubadd.1 resubcl ltsubadd.3 mulgt0 expcom ltsubadd.1 recn addid2 (<) B breq1i 0re ltsubadd.1 ltsubadd.2 ltaddsub bitr3 ltsubadd.2 recn ltsubadd.1 recn ltsubadd.3 recn subdir (0) (<) breq2i 0re ltsubadd.1 ltsubadd.3 remulcl ltsubadd.2 ltsubadd.3 remulcl ltaddsub ltsubadd.1 ltsubadd.3 remulcl recn addid2 (<) (opr B (x.) C) breq1i 3bitr2r 3imtr4g)) thm (ltsub23t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<) C) (br (opr A (-) C) (<) B))) (A B C ltsubaddt A C B ltsubadd2t 3com23 bitr4d)) thm (ltsub13t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<) (opr B (-) C)) (br C (<) (opr B (-) A)))) (A C B ltaddsubt A C B ltaddsub2t bitr3d 3com23)) thm (lt2addt () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (e. C (RR)) (e. D (RR)))) (-> (/\ (br A (<) C) (br B (<) D)) (br (opr A (+) B) (<) (opr C (+) D)))) (A (if (e. A (RR)) A (0)) (<) C breq1 (br B (<) D) anbi1d A (if (e. A (RR)) A (0)) (+) B opreq1 (<) (opr C (+) D) breq1d imbi12d B (if (e. B (RR)) B (0)) (<) D breq1 (br (if (e. A (RR)) A (0)) (<) C) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (+) opreq2 (<) (opr C (+) D) breq1d imbi12d C (if (e. C (RR)) C (0)) (if (e. A (RR)) A (0)) (<) breq2 (br (if (e. B (RR)) B (0)) (<) D) anbi1d C (if (e. C (RR)) C (0)) (+) D opreq1 (opr (if (e. A (RR)) A (0)) (+) (if (e. B (RR)) B (0))) (<) breq2d imbi12d D (if (e. D (RR)) D (0)) (if (e. B (RR)) B (0)) (<) breq2 (br (if (e. A (RR)) A (0)) (<) (if (e. C (RR)) C (0))) anbi2d D (if (e. D (RR)) D (0)) (if (e. C (RR)) C (0)) (+) opreq2 (opr (if (e. A (RR)) A (0)) (+) (if (e. B (RR)) B (0))) (<) breq2d imbi12d 0re A elimel 0re B elimel 0re C elimel 0re D elimel lt2add dedth4h)) thm (le2addt () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (e. C (RR)) (e. D (RR)))) (-> (/\ (br A (<_) C) (br B (<_) D)) (br (opr A (+) B) (<_) (opr C (+) D)))) (A (if (e. A (RR)) A (0)) (<_) C breq1 (br B (<_) D) anbi1d A (if (e. A (RR)) A (0)) (+) B opreq1 (<_) (opr C (+) D) breq1d imbi12d B (if (e. B (RR)) B (0)) (<_) D breq1 (br (if (e. A (RR)) A (0)) (<_) C) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (+) opreq2 (<_) (opr C (+) D) breq1d imbi12d C (if (e. C (RR)) C (0)) (if (e. A (RR)) A (0)) (<_) breq2 (br (if (e. B (RR)) B (0)) (<_) D) anbi1d C (if (e. C (RR)) C (0)) (+) D opreq1 (opr (if (e. A (RR)) A (0)) (+) (if (e. B (RR)) B (0))) (<_) breq2d imbi12d D (if (e. D (RR)) D (0)) (if (e. B (RR)) B (0)) (<_) breq2 (br (if (e. A (RR)) A (0)) (<_) (if (e. C (RR)) C (0))) anbi2d D (if (e. D (RR)) D (0)) (if (e. C (RR)) C (0)) (+) opreq2 (opr (if (e. A (RR)) A (0)) (+) (if (e. B (RR)) B (0))) (<_) breq2d imbi12d 0re A elimel 0re B elimel 0re C elimel 0re D elimel le2add dedth4h)) thm (addgt0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (br (0) (<) (opr A (+) B))) (A (if (e. A (RR)) A (0)) (0) (<) breq2 (br (0) (<) B) anbi1d A (if (e. A (RR)) A (0)) (+) B opreq1 (0) (<) breq2d imbi12d B (if (e. B (RR)) B (0)) (0) (<) breq2 (br (0) (<) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (+) opreq2 (0) (<) breq2d imbi12d 0re A elimel 0re B elimel addgt0 dedth2h imp)) thm (addgegt0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<) B))) (br (0) (<) (opr A (+) B))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 (br (0) (<) B) anbi1d A (if (e. A (RR)) A (0)) (+) B opreq1 (0) (<) breq2d imbi12d B (if (e. B (RR)) B (0)) (0) (<) breq2 (br (0) (<_) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (+) opreq2 (0) (<) breq2d imbi12d 0re A elimel 0re B elimel addgegt0 dedth2h imp)) thm (addgtge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<_) B))) (br (0) (<) (opr A (+) B))) (B A addgegt0t (e. A (RR)) (e. B (RR)) ancom (br (0) (<) A) (br (0) (<_) B) ancom syl2anb A B axaddcom A recnt B recnt syl2an (/\ (br (0) (<) A) (br (0) (<_) B)) adantr breqtrrd)) thm (addge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<_) B))) (br (0) (<_) (opr A (+) B))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 (br (0) (<_) B) anbi1d A (if (e. A (RR)) A (0)) (+) B opreq1 (0) (<_) breq2d imbi12d B (if (e. B (RR)) B (0)) (0) (<_) breq2 (br (0) (<_) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (+) opreq2 (0) (<_) breq2d imbi12d 0re A elimel 0re B elimel addge0 dedth2h imp)) thm (ltaddpost () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<) A) (br B (<) (opr B (+) A)))) (0re (0) A B ltadd2t mp3an1 B recnt B ax0id syl (e. A (RR)) adantl (<) (opr B (+) A) breq1d bitrd)) thm (ltaddpos2t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<) A) (br B (<) (opr A (+) B)))) (A B ltaddpost A B axaddcom A recnt B recnt syl2an B (<) breq2d bitr4d)) thm (subge02t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<_) B) (br (opr A (-) B) (<_) A))) (A B addge01t A B A lesubaddt (e. A (RR)) (e. B (RR)) pm3.26 (e. A (RR)) (e. B (RR)) pm3.27 (e. A (RR)) (e. B (RR)) pm3.26 syl3anc bitr4d)) thm (ltsubpost () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<) A) (br (opr B (-) A) (<) B))) (A B ltaddpost B A B ltsubaddt (e. A (RR)) (e. B (RR)) pm3.27 (e. A (RR)) (e. B (RR)) pm3.26 (e. A (RR)) (e. B (RR)) pm3.27 syl3anc bitr4d)) thm (posdift () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br (0) (<) (opr B (-) A)))) ((opr B (-) A) A ltaddpost B A resubclt ancoms (e. A (RR)) (e. B (RR)) pm3.26 sylanc A B pncan3t A recnt B recnt syl2an A (<) breq2d bitr2d)) thm (ltnegt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br (-u B) (<) (-u A)))) (A (if (e. A (RR)) A (0)) (<) B breq1 A (if (e. A (RR)) A (0)) negeq (-u B) (<) breq2d bibi12d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<) breq2 B (if (e. B (RR)) B (0)) negeq (<) (-u (if (e. A (RR)) A (0))) breq1d bibi12d 0re A elimel 0re B elimel ltneg dedth2h)) thm (ltnegcon1t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (-u A) (<) B) (br (-u B) (<) A))) ((-u A) B ltnegt A renegclt sylan A recnt A negnegt syl (-u B) (<) breq2d (e. B (RR)) adantr bitrd)) thm (lenegt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) B) (br (-u B) (<_) (-u A)))) (A (if (e. A (RR)) A (0)) (<_) B breq1 A (if (e. A (RR)) A (0)) negeq (-u B) (<_) breq2d bibi12d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<_) breq2 B (if (e. B (RR)) B (0)) negeq (<_) (-u (if (e. A (RR)) A (0))) breq1d bibi12d 0re A elimel 0re B elimel leneg dedth2h)) thm (lenegcon1t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (-u A) (<_) B) (br (-u B) (<_) A))) ((-u A) B lenegt A renegclt sylan A recnt A negnegt syl (-u B) (<_) breq2d (e. B (RR)) adantr bitrd)) thm (lenegcon2t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) (-u B)) (br B (<_) (-u A)))) (A (-u B) lenegt B renegclt sylan2 B recnt B negnegt syl (e. A (RR)) adantl (<_) (-u A) breq1d bitrd)) thm (lesub1t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<_) B) (br (opr A (-) C) (<_) (opr B (-) C)))) (A B (-u C) leadd1t C renegclt syl3an3 A C negsubt (e. B (CC)) 3adant2 B C negsubt (e. A (CC)) 3adant1 (<_) breq12d A recnt B recnt C recnt syl3an bitrd)) thm (lesub2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<_) B) (br (opr C (-) B) (<_) (opr C (-) A)))) (A B lenegt (e. C (RR)) 3adant3 (-u B) (-u A) C leadd2t A renegclt syl3an2 B renegclt syl3an1 3com12 C B negsubt (e. A (CC)) 3adant2 C A negsubt (e. B (CC)) 3adant3 (<_) breq12d 3coml A recnt B recnt C recnt syl3an 3bitrd)) thm (ltsub2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<) B) (br (opr C (-) B) (<) (opr C (-) A)))) (A B ltnegt (e. C (RR)) 3adant3 (-u B) (-u A) C ltadd2t A renegclt syl3an2 B renegclt syl3an1 3com12 C B negsubt (e. A (CC)) 3adant2 C A negsubt (e. B (CC)) 3adant3 (<) breq12d 3coml A recnt B recnt C recnt syl3an 3bitrd)) thm (ltaddpos () ((ltaddpos.1 (e. A (RR))) (ltaddpos.2 (e. B (RR)))) (<-> (br (0) (<) A) (br B (<) (opr B (+) A))) (ltaddpos.1 ltaddpos.2 A B ltaddpost mp2an)) thm (posdif () ((ltaddpos.1 (e. A (RR))) (ltaddpos.2 (e. B (RR)))) (<-> (br A (<) B) (br (0) (<) (opr B (-) A))) (ltaddpos.1 ltaddpos.2 A B posdift mp2an)) thm (ltnegcon1 () ((ltaddpos.1 (e. A (RR))) (ltaddpos.2 (e. B (RR)))) (<-> (br (-u A) (<) B) (br (-u B) (<) A)) (ltaddpos.1 ltaddpos.2 A B ltnegcon1t mp2an)) thm (lenegcon1 () ((ltaddpos.1 (e. A (RR))) (ltaddpos.2 (e. B (RR)))) (<-> (br (-u A) (<_) B) (br (-u B) (<_) A)) (ltaddpos.1 ltaddpos.2 A B lenegcon1t mp2an)) thm (lt0neg1t () () (-> (e. A (RR)) (<-> (br A (<) (0)) (br (0) (<) (-u A)))) (0re A (0) ltnegt mpan2 neg0 (<) (-u A) breq1i syl6bb)) thm (lt0neg2t () () (-> (e. A (RR)) (<-> (br (0) (<) A) (br (-u A) (<) (0)))) (0re (0) A ltnegt mpan neg0 (-u A) (<) breq2i syl6bb)) thm (le0neg1t () () (-> (e. A (RR)) (<-> (br A (<_) (0)) (br (0) (<_) (-u A)))) (0re A (0) lenegt mpan2 neg0 (<_) (-u A) breq1i syl6bb)) thm (le0neg2t () () (-> (e. A (RR)) (<-> (br (0) (<_) A) (br (-u A) (<_) (0)))) (0re (0) A lenegt mpan neg0 (-u A) (<_) breq2i syl6bb)) thm (lesub0t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (/\ (br (0) (<_) A) (br B (<_) (opr B (-) A))) (= A (0)))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 A (if (e. A (RR)) A (0)) B (-) opreq2 B (<_) breq2d anbi12d A (if (e. A (RR)) A (0)) (0) eqeq1 bibi12d (= B (if (e. B (RR)) B (0))) id B (if (e. B (RR)) B (0)) (-) (if (e. A (RR)) A (0)) opreq1 (<_) breq12d (br (0) (<_) (if (e. A (RR)) A (0))) anbi2d (= (if (e. A (RR)) A (0)) (0)) bibi1d 0re A elimel 0re B elimel lesub0 dedth2h)) thm (mulge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<_) B))) (br (0) (<_) (opr A (x.) B))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 (br (0) (<_) B) anbi1d A (if (e. A (RR)) A (0)) (x.) B opreq1 (0) (<_) breq2d imbi12d B (if (e. B (RR)) B (0)) (0) (<_) breq2 (br (0) (<_) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (x.) opreq2 (0) (<_) breq2d imbi12d 0re A elimel 0re B elimel mulge0 dedth2h imp)) thm (mulge0tOLD () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (br (0) (<_) (opr A (x.) B)))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 (br (0) (<_) B) anbi1d A (if (e. A (RR)) A (0)) (x.) B opreq1 (0) (<_) breq2d imbi12d B (if (e. B (RR)) B (0)) (0) (<_) breq2 (br (0) (<_) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (x.) opreq2 (0) (<_) breq2d imbi12d 0re A elimel 0re B elimel mulge0 dedth2h)) thm (ltmsqt () () (-> (e. A (RR)) (-> (-. (= A (0))) (br (0) (<) (opr A (x.) A)))) (A (if (e. A (RR)) A (0)) (0) eqeq1 negbid (= A (if (e. A (RR)) A (0))) id (= A (if (e. A (RR)) A (0))) id (x.) opreq12d (0) (<) breq2d imbi12d 0re A elimel msqgt0 dedth)) thm (lt01 () () (br (0) (<) (1)) (ax1ne0 (1) (0) df-ne mpbi ax1re msqgt0 ax-mp 1cn mulid1 breqtr)) thm (eqneg () ((eqneg.1 (e. A (CC)))) (<-> (= A (-u A)) (= A (0))) (ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i (opr (1) (+) (1)) (0) df-ne mpbi A (-u A) A (+) opreq2 eqneg.1 1p1times eqneg.1 negid eqcomi 3eqtr4g ax1re ax1re readdcl recn eqneg.1 mul0or sylib ord mpi (0) df-neg 0cn subid eqtr2 (= A (0)) id A (0) negeq eqeq12d mpbiri impbi)) thm (eqnegt () () (-> (e. A (CC)) (<-> (= A (-u A)) (= A (0)))) ((= A (if (e. A (CC)) A (0))) id A (if (e. A (CC)) A (0)) negeq eqeq12d A (if (e. A (CC)) A (0)) (0) eqeq1 bibi12d 0cn A elimel eqneg dedth)) thm (negeq0t () () (-> (e. A (CC)) (<-> (= A (0)) (= (-u A) (0)))) (A negnegt (-u A) eqeq2d (-u A) A eqcom syl6rbb A eqnegt A negclt (-u A) eqnegt syl 3bitr3d)) thm (negne0 () ((negne0.1 (e. A (CC)))) (<-> (=/= A (0)) (=/= (-u A) (0))) (negne0.1 A negeq0t ax-mp eqneqi)) thm (negn0 () ((negne0.1 (e. A (CC))) (negn0.2 (=/= A (0)))) (=/= (-u A) (0)) (negn0.2 negne0.1 negne0 mpbi)) thm (elimgt0 () () (br (0) (<) (if (br (0) (<) A) A (1))) (A (if (br (0) (<) A) A (1)) (0) (<) breq2 (1) (if (br (0) (<) A) A (1)) (0) (<) breq2 lt01 elimhyp)) thm (elimge0 () () (br (0) (<_) (if (br (0) (<_) A) A (0))) (A (if (br (0) (<_) A) A (0)) (0) (<_) breq2 (0) (if (br (0) (<_) A) A (0)) (0) (<_) breq2 0re leid elimhyp)) thm (ltp1t () () (-> (e. A (RR)) (br A (<) (opr A (+) (1)))) (ax1re lt01 (1) A ltaddpost mpbii mpan)) thm (lep1t () () (-> (e. A (RR)) (br A (<_) (opr A (+) (1)))) (A ltp1t A peano2re A (opr A (+) (1)) ltlet mpdan mpd)) thm (ltp1 () ((ltplus1.1 (e. A (RR)))) (br A (<) (opr A (+) (1))) (ltplus1.1 A ltp1t ax-mp)) thm (recgt0i () ((ltplus1.1 (e. A (RR))) (recgt0i.2 (br (0) (<) A))) (br (0) (<) (opr (1) (/) A)) (1cn ltplus1.1 recn ax1ne0 ltplus1.1 recgt0i.2 gt0ne0i divne0 (opr (1) (/) A) (0) necom mpbi (0) (opr (1) (/) A) df-ne mpbi lt01 0re ax1re ltnsym ax-mp recgt0i.2 ltplus1.1 ltplus1.1 recgt0i.2 gt0ne0i rereccl renegcl ltplus1.1 mulgt0 mpan2 ltplus1.1 ltplus1.1 recgt0i.2 gt0ne0i rereccl recn ltplus1.1 recn mulneg1 ltplus1.1 ltplus1.1 recgt0i.2 gt0ne0i rereccl recn ltplus1.1 recn mulcom ltplus1.1 recn ltplus1.1 recgt0i.2 gt0ne0i recid eqtr negeqi eqtr syl6breq ltplus1.1 ltplus1.1 recgt0i.2 gt0ne0i rereccl (opr (1) (/) A) lt0neg1t ax-mp ax1re (1) lt0neg1t ax-mp 3imtr4 mto pm3.2ni 0re ltplus1.1 ltplus1.1 recgt0i.2 gt0ne0i rereccl (0) (opr (1) (/) A) axlttri mp2an mpbir)) thm (ltm1t () () (-> (e. A (RR)) (br (opr A (-) (1)) (<) A)) (lt01 0re ax1re (0) (1) A ltsub2t mp3an12 mpbii A recnt A subid1t syl breqtrd)) thm (letrp1t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) (br A (<_) (opr B (+) (1)))) (B ltp1t (e. A (RR)) adantl A B (opr B (+) (1)) lelttrt 3expb B peano2re ancli sylan2 mpan2d 3impia A (opr B (+) (1)) ltlet B peano2re sylan2 (br A (<_) B) 3adant3 mpd)) thm (p1let () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (opr A (+) (1)) (<_) B)) (br A (<_) B)) (A lep1t (e. B (RR)) adantr A (opr A (+) (1)) B letrt 3expa A peano2re ancli sylan mpand 3impia)) thm (prodgt0lem () ((prodgt0.1 (e. A (RR))) (prodgt0.2 (e. B (RR))) (prodgt0i.3 (br (0) (<) A))) (-> (br (0) (<) (opr A (x.) B)) (br (0) (<) B)) (prodgt0.1 prodgt0i.3 recgt0i 0re prodgt0.1 prodgt0.2 remulcl prodgt0.1 prodgt0.1 prodgt0i.3 gt0ne0i rereccl ltmullem ax-mp prodgt0.1 prodgt0.1 prodgt0i.3 gt0ne0i rereccl recn mul02 prodgt0.1 recn prodgt0.2 recn prodgt0.1 prodgt0.1 prodgt0i.3 gt0ne0i rereccl recn mul23 prodgt0.1 recn prodgt0.1 prodgt0i.3 gt0ne0i recid (x.) B opreq1i prodgt0.2 recn mulid2 3eqtr 3brtr3g)) thm (prodgt0 () ((prodgt0.1 (e. A (RR))) (prodgt0.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<) (opr A (x.) B))) (br (0) (<) B)) (0re prodgt0.1 leloe A (if (br (0) (<) A) A (1)) (x.) B opreq1 (0) (<) breq2d (br (0) (<) B) imbi1d prodgt0.1 ax1re (br (0) (<) A) keepel prodgt0.2 A elimgt0 prodgt0lem dedth 0re ltnr (0) A (x.) B opreq1 prodgt0.2 recn mul02 syl5eqr (0) (<) breq2d mtbii (br (0) (<) B) pm2.21d jaoi sylbi imp)) thm (prodge0 () ((prodgt0.1 (e. A (RR))) (prodgt0.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<_) (opr A (x.) B))) (br (0) (<_) B)) (prodgt0.1 prodgt0.2 renegcl mulgt0 ex prodgt0.2 B lt0neg1t ax-mp prodgt0.1 prodgt0.2 remulcl (opr A (x.) B) lt0neg1t ax-mp prodgt0.1 recn prodgt0.2 recn mulneg2 (0) (<) breq2i bitr4 3imtr4g con3d 0re prodgt0.1 prodgt0.2 remulcl lenlt 0re prodgt0.2 lenlt 3imtr4g imp)) thm (ltmul1i () ((ltmul1.1 (e. A (RR))) (ltmul1.2 (e. B (RR))) (ltmul1.3 (e. C (RR))) (ltmul1i.4 (br (0) (<) C))) (<-> (br A (<) B) (br (opr A (x.) C) (<) (opr B (x.) C))) (ltmul1i.4 ltmul1.1 ltmul1.2 ltmul1.3 ltmullem ax-mp ltmul1.3 ltmul1i.4 recgt0i ltmul1.1 ltmul1.3 remulcl ltmul1.2 ltmul1.3 remulcl ltmul1.3 ltmul1.3 ltmul1i.4 gt0ne0i rereccl ltmullem ax-mp ltmul1.1 recn ltmul1.3 recn ltmul1.3 recn ltmul1.3 ltmul1i.4 gt0ne0i reccl mulass ltmul1.3 recn ltmul1.3 ltmul1i.4 gt0ne0i recid A (x.) opreq2i ltmul1.1 recn mulid1 3eqtr ltmul1.2 recn ltmul1.3 recn ltmul1.3 recn ltmul1.3 ltmul1i.4 gt0ne0i reccl mulass ltmul1.3 recn ltmul1.3 ltmul1i.4 gt0ne0i recid B (x.) opreq2i ltmul1.2 recn mulid1 3eqtr 3brtr3g impbi)) thm (ltmul1 () ((ltmul1.1 (e. A (RR))) (ltmul1.2 (e. B (RR))) (ltmul1.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br A (<) B) (br (opr A (x.) C) (<) (opr B (x.) C)))) (C (if (br (0) (<) C) C (1)) A (x.) opreq2 C (if (br (0) (<) C) C (1)) B (x.) opreq2 (<) breq12d (br A (<) B) bibi2d ltmul1.1 ltmul1.2 ltmul1.3 ax1re (br (0) (<) C) keepel C elimgt0 ltmul1i dedth)) thm (ltdiv1i () ((ltmul1.1 (e. A (RR))) (ltmul1.2 (e. B (RR))) (ltmul1.3 (e. C (RR))) (ltdivi.4 (br (0) (<) C))) (<-> (br A (<) B) (br (opr A (/) C) (<) (opr B (/) C))) (ltmul1.1 ltmul1.2 ltmul1.3 ltmul1.3 ltdivi.4 gt0ne0i rereccl ltmul1.3 ltdivi.4 recgt0i ltmul1i ltmul1.1 recn ltmul1.3 recn ltmul1.3 ltdivi.4 gt0ne0i divrec ltmul1.2 recn ltmul1.3 recn ltmul1.3 ltdivi.4 gt0ne0i divrec (<) breq12i bitr4)) thm (ltdiv1 () ((ltmul1.1 (e. A (RR))) (ltmul1.2 (e. B (RR))) (ltmul1.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br A (<) B) (br (opr A (/) C) (<) (opr B (/) C)))) (C (if (br (0) (<) C) C (1)) A (/) opreq2 C (if (br (0) (<) C) C (1)) B (/) opreq2 (<) breq12d (br A (<) B) bibi2d ltmul1.1 ltmul1.2 ltmul1.3 ax1re (br (0) (<) C) keepel C elimgt0 ltdiv1i dedth)) thm (ltmuldiv () ((ltmul1.1 (e. A (RR))) (ltmul1.2 (e. B (RR))) (ltmul1.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br (opr A (x.) C) (<) B) (br A (<) (opr B (/) C)))) (ltmul1.1 ltmul1.3 remulcl ltmul1.2 ltmul1.3 ltdiv1 ltmul1.3 gt0ne0 ltmul1.3 recn ltmul1.1 recn divcan4z syl (<) (opr B (/) C) breq1d bitrd)) thm (prodgt0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<) (opr A (x.) B)))) (br (0) (<) B)) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 A (if (e. A (RR)) A (0)) (x.) B opreq1 (0) (<) breq2d anbi12d (br (0) (<) B) imbi1d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (x.) opreq2 (0) (<) breq2d (br (0) (<_) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (0) (<) breq2 imbi12d 0re A elimel 0re B elimel prodgt0 dedth2h imp)) thm (prodgt02t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) B) (br (0) (<) (opr A (x.) B)))) (br (0) (<) A)) (B A prodgt0t ex ancoms A B axmulcom A recnt B recnt syl2an (0) (<) breq2d biimpd sylan2d imp)) thm (prodge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<_) (opr A (x.) B)))) (br (0) (<_) B)) (A (if (e. A (RR)) A (0)) (0) (<) breq2 A (if (e. A (RR)) A (0)) (x.) B opreq1 (0) (<_) breq2d anbi12d (br (0) (<_) B) imbi1d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (x.) opreq2 (0) (<_) breq2d (br (0) (<) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (0) (<_) breq2 imbi12d 0re A elimel 0re B elimel prodge0 dedth2h imp)) thm (prodge02t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) B) (br (0) (<_) (opr A (x.) B)))) (br (0) (<_) A)) (B A prodge0t ex ancoms A B axmulcom A recnt B recnt syl2an (0) (<_) breq2d biimpd sylan2d imp)) thm (ltmul1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<) B) (br (opr A (x.) C) (<) (opr B (x.) C)))) (A (if (e. A (RR)) A (0)) (<) B breq1 A (if (e. A (RR)) A (0)) (x.) C opreq1 (<) (opr B (x.) C) breq1d bibi12d (br (0) (<) C) imbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<) breq2 B (if (e. B (RR)) B (0)) (x.) C opreq1 (opr (if (e. A (RR)) A (0)) (x.) C) (<) breq2d bibi12d (br (0) (<) C) imbi2d C (if (e. C (RR)) C (0)) (0) (<) breq2 C (if (e. C (RR)) C (0)) (if (e. A (RR)) A (0)) (x.) opreq2 C (if (e. C (RR)) C (0)) (if (e. B (RR)) B (0)) (x.) opreq2 (<) breq12d (br (if (e. A (RR)) A (0)) (<) (if (e. B (RR)) B (0))) bibi2d imbi12d 0re A elimel 0re B elimel 0re C elimel ltmul1 dedth3h imp)) thm (ltmul2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<) B) (br (opr C (x.) A) (<) (opr C (x.) B)))) (A B C ltmul1t A C axmulcom A recnt sylan (e. B (RR)) 3adant2 B C axmulcom B recnt sylan (e. A (RR)) 3adant1 (<) breq12d C recnt syl3an3 (br (0) (<) C) adantr bitrd)) thm (lemul1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<_) B) (br (opr A (x.) C) (<_) (opr B (x.) C)))) (A B C ltmul1t A B C mulcan2t A recnt B recnt C recnt 3anim123i (br (0) (<) C) adantr C gt0ne0t (e. A (RR)) adantll (e. B (RR)) 3adantl2 sylanc bicomd orbi12d A B leloet (e. C (RR)) 3adant3 (br (0) (<) C) adantr (opr A (x.) C) (opr B (x.) C) leloet A C axmulrcl (e. B (RR)) 3adant2 B C axmulrcl (e. A (RR)) 3adant1 sylanc (br (0) (<) C) adantr 3bitr4d)) thm (lemul2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<_) B) (br (opr C (x.) A) (<_) (opr C (x.) B)))) (A B C lemul1t A C axmulcom A recnt C recnt syl2an (e. B (RR)) 3adant2 B C axmulcom B recnt C recnt syl2an (e. A (RR)) 3adant1 (<_) breq12d (br (0) (<) C) adantr bitrd)) thm (ltmul2 () ((ltmul2.1 (e. A (RR))) (ltmul2.2 (e. B (RR))) (ltmul2.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br A (<) B) (br (opr C (x.) A) (<) (opr C (x.) B)))) (ltmul2.1 ltmul2.2 ltmul2.3 A B C ltmul2t ex mp3an)) thm (lemul1 () ((ltmul2.1 (e. A (RR))) (ltmul2.2 (e. B (RR))) (ltmul2.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br A (<_) B) (br (opr A (x.) C) (<_) (opr B (x.) C)))) (ltmul2.1 ltmul2.2 ltmul2.3 A B C lemul1t ex mp3an)) thm (lemul2 () ((ltmul2.1 (e. A (RR))) (ltmul2.2 (e. B (RR))) (ltmul2.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br A (<_) B) (br (opr C (x.) A) (<_) (opr C (x.) B)))) (ltmul2.1 ltmul2.2 ltmul2.3 A B C lemul2t ex mp3an)) thm (lemul1it () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<_) C) (br A (<_) B))) (br (opr A (x.) C) (<_) (opr B (x.) C))) (0re (0) C leloet mpan (e. A (RR)) (e. B (RR)) 3ad2ant3 A B C lemul1t biimpd ex (0) C A (x.) opreq2 (0) C B (x.) opreq2 (<_) breq12d 0re leid A mul01t B mul01t (<_) breqan12d A recnt B recnt syl2an mpbiri syl5bi com12 (e. C (RR)) 3adant3 (br A (<_) B) a1dd jaod sylbid imp32)) thm (lemul2it () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<_) C) (br A (<_) B))) (br (opr C (x.) A) (<_) (opr C (x.) B))) (A B C lemul1it A C axmulcom A recnt C recnt syl2an (e. B (RR)) 3adant2 (/\ (br (0) (<_) C) (br A (<_) B)) adantr B C axmulcom B recnt C recnt syl2an (e. A (RR)) 3adant1 (/\ (br (0) (<_) C) (br A (<_) B)) adantr 3brtr3d)) thm (ltmul12it () () (-> (/\ (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br A (<) B))) (/\ (/\ (e. C (RR)) (e. D (RR))) (/\ (br (0) (<_) C) (br C (<) D)))) (br (opr A (x.) C) (<) (opr B (x.) D))) (A B C lemul1it (e. A (RR)) (e. B (RR)) pm3.26 (/\ (e. C (RR)) (e. D (RR))) adantr (e. A (RR)) (e. B (RR)) pm3.27 (/\ (e. C (RR)) (e. D (RR))) adantr (e. C (RR)) (e. D (RR)) pm3.26 (/\ (e. A (RR)) (e. B (RR))) adantl 3jca (/\ (/\ (br (0) (<_) A) (br A (<) B)) (/\ (br (0) (<_) C) (br C (<) D))) adantr (br (0) (<_) C) (br C (<) D) pm3.26 (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (e. C (RR)) (e. D (RR)))) (/\ (br (0) (<_) A) (br A (<) B)) ad2antll A B ltlet imp (br (0) (<_) A) adantrl (/\ (e. C (RR)) (e. D (RR))) (/\ (br (0) (<_) C) (br C (<) D)) ad2ant2r jca sylanc C D B ltmul2t (e. C (RR)) (e. D (RR)) pm3.26 (/\ (e. A (RR)) (e. B (RR))) adantl (e. C (RR)) (e. D (RR)) pm3.27 (/\ (e. A (RR)) (e. B (RR))) adantl (e. A (RR)) (e. B (RR)) pm3.27 (/\ (e. C (RR)) (e. D (RR))) adantr 3jca (/\ (br (0) (<_) A) (br A (<) B)) adantr 0re (0) A B lelttrt mp3an1 imp (/\ (e. C (RR)) (e. D (RR))) adantlr sylanc biimpa anasss (br (0) (<_) C) adantrrl (opr A (x.) C) (opr B (x.) C) (opr B (x.) D) lelttrt A C axmulrcl (e. B (RR)) (e. D (RR)) ad2ant2r B C axmulrcl (e. D (RR)) adantrr (e. A (RR)) adantll B D axmulrcl (e. A (RR)) (e. C (RR)) ad2ant2l syl3anc (/\ (/\ (br (0) (<_) A) (br A (<) B)) (/\ (br (0) (<_) C) (br C (<) D))) adantr mp2and an4s)) thm (lemul12it () () (-> (/\ (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br A (<_) B))) (/\ (/\ (e. C (RR)) (e. D (RR))) (/\ (br (0) (<_) C) (br C (<_) D)))) (br (opr A (x.) C) (<_) (opr B (x.) D))) (A C axmulrcl (e. B (RR)) (e. D (RR)) ad2ant2r (/\ (/\ (br (0) (<_) A) (br A (<_) B)) (/\ (br (0) (<_) C) (br C (<_) D))) adantr A D axmulrcl (/\ (e. B (RR)) (e. C (RR))) adantr an42s (/\ (/\ (br (0) (<_) A) (br A (<_) B)) (/\ (br (0) (<_) C) (br C (<_) D))) adantr B D axmulrcl (e. A (RR)) (e. C (RR)) ad2ant2l (/\ (/\ (br (0) (<_) A) (br A (<_) B)) (/\ (br (0) (<_) C) (br C (<_) D))) adantr C D A lemul2it ex 3expa ancoms (br (0) (<_) A) (br A (<_) B) pm3.26 (br (0) (<_) C) (br C (<_) D) pm3.27 syl2ani (e. B (RR)) adantlr imp (br (0) (<_) A) (br A (<_) B) pm3.27 (/\ (e. A (RR)) (e. B (RR))) a1i 0re (0) C D letrt mp3an1 im2anan9 A B D lemul1it ex ancomsd 3expa (e. C (RR)) adantrl syld imp letrd an4s)) thm (mulgt1t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (1) (<) A) (br (1) (<) B))) (br (1) (<) (opr A (x.) B))) ((br (1) (<) A) (br (1) (<) B) pm3.26 (/\ (e. A (RR)) (e. B (RR))) a1i lt01 0re ax1re (0) (1) A axlttrn mp3an12 mpani (e. B (RR)) adantr ax1re (1) B A ltmul2t biimpd ex mp3an1 ancoms syld imp3a A recnt A ax1id syl (e. B (RR)) adantr (<) (opr A (x.) B) breq1d sylibd jcad A B axmulrcl ax1re (1) A (opr A (x.) B) axlttrn mp3an1 syldan syld imp)) thm (ltmulgt11t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (0) (<) A)) (<-> (br (1) (<) B) (br A (<) (opr A (x.) B)))) (ax1re (1) B A ltmul2t mp3anl1 3impa 3com12 A recnt A ax1id syl (e. B (RR)) (br (0) (<) A) 3ad2ant1 (<) (opr A (x.) B) breq1d bitrd)) thm (ltmulgt12t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (0) (<) A)) (<-> (br (1) (<) B) (br A (<) (opr B (x.) A)))) (A B ltmulgt11t A B axmulcom A recnt B recnt syl2an (br (0) (<) A) 3adant3 A (<) breq2d bitrd)) thm (lemulge11t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (1) (<_) B))) (br A (<_) (opr A (x.) B))) (A recnt A ax1id syl (e. B (RR)) (/\ (br (0) (<_) A) (br (1) (<_) B)) ad2antrr A A (1) B lemul12it (e. A (RR)) (e. A (RR)) pm3.2 pm2.43i (e. B (RR)) (/\ (br (0) (<_) A) (br (1) (<_) B)) ad2antrr (br (0) (<_) A) (br (1) (<_) B) pm3.26 (/\ (e. A (RR)) (e. B (RR))) adantl A leidt (e. B (RR)) (/\ (br (0) (<_) A) (br (1) (<_) B)) ad2antrr jca jca (e. A (RR)) (e. B (RR)) pm3.27 (/\ (br (0) (<_) A) (br (1) (<_) B)) adantr ax1re jctil (br (0) (<_) A) (br (1) (<_) B) pm3.27 (/\ (e. A (RR)) (e. B (RR))) adantl 0re ax1re lt01 ltlei jctil jca sylanc eqbrtrrd)) thm (ltdiv1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<) B) (br (opr A (/) C) (<) (opr B (/) C)))) (A (if (e. A (RR)) A (0)) (<) B breq1 A (if (e. A (RR)) A (0)) (/) C opreq1 (<) (opr B (/) C) breq1d bibi12d (br (0) (<) C) imbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<) breq2 B (if (e. B (RR)) B (0)) (/) C opreq1 (opr (if (e. A (RR)) A (0)) (/) C) (<) breq2d bibi12d (br (0) (<) C) imbi2d C (if (e. C (RR)) C (0)) (0) (<) breq2 C (if (e. C (RR)) C (0)) (if (e. A (RR)) A (0)) (/) opreq2 C (if (e. C (RR)) C (0)) (if (e. B (RR)) B (0)) (/) opreq2 (<) breq12d (br (if (e. A (RR)) A (0)) (<) (if (e. B (RR)) B (0))) bibi2d imbi12d 0re A elimel 0re B elimel 0re C elimel ltdiv1 dedth3h imp)) thm (lediv1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<_) B) (br (opr A (/) C) (<_) (opr B (/) C)))) (B A C ltdiv1t ex 3com12 imp negbid A B lenltt (e. C (RR)) 3adant3 (br (0) (<) C) adantr C gt0ne0t (e. B (RR)) adantll (e. A (RR)) 3adantl1 A C redivclt 3expa (e. B (RR)) 3adantl2 B C redivclt 3expa (e. A (RR)) 3adantl1 jca syldan (opr A (/) C) (opr B (/) C) lenltt syl 3bitr4d)) thm (gt0divt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (0) (<) B)) (<-> (br (0) (<) A) (br (0) (<) (opr A (/) B)))) (0re (0) A B ltdiv1t mp3anl1 3impa B gt0ne0t B div0t B recnt sylan syldan (<) (opr A (/) B) breq1d (e. A (RR)) 3adant1 bitrd)) thm (ge0divt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (0) (<) B)) (<-> (br (0) (<_) A) (br (0) (<_) (opr A (/) B)))) (0re (0) A B lediv1t mp3anl1 3impa B gt0ne0t B div0t B recnt sylan syldan (<_) (opr A (/) B) breq1d (e. A (RR)) 3adant1 bitrd)) thm (divgt0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (br (0) (<) (opr A (/) B))) (A B gt0divt biimpd 3exp com34 imp43)) thm (divge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<) B))) (br (0) (<_) (opr A (/) B))) (A B ge0divt biimpd 3exp com34 imp43)) thm (divgt0 () ((divgt0.1 (e. A (RR))) (divgt0.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (br (0) (<) (opr A (/) B))) (divgt0.1 divgt0.2 A B divgt0t mpanl12)) thm (divge0 () ((divgt0.1 (e. A (RR))) (divgt0.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<) B)) (br (0) (<_) (opr A (/) B))) (divgt0.1 divgt0.2 A B divge0t mpanl12)) thm (divgt0i2 () ((divgt0.1 (e. A (RR))) (divgt0.2 (e. B (RR))) (divgt0i2.3 (br (0) (<) B))) (-> (br (0) (<) A) (br (0) (<) (opr A (/) B))) (divgt0i2.3 divgt0.1 divgt0.2 divgt0 mpan2)) thm (divgt0i () ((divgt0.1 (e. A (RR))) (divgt0.2 (e. B (RR))) (divgt0i.3 (br (0) (<) A)) (divgt0i.4 (br (0) (<) B))) (br (0) (<) (opr A (/) B)) (divgt0i.3 divgt0.1 divgt0.2 divgt0i.4 divgt0i2 ax-mp)) thm (recgt0t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (br (0) (<) (opr (1) (/) A))) (ax1re lt01 (1) A divgt0t mpanr1 mpanl1)) thm (recgt0 () ((recgt0.1 (e. A (RR)))) (-> (br (0) (<) A) (br (0) (<) (opr (1) (/) A))) (recgt0.1 A recgt0t mpan)) thm (ltmuldivt () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (x.) B) (<) C) (br A (<) (opr C (/) B)))) (A (if (e. A (RR)) A (0)) (x.) B opreq1 (<) C breq1d A (if (e. A (RR)) A (0)) (<) (opr C (/) B) breq1 bibi12d (br (0) (<) B) imbi2d B (if (e. B (RR)) B (0)) (0) (<) breq2 B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (x.) opreq2 (<) C breq1d B (if (e. B (RR)) B (0)) C (/) opreq2 (if (e. A (RR)) A (0)) (<) breq2d bibi12d imbi12d C (if (e. C (RR)) C (0)) (opr (if (e. A (RR)) A (0)) (x.) (if (e. B (RR)) B (0))) (<) breq2 C (if (e. C (RR)) C (0)) (/) (if (e. B (RR)) B (0)) opreq1 (if (e. A (RR)) A (0)) (<) breq2d bibi12d (br (0) (<) (if (e. B (RR)) B (0))) imbi2d 0re A elimel 0re C elimel 0re B elimel ltmuldiv dedth3h imp)) thm (ltmuldiv2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) A)) (<-> (br (opr A (x.) B) (<) C) (br B (<) (opr C (/) A)))) (B A axmulcom B recnt A recnt syl2an (e. C (RR)) 3adant3 (br (0) (<) A) adantr (<) C breq1d B A C ltmuldivt bitr3d ex 3com12 imp)) thm (ltdivmult () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (/) B) (<) C) (br A (<) (opr B (x.) C)))) (A (opr B (x.) C) B ltdiv1t (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp1 B C axmulrcl (e. A (RR)) 3adant1 (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp2 3jca sylan B gt0ne0t ex (e. C (RR)) adantr B C divcan3t 3exp imp B recnt C recnt syl2an syld (e. A (RR)) 3adant1 imp (opr A (/) B) (<) breq2d bitr2d)) thm (ledivmult () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (/) B) (<_) C) (br A (<_) (opr B (x.) C)))) (A (opr B (x.) C) B lediv1t (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp1 B C axmulrcl (e. A (RR)) 3adant1 (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp2 3jca sylan B gt0ne0t ex (e. C (RR)) adantr B C divcan3t 3exp imp B recnt C recnt syl2an syld (e. A (RR)) 3adant1 imp (opr A (/) B) (<_) breq2d bitr2d)) thm (ltdivmul2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (/) B) (<) C) (br A (<) (opr C (x.) B)))) (A B C ltdivmult B C axmulcom B recnt C recnt syl2an (e. A (RR)) 3adant1 (br (0) (<) B) adantr A (<) breq2d bitrd)) thm (lt2mul2divt () () (-> (/\/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (e. C (RR)) (e. D (RR))) (/\ (br (0) (<) B) (br (0) (<) D))) (<-> (br (opr A (x.) B) (<) (opr C (x.) D)) (br (opr A (/) D) (<) (opr C (/) B)))) (C D axmulcom C recnt D recnt syl2an (/) B opreq1d (e. B (RR)) (br (0) (<) B) 3ad2ant2 D C B divasst D recnt (e. C (RR)) (br (0) (<) B) ad2antlr (e. B (RR)) 3adant1 C recnt (e. B (RR)) (e. D (RR)) ad2antrl (br (0) (<) B) 3adant3 B recnt (/\ (e. C (RR)) (e. D (RR))) (br (0) (<) B) 3ad2ant1 3jca B gt0ne0t (/\ (e. C (RR)) (e. D (RR))) 3adant2 sylanc eqtrd (br (0) (<) D) 3adant3r (e. A (RR)) 3adant1l A (<) breq2d A B (opr C (x.) D) ltmuldivt (e. A (RR)) (e. B (RR)) pm3.26 (/\ (e. C (RR)) (e. D (RR))) (/\ (br (0) (<) B) (br (0) (<) D)) 3ad2ant1 (e. A (RR)) (e. B (RR)) pm3.27 (/\ (e. C (RR)) (e. D (RR))) (/\ (br (0) (<) B) (br (0) (<) D)) 3ad2ant1 C D axmulrcl (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) B) (br (0) (<) D)) 3ad2ant2 3jca (br (0) (<) B) (br (0) (<) D) pm3.26 (/\ (e. A (RR)) (e. B (RR))) (/\ (e. C (RR)) (e. D (RR))) 3ad2ant3 sylanc A D (opr C (/) B) ltdivmult (e. A (RR)) (e. B (RR)) pm3.26 (/\ (e. C (RR)) (e. D (RR))) (/\ (br (0) (<) B) (br (0) (<) D)) 3ad2ant1 (e. C (RR)) (e. D (RR)) pm3.27 (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) B) (br (0) (<) D)) 3ad2ant2 C B redivclt (e. C (RR)) (e. B (RR)) (br (0) (<) B) 3simp1 (e. C (RR)) (e. B (RR)) (br (0) (<) B) 3simp2 B gt0ne0t (e. C (RR)) 3adant1 syl3anc 3com12 (br (0) (<) D) 3adant3r (e. D (RR)) 3adant2r (e. A (RR)) 3adant1l 3jca (br (0) (<) B) (br (0) (<) D) pm3.27 (/\ (e. A (RR)) (e. B (RR))) (/\ (e. C (RR)) (e. D (RR))) 3ad2ant3 sylanc 3bitr4d)) thm (ledivmul2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (/) B) (<_) C) (br A (<_) (opr C (x.) B)))) (A B C ledivmult B C axmulcom B recnt C recnt syl2an (e. A (RR)) 3adant1 (br (0) (<) B) adantr A (<_) breq2d bitrd)) thm (lemuldivt () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (x.) B) (<_) C) (br A (<_) (opr C (/) B)))) (C B A ltdivmul2t (e. A (RR)) (e. B (RR)) (e. C (RR)) 3anrev sylanb negbid A (opr C (/) B) lenltt (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp1 (br (0) (<) B) adantr B gt0ne0t (e. C (RR)) adantlr C B redivclt 3com12 3expa syldan (e. A (RR)) 3adantl1 sylanc (opr A (x.) B) C lenltt A B axmulrcl (e. C (RR)) 3adant3 (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp3 sylanc (br (0) (<) B) adantr 3bitr4rd)) thm (lemuldiv2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) A)) (<-> (br (opr A (x.) B) (<_) C) (br B (<_) (opr C (/) A)))) (B A axmulcom B recnt A recnt syl2an (e. C (RR)) 3adant3 (br (0) (<) A) adantr (<_) C breq1d B A C lemuldivt bitr3d ex 3com12 imp)) thm (ltreci () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR))) (ltreci.3 (br (0) (<) A)) (ltreci.4 (br (0) (<) B))) (<-> (br A (<) B) (br (opr (1) (/) B) (<) (opr (1) (/) A))) (ltrec.1 recn ltrec.1 ltreci.3 gt0ne0i divid ltrec.2 recn ltrec.2 ltreci.4 gt0ne0i recid eqtr4 ltrec.2 recn ltrec.1 recn ltrec.1 ltreci.3 gt0ne0i divrec (<) breq12i ltrec.1 ltrec.2 ltrec.1 ltreci.3 ltdiv1i ltreci.4 ltrec.2 ltrec.2 ltreci.4 gt0ne0i rereccl ltrec.1 ltrec.1 ltreci.3 gt0ne0i rereccl ltrec.2 ltmul2 ax-mp 3bitr4)) thm (ltrec () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (<-> (br A (<) B) (br (opr (1) (/) B) (<) (opr (1) (/) A)))) (A (if (br (0) (<) A) A (1)) (<) B breq1 A (if (br (0) (<) A) A (1)) (1) (/) opreq2 (opr (1) (/) B) (<) breq2d bibi12d B (if (br (0) (<) B) B (1)) (if (br (0) (<) A) A (1)) (<) breq2 B (if (br (0) (<) B) B (1)) (1) (/) opreq2 (<) (opr (1) (/) (if (br (0) (<) A) A (1))) breq1d bibi12d ltrec.1 ax1re (br (0) (<) A) keepel ltrec.2 ax1re (br (0) (<) B) keepel A elimgt0 B elimgt0 ltreci dedth2h)) thm (lerec () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (<-> (br A (<_) B) (br (opr (1) (/) B) (<_) (opr (1) (/) A)))) (ltrec.1 ltrec.2 ltrec ltrec.1 recn ltrec.2 recn rec11 ltrec.1 gt0ne0 ltrec.2 gt0ne0 syl2an (opr (1) (/) A) (opr (1) (/) B) eqcom syl5rbbr orbi12d ltrec.2 gt0ne0 ltrec.2 rerecclz syl ltrec.1 gt0ne0 ltrec.1 rerecclz syl anim12i ancoms (opr (1) (/) B) (opr (1) (/) A) leloet syl bitr4d ltrec.1 ltrec.2 leloe syl5bb)) thm (lt2msq () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (opr A (x.) A) (<) (opr B (x.) B)))) (ltrec.1 ltrec.2 ltrec.1 ltmul2 ltrec.1 ltrec.2 ltrec.2 ltmul1 bi2anan9 (br A (<) B) anidm syl5bbr ltrec.1 ltrec.1 remulcl ltrec.1 ltrec.2 remulcl ltrec.2 ltrec.2 remulcl lttr syl6bi ltrec.2 ltrec.1 ltrec.2 lemul2 ltrec.2 ltrec.1 ltrec.1 lemul1 bi2anan9r (br B (<_) A) anidm syl5bbr ltrec.2 ltrec.2 remulcl ltrec.2 ltrec.1 remulcl ltrec.1 ltrec.1 remulcl letr syl6bi ltrec.2 ltrec.1 lenlt ltrec.2 ltrec.2 remulcl ltrec.1 ltrec.1 remulcl lenlt 3imtr3g a3d impbid (0) A (<) B breq1 (br (0) (<) B) adantr 0re ltrec.2 ltrec.2 ltmul2 (0) A A (x.) opreq2 (<) (opr B (x.) B) breq1d ltrec.2 recn mul01 ltrec.1 recn mul01 eqtr4 (<) (opr B (x.) B) breq1i syl5bb sylan9bbr bitr3d (0) B (<_) A breq1 (br (0) (<) A) adantl 0re ltrec.1 ltrec.1 lemul2 (0) B B (x.) opreq2 (<_) (opr A (x.) A) breq1d ltrec.1 recn mul01 ltrec.2 recn mul01 eqtr4 (<_) (opr A (x.) A) breq1i syl5bb sylan9bb bitr3d ltrec.2 ltrec.1 lenlt ltrec.2 ltrec.2 remulcl ltrec.1 ltrec.1 remulcl lenlt 3bitr3g con4bid (br A (<) B) (br (opr A (x.) A) (<) (opr B (x.) B)) pm5.21 ltrec.2 ltnr (0) B (<) B breq1 bicomd (0) A (<) B breq1 sylan9bbr mtbii ltrec.2 ltrec.2 remulcl ltnr (0) B B (x.) opreq2 (<) (opr B (x.) B) breq1d bicomd (0) A A (x.) opreq2 (<) (opr B (x.) B) breq1d ltrec.2 recn mul01 ltrec.1 recn mul01 eqtr4 (<) (opr B (x.) B) breq1i syl5bb sylan9bbr mtbii sylanc ccase 0re ltrec.1 leloe 0re ltrec.2 leloe syl2anb)) thm (le2msq () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (opr A (x.) A) (<_) (opr B (x.) B)))) (ltrec.2 ltrec.1 lt2msq negbid ltrec.1 ltrec.2 lenlt ltrec.1 ltrec.1 remulcl ltrec.2 ltrec.2 remulcl lenlt 3bitr4g ancoms)) thm (msq11 () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (opr A (x.) A) (opr B (x.) B)) (= A B))) (ltrec.1 ltrec.2 lt2msq ltrec.2 ltrec.1 lt2msq ancoms orbi12d ltrec.1 ltrec.2 lttri2 ltrec.1 ltrec.1 remulcl ltrec.2 ltrec.2 remulcl lttri2 3bitr4g con4bid bicomd)) thm (ltrect () () (-> (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (RR)) (br (0) (<) B))) (<-> (br A (<) B) (br (opr (1) (/) B) (<) (opr (1) (/) A)))) (A (if (e. A (RR)) A (0)) (0) (<) breq2 (br (0) (<) B) anbi1d A (if (e. A (RR)) A (0)) (<) B breq1 A (if (e. A (RR)) A (0)) (1) (/) opreq2 (opr (1) (/) B) (<) breq2d bibi12d imbi12d B (if (e. B (RR)) B (0)) (0) (<) breq2 (br (0) (<) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<) breq2 B (if (e. B (RR)) B (0)) (1) (/) opreq2 (<) (opr (1) (/) (if (e. A (RR)) A (0))) breq1d bibi12d imbi12d 0re A elimel 0re B elimel ltrec dedth2h imp an4s)) thm (lerect () () (-> (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (RR)) (br (0) (<) B))) (<-> (br A (<_) B) (br (opr (1) (/) B) (<_) (opr (1) (/) A)))) (A (if (e. A (RR)) A (0)) (0) (<) breq2 (br (0) (<) B) anbi1d A (if (e. A (RR)) A (0)) (<_) B breq1 A (if (e. A (RR)) A (0)) (1) (/) opreq2 (opr (1) (/) B) (<_) breq2d bibi12d imbi12d B (if (e. B (RR)) B (0)) (0) (<) breq2 (br (0) (<) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<_) breq2 B (if (e. B (RR)) B (0)) (1) (/) opreq2 (<_) (opr (1) (/) (if (e. A (RR)) A (0))) breq1d bibi12d imbi12d 0re A elimel 0re B elimel lerec dedth2h imp an4s)) thm (lerectOLD () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (<-> (br A (<_) B) (br (opr (1) (/) B) (<_) (opr (1) (/) A)))) (A (if (e. A (RR)) A (0)) (0) (<) breq2 (br (0) (<) B) anbi1d A (if (e. A (RR)) A (0)) (<_) B breq1 A (if (e. A (RR)) A (0)) (1) (/) opreq2 (opr (1) (/) B) (<_) breq2d bibi12d imbi12d B (if (e. B (RR)) B (0)) (0) (<) breq2 (br (0) (<) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<_) breq2 B (if (e. B (RR)) B (0)) (1) (/) opreq2 (<_) (opr (1) (/) (if (e. A (RR)) A (0))) breq1d bibi12d imbi12d 0re A elimel 0re B elimel lerec dedth2h imp)) thm (lt2msqt () () (-> (/\ (/\ (e. A (RR)) (br (0) (<_) A)) (/\ (e. B (RR)) (br (0) (<_) B))) (<-> (br A (<) B) (br (opr A (x.) A) (<) (opr B (x.) B)))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 (br (0) (<_) B) anbi1d A (if (e. A (RR)) A (0)) (<) B breq1 A (if (e. A (RR)) A (0)) A (if (e. A (RR)) A (0)) (x.) opreq12 anidms (<) (opr B (x.) B) breq1d bibi12d imbi12d B (if (e. B (RR)) B (0)) (0) (<_) breq2 (br (0) (<_) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<) breq2 B (if (e. B (RR)) B (0)) B (if (e. B (RR)) B (0)) (x.) opreq12 anidms (opr (if (e. A (RR)) A (0)) (x.) (if (e. A (RR)) A (0))) (<) breq2d bibi12d imbi12d 0re A elimel 0re B elimel lt2msq dedth2h imp an4s)) thm (ltdiv2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\/\ (br (0) (<) A) (br (0) (<) B) (br (0) (<) C))) (<-> (br A (<) B) (br (opr C (/) B) (<) (opr C (/) A)))) ((e. A (RR)) (e. B (RR)) (e. C (RR)) (br (0) (<) A) (br (0) (<) B) (br (0) (<) C) an6 A B ltrect (/\ (e. C (RR)) (br (0) (<) C)) 3adant3 (opr (1) (/) B) (opr (1) (/) A) C ltmul2t A rerecclt syl3anl2 B rerecclt syl3anl1 C B divrect B recnt syl3an2 3expb (/\ (e. A (RR)) (=/= A (0))) 3adant3 C A divrect A recnt syl3an2 3expb (/\ (e. B (RR)) (=/= B (0))) 3adant2 (<) breq12d 3coml C recnt syl3an3 (br (0) (<) C) adantr bitr4d ex 3com12 3exp imp4a 3imp (e. B (RR)) (br (0) (<) B) pm3.26 B gt0ne0t jca syl3an2 (e. A (RR)) (br (0) (<) A) pm3.26 A gt0ne0t jca syl3an1 bitrd sylbi)) thm (ltrec1t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (<-> (br (opr (1) (/) A) (<) B) (br (opr (1) (/) B) (<) A))) ((opr (1) (/) A) B ltrect A gt0ne0t A rerecclt syldan A recgt0t jca (e. B (RR)) (br (0) (<) B) ad2ant2r (/\ (e. B (RR)) (br (0) (<) B)) id (e. A (RR)) (br (0) (<) A) ad2ant2l sylanc A recrect A recnt (br (0) (<) A) adantr A gt0ne0t sylanc (e. B (RR)) (br (0) (<) B) ad2ant2r (opr (1) (/) B) (<) breq2d bitrd)) thm (lerec2t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (<-> (br A (<_) (opr (1) (/) B)) (br B (<_) (opr (1) (/) A)))) (A (opr (1) (/) B) lerectOLD (e. A (RR)) (e. B (RR)) pm3.26 (/\ (br (0) (<) A) (br (0) (<) B)) adantr B gt0ne0t B rerecclt syldan (e. A (RR)) (br (0) (<) A) ad2ant2l jca (br (0) (<) A) (br (0) (<) B) pm3.26 (/\ (e. A (RR)) (e. B (RR))) adantl B recgt0t (e. A (RR)) (br (0) (<) A) ad2ant2l jca sylanc B recrect B recnt (br (0) (<) B) adantr B gt0ne0t sylanc (e. A (RR)) (br (0) (<) A) ad2ant2l (<_) (opr (1) (/) A) breq1d bitrd)) thm (ledivdivt () () (-> (/\ (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (RR)) (br (0) (<) B))) (/\ (/\ (e. C (RR)) (br (0) (<) C)) (/\ (e. D (RR)) (br (0) (<) D)))) (<-> (br (opr A (/) B) (<_) (opr C (/) D)) (br (opr D (/) C) (<_) (opr B (/) A)))) ((opr A (/) B) (opr C (/) D) lerect A B redivclt 3expb (e. B (RR)) (br (0) (<) B) pm3.26 B gt0ne0t jca sylan2 (br (0) (<) A) adantlr A B divgt0t an4s jca C D redivclt 3expb (e. D (RR)) (br (0) (<) D) pm3.26 D gt0ne0t jca sylan2 (br (0) (<) C) adantlr C D divgt0t an4s jca syl2an C D recdivt an4s C recnt (br (0) (<) C) adantr C gt0ne0t jca D recnt (br (0) (<) D) adantr D gt0ne0t jca syl2an A B recdivt an4s A recnt (br (0) (<) A) adantr A gt0ne0t jca B recnt (br (0) (<) B) adantr B gt0ne0t jca syl2an (<_) breqan12rd bitrd)) thm (lediv2t () () (-> (/\/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (RR)) (br (0) (<) B)) (/\ (e. C (RR)) (br (0) (<) C))) (<-> (br A (<_) B) (br (opr C (/) B) (<_) (opr C (/) A)))) ((opr (1) (/) B) (opr (1) (/) A) C lemul2t B gt0ne0t B rerecclt syldan (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. C (RR)) (br (0) (<) C)) 3ad2ant2 A gt0ne0t A rerecclt syldan (/\ (e. B (RR)) (br (0) (<) B)) (/\ (e. C (RR)) (br (0) (<) C)) 3ad2ant1 (e. C (RR)) (br (0) (<) C) pm3.26 (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (RR)) (br (0) (<) B)) 3ad2ant3 3jca (e. C (RR)) (br (0) (<) C) pm3.27 (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (RR)) (br (0) (<) B)) 3ad2ant3 sylanc A B lerect (/\ (e. C (RR)) (br (0) (<) C)) 3adant3 C B divrect 3expb C recnt B recnt (br (0) (<) B) adantr B gt0ne0t jca syl2an (/\ (e. A (RR)) (br (0) (<) A)) 3adant2 C A divrect 3expb C recnt A recnt (br (0) (<) A) adantr A gt0ne0t jca syl2an (/\ (e. B (RR)) (br (0) (<) B)) 3adant3 (<_) breq12d 3coml (br (0) (<) C) 3adant3r 3bitr4d)) thm (ltdiv23t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<) B) (br (0) (<) C))) (<-> (br (opr A (/) B) (<) C) (br (opr A (/) C) (<) B))) ((opr A (/) B) C B ltmul1t B gt0ne0t (e. A (RR)) adantll A B redivclt 3expa syldan (e. C (RR)) 3adantl3 (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp3 (br (0) (<) B) adantr (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp2 (br (0) (<) B) adantr 3jca (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B) pm3.27 sylanc (br (0) (<) C) adantrr B gt0ne0t (e. A (RR)) adantll B A divcan1t 3com12 3expa A recnt B recnt anim12i sylan syldan (e. C (RR)) 3adantl3 (br (0) (<) C) adantrr (<) (opr C (x.) B) breq1d A (opr C (x.) B) C ltdiv1t (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp1 C B axmulrcl ancoms (e. A (RR)) 3adant1 (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp3 3jca sylan C gt0ne0t (e. B (RR)) adantll C B divcan3t B recnt syl3an2 C recnt syl3an1 3com12 3expa syldan (e. A (RR)) 3adantl1 (opr A (/) C) (<) breq2d bitrd (br (0) (<) B) adantrl 3bitrd)) thm (lediv23t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<) B) (br (0) (<) C))) (<-> (br (opr A (/) B) (<_) C) (br (opr A (/) C) (<_) B))) ((opr A (/) B) C B lemul1t B gt0ne0t (e. A (RR)) adantll A B redivclt 3expa syldan (e. C (RR)) 3adantl3 (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp3 (br (0) (<) B) adantr (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp2 (br (0) (<) B) adantr 3jca (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B) pm3.27 sylanc (br (0) (<) C) adantrr B gt0ne0t (e. A (RR)) adantll B A divcan1t 3com12 3expa A recnt B recnt anim12i sylan syldan (e. C (RR)) 3adantl3 (br (0) (<) C) adantrr (<_) (opr C (x.) B) breq1d A (opr C (x.) B) C lediv1t (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp1 C B axmulrcl ancoms (e. A (RR)) 3adant1 (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp3 3jca sylan C gt0ne0t (e. B (RR)) adantll C B divcan3t B recnt syl3an2 C recnt syl3an1 3com12 3expa syldan (e. A (RR)) 3adantl1 (opr A (/) C) (<_) breq2d bitrd (br (0) (<) B) adantrl 3bitrd)) thm (ltdiv23 () ((ltdiv23.1 (e. A (RR))) (ltdiv23.2 (e. B (RR))) (ltdiv23.3 (e. C (RR)))) (-> (/\ (br (0) (<) B) (br (0) (<) C)) (<-> (br (opr A (/) B) (<) C) (br (opr A (/) C) (<) B))) (ltdiv23.1 ltdiv23.2 ltdiv23.3 3pm3.2i A B C ltdiv23t mpan)) thm (ltdiv23i () ((ltdiv23.1 (e. A (RR))) (ltdiv23.2 (e. B (RR))) (ltdiv23.3 (e. C (RR))) (ltdiv23i.4 (br (0) (<) B)) (ltdiv23i.5 (br (0) (<) C))) (<-> (br (opr A (/) B) (<) C) (br (opr A (/) C) (<) B)) (ltdiv23i.4 ltdiv23i.5 ltdiv23.1 ltdiv23.2 ltdiv23.3 ltdiv23 mp2an)) thm (lediv12it () () (-> (/\ (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br A (<_) B))) (/\ (/\ (e. C (RR)) (e. D (RR))) (/\ (br (0) (<) C) (br C (<_) D)))) (br (opr A (/) D) (<_) (opr B (/) C))) (A B (opr (1) (/) D) (opr (1) (/) C) lemul12it D rerecclt (e. C (RR)) (e. D (RR)) pm3.27 (/\ (br (0) (<) C) (br C (<_) D)) adantr D gt0ne0t (e. C (RR)) (e. D (RR)) pm3.27 (/\ (br (0) (<) C) (br C (<_) D)) adantr 0re (0) C D ltletrt mp3an1 imp sylanc sylanc C gt0ne0t C rerecclt syldan (e. D (RR)) (br C (<_) D) ad2ant2r jca D recgt0t (e. C (RR)) (e. D (RR)) pm3.27 (/\ (br (0) (<) C) (br C (<_) D)) adantr 0re (0) C D ltletrt mp3an1 imp sylanc (0) (opr (1) (/) D) ltlet 0re (/\ (/\ (e. C (RR)) (e. D (RR))) (/\ (br (0) (<) C) (br C (<_) D))) a1i D rerecclt (e. C (RR)) (e. D (RR)) pm3.27 (/\ (br (0) (<) C) (br C (<_) D)) adantr D gt0ne0t (e. C (RR)) (e. D (RR)) pm3.27 (/\ (br (0) (<) C) (br C (<_) D)) adantr 0re (0) C D ltletrt mp3an1 imp sylanc sylanc sylanc mpd (br (0) (<) C) (br C (<_) D) pm3.27 (/\ (e. C (RR)) (e. D (RR))) adantl (br (0) (<) C) (br C (<_) D) pm3.26 (/\ (e. C (RR)) (e. D (RR))) adantl 0re (0) C D ltletrt mp3an1 imp jca C D lerectOLD syldan mpbid jca jca sylan2 A D divrect A recnt (/\ (/\ (e. C (RR)) (e. D (RR))) (/\ (br (0) (<) C) (br C (<_) D))) adantr D recnt (e. C (RR)) (/\ (br (0) (<) C) (br C (<_) D)) ad2antlr (e. A (RR)) adantl D gt0ne0t (e. C (RR)) (e. D (RR)) pm3.27 (/\ (br (0) (<) C) (br C (<_) D)) adantr 0re (0) C D ltletrt mp3an1 imp sylanc (e. A (RR)) adantl syl3anc (e. B (RR)) adantlr (/\ (br (0) (<_) A) (br A (<_) B)) adantlr B C divrect B recnt (/\ (e. C (RR)) (br (0) (<) C)) adantr C recnt (e. B (RR)) (br (0) (<) C) ad2antrl C gt0ne0t (e. B (RR)) adantl syl3anc (br C (<_) D) adantrrr (e. D (RR)) adantrlr (e. A (RR)) adantll (/\ (br (0) (<_) A) (br A (<_) B)) adantlr 3brtr4d)) thm (reclt1t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (<-> (br A (<) (1)) (br (1) (<) (opr (1) (/) A)))) (ax1re lt01 pm3.2i A (1) ltrect mpan2 1cn div1 (<) (opr (1) (/) A) breq1i syl6bb)) thm (recgt1t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (<-> (br (1) (<) A) (br (opr (1) (/) A) (<) (1)))) (ax1re lt01 (1) A ltrect mpanl12 1cn div1 (opr (1) (/) A) (<) breq2i syl6bb)) thm (recgt1it () () (-> (/\ (e. A (RR)) (br (1) (<) A)) (/\ (br (0) (<) (opr (1) (/) A)) (br (opr (1) (/) A) (<) (1)))) (lt01 0re ax1re (0) (1) A axlttrn mp3an12 mpani imdistani A recgt0t syl A recgt1t biimpa lt01 0re ax1re (0) (1) A axlttrn mp3an12 mpani imdistani sylan anabss3 jca)) thm (recp1lt1 () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (br (opr A (/) (opr (1) (+) A)) (<) (1))) (A ltp1t A recnt 1cn A (1) axaddcom mpan2 syl breqtrd (br (0) (<_) A) adantr (opr (1) (+) A) A divcan1t ax1re (1) A axaddrcl mpan (br (0) (<_) A) adantr recnd A recnt (br (0) (<_) A) adantr (opr (1) (+) A) gt0ne0t ax1re (1) A axaddrcl mpan (br (0) (<_) A) adantr ax1re lt01 (1) A addgtge0t mpanr1 mpanl1 sylanc syl3anc ax1re (1) A axaddrcl mpan recnd (opr (1) (+) A) mulid2t syl (br (0) (<_) A) adantr 3brtr4d ax1re (opr A (/) (opr (1) (+) A)) (1) (opr (1) (+) A) ltmul1t mp3anl2 A (opr (1) (+) A) redivclt (e. A (RR)) (br (0) (<_) A) pm3.26 ax1re (1) A axaddrcl mpan (br (0) (<_) A) adantr (opr (1) (+) A) gt0ne0t ax1re (1) A axaddrcl mpan (br (0) (<_) A) adantr ax1re lt01 (1) A addgtge0t mpanr1 mpanl1 sylanc syl3anc ax1re (1) A axaddrcl mpan (br (0) (<_) A) adantr jca ax1re lt01 (1) A addgtge0t mpanr1 mpanl1 sylanc mpbird)) thm (recrecltt () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (/\ (br (opr (1) (/) (opr (1) (+) (opr (1) (/) A))) (<) (1)) (br (opr (1) (/) (opr (1) (+) (opr (1) (/) A))) (<) A))) (A recgt0t A gt0ne0t A rerecclt syldan ax1re (opr (1) (/) A) (1) ltaddpost mpan2 syl mpbid (opr (1) (+) (opr (1) (/) A)) recgt1t A gt0ne0t A rerecclt syldan ax1re (1) (opr (1) (/) A) axaddrcl mpan syl A recgt0t A gt0ne0t A rerecclt syldan ax1re (opr (1) (/) A) (1) ltaddpost mpan2 syl mpbid lt01 A gt0ne0t A rerecclt syldan ax1re (1) (opr (1) (/) A) axaddrcl mpan syl 0re ax1re (0) (1) (opr (1) (+) (opr (1) (/) A)) axlttrn mp3an12 syl mpani mpd sylanc mpbid lt01 A gt0ne0t A rerecclt syldan ax1re (1) (opr (1) (/) A) ltaddpost mpan syl mpbii A gt0ne0t A rerecclt syldan recnd 1cn (opr (1) (/) A) (1) axaddcom mpan2 syl breqtrd A (opr (1) (+) (opr (1) (/) A)) ltrec1t (e. A (RR)) (br (0) (<) A) pm3.26 A gt0ne0t A rerecclt syldan ax1re (1) (opr (1) (/) A) axaddrcl mpan syl jca (e. A (RR)) (br (0) (<) A) pm3.27 A recgt0t A gt0ne0t A rerecclt syldan ax1re (opr (1) (/) A) (1) ltaddpost mpan2 syl mpbid lt01 A gt0ne0t A rerecclt syldan ax1re (1) (opr (1) (/) A) axaddrcl mpan syl 0re ax1re (0) (1) (opr (1) (+) (opr (1) (/) A)) axlttrn mp3an12 syl mpani mpd jca sylanc mpbid jca)) thm (le2msqt () () (-> (/\ (/\ (e. A (RR)) (br (0) (<_) A)) (/\ (e. B (RR)) (br (0) (<_) B))) (<-> (br A (<_) B) (br (opr A (x.) A) (<_) (opr B (x.) B)))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 (br (0) (<_) B) anbi1d A (if (e. A (RR)) A (0)) (<_) B breq1 A (if (e. A (RR)) A (0)) A (if (e. A (RR)) A (0)) (x.) opreq12 anidms (<_) (opr B (x.) B) breq1d bibi12d imbi12d B (if (e. B (RR)) B (0)) (0) (<_) breq2 (br (0) (<_) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<_) breq2 B (if (e. B (RR)) B (0)) B (if (e. B (RR)) B (0)) (x.) opreq12 anidms (opr (if (e. A (RR)) A (0)) (x.) (if (e. A (RR)) A (0))) (<_) breq2d bibi12d imbi12d 0re A elimel 0re B elimel le2msq dedth2h imp an4s)) thm (halfpos () ((halfpos.1 (e. A (RR)))) (<-> (br (0) (<) A) (br (opr A (/) (opr (1) (+) (1))) (<) A)) (0re halfpos.1 halfpos.1 ltadd1 halfpos.1 recn addid2 halfpos.1 recn 1p1times eqcomi (<) breq12i halfpos.1 ax1re ax1re readdcl halfpos.1 remulcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i ltdiv1i bitr 1cn 1cn addcl halfpos.1 recn ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i divcan3 (opr A (/) (opr (1) (+) (1))) (<) breq2i 3bitr)) thm (ledivp1t () () (-> (/\ (/\ (e. A (RR)) (br (0) (<_) A)) (/\ (e. B (RR)) (br (0) (<_) B))) (br (opr (opr A (/) (opr B (+) (1))) (x.) B) (<_) A)) (B (opr B (+) (1)) (opr A (/) (opr B (+) (1))) lemul2it (e. B (RR)) (br (0) (<_) B) pm3.26 (/\ (e. A (RR)) (br (0) (<_) A)) adantl B peano2re (/\ (e. A (RR)) (br (0) (<_) A)) (br (0) (<_) B) ad2antrl A (opr B (+) (1)) redivclt (e. A (RR)) (/\ (e. B (RR)) (br (0) (<_) B)) pm3.26 B peano2re (e. A (RR)) (br (0) (<_) B) ad2antrl (opr B (+) (1)) gt0ne0t B peano2re (br (0) (<_) B) adantr B ltp1t B peano2re 0re (0) B (opr B (+) (1)) lelttrt mp3an1 mpdan mpan2d imp sylanc (e. A (RR)) adantl syl3anc (br (0) (<_) A) adantlr 3jca A (opr B (+) (1)) divge0t (e. A (RR)) (br (0) (<_) A) pm3.26 B peano2re (br (0) (<_) B) adantr anim12i (e. A (RR)) (br (0) (<_) A) pm3.27 B ltp1t B peano2re 0re (0) B (opr B (+) (1)) lelttrt mp3an1 mpdan mpan2d imp anim12i sylanc B ltp1t (br (0) (<_) B) adantr B peano2re B (opr B (+) (1)) ltlet mpdan (br (0) (<_) B) adantr mpd (/\ (e. A (RR)) (br (0) (<_) A)) adantl jca sylanc (opr B (+) (1)) A divcan1t B peano2re recnd (/\ (e. A (RR)) (br (0) (<_) A)) (br (0) (<_) B) ad2antrl A recnt (br (0) (<_) A) (/\ (e. B (RR)) (br (0) (<_) B)) ad2antrr (opr B (+) (1)) gt0ne0t B peano2re (br (0) (<_) B) adantr B ltp1t B peano2re 0re (0) B (opr B (+) (1)) lelttrt mp3an1 mpdan mpan2d imp sylanc (/\ (e. A (RR)) (br (0) (<_) A)) adantl syl3anc breqtrd)) thm (ledivp1 () ((ledivp1.1 (e. A (RR))) (ledivp1.2 (e. B (RR))) (ledivp1.3 (e. C (RR)))) (-> (/\/\ (br (0) (<_) A) (br (0) (<_) C) (br A (<_) (opr B (/) (opr C (+) (1))))) (br (opr A (x.) C) (<_) B)) (ledivp1.1 ledivp1.3 remulcl ledivp1.1 ledivp1.3 ax1re readdcl remulcl ledivp1.2 letr ledivp1.3 ledivp1.3 ax1re readdcl ledivp1.3 ltp1 ltlei ledivp1.3 ledivp1.3 ax1re readdcl ledivp1.1 C (opr C (+) (1)) A lemul2it ex mp3an mpan2 (br (0) (<_) C) (br A (<_) (opr B (/) (opr C (+) (1)))) 3ad2ant1 ledivp1.3 ltp1 0re ledivp1.3 ledivp1.3 ax1re readdcl lelttr mpan2 ledivp1.3 ax1re readdcl gt0ne0 ledivp1.2 ledivp1.3 ax1re readdcl redivclz syl ledivp1.3 ax1re readdcl ledivp1.1 A (opr B (/) (opr C (+) (1))) (opr C (+) (1)) lemul1t mp3anl1 mpanl2 mpancom biimpd syl imp ledivp1.3 ltp1 0re ledivp1.3 ledivp1.3 ax1re readdcl lelttr mpan2 ledivp1.3 ax1re readdcl gt0ne0 ledivp1.3 ax1re readdcl recn ledivp1.2 recn divcan1z 3syl (br A (<_) (opr B (/) (opr C (+) (1)))) adantr breqtrd (br (0) (<_) A) 3adant1 sylanc)) thm (ltdivp1 () ((ledivp1.1 (e. A (RR))) (ledivp1.2 (e. B (RR))) (ledivp1.3 (e. C (RR)))) (-> (/\/\ (br (0) (<_) A) (br (0) (<_) C) (br A (<) (opr B (/) (opr C (+) (1))))) (br (opr A (x.) C) (<) B)) (ledivp1.1 ledivp1.3 remulcl ledivp1.1 ledivp1.3 ax1re readdcl remulcl ledivp1.2 lelttr ledivp1.3 ledivp1.3 ax1re readdcl ledivp1.3 ltp1 ltlei ledivp1.3 ledivp1.3 ax1re readdcl ledivp1.1 C (opr C (+) (1)) A lemul2it ex mp3an mpan2 (br (0) (<_) C) (br A (<) (opr B (/) (opr C (+) (1)))) 3ad2ant1 ledivp1.3 ltp1 0re ledivp1.3 ledivp1.3 ax1re readdcl lelttr mpan2 ledivp1.3 ax1re readdcl gt0ne0 ledivp1.2 ledivp1.3 ax1re readdcl redivclz syl ledivp1.3 ax1re readdcl ledivp1.1 A (opr B (/) (opr C (+) (1))) (opr C (+) (1)) ltmul1t mp3anl1 mpanl2 mpancom biimpd syl imp ledivp1.3 ltp1 0re ledivp1.3 ledivp1.3 ax1re readdcl lelttr mpan2 ledivp1.3 ax1re readdcl gt0ne0 ledivp1.3 ax1re readdcl recn ledivp1.2 recn divcan1z 3syl (br A (<) (opr B (/) (opr C (+) (1)))) adantr breqtrd (br (0) (<_) A) 3adant1 sylanc)) thm (posex ((A x) (B x)) ((posex.1 (e. A (RR))) (posex.2 (e. B (RR))) (posex.3 (br (0) (<) A)) (posex.4 (br (0) (<) B))) (E.e. x (RR) (/\ (br (0) (<) (cv x)) (/\ (br (cv x) (<) A) (br (cv x) (<) B)))) (posex.1 posex.2 lttri2 biimp orri (= A B) (br A (<) B) (br B (<) A) or12 mpbi posex.3 posex.1 halfpos mpbi posex.1 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl posex.1 posex.2 lttr mpan posex.3 posex.1 halfpos mpbi jctil posex.1 ax1re ax1re readdcl posex.3 ax1re ax1re lt01 lt01 addgt0i divgt0i jctil posex.1 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl jctil (cv x) (opr A (/) (opr (1) (+) (1))) (0) (<) breq2 (cv x) (opr A (/) (opr (1) (+) (1))) (<) A breq1 (cv x) (opr A (/) (opr (1) (+) (1))) (<) B breq1 anbi12d anbi12d (RR) rcla4ev syl posex.4 posex.2 halfpos mpbi A B (opr B (/) (opr (1) (+) (1))) (<) breq2 mpbiri posex.4 posex.2 halfpos mpbi posex.2 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl posex.2 posex.1 lttr mpan jaoi posex.4 posex.2 halfpos mpbi jctir posex.2 ax1re ax1re readdcl posex.4 ax1re ax1re lt01 lt01 addgt0i divgt0i jctil posex.2 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl jctil (cv x) (opr B (/) (opr (1) (+) (1))) (0) (<) breq2 (cv x) (opr B (/) (opr (1) (+) (1))) (<) A breq1 (cv x) (opr B (/) (opr (1) (+) (1))) (<) B breq1 anbi12d anbi12d (RR) rcla4ev syl jaoi ax-mp)) thm (max1 () () (-> (/\ (e. A (RR)) (e. B (RR))) (br A (<_) (if (br A (<_) B) B A))) ((br A (<_) B) B A iffalse A (<_) breq2d A leidt syl5bir com12 (br A (<_) B) id (br A (<_) B) B A iftrue breqtrrd pm2.61d2 (e. B (RR)) adantr)) thm (max1ALT () () (-> (e. A (RR)) (br A (<_) (if (br A (<_) B) B A))) ((br A (<_) B) B A iffalse A (<_) breq2d A leidt syl5bir com12 (br A (<_) B) id (br A (<_) B) B A iftrue breqtrrd pm2.61d2)) thm (max2 () () (-> (/\ (e. A (RR)) (e. B (RR))) (br B (<_) (if (br A (<_) B) B A))) (B leidt (e. A (RR)) (br A (<_) B) ad2antlr (br A (<_) B) B A iftrue (/\ (e. A (RR)) (e. B (RR))) adantl breqtrrd A B letrit orcanai (br A (<_) B) B A iffalse (/\ (e. A (RR)) (e. B (RR))) adantl breqtrrd pm2.61dan)) thm (min1 () () (-> (/\ (e. A (RR)) (e. B (RR))) (br (if (br A (<_) B) A B) (<_) A)) ((br A (<_) B) A B iftrue (/\ (e. A (RR)) (e. B (RR))) adantl A leidt (e. B (RR)) (br A (<_) B) ad2antrr eqbrtrd (br A (<_) B) A B iffalse (/\ (e. A (RR)) (e. B (RR))) adantl A B letrit orcanai eqbrtrd pm2.61dan)) thm (min2 () () (-> (/\ (e. A (RR)) (e. B (RR))) (br (if (br A (<_) B) A B) (<_) B)) ((br A (<_) B) A B iffalse (<_) B breq1d B leidt syl5bir com12 (br A (<_) B) A B iftrue (br A (<_) B) id eqbrtrd pm2.61d2 (e. A (RR)) adantl)) thm (maxlet () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (if (br A (<_) B) B A) (<_) C) (/\ (br A (<_) C) (br B (<_) C)))) (A B max1 (e. C (RR)) 3adant3 A (if (br A (<_) B) B A) C letrt (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp1 B (RR) A (br A (<_) B) ifcl ancoms (e. C (RR)) 3adant3 (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp3 syl3anc mpand A B max2 (e. C (RR)) 3adant3 B (if (br A (<_) B) B A) C letrt (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp2 B (RR) A (br A (<_) B) ifcl ancoms (e. C (RR)) 3adant3 (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp3 syl3anc mpand jcad (br A (<_) B) B A iftrue (<_) C breq1d biimprd (br A (<_) C) adantld (br A (<_) B) B A iffalse (<_) C breq1d biimprd (br B (<_) C) adantrd pm2.61i (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) a1i impbid)) thm (ltmint () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<) (if (br B (<_) C) B C)) (/\ (br A (<) B) (br A (<) C)))) (B C min1 (e. A (RR)) 3adant1 A (if (br B (<_) C) B C) B ltletrt (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp1 B (RR) C (br B (<_) C) ifcl (e. A (RR)) 3adant1 (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp2 syl3anc mpan2d B C min2 (e. A (RR)) 3adant1 A (if (br B (<_) C) B C) C ltletrt (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp1 B (RR) C (br B (<_) C) ifcl (e. A (RR)) 3adant1 (e. A (RR)) (e. B (RR)) (e. C (RR)) 3simp3 syl3anc mpan2d jcad B (if (br B (<_) C) B C) A (<) breq2 C (if (br B (<_) C) B C) A (<) breq2 ifboth (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) a1i impbid)) thm (squeeze0 ((A x)) () (-> (/\/\ (e. A (RR)) (br (0) (<_) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (br A (<) (cv x))))) (= A (0))) (0re (0) A leloet mpan A ltnrt (= A (0)) pm2.21d com12 (br (0) (<) A) imim2i com13 (cv x) A (0) (<) breq2 (cv x) A A (<) breq2 imbi12d (RR) rcla4v syl5d (= A (0)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (br A (<) (cv x)))) ax-1 eqcoms (e. A (RR)) a1i jaod sylbid 3imp)) thm (peano5nn ((x y) (A x) (A y)) ((peano5nn.1 (e. A (V)))) (-> (/\ (e. (1) A) (A.e. x A (e. (opr (cv x) (+) (1)) A))) (C_ (NN) A)) (peano5nn.1 (cv y) A (1) eleq2 (cv y) A (opr (cv x) (+) (1)) eleq2 x raleqd anbi12d elab A ({|} y (/\ (e. (1) (cv y)) (A.e. x (cv y) (e. (opr (cv x) (+) (1)) (cv y))))) intss1 sylbir y x df-n syl5ss)) thm (nnssre () () (C_ (NN) (RR)) (ax1re (cv x) peano2re rgen reex x peano5nn mp2an)) thm (nnsscn () () (C_ (NN) (CC)) (nnssre axresscn sstri)) thm (nnret () () (-> (e. A (NN)) (e. A (RR))) (nnssre A sseli)) thm (nncnt () () (-> (e. A (NN)) (e. A (CC))) (nnsscn A sseli)) thm (nnre () ((nnre.1 (e. A (NN)))) (e. A (RR)) (nnre.1 A nnret ax-mp)) thm (nncn () ((nnre.1 (e. A (NN)))) (e. A (CC)) (nnre.1 nnre recn)) thm (nnex () () (e. (NN) (V)) (reex nnssre ssexi)) thm (1nn ((x y)) () (e. (1) (NN)) (1cn elisseti x (/\ (e. (1) (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x)))) elintab (e. (1) (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x))) pm3.26 mpgbir x y df-n eleqtrr)) thm (peano2nn ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (NN)) (e. (opr A (+) (1)) (NN))) ((cv z) A (+) (1) opreq1 (NN) eleq1d (cv y) (cv z) (+) (1) opreq1 (cv x) eleq1d (cv x) rcla4cv (e. (1) (cv x)) adantl a2i x 19.20i z visset x (/\ (e. (1) (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x)))) elintab (cv z) (+) (1) oprex x (/\ (e. (1) (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x)))) elintab 3imtr4 x y df-n (cv z) eleq2i x y df-n (opr (cv z) (+) (1)) eleq2i 3imtr4 vtoclga)) thm (dfnn2 ((x y) (x z) (y z)) () (= (NN) (|^| ({|} x (/\/\ (C_ (cv x) (RR)) (e. (1) (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x))))))) ((cv x) (cv z) (1) eleq2 (cv x) (cv z) (opr (cv y) (+) (1)) eleq2 y raleqd anbi12d z y df-n (cv x) eqeq2i (cv x) (NN) (1) eleq2 (cv x) (NN) (opr (cv y) (+) (1)) eleq2 y raleqd anbi12d sylbir z y df-n nnssre eqsstr3 1nn (cv y) peano2nn rgen pm3.2i pm3.2i intabs (C_ (cv x) (RR)) (e. (1) (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x))) 3anass x abbii inteqi x y df-n 3eqtr4r)) thm (nnind ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((nnind.1 (-> (= (cv x) (1)) (<-> ph ps))) (nnind.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (nnind.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th))) (nnind.4 (-> (= (cv x) A) (<-> ph ta))) (nnind.5 ps) (nnind.6 (-> (e. (cv y) (NN)) (-> ch th)))) (-> (e. A (NN)) ta) (nnind.1 (NN) elrab 1nn nnind.5 mpbir2an x (NN) ph ssrab2 (cv y) sseli (cv y) peano2nn (e. (cv y) (NN)) a1d nnind.6 anim12d nnind.2 (NN) elrab nnind.3 (NN) elrab 3imtr4g mpcom rgen nnex x ph rabex y peano5nn mp2an A sseli nnind.4 (NN) elrab sylib pm3.27d)) thm (nn1suc ((x y) (x z) (y z) (A x) (A y) (A z) (ps x) (ch x) (th x) (th z) (ph y) (ph z)) ((nn1suc.1 (-> (= (cv x) (1)) (<-> ph ps))) (nn1suc.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph ch))) (nn1suc.4 (-> (= (cv x) A) (<-> ph th))) (nn1suc.5 ps) (nn1suc.6 (-> (e. (cv y) (NN)) ch))) (-> (e. A (NN)) th) ((cv z) (1) x (-> (e. A (NN)) ph) dfsbcq z y x (-> (e. A (NN)) ph) sbequ (cv z) (opr (cv y) (+) (1)) x (-> (e. A (NN)) ph) dfsbcq (cv z) A x ph dfsbcq A (NN) x elex (e. (cv z) A) x ax-17 (NN) ph hbsbc1 (-> (e. A (NN)) th) x ax-17 hbbi x A ph sbceq1a nn1suc.4 bitr3d (e. A (NN)) imbi2d 19.23ai syl pm5.74rd pm2.43i sylan9bbr expcom pm5.74d (e. A (NN)) x ax-17 z ph sb19.21 syl5bb 1nn elisseti x isseti 1nn elisseti x ph hbsbc1v nn1suc.5 nn1suc.1 mpbiri x (1) ph sbceq1a mpbid 19.23ai ax-mp (e. A (NN)) a1i 1nn elisseti (e. A (NN)) x ax-17 (1) (V) ph sbc19.21g ax-mp mpbir nn1suc.6 (cv y) (+) (1) oprex x isseti ch x ax-17 (cv y) (+) (1) oprex x ph hbsbc1v hbbi nn1suc.3 x (opr (cv y) (+) (1)) ph sbceq1a bitr3d 19.23ai ax-mp sylib (e. A (NN)) a1d (cv y) (+) (1) oprex (e. A (NN)) x ax-17 (opr (cv y) (+) (1)) (V) ph sbc19.21g ax-mp sylibr ([/] (cv y) x (-> (e. A (NN)) ph)) a1d nnind pm2.43i)) thm (nnaddclt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (NN)) (e. B (NN))) (e. (opr A (+) B) (NN))) ((cv x) (1) A (+) opreq2 (NN) eleq1d (e. A (NN)) imbi2d (cv x) (cv y) A (+) opreq2 (NN) eleq1d (e. A (NN)) imbi2d (cv x) (opr (cv y) (+) (1)) A (+) opreq2 (NN) eleq1d (e. A (NN)) imbi2d (cv x) B A (+) opreq2 (NN) eleq1d (e. A (NN)) imbi2d A peano2nn 1cn A (cv y) (1) axaddass mp3an3 A nncnt (cv y) nncnt syl2an (NN) eleq1d (opr A (+) (cv y)) peano2nn syl5bi expcom a2d nnind impcom)) thm (nnmulclt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (NN)) (e. B (NN))) (e. (opr A (x.) B) (NN))) ((cv x) (1) A (x.) opreq2 (NN) eleq1d (e. A (NN)) imbi2d (cv x) (cv y) A (x.) opreq2 (NN) eleq1d (e. A (NN)) imbi2d (cv x) (opr (cv y) (+) (1)) A (x.) opreq2 (NN) eleq1d (e. A (NN)) imbi2d (cv x) B A (x.) opreq2 (NN) eleq1d (e. A (NN)) imbi2d A nncnt A ax1id (NN) eleq1d biimprd mpcom 1cn A (cv y) (1) axdistr mp3an3 A ax1id (opr A (x.) (cv y)) (+) opreq2d (e. (cv y) (CC)) adantr eqtrd A nncnt (cv y) nncnt syl2an (NN) eleq1d (opr A (x.) (cv y)) A nnaddclt ancoms syl5bir exp4b pm2.43b a2d nnind impcom)) thm (nn2get ((A x) (B x)) () (-> (/\ (e. A (NN)) (e. B (NN))) (E.e. x (NN) (/\ (br A (<_) (cv x)) (br B (<_) (cv x))))) (A B letrit A nnret B nnret syl2an B nnret B leidt (br A (<_) B) biantrud biimpd syl anc2li (cv x) B A (<_) breq2 (cv x) B B (<_) breq2 anbi12d (NN) rcla4ev syl6 (e. A (NN)) adantl A nnret A leidt (br B (<_) A) biantrurd biimpd syl anc2li (cv x) A A (<_) breq2 (cv x) A B (<_) breq2 anbi12d (NN) rcla4ev syl6 (e. B (NN)) adantr jaod mpd)) thm (nnge1t ((x y) (A x) (A y)) () (-> (e. A (NN)) (br (1) (<_) A)) ((cv x) (1) (1) (<_) breq2 (cv x) (cv y) (1) (<_) breq2 (cv x) (opr (cv y) (+) (1)) (1) (<_) breq2 (cv x) A (1) (<_) breq2 ax1re leid (cv y) nnret (cv y) recnt (cv y) ax0id syl (1) (<_) breq2d lt01 0re ax1re (0) (1) (cv y) axltadd mp3an12 mpi ax1re (opr (cv y) (+) (0)) (opr (cv y) (+) (1)) (1) axlttrn mp3an3 0re (cv y) (0) axaddrcl mpan2 (cv y) peano2re sylanc mpand con3d 0re (cv y) (0) axaddrcl mpan2 ax1re jctil (1) (opr (cv y) (+) (0)) lenltt syl (cv y) peano2re ax1re jctil (1) (opr (cv y) (+) (1)) lenltt syl 3imtr4d sylbird syl nnind)) thm (nngt1ne1t () () (-> (e. A (NN)) (<-> (br (1) (<) A) (-. (= A (1))))) (ax1re (1) A leltnet mp3an1 A nnret A nnge1t sylanc (1) A eqcom negbii syl6bb)) thm (nnle1eq1t () () (-> (e. A (NN)) (<-> (br A (<_) (1)) (= A (1)))) (A nnge1t (br A (<_) (1)) biantrud A nnret ax1re A (1) letri3t mpan2 syl bitr4d)) thm (nngt0t () () (-> (e. A (NN)) (br (0) (<) A)) (lt01 0re ax1re (0) (1) A ltletrt mp3an12 mpani A nnret A nnge1t sylc)) thm (lt1nnn () () (-> (/\ (e. A (RR)) (br A (<) (1))) (-. (e. A (NN)))) (ax1re A (1) ltnlet mpan2 A nnge1t con3i syl6bi imp)) thm (0nnn () () (-. (e. (0) (NN))) (0re lt01 (0) lt1nnn mp2an)) thm (nnne0t () () (-> (e. A (NN)) (=/= A (0))) (0nnn A (0) (NN) eleq1 mtbiri con2i A (0) df-ne sylibr)) thm (nngt0 () ((nngt0.1 (e. A (NN)))) (br (0) (<) A) (nngt0.1 A nngt0t ax-mp)) thm (nnne0 () ((nngt0.1 (e. A (NN)))) (=/= A (0)) (nngt0.1 nnre nngt0.1 nngt0 gt0ne0i)) thm (nnrecgt0t () () (-> (e. A (NN)) (br (0) (<) (opr (1) (/) A))) (A nnge1t lt01 A nnret 0re ax1re (0) (1) A ltletrt mp3an12 A recgt0t ex syld syl mpani mpd)) thm (nnleltp1t ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B x) (B y) (B z) (B w) (B v)) () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (br A (<_) B) (br A (<) (opr B (+) (1))))) (A B leloet A nnret B nnret syl2an lt01 0re ax1re pm3.2i A (0) B (1) lt2addt an4s mpan2 mpan2i (e. A (RR)) (e. B (RR)) pm3.26 A recnt A ax0id 3syl (<) (opr B (+) (1)) breq1d sylibd A B (<) (opr B (+) (1)) breq1 B ltp1t syl5bir com12 (e. A (RR)) adantl jaod A nnret B nnret syl2an (cv z) A (<) (opr B (+) (1)) breq1 (cv z) A (<) B breq1 (cv z) A B eqeq1 orbi12d imbi12d (NN) rcla4v (cv x) (1) (+) (1) opreq1 (cv z) (<) breq2d (cv x) (1) (cv z) (<) breq2 (cv x) (1) (cv z) eqeq2 orbi12d imbi12d z (NN) ralbidv (cv x) (cv y) (+) (1) opreq1 (cv z) (<) breq2d (cv x) (cv y) (cv z) (<) breq2 (cv x) (cv y) (cv z) eqeq2 orbi12d imbi12d z (NN) ralbidv (cv x) (opr (cv y) (+) (1)) (+) (1) opreq1 (cv z) (<) breq2d (cv x) (opr (cv y) (+) (1)) (cv z) (<) breq2 (cv x) (opr (cv y) (+) (1)) (cv z) eqeq2 orbi12d imbi12d z (NN) ralbidv (cv x) B (+) (1) opreq1 (cv z) (<) breq2d (cv x) B (cv z) (<) breq2 (cv x) B (cv z) eqeq2 orbi12d imbi12d z (NN) ralbidv (cv x) (1) (<) (opr (1) (+) (1)) breq1 (cv x) (1) (<) (1) breq1 (cv x) (1) (1) eqeq1 orbi12d imbi12d (cv x) (opr (cv y) (+) (1)) (<) (opr (1) (+) (1)) breq1 (cv x) (opr (cv y) (+) (1)) (<) (1) breq1 (cv x) (opr (cv y) (+) (1)) (1) eqeq1 orbi12d imbi12d (cv x) (cv z) (<) (opr (1) (+) (1)) breq1 (cv x) (cv z) (<) (1) breq1 (cv x) (cv z) (1) eqeq1 orbi12d imbi12d (1) eqid (-. (br (1) (<) (1))) a1i orri (br (1) (<) (opr (1) (+) (1))) a1i (cv y) nnret ax1re ax1re (cv y) (1) (1) ltadd1t mp3an23 syl (cv y) nnge1t (cv y) nnret ax1re (1) (cv y) lenltt mpan syl mpbid (\/ (br (opr (cv y) (+) (1)) (<) (1)) (= (opr (cv y) (+) (1)) (1))) pm2.21d sylbird nn1suc rgen (cv x) (1) (<) (opr (opr (cv y) (+) (1)) (+) (1)) breq1 (cv x) (1) (<) (opr (cv y) (+) (1)) breq1 (cv x) (1) (opr (cv y) (+) (1)) eqeq1 orbi12d imbi12d (/\ (e. (cv y) (NN)) (A.e. w (NN) (-> (br (cv w) (<) (opr (cv y) (+) (1))) (\/ (br (cv w) (<) (cv y)) (= (cv w) (cv y)))))) imbi2d (cv x) (opr (cv v) (+) (1)) (<) (opr (opr (cv y) (+) (1)) (+) (1)) breq1 (cv x) (opr (cv v) (+) (1)) (<) (opr (cv y) (+) (1)) breq1 (cv x) (opr (cv v) (+) (1)) (opr (cv y) (+) (1)) eqeq1 orbi12d imbi12d (/\ (e. (cv y) (NN)) (A.e. w (NN) (-> (br (cv w) (<) (opr (cv y) (+) (1))) (\/ (br (cv w) (<) (cv y)) (= (cv w) (cv y)))))) imbi2d (cv x) (cv z) (<) (opr (opr (cv y) (+) (1)) (+) (1)) breq1 (cv x) (cv z) (<) (opr (cv y) (+) (1)) breq1 (cv x) (cv z) (opr (cv y) (+) (1)) eqeq1 orbi12d imbi12d (/\ (e. (cv y) (NN)) (A.e. w (NN) (-> (br (cv w) (<) (opr (cv y) (+) (1))) (\/ (br (cv w) (<) (cv y)) (= (cv w) (cv y)))))) imbi2d (cv y) nngt0t (cv y) nnret 0re ax1re (0) (cv y) (1) ltadd1t mp3an13 syl mpbid 1cn addid2 syl5eqbrr (= (1) (opr (cv y) (+) (1))) pm2.21nd orrd (br (1) (<) (opr (opr (cv y) (+) (1)) (+) (1))) a1d (A.e. w (NN) (-> (br (cv w) (<) (opr (cv y) (+) (1))) (\/ (br (cv w) (<) (cv y)) (= (cv w) (cv y))))) adantr (cv w) (cv v) (<) (opr (cv y) (+) (1)) breq1 (cv w) (cv v) (<) (cv y) breq1 (cv w) (cv v) (cv y) eqeq1 orbi12d imbi12d (NN) rcla4va (e. (cv y) (NN)) adantlr ax1re (cv v) (opr (cv y) (+) (1)) (1) ltadd1t mp3an3 (cv v) nnret (cv y) peano2nn (opr (cv y) (+) (1)) nnret syl syl2an (A.e. w (NN) (-> (br (cv w) (<) (opr (cv y) (+) (1))) (\/ (br (cv w) (<) (cv y)) (= (cv w) (cv y))))) adantr ax1re (cv v) (cv y) (1) ltadd1t (cv v) recnt (cv y) recnt (1) recnt 3anim123i (e. (1) (CC)) (e. (cv v) (CC)) (e. (cv y) (CC)) 3anrot sylibr (e. (1) (CC)) (e. (cv v) (CC)) (e. (cv y) (CC)) 3simpc (e. (1) (CC)) (e. (cv v) (CC)) (e. (cv y) (CC)) 3simp1 jca (e. (cv v) (CC)) (e. (cv y) (CC)) (e. (1) (CC)) anandir sylib (cv v) (1) axaddcom (cv y) (1) axaddcom eqeqan12d syl (1) (cv v) (cv y) addcant bitr2d syl orbi12d mp3an3 (cv v) nnret (cv y) nnret syl2an (A.e. w (NN) (-> (br (cv w) (<) (opr (cv y) (+) (1))) (\/ (br (cv w) (<) (cv y)) (= (cv w) (cv y))))) adantr 3imtr3d exp31 imp3a nn1suc exp3a com3l r19.21adv (cv z) (cv w) (<) (opr (cv y) (+) (1)) breq1 (cv z) (cv w) (<) (cv y) breq1 (cv z) (cv w) (cv y) eqeq1 orbi12d imbi12d (NN) cbvralv syl5ib nnind syl5 imp impbid bitrd)) thm (nnltp1let () () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (br A (<) B) (br (opr A (+) (1)) (<_) B))) (ax1re A B (1) ltadd1t mp3an3 A nnret B nnret syl2an (opr A (+) (1)) B nnleltp1t A peano2nn sylan bitr4d)) thm (nnsub ((x y) (A x) (A y) (B x) (B y)) ((nnsub.1 (e. A (NN))) (nnsub.2 (e. B (NN)))) (<-> (br A (<) B) (e. (opr B (-) A) (NN))) (nnsub.2 (cv x) (1) A (<) breq2 (cv x) (1) (-) A opreq1 (NN) eleq1d imbi12d (cv x) (cv y) A (<) breq2 (cv x) (cv y) (-) A opreq1 (NN) eleq1d imbi12d (cv x) (opr (cv y) (+) (1)) A (<) breq2 (cv x) (opr (cv y) (+) (1)) (-) A opreq1 (NN) eleq1d imbi12d (cv x) B A (<) breq2 (cv x) B (-) A opreq1 (NN) eleq1d imbi12d nnsub.1 A nnge1t ax-mp ax1re nnsub.1 nnre lenlt mpbi (e. (opr (1) (-) A) (NN)) pm2.21i (cv y) nnret nnsub.1 nnre jctil A (cv y) leloet syl nnsub.1 A (cv y) nnleltp1t mpan bitr3d (cv y) nncnt nnsub.1 nncn jctir 1cn (cv y) (1) A addsubt mp3an2 syl (NN) eleq1d (opr (cv y) (-) A) peano2nn syl5bir (br A (<) (cv y)) imim2d com23 A (cv y) (+) (1) opreq1 (-) A opreq1d nnsub.1 nncn 1cn nnsub.1 nncn addsub nnsub.1 nncn subid (+) (1) opreq1i 1cn addid2 3eqtr 1nn eqeltr syl6eqelr (-> (br A (<) (cv y)) (e. (opr (cv y) (-) A) (NN))) a1d (e. (cv y) (NN)) a1i jaod sylbird com23 nnind ax-mp (opr B (-) A) nngt0t nnsub.1 nnre nnsub.2 nnre posdif sylibr impbi)) thm (nnsubt () () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (br A (<) B) (e. (opr B (-) A) (NN)))) (A (if (e. A (NN)) A (1)) (<) B breq1 A (if (e. A (NN)) A (1)) B (-) opreq2 (NN) eleq1d bibi12d B (if (e. B (NN)) B (1)) (if (e. A (NN)) A (1)) (<) breq2 B (if (e. B (NN)) B (1)) (-) (if (e. A (NN)) A (1)) opreq1 (NN) eleq1d bibi12d 1nn A elimel 1nn B elimel nnsub dedth2h)) thm (nnaddm1clt () () (-> (/\ (e. A (NN)) (e. B (NN))) (e. (opr (opr A (+) B) (-) (1)) (NN))) (ax1re ax1re ax1re readdcl (1) (opr (1) (+) (1)) (opr A (+) B) ltletrt mp3an12 A B axaddrcl A nnret B nnret syl2an ax1re ax1re readdcl (/\ (e. A (NN)) (e. B (NN))) a1i B nnret ax1re (1) B axaddrcl mpan syl (e. A (NN)) adantl A B axaddrcl A nnret B nnret syl2an B nnge1t B nnret ax1re ax1re (1) B (1) leadd2t mp3an13 syl mpbid (e. A (NN)) adantl A nnge1t (e. B (NN)) adantr ax1re (1) A B leadd1t mp3an1 A nnret B nnret syl2an mpbid letrd ax1re ltp1 jctil sylc A B nnaddclt 1nn (1) (opr A (+) B) nnsubt mpan syl mpbid)) thm (nndivt ((A x) (B x)) () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (E.e. x (NN) (= (opr A (x.) (cv x)) B)) (e. (opr B (/) A) (NN)))) (A nnne0t (e. B (NN)) adantr A (cv x) divcan3t 3com23 3expa (cv x) nncnt sylan2 (e. B (CC)) 3adantl2 eqcomd (opr A (x.) (cv x)) B (/) A opreq1 sylan9eq (/\ (/\/\ (e. A (CC)) (e. B (CC)) (=/= A (0))) (e. (cv x) (NN))) (= (opr A (x.) (cv x)) B) pm3.26 pm3.27d eqeltrrd exp31 r19.23adv A B divcan2t (cv x) (opr B (/) A) A (x.) opreq2 B eqeq1d (NN) rcla4ev expcom syl impbid 3exp imp A nncnt B nncnt syl2an mpd)) thm (nndivtrt () () (-> (/\ (/\/\ (e. A (NN)) (e. B (NN)) (e. C (CC))) (/\ (e. (opr B (/) A) (NN)) (e. (opr C (/) B) (NN)))) (e. (opr C (/) A) (NN))) (B A C B divmul24t B nncnt A nncnt anim12i ancoms (e. C (CC)) 3adant3 B nncnt (e. C (CC)) anim2i ancoms (e. A (NN)) 3adant1 jca A nnne0t B nnne0t anim12i (e. C (CC)) 3adant3 sylanc B dividt B nncnt B nnne0t sylanc (x.) (opr C (/) A) opreq1d (e. A (NN)) (e. C (CC)) 3ad2ant2 C A divclt 3expb A nncnt A nnne0t jca sylan2 ancoms (opr C (/) A) mulid2t syl (e. B (NN)) 3adant2 3eqtrd (NN) eleq1d (opr B (/) A) (opr C (/) B) nnmulclt syl5bi imp)) thm (2re () () (e. (2) (RR)) (df-2 ax1re ax1re readdcl eqeltr)) thm (2cn () () (e. (2) (CC)) (2re recn)) thm (3re () () (e. (3) (RR)) (df-3 2re ax1re readdcl eqeltr)) thm (4re () () (e. (4) (RR)) (df-4 3re ax1re readdcl eqeltr)) thm (5re () () (e. (5) (RR)) (df-5 4re ax1re readdcl eqeltr)) thm (6re () () (e. (6) (RR)) (df-6 5re ax1re readdcl eqeltr)) thm (7re () () (e. (7) (RR)) (df-7 6re ax1re readdcl eqeltr)) thm (8re () () (e. (8) (RR)) (df-8 7re ax1re readdcl eqeltr)) thm (9re () () (e. (9) (RR)) (df-9 8re ax1re readdcl eqeltr)) thm (2pos () () (br (0) (<) (2)) (ax1re ax1re lt01 lt01 addgt0i df-2 breqtrr)) thm (3pos () () (br (0) (<) (3)) (2re ax1re 2pos lt01 addgt0i df-3 breqtrr)) thm (4pos () () (br (0) (<) (4)) (3re ax1re 3pos lt01 addgt0i df-4 breqtrr)) thm (5pos () () (br (0) (<) (5)) (4re ax1re 4pos lt01 addgt0i df-5 breqtrr)) thm (6pos () () (br (0) (<) (6)) (5re ax1re 5pos lt01 addgt0i df-6 breqtrr)) thm (7pos () () (br (0) (<) (7)) (6re ax1re 6pos lt01 addgt0i df-7 breqtrr)) thm (8pos () () (br (0) (<) (8)) (7re ax1re 7pos lt01 addgt0i df-8 breqtrr)) thm (9pos () () (br (0) (<) (9)) (8re ax1re 8pos lt01 addgt0i df-9 breqtrr)) thm (2nn () () (e. (2) (NN)) (df-2 1nn 1nn (1) (1) nnaddclt mp2an eqeltr)) thm (3nn () () (e. (3) (NN)) (df-3 2nn 1nn (2) (1) nnaddclt mp2an eqeltr)) thm (2p2e4 () () (= (opr (2) (+) (2)) (4)) (df-2 (2) (+) opreq2i df-4 df-3 (+) (1) opreq1i 2cn 1cn 1cn addass 3eqtr eqtr4)) thm (4nn () () (e. (4) (NN)) (2p2e4 2nn 2nn (2) (2) nnaddclt mp2an eqeltrr)) thm (2times () ((2times.1 (e. A (CC)))) (= (opr (2) (x.) A) (opr A (+) A)) (df-2 (x.) A opreq1i 2times.1 1p1times eqtr)) thm (2timest () () (-> (e. A (CC)) (= (opr (2) (x.) A) (opr A (+) A))) (A (if (e. A (CC)) A (0)) (2) (x.) opreq2 (= A (if (e. A (CC)) A (0))) id (= A (if (e. A (CC)) A (0))) id (+) opreq12d eqeq12d 0cn A elimel 2times dedth)) thm (times2 () ((times2.1 (e. A (CC)))) (= (opr A (x.) (2)) (opr A (+) A)) (times2.1 2cn mulcom times2.1 2times eqtr)) thm (3p2e5 () () (= (opr (3) (+) (2)) (5)) (df-4 (+) (1) opreq1i df-5 df-2 (3) (+) opreq2i 3re recn 1cn 1cn addass eqtr4 3eqtr4r)) thm (3p3e6 () () (= (opr (3) (+) (3)) (6)) (3p2e5 (+) (1) opreq1i df-3 (3) (+) opreq2i 3re recn 2cn 1cn addass eqtr4 df-6 3eqtr4)) thm (4p2e6 () () (= (opr (4) (+) (2)) (6)) (df-5 (+) (1) opreq1i df-6 df-2 (4) (+) opreq2i 4re recn 1cn 1cn addass eqtr4 3eqtr4r)) thm (4p3e7 () () (= (opr (4) (+) (3)) (7)) (4p2e6 (+) (1) opreq1i df-3 (4) (+) opreq2i 4re recn 2cn 1cn addass eqtr4 df-7 3eqtr4)) thm (4p4e8 () () (= (opr (4) (+) (4)) (8)) (4p3e7 (+) (1) opreq1i df-4 (4) (+) opreq2i 4re recn 3re recn 1cn addass eqtr4 df-8 3eqtr4)) thm (5p2e7 () () (= (opr (5) (+) (2)) (7)) (df-6 (+) (1) opreq1i df-7 df-2 (5) (+) opreq2i 5re recn 1cn 1cn addass eqtr4 3eqtr4r)) thm (5p3e8 () () (= (opr (5) (+) (3)) (8)) (5p2e7 (+) (1) opreq1i df-3 (5) (+) opreq2i 5re recn 2cn 1cn addass eqtr4 df-8 3eqtr4)) thm (5p4e9 () () (= (opr (5) (+) (4)) (9)) (5p3e8 (+) (1) opreq1i df-4 (5) (+) opreq2i 5re recn 3re recn 1cn addass eqtr4 df-9 3eqtr4)) thm (6p2e8 () () (= (opr (6) (+) (2)) (8)) (df-7 (+) (1) opreq1i df-8 df-2 (6) (+) opreq2i 6re recn 1cn 1cn addass eqtr4 3eqtr4r)) thm (6p3e9 () () (= (opr (6) (+) (3)) (9)) (6p2e8 (+) (1) opreq1i df-3 (6) (+) opreq2i 6re recn 2cn 1cn addass eqtr4 df-9 3eqtr4)) thm (7p2e9 () () (= (opr (7) (+) (2)) (9)) (df-8 (+) (1) opreq1i df-9 df-2 (7) (+) opreq2i 7re recn 1cn 1cn addass eqtr4 3eqtr4r)) thm (2t2e4 () () (= (opr (2) (x.) (2)) (4)) (2cn 2times 2p2e4 eqtr)) thm (3t2e6 () () (= (opr (3) (x.) (2)) (6)) (3re recn times2 3p3e6 eqtr)) thm (3t3e9 () () (= (opr (3) (x.) (3)) (9)) (df-3 (3) (x.) opreq2i 3re recn 2cn 1cn adddi 3t2e6 3re recn mulid1 (+) opreq12i eqtr 6p3e9 3eqtr)) thm (4t2e8 () () (= (opr (4) (x.) (2)) (8)) (4re recn times2 4p4e8 eqtr)) thm (4d2e2 () () (= (opr (4) (/) (2)) (2)) (2t2e4 4re recn 2cn 2cn 2re 2pos gt0ne0i divmul mpbir)) thm (1lt2 () () (br (1) (<) (2)) (ax1re ltp1 df-2 breqtrr)) thm (halfgt0 () () (br (0) (<) (opr (1) (/) (2))) (2re 2pos recgt0i)) thm (halflt1 () () (br (opr (1) (/) (2)) (<) (1)) (1cn div1 1lt2 eqbrtr ax1re ax1re 2re lt01 2pos ltdiv23i mpbi)) thm (halfclt () () (-> (e. A (CC)) (e. (opr A (/) (2)) (CC))) (2cn 2re 2pos gt0ne0i A (2) divclt mp3an23)) thm (rehalfclt () () (-> (e. A (RR)) (e. (opr A (/) (2)) (RR))) (2re 2re 2pos gt0ne0i A (2) redivclt mp3an23)) thm (half0t () () (-> (e. A (CC)) (<-> (= (opr A (/) (2)) (0)) (= A (0)))) (2cn 2re 2pos gt0ne0i A (2) diveq0t mp3an23)) thm (halfpost () () (-> (e. A (RR)) (<-> (br (0) (<) A) (br (opr A (/) (2)) (<) A))) (A (if (e. A (RR)) A (0)) (0) (<) breq2 A (if (e. A (RR)) A (0)) (/) (opr (1) (+) (1)) opreq1 (= A (if (e. A (RR)) A (0))) id (<) breq12d bibi12d 0re A elimel halfpos dedth df-2 A (/) opreq2i (<) A breq1i syl6bbr)) thm (halfpos2t () () (-> (e. A (RR)) (<-> (br (0) (<) A) (br (0) (<) (opr A (/) (2))))) (2re 2pos A (2) gt0divt mp3an23)) thm (halfnneg2t () () (-> (e. A (RR)) (<-> (br (0) (<_) A) (br (0) (<_) (opr A (/) (2))))) (2re 2pos A (2) ge0divt mp3an23)) thm (2halvest () () (-> (e. A (CC)) (= (opr (opr A (/) (2)) (+) (opr A (/) (2))) A)) (A 2timest (/) (2) opreq1d 2cn 2re 2pos gt0ne0i (2) A divcan3t mp3an13 2cn 2re 2pos gt0ne0i A A (2) divdirt mpan2 mp3an3 anidms 3eqtr3rd)) thm (nominpos ((x y)) () (-. (E.e. x (RR) (/\ (br (0) (<) (cv x)) (-. (E.e. y (RR) (/\ (br (0) (<) (cv y)) (br (cv y) (<) (cv x)))))))) (2pos 2re (cv x) (2) divgt0t ex mpan2 mpan2i (cv x) halfpost biimpd jcad (cv x) rehalfclt jctild (cv y) (opr (cv x) (/) (2)) (0) (<) breq2 (cv y) (opr (cv x) (/) (2)) (<) (cv x) breq1 anbi12d (RR) rcla4ev syl6 (br (0) (<) (cv x)) (E.e. y (RR) (/\ (br (0) (<) (cv y)) (br (cv y) (<) (cv x)))) iman sylib nrex)) thm (avglet () () (-> (/\ (e. A (RR)) (e. B (RR))) (\/ (br (opr (opr A (+) B) (/) (2)) (<_) A) (br (opr (opr A (+) B) (/) (2)) (<_) B))) ((opr (2) (x.) A) (opr A (+) B) ltnlet 2re (2) A axmulrcl mpan (e. B (RR)) adantr A B axaddrcl sylanc A B A ltadd2t (e. A (RR)) (e. B (RR)) pm3.26 (e. A (RR)) (e. B (RR)) pm3.27 (e. A (RR)) (e. B (RR)) pm3.26 syl3anc A B B ltadd1t (e. A (RR)) (e. B (RR)) pm3.26 (e. A (RR)) (e. B (RR)) pm3.27 (e. A (RR)) (e. B (RR)) pm3.27 syl3anc bitr3d A recnt A 2timest syl (e. B (RR)) adantr (<) (opr A (+) B) breq1d B recnt B 2timest syl (e. A (RR)) adantl (opr A (+) B) (<) breq2d 3bitr4d (opr A (+) B) (opr (2) (x.) B) ltlet A B axaddrcl 2re (2) B axmulrcl mpan (e. A (RR)) adantl sylanc sylbid sylbird orrd 2re 2pos (opr A (+) B) (2) A ledivmult mpan2 mp3an2 A B axaddrcl (e. A (RR)) (e. B (RR)) pm3.26 sylanc 2re 2pos (opr A (+) B) (2) B ledivmult mpan2 mp3an2 A B axaddrcl (e. A (RR)) (e. B (RR)) pm3.27 sylanc orbi12d mpbird)) thm (lbreu ((w x) (w y) (S w) (x y) (S x) (S y)) () (-> (/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (E!e. x S (A.e. y S (br (cv x) (<_) (cv y))))) ((cv y) (cv w) (cv x) (<_) breq2 S rcla4v (cv y) (cv x) (cv w) (<_) breq2 S rcla4v im2anan9r S (RR) (cv x) ssel S (RR) (cv w) ssel anim12d impcom (cv x) (cv w) letri3t syl biimprd ex com23 syld com3r r19.21aivv (E.e. x S (A.e. y S (br (cv x) (<_) (cv y)))) anim2i ancoms (cv x) (cv w) (<_) (cv y) breq1 y S ralbidv S reu4 sylibr)) thm (lbcl ((x y) (S x) (S y)) () (-> (/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (e. (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) S)) (S x y lbreu x S (A.e. y S (br (cv x) (<_) (cv y))) reucl2 x S (A.e. y S (br (cv x) (<_) (cv y))) ssrab2 (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) sseli 3syl)) thm (lble ((w x) (w y) (S w) (x y) (S x) (S y) (A w) (A y)) () (-> (/\/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y)))) (e. A S)) (br (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (<_) A)) (y S (br (cv x) (<_) (cv y)) hbra1 (e. (cv w) S) y ax-17 x hbrab hbuni (e. (cv w) (<_)) y ax-17 (e. (cv w) A) y ax-17 hbbr (cv y) A (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (<_) breq2 S rcla4 imp S x y lbreu (cv x) (cv w) (<_) (cv y) breq1 y S ralbidv S cbvreuv sylib (cv w) (cv x) (<_) (cv y) breq1 y S ralbidv y S (br (cv x) (<_) (cv y)) hbra1 (e. (cv w) S) y ax-17 x hbrab hbuni hbeleq (cv w) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (<_) (cv y) breq1 S ralbid reuuni3 syl sylan2 3impb 3coml)) thm (lbinfm ((w x) (w y) (w z) (v w) (S w) (x y) (x z) (v x) (S x) (y z) (v y) (S y) (v z) (S z) (S v)) () (-> (/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (= (sup S (RR) (`' (<))) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))))) (S x y (cv z) lble 3expa (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (cv z) lenltt S x y lbcl S (RR) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) ssel2 syldan (e. (cv z) S) adantr S (RR) (cv z) ssel2 (E.e. x S (A.e. y S (br (cv x) (<_) (cv y)))) adantlr sylanc mpbid reex S ssex S (V) x (A.e. y S (br (cv x) (<_) (cv y))) rabexg syl ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y)))) (V) uniexg z visset (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (V) (cv z) (V) (<) brcnvg mpan2 3syl negbid (E.e. x S (A.e. y S (br (cv x) (<_) (cv y)))) (e. (cv z) S) ad2antrr mpbird r19.21aiva S x y lbcl (br (cv z) (`' (<)) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y)))))) a1d ancrd (cv w) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (cv z) (`' (<)) breq2 S rcla4ev syl6 (e. (cv z) (RR)) a1d r19.21aiv jca (cv v) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (`' (<)) (cv z) breq1 negbid z S ralbidv (cv v) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (cv z) (`' (<)) breq2 (E.e. w S (br (cv z) (`' (<)) (cv w))) imbi1d z (RR) ralbidv anbi12d (RR) reuuni2 S x y lbcl S (RR) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) ssel (E.e. x S (A.e. y S (br (cv x) (<_) (cv y)))) adantr mpd (cv v) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (`' (<)) (cv z) breq1 negbid z S ralbidv (cv v) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (cv z) (`' (<)) breq2 (E.e. w S (br (cv z) (`' (<)) (cv w))) imbi1d z (RR) ralbidv anbi12d (RR) rcla4ev S x y lbcl S (RR) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) ssel (E.e. x S (A.e. y S (br (cv x) (<_) (cv y)))) adantr mpd S x y (cv z) lble 3expa (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (cv z) lenltt S x y lbcl S (RR) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) ssel2 syldan (e. (cv z) S) adantr S (RR) (cv z) ssel2 (E.e. x S (A.e. y S (br (cv x) (<_) (cv y)))) adantlr sylanc mpbid reex S ssex S (V) x (A.e. y S (br (cv x) (<_) (cv y))) rabexg syl ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y)))) (V) uniexg z visset (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (V) (cv z) (V) (<) brcnvg mpan2 3syl negbid (E.e. x S (A.e. y S (br (cv x) (<_) (cv y)))) (e. (cv z) S) ad2antrr mpbird r19.21aiva S x y lbcl (br (cv z) (`' (<)) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y)))))) a1d ancrd (cv w) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (cv z) (`' (<)) breq2 S rcla4ev syl6 (e. (cv z) (RR)) a1d r19.21aiv jca sylanc ltso (<) (RR) cnvso mpbi v z S w supeu syl sylanc mpbid S (RR) (`' (<)) v z w df-sup syl5eq)) thm (lbinfmcl ((x y) (S x) (S y)) () (-> (/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (e. (sup S (RR) (`' (<))) S)) (S x y lbinfm S x y lbcl eqeltrd)) thm (lbinfmle ((x y) (S x) (S y) (A y)) () (-> (/\/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y)))) (e. A S)) (br (sup S (RR) (`' (<))) (<_) A)) (S x y lbinfm (e. A S) 3adant3 S x y A lble eqbrtrd)) thm (sup2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (=/= A ({/})) (E.e. x (RR) (A.e. y A (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x)))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))))) ((cv x) peano2re (A.e. y A (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x)))) adantr (C_ A (RR)) a1i A (RR) (cv y) ssel (cv y) (cv x) (opr (cv x) (+) (1)) axlttrn 3expb (cv x) peano2re ancli sylan2 (cv x) ltp1t sylan2i exp4b com34 pm2.43d imp (cv x) ltp1t (cv y) (cv x) (<) (opr (cv x) (+) (1)) breq1 biimprcd syl (e. (cv y) (RR)) adantl jaod ex syl6 com23 imp a2d y 19.20dv y A (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x))) df-ral y A (br (cv y) (<) (opr (cv x) (+) (1))) df-ral 3imtr4g ex imp3a jcad (cv x) (+) (1) oprex (cv z) (opr (cv x) (+) (1)) (RR) eleq1 (cv z) (opr (cv x) (+) (1)) (cv y) (<) breq2 y A ralbidv anbi12d cla4ev syl6 x 19.23adv (cv z) (cv x) (RR) eleq1 (cv z) (cv x) (cv y) (<) breq2 y A ralbidv anbi12d cbvexv syl6ib x (RR) (A.e. y A (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x)))) df-rex x (RR) (A.e. y A (br (cv y) (<) (cv x))) df-rex 3imtr4g (=/= A ({/})) adantr imdistani (C_ A (RR)) (=/= A ({/})) (E.e. x (RR) (A.e. y A (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x))))) df-3an (C_ A (RR)) (=/= A ({/})) (E.e. x (RR) (A.e. y A (br (cv y) (<) (cv x)))) df-3an 3imtr4 A x y z axsup syl)) thm (sup3 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))))) (A ({/}) df-ne bicomi (C_ A (RR)) a1i A (RR) (cv y) ssel (cv y) (cv x) leloet expcom syl9 imp31 ralbidva rexbidva anbi12d pm5.32i (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x)))) 3anass (C_ A (RR)) (=/= A ({/})) (E.e. x (RR) (A.e. y A (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x))))) 3anass 3bitr4 A x y z sup2 sylbi)) thm (infm3lem ((x y)) () (-> (e. (cv x) (RR)) (E.e. y (RR) (= (cv x) (-u (cv y))))) ((cv y) (-u (cv x)) negeq (cv x) eqeq2d (RR) rcla4ev (cv x) renegclt (cv x) recnt (cv x) negnegt syl eqcomd sylanc)) thm (infm3 ((x y) (x z) (w x) (v x) (u x) (t x) (A x) (y z) (w y) (v y) (u y) (t y) (A y) (w z) (v z) (u z) (t z) (A z) (v w) (u w) (t w) (A w) (u v) (t v) (A v) (t u) (A u) (A t)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv y) (<) (cv x)))) (A.e. y (RR) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y)))))))) (A (RR) (cv v) ssel pm4.71rd v exbidv v (RR) (e. (cv v) A) df-rex (cv w) renegclt v w infm3lem (cv v) (-u (cv w)) A eleq1 rexxfr bitr3 syl6bb A v n0 w (RR) (e. (-u (cv w)) A) rabn0 3bitr4g A (RR) (cv y) ssel pm4.71rd (br (cv x) (<_) (cv y)) imbi1d (e. (cv y) (RR)) (e. (cv y) A) (br (cv x) (<_) (cv y)) impexp syl6bb y albidv y A (br (cv x) (<_) (cv y)) df-ral (cv v) renegclt y v infm3lem (cv y) (-u (cv v)) A eleq1 (cv y) (-u (cv v)) (cv x) (<_) breq2 imbi12d ralxfr y (RR) (-> (e. (cv y) A) (br (cv x) (<_) (cv y))) df-ral bitr3 3bitr4g x (RR) rexbidv (cv u) renegclt x u infm3lem (cv x) (-u (cv u)) (<_) (-u (cv v)) breq1 (e. (-u (cv v)) A) imbi2d v (RR) ralbidv rexxfr (cv v) (cv u) lenegt ancoms (e. (-u (cv v)) A) imbi2d ralbidva (cv w) (cv v) negeq A eleq1d (RR) elrab (br (cv v) (<_) (cv u)) imbi1i (e. (cv v) (RR)) (e. (-u (cv v)) A) (br (cv v) (<_) (cv u)) impexp bitr v albii v ({e.|} w (RR) (e. (-u (cv w)) A)) (br (cv v) (<_) (cv u)) df-ral v (RR) (-> (e. (-u (cv v)) A) (br (cv v) (<_) (cv u))) df-ral 3bitr4r syl5bbr rexbiia bitr4 syl6bb anbi12d w (RR) (e. (-u (cv w)) A) ssrab2 ({e.|} w (RR) (e. (-u (cv w)) A)) u v t sup3 mp3an1 syl6bi A (RR) (cv y) ssel pm4.71rd (-. (br (cv y) (<) (cv x))) imbi1d (e. (cv y) (RR)) (e. (cv y) A) (-. (br (cv y) (<) (cv x))) impexp syl6bb y albidv y A (-. (br (cv y) (<) (cv x))) df-ral (cv v) renegclt y v infm3lem (cv y) (-u (cv v)) A eleq1 (cv y) (-u (cv v)) (<) (cv x) breq1 negbid imbi12d ralxfr y (RR) (-> (e. (cv y) A) (-. (br (cv y) (<) (cv x)))) df-ral bitr3 3bitr4g A (RR) (cv z) ssel (br (cv z) (<) (-u (cv v))) adantrd pm4.71rd z exbidv z A (br (cv z) (<) (-u (cv v))) df-rex (cv t) renegclt z t infm3lem (cv z) (-u (cv t)) A eleq1 (cv z) (-u (cv t)) (<) (-u (cv v)) breq1 anbi12d rexxfr z (RR) (/\ (e. (cv z) A) (br (cv z) (<) (-u (cv v)))) df-rex bitr3 3bitr4g (br (cv x) (<) (-u (cv v))) imbi2d v (RR) ralbidv (cv v) renegclt y v infm3lem (cv y) (-u (cv v)) (cv x) (<) breq2 (cv y) (-u (cv v)) (cv z) (<) breq2 z A rexbidv imbi12d ralxfr syl5bb anbi12d x (RR) rexbidv (cv u) renegclt x u infm3lem (cv x) (-u (cv u)) (-u (cv v)) (<) breq2 negbid (e. (-u (cv v)) A) imbi2d v (RR) ralbidv (cv x) (-u (cv u)) (<) (-u (cv v)) breq1 (E.e. t (RR) (/\ (e. (-u (cv t)) A) (br (-u (cv t)) (<) (-u (cv v))))) imbi1d v (RR) ralbidv anbi12d rexxfr (cv u) (cv v) ltnegt negbid (e. (-u (cv v)) A) imbi2d ralbidva (cv w) (cv v) negeq A eleq1d (RR) elrab (-. (br (cv u) (<) (cv v))) imbi1i (e. (cv v) (RR)) (e. (-u (cv v)) A) (-. (br (cv u) (<) (cv v))) impexp bitr v albii v ({e.|} w (RR) (e. (-u (cv w)) A)) (-. (br (cv u) (<) (cv v))) df-ral v (RR) (-> (e. (-u (cv v)) A) (-. (br (cv u) (<) (cv v)))) df-ral 3bitr4r syl5bbr (cv v) (cv u) ltnegt ancoms (cv v) (cv t) ltnegt (e. (-u (cv t)) A) anbi2d rexbidva (cv w) (cv t) negeq A eleq1d (RR) elrab (br (cv v) (<) (cv t)) anbi1i (e. (cv t) (RR)) (e. (-u (cv t)) A) (br (cv v) (<) (cv t)) anass bitr rexbii2 syl5bb (e. (cv u) (RR)) adantl imbi12d ralbidva anbi12d rexbiia bitr4 syl6bb sylibrd 3impib)) thm (suprcl ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (e. (sup A (RR) (<)) (RR))) (A x y z sup3 ltso x y A z supcl syl)) thm (suprub ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (e. B A)) (br B (<_) (sup A (RR) (<)))) (A x y z sup3 ltso x y A z B supub syl imp B (sup A (RR) (<)) lenltt (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x)))) 3simp1 B sseld imp A x y suprcl (e. B A) adantr sylanc mpbird)) thm (suprlub ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (/\ (e. B (RR)) (br B (<) (sup A (RR) (<))))) (E.e. z A (br B (<) (cv z)))) (A x y z sup3 ltso x y A z B suplub syl imp)) thm (suprnub ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (/\ (e. B (RR)) (A.e. z A (-. (br B (<) (cv z)))))) (-. (br B (<) (sup A (RR) (<))))) (A x y z sup3 ltso x y A z B supnub syl imp)) thm (suprleub ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (/\ (e. B (RR)) (A.e. z A (br (cv z) (<_) B)))) (br (sup A (RR) (<)) (<_) B)) ((cv z) B lenltt A (RR) (cv z) ssel2 sylan an1rs ralbidva ex pm5.32d (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x)))) 3ad2ant1 A x y B z suprnub (sup A (RR) (<)) B lenltt A x y suprcl sylan (A.e. z A (-. (br B (<) (cv z)))) adantrr mpbird ex sylbid imp)) thm (sup3i ((x y) (x z) (A x) (y z) (A y) (A z)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))))) (sup3i.1 A x y z sup3 ax-mp)) thm (suprcli ((x y) (x z) (A x) (y z) (A y) (A z)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (e. (sup A (RR) (<)) (RR)) (ltso sup3i.1 z sup3i supcli)) thm (suprubi ((x y) (A x) (A y)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (-> (e. B A) (br B (<_) (sup A (RR) (<)))) (sup3i.1 A x y B suprub mpan)) thm (suprlubi ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (-> (/\ (e. B (RR)) (br B (<) (sup A (RR) (<)))) (E.e. z A (br B (<) (cv z)))) (ltso sup3i.1 z sup3i B suplubi)) thm (suprnubi ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (-> (/\ (e. B (RR)) (A.e. z A (-. (br B (<) (cv z))))) (-. (br B (<) (sup A (RR) (<))))) (ltso sup3i.1 z sup3i B supnubi)) thm (suprleubi ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (-> (/\ (e. B (RR)) (A.e. z A (br (cv z) (<_) B))) (br (sup A (RR) (<)) (<_) B)) (sup3i.1 A x y B z suprleub mpan)) thm (reuunineg ((x y) (x z) (y z) (ph y) (ph z) (ps x) (ps z)) ((reuunineg.1 (-> (= (cv x) (-u (cv y))) (<-> ph ps)))) (-> (E!e. x (RR) ph) (= (U. ({e.|} x (RR) ph)) (-u (U. ({e.|} y (RR) ps))))) (z y (RR) ps hbrab1 hbuni hbneg (cv y) renegclt (U. ({e.|} y (RR) ps)) renegclt reuunineg.1 (cv y) (U. ({e.|} y (RR) ps)) negeq (cv x) renegclt (cv x) (cv y) negcon2t (cv x) recnt (cv y) recnt syl2an reuhyp reuunixfr)) thm (dfinfmr ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (RR)) (= (sup A (RR) (`' (<))) (U. ({e.|} x (RR) (/\ (A.e. y A (br (cv x) (<_) (cv y))) (A.e. y (RR) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y)))))))))) ((cv x) (cv y) lenltt x visset y visset (<) brcnv negbii syl6rbbr A (RR) (cv y) ssel2 sylan2 ancoms an1rs ralbidva y visset x visset (<) brcnv y visset z visset (<) brcnv z A rexbii imbi12i y (RR) ralbii (/\ (C_ A (RR)) (e. (cv x) (RR))) a1i anbi12d ex rabbidv unieqd A (RR) (`' (<)) x y z df-sup syl5eq)) thm (infmsup ((x y) (x z) (w x) (v x) (u x) (t x) (A x) (y z) (w y) (v y) (u y) (t y) (A y) (w z) (v z) (u z) (t z) (A z) (v w) (u w) (t w) (A w) (u v) (t v) (A v) (t u) (A u) (A t)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y))))) (= (sup A (RR) (`' (<))) (-u (sup ({e.|} z (RR) (e. (-u (cv z)) A)) (RR) (<))))) (A x y w infm3 3exp (cv x) (-u (cv v)) (cv y) (<) breq2 negbid y A ralbidv (cv x) (-u (cv v)) (<) (cv y) breq1 (E.e. w A (br (cv w) (<) (cv y))) imbi1d y (RR) ralbidv anbi12d reuunineg A (RR) (`' (<)) x y w df-sup x visset y visset (<) brcnv negbii y A ralbii y visset x visset (<) brcnv y visset w visset (<) brcnv w A rexbii imbi12i y (RR) ralbii anbi12i (e. (cv x) (RR)) a1i rabbii unieqi eqtr syl5eq (cv v) (cv u) ltnegt negbid (e. (-u (cv u)) A) imbi2d ralbidva (cv z) (cv u) negeq A eleq1d (RR) elrab (-. (br (cv v) (<) (cv u))) imbi1i (e. (cv u) (RR)) (e. (-u (cv u)) A) (-. (br (cv v) (<) (cv u))) impexp bitr u albii u ({e.|} z (RR) (e. (-u (cv z)) A)) (-. (br (cv v) (<) (cv u))) df-ral u (RR) (-> (e. (-u (cv u)) A) (-. (br (cv v) (<) (cv u)))) df-ral 3bitr4r syl5bbr (cv u) (cv v) ltnegt ancoms (cv u) (cv t) ltnegt (e. (-u (cv t)) A) anbi2d rexbidva (cv z) (cv t) negeq A eleq1d (RR) elrab (br (cv u) (<) (cv t)) anbi1i (e. (cv t) (RR)) (e. (-u (cv t)) A) (br (cv u) (<) (cv t)) anass bitr rexbii2 syl5bb (e. (cv v) (RR)) adantl imbi12d ralbidva anbi12d A (RR) (cv y) ssel pm4.71rd (-. (br (cv y) (<) (-u (cv v)))) imbi1d (e. (cv y) (RR)) (e. (cv y) A) (-. (br (cv y) (<) (-u (cv v)))) impexp syl6rbb y albidv (cv u) renegclt y u infm3lem (cv y) (-u (cv u)) A eleq1 (cv y) (-u (cv u)) (<) (-u (cv v)) breq1 negbid imbi12d ralxfr y (RR) (-> (e. (cv y) A) (-. (br (cv y) (<) (-u (cv v))))) df-ral bitr3 y A (-. (br (cv y) (<) (-u (cv v)))) df-ral 3bitr4g A (RR) (cv w) ssel (br (cv w) (<) (-u (cv u))) adantrd pm4.71rd w exbidv w A (br (cv w) (<) (-u (cv u))) df-rex (cv t) renegclt w t infm3lem (cv w) (-u (cv t)) A eleq1 (cv w) (-u (cv t)) (<) (-u (cv u)) breq1 anbi12d rexxfr w (RR) (/\ (e. (cv w) A) (br (cv w) (<) (-u (cv u)))) df-rex bitr3 3bitr4g (br (-u (cv v)) (<) (-u (cv u))) imbi2d u (RR) ralbidv (cv u) renegclt y u infm3lem (cv y) (-u (cv u)) (-u (cv v)) (<) breq2 (cv y) (-u (cv u)) (cv w) (<) breq2 w A rexbidv imbi12d ralxfr syl5rbb anbi12d sylan9bbr ex rabbidv unieqd ({e.|} z (RR) (e. (-u (cv z)) A)) (RR) (<) v u t df-sup syl5eq negeqd eqcomd sylan9eqr ex ltso (<) (RR) cnvso mpbi x y A w supeu x visset y visset (<) brcnv negbii y A ralbii y visset x visset (<) brcnv y visset w visset (<) brcnv w A rexbii imbi12i y (RR) ralbii anbi12i x (RR) rexbii x visset y visset (<) brcnv negbii y A ralbii y visset x visset (<) brcnv y visset w visset (<) brcnv w A rexbii imbi12i y (RR) ralbii anbi12i x (RR) reubii 3imtr3 syl5 syl6d 3imp)) thm (infmrcl ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y))))) (e. (sup A (RR) (`' (<))) (RR))) (A x y v infmsup v (RR) (e. (-u (cv v)) A) ssrab2 ({e.|} v (RR) (e. (-u (cv v)) A)) z w suprcl mp3an1 A (RR) (cv z) ssel (cv z) renegclt syl6 A (RR) (cv z) ssel2 (cv z) recnt (cv z) negnegt 3syl (C_ A (RR)) (e. (cv z) A) pm3.27 eqeltrd ex jcad (cv v) (-u (cv z)) negeq A eleq1d (RR) elrab (-u (cv z)) ({e.|} v (RR) (e. (-u (cv v)) A)) n0i sylbir syl6 z 19.23adv imp A z n0 sylan2b (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y)))) 3adant3 (cv y) (-u (cv w)) (cv x) (<_) breq2 A rcla4va (e. (cv w) (RR)) adantll (e. (cv x) (RR)) adantll (cv x) (cv w) lenegcon2t (e. (-u (cv w)) A) adantrr (A.e. y A (br (cv x) (<_) (cv y))) adantr mpbid exp31 (cv v) (cv w) negeq A eleq1d (RR) elrab syl5ib com23 r19.21adv (cv x) renegclt jctild (cv z) (-u (cv x)) (cv w) (<_) breq2 w ({e.|} v (RR) (e. (-u (cv v)) A)) ralbidv (RR) rcla4ev syl6 r19.23aiv (C_ A (RR)) (-. (= A ({/}))) 3ad2ant3 sylanc (sup ({e.|} v (RR) (e. (-u (cv v)) A)) (RR) (<)) renegclt syl eqeltrd)) thm (nnunb ((x y) (x z) (y z)) () (-. (E.e. x (RR) (A.e. y (NN) (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x)))))) ((A.e. y (NN) (-. (br (cv x) (<) (cv y)))) pm3.24 (cv x) (-) (1) oprex (cv y) (opr (cv x) (-) (1)) (RR) eleq1 (cv y) (opr (cv x) (-) (1)) (<) (cv x) breq1 (cv y) (opr (cv x) (-) (1)) (<) (cv z) breq1 z (NN) rexbidv imbi12d imbi12d cla4v (cv x) ltm1t syl7 ax1re (cv x) (1) resubclt mpan2 syl5 pm2.43d z (NN) (br (opr (cv x) (-) (1)) (<) (cv z)) df-rex syl6ib com12 ax1re (cv x) (1) (cv z) ltsubaddt mp3an2 (cv z) nnret sylan2 ex pm5.32d z exbidv (cv z) (+) (1) oprex (cv y) (opr (cv z) (+) (1)) (NN) eleq1 (cv y) (opr (cv z) (+) (1)) (cv x) (<) breq2 anbi12d cla4ev (cv z) peano2nn sylan z 19.23aiv syl6bi syld y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z (NN) (br (cv y) (<) (cv z)))) df-ral y (NN) (-. (br (cv x) (<) (cv y))) df-ral y (e. (cv y) (NN)) (br (cv x) (<) (cv y)) alinexa bitr2 con1bii 3imtr4g (A.e. y (NN) (-. (br (cv x) (<) (cv y)))) anim2d mtoi nrex nnssre 1nn (1) (NN) n0i ax-mp (NN) ({/}) df-ne mpbir (NN) x y z sup2 mp3an12 mto)) thm (arch ((x y) (A x) (A y)) () (-> (e. A (RR)) (E.e. x (NN) (br A (<) (cv x)))) ((cv y) A (<) (cv x) breq1 x (NN) rexbidv y x nnunb y (RR) (A.e. x (NN) (\/ (br (cv x) (<) (cv y)) (= (cv x) (cv y)))) ralnex mpbir (cv y) (cv x) axlttri (cv x) nnret sylan2 (cv y) (cv x) eqcom (br (cv x) (<) (cv y)) orbi1i (= (cv x) (cv y)) (br (cv x) (<) (cv y)) orcom bitr negbii syl6bb biimprd r19.22dva x (NN) (\/ (br (cv x) (<) (cv y)) (= (cv x) (cv y))) rexnal syl5ibr r19.20i ax-mp vtoclri)) thm (nnreclt ((A n)) () (-> (/\ (e. A (RR)) (br (0) (<) A)) (E.e. n (NN) (br (opr (1) (/) (cv n)) (<) A))) ((1) A redivclt 3expa (e. A (RR)) (br (0) (<) A) pm3.26 ax1re jctil A gt0ne0t sylanc (opr (1) (/) A) n arch syl (opr (1) (/) A) (cv n) ltrect (1) A redivclt 3expa (e. A (RR)) (br (0) (<) A) pm3.26 ax1re jctil A gt0ne0t sylanc A recgt0t jca (cv n) nnret (cv n) nngt0t jca syl2an A recrect A recnt (br (0) (<) A) adantr A gt0ne0t sylanc (opr (1) (/) (cv n)) (<) breq2d (e. (cv n) (NN)) adantr bitrd rexbidva mpbid)) thm (xrsupexmnf ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR*) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))))) (E.e. x (RR*) (/\ (A.e. y (u. A ({} (-oo))) (-. (br (cv x) (<) (cv y)))) (A.e. y (RR*) (-> (br (cv y) (<) (cv x)) (E.e. z (u. A ({} (-oo))) (br (cv y) (<) (cv z)))))))) ((e. (cv x) (RR*)) (-> (e. (cv y) A) (-. (br (cv x) (<) (cv y)))) pm3.27 (cv y) (-oo) (cv x) (<) breq2 negbid (cv x) nltmnft syl5bir com12 y (-oo) elsn syl5ib (-> (e. (cv y) A) (-. (br (cv x) (<) (cv y)))) adantr jaod (cv y) A ({} (-oo)) elun syl5ib ex r19.20dv2 (cv z) A ({} (-oo)) elun1 (br (cv y) (<) (cv z)) anim1i r19.22i2 (br (cv y) (<) (cv x)) imim2i y (RR*) r19.20si (e. (cv x) (RR*)) a1i anim12d r19.22i)) thm (xrinfmexpnf ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv y) (<) (cv x)))) (A.e. y (RR*) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y))))))) (E.e. x (RR*) (/\ (A.e. y (u. A ({} (+oo))) (-. (br (cv y) (<) (cv x)))) (A.e. y (RR*) (-> (br (cv x) (<) (cv y)) (E.e. z (u. A ({} (+oo))) (br (cv z) (<) (cv y)))))))) ((e. (cv x) (RR*)) (-> (e. (cv y) A) (-. (br (cv y) (<) (cv x)))) pm3.27 (cv y) (+oo) (<) (cv x) breq1 negbid (cv x) pnfnltt syl5bir com12 y (+oo) elsn syl5ib (-> (e. (cv y) A) (-. (br (cv y) (<) (cv x)))) adantr jaod (cv y) A ({} (+oo)) elun syl5ib ex r19.20dv2 (cv z) A ({} (+oo)) elun1 (br (cv z) (<) (cv y)) anim1i r19.22i2 (br (cv x) (<) (cv y)) imim2i y (RR*) r19.20si (e. (cv x) (RR*)) a1i anim12d r19.22i)) thm (xrsupsslem ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (C_ A (RR*)) (\/ (C_ A (RR)) (e. (+oo) A))) (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR*) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))))) (A x y z sup3 (cv x) rexrt (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))) anim1i r19.22i2 syl (/\ (/\ (C_ A (RR)) (-. (= A ({/})))) (e. (cv x) (RR*))) (-> (e. (cv y) (RR)) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))) pm3.27 (cv x) pnfnltt (= (cv y) (+oo)) adantr (cv y) (+oo) (<) (cv x) breq1 negbid (e. (cv x) (RR*)) adantl mpbird (E.e. z A (br (cv y) (<) (cv z))) pm2.21d ex (/\ (C_ A (RR)) (-. (= A ({/})))) (-> (e. (cv y) (RR)) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))) ad2antlr A (RR) (cv z) ssel (cv z) mnfltt syl6 ancld z 19.22dv A z n0 z A (br (-oo) (<) (cv z)) df-rex 3imtr4g imp (br (-oo) (<) (cv x)) a1d (e. (cv x) (RR*)) (= (cv y) (-oo)) ad2antrr (cv y) (-oo) (<) (cv x) breq1 (cv y) (-oo) (<) (cv z) breq1 z A rexbidv imbi12d (/\ (/\ (C_ A (RR)) (-. (= A ({/})))) (e. (cv x) (RR*))) adantl mpbird ex (-> (e. (cv y) (RR)) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))) adantr 3jaod (cv y) elxr syl5ib ex r19.20dv2 (A.e. y A (-. (br (cv x) (<) (cv y)))) anim2d r19.22dva (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x)))) 3adant3 mpd 3expa (cv y) (cv x) letrit ord A (RR) (cv y) ssel2 sylan an1rs r19.22dva y A (br (cv y) (<_) (cv x)) rexnal syl5ibr r19.20dva imp x (RR) (A.e. y A (br (cv y) (<_) (cv x))) ralnex sylan2br (cv y) (cv z) (cv x) (<_) breq2 A cbvrexv x (RR) ralbii sylib A (RR) (cv y) ssel (cv y) rexrt (cv y) pnfnltt syl syl6 r19.21aiv (A.e. x (RR) (E.e. z A (br (cv x) (<_) (cv z)))) adantr (cv y) elxr (cv x) (opr (cv y) (+) (1)) (<_) (cv z) breq1 z A rexbidv (RR) rcla4va (C_ A (RR)) adantrr ancoms (cv y) peano2re sylan2 (cv y) ltp1t (e. (cv z) (RR)) adantr (cv y) (opr (cv y) (+) (1)) (cv z) ltletrt 3expa (cv y) peano2re ancli sylan mpand ancoms A (RR) (cv z) ssel2 sylan an1rs r19.22dva (A.e. x (RR) (E.e. z A (br (cv x) (<_) (cv z)))) adantll mpd exp31 (br (cv y) (<) (+oo)) a1dd com4r pnfxr (+oo) xrltnrt ax-mp (cv y) (+oo) (<) (+oo) breq1 mtbiri (E.e. z A (br (cv y) (<) (cv z))) pm2.21d (C_ A (RR)) a1d (A.e. x (RR) (E.e. z A (br (cv x) (<_) (cv z)))) a1d (cv y) (-oo) (<) (cv z) breq1 z A rexbidv A (RR) (cv z) ssel2 (cv z) mnfltt syl (br (0) (<_) (cv z)) a1d r19.22dva impcom 0re (cv x) (0) (<_) (cv z) breq1 z A rexbidv (RR) rcla4va mpan sylan syl5bir (br (cv y) (<) (+oo)) a1dd exp3a 3jaoi sylbi com13 imp r19.21aiv jca pnfxr jctil (cv x) (+oo) (<) (cv y) breq1 negbid y A ralbidv (cv x) (+oo) (cv y) (<) breq2 (E.e. z A (br (cv y) (<) (cv z))) imbi1d y (RR*) ralbidv anbi12d (RR*) rcla4ev syl syldan (-. (= A ({/}))) adantlr pm2.61dan ex mnfxr y (-. (br (-oo) (<) (cv y))) ral0 (cv y) nltmnft (E.e. z ({/}) (br (cv y) (<) (cv z))) pm2.21d rgen pm3.2i (cv x) (-oo) (<) (cv y) breq1 negbid y ({/}) ralbidv (cv x) (-oo) (cv y) (<) breq2 (E.e. z ({/}) (br (cv y) (<) (cv z))) imbi1d y (RR*) ralbidv anbi12d (RR*) rcla4ev mp2an A ({/}) y (-. (br (cv x) (<) (cv y))) raleq1 A ({/}) z (br (cv y) (<) (cv z)) rexeq1 (br (cv y) (<) (cv x)) imbi2d y (RR*) ralbidv anbi12d x (RR*) rexbidv mpbiri pm2.61d2 (C_ A (RR*)) adantl A (RR*) (cv y) ssel (cv y) pnfnltt syl6 r19.21aiv (cv z) (+oo) (cv y) (<) breq2 A rcla4ev ex (e. (cv y) (RR*)) a1d r19.21aiv anim12i pnfxr jctil (cv x) (+oo) (<) (cv y) breq1 negbid y A ralbidv (cv x) (+oo) (cv y) (<) breq2 (E.e. z A (br (cv y) (<) (cv z))) imbi1d y (RR*) ralbidv anbi12d (RR*) rcla4ev syl jaodan)) thm (xrinfmsslem ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (C_ A (RR*)) (\/ (C_ A (RR)) (e. (-oo) A))) (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv y) (<) (cv x)))) (A.e. y (RR*) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y)))))))) (A x y z infm3 (cv x) rexrt (/\ (A.e. y A (-. (br (cv y) (<) (cv x)))) (A.e. y (RR) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y)))))) anim1i r19.22i2 syl (/\ (/\ (C_ A (RR)) (-. (= A ({/})))) (e. (cv x) (RR*))) (-> (e. (cv y) (RR)) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y))))) pm3.27 A (RR) (cv z) ssel (cv z) ltpnft syl6 ancld z 19.22dv A z n0 z A (br (cv z) (<) (+oo)) df-rex 3imtr4g imp (br (cv x) (<) (+oo)) a1d (e. (cv x) (RR*)) (= (cv y) (+oo)) ad2antrr (cv y) (+oo) (cv x) (<) breq2 (cv y) (+oo) (cv z) (<) breq2 z A rexbidv imbi12d (/\ (/\ (C_ A (RR)) (-. (= A ({/})))) (e. (cv x) (RR*))) adantl mpbird ex (-> (e. (cv y) (RR)) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y))))) adantr (cv x) nltmnft (= (cv y) (-oo)) adantr (cv y) (-oo) (cv x) (<) breq2 negbid (e. (cv x) (RR*)) adantl mpbird (E.e. z A (br (cv z) (<) (cv y))) pm2.21d ex (/\ (C_ A (RR)) (-. (= A ({/})))) (-> (e. (cv y) (RR)) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y))))) ad2antlr 3jaod (cv y) elxr syl5ib ex r19.20dv2 (A.e. y A (-. (br (cv y) (<) (cv x)))) anim2d r19.22dva (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y)))) 3adant3 mpd 3expa (cv x) (cv y) letrit ancoms ord A (RR) (cv y) ssel2 sylan an1rs r19.22dva y A (br (cv x) (<_) (cv y)) rexnal syl5ibr r19.20dva imp x (RR) (A.e. y A (br (cv x) (<_) (cv y))) ralnex sylan2br (cv y) (cv z) (<_) (cv x) breq1 A cbvrexv x (RR) ralbii sylib A (RR) (cv y) ssel (cv y) rexrt (cv y) nltmnft syl syl6 r19.21aiv (A.e. x (RR) (E.e. z A (br (cv z) (<_) (cv x)))) adantr (cv y) elxr (cv x) (opr (cv y) (-) (1)) (cv z) (<_) breq2 z A rexbidv (RR) rcla4va (C_ A (RR)) adantrr ancoms ax1re (cv y) (1) resubclt mpan2 sylan2 (cv y) ltm1t (e. (cv z) (RR)) adantl (cv z) (opr (cv y) (-) (1)) (cv y) lelttrt 3expb ax1re (cv y) (1) resubclt mpan2 ancri sylan2 mpan2d A (RR) (cv z) ssel2 sylan an1rs r19.22dva (A.e. x (RR) (E.e. z A (br (cv z) (<_) (cv x)))) adantll mpd exp31 (br (-oo) (<) (cv y)) a1dd com4r (cv y) (+oo) (cv z) (<) breq2 z A rexbidv A (RR) (cv z) ssel2 (cv z) ltpnft syl (br (cv z) (<_) (0)) a1d r19.22dva impcom 0re (cv x) (0) (cv z) (<_) breq2 z A rexbidv (RR) rcla4va mpan sylan syl5bir (br (-oo) (<) (cv y)) a1dd exp3a mnfxr (-oo) xrltnrt ax-mp (cv y) (-oo) (-oo) (<) breq2 mtbiri (E.e. z A (br (cv z) (<) (cv y))) pm2.21d (C_ A (RR)) a1d (A.e. x (RR) (E.e. z A (br (cv z) (<_) (cv x)))) a1d 3jaoi sylbi com13 imp r19.21aiv jca mnfxr jctil (cv x) (-oo) (cv y) (<) breq2 negbid y A ralbidv (cv x) (-oo) (<) (cv y) breq1 (E.e. z A (br (cv z) (<) (cv y))) imbi1d y (RR*) ralbidv anbi12d (RR*) rcla4ev syl syldan (-. (= A ({/}))) adantlr pm2.61dan ex pnfxr y (-. (br (cv y) (<) (+oo))) ral0 (cv y) pnfnltt (E.e. z ({/}) (br (cv z) (<) (cv y))) pm2.21d rgen pm3.2i (cv x) (+oo) (cv y) (<) breq2 negbid y ({/}) ralbidv (cv x) (+oo) (<) (cv y) breq1 (E.e. z ({/}) (br (cv z) (<) (cv y))) imbi1d y (RR*) ralbidv anbi12d (RR*) rcla4ev mp2an A ({/}) y (-. (br (cv y) (<) (cv x))) raleq1 A ({/}) z (br (cv z) (<) (cv y)) rexeq1 (br (cv x) (<) (cv y)) imbi2d y (RR*) ralbidv anbi12d x (RR*) rexbidv mpbiri pm2.61d2 (C_ A (RR*)) adantl A (RR*) (cv y) ssel (cv y) nltmnft syl6 r19.21aiv (cv z) (-oo) (<) (cv y) breq1 A rcla4ev ex (e. (cv y) (RR*)) a1d r19.21aiv anim12i mnfxr jctil (cv x) (-oo) (cv y) (<) breq2 negbid y A ralbidv (cv x) (-oo) (<) (cv y) breq1 (E.e. z A (br (cv z) (<) (cv y))) imbi1d y (RR*) ralbidv anbi12d (RR*) rcla4ev syl jaodan)) thm (xrsupss ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (RR*)) (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR*) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))))) (A ssxr (C_ A (RR)) (e. (+oo) A) (e. (-oo) A) df-3or sylib A x y z xrsupsslem A ({} (-oo)) difss (\ A ({} (-oo))) A (RR*) sstr mpan (\ A ({} (-oo))) ssxr (\/ (C_ (\ A ({} (-oo))) (RR)) (e. (+oo) (\ A ({} (-oo))))) (e. (-oo) (\ A ({} (-oo)))) orcom (C_ (\ A ({} (-oo))) (RR)) (e. (+oo) (\ A ({} (-oo)))) (e. (-oo) (\ A ({} (-oo)))) df-3or mnfxr elisseti snid (-oo) ({} (-oo)) A elndif ax-mp (e. (-oo) (\ A ({} (-oo)))) (\/ (C_ (\ A ({} (-oo))) (RR)) (e. (+oo) (\ A ({} (-oo))))) biorf ax-mp 3bitr4 sylib (\ A ({} (-oo))) x y z xrsupsslem mpdan syl (e. (-oo) A) adantr mnfxr elisseti A snss ({} (-oo)) A ssundif ({} (-oo)) (\ A ({} (-oo))) uncom A eqeq1i 3bitr (u. (\ A ({} (-oo))) ({} (-oo))) A y (-. (br (cv x) (<) (cv y))) raleq1 (u. (\ A ({} (-oo))) ({} (-oo))) A z (br (cv y) (<) (cv z)) rexeq1 (br (cv y) (<) (cv x)) imbi2d y (RR*) ralbidv anbi12d sylbi x (RR*) rexbidv x y (\ A ({} (-oo))) z xrsupexmnf syl5bi (C_ A (RR*)) adantl mpd jaodan mpdan)) thm (xrinfmss ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (RR*)) (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv y) (<) (cv x)))) (A.e. y (RR*) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y)))))))) (A ssxr (C_ A (RR)) (e. (+oo) A) (e. (-oo) A) df-3or (C_ A (RR)) (e. (+oo) A) (e. (-oo) A) or23 bitr sylib A x y z xrinfmsslem A ({} (+oo)) difss (\ A ({} (+oo))) A (RR*) sstr mpan (\ A ({} (+oo))) ssxr (C_ (\ A ({} (+oo))) (RR)) (e. (+oo) (\ A ({} (+oo)))) (e. (-oo) (\ A ({} (+oo)))) 3orass pnfxr elisseti snid (+oo) ({} (+oo)) A elndif ax-mp (e. (+oo) (\ A ({} (+oo)))) (e. (-oo) (\ A ({} (+oo)))) biorf ax-mp (C_ (\ A ({} (+oo))) (RR)) orbi2i bitr4 sylib (\ A ({} (+oo))) x y z xrinfmsslem mpdan syl (e. (+oo) A) adantr pnfxr elisseti A snss ({} (+oo)) A ssundif ({} (+oo)) (\ A ({} (+oo))) uncom A eqeq1i 3bitr (u. (\ A ({} (+oo))) ({} (+oo))) A y (-. (br (cv y) (<) (cv x))) raleq1 (u. (\ A ({} (+oo))) ({} (+oo))) A z (br (cv z) (<) (cv y)) rexeq1 (br (cv x) (<) (cv y)) imbi2d y (RR*) ralbidv anbi12d sylbi x (RR*) rexbidv x y (\ A ({} (+oo))) z xrinfmexpnf syl5bi (C_ A (RR*)) adantl mpd jaodan mpdan)) thm (xrub ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (C_ A (RR*)) (e. B (RR*))) (<-> (A.e. x (RR) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))) (A.e. x (RR*) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))))) ((e. (cv x) (RR)) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y)))) pm2.27 (/\ (/\ (C_ A (RR*)) (e. B (RR*))) (A.e. z (RR) (-> (br (cv z) (<) B) (E.e. y A (br (cv z) (<) (cv y)))))) a1i (cv x) (+oo) (<) B breq1 negbid B pnfnltt syl5bir (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))) pm2.21 syl6com (C_ A (RR*)) (A.e. z (RR) (-> (br (cv z) (<) B) (E.e. y A (br (cv z) (<) (cv y))))) ad2antlr (-> (e. (cv x) (RR)) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))) a1dd (cv x) (-oo) (<) B breq1 (cv x) (-oo) (<) (cv y) breq1 y A rexbidv imbi12d ax1re B (1) resubclt mpan2 (cv z) (opr B (-) (1)) (<) B breq1 (cv z) (opr B (-) (1)) (<) (cv y) breq1 y A rexbidv imbi12d (RR) rcla4v syl (C_ A (RR*)) adantl (-> (br (opr B (-) (1)) (<) B) (E.e. y A (br (opr B (-) (1)) (<) (cv y)))) id B ltm1t syl5com (C_ A (RR*)) adantl ax1re B (1) resubclt mpan2 (C_ A (RR*)) (e. (cv y) A) ad2antlr (opr B (-) (1)) mnfltt syl mnfxr (-oo) (opr B (-) (1)) (cv y) xrlttrt mp3an1 ax1re B (1) resubclt mpan2 (C_ A (RR*)) (e. (cv y) A) ad2antlr (opr B (-) (1)) rexrt syl A (RR*) (cv y) ssel2 (e. B (RR)) adantlr sylanc mpand r19.22dva syld syld (br (-oo) (<) B) a1dd (-> (br (1) (<) B) (E.e. y A (br (1) (<) (cv y)))) id ax1re (1) ltpnft ax-mp B (+oo) (1) (<) breq2 mpbiri syl5com A (RR*) (cv y) ssel2 ax1re (1) mnfltt ax-mp mnfxr ax1re (1) rexrt ax-mp (-oo) (1) (cv y) xrlttrt mp3an12 mpani syl r19.22dva sylan9r ax1re (cv z) (1) (<) B breq1 (cv z) (1) (<) (cv y) breq1 y A rexbidv imbi12d (RR) rcla4v ax-mp syl5 (br (-oo) (<) B) a1dd mnfxr (-oo) xrltnrt ax-mp B (-oo) (-oo) (<) breq2 mtbiri (C_ A (RR*)) adantl (E.e. y A (br (-oo) (<) (cv y))) pm2.21d (A.e. z (RR) (-> (br (cv z) (<) B) (E.e. y A (br (cv z) (<) (cv y))))) a1d 3jaodan B elxr sylan2b imp syl5bir com12 (-> (e. (cv x) (RR)) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))) a1dd 3jaod (cv x) elxr syl5ib com23 r19.20dv2 ex (cv x) (cv z) (<) B breq1 (cv x) (cv z) (<) (cv y) breq1 y A rexbidv imbi12d (RR) cbvralv syl5ib pm2.43d (cv x) rexrt (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y)))) imim1i r19.20i2 (/\ (C_ A (RR*)) (e. B (RR*))) a1i impbid)) thm (supxrOLD ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (C_ A (RR*)) (e. B (RR*))) (/\ (A.e. x A (-. (br B (<) (cv x)))) (A.e. x (RR*) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))))) (= (sup A (RR*) (<)) B)) (A z x y xrsupss (e. B (RR*)) anim2i ancoms xrltso z x A y supeu (e. B (RR*)) anim2i (cv z) B (<) (cv x) breq1 negbid x A ralbidv (cv z) B (cv x) (<) breq2 (E.e. y A (br (cv x) (<) (cv y))) imbi1d x (RR*) ralbidv anbi12d (RR*) reuuni2 3syl biimpa A (RR*) (<) z x y df-sup syl5eq)) thm (supxr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (C_ A (RR*)) (e. B (RR*))) (/\ (A.e. x A (-. (br B (<) (cv x)))) (A.e. x (RR) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))))) (= (sup A (RR*) (<)) B)) (A B x y xrub biimpd (A.e. x A (-. (br B (<) (cv x)))) anim2d imp A z x y xrsupss (e. B (RR*)) anim2i ancoms xrltso z x A y supeu (e. B (RR*)) anim2i (cv z) B (<) (cv x) breq1 negbid x A ralbidv (cv z) B (cv x) (<) breq2 (E.e. y A (br (cv x) (<) (cv y))) imbi1d x (RR*) ralbidv anbi12d (RR*) reuuni2 3syl biimpa A (RR*) (<) z x y df-sup syl5eq syldan)) thm (supxr2OLD ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (C_ A (RR*)) (e. B (RR*))) (/\ (A.e. x A (br (cv x) (<_) B)) (A.e. x (RR*) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))))) (= (sup A (RR*) (<)) B)) ((cv x) B xrlenltt A (RR*) (cv x) ssel2 sylan an1rs ralbidva (A.e. x (RR*) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))) anbi1d biimpa A B x y supxrOLD syldan)) thm (supxr2 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (C_ A (RR*)) (e. B (RR*))) (/\ (A.e. x A (br (cv x) (<_) B)) (A.e. x (RR) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))))) (= (sup A (RR*) (<)) B)) ((cv x) B xrlenltt A (RR*) (cv x) ssel2 sylan an1rs ralbidva (A.e. x (RR) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))) anbi1d biimpa A B x y supxr syldan)) thm (supxrre ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (= (sup A (RR*) (<)) (sup A (RR) (<)))) ((cv y) rexrt (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))) imim1i (/\ (/\ (C_ A (RR)) (-. (= A ({/})))) (e. (cv x) (RR*))) a1i (/\ (/\ (C_ A (RR)) (-. (= A ({/})))) (e. (cv x) (RR*))) (-> (e. (cv y) (RR)) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))) pm3.27 (cv x) pnfnltt (= (cv y) (+oo)) adantr (cv y) (+oo) (<) (cv x) breq1 negbid (e. (cv x) (RR*)) adantl mpbird (E.e. z A (br (cv y) (<) (cv z))) pm2.21d ex (/\ (C_ A (RR)) (-. (= A ({/})))) (-> (e. (cv y) (RR)) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))) ad2antlr A (RR) (cv z) ssel (cv z) mnfltt syl6 ancld z 19.22dv A z n0 z A (br (-oo) (<) (cv z)) df-rex 3imtr4g imp (br (-oo) (<) (cv x)) a1d (e. (cv x) (RR*)) (= (cv y) (-oo)) ad2antrr (cv y) (-oo) (<) (cv x) breq1 (cv y) (-oo) (<) (cv z) breq1 z A rexbidv imbi12d (/\ (/\ (C_ A (RR)) (-. (= A ({/})))) (e. (cv x) (RR*))) adantl mpbird ex (-> (e. (cv y) (RR)) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))) adantr 3jaod (cv y) elxr syl5ib ex impbid ralbidv2 (A.e. y A (-. (br (cv x) (<) (cv y)))) anbi2d ex rabbidv unieqd (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x)))) 3adant3 ressxr (RR) (RR*) x (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))) reuuniss mp3an1 A x y z sup3 A x y z sup3 ltso x y A z supeu syl (e. (cv x) (RR)) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))) ax-1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. y (RR) (A.e. z A (br (cv z) (<_) (cv y))))) a1i (C_ A (RR)) (-. (= A ({/}))) (E.e. y (RR) (A.e. z A (br (cv z) (<_) (cv y)))) 3simp3 (= (cv x) (+oo)) adantr (cv x) (+oo) (cv y) (<) breq2 (-. (E.e. z A (br (cv y) (<) (cv z)))) anbi1d (cv y) ltpnft (C_ A (RR)) adantl (-. (E.e. z A (br (cv y) (<) (cv z)))) biantrurd (cv z) (cv y) lenltt A (RR) (cv z) ssel2 sylan an1rs ralbidva z A (br (cv y) (<) (cv z)) ralnex syl6rbb bitr3d sylan9bbr an1rs rexbidva (-. (= A ({/}))) adantlr (E.e. y (RR) (A.e. z A (br (cv z) (<_) (cv y)))) 3adantl3 mpbird y (RR) (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))) rexanali sylib (e. (cv x) (RR)) pm2.21d (A.e. y A (-. (br (cv x) (<) (cv y)))) adantld ex A (RR) (cv y) ssel (cv y) mnfltt syl6 (cv x) (-oo) (<) (cv y) breq1 biimprd sylan9 ancld y 19.22dv A y n0 syl5ib imp an1rs y A (br (cv x) (<) (cv y)) df-rex y A (br (cv x) (<) (cv y)) dfrex2 bitr3 sylib (e. (cv x) (RR)) pm2.21d (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))) adantrd ex (E.e. y (RR) (A.e. z A (br (cv z) (<_) (cv y)))) 3adant3 3jaod (cv x) elxr syl5ib imp3a (e. (cv x) (RR*)) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))) pm3.27 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. y (RR) (A.e. z A (br (cv z) (<_) (cv y))))) a1i jcad (cv x) rexrt (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))) anim1i (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. y (RR) (A.e. z A (br (cv z) (<_) (cv y))))) a1i impbid x eubidv x (RR*) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))) df-reu x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))) df-reu 3bitr4g (cv y) (cv z) (<_) (cv x) breq1 A cbvralv x (RR) rexbii (cv x) (cv y) (cv z) (<_) breq2 z A ralbidv (RR) cbvrexv bitr syl3an3b mpbird sylanc eqtr4d A (RR*) (<) x y z df-sup A (RR) (<) x y z df-sup 3eqtr4g)) thm (supxrcl ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (RR*)) (e. (sup A (RR*) (<)) (RR*))) (A x y z xrsupss xrltso x y A z supcl syl)) thm (supxrun ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\/\ (C_ A (RR*)) (C_ B (RR*)) (br (sup A (RR*) (<)) (<_) (sup B (RR*) (<)))) (= (sup (u. A B) (RR*) (<)) (sup B (RR*) (<)))) ((u. A B) (sup B (RR*) (<)) x y supxrOLD A (RR*) B unss biimp (br (sup A (RR*) (<)) (<_) (sup B (RR*) (<))) 3adant3 B supxrcl (C_ A (RR*)) (br (sup A (RR*) (<)) (<_) (sup B (RR*) (<))) 3ad2ant2 jca A x y z xrsupss xrltso x y A z (cv x) supub syl (C_ B (RR*)) (br (sup A (RR*) (<)) (<_) (sup B (RR*) (<))) 3ad2ant1 (sup A (RR*) (<)) (sup B (RR*) (<)) (cv x) xrlelttrt A supxrcl (C_ B (RR*)) (e. (cv x) A) ad2antrr B supxrcl (C_ A (RR*)) (e. (cv x) A) ad2antlr A (RR*) (cv x) ssel2 (C_ B (RR*)) adantlr syl3anc exp3a imp con3d exp41 com34 3imp mpdd B y z w xrsupss xrltso y z B w (cv x) supub syl (C_ A (RR*)) (br (sup A (RR*) (<)) (<_) (sup B (RR*) (<))) 3ad2ant2 jaod (cv x) A B elun syl5ib r19.21aiv B x z y xrsupss xrltso x z B y (cv x) suplub syl (cv y) B A elun2 (br (cv x) (<) (cv y)) anim1i r19.22i2 syl6 exp3a r19.21aiv (C_ A (RR*)) (br (sup A (RR*) (<)) (<_) (sup B (RR*) (<))) 3ad2ant2 jca sylanc)) thm (infmxrcl ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (RR*)) (e. (sup A (RR*) (`' (<))) (RR*))) (A x y z xrinfmss x visset y visset (<) brcnv negbii y A ralbii y visset x visset (<) brcnv y visset z visset (<) brcnv z A rexbii imbi12i y (RR*) ralbii anbi12i x (RR*) rexbii sylibr xrltso (<) (RR*) cnvso mpbi x y A z supcl syl)) thm (supxrmnf () () (-> (C_ A (RR*)) (= (sup (u. A ({} (-oo))) (RR*) (<)) (sup A (RR*) (<)))) (({} (-oo)) A supxrun mnfxr (-oo) (RR*) snssi ax-mp (C_ A (RR*)) a1i (C_ A (RR*)) id A supxrcl (sup A (RR*) (<)) mnflet syl mnfxr xrltso (-oo) supsn ax-mp syl5eqbr syl3anc A ({} (-oo)) uncom (u. A ({} (-oo))) (u. ({} (-oo)) A) (RR*) (<) supeq1 ax-mp syl5eq)) thm (supxrunb1 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (C_ A (RR*)) (<-> (A.e. x (RR) (E.e. y A (br (cv x) (<_) (cv y)))) (= (sup A (RR*) (<)) (+oo)))) (A (RR*) (cv z) ssel (cv z) pnfnltt syl6 r19.21aiv (A.e. x (RR) (E.e. y A (br (cv x) (<_) (cv y)))) adantr (cv x) (opr (cv z) (+) (1)) (<_) (cv y) breq1 y A rexbidv (RR) rcla4va (C_ A (RR*)) adantrr ancoms (cv z) peano2re sylan2 (cv z) ltp1t (e. (cv y) (RR*)) adantr (cv z) (opr (cv z) (+) (1)) (cv y) xrltletrt (opr (cv z) (+) (1)) rexrt syl3an2 (cv z) rexrt syl3an1 3expa (cv z) peano2re ancli sylan mpand ancoms A (RR*) (cv y) ssel2 sylan an1rs r19.22dva (A.e. x (RR) (E.e. y A (br (cv x) (<_) (cv y)))) adantll mpd exp31 (br (cv z) (<) (+oo)) a1dd com4r com13 imp r19.21aiv jca pnfxr A (+oo) z y supxr mpanl2 syldan ex xrltso z w A y (cv x) suplub A z w y xrsupss (e. (cv x) (RR)) (= (sup A (RR*) (<)) (+oo)) ad2antrr (cv x) rexrt (C_ A (RR*)) (= (sup A (RR*) (<)) (+oo)) ad2antlr (sup A (RR*) (<)) (+oo) (cv x) (<) breq2 (cv x) ltpnft syl5bir impcom (C_ A (RR*)) adantll jca sylc ex (cv x) (cv y) xrltlet (cv x) rexrt (C_ A (RR*)) (e. (cv y) A) ad2antlr A (RR*) (cv y) ssel2 (e. (cv x) (RR)) adantlr sylanc r19.22dva syld ex com23 r19.21adv impbid)) thm (supxrunb2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (C_ A (RR*)) (<-> (A.e. x (RR) (E.e. y A (br (cv x) (<) (cv y)))) (= (sup A (RR*) (<)) (+oo)))) (A (RR*) (cv z) ssel (cv z) pnfnltt syl6 r19.21aiv (A.e. x (RR) (E.e. y A (br (cv x) (<) (cv y)))) adantr (cv z) elxr (cv x) (cv z) (<) (cv y) breq1 y A rexbidv (RR) rcla4va (C_ A (RR*)) adantrr ancoms exp31 (br (cv z) (<) (+oo)) a1dd com4r pnfxr (+oo) xrltnrt ax-mp (cv z) (+oo) (<) (+oo) breq1 mtbiri (E.e. y A (br (cv z) (<) (cv y))) pm2.21d (C_ A (RR*)) a1d (A.e. x (RR) (E.e. y A (br (cv x) (<) (cv y)))) a1d (cv z) (-oo) (<) (cv y) breq1 y A rexbidv A (RR*) (cv y) ssel2 0re (0) mnfltt ax-mp mnfxr 0re (0) rexrt ax-mp (-oo) (0) (cv y) xrlttrt mp3an12 mpani syl r19.22dva impcom 0re (cv x) (0) (<) (cv y) breq1 y A rexbidv (RR) rcla4va mpan sylan syl5bir (br (cv z) (<) (+oo)) a1dd exp3a 3jaoi sylbi com13 imp r19.21aiv jca pnfxr A (+oo) z y supxrOLD mpanl2 syldan ex xrltso z w A y (cv x) suplub A z w y xrsupss (e. (cv x) (RR)) (= (sup A (RR*) (<)) (+oo)) ad2antrr (cv x) rexrt (C_ A (RR*)) (= (sup A (RR*) (<)) (+oo)) ad2antlr (sup A (RR*) (<)) (+oo) (cv x) (<) breq2 (cv x) ltpnft syl5bir impcom (C_ A (RR*)) adantll jca sylc exp31 com23 r19.21adv impbid)) thm (supxrbnd ((x y) (A x) (A y)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (br (sup A (RR*) (<)) (<) (+oo))) (e. (sup A (RR*) (<)) (RR))) (A x y supxrre A x y suprcl eqeltrd (C_ A (RR)) (-. (= A ({/}))) (br (sup A (RR*) (<)) (<) (+oo)) 3simp1 (C_ A (RR)) (-. (= A ({/}))) (br (sup A (RR*) (<)) (<) (+oo)) 3simp2 A supxrcl pnfxr (sup A (RR*) (<)) (+oo) xrltnet mpan2 syl (cv y) (cv x) xrlenltt con2bid A (RR*) (cv y) ssel2 (e. (cv x) (RR)) adantlr (cv x) rexrt (C_ A (RR*)) (e. (cv y) A) ad2antlr sylanc rexbidva y A (br (cv y) (<_) (cv x)) rexnal syl6bb ralbidva negbid x (RR) (A.e. y A (br (cv y) (<_) (cv x))) dfrex2 syl6rbbr A x y supxrunb2 negbid bitrd sylibrd imp ressxr A (RR) (RR*) sstr mpan2 sylan (-. (= A ({/}))) 3adant2 syl3anc)) thm (supxrgtmnf () () (-> (/\ (C_ A (RR)) (-. (= A ({/})))) (br (-oo) (<) (sup A (RR*) (<)))) (A supxrbnd 3expa ex con3d ressxr A (RR) (RR*) sstr mpan2 A supxrcl syl (-. (= A ({/}))) adantr (sup A (RR*) (<)) nltpnftt syl sylibrd orrd (sup A (RR*) (<)) mnfltxrt syl)) thm (supxrre1 () () (-> (/\ (C_ A (RR)) (-. (= A ({/})))) (<-> (e. (sup A (RR*) (<)) (RR)) (br (sup A (RR*) (<)) (<) (+oo)))) (ressxr A (RR) (RR*) sstr mpan2 A supxrcl (sup A (RR*) (<)) xrrebndt 3syl (-. (= A ({/}))) adantr A supxrgtmnf (br (sup A (RR*) (<)) (<) (+oo)) biantrurd bitr4d)) thm (supxrre2 () () (-> (/\ (C_ A (RR)) (-. (= A ({/})))) (<-> (e. (sup A (RR*) (<)) (RR)) (-. (= (sup A (RR*) (<)) (+oo))))) (A supxrre1 ressxr A (RR) (RR*) sstr mpan2 A supxrcl (sup A (RR*) (<)) nltpnftt 3syl con2bid (-. (= A ({/}))) adantr bitrd)) thm (xrsup0 ((y z)) () (= (sup ({/}) (RR*) (<)) (-oo)) ((RR*) 0ss mnfxr pm3.2i y (-. (br (-oo) (<) (cv y))) ral0 (cv y) nltmnft (E.e. z ({/}) (br (cv y) (<) (cv z))) pm2.21d rgen pm3.2i ({/}) (-oo) y z supxrOLD mp2an)) thm (supxrub ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (C_ A (RR*)) (e. B A)) (br B (<_) (sup A (RR*) (<)))) (A x y z xrsupss xrltso x y A z B supub syl imp B (sup A (RR*) (<)) xrlenltt A (RR*) B ssel2 A supxrcl (e. B A) adantr sylanc mpbird)) thm (supxrleub ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\/\ (C_ A (RR*)) (e. B (RR*)) (A.e. x A (br (cv x) (<_) B))) (br (sup A (RR*) (<)) (<_) B)) ((cv x) B xrlenltt A (RR*) (cv x) ssel2 sylan an1rs ralbidva ex pm5.32d A y z x xrsupss xrltso y z A x B supnub syl imp (sup A (RR*) (<)) B xrlenltt A supxrcl sylan (A.e. x A (-. (br B (<) (cv x)))) adantrr mpbird ex sylbid 3impib)) thm (elnn0 () () (<-> (e. A (NN0)) (\/ (e. A (NN)) (= A (0)))) (df-n0 A eleq2i A (NN) ({} (0)) elun 0cn elisseti A elsnc2 (e. A (NN)) orbi2i 3bitr)) thm (nnssnn0 () () (C_ (NN) (NN0)) ((NN) ({} (0)) ssun1 df-n0 sseqtr4)) thm (nn0ssre () () (C_ (NN0) (RR)) (df-n0 nnssre 0re (0) (RR) snssi ax-mp unssi eqsstr)) thm (nn0sscn () () (C_ (NN0) (CC)) (nn0ssre axresscn sstri)) thm (nn0ex () () (e. (NN0) (V)) (axcnex nn0sscn ssexi)) thm (nnnn0t () () (-> (e. A (NN)) (e. A (NN0))) (nnssnn0 A sseli)) thm (nnnn0 () ((nnnn0.1 (e. N (NN)))) (e. N (NN0)) (nnnn0.1 N nnnn0t ax-mp)) thm (nn0ret () () (-> (e. A (NN0)) (e. A (RR))) (nn0ssre A sseli)) thm (nn0cnt () () (-> (e. A (NN0)) (e. A (CC))) (nn0sscn A sseli)) thm (nn0re () ((nn0re.1 (e. A (NN0)))) (e. A (RR)) (nn0ssre nn0re.1 sselii)) thm (nn0cn () ((nn0re.1 (e. A (NN0)))) (e. A (CC)) (nn0re.1 nn0re recn)) thm (0nn0 () () (e. (0) (NN0)) ((0) eqid (= (0) (0)) (e. (0) (NN)) olc ax-mp (0) elnn0 mpbir)) thm (1nn0 () () (e. (1) (NN0)) (1nn nnnn0)) thm (2nn0 () () (e. (2) (NN0)) (2nn nnnn0)) thm (lt0nnn0 () () (-> (/\ (e. A (RR)) (br A (<) (0))) (-. (e. A (NN0)))) (lt01 0re ax1re A (0) (1) axlttrn mp3an23 mpan2i A lt1nnn ex syld con2d A (0) A (<) breq2 negbid A ltnrt syl5bi com12 jaod A elnn0 syl5ib con2d imp)) thm (nn0ge0t () () (-> (e. N (NN0)) (br (0) (<_) N)) (N nn0ret N lt0nnn0 ex con2d mpcom 0re (0) N lenltt N nn0ret sylan2 mpan mpbird)) thm (nn0ge0 () ((nn0ge0.1 (e. N (NN0)))) (br (0) (<_) N) (nn0ge0.1 N nn0ge0t ax-mp)) thm (nn0le0eq0t () () (-> (e. N (NN0)) (<-> (br N (<_) (0)) (= N (0)))) (N nn0ge0t (br N (<_) (0)) biantrud N nn0ret 0re N (0) letri3t mpan2 syl bitr4d)) thm (nn0addclt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (e. (opr M (+) N) (NN0))) (M N nnaddclt (opr M (+) N) nnnn0t syl M (0) (+) N opreq1 N addid2t sylan9eq N nncnt sylan2 N nnnn0t (= M (0)) adantl eqeltrd N (0) M (+) opreq2 M ax0id sylan9eqr M nncnt sylan M nnnn0t (= N (0)) adantr eqeltrd M (0) N (0) (+) opreq12 0cn addid1 syl6eq 0nn0 syl6eqel ccase M elnn0 N elnn0 syl2anb)) thm (nn0addcl () ((nn0addcl.1 (e. M (NN0))) (nn0addcl.2 (e. N (NN0)))) (e. (opr M (+) N) (NN0)) (nn0addcl.1 nn0addcl.2 M N nn0addclt mp2an)) thm (nn0mulcl () ((nn0addcl.1 (e. M (NN0))) (nn0addcl.2 (e. N (NN0)))) (e. (opr M (x.) N) (NN0)) (nn0addcl.1 M elnn0 mpbi nn0addcl.2 N elnn0 mpbi M N nnmulclt (opr M (x.) N) nnnn0t syl M (0) (x.) N opreq1 nn0addcl.2 nn0cn mul02 syl6eq 0nn0 syl6eqel N (0) M (x.) opreq2 nn0addcl.1 nn0cn mul01 syl6eq 0nn0 syl6eqel ccase2 mp2an)) thm (nn0mulclt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (e. (opr M (x.) N) (NN0))) (M (if (e. M (NN0)) M (0)) (x.) N opreq1 (NN0) eleq1d N (if (e. N (NN0)) N (0)) (if (e. M (NN0)) M (0)) (x.) opreq2 (NN0) eleq1d 0nn0 M elimel 0nn0 N elimel nn0mulcl dedth2h)) thm (peano2nn0 () () (-> (e. N (NN0)) (e. (opr N (+) (1)) (NN0))) (1nn0 N (1) nn0addclt mpan2)) thm (nnnn0addclt () () (-> (/\ (e. M (NN)) (e. N (NN0))) (e. (opr M (+) N) (NN))) (M N nnaddclt ex N (0) M (+) opreq2 (NN) eleq1d M nncnt M ax0id syl (NN) eleq1d ibir syl5bir com12 jaod N elnn0 syl5ib imp)) thm (nn0nnaddclt () () (-> (/\ (e. M (NN0)) (e. N (NN))) (e. (opr M (+) N) (NN))) (N M axaddcom N nncnt M nn0cnt syl2an N M nnnn0addclt eqeltrrd ancoms)) thm (nn0ltp1let () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<) N) (br (opr M (+) (1)) (<_) N))) (M N nnltp1let M (0) (<) N breq1 N nngt0t N nnge1t 1cn addid2 syl5eqbr 2th pm5.74ri sylan9bb M (0) (+) (1) opreq1 (<_) N breq1d (e. N (NN)) adantr bitr4d N (0) M (<) breq2 (br M (<) (0)) (br (opr M (+) (1)) (<_) (0)) pm5.21 M nngt0t M nnret 0re M (0) ltnsymt mpan2 syl mt2d 0re (0) M (opr M (+) (1)) axlttrn mp3an1 M nnret M nnret M peano2re syl jca M nngt0t M nnret M ltp1t syl jca sylc M nnret M peano2re 0re (0) (opr M (+) (1)) ltnlet mpan 3syl mpbid sylanc sylan9bbr N (0) (opr M (+) (1)) (<_) breq2 (e. M (NN)) adantl bitr4d N (0) M (<) breq2 M (0) (<) (0) breq1 0re ltnr 0re ltp1 0re 0re ax1re readdcl ltnle mpbi 2false syl6bb M (0) (+) (1) opreq1 (<_) (0) breq1d bitr4d sylan9bbr N (0) (opr M (+) (1)) (<_) breq2 (= M (0)) adantl bitr4d ccase M elnn0 N elnn0 syl2anb)) thm (nn0leltp1t () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (br M (<) (opr N (+) (1))))) (ax1re M N (1) leadd1t mp3an3 M nn0ret N nn0ret syl2an M (opr N (+) (1)) nn0ltp1let N peano2nn0 sylan2 bitr4d)) thm (nn0ltlem1t () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1))))) (N nn0cnt 1cn N (1) npcant mpan2 syl (opr M (+) (1)) (<_) breq2d (e. M (NN0)) adantl ax1re M (opr N (-) (1)) (1) leadd1t mp3an3 M nn0ret N nn0ret ax1re N (1) resubclt mpan2 syl syl2an M N nn0ltp1let 3bitr4rd)) thm (nn0addge1t () () (-> (/\ (e. A (RR)) (e. N (NN0))) (br A (<_) (opr A (+) N))) (A N addge01t biimp3a 3expb N nn0ret N nn0ge0t jca sylan2)) thm (nn0addge2t () () (-> (/\ (e. A (RR)) (e. N (NN0))) (br A (<_) (opr N (+) A))) (A N addge02t biimp3a 3expb N nn0ret N nn0ge0t jca sylan2)) thm (nn0addge1 () ((nn0addge1.1 (e. A (RR))) (nn0addge1.2 (e. N (NN0)))) (br A (<_) (opr A (+) N)) (nn0addge1.1 nn0addge1.2 A N nn0addge1t mp2an)) thm (nn0addge2 () ((nn0addge1.1 (e. A (RR))) (nn0addge1.2 (e. N (NN0)))) (br A (<_) (opr N (+) A)) (nn0addge1.1 nn0addge1.2 A N nn0addge2t mp2an)) thm (nn0le2x () ((nn0le2x.1 (e. N (NN0)))) (br N (<_) (opr (2) (x.) N)) (nn0le2x.1 nn0re nn0le2x.1 nn0addge1 nn0le2x.1 nn0cn 2times breqtrr)) thm (nn0lele2x () ((nn0lele2x.1 (e. M (NN0))) (nn0lele2x.2 (e. N (NN0)))) (-> (br N (<_) M) (br N (<_) (opr (2) (x.) M))) (nn0lele2x.1 nn0le2x nn0lele2x.2 nn0re nn0lele2x.1 nn0re 2re nn0lele2x.1 nn0re remulcl letr mpan2)) thm (elz ((N x)) () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/\/ (= N (0)) (e. N (NN)) (e. (-u N) (NN))))) (x df-z N eleq2i (cv x) N (0) eqeq1 (cv x) N (NN) eleq1 (cv x) N negeq (NN) eleq1d 3orbi123d (RR) elrab bitr)) thm (nnnegz () () (-> (e. N (NN)) (e. (-u N) (ZZ))) (N nnret N renegclt syl N nncnt N negnegt (NN) eleq1d biimprd mpcom (e. (-u (-u N)) (NN)) (\/ (= (-u N) (0)) (e. (-u N) (NN))) olc syl (= (-u N) (0)) (e. (-u N) (NN)) (e. (-u (-u N)) (NN)) df-3or sylibr jca (-u N) elz sylibr)) thm (zret () () (-> (e. N (ZZ)) (e. N (RR))) (N elz pm3.26bd)) thm (zcnt () () (-> (e. N (ZZ)) (e. N (CC))) (N zret recnd)) thm (zre () ((zre.1 (e. A (ZZ)))) (e. A (RR)) (zre.1 A zret ax-mp)) thm (zssre () () (C_ (ZZ) (RR)) ((cv x) zret ssriv)) thm (zsscn () () (C_ (ZZ) (CC)) ((cv x) zcnt ssriv)) thm (zex () () (e. (ZZ) (V)) (axcnex zsscn ssexi)) thm (elnnz () () (<-> (e. N (NN)) (/\ (e. N (ZZ)) (br (0) (<) N))) (N nnret (e. N (NN)) (\/ (e. (-u N) (NN)) (= N (0))) orc jca N nngt0t jca (/\ (e. N (RR)) (br (0) (<) N)) (e. N (NN)) idd N lt0neg2t N renegclt 0re (-u N) (0) ltnsymt mpan2 syl sylbid imp (-u N) nngt0t nsyl 0re (0) N ltnet mpan imp N (0) eqcom negbii sylibr jca (e. (-u N) (NN)) (= N (0)) ioran sylibr (e. N (NN)) pm2.21d jaod ex com23 imp31 impbi N elz (= N (0)) (e. N (NN)) (e. (-u N) (NN)) 3orrot (e. N (NN)) (e. (-u N) (NN)) (= N (0)) 3orass bitr (e. N (RR)) anbi2i bitr (br (0) (<) N) anbi1i bitr4)) thm (0z () () (e. (0) (ZZ)) ((0) elz 0re (0) eqid (= (0) (0)) (\/ (e. (0) (NN)) (e. (-u (0)) (NN))) orc ax-mp (= (0) (0)) (e. (0) (NN)) (e. (-u (0)) (NN)) 3orass mpbir mpbir2an)) thm (elnn0z () () (<-> (e. N (NN0)) (/\ (e. N (ZZ)) (br (0) (<_) N))) (N elnn0 N elnnz N (0) eqcom orbi12i (e. N (ZZ)) id 0z (0) N (ZZ) eleq1 mpbii jaoi (e. N (ZZ)) (= (0) N) orc impbi (\/ (br (0) (<) N) (= (0) N)) anbi1i (e. N (ZZ)) (br (0) (<) N) (= (0) N) ordir N zret 0re (0) N leloet mpan syl pm5.32i 3bitr4 3bitr)) thm (elznn0nn () () (<-> (e. N (ZZ)) (\/ (e. N (NN0)) (/\ (e. N (RR)) (e. (-u N) (NN))))) (N elz (= N (0)) (e. N (NN)) (e. (-u N) (NN)) df-3or (e. N (RR)) anbi2i (e. N (RR)) (\/ (= N (0)) (e. N (NN))) (e. (-u N) (NN)) andi N nn0ret pm4.71ri N elnn0 (e. N (NN)) (= N (0)) orcom bitr (e. N (RR)) anbi2i bitr (/\ (e. N (RR)) (e. (-u N) (NN))) orbi1i bitr4 3bitr)) thm (elznn0 () () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/ (e. N (NN0)) (e. (-u N) (NN0))))) (N elz N elnn0 (e. N (RR)) a1i N recnt 0cn N (0) negcon1t mpan2 syl neg0 N eqeq1i (0) N eqcom bitr syl6bb (e. (-u N) (NN)) orbi2d (-u N) elnn0 syl5bb orbi12d (= N (0)) (e. N (NN)) (e. (-u N) (NN)) 3orass (= N (0)) (\/ (e. N (NN)) (e. (-u N) (NN))) orcom (e. N (NN)) (e. (-u N) (NN)) (= N (0)) orordir 3bitrr syl6rbb pm5.32i bitr)) thm (elznn () () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/ (e. N (NN)) (e. (-u N) (NN0))))) (N elz N recnt N negeq0t syl (e. (-u N) (NN)) orbi2d (-u N) elnn0 syl6rbbr (e. N (NN)) orbi2d (= N (0)) (e. N (NN)) (e. (-u N) (NN)) 3orrot (e. N (NN)) (e. (-u N) (NN)) (= N (0)) 3orass bitr syl6rbbr pm5.32i bitr)) thm (nnssz () () (C_ (NN) (ZZ)) ((cv x) elnnz pm3.26bd ssriv)) thm (nn0ssz () () (C_ (NN0) (ZZ)) (df-n0 nnssz 0z 0z elisseti (ZZ) snss mpbi unssi eqsstr)) thm (nnzt () () (-> (e. N (NN)) (e. N (ZZ))) (nnssz N sseli)) thm (nn0zt () () (-> (e. N (NN0)) (e. N (ZZ))) (nn0ssz N sseli)) thm (elnnz1 () () (<-> (e. N (NN)) (/\ (e. N (ZZ)) (br (1) (<_) N))) (N nnzt N nnge1t jca lt01 ax1re (1) lt0neg2t ax-mp mpbi ax1re (1) N lenegt mpan N renegclt ax1re renegcl 0re (-u N) (-u (1)) (0) lelttrt mp3an23 0re (-u N) (0) ltlet mpan2 syld syl exp3a sylbid imp mpi N renegclt 0re (-u N) (0) lenltt mpan2 syl (br (1) (<_) N) adantr mpbid N zret sylan (-u N) nngt0t nsyl lt01 N (0) (1) (<_) breq2 ax1re 0re lenlt syl6bb con2bid mpbii con2i (e. N (ZZ)) adantl jca (e. (-u N) (NN)) (= N (0)) ioran sylibr N elz (e. N (RR)) (\/\/ (= N (0)) (e. N (NN)) (e. (-u N) (NN))) pm3.27 (e. (-u N) (NN)) (= N (0)) (e. N (NN)) 3orrot (e. (-u N) (NN)) (= N (0)) (e. N (NN)) df-3or bitr3 sylib sylbi ord (br (1) (<_) N) adantr mpd impbi)) thm (znnnlt1t () () (-> (e. N (ZZ)) (<-> (-. (e. N (NN))) (br N (<) (1)))) (N elnnz1 baib negbid N zret ax1re N (1) ltnlet mpan2 syl bitr4d)) thm (nnzrab () () (= (NN) ({e.|} x (ZZ) (br (1) (<_) (cv x)))) ((cv x) elnnz1 abbi2i x (ZZ) (br (1) (<_) (cv x)) df-rab eqtr4)) thm (nn0zrab () () (= (NN0) ({e.|} x (ZZ) (br (0) (<_) (cv x)))) ((cv x) elnn0z abbi2i x (ZZ) (br (0) (<_) (cv x)) df-rab eqtr4)) thm (1z () () (e. (1) (ZZ)) (1nn (1) nnzt ax-mp)) thm (2z () () (e. (2) (ZZ)) (2nn (2) nnzt ax-mp)) thm (nn0subt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (e. (opr N (-) M) (NN0)))) (M N nnsubt ex N (0) M (<) breq2 N (0) (-) M opreq1 (NN) eleq1d bibi12d M nnret M lt0neg1t syl M nnnegz (-u M) elnnz baib syl M df-neg (NN) eleq1i syl5rbbr bitrd syl5bir com12 jaod M (0) (<) N breq1 M (0) N (-) opreq2 (NN) eleq1d bibi12d N nnzt N zcnt N subid1t (NN) eleq1d syl N elnnz baib bitr2d syl syl5bir M (0) (<) N breq1 M (0) N (-) opreq2 (NN) eleq1d bibi12d 0re ltnr 0nnn 0cn subid (NN) eleq1i mtbir 2false N (0) (0) (<) breq2 N (0) (-) (0) opreq1 (NN) eleq1d bibi12d mpbiri syl5bir jaod jaoi imp N M subeq0t M N eqcom syl6rbbr ancoms M nncnt 0cn M (0) (CC) eleq1 mpbiri jaoi N nncnt 0cn N (0) (CC) eleq1 mpbiri jaoi syl2an orbi12d M N leloet M nnret 0re M (0) (RR) eleq1 mpbiri jaoi N nnret 0re N (0) (RR) eleq1 mpbiri jaoi syl2an (opr N (-) M) elnn0 (/\ (\/ (e. M (NN)) (= M (0))) (\/ (e. N (NN)) (= N (0)))) a1i 3bitr4d M elnn0 N elnn0 syl2anb)) thm (nn0sub2t () () (-> (/\/\ (e. M (NN0)) (e. N (NN0)) (br M (<_) N)) (e. (opr N (-) M) (NN0))) (M N nn0subt biimp3a)) thm (znegclt () () (-> (e. N (ZZ)) (e. (-u N) (ZZ))) (N elz N (0) negeq neg0 syl6eq 0z syl6eqel N nnnegz (-u N) nnzt 3jaoi (e. N (RR)) adantl sylbi)) thm (nn0negz () () (-> (e. N (NN0)) (e. (-u N) (ZZ))) (N nn0zt N znegclt syl)) thm (zaddclt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (e. (opr M (+) N) (ZZ))) (M N nn0addclt (e. (opr M (+) N) (NN0)) (e. (-u (opr M (+) N)) (NN0)) orc syl (/\ (e. M (RR)) (e. N (RR))) a1i M N axaddrcl jctild (-u M) N letrit M renegclt sylan (/\ (e. (-u M) (NN0)) (e. N (NN0))) adantr (-u M) N nn0subt N M subnegt ancoms M N axaddcom eqtr4d M recnt N recnt syl2an (NN0) eleq1d sylan9bbr N (-u M) nn0subt ancoms M N negdit (-u M) N negsubt M negclt sylan eqtr2d M recnt N recnt syl2an (NN0) eleq1d sylan9bbr orbi12d mpbid ex M N axaddrcl jctild (-u N) M letrit N renegclt sylan ancoms (/\ (e. M (NN0)) (e. (-u N) (NN0))) adantr (-u N) M nn0subt ancoms M N subnegt M recnt N recnt syl2an (NN0) eleq1d sylan9bbr M (-u N) nn0subt (-u M) (-u N) axaddcom M negclt sylan (-u N) M negsubt ancoms eqtr2d N negclt sylan2 M N negdit eqtr4d M recnt N recnt syl2an (NN0) eleq1d sylan9bbr orbi12d mpbid ex M N axaddrcl jctild M N negdit M recnt N recnt syl2an (NN0) eleq1d (-u M) (-u N) nn0addclt syl5bir (e. (-u (opr M (+) N)) (NN0)) (e. (opr M (+) N) (NN0)) olc syl6 M N axaddrcl jctild ccased (opr M (+) N) elznn0 syl6ibr imp an4s M elznn0 N elznn0 syl2anb)) thm (peano2z () () (-> (e. N (ZZ)) (e. (opr N (+) (1)) (ZZ))) (1z N (1) zaddclt mpan2)) thm (peano2zd () ((peano2zd.1 (-> ph (e. N (ZZ))))) (-> ph (e. (opr N (+) (1)) (ZZ))) (peano2zd.1 N peano2z syl)) thm (zsubclt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (e. (opr M (-) N) (ZZ))) (M N negsubt M zcnt N zcnt syl2an M (-u N) zaddclt N znegclt sylan2 eqeltrrd)) thm (peano2zm () () (-> (e. N (ZZ)) (e. (opr N (-) (1)) (ZZ))) (1z N (1) zsubclt mpan2)) thm (zrevaddclt () () (-> (e. N (ZZ)) (<-> (/\ (e. M (CC)) (e. (opr M (+) N) (ZZ))) (e. M (ZZ)))) (M N pncant N zcnt sylan2 ancoms (e. (opr M (+) N) (ZZ)) adantr (opr M (+) N) N zsubclt ancoms (e. M (CC)) adantlr eqeltrrd ex M N zaddclt expcom (e. M (CC)) adantr impbid ex pm5.32d M zcnt pm4.71ri syl6bbr)) thm (elnn0nn () () (<-> (e. N (NN0)) (/\ (e. N (CC)) (e. (opr N (+) (1)) (NN)))) (N peano2nn N (0) (+) (1) opreq1 1cn addid2 syl6eq 1nn syl6eqel jaoi (e. N (RR)) a1i N recnt 1z (1) N zrevaddclt ax-mp biimp ex syl 0re ax1re (0) N (1) leadd1t mp3an13 1cn addid2 (<_) (opr N (+) (1)) breq1i syl6bb 0re (0) N leloet mpan bitr3d biimpd anim12d (e. N (ZZ)) (br (0) (<) N) (= (0) N) andi N elnnz N (0) eqcom 0z (0) N (ZZ) eleq1 mpbii pm4.71ri bitr orbi12i bitr4 syl6ib (opr N (+) (1)) elnnz1 syl5ib impbid N elnn0 syl5bb pm5.32i N nn0ret pm4.71ri 1z (1) N zrevaddclt ax-mp N zret sylbi (opr N (+) (1)) nnzt sylan2 (e. N (CC)) (e. (opr N (+) (1)) (NN)) pm3.27 jca N recnt (e. (opr N (+) (1)) (NN)) anim1i impbi 3bitr4)) thm (nn0p1nnt () () (-> (e. N (NN0)) (e. (opr N (+) (1)) (NN))) (N elnn0nn pm3.27bd)) thm (elnnnn0 () () (<-> (e. N (NN)) (/\ (e. N (CC)) (e. (opr N (-) (1)) (NN0)))) (N nncnt pm4.71ri 1cn N (1) npcant mpan2 (NN) eleq1d 1cn N (1) subclt mpan2 (e. (opr (opr N (-) (1)) (+) (1)) (NN)) biantrurd bitr3d (opr N (-) (1)) elnn0nn syl6bbr pm5.32i bitr)) thm (elnnnn0b () () (<-> (e. N (NN)) (/\ (e. N (NN0)) (br (0) (<) N))) (N nnnn0t N nngt0t jca N elnnz biimpr N nn0zt sylan impbi)) thm (elnnnn0c () () (<-> (e. N (NN)) (/\ (e. N (NN0)) (br (1) (<_) N))) (N nnnn0t N nnge1t jca N elnnz1 biimpr N nn0zt sylan impbi)) thm (znnsubt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (e. (opr N (-) M) (NN)))) (M N posdift M zret N zret syl2an N M zsubclt ancoms (br (0) (<) (opr N (-) M)) biantrurd bitrd (opr N (-) M) elnnz syl6bbr)) thm (znn0subt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (e. (opr N (-) M) (NN0)))) (N M subge0t N zret M zret syl2an N M zsubclt (br (0) (<_) (opr N (-) M)) biantrurd bitr3d ancoms (opr N (-) M) elnn0z syl6bbr)) thm (znn0sub2t () () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (e. (opr N (-) M) (NN0))) (M N znn0subt biimp3a)) thm (zmulclt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (e. (opr M (x.) N) (ZZ))) (M N nn0mulclt (e. (opr M (x.) N) (NN0)) (e. (-u (opr M (x.) N)) (NN0)) orc syl (/\ (e. M (RR)) (e. N (RR))) a1i M N axmulrcl jctild M N mulneg1t M recnt N recnt syl2an (NN0) eleq1d (-u M) N nn0mulclt syl5bi (e. (-u (opr M (x.) N)) (NN0)) (e. (opr M (x.) N) (NN0)) olc syl6 M N axmulrcl jctild M N mulneg2t M recnt N recnt syl2an (NN0) eleq1d M (-u N) nn0mulclt syl5bi (e. (-u (opr M (x.) N)) (NN0)) (e. (opr M (x.) N) (NN0)) olc syl6 M N axmulrcl jctild M N mul2negt M recnt N recnt syl2an (NN0) eleq1d (-u M) (-u N) nn0mulclt syl5bi (e. (opr M (x.) N) (NN0)) (e. (-u (opr M (x.) N)) (NN0)) orc syl6 M N axmulrcl jctild ccased (opr M (x.) N) elznn0 syl6ibr imp an4s M elznn0 N elznn0 syl2anb)) thm (zltp1let () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (br (opr M (+) (1)) (<_) N))) (M N nn0ltp1let (/\ (e. M (RR)) (e. N (RR))) a1i M recnt 0cn M (0) negcon1t mpan2 neg0 M eqeq1i (0) M eqcom bitr syl6bb syl (e. (-u M) (NN)) orbi2d (-u M) elnn0 syl5bb N elnn0 (e. M (RR)) a1i anbi12d (e. N (RR)) adantr M lt0neg1t (-u M) nngt0t syl5bir imp N nngt0t anim12i 0re M (0) N axlttrn mp3an2 N nnret sylan2 (e. (-u M) (NN)) adantlr mpd M peano2re (e. (-u M) (NN)) (e. N (NN)) ad2antrr ax1re (/\ (/\ (e. M (RR)) (e. (-u M) (NN))) (e. N (NN))) a1i N nnret (/\ (e. M (RR)) (e. (-u M) (NN))) adantl M lt0neg1t (-u M) nngt0t syl5bir imp 0re M (0) ltlet mpan2 (e. (-u M) (NN)) adantr mpd 0re ax1re M (0) (1) leadd1t mp3an23 (e. (-u M) (NN)) adantr mpbid 1cn addid2 syl6breq (e. N (NN)) adantr N nnge1t (/\ (e. M (RR)) (e. (-u M) (NN))) adantl letrd jca exp31 imp3a (br M (<) N) (br (opr M (+) (1)) (<_) N) pm5.1 syl6 (e. N (RR)) adantr M (0) (<) N breq1 M (0) (+) (1) opreq1 1cn addid2 syl6eq (<_) N breq1d bibi12d (br (0) (<) N) (br (1) (<_) N) pm5.1 N nngt0t N nnge1t sylanc syl5bir imp (/\ (e. M (RR)) (e. N (RR))) a1i N (0) M (<) breq2 N (0) (opr M (+) (1)) (<_) breq2 bibi12d (-u M) nnnn0t 0nn0 (0) (-u M) nn0ltlem1t mpan syl (e. M (RR)) adantl M lt0neg1t (e. (-u M) (NN)) adantr M peano2re (opr M (+) (1)) le0neg1t syl M recnt 1cn M (1) negdi2t mpan2 syl (0) (<_) breq2d bitrd (e. (-u M) (NN)) adantr 3bitr4d syl5bir com12 ex imp3a (e. N (RR)) adantr 0re ltnr lt01 0re ax1re ltnle mpbi 2false M (0) (<) N breq1 M (0) (+) (1) opreq1 1cn addid2 syl6eq (<_) N breq1d bibi12d N (0) (0) (<) breq2 N (0) (1) (<_) breq2 bibi12d sylan9bb mpbiri (/\ (e. M (RR)) (e. N (RR))) a1i ccased sylbid (e. M (RR)) (e. N (RR)) pm3.27 (/\ (e. M (NN0)) (e. (-u N) (NN0))) adantr 0re (/\ (/\ (e. M (RR)) (e. N (RR))) (/\ (e. M (NN0)) (e. (-u N) (NN0)))) a1i (e. M (RR)) (e. N (RR)) pm3.26 (/\ (e. M (NN0)) (e. (-u N) (NN0))) adantr (-u N) nn0ge0t (e. N (RR)) adantl N le0neg1t (e. (-u N) (NN0)) adantr mpbird (e. M (RR)) (e. M (NN0)) ad2ant2l M nn0ge0t (/\ (e. M (RR)) (e. N (RR))) (e. (-u N) (NN0)) ad2antrl letrd N M lenltt ancoms (/\ (e. M (NN0)) (e. (-u N) (NN0))) adantr mpbid (e. M (RR)) (e. N (RR)) pm3.27 (/\ (e. M (NN0)) (e. (-u N) (NN0))) adantr 0re (/\ (/\ (e. M (RR)) (e. N (RR))) (/\ (e. M (NN0)) (e. (-u N) (NN0)))) a1i (e. M (RR)) (e. N (RR)) pm3.26 (/\ (e. M (NN0)) (e. (-u N) (NN0))) adantr (-u N) nn0ge0t (e. N (RR)) adantl N le0neg1t (e. (-u N) (NN0)) adantr mpbird (e. M (RR)) (e. M (NN0)) ad2ant2l M nn0ge0t (/\ (e. M (RR)) (e. N (RR))) (e. (-u N) (NN0)) ad2antrl letrd M ltp1t (e. N (RR)) (/\ (e. M (NN0)) (e. (-u N) (NN0))) ad2antrr N M (opr M (+) (1)) lelttrt 3expb M peano2re ancli sylan2 ancoms (/\ (e. M (NN0)) (e. (-u N) (NN0))) adantr mp2and N (opr M (+) (1)) ltnlet M peano2re sylan2 ancoms (/\ (e. M (NN0)) (e. (-u N) (NN0))) adantr mpbid jca ex (br M (<) N) (br (opr M (+) (1)) (<_) N) pm5.21 syl6 (-u N) (-u M) nn0ltlem1t ancoms (/\ (e. M (RR)) (e. N (RR))) adantl M N ltnegt (/\ (e. (-u M) (NN0)) (e. (-u N) (NN0))) adantr (opr M (+) (1)) N lenegt M peano2re sylan M recnt 1cn M (1) negdit mpan2 M negclt 1cn (-u M) (1) negsubt mpan2 syl eqtrd syl (-u N) (<_) breq2d (e. N (RR)) adantr bitrd (/\ (e. (-u M) (NN0)) (e. (-u N) (NN0))) adantr 3bitr4d ex ccased imp an4s M elznn0 N elznn0 syl2anb)) thm (zleltp1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (br M (<) (opr N (+) (1))))) (ax1re M N (1) leadd1t mp3an3 M zret N zret syl2an M (opr N (+) (1)) zltp1let N peano2z sylan2 bitr4d)) thm (zlem1ltt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N))) ((opr M (-) (1)) N zltp1let M peano2zm sylan M zcnt 1cn M (1) npcant mpan2 syl (e. N (ZZ)) adantr (<_) N breq1d bitr2d)) thm (zltlem1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1))))) (M (opr N (-) (1)) zleltp1t N peano2zm sylan2 N zcnt 1cn N (1) npcant mpan2 syl (e. M (ZZ)) adantl M (<) breq2d bitr2d)) thm (nn0lem1ltt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N))) (M N zlem1ltt M nn0zt N nn0zt syl2an)) thm (nnlem1ltt () () (-> (/\ (e. M (NN)) (e. N (NN))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N))) (M N zlem1ltt M nnzt N nnzt syl2an)) thm (nnltlem1t () () (-> (/\ (e. M (NN)) (e. N (NN))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1))))) (M N zltlem1t M nnzt N nnzt syl2an)) thm (z2get ((M k) (N k)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (E.e. k (ZZ) (/\ (br M (<_) (cv k)) (br N (<_) (cv k))))) ((cv k) (if (br M (<_) N) N M) M (<_) breq2 (cv k) (if (br M (<_) N) N M) N (<_) breq2 anbi12d (ZZ) rcla4ev N (ZZ) M (br M (<_) N) ifcl ancoms M N max1 M N max2 jca M zret N zret syl2an sylanc)) thm (zextlet ((M k) (N k)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (A.e. k (ZZ) (<-> (br (cv k) (<_) M) (br (cv k) (<_) N)))) (= M N)) (M zret M leidt syl (A.e. k (ZZ) (<-> (br (cv k) (<_) M) (br (cv k) (<_) N))) adantr (cv k) M (<_) M breq1 (cv k) M (<_) N breq1 bibi12d (ZZ) rcla4va mpbid (e. N (ZZ)) adantlr N zret N leidt syl (A.e. k (ZZ) (<-> (br (cv k) (<_) M) (br (cv k) (<_) N))) adantr (cv k) N (<_) M breq1 (cv k) N (<_) N breq1 bibi12d (ZZ) rcla4va mpbird (e. M (ZZ)) adantll jca ex M N letri3t M zret N zret syl2an sylibrd 3impia)) thm (zextltt ((M k) (N k)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (A.e. k (ZZ) (<-> (br (cv k) (<) M) (br (cv k) (<) N)))) (= M N)) ((cv k) M zltlem1t (e. N (ZZ)) adantrr (cv k) N zltlem1t (e. M (ZZ)) adantrl bibi12d ancoms ralbidva (opr M (-) (1)) (opr N (-) (1)) k zextlet 3expa ex M peano2zm N peano2zm syl2an 1cn M N (1) subcan2t mp3an3 M zcnt N zcnt syl2an sylibd sylbid 3impia)) thm (recnzt () () (-> (/\ (e. A (RR)) (br (1) (<) A)) (-. (e. (opr (1) (/) A) (ZZ)))) (A recgt1it pm3.27d A recgt1it pm3.26d 0z (0) (opr (1) (/) A) zltp1let mpan 1cn addid2 (<_) (opr (1) (/) A) breq1i syl6bb biimpcd syl lt01 0re ax1re (0) (1) A axlttrn mp3an12 mpani imdistani A gt0ne0t syl A rerecclt syldan ax1re (1) (opr (1) (/) A) lenltt mpan syl sylibd mt2d)) thm (btwnnzt () () (-> (/\/\ (e. A (ZZ)) (br A (<) B) (br B (<) (opr A (+) (1)))) (-. (e. B (ZZ)))) (A B zltp1let (opr A (+) (1)) B lenltt A peano2z (opr A (+) (1)) zret syl B zret syl2an bitrd biimpd ex com23 imp con2d 3impia)) thm (gtndivt () () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br B (<) A)) (-. (e. (opr B (/) A) (ZZ)))) (0z (0) (opr B (/) A) btwnnzt mp3an1 B A divgt0t B nnret (e. A (RR)) (br B (<) A) 3ad2ant2 (e. A (RR)) (e. B (NN)) (br B (<) A) 3simp1 jca B nngt0t (e. A (RR)) (br B (<) A) 3ad2ant2 B nngt0t (e. A (RR)) adantl 0re (0) B A axlttrn mp3an1 B nnret sylan ancoms mpand 3impia jca sylanc (e. A (RR)) (e. B (NN)) (br B (<) A) 3simp3 ax1re B A (1) ltdivmul2t mp3anl3 B nnret (e. A (RR)) (br B (<) A) 3ad2ant2 (e. A (RR)) (e. B (NN)) (br B (<) A) 3simp1 jca B nngt0t (e. A (RR)) adantl 0re (0) B A axlttrn mp3an1 B nnret sylan ancoms mpand 3impia sylanc A recnt A mulid2t syl B (<) breq2d (e. B (NN)) (br B (<) A) 3ad2ant1 bitrd mpbird 1cn addid2 syl6breqr sylanc)) thm (halfnz () () (-. (e. (opr (1) (/) (2)) (ZZ))) (2re 1lt2 (2) recnzt mp2an)) thm (primet ((A x)) () (-> (e. A (NN)) (<-> (A.e. x (NN) (-> (e. (opr A (/) (cv x)) (NN)) (\/ (= (cv x) (1)) (= (cv x) A)))) (A.e. x (NN) (-> (/\/\ (br (1) (<) (cv x)) (br (cv x) (<_) A) (e. (opr A (/) (cv x)) (NN))) (= (cv x) A))))) ((cv x) nngt1ne1t (e. A (NN)) adantl (e. (opr A (/) (cv x)) (NN)) anbi1d (cv x) A gtndivt 3expa ex (cv x) nnret sylan con2d (cv x) A lenltt (cv x) nnret A nnret syl2an sylibrd ancoms (opr A (/) (cv x)) nnzt syl5 pm4.71rd (br (1) (<) (cv x)) anbi2d (br (1) (<) (cv x)) (br (cv x) (<_) A) (e. (opr A (/) (cv x)) (NN)) 3anass syl6bbr bitr3d (= (cv x) A) imbi1d (-. (= (cv x) (1))) (e. (opr A (/) (cv x)) (NN)) (= (cv x) A) bi2.04 (-. (= (cv x) (1))) (e. (opr A (/) (cv x)) (NN)) (= (cv x) A) impexp (= (cv x) (1)) (= (cv x) A) df-or (e. (opr A (/) (cv x)) (NN)) imbi2i 3bitr4r syl5bb ralbidva)) thm (msqznn () () (-> (/\ (e. A (ZZ)) (=/= A (0))) (e. (opr A (x.) A) (NN))) (A A zmulclt anidms (=/= A (0)) adantr A ltmsqt A (0) df-ne syl5ib imp A zret sylan jca (opr A (x.) A) elnnz sylibr)) thm (nneo ((j k) (N j) (N k)) ((nneo.1 (e. N (NN)))) (<-> (e. (opr N (/) (2)) (NN)) (-. (e. (opr (opr N (+) (1)) (/) (2)) (NN)))) (1lt2 ax1re 2re nneo.1 nnre ltadd2 mpbi 2cn nneo.1 nnre ax1re readdcl recn 2re 2pos gt0ne0i divcan2 2cn nneo.1 nncn 2cn 2re 2pos gt0ne0i divcl 1cn adddi 2cn nneo.1 nncn 2re 2pos gt0ne0i divcan2 2cn mulid1 (+) opreq12i eqtr 3brtr4 2pos nneo.1 nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl nneo.1 nnre 2re 2re 2pos gt0ne0i redivcl ax1re readdcl 2re ltmul2 ax-mp mpbir nneo.1 nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl nneo.1 nnre 2re 2re 2pos gt0ne0i redivcl ax1re readdcl ltnle mpbi nneo.1 nnre ltp1 nneo.1 nnre nneo.1 nnre ax1re readdcl 2re 2pos ltdiv1i mpbi (opr N (/) (2)) (opr (opr N (+) (1)) (/) (2)) nnltp1let mpbii mto (e. (opr N (/) (2)) (NN)) (e. (opr (opr N (+) (1)) (/) (2)) (NN)) imnan mpbir nneo.1 (cv j) (1) (+) (1) opreq1 (/) (2) opreq1d (NN) eleq1d (cv j) (1) (/) (2) opreq1 (NN) eleq1d orbi12d (cv j) (cv k) (+) (1) opreq1 (/) (2) opreq1d (NN) eleq1d (cv j) (cv k) (/) (2) opreq1 (NN) eleq1d orbi12d (cv j) (opr (cv k) (+) (1)) (+) (1) opreq1 (/) (2) opreq1d (NN) eleq1d (cv j) (opr (cv k) (+) (1)) (/) (2) opreq1 (NN) eleq1d orbi12d (cv j) N (+) (1) opreq1 (/) (2) opreq1d (NN) eleq1d (cv j) N (/) (2) opreq1 (NN) eleq1d orbi12d df-2 (/) (2) opreq1i 2cn 2re 2pos gt0ne0i divid eqtr3 1nn eqeltr (e. (opr (1) (/) (2)) (NN)) pm2.21ni orri (cv k) nncnt 1cn 1cn (cv k) (1) (1) axaddass mp3an23 df-2 (cv k) (+) opreq2i syl6eqr (/) (2) opreq1d 2cn 2cn 2re 2pos gt0ne0i (cv k) (2) (2) divdirt mpan2 mp3an23 2cn 2re 2pos gt0ne0i divid (opr (cv k) (/) (2)) (+) opreq2i syl6eq eqtrd syl (NN) eleq1d (opr (cv k) (/) (2)) peano2nn syl5bir (e. (opr (opr (cv k) (+) (1)) (/) (2)) (NN)) orim2d (e. (opr (opr (cv k) (+) (1)) (/) (2)) (NN)) (e. (opr (opr (opr (cv k) (+) (1)) (+) (1)) (/) (2)) (NN)) orcom syl6ib nnind ax-mp ori impbi)) thm (nneot () () (-> (e. N (NN)) (<-> (e. (opr N (/) (2)) (NN)) (-. (e. (opr (opr N (+) (1)) (/) (2)) (NN))))) (N (if (e. N (NN)) N (1)) (/) (2) opreq1 (NN) eleq1d N (if (e. N (NN)) N (1)) (+) (1) opreq1 (/) (2) opreq1d (NN) eleq1d negbid bibi12d 1nn N elimel nneo dedth)) thm (zeot () () (-> (e. N (ZZ)) (\/ (e. (opr N (/) (2)) (ZZ)) (e. (opr (opr N (+) (1)) (/) (2)) (ZZ)))) (N elz N (0) (/) (2) opreq1 2cn 2re 2pos gt0ne0i div0 0z eqeltr syl6eqel (e. (opr (opr N (+) (1)) (/) (2)) (ZZ)) pm2.21nd (e. N (RR)) adantl N nneot biimprd con1d (opr N (/) (2)) nnzt con3i syl5 (opr (opr N (+) (1)) (/) (2)) nnzt syl6 (e. N (RR)) adantl N recnt 2cn 2re 2pos gt0ne0i N (2) divnegt mp3an23 syl (NN) eleq1d (-u (opr N (/) (2))) nnnegz syl6bir N recnt N halfclt (opr N (/) (2)) negnegt 3syl (ZZ) eleq1d sylibd (e. (-u N) (NN)) adantr con3d (-u N) nneot biimprd con1d N recnt 1cn 2cn negsubdi2 df-2 eqcomi 2cn 1cn 1cn subadd mpbir eqtr2 1cn 1cn 2cn subcl negcon2 mpbi (-u N) (+) opreq2i (e. N (CC)) a1i N negclt 1cn 2cn (-u N) (1) (2) addsubasst mp3an23 syl 1cn N (1) negdit mpan2 3eqtr4d (/) (2) opreq1d N negclt (-u N) peano2cn 2cn 2cn 2re 2pos gt0ne0i (opr (-u N) (+) (1)) (2) (2) divsubdirt mpan2 mp3an23 3syl 2cn 2re 2pos gt0ne0i divid eqcomi (opr (opr (-u N) (+) (1)) (/) (2)) (-) opreq2i syl6reqr N peano2cn 2cn 2re 2pos gt0ne0i (opr N (+) (1)) (2) divnegt mp3an23 syl 3eqtr4d syl (ZZ) eleq1d (opr (opr (-u N) (+) (1)) (/) (2)) peano2zm syl5bi (-u (opr (opr N (+) (1)) (/) (2))) znegclt syl6 N peano2re recnd (opr N (+) (1)) halfclt (opr (opr N (+) (1)) (/) (2)) negnegt 3syl (ZZ) eleq1d sylibd (opr (opr (-u N) (+) (1)) (/) (2)) nnzt syl5 sylan9r syld 3jaodan sylbi orrd)) thm (zneo () () (-> (/\ (e. A (ZZ)) (e. B (ZZ))) (-. (= (opr (2) (x.) A) (opr (opr (2) (x.) B) (+) (1))))) (halfnz B zcnt 2cn 2re 2pos gt0ne0i reccl B (opr (1) (/) (2)) axaddcom mpan2 (-) B opreq1d 2cn 2re 2pos gt0ne0i reccl (opr (1) (/) (2)) B pncant mpan eqtrd syl (ZZ) eleq1d mtbiri (opr B (+) (opr (1) (/) (2))) B zsubclt expcom mtod (e. A (ZZ)) adantl 2cn 2re 2pos gt0ne0i (2) A divcan3t mp3an13 2cn (2) B axmulcl mpan 1cn 2cn 2re 2pos gt0ne0i (opr (2) (x.) B) (1) (2) divdirt mpan2 mp3an23 syl 2cn 2re 2pos gt0ne0i (2) B divcan3t mp3an13 (+) (opr (1) (/) (2)) opreq1d eqtrd eqeqan12d (opr (2) (x.) A) (opr (opr (2) (x.) B) (+) (1)) (/) (2) opreq1 syl5bi A zcnt B zcnt syl2an imp (ZZ) eleq1d exp31 com3l (e. A (ZZ)) (e. (opr B (+) (opr (1) (/) (2))) (ZZ)) ibib syl6ibr com3r imp mtod)) thm (peano2uz2 ((A x) (B x)) () (-> (/\ (e. A (ZZ)) (e. B ({e.|} x (ZZ) (br A (<_) (cv x))))) (e. (opr B (+) (1)) ({e.|} x (ZZ) (br A (<_) (cv x))))) (B peano2z (e. A (ZZ)) (br A (<_) B) ad2antrl B lep1t (e. A (RR)) adantl A B (opr B (+) (1)) letrt 3expb B peano2re ancli sylan2 mpan2d A zret B zret syl2an ex imp32 jca (cv x) B A (<_) breq2 (ZZ) elrab (e. A (ZZ)) anbi2i (cv x) (opr B (+) (1)) A (<_) breq2 (ZZ) elrab 3imtr4)) thm (dfuz ((x y) (x z) (w x) (v x) (u x) (m x) (N x) (y z) (w y) (v y) (u y) (m y) (N y) (w z) (v z) (u z) (m z) (N z) (v w) (u w) (m w) (N w) (u v) (m v) (N v) (m u) (N u) (N m)) ((dfuz.1 (e. N (ZZ)))) (= ({e.|} z (ZZ) (br N (<_) (cv z))) (|^| ({|} x (/\ (e. N (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x))))))) ((cv z) (cv m) N (<_) breq2 (ZZ) elrab pm3.26bd zex (cv x) (ZZ) N eleq2 (cv x) (ZZ) (opr (cv y) (+) (1)) eleq2 y raleqd anbi12d dfuz.1 (cv y) peano2z rgen pm3.2i (V) intmin3 ax-mp (cv m) sseli (e. (cv u) (cv m)) z ax-17 (e. (cv u) (opr (cv m) (+) (opr (1) (-) N))) z ax-17 1z dfuz.1 (1) N zsubclt mp2an (cv z) (opr (1) (-) N) zaddclt mpan2 (cv v) (opr (cv z) (+) (opr (1) (-) N)) (1) (<_) breq2 (cv z) (cv m) (+) (opr (1) (-) N) opreq1 rabxfr (cv z) zret dfuz.1 zre ax1re dfuz.1 zre resubcl N (cv z) (opr (1) (-) N) leadd1t mp3an13 syl dfuz.1 zre recn 1cn pncan3 (<_) (opr (cv z) (+) (opr (1) (-) N)) breq1i syl6bb rabbii (cv m) eleq2i syl6bbr (cv v) (opr (cv y) (+) (opr (1) (-) N)) (+) (1) opreq1 (cv w) eleq1d (cv w) rcla4v (cv y) recnt ax1re dfuz.1 zre resubcl recn 1cn (cv y) (opr (1) (-) N) (1) add23t mp3an23 syl (cv w) eleq1d biimpd (cv y) peano2re jctild sylan9r com12 y 19.21aiv (e. (1) (cv w)) anim2i (e. (cv m) (RR)) (e. (opr (cv m) (+) (opr (1) (-) N)) (cv w)) pm3.27 imim12i (e. (cv m) (ZZ)) a1i reex u (e. (opr (cv u) (+) (opr (1) (-) N)) (cv w)) rabex (cv x) ({e.|} u (RR) (e. (opr (cv u) (+) (opr (1) (-) N)) (cv w))) N eleq2 (cv u) N (+) (opr (1) (-) N) opreq1 dfuz.1 zre recn 1cn pncan3 syl6eq (cv w) eleq1d (RR) elrab dfuz.1 zre mpbiran syl6bb (cv x) ({e.|} u (RR) (e. (opr (cv u) (+) (opr (1) (-) N)) (cv w))) (cv y) eleq2 (cv u) (cv y) (+) (opr (1) (-) N) opreq1 (cv w) eleq1d (RR) elrab syl6bb (cv x) ({e.|} u (RR) (e. (opr (cv u) (+) (opr (1) (-) N)) (cv w))) (opr (cv y) (+) (1)) eleq2 (cv u) (opr (cv y) (+) (1)) (+) (opr (1) (-) N) opreq1 (cv w) eleq1d (RR) elrab syl6bb imbi12d y albidv y (cv x) (e. (opr (cv y) (+) (1)) (cv x)) df-ral syl5bb anbi12d (cv x) ({e.|} u (RR) (e. (opr (cv u) (+) (opr (1) (-) N)) (cv w))) (cv m) eleq2 (cv u) (cv m) (+) (opr (1) (-) N) opreq1 (cv w) eleq1d (RR) elrab syl6bb imbi12d cla4v syl5 w 19.21adv (cv y) (opr (cv v) (-) (opr (1) (-) N)) (+) (1) opreq1 (cv x) eleq1d (cv x) rcla4v (cv v) recnt 1cn ax1re dfuz.1 zre resubcl recn (cv v) (1) (opr (1) (-) N) addsubt mp3an23 syl (cv x) eleq1d biimprd (cv v) peano2re jctild sylan9r com12 v 19.21aiv (e. N (cv x)) anim2i (e. (cv m) (ZZ)) a1i (cv m) zcnt ax1re dfuz.1 zre resubcl recn (cv m) (opr (1) (-) N) pncant mpan2 syl (cv x) eleq1d biimpd (e. (opr (cv m) (+) (opr (1) (-) N)) (RR)) adantld imim12d reex u (e. (opr (cv u) (-) (opr (1) (-) N)) (cv x)) rabex (cv w) ({e.|} u (RR) (e. (opr (cv u) (-) (opr (1) (-) N)) (cv x))) (1) eleq2 (cv u) (1) (-) (opr (1) (-) N) opreq1 1cn dfuz.1 zre recn (1) N nncant mp2an syl6eq (cv x) eleq1d (RR) elrab ax1re mpbiran syl6bb (cv w) ({e.|} u (RR) (e. (opr (cv u) (-) (opr (1) (-) N)) (cv x))) (cv v) eleq2 (cv u) (cv v) (-) (opr (1) (-) N) opreq1 (cv x) eleq1d (RR) elrab syl6bb (cv w) ({e.|} u (RR) (e. (opr (cv u) (-) (opr (1) (-) N)) (cv x))) (opr (cv v) (+) (1)) eleq2 (cv u) (opr (cv v) (+) (1)) (-) (opr (1) (-) N) opreq1 (cv x) eleq1d (RR) elrab syl6bb imbi12d v albidv v (cv w) (e. (opr (cv v) (+) (1)) (cv w)) df-ral syl5bb anbi12d (cv w) ({e.|} u (RR) (e. (opr (cv u) (-) (opr (1) (-) N)) (cv x))) (opr (cv m) (+) (opr (1) (-) N)) eleq2 (cv u) (opr (cv m) (+) (opr (1) (-) N)) (-) (opr (1) (-) N) opreq1 (cv x) eleq1d (RR) elrab syl6bb imbi12d cla4v syl5 x 19.21adv impbid m visset x (/\ (e. N (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x)))) elintab (cv m) (+) (opr (1) (-) N) oprex w (/\ (e. (1) (cv w)) (A.e. v (cv w) (e. (opr (cv v) (+) (1)) (cv w)))) elintab 3bitr4g w v df-n v nnzrab eqtr3 (opr (cv m) (+) (opr (1) (-) N)) eleq2i syl6rbb bitr3d pm5.21nii eqriv)) thm (peano5uz ((x y) (A x) (A y) (k x) (N x) (k y) (N y) (N k)) ((peano5uz.1 (e. A (V))) (peano5uz.2 (e. N (ZZ)))) (-> (/\ (e. N A) (A.e. x A (e. (opr (cv x) (+) (1)) A))) (C_ ({e.|} k (ZZ) (br N (<_) (cv k))) A)) (peano5uz.1 (cv y) A N eleq2 (cv y) A (opr (cv x) (+) (1)) eleq2 x raleqd anbi12d elab A ({|} y (/\ (e. N (cv y)) (A.e. x (cv y) (e. (opr (cv x) (+) (1)) (cv y))))) intss1 sylbir peano5uz.2 k y x dfuz syl5ss)) thm (peano5uzt ((A x) (k x) (N x) (N k)) ((peano5uz.1 (e. A (V)))) (-> (e. N (ZZ)) (-> (/\ (e. N A) (A.e. x A (e. (opr (cv x) (+) (1)) A))) (C_ ({e.|} k (ZZ) (br N (<_) (cv k))) A))) (N (if (e. N (ZZ)) N (1)) A eleq1 (A.e. x A (e. (opr (cv x) (+) (1)) A)) anbi1d N (if (e. N (ZZ)) N (1)) (<_) (cv k) breq1 k (ZZ) rabbisdv A sseq1d imbi12d peano5uz.1 1z N elimel x k peano5uz dedth)) thm (uzind ((j w) (N j) (N w) (j ps) (ch j) (j th) (j ta) (k w) (k ph) (ph w) (j k) (M j) (M k) (M w)) ((uzind.1 (-> (= (cv j) M) (<-> ph ps))) (uzind.2 (-> (= (cv j) (cv k)) (<-> ph ch))) (uzind.3 (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th))) (uzind.4 (-> (= (cv j) N) (<-> ph ta))) (uzind.5 (-> (e. M (ZZ)) ps)) (uzind.6 (-> (/\/\ (e. M (ZZ)) (e. (cv k) (ZZ)) (br M (<_) (cv k))) (-> ch th)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) ta) (M zret M leidt syl uzind.5 jca ancli (cv j) M M (<_) breq2 uzind.1 anbi12d (ZZ) elrab sylibr (cv k) peano2z (e. M (ZZ)) a1i (/\ (br M (<_) (cv k)) ch) adantrd (cv k) ltp1t (e. M (RR)) adantl M (cv k) (opr (cv k) (+) (1)) lelttrt 3expb (cv k) peano2re ancli sylan2 mpan2d M (opr (cv k) (+) (1)) ltlet (cv k) peano2re sylan2 syld M zret (cv k) zret syl2an ch adantrd exp4b imp4d uzind.6 3exp imp4d jcad jcad (cv j) (cv k) M (<_) breq2 uzind.2 anbi12d (ZZ) elrab (cv j) (opr (cv k) (+) (1)) M (<_) breq2 uzind.3 anbi12d (ZZ) elrab 3imtr4g r19.21aiv zex j (/\ (br M (<_) (cv j)) ph) rabex M k w peano5uzt mp2and N sseld (cv w) N M (<_) breq2 (ZZ) elrab (cv j) N M (<_) breq2 uzind.4 anbi12d (ZZ) elrab 3imtr3g 3impib pm3.27d pm3.27d)) thm (uzind2 ((N j) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (M k)) ((uzind2.1 (-> (= (cv j) (opr M (+) (1))) (<-> ph ps))) (uzind2.2 (-> (= (cv j) (cv k)) (<-> ph ch))) (uzind2.3 (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th))) (uzind2.4 (-> (= (cv j) N) (<-> ph ta))) (uzind2.5 (-> (e. M (ZZ)) ps)) (uzind2.6 (-> (/\/\ (e. M (ZZ)) (e. (cv k) (ZZ)) (br M (<) (cv k))) (-> ch th)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) ta) (M N zltp1let M peano2z uzind2.1 (e. M (ZZ)) imbi2d uzind2.2 (e. M (ZZ)) imbi2d uzind2.3 (e. M (ZZ)) imbi2d uzind2.4 (e. M (ZZ)) imbi2d uzind2.5 (e. (opr M (+) (1)) (ZZ)) a1i M (cv k) zltp1let uzind2.6 3exp imp sylbird ex com3l imp (e. (opr M (+) (1)) (ZZ)) 3adant1 a2d uzind 3exp syl com34 pm2.43a imp sylbid 3impia)) thm (uzind3 ((j k) (N j) (N k) (j ps) (ch j) (j th) (j ta) (m ph) (j m) (M j) (k m) (M m) (M k)) ((uzind3.1 (-> (= (cv j) M) (<-> ph ps))) (uzind3.2 (-> (= (cv j) (cv m)) (<-> ph ch))) (uzind3.3 (-> (= (cv j) (opr (cv m) (+) (1))) (<-> ph th))) (uzind3.4 (-> (= (cv j) N) (<-> ph ta))) (uzind3.5 (-> (e. M (ZZ)) ps)) (uzind3.6 (-> (/\ (e. M (ZZ)) (e. (cv m) ({e.|} k (ZZ) (br M (<_) (cv k))))) (-> ch th)))) (-> (/\ (e. M (ZZ)) (e. N ({e.|} k (ZZ) (br M (<_) (cv k))))) ta) (uzind3.1 uzind3.2 uzind3.3 uzind3.4 uzind3.5 uzind3.6 (cv k) (cv m) M (<_) breq2 (ZZ) elrab sylan2br 3impb uzind 3expb (cv k) N M (<_) breq2 (ZZ) elrab sylan2b)) thm (uzindOLD ((x z) (w x) (A x) (w z) (A z) (A w) (ps x) (ps z) (ch x) (ch z) (th x) (th z) (ta x) (ta z) (w y) (ph y) (ph w) (x y) (B x) (y z) (B y) (B z) (B w)) ((uzindOLD.1 (-> (= (cv x) B) (<-> ph ps))) (uzindOLD.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (uzindOLD.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th))) (uzindOLD.4 (-> (= (cv x) A) (<-> ph ta))) (uzindOLD.5 ps) (uzindOLD.6 (-> (/\ (/\ (e. (cv y) (ZZ)) (e. B (ZZ))) (br B (<_) (cv y))) (-> ch th)))) (-> (/\ (/\ (e. A (ZZ)) (e. B (ZZ))) (br B (<_) A)) ta) (A B subge0t A B resubclt 0re ax1re (0) (opr A (-) B) (1) leadd1t mp3an13 syl 1cn addid2 (<_) (opr (opr A (-) B) (+) (1)) breq1i syl6bb bitr3d A zret B zret syl2an A B zsubclt peano2zd (opr (opr A (-) B) (+) (1)) elnnz1 (cv z) (1) (ZZ) eleq1 (opr (cv z) (-) (1)) (+) B oprex x isseti (/\ (= (cv z) (1)) (e. B (ZZ))) x ax-17 (opr (cv z) (-) (1)) (+) B oprex x ph hbsbc1v ps x ax-17 hbbi hbim x (opr (opr (cv z) (-) (1)) (+) B) ph sbceq1a (/\ (= (cv z) (1)) (e. B (ZZ))) adantr (cv x) (opr (opr (cv z) (-) (1)) (+) B) B eqtrt (cv z) (1) (-) (1) opreq1 (+) B opreq1d B zcnt B addid2t syl 1cn subid (+) B opreq1i syl5eq sylan9eq sylan2 uzindOLD.1 syl bitr3d ex 19.23ai ax-mp ex (e. A (ZZ)) adantld pm5.74d imbi12d (cv z) (cv w) (ZZ) eleq1 (opr (cv z) (-) (1)) (+) B oprex x isseti (opr (cv w) (-) (1)) (+) B oprex y isseti x y (= (cv x) (opr (opr (cv z) (-) (1)) (+) B)) (= (cv y) (opr (opr (cv w) (-) (1)) (+) B)) eeanv (= (cv z) (cv w)) x ax-17 (opr (cv z) (-) (1)) (+) B oprex x ph hbsbc1v ([/] (opr (opr (cv w) (-) (1)) (+) B) y ch) x ax-17 hbbi hbim (= (cv z) (cv w)) y ax-17 ([/] (opr (opr (cv z) (-) (1)) (+) B) x ph) y ax-17 (opr (cv w) (-) (1)) (+) B oprex y ch hbsbc1v hbbi hbim (cv x) (opr (opr (cv z) (-) (1)) (+) B) (cv y) (opr (opr (cv w) (-) (1)) (+) B) eqeq12 (cv z) (cv w) (-) (1) opreq1 (+) B opreq1d syl5bir uzindOLD.2 syl6 x (opr (opr (cv z) (-) (1)) (+) B) ph sbceq1a y (opr (opr (cv w) (-) (1)) (+) B) ch sbceq1a bi2bian9 sylibd 19.23ai 19.23ai sylbir mp2an (/\ (e. A (ZZ)) (e. B (ZZ))) imbi2d imbi12d (cv z) (opr (cv w) (+) (1)) (ZZ) eleq1 (opr (opr (opr (cv z) (-) (1)) (+) B) (-) (1)) (+) (1) oprex x isseti (opr (opr (opr (cv w) (+) (1)) (-) (1)) (+) B) (-) (1) oprex y isseti x y (= (cv x) (opr (opr (opr (opr (cv z) (-) (1)) (+) B) (-) (1)) (+) (1))) (= (cv y) (opr (opr (opr (opr (cv w) (+) (1)) (-) (1)) (+) B) (-) (1))) eeanv (= (cv z) (opr (cv w) (+) (1))) x ax-17 (opr (opr (opr (cv z) (-) (1)) (+) B) (-) (1)) (+) (1) oprex x ph hbsbc1v ([/] (opr (opr (opr (opr (cv w) (+) (1)) (-) (1)) (+) B) (-) (1)) y th) x ax-17 hbbi hbim (= (cv z) (opr (cv w) (+) (1))) y ax-17 ([/] (opr (opr (opr (opr (cv z) (-) (1)) (+) B) (-) (1)) (+) (1)) x ph) y ax-17 (opr (opr (opr (cv w) (+) (1)) (-) (1)) (+) B) (-) (1) oprex y th hbsbc1v hbbi hbim (cv x) (opr (opr (opr (opr (cv z) (-) (1)) (+) B) (-) (1)) (+) (1)) (opr (cv y) (+) (1)) (opr (opr (opr (opr (opr (cv w) (+) (1)) (-) (1)) (+) B) (-) (1)) (+) (1)) eqeq12 (cv y) (opr (opr (opr (opr (cv w) (+) (1)) (-) (1)) (+) B) (-) (1)) (+) (1) opreq1 sylan2 (cv z) (opr (cv w) (+) (1)) (-) (1) opreq1 (+) B opreq1d (-) (1) opreq1d (+) (1) opreq1d syl5bir uzindOLD.3 syl6 x (opr (opr (opr (opr (cv z) (-) (1)) (+) B) (-) (1)) (+) (1)) ph sbceq1a y (opr (opr (opr (opr (cv w) (+) (1)) (-) (1)) (+) B) (-) (1)) th sbceq1a bi2bian9 sylibd 19.23ai 19.23ai sylbir mp2an (/\ (e. A (ZZ)) (e. B (ZZ))) imbi2d imbi12d (opr (cv z) (-) (1)) B axaddcl 1cn (cv z) (1) subclt mpan2 sylan 1cn (opr (opr (cv z) (-) (1)) (+) B) (1) npcant mpan2 syl (cv z) zcnt B zcnt syl2an ex (e. A (ZZ)) adantld (opr (opr (opr (opr (cv z) (-) (1)) (+) B) (-) (1)) (+) (1)) (opr (opr (cv z) (-) (1)) (+) B) x ph dfsbcq syl6 pm5.74d pm5.74i syl5bbr (cv z) (opr (opr A (-) B) (+) (1)) (ZZ) eleq1 (opr (cv z) (-) (1)) (+) B oprex x isseti (/\ (= (cv z) (opr (opr A (-) B) (+) (1))) (/\ (e. A (ZZ)) (e. B (ZZ)))) x ax-17 (opr (cv z) (-) (1)) (+) B oprex x ph hbsbc1v ta x ax-17 hbbi hbim x (opr (opr (cv z) (-) (1)) (+) B) ph sbceq1a (/\ (= (cv z) (opr (opr A (-) B) (+) (1))) (/\ (e. A (ZZ)) (e. B (ZZ)))) adantr (cv x) (opr (opr (cv z) (-) (1)) (+) B) (opr (opr (opr (opr A (-) B) (+) (1)) (-) (1)) (+) B) eqtrt (cv z) (opr (opr A (-) B) (+) (1)) (-) (1) opreq1 (+) B opreq1d sylan2 (opr A (-) B) (1) B add23t A B subclt 1cn (/\ (e. A (CC)) (e. B (CC))) a1i (e. A (CC)) (e. B (CC)) pm3.27 syl3anc A B npcant (+) (1) opreq1d eqtrd (-) (1) opreq1d (opr (opr A (-) B) (+) (1)) B (1) addsubt A B subclt (opr A (-) B) peano2cn syl (e. A (CC)) (e. B (CC)) pm3.27 1cn (/\ (e. A (CC)) (e. B (CC))) a1i syl3anc 1cn A (1) pncant mpan2 (e. B (CC)) adantr 3eqtr3d A zcnt B zcnt syl2an sylan9eq anasss uzindOLD.4 syl bitr3d ex 19.23ai ax-mp ex pm5.74d imbi12d uzindOLD.5 (/\ (e. A (ZZ)) (e. B (ZZ))) a1i (e. (1) (ZZ)) a1i (cv w) nnzt (e. (opr (cv w) (+) (1)) (ZZ)) a1d (opr (cv w) (-) (1)) (+) B oprex y isseti (/\ (e. (cv w) (NN)) (e. B (ZZ))) y ax-17 (opr (cv w) (-) (1)) (+) B oprex y ch hbsbc1v (opr (cv w) (-) (1)) (+) B oprex y th hbsbc1v hbim hbim uzindOLD.6 (cv y) (opr (opr (cv w) (-) (1)) (+) B) (ZZ) eleq1 (e. B (ZZ)) anbi1d (cv y) (opr (opr (cv w) (-) (1)) (+) B) B (<_) breq2 anbi12d (opr (cv w) (-) (1)) B zaddclt (cv w) peano2zm sylan (cv w) nnzt sylan (e. (cv w) (NN)) (e. B (ZZ)) pm3.27 jca B (opr (opr (cv w) (-) (1)) (+) B) (opr B (-) (1)) lesub1t (e. (cv w) (RR)) (e. B (RR)) pm3.27 (opr (cv w) (-) (1)) B axaddrcl ax1re (cv w) (1) resubclt mpan2 sylan ax1re B (1) resubclt mpan2 (e. (cv w) (RR)) adantl syl3anc 1cn B B (1) subsubt mp3an3 anidms B subidt (+) (1) opreq1d 1cn addid2 syl6eq eqtrd (e. (cv w) (CC)) adantl 1cn (opr (opr (cv w) (-) (1)) (+) B) B (1) subsubt mp3an3 (opr (cv w) (-) (1)) B axaddcl 1cn (cv w) (1) subclt mpan2 sylan (e. (cv w) (CC)) (e. B (CC)) pm3.27 sylanc (opr (cv w) (-) (1)) B pncant 1cn (cv w) (1) subclt mpan2 sylan (+) (1) opreq1d 1cn (cv w) (1) npcant mpan2 (e. B (CC)) adantr 3eqtrd (<_) breq12d (cv w) recnt B recnt syl2an bitrd (cv w) zret B zret syl2an biimpar an1rs (cv w) elnnz1 sylanb jca syl5bir y (opr (opr (cv w) (-) (1)) (+) B) ch sbceq1a y (opr (opr (cv w) (-) (1)) (+) B) th sbceq1a imbi12d biimpd imim12d mpi 19.23ai ax-mp 1cn (opr (opr (cv w) (+) (1)) (-) (1)) B (1) addsubt mp3an3 (cv w) peano2cn 1cn (opr (cv w) (+) (1)) (1) subclt mpan2 syl sylan 1cn (cv w) (1) pncant mpan2 (-) (1) opreq1d (+) B opreq1d (e. B (CC)) adantr eqtrd (cv w) nncnt B zcnt syl2an (opr (opr (opr (opr (cv w) (+) (1)) (-) (1)) (+) B) (-) (1)) (opr (opr (cv w) (-) (1)) (+) B) y th dfsbcq syl sylibrd ex (e. A (ZZ)) adantld a2d imim12d nnind sylbir ex pm2.43a com23 mpcom sylbid imp)) thm (uzind3OLD ((x z) (A x) (A z) (ps x) (ch x) (th x) (ta x) (ph y) (x y) (B x) (y z) (B y) (B z)) ((uzind3OLD.1 (-> (= (cv x) B) (<-> ph ps))) (uzind3OLD.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (uzind3OLD.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th))) (uzind3OLD.4 (-> (= (cv x) A) (<-> ph ta))) (uzind3OLD.5 ps) (uzind3OLD.6 (-> (/\ (e. B (ZZ)) (e. (cv y) ({e.|} z (ZZ) (br B (<_) (cv z))))) (-> ch th)))) (-> (/\ (e. B (ZZ)) (e. A ({e.|} z (ZZ) (br B (<_) (cv z))))) ta) (uzind3OLD.1 uzind3OLD.2 uzind3OLD.3 uzind3OLD.4 uzind3OLD.5 uzind3OLD.6 (cv z) (cv y) B (<_) breq2 (ZZ) elrab sylan2br an1s anassrs uzindOLD anasss an1s (cv z) A B (<_) breq2 (ZZ) elrab sylan2b)) thm (uzwoOLD ((x y) (w x) (v x) (u x) (A x) (w y) (v y) (u y) (A y) (v w) (u w) (A w) (u v) (A v) (A u) (y z) (B y) (w z) (v z) (u z) (B z) (B w) (B v) (B u)) () (-> (/\ (e. B (ZZ)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))) ((cv v) B (<_) (cv y) breq1 y A ralbidv (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (-. (E.e. x A (A.e. y A (br (cv x) (<_) (cv y)))))) imbi2d (cv v) (cv u) (<_) (cv y) breq1 y A ralbidv (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (-. (E.e. x A (A.e. y A (br (cv x) (<_) (cv y)))))) imbi2d (cv v) (opr (cv u) (+) (1)) (<_) (cv y) breq1 y A ralbidv (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (-. (E.e. x A (A.e. y A (br (cv x) (<_) (cv y)))))) imbi2d (cv v) (cv w) (<_) (cv y) breq1 y A ralbidv (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (-. (E.e. x A (A.e. y A (br (cv x) (<_) (cv y)))))) imbi2d A ({e.|} z (ZZ) (br B (<_) (cv z))) (cv y) ssel (cv z) (cv y) B (<_) breq2 (ZZ) elrab pm3.27bd syl6 r19.21aiv (-. (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))) adantr z (ZZ) (br B (<_) (cv z)) ssrab2 (cv u) sseli (cv x) (cv u) (<_) (cv y) breq1 y A ralbidv A rcla4ev expcom con3d com12 (cv u) (cv y) letri3t (cv u) zret (cv y) zret syl2an (cv y) (cv u) zleltp1t (cv y) (opr (cv u) (+) (1)) ltnlet (cv y) zret (cv u) zret (cv u) peano2re syl syl2an bitrd ancoms (br (cv u) (<_) (cv y)) anbi2d bitrd A (ZZ) (cv y) ssel2 sylan2 (cv y) A (cv u) eleq1a (e. (cv u) (ZZ)) (C_ A (ZZ)) ad2antll sylbird exp3a (br (opr (cv u) (+) (1)) (<_) (cv y)) (e. (cv u) A) con1 syl6 com23 exp32 com34 imp41 r19.20dva ex sylan9r pm2.43d exp31 imp3a syl z (ZZ) (br B (<_) (cv z)) ssrab2 A ({e.|} z (ZZ) (br B (<_) (cv z))) (ZZ) sstr mpan2 sylani a2d (e. B (ZZ)) adantl uzind3OLD (cv x) (cv w) (<_) (cv y) breq1 y A ralbidv A rcla4ev expcom con3d com12 (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) adantl sylcom ex A ({e.|} z (ZZ) (br B (<_) (cv z))) (cv w) ssel con3d com12 (-. (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))) adantrd pm2.61d1 exp3a imp w 19.21adv A w eq0 syl6ibr con1d A ({/}) df-ne syl5ib imp anasss)) thm (uzwo2OLD ((x y) (w x) (A x) (w y) (A y) (A w) (y z) (B y) (w z) (B z) (B w)) () (-> (/\ (e. B (ZZ)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (B A z x y uzwoOLD z (ZZ) (br B (<_) (cv z)) ssrab2 zssre sstri A ({e.|} z (ZZ) (br B (<_) (cv z))) (RR) sstr mpan2 (cv y) (cv w) (cv x) (<_) breq2 A rcla4v (cv y) (cv x) (cv w) (<_) breq2 A rcla4v im2anan9 ancoms A (RR) (cv x) ssel A (RR) (cv w) ssel anim12d impcom (cv x) (cv w) letri3t syl biimprd ex com23 syld com3r r19.21aivv syl (e. B (ZZ)) (=/= A ({/})) ad2antrl jca (cv x) (cv w) (<_) (cv y) breq1 y A ralbidv A reu4 sylibr)) thm (nn0ind ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((nn0ind.1 (-> (= (cv x) (0)) (<-> ph ps))) (nn0ind.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (nn0ind.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th))) (nn0ind.4 (-> (= (cv x) A) (<-> ph ta))) (nn0ind.5 ps) (nn0ind.6 (-> (e. (cv y) (NN0)) (-> ch th)))) (-> (e. A (NN0)) ta) (A elnn0z 0z nn0ind.1 nn0ind.2 nn0ind.3 nn0ind.4 nn0ind.5 (e. (0) (ZZ)) a1i (cv y) elnn0z nn0ind.6 sylbir (e. (0) (ZZ)) 3adant1 uzind mp3an1 sylbi)) thm (nn0indALT ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((nn0indALT.6 (-> (e. (cv y) (NN0)) (-> ch th))) (nn0indALT.5 ps) (nn0indALT.1 (-> (= (cv x) (0)) (<-> ph ps))) (nn0indALT.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (nn0indALT.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th))) (nn0indALT.4 (-> (= (cv x) A) (<-> ph ta)))) (-> (e. A (NN0)) ta) (nn0indALT.1 nn0indALT.2 nn0indALT.3 nn0indALT.4 nn0indALT.5 nn0indALT.6 nn0ind)) thm (nn0ind-raph ((x y) (x z) (y z) (A x) (A z) (ps x) (ps z) (ch x) (ch z) (th x) (th z) (ta x) (ta z) (ph y) (ph z)) ((nn0ind-raph.1 (-> (= (cv x) (0)) (<-> ph ps))) (nn0ind-raph.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (nn0ind-raph.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th))) (nn0ind-raph.4 (-> (= (cv x) A) (<-> ph ta))) (nn0ind-raph.5 ps) (nn0ind-raph.6 (-> (e. (cv y) (NN0)) (-> ch th)))) (-> (e. A (NN0)) ta) (A elnn0 (cv z) (1) x ph dfsbcq nn0ind-raph.2 z vsbcint nn0ind-raph.3 z vsbcint nn0ind-raph.4 z vsbcint ax1re elisseti x ph hbsbc1v ax1re elisseti 0nn0 elisseti nn0ind-raph.6 0nn0 (0) (NN0) (cv y) eleq1a ax-mp 0nn0 elisseti nn0ind-raph.5 nn0ind-raph.1 mpbiri (cv y) (0) (cv x) eqeq2 nn0ind-raph.2 syl6bir pm5.74d mpbii com12 vtocle sylc (= (cv x) (1)) adantr (cv y) (0) (+) (1) opreq1 1cn addid2 syl6eq (cv x) eqeq2d nn0ind-raph.3 syl6bir imp mpbird ex vtocle x (1) ph sbceq1a mpbid vtoclef (cv y) nnnn0t nn0ind-raph.6 syl nnind (= (0) A) x ax-17 ta x ax-17 hbim 0nn0 elisseti (cv x) (0) A eqeq1 nn0ind-raph.5 nn0ind-raph.1 bicomd nn0ind-raph.4 sylan9bb mpbii ex sylbird vtoclef eqcoms jaoi sylbi)) thm (btwnz ((x z) (A x) (A z) (A y)) () (-> (e. A (RR)) (/\ (E.e. x (ZZ) (br (cv x) (<) A)) (E.e. y (ZZ) (br A (<) (cv y))))) (A renegclt (-u A) z arch syl A (cv z) ltnegcon1t ex (cv z) nnret syl5 pm5.32d (cv x) (-u (cv z)) (<) A breq1 (ZZ) rcla4ev (cv z) nnnegz sylan syl6bi exp3a r19.23adv mpd A y arch (cv y) nnzt (br A (<) (cv y)) anim1i r19.22i2 syl jca)) thm (uzwo3lem1 ((v z) (B z) (B v) (x y) (v x) (R x) (v y) (R y) (R v) (y z)) ((uzwo3lem.1 (= R ({e.|} z (ZZ) (br B (<_) (cv z)))))) (-> (e. B (RR)) (E!e. x R (A.e. y R (br (cv x) (<_) (cv y))))) (B v z btwnz pm3.26d (cv v) B (cv z) ltletrt (cv v) (cv z) ltlet (e. B (RR)) 3adant2 syld (cv z) zret syl3an3 (cv v) zret syl3an1 exp3a 3exp com34 imp32 r19.21aiv z (ZZ) (br B (<_) (cv z)) (br (cv v) (<_) (cv z)) ss2rab sylibr uzwo3lem.1 syl5ss B v z btwnz pm3.27d B (cv z) ltlet (cv z) zret sylan2 r19.22dva R ({/}) df-ne uzwo3lem.1 ({/}) eqeq1i negbii z (ZZ) (br B (<_) (cv z)) rabn0 3bitrr syl6ib mpd (e. (cv v) (ZZ)) (br (cv v) (<) B) ad2antrl jca ex (cv v) R z x y uzwo2OLD ex syld exp3a com12 r19.23adv mpd)) thm (uzwo3lem2 ((x y) (x z) (w x) (v x) (t x) (y z) (w y) (v y) (t y) (w z) (v z) (t z) (v w) (t w) (t v) (A x) (A y) (B t) (S y) (S t) (R w) (v z) (B z) (B v) (x y) (v x) (R x) (v y) (R y) (R v) (y z)) ((uzwo3lem.1 (= R ({e.|} z (ZZ) (br B (<_) (cv z))))) (uzwo3lem.2 (= S (U. ({e.|} w R (A.e. v R (br (cv w) (<_) (cv v)))))))) (-> (/\ (e. B (RR)) (/\ (C_ A R) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (A R ({e.|} t (ZZ) (br S (<_) (cv t))) sstr2 (cv y) (cv t) S (<_) breq2 R rcla4v uzwo3lem.1 x y uzwo3lem1 (cv x) (cv w) (<_) (cv y) breq1 y R ralbidv (cv y) (cv v) (cv w) (<_) breq2 R cbvralv syl6bb (cv x) (U. ({e.|} w R (A.e. v R (br (cv w) (<_) (cv v))))) (<_) (cv y) breq1 y R ralbidv reuuni3 uzwo3lem.2 (<_) (cv y) breq1i y R ralbii sylibr syl syl5com uzwo3lem.1 (cv t) eleq2i (cv z) (cv t) B (<_) breq2 (ZZ) elrab bitr syl5ibr exp3a r19.21aiv t (ZZ) (br B (<_) (cv t)) (br S (<_) (cv t)) ss2rab sylibr uzwo3lem.1 (cv z) (cv t) B (<_) breq2 (ZZ) cbvrabv eqtr syl5ss syl5com uzwo3lem.1 w v uzwo3lem1 w R (A.e. v R (br (cv w) (<_) (cv v))) reucl syl uzwo3lem.2 syl5eqel uzwo3lem.1 (cv z) (cv t) B (<_) breq2 (ZZ) cbvrabv eqtr S eleq2i (cv t) S B (<_) breq2 (ZZ) elrab bitr pm3.26bd S A t x y uzwo2OLD exp32 3syl syld imp32)) thm (uzwo3 ((x y) (u x) (t x) (A x) (u y) (t y) (A y) (t u) (A u) (A t) (w z) (v z) (u z) (t z) (B z) (v w) (u w) (t w) (B w) (u v) (t v) (B v) (B u) (B t)) () (-> (/\ (e. B (RR)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (({e.|} z (ZZ) (br B (<_) (cv z))) eqid (U. ({e.|} w ({e.|} z (ZZ) (br B (<_) (cv z))) (A.e. v ({e.|} z (ZZ) (br B (<_) (cv z))) (br (cv w) (<_) (cv v))))) eqid A u t uzwo3lem2 (cv t) (cv y) (cv u) (<_) breq2 A cbvralv u A reubii (cv u) (cv x) (<_) (cv y) breq1 y A ralbidv A cbvreuv bitr sylib)) thm (zmin ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br A (<_) (cv x)) (A.e. y (ZZ) (-> (br A (<_) (cv y)) (br (cv x) (<_) (cv y))))))) (({e.|} z (ZZ) (br A (<_) (cv z))) eqid x y uzwo3lem1 (cv z) (cv x) A (<_) breq2 (ZZ) elrab (cv z) (cv y) A (<_) breq2 (ZZ) elrab (br (cv x) (<_) (cv y)) imbi1i (e. (cv y) (ZZ)) (br A (<_) (cv y)) (br (cv x) (<_) (cv y)) impexp bitr ralbii2 anbi12i (e. (cv x) (ZZ)) (br A (<_) (cv x)) (A.e. y (ZZ) (-> (br A (<_) (cv y)) (br (cv x) (<_) (cv y)))) anass bitr x eubii x ({e.|} z (ZZ) (br A (<_) (cv z))) (A.e. y ({e.|} z (ZZ) (br A (<_) (cv z))) (br (cv x) (<_) (cv y))) df-reu x (ZZ) (/\ (br A (<_) (cv x)) (A.e. y (ZZ) (-> (br A (<_) (cv y)) (br (cv x) (<_) (cv y))))) df-reu 3bitr4r sylibr)) thm (zmax ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br (cv x) (<_) A) (A.e. y (ZZ) (-> (br (cv y) (<_) A) (br (cv y) (<_) (cv x))))))) (A renegclt (-u A) z w zmin syl (cv x) A lenegt (cv x) zret sylan ancoms (cv w) znegclt (cv y) (-u (cv w)) (<_) A breq1 (cv y) (-u (cv w)) (<_) (cv x) breq1 imbi12d (ZZ) rcla4v syl (cv w) A lenegcon1t (e. (cv x) (ZZ)) adantrr (cv w) (cv x) lenegcon1t (cv x) zret sylan2 (e. A (RR)) adantrl imbi12d (cv w) zret sylan biimpd ex com23 syld com13 r19.21adv (cv y) znegclt (cv w) (-u (cv y)) (-u A) (<_) breq2 (cv w) (-u (cv y)) (-u (cv x)) (<_) breq2 imbi12d (ZZ) rcla4v syl (cv y) A lenegt (e. (cv x) (ZZ)) adantrr (cv y) (cv x) lenegt (cv x) zret sylan2 (e. A (RR)) adantrl imbi12d (cv y) zret sylan biimprd ex com23 syld com13 r19.21adv impbid anbi12d reubidva (cv x) znegclt (cv z) znegclt (cv z) (cv x) negcon2t (cv z) zcnt (cv x) zcnt syl2an reuhyp (cv z) (-u (cv x)) (-u A) (<_) breq2 (cv z) (-u (cv x)) (<_) (cv w) breq1 (br (-u A) (<_) (cv w)) imbi2d w (ZZ) ralbidv anbi12d reuxfr syl6rbbr mpbid)) thm (zbtwnre ((x y) (A x) (A y)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br A (<_) (cv x)) (br (cv x) (<) (opr A (+) (1)))))) (A x y zmin (opr (cv x) (-) (1)) A (cv y) ltletrt (cv x) zret ax1re (cv x) (1) resubclt mpan2 syl syl3an1 3expa (cv y) zret sylan2 (cv x) (cv y) zlem1ltt (e. A (RR)) adantlr sylibrd exp4b com23 r19.21adv (cv x) zret ax1re (cv x) (1) resubclt mpan2 (opr (cv x) (-) (1)) ltnrt 3syl (cv x) peano2zm (cv x) (opr (cv x) (-) (1)) zlem1ltt mpdan mtbird (e. A (RR)) (A.e. y (ZZ) (-> (br A (<_) (cv y)) (br (cv x) (<_) (cv y)))) ad2antrr A (opr (cv x) (-) (1)) lenltt (cv x) zret ax1re (cv x) (1) resubclt mpan2 syl sylan2 ancoms (A.e. y (ZZ) (-> (br A (<_) (cv y)) (br (cv x) (<_) (cv y)))) adantr (cv x) peano2zm (cv y) (opr (cv x) (-) (1)) A (<_) breq2 (cv y) (opr (cv x) (-) (1)) (cv x) (<_) breq2 imbi12d (ZZ) rcla4v syl imp (e. A (RR)) adantlr sylbird mt3d ex impbid ax1re (cv x) (1) A ltsubaddt mp3an2 (cv x) zret sylan bitr3d ancoms (br A (<_) (cv x)) anbi2d reubidva mpbid)) thm (rebtwnz ((x y) (A x) (A y)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br (cv x) (<_) A) (br A (<) (opr (cv x) (+) (1)))))) (A renegclt (-u A) y zbtwnre syl (cv x) A lenegt ancoms (opr A (-) (1)) (cv x) ltnegt ax1re A (1) resubclt mpan2 sylan ax1re A (1) (cv x) ltsubaddt mp3an2 A recnt 1cn A (1) negsubdit mpan2 syl (e. (cv x) (RR)) adantr (-u (cv x)) (<) breq2d 3bitr3d anbi12d (cv x) zret sylan2 bicomd reubidva (cv x) znegclt (cv y) znegclt (cv y) (cv x) negcon2t (cv y) zcnt (cv x) zcnt syl2an reuhyp (cv y) (-u (cv x)) (-u A) (<_) breq2 (cv y) (-u (cv x)) (<) (opr (-u A) (+) (1)) breq1 anbi12d reuxfr syl5bb mpbid)) thm (flvalt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (RR)) (= (` (floor) A) (U. ({e.|} x (ZZ) (/\ (br (cv x) (<_) A) (br A (<) (opr (cv x) (+) (1)))))))) ((cv y) A (cv x) (<_) breq2 (cv y) A (<) (opr (cv x) (+) (1)) breq1 anbi12d x (ZZ) rabbisdv unieqd y z x df-fl zex x (/\ (br (cv x) (<_) A) (br A (<) (opr (cv x) (+) (1)))) rabex uniex fvopab4)) thm (flclt ((A z)) () (-> (e. A (RR)) (e. (` (floor) A) (ZZ))) (A z flvalt A z rebtwnz z (ZZ) (/\ (br (cv z) (<_) A) (br A (<) (opr (cv z) (+) (1)))) reucl syl eqeltrd)) thm (flleltt ((x y) (A x) (A y)) () (-> (e. A (RR)) (/\ (br (` (floor) A) (<_) A) (br A (<) (opr (` (floor) A) (+) (1))))) (A x rebtwnz (cv x) (cv y) (<_) A breq1 (cv x) (cv y) (+) (1) opreq1 A (<) breq2d anbi12d (cv x) (U. ({e.|} y (ZZ) (/\ (br (cv y) (<_) A) (br A (<) (opr (cv y) (+) (1)))))) (<_) A breq1 (cv x) (U. ({e.|} y (ZZ) (/\ (br (cv y) (<_) A) (br A (<) (opr (cv y) (+) (1)))))) (+) (1) opreq1 A (<) breq2d anbi12d reuuni3 syl A y flvalt (<_) A breq1d A y flvalt (+) (1) opreq1d A (<) breq2d anbi12d mpbird)) thm (fllet () () (-> (e. A (RR)) (br (` (floor) A) (<_) A)) (A flleltt pm3.26d)) thm (flltp1t () () (-> (e. A (RR)) (br A (<) (opr (` (floor) A) (+) (1)))) (A flleltt pm3.27d)) thm (flget () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (br B (<_) A) (br B (<_) (` (floor) A)))) (A flltp1t (e. B (ZZ)) adantl B A (opr (` (floor) A) (+) (1)) lelttrt 3expb B zret A flclt (` (floor) A) zret (` (floor) A) peano2re 3syl ancli syl2an mpan2d B (` (floor) A) zleltp1t A flclt sylan2 sylibrd A fllet (e. B (ZZ)) adantl B (` (floor) A) A letrt 3expb B zret A flclt (` (floor) A) zret syl ancri syl2an mpan2d impbid ancoms)) thm (flltt () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (br A (<) B) (br (` (floor) A) (<) B))) (A B flget B A lenltt B zret sylan ancoms B (` (floor) A) lenltt B zret A flclt (` (floor) A) zret syl syl2an ancoms 3bitr3d con4bid)) thm (flidt () () (-> (e. A (ZZ)) (= (` (floor) A) A)) (A zret A fllet syl A zret A leidt syl A zret A A flget mpancom mpbid jca (` (floor) A) A letri3t A zret A flclt (` (floor) A) zret 3syl A zret sylanc mpbird)) thm (flval2t ((x y) (A x) (A y)) () (-> (e. A (RR)) (= (` (floor) A) (U. ({e.|} x (ZZ) (/\ (br (cv x) (<_) A) (A.e. y (ZZ) (-> (br (cv y) (<_) A) (br (cv y) (<_) (cv x))))))))) (A fllet A (cv y) flget biimpd r19.21aiva jca (cv x) (` (floor) A) (<_) A breq1 (cv x) (` (floor) A) (cv y) (<_) breq2 (br (cv y) (<_) A) imbi2d y (ZZ) ralbidv anbi12d (ZZ) reuuni2 A flclt A x y zmax sylanc mpbid eqcomd)) thm (flval3t ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (RR)) (= (` (floor) A) (sup ({e.|} x (ZZ) (br (cv x) (<_) A)) (RR) (<)))) (({e.|} x (ZZ) (br (cv x) (<_) A)) y z (` (floor) A) suprub x (ZZ) (br (cv x) (<_) A) ssrab2 zssre sstri (e. A (RR)) a1i A flclt A fllet jca (cv x) (` (floor) A) (<_) A breq1 (ZZ) elrab sylibr (` (floor) A) ({e.|} x (ZZ) (br (cv x) (<_) A)) n0i syl (cv y) (` (floor) A) (cv z) (<_) breq2 z ({e.|} x (ZZ) (br (cv x) (<_) A)) ralbidv (RR) rcla4ev A flclt (` (floor) A) zret syl A (cv z) flget biimpd ex imp3a (cv x) (cv z) (<_) A breq1 (ZZ) elrab syl5ib r19.21aiv sylanc 3jca A flclt A fllet jca (cv x) (` (floor) A) (<_) A breq1 (ZZ) elrab sylibr sylanc ({e.|} x (ZZ) (br (cv x) (<_) A)) y z (` (floor) A) w suprnub x (ZZ) (br (cv x) (<_) A) ssrab2 zssre sstri (e. A (RR)) a1i A flclt A fllet jca (cv x) (` (floor) A) (<_) A breq1 (ZZ) elrab sylibr (` (floor) A) ({e.|} x (ZZ) (br (cv x) (<_) A)) n0i syl (cv y) (` (floor) A) (cv z) (<_) breq2 z ({e.|} x (ZZ) (br (cv x) (<_) A)) ralbidv (RR) rcla4ev A flclt (` (floor) A) zret syl A (cv z) flget biimpd ex imp3a (cv x) (cv z) (<_) A breq1 (ZZ) elrab syl5ib r19.21aiv sylanc 3jca A flclt (` (floor) A) zret syl A (cv w) flget (cv w) (` (floor) A) lenltt (cv w) zret A flclt (` (floor) A) zret syl syl2an ancoms bitrd biimpd ex imp3a (cv x) (cv w) (<_) A breq1 (ZZ) elrab syl5ib r19.21aiv jca sylanc jca (` (floor) A) (sup ({e.|} x (ZZ) (br (cv x) (<_) A)) (RR) (<)) eqleltt A flclt (` (floor) A) zret syl ({e.|} x (ZZ) (br (cv x) (<_) A)) y z suprcl x (ZZ) (br (cv x) (<_) A) ssrab2 zssre sstri (e. A (RR)) a1i A flclt A fllet jca (cv x) (` (floor) A) (<_) A breq1 (ZZ) elrab sylibr (` (floor) A) ({e.|} x (ZZ) (br (cv x) (<_) A)) n0i syl (cv y) (` (floor) A) (cv z) (<_) breq2 z ({e.|} x (ZZ) (br (cv x) (<_) A)) ralbidv (RR) rcla4ev A flclt (` (floor) A) zret syl A (cv z) flget biimpd ex imp3a (cv x) (cv z) (<_) A breq1 (ZZ) elrab syl5ib r19.21aiv sylanc syl3anc sylanc mpbird)) thm (flwordit () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) (br (` (floor) A) (<_) (` (floor) B))) (A fllet (e. B (RR)) adantr (` (floor) A) A B letrt A flclt (` (floor) A) zret syl (e. B (RR)) adantr (e. A (RR)) (e. B (RR)) pm3.26 (e. A (RR)) (e. B (RR)) pm3.27 syl3anc mpand B (` (floor) A) flget A flclt sylan2 ancoms sylibd 3impia)) thm (flbit ((A x) (B x)) () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (= (` (floor) A) B) (/\ (br B (<_) A) (br A (<) (opr B (+) (1)))))) (A x flvalt B eqeq1d (e. B (ZZ)) adantr (cv x) B (<_) A breq1 (cv x) B (+) (1) opreq1 A (<) breq2d anbi12d (ZZ) reuuni2 A x rebtwnz sylan2 ancoms bitr4d)) thm (flge0nn0t () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (e. (` (floor) A) (NN0))) (A flclt (br (0) (<_) A) adantr 0z A (0) flget mpan2 biimpa jca (` (floor) A) elnn0z sylibr)) thm (flge1nnt () () (-> (/\ (e. A (RR)) (br (1) (<_) A)) (e. (` (floor) A) (NN))) (A flclt (br (1) (<_) A) adantr 1z A (1) flget mpan2 biimpa jca (` (floor) A) elnnz1 sylibr)) thm (fladdzt () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (= (` (floor) (opr A (+) B)) (opr (` (floor) A) (+) B))) (A fllet (e. B (ZZ)) adantr (` (floor) A) A B leadd1t 3expa A flclt (` (floor) A) zret syl ancri B zret syl2an mpbid A flltp1t (e. B (ZZ)) adantr A (opr (` (floor) A) (+) (1)) B ltadd1t 3expa A flclt (` (floor) A) peano2z (opr (` (floor) A) (+) (1)) zret 3syl ancli B zret syl2an mpbid 1cn (` (floor) A) (1) B add23t mp3an2 A flclt (` (floor) A) zcnt syl B zcnt syl2an breqtrd jca (opr A (+) B) (opr (` (floor) A) (+) B) flbit A B axaddrcl B zret sylan2 (` (floor) A) B zaddclt A flclt sylan sylanc mpbird)) thm (btwnzge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (ZZ))) (/\ (br B (<_) A) (br A (<) (opr B (+) (1))))) (<-> (br (0) (<_) A) (br (0) (<_) B))) (0z A (0) flget mpan2 (e. B (ZZ)) (/\ (br B (<_) A) (br A (<) (opr B (+) (1)))) ad2antrr A B flbit biimpar (0) (<_) breq2d bitrd)) thm (flhalft () () (-> (e. N (ZZ)) (br N (<_) (opr (2) (x.) (` (floor) (opr (opr N (+) (1)) (/) (2)))))) (N zeot (opr N (/) (2)) flidt (2) (x.) opreq2d N zcnt 2cn 2re 2pos gt0ne0i (2) N divcan2t mp3an13 syl sylan9eqr (` (floor) (opr N (/) (2))) (` (floor) (opr (opr N (+) (1)) (/) (2))) (2) lemul2it N zret N rehalfclt syl (opr N (/) (2)) flclt (` (floor) (opr N (/) (2))) zret 3syl N zret N peano2re (opr N (+) (1)) rehalfclt 3syl (opr (opr N (+) (1)) (/) (2)) flclt (` (floor) (opr (opr N (+) (1)) (/) (2))) zret 3syl 2re (e. N (ZZ)) a1i 3jca (opr N (/) (2)) (opr (opr N (+) (1)) (/) (2)) flwordit N zret N rehalfclt syl N zret N peano2re (opr N (+) (1)) rehalfclt 3syl N lep1t (e. (opr N (+) (1)) (RR)) adantr 2re 2pos N (opr N (+) (1)) (2) lediv1t mpan2 mp3an3 mpbid N zret N zret N peano2re syl sylanc syl3anc 0re 2re 2pos ltlei jctil sylanc (e. (opr N (/) (2)) (ZZ)) adantr eqbrtrrd N zret N lep1t syl (e. (opr (opr N (+) (1)) (/) (2)) (ZZ)) adantr (opr (opr N (+) (1)) (/) (2)) flidt (2) (x.) opreq2d N zcnt N peano2cn 2cn 2re 2pos gt0ne0i (2) (opr N (+) (1)) divcan2t mp3an13 3syl sylan9eqr breqtrrd jaodan mpdan)) thm (elq ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (e. A (QQ)) (E.e. x (ZZ) (E.e. y (NN) (= A (opr (cv x) (/) (cv y)))))) (z x y df-q A eleq2i (cv x) (/) (cv y) oprex A (opr (cv x) (/) (cv y)) (V) eleq1 mpbiri (e. (cv y) (NN)) a1i r19.23aiv (e. (cv x) (ZZ)) a1i r19.23aiv (cv z) A (opr (cv x) (/) (cv y)) eqeq1 x (ZZ) y (NN) 2rexbidv elab3 bitr)) thm (znq ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (ZZ)) (e. B (NN))) (e. (opr A (/) B) (QQ))) ((opr A (/) B) eqid (cv x) A (/) (cv y) opreq1 (opr A (/) B) eqeq2d (cv y) B A (/) opreq2 (opr A (/) B) eqeq2d (ZZ) (NN) rcla42ev mpan2 (opr A (/) B) x y elq sylibr)) thm (qret ((x y) (A x) (A y)) () (-> (e. A (QQ)) (e. A (RR))) (A x y elq A (opr (cv x) (/) (cv y)) (RR) eleq1 (cv x) (cv y) redivclt 3expb (cv x) zret (cv y) nnret (cv y) nnne0t jca syl2an syl5bir com12 r19.23aivv sylbi)) thm (zqt ((x y) (A x) (A y)) () (-> (e. A (ZZ)) (e. A (QQ))) ((cv x) zret (cv x) recnt (cv x) div1t 3syl A eqeq2d (cv x) A eqcom syl6rbbr 1nn (cv y) (1) (cv x) (/) opreq2 A eqeq2d (NN) rcla4ev mpan syl6bi r19.22i A (ZZ) x risset A x y elq 3imtr4)) thm (zssq () () (C_ (ZZ) (QQ)) ((cv x) zqt ssriv)) thm (nn0ssq () () (C_ (NN0) (QQ)) (nn0ssz zssq sstri)) thm (nnssq () () (C_ (NN) (QQ)) (nnssz zssq sstri)) thm (qssre () () (C_ (QQ) (RR)) ((cv x) qret ssriv)) thm (qsscn () () (C_ (QQ) (CC)) (qssre axresscn sstri)) thm (nnqt () () (-> (e. A (NN)) (e. A (QQ))) (nnssq A sseli)) thm (qcnt () () (-> (e. A (QQ)) (e. A (CC))) (qsscn A sseli)) thm (qex () () (e. (QQ) (V)) (reex qssre ssexi)) thm (qaddclt ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (+) B) (QQ))) ((cv u) (opr (opr (cv x) (x.) (cv w)) (+) (opr (cv y) (x.) (cv z))) (/) (cv v) opreq1 (opr A (+) B) eqeq2d (cv v) (opr (cv y) (x.) (cv w)) (opr (opr (cv x) (x.) (cv w)) (+) (opr (cv y) (x.) (cv z))) (/) opreq2 (opr A (+) B) eqeq2d (ZZ) (NN) rcla42ev (opr A (+) B) u v elq sylibr (opr (cv x) (x.) (cv w)) (opr (cv y) (x.) (cv z)) zaddclt (cv x) (cv w) zmulclt (cv w) nnzt sylan2 (e. (cv z) (ZZ)) adantrl (e. (cv y) (NN)) adantlr (cv y) (cv z) zmulclt (cv y) nnzt sylan (e. (cv w) (NN)) adantrr (e. (cv x) (ZZ)) adantll sylanc (cv y) (cv w) nnmulclt (e. (cv x) (ZZ)) (e. (cv z) (ZZ)) ad2ant2l jca (/\ (= A (opr (cv x) (/) (cv y))) (= B (opr (cv z) (/) (cv w)))) adantr A (opr (cv x) (/) (cv y)) B (opr (cv z) (/) (cv w)) (+) opreq12 (cv x) (cv y) (cv z) (cv w) divadddivt (cv x) zcnt (cv y) nncnt anim12i (cv z) zcnt (cv w) nncnt anim12i anim12i (cv y) nnne0t (e. (cv x) (ZZ)) adantl (cv w) nnne0t (e. (cv z) (ZZ)) adantl anim12i sylanc sylan9eqr sylanc an4s exp43 r19.23aivv r19.23advv imp A x y elq B z w elq syl2anb)) thm (qnegclt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (QQ)) (e. (-u A) (QQ))) (A x y elq (cv x) (cv y) divnegt 3expb (cv x) zcnt (cv y) nncnt (cv y) nnne0t jca syl2an (-u A) eqeq2d A (opr (cv x) (/) (cv y)) negeq syl5bi (cv x) znegclt (e. (cv y) (NN)) anim1i jctild (cv z) (-u (cv x)) (/) (cv w) opreq1 (-u A) eqeq2d (cv w) (cv y) (-u (cv x)) (/) opreq2 (-u A) eqeq2d (ZZ) (NN) rcla42ev (-u A) z w elq sylibr syl6 r19.23aivv sylbi)) thm (qmulclt ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (x.) B) (QQ))) ((cv v) (opr (cv x) (x.) (cv z)) (/) (cv u) opreq1 (opr A (x.) B) eqeq2d (cv u) (opr (cv y) (x.) (cv w)) (opr (cv x) (x.) (cv z)) (/) opreq2 (opr A (x.) B) eqeq2d (ZZ) (NN) rcla42ev (opr A (x.) B) v u elq sylibr (cv x) (cv z) zmulclt (cv y) (cv w) nnmulclt anim12i an4s (/\ (= A (opr (cv x) (/) (cv y))) (= B (opr (cv z) (/) (cv w)))) adantr A (opr (cv x) (/) (cv y)) B (opr (cv z) (/) (cv w)) (x.) opreq12 (cv x) (cv y) (cv z) (cv w) divmuldivt (cv x) zcnt (cv y) nncnt anim12i (cv z) zcnt (cv w) nncnt anim12i anim12i (cv y) nnne0t (e. (cv x) (ZZ)) adantl (cv w) nnne0t (e. (cv z) (ZZ)) adantl anim12i sylanc sylan9eqr sylanc an4s exp43 r19.23aivv r19.23advv imp A x y elq B z w elq syl2anb)) thm (qsubclt () () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (-) B) (QQ))) (A B negsubt A qcnt B qcnt syl2an A (-u B) qaddclt B qnegclt sylan2 eqeltrrd)) thm (qrecclt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (/\ (e. A (QQ)) (=/= A (0))) (e. (opr (1) (/) A) (QQ))) (A x y elq A (opr (cv x) (/) (cv y)) (0) neeq1 (cv x) (cv y) divne0bt 3expa (cv x) zcnt (cv y) nncnt anim12i sylan bicomd sylan9bbr (cv x) (cv y) zmulclt (cv y) nnzt sylan2 (=/= (cv x) (0)) adantr (cv x) msqznn (e. (cv y) (NN)) adantlr jca (=/= (cv y) (0)) adantlr (= A (opr (cv x) (/) (cv y))) adantlr A (opr (cv x) (/) (cv y)) (1) (/) opreq2 (cv x) dividt (e. (cv y) (CC)) (=/= (cv y) (0)) ad2ant2r (/) (opr (cv x) (/) (cv y)) opreq1d (cv x) (cv x) (cv x) (cv y) divdivdivt (e. (cv x) (CC)) (e. (cv y) (CC)) pm3.26 (e. (cv x) (CC)) (e. (cv y) (CC)) pm3.26 jca ancri (=/= (cv x) (0)) (=/= (cv y) (0)) pm3.26 (=/= (cv x) (0)) (=/= (cv y) (0)) pm3.26 (=/= (cv x) (0)) (=/= (cv y) (0)) pm3.27 3jca syl2an eqtr3d (cv x) zcnt (cv y) nncnt anim12i sylan anassrs an1rs sylan9eqr an1rs jca ex sylbid ex anasss (cv y) nnne0t ancli sylan2 (cv z) (opr (cv x) (x.) (cv y)) (/) (cv w) opreq1 (opr (1) (/) A) eqeq2d (cv w) (opr (cv x) (x.) (cv x)) (opr (cv x) (x.) (cv y)) (/) opreq2 (opr (1) (/) A) eqeq2d (ZZ) (NN) rcla42ev (opr (1) (/) A) z w elq sylibr syl8 r19.23aivv sylbi imp)) thm (qdivclt () () (-> (/\ (/\ (e. A (QQ)) (e. B (QQ))) (=/= B (0))) (e. (opr A (/) B) (QQ))) (A B divrect 3expa A qcnt B qcnt anim12i sylan A (opr (1) (/) B) qmulclt B qrecclt sylan2 anassrs eqeltrd)) thm (qrevaddclt () () (-> (e. B (QQ)) (<-> (/\ (e. A (CC)) (e. (opr A (+) B) (QQ))) (e. A (QQ)))) (A B pncant B qcnt sylan2 ancoms (e. (opr A (+) B) (QQ)) adantr (opr A (+) B) B qsubclt ancoms (e. A (CC)) adantlr eqeltrrd ex A B qaddclt expcom (e. A (CC)) adantr impbid ex pm5.32d A qcnt pm4.71ri syl6bbr)) thm (nnrecqt () () (-> (e. A (NN)) (e. (opr (1) (/) A) (QQ))) (1z (1) A znq mpan)) thm (qbtwnre ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (br A (<) B)) (E.e. x (QQ) (/\ (br A (<) (cv x)) (br (cv x) (<) B)))) (A B posdift (opr B (-) A) y nnreclt B A resubclt sylan ex ancoms sylbid (cv y) B axmulrcl (cv y) nnret sylan ax1re (opr (cv y) (x.) B) (1) resubclt mpan2 syl (e. A (RR)) adantrl (opr (opr (cv y) (x.) B) (-) (1)) z zbtwnre z (ZZ) (/\ (br (opr (opr (cv y) (x.) B) (-) (1)) (<_) (cv z)) (br (cv z) (<) (opr (opr (opr (cv y) (x.) B) (-) (1)) (+) (1)))) reurex 3syl (cv y) B A subdit (cv y) nncnt B recnt A recnt syl3an 3com23 3expb (1) (<) breq2d ax1re (1) (cv y) (opr B (-) A) ltdivmult mp3anl1 exp31 com23 (cv y) nnret (cv y) nngt0t sylc B A resubclt ancoms syl5 imp ax1re (opr (cv y) (x.) A) (opr (cv y) (x.) B) (1) ltsub13t mp3an3 (cv y) A axmulrcl (cv y) B axmulrcl syl2an anandis (cv y) nnret sylan 3bitr4d (e. (cv z) (RR)) adantr (br (opr (opr (cv y) (x.) B) (-) (1)) (<_) (cv z)) anbi1d (opr (cv y) (x.) A) (opr (opr (cv y) (x.) B) (-) (1)) (cv z) ltletrt 3expa (cv y) A axmulrcl (cv y) nnret sylan (e. B (RR)) adantrr (cv y) B axmulrcl (cv y) nnret sylan ax1re (opr (cv y) (x.) B) (1) resubclt mpan2 syl (e. A (RR)) adantrl jca sylan sylbid (cv y) nngt0t (cv y) nnret (cv y) A (cv z) ltmuldiv2t 3exp1 syl com4r mpcom (e. B (RR)) adantrd imp31 sylibd (cv y) B axmulcl (cv y) nncnt B recnt syl2an 1cn (opr (cv y) (x.) B) (1) npcant mpan2 syl (e. (cv z) (RR)) adantr (cv z) (<) breq2d (cv y) nngt0t (e. (cv z) (RR)) (e. B (RR)) 3ad2ant2 (cv z) (cv y) B ltdivmult (cv y) nnret syl3anl2 mpdan 3coml 3expa bitr4d biimpd (e. A (RR)) adantlrl anim12d (cv z) zret sylan2 (cv z) (cv y) znq ancoms (/\ (e. A (RR)) (e. B (RR))) adantlr jctild (cv x) (opr (cv z) (/) (cv y)) A (<) breq2 (cv x) (opr (cv z) (/) (cv y)) (<) B breq1 anbi12d (QQ) rcla4ev syl6 (br (opr (1) (/) (cv y)) (<) (opr B (-) A)) (br (opr (opr (cv y) (x.) B) (-) (1)) (<_) (cv z)) (br (cv z) (<) (opr (opr (opr (cv y) (x.) B) (-) (1)) (+) (1))) anass syl5ibr exp4b com34 r19.23adv mpd expcom r19.23adv syld imp)) thm (monoord ((y z) (A y) (A z) (B y) (x y) (x z) (F x) (F y) (F z)) ((monoord.1 (:--> F (NN) (RR))) (monoord.2 (-> (e. (cv x) (NN)) (br (` F (cv x)) (<_) (` F (opr (cv x) (+) (1))))))) (-> (/\/\ (e. A (NN)) (e. B (NN)) (br A (<_) B)) (br (` F A) (<_) (` F B))) ((cv y) (1) A (<_) breq2 (cv y) (1) F fveq2 (` F A) (<_) breq2d imbi12d (e. A (NN)) imbi2d (cv y) (cv z) A (<_) breq2 (cv y) (cv z) F fveq2 (` F A) (<_) breq2d imbi12d (e. A (NN)) imbi2d (cv y) (opr (cv z) (+) (1)) A (<_) breq2 (cv y) (opr (cv z) (+) (1)) F fveq2 (` F A) (<_) breq2d imbi12d (e. A (NN)) imbi2d (cv y) B A (<_) breq2 (cv y) B F fveq2 (` F A) (<_) breq2d imbi12d (e. A (NN)) imbi2d A nnge1t A nnret ax1re A (1) letri3t mpan2 syl (` F A) (` F (1)) eqlet monoord.1 A ffvrni A (1) F fveq2 syl2an ex sylbird mpan2d A (opr (cv z) (+) (1)) leloet A nnret (cv z) peano2nn (opr (cv z) (+) (1)) nnret syl syl2an (-> (br A (<_) (cv z)) (br (` F A) (<_) (` F (cv z)))) adantr A (cv z) nnleltp1t (-> (br A (<_) (cv z)) (br (` F A) (<_) (` F (cv z)))) adantr monoord.1 A ffvrni (e. (cv z) (NN)) (br (` F A) (<_) (` F (cv z))) ad2antrr monoord.1 (cv z) ffvrni (e. A (NN)) (br (` F A) (<_) (` F (cv z))) ad2antlr (cv z) peano2nn monoord.1 (opr (cv z) (+) (1)) ffvrni syl (e. A (NN)) (br (` F A) (<_) (` F (cv z))) ad2antlr (/\ (e. A (NN)) (e. (cv z) (NN))) (br (` F A) (<_) (` F (cv z))) pm3.27 (cv x) (cv z) F fveq2 (cv x) (cv z) (+) (1) opreq1 F fveq2d (<_) breq12d monoord.2 vtoclga (e. A (NN)) (br (` F A) (<_) (` F (cv z))) ad2antlr letrd ex (br A (<_) (cv z)) imim2d imp sylbird (` F A) (` F (opr (cv z) (+) (1))) eqlet monoord.1 A ffvrni A (opr (cv z) (+) (1)) F fveq2 syl2an ex (e. (cv z) (NN)) (-> (br A (<_) (cv z)) (br (` F A) (<_) (` F (cv z)))) ad2antrr jaod sylbid exp31 com12 a2d nnind com12 3imp)) thm (om2uz0 ((x y) (C x) (C y)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (= (` G ({/})) C) (om2uz.2 ({/}) fveq1i om2uz.1 C (ZZ) ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) fr0t ax-mp eqtr)) thm (om2uzsuc ((x y) (w x) (v x) (w y) (v y) (v w) (G w) (G v) (A w) (A v) (C x) (C y) (C w) (C v)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. A (om)) (= (` G (suc A)) (opr (` G A) (+) (1)))) ((cv w) A suceq G fveq2d (cv w) A G fveq2 (+) (1) opreq1d eqeq12d (` G (cv w)) (+) (1) oprex (e. (cv v) C) x ax-17 (e. (cv v) (cv w)) x ax-17 v x y (= (cv y) (opr (cv x) (+) (1))) hbopab1 (e. (cv v) C) x ax-17 hbrdg (e. (cv v) (om)) x ax-17 hbres (e. (cv v) (cv w)) x ax-17 hbfv (e. (cv v) (+)) x ax-17 (e. (cv v) (1)) x ax-17 hbopr om2uz.2 (cv w) fveq1i (+) (1) opreq1i (cv v) eleq2i om2uz.2 (cv w) fveq1i (+) (1) opreq1i (cv v) eleq2i x albii 3imtr4 om2uz.2 (cv x) (` G (cv w)) (+) (1) opreq1 (V) frsucopab mpan2 vtoclga)) thm (om2uzuz ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (A z) (A w) (A v) (C x) (C y) (C z) (C w) (C v)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. A (om)) (e. (` G A) ({e.|} z (ZZ) (br C (<_) (cv z))))) ((cv v) ({/}) G fveq2 ({e.|} z (ZZ) (br C (<_) (cv z))) eleq1d (cv v) (cv w) G fveq2 ({e.|} z (ZZ) (br C (<_) (cv z))) eleq1d (cv v) (suc (cv w)) G fveq2 ({e.|} z (ZZ) (br C (<_) (cv z))) eleq1d (cv v) A G fveq2 ({e.|} z (ZZ) (br C (<_) (cv z))) eleq1d om2uz.1 om2uz.2 om2uz0 (cv z) C C (<_) breq2 (ZZ) elrab om2uz.1 om2uz.1 zre leid mpbir2an eqeltr om2uz.1 om2uz.2 (cv w) om2uzsuc ({e.|} z (ZZ) (br C (<_) (cv z))) eleq1d om2uz.1 C (` G (cv w)) z peano2uz2 mpan syl5bir finds)) thm (om2uzlt ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (A z) (A w) (A v) (B z) (B w) (B v) (C x) (C y) (C z) (C w) (C v)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (/\ (e. A (om)) (e. B (om))) (-> (e. A B) (br (` G A) (<) (` G B)))) ((cv v) ({/}) A eleq2 (cv v) ({/}) G fveq2 (` G A) (<) breq2d imbi12d (e. A (om)) imbi2d (cv v) (cv w) A eleq2 (cv v) (cv w) G fveq2 (` G A) (<) breq2d imbi12d (e. A (om)) imbi2d (cv v) (suc (cv w)) A eleq2 (cv v) (suc (cv w)) G fveq2 (` G A) (<) breq2d imbi12d (e. A (om)) imbi2d (cv v) B A eleq2 (cv v) B G fveq2 (` G A) (<) breq2d imbi12d (e. A (om)) imbi2d A noel (br (` G A) (<) (` G ({/}))) pm2.21i (e. A (om)) a1i (cv w) (om) A elsuc2g bicomd (e. A (om)) adantl om2uz.1 om2uz.2 (cv w) om2uzsuc (` G A) (<) breq2d (e. A (om)) adantl (` G A) (` G (cv w)) zleltp1t z (ZZ) (br C (<_) (cv z)) ssrab2 (` G A) sseli z (ZZ) (br C (<_) (cv z)) ssrab2 (` G (cv w)) sseli syl2an om2uz.1 om2uz.2 A z om2uzuz om2uz.1 om2uz.2 (cv w) z om2uzuz syl2an (` G A) (` G (cv w)) leloet om2uz.1 om2uz.2 A z om2uzuz z (ZZ) (br C (<_) (cv z)) ssrab2 (` G A) sseli (` G A) zret 3syl om2uz.1 om2uz.2 (cv w) z om2uzuz z (ZZ) (br C (<_) (cv z)) ssrab2 (` G (cv w)) sseli (` G (cv w)) zret 3syl syl2an 3bitr2rd imbi12d (-> (e. A (cv w)) (br (` G A) (<) (` G (cv w)))) id A (cv w) G fveq2 (-> (e. A (cv w)) (br (` G A) (<) (` G (cv w)))) a1i orim12d syl5bi expcom a2d finds impcom)) thm (om2uzlt2 ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (/\ (e. A (om)) (e. B (om))) (<-> (e. A B) (br (` G A) (<) (` G B)))) (om2uz.1 om2uz.2 A B om2uzlt om2uz.1 om2uz.2 B A om2uzlt B A G fveq2 (/\ (e. B (om)) (e. A (om))) a1i orim12d ancoms B A onsseleq B A ontri1 bitr3d B nnont A nnont syl2an ancoms (` G B) (` G A) leloet (` G B) (` G A) lenltt bitr3d om2uz.1 om2uz.2 B z om2uzuz z (ZZ) (br C (<_) (cv z)) ssrab2 zssre sstri (` G B) sseli syl om2uz.1 om2uz.2 A z om2uzuz z (ZZ) (br C (<_) (cv z)) ssrab2 zssre sstri (` G A) sseli syl syl2an ancoms 3imtr3d a3d impbid)) thm (om2uzran ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G z) (G w) (G v) (G u) (C x) (C y) (C z) (C w) (C v) (C u)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (= (ran G) ({e.|} z (ZZ) (br C (<_) (cv z)))) (({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C frfnom om2uz.2 G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)) (om) fneq1 ax-mp mpbir G (om) (cv u) w fvelrn ax-mp (` G (cv w)) (cv u) ({e.|} z (ZZ) (br C (<_) (cv z))) eleq1 om2uz.1 om2uz.2 (cv w) z om2uzuz syl5bi com12 r19.23aiv sylbi (cv z) (cv u) C (<_) breq2 (ZZ) elrab om2uz.1 (cv w) C (ran G) eleq1 (cv w) (cv v) (ran G) eleq1 (cv w) (opr (cv v) (+) (1)) (ran G) eleq1 (cv w) (cv u) (ran G) eleq1 om2uz.1 om2uz.2 om2uz0 ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C frfnom om2uz.2 G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)) (om) fneq1 ax-mp mpbir peano1 G (om) ({/}) fnfvrn mp2an eqeltrr (e. C (ZZ)) a1i ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C frfnom om2uz.2 G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)) (om) fneq1 ax-mp mpbir G (om) (cv v) w fvelrn ax-mp om2uz.1 om2uz.2 (cv w) om2uzsuc (` G (cv w)) (cv v) (+) (1) opreq1 sylan9eq (cv w) peano2 ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C frfnom om2uz.2 G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)) (om) fneq1 ax-mp mpbir G (om) (suc (cv w)) fnfvrn mpan syl (= (` G (cv w)) (cv v)) adantr eqeltrrd ex r19.23aiv sylbi (/\/\ (e. C (ZZ)) (e. (cv v) (ZZ)) (br C (<_) (cv v))) a1i uzind mp3an1 sylbi impbi eqriv)) thm (om2uzf1o ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (C x) (C y) (C z) (C w) (C v)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (:-1-1-onto-> G (om) ({e.|} z (ZZ) (br C (<_) (cv z)))) (G (om) ({e.|} z (ZZ) (br C (<_) (cv z))) f1o5 G (om) ({e.|} z (ZZ) (br C (<_) (cv z))) w v f1fv G (om) ({e.|} z (ZZ) (br C (<_) (cv z))) df-f ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C frfnom om2uz.2 G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)) (om) fneq1 ax-mp mpbir om2uz.1 om2uz.2 z om2uzran (ran G) ({e.|} z (ZZ) (br C (<_) (cv z))) eqimss ax-mp mpbir2an (` G (cv w)) (` G (cv v)) lttri3t om2uz.1 om2uz.2 (cv w) z om2uzuz z (ZZ) (br C (<_) (cv z)) ssrab2 (` G (cv w)) sseli (` G (cv w)) zret 3syl om2uz.1 om2uz.2 (cv v) z om2uzuz z (ZZ) (br C (<_) (cv z)) ssrab2 (` G (cv v)) sseli (` G (cv v)) zret 3syl syl2an (br (` G (cv w)) (<) (` G (cv v))) (br (` G (cv v)) (<) (` G (cv w))) ioran syl6bbr (cv w) (cv v) ordtri3 (cv w) nnord (cv v) nnord syl2an con2bid om2uz.1 om2uz.2 (cv w) (cv v) om2uzlt om2uz.1 om2uz.2 (cv v) (cv w) om2uzlt ancoms orim12d sylbird con1d sylbid rgen2 mpbir2an om2uz.1 om2uz.2 z om2uzran mpbir2an)) thm (uzrdgval ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. B ({e.|} z (ZZ) (br C (<_) (cv z)))) (= (` (o. (rec F A) (`' G)) B) (` (rec F A) (` (`' G) B)))) (om2uz.1 om2uz.2 z om2uzran G df-rn eqtr3 B eleq2i F A rdgfnon (rec F A) (On) fnfun ax-mp om2uz.1 om2uz.2 z om2uzf1o G (om) ({e.|} z (ZZ) (br C (<_) (cv z))) f1ocnv ax-mp (`' G) ({e.|} z (ZZ) (br C (<_) (cv z))) (om) f1ofun ax-mp (rec F A) (`' G) B fvco mp3an12 sylbi)) thm (uzrdgini ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. A B) (= (` (o. (rec F A) (`' G)) C) A)) (A B F rdg0t (cv z) C C (<_) breq2 (ZZ) elrab om2uz.1 om2uz.1 zre leid mpbir2an om2uz.1 om2uz.2 C z F A uzrdgval ax-mp om2uz.1 om2uz.2 om2uz0 om2uz.1 om2uz.2 z om2uzf1o peano1 G (om) ({e.|} z (ZZ) (br C (<_) (cv z))) ({/}) C f1ocnvfv mp2an ax-mp (rec F A) fveq2i eqtr syl5eq)) thm (uzrdgsuc ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. B ({e.|} z (ZZ) (br C (<_) (cv z)))) (= (` (o. (rec F A) (`' G)) (opr B (+) (1))) (` F (` (o. (rec F A) (`' G)) B)))) (om2uz.1 C B z peano2uz2 mpan om2uz.1 om2uz.2 (opr B (+) (1)) z F A uzrdgval syl om2uz.1 om2uz.2 z om2uzf1o G (om) ({e.|} z (ZZ) (br C (<_) (cv z))) f1ocnv ax-mp (`' G) ({e.|} z (ZZ) (br C (<_) (cv z))) (om) f1of ax-mp B ffvrni om2uz.1 om2uz.2 (` (`' G) B) om2uzsuc syl om2uz.1 om2uz.2 z om2uzf1o G (om) ({e.|} z (ZZ) (br C (<_) (cv z))) B f1ocnvfv2 mpan (+) (1) opreq1d eqtrd om2uz.1 om2uz.2 z om2uzf1o G (om) ({e.|} z (ZZ) (br C (<_) (cv z))) f1ocnv ax-mp (`' G) ({e.|} z (ZZ) (br C (<_) (cv z))) (om) f1of ax-mp B ffvrni (` (`' G) B) peano2 om2uz.1 om2uz.2 z om2uzf1o G (om) ({e.|} z (ZZ) (br C (<_) (cv z))) (suc (` (`' G) B)) (opr B (+) (1)) f1ocnvfv mpan 3syl mpd (rec F A) fveq2d om2uz.1 om2uz.2 z om2uzf1o G (om) ({e.|} z (ZZ) (br C (<_) (cv z))) f1ocnv ax-mp (`' G) ({e.|} z (ZZ) (br C (<_) (cv z))) (om) f1of ax-mp B ffvrni (` (`' G) B) nnont (` (`' G) B) F A rdgsuct 3syl 3eqtrd om2uz.1 om2uz.2 B z F A uzrdgval F fveq2d eqtr4d)) thm (uzrdginip1 ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. A B) (= (` (o. (rec F A) (`' G)) (opr C (+) (1))) (` F A))) (om2uz.1 om2uz.2 A B F uzrdgini F fveq2d (cv z) C C (<_) breq2 (ZZ) elrab om2uz.1 om2uz.1 zre leid mpbir2an om2uz.1 om2uz.2 C z F A uzrdgsuc ax-mp syl5eq)) thm (uzrdgfnuz ((x y) (x z) (y z) (G z) (A z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (Fn (o. (rec F A) (`' G)) ({e.|} z (ZZ) (br C (<_) (cv z)))) (F A rdgfnon (rec F A) (On) fnf mpbi om2uz.1 om2uz.2 z om2uzf1o G (om) ({e.|} z (ZZ) (br C (<_) (cv z))) f1ocnv ax-mp (`' G) ({e.|} z (ZZ) (br C (<_) (cv z))) (om) f1of ax-mp omsson (`' G) ({e.|} z (ZZ) (br C (<_) (cv z))) (om) (On) fss mp2an (rec F A) (On) (V) (`' G) ({e.|} z (ZZ) (br C (<_) (cv z))) fco mp2an (o. (rec F A) (`' G)) ({e.|} z (ZZ) (br C (<_) (cv z))) fnf mpbir)) thm (seq1lem1 ((x y) (v x) (v y) (w z) (v z) (u z) (t z) (f z) (v w) (u w) (t w) (f w) (u v) (t v) (f v) (t u) (f u) (f t) (A v) (B w) (B v) (B u) (B t) (B f) (u x) (t x) (f x) (C x) (C v) (C u) (C t) (C f) (G v) (G u) (G t) (G f)) ((seq1lem1.1 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) (1)) (om))))) (-> (e. A (NN)) (= (` (1st) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) A)) A)) ((cv v) (1) (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) fveq2 (1st) fveq2d (= (cv v) (1)) id eqeq12d (cv v) (cv u) (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) fveq2 (1st) fveq2d (= (cv v) (cv u)) id eqeq12d (cv v) (opr (cv u) (+) (1)) (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) fveq2 (1st) fveq2d (= (cv v) (opr (cv u) (+) (1))) id eqeq12d (cv v) A (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) fveq2 (1st) fveq2d (= (cv v) A) id eqeq12d (1) C opex 1z seq1lem1.1 (<,> (1) C) (V) ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) uzrdgini ax-mp (1st) fveq2i 1z elisseti C op1st eqtr v nnzrab (cv u) eleq2i 1z seq1lem1.1 (cv u) v ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C) uzrdgsuc sylbi (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B)) v ax-17 (e. (cv w) (opr (` (1st) (cv v)) (+) (1))) z ax-17 v z (e. (cv t) B) hbs1 w t hbab hbop hbeleq (opr (` (1st) (cv z)) (+) (1)) (opr (` (1st) (cv v)) (+) (1)) B ({|} t ([/] (cv v) z (e. (cv t) B))) opeq12 (cv z) (cv v) (1st) fveq2 (+) (1) opreq1d z v B t sbab sylanc (cv w) eqeq2d w cbvopab1 (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u)) fveq1i (e. (cv f) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u))) v ax-17 (e. (cv f) (<,> (opr (` (1st) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u))) (+) (1)) ({|} t ([/] (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u)) z (e. (cv t) B))))) v ax-17 (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u) fvex (opr (` (1st) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u))) (+) (1)) ({|} t ([/] (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u)) z (e. (cv t) B))) opex (opr (` (1st) (cv v)) (+) (1)) (opr (` (1st) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u))) (+) (1)) ({|} t ([/] (cv v) z (e. (cv t) B))) ({|} t ([/] (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u)) z (e. (cv t) B))) opeq12 (cv v) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u)) (1st) fveq2 (+) (1) opreq1d (cv v) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u)) z (e. (cv t) B) dfsbcq t abbidv sylanc w fvopabf eqtr syl6eq (1st) fveq2d (` (1st) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u))) (+) (1) oprex ({|} t ([/] (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u)) z (e. (cv t) B))) op1st syl6eq (` (1st) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) (cv u))) (cv u) (+) (1) opreq1 sylan9eq ex nnind)) thm (seq1lem2 () () (-> (e. A (NN)) (e. (opr A (+) (1)) (\ (NN) ({} (1))))) ((opr A (+) (1)) nngt1ne1t 1nn elisseti (opr A (+) (1)) elsnc2 negbii syl6bbr pm5.32i (opr A (+) (1)) (NN) ({} (1)) eldif bitr4 biimp A peano2nn A nngt0t A nnret 0re ax1re (0) A (1) ltadd1t mp3an13 1cn addid2 (<) (opr A (+) (1)) breq1i syl6bb syl mpbid sylanc)) thm (peano3nn0 () () (-> (e. N (NN0)) (e. (opr N (+) (1)) (NN))) (N nn0p1nnt)) thm (seq1rval ((A z) (S z) (S w) (w z) (F z) (F w)) ((seq1rval.1 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (seq1rval.2 (e. A (V)))) (= (` H A) (<,> (opr (` (1st) A) (+) (1)) (opr (` (2nd) A) S (` F (opr (` (1st) A) (+) (1)))))) (seq1rval.1 A fveq1i seq1rval.2 (opr (` (1st) A) (+) (1)) (opr (` (2nd) A) S (` F (opr (` (1st) A) (+) (1)))) opex (opr (` (1st) (cv z)) (+) (1)) (opr (` (1st) A) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))) (opr (` (2nd) A) S (` F (opr (` (1st) A) (+) (1)))) opeq12 (cv z) A (1st) fveq2 (+) (1) opreq1d (cv z) A (2nd) fveq2 (cv z) A (1st) fveq2 (+) (1) opreq1d F fveq2d S opreq12d sylanc w fvopab eqtr)) thm (seq1val ((x y) (f x) (g x) (h x) (G x) (f y) (g y) (h y) (G y) (f g) (f h) (G f) (g h) (G g) (G h) (H x) (H y) (H f) (H g) (H h) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (f z) (g z) (h z) (F z) (f w) (g w) (h w) (F w) (F f) (F g) (F h) (S x) (S y) (S z) (S w) (S f) (S g) (S h)) ((seq1val.1 (e. S (V))) (seq1val.2 (e. F (V))) (seq1val.3 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (seq1val.4 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))))) (= (opr S (seq1) F) ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) (cv x))))))) (seq1val.1 seq1val.2 nnex x y (` (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) (cv x))) funopabex2 (cv f) S (` (2nd) (cv z)) (` (cv g) (opr (` (1st) (cv z)) (+) (1))) opreq (opr (` (2nd) (cv z)) (cv f) (` (cv g) (opr (` (1st) (cv z)) (+) (1)))) (opr (` (2nd) (cv z)) S (` (cv g) (opr (` (1st) (cv z)) (+) (1)))) (opr (` (1st) (cv z)) (+) (1)) opeq2 syl (cv w) eqeq2d z w opabbidv ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) (cv f) (` (cv g) (opr (` (1st) (cv z)) (+) (1))))))) ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` (cv g) (opr (` (1st) (cv z)) (+) (1))))))) (<,> (1) (` (cv g) (1))) rdgeq1 syl (`' (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))) coeq1d seq1val.3 G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)) cnveq ax-mp (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` (cv g) (opr (` (1st) (cv z)) (+) (1))))))) (<,> (1) (` (cv g) (1)))) coeq2i syl6eqr (cv x) fveq1d (2nd) fveq2d (cv y) eqeq2d (e. (cv x) (NN)) anbi2d x y opabbidv (cv g) F (opr (` (1st) (cv z)) (+) (1)) fveq1 (` (2nd) (cv z)) S opreq2d (opr (` (2nd) (cv z)) S (` (cv g) (opr (` (1st) (cv z)) (+) (1)))) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))) (opr (` (1st) (cv z)) (+) (1)) opeq2 syl (cv w) eqeq2d z w opabbidv seq1val.4 syl6eqr ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` (cv g) (opr (` (1st) (cv z)) (+) (1))))))) H (<,> (1) (` (cv g) (1))) rdgeq1 syl (cv g) F (1) fveq1 (` (cv g) (1)) (` F (1)) (1) opeq2 (<,> (1) (` (cv g) (1))) (<,> (1) (` F (1))) H rdgeq2 3syl eqtrd (`' G) coeq1d (cv x) fveq1d (2nd) fveq2d (cv y) eqeq2d (e. (cv x) (NN)) anbi2d x y opabbidv f g h x y z w df-seq1 (V) (V) oprabval5 mp2an)) thm (seq1fnlem ((x y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((seq1val.1 (e. S (V))) (seq1val.2 (e. F (V))) (seq1val.3 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (seq1val.4 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))))) (Fn (opr S (seq1) F) (NN)) ((2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) (cv x)) fvex seq1val.1 seq1val.2 seq1val.3 seq1val.4 x y seq1val fnopab2)) thm (seq1val2 ((x y) (A x) (A y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((seq1val.1 (e. S (V))) (seq1val.2 (e. F (V))) (seq1val.3 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (seq1val.4 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))))) (-> (e. A (NN)) (= (` (opr S (seq1) F) A) (` (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) A)))) ((cv x) A (o. (rec H (<,> (1) (` F (1)))) (`' G)) fveq2 (2nd) fveq2d seq1val.1 seq1val.2 seq1val.3 seq1val.4 x y seq1val (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) A) fvex fvopab4)) thm (seq11lem ((w z) (F z) (F w) (S z) (S w)) ((seq1val.1 (e. S (V))) (seq1val.2 (e. F (V))) (seq1val.3 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (seq1val.4 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))))) (= (` (opr S (seq1) F) (1)) (` F (1))) (1nn seq1val.1 seq1val.2 seq1val.3 seq1val.4 (1) seq1val2 ax-mp (1) (` F (1)) opex 1z seq1val.3 (<,> (1) (` F (1))) (V) H uzrdgini ax-mp (2nd) fveq2i 1nn elisseti F (1) fvex op2nd 3eqtr)) thm (seq1suclem ((x y) (A x) (A y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((seq1val.1 (e. S (V))) (seq1val.2 (e. F (V))) (seq1val.3 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (seq1val.4 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))))) (-> (e. A (NN)) (= (` (opr S (seq1) F) (opr A (+) (1))) (opr (` (opr S (seq1) F) A) S (` F (opr A (+) (1)))))) (x nnzrab A eleq2i 1z seq1val.3 A x H (<,> (1) (` F (1))) uzrdgsuc sylbi seq1val.4 (opr (` (1st) (cv x)) (+) (1)) (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv x)) S (` F (opr (` (1st) (cv x)) (+) (1)))) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))) opeq12 (cv x) (cv z) (1st) fveq2 (+) (1) opreq1d (cv x) (cv z) (2nd) fveq2 (cv x) (cv z) (1st) fveq2 (+) (1) opreq1d F fveq2d S opreq12d sylanc (cv y) eqeq2d (cv y) (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))) eqeq1 sylan9bb cbvopabv eqtr4 (o. (rec H (<,> (1) (` F (1)))) (`' G)) A fvex seq1rval syl6eq (2nd) fveq2d (` (1st) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) A)) (+) (1) oprex (` (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) A)) S (` F (opr (` (1st) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) A)) (+) (1))) oprex op2nd syl6eq seq1val.3 A z w (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))) (` F (1)) seq1lem1 seq1val.4 H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))) (<,> (1) (` F (1))) rdgeq1 ax-mp (`' G) coeq1i A fveq1i (1st) fveq2i syl5eq (+) (1) opreq1d F fveq2d (` (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) A)) S opreq2d eqtrd A peano2nn seq1val.1 seq1val.2 seq1val.3 seq1val.4 (opr A (+) (1)) seq1val2 syl seq1val.1 seq1val.2 seq1val.3 seq1val.4 A seq1val2 S (` F (opr A (+) (1))) opreq1d 3eqtr4d)) thm (seq11 ((w z) (S z) (S w) (F z) (F w)) ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (= (` (opr S (seq1) F) (1)) (` F (1))) (seq111.1 seq111.2 (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)) eqid ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))) eqid seq11lem)) thm (seq1p1 ((w z) (S z) (S w) (F z) (F w)) ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (-> (e. A (NN)) (= (` (opr S (seq1) F) (opr A (+) (1))) (opr (` (opr S (seq1) F) A) S (` F (opr A (+) (1)))))) (seq111.1 seq111.2 (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)) eqid ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))) eqid A seq1suclem)) thm (seq1m1 () ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (-> (/\ (e. N (NN)) (br (1) (<) N)) (= (` (opr S (seq1) F) N) (opr (` (opr S (seq1) F) (opr N (-) (1))) S (` F N)))) (1nn (1) N nnsubt mpan biimpa seq111.1 seq111.2 (opr N (-) (1)) seq1p1 syl N nncnt 1cn N (1) npcant mpan2 syl (opr S (seq1) F) fveq2d (br (1) (<) N) adantr N nncnt 1cn N (1) npcant mpan2 syl F fveq2d (` (opr S (seq1) F) (opr N (-) (1))) S opreq2d (br (1) (<) N) adantr 3eqtr3d)) thm (seq1fn ((w z) (S z) (S w) (F z) (F w)) ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (Fn (opr S (seq1) F) (NN)) (seq111.1 seq111.2 (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)) eqid ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))) eqid seq1fnlem)) thm (seq1rn ((x y) (x z) (w x) (v x) (S x) (y z) (w y) (v y) (S y) (w z) (v z) (S z) (v w) (S w) (S v) (F x) (F y) (F z) (F w) (F v) (C x) (C y) (C z) (C w) (C v) (D x) (D y) (D z) (D w) (D v)) ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (-> (/\/\ (e. (` F (1)) C) (:--> (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D) (:--> S (X. C D) C)) (C_ (ran (opr S (seq1) F)) C)) ((cv y) (1) (opr S (seq1) F) fveq2 C eleq1d (/\/\ (e. (` F (1)) C) (:--> (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D) (:--> S (X. C D) C)) imbi2d (cv y) (cv z) (opr S (seq1) F) fveq2 C eleq1d (/\/\ (e. (` F (1)) C) (:--> (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D) (:--> S (X. C D) C)) imbi2d (cv y) (opr (cv z) (+) (1)) (opr S (seq1) F) fveq2 C eleq1d (/\/\ (e. (` F (1)) C) (:--> (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D) (:--> S (X. C D) C)) imbi2d (cv y) (cv x) (opr S (seq1) F) fveq2 C eleq1d (/\/\ (e. (` F (1)) C) (:--> (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D) (:--> S (X. C D) C)) imbi2d (e. (` F (1)) C) (:--> (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D) pm3.26 seq111.1 seq111.2 seq11 syl5eqel (:--> S (X. C D) C) 3adant3 (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D (opr (cv z) (+) (1)) ffvrn (opr (cv z) (+) (1)) (\ (NN) ({} (1))) F fvres D eleq1d (:--> (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D) adantl mpbid (cv z) seq1lem2 sylan2 seq111.1 seq111.2 (cv z) seq1p1 C eleq1d (cv w) (` (opr S (seq1) F) (cv z)) S (cv v) opreq1 C eleq1d (cv v) (` F (opr (cv z) (+) (1))) (` (opr S (seq1) F) (cv z)) S opreq2 C eleq1d C D rcla42v ancoms S C D C w v ffnoprval pm3.27bd syl5 imp syl5bir exp4c (:--> (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D) adantl mpd ex com4r impcom (e. (` F (1)) C) 3adant1 com12 a2d nnind com12 r19.21aiv seq111.1 seq111.2 seq1fn (opr S (seq1) F) (NN) x C fnfvrnss mpan syl)) thm (seq1rn2 () ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (-> (/\ (:--> F (NN) C) (:--> S (X. C C) C)) (C_ (ran (opr S (seq1) F)) C)) (seq111.1 seq111.2 C C seq1rn 3expa 1nn F (NN) C (1) ffvrn mpan2 (NN) ({} (1)) difss F (NN) C (\ (NN) ({} (1))) fssres mpan2 jca sylan)) thm (seq1f () ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (-> (/\ (:--> F (NN) C) (:--> S (X. C C) C)) (:--> (opr S (seq1) F) (NN) C)) (seq111.1 seq111.2 C seq1rn2 seq111.1 seq111.2 seq1fn jctil (opr S (seq1) F) (NN) C df-f sylibr)) thm (seq1cl () ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (-> (/\/\ (e. A (NN)) (:--> F (NN) C) (:--> S (X. C C) C)) (e. (` (opr S (seq1) F) A) C)) ((opr S (seq1) F) (NN) C A ffvrn seq111.1 seq111.2 C seq1f sylan 3impa 3comr)) thm (seq1res ((x y) (x z) (S x) (y z) (S y) (S z) (F x) (F y) (F z)) ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (= (opr S (seq1) (|` F (NN))) (opr S (seq1) F)) (seq111.1 seq111.2 F (V) (NN) resexg ax-mp seq1fn seq111.1 seq111.2 seq1fn (opr S (seq1) (|` F (NN))) (NN) (opr S (seq1) F) (NN) x eqfnfv mp2an (NN) eqid (cv y) (1) (opr S (seq1) (|` F (NN))) fveq2 (cv y) (1) (opr S (seq1) F) fveq2 eqeq12d (cv y) (cv z) (opr S (seq1) (|` F (NN))) fveq2 (cv y) (cv z) (opr S (seq1) F) fveq2 eqeq12d (cv y) (opr (cv z) (+) (1)) (opr S (seq1) (|` F (NN))) fveq2 (cv y) (opr (cv z) (+) (1)) (opr S (seq1) F) fveq2 eqeq12d (cv y) (cv x) (opr S (seq1) (|` F (NN))) fveq2 (cv y) (cv x) (opr S (seq1) F) fveq2 eqeq12d 1nn (1) (NN) F fvres ax-mp seq111.1 seq111.2 F (V) (NN) resexg ax-mp seq11 seq111.1 seq111.2 seq11 3eqtr4 (= (` (opr S (seq1) (|` F (NN))) (cv z)) (` (opr S (seq1) F) (cv z))) id (cv z) peano2nn (opr (cv z) (+) (1)) (NN) F fvres syl S opreqan12rd seq111.1 seq111.2 F (V) (NN) resexg ax-mp (cv z) seq1p1 (= (` (opr S (seq1) (|` F (NN))) (cv z)) (` (opr S (seq1) F) (cv z))) adantr seq111.1 seq111.2 (cv z) seq1p1 (= (` (opr S (seq1) (|` F (NN))) (cv z)) (` (opr S (seq1) F) (cv z))) adantr 3eqtr4d ex nnind rgen mpbir2an)) thm (ser1ft ((F f)) () (-> (:--> F (NN) (CC)) (:--> (opr (+) (seq1) F) (NN) (CC))) (nnex F (NN) (CC) (V) fex mpan2 (cv f) F (NN) (CC) feq1 (cv f) F (+) (seq1) opreq2 (opr (+) (seq1) (cv f)) (opr (+) (seq1) F) (NN) (CC) feq1 syl imbi12d axaddopr addex f visset (CC) seq1rn2 mpan2 addex f visset seq1fn jctil (opr (+) (seq1) (cv f)) (NN) (CC) df-f sylibr (V) vtoclg mpcom)) thm (ser1f () ((ser1f.1 (:--> F (NN) (CC)))) (:--> (opr (+) (seq1) F) (NN) (CC)) (ser1f.1 F ser1ft ax-mp)) thm (ser1cl () ((ser1cl.1 (:--> F (NN) (CC)))) (-> (e. A (NN)) (e. (` (opr (+) (seq1) F) A) (CC))) (ser1cl.1 axaddopr addex ser1cl.1 nnex F (NN) (CC) (V) fex mp2an A (CC) seq1cl mp3an23)) thm (ser1recl ((A x) (x y) (F x) (F y)) ((ser1recl.1 (:--> F (NN) (RR)))) (-> (e. A (NN)) (e. (` (opr (+) (seq1) F) A) (RR))) ((cv x) (1) (opr (+) (seq1) F) fveq2 (RR) eleq1d (cv x) (cv y) (opr (+) (seq1) F) fveq2 (RR) eleq1d (cv x) (opr (cv y) (+) (1)) (opr (+) (seq1) F) fveq2 (RR) eleq1d (cv x) A (opr (+) (seq1) F) fveq2 (RR) eleq1d addex ser1recl.1 nnex F (NN) (RR) (V) fex mp2an seq11 ser1recl.1 1nn F (NN) (RR) (1) ffvrn mp2an eqeltr (` (opr (+) (seq1) F) (cv y)) (` F (opr (cv y) (+) (1))) axaddrcl (cv y) peano2nn ser1recl.1 (opr (cv y) (+) (1)) ffvrni syl sylan2 addex ser1recl.1 nnex F (NN) (RR) (V) fex mp2an (cv y) seq1p1 (RR) eleq1d (e. (` (opr (+) (seq1) F) (cv y)) (RR)) adantl mpbird expcom nnind)) thm (ser1re ((F x)) ((ser1recl.1 (:--> F (NN) (RR)))) (:--> (opr (+) (seq1) F) (NN) (RR)) ((opr (+) (seq1) F) (NN) (RR) x ffnfv addex ser1recl.1 nnex F (NN) (RR) (V) fex mp2an seq1fn ser1recl.1 (cv x) ser1recl rgen mpbir2an)) thm (ser1cl2 ((x y) (A y) (B x)) ((ser1cl2.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A))))) (ser1cl2.2 (A.e. x (NN) (e. A (CC))))) (-> (e. B (NN)) (e. (` (opr (+) (seq1) F) B) (CC))) (ser1cl2.2 ser1cl2.1 (CC) fopab2 mpbi B ser1cl)) thm (ser1f2 ((x y) (x z) (y z) (A y) (F z)) ((ser1cl2.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A))))) (ser1cl2.2 (A.e. x (NN) (e. A (CC))))) (:--> (opr (+) (seq1) F) (NN) (CC)) ((opr (+) (seq1) F) (NN) (CC) z ffnfv addex ser1cl2.1 nnex x y A funopabex2 eqeltr seq1fn ser1cl2.1 ser1cl2.2 (cv z) ser1cl2 rgen mpbir2an)) thm (ser11 ((B y) (x y) (A y) (B x)) ((ser1cl2.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A))))) (ser11.2 (e. B (V))) (ser11.3 (-> (= (cv x) (1)) (= A B)))) (= (` (opr (+) (seq1) F) (1)) B) (addex ser1cl2.1 nnex x y A funopabex2 eqeltr seq11 1nn ser11.3 ser1cl2.1 ser11.2 fvopab4 ax-mp eqtr)) thm (ser1p1 ((B y) (x y) (C x) (C y) (x y) (A y) (B x)) ((ser1cl2.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A))))) (ser1p1.2 (e. C (V))) (ser1p1.3 (-> (= (cv x) (opr B (+) (1))) (= A C)))) (-> (e. B (NN)) (= (` (opr (+) (seq1) F) (opr B (+) (1))) (opr (` (opr (+) (seq1) F) B) (+) C))) (addex ser1cl2.1 nnex x y A funopabex2 eqeltr B seq1p1 B peano2nn ser1p1.3 ser1cl2.1 ser1p1.2 fvopab4 syl (` (opr (+) (seq1) F) B) (+) opreq2d eqtrd)) thm (ser1mono ((A y) (x y) (F x) (F y)) ((ser1mono.1 (:--> F (NN) (RR))) (ser1mono.2 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))))) (-> (e. A (NN)) (br (` (opr (+) (seq1) F) A) (<_) (` (opr (+) (seq1) F) (opr A (+) (1))))) (ser1mono.1 axresscn F (NN) (RR) (CC) fss mp2an A ser1cl (` (opr (+) (seq1) F) A) ax0id syl A peano2nn (cv y) (opr A (+) (1)) F fveq2 (0) (<_) breq2d (cv x) (cv y) F fveq2 (0) (<_) breq2d ser1mono.2 vtoclga vtoclga syl 0re (0) (` F (opr A (+) (1))) (` (opr (+) (seq1) F) A) leadd2t mp3an1 A peano2nn ser1mono.1 (opr A (+) (1)) ffvrni syl ser1mono.1 A ser1recl sylanc mpbid addex ser1mono.1 nnex F (NN) (RR) (V) fex mp2an A seq1p1 breqtrrd eqbrtrrd)) thm (ser1add2 ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (H k) (H j) (H m) (N j) (N k) (N m)) ((ser1add2.1 (:--> F (NN) (CC))) (ser1add2.2 (:--> G (NN) (CC))) (ser1add2.3 (e. H (V))) (ser1add2.4 (-> (/\/\ (e. (cv k) (NN)) (e. N (NN)) (br (cv k) (<_) N)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) H) N) (opr (` (opr (+) (seq1) F) N) (+) (` (opr (+) (seq1) G) N)))) (N nnret N leidt syl (cv j) (1) (<_) N breq1 (e. N (NN)) anbi2d (cv j) (1) (opr (+) (seq1) H) fveq2 (cv j) (1) (opr (+) (seq1) F) fveq2 (cv j) (1) (opr (+) (seq1) G) fveq2 (+) opreq12d eqeq12d imbi12d (cv j) (cv m) (<_) N breq1 (e. N (NN)) anbi2d (cv j) (cv m) (opr (+) (seq1) H) fveq2 (cv j) (cv m) (opr (+) (seq1) F) fveq2 (cv j) (cv m) (opr (+) (seq1) G) fveq2 (+) opreq12d eqeq12d imbi12d (cv j) (opr (cv m) (+) (1)) (<_) N breq1 (e. N (NN)) anbi2d (cv j) (opr (cv m) (+) (1)) (opr (+) (seq1) H) fveq2 (cv j) (opr (cv m) (+) (1)) (opr (+) (seq1) F) fveq2 (cv j) (opr (cv m) (+) (1)) (opr (+) (seq1) G) fveq2 (+) opreq12d eqeq12d imbi12d (cv j) N (<_) N breq1 (e. N (NN)) anbi2d (cv j) N (opr (+) (seq1) H) fveq2 (cv j) N (opr (+) (seq1) F) fveq2 (cv j) N (opr (+) (seq1) G) fveq2 (+) opreq12d eqeq12d imbi12d 1nn (cv k) (1) (<_) N breq1 (e. N (NN)) anbi2d (cv k) (1) H fveq2 (cv k) (1) F fveq2 (cv k) (1) G fveq2 (+) opreq12d eqeq12d imbi12d ser1add2.4 3exp imp3a vtoclga ax-mp addex ser1add2.3 seq11 addex ser1add2.1 nnex F (NN) (CC) (V) fex mp2an seq11 addex ser1add2.2 nnex G (NN) (CC) (V) fex mp2an seq11 (+) opreq12i 3eqtr4g (cv m) lep1t (e. N (RR)) adantr (cv m) (opr (cv m) (+) (1)) N letrt (e. (cv m) (RR)) (e. N (RR)) pm3.26 (cv m) peano2re (e. N (RR)) adantr (e. (cv m) (RR)) (e. N (RR)) pm3.27 syl3anc mpand (cv m) nnret N nnret syl2an imp (e. N (NN)) adantllr (/\ (/\ (e. (cv m) (NN)) (e. N (NN))) (br (opr (cv m) (+) (1)) (<_) N)) (= (` (opr (+) (seq1) H) (cv m)) (opr (` (opr (+) (seq1) F) (cv m)) (+) (` (opr (+) (seq1) G) (cv m)))) pm3.27 (cv m) peano2nn (cv k) (opr (cv m) (+) (1)) (<_) N breq1 (cv k) (opr (cv m) (+) (1)) H fveq2 (cv k) (opr (cv m) (+) (1)) F fveq2 (cv k) (opr (cv m) (+) (1)) G fveq2 (+) opreq12d eqeq12d imbi12d (e. N (NN)) imbi2d ser1add2.4 3exp vtoclga syl imp31 (= (` (opr (+) (seq1) H) (cv m)) (opr (` (opr (+) (seq1) F) (cv m)) (+) (` (opr (+) (seq1) G) (cv m)))) adantr (+) opreq12d addex ser1add2.3 (cv m) seq1p1 (br (opr (cv m) (+) (1)) (<_) N) (= (` (opr (+) (seq1) H) (cv m)) (opr (` (opr (+) (seq1) F) (cv m)) (+) (` (opr (+) (seq1) G) (cv m)))) ad2antrr (e. N (NN)) adantllr addex ser1add2.1 nnex F (NN) (CC) (V) fex mp2an (cv m) seq1p1 addex ser1add2.2 nnex G (NN) (CC) (V) fex mp2an (cv m) seq1p1 (+) opreq12d (` (opr (+) (seq1) F) (cv m)) (` F (opr (cv m) (+) (1))) (` (opr (+) (seq1) G) (cv m)) (` G (opr (cv m) (+) (1))) add4t ser1add2.1 (cv m) ser1cl (cv m) peano2nn ser1add2.1 (opr (cv m) (+) (1)) ffvrni syl jca ser1add2.2 (cv m) ser1cl (cv m) peano2nn ser1add2.2 (opr (cv m) (+) (1)) ffvrni syl jca sylanc eqtrd (e. N (NN)) (= (` (opr (+) (seq1) H) (cv m)) (opr (` (opr (+) (seq1) F) (cv m)) (+) (` (opr (+) (seq1) G) (cv m)))) ad2antrr (br (opr (cv m) (+) (1)) (<_) N) adantlr 3eqtr4d ex (/\ (e. N (NN)) (br (cv m) (<_) N)) imim2d com23 imp an4s anassrs mpdan exp41 pm2.43d imp3a com23 nnind mpan2d pm2.43i)) thm (ser1add ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (H k) (H j) (H m) (N j) (N k) (N m)) ((ser1add.1 (:--> F (NN) (CC))) (ser1add.2 (:--> G (NN) (CC))) (ser1add.3 (e. H (V))) (ser1add.4 (-> (/\ (e. (cv k) (NN)) (br (cv k) (<_) N)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) H) N) (opr (` (opr (+) (seq1) F) N) (+) (` (opr (+) (seq1) G) N)))) (N nnret N leidt syl (cv j) (1) (<_) N breq1 (e. N (NN)) anbi2d (cv j) (1) (opr (+) (seq1) H) fveq2 (cv j) (1) (opr (+) (seq1) F) fveq2 (cv j) (1) (opr (+) (seq1) G) fveq2 (+) opreq12d eqeq12d imbi12d (cv j) (cv m) (<_) N breq1 (e. N (NN)) anbi2d (cv j) (cv m) (opr (+) (seq1) H) fveq2 (cv j) (cv m) (opr (+) (seq1) F) fveq2 (cv j) (cv m) (opr (+) (seq1) G) fveq2 (+) opreq12d eqeq12d imbi12d (cv j) (opr (cv m) (+) (1)) (<_) N breq1 (e. N (NN)) anbi2d (cv j) (opr (cv m) (+) (1)) (opr (+) (seq1) H) fveq2 (cv j) (opr (cv m) (+) (1)) (opr (+) (seq1) F) fveq2 (cv j) (opr (cv m) (+) (1)) (opr (+) (seq1) G) fveq2 (+) opreq12d eqeq12d imbi12d (cv j) N (<_) N breq1 (e. N (NN)) anbi2d (cv j) N (opr (+) (seq1) H) fveq2 (cv j) N (opr (+) (seq1) F) fveq2 (cv j) N (opr (+) (seq1) G) fveq2 (+) opreq12d eqeq12d imbi12d 1nn (cv k) (1) (<_) N breq1 (cv k) (1) H fveq2 (cv k) (1) F fveq2 (cv k) (1) G fveq2 (+) opreq12d eqeq12d imbi12d ser1add.4 ex vtoclga ax-mp addex ser1add.3 seq11 addex ser1add.1 nnex F (NN) (CC) (V) fex mp2an seq11 addex ser1add.2 nnex G (NN) (CC) (V) fex mp2an seq11 (+) opreq12i 3eqtr4g (e. N (NN)) adantl (cv m) lep1t (e. N (RR)) adantr (cv m) (opr (cv m) (+) (1)) N letrt (e. (cv m) (RR)) (e. N (RR)) pm3.26 (cv m) peano2re (e. N (RR)) adantr (e. (cv m) (RR)) (e. N (RR)) pm3.27 syl3anc mpand (cv m) nnret N nnret syl2an imp (/\ (e. (cv m) (NN)) (br (opr (cv m) (+) (1)) (<_) N)) (= (` (opr (+) (seq1) H) (cv m)) (opr (` (opr (+) (seq1) F) (cv m)) (+) (` (opr (+) (seq1) G) (cv m)))) pm3.27 (cv k) (opr (cv m) (+) (1)) (<_) N breq1 (cv k) (opr (cv m) (+) (1)) H fveq2 (cv k) (opr (cv m) (+) (1)) F fveq2 (cv k) (opr (cv m) (+) (1)) G fveq2 (+) opreq12d eqeq12d imbi12d ser1add.4 ex vtoclga imp (cv m) peano2nn sylan (= (` (opr (+) (seq1) H) (cv m)) (opr (` (opr (+) (seq1) F) (cv m)) (+) (` (opr (+) (seq1) G) (cv m)))) adantr (+) opreq12d addex ser1add.3 (cv m) seq1p1 (br (opr (cv m) (+) (1)) (<_) N) (= (` (opr (+) (seq1) H) (cv m)) (opr (` (opr (+) (seq1) F) (cv m)) (+) (` (opr (+) (seq1) G) (cv m)))) ad2antrr addex ser1add.1 nnex F (NN) (CC) (V) fex mp2an (cv m) seq1p1 addex ser1add.2 nnex G (NN) (CC) (V) fex mp2an (cv m) seq1p1 (+) opreq12d (` (opr (+) (seq1) F) (cv m)) (` F (opr (cv m) (+) (1))) (` (opr (+) (seq1) G) (cv m)) (` G (opr (cv m) (+) (1))) add4t ser1add.1 (cv m) ser1cl (cv m) peano2nn ser1add.1 (opr (cv m) (+) (1)) ffvrni syl jca ser1add.2 (cv m) ser1cl (cv m) peano2nn ser1add.2 (opr (cv m) (+) (1)) ffvrni syl jca sylanc eqtrd (br (opr (cv m) (+) (1)) (<_) N) (= (` (opr (+) (seq1) H) (cv m)) (opr (` (opr (+) (seq1) F) (cv m)) (+) (` (opr (+) (seq1) G) (cv m)))) ad2antrr 3eqtr4d ex (/\ (e. N (NN)) (br (cv m) (<_) N)) imim2d com23 imp an4s anassrs mpdan exp31 imp3a com23 nnind mpan2d pm2.43i)) thm (shftfval ((x y) (w x) (f x) (g x) (A x) (w y) (f y) (g y) (A y) (f w) (g w) (A w) (f g) (A f) (A g) (F x) (F y) (F w) (F f) (F g) (B x) (B y)) ((shftfval.1 (e. F (V)))) (-> (e. A B) (= (opr F (shift) A) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (` F (opr (cv x) (-) A))))))) (shftfval.1 axcnex x y (` F (opr (cv x) (-) A)) funopabex2 (cv f) F (opr (cv x) (-) (cv w)) fveq1 (cv y) eqeq2d (e. (cv x) (CC)) anbi2d x y opabbidv (cv w) A (cv x) (-) opreq2 F fveq2d (cv y) eqeq2d (e. (cv x) (CC)) anbi2d x y opabbidv f w g x y df-shft (V) B oprabval5 mpan)) thm (shftfn ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) ((shftfval.1 (e. F (V)))) (-> (e. A B) (Fn (opr F (shift) A) (CC))) (F (opr (cv x) (-) A) fvex ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (` F (opr (cv x) (-) A))))) eqid fnopab2 shftfval.1 A B x y shftfval (opr F (shift) A) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (` F (opr (cv x) (-) A))))) (CC) fneq1 syl mpbiri)) thm (shftres () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A C) (C_ B (CC))) (Fn (|` (opr F (shift) A) B) B)) ((opr F (shift) A) (CC) B fnssres shftfval.1 A C shftfn sylan)) thm (shftresvalt () ((shftfval.1 (e. F (V)))) (-> (e. B C) (= (` (|` (opr F (shift) A) C) B) (` (opr F (shift) A) B))) (B C (opr F (shift) A) fvres)) thm (shftvalt ((x y) (A x) (A y) (F x) (F y) (B x) (B y) (C x) (C y)) ((shftfval.1 (e. F (V)))) (-> (/\ (e. A C) (e. B (CC))) (= (` (opr F (shift) A) B) (` F (opr B (-) A)))) (shftfval.1 A C x y shftfval B fveq1d (cv x) B (-) A opreq1 F fveq2d ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (` F (opr (cv x) (-) A))))) eqid F (opr B (-) A) fvex fvopab4 sylan9eq)) thm (shftval2t () ((shftfval.1 (e. F (V)))) (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (` (opr F (shift) (opr A (-) B)) (opr A (+) C)) (` F (opr B (+) C)))) (shftfval.1 (opr A (-) B) (CC) (opr A (+) C) shftvalt A B subclt (e. C (CC)) 3adant3 A C axaddcl (e. B (CC)) 3adant2 sylanc A C B pnncant 3com23 B C axaddcom (e. A (CC)) 3adant1 eqtr4d F fveq2d eqtrd)) thm (shftval3t () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) (opr A (-) B)) A) (` F B))) (0cn shftfval.1 A B (0) shftval2t mp3an3 A ax0id (e. B (CC)) adantr (opr F (shift) (opr A (-) B)) fveq2d B ax0id (e. A (CC)) adantl F fveq2d 3eqtr3d)) thm (shftval4t () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) (-u A)) B) (` F (opr A (+) B)))) (shftfval.1 (-u A) (CC) B shftvalt A negclt sylan B A subnegt ancoms A B axaddcom eqtr4d F fveq2d eqtrd)) thm (shftval5t () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) A) (opr B (+) A)) (` F B))) (shftfval.1 A (CC) (opr B (+) A) shftvalt (e. B (CC)) (e. A (CC)) pm3.27 B A axaddcl sylanc B A pncant F fveq2d eqtrd ancoms)) thm (shftf ((A x) (F x) (B x) (C x)) ((shftfval.1 (e. F (V)))) (-> (/\/\ (e. A D) (C_ B (CC)) (A.e. x B (e. (` F (opr (cv x) (-) A)) C))) (:--> (|` (opr F (shift) A) B) B C)) (shftfval.1 A (V) B shftres (A.e. x B (e. (` F (opr (cv x) (-) A)) C)) 3adant3 shftfval.1 (cv x) B A shftresvalt (/\ (e. A (V)) (C_ B (CC))) adantl B (CC) (cv x) ssel2 (e. A (V)) anim2i anassrs shftfval.1 A (V) (cv x) shftvalt syl eqtr2d C eleq1d ralbidva biimp3a jca (|` (opr F (shift) A) B) B C x ffnfv sylibr A D elisset syl3an1)) thm (2shft ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr F (shift) A) (shift) B) (opr F (shift) (opr A (+) B)))) (shftfval.1 A (CC) (opr (cv x) (-) B) shftvalt (e. A (CC)) (e. B (CC)) pm3.26 (e. (cv x) (CC)) adantl (cv x) B subclt (e. A (CC)) adantrl sylanc (cv x) A B sub23t (cv x) A B subsub4t eqtr3d 3expb F fveq2d eqtrd ancoms (cv y) eqeq2d ex pm5.32d x y opabbidv F (shift) A oprex B (CC) x y shftfval (e. A (CC)) adantl A B axaddcl shftfval.1 (opr A (+) B) (CC) x y shftfval syl 3eqtr4d)) thm (shftcan2t () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr (opr F (shift) (-u A)) (shift) A) B) (` F B))) (F (shift) (-u A) oprex A (CC) B shftvalt shftfval.1 (-u A) (CC) (opr B (-) A) shftvalt A negclt (e. B (CC)) adantl B A subclt sylanc (opr B (-) A) A subnegt B A subclt (e. B (CC)) (e. A (CC)) pm3.27 sylanc B A npcant eqtrd F fveq2d eqtrd ancoms eqtrd)) thm (shftcan1t () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr (opr F (shift) A) (shift) (-u A)) B) (` F B))) (A negnegt F (shift) opreq2d (shift) (-u A) opreq1d B fveq1d (e. B (CC)) adantr shftfval.1 (-u A) B shftcan2t A negclt sylan eqtr3d)) thm (shftidt () ((shftfval.1 (e. F (V)))) (-> (e. A (CC)) (= (` (opr F (shift) (0)) A) (` F A))) (0cn shftfval.1 (0) A shftcan1t mpan 0cn 0cn negcl shftfval.1 (0) (-u (0)) 2shft mp2an 0cn negid F (shift) opreq2i eqtr A fveq1i syl5eqr)) thm (seq1shftid ((j k) (j n) (S j) (k n) (S k) (S n) (F j) (F k) (F n)) ((seq1shftid.1 (e. S (V))) (seq1shftid.2 (e. F (V)))) (= (opr S (seq1) (opr F (shift) (0))) (opr S (seq1) F)) (seq1shftid.1 F (shift) (0) oprex seq1fn seq1shftid.1 seq1shftid.2 seq1fn (opr S (seq1) (opr F (shift) (0))) (NN) (opr S (seq1) F) (NN) n eqfnfv mp2an (NN) eqid (cv j) (1) (opr S (seq1) (opr F (shift) (0))) fveq2 (cv j) (1) (opr S (seq1) F) fveq2 eqeq12d (cv j) (cv k) (opr S (seq1) (opr F (shift) (0))) fveq2 (cv j) (cv k) (opr S (seq1) F) fveq2 eqeq12d (cv j) (opr (cv k) (+) (1)) (opr S (seq1) (opr F (shift) (0))) fveq2 (cv j) (opr (cv k) (+) (1)) (opr S (seq1) F) fveq2 eqeq12d (cv j) (cv n) (opr S (seq1) (opr F (shift) (0))) fveq2 (cv j) (cv n) (opr S (seq1) F) fveq2 eqeq12d 1cn seq1shftid.2 (1) shftidt ax-mp seq1shftid.1 F (shift) (0) oprex seq11 seq1shftid.1 seq1shftid.2 seq11 3eqtr4 (` (opr S (seq1) (opr F (shift) (0))) (cv k)) (` (opr S (seq1) F) (cv k)) S (` (opr F (shift) (0)) (opr (cv k) (+) (1))) opreq1 (cv k) nncnt (cv k) peano2cn syl seq1shftid.2 (opr (cv k) (+) (1)) shftidt syl (` (opr S (seq1) F) (cv k)) S opreq2d sylan9eqr seq1shftid.1 F (shift) (0) oprex (cv k) seq1p1 (= (` (opr S (seq1) (opr F (shift) (0))) (cv k)) (` (opr S (seq1) F) (cv k))) adantr seq1shftid.1 seq1shftid.2 (cv k) seq1p1 (= (` (opr S (seq1) (opr F (shift) (0))) (cv k)) (` (opr S (seq1) F) (cv k))) adantr 3eqtr4d ex nnind rgen mpbir2an)) thm (uzvalt ((j k) (j y) (N j) (k y) (N k) (N y)) () (-> (e. N (ZZ)) (= (` (ZZ>) N) ({e.|} k (ZZ) (br N (<_) (cv k))))) ((cv j) N (<_) (cv k) breq1 k (ZZ) rabbisdv j y k df-uz zex k (br N (<_) (cv k)) rabex fvopab4)) thm (eluz1t ((N k) (M k)) () (-> (e. M (ZZ)) (<-> (e. N (` (ZZ>) M)) (/\ (e. N (ZZ)) (br M (<_) N)))) (M k uzvalt N eleq2d (cv k) N M (<_) breq2 (ZZ) elrab syl6bb)) thm (eluz2t ((j k) (j y) (N j) (k y) (N k) (N y) (M k)) () (<-> (e. N (` (ZZ>) M)) (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N))) (N (` (ZZ>) M) n0i M (ZZ>) ndmfv nsyl2 zex k (br (cv j) (<_) (cv k)) rabex j y k df-uz dmopab2 syl6eleq (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N) 3simp1 M N eluz1t (e. M (ZZ)) (/\ (e. N (ZZ)) (br M (<_) N)) ibar bitrd (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N) 3anass syl6bbr pm5.21nii)) thm (eluz1 () ((eluz.1 (e. M (ZZ)))) (<-> (e. N (` (ZZ>) M)) (/\ (e. N (ZZ)) (br M (<_) N))) (eluz.1 M N eluz1t ax-mp)) thm (eluzelz () () (-> (e. N (` (ZZ>) M)) (e. N (ZZ))) (N M eluz2t (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N) 3simp2 sylbi)) thm (eluzel2 () () (-> (e. N (` (ZZ>) M)) (e. M (ZZ))) (N M eluz2t (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N) 3simp1 sylbi)) thm (eluzle () () (-> (e. N (` (ZZ>) M)) (br M (<_) N)) (N M eluz2t (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N) 3simp3 sylbi)) thm (eluzt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (e. N (` (ZZ>) M)) (br M (<_) N))) (M N eluz1t N M eluzelz pm4.71ri syl5bbr (e. N (ZZ)) (e. N (` (ZZ>) M)) (br M (<_) N) pm5.32 sylibr imp)) thm (uzidt () () (-> (e. M (ZZ)) (e. M (` (ZZ>) M))) (M zret M leidt syl ancli M M eluz1t mpbird)) thm (uztrn () () (-> (/\ (e. M (` (ZZ>) K)) (e. K (` (ZZ>) N))) (e. M (` (ZZ>) N))) (K N eluzel2 (e. M (` (ZZ>) K)) adantl M K eluzelz (e. K (` (ZZ>) N)) adantr K N eluzle (e. M (` (ZZ>) K)) adantl M K eluzle (e. K (` (ZZ>) N)) adantr N K M letrt N zret K zret M zret syl3an K N eluzel2 (e. M (` (ZZ>) K)) adantl K N eluzelz (e. M (` (ZZ>) K)) adantl M K eluzelz (e. K (` (ZZ>) N)) adantr syl3anc mp2and 3jca M N eluz2t sylibr)) thm (uznegit () () (-> (e. N (` (ZZ>) M)) (e. (-u M) (` (ZZ>) (-u N)))) (N M eluzle M N lenegt M zret N zret syl2an N M eluzel2 N M eluzelz sylanc mpbid (-u N) (-u M) eluzt N znegclt M znegclt syl2an N M eluzelz N M eluzel2 sylanc mpbird)) thm (uzssz ((j k) (j y) (M j) (k y) (M k) (M y)) () (C_ (` (ZZ>) M) (ZZ)) (M k uzvalt k (ZZ) (br M (<_) (cv k)) ssrab2 (e. M (ZZ)) a1i eqsstrd j y k df-uz dmeqi j y (ZZ) (= (cv y) ({e.|} k (ZZ) (br (cv j) (<_) (cv k)))) dmopabss eqsstr M sseli con3i M (ZZ>) ndmfv syl (ZZ) 0ss (-. (e. M (ZZ))) a1i eqsstrd pm2.61i)) thm (uzss ((M k) (N k)) () (-> (e. N (` (ZZ>) M)) (C_ (` (ZZ>) N) (` (ZZ>) M))) (N M eluzle (e. (cv k) (ZZ)) adantr M N (cv k) letrt M zret N zret (cv k) zret syl3an 3expa N M eluzel2 N M eluzelz jca sylan mpand ex imdistand N M eluzelz N (cv k) eluz1t syl N M eluzel2 M (cv k) eluz1t syl 3imtr4d ssrdv)) thm (uz11t () () (-> (e. M (ZZ)) (<-> (= (` (ZZ>) M) (` (ZZ>) N)) (= M N))) ((` (ZZ>) M) (` (ZZ>) N) M eleq2 M N eluzel2 syl6bi M uzidt syl5 impcom (` (ZZ>) M) (` (ZZ>) N) N eleq2 N uzidt syl5bir N M eluzle syl6 (` (ZZ>) M) (` (ZZ>) N) M eleq2 M uzidt syl5bi M N eluzle syl6 anim12d imp anassrs ancoms anassrs M N letri3t M zret N zret syl2an (= (` (ZZ>) M) (` (ZZ>) N)) adantlr mpbird mpdan ex M N (ZZ>) fveq2 (e. M (ZZ)) a1i impbid)) thm (eluzp1m1t () () (-> (/\ (e. M (ZZ)) (e. N (` (ZZ>) (opr M (+) (1))))) (e. (opr N (-) (1)) (` (ZZ>) M))) (N peano2zm (e. M (ZZ)) (br (opr M (+) (1)) (<_) N) ad2antrl ax1re M (1) N leaddsubt mp3an2 M zret N zret syl2an biimpa anasss jca ex M peano2z (opr M (+) (1)) N eluz1t syl M (opr N (-) (1)) eluz1t 3imtr4d imp)) thm (eluzp1lt () () (-> (/\ (e. M (ZZ)) (e. N (` (ZZ>) (opr M (+) (1))))) (br M (<) N)) (N (opr M (+) (1)) eluzle (e. M (ZZ)) adantl M N zltp1let N (opr M (+) (1)) eluzelz sylan2 mpbird)) thm (eluzp1p1t () () (-> (e. N (` (ZZ>) M)) (e. (opr N (+) (1)) (` (ZZ>) (opr M (+) (1))))) (M peano2z (e. N (ZZ)) (br M (<_) N) 3ad2ant1 N peano2z (e. M (ZZ)) (br M (<_) N) 3ad2ant2 ax1re M N (1) leadd1t mp3an3 M zret N zret syl2an biimp3a 3jca N M eluz2t (opr N (+) (1)) (opr M (+) (1)) eluz2t 3imtr4)) thm (nn0uz () () (= (NN0) (` (ZZ>) (0))) (k nn0zrab 0z (0) k uzvalt ax-mp eqtr4)) thm (nnuz () () (= (NN) (` (ZZ>) (1))) (k nnzrab 1z (1) k uzvalt ax-mp eqtr4)) thm (elnnuz () () (<-> (e. N (NN)) (e. N (` (ZZ>) (1)))) (nnuz N eleq2i)) thm (elnn0uz () () (<-> (e. N (NN0)) (e. N (` (ZZ>) (0)))) (nn0uz N eleq2i)) thm (raluz ((M n)) () (-> (e. M (ZZ)) (<-> (A.e. n (` (ZZ>) M) ph) (A.e. n (ZZ) (-> (br M (<_) (cv n)) ph)))) (M (cv n) eluz1t ph imbi1d (e. (cv n) (ZZ)) (br M (<_) (cv n)) ph impexp syl6bb ralbidv2)) thm (raluz2 ((M n)) () (<-> (A.e. n (` (ZZ>) M) ph) (-> (e. M (ZZ)) (A.e. n (ZZ) (-> (br M (<_) (cv n)) ph)))) ((cv n) M eluz2t (e. M (ZZ)) (e. (cv n) (ZZ)) (br M (<_) (cv n)) 3anass bitr ph imbi1i (e. M (ZZ)) (/\ (e. (cv n) (ZZ)) (br M (<_) (cv n))) ph impexp (e. (cv n) (ZZ)) (br M (<_) (cv n)) ph impexp (e. M (ZZ)) imbi2i bitr (e. M (ZZ)) (e. (cv n) (ZZ)) (-> (br M (<_) (cv n)) ph) bi2.04 3bitr ralbii2 n (ZZ) (e. M (ZZ)) (-> (br M (<_) (cv n)) ph) r19.21v bitr)) thm (rexuz ((M n)) () (-> (e. M (ZZ)) (<-> (E.e. n (` (ZZ>) M) ph) (E.e. n (ZZ) (/\ (br M (<_) (cv n)) ph)))) (M (cv n) eluz1t ph anbi1d (e. (cv n) (ZZ)) (br M (<_) (cv n)) ph anass syl6bb rexbidv2)) thm (rexuz2 ((M n)) () (<-> (E.e. n (` (ZZ>) M) ph) (/\ (e. M (ZZ)) (E.e. n (ZZ) (/\ (br M (<_) (cv n)) ph)))) ((cv n) M eluz2t (e. M (ZZ)) (e. (cv n) (ZZ)) (br M (<_) (cv n)) df-3an bitr ph anbi1i (/\ (e. M (ZZ)) (e. (cv n) (ZZ))) (br M (<_) (cv n)) ph anass (e. M (ZZ)) (e. (cv n) (ZZ)) (/\ (br M (<_) (cv n)) ph) anass (e. M (ZZ)) (e. (cv n) (ZZ)) (/\ (br M (<_) (cv n)) ph) an12 bitr 3bitr rexbii2 n (ZZ) (e. M (ZZ)) (/\ (br M (<_) (cv n)) ph) r19.42v bitr)) thm (2rexuz ((m n)) () (<-> (E. m (E.e. n (` (ZZ>) (cv m)) ph)) (E.e. m (ZZ) (E.e. n (ZZ) (/\ (br (cv m) (<_) (cv n)) ph)))) (n (cv m) ph rexuz2 m exbii m (ZZ) (E.e. n (ZZ) (/\ (br (cv m) (<_) (cv n)) ph)) df-rex bitr4)) thm (peano2uz () () (-> (e. N (` (ZZ>) M)) (e. (opr N (+) (1)) (` (ZZ>) M))) ((e. M (ZZ)) (e. N (ZZ)) (br M (<_) N) 3simp1 N peano2z (e. M (ZZ)) (br M (<_) N) 3ad2ant2 M N letrp1t N zret syl3an2 M zret syl3an1 3jca N M eluz2t (opr N (+) (1)) M eluz2t 3imtr4)) thm (peano2uzr () () (-> (/\ (e. M (ZZ)) (e. N (` (ZZ>) (opr M (+) (1))))) (e. N (` (ZZ>) M))) (N (opr M (+) (1)) eluzelz N zcnt 1cn N (1) npcant mpan2 3syl (e. M (ZZ)) adantl M N eluzp1m1t (opr N (-) (1)) M peano2uz syl eqeltrrd)) thm (uzaddclt ((K j) (j k) (M j) (M k) (N j) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (e. K (NN0))) (e. (opr N (+) K) (` (ZZ>) M))) (1cn N (cv k) (1) axaddass mp3an3 N M eluzelz N zcnt syl (cv k) nn0cnt syl2an ancoms (e. (opr N (+) (cv k)) (` (ZZ>) M)) adantr (opr N (+) (cv k)) M peano2uz (/\ (e. (cv k) (NN0)) (e. N (` (ZZ>) M))) adantl eqeltrrd exp31 a2d N M eluzelz N zcnt syl N ax0id syl (` (ZZ>) M) eleq1d ibir (cv j) (0) N (+) opreq2 (` (ZZ>) M) eleq1d (e. N (` (ZZ>) M)) imbi2d (cv j) (cv k) N (+) opreq2 (` (ZZ>) M) eleq1d (e. N (` (ZZ>) M)) imbi2d (cv j) (opr (cv k) (+) (1)) N (+) opreq2 (` (ZZ>) M) eleq1d (e. N (` (ZZ>) M)) imbi2d (cv j) K N (+) opreq2 (` (ZZ>) M) eleq1d (e. N (` (ZZ>) M)) imbi2d nn0indALT impcom)) thm (uzind4 ((j m) (N j) (N m) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (k m) (M k) (M m)) ((uzind4.1 (-> (= (cv j) M) (<-> ph ps))) (uzind4.2 (-> (= (cv j) (cv k)) (<-> ph ch))) (uzind4.3 (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th))) (uzind4.4 (-> (= (cv j) N) (<-> ph ta))) (uzind4.5 (-> (e. M (ZZ)) ps)) (uzind4.6 (-> (e. (cv k) (` (ZZ>) M)) (-> ch th)))) (-> (e. N (` (ZZ>) M)) ta) (uzind4.1 uzind4.2 uzind4.3 uzind4.4 uzind4.5 (cv k) M eluz2t biimpr 3expb (cv m) (cv k) M (<_) breq2 (ZZ) elrab sylan2b uzind4.6 syl uzind3 N M eluzel2 N M eluzelz N M eluzle jca (cv m) N M (<_) breq2 (ZZ) elrab sylibr sylanc)) thm (uzind4ALT ((N j) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (M k)) ((uzind4ALT.5 (-> (e. M (ZZ)) ps)) (uzind4ALT.6 (-> (e. (cv k) (` (ZZ>) M)) (-> ch th))) (uzind4ALT.1 (-> (= (cv j) M) (<-> ph ps))) (uzind4ALT.2 (-> (= (cv j) (cv k)) (<-> ph ch))) (uzind4ALT.3 (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th))) (uzind4ALT.4 (-> (= (cv j) N) (<-> ph ta)))) (-> (e. N (` (ZZ>) M)) ta) (uzind4ALT.1 uzind4ALT.2 uzind4ALT.3 uzind4ALT.4 uzind4ALT.5 uzind4ALT.6 uzind4)) thm (uzind4s ((k m) (j m) (M m) (j k) (M k) (M j) (N j) (j ph) (m ph)) ((uzind4s.1 (-> (e. M (ZZ)) ([/] M k ph))) (uzind4s.2 (-> (e. (cv k) (` (ZZ>) M)) (-> ph ([/] (opr (cv k) (+) (1)) k ph))))) (-> (e. N (` (ZZ>) M)) ([/] N k ph)) ((cv j) M k ph dfsbcq (cv j) (cv m) k ph dfsbcq (cv j) (opr (cv m) (+) (1)) k ph dfsbcq (cv j) N k ph dfsbcq uzind4s.1 (e. (cv m) (` (ZZ>) M)) k ax-17 m visset k ph hbsbc1v (cv m) (+) (1) oprex k ph hbsbc1v hbim hbim (cv k) (cv m) (` (ZZ>) M) eleq1 k m ph sbequ12 (cv k) (cv m) (+) (1) opreq1 (opr (cv k) (+) (1)) (opr (cv m) (+) (1)) k ph dfsbcq syl imbi12d imbi12d uzind4s.2 chvar uzind4)) thm (uzind4s2 ((k m) (k n) (M k) (m n) (M m) (M n) (N m) (k ph) (m ph) (n ph) (j k) (j m) (j n)) ((uzind4s2.1 (-> (e. M (ZZ)) ([/] M j ph))) (uzind4s2.2 (-> (e. (cv k) (` (ZZ>) M)) (-> ([/] (cv k) j ph) ([/] (opr (cv k) (+) (1)) j ph))))) (-> (e. N (` (ZZ>) M)) ([/] N j ph)) ((cv m) M j ph dfsbcq (cv m) (cv n) j ph dfsbcq (cv m) (opr (cv n) (+) (1)) j ph dfsbcq (cv m) N j ph dfsbcq uzind4s2.1 (e. (cv n) (` (ZZ>) M)) k ax-17 ([/] (cv n) j ph) k ax-17 ([/] (opr (cv n) (+) (1)) j ph) k ax-17 hbim hbim (cv k) (cv n) (` (ZZ>) M) eleq1 (cv k) (cv n) j ph dfsbcq (cv k) (cv n) (+) (1) opreq1 (opr (cv k) (+) (1)) (opr (cv n) (+) (1)) j ph dfsbcq syl imbi12d imbi12d uzind4s2.2 chvar uzind4)) thm (uzind4i ((j m) (N j) (N m) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (k m) (M k) (M m)) ((uzind4i.1 (e. M (ZZ))) (uzind4i.2 (-> (= (cv j) M) (<-> ph ps))) (uzind4i.3 (-> (= (cv j) (cv k)) (<-> ph ch))) (uzind4i.4 (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th))) (uzind4i.5 (-> (= (cv j) N) (<-> ph ta))) (uzind4i.6 ps) (uzind4i.7 (-> (e. (cv k) (` (ZZ>) M)) (-> ch th)))) (-> (e. N (` (ZZ>) M)) ta) (uzind4i.1 M m uzvalt ax-mp N eleq2i uzind4i.1 uzind4i.2 uzind4i.3 uzind4i.4 uzind4i.5 uzind4i.6 (e. M (ZZ)) a1i uzind4i.1 M m uzvalt ax-mp (cv k) eleq2i uzind4i.7 sylbir (e. M (ZZ)) adantl uzind3 mpan sylbi)) thm (uzwo ((h t) (h n) (h m) (M h) (n t) (m t) (M t) (m n) (M n) (M m) (h j) (h k) (S h) (j t) (j k) (j m) (j n) (S j) (k t) (S t) (k m) (k n) (S k) (S m) (S n)) () (-> (/\ (C_ S (` (ZZ>) M)) (-. (= S ({/})))) (E.e. j S (A.e. k S (br (cv j) (<_) (cv k))))) ((cv h) M (<_) (cv t) breq1 t S ralbidv (/\ (C_ S (` (ZZ>) M)) (-. (E.e. j S (A.e. t S (br (cv j) (<_) (cv t)))))) imbi2d (cv h) (cv m) (<_) (cv t) breq1 t S ralbidv (/\ (C_ S (` (ZZ>) M)) (-. (E.e. j S (A.e. t S (br (cv j) (<_) (cv t)))))) imbi2d (cv h) (opr (cv m) (+) (1)) (<_) (cv t) breq1 t S ralbidv (/\ (C_ S (` (ZZ>) M)) (-. (E.e. j S (A.e. t S (br (cv j) (<_) (cv t)))))) imbi2d (cv h) (cv n) (<_) (cv t) breq1 t S ralbidv (/\ (C_ S (` (ZZ>) M)) (-. (E.e. j S (A.e. t S (br (cv j) (<_) (cv t)))))) imbi2d S (` (ZZ>) M) (cv t) ssel (cv t) M eluzle syl6 r19.21aiv (-. (E.e. j S (A.e. t S (br (cv j) (<_) (cv t))))) adantr (e. M (ZZ)) a1i (cv m) M eluzelz (cv j) (cv m) (<_) (cv t) breq1 t S ralbidv S rcla4ev expcom con3d com12 (cv m) (cv t) letri3t (cv m) zret (cv t) zret syl2an (cv t) (cv m) zleltp1t (cv t) (opr (cv m) (+) (1)) ltnlet (cv t) zret (cv m) zret (cv m) peano2re syl syl2an bitrd ancoms (br (cv m) (<_) (cv t)) anbi2d bitrd S (ZZ) (cv t) ssel2 sylan2 (cv t) S (cv m) eleq1a (e. (cv m) (ZZ)) (C_ S (ZZ)) ad2antll sylbird exp3a (br (opr (cv m) (+) (1)) (<_) (cv t)) (e. (cv m) S) con1 syl6 com23 exp32 com34 imp41 r19.20dva ex sylan9r pm2.43d exp31 imp3a syl M uzssz S (` (ZZ>) M) (ZZ) sstr mpan2 sylani a2d uzind4 (cv j) (cv n) (<_) (cv t) breq1 t S ralbidv S rcla4ev expcom con3d com12 (C_ S (` (ZZ>) M)) adantl sylcom S (` (ZZ>) M) (cv n) ssel con3d com12 (-. (E.e. j S (A.e. t S (br (cv j) (<_) (cv t))))) adantrd pm2.61i ex n 19.21adv S n eq0 syl6ibr con1d imp (cv t) (cv k) (cv j) (<_) breq2 S cbvralv j S rexbii sylib)) thm (uzwo2 ((j k) (S j) (S k)) () (-> (/\ (C_ S (` (ZZ>) M)) (-. (= S ({/})))) (E!e. j S (A.e. k S (br (cv j) (<_) (cv k))))) (S M j k uzwo S j k lbreu M uzssz zssre sstri S (` (ZZ>) M) (RR) sstr mpan2 sylan syldan)) thm (nnwo ((x y) (A x) (A y)) () (-> (/\ (C_ A (NN)) (=/= A ({/}))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (A (1) x y uzwo nnuz A sseq2i A ({/}) df-ne syl2anb)) thm (nnwof ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (A z)) ((nnwof.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (nnwof.2 (-> (e. (cv z) A) (A. y (e. (cv z) A))))) (-> (/\ (C_ A (NN)) (=/= A ({/}))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (A w v nnwo (e. (cv z) (cv v)) y ax-17 nnwof.2 hbel (br (cv w) (<_) (cv v)) y ax-17 hbim (-> (e. (cv y) A) (br (cv w) (<_) (cv y))) v ax-17 (cv v) (cv y) A eleq1 (cv v) (cv y) (cv w) (<_) breq2 imbi12d cbval (e. (cv w) A) anbi2i w exbii (e. (cv z) (cv w)) x ax-17 nnwof.1 hbel (e. (cv z) (cv y)) x ax-17 nnwof.1 hbel (br (cv w) (<_) (cv y)) x ax-17 hbim y hbal hban (/\ (e. (cv x) A) (A. y (-> (e. (cv y) A) (br (cv x) (<_) (cv y))))) w ax-17 (cv w) (cv x) A eleq1 (cv w) (cv x) (<_) (cv y) breq1 (e. (cv y) A) imbi2d y albidv anbi12d cbvex bitr w A (A. v (-> (e. (cv v) A) (br (cv w) (<_) (cv v)))) df-rex x A (A. y (-> (e. (cv y) A) (br (cv x) (<_) (cv y)))) df-rex 3bitr4 v A (br (cv w) (<_) (cv v)) df-ral w A rexbii y A (br (cv x) (<_) (cv y)) df-ral x A rexbii 3bitr4 sylib)) thm (nnwos ((x y) (x z) (y z) (ph y) (ph z) (ps x)) ((nnwos.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (-> (E.e. x (NN) ph) (E.e. x (NN) (/\ ph (A.e. y (NN) (-> ps (br (cv x) (<_) (cv y))))))) (z x (NN) ph hbrab1 (e. (cv z) ({e.|} x (NN) ph)) y ax-17 nnwof ({e.|} x (NN) ph) ({/}) df-ne x (NN) ph ssrab2 (=/= ({e.|} x (NN) ph) ({/})) biantrur x (NN) ph rabn0 3bitr3 x ({e.|} x (NN) ph) (A.e. y ({e.|} x (NN) ph) (br (cv x) (<_) (cv y))) df-rex x (NN) ph rabid y ({e.|} x (NN) ph) (br (cv x) (<_) (cv y)) df-ral nnwos.1 (NN) elrab (br (cv x) (<_) (cv y)) imbi1i (e. (cv y) (NN)) ps (br (cv x) (<_) (cv y)) impexp bitr y albii bitr anbi12i x exbii y (NN) (-> ps (br (cv x) (<_) (cv y))) df-ral (/\ (e. (cv x) (NN)) ph) anbi2i (e. (cv x) (NN)) ph (A.e. y (NN) (-> ps (br (cv x) (<_) (cv y)))) anass bitr3 x exbii x (NN) (/\ ph (A.e. y (NN) (-> ps (br (cv x) (<_) (cv y))))) df-rex bitr4 3bitr 3imtr3)) thm (indstr ((x y) (ph y) (ps x)) ((indstr.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (indstr.2 (-> (e. (cv x) (NN)) (-> (A.e. y (NN) (-> (br (cv y) (<) (cv x)) ps)) ph)))) (-> (e. (cv x) (NN)) ph) (ph pm3.24 (cv x) (cv y) lenltt (cv x) nnret (cv y) nnret syl2an (-. ps) imbi2d (br (cv y) (<) (cv x)) ps pm4.1 syl6bbr ralbidva indstr.2 sylbid (-. ph) anim2d (-. ph) ph ancom syl6ib mtoi nrex indstr.1 negbid nnwos mto x (NN) ph dfral2 mpbir rspec)) thm (uzinfm ((j k) (M j) (M k)) ((uzinfm.1 (e. M (ZZ)))) (= (sup (` (ZZ>) M) (RR) (`' (<))) M) (M uzssz zssre sstri uzinfm.1 M uzidt ax-mp (cv k) M eluzle rgen (cv j) M (<_) (cv k) breq1 k (` (ZZ>) M) ralbidv (` (ZZ>) M) rcla4ev mp2an (` (ZZ>) M) j k lbinfm mp2an (cv k) M eluzle rgen uzinfm.1 M uzidt ax-mp M uzssz zssre sstri uzinfm.1 M uzidt ax-mp (cv k) M eluzle rgen (cv j) M (<_) (cv k) breq1 k (` (ZZ>) M) ralbidv (` (ZZ>) M) rcla4ev mp2an (` (ZZ>) M) j k lbreu mp2an (cv j) M (<_) (cv k) breq1 k (` (ZZ>) M) ralbidv (` (ZZ>) M) reuuni2 mp2an mpbi eqtr)) thm (nninfm () () (= (sup (NN) (RR) (`' (<))) (1)) (nnuz (NN) (` (ZZ>) (1)) (RR) (`' (<)) supeq1 ax-mp 1z uzinfm eqtr)) thm (nn0infm () () (= (sup (NN0) (RR) (`' (<))) (0)) (nn0uz (NN0) (` (ZZ>) (0)) (RR) (`' (<)) supeq1 ax-mp 0z uzinfm eqtr)) thm (infmssuzcl ((j k) (S j) (S k)) () (-> (/\ (C_ S (` (ZZ>) M)) (-. (= S ({/})))) (e. (sup S (RR) (`' (<))) S)) (S j k lbinfmcl M uzssz zssre sstri S (` (ZZ>) M) (RR) sstr mpan2 (-. (= S ({/}))) adantr S M j k uzwo sylanc)) thm (fzvalt ((k m) (k n) (k z) (M k) (m n) (m z) (M m) (n z) (M n) (M z) (N k) (N m) (N n) (N z)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (= (opr M (...) N) ({e.|} k (ZZ) (/\ (br M (<_) (cv k)) (br (cv k) (<_) N))))) (zex k (/\ (br M (<_) (cv k)) (br (cv k) (<_) N)) rabex (cv m) M (<_) (cv k) breq1 (br (cv k) (<_) (cv n)) anbi1d k (ZZ) rabbisdv (cv n) N (cv k) (<_) breq2 (br M (<_) (cv k)) anbi2d k (ZZ) rabbisdv m n z k df-fz oprabval2)) thm (elfz1t ((K j) (M j) (N j)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (e. K (opr M (...) N)) (/\/\ (e. K (ZZ)) (br M (<_) K) (br K (<_) N)))) (M N j fzvalt K eleq2d (cv j) K M (<_) breq2 (cv j) K (<_) N breq1 anbi12d (ZZ) elrab (e. K (ZZ)) (br M (<_) K) (br K (<_) N) 3anass bitr4 syl6bb)) thm (elfzt () () (-> (/\/\ (e. K (ZZ)) (e. M (ZZ)) (e. N (ZZ))) (<-> (e. K (opr M (...) N)) (/\ (br M (<_) K) (br K (<_) N)))) (M N K elfz1t (e. K (ZZ)) (br M (<_) K) (br K (<_) N) 3anass baib sylan9bb 3impa 3comr)) thm (elfz2t ((k n) (k m) (k z) (M k) (m n) (n z) (M n) (m z) (M m) (M z) (N k) (N n) (N m) (N z)) () (-> (e. N A) (<-> (e. K (opr M (...) N)) (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ))) (/\ (br M (<_) K) (br K (<_) N))))) (M N K elfz1t (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. K (ZZ)) (/\ (br M (<_) K) (br K (<_) N))) ibar (e. K (ZZ)) (br M (<_) K) (br K (<_) N) 3anass syl5bb bitrd (e. N A) adantl zex k (/\ (br (cv m) (<_) (cv k)) (br (cv k) (<_) (cv n))) rabex m n z k df-fz dmoprab2 N A M ndmoprg K eleq2d K noel (/\ (e. M (ZZ)) (e. N (ZZ))) pm2.21i (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. K (ZZ)) (/\ (br M (<_) K) (br K (<_) N))) pm3.26 pm5.21ni (e. N A) adantl bitrd pm2.61dan (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ)) df-3an (/\ (br M (<_) K) (br K (<_) N)) anbi1i (/\ (e. M (ZZ)) (e. N (ZZ))) (e. K (ZZ)) (/\ (br M (<_) K) (br K (<_) N)) anass bitr2 syl6bb)) thm (elfzlem ((k n) (k m) (k z) (M k) (m n) (n z) (M n) (m z) (M m) (M z) (N k) (N n) (N m) (N z)) () (-> (e. K (opr M (...) N)) (/\ (/\ (e. M (ZZ)) (e. K (ZZ))) (/\ (br M (<_) K) (br K (<_) N)))) (N (V) K M elfz2t (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ)) 3simpb (/\ (br M (<_) K) (br K (<_) N)) anim1i syl6bi N M (...) oprprc2 K eleq2d N K (<_) brprc (br M (<_) K) anbi2d (/\ (e. M (ZZ)) (e. K (ZZ))) anbi2d M (V) K M elfz2t (e. M (ZZ)) (e. M (ZZ)) (e. K (ZZ)) 3simpb (br M (<_) K) (br K (<_) M) pm3.26 anim12i syl6bi M (V) K M elfz2t K zret K leidt syl (e. M (ZZ)) (e. M (ZZ)) 3ad2ant3 (/\ (br M (<_) K) (br K (<_) M)) adantr syl6bi jcad (/\ (e. M (ZZ)) (e. K (ZZ))) (br M (<_) K) (br K (<_) K) anass syl6ib (ZZ) (ZZ) relxp zex k (/\ (br (cv m) (<_) (cv k)) (br (cv k) (<_) (cv n))) rabex m n z k df-fz dmoprab2 (dom (...)) (X. (ZZ) (ZZ)) releq ax-mp mpbir M M oprprc1 (/\ (/\ (e. M (ZZ)) (e. K (ZZ))) (/\ (br M (<_) K) (br K (<_) K))) pm2.21nd K (opr M (...) M) n0i syl5 pm2.61i syl5bir sylbid pm2.61i)) thm (elfz5t () () (-> (/\ (e. K (` (ZZ>) M)) (e. N (ZZ))) (<-> (e. K (opr M (...) N)) (br K (<_) N))) (K M N elfzt 3expa K M eluzelz K M eluzel2 jca sylan K M eluzle (br K (<_) N) biantrurd (e. N (ZZ)) adantr bitr4d)) thm (elfz4t () () (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ))) (/\ (br M (<_) K) (br K (<_) N))) (e. K (opr M (...) N))) (M N K elfz1t biimprcd 3exp com4r 3impia imp32)) thm (elfzuzb () () (-> (e. N A) (<-> (e. K (opr M (...) N)) (/\ (e. K (` (ZZ>) M)) (e. N (` (ZZ>) K))))) (N A K M elfz2t (e. M (ZZ)) (e. K (ZZ)) (br M (<_) K) (e. K (ZZ)) (e. N (ZZ)) (br K (<_) N) an6 K M eluz2t N K eluz2t anbi12i (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ)) df-3an (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ)) anandir (e. N (ZZ)) (e. K (ZZ)) ancom (/\ (e. M (ZZ)) (e. K (ZZ))) anbi2i 3bitr (/\ (br M (<_) K) (br K (<_) N)) anbi1i (/\ (e. M (ZZ)) (e. K (ZZ))) (/\ (e. K (ZZ)) (e. N (ZZ))) (/\ (br M (<_) K) (br K (<_) N)) df-3an bitr4 3bitr4r syl6bb)) thm (eluzfzt () () (-> (/\ (e. K (` (ZZ>) M)) (e. N (` (ZZ>) K))) (e. K (opr M (...) N))) (N K eluzelz (e. K (` (ZZ>) M)) adantl N (ZZ) K M elfzuzb biimpar mpancom)) thm (elfzuz3t () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. N (` (ZZ>) K))) (N A K M elfzuzb biimpa pm3.27d)) thm (elfzel2 () ((elfzel2.1 (e. N (V)))) (-> (e. K (opr M (...) N)) (e. N (ZZ))) (elfzel2.1 N (V) K M elfz2t ax-mp (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ)) 3simp2 (/\ (br M (<_) K) (br K (<_) N)) adantr sylbi)) thm (elfzel2g () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. N (ZZ))) (N A K M elfz2t (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ)) 3simp2 (/\ (br M (<_) K) (br K (<_) N)) adantr syl6bi imp)) thm (elfzel1 () () (-> (e. K (opr M (...) N)) (e. M (ZZ))) (K M N elfzlem (e. M (ZZ)) (e. K (ZZ)) pm3.26 (/\ (br M (<_) K) (br K (<_) N)) adantr syl)) thm (elfzelz () () (-> (e. K (opr M (...) N)) (e. K (ZZ))) (K M N elfzlem (e. M (ZZ)) (e. K (ZZ)) pm3.27 (/\ (br M (<_) K) (br K (<_) N)) adantr syl)) thm (elfzle1 () () (-> (e. K (opr M (...) N)) (br M (<_) K)) (K M N elfzlem (br M (<_) K) (br K (<_) N) pm3.26 (/\ (e. M (ZZ)) (e. K (ZZ))) adantl syl)) thm (elfzle2 () () (-> (e. K (opr M (...) N)) (br K (<_) N)) (K M N elfzlem (br M (<_) K) (br K (<_) N) pm3.27 (/\ (e. M (ZZ)) (e. K (ZZ))) adantl syl)) thm (elfzle3 () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (br M (<_) N)) (K M N elfzle1 (e. N A) adantl K M N elfzle2 (e. N A) adantl M K N letrt M zret K zret N zret syl3an K M N elfzel1 (e. N A) adantl K M N elfzelz (e. N A) adantl N A K M elfzel2g syl3anc mp2and)) thm (elfzuz2t () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. N (` (ZZ>) M))) (N A K M elfzuzb (e. K (` (ZZ>) M)) (e. N (` (ZZ>) K)) ancom syl6bb biimpa N K M uztrn syl)) thm (eluzfz1t () () (-> (e. N (` (ZZ>) M)) (e. M (opr M (...) N))) (N M eluz2t M N M elfz4t (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N) 3simp1 (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N) 3simp2 (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N) 3simp1 3jca M zret M leidt syl (e. N (ZZ)) (br M (<_) N) 3ad2ant1 (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N) 3simp3 jca sylanc sylbi)) thm (elfzuzt () () (-> (e. K (opr M (...) N)) (e. K (` (ZZ>) M))) (K M N elfzel1 K M N elfzelz K M N elfzle1 3jca K M eluz2t sylibr)) thm (eluzfz2t () () (-> (e. N (` (ZZ>) M)) (e. N (opr M (...) N))) (M N N elfz4t N M eluzel2 N M eluzelz N M eluzelz 3jca N M eluzle N M eluzelz N zret N leidt 3syl jca sylanc)) thm (eluzfz2b () () (<-> (e. N (` (ZZ>) M)) (e. N (opr M (...) N))) (N M eluzfz2t N M N elfzuzt impbi)) thm (elfz3t () () (-> (e. N (ZZ)) (e. N (opr N (...) N))) (N uzidt N N eluzfz1t syl)) thm (elfz1eqt () () (-> (e. K (opr N (...) N)) (= K N)) (K N N elfzlem N K letri3t N zret K zret syl2an biimpar eqcomd syl)) thm (fznt ((M k) (N k)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br N (<) M) (= (opr M (...) N) ({/})))) (M N lenltt M zret N zret syl2an N M eluz2t N M eluzfz1t sylbir M (opr M (...) N) n0i syl 3expa ex N (ZZ) (cv k) M elfzle3 ex k 19.23adv (opr M (...) N) k n0 syl5ib (e. M (ZZ)) adantl impbid bitr3d con4bid)) thm (elfznnt () () (-> (e. K (opr (1) (...) N)) (e. K (NN))) (K (1) N elfzelz K (1) N elfzle1 jca K elnnz1 sylibr)) thm (elfz2nn0t () () (-> (e. N A) (<-> (e. K (opr (0) (...) N)) (/\/\ (e. K (NN0)) (e. N (NN0)) (br K (<_) N)))) (N A K (0) elfz2t (e. N (ZZ)) (e. K (ZZ)) pm3.27 (br (0) (<_) K) (br K (<_) N) pm3.26 anim12i (e. N (ZZ)) (e. K (ZZ)) pm3.26 (/\ (br (0) (<_) K) (br K (<_) N)) adantr 0re (0) K N letrt mp3an1 ancoms N zret K zret syl2an imp jca (br (0) (<_) K) (br K (<_) N) pm3.27 (/\ (e. N (ZZ)) (e. K (ZZ))) adantl 3jca (e. (0) (ZZ)) 3adantl1 0z (br K (<_) N) a1i (e. N (ZZ)) (br (0) (<_) N) pm3.26 (e. K (ZZ)) (br (0) (<_) K) pm3.26 3anim123i 3com13 (/\ (br (0) (<_) K) (br K (<_) N)) id (e. K (ZZ)) adantll (/\ (e. N (ZZ)) (br (0) (<_) N)) 3adant2 jca impbi K elnn0z N elnn0z (br K (<_) N) pm4.2 3anbi123i bitr4 syl6bb)) thm (elfznn0t () () (-> (e. K (opr (0) (...) N)) (e. K (NN0))) (K (0) N elfzelz K (0) N elfzle1 jca K elnn0z sylibr)) thm (elfz3nn0t () () (-> (/\ (e. N A) (e. K (opr (0) (...) N))) (e. N (NN0))) (N A K elfz2nn0t (e. K (NN0)) (e. N (NN0)) (br K (<_) N) 3simp2 syl6bi imp)) thm (fznn0subt () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. (opr N (-) K) (NN0))) (N A K M elfz2t K N znn0sub2t 3com12 3expa (br M (<_) K) adantrl (e. M (ZZ)) 3adantl1 syl6bi imp)) thm (fznn0sub2t () () (-> (/\ (e. N A) (e. K (opr (0) (...) N))) (e. (opr N (-) K) (opr (0) (...) N))) (N A K (0) fznn0subt N A K elfz3nn0t K (0) N elfzle1 (e. N A) adantl N K subge02t N zret K zret syl2an N A K (0) elfzel2g K (0) N elfzelz (e. N A) adantl sylanc mpbid 3jca N A (opr N (-) K) elfz2nn0t biimpar syldan)) thm (fzaddelt () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. J (opr M (...) N)) (e. (opr J (+) K) (opr (opr M (+) K) (...) (opr N (+) K))))) (J K zaddclt (e. J (ZZ)) a1d (e. J (ZZ)) (e. K (ZZ)) pm3.26 (e. (opr J (+) K) (ZZ)) a1d impbid (/\ (e. M (ZZ)) (e. N (ZZ))) adantl M J K leadd1t M zret J zret K zret syl3an 3expb (e. N (ZZ)) adantlr J N K leadd1t J zret N zret K zret syl3an 3com12 3expb (e. M (ZZ)) adantll 3anbi123d M N J elfz1t (/\ (e. J (ZZ)) (e. K (ZZ))) adantr (opr M (+) K) (opr N (+) K) (opr J (+) K) elfz1t M K zaddclt N K zaddclt syl2an anandirs (e. J (ZZ)) adantrl 3bitr4d)) thm (fzsubelt () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. J (opr M (...) N)) (e. (opr J (-) K) (opr (opr M (-) K) (...) (opr N (-) K))))) (M N J (-u K) fzaddelt K znegclt sylanr2 J K negsubt (/\ (e. M (CC)) (e. N (CC))) adantl M K negsubt N K negsubt (...) opreqan12d anandirs (e. J (CC)) adantrl eleq12d M zcnt N zcnt anim12i J zcnt K zcnt anim12i syl2an bitrd)) thm (fzoptht () () (-> (/\ (e. N (` (ZZ>) M)) (e. K A)) (<-> (= (opr M (...) N) (opr J (...) K)) (/\ (= M J) (= N K)))) ((opr M (...) N) (opr J (...) K) N eleq2 N M eluzfz2t syl5bi (e. K A) anim1d K A N J elfzuz2t ancoms syl6com (opr M (...) N) (opr J (...) K) J eleq2 K J eluzfz1t syl5bir J M N elfzle1 syl6com (e. N (` (ZZ>) M)) adantl (opr M (...) N) (opr J (...) K) M eleq2 N M eluzfz1t syl5bi M J K elfzle1 syl6com (e. K (` (ZZ>) J)) adantr jcad M J letri3t N M eluzel2 M zret syl K J eluzel2 J zret syl syl2an sylibrd (opr M (...) N) (opr J (...) K) N eleq2 N M eluzfz2t syl5bi N J K elfzle2 syl6com (e. K (` (ZZ>) J)) adantr (opr M (...) N) (opr J (...) K) K eleq2 K J eluzfz2t syl5bir K M N elfzle2 syl6com (e. N (` (ZZ>) M)) adantl jcad N K letri3t N M eluzelz N zret syl K J eluzelz K zret syl syl2an sylibrd jcad ex com23 (e. K A) adantr mpdd M J N K (...) opreq12 (/\ (e. N (` (ZZ>) M)) (e. K A)) a1i impbid)) thm (fzss1t ((M k) (N k) (K k)) () (-> (/\ (e. K (` (ZZ>) M)) (e. N (ZZ))) (C_ (opr K (...) N) (opr M (...) N))) (K M eluzle (e. (cv k) (ZZ)) adantr M K (cv k) letrt M zret K zret (cv k) zret syl3an 3expa K M eluzel2 K M eluzelz jca sylan mpand (br (cv k) (<_) N) anim1d ss2rabdv (e. N (ZZ)) adantr K N k fzvalt K M eluzelz sylan M N k fzvalt K M eluzel2 sylan 3sstr4d)) thm (fzss2t ((M k) (N k) (K k)) () (-> (/\ (e. N (` (ZZ>) K)) (e. M (ZZ))) (C_ (opr M (...) K) (opr M (...) N))) (N K eluzle (e. (cv k) (ZZ)) adantl (cv k) K N letrt (cv k) zret K zret N zret syl3an 3expb N K eluzel2 N K eluzelz jca sylan2 mpan2d ancoms (br M (<_) (cv k)) anim2d ss2rabdv (e. M (ZZ)) adantl M K k fzvalt N K eluzel2 sylan2 M N k fzvalt N K eluzelz sylan2 3sstr4d ancoms)) thm (fzssuzt ((M k) (N k)) () (C_ (opr M (...) N) (` (ZZ>) M)) ((cv k) M N elfzuzt ssriv)) thm (fzssp1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (C_ (opr M (...) N) (opr M (...) (opr N (+) (1))))) ((opr N (+) (1)) N M fzss2t N uzidt N N peano2uz syl sylan ancoms)) thm (fzp1sst () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (C_ (opr (opr M (+) (1)) (...) N) (opr M (...) N))) ((opr M (+) (1)) M N fzss1t M uzidt M M peano2uz syl sylan)) thm (fzelp1t () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. K (opr M (...) (opr N (+) (1))))) (M N fzssp1t K M N elfzel1 (e. N A) adantl N A K M elfzel2g sylanc (e. N A) (e. K (opr M (...) N)) pm3.27 sseldd)) thm (fzelp1 () ((fzelp1.1 (e. N (V)))) (-> (e. K (opr M (...) N)) (e. K (opr M (...) (opr N (+) (1))))) (fzelp1.1 N (V) K M fzelp1t mpan)) thm (elfzp1 () () (-> (e. N (` (ZZ>) M)) (<-> (e. K (opr M (...) (opr N (+) (1)))) (\/ (e. K (opr M (...) N)) (= K (opr N (+) (1)))))) (N M eluzel2 (e. K (opr M (...) (opr N (+) (1)))) (-. (= K (opr N (+) (1)))) ad2antrr N M eluzelz (e. K (opr M (...) (opr N (+) (1)))) (-. (= K (opr N (+) (1)))) ad2antrr K M (opr N (+) (1)) elfzelz (e. N (` (ZZ>) M)) (-. (= K (opr N (+) (1)))) ad2antlr 3jca K M (opr N (+) (1)) elfzle1 (e. N (` (ZZ>) M)) (-. (= K (opr N (+) (1)))) ad2antlr K M (opr N (+) (1)) elfzle2 (e. N (` (ZZ>) M)) adantl (-. (= K (opr N (+) (1)))) anim1i K (opr N (+) (1)) ltlent K M (opr N (+) (1)) elfzelz K zret syl (e. N (` (ZZ>) M)) (-. (= K (opr N (+) (1)))) ad2antlr N M eluzelz N zret N peano2re 3syl (e. K (opr M (...) (opr N (+) (1)))) (-. (= K (opr N (+) (1)))) ad2antrr sylanc mpbird K N zleltp1t K M (opr N (+) (1)) elfzelz (e. N (` (ZZ>) M)) (-. (= K (opr N (+) (1)))) ad2antlr N M eluzelz (e. K (opr M (...) (opr N (+) (1)))) (-. (= K (opr N (+) (1)))) ad2antrr sylanc mpbird jca jca ex N (` (ZZ>) M) K M elfz2t (e. K (opr M (...) (opr N (+) (1)))) adantr sylibrd orrd (= K (opr N (+) (1))) (e. K (opr M (...) N)) orcom sylib ex M N fzssp1t N M eluzel2 N M eluzelz sylanc K sseld K (opr N (+) (1)) (opr M (...) (opr N (+) (1))) eleq1 N M peano2uz (opr N (+) (1)) M eluzfz2t syl syl5bir com12 jaod impbid)) thm (fzrevt () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. K (opr (opr J (-) N) (...) (opr J (-) M))) (e. (opr J (-) K) (opr M (...) N)))) (J K N sublet J zret K zret N zret syl3an 3comr 3expb (e. M (ZZ)) adantll M J K lesubt M zret J zret K zret syl3an 3expb (e. N (ZZ)) adantlr anbi12d (br (opr J (-) K) (<_) N) (br M (<_) (opr J (-) K)) ancom syl5rbbr K (opr J (-) N) (opr J (-) M) elfzt (e. J (ZZ)) (e. K (ZZ)) pm3.27 (/\ (e. M (ZZ)) (e. N (ZZ))) adantl J N zsubclt ancoms (e. K (ZZ)) adantrr (e. M (ZZ)) adantll J M zsubclt ancoms (e. N (ZZ)) (e. K (ZZ)) ad2ant2r syl3anc (opr J (-) K) M N elfzt J K zsubclt (/\ (e. M (ZZ)) (e. N (ZZ))) adantl (e. M (ZZ)) (e. N (ZZ)) pm3.26 (/\ (e. J (ZZ)) (e. K (ZZ))) adantr (e. M (ZZ)) (e. N (ZZ)) pm3.27 (/\ (e. J (ZZ)) (e. K (ZZ))) adantr syl3anc 3bitr4d)) thm (fzrev2t () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. K (opr M (...) N)) (e. (opr J (-) K) (opr (opr J (-) N) (...) (opr J (-) M))))) (M N J (opr J (-) K) fzrevt (e. J (ZZ)) (e. K (ZZ)) pm3.26 J K zsubclt jca sylan2 J K nncant J zcnt K zcnt syl2an (/\ (e. M (ZZ)) (e. N (ZZ))) adantl (opr M (...) N) eleq1d bitr2d)) thm (fzrev2it () () (-> (/\/\ (e. N A) (e. J (ZZ)) (e. K (opr M (...) N))) (e. (opr J (-) K) (opr (opr J (-) N) (...) (opr J (-) M)))) ((e. N A) (e. J (ZZ)) (e. K (opr M (...) N)) 3simp3 M N J K fzrev2t K M N elfzel1 (e. N A) adantl N A K M elfzel2g jca (e. J (ZZ)) 3adant2 K M N elfzelz (e. J (ZZ)) anim2i (e. N A) 3adant1 sylanc mpbid)) thm (fzrev3t () () (-> (/\ (e. N A) (e. K (ZZ))) (<-> (e. K (opr M (...) N)) (e. (opr (opr M (+) N) (-) K) (opr M (...) N)))) (K M N elfzelz (/\ (e. N A) (e. K (ZZ))) adantl K M N elfzel1 (/\ (e. N A) (e. K (ZZ))) adantl N A K M elfzel2g (e. K (ZZ)) adantlr 3jca (e. N A) (e. K (ZZ)) pm3.27 (e. (opr (opr M (+) N) (-) K) (opr M (...) N)) adantr (opr (opr M (+) N) (-) K) M N elfzel1 (/\ (e. N A) (e. K (ZZ))) adantl N A (opr (opr M (+) N) (-) K) M elfzel2g (e. K (ZZ)) adantlr 3jca M N pncant M N pncan2t (...) opreq12d M zcnt N zcnt syl2an K eleq2d (e. K (ZZ)) 3adant1 M N (opr M (+) N) K fzrevt (e. K (ZZ)) (e. M (ZZ)) (e. N (ZZ)) 3simpc M N zaddclt (e. K (ZZ)) 3adant1 (e. K (ZZ)) (e. M (ZZ)) (e. N (ZZ)) 3simp1 jca sylanc bitr3d pm5.21nd)) thm (fzrev3it () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. (opr (opr M (+) N) (-) K) (opr M (...) N))) ((e. N A) (e. K (opr M (...) N)) pm3.27 N A K M fzrev3t K M N elfzelz sylan2 mpbid)) thm (fznn0t () () (-> (e. N (NN0)) (<-> (e. K (opr (0) (...) N)) (/\ (e. K (NN0)) (br K (<_) N)))) (N nn0zt 0z jctil (0) N K elfz1t syl (e. K (ZZ)) (br (0) (<_) K) (br K (<_) N) df-3an K elnn0z (br K (<_) N) anbi1i bitr4 syl6bb)) thm (fz1sbct ((N k)) () (-> (e. N (ZZ)) (<-> (A.e. k (opr N (...) N) ph) ([/] N k ph))) (N (ZZ) k ph sbc6g (cv k) N elfz1eqt (e. N (ZZ)) a1i (cv k) N (opr N (...) N) eleq1 N elfz3t syl5bir com12 impbid ph imbi1d k albidv k (opr N (...) N) ph df-ral syl5rbb bitr2d)) thm (fzneuzt () () (-> (/\ (e. N (` (ZZ>) M)) (e. K (ZZ))) (-. (= (opr M (...) N) (` (ZZ>) K)))) ((opr N (+) (1)) (` (ZZ>) K) (opr M (...) N) nelneq2 N K peano2uz (/\ (e. N (` (ZZ>) M)) (e. K (ZZ))) adantl N M eluzelz N zret N ltp1t N peano2re N (opr N (+) (1)) ltnlet mpdan mpbid 3syl (opr N (+) (1)) M N elfzle2 nsyl (e. K (ZZ)) (e. N (` (ZZ>) K)) ad2antrr sylanc (` (ZZ>) K) (opr M (...) N) eqcom negbii sylib N (opr M (...) N) (` (ZZ>) K) nelneq2 N M eluzfz2t (e. K (ZZ)) (-. (e. N (` (ZZ>) K))) ad2antrr (/\ (e. N (` (ZZ>) M)) (e. K (ZZ))) (-. (e. N (` (ZZ>) K))) pm3.27 sylanc pm2.61dan)) thm (fzrevralt ((j k) (j x) (K j) (k x) (K k) (K x) (M j) (M k) (M x) (N j) (N k) (N x) (k ph) (ph x)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ))) (<-> (A.e. j (opr M (...) N) ph) (A.e. k (opr (opr K (-) N) (...) (opr K (-) M)) ([/] (opr K (-) (cv k)) j ph)))) ((/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (e. K (ZZ))) (e. (cv k) (opr (opr K (-) N) (...) (opr K (-) M))) pm3.27 M N K (cv k) fzrevt anassrs (cv k) (opr K (-) N) (opr K (-) M) elfzelz sylan2 mpbid (e. (cv x) (opr K (-) (cv k))) j ax-17 (opr M (...) N) ph ra4sbcf syl ex 3impa com23 r19.21adv (/\ (e. N (ZZ)) (e. K (ZZ))) j ax-17 (e. (cv k) (opr (opr K (-) N) (...) (opr K (-) M))) j ax-17 K (-) (cv k) oprex j ph hbsbc1v hbral N (ZZ) K (cv j) M fzrev2it 3expa (cv k) (opr K (-) (cv j)) K (-) opreq2 (opr K (-) (cv k)) (opr K (-) (opr K (-) (cv j))) j ph dfsbcq syl (opr (opr K (-) N) (...) (opr K (-) M)) rcla4v syl K (cv j) nncant K zcnt (cv j) M N elfzelz (cv j) zcnt syl syl2an eqcomd j (opr K (-) (opr K (-) (cv j))) ph sbceq1a syl (e. N (ZZ)) adantll sylibrd ex com23 r19.21ad (e. M (ZZ)) 3adant1 impbid)) thm (fzrevral2t ((j k) (K j) (K k) (M j) (M k) (N j) (N k) (k ph)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ))) (<-> (A.e. j (opr (opr K (-) N) (...) (opr K (-) M)) ph) (A.e. k (opr M (...) N) ([/] (opr K (-) (cv k)) j ph)))) ((opr K (-) N) (opr K (-) M) K j ph k fzrevralt K N zsubclt (e. M (ZZ)) 3adant2 K M zsubclt (e. N (ZZ)) 3adant3 (e. K (ZZ)) (e. M (ZZ)) (e. N (ZZ)) 3simp1 syl3anc K M nncant (e. N (CC)) 3adant3 K N nncant (e. M (CC)) 3adant2 (...) opreq12d K zcnt M zcnt N zcnt syl3an k ([/] (opr K (-) (cv k)) j ph) raleq1d bitrd 3coml)) thm (fzrevral3t ((j k) (M j) (M k) (N j) (N k) (k ph)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (A.e. j (opr M (...) N) ph) (A.e. k (opr M (...) N) ([/] (opr (opr M (+) N) (-) (cv k)) j ph)))) (M N (opr M (+) N) j ph k fzrevralt (e. M (ZZ)) (e. N (ZZ)) pm3.26 (e. M (ZZ)) (e. N (ZZ)) pm3.27 M N zaddclt syl3anc M N pncant M N pncan2t (...) opreq12d M zcnt N zcnt syl2an k ([/] (opr (opr M (+) N) (-) (cv k)) j ph) raleq1d bitrd)) thm (fsequb ((j k) (j m) (j n) (j x) (j y) (F j) (k m) (k n) (k x) (k y) (F k) (m n) (m x) (m y) (F m) (n x) (n y) (F n) (x y) (F x) (F y) (M j) (M k) (M m) (M n) (M x) (M y) (N j) (N k) (N n) (N x) (N y)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (RR)))) (E.e. x (RR) (A.e. k (opr M (...) N) (br (` F (cv k)) (<) (cv x))))) ((cv j) M M (...) opreq2 n (e. (` F (cv n)) (RR)) raleq1d (cv j) M M (...) opreq2 k (br (` F (cv k)) (<) (cv x)) raleq1d x (RR) rexbidv imbi12d (cv j) (cv m) M (...) opreq2 n (e. (` F (cv n)) (RR)) raleq1d (cv j) (cv m) M (...) opreq2 k (br (` F (cv k)) (<) (cv x)) raleq1d x (RR) rexbidv imbi12d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 n (e. (` F (cv n)) (RR)) raleq1d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k (br (` F (cv k)) (<) (cv x)) raleq1d x (RR) rexbidv imbi12d (cv j) N M (...) opreq2 n (e. (` F (cv n)) (RR)) raleq1d (cv j) N M (...) opreq2 k (br (` F (cv k)) (<) (cv x)) raleq1d x (RR) rexbidv imbi12d M elfz3t (cv n) M F fveq2 (RR) eleq1d (opr M (...) M) rcla4v syl (` F M) peano2re (cv k) M F fveq2 (<) (opr (` F M) (+) (1)) breq1d (` F M) ltp1t syl5bir com12 (cv k) M elfz1eqt syl5 r19.21aiv jca (e. M (ZZ)) a1i (cv x) (opr (` F M) (+) (1)) (` F (cv k)) (<) breq2 k (opr M (...) M) ralbidv (RR) rcla4ev syl6 syld M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv n) sseld (e. (` F (cv n)) (RR)) imim1d r19.20dv2 (E.e. x (RR) (A.e. k (opr M (...) (cv m)) (br (` F (cv k)) (<) (cv x)))) imim1d imp32 (cv n) (opr (cv m) (+) (1)) F fveq2 (RR) eleq1d (opr M (...) (opr (cv m) (+) (1))) rcla4va (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl sylan (-> (A.e. n (opr M (...) (cv m)) (e. (` F (cv n)) (RR))) (E.e. x (RR) (A.e. k (opr M (...) (cv m)) (br (` F (cv k)) (<) (cv x))))) adantrl (cv m) M (cv k) elfzp1 (A.e. n (opr M (...) (opr (cv m) (+) (1))) (e. (` F (cv n)) (RR))) adantr (/\ (e. (cv x) (RR)) (e. (` F (opr (cv m) (+) (1))) (RR))) (-> (e. (cv k) (opr M (...) (cv m))) (br (` F (cv k)) (<) (cv x))) ad2antrr (opr (` F (opr (cv m) (+) (1))) (+) (1)) (cv x) max2 (` F (opr (cv m) (+) (1))) peano2re sylan ancoms (/\ (e. (cv m) (` (ZZ>) M)) (A.e. n (opr M (...) (opr (cv m) (+) (1))) (e. (` F (cv n)) (RR)))) (e. (cv k) (opr M (...) (cv m))) ad2antlr (` F (cv k)) (cv x) (if (br (opr (` F (opr (cv m) (+) (1))) (+) (1)) (<_) (cv x)) (cv x) (opr (` F (opr (cv m) (+) (1))) (+) (1))) ltletrt (cv n) (cv k) F fveq2 (RR) eleq1d (opr M (...) (opr (cv m) (+) (1))) rcla4va (cv m) (` (ZZ>) M) (cv k) M fzelp1t sylan an1rs (/\ (e. (cv x) (RR)) (e. (` F (opr (cv m) (+) (1))) (RR))) adantlr (e. (cv x) (RR)) (e. (` F (opr (cv m) (+) (1))) (RR)) pm3.26 (/\ (e. (cv m) (` (ZZ>) M)) (A.e. n (opr M (...) (opr (cv m) (+) (1))) (e. (` F (cv n)) (RR)))) (e. (cv k) (opr M (...) (cv m))) ad2antlr (cv x) (RR) (opr (` F (opr (cv m) (+) (1))) (+) (1)) (br (opr (` F (opr (cv m) (+) (1))) (+) (1)) (<_) (cv x)) ifcl (` F (opr (cv m) (+) (1))) peano2re sylan2 (/\ (e. (cv m) (` (ZZ>) M)) (A.e. n (opr M (...) (opr (cv m) (+) (1))) (e. (` F (cv n)) (RR)))) (e. (cv k) (opr M (...) (cv m))) ad2antlr syl3anc mpan2d ex a2d imp (cv k) (opr (cv m) (+) (1)) F fveq2 (<) (if (br (opr (` F (opr (cv m) (+) (1))) (+) (1)) (<_) (cv x)) (cv x) (opr (` F (opr (cv m) (+) (1))) (+) (1))) breq1d (e. (cv x) (RR)) (e. (` F (opr (cv m) (+) (1))) (RR)) pm3.27 (` F (opr (cv m) (+) (1))) peano2re (e. (cv x) (RR)) adantl (cv x) (RR) (opr (` F (opr (cv m) (+) (1))) (+) (1)) (br (opr (` F (opr (cv m) (+) (1))) (+) (1)) (<_) (cv x)) ifcl (` F (opr (cv m) (+) (1))) peano2re sylan2 (` F (opr (cv m) (+) (1))) ltp1t (e. (cv x) (RR)) adantl (` F (opr (cv m) (+) (1))) peano2re (opr (` F (opr (cv m) (+) (1))) (+) (1)) (cv x) max1ALT syl (e. (cv x) (RR)) adantl ltletrd syl5bir com12 (/\ (e. (cv m) (` (ZZ>) M)) (A.e. n (opr M (...) (opr (cv m) (+) (1))) (e. (` F (cv n)) (RR)))) (-> (e. (cv k) (opr M (...) (cv m))) (br (` F (cv k)) (<) (cv x))) ad2antlr jaod sylbid ex r19.20dv2 (cv x) (RR) (opr (` F (opr (cv m) (+) (1))) (+) (1)) (br (opr (` F (opr (cv m) (+) (1))) (+) (1)) (<_) (cv x)) ifcl (` F (opr (cv m) (+) (1))) peano2re sylan2 (/\ (e. (cv m) (` (ZZ>) M)) (A.e. n (opr M (...) (opr (cv m) (+) (1))) (e. (` F (cv n)) (RR)))) adantl jctild (cv y) (if (br (opr (` F (opr (cv m) (+) (1))) (+) (1)) (<_) (cv x)) (cv x) (opr (` F (opr (cv m) (+) (1))) (+) (1))) (` F (cv k)) (<) breq2 k (opr M (...) (opr (cv m) (+) (1))) ralbidv (RR) rcla4ev syl6 exp32 com34 r19.23adv imp3a (cv y) (cv x) (` F (cv k)) (<) breq2 k (opr M (...) (opr (cv m) (+) (1))) ralbidv (RR) cbvrexv syl6ib (-> (A.e. n (opr M (...) (cv m)) (e. (` F (cv n)) (RR))) (E.e. x (RR) (A.e. k (opr M (...) (cv m)) (br (` F (cv k)) (<) (cv x))))) adantrl mp2and exp32 uzind4 imp (cv k) (cv n) F fveq2 (RR) eleq1d (opr M (...) N) cbvralv sylan2b)) thm (fsequb2 ((k x) (k y) (F k) (x y) (F x) (F y) (M k) (M x) (M y) (N k) (N x) (N y)) () (-> (/\ (e. N (` (ZZ>) M)) (:--> F (opr M (...) N) (RR))) (E.e. x (RR) (A.e. y (ran F) (br (cv y) (<_) (cv x))))) (N M k F x fsequb F (opr M (...) N) (RR) (cv k) ffvrn r19.21aiva sylan2 F (opr M (...) N) (RR) ffn F (opr M (...) N) (cv y) k fvelrn syl biimpa (` F (cv k)) (cv y) (<) (cv x) breq1 biimpd k (opr M (...) N) r19.22si syl k (opr M (...) N) (br (` F (cv k)) (<) (cv x)) (br (cv y) (<) (cv x)) r19.36av syl (e. N (` (ZZ>) M)) adantll (e. (cv x) (RR)) adantlr (cv y) (cv x) ltlet F (opr M (...) N) (RR) frn (cv y) sseld imp (e. (cv x) (RR)) adantlr (:--> F (opr M (...) N) (RR)) (e. (cv x) (RR)) pm3.27 (e. (cv y) (ran F)) adantr sylanc (e. N (` (ZZ>) M)) adantlll syld ex com23 r19.21adv r19.22dva mpd)) thm (limsupvalt ((k x) (k y) (k z) (F k) (x y) (x z) (F x) (y z) (F y) (F z)) () (-> (e. F A) (= (` (limsup) F) (sup ({|} x (E.e. k (ZZ) (= (cv x) (sup (i^i (" F (` (ZZ>) (cv k))) (RR*)) (RR*) (<))))) (RR*) (`' (<))))) (xrltso (<) (RR*) cnvso mpbi ({|} x (E.e. k (ZZ) (= (cv x) (sup (i^i (" F (` (ZZ>) (cv k))) (RR*)) (RR*) (<))))) supex (cv z) F (` (ZZ>) (cv k)) imaeq1 (RR*) ineq1d (i^i (" (cv z) (` (ZZ>) (cv k))) (RR*)) (i^i (" F (` (ZZ>) (cv k))) (RR*)) (RR*) (<) supeq1 syl (cv x) eqeq2d k (ZZ) rexbidv x abbidv ({|} x (E.e. k (ZZ) (= (cv x) (sup (i^i (" (cv z) (` (ZZ>) (cv k))) (RR*)) (RR*) (<))))) ({|} x (E.e. k (ZZ) (= (cv x) (sup (i^i (" F (` (ZZ>) (cv k))) (RR*)) (RR*) (<))))) (RR*) (`' (<)) supeq1 syl A (V) y fvopabg mpan2 z y x k df-limsup F fveq1i syl5eq)) thm (seq0fval ((f g) (f h) (S f) (g h) (S g) (S h) (F f) (F g) (F h)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (opr S (seq0) F) (|` (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) (NN0))) (seq0val.1 seq0val.2 (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1)) oprex (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) (V) (NN0) resexg ax-mp (cv f) S (seq1) (opr (cv g) (shift) (1)) opreq1 (shift) (-u (1)) opreq1d (opr (opr (cv f) (seq1) (opr (cv g) (shift) (1))) (shift) (-u (1))) (opr (opr S (seq1) (opr (cv g) (shift) (1))) (shift) (-u (1))) (NN0) reseq1 syl (cv g) F (shift) (1) opreq1 S (seq1) opreq2d (shift) (-u (1)) opreq1d (opr (opr S (seq1) (opr (cv g) (shift) (1))) (shift) (-u (1))) (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) (NN0) reseq1 syl f g h df-seq0 (V) (V) oprabval5 mp2an)) thm (seq0valt () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. N (NN0)) (= (` (opr S (seq0) F) N) (` (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) N))) (N (NN0) (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) fvres seq0val.1 seq0val.2 seq0fval N fveq1i syl5eq)) thm (seqzfval ((g h) (g k) (g x) (S g) (h k) (h x) (S h) (k x) (S k) (S x) (F g) (F h) (F k) (F x) (A k) (M g) (M h) (M k) (M x)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. M A) (= (opr (<,> M S) (seq) F) (|` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) ({e.|} k (ZZ) (br M (<_) (cv k)))))) (seq0val.1 M A S (V) op2ndg mpan2 M A S op1stg (1) (-) opreq2d F (shift) opreq2d (seq1) opreq12d M A S op1stg (-) (1) opreq1d (shift) opreq12d (opr (opr (` (2nd) (<,> M S)) (seq1) (opr F (shift) (opr (1) (-) (` (1st) (<,> M S))))) (shift) (opr (` (1st) (<,> M S)) (-) (1))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) ({e.|} k (ZZ) (br (` (1st) (<,> M S)) (<_) (cv k))) reseq1 syl M A S op1stg (<_) (cv k) breq1d k (ZZ) rabbisdv ({e.|} k (ZZ) (br (` (1st) (<,> M S)) (<_) (cv k))) ({e.|} k (ZZ) (br M (<_) (cv k))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) reseq2 syl eqtrd M S opex seq0val.2 (opr (` (2nd) (<,> M S)) (seq1) (opr F (shift) (opr (1) (-) (` (1st) (<,> M S))))) (shift) (opr (` (1st) (<,> M S)) (-) (1)) oprex (opr (opr (` (2nd) (<,> M S)) (seq1) (opr F (shift) (opr (1) (-) (` (1st) (<,> M S))))) (shift) (opr (` (1st) (<,> M S)) (-) (1))) (V) ({e.|} k (ZZ) (br (` (1st) (<,> M S)) (<_) (cv k))) resexg ax-mp (cv x) (<,> M S) (2nd) fveq2 (cv x) (<,> M S) (1st) fveq2 (1) (-) opreq2d (cv g) (shift) opreq2d (seq1) opreq12d (cv x) (<,> M S) (1st) fveq2 (-) (1) opreq1d (shift) opreq12d (opr (opr (` (2nd) (cv x)) (seq1) (opr (cv g) (shift) (opr (1) (-) (` (1st) (cv x))))) (shift) (opr (` (1st) (cv x)) (-) (1))) (opr (opr (` (2nd) (<,> M S)) (seq1) (opr (cv g) (shift) (opr (1) (-) (` (1st) (<,> M S))))) (shift) (opr (` (1st) (<,> M S)) (-) (1))) ({e.|} k (ZZ) (br (` (1st) (cv x)) (<_) (cv k))) reseq1 syl (cv x) (<,> M S) (1st) fveq2 (<_) (cv k) breq1d k (ZZ) rabbisdv ({e.|} k (ZZ) (br (` (1st) (cv x)) (<_) (cv k))) ({e.|} k (ZZ) (br (` (1st) (<,> M S)) (<_) (cv k))) (opr (opr (` (2nd) (<,> M S)) (seq1) (opr (cv g) (shift) (opr (1) (-) (` (1st) (<,> M S))))) (shift) (opr (` (1st) (<,> M S)) (-) (1))) reseq2 syl eqtrd (cv g) F (shift) (opr (1) (-) (` (1st) (<,> M S))) opreq1 (` (2nd) (<,> M S)) (seq1) opreq2d (shift) (opr (` (1st) (<,> M S)) (-) (1)) opreq1d (opr (opr (` (2nd) (<,> M S)) (seq1) (opr (cv g) (shift) (opr (1) (-) (` (1st) (<,> M S))))) (shift) (opr (` (1st) (<,> M S)) (-) (1))) (opr (opr (` (2nd) (<,> M S)) (seq1) (opr F (shift) (opr (1) (-) (` (1st) (<,> M S))))) (shift) (opr (` (1st) (<,> M S)) (-) (1))) ({e.|} k (ZZ) (br (` (1st) (<,> M S)) (<_) (cv k))) reseq1 syl x g h k df-seqz (V) (V) oprabval5 mp2an syl5eq)) thm (seqzfval2 ((S k) (F k) (M k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. M (ZZ)) (= (opr (<,> M S) (seq) F) (|` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) (` (ZZ>) M)))) (seq0val.1 seq0val.2 M (ZZ) k seqzfval M k uzvalt (` (ZZ>) M) ({e.|} k (ZZ) (br M (<_) (cv k))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) reseq2 syl eqtr4d)) thm (seqzfn () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. M (ZZ)) (Fn (opr (<,> M S) (seq) F) (` (ZZ>) M))) (M (-) (1) oprex S (seq1) (opr F (shift) (opr (1) (-) M)) oprex (opr M (-) (1)) (V) shftfn ax-mp M uzssz zsscn sstri (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) (CC) (` (ZZ>) M) fnssres mp2an seq0val.1 seq0val.2 M seqzfval2 (opr (<,> M S) (seq) F) (|` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) (` (ZZ>) M)) (` (ZZ>) M) fneq1 syl mpbiri)) thm (seqzvalt ((S k) (F k) (A k) (M k) (N k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (/\/\ (e. M A) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) N) (` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) N))) (seq0val.1 seq0val.2 M A k seqzfval N fveq1d (e. N (ZZ)) (br M (<_) N) 3ad2ant1 (e. M A) (e. N (ZZ)) (br M (<_) N) 3simpc (cv k) N M (<_) breq2 (ZZ) elrab sylibr N ({e.|} k (ZZ) (br M (<_) (cv k))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) fvres syl eqtrd)) thm (seq1seqz ((S k) (F k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (opr S (seq1) F) (opr (<,> (1) S) (seq) F)) (seq0val.1 F (shift) (0) oprex seq1fn 0cn nnsscn S (seq1) (opr F (shift) (0)) oprex (0) (CC) (NN) shftres mp2an (opr S (seq1) (opr F (shift) (0))) (NN) (|` (opr (opr S (seq1) (opr F (shift) (0))) (shift) (0)) (NN)) (NN) k eqfnfv mp2an (NN) eqid (cv k) (NN) (opr (opr S (seq1) (opr F (shift) (0))) (shift) (0)) fvres (cv k) nncnt S (seq1) (opr F (shift) (0)) oprex (cv k) shftidt syl eqtr2d rgen mpbir2an 1cn subid F (shift) opreq2i S (seq1) opreq2i 1cn subid (shift) opreq12i (opr (opr S (seq1) (opr F (shift) (opr (1) (-) (1)))) (shift) (opr (1) (-) (1))) (opr (opr S (seq1) (opr F (shift) (0))) (shift) (0)) (NN) reseq1 ax-mp eqtr4 seq0val.1 seq0val.2 seq1shftid k nnzrab (NN) ({e.|} k (ZZ) (br (1) (<_) (cv k))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) (1)))) (shift) (opr (1) (-) (1))) reseq2 ax-mp 3eqtr3 1cn seq0val.1 seq0val.2 (1) (CC) k seqzfval ax-mp eqtr4)) thm (seq0seqz ((S k) (F k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (opr S (seq0) F) (opr (<,> (0) S) (seq) F)) (seq0val.1 seq0val.2 seq0fval 1cn subid1 eqcomi F (shift) opreq2i S (seq1) opreq2i (1) df-neg (shift) opreq12i (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) (0)))) (shift) (opr (0) (-) (1))) (NN0) reseq1 ax-mp k nn0zrab (NN0) ({e.|} k (ZZ) (br (0) (<_) (cv k))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) (0)))) (shift) (opr (0) (-) (1))) reseq2 ax-mp 3eqtr 0cn seq0val.1 seq0val.2 (0) (CC) k seqzfval ax-mp eqtr4)) thm (seq1seq02t () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. N (NN)) (= (` (opr S (seq1) (opr F (shift) (1))) N) (` (opr (opr S (seq0) F) (shift) (1)) N))) (N nncnt 1cn S (seq0) F oprex (1) (CC) N shftvalt mpan syl N elnnnn0 pm3.27bd seq0val.1 seq0val.2 (opr N (-) (1)) seq0valt syl N nncnt 1cn N (1) subclt mpan2 1cn S (seq1) (opr F (shift) (1)) oprex (1) (opr N (-) (1)) shftval4t mpan 3syl N nncnt 1cn (1) N pncan3t mpan syl (opr S (seq1) (opr F (shift) (1))) fveq2d eqtrd 3eqtrrd)) thm (seq1seq0t () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. N (NN)) (= (` (opr S (seq1) F) N) (` (opr (opr S (seq0) (opr F (shift) (-u (1)))) (shift) (1)) N))) (seq0val.1 F (shift) (-u (1)) oprex N seq1seq02t 1cn negcl 1cn seq0val.2 (-u (1)) (1) 2shft mp2an 1cn negcl 1cn addcom 1cn negid eqtr F (shift) opreq2i eqtr S (seq1) opreq2i seq0val.1 seq0val.2 seq1shftid eqtr2 N fveq1i syl5eq)) thm (seq1seq0 ((S k) (F k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (opr S (seq1) F) (|` (opr (opr S (seq0) (opr F (shift) (-u (1)))) (shift) (1)) (NN))) (seq0val.1 seq0val.2 seq1fn 1cn nnsscn S (seq0) (opr F (shift) (-u (1))) oprex (1) (CC) (NN) shftres mp2an (opr S (seq1) F) (NN) (|` (opr (opr S (seq0) (opr F (shift) (-u (1)))) (shift) (1)) (NN)) (NN) k eqfnfv mp2an (NN) eqid seq0val.1 seq0val.2 (cv k) seq1seq0t (cv k) (NN) (opr (opr S (seq0) (opr F (shift) (-u (1)))) (shift) (1)) fvres eqtr4d rgen mpbir2an)) thm (seq0fn () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (Fn (opr S (seq0) F) (NN0)) ((1) minusex S (seq1) (opr F (shift) (1)) oprex (-u (1)) (V) shftfn ax-mp nn0sscn (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) (CC) (NN0) fnssres mp2an seq0val.1 seq0val.2 seq0fval (opr S (seq0) F) (|` (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) (NN0)) (NN0) fneq1 ax-mp mpbir)) thm (seqz1 ((S k) (F k) (M k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. M (ZZ)) (= (` (opr (<,> M S) (seq) F) M) (` F M))) (seq0val.1 seq0val.2 M (ZZ) k seqzfval M fveq1d M zret M leidt syl ancli (cv k) M M (<_) breq2 (ZZ) elrab sylibr M ({e.|} k (ZZ) (br M (<_) (cv k))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) fvres syl M zcnt M (-) (1) oprex S (seq1) (opr F (shift) (opr (1) (-) M)) oprex (opr M (-) (1)) (V) M shftvalt mpan syl M zcnt 1cn M (1) nncant mpan2 syl (opr S (seq1) (opr F (shift) (opr (1) (-) M))) fveq2d seq0val.1 F (shift) (opr (1) (-) M) oprex seq11 syl6eq M zcnt 1cn seq0val.2 (1) M shftval3t mpan syl 3eqtrd 3eqtrd)) thm (seqzp1 ((S k) (F k) (M k) (N k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. N (` (ZZ>) M)) (= (` (opr (<,> M S) (seq) F) (opr N (+) (1))) (opr (` (opr (<,> M S) (seq) F) N) S (` F (opr N (+) (1)))))) (N M eluzel2 seq0val.1 seq0val.2 M (ZZ) k seqzfval (opr N (+) (1)) fveq1d syl N M peano2uz (opr N (+) (1)) M eluzel2 M k uzvalt syl (opr N (+) (1)) eleq2d ibi (opr N (+) (1)) ({e.|} k (ZZ) (br M (<_) (cv k))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) fvres 3syl S (seq1) (opr F (shift) (opr (1) (-) M)) oprex (opr M (-) (1)) (CC) (opr N (+) (1)) shftvalt 1cn M (1) subclt mpan2 N peano2cn syl2an 1cn M (1) subclt mpan2 (e. N (CC)) anim2i ancoms 1cn N (1) (opr M (-) (1)) addsubt mp3an2 syl (opr S (seq1) (opr F (shift) (opr (1) (-) M))) fveq2d eqtrd N M eluzel2 M zcnt syl N M eluzelz N zcnt syl sylanc N M eluz2t 1cn N M (1) subsubt mp3an3 N zcnt M zcnt syl2an ancoms (br M (<_) N) 3adant3 M N znn0sub2t (opr N (-) M) nn0p1nnt syl eqeltrd sylbi seq0val.1 F (shift) (opr (1) (-) M) oprex (opr N (-) (opr M (-) (1))) seq1p1 syl seq0val.2 (opr (1) (-) M) (CC) (opr (opr N (-) (opr M (-) (1))) (+) (1)) shftvalt 1cn (1) M subclt mpan (e. N (CC)) adantr N (opr M (-) (1)) subclt 1cn M (1) subclt mpan2 sylan2 ancoms (opr N (-) (opr M (-) (1))) peano2cn syl sylanc 1cn (opr (opr N (-) (opr M (-) (1))) (+) (1)) (1) M subsub2t mp3an2 N (opr M (-) (1)) subclt 1cn M (1) subclt mpan2 sylan2 ancoms (opr N (-) (opr M (-) (1))) peano2cn syl (e. M (CC)) (e. N (CC)) pm3.26 sylanc 1cn M (1) subclt mpan2 (e. N (CC)) anim2i ancoms 1cn N (opr M (-) (1)) (1) nppcant mp3an3 syl eqtrd F fveq2d eqtrd N M eluzel2 M zcnt syl N M eluzelz N zcnt syl sylanc (` (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (opr N (-) (opr M (-) (1)))) S opreq2d 3eqtrd N M eluzel2 seq0val.1 seq0val.2 M (ZZ) k seqzfval N fveq1d syl N M eluzel2 M k uzvalt N eleq2d syl ibi N ({e.|} k (ZZ) (br M (<_) (cv k))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) fvres syl S (seq1) (opr F (shift) (opr (1) (-) M)) oprex (opr M (-) (1)) (ZZ) N shftvalt N M eluzel2 M peano2zm syl N M eluzelz N zcnt syl sylanc 3eqtrrd S (` F (opr N (+) (1))) opreq1d 3eqtrd eqtrd)) thm (seqzp1OLD ((S k) (F k) (M k) (N k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) (opr N (+) (1))) (opr (` (opr (<,> M S) (seq) F) N) S (` F (opr N (+) (1)))))) (seq0val.1 seq0val.2 M (ZZ) k seqzfval (opr N (+) (1)) fveq1d (e. N (ZZ)) (br M (<_) N) 3ad2ant1 N peano2z (e. M (ZZ)) (br M (<_) N) 3ad2ant2 M N letrp1t N zret syl3an2 M zret syl3an1 jca (cv k) (opr N (+) (1)) M (<_) breq2 (ZZ) elrab sylibr (opr N (+) (1)) ({e.|} k (ZZ) (br M (<_) (cv k))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) fvres syl S (seq1) (opr F (shift) (opr (1) (-) M)) oprex (opr M (-) (1)) (CC) (opr N (+) (1)) shftvalt 1cn M (1) subclt mpan2 N peano2cn syl2an 1cn M (1) subclt mpan2 (e. N (CC)) anim2i ancoms 1cn N (1) (opr M (-) (1)) addsubt mp3an2 syl (opr S (seq1) (opr F (shift) (opr (1) (-) M))) fveq2d eqtrd M zcnt N zcnt syl2an (br M (<_) N) 3adant3 1cn N M (1) subsubt mp3an3 N zcnt M zcnt syl2an ancoms (br M (<_) N) 3adant3 M N znn0sub2t (opr N (-) M) nn0p1nnt syl eqeltrd seq0val.1 F (shift) (opr (1) (-) M) oprex (opr N (-) (opr M (-) (1))) seq1p1 syl seq0val.2 (opr (1) (-) M) (CC) (opr (opr N (-) (opr M (-) (1))) (+) (1)) shftvalt 1cn (1) M subclt mpan (e. N (CC)) adantr N (opr M (-) (1)) subclt 1cn M (1) subclt mpan2 sylan2 ancoms (opr N (-) (opr M (-) (1))) peano2cn syl sylanc 1cn (opr (opr N (-) (opr M (-) (1))) (+) (1)) (1) M subsub2t mp3an2 N (opr M (-) (1)) subclt 1cn M (1) subclt mpan2 sylan2 ancoms (opr N (-) (opr M (-) (1))) peano2cn syl (e. M (CC)) (e. N (CC)) pm3.26 sylanc 1cn M (1) subclt mpan2 (e. N (CC)) anim2i ancoms 1cn N (opr M (-) (1)) (1) nppcant mp3an3 syl eqtrd F fveq2d eqtrd M zcnt N zcnt syl2an (` (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (opr N (-) (opr M (-) (1)))) S opreq2d (br M (<_) N) 3adant3 3eqtrd seq0val.1 seq0val.2 M (ZZ) k seqzfval N fveq1d (e. N (ZZ)) (br M (<_) N) 3ad2ant1 (cv k) N M (<_) breq2 (ZZ) elrab biimpr (e. M (ZZ)) 3adant1 N ({e.|} k (ZZ) (br M (<_) (cv k))) (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) fvres syl M peano2zm N zcnt anim12i (br M (<_) N) 3adant3 S (seq1) (opr F (shift) (opr (1) (-) M)) oprex (opr M (-) (1)) (ZZ) N shftvalt syl 3eqtrrd S (` F (opr N (+) (1))) opreq1d 3eqtrd eqtrd)) thm (seqzm1 () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) (= (` (opr (<,> M S) (seq) F) N) (opr (` (opr (<,> M S) (seq) F) (opr N (-) (1))) S (` F N)))) ((opr N (-) (1)) M eluz2t seq0val.1 seq0val.2 (opr N (-) (1)) M seqzp1 sylbir (e. M (ZZ)) (e. N (ZZ)) (br M (<) N) 3simp1 N peano2zm (e. M (ZZ)) (br M (<) N) 3ad2ant2 M N zltlem1t biimp3a syl3anc N zcnt 1cn N (1) npcant mpan2 syl (opr (<,> M S) (seq) F) fveq2d (e. M (ZZ)) (br M (<) N) 3ad2ant2 N zcnt 1cn N (1) npcant mpan2 syl F fveq2d (` (opr (<,> M S) (seq) F) (opr N (-) (1))) S opreq2d (e. M (ZZ)) (br M (<) N) 3ad2ant2 3eqtr3d)) thm (seq00 () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (` (opr S (seq0) F) (0)) (` F (0))) (seq0val.1 seq0val.2 seq0fval (0) fveq1i 0nn0 (0) (NN0) (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) fvres ax-mp eqtr 1cn negcl 0cn S (seq1) (opr F (shift) (1)) oprex (-u (1)) (CC) (0) shftvalt mp2an (-u (1)) df-neg 1cn negneg eqtr3 (opr S (seq1) (opr F (shift) (1))) fveq2i seq0val.1 F (shift) (1) oprex seq11 1cn 1cn seq0val.2 (1) (CC) (1) shftvalt mp2an 1cn subid F fveq2i eqtr 3eqtr 3eqtr)) thm (seq0p1 () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. N (NN0)) (= (` (opr S (seq0) F) (opr N (+) (1))) (opr (` (opr S (seq0) F) N) S (` F (opr N (+) (1)))))) (N peano2nn0 (opr N (+) (1)) (NN0) (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) fvres syl N nn0cnt N peano2cn 1cn negcl S (seq1) (opr F (shift) (1)) oprex (-u (1)) (CC) (opr N (+) (1)) shftvalt mpan 3syl N nn0cnt 1cn 1cn negcl N (1) (-u (1)) addsubt mp3an23 syl (opr S (seq1) (opr F (shift) (1))) fveq2d N nn0cnt 1cn N (1) subnegt mpan2 syl N nn0p1nnt eqeltrd seq0val.1 F (shift) (1) oprex (opr N (-) (-u (1))) seq1p1 syl N nn0cnt 1cn negcl N (-u (1)) subclt mpan2 syl (opr N (-) (-u (1))) peano2cn 1cn seq0val.2 (1) (CC) (opr (opr N (-) (-u (1))) (+) (1)) shftvalt mpan 3syl N nn0cnt 1cn negcl N (-u (1)) subclt mpan2 1cn (opr N (-) (-u (1))) (1) pncant mpan2 syl 1cn N (1) subnegt mpan2 eqtrd syl F fveq2d eqtrd (` (opr S (seq1) (opr F (shift) (1))) (opr N (-) (-u (1)))) S opreq2d eqtrd 3eqtrd N (NN0) (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) fvres N nn0cnt 1cn negcl S (seq1) (opr F (shift) (1)) oprex (-u (1)) (CC) N shftvalt mpan syl eqtrd seq0val.1 seq0val.2 seq0fval N fveq1i syl5req S (` F (opr N (+) (1))) opreq1d 3eqtrd seq0val.1 seq0val.2 seq0fval (opr N (+) (1)) fveq1i syl5eq)) thm (seq01 () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (` (opr S (seq0) F) (1)) (opr (` F (0)) S (` F (1)))) (0nn0 seq0val.1 seq0val.2 (0) seq0p1 ax-mp 1cn addid2 (opr S (seq0) F) fveq2i seq0val.1 seq0val.2 seq00 1cn addid2 F fveq2i S opreq12i 3eqtr3)) thm (seqzval2t () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) N) (` (opr (opr S (seq0) (opr F (shift) (-u M))) (shift) M) N))) (N zcnt M (-) (1) oprex S (seq1) (opr F (shift) (opr (1) (-) M)) oprex (opr M (-) (1)) (V) N shftvalt mpan syl (e. M (ZZ)) (br M (<_) N) 3ad2ant2 M zcnt M negclt 1cn seq0val.2 (-u M) (1) 2shft mpan2 syl M negclt 1cn (-u M) (1) axaddcom mpan2 syl 1cn (1) M negsubt mpan eqtrd F (shift) opreq2d eqtrd syl S (seq1) opreq2d (opr N (-) (opr M (-) (1))) fveq1d (e. N (ZZ)) (br M (<_) N) 3ad2ant1 1cn N M (1) subsubt mp3an3 N zcnt M zcnt syl2an ancoms (br M (<_) N) 3adant3 M N znn0sub2t (opr N (-) M) nn0p1nnt syl eqeltrd seq0val.1 F (shift) (-u M) oprex (opr N (-) (opr M (-) (1))) seq1seq02t syl eqtr3d N (opr M (-) (1)) subclt 1cn M (1) subclt mpan2 sylan2 1cn S (seq0) (opr F (shift) (-u M)) oprex (1) (CC) (opr N (-) (opr M (-) (1))) shftvalt mpan syl 1cn N M (1) nnncant mp3an3 (opr S (seq0) (opr F (shift) (-u M))) fveq2d eqtrd ancoms M zcnt N zcnt syl2an (br M (<_) N) 3adant3 3eqtrd seq0val.1 seq0val.2 M (ZZ) N seqzvalt S (seq0) (opr F (shift) (-u M)) oprex M (ZZ) N shftvalt N zcnt sylan2 (br M (<_) N) 3adant3 3eqtr4d)) thm (seqzfveq ((j k) (j m) (F j) (k m) (F k) (F m) (G j) (G k) (G m) (M j) (M k) (M m) (N j) (N k) (S j) (S m)) ((seqzfveq.1 (e. S (V))) (seqzfveq.2 (e. F (V))) (seqzfveq.3 (e. G (V)))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (= (` F (cv k)) (` G (cv k))))) (= (` (opr (<,> M S) (seq) F) N) (` (opr (<,> M S) (seq) G) N))) ((cv j) M M (...) opreq2 k (= (` F (cv k)) (` G (cv k))) raleq1d (cv j) M (opr (<,> M S) (seq) F) fveq2 (cv j) M (opr (<,> M S) (seq) G) fveq2 eqeq12d imbi12d (cv j) (cv m) M (...) opreq2 k (= (` F (cv k)) (` G (cv k))) raleq1d (cv j) (cv m) (opr (<,> M S) (seq) F) fveq2 (cv j) (cv m) (opr (<,> M S) (seq) G) fveq2 eqeq12d imbi12d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k (= (` F (cv k)) (` G (cv k))) raleq1d (cv j) (opr (cv m) (+) (1)) (opr (<,> M S) (seq) F) fveq2 (cv j) (opr (cv m) (+) (1)) (opr (<,> M S) (seq) G) fveq2 eqeq12d imbi12d (cv j) N M (...) opreq2 k (= (` F (cv k)) (` G (cv k))) raleq1d (cv j) N (opr (<,> M S) (seq) F) fveq2 (cv j) N (opr (<,> M S) (seq) G) fveq2 eqeq12d imbi12d (cv k) M F fveq2 (cv k) M G fveq2 eqeq12d (opr M (...) M) rcla4va M elfz3t sylan seqzfveq.1 seqzfveq.2 M seqz1 (A.e. k (opr M (...) M) (= (` F (cv k)) (` G (cv k)))) adantr seqzfveq.1 seqzfveq.3 M seqz1 (A.e. k (opr M (...) M) (= (` F (cv k)) (` G (cv k)))) adantr 3eqtr4d ex M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (= (` F (cv k)) (` G (cv k))) imim1d r19.20dv2 (= (` (opr (<,> M S) (seq) F) (cv m)) (` (opr (<,> M S) (seq) G) (cv m))) imim1d (= (` (opr (<,> M S) (seq) F) (cv m)) (` (opr (<,> M S) (seq) G) (cv m))) id (cv k) (opr (cv m) (+) (1)) F fveq2 (cv k) (opr (cv m) (+) (1)) G fveq2 eqeq12d (opr M (...) (opr (cv m) (+) (1))) rcla4va (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl sylan S opreqan12rd seqzfveq.1 seqzfveq.2 (cv m) M seqzp1 (A.e. k (opr M (...) (opr (cv m) (+) (1))) (= (` F (cv k)) (` G (cv k)))) (= (` (opr (<,> M S) (seq) F) (cv m)) (` (opr (<,> M S) (seq) G) (cv m))) ad2antrr seqzfveq.1 seqzfveq.3 (cv m) M seqzp1 (A.e. k (opr M (...) (opr (cv m) (+) (1))) (= (` F (cv k)) (` G (cv k)))) (= (` (opr (<,> M S) (seq) F) (cv m)) (` (opr (<,> M S) (seq) G) (cv m))) ad2antrr 3eqtr4d exp31 a2d syld uzind4 imp)) thm (seqzeq ((k m) (F k) (F m) (G k) (G m) (M k) (M m) (S m)) ((seqzfveq.1 (e. S (V))) (seqzfveq.2 (e. F (V))) (seqzfveq.3 (e. G (V)))) (-> (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (= (` F (cv k)) (` G (cv k))))) (= (opr (<,> M S) (seq) F) (opr (<,> M S) (seq) G))) (seqzfveq.1 seqzfveq.2 seqzfveq.3 (cv m) M k seqzfveq (cv k) M (cv m) elfzuzt (= (` F (cv k)) (` G (cv k))) imim1i r19.20i2 sylan2 expcom r19.21aiv (e. M (ZZ)) adantl (` (ZZ>) M) eqid jctil seqzfveq.1 seqzfveq.2 M seqzfn seqzfveq.1 seqzfveq.3 M seqzfn jca (A.e. k (` (ZZ>) M) (= (` F (cv k)) (` G (cv k)))) adantr (opr (<,> M S) (seq) F) (` (ZZ>) M) (opr (<,> M S) (seq) G) (` (ZZ>) M) m eqfnfv syl mpbird)) thm (seqzrn ((j m) (j n) (j v) (j y) (B j) (m n) (m v) (m y) (B m) (n v) (n y) (B n) (v y) (B v) (B y) (C j) (C m) (C n) (C v) (C y) (F j) (F m) (F n) (F v) (F y) (M j) (M m) (M n) (M v) (M y) (S j) (S m) (S n) (S v) (S y)) ((seqzrn.1 (e. S (V))) (seqzrn.2 (e. F (V)))) (-> (/\ (/\ (e. M (ZZ)) (e. (` F M) C)) (/\ (:--> (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B) (:--> S (X. C B) C))) (C_ (ran (opr (<,> M S) (seq) F)) C)) ((opr (<,> M S) (seq) F) (` (ZZ>) M) n C fnfvrnss seqzrn.1 seqzrn.2 M seqzfn (cv j) M (opr (<,> M S) (seq) F) fveq2 C eleq1d (/\ (e. (` F M) C) (/\ (:--> (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B) (:--> S (X. C B) C))) imbi2d (cv j) (cv m) (opr (<,> M S) (seq) F) fveq2 C eleq1d (/\ (e. (` F M) C) (/\ (:--> (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B) (:--> S (X. C B) C))) imbi2d (cv j) (opr (cv m) (+) (1)) (opr (<,> M S) (seq) F) fveq2 C eleq1d (/\ (e. (` F M) C) (/\ (:--> (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B) (:--> S (X. C B) C))) imbi2d (cv j) (cv n) (opr (<,> M S) (seq) F) fveq2 C eleq1d (/\ (e. (` F M) C) (/\ (:--> (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B) (:--> S (X. C B) C))) imbi2d seqzrn.1 seqzrn.2 M seqz1 C eleq1d (e. (` F M) C) (/\ (:--> (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B) (:--> S (X. C B) C)) pm3.26 syl5bir (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B (opr (cv m) (+) (1)) ffvrn (opr (cv m) (+) (1)) (` (ZZ>) (opr M (+) (1))) F fvres B eleq1d (:--> (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B) adantl mpbid (cv m) M eluzp1p1t sylan2 seqzrn.1 seqzrn.2 (cv m) M seqzp1 C eleq1d (cv y) (` (opr (<,> M S) (seq) F) (cv m)) S (cv v) opreq1 C eleq1d (cv v) (` F (opr (cv m) (+) (1))) (` (opr (<,> M S) (seq) F) (cv m)) S opreq2 C eleq1d C B rcla42v ancoms S C B C y v ffnoprval pm3.27bd syl5 imp syl5bir exp4c (:--> (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B) adantl mpd ex com4r impcom (e. (` F M) C) adantl com12 a2d uzind4 com12 r19.21aiv syl2an anassrs)) thm (seqzrn2 () ((seqzrn.1 (e. S (V))) (seqzrn.2 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (:--> F (` (ZZ>) M) C) (:--> S (X. C C) C)) (C_ (ran (opr (<,> M S) (seq) F)) C)) (seqzrn.1 seqzrn.2 M C C seqzrn exp43 imp4c F (` (ZZ>) M) C M ffvrn M uzidt sylan2 F (` (ZZ>) M) C (` (ZZ>) (opr M (+) (1))) fssres M uzidt M M peano2uz syl (opr M (+) (1)) M uzss syl sylan2 jca expcom syland 3impib)) thm (seqzcl () ((seqzrn.1 (e. S (V))) (seqzrn.2 (e. F (V)))) (-> (/\/\ (e. N (` (ZZ>) M)) (:--> F (` (ZZ>) M) C) (:--> S (X. C C) C)) (e. (` (opr (<,> M S) (seq) F) N) C)) (seqzrn.1 seqzrn.2 M C seqzrn2 N M eluzel2 syl3an1 N M eluzel2 seqzrn.1 seqzrn.2 M seqzfn syl (opr (<,> M S) (seq) F) (` (ZZ>) M) N fnfvrn mpancom (:--> F (` (ZZ>) M) C) (:--> S (X. C C) C) 3ad2ant1 sseldd)) thm (seqzresval ((F m) (M m) (N m) (S m)) ((seqzrn.1 (e. S (V))) (seqzrn.2 (e. F (V)))) (-> (e. N (` (ZZ>) M)) (= (` (opr (<,> M S) (seq) (|` F (opr M (...) N))) N) (` (opr (<,> M S) (seq) F) N))) ((cv m) (opr M (...) N) F fvres rgen seqzrn.1 seqzrn.2 F (V) (opr M (...) N) resexg ax-mp seqzrn.2 N M m seqzfveq mpan2)) thm (seqzres ((m n) (F m) (F n) (M m) (M n) (S m) (S n)) ((seqzrn.1 (e. S (V))) (seqzrn.2 (e. F (V)))) (-> (e. M (ZZ)) (= (opr (<,> M S) (seq) (|` F (` (ZZ>) M))) (opr (<,> M S) (seq) F))) ((` (ZZ>) M) eqid (cv m) M (cv n) elfzuzt (cv m) (` (ZZ>) M) F fvres syl rgen seqzrn.1 seqzrn.2 F (V) (` (ZZ>) M) resexg ax-mp seqzrn.2 (cv n) M m seqzfveq mpan2 rgen pm3.2i (opr (<,> M S) (seq) (|` F (` (ZZ>) M))) (` (ZZ>) M) (opr (<,> M S) (seq) F) (` (ZZ>) M) n eqfnfv mpbiri seqzrn.1 seqzrn.2 F (V) (` (ZZ>) M) resexg ax-mp M seqzfn seqzrn.1 seqzrn.2 M seqzfn sylanc)) thm (seqzres2 ((n v) (n y) (v y) (k n) (k v) (k y) (F k) (F n) (F v) (F y) (M n) (M v) (M y) (S n) (S v) (S y)) ((seqzrn.1 (e. S (V))) (seqzrn.2 (e. F (V)))) (-> (e. M (ZZ)) (= (opr (<,> M S) (seq) (|` ({<,>|} k y (= (cv y) (` F (cv k)))) (ZZ))) (opr (<,> M S) (seq) F))) ((` (ZZ>) M) eqid (cv v) M (cv n) elfzelz (cv k) (cv v) F fveq2 ({<,>|} k y (/\ (e. (cv k) (ZZ)) (= (cv y) (` F (cv k))))) eqid F (cv v) fvex fvopab4 syl rgen seqzrn.1 zex k y (` F (cv k)) funopabex2 seqzrn.2 (cv n) M v seqzfveq mpan2 rgen pm3.2i (opr (<,> M S) (seq) ({<,>|} k y (/\ (e. (cv k) (ZZ)) (= (cv y) (` F (cv k)))))) (` (ZZ>) M) (opr (<,> M S) (seq) F) (` (ZZ>) M) n eqfnfv mpbiri seqzrn.1 zex k y (` F (cv k)) funopabex2 M seqzfn seqzrn.1 seqzrn.2 M seqzfn sylanc k y (= (cv y) (` F (cv k))) (ZZ) resopab (<,> M S) (seq) opreq2i syl5eq)) thm (serzcl1OLD () ((serzcl.1 (:--> F (` (ZZ>) M) (CC)))) (-> (e. N (` (ZZ>) M)) (e. (` (opr (<,> M (+)) (seq) F) N) (CC))) (serzcl.1 axaddopr addex serzcl.1 (ZZ>) M fvex F (` (ZZ>) M) (CC) (V) fex mp2an N M (CC) seqzcl mp3an23)) thm (serzreclOLD ((j k) (F j) (F k) (M j) (M k) (N j)) ((serzrecli.1 (e. M (ZZ))) (serzrecli.2 (:--> F (` (ZZ>) M) (RR)))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (e. (` (opr (<,> M (+)) (seq) F) N) (RR))) (serzrecli.1 (cv j) M (opr (<,> M (+)) (seq) F) fveq2 (RR) eleq1d (cv j) (cv k) (opr (<,> M (+)) (seq) F) fveq2 (RR) eleq1d (cv j) (opr (cv k) (+) (1)) (opr (<,> M (+)) (seq) F) fveq2 (RR) eleq1d (cv j) N (opr (<,> M (+)) (seq) F) fveq2 (RR) eleq1d addex serzrecli.2 (ZZ>) M fvex F (` (ZZ>) M) (RR) (V) fex mp2an M seqz1 serzrecli.1 M uzidt ax-mp serzrecli.2 M ffvrni ax-mp syl6eqel (` (opr (<,> M (+)) (seq) F) (cv k)) (` F (opr (cv k) (+) (1))) axaddrcl serzrecli.1 (cv k) eluz1 (cv k) M peano2uz sylbir serzrecli.2 (opr (cv k) (+) (1)) ffvrni syl sylan2 serzrecli.1 addex serzrecli.2 (ZZ>) M fvex F (` (ZZ>) M) (RR) (V) fex mp2an M (cv k) seqzp1OLD mp3an1 (RR) eleq1d (e. (` (opr (<,> M (+)) (seq) F) (cv k)) (RR)) adantl mpbird expcom (e. M (ZZ)) 3adant1 uzind mp3an1)) thm (serzref ((F k) (M k)) ((serzrecli.1 (e. M (ZZ))) (serzrecli.2 (:--> F (` (ZZ>) M) (RR)))) (:--> (opr (<,> M (+)) (seq) F) (` (ZZ>) M) (RR)) ((opr (<,> M (+)) (seq) F) (` (ZZ>) M) (RR) k ffnfv serzrecli.1 addex serzrecli.2 (ZZ>) M fvex F (` (ZZ>) M) (RR) (V) fex mp2an M seqzfn ax-mp serzrecli.1 (cv k) eluz1 serzrecli.1 serzrecli.2 (cv k) serzreclOLD sylbi rgen mpbir2an)) thm (dfseq0 ((f g) (f h) (f x) (f y) (g h) (g x) (g y) (h x) (h y) (x y)) () (= (seq0) ({<<,>,>|} f g h (= (cv h) (opr (<,> (0) (cv f)) (seq) (cv g))))) ((opr (cv f) (seq1) (opr (cv g) (shift) (1))) (shift) (-u (1)) oprex (opr (opr (cv f) (seq1) (opr (cv g) (shift) (1))) (shift) (-u (1))) (V) (NN0) resexg ax-mp f visset g visset pm3.2i (= (cv h) (|` (opr (opr (cv f) (seq1) (opr (cv g) (shift) (1))) (shift) (-u (1))) (NN0))) biantrur f g h oprabbii fnoprab2 f g h df-seq0 (seq0) ({<<,>,>|} f g h (= (cv h) (|` (opr (opr (cv f) (seq1) (opr (cv g) (shift) (1))) (shift) (-u (1))) (NN0)))) (X. (V) (V)) fneq1 ax-mp mpbir (<,> (0) (cv f)) (seq) (cv g) oprex f visset g visset pm3.2i (= (cv h) (opr (<,> (0) (cv f)) (seq) (cv g))) biantrur f g h oprabbii fnoprab2 (seq0) (V) (V) ({<<,>,>|} f g h (= (cv h) (opr (<,> (0) (cv f)) (seq) (cv g)))) (V) (V) x y eqfnoprval mp2an (X. (V) (V)) eqid (<,> (0) (cv x)) (seq) (cv y) oprex (cv f) (cv x) (0) opeq2 (seq) (cv g) opreq1d (cv g) (cv y) (<,> (0) (cv x)) (seq) opreq2 ({<<,>,>|} f g h (= (cv h) (opr (<,> (0) (cv f)) (seq) (cv g)))) eqid (V) (V) oprabval5 x visset y visset seq0seqz syl6reqr rgen2 mpbir2an)) thm (ser0cl ((N j) (j k) (F j) (F k)) ((ser0cl.1 (:--> F (NN0) (CC)))) (-> (e. N (NN0)) (e. (` (opr (+) (seq0) F) N) (CC))) ((cv j) (0) (opr (+) (seq0) F) fveq2 (CC) eleq1d (cv j) (cv k) (opr (+) (seq0) F) fveq2 (CC) eleq1d (cv j) (opr (cv k) (+) (1)) (opr (+) (seq0) F) fveq2 (CC) eleq1d (cv j) N (opr (+) (seq0) F) fveq2 (CC) eleq1d addex ser0cl.1 nn0ex F (NN0) (CC) (V) fex mp2an seq00 0nn0 ser0cl.1 (0) ffvrni ax-mp eqeltr addex ser0cl.1 nn0ex F (NN0) (CC) (V) fex mp2an (cv k) seq0p1 (e. (` (opr (+) (seq0) F) (cv k)) (CC)) adantr (` (opr (+) (seq0) F) (cv k)) (` F (opr (cv k) (+) (1))) axaddcl (cv k) peano2nn0 ser0cl.1 (opr (cv k) (+) (1)) ffvrni syl sylan2 ancoms eqeltrd ex nn0ind)) thm (ser0f ((F j)) ((ser0cl.1 (:--> F (NN0) (CC)))) (:--> (opr (+) (seq0) F) (NN0) (CC)) ((opr (+) (seq0) F) (NN0) (CC) j ffnfv addex ser0cl.1 nn0ex F (NN0) (CC) (V) fex mp2an seq0fn ser0cl.1 (cv j) ser0cl rgen mpbir2an)) thm (ser00 ((A y) (k y) (B k) (B y)) ((ser00.1 (= F ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) A))))) (ser00.2 (e. B (V))) (ser00.3 (-> (= (cv k) (0)) (= A B)))) (= (` (opr (+) (seq0) F) (0)) B) (addex ser00.1 nn0ex k y A funopabex2 eqeltr seq00 0nn0 ser00.3 ser00.1 ser00.2 fvopab4 ax-mp eqtr)) thm (ser0p1 ((A y) (k y) (B k) (B y) (N k) (N y)) ((ser00.1 (= F ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) A))))) (ser00.2 (e. B (V))) (ser0p1.3 (-> (= (cv k) (opr N (+) (1))) (= A B)))) (-> (e. N (NN0)) (= (` (opr (+) (seq0) F) (opr N (+) (1))) (opr (` (opr (+) (seq0) F) N) (+) B))) (addex ser00.1 nn0ex k y A funopabex2 eqeltr N seq0p1 N peano2nn0 ser0p1.3 ser00.1 ser00.2 fvopab4 syl (` (opr (+) (seq0) F) N) (+) opreq2d eqtrd)) thm (serzmulc ((j k) (j m) (C j) (k m) (C k) (C m) (F j) (F k) (F m) (G j) (G k) (G m) (M j) (M k) (M m) (N j)) ((serzmulc.1 (e. C (CC))) (serzmulc.2 (:--> F (` (ZZ>) M) (CC))) (serzmulc.3 (e. G (V))) (serzmulc.4 (-> (e. (cv k) (` (ZZ>) M)) (= (` G (cv k)) (opr C (x.) (` F (cv k))))))) (-> (e. N (` (ZZ>) M)) (= (` (opr (<,> M (+)) (seq) G) N) (opr C (x.) (` (opr (<,> M (+)) (seq) F) N)))) ((cv j) M (opr (<,> M (+)) (seq) G) fveq2 (cv j) M (opr (<,> M (+)) (seq) F) fveq2 C (x.) opreq2d eqeq12d (cv j) (cv m) (opr (<,> M (+)) (seq) G) fveq2 (cv j) (cv m) (opr (<,> M (+)) (seq) F) fveq2 C (x.) opreq2d eqeq12d (cv j) (opr (cv m) (+) (1)) (opr (<,> M (+)) (seq) G) fveq2 (cv j) (opr (cv m) (+) (1)) (opr (<,> M (+)) (seq) F) fveq2 C (x.) opreq2d eqeq12d (cv j) N (opr (<,> M (+)) (seq) G) fveq2 (cv j) N (opr (<,> M (+)) (seq) F) fveq2 C (x.) opreq2d eqeq12d M uzidt (cv k) M G fveq2 (cv k) M F fveq2 C (x.) opreq2d eqeq12d serzmulc.4 vtoclga syl addex serzmulc.3 M seqz1 addex serzmulc.2 (ZZ>) M fvex F (` (ZZ>) M) (CC) (V) fex mp2an M seqz1 C (x.) opreq2d 3eqtr4d addex serzmulc.3 (cv m) M seqzp1 (= (` (opr (<,> M (+)) (seq) G) (cv m)) (opr C (x.) (` (opr (<,> M (+)) (seq) F) (cv m)))) adantr (` (opr (<,> M (+)) (seq) G) (cv m)) (opr C (x.) (` (opr (<,> M (+)) (seq) F) (cv m))) (+) (` G (opr (cv m) (+) (1))) opreq1 (e. (cv m) (` (ZZ>) M)) adantl serzmulc.1 C (` (opr (<,> M (+)) (seq) F) (cv m)) (` F (opr (cv m) (+) (1))) axdistr mp3an1 serzmulc.2 (cv m) serzcl1OLD (cv m) M peano2uz serzmulc.2 (opr (cv m) (+) (1)) ffvrni syl sylanc addex serzmulc.2 (ZZ>) M fvex F (` (ZZ>) M) (CC) (V) fex mp2an (cv m) M seqzp1 C (x.) opreq2d (cv m) M peano2uz (cv k) (opr (cv m) (+) (1)) G fveq2 (cv k) (opr (cv m) (+) (1)) F fveq2 C (x.) opreq2d eqeq12d serzmulc.4 vtoclga syl (opr C (x.) (` (opr (<,> M (+)) (seq) F) (cv m))) (+) opreq2d 3eqtr4rd (= (` (opr (<,> M (+)) (seq) G) (cv m)) (opr C (x.) (` (opr (<,> M (+)) (seq) F) (cv m)))) adantr 3eqtrd ex uzind4)) thm (ser0mulc ((F k) (G k) (C k)) ((ser0mulc.1 (e. C (CC))) (ser0mulc.2 (:--> F (NN0) (CC))) (ser0mulc.3 (e. G (V))) (ser0mulc.4 (-> (e. (cv k) (NN0)) (= (` G (cv k)) (opr C (x.) (` F (cv k))))))) (-> (e. N (NN0)) (= (` (opr (+) (seq0) G) N) (opr C (x.) (` (opr (+) (seq0) F) N)))) (N elnn0uz ser0mulc.1 ser0mulc.2 nn0uz (NN0) (` (ZZ>) (0)) F (CC) feq2 ax-mp mpbi ser0mulc.3 (cv k) elnn0uz ser0mulc.4 sylbir N serzmulc sylbi addex ser0mulc.3 seq0seqz N fveq1i addex ser0mulc.2 nn0ex F (NN0) (CC) (V) fex mp2an seq0seqz N fveq1i C (x.) opreq2i 3eqtr4g)) thm (ser1mulc ((F k) (G k) (C k)) ((ser1mulc.1 (e. C (CC))) (ser1mulc.2 (:--> F (NN) (CC))) (ser1mulc.3 (e. G (V))) (ser1mulc.4 (-> (e. (cv k) (NN)) (= (` G (cv k)) (opr C (x.) (` F (cv k))))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) G) N) (opr C (x.) (` (opr (+) (seq1) F) N)))) (N elnnuz ser1mulc.1 ser1mulc.2 nnuz (NN) (` (ZZ>) (1)) F (CC) feq2 ax-mp mpbi ser1mulc.3 (cv k) elnnuz ser1mulc.4 sylbir N serzmulc sylbi addex ser1mulc.3 seq1seqz N fveq1i addex ser1mulc.2 nnex F (NN) (CC) (V) fex mp2an seq1seqz N fveq1i C (x.) opreq2i 3eqtr4g)) thm (expvalt ((x y) (x z) (A x) (y z) (A y) (A z) (N x) (N y) (N z)) () (-> (/\ (e. A (CC)) (e. N (NN0))) (= (opr A (^) N) (if (= N (0)) (1) (` (opr (x.) (seq1) (X. (NN) ({} A))) N)))) (1nn0 elisseti (opr (x.) (seq1) (X. (NN) ({} A))) N fvex (= N (0)) ifex (cv x) A sneq ({} (cv x)) ({} A) (NN) xpeq2 syl (x.) (seq1) opreq2d (cv y) fveq1d (= (cv y) (0)) (1) ifeq2d (cv y) N (opr (x.) (seq1) (X. (NN) ({} A))) fveq2 (= (cv y) (0)) (1) ifeq2d (cv y) N (0) eqeq1 (1) (` (opr (x.) (seq1) (X. (NN) ({} A))) N) ifbid eqtrd x y z df-exp oprabval2)) thm (exp0t () () (-> (e. A (CC)) (= (opr A (^) (0)) (1))) (0nn0 A (0) expvalt mpan2 (0) eqid (= (0) (0)) (1) (` (opr (x.) (seq1) (X. (NN) ({} A))) (0)) iftrue ax-mp syl6eq)) thm (expnnvalt () () (-> (/\ (e. A (CC)) (e. B (NN))) (= (opr A (^) B) (` (opr (x.) (seq1) (X. (NN) ({} A))) B))) (A B expvalt B nnnn0t sylan2 B nnne0t B (0) df-ne sylib (= B (0)) (1) (` (opr (x.) (seq1) (X. (NN) ({} A))) B) iffalse syl (e. A (CC)) adantl eqtrd)) thm (exp1t () () (-> (e. A (CC)) (= (opr A (^) (1)) A)) (1nn A (1) expnnvalt mpan2 A (CC) (NN) fconstg 1nn (X. (NN) ({} A)) (NN) A (1) fvconst mpan2 syl mulex nnex A snex xpex seq11 syl5eq eqtrd)) thm (expp1t () () (-> (/\ (e. A (CC)) (e. N (NN0))) (= (opr A (^) (opr N (+) (1))) (opr (opr A (^) N) (x.) A))) (mulex nnex A snex xpex N seq1p1 (e. A (CC)) adantl (X. (NN) ({} A)) (NN) A (opr N (+) (1)) fvconst A (CC) (NN) fconstg N peano2nn syl2an (` (opr (x.) (seq1) (X. (NN) ({} A))) N) (x.) opreq2d eqtrd A (opr N (+) (1)) expnnvalt N peano2nn sylan2 A N expnnvalt (x.) A opreq1d 3eqtr4d ex N (0) (+) (1) opreq1 A (^) opreq2d N (0) A (^) opreq2 (x.) A opreq1d eqeq12d A mulid2t A exp0t (x.) A opreq1d A exp1t 1cn addid2 A (^) opreq2i syl5eq 3eqtr4rd syl5bir com12 jaod N elnn0 syl5ib imp)) thm (expcllem ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B z) (F x) (F y) (F z) (F w)) ((expcllem.1 (C_ F (CC))) (expcllem.2 (-> (/\ (e. (cv x) F) (e. (cv y) F)) (e. (opr (cv x) (x.) (cv y)) F))) (expcllem.3 (e. (1) F))) (-> (/\ (e. A F) (e. B (NN0))) (e. (opr A (^) B) F)) ((cv z) (1) A (^) opreq2 F eleq1d (e. A F) imbi2d (cv z) (cv w) A (^) opreq2 F eleq1d (e. A F) imbi2d (cv z) (opr (cv w) (+) (1)) A (^) opreq2 F eleq1d (e. A F) imbi2d (cv z) B A (^) opreq2 F eleq1d (e. A F) imbi2d expcllem.1 A sseli A exp1t syl F eleq1d ibir (cv x) (opr A (^) (cv w)) (x.) (cv y) opreq1 F eleq1d (cv y) A (opr A (^) (cv w)) (x.) opreq2 F eleq1d expcllem.2 vtocl2ga ancoms (e. (cv w) (NN)) adantlr A (cv w) expp1t (cv w) nnnn0t sylan2 expcllem.1 A sseli sylan F eleq1d (e. (opr A (^) (cv w)) F) adantr mpbird exp31 com12 a2d nnind impcom B (0) A (^) opreq2 expcllem.1 A sseli A exp0t syl sylan9eqr expcllem.3 syl6eqel jaodan B elnn0 sylan2b)) thm (nnexpclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (NN)) (e. N (NN0))) (e. (opr A (^) N) (NN))) (nnsscn (cv x) (cv y) nnmulclt 1nn A N expcllem)) thm (nn0expclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (NN0)) (e. N (NN0))) (e. (opr A (^) N) (NN0))) (nn0sscn (cv x) (cv y) nn0mulclt 1nn0 A N expcllem)) thm (zexpclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (ZZ)) (e. N (NN0))) (e. (opr A (^) N) (ZZ))) (zsscn (cv x) (cv y) zmulclt 1z A N expcllem)) thm (qexpclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (QQ)) (e. N (NN0))) (e. (opr A (^) N) (QQ))) (qsscn (cv x) (cv y) qmulclt 1z (1) zqt ax-mp A N expcllem)) thm (reexpclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (RR)) (e. N (NN0))) (e. (opr A (^) N) (RR))) (axresscn (cv x) (cv y) axmulrcl ax1re A N expcllem)) thm (expclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (CC)) (e. N (NN0))) (e. (opr A (^) N) (CC))) ((CC) ssid (cv x) (cv y) axmulcl 1cn A N expcllem)) thm (expm1t () () (-> (/\ (e. A (CC)) (e. N (NN))) (= (opr A (^) N) (opr (opr A (^) (opr N (-) (1))) (x.) A))) (N nncnt 1cn N (1) npcant mpan2 syl A (^) opreq2d (e. A (CC)) adantl A (opr N (-) (1)) expp1t N elnnnn0 pm3.27bd sylan2 eqtr3d)) thm (1expt ((j k) (N j) (N k)) () (-> (e. N (NN0)) (= (opr (1) (^) N) (1))) ((cv j) (0) (1) (^) opreq2 (1) eqeq1d (cv j) (cv k) (1) (^) opreq2 (1) eqeq1d (cv j) (opr (cv k) (+) (1)) (1) (^) opreq2 (1) eqeq1d (cv j) N (1) (^) opreq2 (1) eqeq1d 1cn (1) exp0t ax-mp 1cn (1) (cv k) expp1t mpan (1) eqeq1d (opr (1) (^) (cv k)) (1) (x.) (1) opreq1 1cn mulid1 syl6eq syl5bir nn0ind)) thm (expeq0t ((j k) (A j) (A k) (N j) (N k)) () (-> (/\ (e. A (CC)) (e. N (NN))) (<-> (= (opr A (^) N) (0)) (= A (0)))) ((cv j) (1) A (^) opreq2 (0) eqeq1d (= A (0)) bibi1d (e. A (CC)) imbi2d (cv j) (cv k) A (^) opreq2 (0) eqeq1d (= A (0)) bibi1d (e. A (CC)) imbi2d (cv j) (opr (cv k) (+) (1)) A (^) opreq2 (0) eqeq1d (= A (0)) bibi1d (e. A (CC)) imbi2d (cv j) N A (^) opreq2 (0) eqeq1d (= A (0)) bibi1d (e. A (CC)) imbi2d A exp1t (0) eqeq1d A (cv k) expp1t (0) eqeq1d (opr A (^) (cv k)) A mul0ort A (cv k) expclt (e. A (CC)) (e. (cv k) (NN0)) pm3.26 sylanc bitrd (cv k) nnnn0t sylan2 (= (opr A (^) (cv k)) (0)) (= A (0)) bi1 (<-> (= (opr A (^) (cv k)) (0)) (= A (0))) (= A (0)) idd jaod (= A (0)) (= (opr A (^) (cv k)) (0)) olc jctir (\/ (= (opr A (^) (cv k)) (0)) (= A (0))) (= A (0)) bi sylibr sylan9bb exp31 com12 a2d nnind impcom)) thm (expne0t () () (-> (/\ (e. A (CC)) (e. N (NN))) (<-> (=/= A (0)) (=/= (opr A (^) N) (0)))) (A N expeq0t bicomd eqneqd)) thm (expne0it () () (-> (/\/\ (e. A (CC)) (e. N (NN0)) (=/= A (0))) (=/= (opr A (^) N) (0))) (A N expne0t biimpd ex ax1ne0 N (0) A (^) opreq2 A exp0t sylan9eqr (0) neeq1d mpbiri ex (=/= A (0)) a1dd jaod N elnn0 syl5ib 3imp)) thm (expgt0t ((j k) (A j) (A k) (N j) (N k)) () (-> (/\/\ (e. A (RR)) (e. N (NN0)) (br (0) (<) A)) (br (0) (<) (opr A (^) N))) ((cv j) (0) A (^) opreq2 (0) (<) breq2d (/\ (e. A (RR)) (br (0) (<) A)) imbi2d (cv j) (cv k) A (^) opreq2 (0) (<) breq2d (/\ (e. A (RR)) (br (0) (<) A)) imbi2d (cv j) (opr (cv k) (+) (1)) A (^) opreq2 (0) (<) breq2d (/\ (e. A (RR)) (br (0) (<) A)) imbi2d (cv j) N A (^) opreq2 (0) (<) breq2d (/\ (e. A (RR)) (br (0) (<) A)) imbi2d A recnt A exp0t syl lt01 syl5breqr (br (0) (<) A) adantr (opr A (^) (cv k)) A axmulgt0 A (cv k) reexpclt (e. A (RR)) (e. (cv k) (NN0)) pm3.26 sylanc A (cv k) expp1t A recnt sylan (0) (<) breq2d sylibrd exp4b com12 com34 imp3a a2d nn0ind exp3a com12 3imp)) thm (0expt () () (-> (e. N (NN)) (= (opr (0) (^) N) (0))) ((0) eqid 0cn (0) N expeq0t mpan mpbiri)) thm (expge0t () () (-> (/\/\ (e. A (RR)) (e. N (NN0)) (br (0) (<_) A)) (br (0) (<_) (opr A (^) N))) (A N expgt0t N nnnn0t syl3an2 3expa ex (0) A (^) N opreq1 (0) eqeq2d N 0expt eqcomd syl5bi com12 (e. A (RR)) adantl orim12d 0re (0) A leloet mpan (e. N (NN)) adantr A N reexpclt N nnnn0t sylan2 0re (0) (opr A (^) N) leloet mpan syl 3imtr4d ex N (0) A (^) opreq2 (0) (<_) breq2d A recnt A exp0t syl 0re ax1re lt01 ltlei syl5breqr syl5bir com12 (br (0) (<_) A) a1dd jaod N elnn0 syl5ib 3imp)) thm (expgt1t ((x y) (A x) (A y) (N x)) () (-> (/\/\ (e. A (RR)) (e. N (NN)) (br (1) (<) A)) (br (1) (<) (opr A (^) N))) ((cv x) (1) A (^) opreq2 (1) (<) breq2d (/\ (e. A (RR)) (br (1) (<) A)) imbi2d (cv x) (cv y) A (^) opreq2 (1) (<) breq2d (/\ (e. A (RR)) (br (1) (<) A)) imbi2d (cv x) (opr (cv y) (+) (1)) A (^) opreq2 (1) (<) breq2d (/\ (e. A (RR)) (br (1) (<) A)) imbi2d (cv x) N A (^) opreq2 (1) (<) breq2d (/\ (e. A (RR)) (br (1) (<) A)) imbi2d A recnt A exp1t syl (1) (<) breq2d biimpar (opr A (^) (cv y)) A mulgt1t A (cv y) reexpclt (e. A (RR)) (e. (cv y) (NN0)) pm3.26 jca sylan ex A (cv y) expp1t A recnt sylan (1) (<) breq2d sylibrd (cv y) nnnn0t sylan2 exp4b com12 com34 imp3a a2d nnind exp3a com12 3imp)) thm (expge1t () () (-> (/\/\ (e. A (RR)) (e. N (NN)) (br (1) (<_) A)) (br (1) (<_) (opr A (^) N))) (A N expgt1t 3expa ex (1) A (^) N opreq1 (1) eqeq2d N nnnn0t N 1expt syl eqcomd syl5bi com12 (e. A (RR)) adantl orim12d ax1re (1) A leloet mpan (e. N (NN)) adantr A N reexpclt N nnnn0t sylan2 ax1re (1) (opr A (^) N) leloet mpan syl 3imtr4d 3impia)) thm (mulexpt ((j k) (A j) (A k) (B j) (B k) (N j)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. N (NN0))) (= (opr (opr A (x.) B) (^) N) (opr (opr A (^) N) (x.) (opr B (^) N)))) ((cv j) (0) (opr A (x.) B) (^) opreq2 (cv j) (0) A (^) opreq2 (cv j) (0) B (^) opreq2 (x.) opreq12d eqeq12d (/\ (e. A (CC)) (e. B (CC))) imbi2d (cv j) (cv k) (opr A (x.) B) (^) opreq2 (cv j) (cv k) A (^) opreq2 (cv j) (cv k) B (^) opreq2 (x.) opreq12d eqeq12d (/\ (e. A (CC)) (e. B (CC))) imbi2d (cv j) (opr (cv k) (+) (1)) (opr A (x.) B) (^) opreq2 (cv j) (opr (cv k) (+) (1)) A (^) opreq2 (cv j) (opr (cv k) (+) (1)) B (^) opreq2 (x.) opreq12d eqeq12d (/\ (e. A (CC)) (e. B (CC))) imbi2d (cv j) N (opr A (x.) B) (^) opreq2 (cv j) N A (^) opreq2 (cv j) N B (^) opreq2 (x.) opreq12d eqeq12d (/\ (e. A (CC)) (e. B (CC))) imbi2d A B axmulcl (opr A (x.) B) exp0t syl A exp0t B exp0t (x.) opreqan12d 1cn mulid1 syl6eq eqtr4d (opr A (x.) B) (cv k) expp1t A B axmulcl sylan (= (opr (opr A (x.) B) (^) (cv k)) (opr (opr A (^) (cv k)) (x.) (opr B (^) (cv k)))) adantr (opr (opr A (x.) B) (^) (cv k)) (opr (opr A (^) (cv k)) (x.) (opr B (^) (cv k))) (x.) (opr A (x.) B) opreq1 (/\ (/\ (e. A (CC)) (e. B (CC))) (e. (cv k) (NN0))) adantl (opr A (^) (cv k)) (opr B (^) (cv k)) A B mul4t A (cv k) expclt B (cv k) expclt anim12i anandirs (/\ (e. A (CC)) (e. B (CC))) (e. (cv k) (NN0)) pm3.26 sylanc A (cv k) expp1t (e. B (CC)) adantlr B (cv k) expp1t (e. A (CC)) adantll (x.) opreq12d eqtr4d (= (opr (opr A (x.) B) (^) (cv k)) (opr (opr A (^) (cv k)) (x.) (opr B (^) (cv k)))) adantr 3eqtrd exp31 com12 a2d nn0ind exp3a com3l 3imp)) thm (recexpt ((j k) (A j) (A k) (N j)) () (-> (/\/\ (e. A (CC)) (e. N (NN0)) (=/= A (0))) (= (opr (opr (1) (/) A) (^) N) (opr (1) (/) (opr A (^) N)))) ((cv j) (0) (opr (1) (/) A) (^) opreq2 (cv j) (0) A (^) opreq2 (1) (/) opreq2d eqeq12d (/\ (e. A (CC)) (=/= A (0))) imbi2d (cv j) (cv k) (opr (1) (/) A) (^) opreq2 (cv j) (cv k) A (^) opreq2 (1) (/) opreq2d eqeq12d (/\ (e. A (CC)) (=/= A (0))) imbi2d (cv j) (opr (cv k) (+) (1)) (opr (1) (/) A) (^) opreq2 (cv j) (opr (cv k) (+) (1)) A (^) opreq2 (1) (/) opreq2d eqeq12d (/\ (e. A (CC)) (=/= A (0))) imbi2d (cv j) N (opr (1) (/) A) (^) opreq2 (cv j) N A (^) opreq2 (1) (/) opreq2d eqeq12d (/\ (e. A (CC)) (=/= A (0))) imbi2d A recclt (opr (1) (/) A) exp0t syl A exp0t (1) (/) opreq2d 1cn div1 syl6eq (=/= A (0)) adantr eqtr4d (opr (opr (1) (/) A) (^) (cv k)) (opr (1) (/) (opr A (^) (cv k))) (x.) (opr (1) (/) A) opreq1 (/\ (e. A (CC)) (e. (cv k) (NN0))) (=/= A (0)) ad2antll (opr (1) (/) A) (cv k) expp1t A recclt sylan an1rs (= (opr (opr (1) (/) A) (^) (cv k)) (opr (1) (/) (opr A (^) (cv k)))) adantrr A (cv k) expp1t (1) (/) opreq2d (=/= A (0)) adantr (1) (opr A (^) (cv k)) (1) A divmuldivt A (cv k) expclt 1cn jctil (e. A (CC)) (e. (cv k) (NN0)) pm3.26 1cn jctil jca (=/= A (0)) adantr A (cv k) expne0it 3expa (/\ (e. A (CC)) (e. (cv k) (NN0))) (=/= A (0)) pm3.27 jca sylanc 1cn mulid1 (/) (opr (opr A (^) (cv k)) (x.) A) opreq1i syl6req eqtrd (= (opr (opr (1) (/) A) (^) (cv k)) (opr (1) (/) (opr A (^) (cv k)))) adantrr 3eqtr4d exp43 com12 imp3a a2d nn0ind exp3a com12 3imp)) thm (expaddt ((j k) (A j) (A k) (M j) (M k) (N j)) () (-> (/\/\ (e. A (CC)) (e. M (NN0)) (e. N (NN0))) (= (opr A (^) (opr M (+) N)) (opr (opr A (^) M) (x.) (opr A (^) N)))) ((cv j) (0) M (+) opreq2 A (^) opreq2d (cv j) (0) A (^) opreq2 (opr A (^) M) (x.) opreq2d eqeq12d (/\ (e. A (CC)) (e. M (NN0))) imbi2d (cv j) (cv k) M (+) opreq2 A (^) opreq2d (cv j) (cv k) A (^) opreq2 (opr A (^) M) (x.) opreq2d eqeq12d (/\ (e. A (CC)) (e. M (NN0))) imbi2d (cv j) (opr (cv k) (+) (1)) M (+) opreq2 A (^) opreq2d (cv j) (opr (cv k) (+) (1)) A (^) opreq2 (opr A (^) M) (x.) opreq2d eqeq12d (/\ (e. A (CC)) (e. M (NN0))) imbi2d (cv j) N M (+) opreq2 A (^) opreq2d (cv j) N A (^) opreq2 (opr A (^) M) (x.) opreq2d eqeq12d (/\ (e. A (CC)) (e. M (NN0))) imbi2d A M expclt (opr A (^) M) ax1id syl A exp0t (e. M (NN0)) adantr (opr A (^) M) (x.) opreq2d M nn0cnt M ax0id syl (e. A (CC)) adantl A (^) opreq2d 3eqtr4rd 1cn M (cv k) (1) axaddass mp3an3 M nn0cnt (cv k) nn0cnt syl2an (e. A (CC)) adantll A (^) opreq2d A (opr M (+) (cv k)) expp1t (e. A (CC)) (e. M (NN0)) pm3.26 (e. (cv k) (NN0)) adantr M (cv k) nn0addclt (e. A (CC)) adantll sylanc eqtr3d A (cv k) expp1t (e. M (NN0)) adantlr (opr A (^) M) (x.) opreq2d (opr A (^) M) (opr A (^) (cv k)) A axmulass A M expclt (e. (cv k) (NN0)) adantr A (cv k) expclt (e. M (NN0)) adantlr (e. A (CC)) (e. M (NN0)) pm3.26 (e. (cv k) (NN0)) adantr syl3anc eqtr4d eqeq12d (opr A (^) (opr M (+) (cv k))) (opr (opr A (^) M) (x.) (opr A (^) (cv k))) (x.) A opreq1 syl5bir expcom a2d nn0ind exp3a com3l 3imp)) thm (expmult ((j k) (A j) (A k) (M j) (M k) (N j)) () (-> (/\/\ (e. A (CC)) (e. M (NN0)) (e. N (NN0))) (= (opr A (^) (opr M (x.) N)) (opr (opr A (^) M) (^) N))) ((cv j) (0) M (x.) opreq2 A (^) opreq2d (cv j) (0) (opr A (^) M) (^) opreq2 eqeq12d (/\ (e. A (CC)) (e. M (NN0))) imbi2d (cv j) (cv k) M (x.) opreq2 A (^) opreq2d (cv j) (cv k) (opr A (^) M) (^) opreq2 eqeq12d (/\ (e. A (CC)) (e. M (NN0))) imbi2d (cv j) (opr (cv k) (+) (1)) M (x.) opreq2 A (^) opreq2d (cv j) (opr (cv k) (+) (1)) (opr A (^) M) (^) opreq2 eqeq12d (/\ (e. A (CC)) (e. M (NN0))) imbi2d (cv j) N M (x.) opreq2 A (^) opreq2d (cv j) N (opr A (^) M) (^) opreq2 eqeq12d (/\ (e. A (CC)) (e. M (NN0))) imbi2d M nn0cnt M mul01t syl A (^) opreq2d A exp0t sylan9eqr A M expclt (opr A (^) M) exp0t syl eqtr4d 1cn M (cv k) (1) axdistr mp3an3 M ax1id (e. (cv k) (CC)) adantr (opr M (x.) (cv k)) (+) opreq2d eqtrd M nn0cnt (cv k) nn0cnt syl2an (e. A (CC)) adantll A (^) opreq2d A (opr M (x.) (cv k)) M expaddt (e. A (CC)) (e. M (NN0)) pm3.26 (e. (cv k) (NN0)) adantr M (cv k) nn0mulclt (e. A (CC)) adantll (e. A (CC)) (e. M (NN0)) pm3.27 (e. (cv k) (NN0)) adantr syl3anc eqtrd (opr A (^) M) (cv k) expp1t A M expclt sylan eqeq12d (opr A (^) (opr M (x.) (cv k))) (opr (opr A (^) M) (^) (cv k)) (x.) (opr A (^) M) opreq1 syl5bir expcom a2d nn0ind exp3a com3l 3imp)) thm (expsubt () () (-> (/\ (/\/\ (e. A (CC)) (e. M (NN0)) (e. N (NN0))) (/\ (=/= A (0)) (br N (<_) M))) (= (opr A (^) (opr M (-) N)) (opr (opr A (^) M) (/) (opr A (^) N)))) (A N (opr M (-) N) expaddt (e. A (CC)) (e. M (NN0)) (e. N (NN0)) 3simp1 (br N (<_) M) adantr (e. A (CC)) (e. M (NN0)) (e. N (NN0)) 3simp3 (br N (<_) M) adantr N M nn0subt ancoms biimpa (e. A (CC)) 3adantl1 syl3anc N M pncan3t N nn0cnt M nn0cnt syl2an ancoms A (^) opreq2d (e. A (CC)) 3adant1 (br N (<_) M) adantr eqtr3d (=/= A (0)) adantrl (opr A (^) M) (opr A (^) N) (opr A (^) (opr M (-) N)) divmult A M expclt (e. N (NN0)) 3adant3 (/\ (=/= A (0)) (br N (<_) M)) adantr A N expclt (e. M (NN0)) 3adant2 (/\ (=/= A (0)) (br N (<_) M)) adantr A (opr M (-) N) expclt (e. A (CC)) (e. M (NN0)) (e. N (NN0)) 3simp1 (br N (<_) M) adantr N M nn0subt ancoms biimpa (e. A (CC)) 3adantl1 sylanc (=/= A (0)) adantrl 3jca A N expne0it 3expa (e. M (NN0)) 3adantl2 (br N (<_) M) adantrr sylanc mpbird eqcomd)) thm (expordit () () (-> (/\ (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) (/\ (br (1) (<) A) (br M (<) N))) (br (opr A (^) M) (<) (opr A (^) N))) (A (opr N (-) M) expgt1t (e. A (RR)) (e. M (NN0)) (e. N (NN0)) 3simp1 (/\ (br (1) (<) A) (br M (<) N)) adantr M N znnsubt M nn0zt N nn0zt syl2an (e. A (RR)) 3adant1 biimpa (br (1) (<) A) adantrl (br (1) (<) A) (br M (<) N) pm3.26 (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) adantl syl3anc (1) (opr A (^) (opr N (-) M)) (opr A (^) M) ltmul1t ax1re (/\ (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) (/\ (br (1) (<) A) (br M (<) N))) a1i A (opr N (-) M) reexpclt (e. A (RR)) (e. M (NN0)) (e. N (NN0)) 3simp1 (br M (<) N) adantr M N znnsubt M nn0zt N nn0zt syl2an (e. A (RR)) 3adant1 biimpa (opr N (-) M) nnnn0t syl sylanc (br (1) (<) A) adantrl A M reexpclt (e. N (NN0)) 3adant3 (/\ (br (1) (<) A) (br M (<) N)) adantr 3jca lt01 0re ax1re (0) (1) A axlttrn mp3an12 mpani (e. M (NN0)) adantr A M expgt0t 3expa ex syld (e. N (NN0)) 3adant3 imp (br M (<) N) adantrr sylanc mpbid A M expclt A recnt sylan (opr A (^) M) mulid2t syl (e. N (NN0)) 3adant3 (/\ (br (1) (<) A) (br M (<) N)) adantr (e. A (RR)) (e. M (NN0)) (e. N (NN0)) 3simp1 recnd (br M (<) N) adantr M N znnsubt M nn0zt N nn0zt syl2an (e. A (RR)) 3adant1 biimpa (e. A (RR)) (e. M (NN0)) (e. N (NN0)) 3simp2 (br M (<) N) adantr 3jca (br (1) (<) A) adantrl A (opr N (-) M) M expaddt (opr N (-) M) nnnn0t syl3an2 syl N M npcant N nn0cnt M nn0cnt syl2an ancoms (e. A (RR)) 3adant1 A (^) opreq2d (/\ (br (1) (<) A) (br M (<) N)) adantr eqtr3d 3brtr3d)) thm (expcant () () (-> (/\ (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) (br (1) (<) A)) (<-> (= (opr A (^) M) (opr A (^) N)) (= M N))) (A M N expordit exp32 imp A N M expordit exp32 3com23 imp orim12d M N lttri2t M nn0ret N nn0ret syl2an (e. A (RR)) 3adant1 (br (1) (<) A) adantr (opr A (^) M) (opr A (^) N) lttri2t A M reexpclt A N reexpclt syl2an 3impdi (br (1) (<) A) adantr 3imtr4d a3d M N A (^) opreq2 (/\ (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) (br (1) (<) A)) a1i impbid)) thm (expordt () () (-> (/\ (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) (br (1) (<) A)) (<-> (br M (<) N) (br (opr A (^) M) (<) (opr A (^) N)))) (A M N expordit exp32 imp A M N expcant biimprd A N M expordit exp32 3com23 imp orim12d con3d (opr A (^) M) (opr A (^) N) axlttri A M reexpclt A N reexpclt syl2an 3impdi (br (1) (<) A) adantr M N axlttri M nn0ret N nn0ret syl2an (e. A (RR)) 3adant1 (br (1) (<) A) adantr 3imtr4d impbid)) thm (expwordit () () (-> (/\ (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) (/\ (br (1) (<_) A) (br M (<_) N))) (br (opr A (^) M) (<_) (opr A (^) N))) (ax1re (1) A leloet mpan (e. M (NN0)) (e. N (NN0)) 3ad2ant1 A M N expordt biimpd M N A (^) opreq2 (/\ (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) (br (1) (<) A)) a1i orim12d M N leloet M nn0ret N nn0ret syl2an (e. A (RR)) 3adant1 (br (1) (<) A) adantr (opr A (^) M) (opr A (^) N) leloet A M reexpclt (e. N (NN0)) 3adant3 A N reexpclt (e. M (NN0)) 3adant2 sylanc (br (1) (<) A) adantr 3imtr4d ex (1) A (^) M opreq1 (1) A (^) N opreq1 (<_) breq12d ax1re leid M 1expt N 1expt (<_) breqan12d mpbiri syl5bi com12 (e. A (RR)) 3adant1 (br M (<_) N) a1dd jaod sylbid imp32)) thm (expmwordit ((j k) (A j) (A k) (B j) (B k) (N j) (N k)) () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. N (NN0))) (/\ (br (0) (<_) A) (br A (<_) B))) (br (opr A (^) N) (<_) (opr B (^) N))) ((opr A (^) (cv k)) (opr B (^) (cv k)) A B lemul12it A (cv k) reexpclt (e. B (RR)) adantlr (/\ (br (0) (<_) A) (br A (<_) B)) adantlr (br (opr A (^) (cv k)) (<_) (opr B (^) (cv k))) adantr B (cv k) reexpclt (e. A (RR)) adantll (/\ (br (0) (<_) A) (br A (<_) B)) adantlr (br (opr A (^) (cv k)) (<_) (opr B (^) (cv k))) adantr jca A (cv k) expge0t 3expa an1rs (e. B (RR)) adantllr (br A (<_) B) adantlrr (br (opr A (^) (cv k)) (<_) (opr B (^) (cv k))) anim1i jca (e. A (RR)) (e. B (RR)) pm3.26 (/\ (br (0) (<_) A) (br A (<_) B)) adantr (e. (cv k) (NN0)) (br (opr A (^) (cv k)) (<_) (opr B (^) (cv k))) ad2antrr (e. A (RR)) (e. B (RR)) pm3.27 (/\ (br (0) (<_) A) (br A (<_) B)) adantr (e. (cv k) (NN0)) (br (opr A (^) (cv k)) (<_) (opr B (^) (cv k))) ad2antrr jca (br (0) (<_) A) (br A (<_) B) pm3.26 (/\ (e. A (RR)) (e. B (RR))) adantl (e. (cv k) (NN0)) (br (opr A (^) (cv k)) (<_) (opr B (^) (cv k))) ad2antrr (br (0) (<_) A) (br A (<_) B) pm3.27 (/\ (e. A (RR)) (e. B (RR))) adantl (e. (cv k) (NN0)) (br (opr A (^) (cv k)) (<_) (opr B (^) (cv k))) ad2antrr jca jca sylanc A (cv k) expp1t A recnt sylan (e. B (RR)) adantlr (/\ (br (0) (<_) A) (br A (<_) B)) adantlr (br (opr A (^) (cv k)) (<_) (opr B (^) (cv k))) adantr B (cv k) expp1t B recnt sylan (e. A (RR)) adantll (/\ (br (0) (<_) A) (br A (<_) B)) adantlr (br (opr A (^) (cv k)) (<_) (opr B (^) (cv k))) adantr 3brtr4d ex expcom a2d A exp0t (e. B (CC)) adantr ax1re leid syl6eqbr B exp0t (e. A (CC)) adantl breqtrrd A recnt B recnt syl2an (/\ (br (0) (<_) A) (br A (<_) B)) adantr (cv j) (0) A (^) opreq2 (cv j) (0) B (^) opreq2 (<_) breq12d (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br A (<_) B))) imbi2d (cv j) (cv k) A (^) opreq2 (cv j) (cv k) B (^) opreq2 (<_) breq12d (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br A (<_) B))) imbi2d (cv j) (opr (cv k) (+) (1)) A (^) opreq2 (cv j) (opr (cv k) (+) (1)) B (^) opreq2 (<_) breq12d (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br A (<_) B))) imbi2d (cv j) N A (^) opreq2 (cv j) N B (^) opreq2 (<_) breq12d (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br A (<_) B))) imbi2d nn0indALT exp4c com3l 3imp1)) thm (expbndt () () (-> (/\/\ (e. A (RR)) (e. N (NN0)) (br (2) (<_) A)) (br (opr A (^) N) (<_) (opr (opr (2) (^) N) (x.) (opr (opr A (-) (1)) (^) N)))) (A (opr (2) (x.) (opr A (-) (1))) N expmwordit (e. A (RR)) (e. N (NN0)) (br (2) (<_) A) 3simp1 ax1re A (1) resubclt mpan2 2re (2) (opr A (-) (1)) axmulrcl mpan syl (e. N (NN0)) (br (2) (<_) A) 3ad2ant1 (e. A (RR)) (e. N (NN0)) (br (2) (<_) A) 3simp2 3jca 0re 2re 2pos ltlei 0re 2re (0) (2) A letrt mp3an12 mpani imp 2re A (2) resubclt mpan2 2re (2) A (opr A (-) (2)) leadd2t mp3an1 mpdan biimpa A recnt 2cn A (2) npcant mpan2 syl (br (2) (<_) A) adantr A recnt 2cn 1cn (2) A (1) subdit mp3an13 A 2timest 2cn mulid1 (e. A (CC)) a1i (-) opreq12d 2cn A A (2) addsubt mp3an3 anidms 3eqtrrd syl (br (2) (<_) A) adantr 3brtr3d jca (e. N (NN0)) 3adant2 sylanc 2cn (2) (opr A (-) (1)) N mulexpt mp3an1 ax1re A (1) resubclt mpan2 recnd sylan (br (2) (<_) A) 3adant3 breqtrd)) thm (sqvalt () () (-> (e. A (CC)) (= (opr A (^) (2)) (opr A (x.) A))) (1nn0 A (1) expp1t mpan2 df-2 A (^) opreq2i syl5eq A exp1t (x.) A opreq1d eqtrd)) thm (sqclt () () (-> (e. A (CC)) (e. (opr A (^) (2)) (CC))) (A sqvalt A A axmulcl anidms eqeltrd)) thm (sqval () ((sqval.1 (e. A (CC)))) (= (opr A (^) (2)) (opr A (x.) A)) (sqval.1 A sqvalt ax-mp)) thm (sqcl () ((sqval.1 (e. A (CC)))) (e. (opr A (^) (2)) (CC)) (sqval.1 A sqclt ax-mp)) thm (sq00 () ((sqval.1 (e. A (CC)))) (<-> (= (opr A (^) (2)) (0)) (= A (0))) (sqval.1 sqval (0) eqeq1i sqval.1 msq0 bitr)) thm (sqmul () ((sqval.1 (e. A (CC))) (sqmul.2 (e. B (CC)))) (= (opr (opr A (x.) B) (^) (2)) (opr (opr A (^) (2)) (x.) (opr B (^) (2)))) (sqval.1 sqmul.2 2nn0 A B (2) mulexpt mp3an)) thm (sqdiv () ((sqval.1 (e. A (CC))) (sqmul.2 (e. B (CC))) (sqdiv.3 (=/= B (0)))) (= (opr (opr A (/) B) (^) (2)) (opr (opr A (^) (2)) (/) (opr B (^) (2)))) (sqval.1 sqmul.2 sqval.1 sqmul.2 sqdiv.3 sqdiv.3 divmuldiv sqval.1 sqmul.2 sqdiv.3 divcl sqval sqval.1 sqval sqmul.2 sqval (/) opreq12i 3eqtr4)) thm (sqreci () ((sqval.1 (e. A (CC))) (sqreci.1 (=/= A (0)))) (= (opr (opr (1) (/) A) (^) (2)) (opr (1) (/) (opr A (^) (2)))) (1cn sqval.1 1cn sqval.1 sqreci.1 sqreci.1 divmuldiv 1cn mulid1 (/) (opr A (x.) A) opreq1i eqtr sqval.1 sqreci.1 reccl sqval sqval.1 sqval (1) (/) opreq2i 3eqtr4)) thm (sq0t () () (-> (e. A (CC)) (<-> (= (opr A (^) (2)) (0)) (= A (0)))) (A (if (e. A (CC)) A (0)) (^) (2) opreq1 (0) eqeq1d A (if (e. A (CC)) A (0)) (0) eqeq1 bibi12d 0cn A elimel sq00 dedth)) thm (sqne0t () () (-> (e. A (CC)) (<-> (=/= (opr A (^) (2)) (0)) (=/= A (0)))) (A sq0t eqneqd)) thm (resqclt () () (-> (e. A (RR)) (e. (opr A (^) (2)) (RR))) (2nn0 A (2) reexpclt mpan2)) thm (resqcl () ((resqcl.1 (e. A (RR)))) (e. (opr A (^) (2)) (RR)) (resqcl.1 A resqclt ax-mp)) thm (lt2sq () ((resqcl.1 (e. A (RR))) (lt2sq.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (opr A (^) (2)) (<) (opr B (^) (2))))) (resqcl.1 lt2sq.2 lt2msq resqcl.1 recn sqval lt2sq.2 recn sqval (<) breq12i syl6bbr)) thm (le2sq () ((resqcl.1 (e. A (RR))) (lt2sq.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (opr A (^) (2)) (<_) (opr B (^) (2))))) (resqcl.1 lt2sq.2 le2msq resqcl.1 recn sqval lt2sq.2 recn sqval (<_) breq12i syl6bbr)) thm (sq11 () ((resqcl.1 (e. A (RR))) (lt2sq.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (opr A (^) (2)) (opr B (^) (2))) (= A B))) (resqcl.1 lt2sq.2 msq11 resqcl.1 recn sqval lt2sq.2 recn sqval eqeq12i syl5bb)) thm (sqgt0 () ((resqcl.1 (e. A (RR)))) (-> (=/= A (0)) (br (0) (<) (opr A (^) (2)))) (resqcl.1 msqgt0 A (0) df-ne resqcl.1 recn sqval (0) (<) breq2i 3imtr4)) thm (sqge0 () ((resqcl.1 (e. A (RR)))) (br (0) (<_) (opr A (^) (2))) (resqcl.1 msqge0 resqcl.1 recn sqval breqtrr)) thm (sq11t () () (-> (/\ (/\ (e. A (RR)) (br (0) (<_) A)) (/\ (e. B (RR)) (br (0) (<_) B))) (<-> (= (opr A (^) (2)) (opr B (^) (2))) (= A B))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 (br (0) (<_) B) anbi1d A (if (e. A (RR)) A (0)) (^) (2) opreq1 (opr B (^) (2)) eqeq1d A (if (e. A (RR)) A (0)) B eqeq1 bibi12d imbi12d B (if (e. B (RR)) B (0)) (0) (<_) breq2 (br (0) (<_) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (^) (2) opreq1 (opr (if (e. A (RR)) A (0)) (^) (2)) eqeq2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) eqeq2 bibi12d imbi12d 0re A elimel 0re B elimel sq11 dedth2h imp an4s)) thm (lt2sqt () () (-> (/\ (/\ (e. A (RR)) (br (0) (<_) A)) (/\ (e. B (RR)) (br (0) (<_) B))) (<-> (br A (<) B) (br (opr A (^) (2)) (<) (opr B (^) (2))))) (A B lt2msqt A sqvalt B sqvalt (<) breqan12d A recnt B recnt syl2an (br (0) (<_) A) (br (0) (<_) B) ad2ant2r bitr4d)) thm (lt2sqtOLD () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (opr A (^) (2)) (<) (opr B (^) (2)))))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 (br (0) (<_) B) anbi1d A (if (e. A (RR)) A (0)) (<) B breq1 A (if (e. A (RR)) A (0)) (^) (2) opreq1 (<) (opr B (^) (2)) breq1d bibi12d imbi12d B (if (e. B (RR)) B (0)) (0) (<_) breq2 (br (0) (<_) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<) breq2 B (if (e. B (RR)) B (0)) (^) (2) opreq1 (opr (if (e. A (RR)) A (0)) (^) (2)) (<) breq2d bibi12d imbi12d 0re A elimel 0re B elimel lt2sq dedth2h)) thm (le2sqt () () (-> (/\ (/\ (e. A (RR)) (br (0) (<_) A)) (/\ (e. B (RR)) (br (0) (<_) B))) (<-> (br A (<_) B) (br (opr A (^) (2)) (<_) (opr B (^) (2))))) (A B le2msqt A sqvalt B sqvalt (<_) breqan12d A recnt B recnt syl2an (br (0) (<_) A) (br (0) (<_) B) ad2ant2r bitr4d)) thm (sqge0t () () (-> (e. A (RR)) (br (0) (<_) (opr A (^) (2)))) (A (if (e. A (RR)) A (0)) (^) (2) opreq1 (0) (<_) breq2d 0re A elimel sqge0 dedth)) thm (sumsqne0 () ((sumsqne0.1 (e. A (RR))) (sumsqne0.2 (e. B (RR)))) (<-> (\/ (=/= A (0)) (=/= B (0))) (=/= (opr (opr A (^) (2)) (+) (opr B (^) (2))) (0))) (sumsqne0.1 sqgt0 sumsqne0.2 sqge0 sumsqne0.1 resqcl sumsqne0.2 resqcl (opr A (^) (2)) (opr B (^) (2)) addge01t mp2an mpbi 0re sumsqne0.1 resqcl sumsqne0.1 resqcl sumsqne0.2 resqcl readdcl ltletr mpan2 sumsqne0.1 resqcl sumsqne0.2 resqcl readdcl gt0ne0 3syl sumsqne0.2 sqgt0 sumsqne0.1 sqge0 sumsqne0.2 resqcl sumsqne0.1 resqcl (opr B (^) (2)) (opr A (^) (2)) addge02t mp2an mpbi 0re sumsqne0.2 resqcl sumsqne0.1 resqcl sumsqne0.2 resqcl readdcl ltletr mpan2 sumsqne0.1 resqcl sumsqne0.2 resqcl readdcl gt0ne0 3syl jaoi A (0) (^) (2) opreq1 B (0) (^) (2) opreq1 (+) opreqan12d 2nn (2) 0expt ax-mp 2nn (2) 0expt ax-mp (+) opreq12i 0cn addid1 eqtr syl6eq con3i (opr (opr A (^) (2)) (+) (opr B (^) (2))) (0) df-ne A (0) df-ne B (0) df-ne orbi12i (= A (0)) (= B (0)) ianor bitr4 3imtr4 impbi)) thm (sq0 () () (= (opr (0) (^) (2)) (0)) ((0) eqid 0cn sq00 mpbir)) thm (sq1 () () (= (opr (1) (^) (2)) (1)) (1cn sqval 1cn mulid1 eqtr)) thm (sq2 () () (= (opr (2) (^) (2)) (4)) (2cn sqval 2t2e4 eqtr)) thm (sq3 () () (= (opr (3) (^) (2)) (9)) (3re recn sqval 3t3e9 eqtr)) thm (cu2 () () (= (opr (2) (^) (3)) (8)) (df-3 (2) (^) opreq2i 2cn 2nn0 (2) (2) expp1t mp2an sq2 (x.) (2) opreq1i 4t2e8 eqtr 3eqtr)) thm (binom2 () ((binom2.1 (e. A (CC))) (binom2.2 (e. B (CC)))) (= (opr (opr A (+) B) (^) (2)) (opr (opr (opr A (^) (2)) (+) (opr (2) (x.) (opr A (x.) B))) (+) (opr B (^) (2)))) (binom2.1 binom2.2 addcl binom2.1 binom2.2 adddi binom2.1 binom2.2 binom2.1 adddir binom2.2 binom2.1 mulcom (opr A (x.) A) (+) opreq2i eqtr binom2.1 binom2.2 binom2.2 adddir (+) opreq12i binom2.1 binom2.1 mulcl binom2.1 binom2.2 mulcl addcl binom2.1 binom2.2 mulcl binom2.2 binom2.2 mulcl addass binom2.1 binom2.1 mulcl binom2.1 binom2.2 mulcl binom2.1 binom2.2 mulcl addass (+) (opr B (x.) B) opreq1i eqtr3 3eqtr binom2.1 binom2.2 addcl sqval binom2.1 sqval binom2.1 binom2.2 mulcl 2times (+) opreq12i binom2.2 sqval (+) opreq12i 3eqtr4)) thm (binom2a () ((binom2.1 (e. A (CC))) (binom2.2 (e. B (CC)))) (= (opr (opr A (+) B) (x.) (opr A (-) B)) (opr (opr A (^) (2)) (-) (opr B (^) (2)))) (binom2.1 binom2.2 binom2.1 binom2.2 subcl adddir binom2.1 binom2.1 binom2.2 subdi binom2.1 sqval (-) (opr A (x.) B) opreq1i eqtr4 binom2.2 binom2.1 binom2.2 subdi binom2.1 binom2.2 mulcom binom2.2 sqval (-) opreq12i eqtr4 (+) opreq12i binom2.1 sqcl binom2.1 binom2.2 mulcl subcl binom2.1 binom2.2 mulcl binom2.2 sqcl addsubass binom2.1 sqcl binom2.1 binom2.2 mulcl subcl binom2.1 binom2.2 mulcl addcom binom2.1 binom2.2 mulcl binom2.1 sqcl pncan3 eqtr (-) (opr B (^) (2)) opreq1i eqtr3 3eqtr)) thm (sqeqor () ((binom2.1 (e. A (CC))) (binom2.2 (e. B (CC)))) (<-> (= (opr A (^) (2)) (opr B (^) (2))) (\/ (= A B) (= A (-u B)))) (binom2.1 binom2.2 binom2a (0) eqeq1i binom2.1 binom2.2 addcl binom2.1 binom2.2 subcl mul0or binom2.1 sqcl binom2.2 sqcl subeq0 3bitr3r (= (opr A (+) B) (0)) (= (opr A (-) B) (0)) orcom binom2.1 binom2.2 subeq0 binom2.1 binom2.2 subneg (0) eqeq1i binom2.1 binom2.2 negcl subeq0 bitr3 orbi12i 3bitr)) thm (sq01t () () (-> (e. A (CC)) (<-> (= (opr A (^) (2)) A) (\/ (= A (0)) (= A (1))))) (A sqvalt A ax1id eqcomd eqeq12d (=/= A (0)) adantr 1cn A A (1) mulcant mp3anl3 anabsan bitrd biimpd ex com23 imp A (0) df-ne syl5ibr orrd ex sq0 A (0) (^) (2) opreq1 (= A (0)) id eqeq12d mpbiri sq1 A (1) (^) (2) opreq1 (= A (1)) id eqeq12d mpbiri jaoi (e. A (CC)) a1i impbid)) thm (bernneqOLD ((x y) (A x) (A y) (B x) (B y)) () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (-u (1)) (<_) A)) (br (opr (1) (+) (opr A (x.) B)) (<_) (opr (opr (1) (+) A) (^) B))) ((cv x) (1) A (x.) opreq2 (1) (+) opreq2d (cv x) (1) (opr (1) (+) A) (^) opreq2 (<_) breq12d (/\ (e. A (RR)) (br (-u (1)) (<_) A)) imbi2d (cv x) (cv y) A (x.) opreq2 (1) (+) opreq2d (cv x) (cv y) (opr (1) (+) A) (^) opreq2 (<_) breq12d (/\ (e. A (RR)) (br (-u (1)) (<_) A)) imbi2d (cv x) (opr (cv y) (+) (1)) A (x.) opreq2 (1) (+) opreq2d (cv x) (opr (cv y) (+) (1)) (opr (1) (+) A) (^) opreq2 (<_) breq12d (/\ (e. A (RR)) (br (-u (1)) (<_) A)) imbi2d (cv x) B A (x.) opreq2 (1) (+) opreq2d (cv x) B (opr (1) (+) A) (^) opreq2 (<_) breq12d (/\ (e. A (RR)) (br (-u (1)) (<_) A)) imbi2d ax1re (1) A axaddrcl mpan (opr (1) (+) A) leidt syl A recnt A ax1id syl (1) (+) opreq2d A recnt 1cn (1) A axaddcl mpan (opr (1) (+) A) exp1t 3syl 3brtr4d (br (-u (1)) (<_) A) adantr (opr (1) (+) (opr A (x.) (cv y))) A axaddrcl A (cv y) axmulrcl (cv y) nnret sylan2 ax1re (1) (opr A (x.) (cv y)) axaddrcl mpan syl (e. A (RR)) (e. (cv y) (NN)) pm3.26 sylanc (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv y))) (<_) (opr (opr (1) (+) A) (^) (cv y)))) adantr (opr (1) (+) (opr A (x.) (cv y))) (opr (1) (+) A) axmulrcl A (cv y) axmulrcl (cv y) nnret sylan2 ax1re (1) (opr A (x.) (cv y)) axaddrcl mpan syl ax1re (1) A axaddrcl mpan (e. (cv y) (NN)) adantr sylanc (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv y))) (<_) (opr (opr (1) (+) A) (^) (cv y)))) adantr (opr (opr (1) (+) A) (^) (cv y)) (opr (1) (+) A) axmulrcl (opr (1) (+) A) (cv y) reexpclt ax1re (1) A axaddrcl mpan (cv y) nnnn0t syl2an ax1re (1) A axaddrcl mpan (e. (cv y) (NN)) adantr sylanc (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv y))) (<_) (opr (opr (1) (+) A) (^) (cv y)))) adantr (opr A (x.) A) (cv y) mulge0t A A axmulrcl anidms (cv y) nnret anim12i A sqge0t A recnt A sqvalt syl breqtrd 0re (0) (cv y) ltlet mpan (cv y) nnret (cv y) nngt0t sylc anim12i sylanc A A (cv y) mul23t A recnt (e. (cv y) (NN)) adantr A recnt (e. (cv y) (NN)) adantr (cv y) nncnt (e. A (RR)) adantl syl3anc breqtrd (opr (opr (1) (+) (opr A (x.) (cv y))) (+) A) (opr (opr A (x.) (cv y)) (x.) A) addge01t (opr (1) (+) (opr A (x.) (cv y))) A axaddrcl A (cv y) axmulrcl (cv y) nnret sylan2 ax1re (1) (opr A (x.) (cv y)) axaddrcl mpan syl (e. A (RR)) (e. (cv y) (NN)) pm3.26 sylanc (opr A (x.) (cv y)) A axmulrcl A (cv y) axmulrcl (e. A (RR)) (e. (cv y) (RR)) pm3.26 sylanc (cv y) nnret sylan2 sylanc mpbid (opr (1) (+) (opr A (x.) (cv y))) A (opr (opr A (x.) (cv y)) (x.) A) axaddass A (cv y) axmulcl 1cn (1) (opr A (x.) (cv y)) axaddcl mpan syl (e. A (CC)) (e. (cv y) (CC)) pm3.26 (opr A (x.) (cv y)) A axmulcl A (cv y) axmulcl (e. A (CC)) (e. (cv y) (CC)) pm3.26 sylanc syl3anc (opr A (x.) (cv y)) A muladd11t A (cv y) axmulcl (e. A (CC)) (e. (cv y) (CC)) pm3.26 sylanc eqtr4d A recnt (cv y) nncnt syl2an breqtrd (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv y))) (<_) (opr (opr (1) (+) A) (^) (cv y)))) adantr (opr (1) (+) (opr A (x.) (cv y))) (opr (opr (1) (+) A) (^) (cv y)) (opr (1) (+) A) lemul1it A (cv y) axmulrcl (cv y) nnret sylan2 ax1re (1) (opr A (x.) (cv y)) axaddrcl mpan syl (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv y))) (<_) (opr (opr (1) (+) A) (^) (cv y)))) adantr (opr (1) (+) A) (cv y) reexpclt ax1re (1) A axaddrcl mpan (cv y) nnnn0t syl2an (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv y))) (<_) (opr (opr (1) (+) A) (^) (cv y)))) adantr ax1re (1) A axaddrcl mpan (e. (cv y) (NN)) (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv y))) (<_) (opr (opr (1) (+) A) (^) (cv y)))) ad2antrr 3jca ax1re renegcl ax1re (-u (1)) A (1) leadd2t mp3an13 1cn negid (<_) (opr (1) (+) A) breq1i syl6bb biimpa (e. (cv y) (NN)) (br (opr (1) (+) (opr A (x.) (cv y))) (<_) (opr (opr (1) (+) A) (^) (cv y))) ad2ant2r (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv y))) (<_) (opr (opr (1) (+) A) (^) (cv y))) pm3.27 (/\ (e. A (RR)) (e. (cv y) (NN))) adantl jca sylanc letrd 1cn A (cv y) (1) axdistr mp3an3 A ax1id (e. (cv y) (CC)) adantr (opr A (x.) (cv y)) (+) opreq2d eqtrd (1) (+) opreq2d 1cn (1) (opr A (x.) (cv y)) A axaddass mp3an1 A (cv y) axmulcl (e. A (CC)) (e. (cv y) (CC)) pm3.26 sylanc eqtr4d A recnt (cv y) nncnt syl2an (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv y))) (<_) (opr (opr (1) (+) A) (^) (cv y)))) adantr (opr (1) (+) A) (cv y) expp1t A recnt 1cn (1) A axaddcl mpan syl (cv y) nnnn0t syl2an (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv y))) (<_) (opr (opr (1) (+) A) (^) (cv y)))) adantr 3brtr4d exp43 com12 imp3a a2d nnind exp3a com12 3imp)) thm (bernneq ((j k) (A j) (A k) (N j) (N k)) () (-> (/\/\ (e. A (RR)) (e. N (NN0)) (br (-u (1)) (<_) A)) (br (opr (1) (+) (opr A (x.) N)) (<_) (opr (opr (1) (+) A) (^) N))) ((cv j) (0) A (x.) opreq2 (1) (+) opreq2d (cv j) (0) (opr (1) (+) A) (^) opreq2 (<_) breq12d (/\ (e. A (RR)) (br (-u (1)) (<_) A)) imbi2d (cv j) (cv k) A (x.) opreq2 (1) (+) opreq2d (cv j) (cv k) (opr (1) (+) A) (^) opreq2 (<_) breq12d (/\ (e. A (RR)) (br (-u (1)) (<_) A)) imbi2d (cv j) (opr (cv k) (+) (1)) A (x.) opreq2 (1) (+) opreq2d (cv j) (opr (cv k) (+) (1)) (opr (1) (+) A) (^) opreq2 (<_) breq12d (/\ (e. A (RR)) (br (-u (1)) (<_) A)) imbi2d (cv j) N A (x.) opreq2 (1) (+) opreq2d (cv j) N (opr (1) (+) A) (^) opreq2 (<_) breq12d (/\ (e. A (RR)) (br (-u (1)) (<_) A)) imbi2d A recnt A mul01t (1) (+) opreq2d 1cn addid1 syl6eq 1cn (1) A axaddcl mpan (opr (1) (+) A) exp0t syl ax1re leid syl5breqr eqbrtrd syl (br (-u (1)) (<_) A) adantr (opr (1) (+) (opr A (x.) (cv k))) A axaddrcl A (cv k) axmulrcl (cv k) nn0ret sylan2 ax1re (1) (opr A (x.) (cv k)) axaddrcl mpan syl (e. A (RR)) (e. (cv k) (NN0)) pm3.26 sylanc (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv k))) (<_) (opr (opr (1) (+) A) (^) (cv k)))) adantr (opr (1) (+) (opr A (x.) (cv k))) (opr (1) (+) A) axmulrcl A (cv k) axmulrcl (cv k) nn0ret sylan2 ax1re (1) (opr A (x.) (cv k)) axaddrcl mpan syl ax1re (1) A axaddrcl mpan (e. (cv k) (NN0)) adantr sylanc (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv k))) (<_) (opr (opr (1) (+) A) (^) (cv k)))) adantr (opr (opr (1) (+) A) (^) (cv k)) (opr (1) (+) A) axmulrcl (opr (1) (+) A) (cv k) reexpclt ax1re (1) A axaddrcl mpan sylan ax1re (1) A axaddrcl mpan (e. (cv k) (NN0)) adantr sylanc (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv k))) (<_) (opr (opr (1) (+) A) (^) (cv k)))) adantr (opr A (x.) A) (cv k) mulge0t A A axmulrcl anidms (cv k) nn0ret anim12i A sqge0t A recnt A sqvalt syl breqtrd (cv k) nn0ge0t anim12i sylanc A A (cv k) mul23t A recnt (e. (cv k) (NN0)) adantr A recnt (e. (cv k) (NN0)) adantr (cv k) nn0cnt (e. A (RR)) adantl syl3anc breqtrd (opr (opr (1) (+) (opr A (x.) (cv k))) (+) A) (opr (opr A (x.) (cv k)) (x.) A) addge01t (opr (1) (+) (opr A (x.) (cv k))) A axaddrcl A (cv k) axmulrcl (cv k) nn0ret sylan2 ax1re (1) (opr A (x.) (cv k)) axaddrcl mpan syl (e. A (RR)) (e. (cv k) (NN0)) pm3.26 sylanc (opr A (x.) (cv k)) A axmulrcl A (cv k) axmulrcl (e. A (RR)) (e. (cv k) (RR)) pm3.26 sylanc (cv k) nn0ret sylan2 sylanc mpbid (opr (1) (+) (opr A (x.) (cv k))) A (opr (opr A (x.) (cv k)) (x.) A) axaddass A (cv k) axmulcl 1cn (1) (opr A (x.) (cv k)) axaddcl mpan syl (e. A (CC)) (e. (cv k) (CC)) pm3.26 (opr A (x.) (cv k)) A axmulcl A (cv k) axmulcl (e. A (CC)) (e. (cv k) (CC)) pm3.26 sylanc syl3anc (opr A (x.) (cv k)) A muladd11t A (cv k) axmulcl (e. A (CC)) (e. (cv k) (CC)) pm3.26 sylanc eqtr4d A recnt (cv k) nn0cnt syl2an breqtrd (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv k))) (<_) (opr (opr (1) (+) A) (^) (cv k)))) adantr (opr (1) (+) (opr A (x.) (cv k))) (opr (opr (1) (+) A) (^) (cv k)) (opr (1) (+) A) lemul1it A (cv k) axmulrcl (cv k) nn0ret sylan2 ax1re (1) (opr A (x.) (cv k)) axaddrcl mpan syl (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv k))) (<_) (opr (opr (1) (+) A) (^) (cv k)))) adantr (opr (1) (+) A) (cv k) reexpclt ax1re (1) A axaddrcl mpan sylan (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv k))) (<_) (opr (opr (1) (+) A) (^) (cv k)))) adantr ax1re (1) A axaddrcl mpan (e. (cv k) (NN0)) (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv k))) (<_) (opr (opr (1) (+) A) (^) (cv k)))) ad2antrr 3jca ax1re renegcl ax1re (-u (1)) A (1) leadd2t mp3an13 1cn negid (<_) (opr (1) (+) A) breq1i syl6bb biimpa (e. (cv k) (NN0)) (br (opr (1) (+) (opr A (x.) (cv k))) (<_) (opr (opr (1) (+) A) (^) (cv k))) ad2ant2r (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv k))) (<_) (opr (opr (1) (+) A) (^) (cv k))) pm3.27 (/\ (e. A (RR)) (e. (cv k) (NN0))) adantl jca sylanc letrd 1cn A (cv k) (1) axdistr mp3an3 A ax1id (e. (cv k) (CC)) adantr (opr A (x.) (cv k)) (+) opreq2d eqtrd (1) (+) opreq2d 1cn (1) (opr A (x.) (cv k)) A axaddass mp3an1 A (cv k) axmulcl (e. A (CC)) (e. (cv k) (CC)) pm3.26 sylanc eqtr4d A recnt (cv k) nn0cnt syl2an (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv k))) (<_) (opr (opr (1) (+) A) (^) (cv k)))) adantr (opr (1) (+) A) (cv k) expp1t A recnt 1cn (1) A axaddcl mpan syl sylan (/\ (br (-u (1)) (<_) A) (br (opr (1) (+) (opr A (x.) (cv k))) (<_) (opr (opr (1) (+) A) (^) (cv k)))) adantr 3brtr4d exp43 com12 imp3a a2d nn0ind exp3a com12 3imp)) thm (discrlem1 () ((discrlem.1 (e. A (RR))) (discrlem.2 (e. B (RR))) (discrlem.3 (e. C (RR))) (discrlem1.4 (= D (-u (opr B (/) (opr (2) (x.) A))))) (discrlem1.5 (br (0) (<) A))) (<-> (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C)) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))) (4re recn discrlem.1 recn mulcl mul01 4re recn discrlem.1 recn mulcl discrlem.1 recn discrlem1.4 discrlem.2 2re discrlem.1 remulcl 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 redivcl renegcl eqeltr recn sqcl mulcl discrlem.2 recn discrlem1.4 discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl negcl eqeltr mulcl addcl discrlem.3 recn adddi (<_) breq12i 4re discrlem.1 4pos discrlem1.5 mulgt0i 0re discrlem.1 discrlem1.4 discrlem.2 2re discrlem.1 remulcl 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 redivcl renegcl eqeltr resqcl remulcl discrlem.2 discrlem1.4 discrlem.2 2re discrlem.1 remulcl 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 redivcl renegcl eqeltr remulcl readdcl discrlem.3 readdcl 4re discrlem.1 remulcl lemul2 ax-mp neg0 discrlem.2 resqcl recn 4re recn discrlem.1 recn discrlem.3 recn mulcl mulcl negsubdi 2re discrlem.1 remulcl recn 2cn 2re 2pos gt0ne0i divcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl mulass 2re discrlem.1 remulcl recn 2cn discrlem.2 recn 2re discrlem.1 remulcl recn 2re 2pos gt0ne0i 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divmul13 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divid (opr B (/) (2)) (x.) opreq2i discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl 1cn mulcom 3eqtr (x.) (opr B (/) (opr (2) (x.) A)) opreq1i 2cn discrlem.1 recn 2re 2pos gt0ne0i divcan3 discrlem1.4 (^) (2) opreq1i discrlem.2 2re discrlem.1 remulcl 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 redivcl renegcl recn sqval discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl mul2neg 3eqtrr (x.) opreq12i 3eqtr3r 2cn negneg (x.) (opr B (/) (2)) opreq1i 2cn negcl discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl mulneg1 2cn discrlem.2 recn 2re 2pos gt0ne0i divcan2 3eqtr3r discrlem1.4 (x.) opreq12i 2cn negcl discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl mulcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl mul2neg eqtr (+) opreq12i df-2 negeqi 1cn 1cn negdi eqtr (1) (+) opreq2i 1cn negid (+) (-u (1)) opreq1i 1cn 1cn negcl 1cn negcl addass 1cn negcl addid2 3eqtr3 eqtr (x.) (opr B (/) (2)) opreq1i 1cn 2cn negcl discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl adddir discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl mulm1 3eqtr3 (x.) (opr B (/) (opr (2) (x.) A)) opreq1i 1cn discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl mulcl 2cn negcl discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl mulcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl adddir discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl mulneg1 discrlem.2 recn sqval 2t2e4 (x.) A opreq1i 2cn 2cn discrlem.1 recn mulass eqtr3 (/) opreq12i discrlem.2 recn 2cn discrlem.2 recn 2re discrlem.1 remulcl recn 2re 2pos gt0ne0i 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divmuldiv eqtr4 negeqi eqtr4 3eqtr3 eqtr (opr (4) (x.) A) (x.) opreq2i 4re discrlem.1 remulcl recn discrlem.2 resqcl 4re discrlem.1 remulcl 4re recn discrlem.1 recn 4re 4pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 redivcl recn mulneg2 4re discrlem.1 remulcl recn discrlem.2 resqcl recn 4re recn discrlem.1 recn 4re 4pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcan2 negeqi 3eqtr 4re recn discrlem.1 recn discrlem.3 recn mulass (+) opreq12i eqtr4 (<_) breq12i 3bitr4 discrlem.2 resqcl 4re discrlem.1 discrlem.3 remulcl remulcl resubcl 0re leneg bitr4)) thm (discrlem2 () ((discrlem.1 (e. A (RR))) (discrlem.2 (e. B (RR))) (discrlem.3 (e. C (RR))) (discrlem1.4 (= D (-u (opr B (/) (opr (2) (x.) A))))) (discrlem2.5 (-> (e. D (RR)) (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C))))) (-> (br (0) (<) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))) (2pos 2re discrlem.1 mulgt0 mpan 2re discrlem.1 remulcl gt0ne0 syl discrlem.2 2re discrlem.1 remulcl redivclz (opr B (/) (opr (2) (x.) A)) renegclt 3syl discrlem1.4 syl5eqel discrlem2.5 syl (= A (if (br (0) (<) A) A (1))) id A (if (br (0) (<) A) A (1)) (2) (x.) opreq2 B (/) opreq2d negeqd discrlem1.4 syl5eq (^) (2) opreq1d (x.) opreq12d A (if (br (0) (<) A) A (1)) (2) (x.) opreq2 B (/) opreq2d negeqd discrlem1.4 syl5eq B (x.) opreq2d (+) opreq12d (+) C opreq1d (0) (<_) breq2d A (if (br (0) (<) A) A (1)) (x.) C opreq1 (4) (x.) opreq2d (opr B (^) (2)) (-) opreq2d (<_) (0) breq1d bibi12d discrlem.1 ax1re (br (0) (<) A) keepel discrlem.2 discrlem.3 (-u (opr B (/) (opr (2) (x.) (if (br (0) (<) A) A (1))))) eqid A elimgt0 discrlem1 dedth mpbid)) thm (discrlem3 () ((discrlem.1 (e. A (RR))) (discrlem.2 (e. B (RR))) (discrlem.3 (e. C (RR))) (discrlem3.4 (= D (opr (opr C (+) (1)) (/) (-u B)))) (discrlem3.5 (-> (e. D (RR)) (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C))))) (-> (= (0) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))) (discrlem.3 ltp1 B (0) df-ne discrlem.2 recn negne0 bitr3 discrlem.3 ax1re readdcl discrlem.2 renegcl redivclz discrlem3.4 syl5eqel discrlem3.5 syl (= (0) A) adantr (0) A (x.) (opr D (^) (2)) opreq1 eqcomd discrlem.3 ax1re readdcl discrlem.2 renegcl redivclz discrlem3.4 syl5eqel recnd D sqclt (opr D (^) (2)) mul02t 3syl sylan9eqr (+) (opr B (x.) D) opreq1d discrlem.3 ax1re readdcl discrlem.2 renegcl redivclz discrlem3.4 syl5eqel discrlem.2 jctil B D axmulrcl syl recnd (opr B (x.) D) addid2t syl (= (0) A) adantr eqtrd (+) C opreq1d breqtrd discrlem.3 ax1re readdcl discrlem.2 renegcl redivclz discrlem3.4 syl5eqel discrlem.2 jctil B D axmulrcl 0re discrlem.3 (0) (opr B (x.) D) C lesubadd2t mp3an13 3syl (= (0) A) adantr mpbird discrlem.3 ax1re readdcl discrlem.2 renegcl redivclz discrlem3.4 syl5eqel discrlem.2 jctil B recnt D recnt anim12i B D mulneg1t 3syl eqcomd (opr B (x.) D) df-neg discrlem3.4 (-u B) (x.) opreq2i 3eqtr3g discrlem.2 recn negcl discrlem.3 ax1re readdcl recn divcan2z eqtrd (<_) C breq1d (= (0) A) adantr mpbid discrlem.3 ax1re readdcl discrlem.3 lenlt sylib ex sylbi mt2i a3i (x.) B opreq1d discrlem.2 recn mul02 syl6eq discrlem.2 recn sqval syl5eq (0) A (x.) C opreq1 discrlem.3 recn mul02 syl5reqr (4) (x.) opreq2d 4re recn mul01 syl6eq (-) opreq12d 0cn subid syl6eq discrlem.2 resqcl 4re discrlem.1 discrlem.3 remulcl remulcl resubcl 0re eqle syl)) thm (discrlem ((A x) (B x) (C x)) ((discrlem.1 (e. A (RR))) (discrlem.2 (e. B (RR))) (discrlem.3 (e. C (RR))) (discrlem.4 (A.e. x (RR) (br (0) (<_) (opr (opr (opr A (x.) (opr (cv x) (^) (2))) (+) (opr B (x.) (cv x))) (+) C))))) (-> (br (0) (<_) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))) (0re discrlem.1 leloe discrlem.1 discrlem.2 discrlem.3 (-u (opr B (/) (opr (2) (x.) A))) eqid (cv x) (-u (opr B (/) (opr (2) (x.) A))) (^) (2) opreq1 A (x.) opreq2d (cv x) (-u (opr B (/) (opr (2) (x.) A))) B (x.) opreq2 (+) opreq12d (+) C opreq1d (0) (<_) breq2d discrlem.4 vtoclri discrlem2 discrlem.1 discrlem.2 discrlem.3 (opr (opr C (+) (1)) (/) (-u B)) eqid (cv x) (opr (opr C (+) (1)) (/) (-u B)) (^) (2) opreq1 A (x.) opreq2d (cv x) (opr (opr C (+) (1)) (/) (-u B)) B (x.) opreq2 (+) opreq12d (+) C opreq1d (0) (<_) breq2d discrlem.4 vtoclri discrlem3 jaoi sylbi)) thm (nnsqcl () ((nnsqcl.1 (e. N (NN)))) (e. (opr N (^) (2)) (NN)) (nnsqcl.1 2nn0 N (2) nnexpclt mp2an)) thm (nnlesq () ((nnsqcl.1 (e. N (NN)))) (br N (<_) (opr N (^) (2))) (nnsqcl.1 nncn mulid1 nnsqcl.1 N nnge1t ax-mp nnsqcl.1 nngt0 ax1re nnsqcl.1 nnre nnsqcl.1 nnre lemul2 ax-mp mpbi nnsqcl.1 nncn sqval breqtrr eqbrtrr)) thm (nnesq () ((nnsqcl.1 (e. N (NN)))) (<-> (e. (opr N (/) (2)) (NN)) (e. (opr (opr N (^) (2)) (/) (2)) (NN))) ((opr N (/) (2)) (opr N (/) (2)) nnmulclt anidms 2nn (2) (opr (opr N (/) (2)) (x.) (opr N (/) (2))) nnmulclt mpan 2cn nnsqcl.1 nncn 2cn 2re 2pos gt0ne0i divcl nnsqcl.1 nncn 2cn 2re 2pos gt0ne0i divcl mulass nnsqcl.1 nncn nnsqcl.1 nncn 2cn 2re 2pos gt0ne0i divass nnsqcl.1 nncn sqval (/) (2) opreq1i 2cn nnsqcl.1 nncn 2re 2pos gt0ne0i divcan2 (x.) (opr N (/) (2)) opreq1i 3eqtr4r eqtr3 syl5eqelr syl (opr (opr N (+) (1)) (/) (2)) (opr (opr N (+) (1)) (/) (2)) nnmulclt anidms 2nn (2) (opr (opr (opr N (+) (1)) (/) (2)) (x.) (opr (opr N (+) (1)) (/) (2))) nnmulclt mpan 2cn nnsqcl.1 nncn 1cn addcl 2re 2pos gt0ne0i divcan2 (x.) (opr (opr N (+) (1)) (/) (2)) opreq1i 2cn nnsqcl.1 nncn 1cn addcl 2cn 2re 2pos gt0ne0i divcl nnsqcl.1 nncn 1cn addcl 2cn 2re 2pos gt0ne0i divcl mulass nnsqcl.1 nncn 1cn addcl nnsqcl.1 nncn 1cn addcl 2cn 2re 2pos gt0ne0i divass nnsqcl.1 nncn 1cn addcl sqval nnsqcl.1 nncn 1cn binom2 nnsqcl.1 nncn mulid1 (2) (x.) opreq2i (opr N (^) (2)) (+) opreq2i sq1 (+) opreq12i nnsqcl.1 nnsqcl nncn 2cn nnsqcl.1 nncn mulcl 1cn add23 3eqtr eqtr3 (/) (2) opreq1i nnsqcl.1 nnsqcl nncn 1cn addcl 2cn nnsqcl.1 nncn mulcl 2cn 2re 2pos gt0ne0i divdir 2cn nnsqcl.1 nncn 2re 2pos gt0ne0i divcan3 (opr (opr (opr N (^) (2)) (+) (1)) (/) (2)) (+) opreq2i 3eqtr eqtr3 3eqtr3 (NN) eleq1i nnsqcl.1 nncn addid2 nnsqcl.1 nnsqcl nnre ax1re readdcl 2re nnsqcl.1 nnsqcl nnre ax1re nnsqcl.1 nnsqcl nngt0 lt01 addgt0i 2pos divgt0i 0re nnsqcl.1 nnsqcl nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl nnsqcl.1 nnre ltadd1 mpbi eqbrtrr nnsqcl.1 N (opr (opr (opr (opr N (^) (2)) (+) (1)) (/) (2)) (+) N) nnsubt mpan mpbii nnsqcl.1 nnsqcl nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl recn nnsqcl.1 nncn nnsqcl.1 nncn addsubass nnsqcl.1 nncn subid (opr (opr (opr N (^) (2)) (+) (1)) (/) (2)) (+) opreq2i nnsqcl.1 nnsqcl nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl recn addid1 3eqtr syl5eqelr sylbi 3syl con3i nnsqcl.1 nnsqcl nneo nnsqcl.1 nneo 3imtr4 impbi)) thm (nn0le2msqt () ((nn0le2msqt.1 (e. A (NN0))) (nn0le2msqt.2 (e. B (NN0)))) (<-> (br A (<_) B) (br (opr A (x.) A) (<_) (opr B (x.) B))) (nn0le2msqt.1 nn0ge0 nn0le2msqt.2 nn0ge0 nn0le2msqt.1 nn0re nn0le2msqt.2 nn0re le2sq mp2an nn0le2msqt.1 nn0cn sqval nn0le2msqt.2 nn0cn sqval (<_) breq12i bitr)) thm (nn0opthlem1 () ((nn0opthlem1.1 (e. A (NN0))) (nn0opthlem1.2 (e. C (NN0)))) (<-> (br A (<) C) (br (opr (opr A (x.) A) (+) (opr (2) (x.) A)) (<) (opr C (x.) C))) (nn0opthlem1.1 1nn0 nn0addcl nn0opthlem1.2 nn0le2msqt nn0opthlem1.1 nn0opthlem1.2 A C nn0ltp1let mp2an nn0opthlem1.1 nn0opthlem1.1 nn0mulcl 2nn0 nn0opthlem1.1 nn0mulcl nn0addcl nn0opthlem1.2 nn0opthlem1.2 nn0mulcl (opr (opr A (x.) A) (+) (opr (2) (x.) A)) (opr C (x.) C) nn0ltp1let mp2an nn0opthlem1.1 nn0cn 1cn binom2 nn0opthlem1.1 nn0cn 1cn addcl sqval nn0opthlem1.1 nn0cn sqval (+) (opr (2) (x.) (opr A (x.) (1))) opreq1i 1cn sqval (+) opreq12i 3eqtr3 nn0opthlem1.1 nn0cn mulid1 (2) (x.) opreq2i (opr A (x.) A) (+) opreq2i 1cn mulid1 (+) opreq12i eqtr (<_) (opr C (x.) C) breq1i bitr4 3bitr4)) thm (nn0opthlem2 () ((nn0opth.1 (e. A (NN0))) (nn0opth.2 (e. B (NN0))) (nn0opth.3 (e. C (NN0))) (nn0opth.4 (e. D (NN0)))) (-> (/\ (br B (<_) A) (br D (<_) C)) (-> (br A (<) C) (-. (= (opr (opr A (x.) A) (+) B) (opr (opr C (x.) C) (+) D))))) (nn0opth.1 nn0re nn0opth.1 nn0re remulcl nn0opth.2 nn0re readdcl nn0opth.1 nn0re nn0opth.1 nn0re remulcl 2re nn0opth.1 nn0re remulcl readdcl nn0opth.3 nn0re nn0opth.3 nn0re remulcl lelttr nn0opth.1 nn0opth.2 nn0lele2x nn0opth.2 nn0re 2re nn0opth.1 nn0re remulcl nn0opth.1 nn0re nn0opth.1 nn0re remulcl leadd2 sylib nn0opth.1 nn0opth.3 nn0opthlem1 biimp syl2an nn0opth.3 nn0re nn0opth.3 nn0re remulcl nn0opth.4 nn0addge1 nn0opth.1 nn0re nn0opth.1 nn0re remulcl nn0opth.2 nn0re readdcl nn0opth.3 nn0re nn0opth.3 nn0re remulcl nn0opth.3 nn0re nn0opth.3 nn0re remulcl nn0opth.4 nn0re readdcl ltletr mpan2 syl ex (br D (<_) C) adantr nn0opth.1 nn0re nn0opth.1 nn0re remulcl nn0opth.2 nn0re readdcl nn0opth.3 nn0re nn0opth.3 nn0re remulcl nn0opth.4 nn0re readdcl ltne syl6)) thm (nn0opth () ((nn0opth.1 (e. A (NN0))) (nn0opth.2 (e. B (NN0))) (nn0opth.3 (e. C (NN0))) (nn0opth.4 (e. D (NN0)))) (<-> (= (opr (opr (opr A (+) B) (x.) (opr A (+) B)) (+) B) (opr (opr (opr C (+) D) (x.) (opr C (+) D)) (+) D)) (/\ (= A C) (= B D))) (nn0opth.2 nn0re nn0opth.1 nn0addge2 nn0opth.4 nn0re nn0opth.3 nn0addge2 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opthlem2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opthlem2 ancoms (opr (opr (opr A (+) B) (x.) (opr A (+) B)) (+) B) (opr (opr (opr C (+) D) (x.) (opr C (+) D)) (+) D) eqcom negbii syl6ibr jaod nn0opth.1 nn0re nn0opth.2 nn0re readdcl nn0opth.3 nn0opth.4 nn0addcl nn0re lttri2 syl5ib a3d mp2an (= (opr (opr (opr A (+) B) (x.) (opr A (+) B)) (+) B) (opr (opr (opr C (+) D) (x.) (opr C (+) D)) (+) D)) id nn0opth.2 nn0re nn0opth.1 nn0addge2 nn0opth.4 nn0re nn0opth.3 nn0addge2 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opthlem2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opthlem2 ancoms (opr (opr (opr A (+) B) (x.) (opr A (+) B)) (+) B) (opr (opr (opr C (+) D) (x.) (opr C (+) D)) (+) D) eqcom negbii syl6ibr jaod nn0opth.1 nn0re nn0opth.2 nn0re readdcl nn0opth.3 nn0opth.4 nn0addcl nn0re lttri2 syl5ib a3d mp2an nn0opth.2 nn0re nn0opth.1 nn0addge2 nn0opth.4 nn0re nn0opth.3 nn0addge2 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opthlem2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opthlem2 ancoms (opr (opr (opr A (+) B) (x.) (opr A (+) B)) (+) B) (opr (opr (opr C (+) D) (x.) (opr C (+) D)) (+) D) eqcom negbii syl6ibr jaod nn0opth.1 nn0re nn0opth.2 nn0re readdcl nn0opth.3 nn0opth.4 nn0addcl nn0re lttri2 syl5ib a3d mp2an (x.) opreq12d (+) D opreq1d eqtr4d nn0opth.1 nn0cn nn0opth.2 nn0cn addcl nn0opth.1 nn0cn nn0opth.2 nn0cn addcl mulcl nn0opth.2 nn0cn nn0opth.4 nn0cn addcan sylib C (+) opreq2d eqtr4d nn0opth.1 nn0cn nn0opth.3 nn0cn nn0opth.2 nn0cn addcan2 sylib (= (opr (opr (opr A (+) B) (x.) (opr A (+) B)) (+) B) (opr (opr (opr C (+) D) (x.) (opr C (+) D)) (+) D)) id nn0opth.2 nn0re nn0opth.1 nn0addge2 nn0opth.4 nn0re nn0opth.3 nn0addge2 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opthlem2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opthlem2 ancoms (opr (opr (opr A (+) B) (x.) (opr A (+) B)) (+) B) (opr (opr (opr C (+) D) (x.) (opr C (+) D)) (+) D) eqcom negbii syl6ibr jaod nn0opth.1 nn0re nn0opth.2 nn0re readdcl nn0opth.3 nn0opth.4 nn0addcl nn0re lttri2 syl5ib a3d mp2an nn0opth.2 nn0re nn0opth.1 nn0addge2 nn0opth.4 nn0re nn0opth.3 nn0addge2 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opthlem2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opthlem2 ancoms (opr (opr (opr A (+) B) (x.) (opr A (+) B)) (+) B) (opr (opr (opr C (+) D) (x.) (opr C (+) D)) (+) D) eqcom negbii syl6ibr jaod nn0opth.1 nn0re nn0opth.2 nn0re readdcl nn0opth.3 nn0opth.4 nn0addcl nn0re lttri2 syl5ib a3d mp2an (x.) opreq12d (+) D opreq1d eqtr4d nn0opth.1 nn0cn nn0opth.2 nn0cn addcl nn0opth.1 nn0cn nn0opth.2 nn0cn addcl mulcl nn0opth.2 nn0cn nn0opth.4 nn0cn addcan sylib jca A C B D (+) opreq12 A C B D (+) opreq12 (x.) opreq12d (= A C) (= B D) pm3.27 (+) opreq12d impbi)) thm (nn0opth2 () ((nn0opth.1 (e. A (NN0))) (nn0opth.2 (e. B (NN0))) (nn0opth.3 (e. C (NN0))) (nn0opth.4 (e. D (NN0)))) (<-> (= (opr (opr (opr A (+) B) (^) (2)) (+) B) (opr (opr (opr C (+) D) (^) (2)) (+) D)) (/\ (= A C) (= B D))) (nn0opth.1 nn0cn nn0opth.2 nn0cn addcl sqval (+) B opreq1i nn0opth.3 nn0cn nn0opth.4 nn0cn addcl sqval (+) D opreq1i eqeq12i nn0opth.1 nn0opth.2 nn0opth.3 nn0opth.4 nn0opth bitr)) thm (nn0opth2t () () (-> (/\ (/\ (e. A (NN0)) (e. B (NN0))) (/\ (e. C (NN0)) (e. D (NN0)))) (<-> (= (opr (opr (opr A (+) B) (^) (2)) (+) B) (opr (opr (opr C (+) D) (^) (2)) (+) D)) (/\ (= A C) (= B D)))) (A (if (e. A (NN0)) A (0)) (+) B opreq1 (^) (2) opreq1d (+) B opreq1d (opr (opr (opr C (+) D) (^) (2)) (+) D) eqeq1d A (if (e. A (NN0)) A (0)) C eqeq1 (= B D) anbi1d bibi12d B (if (e. B (NN0)) B (0)) (if (e. A (NN0)) A (0)) (+) opreq2 (^) (2) opreq1d (= B (if (e. B (NN0)) B (0))) id (+) opreq12d (opr (opr (opr C (+) D) (^) (2)) (+) D) eqeq1d B (if (e. B (NN0)) B (0)) D eqeq1 (= (if (e. A (NN0)) A (0)) C) anbi2d bibi12d C (if (e. C (NN0)) C (0)) (+) D opreq1 (^) (2) opreq1d (+) D opreq1d (opr (opr (opr (if (e. A (NN0)) A (0)) (+) (if (e. B (NN0)) B (0))) (^) (2)) (+) (if (e. B (NN0)) B (0))) eqeq2d C (if (e. C (NN0)) C (0)) (if (e. A (NN0)) A (0)) eqeq2 (= (if (e. B (NN0)) B (0)) D) anbi1d bibi12d D (if (e. D (NN0)) D (0)) (if (e. C (NN0)) C (0)) (+) opreq2 (^) (2) opreq1d (= D (if (e. D (NN0)) D (0))) id (+) opreq12d (opr (opr (opr (if (e. A (NN0)) A (0)) (+) (if (e. B (NN0)) B (0))) (^) (2)) (+) (if (e. B (NN0)) B (0))) eqeq2d D (if (e. D (NN0)) D (0)) (if (e. B (NN0)) B (0)) eqeq2 (= (if (e. A (NN0)) A (0)) (if (e. C (NN0)) C (0))) anbi2d bibi12d 0nn0 A elimel 0nn0 B elimel 0nn0 C elimel 0nn0 D elimel nn0opth2 dedth4h)) thm (sqrval ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (= (` (sqr) A) (sup ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))) (RR) (<)))) ((cv w) A (0) (<_) breq2 (RR) elrab (cv y) A (opr (cv x) (x.) (cv x)) (<_) breq2 (br (0) (<_) (cv x)) anbi2d x (RR) rabbisdv ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) (cv y)))) ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))) (RR) (<) supeq1 syl y z x df-sqr (cv w) (cv y) (0) (<_) breq2 (RR) elrab (= (cv z) (sup ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) (cv y)))) (RR) (<))) anbi1i y z opabbii eqtr4 ltso ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))) supex fvopab4 sylbir)) thm (sqr0 () () (= (` (sqr) (0)) (0)) (0re 0re leid (0) x sqrval mp2an (cv x) (cv x) axmulrcl anidms 0re jctir (opr (cv x) (x.) (cv x)) (0) lenltt syl (cv x) ltmsqt con1d sylbid (br (0) (<_) (cv x)) adantld imp 0re 0re leid 0cn mul01 0re 0re remulcl 0re eqle ax-mp pm3.2i pm3.2i (cv x) (0) (RR) eleq1 (cv x) (0) (0) (<_) breq2 (cv x) (0) (cv x) (0) (x.) opreq12 anidms (<_) (0) breq1d anbi12d anbi12d mpbiri impbi x abbii x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) (0))) df-rab (0) x df-sn 3eqtr4 ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) (0)))) ({} (0)) (RR) (<) supeq1 ax-mp 0re ltso (0) supsn ax-mp 3eqtr)) thm (sqrlem1 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (br A (<) (opr (opr (1) (+) A) (x.) (opr (1) (+) A))) (sqrlem1.1 recn addid2 lt01 0re ax1re sqrlem1.1 ltadd1 mpbi eqbrtrr ax1re sqrlem1.1 readdcl recn mulid2 1cn addid1 sqrlem1.2 0re sqrlem1.1 ax1re ltadd2 mpbi eqbrtrr ax1re ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i ltmul1i mpbi eqbrtrr sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl lttr mp2an)) thm (sqrlem2 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (br (opr (opr A (/) (opr (1) (+) A)) (x.) (opr A (/) (opr (1) (+) A))) (<) A) (sqrlem1.1 sqrlem1.2 sqrlem1 sqrlem1.2 sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl sqrlem1.1 ltmul2 ax-mp mpbi sqrlem1.1 sqrlem1.1 remulcl sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl remulcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl sqrlem1.2 sqrlem1.1 sqrlem1.2 sqrlem1 0re sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl lttr mp2an ltdiv1i mpbi sqrlem1.1 recn ax1re sqrlem1.1 readdcl recn sqrlem1.1 recn ax1re sqrlem1.1 readdcl recn ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i divmuldiv sqrlem1.1 recn ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl recn ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl recn ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl sqrlem1.2 sqrlem1.1 sqrlem1.2 sqrlem1 0re sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl lttr mp2an gt0ne0i divass ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl recn ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl sqrlem1.2 sqrlem1.1 sqrlem1.2 sqrlem1 0re sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl lttr mp2an gt0ne0i divid A (x.) opreq2i sqrlem1.1 recn mulid1 3eqtrr 3brtr4)) thm (sqrlem3 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (br (0) (<) (opr A (/) (opr (1) (+) A))) (sqrlem1.1 ax1re sqrlem1.1 readdcl sqrlem1.2 ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i divgt0i)) thm (sqrlem4 ((A x) (S x) (D x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem4.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (<-> (e. D S) (/\ (e. D (RR)) (/\ (br (0) (<_) D) (br (opr D (x.) D) (<_) A)))) (sqrlem4.3 D eleq2i (cv x) D (0) (<_) breq2 (cv x) D (cv x) D (x.) opreq12 anidms (<_) A breq1d anbi12d (RR) elrab bitr)) thm (sqrlem5 ((A x) (S x) (D x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem4.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-> (/\ (e. D (RR)) (/\ (br (0) (<) D) (br (opr D (x.) D) (<) A))) (e. D S)) (0re (0) D ltlet mpan sqrlem1.1 (opr D (x.) D) A ltlet D D axmulrcl anidms sylan mpan2 anim12d imdistani sqrlem1.1 sqrlem1.2 sqrlem4.3 D sqrlem4 sylibr)) thm (sqrlem6 ((x y) (A x) (A y) (S x) (S y) (x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem4.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (/\/\ (C_ S (RR)) (=/= S ({/})) (E.e. x (RR) (A.e. y S (br (cv y) (<) (cv x))))) (sqrlem4.3 x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)) ssrab2 eqsstr sqrlem1.1 sqrlem1.2 sqrlem4.3 (0) sqrlem4 0re 0re leid 0cn mul01 0re sqrlem1.1 sqrlem1.2 ltlei eqbrtr pm3.2i mpbir2an (0) S n0i ax-mp S ({/}) df-ne mpbir ax1re sqrlem1.1 readdcl sqrlem1.1 sqrlem1.2 sqrlem4.3 (cv y) sqrlem4 0re (0) (cv y) leloet mpan (cv y) (if (e. (cv y) (RR)) (cv y) (1)) (0) (<) breq2 (cv y) (if (e. (cv y) (RR)) (cv y) (1)) (cv y) (if (e. (cv y) (RR)) (cv y) (1)) (x.) opreq12 anidms (<) (opr (opr (1) (+) A) (x.) (opr (1) (+) A)) breq1d (cv y) (if (e. (cv y) (RR)) (cv y) (1)) (<) (opr (1) (+) A) breq1 imbi12d imbi12d ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i ax1re (cv y) elimel ax1re sqrlem1.1 readdcl lt2msq 0re ax1re (cv y) elimel ltle 0re ax1re sqrlem1.1 readdcl ltle syl2an mpan2 biimprd dedth ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i (0) (cv y) (<) (opr (1) (+) A) breq1 mpbii (br (opr (cv y) (x.) (cv y)) (<) (opr (opr (1) (+) A) (x.) (opr (1) (+) A))) a1d (e. (cv y) (RR)) a1i jaod sylbid (cv y) (cv y) axmulrcl anidms sqrlem1.1 (opr (cv y) (x.) (cv y)) A leloet sqrlem1.1 sqrlem1.2 sqrlem1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl (opr (cv y) (x.) (cv y)) A (opr (opr (1) (+) A) (x.) (opr (1) (+) A)) axlttrn mp3an3 mpan2i sqrlem1.1 sqrlem1.2 sqrlem1 (opr (cv y) (x.) (cv y)) A (<) (opr (opr (1) (+) A) (x.) (opr (1) (+) A)) breq1 mpbiri (/\ (e. (opr (cv y) (x.) (cv y)) (RR)) (e. A (RR))) a1i jaod sylbid mpan2 syl syl5d imp32 sylbi rgen (cv x) (opr (1) (+) A) (cv y) (<) breq2 y S ralbidv (RR) rcla4ev mp2an 3pm3.2i)) thm (sqrlem7 ((x y) (x z) (S x) (y z) (S y) (S z) (A z) (x y) (A x) (A y) (S x) (S y) (x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem4.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (sqrlem7.4 (= B (sup S (RR) (<))))) (e. B (RR)) (sqrlem7.4 ltso sqrlem1.1 sqrlem1.2 sqrlem4.3 y sqrlem6 S x y z axsup ax-mp supcli eqeltr)) thm (sqrlem8 ((x y) (x z) (S x) (y z) (S y) (S z) (A z) (x y) (A x) (A y) (S x) (S y) (x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem4.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (sqrlem7.4 (= B (sup S (RR) (<))))) (br (0) (<) B) (sqrlem1.1 sqrlem1.2 sqrlem3 sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i redivcl sqrlem1.1 sqrlem1.2 sqrlem3 sqrlem1.1 sqrlem1.2 sqrlem2 pm3.2i sqrlem1.1 sqrlem1.2 sqrlem4.3 (opr A (/) (opr (1) (+) A)) sqrlem5 mp2an ltso sqrlem1.1 sqrlem1.2 sqrlem4.3 y sqrlem6 S x y z axsup ax-mp (opr A (/) (opr (1) (+) A)) supubi ax-mp sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i redivcl sqrlem7.4 sqrlem1.1 sqrlem1.2 sqrlem4.3 sqrlem7.4 sqrlem7 eqeltrr lenlt mpbir 0re sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i redivcl sqrlem7.4 sqrlem1.1 sqrlem1.2 sqrlem4.3 sqrlem7.4 sqrlem7 eqeltrr ltletr mp2an sqrlem7.4 breqtrr)) thm (sqrlem9 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1)))))) (br (0) (<) C) (sqrlem9.3 sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl readdcl ax1re ax1re readdcl sqrlem9.3 sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl sqrlem9.5 sqrlem1.1 sqrlem9.3 sqrlem1.2 sqrlem9.5 divgt0i addgt0i ax1re ax1re lt01 lt01 addgt0i divgt0i sqrlem9.7 breqtrr)) thm (sqrlem10 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1)))))) (br C (<) B) (sqrlem9.7 sqrlem9.6 sqrlem1.1 sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.5 ltdiv1i mpbi sqrlem9.3 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcan3 breqtr sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl sqrlem9.3 sqrlem9.3 ltadd2 mpbi sqrlem9.3 recn 1cn 1cn addcl mulcom sqrlem9.3 recn 1p1times eqtr breqtrr sqrlem9.3 sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl readdcl sqrlem9.3 ax1re ax1re readdcl remulcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i ltdiv1i mpbi 1cn 1cn addcl sqrlem9.3 recn ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i divcan4 breqtr eqbrtr)) thm (sqrlem11 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1)))))) (br A (<) (opr C (x.) C)) (sqrlem9.3 sqrlem9.3 remulcl recn 1cn 1cn addcl negcl sqrlem1.1 renegcl recn mul12 1cn 1cn addcl sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl remulcl recn mulneg1 sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl remulcl recn 1p1times negeqi (opr (opr (opr B (x.) B) (x.) (-u A)) (+) (opr (opr B (x.) B) (x.) (-u A))) df-neg eqtr 3eqtr sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl readdcl sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl readdcl sqrlem1.1 recn negid sqrlem9.6 sqrlem1.1 sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl ltadd1 mpbi eqbrtrr sqrlem1.1 recn negid sqrlem9.6 sqrlem1.1 sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl ltadd1 mpbi eqbrtrr mulgt0i sqrlem9.3 sqrlem9.3 remulcl recn sqrlem1.1 renegcl recn sqrlem9.3 sqrlem9.3 remulcl recn sqrlem1.1 renegcl recn muladd breqtr 0re sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl remulcl readdcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl remulcl sqrlem1.1 renegcl sqrlem1.1 renegcl remulcl readdcl ltsubadd mpbir eqbrtr sqrlem9.3 sqrlem9.3 remulcl ax1re ax1re readdcl renegcl sqrlem1.1 renegcl remulcl remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl remulcl sqrlem1.1 renegcl sqrlem1.1 renegcl remulcl readdcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 sqrlem9.5 sqrlem9.5 mulgt0i ltdiv1i mpbi sqrlem9.3 sqrlem9.3 remulcl recn 1cn 1cn addcl negcl sqrlem1.1 renegcl recn mulcl sqrlem9.3 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i sqrlem9.3 sqrlem9.5 gt0ne0i muln0 divcan3 1cn 1cn addcl sqrlem1.1 recn mul2neg eqtr sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl remulcl recn sqrlem1.1 renegcl sqrlem1.1 renegcl remulcl recn sqrlem9.3 sqrlem9.3 remulcl recn sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 sqrlem9.5 sqrlem9.5 mulgt0i gt0ne0i divdir 3brtr3 ax1re ax1re readdcl sqrlem1.1 remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 sqrlem9.5 sqrlem9.5 mulgt0i gt0ne0i redivcl sqrlem1.1 renegcl sqrlem1.1 renegcl remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 sqrlem9.5 sqrlem9.5 mulgt0i gt0ne0i redivcl readdcl ax1re ax1re readdcl sqrlem1.1 remulcl ltadd1 mpbi sqrlem1.1 recn 1cn 1cn addcl 1cn 1cn addcl mulcl mulcom 1cn 1cn addcl 1cn 1cn addcl sqrlem1.1 recn mulass 1cn 1cn addcl sqrlem1.1 recn mulcl 1p1times 3eqtr sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcl sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcl muladd sqrlem9.3 sqrlem9.3 remulcl recn sqrlem9.3 sqrlem9.3 remulcl recn sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 sqrlem9.5 sqrlem9.5 mulgt0i gt0ne0i divcan4 eqcomi sqrlem1.1 recn sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i sqrlem9.3 sqrlem9.5 gt0ne0i divmuldiv sqrlem1.1 recn sqrlem1.1 recn mul2neg (/) (opr B (x.) B) opreq1i eqtr4 (+) opreq12i sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcl mulcl 1p1times sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcan2 (opr (1) (+) (1)) (x.) opreq2i eqtr3 (+) opreq12i eqtr 3brtr4 sqrlem1.1 ax1re ax1re readdcl ax1re ax1re readdcl remulcl remulcl sqrlem9.3 sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl readdcl sqrlem9.3 sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl readdcl remulcl ax1re ax1re readdcl ax1re ax1re readdcl remulcl ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i ax1re ax1re lt01 lt01 addgt0i mulgt0i ltdiv1i mpbi ax1re ax1re readdcl ax1re ax1re readdcl remulcl recn sqrlem1.1 recn ax1re ax1re readdcl ax1re ax1re readdcl remulcl ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i ax1re ax1re lt01 lt01 addgt0i mulgt0i gt0ne0i divcan4 sqrlem9.7 sqrlem9.7 (x.) opreq12i sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcl addcl 1cn 1cn addcl sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcl addcl 1cn 1cn addcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i divmuldiv eqtr2 3brtr3)) thm (sqrlem12 ((A x) (S x) (D x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1))))) (sqrlem12.8 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-> (e. D S) (br D (<) C)) (sqrlem1.1 sqrlem1.2 sqrlem12.8 D sqrlem4 pm3.27bd pm3.26d sqrlem1.1 sqrlem1.2 sqrlem12.8 D sqrlem4 pm3.26bd 0re (0) D leloet mpan syl mpbid sqrlem1.1 sqrlem1.2 sqrlem12.8 D sqrlem4 pm3.27bd pm3.27d sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem11 sqrlem1.1 sqrlem1.2 sqrlem12.8 D sqrlem4 pm3.26bd D D axmulrcl anidms sqrlem1.1 sqrlem9.4 sqrlem9.4 remulcl (opr D (x.) D) A (opr C (x.) C) lelttrt mp3an23 3syl mpan2i mpd (br (0) (<) D) adantr sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem9 sqrlem1.1 sqrlem1.2 sqrlem12.8 D sqrlem4 pm3.26bd D (if (e. D (RR)) D (0)) (0) (<) breq2 (br (0) (<) C) anbi1d D (if (e. D (RR)) D (0)) (<) C breq1 D (if (e. D (RR)) D (0)) D (if (e. D (RR)) D (0)) (x.) opreq12 anidms (<) (opr C (x.) C) breq1d bibi12d imbi12d 0re D elimel sqrlem9.4 lt2msq 0re 0re D elimel ltle 0re sqrlem9.4 ltle syl2an dedth syl mpan2i imp mpbird ex sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem9 (0) D (<) C breq1 mpbii (e. D S) a1i jaod mpd)) thm (sqrlem13 ((x y) (A x) (A y) (x z) (S x) (y z) (S y) (S z) (C y) (C z) (x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1))))) (sqrlem12.8 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-> (= B (sup S (RR) (<))) (-. (br C (<) B))) (sqrlem9.4 sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem12.8 (cv z) sqrlem12 sqrlem1.1 sqrlem1.2 sqrlem12.8 (cv z) sqrlem4 pm3.26bd sqrlem9.4 (cv z) C axlttri mpan2 syl mpbid (= (cv z) C) (br C (<) (cv z)) ioran sylib pm3.27d rgen ltso sqrlem1.1 sqrlem1.2 sqrlem12.8 y sqrlem6 S x y z axsup ax-mp C supnubi mp2an B (sup S (RR) (<)) C (<) breq2 mtbiri)) thm (sqrlem14 ((A x) (S x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1))))) (sqrlem12.8 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-. (= B (sup S (RR) (<)))) (sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem10 sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem12.8 sqrlem13 mt2)) thm (sqrlem15 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem15.5 (e. C (RR))) (sqrlem15.6 (br (0) (<) C))) (br B (<) (opr B (+) C)) (sqrlem15.3 recn addid1 sqrlem15.6 0re sqrlem15.5 sqrlem15.3 ltadd2 mpbi eqbrtrr)) thm (sqrlem16 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem15.5 (e. C (RR))) (sqrlem15.6 (br (0) (<) C)) (sqrlem16.7 (br C (<) B))) (-> (br C (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B))) (br (opr (opr B (+) C) (x.) (opr B (+) C)) (<) A)) (1cn 1cn addcl 1cn addcl sqrlem15.3 recn sqrlem15.5 recn mulass 1cn 1cn addcl 1cn addcl sqrlem15.3 recn mulcl sqrlem1.1 sqrlem15.3 sqrlem15.3 remulcl resubcl recn ax1re ax1re readdcl ax1re readdcl recn sqrlem15.3 recn ax1re ax1re readdcl ax1re readdcl ax1re ax1re readdcl ax1re ax1re ax1re lt01 lt01 addgt0i lt01 addgt0i gt0ne0i sqrlem15.3 sqrlem15.4 gt0ne0i muln0 divcan2 (<) breq12i ax1re ax1re readdcl ax1re readdcl sqrlem15.3 ax1re ax1re readdcl ax1re ax1re ax1re lt01 lt01 addgt0i lt01 addgt0i sqrlem15.4 mulgt0i sqrlem15.5 sqrlem1.1 sqrlem15.3 sqrlem15.3 remulcl resubcl ax1re ax1re readdcl ax1re readdcl sqrlem15.3 remulcl ax1re ax1re readdcl ax1re readdcl recn sqrlem15.3 recn ax1re ax1re readdcl ax1re readdcl ax1re ax1re readdcl ax1re ax1re ax1re lt01 lt01 addgt0i lt01 addgt0i gt0ne0i sqrlem15.3 sqrlem15.4 gt0ne0i muln0 redivcl ax1re ax1re readdcl ax1re readdcl sqrlem15.3 remulcl ltmul2 ax-mp ax1re ax1re readdcl ax1re readdcl sqrlem15.3 sqrlem15.5 remulcl remulcl sqrlem15.3 sqrlem15.3 remulcl sqrlem1.1 ltaddsub 3bitr4 sqrlem16.7 sqrlem15.6 sqrlem15.5 sqrlem15.3 sqrlem15.5 ltmul2 ax-mp mpbi sqrlem15.5 sqrlem15.5 remulcl sqrlem15.5 sqrlem15.3 remulcl sqrlem15.3 sqrlem15.3 remulcl ltadd2 mpbi sqrlem15.3 sqrlem15.3 remulcl sqrlem15.5 sqrlem15.5 remulcl readdcl sqrlem15.3 sqrlem15.3 remulcl sqrlem15.5 sqrlem15.3 remulcl readdcl sqrlem15.3 sqrlem15.5 remulcl sqrlem15.3 sqrlem15.5 remulcl readdcl ltadd1 mpbi sqrlem15.3 recn sqrlem15.5 recn sqrlem15.3 recn sqrlem15.5 recn muladd sqrlem15.5 sqrlem15.3 remulcl sqrlem15.3 sqrlem15.5 remulcl sqrlem15.3 sqrlem15.5 remulcl readdcl readdcl recn sqrlem15.3 sqrlem15.3 remulcl recn addcom 1cn 1cn addcl 1cn sqrlem15.3 sqrlem15.5 remulcl recn adddir sqrlem15.3 sqrlem15.5 remulcl recn 1p1times sqrlem15.3 sqrlem15.5 remulcl recn mulid2 sqrlem15.3 recn sqrlem15.5 recn mulcom eqtr (+) opreq12i sqrlem15.3 sqrlem15.5 remulcl sqrlem15.3 sqrlem15.5 remulcl readdcl recn sqrlem15.5 sqrlem15.3 remulcl recn addcom 3eqtr (+) (opr B (x.) B) opreq1i sqrlem15.3 sqrlem15.3 remulcl recn sqrlem15.5 sqrlem15.3 remulcl recn sqrlem15.3 sqrlem15.5 remulcl sqrlem15.3 sqrlem15.5 remulcl readdcl recn addass 3eqtr4 3brtr4 sqrlem15.3 sqrlem15.5 readdcl sqrlem15.3 sqrlem15.5 readdcl remulcl ax1re ax1re readdcl ax1re readdcl sqrlem15.3 sqrlem15.5 remulcl remulcl sqrlem15.3 sqrlem15.3 remulcl readdcl sqrlem1.1 lttr mpan sylbi)) thm (sqrlem17 ((A x) (B x) (C x) (S x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem15.5 (e. C (RR))) (sqrlem15.6 (br (0) (<) C)) (sqrlem16.7 (br C (<) B)) (sqrlem17.8 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-> (br C (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B))) (e. (opr B (+) C) S)) (sqrlem1.1 sqrlem1.2 sqrlem15.3 sqrlem15.4 sqrlem15.5 sqrlem15.6 sqrlem16.7 sqrlem16 sqrlem15.3 sqrlem15.5 sqrlem15.4 sqrlem15.6 addgt0i jctil sqrlem15.3 sqrlem15.5 readdcl sqrlem1.1 sqrlem1.2 sqrlem17.8 (opr B (+) C) sqrlem5 mpan syl)) thm (sqrlem18 ((x y) (A x) (A y) (x z) (B x) (y z) (B y) (B z) (C x) (S x) (S y) (S z) (x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem15.5 (e. C (RR))) (sqrlem15.6 (br (0) (<) C)) (sqrlem16.7 (br C (<) B)) (sqrlem17.8 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-> (br C (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B))) (-. (= B (sup S (RR) (<))))) (sqrlem1.1 sqrlem1.2 sqrlem15.3 sqrlem15.4 sqrlem15.5 sqrlem15.6 sqrlem15 B (sup S (RR) (<)) (<) (opr B (+) C) breq1 negbid sqrlem1.1 sqrlem1.2 sqrlem15.3 sqrlem15.4 sqrlem15.5 sqrlem15.6 sqrlem16.7 sqrlem17.8 sqrlem17 ltso sqrlem1.1 sqrlem1.2 sqrlem17.8 y sqrlem6 S x y z axsup ax-mp (opr B (+) C) supubi syl syl5bir com12 mt2i)) thm (sqrlem19 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem19.5 (br (opr B (x.) B) (<) A))) (br (0) (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B))) (sqrlem1.1 sqrlem15.3 sqrlem15.3 remulcl resubcl ax1re ax1re readdcl ax1re readdcl sqrlem15.3 remulcl sqrlem15.3 sqrlem15.3 remulcl recn addid2 sqrlem19.5 eqbrtr 0re sqrlem15.3 sqrlem15.3 remulcl sqrlem1.1 ltaddsub mpbi ax1re ax1re readdcl ax1re readdcl sqrlem15.3 ax1re ax1re readdcl ax1re ax1re ax1re lt01 lt01 addgt0i lt01 addgt0i sqrlem15.4 mulgt0i divgt0i)) thm (sqrlem20 ((w x) (A x) (A w) (B x) (B w) (S x) (S w) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem19.5 (br (opr B (x.) B) (<) A)) (sqrlem20.6 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-. (= B (sup S (RR) (<)))) (sqrlem15.3 sqrlem1.1 sqrlem15.3 sqrlem15.3 remulcl resubcl ax1re ax1re readdcl ax1re readdcl sqrlem15.3 remulcl ax1re ax1re readdcl ax1re readdcl recn sqrlem15.3 recn ax1re ax1re readdcl ax1re readdcl ax1re ax1re readdcl ax1re ax1re ax1re lt01 lt01 addgt0i lt01 addgt0i gt0ne0i sqrlem15.3 sqrlem15.4 gt0ne0i muln0 redivcl sqrlem15.4 sqrlem1.1 sqrlem1.2 sqrlem15.3 sqrlem15.4 sqrlem19.5 sqrlem19 w posex (cv w) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B)) breq1 (-. (= B (sup S (RR) (<)))) imbi1d sqrlem1.1 sqrlem1.2 sqrlem15.3 sqrlem15.4 (cv w) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (RR) eleq1 (cv w) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (0) (<) breq2 (cv w) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (<) B breq1 3anbi123d (opr B (/) (opr (1) (+) (1))) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (RR) eleq1 (opr B (/) (opr (1) (+) (1))) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (0) (<) breq2 (opr B (/) (opr (1) (+) (1))) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (<) B breq1 3anbi123d sqrlem15.3 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl sqrlem15.3 ax1re ax1re readdcl sqrlem15.4 ax1re ax1re lt01 lt01 addgt0i divgt0i sqrlem15.4 sqrlem15.3 halfpos mpbi 3pm3.2i elimhyp 3simp1i (cv w) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (RR) eleq1 (cv w) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (0) (<) breq2 (cv w) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (<) B breq1 3anbi123d (opr B (/) (opr (1) (+) (1))) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (RR) eleq1 (opr B (/) (opr (1) (+) (1))) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (0) (<) breq2 (opr B (/) (opr (1) (+) (1))) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (<) B breq1 3anbi123d sqrlem15.3 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl sqrlem15.3 ax1re ax1re readdcl sqrlem15.4 ax1re ax1re lt01 lt01 addgt0i divgt0i sqrlem15.4 sqrlem15.3 halfpos mpbi 3pm3.2i elimhyp 3simp2i (cv w) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (RR) eleq1 (cv w) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (0) (<) breq2 (cv w) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (<) B breq1 3anbi123d (opr B (/) (opr (1) (+) (1))) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (RR) eleq1 (opr B (/) (opr (1) (+) (1))) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (0) (<) breq2 (opr B (/) (opr (1) (+) (1))) (if (/\/\ (e. (cv w) (RR)) (br (0) (<) (cv w)) (br (cv w) (<) B)) (cv w) (opr B (/) (opr (1) (+) (1)))) (<) B breq1 3anbi123d sqrlem15.3 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl sqrlem15.3 ax1re ax1re readdcl sqrlem15.4 ax1re ax1re lt01 lt01 addgt0i divgt0i sqrlem15.4 sqrlem15.3 halfpos mpbi 3pm3.2i elimhyp 3simp3i sqrlem20.6 sqrlem18 dedth 3exp imp4d r19.23aiv ax-mp)) thm (sqrlem21 ((A x) (B x) (S x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem21.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (sqrlem21.4 (= B (sup S (RR) (<))))) (-. (br A (<) (opr B (x.) B))) (sqrlem21.4 B (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A)) (sup S (RR) (<)) eqeq1 negbid sqrlem1.1 sqrlem1.2 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 ax1re sqrlem1.1 readdcl (br A (<) (opr B (x.) B)) keepel (= B (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A))) id B (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A)) A (/) opreq2 (+) opreq12d (/) (opr (1) (+) (1)) opreq1d (RR) eleq1d (= (opr (1) (+) A) (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A))) id (opr (1) (+) A) (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A)) A (/) opreq2 (+) opreq12d (/) (opr (1) (+) (1)) opreq1d (RR) eleq1d sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 sqrlem1.1 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem8 gt0ne0i redivcl readdcl ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl ax1re sqrlem1.1 readdcl sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i redivcl readdcl ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl keephyp B (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A)) (0) (<) breq2 (opr (1) (+) A) (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A)) (0) (<) breq2 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem8 ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i keephyp B (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A)) B (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A)) (x.) opreq12 anidms A (<) breq2d (opr (1) (+) A) (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A)) (opr (1) (+) A) (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A)) (x.) opreq12 anidms A (<) breq2d sqrlem1.1 sqrlem1.2 sqrlem1 elimhyp (opr (opr (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A)) (+) (opr A (/) (if (br A (<) (opr B (x.) B)) B (opr (1) (+) A)))) (/) (opr (1) (+) (1))) eqid sqrlem21.3 sqrlem14 dedth mt2)) thm (sqrlem22 ((A x) (B x) (S x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem21.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (sqrlem21.4 (= B (sup S (RR) (<))))) (-. (br (opr B (x.) B) (<) A)) (sqrlem21.4 B (if (br (opr B (x.) B) (<) A) B (opr A (/) (opr (1) (+) A))) (sup S (RR) (<)) eqeq1 negbid sqrlem1.1 sqrlem1.2 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i redivcl (br (opr B (x.) B) (<) A) keepel B (if (br (opr B (x.) B) (<) A) B (opr A (/) (opr (1) (+) A))) (0) (<) breq2 (opr A (/) (opr (1) (+) A)) (if (br (opr B (x.) B) (<) A) B (opr A (/) (opr (1) (+) A))) (0) (<) breq2 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem8 sqrlem1.1 sqrlem1.2 sqrlem3 keephyp B (if (br (opr B (x.) B) (<) A) B (opr A (/) (opr (1) (+) A))) B (if (br (opr B (x.) B) (<) A) B (opr A (/) (opr (1) (+) A))) (x.) opreq12 anidms (<) A breq1d (opr A (/) (opr (1) (+) A)) (if (br (opr B (x.) B) (<) A) B (opr A (/) (opr (1) (+) A))) (opr A (/) (opr (1) (+) A)) (if (br (opr B (x.) B) (<) A) B (opr A (/) (opr (1) (+) A))) (x.) opreq12 anidms (<) A breq1d sqrlem1.1 sqrlem1.2 sqrlem2 elimhyp sqrlem21.3 sqrlem20 dedth mt2)) thm (sqrlem23 ((A x) (B x) (S x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem21.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (sqrlem21.4 (= B (sup S (RR) (<))))) (= (opr B (x.) B) A) (sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 remulcl sqrlem1.1 lttri3 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem22 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem21 mpbir2an)) thm (sqrlem24 ((x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (e. (` (sqr) A) (RR)) (sqrlem1.1 sqrlem1.2 (cv x) (cv y) (0) (<_) breq2 (cv x) (cv y) (cv x) (cv y) (x.) opreq12 anidms (<_) A breq1d anbi12d (RR) cbvrabv sqrlem1.1 0re sqrlem1.1 sqrlem1.2 ltlei A x sqrval mp2an sqrlem7)) thm (sqrgt0i ((x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (br (0) (<) (` (sqr) A)) (sqrlem1.1 sqrlem1.2 (cv x) (cv y) (0) (<_) breq2 (cv x) (cv y) (cv x) (cv y) (x.) opreq12 anidms (<_) A breq1d anbi12d (RR) cbvrabv sqrlem1.1 0re sqrlem1.1 sqrlem1.2 ltlei A x sqrval mp2an sqrlem8)) thm (sqrlem26 ((x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (= (opr (` (sqr) A) (x.) (` (sqr) A)) A) (sqrlem1.1 sqrlem1.2 (cv x) (cv y) (0) (<_) breq2 (cv x) (cv y) (cv x) (cv y) (x.) opreq12 anidms (<_) A breq1d anbi12d (RR) cbvrabv sqrlem1.1 0re sqrlem1.1 sqrlem1.2 ltlei A x sqrval mp2an sqrlem23)) thm (sqrth () ((sqrth.1 (e. A (RR)))) (-> (br (0) (<_) A) (= (opr (` (sqr) A) (x.) (` (sqr) A)) A)) (0re sqrth.1 leloe A (if (br (0) (<) A) A (1)) (sqr) fveq2 A (if (br (0) (<) A) A (1)) (sqr) fveq2 (x.) opreq12d (= A (if (br (0) (<) A) A (1))) id eqeq12d sqrth.1 ax1re (br (0) (<) A) keepel A elimgt0 sqrlem26 dedth sqr0 sqr0 (x.) opreq12i 0cn mul01 eqtr (0) A (sqr) fveq2 (0) A (sqr) fveq2 (x.) opreq12d (= (0) A) id eqeq12d mpbii jaoi sylbi)) thm (sqrcl () ((sqrth.1 (e. A (RR)))) (-> (br (0) (<_) A) (e. (` (sqr) A) (RR))) (0re sqrth.1 leloe A (if (br (0) (<) A) A (1)) (sqr) fveq2 (RR) eleq1d sqrth.1 ax1re (br (0) (<) A) keepel A elimgt0 sqrlem24 dedth (0) A (sqr) fveq2 sqr0 syl5eqr 0re syl6eqelr jaoi sylbi)) thm (sqrgt0 () ((sqrth.1 (e. A (RR)))) (-> (br (0) (<) A) (br (0) (<) (` (sqr) A))) (A (if (br (0) (<) A) A (1)) (sqr) fveq2 (0) (<) breq2d sqrth.1 ax1re (br (0) (<) A) keepel A elimgt0 sqrgt0i dedth)) thm (sqrge0 () ((sqrth.1 (e. A (RR)))) (-> (br (0) (<_) A) (br (0) (<_) (` (sqr) A))) (0re sqrth.1 leloe 0re sqrth.1 ltle sqrth.1 sqrcl syl 0re (0) (` (sqr) A) leloet mpan sqrth.1 sqrgt0 (br (0) (<) (` (sqr) A)) (= (0) (` (sqr) A)) orc syl syl5bir mpcom (0) A (sqr) fveq2 sqr0 syl5eqr 0re leid syl5breq jaoi sylbi)) thm (sqr11 () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (` (sqr) A) (` (sqr) B)) (= A B))) ((` (sqr) A) (` (sqr) B) (` (sqr) A) (` (sqr) B) (x.) opreq12 anidms (/\ (br (0) (<_) A) (br (0) (<_) B)) adantl sqrth.1 sqrth (br (0) (<_) B) (= (` (sqr) A) (` (sqr) B)) ad2antrr sqr11.1 sqrth (br (0) (<_) A) (= (` (sqr) A) (` (sqr) B)) ad2antlr 3eqtr3d ex A B (sqr) fveq2 (/\ (br (0) (<_) A) (br (0) (<_) B)) a1i impbid)) thm (sqrmuli () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR))) (sqrmuli.1 (br (0) (<_) A)) (sqrmuli.2 (br (0) (<_) B))) (= (` (sqr) (opr A (x.) B)) (opr (` (sqr) A) (x.) (` (sqr) B))) (sqrmuli.1 sqrth.1 sqrth ax-mp sqrmuli.2 sqr11.1 sqrth ax-mp (x.) opreq12i sqrmuli.1 sqrth.1 sqrcl ax-mp recn sqrmuli.2 sqr11.1 sqrcl ax-mp recn sqrmuli.1 sqrth.1 sqrcl ax-mp recn sqrmuli.2 sqr11.1 sqrcl ax-mp recn mul4 sqrmuli.1 sqrmuli.2 sqrth.1 sqr11.1 mulge0 mp2an sqrth.1 sqr11.1 remulcl sqrth ax-mp 3eqtr4r sqrmuli.1 sqrmuli.2 sqrth.1 sqr11.1 mulge0 mp2an sqrth.1 sqr11.1 remulcl sqrge0 ax-mp sqrmuli.1 sqrth.1 sqrge0 ax-mp sqrmuli.2 sqr11.1 sqrge0 ax-mp sqrmuli.1 sqrth.1 sqrcl ax-mp sqrmuli.2 sqr11.1 sqrcl ax-mp mulge0 mp2an sqrmuli.1 sqrmuli.2 sqrth.1 sqr11.1 mulge0 mp2an sqrth.1 sqr11.1 remulcl sqrcl ax-mp sqrmuli.1 sqrth.1 sqrcl ax-mp sqrmuli.2 sqr11.1 sqrcl ax-mp remulcl msq11 mp2an mpbi)) thm (sqrmul () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (= (` (sqr) (opr A (x.) B)) (opr (` (sqr) A) (x.) (` (sqr) B)))) (A (if (br (0) (<_) A) A (0)) (x.) B opreq1 (sqr) fveq2d A (if (br (0) (<_) A) A (0)) (sqr) fveq2 (x.) (` (sqr) B) opreq1d eqeq12d B (if (br (0) (<_) B) B (0)) (if (br (0) (<_) A) A (0)) (x.) opreq2 (sqr) fveq2d B (if (br (0) (<_) B) B (0)) (sqr) fveq2 (` (sqr) (if (br (0) (<_) A) A (0))) (x.) opreq2d eqeq12d sqrth.1 0re (br (0) (<_) A) keepel sqr11.1 0re (br (0) (<_) B) keepel A elimge0 B elimge0 sqrmuli dedth2h)) thm (sqrmsq2 () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (` (sqr) A) B) (= A (opr B (x.) B)))) (sqr11.1 sqr11.1 sqrmul anidms sqr11.1 sqrth eqtrd (` (sqr) A) eqeq2d (br (0) (<_) A) adantl sqr11.1 msqge0 sqrth.1 sqr11.1 sqr11.1 remulcl sqr11 mpan2 (br (0) (<_) B) adantr bitr3d)) thm (sqrle () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (` (sqr) A) (<_) (` (sqr) B)))) ((` (sqr) A) (` (sqr) B) le2msqt sqrth.1 sqrcl sqrth.1 sqrge0 jca sqr11.1 sqrcl sqr11.1 sqrge0 jca syl2an sqrth.1 sqrth sqr11.1 sqrth (<_) breqan12d bitr2d)) thm (sqrlt () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (` (sqr) A) (<) (` (sqr) B)))) ((` (sqr) A) (` (sqr) B) lt2msqt sqrth.1 sqrcl sqrth.1 sqrge0 jca sqr11.1 sqrcl sqr11.1 sqrge0 jca syl2an sqrth.1 sqrth sqr11.1 sqrth (<) breqan12d bitr2d)) thm (sqrmsq () ((sqrth.1 (e. A (RR)))) (-> (br (0) (<_) A) (= (` (sqr) (opr A (x.) A)) A)) (sqrth.1 msqge0 (opr A (x.) A) eqid sqrth.1 sqrth.1 remulcl sqrth.1 sqrmsq2 mpbiri mpan)) thm (sqrclt () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (e. (` (sqr) A) (RR))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 A (if (e. A (RR)) A (0)) (sqr) fveq2 (RR) eleq1d imbi12d 0re A elimel sqrcl dedth imp)) thm (sqrgt0t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (br (0) (<) (` (sqr) A))) (A (if (e. A (RR)) A (0)) (0) (<) breq2 A (if (e. A (RR)) A (0)) (sqr) fveq2 (0) (<) breq2d imbi12d 0re A elimel sqrgt0 dedth imp)) thm (sqrge0t () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (br (0) (<_) (` (sqr) A))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 A (if (e. A (RR)) A (0)) (sqr) fveq2 (0) (<_) breq2d imbi12d 0re A elimel sqrge0 dedth imp)) thm (sqrlet () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<_) B))) (<-> (br A (<_) B) (br (` (sqr) A) (<_) (` (sqr) B)))) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 (br (0) (<_) B) anbi1d A (if (e. A (RR)) A (0)) (<_) B breq1 A (if (e. A (RR)) A (0)) (sqr) fveq2 (<_) (` (sqr) B) breq1d bibi12d imbi12d B (if (e. B (RR)) B (0)) (0) (<_) breq2 (br (0) (<_) (if (e. A (RR)) A (0))) anbi2d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<_) breq2 B (if (e. B (RR)) B (0)) (sqr) fveq2 (` (sqr) (if (e. A (RR)) A (0))) (<_) breq2d bibi12d imbi12d 0re A elimel 0re B elimel sqrle dedth2h imp)) thm (sqr00t () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (<-> (= (` (sqr) A) (0)) (= A (0)))) (0re leid A (if (e. A (RR)) A (0)) (0) (<_) breq2 (br (0) (<_) (0)) anbi1d A (if (e. A (RR)) A (0)) (sqr) fveq2 (` (sqr) (0)) eqeq1d A (if (e. A (RR)) A (0)) (0) eqeq1 bibi12d imbi12d 0re A elimel 0re sqr11 dedth mpan2i imp sqr0 (` (sqr) A) eqeq2i syl5bbr)) thm (sqr1 () () (= (` (sqr) (1)) (1)) (1cn mulid1 (sqr) fveq2i 0re ax1re lt01 ltlei ax1re sqrmsq ax-mp eqtr3)) thm (sqr4 () () (= (` (sqr) (4)) (2)) (2t2e4 eqcomi 4pos 2pos 4re 2re sqrmsq2 0re 4re ltle 0re 2re ltle syl2an mp2an mpbir)) thm (sqr9 () () (= (` (sqr) (9)) (3)) (3t3e9 eqcomi 9pos 3pos 9re 3re sqrmsq2 0re 9re ltle 0re 3re ltle syl2an mp2an mpbir)) thm (sqr2gt1lt2 () () (/\ (br (1) (<) (` (sqr) (2))) (br (` (sqr) (2)) (<) (2))) (sqr1 1lt2 0re ax1re lt01 ltlei 0re 2re 2pos ltlei ax1re 2re sqrlt mp2an mpbi eqbrtrr 2re 2pos 2re 2pos 2re 2pos sqrlem15 2p2e4 breqtr 0re 2re 2pos ltlei 0re 4re 4pos ltlei 2re 4re sqrlt mp2an mpbi sqr4 breqtr pm3.2i)) thm (sqrsq () ((sqrsq.1 (e. A (RR)))) (-> (br (0) (<_) A) (= (` (sqr) (opr A (^) (2))) A)) (sqrsq.1 sqrmsq sqrsq.1 recn sqval (sqr) fveq2i syl5eq)) thm (sqsqr () ((sqrsq.1 (e. A (RR)))) (-> (br (0) (<_) A) (= (opr (` (sqr) A) (^) (2)) A)) (sqrsq.1 sqrcl recnd (` (sqr) A) sqvalt syl sqrsq.1 sqrth eqtrd)) thm (sqsqrt () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (= (opr (` (sqr) A) (^) (2)) A)) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 A (if (e. A (RR)) A (0)) (sqr) fveq2 (^) (2) opreq1d (= A (if (e. A (RR)) A (0))) id eqeq12d imbi12d 0re A elimel sqsqr dedth imp)) thm (sqr2irrlem1 () ((sqr2irrlem1.1 (e. A (NN))) (sqr2irrlem1.2 (e. B (NN)))) (-> (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2)))) (/\ (/\ (br B (<) A) (e. (opr A (/) (2)) (NN))) (= (opr B (^) (2)) (opr (2) (x.) (opr (opr A (/) (2)) (^) (2)))))) (sqr2irrlem1.2 nnre resqcl recn mulid2 1lt2 ax1re 2re sqr2irrlem1.2 nnre resqcl sqr2irrlem1.2 nnsqcl nngt0 ltmul1i mpbi eqbrtrr (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2))) (opr B (^) (2)) (<) breq2 mpbiri 0re sqr2irrlem1.2 nnre sqr2irrlem1.2 nngt0 ltlei 0re sqr2irrlem1.1 nnre sqr2irrlem1.1 nngt0 ltlei sqr2irrlem1.2 nnre sqr2irrlem1.1 nnre lt2sq mp2an sylibr sqr2irrlem1.1 nnre resqcl recn 2cn sqr2irrlem1.2 nnre resqcl recn 2re 2pos gt0ne0i divmul sqr2irrlem1.2 nnsqcl (opr (opr A (^) (2)) (/) (2)) (opr B (^) (2)) (NN) eleq1 mpbiri sqr2irrlem1.1 nnesq sylibr sylbir eqcoms jca 2cn 2re sqr2irrlem1.1 nnre 2re 2re 2pos gt0ne0i redivcl resqcl remulcl recn sqr2irrlem1.2 nnre resqcl recn 2re 2pos gt0ne0i mulcan sqr2irrlem1.1 nncn 2cn 2re 2pos gt0ne0i sqdiv 2cn sqval (opr A (^) (2)) (/) opreq2i eqtr (opr (2) (x.) (2)) (x.) opreq2i 2cn 2cn sqr2irrlem1.1 nnre 2re 2re 2pos gt0ne0i redivcl resqcl recn mulass 2cn 2cn mulcl sqr2irrlem1.1 nnre resqcl recn 2cn 2cn 2re 2pos gt0ne0i 2re 2pos gt0ne0i muln0 divcan2 3eqtr3 (opr (2) (x.) (opr B (^) (2))) eqeq1i bitr3 biimpr eqcomd jca)) thm (sqr2irrlem2 () () (-> (/\ (e. A (NN)) (e. B (NN))) (-> (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2)))) (/\ (/\ (br B (<) A) (e. (opr A (/) (2)) (NN))) (= (opr B (^) (2)) (opr (2) (x.) (opr (opr A (/) (2)) (^) (2))))))) (A (if (e. A (NN)) A (2)) (^) (2) opreq1 (opr (2) (x.) (opr B (^) (2))) eqeq1d A (if (e. A (NN)) A (2)) B (<) breq2 A (if (e. A (NN)) A (2)) (/) (2) opreq1 (NN) eleq1d anbi12d A (if (e. A (NN)) A (2)) (/) (2) opreq1 (^) (2) opreq1d (2) (x.) opreq2d (opr B (^) (2)) eqeq2d anbi12d imbi12d B (if (e. B (NN)) B (2)) (^) (2) opreq1 (2) (x.) opreq2d (opr (if (e. A (NN)) A (2)) (^) (2)) eqeq2d B (if (e. B (NN)) B (2)) (<) (if (e. A (NN)) A (2)) breq1 (e. (opr (if (e. A (NN)) A (2)) (/) (2)) (NN)) anbi1d B (if (e. B (NN)) B (2)) (^) (2) opreq1 (opr (2) (x.) (opr (opr (if (e. A (NN)) A (2)) (/) (2)) (^) (2))) eqeq1d anbi12d imbi12d 2nn A elimel 2nn B elimel sqr2irrlem1 dedth2h)) thm (sqr2irrlem3 ((x y) (x z) (w x) (y z) (w y) (w z)) () (-. (E.e. x (NN) (E.e. y (NN) (= (opr (cv x) (^) (2)) (opr (2) (x.) (opr (cv y) (^) (2))))))) ((cv x) (cv z) (^) (2) opreq1 (opr (2) (x.) (opr (cv y) (^) (2))) eqeq1d negbid y (NN) ralbidv (cv y) (cv w) (^) (2) opreq1 (2) (x.) opreq2d (opr (cv z) (^) (2)) eqeq2d negbid (NN) cbvralv syl6bb (cv z) (cv y) (<) (cv x) breq1 (cv z) (cv y) (^) (2) opreq1 (opr (2) (x.) (opr (cv w) (^) (2))) eqeq1d negbid w (NN) ralbidv imbi12d (NN) rcla4cva (cv w) (opr (cv x) (/) (2)) (^) (2) opreq1 (2) (x.) opreq2d (opr (cv y) (^) (2)) eqeq2d negbid (NN) rcla4cv syl6 imp3a (/\ (br (cv y) (<) (cv x)) (e. (opr (cv x) (/) (2)) (NN))) (= (opr (cv y) (^) (2)) (opr (2) (x.) (opr (opr (cv x) (/) (2)) (^) (2)))) imnan sylib (e. (cv x) (NN)) adantll (cv x) (cv y) sqr2irrlem2 (A.e. z (NN) (-> (br (cv z) (<) (cv x)) (A.e. w (NN) (-. (= (opr (cv z) (^) (2)) (opr (2) (x.) (opr (cv w) (^) (2)))))))) adantlr mtod exp31 r19.21adv indstr y (NN) (= (opr (cv x) (^) (2)) (opr (2) (x.) (opr (cv y) (^) (2)))) ralnex sylib nrex)) thm (sqr2irrlem4 () ((sqr2irrlem4.1 (e. A (NN))) (sqr2irrlem4.2 (e. B (NN)))) (<-> (= (` (sqr) (2)) (opr A (/) B)) (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2))))) (sqr2irrlem4.1 nncn sqr2irrlem4.2 nncn sqr2irrlem4.2 nnne0 sqdiv eqcomi sqr2irrlem4.2 nnsqcl nncn 2cn sqr2irrlem4.2 nnsqcl nnne0 divcan4 eqeq12i sqr2irrlem4.1 nnre resqcl recn 2re sqr2irrlem4.2 nnsqcl nnre remulcl recn sqr2irrlem4.2 nnsqcl nncn sqr2irrlem4.2 nnsqcl nnne0 div11 (opr (opr A (/) B) (^) (2)) (2) eqcom 3bitr3 0re 2re 2pos ltlei sqr2irrlem4.1 nnre sqr2irrlem4.2 nnre sqr2irrlem4.2 nnne0 redivcl sqge0 2re sqr2irrlem4.1 nnre sqr2irrlem4.2 nnre sqr2irrlem4.2 nnne0 redivcl resqcl sqr11 mp2an 0re sqr2irrlem4.1 nnre sqr2irrlem4.2 nnre sqr2irrlem4.2 nnne0 redivcl sqr2irrlem4.1 nnre sqr2irrlem4.2 nnre sqr2irrlem4.1 nngt0 sqr2irrlem4.2 nngt0 divgt0i ltlei sqr2irrlem4.1 nnre sqr2irrlem4.2 nnre sqr2irrlem4.2 nnne0 redivcl sqrsq ax-mp (` (sqr) (2)) eqeq2i 3bitr2r)) thm (sqr2irrlem5 () () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (= (` (sqr) (2)) (opr A (/) B)) (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2)))))) (A (if (e. A (NN)) A (2)) (/) B opreq1 (` (sqr) (2)) eqeq2d A (if (e. A (NN)) A (2)) (^) (2) opreq1 (opr (2) (x.) (opr B (^) (2))) eqeq1d bibi12d B (if (e. B (NN)) B (2)) (if (e. A (NN)) A (2)) (/) opreq2 (` (sqr) (2)) eqeq2d B (if (e. B (NN)) B (2)) (^) (2) opreq1 (2) (x.) opreq2d (opr (if (e. A (NN)) A (2)) (^) (2)) eqeq2d bibi12d 2nn A elimel 2nn B elimel sqr2irrlem4 dedth2h)) thm (sqr2irr ((x y)) () (e/ (` (sqr) (2)) (QQ)) (x y sqr2irrlem3 (cv x) (cv y) sqr2irrlem5 2rexbiia mtbir (cv y) nngt0t (e. (cv x) (ZZ)) adantr 0re (0) (cv y) (cv x) ltmuldivt ex mp3an1 (cv y) nnret (cv x) zret syl2an mpd ancoms 2re 2pos sqrgt0i (` (sqr) (2)) (opr (cv x) (/) (cv y)) (0) (<) breq2 mpbii syl5bir (cv y) nncnt (cv y) mul02t syl (<) (cv x) breq1d (e. (cv x) (ZZ)) adantl sylibd ex r19.23adv anc2li (cv x) elnnz syl6ibr impac r19.22i2 mto (` (sqr) (2)) x y elq mtbir (` (sqr) (2)) (QQ) df-nel mpbir)) thm (sqr2re () () (e. (` (sqr) (2)) (RR)) (2re 2pos sqrlem24)) thm (ine0 () () (=/= (i) (0)) (ax1ne0 (1) (0) df-ne mpbi (i) (0) (i) (x.) opreq2 axicn mul01 syl6req (+) (1) opreq1d 1cn addid2 axi2m1 3eqtr3g mto (i) (0) df-ne mpbir)) thm (isqm1 () () (= (opr (i) (^) (2)) (-u (1))) (axicn sqval axicn axicn mulcl 1cn 1cn negcl addass axi2m1 (+) (-u (1)) opreq1i 1cn negid (opr (i) (x.) (i)) (+) opreq2i 3eqtr3r axicn axicn mulcl addid1 1cn negcl addid2 3eqtr3 eqtr)) thm (itimesi () () (= (opr (i) (x.) (i)) (-u (1))) (axicn axicn mulcl 1cn 1cn negcl addass axi2m1 (+) (-u (1)) opreq1i 1cn negid (opr (i) (x.) (i)) (+) opreq2i 3eqtr3r axicn axicn mulcl addid1 1cn negcl addid2 3eqtr3)) thm (irec () () (= (opr (1) (/) (i)) (-u (i))) (axicn axicn mulneg2 itimesi 1cn axicn axicn mulcl negcon2 mpbir eqtr4 1cn axicn axicn negcl ine0 divmul mpbir)) thm (inelr () () (-. (e. (i) (RR))) (ine0 (i) (0) df-ne mpbi lt01 0re ax1re ltnsym ax-mp 0re itimesi ax1re renegcl eqeltr ax1re ltadd1 1cn addid2 axi2m1 (<) breq12i bitr mtbir (i) ltmsqt mt3i mto)) thm (crulem () ((cru.1 (e. A (RR))) (cru.2 (e. B (RR))) (cru.3 (e. C (RR))) (cru.4 (e. D (RR)))) (-> (= (opr A (+) (opr B (x.) (i))) (opr C (+) (opr D (x.) (i)))) (= B D)) (inelr cru.3 cru.1 renegcl readdcl recn cru.2 recn cru.4 recn subcl axicn divmulz (opr A (+) (opr B (x.) (i))) (opr C (+) (opr D (x.) (i))) (+) (opr (-u A) (+) (-u (opr D (x.) (i)))) opreq1 cru.1 recn cru.2 recn axicn mulcl cru.1 renegcl recn cru.4 recn axicn mulcl negcl add4 cru.1 recn negid (+) (opr (opr B (x.) (i)) (+) (-u (opr D (x.) (i)))) opreq1i cru.2 recn axicn mulcl cru.4 recn axicn mulcl negcl addcl addid2 3eqtr cru.3 recn cru.4 recn axicn mulcl cru.1 renegcl recn cru.4 recn axicn mulcl negcl add4 cru.4 recn axicn mulcl negid (opr C (+) (-u A)) (+) opreq2i cru.3 cru.1 renegcl readdcl recn addid1 3eqtr 3eqtr3g cru.2 recn cru.4 recn axicn subdir cru.2 recn axicn mulcl cru.4 recn axicn mulcl negsub eqtr4 syl5eq syl5bir imp (RR) eleq1d cru.3 cru.1 renegcl readdcl cru.2 cru.4 resubcl redivclz syl5bi ex pm2.43b (opr B (-) D) (0) df-ne syl5ibr mt3i cru.2 recn cru.4 recn subeq0 sylib)) thm (cru () ((cru.1 (e. A (RR))) (cru.2 (e. B (RR))) (cru.3 (e. C (RR))) (cru.4 (e. D (RR)))) (<-> (= (opr A (+) (opr B (x.) (i))) (opr C (+) (opr D (x.) (i)))) (/\ (= A C) (= B D))) (cru.1 cru.2 cru.3 cru.4 crulem B D (x.) (i) opreq1 C (+) opreq2d (opr A (+) (opr B (x.) (i))) eqeq2d cru.1 recn cru.2 recn axicn mulcl addcom cru.3 recn cru.2 recn axicn mulcl addcom eqeq12i cru.2 recn axicn mulcl cru.1 recn cru.3 recn addcan biimp sylbi syl6bir mpcom cru.1 cru.2 cru.3 cru.4 crulem jca (= A C) id B D (x.) (i) opreq1 (+) opreqan12d impbi)) thm (crut () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (e. C (RR)) (e. D (RR)))) (<-> (= (opr A (+) (opr B (x.) (i))) (opr C (+) (opr D (x.) (i)))) (/\ (= A C) (= B D)))) (A (if (e. A (RR)) A (0)) (+) (opr B (x.) (i)) opreq1 (opr C (+) (opr D (x.) (i))) eqeq1d A (if (e. A (RR)) A (0)) C eqeq1 (= B D) anbi1d bibi12d B (if (e. B (RR)) B (0)) (x.) (i) opreq1 (if (e. A (RR)) A (0)) (+) opreq2d (opr C (+) (opr D (x.) (i))) eqeq1d B (if (e. B (RR)) B (0)) D eqeq1 (= (if (e. A (RR)) A (0)) C) anbi2d bibi12d C (if (e. C (RR)) C (0)) (+) (opr D (x.) (i)) opreq1 (opr (if (e. A (RR)) A (0)) (+) (opr (if (e. B (RR)) B (0)) (x.) (i))) eqeq2d C (if (e. C (RR)) C (0)) (if (e. A (RR)) A (0)) eqeq2 (= (if (e. B (RR)) B (0)) D) anbi1d bibi12d D (if (e. D (RR)) D (0)) (x.) (i) opreq1 (if (e. C (RR)) C (0)) (+) opreq2d (opr (if (e. A (RR)) A (0)) (+) (opr (if (e. B (RR)) B (0)) (x.) (i))) eqeq2d D (if (e. D (RR)) D (0)) (if (e. B (RR)) B (0)) eqeq2 (= (if (e. A (RR)) A (0)) (if (e. C (RR)) C (0))) anbi2d bibi12d 0re A elimel 0re B elimel 0re C elimel 0re D elimel cru dedth4h)) thm (crne0 () ((crne0.1 (e. A (RR))) (crne0.2 (e. B (RR)))) (<-> (\/ (=/= A (0)) (=/= B (0))) (=/= (opr A (+) (opr B (x.) (i))) (0))) (A (0) df-ne B (0) df-ne orbi12i (opr A (+) (opr B (x.) (i))) (0) df-ne (= (opr A (+) (opr B (x.) (i))) (0)) id 0cn axicn mulcl addid2 axicn mul02 eqtr2 syl6eq crne0.1 crne0.2 0re 0re cru sylib (= A (0)) id B (0) (x.) (i) opreq1 (+) opreqan12d 0cn axicn mulcl addid2 axicn mul02 eqtr syl6eq impbi negbii (= A (0)) (= B (0)) ianor 3bitrr bitr)) thm (crmul () ((crmul.1 (e. A (CC))) (crmul.2 (e. B (CC))) (crmul.3 (e. C (CC))) (crmul.4 (e. D (CC)))) (= (opr (opr A (+) (opr B (x.) (i))) (x.) (opr C (+) (opr D (x.) (i)))) (opr (opr (opr A (x.) C) (-) (opr B (x.) D)) (+) (opr (opr (opr A (x.) D) (+) (opr B (x.) C)) (x.) (i)))) (crmul.1 crmul.2 axicn mulcl crmul.3 crmul.4 axicn mulcl addcl adddir crmul.1 crmul.3 mulcl crmul.1 crmul.4 mulcl axicn mulcl crmul.2 crmul.4 mulcl negcl crmul.2 crmul.3 mulcl axicn mulcl add4 crmul.1 crmul.3 crmul.4 axicn mulcl adddi crmul.1 crmul.4 axicn mulass (opr A (x.) C) (+) opreq2i eqtr4 crmul.2 axicn mulcl crmul.3 crmul.4 axicn mulcl adddi crmul.2 axicn crmul.3 mul23 crmul.2 axicn crmul.4 axicn mul4 itimesi (opr B (x.) D) (x.) opreq2i crmul.2 crmul.4 mulcl 1cn negcl mulcom 1cn crmul.2 crmul.4 mulcl mulneg1 crmul.2 crmul.4 mulcl mulid2 negeqi 3eqtr 3eqtr (+) opreq12i crmul.2 crmul.3 mulcl axicn mulcl crmul.2 crmul.4 mulcl negcl addcom 3eqtr (+) opreq12i crmul.1 crmul.4 mulcl crmul.2 crmul.3 mulcl axicn adddir (opr (opr A (x.) C) (+) (-u (opr B (x.) D))) (+) opreq2i 3eqtr4 crmul.1 crmul.3 mulcl crmul.2 crmul.4 mulcl negsub (+) (opr (opr (opr A (x.) D) (+) (opr B (x.) C)) (x.) (i)) opreq1i 3eqtr)) thm (crrecz () ((crrecz.1 (e. A (RR))) (crrecz.2 (e. B (RR)))) (-> (\/ (=/= A (0)) (=/= B (0))) (= (opr (1) (/) (opr A (+) (opr B (x.) (i)))) (opr (opr A (-) (opr B (x.) (i))) (/) (opr (opr A (^) (2)) (+) (opr B (^) (2)))))) (crrecz.1 crrecz.2 renegcl crne0 crrecz.2 recn negne0 (=/= A (0)) orbi2i crrecz.2 recn axicn mulneg1 A (+) opreq2i crrecz.1 recn crrecz.2 recn axicn mulcl negsub eqtr2 (0) neeq1i 3bitr4 crrecz.1 recn crrecz.2 recn axicn mulcl subcl (opr A (-) (opr B (x.) (i))) dividt mpan sylbi (opr (1) (/) (opr A (+) (opr B (x.) (i)))) (x.) opreq2d 1cn crrecz.1 recn crrecz.2 recn axicn mulcl addcl pm3.2i crrecz.1 recn crrecz.2 recn axicn mulcl subcl crrecz.1 recn crrecz.2 recn axicn mulcl subcl pm3.2i (1) (opr A (+) (opr B (x.) (i))) (opr A (-) (opr B (x.) (i))) (opr A (-) (opr B (x.) (i))) divmuldivt mpanl12 crrecz.1 crrecz.2 crne0 crrecz.1 crrecz.2 renegcl crne0 crrecz.2 recn negne0 (=/= A (0)) orbi2i crrecz.2 recn axicn mulneg1 A (+) opreq2i crrecz.1 recn crrecz.2 recn axicn mulcl negsub eqtr2 (0) neeq1i 3bitr4 sylancb crrecz.1 crrecz.2 crne0 crrecz.1 recn crrecz.2 recn axicn mulcl addcl recclz sylbi (opr (1) (/) (opr A (+) (opr B (x.) (i)))) ax1id syl 3eqtr3rd crrecz.1 recn crrecz.2 recn axicn mulcl subcl mulid2 crrecz.1 recn crrecz.2 recn axicn mulcl binom2a crrecz.1 recn sqcl crrecz.2 recn axicn mulcl sqcl negsub crrecz.2 recn axicn sqmul isqm1 (opr B (^) (2)) (x.) opreq2i crrecz.2 recn sqcl 1cn mulneg2 3eqtr negeqi crrecz.2 recn sqcl 1cn mulcl negneg crrecz.2 recn sqcl mulid1 3eqtr (opr A (^) (2)) (+) opreq2i 3eqtr2 (/) opreq12i syl6eq)) thm (creur ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (E!e. x (RR) (E.e. y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))))) (A x y axcnre (cv x) (cv y) (cv z) (cv w) crut (= (cv x) (cv z)) (= (cv y) (cv w)) pm3.26 syl6bi A (opr (cv x) (+) (opr (cv y) (x.) (i))) (opr (cv z) (+) (opr (cv w) (x.) (i))) eqtr2t syl5 an4s ex imp3a (e. (cv y) (RR)) (= A (opr (cv x) (+) (opr (cv y) (x.) (i)))) (e. (cv w) (RR)) (= A (opr (cv z) (+) (opr (cv w) (x.) (i)))) an4 syl5ib y w 19.23advv y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i)))) df-rex w (RR) (= A (opr (cv z) (+) (opr (cv w) (x.) (i)))) df-rex anbi12i y w (/\ (e. (cv y) (RR)) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))) (/\ (e. (cv w) (RR)) (= A (opr (cv z) (+) (opr (cv w) (x.) (i))))) eeanv bitr4 syl5ib rgen2 jctir (cv x) (cv z) (+) (opr (cv y) (x.) (i)) opreq1 A eqeq2d y (RR) rexbidv (cv y) (cv w) (x.) (i) opreq1 (cv z) (+) opreq2d A eqeq2d (RR) cbvrexv syl6bb (RR) reu4 sylibr)) thm (creui ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (E!e. y (RR) (E.e. x (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))))) (A x y axcnre x (RR) y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i)))) rexcom sylib (cv x) (cv y) (cv z) (cv w) crut (= (cv x) (cv z)) (= (cv y) (cv w)) pm3.27 syl6bi A (opr (cv x) (+) (opr (cv y) (x.) (i))) (opr (cv z) (+) (opr (cv w) (x.) (i))) eqtr2t syl5 an4s expcom imp3a (e. (cv x) (RR)) (= A (opr (cv x) (+) (opr (cv y) (x.) (i)))) (e. (cv z) (RR)) (= A (opr (cv z) (+) (opr (cv w) (x.) (i)))) an4 syl5ib x z 19.23advv x (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i)))) df-rex z (RR) (= A (opr (cv z) (+) (opr (cv w) (x.) (i)))) df-rex anbi12i x z (/\ (e. (cv x) (RR)) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))) (/\ (e. (cv z) (RR)) (= A (opr (cv z) (+) (opr (cv w) (x.) (i))))) eeanv bitr4 syl5ib rgen2 jctir (cv y) (cv w) (x.) (i) opreq1 (cv x) (+) opreq2d A eqeq2d x (RR) rexbidv (cv x) (cv z) (+) (opr (cv w) (x.) (i)) opreq1 A eqeq2d (RR) cbvrexv syl6bb (RR) reu4 sylibr)) thm (rimul () () (-> (/\ (e. A (RR)) (e. (opr A (x.) (i)) (RR))) (= A (0))) ((opr A (x.) (i)) recnt 0cn axicn mulcl (opr (0) (x.) (i)) (opr A (x.) (i)) axaddcom mpan axicn mul02 (+) (opr A (x.) (i)) opreq1i syl5eqr syl (e. A (RR)) adantl 0re 0re (0) A (opr A (x.) (i)) (0) crut mpanr2 mpanl1 mpbid pm3.27d)) thm (nthruc () () (/\ (/\ (C: (NN) (ZZ)) (C: (ZZ) (QQ))) (/\ (C: (QQ) (RR)) (C: (RR) (CC)))) (nnssz 0z 0nnn pm3.2i (NN) (ZZ) (0) ssnelpss mp2 zssq 1z 2nn (1) (2) znq mp2an halfnz pm3.2i (ZZ) (QQ) (opr (1) (/) (2)) ssnelpss mp2 pm3.2i qssre sqr2re sqr2irr (` (sqr) (2)) (QQ) df-nel mpbi pm3.2i (QQ) (RR) (` (sqr) (2)) ssnelpss mp2 axresscn axicn inelr pm3.2i (RR) (CC) (i) ssnelpss mp2 pm3.2i pm3.2i)) thm (nthruz () () (/\ (C: (NN) (NN0)) (C: (NN0) (ZZ))) (nnssnn0 0nn0 0nnn pm3.2i (NN) (NN0) (0) ssnelpss mp2 nn0ssz 1nn (1) nnnegz ax-mp ax1re renegcl neg0 lt01 eqbrtr ax1re 0re ltnegcon1 mpbir (-u (1)) lt0nnn0 mp2an pm3.2i (NN0) (ZZ) (-u (1)) ssnelpss mp2 pm3.2i)) thm (revalt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (= (` (Re) A) (U. ({e.|} x (RR) (E.e. y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))))))) ((cv z) A (opr (cv x) (+) (opr (cv y) (x.) (i))) eqeq1 y (RR) rexbidv x (RR) rabbisdv unieqd z w x y df-re reex x (E.e. y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))) rabex uniex fvopab4)) thm (imvalt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (= (` (Im) A) (U. ({e.|} y (RR) (E.e. x (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))))))) ((cv z) A (opr (cv x) (+) (opr (cv y) (x.) (i))) eqeq1 x (RR) rexbidv y (RR) rabbisdv unieqd z w y x df-im reex y (E.e. x (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))) rabex uniex fvopab4)) thm (reclt ((x y) (A x) (A y)) () (-> (e. A (CC)) (e. (` (Re) A) (RR))) (A x y revalt A x y creur x (RR) (E.e. y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))) reucl syl eqeltrd)) thm (imclt ((x y) (A x) (A y)) () (-> (e. A (CC)) (e. (` (Im) A) (RR))) (A y x imvalt A y x creui y (RR) (E.e. x (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))) reucl syl eqeltrd)) thm (replimt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= A (opr (` (Re) A) (+) (opr (` (Im) A) (x.) (i))))) ((` (Im) A) eqid A imclt (cv y) (` (Im) A) (x.) (i) opreq1 (cv x) (+) opreq2d A eqeq2d x (RR) rexbidv (cv y) (` (Im) A) (` (Im) A) eqeq2 bibi12d (e. A (CC)) imbi2d y (RR) (E.e. x (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))) reuuni1 A y x creui sylan2 A y x imvalt (cv y) eqeq1d (e. (cv y) (RR)) adantl bitr4d ex vtoclga mpcom mpbiri x (RR) (= A (opr (cv x) (+) (opr (` (Im) A) (x.) (i)))) df-rex sylib (` (Re) A) eqid A reclt (cv x) (` (Re) A) (+) (opr (cv y) (x.) (i)) opreq1 A eqeq2d y (RR) rexbidv (cv x) (` (Re) A) (` (Re) A) eqeq2 bibi12d (e. A (CC)) imbi2d x (RR) (E.e. y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))) reuuni1 A x y creur sylan2 A x y revalt (cv x) eqeq1d (e. (cv x) (RR)) adantl bitr4d ex vtoclga mpcom mpbiri y (RR) (= A (opr (` (Re) A) (+) (opr (cv y) (x.) (i)))) df-rex sylib jca (e. (cv x) (RR)) (e. (cv y) (RR)) (= A (opr (cv x) (+) (opr (` (Im) A) (x.) (i)))) (= A (opr (` (Re) A) (+) (opr (cv y) (x.) (i)))) an4 x y 2exbii x y (/\ (e. (cv x) (RR)) (= A (opr (cv x) (+) (opr (` (Im) A) (x.) (i))))) (/\ (e. (cv y) (RR)) (= A (opr (` (Re) A) (+) (opr (cv y) (x.) (i))))) eeanv bitr sylibr (= A (opr (cv x) (+) (opr (` (Im) A) (x.) (i)))) (= A (opr (` (Re) A) (+) (opr (cv y) (x.) (i)))) pm3.26 (e. A (CC)) (/\ (e. (cv x) (RR)) (e. (cv y) (RR))) ad2antll A imclt (e. (cv x) (RR)) a1d ancld A reclt (e. (cv y) (RR)) a1d ancrd anim12d imp (cv x) (` (Im) A) (` (Re) A) (cv y) crut syl A (opr (cv x) (+) (opr (` (Im) A) (x.) (i))) (opr (` (Re) A) (+) (opr (cv y) (x.) (i))) eqtr2t syl5bi ex imp32 pm3.26d (+) (opr (` (Im) A) (x.) (i)) opreq1d eqtrd ex x y 19.23advv mpd)) thm (cjvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (*) A) (opr (` (Re) A) (-) (opr (` (Im) A) (x.) (i))))) ((cv x) A (Re) fveq2 (cv x) A (Im) fveq2 (x.) (i) opreq1d (-) opreq12d x y df-cj (` (Re) A) (-) (opr (` (Im) A) (x.) (i)) oprex fvopab4)) thm (absvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (abs) A) (` (sqr) (opr A (x.) (` (*) A))))) ((= (cv x) A) id (cv x) A (*) fveq2 (x.) opreq12d (sqr) fveq2d x y df-abs (sqr) (opr A (x.) (` (*) A)) fvex fvopab4)) thm (cjclt () () (-> (e. A (CC)) (e. (` (*) A) (CC))) (A cjvalt (` (Re) A) (opr (` (Im) A) (x.) (i)) subclt A reclt recnd (` (Im) A) (i) axmulcl A imclt recnd axicn (e. A (CC)) a1i sylanc sylanc eqeltrd)) thm (recl () ((recl.1 (e. A (CC)))) (e. (` (Re) A) (RR)) (recl.1 A reclt ax-mp)) thm (imcl () ((recl.1 (e. A (CC)))) (e. (` (Im) A) (RR)) (recl.1 A imclt ax-mp)) thm (cjcl () ((recl.1 (e. A (CC)))) (e. (` (*) A) (CC)) (recl.1 A cjclt ax-mp)) thm (replim () ((recl.1 (e. A (CC)))) (= A (opr (` (Re) A) (+) (opr (` (Im) A) (x.) (i)))) (recl.1 A replimt ax-mp)) thm (crret () () (-> (/\ (e. A (RR)) (e. B (RR))) (= (` (Re) (opr A (+) (opr B (x.) (i)))) A)) (A (opr B (x.) (i)) axaddcl A recnt B recnt axicn B (i) axmulcl mpan2 syl syl2an (opr A (+) (opr B (x.) (i))) replimt syl eqcomd A (opr B (x.) (i)) axaddcl A recnt B recnt axicn B (i) axmulcl mpan2 syl syl2an (opr A (+) (opr B (x.) (i))) reclt syl A (opr B (x.) (i)) axaddcl A recnt B recnt axicn B (i) axmulcl mpan2 syl syl2an (opr A (+) (opr B (x.) (i))) imclt syl jca (` (Re) (opr A (+) (opr B (x.) (i)))) (` (Im) (opr A (+) (opr B (x.) (i)))) A B crut mpancom mpbid pm3.26d)) thm (crimt () () (-> (/\ (e. A (RR)) (e. B (RR))) (= (` (Im) (opr A (+) (opr B (x.) (i)))) B)) (A (opr B (x.) (i)) axaddcl A recnt B recnt axicn B (i) axmulcl mpan2 syl syl2an (opr A (+) (opr B (x.) (i))) replimt syl eqcomd A (opr B (x.) (i)) axaddcl A recnt B recnt axicn B (i) axmulcl mpan2 syl syl2an (opr A (+) (opr B (x.) (i))) reclt syl A (opr B (x.) (i)) axaddcl A recnt B recnt axicn B (i) axmulcl mpan2 syl syl2an (opr A (+) (opr B (x.) (i))) imclt syl jca (` (Re) (opr A (+) (opr B (x.) (i)))) (` (Im) (opr A (+) (opr B (x.) (i)))) A B crut mpancom mpbid pm3.27d)) thm (crre () ((crre.1 (e. A (RR))) (crre.2 (e. B (RR)))) (= (` (Re) (opr A (+) (opr B (x.) (i)))) A) (crre.1 crre.2 A B crret mp2an)) thm (crim () ((crre.1 (e. A (RR))) (crre.2 (e. B (RR)))) (= (` (Im) (opr A (+) (opr B (x.) (i)))) B) (crre.1 crre.2 A B crimt mp2an)) thm (imret () () (-> (e. A (CC)) (= (` (Im) A) (` (Re) (opr (-u (i)) (x.) A)))) (A replimt (-u (i)) (x.) opreq2d axicn negcl (-u (i)) (` (Re) A) (opr (` (Im) A) (x.) (i)) axdistr mp3an1 A reclt recnd A imclt recnd axicn (` (Im) A) (i) axmulcl mpan2 syl sylanc A reclt recnd axicn negcl (-u (i)) (` (Re) A) axmulcom mpan syl A reclt recnd axicn (` (Re) A) (i) mulneg12t mpan2 syl eqtr4d A imclt recnd axicn negcl axicn (-u (i)) (` (Im) A) (i) mul12t mp3an13 syl A imclt recnd (` (Im) A) ax1id syl axicn axicn mulneg1 itimesi negeqi 1cn negneg 3eqtr (` (Im) A) (x.) opreq2i syl5eq eqtrd (+) opreq12d (opr (-u (` (Re) A)) (x.) (i)) (` (Im) A) axaddcom A reclt (` (Re) A) renegclt syl recnd axicn (-u (` (Re) A)) (i) axmulcl mpan2 syl A imclt recnd sylanc eqtrd 3eqtrd (Re) fveq2d (` (Im) A) (-u (` (Re) A)) crret A imclt A reclt (` (Re) A) renegclt syl sylanc eqtr2d)) thm (reim0t () () (-> (e. A (RR)) (= (` (Im) A) (0))) ((` (Im) A) rimul A recnt A imclt syl A recnt A replimt syl eqcomd A (` (Re) A) (opr (` (Im) A) (x.) (i)) subaddt A recnt A recnt A reclt syl (` (Re) A) recnt syl A recnt A imclt syl (` (Im) A) recnt syl axicn (` (Im) A) (i) axmulcl mpan2 syl syl3anc mpbird A recnt A reclt syl A (` (Re) A) resubclt mpdan eqeltrrd sylanc)) thm (reim0bt () () (-> (e. A (CC)) (<-> (e. A (RR)) (= (` (Im) A) (0)))) (A reim0t (e. A (CC)) a1i A replimt (= (` (Im) A) (0)) adantr (` (Im) A) (0) (x.) (i) opreq1 axicn mul02 syl6eq (` (Re) A) (+) opreq2d A reclt recnd (` (Re) A) ax0id syl sylan9eqr eqtrd A reclt (= (` (Im) A) (0)) adantr eqeltrd ex impbid)) thm (cjcj () ((cjcj.1 (e. A (CC)))) (= (` (*) (` (*) A)) A) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negsub cjcj.1 imcl recn axicn mulneg1 (` (Re) A) (+) opreq2i cjcj.1 A cjvalt ax-mp 3eqtr4r (Re) fveq2i cjcj.1 recl cjcj.1 imcl renegcl crre eqtr cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negsub cjcj.1 imcl recn axicn mulneg1 (` (Re) A) (+) opreq2i cjcj.1 A cjvalt ax-mp 3eqtr4r (Im) fveq2i (x.) (i) opreq1i cjcj.1 recl cjcj.1 imcl renegcl crim (x.) (i) opreq1i cjcj.1 imcl recn axicn mulneg1 3eqtr (-) opreq12i cjcj.1 cjcl (` (*) A) cjvalt ax-mp cjcj.1 replim cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl subneg eqtr4 3eqtr4)) thm (reim0b () ((cjcj.1 (e. A (CC)))) (<-> (e. A (RR)) (= (` (Im) A) (0))) (cjcj.1 A reim0bt ax-mp)) thm (rereb () ((cjcj.1 (e. A (CC)))) (<-> (e. A (RR)) (= (` (Re) A) A)) (cjcj.1 reim0b (` (Im) A) (0) (x.) (i) opreq1 sylbi axicn mul02 syl6eq (` (Re) A) (+) opreq2d cjcj.1 replim syl5req cjcj.1 recl recn addid1 syl5eqr cjcj.1 recl (` (Re) A) A (RR) eleq1 mpbii impbi)) thm (cjreb () ((cjcj.1 (e. A (CC)))) (<-> (e. A (RR)) (= (` (*) A) A)) (cjcj.1 reim0b cjcj.1 imcl recn eqneg axicn cjcj.1 imcl recn mulcom axicn cjcj.1 imcl recn negcl mulcom cjcj.1 imcl recn axicn mulneg1 eqtr eqeq12i axicn cjcj.1 imcl recn cjcj.1 imcl recn negcl ine0 mulcan cjcj.1 replim cjcj.1 A cjvalt ax-mp cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negsub eqtr4 eqeq12i cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl cjcj.1 imcl recn axicn mulcl negcl addcan bitr2 3bitr3 bitr3 A (` (*) A) eqcom 3bitr)) thm (recj () ((cjcj.1 (e. A (CC)))) (= (` (Re) (` (*) A)) (` (Re) A)) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negsub cjcj.1 imcl recn axicn mulneg1 (` (Re) A) (+) opreq2i cjcj.1 A cjvalt ax-mp 3eqtr4r (Re) fveq2i cjcj.1 recl cjcj.1 imcl renegcl crre eqtr)) thm (imcj () ((cjcj.1 (e. A (CC)))) (= (` (Im) (` (*) A)) (-u (` (Im) A))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negsub cjcj.1 imcl recn axicn mulneg1 (` (Re) A) (+) opreq2i cjcj.1 A cjvalt ax-mp 3eqtr4r (Im) fveq2i cjcj.1 recl cjcj.1 imcl renegcl crim eqtr)) thm (readd () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (Re) (opr A (+) B)) (opr (` (Re) A) (+) (` (Re) B))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl readd.2 recl recn readd.2 imcl recn axicn mulcl add4 cjcj.1 replim readd.2 replim (+) opreq12i cjcj.1 imcl recn readd.2 imcl recn axicn adddir (opr (` (Re) A) (+) (` (Re) B)) (+) opreq2i 3eqtr4 (Re) fveq2i cjcj.1 recl readd.2 recl readdcl cjcj.1 imcl readd.2 imcl readdcl crre eqtr)) thm (imadd () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (Im) (opr A (+) B)) (opr (` (Im) A) (+) (` (Im) B))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl readd.2 recl recn readd.2 imcl recn axicn mulcl add4 cjcj.1 replim readd.2 replim (+) opreq12i cjcj.1 imcl recn readd.2 imcl recn axicn adddir (opr (` (Re) A) (+) (` (Re) B)) (+) opreq2i 3eqtr4 (Im) fveq2i cjcj.1 recl readd.2 recl readdcl cjcj.1 imcl readd.2 imcl readdcl crim eqtr)) thm (remul () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (Re) (opr A (x.) B)) (opr (opr (` (Re) A) (x.) (` (Re) B)) (-) (opr (` (Im) A) (x.) (` (Im) B)))) (cjcj.1 replim readd.2 replim (x.) opreq12i cjcj.1 recl recn cjcj.1 imcl recn readd.2 recl recn readd.2 imcl recn crmul eqtr (Re) fveq2i cjcj.1 recl readd.2 recl remulcl cjcj.1 imcl readd.2 imcl remulcl resubcl cjcj.1 recl readd.2 imcl remulcl cjcj.1 imcl readd.2 recl remulcl readdcl crre eqtr)) thm (immul () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (Im) (opr A (x.) B)) (opr (opr (` (Re) A) (x.) (` (Im) B)) (+) (opr (` (Im) A) (x.) (` (Re) B)))) (cjcj.1 replim readd.2 replim (x.) opreq12i cjcj.1 recl recn cjcj.1 imcl recn readd.2 recl recn readd.2 imcl recn crmul eqtr (Im) fveq2i cjcj.1 recl readd.2 recl remulcl cjcj.1 imcl readd.2 imcl remulcl resubcl cjcj.1 recl readd.2 imcl remulcl cjcj.1 imcl readd.2 recl remulcl readdcl crim eqtr)) thm (cjadd () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (*) (opr A (+) B)) (opr (` (*) A) (+) (` (*) B))) (cjcj.1 readd.2 readd cjcj.1 readd.2 imadd (x.) (i) opreq1i cjcj.1 imcl recn readd.2 imcl recn axicn adddir eqtr (-) opreq12i cjcj.1 recl recn readd.2 recl recn cjcj.1 imcl recn axicn mulcl readd.2 imcl recn axicn mulcl sub4 eqtr cjcj.1 readd.2 addcl (opr A (+) B) cjvalt ax-mp cjcj.1 A cjvalt ax-mp readd.2 B cjvalt ax-mp (+) opreq12i 3eqtr4)) thm (cjmul () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (*) (opr A (x.) B)) (opr (` (*) A) (x.) (` (*) B))) (cjcj.1 recl readd.2 recl remulcl cjcj.1 imcl readd.2 imcl remulcl resubcl recn cjcj.1 recl readd.2 imcl remulcl cjcj.1 imcl readd.2 recl remulcl readdcl recn axicn mulcl negsub cjcj.1 imcl recn readd.2 imcl recn mul2neg eqcomi (opr (` (Re) A) (x.) (` (Re) B)) (-) opreq2i cjcj.1 recl readd.2 imcl remulcl cjcj.1 imcl readd.2 recl remulcl readdcl recn axicn mulneg1 cjcj.1 recl readd.2 imcl remulcl recn cjcj.1 imcl readd.2 recl remulcl recn negdi cjcj.1 recl recn readd.2 imcl recn mulneg2 cjcj.1 imcl recn readd.2 recl recn mulneg1 (+) opreq12i eqtr4 (x.) (i) opreq1i eqtr3 (+) opreq12i eqtr3 cjcj.1 readd.2 remul cjcj.1 readd.2 immul (x.) (i) opreq1i (-) opreq12i cjcj.1 recl recn cjcj.1 imcl recn negcl readd.2 recl recn readd.2 imcl recn negcl crmul 3eqtr4 cjcj.1 readd.2 mulcl (opr A (x.) B) cjvalt ax-mp cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negsub cjcj.1 imcl recn axicn mulneg1 (` (Re) A) (+) opreq2i cjcj.1 A cjvalt ax-mp 3eqtr4r readd.2 recl recn readd.2 imcl recn axicn mulcl negsub readd.2 imcl recn axicn mulneg1 (` (Re) B) (+) opreq2i readd.2 B cjvalt ax-mp 3eqtr4r (x.) opreq12i 3eqtr4)) thm (ipcn () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (Re) (opr A (x.) (` (*) B))) (opr (opr (` (Re) A) (x.) (` (Re) B)) (+) (opr (` (Im) A) (x.) (` (Im) B)))) (cjcj.1 readd.2 cjcl remul cjcj.1 recl readd.2 cjcl recl remulcl recn cjcj.1 imcl readd.2 cjcl imcl remulcl recn negsub readd.2 recj (` (Re) A) (x.) opreq2i readd.2 imcj readd.2 cjcl imcl recn readd.2 imcl recn negcon2 mpbi (` (Im) A) (x.) opreq2i cjcj.1 imcl recn readd.2 cjcl imcl recn mulneg2 eqtr2 (+) opreq12i 3eqtr2)) thm (cjmulrcl () ((cjcj.1 (e. A (CC)))) (e. (opr A (x.) (` (*) A)) (RR)) (cjcj.1 cjcj (` (*) A) (x.) opreq2i cjcj.1 cjcj.1 cjcl cjmul cjcj.1 cjcj.1 cjcl mulcom 3eqtr4 cjcj.1 cjcj.1 cjcl mulcl cjreb mpbir)) thm (cjmulval () ((cjcj.1 (e. A (CC)))) (= (opr A (x.) (` (*) A)) (opr (opr (` (Re) A) (^) (2)) (+) (opr (` (Im) A) (^) (2)))) (cjcj.1 recl recn sqval cjcj.1 imcl recn sqval (+) opreq12i cjcj.1 cjcj.1 ipcn cjcj.1 cjmulrcl cjcj.1 cjmulrcl recn rereb mpbi 3eqtr2r)) thm (cjmulge0 () ((cjcj.1 (e. A (CC)))) (br (0) (<_) (opr A (x.) (` (*) A))) (cjcj.1 recl sqge0 cjcj.1 imcl sqge0 cjcj.1 recl resqcl cjcj.1 imcl resqcl addge0 mp2an cjcj.1 cjmulval breqtrr)) thm (reneg () ((cjcj.1 (e. A (CC)))) (= (` (Re) (-u A)) (-u (` (Re) A))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negdi cjcj.1 replim negeqi cjcj.1 imcl recn axicn mulneg1 (-u (` (Re) A)) (+) opreq2i 3eqtr4 (Re) fveq2i cjcj.1 recl renegcl cjcj.1 imcl renegcl crre eqtr)) thm (negreb () ((cjcj.1 (e. A (CC)))) (<-> (e. (-u A) (RR)) (e. A (RR))) (cjcj.1 negcl rereb cjcj.1 reneg (-u A) eqeq1i cjcj.1 recl recn cjcj.1 neg11 3bitr cjcj.1 recl (` (Re) A) A (RR) eleq1 mpbii sylbi A renegclt impbi)) thm (imneg () ((cjcj.1 (e. A (CC)))) (= (` (Im) (-u A)) (-u (` (Im) A))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negdi cjcj.1 replim negeqi cjcj.1 imcl recn axicn mulneg1 (-u (` (Re) A)) (+) opreq2i 3eqtr4 (Im) fveq2i cjcj.1 recl renegcl cjcj.1 imcl renegcl crim eqtr)) thm (cjneg () ((cjcj.1 (e. A (CC)))) (= (` (*) (-u A)) (-u (` (*) A))) (cjcj.1 recl recn negcl cjcj.1 imcl recn axicn mulcl negcl negsub cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negcl negdi cjcj.1 reneg cjcj.1 imneg (x.) (i) opreq1i cjcj.1 imcl recn axicn mulneg1 eqtr (-) opreq12i 3eqtr4r cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negsub negeqi eqtr cjcj.1 negcl (-u A) cjvalt ax-mp cjcj.1 A cjvalt ax-mp negeqi 3eqtr4)) thm (addcj () ((cjcj.1 (e. A (CC)))) (= (opr A (+) (` (*) A)) (opr (2) (x.) (` (Re) A))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negcl add4 cjcj.1 imcl recn axicn mulcl negid (opr (` (Re) A) (+) (` (Re) A)) (+) opreq2i cjcj.1 recl recn cjcj.1 recl recn addcl addid1 3eqtr cjcj.1 replim cjcj.1 A cjvalt ax-mp cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negsub eqtr4 (+) opreq12i cjcj.1 recl recn 2times 3eqtr4)) thm (cjrebt () () (-> (e. A (CC)) (<-> (e. A (RR)) (= (` (*) A) A))) (A (if (e. A (CC)) A (0)) (RR) eleq1 A (if (e. A (CC)) A (0)) (*) fveq2 (= A (if (e. A (CC)) A (0))) id eqeq12d bibi12d 0cn A elimel cjreb dedth)) thm (cjmulrclt () () (-> (e. A (CC)) (e. (opr A (x.) (` (*) A)) (RR))) ((= A (if (e. A (CC)) A (0))) id A (if (e. A (CC)) A (0)) (*) fveq2 (x.) opreq12d (RR) eleq1d 0cn A elimel cjmulrcl dedth)) thm (cjmulge0t () () (-> (e. A (CC)) (br (0) (<_) (opr A (x.) (` (*) A)))) ((= A (if (e. A (CC)) A (0))) id A (if (e. A (CC)) A (0)) (*) fveq2 (x.) opreq12d (0) (<_) breq2d 0cn A elimel cjmulge0 dedth)) thm (renegt () () (-> (e. A (CC)) (= (` (Re) (-u A)) (-u (` (Re) A)))) (A (if (e. A (CC)) A (0)) negeq (Re) fveq2d A (if (e. A (CC)) A (0)) (Re) fveq2 negeqd eqeq12d 0cn A elimel reneg dedth)) thm (readdt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (Re) (opr A (+) B)) (opr (` (Re) A) (+) (` (Re) B)))) (A (if (e. A (CC)) A (0)) (+) B opreq1 (Re) fveq2d A (if (e. A (CC)) A (0)) (Re) fveq2 (+) (` (Re) B) opreq1d eqeq12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (+) opreq2 (Re) fveq2d B (if (e. B (CC)) B (0)) (Re) fveq2 (` (Re) (if (e. A (CC)) A (0))) (+) opreq2d eqeq12d 0cn A elimel 0cn B elimel readd dedth2h)) thm (resubt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (Re) (opr A (-) B)) (opr (` (Re) A) (-) (` (Re) B)))) (A (-u B) readdt B negclt sylan2 B renegt (e. A (CC)) adantl (` (Re) A) (+) opreq2d eqtrd A B negsubt (Re) fveq2d (` (Re) A) (` (Re) B) negsubt A reclt recnd B reclt recnd syl2an 3eqtr3d)) thm (imnegt () () (-> (e. A (CC)) (= (` (Im) (-u A)) (-u (` (Im) A)))) (A (if (e. A (CC)) A (0)) negeq (Im) fveq2d A (if (e. A (CC)) A (0)) (Im) fveq2 negeqd eqeq12d 0cn A elimel imneg dedth)) thm (imaddt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (Im) (opr A (+) B)) (opr (` (Im) A) (+) (` (Im) B)))) (A (if (e. A (CC)) A (0)) (+) B opreq1 (Im) fveq2d A (if (e. A (CC)) A (0)) (Im) fveq2 (+) (` (Im) B) opreq1d eqeq12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (+) opreq2 (Im) fveq2d B (if (e. B (CC)) B (0)) (Im) fveq2 (` (Im) (if (e. A (CC)) A (0))) (+) opreq2d eqeq12d 0cn A elimel 0cn B elimel imadd dedth2h)) thm (imsubt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (Im) (opr A (-) B)) (opr (` (Im) A) (-) (` (Im) B)))) (A (-u B) imaddt B negclt sylan2 B imnegt (e. A (CC)) adantl (` (Im) A) (+) opreq2d eqtrd A B negsubt (Im) fveq2d (` (Im) A) (` (Im) B) negsubt A imclt recnd B imclt recnd syl2an 3eqtr3d)) thm (cjret () () (-> (e. A (RR)) (= (` (*) A) A)) (A recnt A cjrebt biimpd mpcom)) thm (cjcjt () () (-> (e. A (CC)) (= (` (*) (` (*) A)) A)) (A (if (e. A (CC)) A (0)) (*) fveq2 (*) fveq2d (= A (if (e. A (CC)) A (0))) id eqeq12d 0cn A elimel cjcj dedth)) thm (cjaddt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (*) (opr A (+) B)) (opr (` (*) A) (+) (` (*) B)))) (A (if (e. A (CC)) A (0)) (+) B opreq1 (*) fveq2d A (if (e. A (CC)) A (0)) (*) fveq2 (+) (` (*) B) opreq1d eqeq12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (+) opreq2 (*) fveq2d B (if (e. B (CC)) B (0)) (*) fveq2 (` (*) (if (e. A (CC)) A (0))) (+) opreq2d eqeq12d 0cn A elimel 0cn B elimel cjadd dedth2h)) thm (cjmult () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (*) (opr A (x.) B)) (opr (` (*) A) (x.) (` (*) B)))) (A (if (e. A (CC)) A (0)) (x.) B opreq1 (*) fveq2d A (if (e. A (CC)) A (0)) (*) fveq2 (x.) (` (*) B) opreq1d eqeq12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (x.) opreq2 (*) fveq2d B (if (e. B (CC)) B (0)) (*) fveq2 (` (*) (if (e. A (CC)) A (0))) (x.) opreq2d eqeq12d 0cn A elimel 0cn B elimel cjmul dedth2h)) thm (cjnegt () () (-> (e. A (CC)) (= (` (*) (-u A)) (-u (` (*) A)))) (A (if (e. A (CC)) A (0)) negeq (*) fveq2d A (if (e. A (CC)) A (0)) (*) fveq2 negeqd eqeq12d 0cn A elimel cjneg dedth)) thm (cjsubt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (*) (opr A (-) B)) (opr (` (*) A) (-) (` (*) B)))) (A (-u B) cjaddt B negclt sylan2 A B negsubt (*) fveq2d B cjnegt (e. A (CC)) adantl (` (*) A) (+) opreq2d (` (*) A) (` (*) B) negsubt A cjclt B cjclt syl2an eqtrd 3eqtr3d)) thm (cjexpt ((j k) (A j) (A k) (N j) (N k)) () (-> (/\ (e. A (CC)) (e. N (NN0))) (= (` (*) (opr A (^) N)) (opr (` (*) A) (^) N))) ((cv j) (0) A (^) opreq2 (*) fveq2d (cv j) (0) (` (*) A) (^) opreq2 eqeq12d (e. A (CC)) imbi2d (cv j) (cv k) A (^) opreq2 (*) fveq2d (cv j) (cv k) (` (*) A) (^) opreq2 eqeq12d (e. A (CC)) imbi2d (cv j) (opr (cv k) (+) (1)) A (^) opreq2 (*) fveq2d (cv j) (opr (cv k) (+) (1)) (` (*) A) (^) opreq2 eqeq12d (e. A (CC)) imbi2d (cv j) N A (^) opreq2 (*) fveq2d (cv j) N (` (*) A) (^) opreq2 eqeq12d (e. A (CC)) imbi2d A exp0t (*) fveq2d A cjclt (` (*) A) exp0t ax1re (1) cjret ax-mp syl6eqr syl eqtr4d A (cv k) expp1t (*) fveq2d (opr A (^) (cv k)) A cjmult A (cv k) expclt (e. A (CC)) (e. (cv k) (NN0)) pm3.26 sylanc eqtrd (= (` (*) (opr A (^) (cv k))) (opr (` (*) A) (^) (cv k))) adantr (` (*) (opr A (^) (cv k))) (opr (` (*) A) (^) (cv k)) (x.) (` (*) A) opreq1 (` (*) A) (cv k) expp1t A cjclt sylan eqcomd sylan9eqr eqtrd exp31 com12 a2d nn0ind impcom)) thm (recjt () () (-> (e. A (CC)) (= (` (Re) A) (opr (opr A (+) (` (*) A)) (/) (2)))) (2cn 2re 2pos gt0ne0i (` (Re) A) (` (Re) A) (2) divdirt mpan2 mp3an3 A reclt recnd A reclt recnd sylanc A replimt A cjvalt (+) opreq12d (` (Re) A) (opr (` (Im) A) (x.) (i)) negsubt A reclt recnd A imclt recnd axicn (` (Im) A) (i) axmulcl mpan2 syl sylanc (opr (` (Re) A) (+) (opr (` (Im) A) (x.) (i))) (+) opreq2d A imclt recnd axicn (` (Im) A) (i) axmulcl mpan2 (opr (` (Im) A) (x.) (i)) negidt 3syl (opr (` (Re) A) (+) (` (Re) A)) (+) opreq2d (` (Re) A) (` (Re) A) (opr (` (Im) A) (x.) (i)) (-u (opr (` (Im) A) (x.) (i))) add4t A reclt recnd A reclt recnd jca A imclt recnd axicn (` (Im) A) (i) axmulcl mpan2 syl A imclt recnd axicn (` (Im) A) (i) axmulcl mpan2 (opr (` (Im) A) (x.) (i)) negclt 3syl jca sylanc A reclt recnd A reclt recnd jca (` (Re) A) (` (Re) A) axaddcl (opr (` (Re) A) (+) (` (Re) A)) ax0id 3syl 3eqtr3d 3eqtr2rd (/) (2) opreq1d A reclt recnd (` (Re) A) 2halvest syl 3eqtr3rd)) thm (imcjt () () (-> (e. A (CC)) (= (` (Im) A) (opr (opr A (-) (` (*) A)) (/) (opr (2) (x.) (i))))) (A imclt recnd 2cn axicn mulcl 2cn axicn 2re 2pos gt0ne0i ine0 muln0 (opr (2) (x.) (i)) (` (Im) A) divcan4t mp3an13 syl (opr (` (Re) A) (+) (opr (` (Im) A) (x.) (i))) (` (Re) A) (opr (` (Im) A) (x.) (i)) subsubt (` (Re) A) (opr (` (Im) A) (x.) (i)) axaddcl A reclt recnd A imclt recnd axicn (` (Im) A) (i) axmulcl mpan2 syl sylanc A reclt recnd A imclt recnd axicn (` (Im) A) (i) axmulcl mpan2 syl syl3anc A replimt A cjvalt (-) opreq12d A imclt recnd axicn (` (Im) A) (i) axmulcl mpan2 (opr (` (Im) A) (x.) (i)) 2timest 3syl A imclt recnd 2cn axicn (` (Im) A) (2) (i) mul12t mp3an23 syl (` (Re) A) (opr (` (Im) A) (x.) (i)) pncan2t A reclt recnd A imclt recnd axicn (` (Im) A) (i) axmulcl mpan2 syl sylanc (+) (opr (` (Im) A) (x.) (i)) opreq1d 3eqtr4d 3eqtr4rd (/) (opr (2) (x.) (i)) opreq1d eqtr3d)) thm (re0 () () (= (` (Re) (0)) (0)) (0re 0cn rereb mpbi)) thm (im0 () () (= (` (Im) (0)) (0)) (0re 0cn reim0b mpbi)) thm (re1 () () (= (` (Re) (1)) (1)) (ax1re 1cn rereb mpbi)) thm (im1 () () (= (` (Im) (1)) (0)) (ax1re 1cn reim0b mpbi)) thm (rei () () (= (` (Re) (i)) (0)) (1cn axicn mulcl addid2 (Re) fveq2i 0re ax1re crre axicn mulid2 (Re) fveq2i 3eqtr3r)) thm (imi () () (= (` (Im) (i)) (1)) (1cn axicn mulcl addid2 eqcomi (Im) fveq2i axicn mulid2 (Im) fveq2i 0re ax1re crim 3eqtr3)) thm (cj0 () () (= (` (*) (0)) (0)) (0re 0cn cjreb mpbi)) thm (cji () () (= (` (*) (i)) (-u (i))) (rei imi (x.) (i) opreq1i axicn mulid2 eqtr (-) opreq12i axicn (i) cjvalt ax-mp (i) df-neg 3eqtr4)) thm (cjreimt () () (-> (/\ (e. A (RR)) (e. B (RR))) (= (` (*) (opr A (+) (opr (i) (x.) B))) (opr A (-) (opr (i) (x.) B)))) (A (opr (i) (x.) B) cjaddt A recnt B recnt axicn (i) B axmulcl mpan syl syl2an A cjret B recnt axicn (i) B cjmult mpan syl cji (e. B (RR)) a1i B cjret (x.) opreq12d B recnt axicn (i) B mulneg1t mpan syl 3eqtrd (+) opreqan12d A (opr (i) (x.) B) negsubt A recnt B recnt axicn (i) B axmulcl mpan syl syl2an 3eqtrd)) thm (cjreim2t () () (-> (/\ (e. A (RR)) (e. B (RR))) (= (` (*) (opr A (-) (opr (i) (x.) B))) (opr A (+) (opr (i) (x.) B)))) (A (opr (i) (x.) B) cjsubt A recnt B recnt axicn (i) B axmulcl mpan syl syl2an A cjret B recnt axicn (i) B cjmult mpan syl cji (e. B (RR)) a1i B cjret (x.) opreq12d B recnt axicn (i) B mulneg1t mpan syl 3eqtrd (-) opreqan12d A (opr (i) (x.) B) subnegt A recnt B recnt axicn (i) B axmulcl mpan syl syl2an 3eqtrd)) thm (cj11t () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= (` (*) A) (` (*) B)) (= A B))) ((` (Im) A) (` (Im) B) neg11t A imclt recnd B imclt recnd syl2an (= (` (Re) A) (` (Re) B)) anbi2d (` (Re) A) (-u (` (Im) A)) (` (Re) B) (-u (` (Im) B)) crut A reclt A imclt (` (Im) A) renegclt syl jca B reclt B imclt (` (Im) B) renegclt syl jca syl2an (` (Re) A) (` (Im) A) (` (Re) B) (` (Im) B) crut A reclt A imclt jca B reclt B imclt jca syl2an 3bitr4d (` (Re) A) (opr (` (Im) A) (x.) (i)) negsubt A reclt recnd A imclt recnd axicn (` (Im) A) (i) axmulcl mpan2 syl sylanc A imclt recnd axicn jctir (` (Im) A) (i) mulneg1t syl (` (Re) A) (+) opreq2d A cjvalt 3eqtr4rd (` (Re) B) (opr (` (Im) B) (x.) (i)) negsubt B reclt recnd B imclt recnd axicn (` (Im) B) (i) axmulcl mpan2 syl sylanc B imclt recnd axicn jctir (` (Im) B) (i) mulneg1t syl (` (Re) B) (+) opreq2d B cjvalt 3eqtr4rd eqeqan12d A replimt B replimt eqeqan12d 3bitr4d)) thm (cjne0t () () (-> (e. A (CC)) (<-> (=/= A (0)) (=/= (` (*) A) (0)))) (0cn A (0) cj11t mpan2 cj0 (` (*) A) eqeq2i syl5rbbr eqneqd)) thm (absnegt () () (-> (e. A (CC)) (= (` (abs) (-u A)) (` (abs) A))) (A cjnegt (-u A) (x.) opreq2d A cjclt A (` (*) A) mul2negt mpdan eqtrd (sqr) fveq2d A negclt (-u A) absvalt syl A absvalt 3eqtr4d)) thm (absclt () () (-> (e. A (CC)) (e. (` (abs) A) (RR))) (A absvalt (opr A (x.) (` (*) A)) sqrclt A cjmulrclt A cjmulge0t sylanc eqeltrd)) thm (abscjt () () (-> (e. A (CC)) (= (` (abs) (` (*) A)) (` (abs) A))) (A cjclt A (` (*) A) axmulcom mpdan A cjcjt (` (*) A) (x.) opreq2d eqtr4d (sqr) fveq2d A absvalt A cjclt (` (*) A) absvalt syl 3eqtr4rd)) thm (absval2 () ((absval2.1 (e. A (CC)))) (= (` (abs) A) (` (sqr) (opr (opr (` (Re) A) (^) (2)) (+) (opr (` (Im) A) (^) (2))))) (absval2.1 A absvalt ax-mp absval2.1 cjmulval (sqr) fveq2i eqtr)) thm (absvalsq () ((absval2.1 (e. A (CC)))) (= (opr (` (abs) A) (^) (2)) (opr A (x.) (` (*) A))) (absval2.1 A absvalt ax-mp (^) (2) opreq1i absval2.1 cjmulge0 absval2.1 cjmulrcl sqsqr ax-mp eqtr)) thm (absvalsq2 () ((absval2.1 (e. A (CC)))) (= (opr (` (abs) A) (^) (2)) (opr (opr (` (Re) A) (^) (2)) (+) (opr (` (Im) A) (^) (2)))) (absval2.1 absvalsq absval2.1 cjmulval eqtr)) thm (abs00 () ((absval2.1 (e. A (CC)))) (<-> (= (` (abs) A) (0)) (= A (0))) (absval2.1 absval2 (0) eqeq1i absval2.1 recl resqcl absval2.1 imcl resqcl readdcl absval2.1 recl sqge0 absval2.1 imcl sqge0 absval2.1 recl resqcl absval2.1 imcl resqcl addge0 mp2an (opr (opr (` (Re) A) (^) (2)) (+) (opr (` (Im) A) (^) (2))) sqr00t mp2an absval2.1 recl sqge0 absval2.1 imcl sqge0 absval2.1 recl resqcl absval2.1 imcl resqcl add20 mp2an absval2.1 recl recn sq00 absval2.1 imcl recn sq00 anbi12i (` (Re) A) (0) (+) (opr (` (Im) A) (x.) (i)) opreq1 (` (Im) A) (0) (x.) (i) opreq1 (0) (+) opreq2d sylan9eq 0cn axicn mulcl addid2 axicn mul02 eqtr syl6eq absval2.1 replim syl5eq A (0) (Re) fveq2 re0 syl6eq A (0) (Im) fveq2 im0 syl6eq jca impbi 3bitr 3bitr)) thm (abscl () ((absval2.1 (e. A (CC)))) (e. (` (abs) A) (RR)) (absval2.1 A absclt ax-mp)) thm (absge0 () ((absval2.1 (e. A (CC)))) (br (0) (<_) (` (abs) A)) (absval2.1 cjmulge0 absval2.1 cjmulrcl sqrge0 ax-mp absval2.1 A absvalt ax-mp breqtrr)) thm (absgt0 () ((absval2.1 (e. A (CC)))) (<-> (=/= A (0)) (br (0) (<) (` (abs) A))) (A (0) df-ne absval2.1 abs00 (` (abs) A) (0) eqcom bitr3 negbii absval2.1 absge0 0re absval2.1 abscl leloe mpbi (br (0) (<) (` (abs) A)) (= (0) (` (abs) A)) orcom mpbi ori 0re absval2.1 abscl ltne impbi 3bitr)) thm (absneg () ((absval2.1 (e. A (CC)))) (= (` (abs) (-u A)) (` (abs) A)) (absval2.1 A absnegt ax-mp)) thm (abscj () ((absval2.1 (e. A (CC)))) (= (` (abs) (` (*) A)) (` (abs) A)) (absval2.1 A abscjt ax-mp)) thm (abssub () ((absval2.1 (e. A (CC))) (abssub.2 (e. B (CC)))) (= (` (abs) (opr A (-) B)) (` (abs) (opr B (-) A))) (abssub.2 absval2.1 negsubdi2 (abs) fveq2i abssub.2 absval2.1 subcl absneg eqtr3)) thm (absmul () ((absval2.1 (e. A (CC))) (abssub.2 (e. B (CC)))) (= (` (abs) (opr A (x.) B)) (opr (` (abs) A) (x.) (` (abs) B))) (absval2.1 abssub.2 cjmul (opr A (x.) B) (x.) opreq2i absval2.1 abssub.2 absval2.1 cjcl abssub.2 cjcl mul4 eqtr (sqr) fveq2i absval2.1 cjmulrcl abssub.2 cjmulrcl absval2.1 cjmulge0 abssub.2 cjmulge0 sqrmuli eqtr absval2.1 abssub.2 mulcl (opr A (x.) B) absvalt ax-mp absval2.1 A absvalt ax-mp abssub.2 B absvalt ax-mp (x.) opreq12i 3eqtr4)) thm (sqabsadd () ((absval2.1 (e. A (CC))) (abssub.2 (e. B (CC)))) (= (opr (` (abs) (opr A (+) B)) (^) (2)) (opr (opr (opr (` (abs) A) (^) (2)) (+) (opr (` (abs) B) (^) (2))) (+) (opr (2) (x.) (` (Re) (opr A (x.) (` (*) B)))))) (absval2.1 abssub.2 cjadd (opr A (+) B) (x.) opreq2i absval2.1 abssub.2 absval2.1 cjcl abssub.2 cjcl muladd eqtr absval2.1 abssub.2 addcl absvalsq absval2.1 absvalsq abssub.2 absvalsq abssub.2 abssub.2 cjcl mulcom eqtr (+) opreq12i absval2.1 abssub.2 cjcl mulcl addcj absval2.1 abssub.2 cjcl cjmul abssub.2 cjcj (` (*) A) (x.) opreq2i eqtr (opr A (x.) (` (*) B)) (+) opreq2i eqtr3 (+) opreq12i 3eqtr4)) thm (absvalsqt () () (-> (e. A (CC)) (= (opr (` (abs) A) (^) (2)) (opr A (x.) (` (*) A)))) (A (if (e. A (CC)) A (2)) (abs) fveq2 (^) (2) opreq1d (= A (if (e. A (CC)) A (2))) id A (if (e. A (CC)) A (2)) (*) fveq2 (x.) opreq12d eqeq12d 2cn A elimel absvalsq dedth)) thm (absval2t () () (-> (e. A (CC)) (= (` (abs) A) (` (sqr) (opr (opr (` (Re) A) (^) (2)) (+) (opr (` (Im) A) (^) (2)))))) (A (if (e. A (CC)) A (0)) (abs) fveq2 A (if (e. A (CC)) A (0)) (Re) fveq2 (^) (2) opreq1d A (if (e. A (CC)) A (0)) (Im) fveq2 (^) (2) opreq1d (+) opreq12d (sqr) fveq2d eqeq12d 0cn A elimel absval2 dedth)) thm (absreimt () () (-> (/\ (e. A (RR)) (e. B (RR))) (= (` (abs) (opr A (+) (opr (i) (x.) B))) (` (sqr) (opr (opr A (^) (2)) (+) (opr B (^) (2)))))) (A (opr (i) (x.) B) axaddcl A recnt B recnt axicn (i) B axmulcl mpan syl syl2an (opr A (+) (opr (i) (x.) B)) absval2t syl B recnt axicn (i) B axmulcom mpan syl (e. A (RR)) adantl A (+) opreq2d (Re) fveq2d A B crret eqtrd (^) (2) opreq1d B recnt axicn (i) B axmulcom mpan syl (e. A (RR)) adantl A (+) opreq2d (Im) fveq2d A B crimt eqtrd (^) (2) opreq1d (+) opreq12d (sqr) fveq2d eqtrd)) thm (abs00t () () (-> (e. A (CC)) (<-> (= (` (abs) A) (0)) (= A (0)))) (A (if (e. A (CC)) A (0)) (abs) fveq2 (0) eqeq1d A (if (e. A (CC)) A (0)) (0) eqeq1 bibi12d 0cn A elimel abs00 dedth)) thm (absge0t () () (-> (e. A (CC)) (br (0) (<_) (` (abs) A))) (A (if (e. A (CC)) A (0)) (abs) fveq2 (0) (<_) breq2d 0cn A elimel absge0 dedth)) thm (absmult () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (abs) (opr A (x.) B)) (opr (` (abs) A) (x.) (` (abs) B)))) (A (if (e. A (CC)) A (0)) (x.) B opreq1 (abs) fveq2d A (if (e. A (CC)) A (0)) (abs) fveq2 (x.) (` (abs) B) opreq1d eqeq12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (x.) opreq2 (abs) fveq2d B (if (e. B (CC)) B (0)) (abs) fveq2 (` (abs) (if (e. A (CC)) A (0))) (x.) opreq2d eqeq12d 0cn A elimel 0cn B elimel absmul dedth2h)) thm (absdivz () ((absdivz.1 (e. A (CC))) (absdivz.2 (e. B (CC)))) (-> (=/= B (0)) (= (` (abs) (opr A (/) B)) (opr (` (abs) A) (/) (` (abs) B)))) (absdivz.1 absdivz.2 divclz absdivz.2 B (opr A (/) B) absmult mpan syl absdivz.2 absdivz.1 divcan2z (abs) fveq2d eqtr3d (` (abs) A) (` (abs) B) (` (abs) (opr A (/) B)) divmult absdivz.1 abscl recn (=/= B (0)) a1i absdivz.2 abscl recn (=/= B (0)) a1i absdivz.1 absdivz.2 divclz (opr A (/) B) absclt (` (abs) (opr A (/) B)) recnt 3syl 3jca absdivz.2 abs00 eqneqi biimpr sylanc mpbird eqcomd)) thm (absdivt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (= (` (abs) (opr A (/) B)) (opr (` (abs) A) (/) (` (abs) B)))) (A (if (e. A (CC)) A (0)) (/) B opreq1 (abs) fveq2d A (if (e. A (CC)) A (0)) (abs) fveq2 (/) (` (abs) B) opreq1d eqeq12d (=/= B (0)) imbi2d B (if (e. B (CC)) B (0)) (0) neeq1 B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (/) opreq2 (abs) fveq2d B (if (e. B (CC)) B (0)) (abs) fveq2 (` (abs) (if (e. A (CC)) A (0))) (/) opreq2d eqeq12d imbi12d 0cn A elimel 0cn B elimel absdivz dedth2h 3impia)) thm (absid () ((absid.1 (e. A (RR)))) (-> (br (0) (<_) A) (= (` (abs) A) A)) (absid.1 sqrmsq absid.1 recn A absvalt ax-mp absid.1 absid.1 recn cjreb mpbi A (x.) opreq2i (sqr) fveq2i eqtr syl5eq)) thm (absidt () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (= (` (abs) A) A)) (A (if (e. A (RR)) A (0)) (0) (<_) breq2 A (if (e. A (RR)) A (0)) (abs) fveq2 (= A (if (e. A (RR)) A (0))) id eqeq12d imbi12d 0re A elimel absid dedth imp)) thm (absnidt () () (-> (/\ (e. A (RR)) (br A (<_) (0))) (= (` (abs) A) (-u A))) (A le0neg1t A recnt A absnegt syl (br (0) (<_) (-u A)) adantr (-u A) absidt A renegclt sylan eqtr3d ex sylbid imp)) thm (leabst () () (-> (e. A (RR)) (br A (<_) (` (abs) A))) (0re (0) A letrit mpan A absidt A (` (abs) A) eqlet (` (abs) A) A eqcom sylan2b syldan ex A recnt A absge0t syl A recnt A absclt syl 0re A (0) (` (abs) A) letrt mp3an2 mpdan mpan2d jaod mpd)) thm (absort () () (-> (e. A (RR)) (\/ (= (` (abs) A) A) (= (` (abs) A) (-u A)))) (0re (0) A letrit mpan A absidt ex A absnidt ex orim12d mpd)) thm (absret () () (-> (e. A (RR)) (= (` (abs) A) (` (sqr) (opr A (^) (2))))) (A recnt A absvalt syl A recnt A sqvalt syl A cjret A (x.) opreq2d eqtr4d (sqr) fveq2d eqtr4d)) thm (absresqt () () (-> (e. A (RR)) (= (opr (` (abs) A) (^) (2)) (opr A (^) (2)))) (A cjret A (x.) opreq2d A recnt A absvalsqt syl A recnt A sqvalt syl 3eqtr4d)) thm (absexpt ((j k) (A j) (A k) (N j) (N k)) () (-> (/\ (e. A (CC)) (e. N (NN0))) (= (` (abs) (opr A (^) N)) (opr (` (abs) A) (^) N))) ((cv j) (0) A (^) opreq2 (abs) fveq2d (cv j) (0) (` (abs) A) (^) opreq2 eqeq12d (e. A (CC)) imbi2d (cv j) (cv k) A (^) opreq2 (abs) fveq2d (cv j) (cv k) (` (abs) A) (^) opreq2 eqeq12d (e. A (CC)) imbi2d (cv j) (opr (cv k) (+) (1)) A (^) opreq2 (abs) fveq2d (cv j) (opr (cv k) (+) (1)) (` (abs) A) (^) opreq2 eqeq12d (e. A (CC)) imbi2d (cv j) N A (^) opreq2 (abs) fveq2d (cv j) N (` (abs) A) (^) opreq2 eqeq12d (e. A (CC)) imbi2d A exp0t (abs) fveq2d 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq A absclt recnd (` (abs) A) exp0t syl eqtr4d (` (abs) (opr A (^) (cv k))) (opr (` (abs) A) (^) (cv k)) (x.) (` (abs) A) opreq1 (/\ (e. A (CC)) (e. (cv k) (NN0))) adantl A (cv k) expp1t (abs) fveq2d (opr A (^) (cv k)) A absmult A (cv k) expclt (e. A (CC)) (e. (cv k) (NN0)) pm3.26 sylanc eqtrd (= (` (abs) (opr A (^) (cv k))) (opr (` (abs) A) (^) (cv k))) adantr (` (abs) A) (cv k) expp1t A absclt recnd sylan (= (` (abs) (opr A (^) (cv k))) (opr (` (abs) A) (^) (cv k))) adantr 3eqtr4d exp31 com12 a2d nn0ind impcom)) thm (absrelet () () (-> (e. A (CC)) (br (` (abs) (` (Re) A)) (<_) (` (abs) A))) (A imclt (` (Im) A) sqge0t syl (opr (` (Re) A) (^) (2)) (opr (` (Im) A) (^) (2)) addge01t A reclt (` (Re) A) resqclt syl A imclt (` (Im) A) resqclt syl sylanc mpbid (opr (` (Re) A) (^) (2)) (opr (opr (` (Re) A) (^) (2)) (+) (opr (` (Im) A) (^) (2))) sqrlet A reclt (` (Re) A) resqclt syl (opr (` (Re) A) (^) (2)) (opr (` (Im) A) (^) (2)) axaddrcl A reclt (` (Re) A) resqclt syl A imclt (` (Im) A) resqclt syl sylanc jca A reclt (` (Re) A) sqge0t syl (opr (` (Re) A) (^) (2)) (opr (` (Im) A) (^) (2)) addge0t A reclt (` (Re) A) resqclt syl A imclt (` (Im) A) resqclt syl jca A reclt (` (Re) A) sqge0t syl A imclt (` (Im) A) sqge0t syl jca sylanc jca sylanc mpbid A reclt (` (Re) A) absret syl A absval2t 3brtr4d)) thm (absimlet () () (-> (e. A (CC)) (br (` (abs) (` (Im) A)) (<_) (` (abs) A))) ((opr (` (Im) A) (^) (2)) (opr (` (Re) A) (^) (2)) addge02t biimp3a A imclt (` (Im) A) resqclt syl A reclt (` (Re) A) resqclt syl A reclt (` (Re) A) sqge0t syl syl3anc (opr (` (Im) A) (^) (2)) (opr (opr (` (Re) A) (^) (2)) (+) (opr (` (Im) A) (^) (2))) sqrlet A imclt (` (Im) A) resqclt syl (opr (` (Re) A) (^) (2)) (opr (` (Im) A) (^) (2)) axaddrcl A reclt (` (Re) A) resqclt syl A imclt (` (Im) A) resqclt syl sylanc jca A imclt (` (Im) A) sqge0t syl (opr (` (Re) A) (^) (2)) (opr (` (Im) A) (^) (2)) addge0t A reclt (` (Re) A) resqclt syl A imclt (` (Im) A) resqclt syl jca A reclt (` (Re) A) sqge0t syl A imclt (` (Im) A) sqge0t syl jca sylanc jca sylanc mpbid A imclt (` (Im) A) absret syl A absval2t 3brtr4d)) thm (absnid () ((absnid.1 (e. A (RR)))) (-> (br A (<_) (0)) (= (` (abs) A) (-u A))) (absnid.1 A absnidt mpan)) thm (leabs () ((absnid.1 (e. A (RR)))) (br A (<_) (` (abs) A)) (0re absnid.1 letri absnid.1 absid absnid.1 absnid.1 recn abscl eqle eqcoms syl absnid.1 recn absge0 absnid.1 0re absnid.1 recn abscl letr mpan2 jaoi ax-mp)) thm (absor () ((absnid.1 (e. A (RR)))) (\/ (= (` (abs) A) A) (= (` (abs) A) (-u A))) (absnid.1 A absort ax-mp)) thm (absre () ((absnid.1 (e. A (RR)))) (= (` (abs) A) (` (sqr) (opr A (^) (2)))) (absnid.1 A absret ax-mp)) thm (abslt () ((absnid.1 (e. A (RR))) (abslt.1 (e. B (RR)))) (<-> (br (` (abs) A) (<) B) (/\ (br (-u B) (<) A) (br A (<) B))) (absnid.1 renegcl leabs absnid.1 recn absneg breqtr absnid.1 renegcl absnid.1 recn abscl abslt.1 lelttr mpan absnid.1 leabs absnid.1 absnid.1 recn abscl abslt.1 lelttr mpan jca absnid.1 absor (` (abs) A) A (<) B breq1 biimprcd (` (abs) A) (-u A) (<) B breq1 biimprcd jaao mpi ancoms impbi absnid.1 abslt.1 ltnegcon1 (br A (<) B) anbi1i bitr)) thm (absle () ((absnid.1 (e. A (RR))) (abslt.1 (e. B (RR)))) (<-> (br (` (abs) A) (<_) B) (/\ (br (-u B) (<_) A) (br A (<_) B))) (absnid.1 renegcl leabs absnid.1 recn absneg breqtr absnid.1 renegcl absnid.1 recn abscl abslt.1 letr mpan absnid.1 leabs absnid.1 absnid.1 recn abscl abslt.1 letr mpan jca absnid.1 absor (` (abs) A) A (<_) B breq1 biimprcd (` (abs) A) (-u A) (<_) B breq1 biimprcd jaao mpi ancoms impbi absnid.1 abslt.1 lenegcon1 (br A (<_) B) anbi1i bitr)) thm (absltOLD () ((absnid.1 (e. A (RR))) (absltOLD.1 (e. B (RR)))) (<-> (br (` (abs) A) (<) B) (/\ (br A (<) B) (br (-u A) (<) B))) (absnid.1 leabs absnid.1 absnid.1 recn abscl absltOLD.1 lelttr mpan absnid.1 renegcl leabs absnid.1 recn absneg breqtr absnid.1 renegcl absnid.1 recn abscl absltOLD.1 lelttr mpan jca absnid.1 absor (` (abs) A) A (<) B breq1 biimprcd (` (abs) A) (-u A) (<) B breq1 biimprcd jaao mpi impbi)) thm (absleOLD () ((absnid.1 (e. A (RR))) (absltOLD.1 (e. B (RR)))) (<-> (br (` (abs) A) (<_) B) (/\ (br A (<_) B) (br (-u A) (<_) B))) (absnid.1 leabs absnid.1 absnid.1 recn abscl absltOLD.1 letr mpan absnid.1 renegcl leabs absnid.1 recn absneg breqtr absnid.1 renegcl absnid.1 recn abscl absltOLD.1 letr mpan jca absnid.1 absor (` (abs) A) A (<_) B breq1 biimprcd (` (abs) A) (-u A) (<_) B breq1 biimprcd jaao mpi impbi)) thm (abs0 () () (= (` (abs) (0)) (0)) ((0) eqid 0cn abs00 mpbir)) thm (absi () () (= (` (abs) (i)) (1)) (axicn (i) absvalt ax-mp axicn axicn mulneg2 cji (i) (x.) opreq2i itimesi axicn axicn mulcl 1cn negcon2 mpbi 3eqtr4 (sqr) fveq2i sqr1 3eqtr)) thm (nn0absclt () () (-> (e. A (ZZ)) (e. (` (abs) A) (NN0))) (A zret A absort syl (` (abs) A) A (ZZ) eleq1 biimprd (` (abs) A) (-u A) (ZZ) eleq1 A znegclt syl5bir jaoi mpcom A zcnt A absge0t syl jca (` (abs) A) elnn0z sylibr)) thm (absltt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (` (abs) A) (<) B) (/\ (br (-u B) (<) A) (br A (<) B)))) (A (if (e. A (RR)) A (0)) (abs) fveq2 (<) B breq1d A (if (e. A (RR)) A (0)) (-u B) (<) breq2 A (if (e. A (RR)) A (0)) (<) B breq1 anbi12d bibi12d B (if (e. B (RR)) B (0)) (` (abs) (if (e. A (RR)) A (0))) (<) breq2 B (if (e. B (RR)) B (0)) negeq (<) (if (e. A (RR)) A (0)) breq1d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<) breq2 anbi12d bibi12d 0re A elimel 0re B elimel abslt dedth2h)) thm (abslttOLD () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (` (abs) A) (<) B) (/\ (br A (<) B) (br (-u A) (<) B)))) (A (if (e. A (RR)) A (0)) (abs) fveq2 (<) B breq1d A (if (e. A (RR)) A (0)) (<) B breq1 A (if (e. A (RR)) A (0)) negeq (<) B breq1d anbi12d bibi12d B (if (e. B (RR)) B (0)) (` (abs) (if (e. A (RR)) A (0))) (<) breq2 B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<) breq2 B (if (e. B (RR)) B (0)) (-u (if (e. A (RR)) A (0))) (<) breq2 anbi12d bibi12d 0re A elimel 0re B elimel absltOLD dedth2h)) thm (abslet () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (` (abs) A) (<_) B) (/\ (br (-u B) (<_) A) (br A (<_) B)))) (A (if (e. A (RR)) A (0)) (abs) fveq2 (<_) B breq1d A (if (e. A (RR)) A (0)) (-u B) (<_) breq2 A (if (e. A (RR)) A (0)) (<_) B breq1 anbi12d bibi12d B (if (e. B (RR)) B (0)) (` (abs) (if (e. A (RR)) A (0))) (<_) breq2 B (if (e. B (RR)) B (0)) negeq (<_) (if (e. A (RR)) A (0)) breq1d B (if (e. B (RR)) B (0)) (if (e. A (RR)) A (0)) (<_) breq2 anbi12d bibi12d 0re A elimel 0re B elimel absle dedth2h)) thm (lenegsqt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (0) (<_) B)) (<-> (/\ (br A (<_) B) (br (-u A) (<_) B)) (br (opr A (^) (2)) (<_) (opr B (^) (2))))) ((` (abs) A) B le2sqt A recnt A absclt A absge0t jca syl sylan A B abslet A B lenegcon1t (br A (<_) B) anbi1d (br (-u A) (<_) B) (br A (<_) B) ancom syl5rbbr bitrd (br (0) (<_) B) adantrr A absresqt (<_) (opr B (^) (2)) breq1d (/\ (e. B (RR)) (br (0) (<_) B)) adantr 3bitr3d 3impb)) thm (releabst () () (-> (e. A (CC)) (br (` (Re) A) (<_) (` (abs) A))) (A reclt A reclt (` (Re) A) recnt (` (Re) A) absclt 3syl A absclt A reclt (` (Re) A) leabst syl A absrelet letrd)) thm (recvalz () ((recvalz.1 (e. A (CC)))) (-> (=/= A (0)) (= (opr (1) (/) A) (opr (` (*) A) (/) (opr (` (abs) A) (^) (2))))) (recvalz.1 cjcl recvalz.1 mulcom recvalz.1 absvalsq eqtr4 recvalz.1 A cjne0t ax-mp recvalz.1 abscl resqcl recn recvalz.1 cjcl recvalz.1 divmulz sylbi mpbiri (1) (/) opreq2d recvalz.1 abscl resqcl recn recvalz.1 cjcl (opr (` (abs) A) (^) (2)) (` (*) A) recdivt mpanl12 recvalz.1 abscl gt0ne0 recvalz.1 absgt0 recvalz.1 abscl recn sq00 eqneqi 3imtr4 recvalz.1 A cjne0t ax-mp biimp sylanc eqtr3d)) thm (cjdiv () ((cjdiv.1 (e. A (CC))) (cjdiv.2 (e. B (CC)))) (-> (=/= B (0)) (= (` (*) (opr A (/) B)) (opr (` (*) A) (/) (` (*) B)))) (cjdiv.1 cjdiv.2 divrecz (*) fveq2d cjdiv.2 recclz cjdiv.1 A (opr (1) (/) B) cjmult mpan syl cjdiv.2 B cjne0t ax-mp cjdiv.1 cjcl cjdiv.2 cjcl divrecz sylbi cjdiv.2 abscl gt0ne0 cjdiv.2 absgt0 cjdiv.2 abscl recn sq00 eqneqi 3imtr4 cjdiv.2 abscl resqcl rerecclz (opr (1) (/) (opr (` (abs) B) (^) (2))) cjret 3syl (` (*) (` (*) B)) (x.) opreq2d (` (*) B) (opr (1) (/) (opr (` (abs) B) (^) (2))) cjmult cjdiv.2 cjcl (=/= B (0)) a1i cjdiv.2 abscl gt0ne0 cjdiv.2 absgt0 cjdiv.2 abscl recn sq00 eqneqi 3imtr4 cjdiv.2 abscl resqcl recn recclz syl sylanc cjdiv.2 abscl gt0ne0 cjdiv.2 absgt0 cjdiv.2 abscl recn sq00 eqneqi 3imtr4 cjdiv.2 cjcl cjcl cjdiv.2 abscl resqcl recn divrecz syl cjdiv.2 abscj (^) (2) opreq1i (` (*) (` (*) B)) (/) opreq2i syl5eq 3eqtr4rd cjdiv.2 B cjne0t ax-mp cjdiv.2 cjcl recvalz sylbi cjdiv.2 recvalz cjdiv.2 abscl gt0ne0 cjdiv.2 absgt0 cjdiv.2 abscl recn sq00 eqneqi 3imtr4 cjdiv.2 cjcl cjdiv.2 abscl resqcl recn divrecz syl eqtrd (*) fveq2d 3eqtr4d (` (*) A) (x.) opreq2d eqtr2d 3eqtrd)) thm (cjdivt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (= (` (*) (opr A (/) B)) (opr (` (*) A) (/) (` (*) B)))) (A (if (e. A (CC)) A (0)) (/) B opreq1 (*) fveq2d A (if (e. A (CC)) A (0)) (*) fveq2 (/) (` (*) B) opreq1d eqeq12d (=/= B (0)) imbi2d B (if (e. B (CC)) B (0)) (0) neeq1 B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (/) opreq2 (*) fveq2d B (if (e. B (CC)) B (0)) (*) fveq2 (` (*) (if (e. A (CC)) A (0))) (/) opreq2d eqeq12d imbi12d 0cn A elimel 0cn B elimel cjdiv dedth2h 3impia)) thm (releabs () ((releabs.1 (e. A (CC)))) (br (` (Re) A) (<_) (` (abs) A)) (releabs.1 A releabst ax-mp)) thm (abstri () ((releabs.1 (e. A (CC))) (abstri.2 (e. B (CC)))) (br (` (abs) (opr A (+) B)) (<_) (opr (` (abs) A) (+) (` (abs) B))) (releabs.1 abstri.2 cjcl mulcl releabs releabs.1 abstri.2 cjcl absmul abstri.2 abscj (` (abs) A) (x.) opreq2i eqtr breqtr 2pos releabs.1 abstri.2 cjcl mulcl recl releabs.1 abscl abstri.2 abscl remulcl 2re lemul2 ax-mp mpbi 2re releabs.1 abstri.2 cjcl mulcl recl remulcl 2re releabs.1 abscl abstri.2 abscl remulcl remulcl releabs.1 abscl resqcl abstri.2 abscl resqcl readdcl leadd2 mpbi releabs.1 abstri.2 sqabsadd releabs.1 abscl recn abstri.2 abscl recn binom2 releabs.1 abscl recn sqcl 2re releabs.1 abscl abstri.2 abscl remulcl remulcl recn abstri.2 abscl recn sqcl add23 eqtr 3brtr4 releabs.1 abstri.2 addcl absge0 releabs.1 absge0 abstri.2 absge0 releabs.1 abscl abstri.2 abscl addge0 mp2an releabs.1 abstri.2 addcl abscl releabs.1 abscl abstri.2 abscl readdcl le2sq mp2an mpbir)) thm (absidmt () () (-> (e. A (CC)) (= (` (abs) (` (abs) A)) (` (abs) A))) ((` (abs) A) absidt A absclt A absge0t sylanc)) thm (absgt0t () () (-> (e. A (CC)) (<-> (=/= A (0)) (br (0) (<) (` (abs) A)))) (A (if (e. A (CC)) A (0)) (0) neeq1 A (if (e. A (CC)) A (0)) (abs) fveq2 (0) (<) breq2d bibi12d 0cn A elimel absgt0 dedth)) thm (abssubt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (abs) (opr A (-) B)) (` (abs) (opr B (-) A)))) (A (if (e. A (CC)) A (0)) (-) B opreq1 (abs) fveq2d A (if (e. A (CC)) A (0)) B (-) opreq2 (abs) fveq2d eqeq12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (-) opreq2 (abs) fveq2d B (if (e. B (CC)) B (0)) (-) (if (e. A (CC)) A (0)) opreq1 (abs) fveq2d eqeq12d 0cn A elimel 0cn B elimel abssub dedth2h)) thm (abstrit () () (-> (/\ (e. A (CC)) (e. B (CC))) (br (` (abs) (opr A (+) B)) (<_) (opr (` (abs) A) (+) (` (abs) B)))) (A (if (e. A (CC)) A (0)) (+) B opreq1 (abs) fveq2d A (if (e. A (CC)) A (0)) (abs) fveq2 (+) (` (abs) B) opreq1d (<_) breq12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (+) opreq2 (abs) fveq2d B (if (e. B (CC)) B (0)) (abs) fveq2 (` (abs) (if (e. A (CC)) A (0))) (+) opreq2d (<_) breq12d 0cn A elimel 0cn B elimel abstri dedth2h)) thm (abs3dift () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (br (` (abs) (opr A (-) B)) (<_) (opr (` (abs) (opr A (-) C)) (+) (` (abs) (opr C (-) B))))) (A C B npncant 3com23 (abs) fveq2d (opr A (-) C) (opr C (-) B) abstrit A C subclt (e. B (CC)) 3adant2 C B subclt ancoms (e. A (CC)) 3adant1 sylanc eqbrtrrd)) thm (abs3dif () ((abs3dif.1 (e. A (CC))) (abs3dif.2 (e. B (CC))) (abs3dif.3 (e. C (CC)))) (br (` (abs) (opr A (-) B)) (<_) (opr (` (abs) (opr A (-) C)) (+) (` (abs) (opr C (-) B)))) (abs3dif.1 abs3dif.2 abs3dif.3 A B C abs3dift mp3an)) thm (abs3lem () ((abs3dif.1 (e. A (CC))) (abs3dif.2 (e. B (CC))) (abs3dif.3 (e. C (CC))) (abs3lem.4 (e. D (RR)))) (-> (/\ (br (` (abs) (opr A (-) C)) (<) (opr D (/) (2))) (br (` (abs) (opr C (-) B)) (<) (opr D (/) (2)))) (br (` (abs) (opr A (-) B)) (<) D)) (abs3dif.1 abs3dif.3 subcl abscl abs3dif.3 abs3dif.2 subcl abscl abs3lem.4 2re 2re 2pos gt0ne0i redivcl abs3lem.4 2re 2re 2pos gt0ne0i redivcl lt2add abs3dif.1 abs3dif.2 abs3dif.3 abs3dif abs3dif.1 abs3dif.2 subcl abscl abs3dif.1 abs3dif.3 subcl abscl abs3dif.3 abs3dif.2 subcl abscl readdcl abs3lem.4 2re 2re 2pos gt0ne0i redivcl abs3lem.4 2re 2re 2pos gt0ne0i redivcl readdcl lelttr mpan syl abs3lem.4 recn 2cn 2re 2pos gt0ne0i divcl 2times 2cn abs3lem.4 recn 2re 2pos gt0ne0i divcan2 eqtr3 syl6breq)) thm (abs1m ((A x)) ((abs1m.1 (e. A (CC)))) (E.e. x (CC) (/\ (= (` (abs) (cv x)) (1)) (= (` (abs) A) (opr (cv x) (x.) A)))) (abs1m.1 abs00 biimpr A (0) (1) (x.) opreq2 0cn mulid2 syl6eq eqtr4d 0re ax1re lt01 ltlei ax1re absid ax-mp jctil 1cn jctil (cv x) (1) (abs) fveq2 (1) eqeq1d (cv x) (1) (x.) A opreq1 (` (abs) A) eqeq2d anbi12d (CC) rcla4ev syl A (0) df-ne abs1m.1 abs00 eqneqi abs1m.1 cjcl abs1m.1 abscl recn divclz abs1m.1 cjcl abs1m.1 abscl recn absdivz abs1m.1 abscl recn (` (abs) A) dividt mpan abs1m.1 abscj abs1m.1 A absidmt ax-mp (/) opreq12i syl5eq eqtrd abs1m.1 absvalsq abs1m.1 abscl recn sqval abs1m.1 abs1m.1 cjcl mulcom 3eqtr3 abs1m.1 cjcl abs1m.1 mulcl abs1m.1 abscl recn abs1m.1 abscl recn divmulz mpbiri abs1m.1 cjcl abs1m.1 abs1m.1 abscl recn 3pm3.2i (` (*) A) A (` (abs) A) div23t mpan eqtr3d jca jca sylbir (cv x) (opr (` (*) A) (/) (` (abs) A)) (abs) fveq2 (1) eqeq1d (cv x) (opr (` (*) A) (/) (` (abs) A)) (x.) A opreq1 (` (abs) A) eqeq2d anbi12d (CC) rcla4ev syl sylbir pm2.61i)) thm (recant ((A x) (B x)) () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (A.e. x (CC) (= (` (Re) (opr (cv x) (x.) A)) (` (Re) (opr (cv x) (x.) B)))) (= A B))) (A replimt A mulid2t eqcomd (Re) fveq2d A imret (x.) (i) opreq1d (+) opreq12d eqtrd B replimt B mulid2t eqcomd (Re) fveq2d B imret (x.) (i) opreq1d (+) opreq12d eqtrd eqeqan12d 1cn (cv x) (1) (x.) A opreq1 (Re) fveq2d (cv x) (1) (x.) B opreq1 (Re) fveq2d eqeq12d (CC) rcla4v ax-mp axicn negcl (cv x) (-u (i)) (x.) A opreq1 (Re) fveq2d (cv x) (-u (i)) (x.) B opreq1 (Re) fveq2d eqeq12d (CC) rcla4v ax-mp (x.) (i) opreq1d (+) opreq12d syl5bir A B (cv x) (x.) opreq2 (Re) fveq2d (e. (cv x) (CC)) a1d r19.21aiv (/\ (e. A (CC)) (e. B (CC))) a1i impbid)) thm (abs3lemt () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (RR)))) (-> (/\ (br (` (abs) (opr A (-) C)) (<) (opr D (/) (2))) (br (` (abs) (opr C (-) B)) (<) (opr D (/) (2)))) (br (` (abs) (opr A (-) B)) (<) D))) (A (if (e. A (CC)) A (0)) (-) C opreq1 (abs) fveq2d (<) (opr D (/) (2)) breq1d (br (` (abs) (opr C (-) B)) (<) (opr D (/) (2))) anbi1d A (if (e. A (CC)) A (0)) (-) B opreq1 (abs) fveq2d (<) D breq1d imbi12d B (if (e. B (CC)) B (0)) C (-) opreq2 (abs) fveq2d (<) (opr D (/) (2)) breq1d (br (` (abs) (opr (if (e. A (CC)) A (0)) (-) C)) (<) (opr D (/) (2))) anbi2d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (-) opreq2 (abs) fveq2d (<) D breq1d imbi12d C (if (e. C (CC)) C (0)) (if (e. A (CC)) A (0)) (-) opreq2 (abs) fveq2d (<) (opr D (/) (2)) breq1d C (if (e. C (CC)) C (0)) (-) (if (e. B (CC)) B (0)) opreq1 (abs) fveq2d (<) (opr D (/) (2)) breq1d anbi12d (br (` (abs) (opr (if (e. A (CC)) A (0)) (-) (if (e. B (CC)) B (0)))) (<) D) imbi1d D (if (e. D (RR)) D (0)) (/) (2) opreq1 (` (abs) (opr (if (e. A (CC)) A (0)) (-) (if (e. C (CC)) C (0)))) (<) breq2d D (if (e. D (RR)) D (0)) (/) (2) opreq1 (` (abs) (opr (if (e. C (CC)) C (0)) (-) (if (e. B (CC)) B (0)))) (<) breq2d anbi12d D (if (e. D (RR)) D (0)) (` (abs) (opr (if (e. A (CC)) A (0)) (-) (if (e. B (CC)) B (0)))) (<) breq2 imbi12d 0cn A elimel 0cn B elimel 0cn C elimel 0re D elimel abs3lem dedth4h)) thm (abslem2i () ((abslem2.1 (e. A (CC))) (abslem2i.2 (=/= A (0)))) (= (opr (opr (` (*) (opr A (/) (` (abs) A))) (x.) A) (+) (opr (opr A (/) (` (abs) A)) (x.) (` (*) A))) (opr (2) (x.) (` (abs) A))) (abslem2.1 abscl recn 2times abslem2.1 abscl abslem2.1 abscl recn cjreb mpbi abslem2.1 abscl recn sqval abslem2.1 absvalsq eqtr3 (/) (` (abs) A) opreq1i abslem2.1 abscl recn abslem2.1 abscl recn abslem2i.2 abslem2.1 abs00 eqneqi mpbir divcan4 abslem2.1 abslem2.1 cjcl abslem2.1 abscl recn abslem2i.2 abslem2.1 abs00 eqneqi mpbir div23 3eqtr3 (*) fveq2i eqtr3 abslem2.1 abslem2.1 abscl recn abslem2i.2 abslem2.1 abs00 eqneqi mpbir divcl abslem2.1 cjcl cjmul abslem2.1 cjcj (` (*) (opr A (/) (` (abs) A))) (x.) opreq2i 3eqtr abslem2.1 abscl recn sqval abslem2.1 absvalsq eqtr3 (/) (` (abs) A) opreq1i abslem2.1 abscl recn abslem2.1 abscl recn abslem2i.2 abslem2.1 abs00 eqneqi mpbir divcan4 abslem2.1 abslem2.1 cjcl abslem2.1 abscl recn abslem2i.2 abslem2.1 abs00 eqneqi mpbir div23 3eqtr3 (+) opreq12i eqtr2)) thm (abslem2 () ((abslem2.1 (e. A (CC)))) (-> (=/= A (0)) (= (opr (opr (` (*) (opr A (/) (` (abs) A))) (x.) A) (+) (opr (opr A (/) (` (abs) A)) (x.) (` (*) A))) (opr (2) (x.) (` (abs) A)))) ((= A (if (=/= A (0)) A (1))) id A (if (=/= A (0)) A (1)) (abs) fveq2 (/) opreq12d (*) fveq2d (= A (if (=/= A (0)) A (1))) id (x.) opreq12d (= A (if (=/= A (0)) A (1))) id A (if (=/= A (0)) A (1)) (abs) fveq2 (/) opreq12d A (if (=/= A (0)) A (1)) (*) fveq2 (x.) opreq12d (+) opreq12d A (if (=/= A (0)) A (1)) (abs) fveq2 (2) (x.) opreq2d eqeq12d abslem2.1 1cn (=/= A (0)) keepel A elimne0 abslem2i dedth)) thm (seq1bnd ((x y) (x z) (A x) (y z) (A y) (A z) (w x) (F x) (w y) (F y) (w z) (F z) (F w)) ((seq1bnd.1 (:--> F (NN) (CC)))) (-> (e. A (NN)) (E.e. x (RR) (A.e. y (NN) (-> (br (cv y) (<_) A) (br (` (abs) (` F (cv y))) (<) (cv x)))))) ((cv z) (1) (cv y) (<_) breq2 (br (` (abs) (` F (cv y))) (<) (cv x)) imbi1d x (RR) y (NN) rexralbidv (cv z) (cv w) (cv y) (<_) breq2 (br (` (abs) (` F (cv y))) (<) (cv x)) imbi1d x (RR) y (NN) rexralbidv (cv z) (opr (cv w) (+) (1)) (cv y) (<_) breq2 (br (` (abs) (` F (cv y))) (<) (cv x)) imbi1d x (RR) y (NN) rexralbidv (cv z) A (cv y) (<_) breq2 (br (` (abs) (` F (cv y))) (<) (cv x)) imbi1d x (RR) y (NN) rexralbidv seq1bnd.1 1nn F (NN) (CC) (1) ffvrn mp2an abscl ax1re readdcl (cv y) nnle1eq1t (cv y) (1) F fveq2 (abs) fveq2d seq1bnd.1 1nn F (NN) (CC) (1) ffvrn mp2an abscl ltp1 syl6eqbr syl6bi rgen (cv x) (opr (` (abs) (` F (1))) (+) (1)) (` (abs) (` F (cv y))) (<) breq2 (br (cv y) (<_) (1)) imbi2d y (NN) ralbidv (RR) rcla4ev mp2an (cv y) (opr (cv w) (+) (1)) leloet (cv y) nnret (cv w) peano2nn (opr (cv w) (+) (1)) nnret syl syl2an (cv y) (cv w) nnleltp1t (= (cv y) (opr (cv w) (+) (1))) orbi1d bitr4d ancoms (e. (cv z) (RR)) adantlr (-> (br (cv y) (<_) (cv w)) (br (` (abs) (` F (cv y))) (<) (cv z))) adantr (cv z) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1)) max1ALT (e. (cv w) (NN)) (e. (cv y) (NN)) ad2antlr (` (abs) (` F (cv y))) (cv z) (if (br (cv z) (<_) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1))) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1)) (cv z)) ltletrt seq1bnd.1 (cv y) ffvrni (` F (cv y)) absclt syl (/\ (e. (cv w) (NN)) (e. (cv z) (RR))) adantl (e. (cv w) (NN)) (e. (cv z) (RR)) pm3.27 (e. (cv y) (NN)) adantr (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1)) (RR) (cv z) (br (cv z) (<_) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1))) ifcl (cv w) peano2nn seq1bnd.1 (opr (cv w) (+) (1)) ffvrni syl (` F (opr (cv w) (+) (1))) absclt (` (abs) (` F (opr (cv w) (+) (1)))) peano2re 3syl sylan (e. (cv y) (NN)) adantr syl3anc mpan2d (br (cv y) (<_) (cv w)) imim2d imp (cv y) (opr (cv w) (+) (1)) F fveq2 (abs) fveq2d (<) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1)) breq1d (cv w) peano2nn seq1bnd.1 (opr (cv w) (+) (1)) ffvrni syl (` F (opr (cv w) (+) (1))) absclt (` (abs) (` F (opr (cv w) (+) (1)))) ltp1t 3syl syl5bir com12 (e. (cv z) (RR)) (e. (cv y) (NN)) ad2antrr (cv z) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1)) max2 (cv w) peano2nn seq1bnd.1 (opr (cv w) (+) (1)) ffvrni syl (` F (opr (cv w) (+) (1))) absclt (` (abs) (` F (opr (cv w) (+) (1)))) peano2re 3syl sylan2 ancoms (e. (cv y) (NN)) adantr (` (abs) (` F (cv y))) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1)) (if (br (cv z) (<_) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1))) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1)) (cv z)) ltletrt seq1bnd.1 (cv y) ffvrni (` F (cv y)) absclt syl (/\ (e. (cv w) (NN)) (e. (cv z) (RR))) adantl (cv w) peano2nn seq1bnd.1 (opr (cv w) (+) (1)) ffvrni syl (` F (opr (cv w) (+) (1))) absclt (` (abs) (` F (opr (cv w) (+) (1)))) peano2re 3syl (e. (cv z) (RR)) (e. (cv y) (NN)) ad2antrr (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1)) (RR) (cv z) (br (cv z) (<_) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1))) ifcl (cv w) peano2nn seq1bnd.1 (opr (cv w) (+) (1)) ffvrni syl (` F (opr (cv w) (+) (1))) absclt (` (abs) (` F (opr (cv w) (+) (1)))) peano2re 3syl sylan (e. (cv y) (NN)) adantr syl3anc mpan2d syld (-> (br (cv y) (<_) (cv w)) (br (` (abs) (` F (cv y))) (<) (cv z))) adantr jaod sylbid ex r19.20dva (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1)) (RR) (cv z) (br (cv z) (<_) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1))) ifcl (cv w) peano2nn seq1bnd.1 (opr (cv w) (+) (1)) ffvrni syl (` F (opr (cv w) (+) (1))) absclt (` (abs) (` F (opr (cv w) (+) (1)))) peano2re 3syl sylan jctild (cv x) (if (br (cv z) (<_) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1))) (opr (` (abs) (` F (opr (cv w) (+) (1)))) (+) (1)) (cv z)) (` (abs) (` F (cv y))) (<) breq2 (br (cv y) (<_) (opr (cv w) (+) (1))) imbi2d y (NN) ralbidv (RR) rcla4ev syl6 ex r19.23adv (cv x) (cv z) (` (abs) (` F (cv y))) (<) breq2 (br (cv y) (<_) (cv w)) imbi2d y (NN) ralbidv (RR) cbvrexv syl5ib nnind)) thm (seq1ublem ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (F y) (F z) (F w) (F v)) ((seq1ub.1 (:--> F (NN) (RR)))) (-> (e. B (NN)) (/\/\ (C_ (ran (|` F ({e.|} x (NN) (br (cv x) (<_) B)))) (RR)) (-. (= (ran (|` F ({e.|} x (NN) (br (cv x) (<_) B)))) ({/}))) (E.e. y (RR) (A.e. z (ran (|` F ({e.|} x (NN) (br (cv x) (<_) B)))) (br (cv z) (<_) (cv y)))))) (seq1ub.1 x (NN) (br (cv x) (<_) B) ssrab2 F (NN) (RR) ({e.|} x (NN) (br (cv x) (<_) B)) fssres mp2an (|` F ({e.|} x (NN) (br (cv x) (<_) B))) ({e.|} x (NN) (br (cv x) (<_) B)) (RR) frn ax-mp (e. B (NN)) a1i B nnge1t 1nn jctil (cv x) (1) (<_) B breq1 (NN) elrab sylibr seq1ub.1 F (NN) (RR) ffn ax-mp x (NN) (br (cv x) (<_) B) ssrab2 F (NN) ({e.|} x (NN) (br (cv x) (<_) B)) fnssres mp2an jctil (|` F ({e.|} x (NN) (br (cv x) (<_) B))) ({e.|} x (NN) (br (cv x) (<_) B)) (1) fnfvrn syl (` (|` F ({e.|} x (NN) (br (cv x) (<_) B))) (1)) (ran (|` F ({e.|} x (NN) (br (cv x) (<_) B)))) n0i syl seq1ub.1 axresscn F (NN) (RR) (CC) fss mp2an B y w seq1bnd (cv w) (cv v) (<_) B breq1 (cv w) (cv v) F fveq2 (abs) fveq2d (<) (cv y) breq1d imbi12d (NN) rcla4cv imp32 (e. (cv y) (RR)) adantll (= (` (|` F ({e.|} x (NN) (br (cv x) (<_) B))) (cv v)) (cv z)) adantr (` F (cv v)) leabst (e. (cv y) (RR)) adantr (` F (cv v)) (` (abs) (` F (cv v))) (cv y) lelttrt 3expa (` F (cv v)) recnt (` F (cv v)) absclt syl ancli sylan mpand (` F (cv v)) (cv y) ltlet syld seq1ub.1 (cv v) ffvrni sylan ancoms (br (cv v) (<_) B) adantrr (cv x) (cv v) (<_) B breq1 (NN) elrab (cv v) ({e.|} x (NN) (br (cv x) (<_) B)) F fvres sylbir (<_) (cv y) breq1d (e. (cv y) (RR)) adantl sylibrd (` (|` F ({e.|} x (NN) (br (cv x) (<_) B))) (cv v)) (cv z) (<_) (cv y) breq1 biimpd sylan9 (A.e. w (NN) (-> (br (cv w) (<_) B) (br (` (abs) (` F (cv w))) (<) (cv y)))) adantllr mpd exp31 (cv x) (cv v) (<_) B breq1 (NN) elrab syl5ib r19.23adv seq1ub.1 F (NN) (RR) ffn ax-mp x (NN) (br (cv x) (<_) B) ssrab2 F (NN) ({e.|} x (NN) (br (cv x) (<_) B)) fnssres mp2an (|` F ({e.|} x (NN) (br (cv x) (<_) B))) ({e.|} x (NN) (br (cv x) (<_) B)) (cv z) v fvelrn ax-mp syl5ib ex r19.21adv r19.22i syl 3jca)) thm (seq1ub ((A y) (x y) (x z) (B x) (y z) (B y) (B z) (F y) (F z)) ((seq1ub.1 (:--> F (NN) (RR)))) (-> (/\/\ (e. A (NN)) (e. B (NN)) (br A (<_) B)) (br (` F A) (<_) (sup (ran (|` F ({e.|} x (NN) (br (cv x) (<_) B)))) (RR) (<)))) ((ran (|` F ({e.|} x (NN) (br (cv x) (<_) B)))) y z (` F A) suprub seq1ub.1 B x y z seq1ublem (e. A (NN)) (br A (<_) B) 3ad2ant2 (cv x) (cv y) (<_) B breq1 (NN) cbvrabv A eleq2i (cv y) A (<_) B breq1 (NN) elrab bitr A ({e.|} x (NN) (br (cv x) (<_) B)) F fvres sylbir (cv x) (cv y) (<_) B breq1 (NN) cbvrabv A eleq2i (cv y) A (<_) B breq1 (NN) elrab bitr biimpr seq1ub.1 F (NN) (RR) ffn ax-mp x (NN) (br (cv x) (<_) B) ssrab2 F (NN) ({e.|} x (NN) (br (cv x) (<_) B)) fnssres mp2an jctil (|` F ({e.|} x (NN) (br (cv x) (<_) B))) ({e.|} x (NN) (br (cv x) (<_) B)) A fnfvrn syl eqeltrrd (e. B (NN)) 3adant2 sylanc)) thm (cvgannn ((j k) (j m) (j n) (k m) (k n) (m n) (j ph) (m ph) (n ph) (j ps) (m ps) (n ps)) () (<-> (/\ (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) ph))) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) ps)))) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (/\ ph ps))))) (m (NN) n (NN) (A.e. k (NN) (-> (br (cv m) (<_) (cv k)) ph)) (A.e. k (NN) (-> (br (cv n) (<_) (cv k)) ps)) reeanv (cv m) (cv n) nn0addge1t (cv m) nnret (cv n) nnnn0t syl2an (e. (cv k) (NN)) adantr (cv m) (opr (cv m) (+) (cv n)) (cv k) letrt (cv m) nnret (e. (cv n) (NN)) (e. (cv k) (NN)) ad2antrr (cv m) (cv n) nnaddclt (opr (cv m) (+) (cv n)) nnret syl (e. (cv k) (NN)) adantr (cv k) nnret (/\ (e. (cv m) (NN)) (e. (cv n) (NN))) adantl syl3anc mpand (cv n) (cv m) nn0addge2t (cv n) nnret (cv m) nnnn0t syl2an ancoms (e. (cv k) (NN)) adantr (cv n) (opr (cv m) (+) (cv n)) (cv k) letrt (cv n) nnret (e. (cv m) (NN)) (e. (cv k) (NN)) ad2antlr (cv m) (cv n) nnaddclt (opr (cv m) (+) (cv n)) nnret syl (e. (cv k) (NN)) adantr (cv k) nnret (/\ (e. (cv m) (NN)) (e. (cv n) (NN))) adantl syl3anc mpand jcad (br (cv m) (<_) (cv k)) ph (br (cv n) (<_) (cv k)) ps prth syl9 r19.20dva k (NN) (-> (br (cv m) (<_) (cv k)) ph) (-> (br (cv n) (<_) (cv k)) ps) r19.26 syl5ibr (cv m) (cv n) nnaddclt jctild (cv j) (opr (cv m) (+) (cv n)) (<_) (cv k) breq1 (/\ ph ps) imbi1d k (NN) ralbidv (NN) rcla4ev syl6 r19.23aivv sylbir (cv j) (cv m) (<_) (cv k) breq1 ph imbi1d k (NN) ralbidv (NN) cbvrexv (cv j) (cv n) (<_) (cv k) breq1 ps imbi1d k (NN) ralbidv (NN) cbvrexv syl2anb ph ps pm3.26 (br (cv j) (<_) (cv k)) imim2i k (NN) r19.20si j (NN) r19.22si ph ps pm3.27 (br (cv j) (<_) (cv k)) imim2i k (NN) r19.20si j (NN) r19.22si jca impbi)) thm (cvgannn0 ((j k) (j m) (j n) (k m) (k n) (m n) (j ph) (m ph) (n ph) (j ps) (m ps) (n ps)) () (<-> (/\ (E.e. j (NN0) (A.e. k (NN0) (-> (br (cv j) (<_) (cv k)) ph))) (E.e. j (NN0) (A.e. k (NN0) (-> (br (cv j) (<_) (cv k)) ps)))) (E.e. j (NN0) (A.e. k (NN0) (-> (br (cv j) (<_) (cv k)) (/\ ph ps))))) (m (NN0) n (NN0) (A.e. k (NN0) (-> (br (cv m) (<_) (cv k)) ph)) (A.e. k (NN0) (-> (br (cv n) (<_) (cv k)) ps)) reeanv (cv m) (cv n) nn0addge1t (cv m) nn0ret sylan (e. (cv k) (NN0)) adantr (cv m) (opr (cv m) (+) (cv n)) (cv k) letrt (cv m) nn0ret (e. (cv n) (NN0)) (e. (cv k) (NN0)) ad2antrr (cv m) (cv n) nn0addclt (opr (cv m) (+) (cv n)) nn0ret syl (e. (cv k) (NN0)) adantr (cv k) nn0ret (/\ (e. (cv m) (NN0)) (e. (cv n) (NN0))) adantl syl3anc mpand (cv n) (cv m) nn0addge2t (cv n) nn0ret sylan ancoms (e. (cv k) (NN0)) adantr (cv n) (opr (cv m) (+) (cv n)) (cv k) letrt (cv n) nn0ret (e. (cv m) (NN0)) (e. (cv k) (NN0)) ad2antlr (cv m) (cv n) nn0addclt (opr (cv m) (+) (cv n)) nn0ret syl (e. (cv k) (NN0)) adantr (cv k) nn0ret (/\ (e. (cv m) (NN0)) (e. (cv n) (NN0))) adantl syl3anc mpand jcad (br (cv m) (<_) (cv k)) ph (br (cv n) (<_) (cv k)) ps prth syl9 r19.20dva k (NN0) (-> (br (cv m) (<_) (cv k)) ph) (-> (br (cv n) (<_) (cv k)) ps) r19.26 syl5ibr (cv m) (cv n) nn0addclt jctild (cv j) (opr (cv m) (+) (cv n)) (<_) (cv k) breq1 (/\ ph ps) imbi1d k (NN0) ralbidv (NN0) rcla4ev syl6 r19.23aivv sylbir (cv j) (cv m) (<_) (cv k) breq1 ph imbi1d k (NN0) ralbidv (NN0) cbvrexv (cv j) (cv n) (<_) (cv k) breq1 ps imbi1d k (NN0) ralbidv (NN0) cbvrexv syl2anb ph ps pm3.26 (br (cv j) (<_) (cv k)) imim2i k (NN0) r19.20si j (NN0) r19.22si ph ps pm3.27 (br (cv j) (<_) (cv k)) imim2i k (NN0) r19.20si j (NN0) r19.22si jca impbi)) thm (cau2 ((x y) (x z) (y z) (R y) (R z)) ((cau2.1 (:--> F (NN) (CC))) (cau2.2 (-> (e. (cv x) (RR)) (-> (br (0) (<) (cv x)) (br (0) R (cv x)))))) (<-> (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) R (cv x))))))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) R (cv x)))))))) (cau2.2 imp (cv y) (cv z) F fveq2 (` F (cv z)) (-) opreq2d cau2.1 (cv z) ffvrni (` F (cv z)) subidt syl sylan9eqr (abs) fveq2d abs0 syl6eq R (cv x) breq1d biimprcd exp3a syl imp (e. (cv y) (NN)) adantlr (-> (br (cv y) (<) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) R (cv x))) biantrud (br (cv y) (<) (cv z)) (= (cv y) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) R (cv x)) jaob syl6bbr (cv y) (cv z) leloet (cv y) nnret (cv z) nnret syl2an (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) adantll (br (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) R (cv x)) imbi1d bitr4d ralbidva rexbidva ex pm5.74d ralbiia)) thm (cvg1 ((y z) (w y) (w z) (ps y) (ps w)) () (<-> (A.e. x F (-> ph (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<) (cv z)) ps))))) (A.e. x F (-> ph (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) ps)))))) ((cv y) (cv z) nnltp1let ps imbi1d ralbidva biimpd (cv y) peano2nn jctild (cv w) (opr (cv y) (+) (1)) (<_) (cv z) breq1 ps imbi1d z (NN) ralbidv (NN) rcla4ev syl6 r19.23aiv (cv w) (cv y) (<_) (cv z) breq1 ps imbi1d z (NN) ralbidv (NN) cbvrexv sylib (cv y) (cv z) ltlet (cv y) nnret (cv z) nnret syl2an ps imim1d r19.20dva r19.22i impbi ph imbi2i x F ralbii)) thm (cvg2 ((x y) (x z) (w x) (y z) (w y) (w z) (A x) (A w) (F x) (F z) (F w) (G x) (G w) (ph x) (ph w)) ((cvg2.1 (-> (/\ (e. (cv y) F) (e. (cv z) G)) (e. A (RR))))) (<-> (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y F (A.e. z G (-> ph (br A (<) (cv x))))))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y F (A.e. z G (-> ph (br A (<_) (cv x)))))))) (A (cv x) ltlet ancoms cvg2.1 sylan2 anassrs ph imim2d r19.20dva r19.22dva (br (0) (<) (cv x)) imim2d r19.20i (cv x) (opr (cv w) (/) (2)) (0) (<) breq2 (cv x) (opr (cv w) (/) (2)) A (<_) breq2 ph imbi2d y F z G rexralbidv imbi12d (RR) rcla4cv (/\ (e. (opr (cv w) (/) (2)) (RR)) (br (0) (<) (opr (cv w) (/) (2)))) (E.e. y F (A.e. z G (-> ph (br A (<_) (opr (cv w) (/) (2)))))) biimt (cv w) rehalfclt (br (0) (<) (cv w)) adantr (cv w) halfpos2t biimpa sylanc (e. (opr (cv w) (/) (2)) (RR)) (br (0) (<) (opr (cv w) (/) (2))) (E.e. y F (A.e. z G (-> ph (br A (<_) (opr (cv w) (/) (2)))))) impexp syl6bb (cv w) halfpost biimpa (e. A (RR)) adantr A (opr (cv w) (/) (2)) (cv w) lelttrt (/\ (e. (cv w) (RR)) (br (0) (<) (cv w))) (e. A (RR)) pm3.27 (cv w) rehalfclt (br (0) (<) (cv w)) (e. A (RR)) ad2antrr (e. (cv w) (RR)) (br (0) (<) (cv w)) pm3.26 (e. A (RR)) adantr syl3anc mpan2d cvg2.1 sylan2 anassrs ph imim2d r19.20dva r19.22dva sylbird com12 exp3a syl r19.21aiv (cv w) (cv x) (0) (<) breq2 (cv w) (cv x) A (<) breq2 ph imbi2d y F z G rexralbidv imbi12d (RR) cbvralv sylib impbi)) thm (caubnd ((x y) (x z) (w x) (u x) (t x) (F x) (y z) (w y) (u y) (t y) (F y) (w z) (u z) (t z) (F z) (u w) (t w) (F w) (t u) (F u) (F t)) ((caubnd.1 (:--> F (NN) (CC))) (caubnd.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))))) (E.e. x (RR) (A.e. y (NN) (br (` (abs) (` F (cv y))) (<) (cv x)))) (lt01 ax1re caubnd.2 (cv z) (1) (0) (<) breq2 (cv z) (1) (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) breq2 (br (cv w) (<) (cv y)) imbi2d w (NN) y (NN) rexralbidv imbi12d (RR) rcla4v mp2 ax-mp ax1re (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (1) (` (abs) (` F (cv w))) ltadd1t mp3an2 (` F (cv y)) (` F (cv w)) subclt caubnd.1 (cv y) ffvrni caubnd.1 (cv w) ffvrni syl2an (opr (` F (cv y)) (-) (` F (cv w))) absclt syl caubnd.1 (cv w) ffvrni (` F (cv w)) absclt syl (e. (cv y) (NN)) adantl sylanc (` F (cv y)) (` F (cv w)) npcant caubnd.1 (cv y) ffvrni caubnd.1 (cv w) ffvrni syl2an (abs) fveq2d (opr (` F (cv y)) (-) (` F (cv w))) (` F (cv w)) abstrit (` F (cv y)) (` F (cv w)) subclt caubnd.1 (cv y) ffvrni caubnd.1 (cv w) ffvrni syl2an caubnd.1 (cv w) ffvrni (e. (cv y) (NN)) adantl sylanc eqbrtrrd (` (abs) (` F (cv y))) (opr (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (+) (` (abs) (` F (cv w)))) (opr (1) (+) (` (abs) (` F (cv w)))) lelttrt caubnd.1 (cv y) ffvrni (` F (cv y)) absclt syl (e. (cv w) (NN)) adantr (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (` (abs) (` F (cv w))) axaddrcl (` F (cv y)) (` F (cv w)) subclt caubnd.1 (cv y) ffvrni caubnd.1 (cv w) ffvrni syl2an (opr (` F (cv y)) (-) (` F (cv w))) absclt syl caubnd.1 (cv w) ffvrni (` F (cv w)) absclt syl (e. (cv y) (NN)) adantl sylanc caubnd.1 (cv w) ffvrni (` F (cv w)) absclt syl ax1re (1) (` (abs) (` F (cv w))) axaddrcl mpan syl (e. (cv y) (NN)) adantl syl3anc mpand sylbid ancoms (br (cv w) (<) (cv y)) imim2d r19.20dva caubnd.1 (cv w) ffvrni (` F (cv w)) absclt syl ax1re (1) (` (abs) (` F (cv w))) axaddrcl mpan syl jctild (cv u) (opr (1) (+) (` (abs) (` F (cv w)))) (` (abs) (` F (cv y))) (<) breq2 (br (cv w) (<) (cv y)) imbi2d y (NN) ralbidv (RR) rcla4ev syl6 r19.22i ax-mp caubnd.1 (cv w) t y seq1bnd (cv y) (cv w) lelttrit (cv y) nnret (cv w) nnret syl2an ancoms (e. (cv t) (RR)) adantlr (e. (cv u) (RR)) adantlr (cv u) (cv t) max2 ancoms (e. (cv y) (NN)) adantr (` (abs) (` F (cv y))) (cv t) (if (br (cv u) (<_) (cv t)) (cv t) (cv u)) ltletrt caubnd.1 (cv y) ffvrni (` F (cv y)) absclt syl (/\ (e. (cv t) (RR)) (e. (cv u) (RR))) adantl (e. (cv t) (RR)) (e. (cv u) (RR)) pm3.26 (e. (cv y) (NN)) adantr (cv t) (RR) (cv u) (br (cv u) (<_) (cv t)) ifcl (e. (cv y) (NN)) adantr syl3anc mpan2d (br (cv y) (<_) (cv w)) imim2d (cv u) (cv t) max1ALT (e. (cv t) (RR)) (e. (cv y) (NN)) ad2antlr (` (abs) (` F (cv y))) (cv u) (if (br (cv u) (<_) (cv t)) (cv t) (cv u)) ltletrt caubnd.1 (cv y) ffvrni (` F (cv y)) absclt syl (/\ (e. (cv t) (RR)) (e. (cv u) (RR))) adantl (e. (cv t) (RR)) (e. (cv u) (RR)) pm3.27 (e. (cv y) (NN)) adantr (cv t) (RR) (cv u) (br (cv u) (<_) (cv t)) ifcl (e. (cv y) (NN)) adantr syl3anc mpan2d (br (cv w) (<) (cv y)) imim2d anim12d (br (cv y) (<_) (cv w)) (br (cv w) (<) (cv y)) (br (` (abs) (` F (cv y))) (<) (if (br (cv u) (<_) (cv t)) (cv t) (cv u))) jaob syl6ibr (e. (cv w) (NN)) adantlll mpid r19.20dva (cv t) (RR) (cv u) (br (cv u) (<_) (cv t)) ifcl (e. (cv w) (NN)) adantll jctild (cv x) (if (br (cv u) (<_) (cv t)) (cv t) (cv u)) (` (abs) (` F (cv y))) (<) breq2 y (NN) ralbidv (RR) rcla4ev syl6 y (NN) (-> (br (cv y) (<_) (cv w)) (br (` (abs) (` F (cv y))) (<) (cv t))) (-> (br (cv w) (<) (cv y)) (br (` (abs) (` F (cv y))) (<) (cv u))) r19.26 syl5ibr exp4b com23 imp r19.23adv exp31 r19.23adv mpd r19.23aiv ax-mp)) thm (caure ((y z) (w y) (w z) (x y) (w x) (F x) (G x)) ((caure.1 (:--> F (NN) (CC))) (caure.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caure.3 (Fn G (NN))) (caure.4 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` F (cv x))))))) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` G (cv y)) (-) (` G (cv w)))) (<) (cv z))))))) (caure.2 (cv x) (cv y) G fveq2 (cv x) (cv y) F fveq2 (Re) fveq2d eqeq12d caure.4 vtoclga (cv x) (cv w) G fveq2 (cv x) (cv w) F fveq2 (Re) fveq2d eqeq12d caure.4 vtoclga (-) opreqan12d (` F (cv y)) (` F (cv w)) resubt caure.1 (cv y) ffvrni caure.1 (cv w) ffvrni syl2an eqtr4d (abs) fveq2d (` F (cv y)) (` F (cv w)) subclt caure.1 (cv y) ffvrni caure.1 (cv w) ffvrni syl2an (opr (` F (cv y)) (-) (` F (cv w))) absrelet syl eqbrtrd (e. (cv z) (RR)) 3adant3 (` (abs) (opr (` G (cv y)) (-) (` G (cv w)))) (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (cv z) lelttrt (` G (cv y)) (` G (cv w)) resubclt G (NN) (RR) x ffnfv caure.3 caure.4 caure.1 (cv x) ffvrni (` F (cv x)) reclt syl eqeltrd rgen mpbir2an (cv y) ffvrni G (NN) (RR) x ffnfv caure.3 caure.4 caure.1 (cv x) ffvrni (` F (cv x)) reclt syl eqeltrd rgen mpbir2an (cv w) ffvrni syl2an recnd (opr (` G (cv y)) (-) (` G (cv w))) absclt syl (e. (cv z) (RR)) 3adant3 (` F (cv y)) (` F (cv w)) subclt caure.1 (cv y) ffvrni caure.1 (cv w) ffvrni syl2an (opr (` F (cv y)) (-) (` F (cv w))) absclt syl (e. (cv z) (RR)) 3adant3 (e. (cv y) (NN)) (e. (cv w) (NN)) (e. (cv z) (RR)) 3simp3 syl3anc mpand 3com13 3expa (br (cv w) (<) (cv y)) imim2d r19.20dva r19.22dva (br (0) (<) (cv z)) imim2d r19.20i ax-mp)) thm (cauim ((y z) (w y) (w z) (x y) (w x) (F x) (G x)) ((cauim.1 (:--> F (NN) (CC))) (cauim.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (cauim.3 (Fn G (NN))) (cauim.4 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Im) (` F (cv x))))))) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` G (cv y)) (-) (` G (cv w)))) (<) (cv z))))))) (cauim.2 (cv x) (cv y) G fveq2 (cv x) (cv y) F fveq2 (Im) fveq2d eqeq12d cauim.4 vtoclga (cv x) (cv w) G fveq2 (cv x) (cv w) F fveq2 (Im) fveq2d eqeq12d cauim.4 vtoclga (-) opreqan12d (` F (cv y)) (` F (cv w)) imsubt cauim.1 (cv y) ffvrni cauim.1 (cv w) ffvrni syl2an eqtr4d (abs) fveq2d (` F (cv y)) (` F (cv w)) subclt cauim.1 (cv y) ffvrni cauim.1 (cv w) ffvrni syl2an (opr (` F (cv y)) (-) (` F (cv w))) absimlet syl eqbrtrd (e. (cv z) (RR)) 3adant3 (` (abs) (opr (` G (cv y)) (-) (` G (cv w)))) (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (cv z) lelttrt (` G (cv y)) (` G (cv w)) resubclt G (NN) (RR) x ffnfv cauim.3 cauim.4 cauim.1 (cv x) ffvrni (` F (cv x)) imclt syl eqeltrd rgen mpbir2an (cv y) ffvrni G (NN) (RR) x ffnfv cauim.3 cauim.4 cauim.1 (cv x) ffvrni (` F (cv x)) imclt syl eqeltrd rgen mpbir2an (cv w) ffvrni syl2an recnd (opr (` G (cv y)) (-) (` G (cv w))) absclt syl (e. (cv z) (RR)) 3adant3 (` F (cv y)) (` F (cv w)) subclt cauim.1 (cv y) ffvrni cauim.1 (cv w) ffvrni syl2an (opr (` F (cv y)) (-) (` F (cv w))) absclt syl (e. (cv z) (RR)) 3adant3 (e. (cv y) (NN)) (e. (cv w) (NN)) (e. (cv z) (RR)) 3simp3 syl3anc mpand 3com13 3expa (br (cv w) (<) (cv y)) imim2d r19.20dva r19.22dva (br (0) (<) (cv z)) imim2d r19.20i ax-mp)) thm (ser1absdiflem ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z) (F x) (F y) (F z) (G x) (G y) (G z)) ((ser1absdif.1 (:--> F (NN) (CC))) (ser1absdif.2 (Fn G (NN))) (ser1absdif.3 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (abs) (` F (cv x))))))) (-> (/\ (e. A (NN)) (e. B (NN))) (br (` (abs) (opr (` (opr (+) (seq1) F) (opr A (+) B)) (-) (` (opr (+) (seq1) F) A))) (<_) (opr (` (opr (+) (seq1) G) (opr A (+) B)) (-) (` (opr (+) (seq1) G) A)))) ((cv y) (1) A (+) opreq2 (opr (+) (seq1) F) fveq2d (-) (` (opr (+) (seq1) F) A) opreq1d (abs) fveq2d (cv y) (1) A (+) opreq2 (opr (+) (seq1) G) fveq2d (-) (` (opr (+) (seq1) G) A) opreq1d (<_) breq12d (e. A (NN)) imbi2d (cv y) (cv z) A (+) opreq2 (opr (+) (seq1) F) fveq2d (-) (` (opr (+) (seq1) F) A) opreq1d (abs) fveq2d (cv y) (cv z) A (+) opreq2 (opr (+) (seq1) G) fveq2d (-) (` (opr (+) (seq1) G) A) opreq1d (<_) breq12d (e. A (NN)) imbi2d (cv y) (opr (cv z) (+) (1)) A (+) opreq2 (opr (+) (seq1) F) fveq2d (-) (` (opr (+) (seq1) F) A) opreq1d (abs) fveq2d (cv y) (opr (cv z) (+) (1)) A (+) opreq2 (opr (+) (seq1) G) fveq2d (-) (` (opr (+) (seq1) G) A) opreq1d (<_) breq12d (e. A (NN)) imbi2d (cv y) B A (+) opreq2 (opr (+) (seq1) F) fveq2d (-) (` (opr (+) (seq1) F) A) opreq1d (abs) fveq2d (cv y) B A (+) opreq2 (opr (+) (seq1) G) fveq2d (-) (` (opr (+) (seq1) G) A) opreq1d (<_) breq12d (e. A (NN)) imbi2d A peano2nn ser1absdif.1 (opr A (+) (1)) ffvrni syl (` F (opr A (+) (1))) absclt syl (` (abs) (` F (opr A (+) (1)))) leidt syl addex ser1absdif.1 nnex F (NN) (CC) (V) fex mp2an A seq1p1 (-) (` (opr (+) (seq1) F) A) opreq1d (` (opr (+) (seq1) F) A) (` F (opr A (+) (1))) pncan2t ser1absdif.1 A ser1cl A peano2nn ser1absdif.1 (opr A (+) (1)) ffvrni syl sylanc eqtrd (abs) fveq2d addex ser1absdif.2 nnex G (NN) (V) fnex mp2an A seq1p1 (-) (` (opr (+) (seq1) G) A) opreq1d (` (opr (+) (seq1) G) A) (` G (opr A (+) (1))) pncan2t G (NN) (RR) x ffnfv ser1absdif.2 ser1absdif.3 ser1absdif.1 (cv x) ffvrni (` F (cv x)) absclt syl eqeltrd rgen mpbir2an axresscn G (NN) (RR) (CC) fss mp2an A ser1cl A peano2nn (cv x) (opr A (+) (1)) G fveq2 (cv x) (opr A (+) (1)) F fveq2 (abs) fveq2d eqeq12d ser1absdif.3 vtoclga ser1absdif.1 (opr A (+) (1)) ffvrni (` F (opr A (+) (1))) absclt syl eqeltrd recnd syl sylanc A peano2nn (cv x) (opr A (+) (1)) G fveq2 (cv x) (opr A (+) (1)) F fveq2 (abs) fveq2d eqeq12d ser1absdif.3 vtoclga syl 3eqtrd 3brtr4d (opr (` (opr (+) (seq1) F) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) F) A)) (` F (opr (opr A (+) (cv z)) (+) (1))) abstrit (` (opr (+) (seq1) F) (opr A (+) (cv z))) (` (opr (+) (seq1) F) A) subclt A (cv z) nnaddclt ser1absdif.1 (opr A (+) (cv z)) ser1cl syl ser1absdif.1 A ser1cl (e. (cv z) (NN)) adantr sylanc A (cv z) nnaddclt (opr A (+) (cv z)) peano2nn syl ser1absdif.1 (opr (opr A (+) (cv z)) (+) (1)) ffvrni syl sylanc (` (abs) (opr (opr (` (opr (+) (seq1) F) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) F) A)) (+) (` F (opr (opr A (+) (cv z)) (+) (1))))) (opr (` (abs) (opr (` (opr (+) (seq1) F) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) F) A))) (+) (` (abs) (` F (opr (opr A (+) (cv z)) (+) (1))))) (opr (opr (` (opr (+) (seq1) G) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) G) A)) (+) (` (abs) (` F (opr (opr A (+) (cv z)) (+) (1))))) letrt (opr (` (opr (+) (seq1) F) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) F) A)) (` F (opr (opr A (+) (cv z)) (+) (1))) axaddcl (` (opr (+) (seq1) F) (opr A (+) (cv z))) (` (opr (+) (seq1) F) A) subclt A (cv z) nnaddclt ser1absdif.1 (opr A (+) (cv z)) ser1cl syl ser1absdif.1 A ser1cl (e. (cv z) (NN)) adantr sylanc A (cv z) nnaddclt (opr A (+) (cv z)) peano2nn syl ser1absdif.1 (opr (opr A (+) (cv z)) (+) (1)) ffvrni syl sylanc (opr (opr (` (opr (+) (seq1) F) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) F) A)) (+) (` F (opr (opr A (+) (cv z)) (+) (1)))) absclt syl (` (abs) (opr (` (opr (+) (seq1) F) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) F) A))) (` (abs) (` F (opr (opr A (+) (cv z)) (+) (1)))) axaddrcl (` (opr (+) (seq1) F) (opr A (+) (cv z))) (` (opr (+) (seq1) F) A) subclt A (cv z) nnaddclt ser1absdif.1 (opr A (+) (cv z)) ser1cl syl ser1absdif.1 A ser1cl (e. (cv z) (NN)) adantr sylanc (opr (` (opr (+) (seq1) F) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) F) A)) absclt syl A (cv z) nnaddclt (opr A (+) (cv z)) peano2nn syl ser1absdif.1 (opr (opr A (+) (cv z)) (+) (1)) ffvrni syl (` F (opr (opr A (+) (cv z)) (+) (1))) absclt syl sylanc (opr (` (opr (+) (seq1) G) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) G) A)) (` (abs) (` F (opr (opr A (+) (cv z)) (+) (1)))) axaddrcl (` (opr (+) (seq1) G) (opr A (+) (cv z))) (` (opr (+) (seq1) G) A) resubclt A (cv z) nnaddclt G (NN) (RR) x ffnfv ser1absdif.2 ser1absdif.3 ser1absdif.1 (cv x) ffvrni (` F (cv x)) absclt syl eqeltrd rgen mpbir2an (opr A (+) (cv z)) ser1recl syl G (NN) (RR) x ffnfv ser1absdif.2 ser1absdif.3 ser1absdif.1 (cv x) ffvrni (` F (cv x)) absclt syl eqeltrd rgen mpbir2an A ser1recl (e. (cv z) (NN)) adantr sylanc A (cv z) nnaddclt (opr A (+) (cv z)) peano2nn syl ser1absdif.1 (opr (opr A (+) (cv z)) (+) (1)) ffvrni syl (` F (opr (opr A (+) (cv z)) (+) (1))) absclt syl sylanc syl3anc mpand (` (abs) (opr (` (opr (+) (seq1) F) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) F) A))) (opr (` (opr (+) (seq1) G) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) G) A)) (` (abs) (` F (opr (opr A (+) (cv z)) (+) (1)))) leadd1t (` (opr (+) (seq1) F) (opr A (+) (cv z))) (` (opr (+) (seq1) F) A) subclt A (cv z) nnaddclt ser1absdif.1 (opr A (+) (cv z)) ser1cl syl ser1absdif.1 A ser1cl (e. (cv z) (NN)) adantr sylanc (opr (` (opr (+) (seq1) F) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) F) A)) absclt syl (` (opr (+) (seq1) G) (opr A (+) (cv z))) (` (opr (+) (seq1) G) A) resubclt A (cv z) nnaddclt G (NN) (RR) x ffnfv ser1absdif.2 ser1absdif.3 ser1absdif.1 (cv x) ffvrni (` F (cv x)) absclt syl eqeltrd rgen mpbir2an (opr A (+) (cv z)) ser1recl syl G (NN) (RR) x ffnfv ser1absdif.2 ser1absdif.3 ser1absdif.1 (cv x) ffvrni (` F (cv x)) absclt syl eqeltrd rgen mpbir2an A ser1recl (e. (cv z) (NN)) adantr sylanc A (cv z) nnaddclt (opr A (+) (cv z)) peano2nn syl ser1absdif.1 (opr (opr A (+) (cv z)) (+) (1)) ffvrni syl (` F (opr (opr A (+) (cv z)) (+) (1))) absclt syl syl3anc 1cn A (cv z) (1) axaddass mp3an3 A nncnt (cv z) nncnt syl2an (opr (+) (seq1) F) fveq2d A (cv z) nnaddclt addex ser1absdif.1 nnex F (NN) (CC) (V) fex mp2an (opr A (+) (cv z)) seq1p1 syl eqtr3d (-) (` (opr (+) (seq1) F) A) opreq1d (` (opr (+) (seq1) F) (opr A (+) (cv z))) (` F (opr (opr A (+) (cv z)) (+) (1))) (` (opr (+) (seq1) F) A) addsubt A (cv z) nnaddclt ser1absdif.1 (opr A (+) (cv z)) ser1cl syl A (cv z) nnaddclt (opr A (+) (cv z)) peano2nn syl ser1absdif.1 (opr (opr A (+) (cv z)) (+) (1)) ffvrni syl ser1absdif.1 A ser1cl (e. (cv z) (NN)) adantr syl3anc eqtrd (abs) fveq2d 1cn A (cv z) (1) axaddass mp3an3 A nncnt (cv z) nncnt syl2an (opr (+) (seq1) G) fveq2d A (cv z) nnaddclt addex ser1absdif.2 nnex G (NN) (V) fnex mp2an (opr A (+) (cv z)) seq1p1 syl eqtr3d (-) (` (opr (+) (seq1) G) A) opreq1d (` (opr (+) (seq1) G) (opr A (+) (cv z))) (` G (opr (opr A (+) (cv z)) (+) (1))) (` (opr (+) (seq1) G) A) addsubt A (cv z) nnaddclt G (NN) (RR) x ffnfv ser1absdif.2 ser1absdif.3 ser1absdif.1 (cv x) ffvrni (` F (cv x)) absclt syl eqeltrd rgen mpbir2an axresscn G (NN) (RR) (CC) fss mp2an (opr A (+) (cv z)) ser1cl syl A (cv z) nnaddclt (opr A (+) (cv z)) peano2nn syl (cv x) (opr (opr A (+) (cv z)) (+) (1)) G fveq2 (cv x) (opr (opr A (+) (cv z)) (+) (1)) F fveq2 (abs) fveq2d eqeq12d ser1absdif.3 vtoclga ser1absdif.1 (opr (opr A (+) (cv z)) (+) (1)) ffvrni (` F (opr (opr A (+) (cv z)) (+) (1))) absclt syl eqeltrd recnd syl G (NN) (RR) x ffnfv ser1absdif.2 ser1absdif.3 ser1absdif.1 (cv x) ffvrni (` F (cv x)) absclt syl eqeltrd rgen mpbir2an axresscn G (NN) (RR) (CC) fss mp2an A ser1cl (e. (cv z) (NN)) adantr syl3anc A (cv z) nnaddclt (opr A (+) (cv z)) peano2nn syl (cv x) (opr (opr A (+) (cv z)) (+) (1)) G fveq2 (cv x) (opr (opr A (+) (cv z)) (+) (1)) F fveq2 (abs) fveq2d eqeq12d ser1absdif.3 vtoclga syl (opr (` (opr (+) (seq1) G) (opr A (+) (cv z))) (-) (` (opr (+) (seq1) G) A)) (+) opreq2d 3eqtrd (<_) breq12d 3imtr4d expcom a2d nnind impcom)) thm (ser1absdif ((A x) (F x) (G x)) ((ser1absdif.1 (:--> F (NN) (CC))) (ser1absdif.2 (Fn G (NN))) (ser1absdif.3 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (abs) (` F (cv x))))))) (-> (/\/\ (e. A (NN)) (e. B (NN)) (br A (<) B)) (br (` (abs) (opr (` (opr (+) (seq1) F) B) (-) (` (opr (+) (seq1) F) A))) (<_) (opr (` (opr (+) (seq1) G) B) (-) (` (opr (+) (seq1) G) A)))) (ser1absdif.1 ser1absdif.2 ser1absdif.3 A (opr B (-) A) ser1absdiflem (e. A (NN)) (e. B (NN)) (br A (<) B) 3simp1 A B nnsubt biimp3a sylanc A B pncan3t A nncnt B nncnt syl2an (opr (+) (seq1) F) fveq2d (-) (` (opr (+) (seq1) F) A) opreq1d (abs) fveq2d (br A (<) B) 3adant3 A B pncan3t A nncnt B nncnt syl2an (opr (+) (seq1) G) fveq2d (-) (` (opr (+) (seq1) G) A) opreq1d (br A (<) B) 3adant3 3brtr3d)) thm (ser1re2 ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (N j) (N m)) ((ser1re2.1 (:--> F (NN) (CC))) (ser1re2.2 (e. G (V))) (ser1re2.3 (-> (e. (cv k) (NN)) (= (` G (cv k)) (` (Re) (` F (cv k))))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) G) N) (` (Re) (` (opr (+) (seq1) F) N)))) ((cv j) (1) (opr (+) (seq1) G) fveq2 (cv j) (1) (opr (+) (seq1) F) fveq2 (Re) fveq2d eqeq12d (cv j) (cv m) (opr (+) (seq1) G) fveq2 (cv j) (cv m) (opr (+) (seq1) F) fveq2 (Re) fveq2d eqeq12d (cv j) (opr (cv m) (+) (1)) (opr (+) (seq1) G) fveq2 (cv j) (opr (cv m) (+) (1)) (opr (+) (seq1) F) fveq2 (Re) fveq2d eqeq12d (cv j) N (opr (+) (seq1) G) fveq2 (cv j) N (opr (+) (seq1) F) fveq2 (Re) fveq2d eqeq12d 1nn (cv k) (1) G fveq2 (cv k) (1) F fveq2 (Re) fveq2d eqeq12d ser1re2.3 vtoclga ax-mp addex ser1re2.2 seq11 addex ser1re2.1 nnex F (NN) (CC) (V) fex mp2an seq11 (Re) fveq2i 3eqtr4 addex ser1re2.2 (cv m) seq1p1 (= (` (opr (+) (seq1) G) (cv m)) (` (Re) (` (opr (+) (seq1) F) (cv m)))) adantr (` (opr (+) (seq1) G) (cv m)) (` (Re) (` (opr (+) (seq1) F) (cv m))) (+) (` G (opr (cv m) (+) (1))) opreq1 (e. (cv m) (NN)) adantl (` (opr (+) (seq1) F) (cv m)) (` F (opr (cv m) (+) (1))) readdt ser1re2.1 (cv m) ser1cl (cv m) peano2nn ser1re2.1 (opr (cv m) (+) (1)) ffvrni syl sylanc addex ser1re2.1 nnex F (NN) (CC) (V) fex mp2an (cv m) seq1p1 (Re) fveq2d (cv m) peano2nn (cv k) (opr (cv m) (+) (1)) G fveq2 (cv k) (opr (cv m) (+) (1)) F fveq2 (Re) fveq2d eqeq12d ser1re2.3 vtoclga syl (` (Re) (` (opr (+) (seq1) F) (cv m))) (+) opreq2d 3eqtr4rd (= (` (opr (+) (seq1) G) (cv m)) (` (Re) (` (opr (+) (seq1) F) (cv m)))) adantr 3eqtrd ex nnind)) thm (ser0cj ((j k) (j m) (F j) (k m) (F k) (F m) (G j) (G k) (G m) (A j) (A m)) ((ser0cj.1 (:--> F (NN0) (CC))) (ser0cj.2 (e. G (V))) (ser0cj.3 (-> (e. (cv k) (NN0)) (= (` G (cv k)) (` (*) (` F (cv k))))))) (-> (e. A (NN0)) (= (` (opr (+) (seq0) G) A) (` (*) (` (opr (+) (seq0) F) A)))) ((cv j) (0) (opr (+) (seq0) G) fveq2 (cv j) (0) (opr (+) (seq0) F) fveq2 (*) fveq2d eqeq12d (cv j) (cv m) (opr (+) (seq0) G) fveq2 (cv j) (cv m) (opr (+) (seq0) F) fveq2 (*) fveq2d eqeq12d (cv j) (opr (cv m) (+) (1)) (opr (+) (seq0) G) fveq2 (cv j) (opr (cv m) (+) (1)) (opr (+) (seq0) F) fveq2 (*) fveq2d eqeq12d (cv j) A (opr (+) (seq0) G) fveq2 (cv j) A (opr (+) (seq0) F) fveq2 (*) fveq2d eqeq12d 0nn0 (cv k) (0) G fveq2 (cv k) (0) F fveq2 (*) fveq2d eqeq12d ser0cj.3 vtoclga ax-mp addex ser0cj.2 seq00 addex ser0cj.1 nn0ex F (NN0) (CC) (V) fex mp2an seq00 (*) fveq2i 3eqtr4 addex ser0cj.2 (cv m) seq0p1 (` (opr (+) (seq0) G) (cv m)) (` (*) (` (opr (+) (seq0) F) (cv m))) (+) (` G (opr (cv m) (+) (1))) opreq1 sylan9eq (` (opr (+) (seq0) F) (cv m)) (` F (opr (cv m) (+) (1))) cjaddt ser0cj.1 (cv m) ser0cl (cv m) peano2nn0 ser0cj.1 (opr (cv m) (+) (1)) ffvrni syl sylanc addex ser0cj.1 nn0ex F (NN0) (CC) (V) fex mp2an (cv m) seq0p1 (*) fveq2d (cv m) peano2nn0 (cv k) (opr (cv m) (+) (1)) G fveq2 (cv k) (opr (cv m) (+) (1)) F fveq2 (*) fveq2d eqeq12d ser0cj.3 vtoclga syl (` (*) (` (opr (+) (seq0) F) (cv m))) (+) opreq2d 3eqtr4rd (= (` (opr (+) (seq0) G) (cv m)) (` (*) (` (opr (+) (seq0) F) (cv m)))) adantr eqtrd ex nn0ind)) thm (ser1cj ((x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z) (A y) (A z)) ((ser1cj.1 (:--> F (NN) (CC))) (ser1cj.2 (e. G (V))) (ser1cj.3 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (*) (` F (cv x))))))) (-> (e. A (NN)) (= (` (opr (+) (seq1) G) A) (` (*) (` (opr (+) (seq1) F) A)))) ((cv y) (1) (opr (+) (seq1) G) fveq2 (cv y) (1) (opr (+) (seq1) F) fveq2 (*) fveq2d eqeq12d (cv y) (cv z) (opr (+) (seq1) G) fveq2 (cv y) (cv z) (opr (+) (seq1) F) fveq2 (*) fveq2d eqeq12d (cv y) (opr (cv z) (+) (1)) (opr (+) (seq1) G) fveq2 (cv y) (opr (cv z) (+) (1)) (opr (+) (seq1) F) fveq2 (*) fveq2d eqeq12d (cv y) A (opr (+) (seq1) G) fveq2 (cv y) A (opr (+) (seq1) F) fveq2 (*) fveq2d eqeq12d 1nn (cv x) (1) G fveq2 (cv x) (1) F fveq2 (*) fveq2d eqeq12d ser1cj.3 vtoclga ax-mp addex ser1cj.2 seq11 addex ser1cj.1 nnex F (NN) (CC) (V) fex mp2an seq11 (*) fveq2i 3eqtr4 addex ser1cj.2 (cv z) seq1p1 (` (opr (+) (seq1) G) (cv z)) (` (*) (` (opr (+) (seq1) F) (cv z))) (+) (` G (opr (cv z) (+) (1))) opreq1 sylan9eq (` (opr (+) (seq1) F) (cv z)) (` F (opr (cv z) (+) (1))) cjaddt ser1cj.1 (cv z) ser1cl (cv z) peano2nn ser1cj.1 (opr (cv z) (+) (1)) ffvrni syl sylanc addex ser1cj.1 nnex F (NN) (CC) (V) fex mp2an (cv z) seq1p1 (*) fveq2d (cv z) peano2nn (cv x) (opr (cv z) (+) (1)) G fveq2 (cv x) (opr (cv z) (+) (1)) F fveq2 (*) fveq2d eqeq12d ser1cj.3 vtoclga syl (` (*) (` (opr (+) (seq1) F) (cv z))) (+) opreq2d 3eqtr4rd (= (` (opr (+) (seq1) G) (cv z)) (` (*) (` (opr (+) (seq1) F) (cv z)))) adantr eqtrd ex nnind)) thm (facnnt () () (-> (e. N (NN)) (= (` (!) N) (` (opr (x.) (seq1) (|` (I) (NN))) N))) (df-fac (!) (u. (opr (x.) (seq1) (|` (I) (NN))) ({} (<,> (0) (1)))) (NN) reseq1 ax-mp (opr (x.) (seq1) (|` (I) (NN))) ({} (<,> (0) (1))) (NN) resundir ({} (0)) (NN) incom 0nnn (NN) (0) disjsn mpbir eqtr 0nn0 elisseti 1nn elisseti f1osn ({} (<,> (0) (1))) ({} (0)) ({} (1)) f1ofn ax-mp ({} (<,> (0) (1))) ({} (0)) (NN) fnresdisj ax-mp mpbi (|` (opr (x.) (seq1) (|` (I) (NN))) (NN)) uneq2i eqtr (|` (opr (x.) (seq1) (|` (I) (NN))) (NN)) un0 3eqtr N fveq1i (e. N (NN)) a1i N (NN) (!) fvres N (NN) (opr (x.) (seq1) (|` (I) (NN))) fvres 3eqtr3d)) thm (fac0 () () (= (` (!) (0)) (1)) (df-fac (!) (u. (opr (x.) (seq1) (|` (I) (NN))) ({} (<,> (0) (1)))) ({} (0)) reseq1 ax-mp (opr (x.) (seq1) (|` (I) (NN))) ({} (<,> (0) (1))) ({} (0)) resundir 0nnn (NN) (0) disjsn mpbir mulex funi nnex (I) (NN) (V) resfunexg mp2an seq1fn (opr (x.) (seq1) (|` (I) (NN))) (NN) ({} (0)) fnresdisj ax-mp mpbi 0nn0 elisseti (1) relsn (0) (1) dmsnop ({} (0)) ssid eqsstr ({} (<,> (0) (1))) ({} (0)) relssres mp2an uneq12i ({/}) ({} (<,> (0) (1))) uncom ({} (<,> (0) (1))) un0 3eqtr 3eqtr (0) fveq1i 0nn0 elisseti snid (0) ({} (0)) (!) fvres ax-mp 0nn0 elisseti 1nn0 elisseti fvsn 3eqtr3)) thm (fac1 () () (= (` (!) (1)) (1)) (1nn (1) facnnt ax-mp mulex funi nnex (I) (NN) (V) resfunexg mp2an seq11 1nn (1) (NN) (I) fvres ax-mp 1nn (1) (NN) fvi ax-mp eqtr 3eqtr)) thm (facp1t () () (-> (e. N (NN0)) (= (` (!) (opr N (+) (1))) (opr (` (!) N) (x.) (opr N (+) (1))))) (N elnn0 N peano2nn (opr N (+) (1)) facnnt syl mulex funi nnex (I) (NN) (V) resfunexg mp2an N seq1p1 N peano2nn (opr N (+) (1)) (NN) (I) fvres syl N (+) (1) oprex (opr N (+) (1)) (V) fvi ax-mp syl6eq (` (opr (x.) (seq1) (|` (I) (NN))) N) (x.) opreq2d N facnnt (x.) (opr N (+) (1)) opreq1d eqtr4d 3eqtrd N (0) (+) (1) opreq1 (!) fveq2d 1cn addid2 (!) fveq2i fac1 eqtr syl6eq N (0) (!) fveq2 N (0) (+) (1) opreq1 (x.) opreq12d fac0 1cn addid2 (x.) opreq12i 1cn mulid1 eqtr syl6eq eqtr4d jaoi sylbi)) thm (fac2 () () (= (` (!) (2)) (2)) (df-2 (!) fveq2i 1nn0 (1) facp1t ax-mp fac1 df-2 eqcomi (x.) opreq12i 2cn mulid2 eqtr 3eqtr)) thm (fac3 () () (= (` (!) (3)) (6)) (df-3 (!) fveq2i 2nn0 (2) facp1t ax-mp fac2 df-3 eqcomi (x.) opreq12i 2cn 3re recn mulcom 3t2e6 3eqtr 3eqtr)) thm (facnn2t () () (-> (e. N (NN)) (= (` (!) N) (opr (` (!) (opr N (-) (1))) (x.) N))) (N elnnnn0 (opr N (-) (1)) facp1t (e. N (CC)) adantl 1cn N (1) npcant mpan2 (!) fveq2d (e. (opr N (-) (1)) (NN0)) adantr 1cn N (1) npcant mpan2 (` (!) (opr N (-) (1))) (x.) opreq2d (e. (opr N (-) (1)) (NN0)) adantr 3eqtr3d sylbi)) thm (facclt ((j k) (N j) (N k)) () (-> (e. N (NN0)) (e. (` (!) N) (NN))) ((cv j) (0) (!) fveq2 (NN) eleq1d (cv j) (cv k) (!) fveq2 (NN) eleq1d (cv j) (opr (cv k) (+) (1)) (!) fveq2 (NN) eleq1d (cv j) N (!) fveq2 (NN) eleq1d fac0 1nn eqeltr (cv k) facp1t (e. (` (!) (cv k)) (NN)) adantl (` (!) (cv k)) (opr (cv k) (+) (1)) nnmulclt (cv k) nn0p1nnt sylan2 eqeltrd expcom nn0ind)) thm (facne0t () () (-> (e. N (NN0)) (=/= (` (!) N) (0))) (N facclt (` (!) N) nnne0t syl)) thm (facdivt ((M j) (j k) (N j) (N k)) () (-> (/\/\ (e. M (NN0)) (e. N (NN)) (br N (<_) M)) (e. (opr (` (!) M) (/) N) (NN))) ((cv j) (0) N (<_) breq2 (cv j) (0) (!) fveq2 (/) N opreq1d (NN) eleq1d imbi12d (e. N (NN)) imbi2d (cv j) (cv k) N (<_) breq2 (cv j) (cv k) (!) fveq2 (/) N opreq1d (NN) eleq1d imbi12d (e. N (NN)) imbi2d (cv j) (opr (cv k) (+) (1)) N (<_) breq2 (cv j) (opr (cv k) (+) (1)) (!) fveq2 (/) N opreq1d (NN) eleq1d imbi12d (e. N (NN)) imbi2d (cv j) M N (<_) breq2 (cv j) M (!) fveq2 (/) N opreq1d (NN) eleq1d imbi12d (e. N (NN)) imbi2d N nngt0t N nnret 0re (0) N ltnlet mpan syl mpbid (e. (opr (` (!) (0)) (/) N) (NN)) pm2.21d N (opr (cv k) (+) (1)) leloet N nnret (cv k) peano2nn0 (opr (cv k) (+) (1)) nn0ret syl syl2an N (cv k) nn0leltp1t N nnnn0t sylan (opr (` (!) (cv k)) (/) N) (opr (cv k) (+) (1)) nnmulclt (cv k) nn0p1nnt sylan2 expcom (e. N (NN)) adantl (` (!) (cv k)) (opr (cv k) (+) (1)) N div23t (cv k) facclt (` (!) (cv k)) nncnt syl (e. N (NN)) adantl (cv k) peano2nn0 (opr (cv k) (+) (1)) nn0ret syl recnd (e. N (NN)) adantl N nncnt (e. (cv k) (NN0)) adantr 3jca N nnne0t (e. (cv k) (NN0)) adantr sylanc (NN) eleq1d sylibrd (br N (<_) (cv k)) imim2d com23 sylbird N (opr (cv k) (+) (1)) (` (!) (cv k)) (x.) opreq2 (/) N opreq1d (NN) eleq1d N (` (!) (cv k)) divcan4t N nncnt (e. (cv k) (NN0)) adantr (cv k) facclt (` (!) (cv k)) nncnt syl (e. N (NN)) adantl N nnne0t (e. (cv k) (NN0)) adantr syl3anc (cv k) facclt (e. N (NN)) adantl eqeltrd syl5bi com12 (-> (br N (<_) (cv k)) (e. (opr (` (!) (cv k)) (/) N) (NN))) a1dd jaod sylbid ex com34 com12 imp4d (cv k) facp1t (/) N opreq1d (NN) eleq1d sylibrd exp4d a2d nn0ind 3imp)) thm (facndivt () () (-> (/\ (/\ (e. M (NN0)) (e. N (NN))) (/\ (br (1) (<) N) (br N (<_) M))) (-. (e. (opr (opr (` (!) M) (+) (1)) (/) N) (ZZ)))) (N recnzt N nnret sylan (br N (<_) M) adantrr (e. M (NN0)) adantll (opr (opr (` (!) M) (+) (1)) (/) N) (opr (` (!) M) (/) N) zsubclt ex M N facdivt 3expa (opr (` (!) M) (/) N) nnzt syl (br (1) (<) N) adantrl syl5com (opr (` (!) M) (+) (1)) (` (!) M) N divsubdirt M facclt (` (!) M) nncnt syl (` (!) M) peano2cn syl (e. N (NN)) (/\ (br (1) (<) N) (br N (<_) M)) ad2antrr M facclt (` (!) M) nncnt syl (e. N (NN)) (/\ (br (1) (<) N) (br N (<_) M)) ad2antrr N nncnt (e. M (NN0)) (/\ (br (1) (<) N) (br N (<_) M)) ad2antlr 3jca N nnne0t (e. M (NN0)) (/\ (br (1) (<) N) (br N (<_) M)) ad2antlr sylanc M facclt (` (!) M) nncnt syl 1cn (` (!) M) (1) pncan2t mpan2 syl (/) N opreq1d (e. N (NN)) (/\ (br (1) (<) N) (br N (<_) M)) ad2antrr eqtr3d (ZZ) eleq1d sylibd mtod)) thm (facwordit ((j k) (M j) (M k) (N j)) () (-> (/\/\ (e. M (NN0)) (e. N (NN0)) (br M (<_) N)) (br (` (!) M) (<_) (` (!) N))) ((cv j) (0) M (<_) breq2 (e. M (NN0)) anbi2d (cv j) (0) (!) fveq2 (` (!) M) (<_) breq2d imbi12d (cv j) (cv k) M (<_) breq2 (e. M (NN0)) anbi2d (cv j) (cv k) (!) fveq2 (` (!) M) (<_) breq2d imbi12d (cv j) (opr (cv k) (+) (1)) M (<_) breq2 (e. M (NN0)) anbi2d (cv j) (opr (cv k) (+) (1)) (!) fveq2 (` (!) M) (<_) breq2d imbi12d (cv j) N M (<_) breq2 (e. M (NN0)) anbi2d (cv j) N (!) fveq2 (` (!) M) (<_) breq2d imbi12d M nn0le0eq0t biimpa (!) fveq2d fac0 ax1re eqeltr leid syl6eqbr M (opr (cv k) (+) (1)) leloet M nn0ret (cv k) nn0ret (cv k) peano2re syl syl2an M (cv k) nn0leltp1t (cv k) facclt (` (!) (cv k)) nncnt (` (!) (cv k)) ax1id 3syl ax1re (1) (opr (cv k) (+) (1)) (` (!) (cv k)) lemul2it mp3anl1 (cv k) nn0ret (cv k) peano2re syl (cv k) facclt (` (!) (cv k)) nnret syl jca (cv k) facclt (` (!) (cv k)) nnnn0t (` (!) (cv k)) nn0ge0t 3syl (cv k) nn0ge0t (cv k) nn0ret 0re ax1re (0) (cv k) (1) leadd1t mp3an13 syl mpbid 1cn addid2 syl5eqbrr jca sylanc eqbrtrrd (cv k) facp1t breqtrrd (e. M (NN0)) adantl (` (!) M) (` (!) (cv k)) (` (!) (opr (cv k) (+) (1))) letrt M facclt (` (!) M) nnret syl (e. (cv k) (NN0)) adantr (cv k) facclt (` (!) (cv k)) nnret syl (e. M (NN0)) adantl (cv k) peano2nn0 (opr (cv k) (+) (1)) facclt (` (!) (opr (cv k) (+) (1))) nnret 3syl (e. M (NN0)) adantl syl3anc mpan2d (br M (<_) (cv k)) imim2d com23 sylbird M facclt (` (!) M) nnret syl (` (!) M) leidt (` (!) M) (` (!) (opr (cv k) (+) (1))) (` (!) M) (<_) breq2 biimpcd 3syl M (opr (cv k) (+) (1)) (!) fveq2 syl5 (e. (cv k) (NN0)) adantr (-> (br M (<_) (cv k)) (br (` (!) M) (<_) (` (!) (cv k)))) a1dd jaod sylbid ex com13 com4l a2d imp4a (e. M (NN0)) (br M (<_) (cv k)) (br (` (!) M) (<_) (` (!) (cv k))) impexp syl5ib nn0ind 3impib 3com12)) thm (faclbnd ((j k) (M j) (M k) (N j)) () (-> (/\ (e. M (NN0)) (e. N (NN0))) (br (opr M (^) (opr N (+) (1))) (<_) (opr (opr M (^) M) (x.) (` (!) N)))) ((cv j) (0) (+) (1) opreq1 M (^) opreq2d (cv j) (0) (!) fveq2 (opr M (^) M) (x.) opreq2d (<_) breq12d (e. M (NN)) imbi2d (cv j) (cv k) (+) (1) opreq1 M (^) opreq2d (cv j) (cv k) (!) fveq2 (opr M (^) M) (x.) opreq2d (<_) breq12d (e. M (NN)) imbi2d (cv j) (opr (cv k) (+) (1)) (+) (1) opreq1 M (^) opreq2d (cv j) (opr (cv k) (+) (1)) (!) fveq2 (opr M (^) M) (x.) opreq2d (<_) breq12d (e. M (NN)) imbi2d (cv j) N (+) (1) opreq1 M (^) opreq2d (cv j) N (!) fveq2 (opr M (^) M) (x.) opreq2d (<_) breq12d (e. M (NN)) imbi2d M (1) M expwordit M nnret 1nn0 (e. M (NN)) a1i M nnnn0t 3jca M nnge1t M nnge1t jca sylanc 1cn addid2 M (^) opreq2i (e. M (NN)) a1i M M reexpclt M nnret M nnnn0t sylanc recnd (opr M (^) M) ax1id syl fac0 (opr M (^) M) (x.) opreq2i syl5eq 3brtr4d M (opr (cv k) (+) (1)) lelttrit M nnret (cv k) nn0ret (cv k) peano2re syl syl2an (opr M (^) (opr (cv k) (+) (1))) (opr (opr M (^) M) (x.) (` (!) (cv k))) M (opr (cv k) (+) (1)) lemul12it M (opr (cv k) (+) (1)) reexpclt M nnret (cv k) peano2nn0 syl2an (br (opr M (^) (opr (cv k) (+) (1))) (<_) (opr (opr M (^) M) (x.) (` (!) (cv k)))) adantr (opr M (^) M) (` (!) (cv k)) axmulrcl M M reexpclt M nnret M nnnn0t sylanc (cv k) facclt (` (!) (cv k)) nnret syl syl2an (br (opr M (^) (opr (cv k) (+) (1))) (<_) (opr (opr M (^) M) (x.) (` (!) (cv k)))) adantr jca M (opr (cv k) (+) (1)) expge0t M nnret (e. (cv k) (NN0)) adantr (cv k) peano2nn0 (e. M (NN)) adantl M nnnn0t M nn0ge0t syl (e. (cv k) (NN0)) adantr syl3anc (br (opr M (^) (opr (cv k) (+) (1))) (<_) (opr (opr M (^) M) (x.) (` (!) (cv k)))) anim1i jca M nnret (e. (cv k) (NN0)) (br M (<_) (opr (cv k) (+) (1))) ad2antrr (cv k) nn0ret (cv k) peano2re syl (e. M (NN)) (br M (<_) (opr (cv k) (+) (1))) ad2antlr jca M nnnn0t M nn0ge0t syl (e. (cv k) (NN0)) adantr (br M (<_) (opr (cv k) (+) (1))) anim1i jca syl2an anandis M (opr (cv k) (+) (1)) expp1t M nncnt (cv k) peano2nn0 syl2an (/\ (br (opr M (^) (opr (cv k) (+) (1))) (<_) (opr (opr M (^) M) (x.) (` (!) (cv k)))) (br M (<_) (opr (cv k) (+) (1)))) adantr (cv k) facp1t (e. M (NN)) adantl (opr M (^) M) (x.) opreq2d (opr M (^) M) (` (!) (cv k)) (opr (cv k) (+) (1)) axmulass M M reexpclt M nnret M nnnn0t sylanc recnd (e. (cv k) (NN0)) adantr (cv k) facclt (` (!) (cv k)) nnret syl recnd (e. M (NN)) adantl (cv k) nn0cnt (cv k) peano2cn syl (e. M (NN)) adantl syl3anc eqtr4d (/\ (br (opr M (^) (opr (cv k) (+) (1))) (<_) (opr (opr M (^) M) (x.) (` (!) (cv k)))) (br M (<_) (opr (cv k) (+) (1)))) adantr 3brtr4d exp32 com23 (opr (cv k) (+) (1)) M nn0ltp1let (cv k) peano2nn0 M nnnn0t syl2an ancoms M (opr (opr (cv k) (+) (1)) (+) (1)) reexpclt M nnret (cv k) peano2nn0 (opr (cv k) (+) (1)) peano2nn0 syl syl2an (br (opr (opr (cv k) (+) (1)) (+) (1)) (<_) M) adantr M M reexpclt M nnret M nnnn0t sylanc (e. (cv k) (NN0)) (br (opr (opr (cv k) (+) (1)) (+) (1)) (<_) M) ad2antrr (opr M (^) M) (` (!) (opr (cv k) (+) (1))) axmulrcl M M reexpclt M nnret M nnnn0t sylanc (cv k) peano2nn0 (opr (cv k) (+) (1)) facclt (` (!) (opr (cv k) (+) (1))) nnret 3syl syl2an (br (opr (opr (cv k) (+) (1)) (+) (1)) (<_) M) adantr M (opr (opr (cv k) (+) (1)) (+) (1)) M expwordit M nnret (e. (cv k) (NN0)) (br (opr (opr (cv k) (+) (1)) (+) (1)) (<_) M) ad2antrr (cv k) peano2nn0 (opr (cv k) (+) (1)) peano2nn0 syl (e. M (NN)) (br (opr (opr (cv k) (+) (1)) (+) (1)) (<_) M) ad2antlr M nnnn0t (e. (cv k) (NN0)) (br (opr (opr (cv k) (+) (1)) (+) (1)) (<_) M) ad2antrr 3jca M nnge1t (e. (cv k) (NN0)) adantr (br (opr (opr (cv k) (+) (1)) (+) (1)) (<_) M) anim1i sylanc (opr M (^) M) (` (!) (opr (cv k) (+) (1))) lemulge11t M M reexpclt M nnret M nnnn0t sylanc (cv k) peano2nn0 (opr (cv k) (+) (1)) facclt (` (!) (opr (cv k) (+) (1))) nnret 3syl anim12i M nnnn0t M M expge0t M nn0ret (e. M (NN0)) id M nn0ge0t syl3anc syl (cv k) peano2nn0 (opr (cv k) (+) (1)) facclt (` (!) (opr (cv k) (+) (1))) nnge1t 3syl anim12i sylanc (br (opr (opr (cv k) (+) (1)) (+) (1)) (<_) M) adantr letrd ex sylbid (br (opr M (^) (opr (cv k) (+) (1))) (<_) (opr (opr M (^) M) (x.) (` (!) (cv k)))) a1dd jaod mpd expcom a2d nn0ind impcom M (0) (^) (opr N (+) (1)) opreq1 M (0) M (0) (^) opreq12 anidms (x.) (` (!) N) opreq1d (<_) breq12d N facclt (` (!) N) nnnn0t (` (!) N) nn0ge0t 3syl N nn0p1nnt (opr N (+) (1)) 0expt syl N facclt (` (!) N) nncnt (` (!) N) mulid2t 3syl 0cn (0) exp0t ax-mp (x.) (` (!) N) opreq1i syl5eq 3brtr4d syl5bir imp jaoian M elnn0 sylanb)) thm (faclbnd2 () () (-> (e. N (NN0)) (br (opr (opr (2) (^) N) (/) (2)) (<_) (` (!) N))) (2cn (2) N expp1t mpan (/) (opr (2) (x.) (2)) opreq1d sq2 2t2e4 eqtr4 (opr (2) (^) (opr N (+) (1))) (/) opreq2i syl5eq 2cn (2) N expclt mpan 2cn 2cn 2cn pm3.2i 2re 2pos gt0ne0i 2re 2pos gt0ne0i pm3.2i (opr (2) (^) N) (2) (2) (2) divmuldivt mpan2 mpan2 mpan2 syl 2cn (2) N expclt mpan (opr (2) (^) N) halfclt (opr (opr (2) (^) N) (/) (2)) ax1id 3syl 2cn 2re 2pos gt0ne0i divid (opr (opr (2) (^) N) (/) (2)) (x.) opreq2i syl5eq 3eqtr2rd 2nn0 (2) N faclbnd mpan sq2 4re eqeltr 4pos sq2 breqtrr (opr (2) (^) (opr N (+) (1))) (opr (2) (^) (2)) (` (!) N) ledivmult mpan2 mp3an2 N peano2nn0 2re (2) (opr N (+) (1)) reexpclt mpan syl N facclt (` (!) N) nnret syl sylanc mpbird eqbrtrd)) thm (faclbnd3 () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (br (opr M (^) N) (<_) (opr (opr M (^) M) (x.) (` (!) N)))) (M N (opr N (+) (1)) expwordit M nnret (e. N (NN0)) adantr (e. M (NN)) (e. N (NN0)) pm3.27 N peano2nn0 (e. M (NN)) adantl 3jca M nnge1t N N letrp1t N nn0ret N nn0ret N nn0ret N leidt syl syl3anc anim12i sylanc M N faclbnd M nnnn0t sylan (opr M (^) N) (opr M (^) (opr N (+) (1))) (opr (opr M (^) M) (x.) (` (!) N)) letrt M N reexpclt M nn0ret sylan M (opr N (+) (1)) reexpclt M nn0ret N peano2nn0 syl2an (opr M (^) M) (` (!) N) axmulrcl M nn0ret M M reexpclt mpancom N facclt (` (!) N) nnret syl syl2an syl3anc M nnnn0t sylan mp2and N elnn0 N 0expt 0re ax1re lt01 ltlei syl6eqbr N (0) (0) (^) opreq2 0cn (0) exp0t ax-mp ax1re leid eqbrtr syl6eqbr jaoi sylbi N facclt 1nn (1) (` (!) N) nnmulclt mpan syl (opr (1) (x.) (` (!) N)) nnge1t syl ax1re (opr (0) (^) N) (1) (opr (1) (x.) (` (!) N)) letrt mp3an2 0re (0) N reexpclt mpan N facclt (` (!) N) nnret syl ax1re (1) (` (!) N) axmulrcl mpan syl sylanc mp2and (= M (0)) adantl M (0) (^) N opreq1 M (0) M (0) (^) opreq12 anidms 0cn (0) exp0t ax-mp syl6eq (x.) (` (!) N) opreq1d (<_) breq12d (e. N (NN0)) adantr mpbird jaoian M elnn0 sylanb)) thm (faclbnd4lem1 () ((faclbnd4lem1.1 (e. N (NN))) (faclbnd4lem1.2 (e. K (NN0))) (faclbnd4lem1.3 (e. M (NN0)))) (-> (br (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1))))) (br (opr (opr N (^) (opr K (+) (1))) (x.) (opr M (^) N)) (<_) (opr (opr (opr (2) (^) (opr (opr K (+) (1)) (^) (2))) (x.) (opr M (^) (opr M (+) (opr K (+) (1))))) (x.) (` (!) N)))) (faclbnd4lem1.1 nnre ax1re N (1) lelttrit mp2an faclbnd4lem1.1 nnre ax1re letri3 faclbnd4lem1.1 N nnge1t ax-mp mpbiran2 ax1re 2re faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnnn0 2nn0 (opr K (+) (1)) (2) nn0expclt mp2an (2) (opr (opr K (+) (1)) (^) (2)) reexpclt mp2an pm3.2i 0re ax1re lt01 ltlei 2cn (2) exp0t ax-mp 2re 0nn0 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnnn0 2nn0 (opr K (+) (1)) (2) nn0expclt mp2an 3pm3.2i ax1re 2re 1lt2 ltlei faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnnn0 2nn0 (opr K (+) (1)) (2) nn0expclt mp2an nn0ge0 pm3.2i (2) (0) (opr (opr K (+) (1)) (^) (2)) expwordit mp2an eqbrtrr pm3.2i pm3.2i faclbnd4lem1.3 nn0re faclbnd4lem1.3 faclbnd4lem1.3 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an M (opr K (+) (1)) nn0nnaddclt mp2an nnnn0 M (opr M (+) (opr K (+) (1))) nn0expclt mp2an nn0re pm3.2i faclbnd4lem1.3 nn0ge0 faclbnd4lem1.3 M elnn0 M nncnt M exp1t syl M nnge1t faclbnd4lem1.3 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an M (opr K (+) (1)) nn0nnaddclt mp2an (opr M (+) (opr K (+) (1))) nnge1t ax-mp faclbnd4lem1.3 nn0re 1nn0 faclbnd4lem1.3 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an M (opr K (+) (1)) nn0nnaddclt mp2an nnnn0 3pm3.2i M (1) (opr M (+) (opr K (+) (1))) expwordit mpan mpan2 syl eqbrtrrd faclbnd4lem1.3 faclbnd4lem1.3 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an M (opr K (+) (1)) nn0nnaddclt mp2an nnnn0 M (opr M (+) (opr K (+) (1))) nn0expclt mp2an nn0ge0 M (0) (<_) (opr M (^) (opr M (+) (opr K (+) (1)))) breq1 mpbiri jaoi sylbi ax-mp pm3.2i pm3.2i (1) (opr (2) (^) (opr (opr K (+) (1)) (^) (2))) M (opr M (^) (opr M (+) (opr K (+) (1)))) lemul12it mp2an N (1) (^) (opr K (+) (1)) opreq1 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnnn0 (opr K (+) (1)) 1expt ax-mp syl6eq N (1) M (^) opreq2 faclbnd4lem1.3 nn0cn M exp1t ax-mp syl6eq (x.) opreq12d N (1) (!) fveq2 fac1 syl6eq (opr (opr (2) (^) (opr (opr K (+) (1)) (^) (2))) (x.) (opr M (^) (opr M (+) (opr K (+) (1))))) (x.) opreq2d 2re faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnnn0 2nn0 (opr K (+) (1)) (2) nn0expclt mp2an (2) (opr (opr K (+) (1)) (^) (2)) reexpclt mp2an recn faclbnd4lem1.3 faclbnd4lem1.3 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an M (opr K (+) (1)) nn0nnaddclt mp2an nnnn0 M (opr M (+) (opr K (+) (1))) nn0expclt mp2an nn0cn mulcl mulid1 syl6eq (<_) breq12d mpbiri sylbir (br (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1))))) adantr faclbnd4lem1.1 nnre faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnnn0 N (opr K (+) (1)) reexpclt mp2an faclbnd4lem1.3 nn0re faclbnd4lem1.1 nnnn0 M N reexpclt mp2an remulcl (/\ (br (1) (<) N) (br (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1)))))) a1i 2re faclbnd4lem1.2 2nn0 K (2) nn0expclt mp2an (2) (opr K (^) (2)) reexpclt mp2an 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an nn0re remulcl faclbnd4lem1.3 faclbnd4lem1.3 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an M (opr K (+) (1)) nn0nnaddclt mp2an nnnn0 M (opr M (+) (opr K (+) (1))) nn0expclt mp2an faclbnd4lem1.1 nnnn0 N facclt ax-mp nnnn0 nn0mulcl nn0re remulcl (/\ (br (1) (<) N) (br (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1)))))) a1i 2re faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnnn0 2nn0 (opr K (+) (1)) (2) nn0expclt mp2an (2) (opr (opr K (+) (1)) (^) (2)) reexpclt mp2an faclbnd4lem1.3 faclbnd4lem1.3 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an M (opr K (+) (1)) nn0nnaddclt mp2an nnnn0 M (opr M (+) (opr K (+) (1))) nn0expclt mp2an faclbnd4lem1.1 nnnn0 N facclt ax-mp nnnn0 nn0mulcl nn0re remulcl (/\ (br (1) (<) N) (br (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1)))))) a1i faclbnd4lem1.1 nnre faclbnd4lem1.2 N K reexpclt mp2an faclbnd4lem1.1 nnnn0 faclbnd4lem1.3 faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i M (opr N (-) (1)) nn0expclt mp2an faclbnd4lem1.3 nn0mulcl nn0mulcl nn0re remulcl (/\ (br (1) (<) N) (br (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1)))))) a1i faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i nn0re faclbnd4lem1.2 (opr N (-) (1)) K reexpclt mp2an faclbnd4lem1.3 faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i M (opr N (-) (1)) nn0expclt mp2an nn0re remulcl 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an faclbnd4lem1.1 nnnn0 nn0mulcl faclbnd4lem1.3 nn0mulcl nn0re remulcl (/\ (br (1) (<) N) (br (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1)))))) a1i 2re faclbnd4lem1.2 2nn0 K (2) nn0expclt mp2an (2) (opr K (^) (2)) reexpclt mp2an faclbnd4lem1.3 nn0re faclbnd4lem1.3 faclbnd4lem1.2 nn0addcl M (opr M (+) K) reexpclt mp2an remulcl faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i (opr N (-) (1)) facclt ax-mp nnre remulcl 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an faclbnd4lem1.1 nnnn0 nn0mulcl faclbnd4lem1.3 nn0mulcl nn0re remulcl (/\ (br (1) (<) N) (br (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1)))))) a1i 1nn faclbnd4lem1.1 (1) N nnltp1let mp2an df-2 (<_) N breq1i bitr4 faclbnd4lem1.1 nnre faclbnd4lem1.2 N K expbndt mp3an12 sylbi faclbnd4lem1.1 nnnn0 faclbnd4lem1.3 faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i M (opr N (-) (1)) nn0expclt mp2an faclbnd4lem1.3 nn0mulcl nn0mulcl nn0ge0 faclbnd4lem1.1 nnre faclbnd4lem1.2 N K reexpclt mp2an 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an nn0re faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i nn0re faclbnd4lem1.2 (opr N (-) (1)) K reexpclt mp2an remulcl faclbnd4lem1.1 nnnn0 faclbnd4lem1.3 faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i M (opr N (-) (1)) nn0expclt mp2an faclbnd4lem1.3 nn0mulcl nn0mulcl nn0re 3pm3.2i (opr N (^) K) (opr (opr (2) (^) K) (x.) (opr (opr N (-) (1)) (^) K)) (opr N (x.) (opr (opr M (^) (opr N (-) (1))) (x.) M)) lemul1it mpan mpan syl 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an nn0cn faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i nn0re faclbnd4lem1.2 (opr N (-) (1)) K reexpclt mp2an recn faclbnd4lem1.1 nncn faclbnd4lem1.3 faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i M (opr N (-) (1)) nn0expclt mp2an faclbnd4lem1.3 nn0mulcl nn0cn mul4 faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i nn0re faclbnd4lem1.2 (opr N (-) (1)) K reexpclt mp2an recn faclbnd4lem1.3 faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i M (opr N (-) (1)) nn0expclt mp2an nn0cn faclbnd4lem1.3 nn0cn mulass (opr (opr (2) (^) K) (x.) N) (x.) opreq2i 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an faclbnd4lem1.1 nnnn0 nn0mulcl nn0cn faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i nn0re faclbnd4lem1.2 (opr N (-) (1)) K reexpclt mp2an faclbnd4lem1.3 faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i M (opr N (-) (1)) nn0expclt mp2an nn0re remulcl recn faclbnd4lem1.3 nn0cn mul12 3eqtr2 syl6breq (br (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1))))) adantr 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an faclbnd4lem1.1 nnnn0 nn0mulcl faclbnd4lem1.3 nn0mulcl nn0ge0 faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i nn0re faclbnd4lem1.2 (opr N (-) (1)) K reexpclt mp2an faclbnd4lem1.3 faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i M (opr N (-) (1)) nn0expclt mp2an nn0re remulcl 2re faclbnd4lem1.2 2nn0 K (2) nn0expclt mp2an (2) (opr K (^) (2)) reexpclt mp2an faclbnd4lem1.3 nn0re faclbnd4lem1.3 faclbnd4lem1.2 nn0addcl M (opr M (+) K) reexpclt mp2an remulcl faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i (opr N (-) (1)) facclt ax-mp nnre remulcl 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an faclbnd4lem1.1 nnnn0 nn0mulcl faclbnd4lem1.3 nn0mulcl nn0re 3pm3.2i (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1)))) (opr (opr (opr (2) (^) K) (x.) N) (x.) M) lemul1it mpan mpan (br (1) (<) N) adantl letrd 2re faclbnd4lem1.2 2nn0 K (2) nn0expclt mp2an (2) (opr K (^) (2)) reexpclt mp2an faclbnd4lem1.3 nn0re faclbnd4lem1.3 faclbnd4lem1.2 nn0addcl M (opr M (+) K) reexpclt mp2an remulcl recn faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i (opr N (-) (1)) facclt ax-mp nncn 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an nn0cn faclbnd4lem1.3 nn0cn mulcl faclbnd4lem1.1 nncn mul4 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an nn0cn faclbnd4lem1.1 nncn faclbnd4lem1.3 nn0cn mul23 (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1)))) (x.) opreq2i faclbnd4lem1.3 nn0cn faclbnd4lem1.3 faclbnd4lem1.2 nn0addcl M (opr M (+) K) expp1t mp2an faclbnd4lem1.3 nn0cn faclbnd4lem1.2 nn0cn 1cn addass M (^) opreq2i eqtr3 (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr (2) (^) K)) (x.) opreq2i 2re faclbnd4lem1.2 2nn0 K (2) nn0expclt mp2an (2) (opr K (^) (2)) reexpclt mp2an recn 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an nn0cn faclbnd4lem1.3 nn0re faclbnd4lem1.3 faclbnd4lem1.2 nn0addcl M (opr M (+) K) reexpclt mp2an recn faclbnd4lem1.3 nn0cn mul4 eqtr3 faclbnd4lem1.1 N facnn2t ax-mp (x.) opreq12i 3eqtr4 2re faclbnd4lem1.2 2nn0 K (2) nn0expclt mp2an (2) (opr K (^) (2)) reexpclt mp2an 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an nn0re remulcl recn faclbnd4lem1.3 faclbnd4lem1.3 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an M (opr K (+) (1)) nn0nnaddclt mp2an nnnn0 M (opr M (+) (opr K (+) (1))) nn0expclt mp2an nn0cn faclbnd4lem1.1 nnnn0 N facclt ax-mp nncn mulass eqtr syl6breq faclbnd4lem1.1 nncn faclbnd4lem1.2 N K expp1t mp2an faclbnd4lem1.3 nn0cn faclbnd4lem1.1 M N expm1t mp2an (x.) opreq12i faclbnd4lem1.1 nnre faclbnd4lem1.2 N K reexpclt mp2an recn faclbnd4lem1.1 nncn faclbnd4lem1.3 faclbnd4lem1.1 N elnnnn0 mpbi pm3.27i M (opr N (-) (1)) nn0expclt mp2an faclbnd4lem1.3 nn0mulcl nn0cn mulass eqtr syl5eqbr 2re faclbnd4lem1.2 2nn0 K (2) nn0expclt mp2an (2) (opr K (^) (2)) reexpclt mp2an 2nn0 faclbnd4lem1.2 (2) K nn0expclt mp2an nn0re remulcl 2re faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnnn0 2nn0 (opr K (+) (1)) (2) nn0expclt mp2an (2) (opr (opr K (+) (1)) (^) (2)) reexpclt mp2an faclbnd4lem1.3 faclbnd4lem1.3 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an M (opr K (+) (1)) nn0nnaddclt mp2an nnnn0 M (opr M (+) (opr K (+) (1))) nn0expclt mp2an faclbnd4lem1.1 nnnn0 N facclt ax-mp nnnn0 nn0mulcl nn0re 3pm3.2i faclbnd4lem1.3 faclbnd4lem1.3 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an M (opr K (+) (1)) nn0nnaddclt mp2an nnnn0 M (opr M (+) (opr K (+) (1))) nn0expclt mp2an faclbnd4lem1.1 nnnn0 N facclt ax-mp nnnn0 nn0mulcl nn0ge0 2cn faclbnd4lem1.2 2nn0 K (2) nn0expclt mp2an faclbnd4lem1.2 (2) (opr K (^) (2)) K expaddt mp3an 2re faclbnd4lem1.2 2nn0 K (2) nn0expclt mp2an faclbnd4lem1.2 nn0addcl faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnnn0 2nn0 (opr K (+) (1)) (2) nn0expclt mp2an 3pm3.2i ax1re 2re 1lt2 ltlei faclbnd4lem1.2 nn0re faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnre faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnre 3pm3.2i faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnnn0 nn0ge0 faclbnd4lem1.2 nn0re 1nn0 nn0addge1 pm3.2i K (opr K (+) (1)) (opr K (+) (1)) lemul1it mp2an faclbnd4lem1.2 nn0cn sqval faclbnd4lem1.2 nn0cn mulid1 eqcomi (+) opreq12i faclbnd4lem1.2 nn0cn faclbnd4lem1.2 nn0cn 1cn adddi eqtr4 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nncn sqval 3brtr4 pm3.2i (2) (opr (opr K (^) (2)) (+) K) (opr (opr K (+) (1)) (^) (2)) expwordit mp2an eqbrtrr pm3.2i (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr (2) (^) K)) (opr (2) (^) (opr (opr K (+) (1)) (^) (2))) (opr (opr M (^) (opr M (+) (opr K (+) (1)))) (x.) (` (!) N)) lemul1it mp2an (/\ (br (1) (<) N) (br (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1)))))) a1i letrd 2re faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an nnnn0 2nn0 (opr K (+) (1)) (2) nn0expclt mp2an (2) (opr (opr K (+) (1)) (^) (2)) reexpclt mp2an recn faclbnd4lem1.3 faclbnd4lem1.3 faclbnd4lem1.2 1nn K (1) nn0nnaddclt mp2an M (opr K (+) (1)) nn0nnaddclt mp2an nnnn0 M (opr M (+) (opr K (+) (1))) nn0expclt mp2an nn0cn faclbnd4lem1.1 nnnn0 N facclt ax-mp nncn mulass syl6breqr jaoian mpan)) thm (faclbnd4lem2 () () (-> (/\/\ (e. M (NN0)) (e. K (NN0)) (e. N (NN))) (-> (br (opr (opr (opr N (-) (1)) (^) K) (x.) (opr M (^) (opr N (-) (1)))) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) (opr N (-) (1))))) (br (opr (opr N (^) (opr K (+) (1))) (x.) (opr M (^) N)) (<_) (opr (opr (opr (2) (^) (opr (opr K (+) (1)) (^) (2))) (x.) (opr M (^) (opr M (+) (opr K (+) (1))))) (x.) (` (!) N))))) (M (if (e. M (NN0)) M (1)) (^) (opr N (-) (1)) opreq1 (opr (opr N (-) (1)) (^) K) (x.) opreq2d (= M (if (e. M (NN0)) M (1))) id M (if (e. M (NN0)) M (1)) (+) K opreq1 (^) opreq12d (opr (2) (^) (opr K (^) (2))) (x.) opreq2d (x.) (` (!) (opr N (-) (1))) opreq1d (<_) breq12d M (if (e. M (NN0)) M (1)) (^) N opreq1 (opr N (^) (opr K (+) (1))) (x.) opreq2d (= M (if (e. M (NN0)) M (1))) id M (if (e. M (NN0)) M (1)) (+) (opr K (+) (1)) opreq1 (^) opreq12d (opr (2) (^) (opr (opr K (+) (1)) (^) (2))) (x.) opreq2d (x.) (` (!) N) opreq1d (<_) breq12d imbi12d K (if (e. K (NN0)) K (1)) (opr N (-) (1)) (^) opreq2 (x.) (opr (if (e. M (NN0)) M (1)) (^) (opr N (-) (1))) opreq1d K (if (e. K (NN0)) K (1)) (^) (2) opreq1 (2) (^) opreq2d K (if (e. K (NN0)) K (1)) (if (e. M (NN0)) M (1)) (+) opreq2 (if (e. M (NN0)) M (1)) (^) opreq2d (x.) opreq12d (x.) (` (!) (opr N (-) (1))) opreq1d (<_) breq12d K (if (e. K (NN0)) K (1)) (+) (1) opreq1 N (^) opreq2d (x.) (opr (if (e. M (NN0)) M (1)) (^) N) opreq1d K (if (e. K (NN0)) K (1)) (+) (1) opreq1 (^) (2) opreq1d (2) (^) opreq2d K (if (e. K (NN0)) K (1)) (+) (1) opreq1 (if (e. M (NN0)) M (1)) (+) opreq2d (if (e. M (NN0)) M (1)) (^) opreq2d (x.) opreq12d (x.) (` (!) N) opreq1d (<_) breq12d imbi12d N (if (e. N (NN)) N (1)) (-) (1) opreq1 (^) (if (e. K (NN0)) K (1)) opreq1d N (if (e. N (NN)) N (1)) (-) (1) opreq1 (if (e. M (NN0)) M (1)) (^) opreq2d (x.) opreq12d N (if (e. N (NN)) N (1)) (-) (1) opreq1 (!) fveq2d (opr (opr (2) (^) (opr (if (e. K (NN0)) K (1)) (^) (2))) (x.) (opr (if (e. M (NN0)) M (1)) (^) (opr (if (e. M (NN0)) M (1)) (+) (if (e. K (NN0)) K (1))))) (x.) opreq2d (<_) breq12d N (if (e. N (NN)) N (1)) (^) (opr (if (e. K (NN0)) K (1)) (+) (1)) opreq1 N (if (e. N (NN)) N (1)) (if (e. M (NN0)) M (1)) (^) opreq2 (x.) opreq12d N (if (e. N (NN)) N (1)) (!) fveq2 (opr (opr (2) (^) (opr (opr (if (e. K (NN0)) K (1)) (+) (1)) (^) (2))) (x.) (opr (if (e. M (NN0)) M (1)) (^) (opr (if (e. M (NN0)) M (1)) (+) (opr (if (e. K (NN0)) K (1)) (+) (1))))) (x.) opreq2d (<_) breq12d imbi12d 1nn N elimel 1nn0 K elimel 1nn0 M elimel faclbnd4lem1 dedth3h)) thm (faclbnd4lem3 () () (-> (/\ (/\ (e. M (NN0)) (e. K (NN0))) (= N (0))) (br (opr (opr N (^) K) (x.) (opr M (^) N)) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) N)))) (K 0expt (e. M (NN0)) adantl (opr (2) (^) (opr K (^) (2))) (opr M (^) (opr M (+) K)) nn0mulclt 2nn0 K (2) nn0expclt mpan2 2nn0 (2) (opr K (^) (2)) nn0expclt mpan syl (e. M (NN0)) adantl M K nn0addclt M (opr M (+) K) nn0expclt syldan sylanc K nnnn0t sylan2 (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) nn0ge0t syl eqbrtrd M elnn0 M nnnn0t 0nn0 M (0) nn0addclt mpan2 syl M (opr M (+) (0)) nnexpclt mpdan (= M (0)) id M (0) (+) (0) opreq1 0cn addid1 syl6eq (^) opreq12d 0cn (0) exp0t ax-mp syl6eq 1nn syl6eqel jaoi sylbi 1nn (1) (opr M (^) (opr M (+) (0))) nnmulclt mpan (opr (1) (x.) (opr M (^) (opr M (+) (0)))) nnge1t 3syl (= K (0)) adantr K (0) (0) (^) opreq2 0cn (0) exp0t ax-mp syl6eq K (0) (^) (2) opreq1 sq0 syl6eq (2) (^) opreq2d 2cn (2) exp0t ax-mp syl6eq K (0) M (+) opreq2 M (^) opreq2d (x.) opreq12d (<_) breq12d (e. M (NN0)) adantl mpbird jaodan K elnn0 sylan2b M nn0cnt M exp0t syl (opr (0) (^) K) (x.) opreq2d 0nn0 (0) K nn0expclt mpan (opr (0) (^) K) nn0cnt (opr (0) (^) K) ax1id 3syl sylan9eq (opr (2) (^) (opr K (^) (2))) (opr M (^) (opr M (+) K)) nn0mulclt 2nn0 K (2) nn0expclt mpan2 2nn0 (2) (opr K (^) (2)) nn0expclt mpan syl (e. M (NN0)) adantl M K nn0addclt M (opr M (+) K) nn0expclt syldan sylanc (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) nn0cnt (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) ax1id 3syl 3brtr4d (= N (0)) adantr N (0) (^) K opreq1 N (0) M (^) opreq2 (x.) opreq12d N (0) (!) fveq2 fac0 syl6eq (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) opreq2d (<_) breq12d (/\ (e. M (NN0)) (e. K (NN0))) adantl mpbird)) thm (faclbnd4lem4 ((j m) (j n) (M j) (m n) (M m) (M n) (K j) (K m) (K n) (N j) (N n)) () (-> (/\/\ (e. N (NN)) (e. K (NN0)) (e. M (NN0))) (br (opr (opr N (^) K) (x.) (opr M (^) N)) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) N)))) ((cv n) N (^) K opreq1 (cv n) N M (^) opreq2 (x.) opreq12d (cv n) N (!) fveq2 (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) opreq2d (<_) breq12d (NN) rcla4va M (cv j) (opr (cv n) (-) (1)) faclbnd4lem3 (cv n) nnge1t (br (cv n) (<_) (1)) adantr (cv n) nnret ax1re (cv n) (1) letri3t mpan2 syl biimpar anassrs mpdan (cv n) (1) (-) (1) opreq1 1cn subid syl6eq syl sylan2 (A.e. m (NN) (br (opr (opr (cv m) (^) (cv j)) (x.) (opr M (^) (cv m))) (<_) (opr (opr (opr (2) (^) (opr (cv j) (^) (2))) (x.) (opr M (^) (opr M (+) (cv j)))) (x.) (` (!) (cv m))))) a1d 1nn (1) (cv n) nnsubt mpan biimpa (cv m) (opr (cv n) (-) (1)) (^) (cv j) opreq1 (cv m) (opr (cv n) (-) (1)) M (^) opreq2 (x.) opreq12d (cv m) (opr (cv n) (-) (1)) (!) fveq2 (opr (opr (2) (^) (opr (cv j) (^) (2))) (x.) (opr M (^) (opr M (+) (cv j)))) (x.) opreq2d (<_) breq12d (NN) rcla4v syl (/\ (e. M (NN0)) (e. (cv j) (NN0))) adantl jaodan (cv n) nnret ax1re (cv n) (1) lelttrit mpan2 syl ancli (e. (cv n) (NN)) (br (cv n) (<_) (1)) (br (1) (<) (cv n)) andi sylib sylan2 M (cv j) (cv n) faclbnd4lem2 3expa syld ex com23 r19.21adv (cv n) (cv m) (^) (cv j) opreq1 (cv n) (cv m) M (^) opreq2 (x.) opreq12d (cv n) (cv m) (!) fveq2 (opr (opr (2) (^) (opr (cv j) (^) (2))) (x.) (opr M (^) (opr M (+) (cv j)))) (x.) opreq2d (<_) breq12d (NN) cbvralv syl5ib expcom a2d M (cv n) faclbnd3 (cv n) nnnn0t sylan2 (cv n) nncnt (cv n) exp0t syl (x.) (opr M (^) (cv n)) opreq1d (e. M (NN0)) adantl M (cv n) expclt M nn0cnt (cv n) nnnn0t syl2an (opr M (^) (cv n)) mulid2t syl eqtrd sq0 (2) (^) opreq2i 2cn (2) exp0t ax-mp eqtr (e. M (NN0)) a1i M nn0cnt M ax0id syl M (^) opreq2d (x.) opreq12d M nn0cnt M M expclt mpancom (opr M (^) M) mulid2t syl eqtrd (x.) (` (!) (cv n)) opreq1d (e. (cv n) (NN)) adantr 3brtr4d r19.21aiva (cv m) (0) (cv n) (^) opreq2 (x.) (opr M (^) (cv n)) opreq1d (cv m) (0) (^) (2) opreq1 (2) (^) opreq2d (cv m) (0) M (+) opreq2 M (^) opreq2d (x.) opreq12d (x.) (` (!) (cv n)) opreq1d (<_) breq12d n (NN) ralbidv (e. M (NN0)) imbi2d (cv m) (cv j) (cv n) (^) opreq2 (x.) (opr M (^) (cv n)) opreq1d (cv m) (cv j) (^) (2) opreq1 (2) (^) opreq2d (cv m) (cv j) M (+) opreq2 M (^) opreq2d (x.) opreq12d (x.) (` (!) (cv n)) opreq1d (<_) breq12d n (NN) ralbidv (e. M (NN0)) imbi2d (cv m) (opr (cv j) (+) (1)) (cv n) (^) opreq2 (x.) (opr M (^) (cv n)) opreq1d (cv m) (opr (cv j) (+) (1)) (^) (2) opreq1 (2) (^) opreq2d (cv m) (opr (cv j) (+) (1)) M (+) opreq2 M (^) opreq2d (x.) opreq12d (x.) (` (!) (cv n)) opreq1d (<_) breq12d n (NN) ralbidv (e. M (NN0)) imbi2d (cv m) K (cv n) (^) opreq2 (x.) (opr M (^) (cv n)) opreq1d (cv m) K (^) (2) opreq1 (2) (^) opreq2d (cv m) K M (+) opreq2 M (^) opreq2d (x.) opreq12d (x.) (` (!) (cv n)) opreq1d (<_) breq12d n (NN) ralbidv (e. M (NN0)) imbi2d nn0indALT imp sylan2 3impb)) thm (faclbnd4 () () (-> (/\/\ (e. N (NN0)) (e. K (NN0)) (e. M (NN0))) (br (opr (opr N (^) K) (x.) (opr M (^) N)) (<_) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) N)))) (N K M faclbnd4lem4 3com13 3expa M K N faclbnd4lem3 jaodan N elnn0 sylan2b 3impa 3com13)) thm (faclbnd5 () () (-> (/\/\ (e. N (NN0)) (e. K (NN0)) (e. M (NN))) (br (opr (opr N (^) K) (x.) (opr M (^) N)) (<) (opr (opr (2) (x.) (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K)))) (x.) (` (!) N)))) ((opr N (^) K) (opr M (^) N) axmulrcl N K reexpclt N nn0ret sylan ancoms M N reexpclt M nnret sylan syl2an anandirs (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (` (!) N) axmulrcl (opr (2) (^) (opr K (^) (2))) (opr M (^) (opr M (+) K)) nnmulclt 2nn0 K (2) nn0expclt mpan2 2nn (2) (opr K (^) (2)) nnexpclt mpan syl M (opr M (+) K) nnexpclt M K nn0addclt ancoms M nnnn0t sylan2 sylan2 anabss7 syl2an anabss5 (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) nnret syl N facclt (` (!) N) nnret syl syl2an (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (` (!) N) axmulrcl (opr (2) (^) (opr K (^) (2))) (opr M (^) (opr M (+) K)) nnmulclt 2nn0 K (2) nn0expclt mpan2 2nn (2) (opr K (^) (2)) nnexpclt mpan syl M (opr M (+) K) nnexpclt M K nn0addclt ancoms M nnnn0t sylan2 sylan2 anabss7 syl2an anabss5 (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) nnret syl N facclt (` (!) N) nnret syl syl2an 2re (2) (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) N)) axmulrcl mpan syl N K M faclbnd4 M nnnn0t syl3an3 3coml 3expa 1lt2 2re (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) N)) (2) ltmulgt12t mp3an2 (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (` (!) N) axmulrcl (opr (2) (^) (opr K (^) (2))) (opr M (^) (opr M (+) K)) nnmulclt 2nn0 K (2) nn0expclt mpan2 2nn (2) (opr K (^) (2)) nnexpclt mpan syl M (opr M (+) K) nnexpclt M K nn0addclt ancoms M nnnn0t sylan2 sylan2 anabss7 syl2an anabss5 (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) nnret syl N facclt (` (!) N) nnret syl syl2an (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (` (!) N) nnmulclt (opr (2) (^) (opr K (^) (2))) (opr M (^) (opr M (+) K)) nnmulclt 2nn0 K (2) nn0expclt mpan2 2nn (2) (opr K (^) (2)) nnexpclt mpan syl M (opr M (+) K) nnexpclt M K nn0addclt ancoms M nnnn0t sylan2 sylan2 anabss7 syl2an anabss5 N facclt syl2an (opr (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (x.) (` (!) N)) nngt0t syl sylanc mpbii lelttrd 2cn (2) (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) (` (!) N) axmulass mp3an1 (opr (2) (^) (opr K (^) (2))) (opr M (^) (opr M (+) K)) nnmulclt 2nn0 K (2) nn0expclt mpan2 2nn (2) (opr K (^) (2)) nnexpclt mpan syl M (opr M (+) K) nnexpclt M K nn0addclt ancoms M nnnn0t sylan2 sylan2 anabss7 syl2an anabss5 (opr (opr (2) (^) (opr K (^) (2))) (x.) (opr M (^) (opr M (+) K))) nnret syl recnd N facclt (` (!) N) nnret syl recnd syl2an breqtrrd 3impa 3comr)) thm (facavgt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (br (` (!) (` (floor) (opr (opr M (+) N) (/) (2)))) (<_) (opr (` (!) M) (x.) (` (!) N)))) (M N avglet M nn0ret N nn0ret syl2an M N nn0addclt (opr M (+) N) nn0ret syl (opr M (+) N) rehalfclt (opr (opr M (+) N) (/) (2)) fllet 3syl (` (floor) (opr (opr M (+) N) (/) (2))) (opr (opr M (+) N) (/) (2)) M letrt M N nn0addclt (opr M (+) N) nn0ret syl (opr M (+) N) rehalfclt (opr (opr M (+) N) (/) (2)) flclt (` (floor) (opr (opr M (+) N) (/) (2))) zret syl 3syl M N nn0addclt (opr M (+) N) nn0ret (opr M (+) N) rehalfclt 3syl M nn0ret (e. N (NN0)) adantr syl3anc mpand (` (floor) (opr (opr M (+) N) (/) (2))) M facwordit 3exp (opr (opr M (+) N) (/) (2)) flge0nn0t M N nn0addclt (opr M (+) N) nn0ret (opr M (+) N) rehalfclt 3syl M N nn0addclt (opr M (+) N) nn0ge0t syl M N nn0addclt (opr M (+) N) nn0ret (opr M (+) N) halfnneg2t 3syl mpbid sylanc (e. M (NN0)) (e. N (NN0)) pm3.26 sylc syld M facclt (` (!) M) nncnt (` (!) M) ax1id 3syl (e. N (NN0)) adantr ax1re (1) (` (!) N) (` (!) M) lemul2it mp3anl1 N facclt (` (!) N) nnret syl M facclt (` (!) M) nnret syl anim12i ancoms M facclt (` (!) M) nnnn0t (` (!) M) nn0ge0t 3syl N facclt (` (!) N) nnge1t syl anim12i sylanc eqbrtrrd (` (!) (` (floor) (opr (opr M (+) N) (/) (2)))) (` (!) M) (opr (` (!) M) (x.) (` (!) N)) letrt (opr (opr M (+) N) (/) (2)) flge0nn0t M N nn0addclt (opr M (+) N) nn0ret (opr M (+) N) rehalfclt 3syl M N nn0addclt (opr M (+) N) nn0ge0t syl M N nn0addclt (opr M (+) N) nn0ret (opr M (+) N) halfnneg2t 3syl mpbid sylanc (` (floor) (opr (opr M (+) N) (/) (2))) facclt (` (!) (` (floor) (opr (opr M (+) N) (/) (2)))) nnret 3syl M facclt (` (!) M) nnret syl (e. N (NN0)) adantr (` (!) M) (` (!) N) axmulrcl M facclt (` (!) M) nnret syl N facclt (` (!) N) nnret syl syl2an syl3anc mpan2d syld M N nn0addclt (opr M (+) N) nn0ret syl (opr M (+) N) rehalfclt (opr (opr M (+) N) (/) (2)) fllet 3syl (` (floor) (opr (opr M (+) N) (/) (2))) (opr (opr M (+) N) (/) (2)) N letrt M N nn0addclt (opr M (+) N) nn0ret syl (opr M (+) N) rehalfclt (opr (opr M (+) N) (/) (2)) flclt (` (floor) (opr (opr M (+) N) (/) (2))) zret syl 3syl M N nn0addclt (opr M (+) N) nn0ret (opr M (+) N) rehalfclt 3syl N nn0ret (e. M (NN0)) adantl syl3anc mpand (` (floor) (opr (opr M (+) N) (/) (2))) N facwordit 3exp (opr (opr M (+) N) (/) (2)) flge0nn0t M N nn0addclt (opr M (+) N) nn0ret (opr M (+) N) rehalfclt 3syl M N nn0addclt (opr M (+) N) nn0ge0t syl M N nn0addclt (opr M (+) N) nn0ret (opr M (+) N) halfnneg2t 3syl mpbid sylanc (e. M (NN0)) (e. N (NN0)) pm3.27 sylc syld N facclt (` (!) N) nncnt (` (!) N) mulid2t 3syl (e. M (NN0)) adantl ax1re (1) (` (!) M) (` (!) N) lemul1it mp3anl1 M facclt (` (!) M) nnret syl N facclt (` (!) N) nnret syl anim12i N facclt (` (!) N) nnnn0t (` (!) N) nn0ge0t 3syl M facclt (` (!) M) nnge1t syl anim12i ancoms sylanc eqbrtrrd (` (!) (` (floor) (opr (opr M (+) N) (/) (2)))) (` (!) N) (opr (` (!) M) (x.) (` (!) N)) letrt (opr (opr M (+) N) (/) (2)) flge0nn0t M N nn0addclt (opr M (+) N) nn0ret (opr M (+) N) rehalfclt 3syl M N nn0addclt (opr M (+) N) nn0ge0t syl M N nn0addclt (opr M (+) N) nn0ret (opr M (+) N) halfnneg2t 3syl mpbid sylanc (` (floor) (opr (opr M (+) N) (/) (2))) facclt (` (!) (` (floor) (opr (opr M (+) N) (/) (2)))) nnret 3syl N facclt (` (!) N) nnret syl (e. M (NN0)) adantl (` (!) M) (` (!) N) axmulrcl M facclt (` (!) M) nnret syl N facclt (` (!) N) nnret syl syl2an syl3anc mpan2d syld jaod mpd)) thm (bcvalt ((k n) (m n) (N n) (k m) (N k) (N m) (K n) (K k) (K m)) () (-> (/\ (e. N (NN0)) (e. K (ZZ))) (= (opr N (C.) K) (if (/\ (br (0) (<_) K) (br K (<_) N)) (opr (` (!) N) (/) (opr (` (!) (opr N (-) K)) (x.) (` (!) K))) (0)))) ((` (!) N) (/) (opr (` (!) (opr N (-) K)) (x.) (` (!) K)) oprex 0nn0 elisseti (/\ (br (0) (<_) K) (br K (<_) N)) ifex (cv n) N (cv k) (<_) breq2 (br (0) (<_) (cv k)) anbi2d (opr (` (!) (cv n)) (/) (opr (` (!) (opr (cv n) (-) (cv k))) (x.) (` (!) (cv k)))) (0) ifbid (cv n) N (!) fveq2 (cv n) N (-) (cv k) opreq1 (!) fveq2d (x.) (` (!) (cv k)) opreq1d (/) opreq12d (/\ (br (0) (<_) (cv k)) (br (cv k) (<_) N)) (0) ifeq1d eqtrd (cv k) K (0) (<_) breq2 (cv k) K (<_) N breq1 anbi12d (opr (` (!) N) (/) (opr (` (!) (opr N (-) (cv k))) (x.) (` (!) (cv k)))) (0) ifbid (cv k) K N (-) opreq2 (!) fveq2d (cv k) K (!) fveq2 (x.) opreq12d (` (!) N) (/) opreq2d (/\ (br (0) (<_) K) (br K (<_) N)) (0) ifeq1d eqtrd n k m df-bc oprabval2)) thm (bcval3tOLD () () (-> (/\ (/\ (e. N (NN0)) (e. K (ZZ))) (/\ (br (0) (<_) K) (br K (<_) N))) (= (opr N (C.) K) (opr (` (!) N) (/) (opr (` (!) (opr N (-) K)) (x.) (` (!) K))))) (N K bcvalt (/\ (br (0) (<_) K) (br K (<_) N)) (opr (` (!) N) (/) (opr (` (!) (opr N (-) K)) (x.) (` (!) K))) (0) iftrue sylan9eq)) thm (bcval2t () () (-> (/\/\ (e. N (NN0)) (e. K (NN0)) (br K (<_) N)) (= (opr N (C.) K) (opr (` (!) N) (/) (opr (` (!) (opr N (-) K)) (x.) (` (!) K))))) (N K bcval3tOLD K nn0zt (e. N (NN0)) anim2i (br K (<_) N) 3adant3 K nn0ge0t (br K (<_) N) anim1i (e. N (NN0)) 3adant1 sylanc)) thm (bcval3t () () (-> (/\ (e. N (NN0)) (e. K (opr (0) (...) N))) (= (opr N (C.) K) (opr (` (!) N) (/) (opr (` (!) (opr N (-) K)) (x.) (` (!) K))))) (N K bcval2t N (NN0) K elfz3nn0t K N elfznn0t (e. N (NN0)) adantl K (0) N elfzle2 (e. N (NN0)) adantl syl3anc)) thm (bcval4t () () (-> (/\/\ (e. N (NN0)) (e. K (ZZ)) (\/ (br K (<) (0)) (br N (<) K))) (= (opr N (C.) K) (0))) (0re K (0) ltnlet mpan2 (e. N (RR)) adantl N K ltnlet orbi12d N nn0ret K zret syl2an biimpa N K bcvalt (/\ (br (0) (<_) K) (br K (<_) N)) (opr (` (!) N) (/) (opr (` (!) (opr N (-) K)) (x.) (` (!) K))) (0) iffalse sylan9eq (br (0) (<_) K) (br K (<_) N) ianor sylan2br syldan 3impa)) thm (bccmplt () () (-> (/\/\ (e. N (NN0)) (e. K (NN0)) (br K (<_) N)) (= (opr N (C.) K) (opr N (C.) (opr N (-) K)))) ((` (!) (opr N (-) K)) (` (!) K) axmulcom K N nn0subt ancoms biimp3a (opr N (-) K) facclt (` (!) (opr N (-) K)) nncnt 3syl K facclt (` (!) K) nncnt syl (e. N (NN0)) (br K (<_) N) 3ad2ant2 sylanc N K nncant N nn0cnt K nn0cnt syl2an (!) fveq2d (br K (<_) N) 3adant3 (x.) (` (!) (opr N (-) K)) opreq1d eqtr4d (` (!) N) (/) opreq2d N K bcval2t N (opr N (-) K) bcval2t (e. N (NN0)) (e. K (NN0)) (br K (<_) N) 3simp1 K N nn0subt ancoms biimp3a N K nn0addge1t N K N lesubaddt (e. N (RR)) (e. K (RR)) pm3.26 (e. N (RR)) (e. K (RR)) pm3.27 (e. N (RR)) (e. K (RR)) pm3.26 syl3anc K nn0ret sylan2 mpbird N nn0ret sylan (br K (<_) N) 3adant3 syl3anc 3eqtr4d)) thm (bcn0t () () (-> (e. N (NN0)) (= (opr N (C.) (0)) (1))) (N nn0ge0t 0nn0 N (0) bcval2t mp3an2 mpdan N nn0cnt N subid1t syl (!) fveq2d fac0 (e. N (NN0)) a1i (x.) opreq12d N facclt (` (!) N) nncnt syl (` (!) N) ax1id syl eqtrd (` (!) N) (/) opreq2d (` (!) N) dividt N facclt (` (!) N) nncnt syl N facne0t sylanc 3eqtrd)) thm (bcnnt () () (-> (e. N (NN0)) (= (opr N (C.) N) (1))) (N nn0ge0t 0nn0 N (0) bccmplt mp3an2 mpdan N bcn0t N nn0cnt N subid1t syl N (C.) opreq2d 3eqtr3rd)) thm (bcnp11t () () (-> (e. N (NN0)) (= (opr (opr N (+) (1)) (C.) (1)) (opr N (+) (1)))) (1nn0 (opr N (+) (1)) (1) bcval2t mp3an2 N peano2nn0 ax1re (1) N nn0addge2t mpan sylanc N nn0cnt 1cn N (1) pncant mpan2 syl (!) fveq2d fac1 (e. N (NN0)) a1i (x.) opreq12d N facclt (` (!) N) nncnt syl (` (!) N) ax1id syl eqtrd (` (!) (opr N (+) (1))) (/) opreq2d N facp1t eqcomd (` (!) (opr N (+) (1))) (` (!) N) (opr N (+) (1)) divmult N peano2nn0 (opr N (+) (1)) facclt (` (!) (opr N (+) (1))) nncnt syl syl N facclt (` (!) N) nncnt syl N peano2nn0 (opr N (+) (1)) nn0cnt syl 3jca N facne0t sylanc mpbird 3eqtrd)) thm (bcnp1nt () () (-> (e. N (NN0)) (= (opr (opr N (+) (1)) (C.) N) (opr N (+) (1)))) (1nn0 (opr N (+) (1)) (1) bccmplt mp3an2 N peano2nn0 ax1re (1) N nn0addge2t mpan sylanc N bcnp11t N nn0cnt 1cn N (1) pncant mpan2 syl (opr N (+) (1)) (C.) opreq2d 3eqtr3rd)) thm (bcpasc2 () ((bcpasc2.1 (e. N (NN))) (bcpasc2.2 (e. K (NN))) (bcpasc2.3 (br K (<_) N))) (= (opr (opr N (C.) K) (+) (opr N (C.) (opr K (-) (1)))) (opr (opr N (+) (1)) (C.) K)) (1cn bcpasc2.2 nncn 1cn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl bcpasc2.2 nnne0 bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) nn0p1nnt ax-mp nnne0 divadddiv bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl mulid2 bcpasc2.2 nncn mulid1 (+) opreq12i bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn bcpasc2.2 nncn add23 bcpasc2.1 nncn bcpasc2.2 nncn N K npcant mp2an (+) (1) opreq1i 3eqtr (/) (opr K (x.) (opr (opr N (-) K) (+) (1))) opreq1i eqtr (opr (` (!) N) (/) (opr (` (!) (opr K (-) (1))) (x.) (` (!) (opr N (-) K)))) (x.) opreq2i bcpasc2.1 nnnn0 N facclt ax-mp nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp (` (!) (opr K (-) (1))) (` (!) (opr N (-) K)) nnmulclt mp2an nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp (` (!) (opr K (-) (1))) (` (!) (opr N (-) K)) nnmulclt mp2an nnne0 divcl bcpasc2.2 nncn bcpasc2.2 nnne0 reccl bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) nn0p1nnt ax-mp nnne0 reccl adddi bcpasc2.1 nnnn0 N facclt ax-mp nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp (` (!) (opr K (-) (1))) (` (!) (opr N (-) K)) nnmulclt mp2an nncn bcpasc2.1 nnnn0 1nn0 nn0addcl nn0cn bcpasc2.2 nncn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl mulcl bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp (` (!) (opr K (-) (1))) (` (!) (opr N (-) K)) nnmulclt mp2an nnne0 bcpasc2.2 nncn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl bcpasc2.2 nnne0 bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) nn0p1nnt ax-mp nnne0 muln0 divmuldiv bcpasc2.1 nnnn0 N facp1t ax-mp bcpasc2.2 nnre bcpasc2.1 nnnn0 1nn0 nn0addcl nn0re bcpasc2.3 bcpasc2.2 bcpasc2.1 K N nnleltp1t mp2an mpbi ltlei bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 1nn0 nn0addcl K (opr N (+) (1)) nn0subt mp2an mpbi (opr (opr N (+) (1)) (-) K) facclt ax-mp nncn bcpasc2.2 nnnn0 K facclt ax-mp nncn mulcom bcpasc2.2 K facnn2t ax-mp bcpasc2.1 nncn 1cn bcpasc2.2 nncn addsub (!) fveq2i bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facp1t ax-mp eqtr (x.) opreq12i bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp nncn bcpasc2.2 nncn bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp nncn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl mul4 3eqtr (/) opreq12i eqtr4 3eqtr3 bcpasc2.1 nnnn0 N facclt ax-mp nncn mulid1 bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp nncn bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp nncn bcpasc2.2 nncn mul23 bcpasc2.2 K facnn2t ax-mp (x.) (` (!) (opr N (-) K)) opreq1i bcpasc2.2 nnnn0 K facclt ax-mp nncn bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp nncn mulcom 3eqtr2 (/) opreq12i bcpasc2.1 nnnn0 N facclt ax-mp nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp (` (!) (opr K (-) (1))) (` (!) (opr N (-) K)) nnmulclt mp2an nncn 1cn bcpasc2.2 nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp (` (!) (opr K (-) (1))) (` (!) (opr N (-) K)) nnmulclt mp2an nnne0 bcpasc2.2 nnne0 divmuldiv bcpasc2.1 nnnn0 bcpasc2.2 nnnn0 bcpasc2.3 N K bcval2t mp3an 3eqtr4r bcpasc2.1 nnnn0 N facclt ax-mp nncn mulid1 bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp nncn bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp nncn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl mulass bcpasc2.1 nncn bcpasc2.2 nncn 1cn N K (1) subsubt mp3an (!) fveq2i bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facp1t ax-mp eqtr2 (` (!) (opr K (-) (1))) (x.) opreq2i bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp nncn bcpasc2.2 nnre bcpasc2.1 nnnn0 1nn0 nn0addcl nn0re bcpasc2.3 bcpasc2.2 bcpasc2.1 K N nnleltp1t mp2an mpbi ltlei bcpasc2.2 nnre ax1re bcpasc2.1 nnre lesubadd mpbir bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi bcpasc2.1 nnnn0 (opr K (-) (1)) N nn0subt mp2an mpbi (opr N (-) (opr K (-) (1))) facclt ax-mp nncn mulcom 3eqtr (/) opreq12i bcpasc2.1 nnnn0 N facclt ax-mp nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp (` (!) (opr K (-) (1))) (` (!) (opr N (-) K)) nnmulclt mp2an nncn 1cn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi (opr K (-) (1)) facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) facclt ax-mp (` (!) (opr K (-) (1))) (` (!) (opr N (-) K)) nnmulclt mp2an nnne0 bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi (opr N (-) K) nn0p1nnt ax-mp nnne0 divmuldiv bcpasc2.1 nnnn0 bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 (1) K nn0subt mp2an mpbi bcpasc2.2 nnre bcpasc2.1 nnnn0 1nn0 nn0addcl nn0re bcpasc2.3 bcpasc2.2 bcpasc2.1 K N nnleltp1t mp2an mpbi ltlei bcpasc2.2 nnre ax1re bcpasc2.1 nnre lesubadd mpbir N (opr K (-) (1)) bcval2t mp3an 3eqtr4r (+) opreq12i bcpasc2.1 nnnn0 1nn0 nn0addcl bcpasc2.2 nnnn0 bcpasc2.2 nnre bcpasc2.1 nnnn0 1nn0 nn0addcl nn0re bcpasc2.3 bcpasc2.2 bcpasc2.1 K N nnleltp1t mp2an mpbi ltlei (opr N (+) (1)) K bcval2t mp3an 3eqtr4)) thm (bcpasc2t () () (-> (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) (= (opr (opr N (C.) K) (+) (opr N (C.) (opr K (-) (1)))) (opr (opr N (+) (1)) (C.) K))) (N (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (C.) K opreq1 N (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (C.) (opr K (-) (1)) opreq1 (+) opreq12d N (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (+) (1) opreq1 (C.) K opreq1d eqeq12d K (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (C.) opreq2 K (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (-) (1) opreq1 (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (C.) opreq2d (+) opreq12d K (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (opr (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (+) (1)) (C.) opreq2 eqeq12d N (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN) eleq1 N (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) K (<_) breq2 (e. K (NN)) 3anbi13d K (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (NN) eleq1 K (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (<_) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) breq1 (e. (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN)) 3anbi23d (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN) eleq1 (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (1) (<_) breq2 (e. (1) (NN)) 3anbi13d (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (NN) eleq1 (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (<_) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) breq1 (e. (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN)) 3anbi23d 1nn 1nn ax1re leid 3pm3.2i elimhyp2v 3simp1i N (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN) eleq1 N (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) K (<_) breq2 (e. K (NN)) 3anbi13d K (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (NN) eleq1 K (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (<_) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) breq1 (e. (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN)) 3anbi23d (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN) eleq1 (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (1) (<_) breq2 (e. (1) (NN)) 3anbi13d (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (NN) eleq1 (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (<_) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) breq1 (e. (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN)) 3anbi23d 1nn 1nn ax1re leid 3pm3.2i elimhyp2v 3simp2i N (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN) eleq1 N (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) K (<_) breq2 (e. K (NN)) 3anbi13d K (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (NN) eleq1 K (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (<_) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) breq1 (e. (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN)) 3anbi23d (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN) eleq1 (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (1) (<_) breq2 (e. (1) (NN)) 3anbi13d (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (NN) eleq1 (1) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) K (1)) (<_) (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) breq1 (e. (if (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) N (1)) (NN)) 3anbi23d 1nn 1nn ax1re leid 3pm3.2i elimhyp2v 3simp3i bcpasc2 dedth2v)) thm (bcpasc () ((bcpasc.1 (e. N (NN0))) (bcpasc.2 (e. K (ZZ)))) (= (opr (opr N (C.) K) (+) (opr N (C.) (opr K (-) (1)))) (opr (opr N (+) (1)) (C.) K)) (bcpasc.1 N elnn0 mpbi ori 1cn addid1 K (0) (0) (C.) opreq2 0nn0 (0) bcn0t ax-mp syl6eq K (0) (-) (1) opreq1 (1) df-neg syl6eqr (0) (C.) opreq2d 0nn0 1z (1) znegclt ax-mp lt01 ax1re (1) lt0neg2t ax-mp mpbi (br (-u (1)) (<) (0)) (br (0) (<) (-u (1))) orc ax-mp (0) (-u (1)) bcval4t mp3an syl6eq (+) opreq12d K (0) (1) (C.) opreq2 1nn0 (1) bcn0t ax-mp syl6eq eqeq12d mpbiri bcpasc.2 zre 0re lttri2 0cn addid1 (br K (<) (0)) a1i 0nn0 bcpasc.2 (0) K bcval4t mp3an12 orci bcpasc.2 zre leid bcpasc.2 bcpasc.2 K K zlem1ltt mp2an mpbi bcpasc.2 zre ax1re resubcl bcpasc.2 zre 0re lttr mpan (br (opr K (-) (1)) (<) (0)) (br (0) (<) (opr K (-) (1))) orc 0nn0 bcpasc.2 1z K (1) zsubclt mp2an (0) (opr K (-) (1)) bcval4t mp3an12 3syl (+) opreq12d 1nn0 bcpasc.2 (1) K bcval4t mp3an12 orci 3eqtr4d 0nn0 bcpasc.2 (0) K bcval4t mp3an12 olci (+) (opr (0) (C.) (opr K (-) (1))) opreq1d 0z bcpasc.2 (0) K zltp1let mp2an 1cn addid2 (<_) K breq1i ax1re bcpasc.2 zre leloe 3bitr 1nn0 bcpasc.2 (1) K bcval4t mp3an12 olci ax1re bcpasc.2 zre posdif (br (0) (<) (opr K (-) (1))) (br (opr K (-) (1)) (<) (0)) olc sylbi 0nn0 bcpasc.2 1z K (1) zsubclt mp2an (0) (opr K (-) (1)) bcval4t mp3an12 syl eqcomd (0) (+) opreq2d 0cn addid1 syl5eqr eqtr2d (1) K (-) (1) opreq1 1cn subid syl5eqr (0) (C.) opreq2d 0nn0 (0) bcn0t ax-mp syl5eqr (0) (+) opreq2d 1cn addid2 syl5eqr (1) K (1) (C.) opreq2 1nn0 (1) bcnnt ax-mp syl5eqr eqtr3d jaoi sylbi eqtrd jaoi sylbi pm2.61i N (0) (C.) K opreq1 N (0) (C.) (opr K (-) (1)) opreq1 (+) opreq12d N (0) (+) (1) opreq1 1cn addid2 syl6eq (C.) K opreq1d eqeq12d mpbiri syl bcpasc.2 K znnnlt1t ax-mp bcpasc.2 0z K (0) zleltp1t mp2an bcpasc.2 zre 0re leloe 1cn addid2 K (<) breq2i 3bitr3r bitr bcpasc.1 bcpasc.2 N K bcval4t mp3an12 orci bcpasc.2 zre leid bcpasc.2 bcpasc.2 K K zlem1ltt mp2an mpbi bcpasc.2 zre ax1re resubcl bcpasc.2 zre 0re lttr mpan (br (opr K (-) (1)) (<) (0)) (br N (<) (opr K (-) (1))) orc bcpasc.1 bcpasc.2 1z K (1) zsubclt mp2an N (opr K (-) (1)) bcval4t mp3an12 3syl (+) opreq12d 0cn addid1 syl6eq bcpasc.1 1nn0 nn0addcl bcpasc.2 (opr N (+) (1)) K bcval4t mp3an12 orci eqtr4d K (0) N (C.) opreq2 bcpasc.1 N bcn0t ax-mp syl6eq K (0) (-) (1) opreq1 (1) df-neg lt01 ax1re (1) lt0neg2t ax-mp mpbi eqbrtrr syl6eqbr (br (opr K (-) (1)) (<) (0)) (br N (<) (opr K (-) (1))) orc bcpasc.1 bcpasc.2 1z K (1) zsubclt mp2an N (opr K (-) (1)) bcval4t mp3an12 3syl (+) opreq12d 1cn addid1 syl6eq K (0) (opr N (+) (1)) (C.) opreq2 bcpasc.1 1nn0 nn0addcl (opr N (+) (1)) bcn0t ax-mp syl6req eqtrd jaoi sylbi bcpasc.1 nn0re bcpasc.2 zre ltnle bcpasc.1 bcpasc.2 N K bcval4t mp3an12 olci (+) (opr N (C.) (opr K (-) (1))) opreq1d bcpasc.1 N nn0zt ax-mp bcpasc.2 N K zltp1let mp2an bcpasc.1 nn0re ax1re readdcl bcpasc.2 zre leloe bitr bcpasc.1 nn0re ax1re bcpasc.2 zre ltaddsub (br N (<) (opr K (-) (1))) (br (opr K (-) (1)) (<) (0)) olc sylbi bcpasc.1 bcpasc.2 1z K (1) zsubclt mp2an N (opr K (-) (1)) bcval4t mp3an12 syl (0) (+) opreq2d 0cn addid1 syl6eq bcpasc.1 1nn0 nn0addcl bcpasc.2 (opr N (+) (1)) K bcval4t mp3an12 olci eqtr4d (opr N (+) (1)) K (-) (1) opreq1 bcpasc.1 nn0cn 1cn N (1) pncant mp2an syl5eqr N (C.) opreq2d bcpasc.1 N bcnnt ax-mp syl5eqr (0) (+) opreq2d 1cn addid2 syl5eqr (opr N (+) (1)) K (opr N (+) (1)) (C.) opreq2 bcpasc.1 1nn0 nn0addcl (opr N (+) (1)) bcnnt ax-mp syl5eqr eqtr3d jaoi sylbi eqtrd sylbir N K bcpasc2t 3ecase)) thm (bcpasct () () (-> (/\ (e. N (NN0)) (e. K (ZZ))) (= (opr (opr N (C.) K) (+) (opr N (C.) (opr K (-) (1)))) (opr (opr N (+) (1)) (C.) K))) (N (if (e. N (NN0)) N (1)) (C.) K opreq1 N (if (e. N (NN0)) N (1)) (C.) (opr K (-) (1)) opreq1 (+) opreq12d N (if (e. N (NN0)) N (1)) (+) (1) opreq1 (C.) K opreq1d eqeq12d K (if (e. K (ZZ)) K (1)) (if (e. N (NN0)) N (1)) (C.) opreq2 K (if (e. K (ZZ)) K (1)) (-) (1) opreq1 (if (e. N (NN0)) N (1)) (C.) opreq2d (+) opreq12d K (if (e. K (ZZ)) K (1)) (opr (if (e. N (NN0)) N (1)) (+) (1)) (C.) opreq2 eqeq12d 1nn0 N elimel 1z K elimel bcpasc dedth2h)) thm (bccl2t ((k n) (N k) (N n) (j n) (j k) (K k)) () (-> (/\ (e. N (NN0)) (e. K (opr (0) (...) N))) (e. (opr N (C.) K) (NN))) ((cv k) K (<_) N breq1 (cv k) K N (C.) opreq2 (NN) eleq1d imbi12d (NN) rcla4v (cv n) (1) (cv k) (<_) breq2 (cv n) (1) (C.) (cv k) opreq1 (NN) eleq1d imbi12d k (NN) ralbidv (cv n) (cv j) (cv k) (<_) breq2 (cv n) (cv j) (C.) (cv k) opreq1 (NN) eleq1d imbi12d k (NN) ralbidv (cv n) (opr (cv j) (+) (1)) (cv k) (<_) breq2 (cv n) (opr (cv j) (+) (1)) (C.) (cv k) opreq1 (NN) eleq1d imbi12d k (NN) ralbidv (cv n) N (cv k) (<_) breq2 (cv n) N (C.) (cv k) opreq1 (NN) eleq1d imbi12d k (NN) ralbidv (cv k) nnle1eq1t (cv k) (1) (1) (C.) opreq2 1nn0 (1) bcnnt ax-mp 1nn eqeltr syl6eqel syl6bi rgen (e. (cv j) (NN)) k ax-17 k (NN) (-> (br (cv k) (<_) (cv j)) (e. (opr (cv j) (C.) (cv k)) (NN))) hbra1 (cv k) (opr (cv j) (+) (1)) leloet (cv k) nnret (cv j) peano2nn (opr (cv j) (+) (1)) nnret syl syl2an (cv k) (cv j) nnleltp1t (= (cv k) (opr (cv j) (+) (1))) orbi1d bitr4d (cv k) nnge1t (cv k) nnret ax1re (1) (cv k) leloet mpan syl mpbid (e. (cv j) (NN)) adantr (opr (cv j) (C.) (cv k)) (opr (cv j) (C.) (opr (cv k) (-) (1))) nnaddclt k (NN) (-> (br (cv k) (<_) (cv j)) (e. (opr (cv j) (C.) (cv k)) (NN))) ra4 imp32 ancoms (e. (cv j) (NN)) adantllr (br (1) (<) (cv k)) adantlrl (cv k) nnzt (cv k) peano2zm syl 1nn (1) (cv k) nnltlem1t mpan biimpd (e. (opr (cv k) (-) (1)) (ZZ)) anim2d (opr (cv k) (-) (1)) elnnz1 syl6ibr mpand (cv n) (opr (cv k) (-) (1)) (<_) (cv j) breq1 (cv n) (opr (cv k) (-) (1)) (cv j) (C.) opreq2 (NN) eleq1d imbi12d (NN) rcla4v (cv k) (cv n) (<_) (cv j) breq1 (cv k) (cv n) (cv j) (C.) opreq2 (NN) eleq1d imbi12d (NN) cbvralv syl5ib com23 syl6 (e. (cv j) (NN)) adantr (cv k) lep1t ax1re (cv k) (1) (cv k) lesubaddt mp3an2 anidms mpbird (e. (cv j) (RR)) adantr (opr (cv k) (-) (1)) (cv k) (cv j) letrt 3expa ax1re (cv k) (1) resubclt mpan2 ancri sylan mpand (cv k) nnret (cv j) nnret syl2an syl5d imp42 sylanc (cv j) (cv k) bcpasc2t 3com12 3expa (NN) eleq1d (br (1) (<) (cv k)) adantrl (A.e. k (NN) (-> (br (cv k) (<_) (cv j)) (e. (opr (cv j) (C.) (cv k)) (NN)))) adantr mpbid exp31 com12 (1) (cv k) (opr (cv j) (+) (1)) (C.) opreq2 (NN) eleq1d (cv j) nnnn0t (cv j) bcnp11t syl (cv j) peano2nn eqeltrd syl5bi (e. (cv k) (NN)) adantld (A.e. k (NN) (-> (br (cv k) (<_) (cv j)) (e. (opr (cv j) (C.) (cv k)) (NN)))) a1dd (cv k) (opr (cv j) (+) (1)) (C.) (cv k) opreq1 (NN) eleq1d (cv k) nnnn0t (cv k) bcnnt syl 1nn syl6eqel syl5bi (e. (cv j) (NN)) adantrd (A.e. k (NN) (-> (br (cv k) (<_) (cv j)) (e. (opr (cv j) (C.) (cv k)) (NN)))) a1dd ccase2 com12 mpand sylbid ex com23 com4t r19.21ad nnind syl5com imp N (0) K (<_) breq2 negbid K nngt0t K nnret 0re (0) K ltnlet mpan syl mpbid syl5bir imp (e. (opr N (C.) K) (NN)) pm2.21d N nnnn0t N bcn0t syl 1nn syl6eqel (= K (0)) adantr K (0) N (C.) opreq2 (NN) eleq1d (e. N (NN)) adantl mpbird (br K (<_) N) a1d N (0) K (0) (C.) opreq12 0nn0 (0) bcn0t ax-mp 1nn eqeltr syl6eqel (br K (<_) N) a1d ccase N elnn0 K elnn0 syl2anb 3impia N (NN0) K elfz3nn0t K N elfznn0t (e. N (NN0)) adantl K (0) N elfzle2 (e. N (NN0)) adantl syl3anc)) thm (bcclt () () (-> (/\ (e. N (NN0)) (e. K (ZZ))) (e. (opr N (C.) K) (NN0))) (N K bccl2t (e. N (NN0)) (e. K (ZZ)) pm3.26 (/\ (br (0) (<_) K) (br K (<_) N)) adantr N K fznn0t biimprd exp3a K elnn0z syl5ibr exp3a imp43 sylanc (opr N (C.) K) nnnn0t syl N K bcvalt (/\ (br (0) (<_) K) (br K (<_) N)) (opr (` (!) N) (/) (opr (` (!) (opr N (-) K)) (x.) (` (!) K))) (0) iffalse sylan9eq 0nn0 syl6eqel pm2.61dan)) thm (climrel ((j k) (j x) (j y) (f j) (k x) (k y) (f k) (x y) (f x) (f y)) () (Rel (~~>)) (f y (/\ (e. (cv y) (CC)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` (cv f) (cv k)) (CC)) (br (` (abs) (opr (` (cv f) (cv k)) (-) (cv y))) (<) (cv x))))))))) relopab f y x j k df-clim (~~>) ({<,>|} f y (/\ (e. (cv y) (CC)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` (cv f) (cv k)) (CC)) (br (` (abs) (opr (` (cv f) (cv k)) (-) (cv y))) (<) (cv x)))))))))) releq ax-mp mpbir)) thm (clim ((j k) (j x) (j y) (f j) (F j) (k x) (k y) (f k) (F k) (x y) (f x) (F x) (f y) (F y) (F f) (A j) (A k) (A x) (A y) (A f)) () (-> (/\ (e. F C) (e. A D)) (<-> (br F (~~>) A) (/\ (e. A (CC)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))) ((cv f) F (cv k) fveq1 (CC) eleq1d (cv f) F (cv k) fveq1 (-) (cv y) opreq1d (abs) fveq2d (<) (cv x) breq1d anbi12d (br (cv j) (<_) (cv k)) imbi2d j (ZZ) k (ZZ) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv (e. (cv y) (CC)) anbi2d (cv y) A (CC) eleq1 (cv y) A (` F (cv k)) (-) opreq2 (abs) fveq2d (<) (cv x) breq1d (e. (` F (cv k)) (CC)) anbi2d (br (cv j) (<_) (cv k)) imbi2d j (ZZ) k (ZZ) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv anbi12d f y x j k df-clim C D brabg)) thm (climcl ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) () (-> (/\ (e. A C) (br F (~~>) A)) (e. A (CC))) (F (V) A C x j k clim biimpa pm3.26d ex climrel F A brrelexi con3i (e. A (CC)) pm2.21d (e. A C) adantr pm2.61ian imp)) thm (sumex ((k m) (k n) (k x) (k y) (m n) (m x) (m y) (n x) (n y) (x y) (A m) (A n) (A x) (A y) (B m) (B n) (B x) (B y)) () (e. (sum_ k A B) (V)) (k A B x m n y df-sum m n (/\ (= A (opr (cv m) (...) (cv n))) (e. (cv x) (` (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) B)) (ZZ))) (cv n)))) 2rexuz x abbii zex zex (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) B)) (ZZ))) (cv n) fvex (br (cv m) (<_) (cv n)) (= A (opr (cv m) (...) (cv n))) (e. (cv x) (` (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) B)) (ZZ))) (cv n))) anass (/\ (br (cv m) (<_) (cv n)) (= A (opr (cv m) (...) (cv n)))) (e. (cv x) (` (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) B)) (ZZ))) (cv n))) ancom bitr3 x abbii x (` (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) B)) (ZZ))) (cv n)) (/\ (br (cv m) (<_) (cv n)) (= A (opr (cv m) (...) (cv n)))) ssab2 eqsstr ssexi n abrexex2 m abrexex2 eqeltr zex x (CC) abid2 axcnex eqeltr x visset (cv x) (V) (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) B)) (ZZ))) climcl mpan (= A (` (ZZ>) (cv m))) adantl x ss2abi ssexi m abrexex2 uniex unex eqeltr)) thm (sumeq1 ((k m) (k n) (k x) (k y) (m n) (m x) (m y) (n x) (n y) (x y) (A m) (A n) (A x) (A y) (B m) (B n) (B x) (B y) (C m) (C n) (C x) (C y)) () (-> (= A B) (= (sum_ k A C) (sum_ k B C))) (A B (opr (cv m) (...) (cv n)) eqeq1 (e. (cv x) (` (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) C)) (ZZ))) (cv n))) anbi1d n (` (ZZ>) (cv m)) rexbidv m exbidv x abbidv A B (` (ZZ>) (cv m)) eqeq1 (br (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) C)) (ZZ))) (~~>) (cv x)) anbi1d m (ZZ) rexbidv x abbidv unieqd uneq12d k A C x m n y df-sum k B C x m n y df-sum 3eqtr4g)) thm (hbsum1 ((m n) (m x) (m y) (m z) (A m) (n x) (n y) (n z) (A n) (x y) (x z) (A x) (y z) (A y) (A z) (B m) (B n) (B x) (B y) (B z) (k m) (k n) (k x) (k y) (k z)) ((hbsum1.1 (-> (e. (cv x) A) (A. k (e. (cv x) A))))) (-> (e. (cv x) (sum_ k A B)) (A. k (e. (cv x) (sum_ k A B)))) ((e. (cv n) (` (ZZ>) (cv m))) k ax-17 hbsum1.1 (e. (cv x) (opr (cv m) (...) (cv n))) k ax-17 hbeq (e. (cv y) (<,> (cv m) (+))) k ax-17 (e. (cv y) (seq)) k ax-17 y k z (= (cv z) B) hbopab1 (e. (cv y) (ZZ)) k ax-17 hbres hbopr (e. (cv y) (cv n)) k ax-17 hbfv hban hbrex m hbex x y hbab (e. (cv m) (ZZ)) k ax-17 hbsum1.1 (e. (cv x) (` (ZZ>) (cv m))) k ax-17 hbeq (e. (cv n) (<,> (cv m) (+))) k ax-17 (e. (cv n) (seq)) k ax-17 n k z (= (cv z) B) hbopab1 (e. (cv n) (ZZ)) k ax-17 hbres hbopr (e. (cv n) (~~>)) k ax-17 (e. (cv n) (cv y)) k ax-17 hbbr hban hbrex x y hbab hbuni hbun k A B y m n z df-sum (cv x) eleq2i k A B y m n z df-sum (cv x) eleq2i k albii 3imtr4)) thm (hbsum ((m n) (m w) (m y) (m z) (A m) (n w) (n y) (n z) (A n) (w y) (w z) (A w) (y z) (A y) (A z) (B m) (B n) (B w) (B y) (B z) (k m) (k n) (k w) (k x) (k y) (k z) (m x) (n x) (w x) (x y) (x z)) ((hbsum.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbsum.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (sum_ k A B)) (A. x (e. (cv y) (sum_ k A B)))) ((e. (cv n) (` (ZZ>) (cv m))) x ax-17 hbsum.1 (e. (cv y) (opr (cv m) (...) (cv n))) x ax-17 hbeq (e. (cv z) (<,> (cv m) (+))) x ax-17 (e. (cv z) (seq)) x ax-17 (e. (cv y) (cv w)) x ax-17 hbsum.2 hbeq z k w hbopab (e. (cv z) (ZZ)) x ax-17 hbres hbopr (e. (cv z) (cv n)) x ax-17 hbfv hban hbrex m hbex y z hbab (e. (cv m) (ZZ)) x ax-17 hbsum.1 (e. (cv y) (` (ZZ>) (cv m))) x ax-17 hbeq (e. (cv y) (<,> (cv m) (+))) x ax-17 (e. (cv y) (seq)) x ax-17 (e. (cv y) (cv w)) x ax-17 hbsum.2 hbeq y k w hbopab (e. (cv y) (ZZ)) x ax-17 hbres hbopr (e. (cv y) (~~>)) x ax-17 (e. (cv y) (cv z)) x ax-17 hbbr hban hbrex y z hbab hbuni hbun k A B z m n w df-sum (cv y) eleq2i k A B z m n w df-sum (cv y) eleq2i x albii 3imtr4)) thm (sumeq2 ((k m) (k n) (k x) (k y) (A k) (m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y) (B m) (B n) (B x) (B y) (C m) (C n) (C x) (C y)) () (-> (A.e. k A (= B C)) (= (sum_ k A B) (sum_ k A C))) (k A (= B C) hbra1 (A.e. k A (= B C)) y ax-17 k A (= B C) ra4 imp (cv y) eqeq2d ex pm5.32d opabbid k y (= (cv y) B) A resopab k y (= (cv y) C) A resopab 3eqtr4g (= A (opr (cv m) (...) (cv n))) adantr A (opr (cv m) (...) (cv n)) ({<,>|} k y (= (cv y) B)) reseq2 (A.e. k A (= B C)) adantl A (opr (cv m) (...) (cv n)) ({<,>|} k y (= (cv y) C)) reseq2 (A.e. k A (= B C)) adantl 3eqtr3d (<,> (cv m) (+)) (seq) opreq2d (cv n) fveq1d (e. (cv n) (` (ZZ>) (cv m))) adantlr (cv x) (cv m) (cv n) elfzelz (cv x) (ZZ) ({<,>|} k y (= (cv y) B)) fvres syl (cv x) (opr (cv m) (...) (cv n)) ({<,>|} k y (= (cv y) B)) fvres eqtr4d rgen addex k y B funopabeq zex ({<,>|} k y (= (cv y) B)) (ZZ) (V) resfunexg mp2an k y B funopabeq (cv m) (...) (cv n) oprex ({<,>|} k y (= (cv y) B)) (opr (cv m) (...) (cv n)) (V) resfunexg mp2an (cv n) (cv m) x seqzfveq mpan2 (A.e. k A (= B C)) (= A (opr (cv m) (...) (cv n))) ad2antlr (cv x) (cv m) (cv n) elfzelz (cv x) (ZZ) ({<,>|} k y (= (cv y) C)) fvres syl (cv x) (opr (cv m) (...) (cv n)) ({<,>|} k y (= (cv y) C)) fvres eqtr4d rgen addex k y C funopabeq zex ({<,>|} k y (= (cv y) C)) (ZZ) (V) resfunexg mp2an k y C funopabeq (cv m) (...) (cv n) oprex ({<,>|} k y (= (cv y) C)) (opr (cv m) (...) (cv n)) (V) resfunexg mp2an (cv n) (cv m) x seqzfveq mpan2 (A.e. k A (= B C)) (= A (opr (cv m) (...) (cv n))) ad2antlr 3eqtr4d (cv x) eleq2d ex pm5.32d rexbidva m exbidv x abbidv k A (= B C) hbra1 (A.e. k A (= B C)) y ax-17 k A (= B C) ra4 imp (cv y) eqeq2d ex pm5.32d opabbid k y (= (cv y) B) A resopab k y (= (cv y) C) A resopab 3eqtr4g (= A (` (ZZ>) (cv m))) adantr A (` (ZZ>) (cv m)) ({<,>|} k y (= (cv y) B)) reseq2 (A.e. k A (= B C)) adantl A (` (ZZ>) (cv m)) ({<,>|} k y (= (cv y) C)) reseq2 (A.e. k A (= B C)) adantl 3eqtr3d (<,> (cv m) (+)) (seq) opreq2d (e. (cv m) (ZZ)) adantlr (cv x) (cv m) eluzelz (cv x) (ZZ) ({<,>|} k y (= (cv y) B)) fvres syl (cv x) (` (ZZ>) (cv m)) ({<,>|} k y (= (cv y) B)) fvres eqtr4d rgen addex k y B funopabeq zex ({<,>|} k y (= (cv y) B)) (ZZ) (V) resfunexg mp2an k y B funopabeq (ZZ>) (cv m) fvex ({<,>|} k y (= (cv y) B)) (` (ZZ>) (cv m)) (V) resfunexg mp2an (cv m) x seqzeq mpan2 (A.e. k A (= B C)) (= A (` (ZZ>) (cv m))) ad2antlr (cv x) (cv m) eluzelz (cv x) (ZZ) ({<,>|} k y (= (cv y) C)) fvres syl (cv x) (` (ZZ>) (cv m)) ({<,>|} k y (= (cv y) C)) fvres eqtr4d rgen addex k y C funopabeq zex ({<,>|} k y (= (cv y) C)) (ZZ) (V) resfunexg mp2an k y C funopabeq (ZZ>) (cv m) fvex ({<,>|} k y (= (cv y) C)) (` (ZZ>) (cv m)) (V) resfunexg mp2an (cv m) x seqzeq mpan2 (A.e. k A (= B C)) (= A (` (ZZ>) (cv m))) ad2antlr 3eqtr4d (~~>) (cv x) breq1d ex pm5.32d rexbidva x abbidv unieqd uneq12d k A B x m n y df-sum k A C x m n y df-sum 3eqtr4g)) thm (cbvsum ((m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y) (B m) (B n) (B x) (B y) (C m) (C n) (C x) (C y) (j k) (j m) (j n) (j x) (j y) (k m) (k n) (k x) (k y)) ((cbvsum.1 (-> (e. (cv x) B) (A. k (e. (cv x) B)))) (cbvsum.2 (-> (e. (cv x) C) (A. j (e. (cv x) C)))) (cbvsum.3 (-> (= (cv j) (cv k)) (= B C)))) (= (sum_ j A B) (sum_ k A C)) (cbvsum.1 hbeleq cbvsum.2 hbeleq cbvsum.3 (cv x) eqeq2d x cbvopab1 ({<,>|} j x (= (cv x) B)) ({<,>|} k x (= (cv x) C)) (ZZ) reseq1 ax-mp (<,> (cv m) (+)) (seq) opreq2i (cv n) fveq1i (cv y) eleq2i (= A (opr (cv m) (...) (cv n))) anbi2i n (` (ZZ>) (cv m)) rexbii m exbii y abbii cbvsum.1 hbeleq cbvsum.2 hbeleq cbvsum.3 (cv x) eqeq2d x cbvopab1 ({<,>|} j x (= (cv x) B)) ({<,>|} k x (= (cv x) C)) (ZZ) reseq1 ax-mp (<,> (cv m) (+)) (seq) opreq2i (~~>) (cv y) breq1i (= A (` (ZZ>) (cv m))) anbi2i m (ZZ) rexbii y abbii unieqi uneq12i j A B y m n x df-sum k A C y m n x df-sum 3eqtr4)) thm (sumeq1i () ((sumeq1i.1 (= A B))) (= (sum_ k A C) (sum_ k B C)) (sumeq1i.1 A B k C sumeq1 ax-mp)) thm (sumeq2i ((A k)) ((sumeq2i.1 (-> (e. (cv k) A) (= B C)))) (= (sum_ k A B) (sum_ k A C)) (k A B C sumeq2 sumeq2i.1 mprg)) thm (sumeq1d () ((sumeq1d.1 (-> ph (= A B)))) (-> ph (= (sum_ k A C) (sum_ k B C))) (sumeq1d.1 A B k C sumeq1 syl)) thm (sumeq2d ((A k)) ((sumeq2d.1 (-> ph (A.e. k A (= B C))))) (-> ph (= (sum_ k A B) (sum_ k A C))) (sumeq2d.1 k A B C sumeq2 syl)) thm (sumeq2dv ((A k) (k ph)) ((sumeq2dv.1 (-> (/\ ph (e. (cv k) A)) (= B C)))) (-> ph (= (sum_ k A B) (sum_ k A C))) (sumeq2dv.1 r19.21aiva k A B C sumeq2 syl)) thm (sumeq2sdv ((A k) (k ph)) ((sumeq2sdv.1 (-> ph (= B C)))) (-> ph (= (sum_ k A B) (sum_ k A C))) (sumeq2sdv.1 (e. (cv k) A) adantr sumeq2dv)) thm (2sumeq2d ((A j) (B k)) ((2sumeq2d.1 (-> ph (A.e. j A (A.e. k B (= C D)))))) (-> ph (= (sum_ j A (sum_ k B C)) (sum_ j A (sum_ k B D)))) (2sumeq2d.1 k B C D sumeq2 j A r19.20si syl j A (sum_ k B C) (sum_ k B D) sumeq2 syl)) thm (2sumeq2dv ((j k) (A j) (A k) (B k) (j ph) (k ph)) ((2sumeq2dv.1 (-> (/\/\ ph (e. (cv j) A) (e. (cv k) B)) (= C D)))) (-> ph (= (sum_ j A (sum_ k B C)) (sum_ j A (sum_ k B D)))) (2sumeq2dv.1 3expa r19.21aiva r19.21aiva k B C D sumeq2 j A r19.20si syl j A (sum_ k B C) (sum_ k B D) sumeq2 syl)) thm (sumeq12d ((A k)) ((sumeq12d.1 (-> ph (= A B))) (sumeq12d.2 (-> ph (A.e. k A (= C D))))) (-> ph (= (sum_ k A C) (sum_ k B D))) (sumeq12d.2 sumeq2d sumeq12d.1 k D sumeq1d eqtrd)) thm (sumeq12rd ((B k)) ((sumeq12rd.1 (-> ph (= A B))) (sumeq12rd.2 (-> ph (A.e. k B (= C D))))) (-> ph (= (sum_ k A C) (sum_ k B D))) (sumeq12rd.1 k C sumeq1d sumeq12rd.2 sumeq2d eqtrd)) thm (sumeqfv ((A y) (k y) (B k) (B y) (C k)) ((sumeqfv.1 (e. A (V))) (sumeqfv.2 (= F ({<,>|} k y (/\ (e. (cv k) B) (= (cv y) A)))))) (-> (C_ C B) (= (sum_ k C (` F (cv k))) (sum_ k C A))) (C B (cv k) ssel2 sumeqfv.1 k B A (V) y fvopab2 mpan2 sumeqfv.2 (cv k) fveq1i syl5eq syl sumeq2dv)) thm (dffsum ((m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y) (M m) (M n) (M x) (M y) (N m) (N n) (N x) (N y) (k m) (k n) (k x) (k y)) () (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) N) A) (` (opr (<,> M (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) N))) (n visset N M (cv n) (V) (cv m) fzoptht mpan2 M (cv m) eqcom N (cv n) eqcom anbi12i syl6bb (e. (cv x) (` (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (cv n))) anbi1d n (` (ZZ>) (cv m)) rexbidv m exbidv (cv m) M (<_) (cv n) breq1 (cv m) M (+) opeq1 (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ)) opreq1d (cv n) fveq1d (cv x) eleq2d anbi12d (cv n) N M (<_) breq2 (cv n) N (opr (<,> M (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) fveq2 (cv x) eleq2d anbi12d (ZZ) (ZZ) ceqsrex2v N M eluzel2 N M eluzelz sylanc N M eluzle (e. (cv x) (` (opr (<,> M (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) N)) biantrurd bitr4d m n (/\ (/\ (= (cv m) M) (= (cv n) N)) (e. (cv x) (` (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (cv n)))) 2rexuz (br (cv m) (<_) (cv n)) (/\ (= (cv m) M) (= (cv n) N)) (e. (cv x) (` (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (cv n))) an12 m (ZZ) n (ZZ) 2rexbii bitr syl5bb bitrd abbi1dv N M (cv m) fzneuzt (br (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (~~>) (cv x)) intnanrd nrexdv x nexdv x (E.e. m (ZZ) (/\ (= (opr M (...) N) (` (ZZ>) (cv m))) (br (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (~~>) (cv x)))) abn0 con1bii sylib unieqd uni0 syl6eq uneq12d (` (opr (<,> M (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) N) un0 syl6eq k (opr M (...) N) A x m n y df-sum syl5eq)) thm (fsumserz ((k y) (F k) (F y) (M y) (N y)) ((fsumserz.1 (e. F (V)))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) N) (` F (cv k))) (` (opr (<,> M (+)) (seq) F) N))) (N M k (` F (cv k)) y dffsum N M eluzel2 addex fsumserz.1 M k y seqzres2 syl N fveq1d eqtrd)) thm (fsumserzf ((j x) (F j) (F x) (M j) (M x) (N x) (j k) (k x)) ((fsumserzf.1 (e. F (V))) (fsumserzf.2 (-> (e. (cv x) F) (A. k (e. (cv x) F))))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) N) (` F (cv k))) (` (opr (<,> M (+)) (seq) F) N))) (fsumserzf.1 N M j fsumserz (e. (cv x) (` F (cv k))) j ax-17 fsumserzf.2 (e. (cv x) (cv j)) k ax-17 hbfv (cv k) (cv j) F fveq2 (opr M (...) N) cbvsum syl5eq)) thm (fsumser0f ((F x) (N x) (k x)) ((fsumserzf.1 (e. F (V))) (fsumserzf.2 (-> (e. (cv x) F) (A. k (e. (cv x) F))))) (-> (e. N (NN0)) (= (sum_ k (opr (0) (...) N) (` F (cv k))) (` (opr (+) (seq0) F) N))) (fsumserzf.1 fsumserzf.2 N (0) fsumserzf N elnn0uz addex fsumserzf.1 seq0seqz N fveq1i (sum_ k (opr (0) (...) N) (` F (cv k))) eqeq2i 3imtr4)) thm (fsumser1f ((F x) (N x) (k x)) ((fsumserzf.1 (e. F (V))) (fsumserzf.2 (-> (e. (cv x) F) (A. k (e. (cv x) F))))) (-> (e. N (NN)) (= (sum_ k (opr (1) (...) N) (` F (cv k))) (` (opr (+) (seq1) F) N))) (fsumserzf.1 fsumserzf.2 N (1) fsumserzf N elnnuz addex fsumserzf.1 seq1seqz N fveq1i (sum_ k (opr (1) (...) N) (` F (cv k))) eqeq2i 3imtr4)) thm (fsumserz2 ((A y) (F x) (k x) (k y) (M k) (x y) (M x) (M y) (N k) (N x)) ((fsumserz2.1 (e. A (V))) (fsumserz2.2 (= F ({<,>|} k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A)))))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) N) A) (` (opr (<,> M (+)) (seq) F) N))) (fsumserz2.2 (ZZ>) M fvex k y A funopabex2 eqeltr x k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A)) hbopab1 fsumserz2.2 (cv x) eleq2i fsumserz2.2 (cv x) eleq2i k albii 3imtr4 N M fsumserzf M N fzssuzt fsumserz2.1 fsumserz2.2 (opr M (...) N) sumeqfv ax-mp syl5eqr)) thm (serzfsum ((A y) (n x) (n z) (F n) (x z) (F x) (F z) (k n) (k x) (k y) (k z) (M k) (n y) (M n) (x y) (M x) (y z) (M y) (M z)) ((serzfsum.1 (e. A (V))) (serzfsum.2 (= F ({<,>|} k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A))))) (serzfsum.3 (= G ({<,>|} n z (/\ (e. (cv n) (` (ZZ>) M)) (= (cv z) (sum_ k (opr M (...) (cv n)) A))))))) (-> (e. M (ZZ)) (= (opr (<,> M (+)) (seq) F) G)) (addex serzfsum.2 (ZZ>) M fvex k y A funopabex2 eqeltr M seqzfn (opr (<,> M (+)) (seq) F) (` (ZZ>) M) n z fnopabfv sylib serzfsum.3 (cv k) M (cv n) elfzuzt serzfsum.1 k (` (ZZ>) M) A (V) y fvopab2 mpan2 syl (e. (cv n) (` (ZZ>) M)) adantl serzfsum.2 (cv k) fveq1i syl5eq sumeq2dv serzfsum.2 (ZZ>) M fvex k y A funopabex2 eqeltr x k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A)) hbopab1 serzfsum.2 (cv x) eleq2i serzfsum.2 (cv x) eleq2i k albii 3imtr4 (cv n) M fsumserzf eqtr3d (cv z) eqeq2d pm5.32i n z opabbii eqtr syl6eqr)) thm (fsum1 ((x y) (A x) (A y) (k y) (B k) (B y) (k x) (M k) (M x) (M y)) ((fsum1.1 (-> (= (cv k) M) (= A B)))) (-> (/\ (e. B C) (e. M (ZZ))) (= (sum_ k (opr M (...) M) A) B)) (B C elisset fsum1.1 (V) eleq1d biimprcd (cv k) M elfz1eqt syl5 r19.21aiv (cv k) M M elfzelz k (ZZ) A (V) y fvopab2 ex syl r19.20i sumeq2d 3syl (e. M (ZZ)) adantr M uzidt zex k y A funopabex2 x k y (/\ (e. (cv k) (ZZ)) (= (cv y) A)) hbopab1 M M fsumserzf syl (e. B C) adantl eqtr3d addex zex k y A funopabex2 M seqz1 (e. B C) adantl fsum1.1 ({<,>|} k y (/\ (e. (cv k) (ZZ)) (= (cv y) A))) eqid C fvopab4g ancoms 3eqtrd)) thm (fsump1 ((x y) (A x) (A y) (k y) (B k) (B y) (k x) (M k) (M x) (N k) (N x) (N y)) ((fsump1.1 (e. A (V))) (fsump1.2 (e. B (V))) (fsump1.3 (-> (= (cv k) (opr N (+) (1))) (= A B)))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) (opr N (+) (1))) A) (opr (sum_ k (opr M (...) N) A) (+) B))) (N M peano2uz zex k y A funopabex2 x k y (/\ (e. (cv k) (ZZ)) (= (cv y) A)) hbopab1 (opr N (+) (1)) M fsumserzf syl k (opr M (...) (opr N (+) (1))) (` ({<,>|} k y (/\ (e. (cv k) (ZZ)) (= (cv y) A))) (cv k)) A sumeq2 (cv k) M (opr N (+) (1)) elfzelz fsump1.1 k (ZZ) A (V) y fvopab2 mpan2 syl mprg syl5eqr addex zex k y A funopabex2 N M seqzp1 zex k y A funopabex2 x k y (/\ (e. (cv k) (ZZ)) (= (cv y) A)) hbopab1 N M fsumserzf k (opr M (...) N) (` ({<,>|} k y (/\ (e. (cv k) (ZZ)) (= (cv y) A))) (cv k)) A sumeq2 (cv k) M N elfzelz fsump1.1 k (ZZ) A (V) y fvopab2 mpan2 syl mprg syl5reqr N M eluzelz N peano2z fsump1.3 ({<,>|} k y (/\ (e. (cv k) (ZZ)) (= (cv y) A))) eqid fsump1.2 fvopab4 3syl (+) opreq12d 3eqtrd)) thm (fsum1f ((j y) (j z) (A j) (y z) (A y) (A z) (j x) (B j) (B x) (j k) (M j) (k z) (M k) (M z) (k x) (k y) (x y)) ((fxm1f.3 (-> (e. (cv x) B) (A. k (e. (cv x) B)))) (fxm1f.4 (-> (= (cv k) M) (= A B)))) (-> (/\ (e. B C) (e. M (ZZ))) (= (sum_ k (opr M (...) M) A) B)) (j visset M k eqvinc j visset k (e. (cv y) A) hbsbc1v z y hbab fxm1f.3 hbeq k j A y sbab eqcomd fxm1f.4 sylan9eq 19.23ai sylbi C fsum1 (e. (cv z) A) j ax-17 j visset k (e. (cv y) A) hbsbc1v z y hbab k j A y sbab (opr M (...) M) cbvsum syl5eq)) thm (fsum1slem ((x y) (A x) (A y) (M x) (k x) (k y)) ((fsum1slem.1 (e. A (V)))) (-> (e. M (ZZ)) (= (sum_ k (opr M (...) M) A) ([_/]_ M k A))) ((cv x) M (cv x) M (...) opreq12 anidms k A sumeq1d (cv x) M k A csbeq1 eqeq12d x visset fsum1slem.1 k ax-gen (cv x) (V) k A (V) csbexg mp2an x visset (e. (cv y) (cv x)) k ax-17 A hbcsb1 k (cv x) A csbeq1a (V) fsum1f mpan vtoclga)) thm (fsum1s ((k x) (M k) (M x) (A x) (B x)) () (-> (/\ (e. M (ZZ)) (A.e. k (opr M (...) M) (e. A B))) (= (sum_ k (opr M (...) M) A) ([_/]_ M k A))) (x A class2set M k fsum1slem (A.e. k (opr M (...) M) (e. A (V))) adantr A (V) x class2seteq k (opr M (...) M) r19.20si sumeq2d (e. M (ZZ)) adantl M k (e. A (V)) fz1sbct x equid A (V) x class2seteq (= (cv x) (cv x)) a1i M (ZZ) k sbc19.20dv mpan sylbid M (ZZ) k ({e.|} x A (e. A (V))) A sbceqdig sylibd imp 3eqtr3d A B elisset k (opr M (...) M) r19.20si sylan2)) thm (fsum1s2 ((M k)) () (-> (/\ (e. M (ZZ)) (e. ([_/]_ M k A) B)) (= (sum_ k (opr M (...) M) A) ([_/]_ M k A))) (M k (e. A (V)) fz1sbct M (ZZ) k A (V) sbcel1g bitrd ([_/]_ M k A) B elisset syl5bir imp M k A (V) fsum1s syldan)) thm (fsump1f ((j y) (j z) (A j) (y z) (A y) (A z) (j x) (B j) (B x) (M j) (M z) (j k) (N j) (k z) (N k) (N z) (k x) (k y) (x y)) ((fsump1f.1 (e. A (V))) (fsump1f.2 (e. B (V))) (fsump1f.3 (-> (e. (cv x) B) (A. k (e. (cv x) B)))) (fsump1f.4 (-> (= (cv k) (opr N (+) (1))) (= A B)))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) (opr N (+) (1))) A) (opr (sum_ k (opr M (...) N) A) (+) B))) ((cv j) k A y df-csb j visset fsump1f.1 k ax-gen (cv j) (V) k A (V) csbexg mp2an eqeltrr fsump1f.2 j visset (opr N (+) (1)) k eqvinc j visset k (e. (cv y) A) hbsbc1v z y hbab fsump1f.3 hbeq k j A y sbab eqcomd fsump1f.4 sylan9eq 19.23ai sylbi M fsump1 (e. (cv z) A) j ax-17 j visset k (e. (cv y) A) hbsbc1v z y hbab k j A y sbab (opr M (...) (opr N (+) (1))) cbvsum (e. (cv z) A) j ax-17 j visset k (e. (cv y) A) hbsbc1v z y hbab k j A y sbab (opr M (...) N) cbvsum (+) B opreq1i 3eqtr4g)) thm (fsump1slem ((j x) (A j) (A x) (M j) (N j) (j k) (k x)) ((fsump1slem.1 (e. A (V)))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) (opr N (+) (1))) A) (opr (sum_ k (opr M (...) N) A) (+) ([_/]_ (opr N (+) (1)) k A)))) ((cv j) N (+) (1) opreq1 M (...) opreq2d k A sumeq1d (cv j) N M (...) opreq2 k A sumeq1d (cv j) N (+) (1) opreq1 k A csbeq1d (+) opreq12d eqeq12d fsump1slem.1 (cv j) (+) (1) oprex fsump1slem.1 k ax-gen (opr (cv j) (+) (1)) (V) k A (V) csbexg mp2an (cv j) (+) (1) oprex (e. (cv x) (opr (cv j) (+) (1))) k ax-17 A hbcsb1 k (opr (cv j) (+) (1)) A csbeq1a M fsump1f vtoclga)) thm (fsump1s ((k x) (M k) (M x) (N k) (N x) (A x) (B x)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) (opr N (+) (1))) (e. A B))) (= (sum_ k (opr M (...) (opr N (+) (1))) A) (opr (sum_ k (opr M (...) N) A) (+) ([_/]_ (opr N (+) (1)) k A)))) (x A class2set N M k fsump1slem (A.e. k (opr M (...) (opr N (+) (1))) (e. A (V))) adantr A (V) x class2seteq k (opr M (...) (opr N (+) (1))) r19.20si sumeq2d (e. N (` (ZZ>) M)) adantl M N fzssp1t N M eluzel2 N M eluzelz sylanc (cv k) sseld A (V) x class2seteq (e. N (` (ZZ>) M)) a1i imim12d r19.20dv2 imp sumeq2d (opr N (+) (1)) (opr M (...) (opr N (+) (1))) k (e. A (V)) ra4sbca N M peano2uz (opr N (+) (1)) M eluzfz2t syl sylan x equid N (+) (1) oprex A (V) x class2seteq (= (cv x) (cv x)) a1i (opr N (+) (1)) (V) k sbc19.20dv mp2an syl N (+) (1) oprex (opr N (+) (1)) (V) k ({e.|} x A (e. A (V))) A sbceqdig ax-mp sylib (+) opreq12d 3eqtr3d A B elisset k (opr M (...) (opr N (+) (1))) r19.20si sylan2)) thm (fsumcllem ((j m) (j x) (j y) (A j) (m x) (m y) (A m) (x y) (A x) (A y) (j k) (C j) (k m) (k x) (k y) (C k) (C m) (C x) (C y) (M j) (M k) (M m) (M x) (M y) (N j) (N k)) ((fsumcllem.1 (C_ C (CC))) (fsumcllem.2 (-> (/\ (e. (cv x) C) (e. (cv y) C)) (e. (opr (cv x) (+) (cv y)) C)))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. A C))) (e. (sum_ k (opr M (...) N) A) C)) ((cv j) M M (...) opreq2 k (e. A C) raleq1d (cv j) M M (...) opreq2 k A sumeq1d C eleq1d imbi12d (cv j) (cv m) M (...) opreq2 k (e. A C) raleq1d (cv j) (cv m) M (...) opreq2 k A sumeq1d C eleq1d imbi12d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k (e. A C) raleq1d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k A sumeq1d C eleq1d imbi12d (cv j) N M (...) opreq2 k (e. A C) raleq1d (cv j) N M (...) opreq2 k A sumeq1d C eleq1d imbi12d M k A C fsum1s M (opr M (...) M) k A C ra4csbela M elfz3t sylan eqeltrd ex (cv m) M k A C fsump1s (-> (A.e. k (opr M (...) (cv m)) (e. A C)) (e. (sum_ k (opr M (...) (cv m)) A) C)) adantrl fsumcllem.2 (sum_ k (opr M (...) (cv m)) A) ([_/]_ (opr (cv m) (+) (1)) k A) caoprcl M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A C) imim1d r19.20dv2 (e. (sum_ k (opr M (...) (cv m)) A) C) imim1d imp32 (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k A C ra4csbela (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl sylan (-> (A.e. k (opr M (...) (cv m)) (e. A C)) (e. (sum_ k (opr M (...) (cv m)) A) C)) adantrl sylanc eqeltrd exp32 uzind4 imp)) thm (fsumclt ((x y) (A x) (A y) (k x) (M x) (k y) (M y) (M k) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. A (CC)))) (e. (sum_ k (opr M (...) N) A) (CC))) ((CC) ssid (cv x) (cv y) axaddcl N M k A fsumcllem)) thm (fsum0clt ((N k)) () (-> (/\ (e. N (NN0)) (A.e. k (opr (0) (...) N) (e. A (CC)))) (e. (sum_ k (opr (0) (...) N) A) (CC))) (N (0) k A fsumclt N elnn0uz sylanb)) thm (fsumreclt ((x y) (A x) (A y) (k x) (M x) (k y) (M y) (M k) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. A (RR)))) (e. (sum_ k (opr M (...) N) A) (RR))) (axresscn (cv x) (cv y) axaddrcl N M k A fsumcllem)) thm (fsum1ps ((j m) (A j) (A m) (j k) (M j) (k m) (M k) (M m) (N j) (N k)) () (-> (/\/\ (e. N (` (ZZ>) M)) (br M (<) N) (A.e. k (opr M (...) N) (e. A (CC)))) (= (sum_ k (opr M (...) N) A) (opr ([_/]_ M k A) (+) (sum_ k (opr (opr M (+) (1)) (...) N) A)))) ((cv j) (opr M (+) (1)) M (...) opreq2 k (e. A (CC)) raleq1d (cv j) (opr M (+) (1)) M (...) opreq2 k A sumeq1d (cv j) (opr M (+) (1)) (opr M (+) (1)) (...) opreq2 k A sumeq1d ([_/]_ M k A) (+) opreq2d eqeq12d imbi12d (cv j) (cv m) M (...) opreq2 k (e. A (CC)) raleq1d (cv j) (cv m) M (...) opreq2 k A sumeq1d (cv j) (cv m) (opr M (+) (1)) (...) opreq2 k A sumeq1d ([_/]_ M k A) (+) opreq2d eqeq12d imbi12d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k (e. A (CC)) raleq1d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k A sumeq1d (cv j) (opr (cv m) (+) (1)) (opr M (+) (1)) (...) opreq2 k A sumeq1d ([_/]_ M k A) (+) opreq2d eqeq12d imbi12d (cv j) N M (...) opreq2 k (e. A (CC)) raleq1d (cv j) N M (...) opreq2 k A sumeq1d (cv j) N (opr M (+) (1)) (...) opreq2 k A sumeq1d ([_/]_ M k A) (+) opreq2d eqeq12d imbi12d M M k A (CC) fsump1s M uzidt sylan M M fzssp1t anidms (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp M k A (CC) fsum1s syldan eqcomd M peano2z M (opr M (+) (1)) fzp1sst mpdan (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (opr M (+) (1)) k A (CC) fsum1s M peano2z sylan syldan (+) opreq12d eqtr4d ex M (cv m) fzssp1t (br M (<) (cv m)) 3adant3 (cv k) sseld (e. A (CC)) imim1d r19.20dv2 (= (sum_ k (opr M (...) (cv m)) A) (opr ([_/]_ M k A) (+) (sum_ k (opr (opr M (+) (1)) (...) (cv m)) A))) imim1d (A.e. k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC))) (= (sum_ k (opr M (...) (cv m)) A) (opr ([_/]_ M k A) (+) (sum_ k (opr (opr M (+) (1)) (...) (cv m)) A))) pm3.35 ancoms (+) ([_/]_ (opr (cv m) (+) (1)) k A) opreq1d (/\/\ (e. M (ZZ)) (e. (cv m) (ZZ)) (br M (<) (cv m))) adantll (cv m) M k A (CC) fsump1s M (cv m) ltlet M zret (cv m) zret syl2an M (cv m) eluzt sylibrd 3impia sylan (-> (A.e. k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC))) (= (sum_ k (opr M (...) (cv m)) A) (opr ([_/]_ M k A) (+) (sum_ k (opr (opr M (+) (1)) (...) (cv m)) A)))) adantlr M (opr (cv m) (+) (1)) fzp1sst (cv m) peano2z sylan2 (br M (<) (cv m)) 3adant3 (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (cv m) (opr M (+) (1)) k A (CC) fsump1s M (cv m) zltp1let (opr M (+) (1)) (cv m) eluzt M peano2z sylan bitr4d biimp3a sylan syldan ([_/]_ M k A) (+) opreq2d ([_/]_ M k A) (sum_ k (opr (opr M (+) (1)) (...) (cv m)) A) ([_/]_ (opr (cv m) (+) (1)) k A) axaddass M (opr M (...) (opr (cv m) (+) (1))) k A (CC) ra4csbela M (cv m) ltlet M zret (cv m) zret syl2an M (cv m) eluzt sylibrd 3impia (cv m) M peano2uz syl (opr (cv m) (+) (1)) M eluzfz1t syl sylan M (cv m) fzp1sst M (cv m) fzssp1t sstrd (br M (<) (cv m)) 3adant3 (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (cv m) (opr M (+) (1)) k A fsumclt M (cv m) zltp1let (opr M (+) (1)) (cv m) eluzt M peano2z sylan bitr4d biimp3a sylan syldan (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k A (CC) ra4csbela M (cv m) ltlet M zret (cv m) zret syl2an M (cv m) eluzt sylibrd 3impia (cv m) M peano2uz syl (opr (cv m) (+) (1)) M eluzfz2b sylib sylan syl3anc eqtr4d (-> (A.e. k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC))) (= (sum_ k (opr M (...) (cv m)) A) (opr ([_/]_ M k A) (+) (sum_ k (opr (opr M (+) (1)) (...) (cv m)) A)))) adantlr 3eqtr4d exp31 syld uzind2 3expa N M eluzel2 N M eluzelz jca sylan 3impia)) thm (fsum1p ((k x) (B x) (B k) (M x) (M k) (N k)) ((fsum1p.1 (-> (= (cv k) M) (= A B)))) (-> (/\/\ (e. N (` (ZZ>) M)) (br M (<) N) (A.e. k (opr M (...) N) (e. A (CC)))) (= (sum_ k (opr M (...) N) A) (opr B (+) (sum_ k (opr (opr M (+) (1)) (...) N) A)))) (N M k A fsum1ps N M eluzel2 (e. (cv x) B) k ax-17 (e. M (ZZ)) a1i fsum1p.1 csbiegf syl (+) (sum_ k (opr (opr M (+) (1)) (...) N) A) opreq1d (br M (<) N) (A.e. k (opr M (...) N) (e. A (CC))) 3ad2ant1 eqtrd)) thm (fsumsplit ((j m) (A j) (A m) (K j) (j k) (M j) (k m) (M k) (M m) (N j) (N k) (N m)) () (-> (/\/\ (e. N (ZZ)) (e. K (opr M (...) (opr N (-) (1)))) (A.e. k (opr M (...) N) (e. A (CC)))) (= (sum_ k (opr M (...) N) A) (opr (sum_ k (opr M (...) K) A) (+) (sum_ k (opr (opr K (+) (1)) (...) N) A)))) ((cv j) M (<) N breq1 (e. N (ZZ)) anbi2d (A.e. k (opr M (...) N) (e. A (CC))) anbi1d (cv j) M M (...) opreq2 k A sumeq1d (cv j) M (+) (1) opreq1 (...) N opreq1d k A sumeq1d (+) opreq12d (sum_ k (opr M (...) N) A) eqeq2d imbi12d (cv j) (cv m) (<) N breq1 (e. N (ZZ)) anbi2d (A.e. k (opr M (...) N) (e. A (CC))) anbi1d (cv j) (cv m) M (...) opreq2 k A sumeq1d (cv j) (cv m) (+) (1) opreq1 (...) N opreq1d k A sumeq1d (+) opreq12d (sum_ k (opr M (...) N) A) eqeq2d imbi12d (cv j) (opr (cv m) (+) (1)) (<) N breq1 (e. N (ZZ)) anbi2d (A.e. k (opr M (...) N) (e. A (CC))) anbi1d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k A sumeq1d (cv j) (opr (cv m) (+) (1)) (+) (1) opreq1 (...) N opreq1d k A sumeq1d (+) opreq12d (sum_ k (opr M (...) N) A) eqeq2d imbi12d (cv j) K (<) N breq1 (e. N (ZZ)) anbi2d (A.e. k (opr M (...) N) (e. A (CC))) anbi1d (cv j) K M (...) opreq2 k A sumeq1d (cv j) K (+) (1) opreq1 (...) N opreq1d k A sumeq1d (+) opreq12d (sum_ k (opr M (...) N) A) eqeq2d imbi12d N M k A fsum1ps 3expa N M eluzel2 N M M fzss2t mpdan (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp M k A (CC) fsum1s N M eluzel2 sylan syldan (br M (<) N) adantlr (+) (sum_ k (opr (opr M (+) (1)) (...) N) A) opreq1d eqtr4d M N ltlet M zret N zret syl2an M N eluzt sylibrd impac sylan exp41 imp4c (cv m) M eluzelz (cv m) ltp1t (e. N (RR)) adantr (cv m) (opr (cv m) (+) (1)) N axlttrn 3expa (cv m) peano2re ancli sylan mpand (cv m) zret N zret syl2an ex syl imdistand (A.e. k (opr M (...) N) (e. A (CC))) anim1d (= (sum_ k (opr M (...) N) A) (opr (sum_ k (opr M (...) (cv m)) A) (+) (sum_ k (opr (opr (cv m) (+) (1)) (...) N) A))) imim1d (= (sum_ k (opr M (...) N) A) (opr (sum_ k (opr M (...) (cv m)) A) (+) (sum_ k (opr (opr (cv m) (+) (1)) (...) N) A))) id (cv m) M k A (CC) fsump1s (e. (cv m) (` (ZZ>) M)) (e. N (ZZ)) pm3.26 (br (opr (cv m) (+) (1)) (<) N) (A.e. k (opr M (...) N) (e. A (CC))) ad2antrr N (opr (cv m) (+) (1)) M fzss2t (opr (cv m) (+) (1)) N ltlet (opr (cv m) (+) (1)) zret N zret syl2an imp (opr (cv m) (+) (1)) N eluzt (br (opr (cv m) (+) (1)) (<) N) adantr mpbird (cv m) M eluzelz peano2zd sylanl1 (cv m) M eluzel2 (e. N (ZZ)) (br (opr (cv m) (+) (1)) (<) N) ad2antrr sylanc (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp sylanc N (opr (cv m) (+) (1)) k A fsum1ps (opr (cv m) (+) (1)) N ltlet (opr (cv m) (+) (1)) zret N zret syl2an (opr (cv m) (+) (1)) N eluzt sylibrd (cv m) M eluzelz peano2zd sylan imp (A.e. k (opr M (...) N) (e. A (CC))) adantr (/\ (e. (cv m) (` (ZZ>) M)) (e. N (ZZ))) (br (opr (cv m) (+) (1)) (<) N) pm3.27 (A.e. k (opr M (...) N) (e. A (CC))) adantr (opr (cv m) (+) (1)) M N fzss1t (cv m) M peano2uz sylan (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (br (opr (cv m) (+) (1)) (<) N) adantlr syl3anc eqcomd (sum_ k (opr (opr (cv m) (+) (1)) (...) N) A) ([_/]_ (opr (cv m) (+) (1)) k A) (sum_ k (opr (opr (opr (cv m) (+) (1)) (+) (1)) (...) N) A) subaddt (opr (cv m) (+) (1)) M N fzss1t (cv m) M peano2uz sylan (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (br (opr (cv m) (+) (1)) (<) N) adantlr N (opr (cv m) (+) (1)) k A fsumclt (opr (cv m) (+) (1)) N ltlet (opr (cv m) (+) (1)) zret N zret syl2an imp (opr (cv m) (+) (1)) N eluzt (br (opr (cv m) (+) (1)) (<) N) adantr mpbird (cv m) M eluzelz peano2zd sylanl1 sylan syldan (opr (cv m) (+) (1)) (opr M (...) N) k A (CC) ra4csbela (opr (cv m) (+) (1)) M N eluzfzt (cv m) M peano2uz (e. N (ZZ)) (br (opr (cv m) (+) (1)) (<) N) ad2antrr (opr (cv m) (+) (1)) N ltlet (opr (cv m) (+) (1)) zret N zret syl2an imp (opr (cv m) (+) (1)) N eluzt (br (opr (cv m) (+) (1)) (<) N) adantr mpbird (cv m) M eluzelz peano2zd sylanl1 sylanc sylan (opr (opr (cv m) (+) (1)) (+) (1)) M N fzss1t (opr (cv m) (+) (1)) M peano2uz sylan (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (br (opr (opr (cv m) (+) (1)) (+) (1)) (<_) N) adantlr N (opr (opr (cv m) (+) (1)) (+) (1)) k A fsumclt (opr (opr (cv m) (+) (1)) (+) (1)) N eluzt (opr (cv m) (+) (1)) M eluzelz peano2zd sylan biimpar sylan syldan (/\ (e. (opr (cv m) (+) (1)) (` (ZZ>) M)) (e. N (ZZ))) (br (opr (cv m) (+) (1)) (<) N) pm3.26 (opr (cv m) (+) (1)) N zltp1let (opr (cv m) (+) (1)) M eluzelz sylan biimpa jca (cv m) M peano2uz sylanl1 sylan syl3anc mpbird eqcomd (+) opreq12d (sum_ k (opr M (...) (cv m)) A) ([_/]_ (opr (cv m) (+) (1)) k A) (sum_ k (opr (opr (cv m) (+) (1)) (...) N) A) ppncant N (cv m) M fzss2t (cv m) ltp1t (e. N (RR)) adantr (cv m) (opr (cv m) (+) (1)) N axlttrn 3expa (cv m) peano2re ancli sylan mpand (cv m) N ltlet syld (cv m) zret N zret syl2an imp (cv m) N eluzt biimpar syldan (cv m) M eluzelz sylanl1 (cv m) M eluzel2 (e. N (ZZ)) (br (opr (cv m) (+) (1)) (<) N) ad2antrr sylanc (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (cv m) M k A fsumclt (e. (cv m) (` (ZZ>) M)) (e. N (ZZ)) pm3.26 (br (opr (cv m) (+) (1)) (<) N) adantr sylan syldan (opr (cv m) (+) (1)) (opr M (...) N) k A (CC) ra4csbela (opr (cv m) (+) (1)) M N eluzfzt (cv m) M peano2uz (e. N (ZZ)) (br (opr (cv m) (+) (1)) (<) N) ad2antrr (opr (cv m) (+) (1)) N ltlet (opr (cv m) (+) (1)) zret N zret syl2an imp (opr (cv m) (+) (1)) N eluzt (br (opr (cv m) (+) (1)) (<) N) adantr mpbird (cv m) M eluzelz peano2zd sylanl1 sylanc sylan (opr (cv m) (+) (1)) M N fzss1t (cv m) M peano2uz sylan (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (br (opr (cv m) (+) (1)) (<) N) adantlr N (opr (cv m) (+) (1)) k A fsumclt (opr (cv m) (+) (1)) N ltlet (opr (cv m) (+) (1)) zret N zret syl2an imp (opr (cv m) (+) (1)) N eluzt (br (opr (cv m) (+) (1)) (<) N) adantr mpbird (cv m) M eluzelz peano2zd sylanl1 sylan syldan syl3anc eqtr2d sylan9eqr ex exp41 imp4c a2d syld uzind4 exp3a imp31 K M (opr N (-) (1)) elfzuzt (e. N (ZZ)) adantl (e. N (ZZ)) (e. K (opr M (...) (opr N (-) (1)))) pm3.26 K N zltlem1t ancoms biimpar anasss K M (opr N (-) (1)) elfzelz K M (opr N (-) (1)) elfzle2 jca sylan2 jca jca sylan 3impa)) thm (fsum0split ((N k)) () (-> (/\/\ (e. N (ZZ)) (e. K (opr (1) (...) N)) (A.e. k (opr (0) (...) N) (e. A (CC)))) (= (sum_ k (opr (0) (...) N) A) (opr (sum_ k (opr (0) (...) (opr N (-) K)) A) (+) (sum_ k (opr (opr (opr N (-) K) (+) (1)) (...) N) A)))) (N (opr N (-) K) (0) k A fsumsplit (e. N (ZZ)) (e. K (opr (1) (...) N)) (A.e. k (opr (0) (...) N) (e. A (CC))) 3simp1 N (ZZ) N K (1) fzrev2it N (ZZ) K (1) elfzel2g N (ZZ) K (1) elfzel2g (e. N (ZZ)) (e. K (opr (1) (...) N)) pm3.27 syl3anc N zcnt N subidt syl (...) (opr N (-) (1)) opreq1d (e. K (opr (1) (...) N)) adantr eleqtrd (A.e. k (opr (0) (...) N) (e. A (CC))) 3adant3 (e. N (ZZ)) (e. K (opr (1) (...) N)) (A.e. k (opr (0) (...) N) (e. A (CC))) 3simp3 syl3anc)) thm (fsumadd ((j m) (A j) (A m) (B j) (B m) (j k) (M j) (k m) (M k) (M m) (N j) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (/\ (e. A (CC)) (e. B (CC))))) (= (sum_ k (opr M (...) N) (opr A (+) B)) (opr (sum_ k (opr M (...) N) A) (+) (sum_ k (opr M (...) N) B)))) ((cv j) M M (...) opreq2 k (/\ (e. A (CC)) (e. B (CC))) raleq1d (cv j) M M (...) opreq2 k (opr A (+) B) sumeq1d (cv j) M M (...) opreq2 k A sumeq1d (cv j) M M (...) opreq2 k B sumeq1d (+) opreq12d eqeq12d imbi12d (cv j) (cv m) M (...) opreq2 k (/\ (e. A (CC)) (e. B (CC))) raleq1d (cv j) (cv m) M (...) opreq2 k (opr A (+) B) sumeq1d (cv j) (cv m) M (...) opreq2 k A sumeq1d (cv j) (cv m) M (...) opreq2 k B sumeq1d (+) opreq12d eqeq12d imbi12d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k (/\ (e. A (CC)) (e. B (CC))) raleq1d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k (opr A (+) B) sumeq1d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k A sumeq1d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k B sumeq1d (+) opreq12d eqeq12d imbi12d (cv j) N M (...) opreq2 k (/\ (e. A (CC)) (e. B (CC))) raleq1d (cv j) N M (...) opreq2 k (opr A (+) B) sumeq1d (cv j) N M (...) opreq2 k A sumeq1d (cv j) N M (...) opreq2 k B sumeq1d (+) opreq12d eqeq12d imbi12d M (ZZ) k A (+) B csbopr12g (A.e. k (opr M (...) M) (/\ (e. A (CC)) (e. B (CC)))) adantr M k (opr A (+) B) (CC) fsum1s A B axaddcl k (opr M (...) M) r19.20si sylan2 M k A (CC) fsum1s (e. A (CC)) (e. B (CC)) pm3.26 k (opr M (...) M) r19.20si sylan2 M k B (CC) fsum1s (e. A (CC)) (e. B (CC)) pm3.27 k (opr M (...) M) r19.20si sylan2 (+) opreq12d 3eqtr4d ex M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (/\ (e. A (CC)) (e. B (CC))) imim1d r19.20dv2 (= (sum_ k (opr M (...) (cv m)) (opr A (+) B)) (opr (sum_ k (opr M (...) (cv m)) A) (+) (sum_ k (opr M (...) (cv m)) B))) imim1d (sum_ k (opr M (...) (cv m)) (opr A (+) B)) (opr (sum_ k (opr M (...) (cv m)) A) (+) (sum_ k (opr M (...) (cv m)) B)) (+) ([_/]_ (opr (cv m) (+) (1)) k (opr A (+) B)) opreq1 (sum_ k (opr M (...) (cv m)) A) (sum_ k (opr M (...) (cv m)) B) ([_/]_ (opr (cv m) (+) (1)) k A) ([_/]_ (opr (cv m) (+) (1)) k B) add4t M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (/\ (e. A (CC)) (e. B (CC))) imim1d r19.20dv2 imp (cv m) M k A fsumclt (e. A (CC)) (e. B (CC)) pm3.26 k (opr M (...) (cv m)) r19.20si sylan2 (cv m) M k B fsumclt (e. A (CC)) (e. B (CC)) pm3.27 k (opr M (...) (cv m)) r19.20si sylan2 jca syldan (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k A (CC) ra4csbela (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl (e. A (CC)) (e. B (CC)) pm3.26 k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k B (CC) ra4csbela (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl (e. A (CC)) (e. B (CC)) pm3.27 k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an jca sylanc (cv m) (+) (1) oprex (opr (cv m) (+) (1)) (V) k A (+) B csbopr12g ax-mp (opr (sum_ k (opr M (...) (cv m)) A) (+) (sum_ k (opr M (...) (cv m)) B)) (+) opreq2i syl5eq sylan9eqr (cv m) M k (opr A (+) B) (CC) fsump1s A B axaddcl k (opr M (...) (opr (cv m) (+) (1))) r19.20si sylan2 (= (sum_ k (opr M (...) (cv m)) (opr A (+) B)) (opr (sum_ k (opr M (...) (cv m)) A) (+) (sum_ k (opr M (...) (cv m)) B))) adantr (cv m) M k A (CC) fsump1s (e. A (CC)) (e. B (CC)) pm3.26 k (opr M (...) (opr (cv m) (+) (1))) r19.20si sylan2 (cv m) M k B (CC) fsump1s (e. A (CC)) (e. B (CC)) pm3.27 k (opr M (...) (opr (cv m) (+) (1))) r19.20si sylan2 (+) opreq12d (= (sum_ k (opr M (...) (cv m)) (opr A (+) B)) (opr (sum_ k (opr M (...) (cv m)) A) (+) (sum_ k (opr M (...) (cv m)) B))) adantr 3eqtr4d exp31 a2d syld uzind4 imp)) thm (csbfsumlem ((k x) (m x) (n x) (A x) (k m) (k n) (A k) (m n) (A m) (A n) (B m) (B n) (M x) (M k) (M m) (M n) (N x) (N n)) ((csbfsumlem.1 (e. A (V))) (csbfsumlem.2 (e. B (V)))) (-> (e. N (` (ZZ>) M)) (= ([_/]_ A x (sum_ k (opr M (...) N) B)) (sum_ k (opr M (...) N) ([_/]_ A x B)))) (csbfsumlem.1 (cv n) M M (...) opreq2 k B sumeq1d A (V) x csbeq2dv mpan2 (cv n) M M (...) opreq2 k ([_/]_ A x B) sumeq1d eqeq12d csbfsumlem.1 (cv n) (cv m) M (...) opreq2 k B sumeq1d A (V) x csbeq2dv mpan2 (cv n) (cv m) M (...) opreq2 k ([_/]_ A x B) sumeq1d eqeq12d csbfsumlem.1 (cv n) (opr (cv m) (+) (1)) M (...) opreq2 k B sumeq1d A (V) x csbeq2dv mpan2 (cv n) (opr (cv m) (+) (1)) M (...) opreq2 k ([_/]_ A x B) sumeq1d eqeq12d csbfsumlem.1 (cv n) N M (...) opreq2 k B sumeq1d A (V) x csbeq2dv mpan2 (cv n) N M (...) opreq2 k ([_/]_ A x B) sumeq1d eqeq12d csbfsumlem.1 A (V) M (ZZ) x k B csbcomg mpan csbfsumlem.1 csbfsumlem.2 M k fsum1slem A (V) x csbeq2dv mpan2 csbfsumlem.1 csbfsumlem.2 x ax-gen A (V) x B (V) csbexg mp2an M k fsum1slem 3eqtr4d (e. (cv m) (` (ZZ>) M)) (= ([_/]_ A x (sum_ k (opr M (...) (cv m)) B)) (sum_ k (opr M (...) (cv m)) ([_/]_ A x B))) pm3.27 (+) ([_/]_ (opr (cv m) (+) (1)) k ([_/]_ A x B)) opreq1d csbfsumlem.1 csbfsumlem.2 (cv m) M k fsump1slem A (V) x csbeq2dv mpan2 csbfsumlem.1 A (V) x (sum_ k (opr M (...) (cv m)) B) (+) ([_/]_ (opr (cv m) (+) (1)) k B) csbopr12g ax-mp csbfsumlem.1 (cv m) (+) (1) oprex A (V) (opr (cv m) (+) (1)) (V) x k B csbcomg mp2an ([_/]_ A x (sum_ k (opr M (...) (cv m)) B)) (+) opreq2i eqtr syl6eq (= ([_/]_ A x (sum_ k (opr M (...) (cv m)) B)) (sum_ k (opr M (...) (cv m)) ([_/]_ A x B))) adantr csbfsumlem.1 csbfsumlem.2 x ax-gen A (V) x B (V) csbexg mp2an (cv m) M k fsump1slem (= ([_/]_ A x (sum_ k (opr M (...) (cv m)) B)) (sum_ k (opr M (...) (cv m)) ([_/]_ A x B))) adantr 3eqtr4d ex uzind4)) thm (csbfsum ((k y) (A k) (A y) (B y) (k x) (x y) (M x) (M k) (M y) (N x) (N k) (N y)) () (-> (/\/\ (e. A C) (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. ([_/]_ A x B) (CC)))) (= ([_/]_ A x (sum_ k (opr M (...) N) B)) (sum_ k (opr M (...) N) ([_/]_ A x B)))) ((cv y) A x B csbeq1 (CC) eleq1d k (opr M (...) N) ralbidv (e. N (` (ZZ>) M)) anbi2d (cv y) A x (sum_ k (opr M (...) N) B) csbeq1 (cv y) A x B csbeq1 k (opr M (...) N) sumeq2sdv eqeq12d imbi12d y visset B (+) (0) oprex N M x k csbfsumlem (A.e. k (opr M (...) N) (e. ([_/]_ (cv y) x B) (CC))) adantr B ax0id k (opr M (...) N) r19.20si sumeq2d y x sbimi y visset (cv y) (V) x k (opr M (...) N) (e. B (CC)) sbcral ax-mp y visset (cv y) (V) x B (CC) sbcel1g ax-mp k (opr M (...) N) ralbii bitr y visset (cv y) (V) x (sum_ k (opr M (...) N) (opr B (+) (0))) (sum_ k (opr M (...) N) B) sbceqdig ax-mp 3imtr3 (e. N (` (ZZ>) M)) adantl ([_/]_ (cv y) x B) ax0id y visset (cv y) (V) x B (+) (0) csbopr1g ax-mp syl5eq k (opr M (...) N) r19.20si sumeq2d (e. N (` (ZZ>) M)) adantl 3eqtr3d C vtoclg 3impib)) thm (fsumcom ((k n) (A k) (A n) (j k) (j m) (j n) (J j) (k m) (J k) (m n) (J m) (J n) (K j) (K k) (K m) (M j) (M k) (M m) (M n) (N j) (N k) (N m) (N n)) () (-> (/\/\ (e. K (` (ZZ>) J)) (e. N (` (ZZ>) M)) (A.e. j (opr J (...) K) (A.e. m (opr M (...) N) (e. A (CC))))) (= (sum_ j (opr J (...) K) (sum_ m (opr M (...) N) A)) (sum_ m (opr M (...) N) (sum_ j (opr J (...) K) A)))) ((cv k) J J (...) opreq2 j (A.e. m (opr M (...) N) (e. A (CC))) raleq1d (e. N (` (ZZ>) M)) anbi2d (cv k) J J (...) opreq2 j (sum_ m (opr M (...) N) A) sumeq1d (cv k) J J (...) opreq2 j A sumeq1d m (opr M (...) N) sumeq2sdv eqeq12d imbi12d (cv k) (cv n) J (...) opreq2 j (A.e. m (opr M (...) N) (e. A (CC))) raleq1d (e. N (` (ZZ>) M)) anbi2d (cv k) (cv n) J (...) opreq2 j (sum_ m (opr M (...) N) A) sumeq1d (cv k) (cv n) J (...) opreq2 j A sumeq1d m (opr M (...) N) sumeq2sdv eqeq12d imbi12d (cv k) (opr (cv n) (+) (1)) J (...) opreq2 j (A.e. m (opr M (...) N) (e. A (CC))) raleq1d (e. N (` (ZZ>) M)) anbi2d (cv k) (opr (cv n) (+) (1)) J (...) opreq2 j (sum_ m (opr M (...) N) A) sumeq1d (cv k) (opr (cv n) (+) (1)) J (...) opreq2 j A sumeq1d m (opr M (...) N) sumeq2sdv eqeq12d imbi12d (cv k) K J (...) opreq2 j (A.e. m (opr M (...) N) (e. A (CC))) raleq1d (e. N (` (ZZ>) M)) anbi2d (cv k) K J (...) opreq2 j (sum_ m (opr M (...) N) A) sumeq1d (cv k) K J (...) opreq2 j A sumeq1d m (opr M (...) N) sumeq2sdv eqeq12d imbi12d J (ZZ) N M m j A csbfsum (e. J (ZZ)) (/\ (e. N (` (ZZ>) M)) (A.e. j (opr J (...) J) (A.e. m (opr M (...) N) (e. A (CC))))) pm3.26 (e. N (` (ZZ>) M)) (A.e. j (opr J (...) J) (A.e. m (opr M (...) N) (e. A (CC)))) pm3.26 (e. J (ZZ)) adantl J (opr J (...) J) j A (CC) ra4csbela J elfz3t sylan ex m (opr M (...) N) r19.20sdv imp j (opr J (...) J) m (opr M (...) N) (e. A (CC)) ralcom sylan2b (e. N (` (ZZ>) M)) adantrl syl3anc J j (sum_ m (opr M (...) N) A) (CC) fsum1s N M m A fsumclt ex j (opr J (...) J) r19.20sdv imp sylan2 J j A (CC) fsum1s ex m (opr M (...) N) r19.20sdv imp sumeq2d j (opr J (...) J) m (opr M (...) N) (e. A (CC)) ralcom sylan2b (e. N (` (ZZ>) M)) adantrl 3eqtr4d ex J (cv n) fzssp1t (cv n) J eluzel2 (cv n) J eluzelz sylanc (cv j) sseld (A.e. m (opr M (...) N) (e. A (CC))) imim1d r19.20dv2 (e. N (` (ZZ>) M)) anim2d (= (sum_ j (opr J (...) (cv n)) (sum_ m (opr M (...) N) A)) (sum_ m (opr M (...) N) (sum_ j (opr J (...) (cv n)) A))) imim1d (= (sum_ j (opr J (...) (cv n)) (sum_ m (opr M (...) N) A)) (sum_ m (opr M (...) N) (sum_ j (opr J (...) (cv n)) A))) id (cv n) (+) (1) oprex (opr (cv n) (+) (1)) (V) N M m j A csbfsum mp3an1 (e. (cv n) (` (ZZ>) J)) (e. N (` (ZZ>) M)) pm3.27 (A.e. j (opr J (...) (opr (cv n) (+) (1))) (A.e. m (opr M (...) N) (e. A (CC)))) adantr (opr (cv n) (+) (1)) (opr J (...) (opr (cv n) (+) (1))) j A (CC) ra4csbela (cv n) J peano2uz (opr (cv n) (+) (1)) J eluzfz2t syl sylan ex m (opr M (...) N) r19.20sdv imp j (opr J (...) (opr (cv n) (+) (1))) m (opr M (...) N) (e. A (CC)) ralcom sylan2b (e. N (` (ZZ>) M)) adantlr sylanc (+) opreqan12rd anasss (cv n) J j (sum_ m (opr M (...) N) A) (CC) fsump1s (e. (cv n) (` (ZZ>) J)) (e. N (` (ZZ>) M)) pm3.26 (A.e. j (opr J (...) (opr (cv n) (+) (1))) (A.e. m (opr M (...) N) (e. A (CC)))) adantr N M m A fsumclt ex (e. (cv n) (` (ZZ>) J)) adantl j (opr J (...) (opr (cv n) (+) (1))) r19.20sdv imp sylanc (= (sum_ j (opr J (...) (cv n)) (sum_ m (opr M (...) N) A)) (sum_ m (opr M (...) N) (sum_ j (opr J (...) (cv n)) A))) adantrr (cv n) J j A (CC) fsump1s ex m (opr M (...) N) r19.20sdv imp sumeq2d (e. N (` (ZZ>) M)) adantlr N M m (sum_ j (opr J (...) (cv n)) A) ([_/]_ (opr (cv n) (+) (1)) j A) fsumadd (e. (cv n) (` (ZZ>) J)) (e. N (` (ZZ>) M)) pm3.27 (A.e. m (opr M (...) N) (A.e. j (opr J (...) (opr (cv n) (+) (1))) (e. A (CC)))) adantr J (cv n) fzssp1t (cv n) J eluzel2 (cv n) J eluzelz sylanc (cv j) sseld (e. A (CC)) imim1d r19.20dv2 (cv n) J j A fsumclt ex syld (opr (cv n) (+) (1)) (opr J (...) (opr (cv n) (+) (1))) j A (CC) ra4csbela (cv n) J peano2uz (opr (cv n) (+) (1)) J eluzfz2t syl sylan ex jcad m (opr M (...) N) r19.20sdv imp (e. N (` (ZZ>) M)) adantlr sylanc eqtrd j (opr J (...) (opr (cv n) (+) (1))) m (opr M (...) N) (e. A (CC)) ralcom sylan2b (= (sum_ j (opr J (...) (cv n)) (sum_ m (opr M (...) N) A)) (sum_ m (opr M (...) N) (sum_ j (opr J (...) (cv n)) A))) adantrr 3eqtr4d exp43 imp3a a2d syld uzind4 3impib)) thm (fsumrev ((k m) (k n) (A k) (m n) (A m) (A n) (j k) (j m) (j n) (K j) (K k) (K m) (K n) (M j) (M k) (M m) (M n) (N j) (N k) (N m)) () (-> (/\/\ (e. N (` (ZZ>) M)) (e. K (ZZ)) (A.e. j (opr M (...) N) (e. A (CC)))) (= (sum_ j (opr M (...) N) A) (sum_ k (opr (opr K (-) N) (...) (opr K (-) M)) ([_/]_ (opr K (-) (cv k)) j A)))) (K M nncant K zcnt M zcnt syl2an j A csbeq1d (A.e. j (opr M (...) M) (e. A (CC))) adantr M M K j (e. A (CC)) k fzrevralt (e. K (ZZ)) (e. M (ZZ)) pm3.27 (e. K (ZZ)) (e. M (ZZ)) pm3.27 (e. K (ZZ)) (e. M (ZZ)) pm3.26 syl3anc K (-) (cv k) oprex (opr K (-) (cv k)) (V) j A (CC) sbcel1g ax-mp k (opr (opr K (-) M) (...) (opr K (-) M)) ralbii syl6bb biimpa (opr K (-) M) k ([_/]_ (opr K (-) (cv k)) j A) (CC) fsum1s K (-) M oprex K (-) (cv k) oprex k ax-gen (cv k) (opr K (-) M) K (-) opreq2 (V) (V) j A csbco3g mp2an syl6eq K M zsubclt sylan syldan M j A (CC) fsum1s (e. K (ZZ)) adantll 3eqtr4rd anasss an1s ex M (cv n) fzssp1t (cv n) M eluzel2 (cv n) M eluzelz sylanc (cv j) sseld (e. A (CC)) imim1d r19.20dv2 (e. K (ZZ)) anim2d (= (sum_ j (opr M (...) (cv n)) A) (sum_ k (opr (opr K (-) (cv n)) (...) (opr K (-) M)) ([_/]_ (opr K (-) (cv k)) j A))) imim1d K (opr (cv n) (+) (1)) nncant K zcnt (cv n) M eluzelz (cv n) zcnt (cv n) peano2cn 3syl syl2an j A csbeq1d K (-) (opr (cv n) (+) (1)) oprex K (-) (cv k) oprex k ax-gen (cv k) (opr K (-) (opr (cv n) (+) (1))) K (-) opreq2 (V) (V) j A csbco3g mp2an syl5req (A.e. j (opr M (...) (opr (cv n) (+) (1))) (e. A (CC))) adantr (= (sum_ j (opr M (...) (cv n)) A) (sum_ k (opr (opr K (-) (cv n)) (...) (opr K (-) M)) ([_/]_ (opr K (-) (cv k)) j A))) id (+) opreqan12d (cv n) M j A (CC) fsump1s (sum_ j (opr M (...) (cv n)) A) ([_/]_ (opr (cv n) (+) (1)) j A) axaddcom M (cv n) fzssp1t (cv n) M eluzel2 (cv n) M eluzelz sylanc (cv j) sseld (e. A (CC)) imim1d r19.20dv2 imp (cv n) M j A fsumclt syldan (opr (cv n) (+) (1)) (opr M (...) (opr (cv n) (+) (1))) j A (CC) ra4csbela (cv n) M peano2uz (opr (cv n) (+) (1)) M eluzfz2t syl sylan sylanc eqtrd (e. K (ZZ)) adantll (= (sum_ j (opr M (...) (cv n)) A) (sum_ k (opr (opr K (-) (cv n)) (...) (opr K (-) M)) ([_/]_ (opr K (-) (cv k)) j A))) adantr (opr K (-) M) (opr K (-) (opr (cv n) (+) (1))) k ([_/]_ (opr K (-) (cv k)) j A) fsum1ps M (cv n) letrp1t (cv n) M eluzel2 M zret syl (cv n) M eluzelz (cv n) zret syl (cv n) M eluzle syl3anc (e. K (ZZ)) adantl M (opr (cv n) (+) (1)) K lesub2t M zret (opr (cv n) (+) (1)) zret K zret syl3an (cv n) M eluzel2 (e. K (ZZ)) adantl (cv n) M eluzelz peano2zd (e. K (ZZ)) adantl (e. K (ZZ)) (e. (cv n) (` (ZZ>) M)) pm3.26 syl3anc mpbid (opr K (-) (opr (cv n) (+) (1))) (opr K (-) M) eluzt K (opr (cv n) (+) (1)) zsubclt (cv n) M eluzelz peano2zd sylan2 K M zsubclt (cv n) M eluzel2 sylan2 sylanc mpbird (A.e. j (opr M (...) (opr (cv n) (+) (1))) (e. A (CC))) adantr (cv n) M eluzle M (cv n) zleltp1t (cv n) M eluzel2 (cv n) M eluzelz sylanc mpbid (e. K (ZZ)) adantl M (opr (cv n) (+) (1)) K ltsub2t M zret (opr (cv n) (+) (1)) zret K zret syl3an (cv n) M eluzel2 (e. K (ZZ)) adantl (cv n) M eluzelz peano2zd (e. K (ZZ)) adantl (e. K (ZZ)) (e. (cv n) (` (ZZ>) M)) pm3.26 syl3anc mpbid (A.e. j (opr M (...) (opr (cv n) (+) (1))) (e. A (CC))) adantr M (opr (cv n) (+) (1)) K j (e. A (CC)) k fzrevralt (cv n) M eluzel2 (e. K (ZZ)) adantl (cv n) M eluzelz peano2zd (e. K (ZZ)) adantl (e. K (ZZ)) (e. (cv n) (` (ZZ>) M)) pm3.26 syl3anc K (-) (cv k) oprex (opr K (-) (cv k)) (V) j A (CC) sbcel1g ax-mp k (opr (opr K (-) (opr (cv n) (+) (1))) (...) (opr K (-) M)) ralbii syl6bb biimpa syl3anc 1cn K (cv n) (1) nppcan2t mp3an3 K zcnt (cv n) M eluzelz (cv n) zcnt syl syl2an (...) (opr K (-) M) opreq1d k ([_/]_ (opr K (-) (cv k)) j A) sumeq1d ([_/]_ (opr K (-) (opr (cv n) (+) (1))) k ([_/]_ (opr K (-) (cv k)) j A)) (+) opreq2d (A.e. j (opr M (...) (opr (cv n) (+) (1))) (e. A (CC))) adantr eqtrd (= (sum_ j (opr M (...) (cv n)) A) (sum_ k (opr (opr K (-) (cv n)) (...) (opr K (-) M)) ([_/]_ (opr K (-) (cv k)) j A))) adantr 3eqtr4d exp41 com12 imp3a a2d syld (cv m) M M (...) opreq2 j (e. A (CC)) raleq1d (e. K (ZZ)) anbi2d (cv m) M M (...) opreq2 j A sumeq1d (cv m) M K (-) opreq2 (...) (opr K (-) M) opreq1d k ([_/]_ (opr K (-) (cv k)) j A) sumeq1d eqeq12d imbi12d (cv m) (cv n) M (...) opreq2 j (e. A (CC)) raleq1d (e. K (ZZ)) anbi2d (cv m) (cv n) M (...) opreq2 j A sumeq1d (cv m) (cv n) K (-) opreq2 (...) (opr K (-) M) opreq1d k ([_/]_ (opr K (-) (cv k)) j A) sumeq1d eqeq12d imbi12d (cv m) (opr (cv n) (+) (1)) M (...) opreq2 j (e. A (CC)) raleq1d (e. K (ZZ)) anbi2d (cv m) (opr (cv n) (+) (1)) M (...) opreq2 j A sumeq1d (cv m) (opr (cv n) (+) (1)) K (-) opreq2 (...) (opr K (-) M) opreq1d k ([_/]_ (opr K (-) (cv k)) j A) sumeq1d eqeq12d imbi12d (cv m) N M (...) opreq2 j (e. A (CC)) raleq1d (e. K (ZZ)) anbi2d (cv m) N M (...) opreq2 j A sumeq1d (cv m) N K (-) opreq2 (...) (opr K (-) M) opreq1d k ([_/]_ (opr K (-) (cv k)) j A) sumeq1d eqeq12d imbi12d uzind4ALT 3impib)) thm (fsumshft ((k m) (A k) (A m) (j k) (j m) (K j) (K k) (K m) (M j) (M k) (M m) (N j) (N k) (N m)) () (-> (/\/\ (e. N (` (ZZ>) M)) (e. K (ZZ)) (A.e. j (opr M (...) N) (e. A (CC)))) (= (sum_ j (opr M (...) N) A) (sum_ k (opr (opr M (+) K) (...) (opr N (+) K)) ([_/]_ (opr (cv k) (-) K) j A)))) (0z N M (0) j A m fsumrev mp3an2 (e. K (ZZ)) 3adant2 (opr (0) (-) M) (opr (0) (-) N) K m ([_/]_ (opr (0) (-) (cv m)) j A) k fsumrev N M uznegit M df-neg N df-neg (ZZ>) fveq2i eleq12i sylib (e. K (ZZ)) (A.e. j (opr M (...) N) (e. A (CC))) 3ad2ant1 (e. N (` (ZZ>) M)) (e. K (ZZ)) (A.e. j (opr M (...) N) (e. A (CC))) 3simp2 0z M N (0) j (e. A (CC)) m fzrevralt mp3an3 N M eluzel2 N M eluzelz sylanc (0) (-) (cv m) oprex (opr (0) (-) (cv m)) (V) j A (CC) sbcel1g ax-mp m (opr (opr (0) (-) N) (...) (opr (0) (-) M)) ralbii syl6bb biimpa (e. K (ZZ)) 3adant2 syl3anc K M subnegt K M axaddcom eqtrd M df-neg K (-) opreq2i syl5eqr (e. N (CC)) adantrr K N subnegt K N axaddcom eqtrd N df-neg K (-) opreq2i syl5eqr (e. M (CC)) adantrl (...) opreq12d K zcnt N M eluzel2 M zcnt syl N M eluzelz N zcnt syl jca syl2an ancoms K (cv k) negsubdi2t K zcnt (cv k) zcnt syl2an (opr K (-) (cv k)) df-neg syl5eqr j A csbeq1d K (-) (cv k) oprex (0) (-) (cv m) oprex m ax-gen (cv m) (opr K (-) (cv k)) (0) (-) opreq2 (V) (V) j A csbco3g mp2an syl5eq ex (cv k) (opr K (-) (opr (0) (-) M)) (opr K (-) (opr (0) (-) N)) elfzelz syl5 r19.21aiv (e. N (` (ZZ>) M)) adantl sumeq12d (A.e. j (opr M (...) N) (e. A (CC))) 3adant3 3eqtrd)) thm (fsummulc1 ((j k) (j m) (C j) (k m) (C k) (C m) (A j) (A m) (M j) (M k) (M m) (N j) (N k)) () (-> (/\/\ (e. N (` (ZZ>) M)) (e. C (CC)) (A.e. k (opr M (...) N) (e. A (CC)))) (= (opr C (x.) (sum_ k (opr M (...) N) A)) (sum_ k (opr M (...) N) (opr C (x.) A)))) (M (ZZ) k C (x.) A csbopr2g (/\ (e. C (CC)) (A.e. k (opr M (...) M) (e. A (CC)))) adantr M k (opr C (x.) A) (CC) fsum1s (e. C (CC)) k (opr M (...) M) (e. A (CC)) r19.28av C A axmulcl k (opr M (...) M) r19.20si syl sylan2 M k A (CC) fsum1s (e. C (CC)) adantrl C (x.) opreq2d 3eqtr4rd ex M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A (CC)) imim1d r19.20dv2 (e. C (CC)) anim2d (= (opr C (x.) (sum_ k (opr M (...) (cv m)) A)) (sum_ k (opr M (...) (cv m)) (opr C (x.) A))) imim1d C (sum_ k (opr M (...) (cv m)) A) ([_/]_ (opr (cv m) (+) (1)) k A) axdistr (e. C (CC)) (A.e. k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC))) pm3.26 (e. (cv m) (` (ZZ>) M)) adantl M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (cv m) M k A fsumclt syldan (e. C (CC)) adantrl (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k A (CC) ra4csbela (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl sylan (e. C (CC)) adantrl syl3anc (-> (/\ (e. C (CC)) (A.e. k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC)))) (= (opr C (x.) (sum_ k (opr M (...) (cv m)) A)) (sum_ k (opr M (...) (cv m)) (opr C (x.) A)))) adantlr (-> (/\ (e. C (CC)) (A.e. k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC)))) (= (opr C (x.) (sum_ k (opr M (...) (cv m)) A)) (sum_ k (opr M (...) (cv m)) (opr C (x.) A)))) id imp (cv m) (+) (1) oprex (opr (cv m) (+) (1)) (V) k C (x.) A csbopr2g ax-mp eqcomi (/\ (-> (/\ (e. C (CC)) (A.e. k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC)))) (= (opr C (x.) (sum_ k (opr M (...) (cv m)) A)) (sum_ k (opr M (...) (cv m)) (opr C (x.) A)))) (/\ (e. C (CC)) (A.e. k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC))))) a1i (+) opreq12d (e. (cv m) (` (ZZ>) M)) adantll eqtrd (cv m) M k A (CC) fsump1s (e. C (CC)) adantrl C (x.) opreq2d (-> (/\ (e. C (CC)) (A.e. k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC)))) (= (opr C (x.) (sum_ k (opr M (...) (cv m)) A)) (sum_ k (opr M (...) (cv m)) (opr C (x.) A)))) adantlr (cv m) M k (opr C (x.) A) (CC) fsump1s (e. C (CC)) k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC)) r19.28av C A axmulcl k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl sylan2 (-> (/\ (e. C (CC)) (A.e. k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC)))) (= (opr C (x.) (sum_ k (opr M (...) (cv m)) A)) (sum_ k (opr M (...) (cv m)) (opr C (x.) A)))) adantlr 3eqtr4d exp31 syld (cv j) M M (...) opreq2 k (e. A (CC)) raleq1d (e. C (CC)) anbi2d (cv j) M M (...) opreq2 k A sumeq1d C (x.) opreq2d (cv j) M M (...) opreq2 k (opr C (x.) A) sumeq1d eqeq12d imbi12d (cv j) (cv m) M (...) opreq2 k (e. A (CC)) raleq1d (e. C (CC)) anbi2d (cv j) (cv m) M (...) opreq2 k A sumeq1d C (x.) opreq2d (cv j) (cv m) M (...) opreq2 k (opr C (x.) A) sumeq1d eqeq12d imbi12d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k (e. A (CC)) raleq1d (e. C (CC)) anbi2d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k A sumeq1d C (x.) opreq2d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k (opr C (x.) A) sumeq1d eqeq12d imbi12d (cv j) N M (...) opreq2 k (e. A (CC)) raleq1d (e. C (CC)) anbi2d (cv j) N M (...) opreq2 k A sumeq1d C (x.) opreq2d (cv j) N M (...) opreq2 k (opr C (x.) A) sumeq1d eqeq12d imbi12d uzind4ALT 3impib)) thm (fsummulc2 ((C k) (M k) (N k)) () (-> (/\/\ (e. N (` (ZZ>) M)) (e. C (CC)) (A.e. k (opr M (...) N) (e. A (CC)))) (= (opr (sum_ k (opr M (...) N) A) (x.) C) (sum_ k (opr M (...) N) (opr A (x.) C)))) (N M C k A fsummulc1 C (sum_ k (opr M (...) N) A) axmulcom N M k A fsumclt sylan2 3impb 3com12 C A axmulcom ex k (opr M (...) N) r19.20sdv imp sumeq2d (e. N (` (ZZ>) M)) 3adant1 3eqtr3d)) thm (fsumdivc ((C k) (M k) (N k)) () (-> (/\ (/\ (e. N (` (ZZ>) M)) (e. C (CC))) (/\ (=/= C (0)) (A.e. k (opr M (...) N) (e. A (CC))))) (= (opr (sum_ k (opr M (...) N) A) (/) C) (sum_ k (opr M (...) N) (opr A (/) C)))) (N M (opr (1) (/) C) k A fsummulc2 (e. N (` (ZZ>) M)) (e. C (CC)) pm3.26 (/\ (=/= C (0)) (A.e. k (opr M (...) N) (e. A (CC)))) adantr C recclt (e. N (` (ZZ>) M)) adantll (A.e. k (opr M (...) N) (e. A (CC))) adantrr (=/= C (0)) (A.e. k (opr M (...) N) (e. A (CC))) pm3.27 (/\ (e. N (` (ZZ>) M)) (e. C (CC))) adantl syl3anc (sum_ k (opr M (...) N) A) C divrect N M k A fsumclt (=/= C (0)) adantrl (e. C (CC)) adantlr (e. N (` (ZZ>) M)) (e. C (CC)) pm3.27 (/\ (=/= C (0)) (A.e. k (opr M (...) N) (e. A (CC)))) adantr (=/= C (0)) (A.e. k (opr M (...) N) (e. A (CC))) pm3.26 (/\ (e. N (` (ZZ>) M)) (e. C (CC))) adantl syl3anc A C divrect 3expb expcom k (opr M (...) N) r19.20sdv imp anasss sumeq2d (e. N (` (ZZ>) M)) adantll 3eqtr4d)) thm (fsum2mul ((A m) (B j) (J j) (K j) (j m) (M j) (M m) (N j) (N m)) () (-> (/\ (/\ (e. K (` (ZZ>) J)) (A.e. j (opr J (...) K) (e. A (CC)))) (/\ (e. N (` (ZZ>) M)) (A.e. m (opr M (...) N) (e. B (CC))))) (= (sum_ j (opr J (...) K) (sum_ m (opr M (...) N) (opr A (x.) B))) (opr (sum_ j (opr J (...) K) A) (x.) (sum_ m (opr M (...) N) B)))) (K J (sum_ m (opr M (...) N) B) j A fsummulc2 3expa an1rs N M m B fsumclt sylan2 N M A m B fsummulc1 3expa an1rs ex j (opr J (...) K) r19.20sdv impcom sumeq2d (e. K (` (ZZ>) J)) adantll eqtr2d)) thm (fsumconst ((j k) (j m) (A j) (k m) (A k) (A m) (M j) (M k) (M m) (N j) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (e. A (CC))) (= (sum_ k (opr M (...) N) A) (opr (opr (opr N (-) M) (+) (1)) (x.) A))) ((cv j) M M (...) opreq2 k A sumeq1d (cv j) M (-) M opreq1 (+) (1) opreq1d (x.) A opreq1d eqeq12d (e. A (CC)) imbi2d (cv j) (cv m) M (...) opreq2 k A sumeq1d (cv j) (cv m) (-) M opreq1 (+) (1) opreq1d (x.) A opreq1d eqeq12d (e. A (CC)) imbi2d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k A sumeq1d (cv j) (opr (cv m) (+) (1)) (-) M opreq1 (+) (1) opreq1d (x.) A opreq1d eqeq12d (e. A (CC)) imbi2d (cv j) N M (...) opreq2 k A sumeq1d (cv j) N (-) M opreq1 (+) (1) opreq1d (x.) A opreq1d eqeq12d (e. A (CC)) imbi2d A eqid (= (cv k) M) a1i (CC) fsum1 ancoms M subidt (+) (1) opreq1d 1cn addid2 syl6eq (x.) A opreq1d A mulid2t sylan9eq M zcnt sylan eqtr4d ex (cv m) M k A (CC) fsump1s (e. A (CC)) (e. (cv k) (opr M (...) (opr (cv m) (+) (1)))) ax-1 r19.21aiv sylan2 (cv m) (+) (1) oprex (e. (cv j) A) k ax-17 (opr (cv m) (+) (1)) (V) csbconstgf ax-mp (sum_ k (opr M (...) (cv m)) A) (+) opreq2i syl6eq (sum_ k (opr M (...) (cv m)) A) (opr (opr (opr (cv m) (-) M) (+) (1)) (x.) A) (+) A opreq1 sylan9eq 1cn (cv m) (1) M addsubt mp3an2 (e. A (CC)) adantr (+) (1) opreq1d (x.) A opreq1d 1cn (opr (opr (cv m) (-) M) (+) (1)) (1) A adddirt mp3an2 (cv m) M subclt (opr (cv m) (-) M) peano2cn syl sylan A mulid2t (opr (opr (opr (cv m) (-) M) (+) (1)) (x.) A) (+) opreq2d (/\ (e. (cv m) (CC)) (e. M (CC))) adantl 3eqtrd (cv m) M eluzelz (cv m) zcnt syl (cv m) M eluzel2 M zcnt syl jca sylan (= (sum_ k (opr M (...) (cv m)) A) (opr (opr (opr (cv m) (-) M) (+) (1)) (x.) A)) adantr eqtr4d exp31 a2d uzind4 imp)) thm (fsum0 ((M k) (N k)) () (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) N) (0)) (0))) (0cn N M (0) k fsumconst mpan2 N M zsubclt N M eluzelz N M eluzel2 sylanc (opr N (-) M) zcnt syl (opr N (-) M) peano2cn (opr (opr N (-) M) (+) (1)) mul01t 3syl eqtrd)) thm (fsumcmp ((j m) (A j) (A m) (B j) (B m) (j k) (M j) (k m) (M k) (M m) (N j) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)))) (br (sum_ k (opr M (...) N) A) (<_) (sum_ k (opr M (...) N) B))) (M (opr M (...) M) k (br A (<_) B) ra4sbca M elfz3t (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp3 k (opr M (...) M) r19.20si syl2an M (ZZ) k A (<_) B sbcbr12g (A.e. k (opr M (...) M) (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B))) adantr mpbid M k A (RR) fsum1s (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp1 k (opr M (...) M) r19.20si sylan2 M k B (RR) fsum1s (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp2 k (opr M (...) M) r19.20si sylan2 3brtr4d ex M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) imim1d r19.20dv2 (br (sum_ k (opr M (...) (cv m)) A) (<_) (sum_ k (opr M (...) (cv m)) B)) imim1d (cv m) M k A (RR) fsump1s (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp1 k (opr M (...) (opr (cv m) (+) (1))) r19.20si sylan2 (br (sum_ k (opr M (...) (cv m)) A) (<_) (sum_ k (opr M (...) (cv m)) B)) adantr (/\ (e. (cv m) (` (ZZ>) M)) (A.e. k (opr M (...) (opr (cv m) (+) (1))) (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)))) (br (sum_ k (opr M (...) (cv m)) A) (<_) (sum_ k (opr M (...) (cv m)) B)) pm3.27 (sum_ k (opr M (...) (cv m)) A) (sum_ k (opr M (...) (cv m)) B) ([_/]_ (opr (cv m) (+) (1)) k A) leadd1t M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp1 (e. (cv m) (` (ZZ>) M)) a1i imim12d r19.20dv2 imp (cv m) M k A fsumreclt syldan (br (sum_ k (opr M (...) (cv m)) A) (<_) (sum_ k (opr M (...) (cv m)) B)) adantr M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp2 (e. (cv m) (` (ZZ>) M)) a1i imim12d r19.20dv2 imp (cv m) M k B fsumreclt syldan (br (sum_ k (opr M (...) (cv m)) A) (<_) (sum_ k (opr M (...) (cv m)) B)) adantr (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k A (RR) ra4csbela (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp1 k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an (br (sum_ k (opr M (...) (cv m)) A) (<_) (sum_ k (opr M (...) (cv m)) B)) adantr syl3anc mpbid eqbrtrd (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k (br A (<_) B) ra4sbca (cv m) (+) (1) oprex (opr (cv m) (+) (1)) (V) k A (<_) B sbcbr12g ax-mp sylib (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp3 k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an ([_/]_ (opr (cv m) (+) (1)) k A) ([_/]_ (opr (cv m) (+) (1)) k B) (sum_ k (opr M (...) (cv m)) B) leadd2t (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k A (RR) ra4csbela (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp1 k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k B (RR) ra4csbela (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp2 k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp2 (e. (cv m) (` (ZZ>) M)) a1i imim12d r19.20dv2 imp (cv m) M k B fsumreclt syldan syl3anc mpbid (cv m) M k B (RR) fsump1s (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp2 k (opr M (...) (opr (cv m) (+) (1))) r19.20si sylan2 breqtrrd (br (sum_ k (opr M (...) (cv m)) A) (<_) (sum_ k (opr M (...) (cv m)) B)) adantr (sum_ k (opr M (...) (opr (cv m) (+) (1))) A) (opr (sum_ k (opr M (...) (cv m)) B) (+) ([_/]_ (opr (cv m) (+) (1)) k A)) (sum_ k (opr M (...) (opr (cv m) (+) (1))) B) letrt (opr (cv m) (+) (1)) M k A fsumreclt (cv m) M peano2uz (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp1 k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an (sum_ k (opr M (...) (cv m)) B) ([_/]_ (opr (cv m) (+) (1)) k A) axaddrcl M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp2 (e. (cv m) (` (ZZ>) M)) a1i imim12d r19.20dv2 imp (cv m) M k B fsumreclt syldan (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k A (RR) ra4csbela (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp1 k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an sylanc (opr (cv m) (+) (1)) M k B fsumreclt (cv m) M peano2uz (e. A (RR)) (e. B (RR)) (br A (<_) B) 3simp2 k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an syl3anc (br (sum_ k (opr M (...) (cv m)) A) (<_) (sum_ k (opr M (...) (cv m)) B)) adantr mp2and exp31 a2d syld (cv j) M M (...) opreq2 k (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) raleq1d (cv j) M M (...) opreq2 k A sumeq1d (cv j) M M (...) opreq2 k B sumeq1d (<_) breq12d imbi12d (cv j) (cv m) M (...) opreq2 k (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) raleq1d (cv j) (cv m) M (...) opreq2 k A sumeq1d (cv j) (cv m) M (...) opreq2 k B sumeq1d (<_) breq12d imbi12d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) raleq1d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k A sumeq1d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k B sumeq1d (<_) breq12d imbi12d (cv j) N M (...) opreq2 k (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) raleq1d (cv j) N M (...) opreq2 k A sumeq1d (cv j) N M (...) opreq2 k B sumeq1d (<_) breq12d imbi12d uzind4ALT imp)) thm (fsumabs ((j m) (A j) (A m) (j k) (M j) (k m) (M k) (M m) (N j) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. A (CC)))) (br (` (abs) (sum_ k (opr M (...) N) A)) (<_) (sum_ k (opr M (...) N) (` (abs) A)))) ((cv j) M M (...) opreq2 k (e. A (CC)) raleq1d (cv j) M M (...) opreq2 k A sumeq1d (abs) fveq2d (cv j) M M (...) opreq2 k (` (abs) A) sumeq1d (<_) breq12d imbi12d (cv j) (cv m) M (...) opreq2 k (e. A (CC)) raleq1d (cv j) (cv m) M (...) opreq2 k A sumeq1d (abs) fveq2d (cv j) (cv m) M (...) opreq2 k (` (abs) A) sumeq1d (<_) breq12d imbi12d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k (e. A (CC)) raleq1d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k A sumeq1d (abs) fveq2d (cv j) (opr (cv m) (+) (1)) M (...) opreq2 k (` (abs) A) sumeq1d (<_) breq12d imbi12d (cv j) N M (...) opreq2 k (e. A (CC)) raleq1d (cv j) N M (...) opreq2 k A sumeq1d (abs) fveq2d (cv j) N M (...) opreq2 k (` (abs) A) sumeq1d (<_) breq12d imbi12d M (ZZ) k (abs) A csbfv2g (A.e. k (opr M (...) M) (e. A (CC))) adantr M (opr M (...) M) k (` (abs) A) (RR) ra4csbela M elfz3t A absclt k (opr M (...) M) r19.20si syl2an ([_/]_ M k (` (abs) A)) leidt syl eqbrtrrd M k A (CC) fsum1s (abs) fveq2d M k (` (abs) A) (CC) fsum1s A absclt recnd k (opr M (...) M) r19.20si sylan2 3brtr4d ex M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A (CC)) imim1d r19.20dv2 (br (` (abs) (sum_ k (opr M (...) (cv m)) A)) (<_) (sum_ k (opr M (...) (cv m)) (` (abs) A))) imim1d (sum_ k (opr M (...) (cv m)) A) ([_/]_ (opr (cv m) (+) (1)) k A) abstrit M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (cv m) M k A fsumclt syldan (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k A (CC) ra4csbela (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl sylan sylanc (cv m) M k A (CC) fsump1s (abs) fveq2d (cv m) (+) (1) oprex (opr (cv m) (+) (1)) (V) k (abs) A csbfv2g ax-mp (/\ (e. (cv m) (` (ZZ>) M)) (A.e. k (opr M (...) (opr (cv m) (+) (1))) (e. A (CC)))) a1i (` (abs) (sum_ k (opr M (...) (cv m)) A)) (+) opreq2d 3brtr4d (br (` (abs) (sum_ k (opr M (...) (cv m)) A)) (<_) (sum_ k (opr M (...) (cv m)) (` (abs) A))) adantr (` (abs) (sum_ k (opr M (...) (cv m)) A)) (sum_ k (opr M (...) (cv m)) (` (abs) A)) ([_/]_ (opr (cv m) (+) (1)) k (` (abs) A)) leadd1t M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (cv m) M k A fsumclt syldan (sum_ k (opr M (...) (cv m)) A) absclt syl M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp A absclt k (opr M (...) (cv m)) r19.20si syl (cv m) M k (` (abs) A) fsumreclt syldan (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k (` (abs) A) (RR) ra4csbela (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl A absclt k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an syl3anc biimpa (cv m) M k (` (abs) A) (CC) fsump1s A absclt recnd k (opr M (...) (opr (cv m) (+) (1))) r19.20si sylan2 (br (` (abs) (sum_ k (opr M (...) (cv m)) A)) (<_) (sum_ k (opr M (...) (cv m)) (` (abs) A))) adantr breqtrrd (` (abs) (sum_ k (opr M (...) (opr (cv m) (+) (1))) A)) (opr (` (abs) (sum_ k (opr M (...) (cv m)) A)) (+) ([_/]_ (opr (cv m) (+) (1)) k (` (abs) A))) (sum_ k (opr M (...) (opr (cv m) (+) (1))) (` (abs) A)) letrt (opr (cv m) (+) (1)) M k A fsumclt (cv m) M peano2uz sylan (sum_ k (opr M (...) (opr (cv m) (+) (1))) A) absclt syl (` (abs) (sum_ k (opr M (...) (cv m)) A)) ([_/]_ (opr (cv m) (+) (1)) k (` (abs) A)) axaddrcl M (cv m) fzssp1t (cv m) M eluzel2 (cv m) M eluzelz sylanc (cv k) sseld (e. A (CC)) imim1d r19.20dv2 imp (cv m) M k A fsumclt syldan (sum_ k (opr M (...) (cv m)) A) absclt syl (opr (cv m) (+) (1)) (opr M (...) (opr (cv m) (+) (1))) k (` (abs) A) (RR) ra4csbela (cv m) M peano2uz (opr (cv m) (+) (1)) M eluzfz2t syl A absclt k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an sylanc (opr (cv m) (+) (1)) M k (` (abs) A) fsumreclt (cv m) M peano2uz A absclt k (opr M (...) (opr (cv m) (+) (1))) r19.20si syl2an syl3anc (br (` (abs) (sum_ k (opr M (...) (cv m)) A)) (<_) (sum_ k (opr M (...) (cv m)) (` (abs) A))) adantr mp2and exp31 a2d syld uzind4 imp)) thm (serzclt ((F k) (M k) (N k)) ((serzclt.1 (e. F (V)))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (e. (` (opr (<,> M (+)) (seq) F) N) (CC))) (serzclt.1 N M k fsumserz (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC))) adantr N M k (` F (cv k)) fsumclt eqeltrrd)) thm (ser0clt ((F k) (N k)) ((serzclt.1 (e. F (V)))) (-> (/\ (e. N (NN0)) (A.e. k (opr (0) (...) N) (e. (` F (cv k)) (CC)))) (e. (` (opr (+) (seq0) F) N) (CC))) (serzclt.1 N (0) k serzclt N elnn0uz sylanb addex serzclt.1 seq0seqz N fveq1i syl5eqel)) thm (ser1clt ((F k) (N k)) ((serzclt.1 (e. F (V)))) (-> (/\ (e. N (NN)) (A.e. k (opr (1) (...) N) (e. (` F (cv k)) (CC)))) (e. (` (opr (+) (seq1) F) N) (CC))) (serzclt.1 N (1) k serzclt N elnnuz sylanb addex serzclt.1 seq1seqz N fveq1i syl5eqel)) thm (serzcl2t ((j k) (F j) (F k) (M j) (M k) (N j)) ((serzcl2t.1 (e. F (V)))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (CC)))) (e. (` (opr (<,> M (+)) (seq) F) N) (CC))) (serzcl2t.1 N M j serzclt (cv k) (cv j) F fveq2 (CC) eleq1d (` (ZZ>) M) rcla4cv (cv j) M N elfzuzt syl5 r19.21aiv sylan2)) thm (serzreclt ((F k) (M k) (N k)) ((serzreclt.1 (e. F (V)))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (RR)))) (e. (` (opr (<,> M (+)) (seq) F) N) (RR))) (serzreclt.1 N M k fsumserz (A.e. k (opr M (...) N) (e. (` F (cv k)) (RR))) adantr N M k (` F (cv k)) fsumreclt eqeltrrd)) thm (serz1p ((F k) (M k) (N k)) ((serz1p.1 (e. F (V)))) (-> (/\/\ (e. N (` (ZZ>) M)) (br M (<) N) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (= (` (opr (<,> M (+)) (seq) F) N) (opr (` F M) (+) (` (opr (<,> (opr M (+) (1)) (+)) (seq) F) N)))) ((cv k) M F fveq2 N fsum1p serz1p.1 N M k fsumserz (br M (<) N) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC))) 3ad2ant1 (opr M (+) (1)) N eluzt M peano2z sylan M N zltp1let bitr4d N M eluzel2 N M eluzelz sylanc biimpar serz1p.1 N (opr M (+) (1)) k fsumserz syl (` F M) (+) opreq2d (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC))) 3adant3 3eqtr3d)) thm (serz0 ((M k) (N k)) () (-> (e. N (` (ZZ>) M)) (= (` (opr (<,> M (+)) (seq) (X. (` (ZZ>) M) ({} (0)))) N) (0))) ((ZZ>) M fvex (0) snex xpex N M k fsumserz (cv k) M N elfzuzt 0cn elisseti (cv k) (` (ZZ>) M) fvconst2 syl rgen k (opr M (...) N) (` (X. (` (ZZ>) M) ({} (0))) (cv k)) (0) sumeq2 N M k fsum0 sylan9eqr mpan2 eqtr3d)) thm (serzcmp ((F k) (G k) (M k) (N k)) ((serzcmp.1 (e. F (V))) (serzcmp.2 (e. G (V)))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (/\/\ (e. (` F (cv k)) (RR)) (e. (` G (cv k)) (RR)) (br (` F (cv k)) (<_) (` G (cv k)))))) (br (` (opr (<,> M (+)) (seq) F) N) (<_) (` (opr (<,> M (+)) (seq) G) N))) (N M k (` F (cv k)) (` G (cv k)) fsumcmp serzcmp.1 N M k fsumserz (A.e. k (opr M (...) N) (/\/\ (e. (` F (cv k)) (RR)) (e. (` G (cv k)) (RR)) (br (` F (cv k)) (<_) (` G (cv k))))) adantr serzcmp.2 N M k fsumserz (A.e. k (opr M (...) N) (/\/\ (e. (` F (cv k)) (RR)) (e. (` G (cv k)) (RR)) (br (` F (cv k)) (<_) (` G (cv k))))) adantr 3brtr3d)) thm (serzcmp0 ((F k) (M k) (N k)) ((serzcmp0.1 (e. F (V)))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k)))))) (br (0) (<_) (` (opr (<,> M (+)) (seq) F) N))) (N M serz0 (A.e. k (opr M (...) N) (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k))))) adantr (ZZ>) M fvex (0) snex xpex serzcmp0.1 N M k serzcmp (cv k) M N elfzuzt 0re elisseti (cv k) (` (ZZ>) M) fvconst2 syl 0re syl6eqel (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k)))) adantr (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k))) pm3.26 (e. (cv k) (opr M (...) N)) adantl (cv k) M N elfzuzt 0re elisseti (cv k) (` (ZZ>) M) fvconst2 syl (<_) (` F (cv k)) breq1d biimpar (e. (` F (cv k)) (RR)) adantrl 3jca ex r19.20i sylan2 eqbrtrrd)) thm (serzsplit ((F k) (M k) (N k)) ((serzsplit.1 (e. F (V)))) (-> (/\/\ (e. N (ZZ)) (e. K (opr M (...) (opr N (-) (1)))) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (= (` (opr (<,> M (+)) (seq) F) N) (opr (` (opr (<,> M (+)) (seq) F) K) (+) (` (opr (<,> (opr K (+) (1)) (+)) (seq) F) N)))) (N K M k (` F (cv k)) fsumsplit N zcnt 1cn N (1) npcant mpan2 syl (e. K (opr M (...) (opr N (-) (1)))) adantr N (-) (1) oprex (opr N (-) (1)) (V) K M elfzuz2t mpan (opr N (-) (1)) M peano2uz syl (e. N (ZZ)) adantl eqeltrrd serzsplit.1 N M k fsumserz syl (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC))) 3adant3 K M (opr N (-) (1)) elfzuzt (e. N (ZZ)) adantl serzsplit.1 K M k fsumserz syl N zcnt 1cn N (1) npcant mpan2 syl (e. K (opr M (...) (opr N (-) (1)))) adantr N (-) (1) oprex (opr N (-) (1)) (V) K M elfzuz3t mpan (opr N (-) (1)) K eluzp1p1t syl (e. N (ZZ)) adantl eqeltrrd serzsplit.1 N (opr K (+) (1)) k fsumserz syl (+) opreq12d (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC))) 3adant3 3eqtr3d)) thm (binomlem1 ((k x) (A k) (A x) (B k) (B x) (N k) (N x)) ((binomlem.1 (e. A (CC))) (binomlem.2 (e. B (CC)))) (-> (e. N (NN)) (= (opr (sum_ k (opr (0) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))))) (x.) A) (opr (opr A (^) (opr N (+) (1))) (+) (sum_ k (opr (1) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))))))) (N nnnn0t binomlem.1 N (0) A k (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k)))) fsummulc2 mp3an2 N elnn0uz biimp (opr N (C.) (cv k)) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl N (cv k) bcclt (opr N (C.) (cv k)) nn0cnt syl (cv k) (0) N elfzelz sylan2 (opr A (^) (opr N (-) (cv k))) (opr B (^) (cv k)) axmulcl N (NN0) (cv k) (0) fznn0subt binomlem.1 A (opr N (-) (cv k)) expclt mpan syl (cv k) N elfznn0t binomlem.2 B (cv k) expclt mpan syl (e. N (NN0)) adantl sylanc sylanc r19.21aiva sylanc binomlem.1 (opr N (C.) (cv k)) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))) A axmulass mp3an3 N (cv k) bcclt (opr N (C.) (cv k)) nn0cnt syl (cv k) (0) N elfzelz sylan2 (opr A (^) (opr N (-) (cv k))) (opr B (^) (cv k)) axmulcl N (NN0) (cv k) (0) fznn0subt binomlem.1 A (opr N (-) (cv k)) expclt mpan syl (cv k) N elfznn0t binomlem.2 B (cv k) expclt mpan syl (e. N (NN0)) adantl sylanc sylanc 1cn N (1) (cv k) addsubt mp3an2 N nn0cnt (cv k) (0) N elfzelz (cv k) zcnt syl syl2an A (^) opreq2d N (NN0) (cv k) (0) fznn0subt binomlem.1 A (opr N (-) (cv k)) expp1t mpan syl eqtrd (x.) (opr B (^) (cv k)) opreq1d binomlem.1 (opr A (^) (opr N (-) (cv k))) A (opr B (^) (cv k)) mul23t mp3an2 N (NN0) (cv k) (0) fznn0subt binomlem.1 A (opr N (-) (cv k)) expclt mpan syl (cv k) N elfznn0t binomlem.2 B (cv k) expclt mpan syl (e. N (NN0)) adantl sylanc eqtr2d (opr N (C.) (cv k)) (x.) opreq2d eqtrd sumeq2dv eqtrd syl N (0) k (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))) fsum1ps N nnnn0t nn0uz syl6eleq N nngt0t (opr N (C.) (cv k)) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl N (cv k) bcclt (opr N (C.) (cv k)) nn0cnt syl (cv k) (0) N elfzelz sylan2 (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (opr B (^) (cv k)) axmulcl (opr N (+) (1)) (NN0) (cv k) (0) fznn0subt N peano2nn0 (e. (cv k) (opr (0) (...) N)) adantr N (NN0) (cv k) (0) fzelp1t sylanc binomlem.1 A (opr (opr N (+) (1)) (-) (cv k)) expclt mpan syl (cv k) N elfznn0t binomlem.2 B (cv k) expclt mpan syl (e. N (NN0)) adantl sylanc sylanc N nnnn0t sylan r19.21aiva syl3anc N nnnn0t N bcn0t N nn0cnt N peano2cn (opr N (+) (1)) subid1t 3syl A (^) opreq2d (x.) (1) opreq1d N peano2nn0 binomlem.1 A (opr N (+) (1)) expclt mpan (opr A (^) (opr N (+) (1))) ax1id 3syl eqtrd (x.) opreq12d N peano2nn0 binomlem.1 A (opr N (+) (1)) expclt mpan (opr A (^) (opr N (+) (1))) mulid2t 3syl eqtrd 0cn elisseti (e. (cv x) (opr (opr N (C.) (0)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (0))) (x.) (1)))) k ax-17 (cv k) (0) N (C.) opreq2 (cv k) (0) (opr N (+) (1)) (-) opreq2 A (^) opreq2d (cv k) (0) B (^) opreq2 binomlem.2 B exp0t ax-mp syl6eq (x.) opreq12d (x.) opreq12d csbief syl5eq 1cn addid2 (...) N opreq1i k (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))) sumeq1i (e. N (NN0)) a1i (+) opreq12d syl 3eqtrd)) thm (binomlem2 ((j k) (j x) (A j) (k x) (A k) (A x) (B j) (B k) (B x) (N j) (N k) (N x)) ((binomlem.1 (e. A (CC))) (binomlem.2 (e. B (CC)))) (-> (e. N (NN)) (= (opr (sum_ k (opr (0) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))))) (x.) B) (opr (sum_ k (opr (1) (...) N) (opr (opr N (C.) (opr (cv k) (-) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (+) (opr B (^) (opr N (+) (1)))))) (N nnnn0t binomlem.2 N (0) B k (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k)))) fsummulc2 mp3an2 N elnn0uz biimp (opr N (C.) (cv k)) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl N (cv k) bcclt (opr N (C.) (cv k)) nn0cnt syl (cv k) (0) N elfzelz sylan2 (opr A (^) (opr N (-) (cv k))) (opr B (^) (cv k)) axmulcl N (NN0) (cv k) (0) fznn0subt binomlem.1 A (opr N (-) (cv k)) expclt mpan syl (cv k) N elfznn0t binomlem.2 B (cv k) expclt mpan syl (e. N (NN0)) adantl sylanc sylanc r19.21aiva sylanc binomlem.2 (opr N (C.) (cv k)) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))) B axmulass mp3an3 N (cv k) bcclt (opr N (C.) (cv k)) nn0cnt syl (cv k) (0) N elfzelz sylan2 (opr A (^) (opr N (-) (cv k))) (opr B (^) (cv k)) axmulcl N (NN0) (cv k) (0) fznn0subt binomlem.1 A (opr N (-) (cv k)) expclt mpan syl (cv k) N elfznn0t binomlem.2 B (cv k) expclt mpan syl (e. N (NN0)) adantl sylanc sylanc binomlem.2 (opr A (^) (opr N (-) (cv k))) (opr B (^) (cv k)) B axmulass mp3an3 N (NN0) (cv k) (0) fznn0subt binomlem.1 A (opr N (-) (cv k)) expclt mpan syl (cv k) N elfznn0t binomlem.2 B (cv k) expclt mpan syl (e. N (NN0)) adantl sylanc (cv k) N elfznn0t binomlem.2 B (cv k) expp1t mpan syl (e. N (NN0)) adantl (opr A (^) (opr N (-) (cv k))) (x.) opreq2d eqtr4d (opr N (C.) (cv k)) (x.) opreq2d eqtrd sumeq2dv eqtrd 1z N (0) (1) j (opr (opr N (C.) (cv j)) (x.) (opr (opr A (^) (opr N (-) (cv j))) (x.) (opr B (^) (opr (cv j) (+) (1))))) k fsumshft mp3an2 N elnn0uz biimp (opr N (C.) (cv j)) (opr (opr A (^) (opr N (-) (cv j))) (x.) (opr B (^) (opr (cv j) (+) (1)))) axmulcl N (cv j) bcclt (opr N (C.) (cv j)) nn0cnt syl (cv j) (0) N elfzelz sylan2 (opr A (^) (opr N (-) (cv j))) (opr B (^) (opr (cv j) (+) (1))) axmulcl N (NN0) (cv j) (0) fznn0subt binomlem.1 A (opr N (-) (cv j)) expclt mpan syl (cv j) N elfznn0t (cv j) peano2nn0 syl binomlem.2 B (opr (cv j) (+) (1)) expclt mpan syl (e. N (NN0)) adantl sylanc sylanc r19.21aiva sylanc (e. (cv x) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (opr (cv k) (+) (1)))))) j ax-17 (e. (cv x) (opr (opr N (C.) (cv j)) (x.) (opr (opr A (^) (opr N (-) (cv j))) (x.) (opr B (^) (opr (cv j) (+) (1)))))) k ax-17 (cv k) (cv j) N (C.) opreq2 (cv k) (cv j) N (-) opreq2 A (^) opreq2d (cv k) (cv j) (+) (1) opreq1 B (^) opreq2d (x.) opreq12d (x.) opreq12d (opr (0) (...) N) cbvsum syl5eq 1cn addid2 (...) (opr N (+) (1)) opreq1i (e. N (NN0)) a1i 1cn N (cv k) (1) subsub3t mp3an3 A (^) opreq2d 1cn (cv k) (1) npcant mpan2 B (^) opreq2d (e. N (CC)) adantl (x.) opreq12d N nn0cnt (cv k) (1) (opr N (+) (1)) elfzelz (cv k) zcnt syl syl2an (opr N (C.) (opr (cv k) (-) (1))) (x.) opreq2d (cv k) (-) (1) oprex (e. (cv x) (opr (opr N (C.) (opr (cv k) (-) (1))) (x.) (opr (opr A (^) (opr N (-) (opr (cv k) (-) (1)))) (x.) (opr B (^) (opr (opr (cv k) (-) (1)) (+) (1)))))) j ax-17 (cv j) (opr (cv k) (-) (1)) N (C.) opreq2 (cv j) (opr (cv k) (-) (1)) N (-) opreq2 A (^) opreq2d (cv j) (opr (cv k) (-) (1)) (+) (1) opreq1 B (^) opreq2d (x.) opreq12d (x.) opreq12d csbief syl5eq r19.21aiva sumeq12rd 3eqtrd syl N elnnuz (opr N (C.) (opr (cv k) (-) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) oprex (opr N (C.) (opr (opr N (+) (1)) (-) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (opr N (+) (1)))) (x.) (opr B (^) (opr N (+) (1)))) oprex (cv k) (opr N (+) (1)) (-) (1) opreq1 N (C.) opreq2d (cv k) (opr N (+) (1)) (opr N (+) (1)) (-) opreq2 A (^) opreq2d (cv k) (opr N (+) (1)) B (^) opreq2 (x.) opreq12d (x.) opreq12d (1) fsump1 sylbi N nnnn0t N nn0cnt 1cn N (1) pncant mpan2 syl N (C.) opreq2d N bcnnt eqtrd N nn0cnt N peano2cn (opr N (+) (1)) subidt 3syl A (^) opreq2d binomlem.1 A exp0t ax-mp syl6eq (x.) (opr B (^) (opr N (+) (1))) opreq1d N peano2nn0 binomlem.2 B (opr N (+) (1)) expclt mpan (opr B (^) (opr N (+) (1))) mulid2t 3syl eqtrd (x.) opreq12d N peano2nn0 binomlem.2 B (opr N (+) (1)) expclt mpan (opr B (^) (opr N (+) (1))) mulid2t 3syl eqtrd syl (sum_ k (opr (1) (...) N) (opr (opr N (C.) (opr (cv k) (-) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (+) opreq2d 3eqtrd)) thm (binomlem3 ((A k) (B k) (N k)) ((binomlem.1 (e. A (CC))) (binomlem.2 (e. B (CC)))) (-> (e. N (NN)) (= (sum_ k (opr (1) (...) (opr N (+) (1))) (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (opr (sum_ k (opr (1) (...) N) (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (+) (opr B (^) (opr N (+) (1)))))) (N elnnuz (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) oprex (opr (opr N (+) (1)) (C.) (opr N (+) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (opr N (+) (1)))) (x.) (opr B (^) (opr N (+) (1)))) oprex (cv k) (opr N (+) (1)) (opr N (+) (1)) (C.) opreq2 (cv k) (opr N (+) (1)) (opr N (+) (1)) (-) opreq2 A (^) opreq2d (cv k) (opr N (+) (1)) B (^) opreq2 (x.) opreq12d (x.) opreq12d (1) fsump1 sylbi N nnnn0t N peano2nn0 (opr N (+) (1)) bcnnt (opr N (+) (1)) nn0cnt (opr N (+) (1)) subidt syl A (^) opreq2d binomlem.1 A exp0t ax-mp syl6eq (x.) (opr B (^) (opr N (+) (1))) opreq1d binomlem.2 B (opr N (+) (1)) expclt mpan (opr B (^) (opr N (+) (1))) mulid2t syl eqtrd (x.) opreq12d binomlem.2 B (opr N (+) (1)) expclt mpan (opr B (^) (opr N (+) (1))) mulid2t syl eqtrd 3syl (sum_ k (opr (1) (...) N) (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (+) opreq2d eqtrd)) thm (binomlem4 ((k x) (A k) (A x) (B k) (B x) (N k) (N x)) ((binomlem.1 (e. A (CC))) (binomlem.2 (e. B (CC)))) (-> (e. N (NN0)) (= (sum_ k (opr (0) (...) (opr N (+) (1))) (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (opr (opr A (^) (opr N (+) (1))) (+) (sum_ k (opr (1) (...) (opr N (+) (1))) (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))))))) ((opr N (+) (1)) (0) k (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))) fsum1ps N peano2nn0 nn0uz syl6eleq N nn0p1nnt (opr N (+) (1)) nngt0t syl (opr (opr N (+) (1)) (C.) (cv k)) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl (opr N (+) (1)) (cv k) bcclt (opr (opr N (+) (1)) (C.) (cv k)) nn0cnt syl N peano2nn0 (cv k) (0) (opr N (+) (1)) elfzelz syl2an (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (opr B (^) (cv k)) axmulcl N (+) (1) oprex (opr N (+) (1)) (V) (cv k) (0) fznn0subt mpan binomlem.1 A (opr (opr N (+) (1)) (-) (cv k)) expclt mpan syl (e. N (NN0)) adantl (cv k) (opr N (+) (1)) elfznn0t binomlem.2 B (cv k) expclt mpan syl (e. N (NN0)) adantl sylanc sylanc r19.21aiva syl3anc N peano2nn0 (opr N (+) (1)) bcn0t syl N peano2nn0 (opr N (+) (1)) nn0cnt (opr N (+) (1)) subid1t 3syl A (^) opreq2d (x.) (1) opreq1d N peano2nn0 binomlem.1 A (opr N (+) (1)) expclt mpan (opr A (^) (opr N (+) (1))) ax1id 3syl eqtrd (x.) opreq12d N peano2nn0 binomlem.1 A (opr N (+) (1)) expclt mpan (opr A (^) (opr N (+) (1))) mulid2t 3syl eqtrd 0cn elisseti (e. (cv x) (opr (opr (opr N (+) (1)) (C.) (0)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (0))) (x.) (1)))) k ax-17 (cv k) (0) (opr N (+) (1)) (C.) opreq2 (cv k) (0) (opr N (+) (1)) (-) opreq2 A (^) opreq2d (cv k) (0) B (^) opreq2 binomlem.2 B exp0t ax-mp syl6eq (x.) opreq12d (x.) opreq12d csbief syl5eq 1cn addid2 (...) (opr N (+) (1)) opreq1i k (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))) sumeq1i (e. N (NN0)) a1i (+) opreq12d eqtrd)) thm (binomlem5 ((A k) (B k) (N k)) ((binomlem.1 (e. A (CC))) (binomlem.2 (e. B (CC)))) (-> (e. N (NN)) (= (opr (sum_ k (opr (0) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))))) (x.) (opr A (+) B)) (sum_ k (opr (0) (...) (opr N (+) (1))) (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))))) ((opr A (^) (opr N (+) (1))) (sum_ k (opr (1) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (opr (sum_ k (opr (1) (...) N) (opr (opr N (C.) (opr (cv k) (-) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (+) (opr B (^) (opr N (+) (1)))) axaddass N peano2nn binomlem.1 A (opr N (+) (1)) expclt (opr N (+) (1)) nnnn0t sylan2 mpan syl (opr N (C.) (cv k)) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl N (cv k) bcclt (opr N (C.) (cv k)) nn0cnt syl N nnnn0t (cv k) (1) N elfzelz syl2an (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (opr B (^) (cv k)) axmulcl N (NN) (cv k) (1) fzelp1t N (+) (1) oprex (opr N (+) (1)) (V) (cv k) (1) fznn0subt mpan binomlem.1 A (opr (opr N (+) (1)) (-) (cv k)) expclt mpan 3syl (cv k) N elfznnt binomlem.2 B (cv k) expclt (cv k) nnnn0t sylan2 mpan syl (e. N (NN)) adantl sylanc sylanc r19.21aiva N (1) k (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))) fsumclt N elnnuz sylanb mpdan (sum_ k (opr (1) (...) N) (opr (opr N (C.) (opr (cv k) (-) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (opr B (^) (opr N (+) (1))) axaddcl (opr N (C.) (opr (cv k) (-) (1))) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl N (opr (cv k) (-) (1)) bcclt (opr N (C.) (opr (cv k) (-) (1))) nn0cnt syl N nnnn0t (cv k) (1) N elfzelz (cv k) peano2zm syl syl2an (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (opr B (^) (cv k)) axmulcl N (NN) (cv k) (1) fzelp1t N (+) (1) oprex (opr N (+) (1)) (V) (cv k) (1) fznn0subt mpan binomlem.1 A (opr (opr N (+) (1)) (-) (cv k)) expclt mpan 3syl (cv k) N elfznnt binomlem.2 B (cv k) expclt (cv k) nnnn0t sylan2 mpan syl (e. N (NN)) adantl sylanc sylanc r19.21aiva N (1) k (opr (opr N (C.) (opr (cv k) (-) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))) fsumclt N elnnuz sylanb mpdan N peano2nn binomlem.2 B (opr N (+) (1)) expclt (opr N (+) (1)) nnnn0t sylan2 mpan syl sylanc syl3anc binomlem.1 binomlem.2 N k binomlem1 binomlem.1 binomlem.2 N k binomlem2 (+) opreq12d binomlem.1 binomlem.2 N k binomlem3 (opr N (C.) (cv k)) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl N (cv k) bcclt N nnnn0t (cv k) (1) N elfzelz syl2an (opr N (C.) (cv k)) nn0cnt syl (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (opr B (^) (cv k)) axmulcl N (NN) (cv k) (1) fzelp1t N (+) (1) oprex (opr N (+) (1)) (V) (cv k) (1) fznn0subt mpan binomlem.1 A (opr (opr N (+) (1)) (-) (cv k)) expclt mpan 3syl (cv k) N elfznnt binomlem.2 B (cv k) expclt (cv k) nnnn0t sylan2 mpan syl (e. N (NN)) adantl sylanc sylanc (opr N (C.) (opr (cv k) (-) (1))) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl N (opr (cv k) (-) (1)) bcclt N nnnn0t (cv k) (1) N elfzelz (cv k) peano2zm syl syl2an (opr N (C.) (opr (cv k) (-) (1))) nn0cnt syl (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (opr B (^) (cv k)) axmulcl N (NN) (cv k) (1) fzelp1t N (+) (1) oprex (opr N (+) (1)) (V) (cv k) (1) fznn0subt mpan binomlem.1 A (opr (opr N (+) (1)) (-) (cv k)) expclt mpan 3syl (cv k) N elfznnt binomlem.2 B (cv k) expclt (cv k) nnnn0t sylan2 mpan syl (e. N (NN)) adantl sylanc sylanc jca r19.21aiva N (1) k (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))) (opr (opr N (C.) (opr (cv k) (-) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))) fsumadd N elnnuz sylanb mpdan (opr N (C.) (cv k)) (opr N (C.) (opr (cv k) (-) (1))) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) adddirt N (cv k) bcclt N nnnn0t (cv k) (1) N elfzelz syl2an (opr N (C.) (cv k)) nn0cnt syl N (opr (cv k) (-) (1)) bcclt N nnnn0t (cv k) (1) N elfzelz (cv k) peano2zm syl syl2an (opr N (C.) (opr (cv k) (-) (1))) nn0cnt syl (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (opr B (^) (cv k)) axmulcl N (NN) (cv k) (1) fzelp1t N (+) (1) oprex (opr N (+) (1)) (V) (cv k) (1) fznn0subt mpan binomlem.1 A (opr (opr N (+) (1)) (-) (cv k)) expclt mpan 3syl (cv k) N elfznnt binomlem.2 B (cv k) expclt (cv k) nnnn0t sylan2 mpan syl (e. N (NN)) adantl sylanc syl3anc N (cv k) bcpasct N nnnn0t (cv k) (1) N elfzelz syl2an (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) opreq1d eqtr3d sumeq2dv eqtr3d (+) (opr B (^) (opr N (+) (1))) opreq1d (sum_ k (opr (1) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (sum_ k (opr (1) (...) N) (opr (opr N (C.) (opr (cv k) (-) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (opr B (^) (opr N (+) (1))) axaddass (opr N (C.) (cv k)) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl N (cv k) bcclt (opr N (C.) (cv k)) nn0cnt syl N nnnn0t (cv k) (1) N elfzelz syl2an (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (opr B (^) (cv k)) axmulcl N (NN) (cv k) (1) fzelp1t N (+) (1) oprex (opr N (+) (1)) (V) (cv k) (1) fznn0subt mpan binomlem.1 A (opr (opr N (+) (1)) (-) (cv k)) expclt mpan 3syl (cv k) N elfznnt binomlem.2 B (cv k) expclt (cv k) nnnn0t sylan2 mpan syl (e. N (NN)) adantl sylanc sylanc r19.21aiva N (1) k (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))) fsumclt N elnnuz sylanb mpdan (opr N (C.) (opr (cv k) (-) (1))) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl N (opr (cv k) (-) (1)) bcclt (opr N (C.) (opr (cv k) (-) (1))) nn0cnt syl N nnnn0t (cv k) (1) N elfzelz (cv k) peano2zm syl syl2an (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (opr B (^) (cv k)) axmulcl N (NN) (cv k) (1) fzelp1t N (+) (1) oprex (opr N (+) (1)) (V) (cv k) (1) fznn0subt mpan binomlem.1 A (opr (opr N (+) (1)) (-) (cv k)) expclt mpan 3syl (cv k) N elfznnt binomlem.2 B (cv k) expclt (cv k) nnnn0t sylan2 mpan syl (e. N (NN)) adantl sylanc sylanc r19.21aiva N (1) k (opr (opr N (C.) (opr (cv k) (-) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))) fsumclt N elnnuz sylanb mpdan N peano2nn binomlem.2 B (opr N (+) (1)) expclt (opr N (+) (1)) nnnn0t sylan2 mpan syl syl3anc 3eqtr2d (opr A (^) (opr N (+) (1))) (+) opreq2d 3eqtr4d N nnnn0t nn0uz syl6eleq (opr N (C.) (cv k)) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl N (cv k) bcclt (opr N (C.) (cv k)) nn0cnt syl N elnn0uz biimpr (cv k) (0) N elfzelz syl2an (opr A (^) (opr N (-) (cv k))) (opr B (^) (cv k)) axmulcl N (` (ZZ>) (0)) (cv k) (0) fznn0subt binomlem.1 A (opr N (-) (cv k)) expclt mpan syl (cv k) N elfznn0t binomlem.2 B (cv k) expclt mpan syl (e. N (` (ZZ>) (0))) adantl sylanc sylanc r19.21aiva N (0) k (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k)))) fsumclt mpdan binomlem.1 binomlem.2 (sum_ k (opr (0) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))))) A B axdistr mp3an23 3syl N nnnn0t binomlem.1 binomlem.2 N k binomlem4 syl 3eqtr4d)) thm (binomlem6 ((j k) (j x) (A j) (k x) (A k) (A x) (B j) (B k) (B x) (N j) (N k) (N x)) ((binomlem.1 (e. A (CC))) (binomlem.2 (e. B (CC)))) (-> (e. N (NN)) (= (opr (opr A (+) B) (^) N) (sum_ k (opr (0) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))))))) ((cv x) (1) (opr A (+) B) (^) opreq2 (cv x) (1) (0) (...) opreq2 (cv x) (1) (C.) (cv k) opreq1 (cv x) (1) (-) (cv k) opreq1 A (^) opreq2d (x.) (opr B (^) (cv k)) opreq1d (x.) opreq12d (e. (cv k) (opr (0) (...) (1))) a1d r19.21aiv sumeq12rd eqeq12d (cv x) (cv j) (opr A (+) B) (^) opreq2 (cv x) (cv j) (0) (...) opreq2 (cv x) (cv j) (C.) (cv k) opreq1 (cv x) (cv j) (-) (cv k) opreq1 A (^) opreq2d (x.) (opr B (^) (cv k)) opreq1d (x.) opreq12d (e. (cv k) (opr (0) (...) (cv j))) a1d r19.21aiv sumeq12rd eqeq12d (cv x) (opr (cv j) (+) (1)) (opr A (+) B) (^) opreq2 (cv x) (opr (cv j) (+) (1)) (0) (...) opreq2 (cv x) (opr (cv j) (+) (1)) (C.) (cv k) opreq1 (cv x) (opr (cv j) (+) (1)) (-) (cv k) opreq1 A (^) opreq2d (x.) (opr B (^) (cv k)) opreq1d (x.) opreq12d (e. (cv k) (opr (0) (...) (opr (cv j) (+) (1)))) a1d r19.21aiv sumeq12rd eqeq12d (cv x) N (opr A (+) B) (^) opreq2 (cv x) N (0) (...) opreq2 (cv x) N (C.) (cv k) opreq1 (cv x) N (-) (cv k) opreq1 A (^) opreq2d (x.) (opr B (^) (cv k)) opreq1d (x.) opreq12d (e. (cv k) (opr (0) (...) N)) a1d r19.21aiv sumeq12rd eqeq12d binomlem.1 binomlem.2 addcl (opr A (+) B) exp1t ax-mp 0z (0) uzidt ax-mp (opr (1) (C.) (cv k)) (x.) (opr (opr A (^) (opr (1) (-) (cv k))) (x.) (opr B (^) (cv k))) oprex binomlem.2 elisseti 1cn addid2 (cv k) eqeq2i (cv k) (1) (1) (C.) opreq2 1nn0 (1) bcnnt ax-mp syl6eq (cv k) (1) (1) (-) opreq2 1cn subid syl6eq A (^) opreq2d binomlem.1 A exp0t ax-mp syl6eq (cv k) (1) B (^) opreq2 binomlem.2 B exp1t ax-mp syl6eq (x.) opreq12d binomlem.2 mulid2 syl6eq (x.) opreq12d binomlem.2 mulid2 syl6eq sylbi (0) fsump1 ax-mp binomlem.1 0z (cv k) (0) (1) (C.) opreq2 1nn0 (1) bcn0t ax-mp syl6eq (cv k) (0) (1) (-) opreq2 1cn subid1 syl6eq A (^) opreq2d binomlem.1 A exp1t ax-mp syl6eq (cv k) (0) B (^) opreq2 binomlem.2 B exp0t ax-mp syl6eq (x.) opreq12d binomlem.1 mulid1 syl6eq (x.) opreq12d binomlem.1 mulid2 syl6eq (CC) fsum1 mp2an (+) B opreq1i eqtr2 1cn addid2 (0) (...) opreq2i k (opr (opr (1) (C.) (cv k)) (x.) (opr (opr A (^) (opr (1) (-) (cv k))) (x.) (opr B (^) (cv k)))) sumeq1i 3eqtr binomlem.1 binomlem.2 addcl (opr A (+) B) (cv j) expp1t (cv j) nnnn0t sylan2 mpan (= (opr (opr A (+) B) (^) (cv j)) (sum_ k (opr (0) (...) (cv j)) (opr (opr (cv j) (C.) (cv k)) (x.) (opr (opr A (^) (opr (cv j) (-) (cv k))) (x.) (opr B (^) (cv k)))))) adantr (e. (cv j) (NN)) (= (opr (opr A (+) B) (^) (cv j)) (sum_ k (opr (0) (...) (cv j)) (opr (opr (cv j) (C.) (cv k)) (x.) (opr (opr A (^) (opr (cv j) (-) (cv k))) (x.) (opr B (^) (cv k)))))) pm3.27 (x.) (opr A (+) B) opreq1d binomlem.1 binomlem.2 (cv j) k binomlem5 (= (opr (opr A (+) B) (^) (cv j)) (sum_ k (opr (0) (...) (cv j)) (opr (opr (cv j) (C.) (cv k)) (x.) (opr (opr A (^) (opr (cv j) (-) (cv k))) (x.) (opr B (^) (cv k)))))) adantr 3eqtrd ex nnind)) thm (binom ((A k) (B k) (N k)) ((binomlem.1 (e. A (CC))) (binomlem.2 (e. B (CC)))) (-> (e. N (NN0)) (= (opr (opr A (+) B) (^) N) (sum_ k (opr (0) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))))))) (N elnn0 binomlem.1 binomlem.2 N k binomlem6 1cn 0z (cv k) (0) (0) (C.) opreq2 0nn0 (0) bcn0t ax-mp syl6eq (cv k) (0) (0) (-) opreq2 0cn subid syl6eq A (^) opreq2d binomlem.1 A exp0t ax-mp syl6eq (cv k) (0) B (^) opreq2 binomlem.2 B exp0t ax-mp syl6eq (x.) opreq12d 1cn mulid1 syl6eq (x.) opreq12d 1cn mulid1 syl6eq (CC) fsum1 mp2an eqcomi (= N (0)) a1i N (0) (opr A (+) B) (^) opreq2 binomlem.1 binomlem.2 addcl (opr A (+) B) exp0t ax-mp syl6eq N (0) (0) (...) opreq2 N (0) (C.) (cv k) opreq1 N (0) (-) (cv k) opreq1 A (^) opreq2d (x.) (opr B (^) (cv k)) opreq1d (x.) opreq12d (e. (cv k) (opr (0) (...) (0))) a1d r19.21aiv sumeq12rd 3eqtr4d jaoi sylbi)) thm (climcvgc1 ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x) (B j) (B k) (B x)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) ((cv x) B (0) (<) breq2 (cv x) B (` (abs) (opr (` F (cv k)) (-) A)) (<) breq2 (e. (` F (cv k)) (CC)) anbi2d (br (cv j) (<_) (cv k)) imbi2d j (ZZ) k (ZZ) rexralbidv imbi12d (RR) rcla4v F (V) A C x j k clim biimpa pm3.27d syl5com exp31 climrel F A brrelexi con3i (-> (e. B (RR)) (-> (br (0) (<) B) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))))) pm2.21d (e. A C) a1d pm2.61i imp43)) thm (climcvg1 ((j k) (F j) (F k) (A j) (A k) (B j) (B k)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))) (A C F B j k climcvgc1 (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) pm3.27 (br (cv j) (<_) (cv k)) imim2i k (ZZ) r19.20si j (ZZ) r19.22si syl)) thm (clim1 ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) ((clim.1 (e. F (V)))) (-> (e. A (CC)) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) (clim.1 F (V) A (CC) x j k clim mpan (e. A (CC)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))) ibar bitr4d)) thm (clim1a ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x)) ((clim.1 (e. F (V)))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (clim.1 A x n k clim1 (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) pm3.27 (br (cv n) (<_) (cv k)) imim2i k (ZZ) r19.20si n (ZZ) r19.22si (cv n) (cv j) (<_) (cv k) breq1 (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) imbi1d k (ZZ) ralbidv (ZZ) cbvrexv sylib (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC))))) a1i (cv j) (ZZ) (cv m) (br (cv m) (<_) (cv j)) ifcl expcom (A.e. k (ZZ) (/\ (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC))) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))) adantr (cv m) (cv j) max1 (e. (cv k) (RR)) adantr (cv m) (if (br (cv m) (<_) (cv j)) (cv j) (cv m)) (cv k) letrt 3expa (e. (cv m) (RR)) (e. (cv j) (RR)) pm3.26 (cv j) (RR) (cv m) (br (cv m) (<_) (cv j)) ifcl ancoms jca sylan mpand (cv m) (cv j) max2 (e. (cv k) (RR)) adantr (cv j) (if (br (cv m) (<_) (cv j)) (cv j) (cv m)) (cv k) letrt 3expa (e. (cv m) (RR)) (e. (cv j) (RR)) pm3.27 (cv j) (RR) (cv m) (br (cv m) (<_) (cv j)) ifcl ancoms jca sylan mpand jcad (cv m) zret (cv j) zret anim12i (cv k) zret syl2an (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)) (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) prth syl9 r19.20dva ex com23 imp jcad (cv n) (if (br (cv m) (<_) (cv j)) (cv j) (cv m)) (<_) (cv k) breq1 (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))) imbi1d k (ZZ) ralbidv (ZZ) rcla4ev syl6 k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC))) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))) r19.26 sylan2br exp32 imp com23 r19.23adv ex r19.23aiv impbid (br (0) (<) (cv x)) imbi2d x (RR) ralbidv sylan9bb)) thm (clim2a ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x) (M j) (M k) (M m) (M n) (M x)) ((clim2.1 (e. F (V))) (clim2.2 (e. M (ZZ)))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) (clim2.1 A m k x n clim1a clim2.2 zre M (cv n) max2 mpan (e. (cv k) (RR)) adantr (cv n) (if (br M (<_) (cv n)) (cv n) M) (cv k) letrt (e. (cv n) (RR)) (e. (cv k) (RR)) pm3.26 clim2.2 zre (cv n) (RR) M (br M (<_) (cv n)) ifcl mpan2 (e. (cv k) (RR)) adantr (e. (cv n) (RR)) (e. (cv k) (RR)) pm3.27 syl3anc mpand (cv n) zret (cv k) zret syl2an (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) imim1d r19.20dva (cv n) zret clim2.2 zre M (cv n) max1 mpan syl jctild clim2.2 (cv n) (ZZ) M (br M (<_) (cv n)) ifcl mpan2 jctild (cv j) (if (br M (<_) (cv n)) (cv n) M) M (<_) breq2 (cv j) (if (br M (<_) (cv n)) (cv n) M) (<_) (cv k) breq1 (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) imbi1d k (ZZ) ralbidv anbi12d (ZZ) rcla4ev syl6 r19.23aiv (br (0) (<) (cv x)) imim2i x (RR) r19.20si syl6bi clim2.1 A m k x j clim1a (br M (<_) (cv j)) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))) pm3.27 j (ZZ) r19.22si (br (0) (<) (cv x)) imim2i x (RR) r19.20si syl5bir impbid)) thm (clim2 ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x)) ((clim2.1 (e. F (V))) (clim2.2 (e. M (ZZ)))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (br M (<_) (cv k)) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) (clim2.1 clim2.2 A m k x j clim2a clim2.2 (cv m) M (<_) (cv k) breq1 (e. (` F (cv k)) (CC)) imbi1d k (ZZ) ralbidv (ZZ) rcla4ev mpan sylan2)) thm (climcvg2 ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) ((climcvg2.1 (e. M (ZZ)))) (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) (A C F B m k climcvg1 climcvg2.1 zre M (cv m) max2 mpan (e. (cv k) (RR)) adantr (cv m) (if (br M (<_) (cv m)) (cv m) M) (cv k) letrt (e. (cv m) (RR)) (e. (cv k) (RR)) pm3.26 climcvg2.1 zre (cv m) (RR) M (br M (<_) (cv m)) ifcl mpan2 (e. (cv k) (RR)) adantr (e. (cv m) (RR)) (e. (cv k) (RR)) pm3.27 syl3anc mpand (cv m) zret (cv k) zret syl2an (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) imim1d r19.20dva (cv m) zret climcvg2.1 zre M (cv m) max1 mpan syl jctild climcvg2.1 (cv m) (ZZ) M (br M (<_) (cv m)) ifcl mpan2 jctild (cv j) (if (br M (<_) (cv m)) (cv m) M) M (<_) breq2 (cv j) (if (br M (<_) (cv m)) (cv m) M) (<_) (cv k) breq1 (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) imbi1d k (ZZ) ralbidv anbi12d (ZZ) rcla4ev syl6 r19.23aiv syl)) thm (climcvg2z ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) () (-> (/\ (e. M (ZZ)) (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B)))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) (cv j)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))) (M (cv m) max2 (e. (cv k) (RR)) 3adant3 (cv m) (if (br M (<_) (cv m)) (cv m) M) (cv k) letrt (e. M (RR)) (e. (cv m) (RR)) (e. (cv k) (RR)) 3simp2 (cv m) (RR) M (br M (<_) (cv m)) ifcl ancoms (e. (cv k) (RR)) 3adant3 (e. M (RR)) (e. (cv m) (RR)) (e. (cv k) (RR)) 3simp3 syl3anc mpand M zret (cv m) zret (cv k) zret syl3an 3expa (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) imim1d r19.20dva M (cv m) max1 M zret (cv m) zret syl2an jctild (cv m) (ZZ) M (br M (<_) (cv m)) ifcl ancoms jctild M j (A.e. k (` (ZZ>) (cv j)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)) rexuz (cv j) (if (br M (<_) (cv m)) (cv m) M) M (<_) breq2 (cv j) (if (br M (<_) (cv m)) (cv m) M) (<_) (cv k) breq1 (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) imbi1d k (ZZ) ralbidv anbi12d (ZZ) rcla4ev (cv j) k (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) raluz (br M (<_) (cv j)) anbi2d rexbiia sylibr syl5bir (e. (cv m) (ZZ)) adantr syld ex r19.23adv A C F B m k climcvg1 syl5 imp)) thm (climcvgc2z ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) (M (cv m) max2 (e. (cv k) (RR)) 3adant3 (cv m) (if (br M (<_) (cv m)) (cv m) M) (cv k) letrt (e. M (RR)) (e. (cv m) (RR)) (e. (cv k) (RR)) 3simp2 (cv m) (RR) M (br M (<_) (cv m)) ifcl ancoms (e. (cv k) (RR)) 3adant3 (e. M (RR)) (e. (cv m) (RR)) (e. (cv k) (RR)) 3simp3 syl3anc mpand M zret (cv m) zret (cv k) zret syl3an 3expa (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)) imim1d r19.20dva M (cv m) max1 M zret (cv m) zret syl2an jctild (cv m) (ZZ) M (br M (<_) (cv m)) ifcl ancoms jctild M j (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))) rexuz (cv j) (if (br M (<_) (cv m)) (cv m) M) M (<_) breq2 (cv j) (if (br M (<_) (cv m)) (cv m) M) (<_) (cv k) breq1 (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)) imbi1d k (ZZ) ralbidv anbi12d (ZZ) rcla4ev syl5bir (e. (cv m) (ZZ)) adantr syld ex r19.23adv A C F B m k climcvgc1 syl5 exp4c 3imp1)) thm (climcvg2zb ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))) (M A C F B j k climcvgc2z (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) pm3.27 (br (cv j) (<_) (cv k)) imim2i k (ZZ) r19.20si j (` (ZZ>) M) r19.22si syl)) thm (climcvg2zbOLD ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) () (-> (/\ (e. M (ZZ)) (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B)))) (E.e. j (` (ZZ>) M) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))) (M (cv m) max2 (e. (cv k) (RR)) 3adant3 (cv m) (if (br M (<_) (cv m)) (cv m) M) (cv k) letrt (e. M (RR)) (e. (cv m) (RR)) (e. (cv k) (RR)) 3simp2 (cv m) (RR) M (br M (<_) (cv m)) ifcl ancoms (e. (cv k) (RR)) 3adant3 (e. M (RR)) (e. (cv m) (RR)) (e. (cv k) (RR)) 3simp3 syl3anc mpand M zret (cv m) zret (cv k) zret syl3an 3expa (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) imim1d r19.20dva M (cv m) max1 M zret (cv m) zret syl2an jctild (cv m) (ZZ) M (br M (<_) (cv m)) ifcl ancoms jctild M j (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))) rexuz (cv j) (if (br M (<_) (cv m)) (cv m) M) M (<_) breq2 (cv j) (if (br M (<_) (cv m)) (cv m) M) (<_) (cv k) breq1 (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) imbi1d k (ZZ) ralbidv anbi12d (ZZ) rcla4ev syl5bir (e. (cv m) (ZZ)) adantr syld ex r19.23adv A C F B m k climcvg1 syl5 imp)) thm (clim2az ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x)) ((clim2x.1 (e. F (V)))) (-> (/\ (e. M (ZZ)) (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC))))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (` (ZZ>) M) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (M A (CC) F (cv x) j k climcvg2zbOLD exp44 imp exp4a r19.21adv (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC))))) adantrr clim2x.1 A m k x j clim1a (cv j) M eluzelz (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))) anim1i r19.22i2 (br (0) (<) (cv x)) imim2i x (RR) r19.20si syl5bir (e. M (ZZ)) adantl impbid)) thm (clim3az ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x) (M j) (M k) (M m) (M n) (M x)) ((clim3az.1 (e. F (V)))) (-> (/\ (e. M (ZZ)) (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC))))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) M) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (clim3az.1 A m k x j clim1a (cv k) M eluzelz (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))) imim1i r19.20i2 j (ZZ) r19.22si (br (0) (<) (cv x)) imim2i x (RR) r19.20si syl6bi (e. M (ZZ)) adantl M k (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))) raluz (e. (cv j) (ZZ)) adantr M (cv j) max1 (e. (cv k) (RR)) adantr M (if (br M (<_) (cv j)) (cv j) M) (cv k) letrt (e. M (RR)) (e. (cv j) (RR)) pm3.26 (e. (cv k) (RR)) adantr (cv j) (RR) M (br M (<_) (cv j)) ifcl ancoms (e. (cv k) (RR)) adantr (/\ (e. M (RR)) (e. (cv j) (RR))) (e. (cv k) (RR)) pm3.27 syl3anc mpand M (cv j) max2 (e. (cv k) (RR)) adantr (cv j) (if (br M (<_) (cv j)) (cv j) M) (cv k) letrt (e. M (RR)) (e. (cv j) (RR)) pm3.27 (e. (cv k) (RR)) adantr (cv j) (RR) M (br M (<_) (cv j)) ifcl ancoms (e. (cv k) (RR)) adantr (/\ (e. M (RR)) (e. (cv j) (RR))) (e. (cv k) (RR)) pm3.27 syl3anc mpand jcad M zret (cv j) zret anim12i (cv k) zret syl2an (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) imim1d (br M (<_) (cv k)) (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) impexp syl5ibr r19.20dva sylbid (cv j) (ZZ) M (br M (<_) (cv j)) ifcl ancoms jctild (cv n) (if (br M (<_) (cv j)) (cv j) M) (<_) (cv k) breq1 (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) imbi1d k (ZZ) ralbidv (ZZ) rcla4ev syl6 ex r19.23adv (br (0) (<) (cv x)) imim2d x (RR) r19.20sdv clim3az.1 A m k x n clim1a biimprd sylan9 impbid)) thm (clim3z ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x)) ((clim3az.1 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. A (CC)) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (CC)))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) M) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (M k (e. (` F (cv k)) (CC)) raluz (cv m) M (<_) (cv k) breq1 (e. (` F (cv k)) (CC)) imbi1d k (ZZ) ralbidv (ZZ) rcla4ev ex sylbid (e. A (CC)) anim2d imp clim3az.1 M A m k x j clim3az syldan 3impb)) thm (clim3zOLD ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x)) ((clim3az.1 (e. F (V)))) (-> (/\ (e. M (ZZ)) (/\ (e. A (CC)) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) M) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (M k (e. (` F (cv k)) (CC)) raluz (cv m) M (<_) (cv k) breq1 (e. (` F (cv k)) (CC)) imbi1d k (ZZ) ralbidv (ZZ) rcla4ev ex sylbid (e. A (CC)) anim2d imp clim3az.1 M A m k x j clim3az syldan)) thm (clim3a ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x) (N j) (N k) (N m) (N n) (N x)) ((clim3.1 (e. F (V))) (clim3.2 (e. N (ZZ)))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (clim3.1 A m k x j clim1a (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) (br N (<_) (cv k)) pm3.42 k (ZZ) r19.20si j (ZZ) r19.22si (br (0) (<) (cv x)) imim2i x (RR) r19.20si syl6bi clim3.1 A m k x n clim1a clim3.2 zre N (cv j) max1 mpan (e. (cv k) (RR)) adantr clim3.2 zre N (if (br N (<_) (cv j)) (cv j) N) (cv k) letrt mp3an1 clim3.2 zre (cv j) (RR) N (br N (<_) (cv j)) ifcl mpan2 sylan mpand clim3.2 zre N (cv j) max2 mpan (e. (cv k) (RR)) adantr (cv j) (if (br N (<_) (cv j)) (cv j) N) (cv k) letrt (e. (cv j) (RR)) (e. (cv k) (RR)) pm3.26 clim3.2 zre (cv j) (RR) N (br N (<_) (cv j)) ifcl mpan2 (e. (cv k) (RR)) adantr (e. (cv j) (RR)) (e. (cv k) (RR)) pm3.27 syl3anc mpand jcad (cv j) zret (cv k) zret syl2an (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) imim1d r19.20dva clim3.2 (cv j) (ZZ) N (br N (<_) (cv j)) ifcl mpan2 jctild (cv n) (if (br N (<_) (cv j)) (cv j) N) (<_) (cv k) breq1 (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) imbi1d k (ZZ) ralbidv (ZZ) rcla4ev syl6 r19.23aiv (br (0) (<) (cv x)) imim2i x (RR) r19.20si syl5bir impbid)) thm (clim3 ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (N j) (N k) (N m) (N x)) ((clim3.1 (e. F (V))) (clim3.2 (e. N (ZZ)))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (br N (<_) (cv k)) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (clim3.1 clim3.2 A m k x j clim3a clim3.2 (cv m) N (<_) (cv k) breq1 (e. (` F (cv k)) (CC)) imbi1d k (ZZ) ralbidv (ZZ) rcla4ev mpan sylan2)) thm (clim3b ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (N j) (N k) (N m) (N x)) ((clim3.1 (e. F (V))) (clim3.2 (e. N (ZZ)))) (-> (e. A (CC)) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) (clim3.1 A x j k clim1 (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))) (br N (<_) (cv k)) pm3.42 k (ZZ) r19.20si j (ZZ) r19.22si (br (0) (<) (cv x)) imim2i x (RR) r19.20si syl6bi clim3.1 A x m k clim1 clim3.2 zre N (cv j) max1 mpan (e. (cv k) (RR)) adantr clim3.2 zre N (if (br N (<_) (cv j)) (cv j) N) (cv k) letrt mp3an1 clim3.2 zre (cv j) (RR) N (br N (<_) (cv j)) ifcl mpan2 sylan mpand clim3.2 zre N (cv j) max2 mpan (e. (cv k) (RR)) adantr (cv j) (if (br N (<_) (cv j)) (cv j) N) (cv k) letrt (e. (cv j) (RR)) (e. (cv k) (RR)) pm3.26 clim3.2 zre (cv j) (RR) N (br N (<_) (cv j)) ifcl mpan2 (e. (cv k) (RR)) adantr (e. (cv j) (RR)) (e. (cv k) (RR)) pm3.27 syl3anc mpand jcad (cv j) zret (cv k) zret syl2an (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))) imim1d r19.20dva clim3.2 (cv j) (ZZ) N (br N (<_) (cv j)) ifcl mpan2 jctild (cv m) (if (br N (<_) (cv j)) (cv j) N) (<_) (cv k) breq1 (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))) imbi1d k (ZZ) ralbidv (ZZ) rcla4ev syl6 r19.23aiv (br (0) (<) (cv x)) imim2i x (RR) r19.20si syl5bir impbid)) thm (climcvg3 ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (N j) (N k)) ((climcvg3.1 (e. N (ZZ)))) (-> (/\ (/\ (e. A (CC)) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))) (A (CC) F B j k climcvg1 (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) (br N (<_) (cv k)) pm3.42 k (ZZ) r19.20si j (ZZ) r19.22si syl)) thm (climcvg3z ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (N j) (N k)) () (-> (/\ (/\ (e. A (CC)) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))) (A (CC) F B j k climcvg1 (cv k) N eluzelz (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)) imim1i r19.20i2 j (ZZ) r19.22si syl)) thm (clim4a ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x) (N j) (N k) (N m) (N x)) ((clim4.1 (e. F (V))) (clim4.2 (e. M (ZZ))) (clim4.3 (e. N (ZZ)))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) (clim4.1 clim4.2 A m k x j clim2a (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) (br N (<_) (cv k)) pm3.42 k (ZZ) r19.20si (br M (<_) (cv j)) anim2i j (ZZ) r19.22si (br (0) (<) (cv x)) imim2i x (RR) r19.20si syl6bi clim4.1 clim4.3 A m k x j clim3a (br M (<_) (cv j)) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))) pm3.27 j (ZZ) r19.22si (br (0) (<) (cv x)) imim2i x (RR) r19.20si syl5bir impbid)) thm (clim4 ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x) (N j) (N k) (N m) (N x)) ((clim4.1 (e. F (V))) (clim4.2 (e. M (ZZ))) (clim4.3 (e. N (ZZ)))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (/\ (br M (<_) (cv k)) (br N (<_) (cv k))) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) (clim4.1 clim4.2 clim4.3 A m k x j clim4a (cv k) zret clim4.2 zre clim4.3 zre M N max1 mp2an clim4.2 zre clim4.3 clim4.2 (br M (<_) N) keepel zre M (if (br M (<_) N) N M) (cv k) letrt mp3an12 mpani clim4.2 zre clim4.3 zre M N max2 mp2an clim4.3 zre clim4.3 clim4.2 (br M (<_) N) keepel zre N (if (br M (<_) N) N M) (cv k) letrt mp3an12 mpani jcad syl (e. (` F (cv k)) (CC)) imim1d r19.20i clim4.3 clim4.2 (br M (<_) N) keepel jctil (cv m) (if (br M (<_) N) N M) (<_) (cv k) breq1 (e. (` F (cv k)) (CC)) imbi1d k (ZZ) ralbidv (ZZ) rcla4ev syl sylan2)) thm (climcvg4 ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) ((climcvg4.1 (e. M (ZZ)))) (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) (climcvg4.1 A C F B j k climcvg2 (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) (br N (<_) (cv k)) pm3.42 k (ZZ) r19.20si (br M (<_) (cv j)) anim2i j (ZZ) r19.22si syl)) thm (climcvgc4z ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) (M A C F B j k climcvgc2z (cv k) N eluzelz (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))) imim1i r19.20i2 j (` (ZZ>) M) r19.22si syl)) thm (climcvg4z ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))) (M A C F B j k climcvg2zb (cv k) N eluzelz (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)) imim1i r19.20i2 j (` (ZZ>) M) r19.22si syl)) thm (clim0cvg4z ((j k) (F j) (F k) (A j) (A k) (M j) (M k)) () (-> (/\ (/\ (e. M (ZZ)) (br F (~~>) (0))) (/\ (e. A (RR)) (br (0) (<) A))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (` F (cv k))) (<) A))))) (0cn M (0) (CC) F A j k N climcvgc4z mp3anl2 (` F (cv k)) subid1t (abs) fveq2d (<) A breq1d biimpa (br (cv j) (<_) (cv k)) imim2i k (` (ZZ>) N) r19.20si j (` (ZZ>) M) r19.22si syl)) thm (climcvg4uzOLD ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) () (-> (/\ (e. M (ZZ)) (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B)))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))) (M A C F B j k climcvg2zbOLD (cv k) N eluzelz (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)) imim1i r19.20i2 j (` (ZZ>) M) r19.22si syl)) thm (climcvgc5z ((j k) (A j) (A k) (B j) (B k) (F j) (F k) (K j) (K k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. K (ZZ)) (e. A C)) (/\/\ (br F (~~>) A) (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (/\ (br K (<_) (cv j)) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))))) ((if (br K (<_) M) M K) A C F B j k N climcvgc4z M (ZZ) K (br K (<_) M) ifcl (e. A C) 3adant3 (/\/\ (br F (~~>) A) (e. B (RR)) (br (0) (<) B)) adantr (e. M (ZZ)) (e. K (ZZ)) (e. A C) 3simp3 (/\/\ (br F (~~>) A) (e. B (RR)) (br (0) (<) B)) adantr (br F (~~>) A) (e. B (RR)) (br (0) (<) B) 3simp1 (/\/\ (e. M (ZZ)) (e. K (ZZ)) (e. A C)) adantl 3jca (br F (~~>) A) (e. B (RR)) (br (0) (<) B) 3simpc (/\/\ (e. M (ZZ)) (e. K (ZZ)) (e. A C)) adantl sylanc K M (cv j) maxlet K zret M zret (cv j) zret syl3an 3com12 3expa ex pm5.32d (e. (cv j) (ZZ)) (br M (<_) (cv j)) (br K (<_) (cv j)) anass (br M (<_) (cv j)) (br K (<_) (cv j)) ancom (e. (cv j) (ZZ)) anbi2i bitr syl6rbbr M (cv j) eluz1t (e. K (ZZ)) adantr (br K (<_) (cv j)) anbi1d M (ZZ) K (br K (<_) M) ifcl (if (br K (<_) M) M K) (cv j) eluz1t syl 3bitr4d (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))) anbi1d (e. (cv j) (` (ZZ>) M)) (br K (<_) (cv j)) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))) anass syl5rbbr rexbidv2 (e. A C) 3adant3 (/\/\ (br F (~~>) A) (e. B (RR)) (br (0) (<) B)) adantr mpbid)) thm (climcvg5z ((j k) (A j) (A k) (B j) (B k) (F j) (F k) (K j) (K k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. K (ZZ)) (e. A C)) (/\/\ (br F (~~>) A) (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (/\ (br K (<_) (cv j)) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) (M K A C F B j k N climcvgc5z (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B) pm3.27 (br (cv j) (<_) (cv k)) imim2i k (` (ZZ>) N) r19.20si (br K (<_) (cv j)) anim2i j (` (ZZ>) M) r19.22si syl)) thm (clim0cvg5z ((j k) (A j) (A k) (F j) (F k) (K j) (K k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. K (ZZ)) (br F (~~>) (0))) (/\ (e. A (RR)) (br (0) (<) A))) (E.e. j (` (ZZ>) M) (/\ (br K (<_) (cv j)) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (` F (cv k))) (<) A)))))) (0cn M K (0) (CC) F A j k N climcvgc5z mp3anl3 (e. M (ZZ)) (e. K (ZZ)) (br F (~~>) (0)) 3simpa (/\ (e. A (RR)) (br (0) (<) A)) adantr (/\/\ (br F (~~>) (0)) (e. A (RR)) (br (0) (<) A)) id 3expb (e. K (ZZ)) adantll (e. M (ZZ)) 3adantl1 sylanc (` F (cv k)) subid1t (abs) fveq2d (<) A breq1d biimpa (br (cv j) (<_) (cv k)) imim2i k (` (ZZ>) N) r19.20si (br K (<_) (cv j)) anim2i j (` (ZZ>) M) r19.22si syl)) thm (climnn0 ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) ((climnn0.1 (e. F (V)))) (-> (/\ (e. A (CC)) (A.e. k (NN0) (e. (` F (cv k)) (CC)))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN0) (A.e. k (NN0) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (climnn0.1 0z 0z A k x j clim4 (cv j) elnn0z (e. (cv k) (ZZ)) (br (0) (<_) (cv k)) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))) impexp (cv k) elnn0z (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))) imbi1i (br (0) (<_) (cv k)) (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) impexp (e. (cv k) (ZZ)) imbi2i 3bitr4 ralbii2 anbi12i (e. (cv j) (ZZ)) (br (0) (<_) (cv j)) (A.e. k (ZZ) (-> (/\ (br (0) (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))) anass bitr rexbii2 (br (0) (<) (cv x)) imbi2i x (RR) ralbii syl6bbr (cv k) elnn0z (br (0) (<_) (cv k)) anidm (e. (cv k) (ZZ)) anbi2i bitr4 (e. (` F (cv k)) (CC)) imbi1i (e. (cv k) (ZZ)) (/\ (br (0) (<_) (cv k)) (br (0) (<_) (cv k))) (e. (` F (cv k)) (CC)) impexp bitr ralbii2 sylan2b)) thm (climnn ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) ((climnn0.1 (e. F (V)))) (-> (/\ (e. A (CC)) (A.e. k (NN) (e. (` F (cv k)) (CC)))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (climnn0.1 1z 1z A k x j clim4 (cv j) elnnz1 (e. (cv k) (ZZ)) (br (1) (<_) (cv k)) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))) impexp (cv k) elnnz1 (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))) imbi1i (br (1) (<_) (cv k)) (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) impexp (e. (cv k) (ZZ)) imbi2i 3bitr4 ralbii2 anbi12i (e. (cv j) (ZZ)) (br (1) (<_) (cv j)) (A.e. k (ZZ) (-> (/\ (br (1) (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))) anass bitr rexbii2 (br (0) (<) (cv x)) imbi2i x (RR) ralbii syl6bbr (cv k) elnnz1 (br (1) (<_) (cv k)) anidm (e. (cv k) (ZZ)) anbi2i bitr4 (e. (` F (cv k)) (CC)) imbi1i (e. (cv k) (ZZ)) (/\ (br (1) (<_) (cv k)) (br (1) (<_) (cv k))) (e. (` F (cv k)) (CC)) impexp bitr ralbii2 sylan2b)) thm (clim0nn ((j k) (j x) (F j) (k x) (F k) (F x)) ((climnn0.1 (e. F (V)))) (-> (A.e. k (NN) (e. (` F (cv k)) (CC))) (<-> (br F (~~>) (0)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (` F (cv k))) (<) (cv x))))))))) (0cn climnn0.1 (0) k x j climnn mpan k (NN) (e. (` F (cv k)) (CC)) hbra1 k (NN) (e. (` F (cv k)) (CC)) ra4 imp (` F (cv k)) subid1t (abs) fveq2d (<) (cv x) breq1d (br (cv j) (<_) (cv k)) imbi2d syl ralbida j (NN) rexbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv bitrd)) thm (climnn0le ((j k) (j x) (j y) (F j) (k x) (k y) (F k) (x y) (F x) (F y) (A j) (A k) (A x) (A y)) ((climnn0.1 (e. F (V)))) (-> (/\ (e. A (CC)) (A.e. k (NN0) (e. (` F (cv k)) (CC)))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN0) (A.e. k (NN0) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<_) (cv x))))))))) (climnn0.1 A k x j climnn0 (e. A (CC)) k ax-17 k (NN0) (e. (` F (cv k)) (CC)) hbra1 hban (e. (cv x) (RR)) k ax-17 hban (` (abs) (opr (` F (cv k)) (-) A)) (cv x) ltlet (` F (cv k)) A subclt ancoms (opr (` F (cv k)) (-) A) absclt syl k (NN0) (e. (` F (cv k)) (CC)) ra4 imp sylan2 anassrs (e. (cv x) (RR)) adantlr (/\ (e. A (CC)) (A.e. k (NN0) (e. (` F (cv k)) (CC)))) (e. (cv x) (RR)) pm3.27 (e. (cv k) (NN0)) adantr sylanc (br (cv j) (<_) (cv k)) imim2d r19.20da j (NN0) r19.22sdv (br (0) (<) (cv x)) imim2d r19.20dva sylbid (cv y) rehalfclt (cv x) (opr (cv y) (/) (2)) (0) (<) breq2 (cv x) (opr (cv y) (/) (2)) (` (abs) (opr (` F (cv k)) (-) A)) (<_) breq2 (br (cv j) (<_) (cv k)) imbi2d j (NN0) k (NN0) rexralbidv imbi12d (RR) rcla4v syl (br (0) (<) (cv y)) (/\ (e. A (CC)) (A.e. k (NN0) (e. (` F (cv k)) (CC)))) ad2antrr (cv y) halfpos2t biimpa (/\ (e. A (CC)) (A.e. k (NN0) (e. (` F (cv k)) (CC)))) adantr (/\ (e. (cv y) (RR)) (br (0) (<) (cv y))) k ax-17 (e. A (CC)) k ax-17 k (NN0) (e. (` F (cv k)) (CC)) hbra1 hban hban (cv y) halfpost biimpa (/\ (e. A (CC)) (e. (` F (cv k)) (CC))) adantr (` (abs) (opr (` F (cv k)) (-) A)) (opr (cv y) (/) (2)) (cv y) lelttrt 3expb (cv y) rehalfclt (br (0) (<) (cv y)) adantr (e. (cv y) (RR)) (br (0) (<) (cv y)) pm3.26 jca sylan2 ancoms (` F (cv k)) A subclt ancoms (opr (` F (cv k)) (-) A) absclt syl sylan2 mpan2d (br (cv j) (<_) (cv k)) imim2d k (NN0) (e. (` F (cv k)) (CC)) ra4 imp (e. A (CC)) anim2i anassrs sylan2 anassrs r19.20da j (NN0) r19.22sdv (br (0) (<) (opr (cv y) (/) (2))) imim2d mpid syld exp31 com4t r19.21adv climnn0.1 A k y j climnn0 sylibrd impbid)) thm (climfnn ((f j) (f k) (f x) (A f) (j k) (j x) (A j) (k x) (A k) (A x) (F f) (F j) (F k) (F x)) () (-> (/\ (:--> F (NN) (CC)) (e. A (CC))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (nnex F (NN) (CC) (V) fex mpan2 (e. A (CC)) adantr (cv f) F (NN) (CC) feq1 (e. A (CC)) anbi1d (cv f) F (~~>) A breq1 (cv f) F (cv k) fveq1 (-) A opreq1d (abs) fveq2d (<) (cv x) breq1d (br (cv j) (<_) (cv k)) imbi2d j (NN) k (NN) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv bibi12d imbi12d f visset A k x j climnn (cv f) (NN) (CC) (cv k) ffvrn r19.21aiva sylan2 ancoms (V) vtoclg mpcom)) thm (climcvgnn ((j k) (A j) (A k) (B j) (B k) (F j) (F k)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))) (1z (1) A C F B j k (1) climcvg4uzOLD mpan nnuz (NN) (` (ZZ>) (1)) k (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)) raleq1 j rexeqd ax-mp sylibr)) thm (climcvgnn0 ((j k) (A j) (A k) (B j) (B k) (F j) (F k)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (NN0) (A.e. k (NN0) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))) (0z (0) A C F B j k (0) climcvg4uzOLD mpan nn0uz (NN0) (` (ZZ>) (0)) k (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)) raleq1 j rexeqd ax-mp sylibr)) thm (climconst ((j k) (j x) (A j) (k x) (A k) (A x) (F j) (F k) (F x) (M j) (M k) (M x)) ((climconst.1 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. A (CC)) (A.e. k (` (ZZ>) M) (= (` F (cv k)) A))) (br F (~~>) A)) ((cv j) M (<_) (cv k) breq1 (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) imbi1d k (` (ZZ>) M) ralbidv (ZZ) rcla4ev (e. M (ZZ)) (e. A (CC)) pm3.26 (A.e. k (` (ZZ>) M) (= (` F (cv k)) A)) (br (0) (<) (cv x)) ad2antrr (` F (cv k)) A (-) A opreq1 A subidt sylan9eqr (abs) fveq2d abs0 syl6eq (<) (cv x) breq1d biimpar an1rs ex (br M (<_) (cv k)) a1dd k (` (ZZ>) M) r19.20sdv imp an1rs (e. M (ZZ)) adantlll sylanc exp31 (e. (cv x) (RR)) a1dd r19.21adv imp anasss climconst.1 M A k x j clim3zOLD A (CC) (` F (cv k)) eleq1a k (` (ZZ>) M) r19.20sdv imdistani sylan2 mpbird 3impb)) thm (climconstyOLD ((j k) (j x) (A j) (k x) (A k) (A x) (F j) (F k) (F x) (N j) (N k) (N x)) ((climcnst.1 (e. F (V))) (climcnst.2 (e. N (ZZ)))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (br N (<_) (cv k)) (= (` F (cv k)) A)))) (br F (~~>) A)) ((br N (<_) (cv k)) (br N (<_) (cv k)) pm3.26 (/\ (e. A (CC)) (br (0) (<) (cv x))) a1i (` F (cv k)) A (-) A opreq1 A subidt sylan9eqr (abs) fveq2d abs0 syl6eq (<) (cv x) breq1d biimpar an1rs ex imim12d k (ZZ) r19.20sdv imp an1rs climcnst.2 jctil (cv j) N (<_) (cv k) breq1 (br N (<_) (cv k)) anbi2d (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) imbi1d k (ZZ) ralbidv (ZZ) rcla4ev syl exp31 (e. (cv x) (RR)) a1dd r19.21adv imp A (CC) (` F (cv k)) eleq1a (br N (<_) (cv k)) imim2d k (ZZ) r19.20sdv imdistani climcnst.1 climcnst.2 A k x j clim3 syl mpbird)) thm (climconst2 ((A k)) () (-> (e. A (CC)) (br (X. (ZZ) ({} A)) (~~>) A)) (A (CC) (cv k) (ZZ) fvconst2g ex (br (0) (<_) (cv k)) a1dd r19.21aiv zex A snex xpex 0z A k climconstyOLD mpdan)) thm (clim0 () () (br (X. (ZZ) ({} (0))) (~~>) (0)) (0cn (0) climconst2 ax-mp)) thm (climunii ((j k) (j m) (A j) (k m) (A k) (A m) (B j) (B k) (B m) (F j) (F k) (F m)) ((climuni.1 (e. A (V))) (climuni.2 (e. B (V))) (climunii.3 (/\ (br F (~~>) A) (br F (~~>) B)))) (= A B) ((cv k) (cv m) j z2get rgen2 climuni.1 climunii.3 pm3.26i A (V) F climcl mp2an climuni.2 climunii.3 pm3.27i B (V) F climcl mp2an subcl abscl 2re 2pos divgt0i2 climuni.1 climunii.3 pm3.26i A (V) F climcl mp2an climuni.2 climunii.3 pm3.27i B (V) F climcl mp2an subcl abscl 2re 2re 2pos gt0ne0i redivcl climuni.1 climunii.3 pm3.26i A (V) F (opr (` (abs) (opr A (-) B)) (/) (2)) k j climcvgc1 mpanl12 mpan climuni.1 climunii.3 pm3.26i A (V) F climcl mp2an climuni.2 climunii.3 pm3.27i B (V) F climcl mp2an subcl abscl 2re 2re 2pos gt0ne0i redivcl climuni.2 climunii.3 pm3.27i B (V) F (opr (` (abs) (opr A (-) B)) (/) (2)) m j climcvgc1 mpanl12 mpan jca j (ZZ) (-> (br (cv k) (<_) (cv j)) (/\ (e. (` F (cv j)) (CC)) (br (` (abs) (opr (` F (cv j)) (-) A)) (<) (opr (` (abs) (opr A (-) B)) (/) (2))))) (-> (br (cv m) (<_) (cv j)) (/\ (e. (` F (cv j)) (CC)) (br (` (abs) (opr (` F (cv j)) (-) B)) (<) (opr (` (abs) (opr A (-) B)) (/) (2))))) r19.26 climuni.1 climunii.3 pm3.26i A (V) F climcl mp2an climuni.2 climunii.3 pm3.27i B (V) F climcl mp2an subcl abscl ltnr (e. (` F (cv j)) (CC)) (br (` (abs) (opr (` F (cv j)) (-) A)) (<) (opr (` (abs) (opr A (-) B)) (/) (2))) (br (` (abs) (opr (` F (cv j)) (-) B)) (<) (opr (` (abs) (opr A (-) B)) (/) (2))) anandi climuni.1 climunii.3 pm3.26i A (V) F climcl mp2an (` F (cv j)) A abssubt mpan2 (<) (opr (` (abs) (opr A (-) B)) (/) (2)) breq1d (br (` (abs) (opr (` F (cv j)) (-) B)) (<) (opr (` (abs) (opr A (-) B)) (/) (2))) anbi1d climuni.1 climunii.3 pm3.26i A (V) F climcl mp2an climuni.2 climunii.3 pm3.27i B (V) F climcl mp2an subcl abscl climuni.1 climunii.3 pm3.26i A (V) F climcl mp2an climuni.2 climunii.3 pm3.27i B (V) F climcl mp2an A B (` F (cv j)) (` (abs) (opr A (-) B)) abs3lemt mpanl12 mpan2 sylbid imp sylbir mto (br (cv k) (<_) (cv j)) (/\ (e. (` F (cv j)) (CC)) (br (` (abs) (opr (` F (cv j)) (-) A)) (<) (opr (` (abs) (opr A (-) B)) (/) (2)))) (br (cv m) (<_) (cv j)) (/\ (e. (` F (cv j)) (CC)) (br (` (abs) (opr (` F (cv j)) (-) B)) (<) (opr (` (abs) (opr A (-) B)) (/) (2)))) prth mtoi j (ZZ) r19.20si sylbir m (ZZ) r19.22si k (ZZ) r19.22si k (ZZ) m (ZZ) (A.e. j (ZZ) (-> (br (cv k) (<_) (cv j)) (/\ (e. (` F (cv j)) (CC)) (br (` (abs) (opr (` F (cv j)) (-) A)) (<) (opr (` (abs) (opr A (-) B)) (/) (2)))))) (A.e. j (ZZ) (-> (br (cv m) (<_) (cv j)) (/\ (e. (` F (cv j)) (CC)) (br (` (abs) (opr (` F (cv j)) (-) B)) (<) (opr (` (abs) (opr A (-) B)) (/) (2)))))) reeanv j (ZZ) (/\ (br (cv k) (<_) (cv j)) (br (cv m) (<_) (cv j))) ralnex m (ZZ) rexbii m (ZZ) (E.e. j (ZZ) (/\ (br (cv k) (<_) (cv j)) (br (cv m) (<_) (cv j)))) rexnal bitr k (ZZ) rexbii k (ZZ) (A.e. m (ZZ) (E.e. j (ZZ) (/\ (br (cv k) (<_) (cv j)) (br (cv m) (<_) (cv j))))) rexnal bitr 3imtr3 3syl mt2 (opr A (-) B) (0) df-ne climuni.1 climunii.3 pm3.26i A (V) F climcl mp2an climuni.2 climunii.3 pm3.27i B (V) F climcl mp2an subcl absgt0 bitr3 con1bii mpbi climuni.1 climunii.3 pm3.26i A (V) F climcl mp2an climuni.2 climunii.3 pm3.27i B (V) F climcl mp2an subeq0 mpbi)) thm (climuni () ((climuni.1 (e. A (V))) (climuni.2 (e. B (V)))) (-> (/\ (br F (~~>) A) (br F (~~>) B)) (= A B)) (A (if (/\ (br F (~~>) A) (br F (~~>) B)) A (0)) B eqeq1 B (if (/\ (br F (~~>) A) (br F (~~>) B)) B (0)) (if (/\ (br F (~~>) A) (br F (~~>) B)) A (0)) eqeq2 climuni.1 0cn elisseti (/\ (br F (~~>) A) (br F (~~>) B)) ifex climuni.2 0cn elisseti (/\ (br F (~~>) A) (br F (~~>) B)) ifex A (if (/\ (br F (~~>) A) (br F (~~>) B)) A (0)) F (~~>) breq2 (br F (~~>) B) anbi1d B (if (/\ (br F (~~>) A) (br F (~~>) B)) B (0)) F (~~>) breq2 (br F (~~>) (if (/\ (br F (~~>) A) (br F (~~>) B)) A (0))) anbi2d F (if (/\ (br F (~~>) A) (br F (~~>) B)) F (X. (ZZ) ({} (0)))) (~~>) (if (/\ (br F (~~>) A) (br F (~~>) B)) A (0)) breq1 F (if (/\ (br F (~~>) A) (br F (~~>) B)) F (X. (ZZ) ({} (0)))) (~~>) (if (/\ (br F (~~>) A) (br F (~~>) B)) B (0)) breq1 anbi12d (0) (if (/\ (br F (~~>) A) (br F (~~>) B)) A (0)) (X. (ZZ) ({} (0))) (~~>) breq2 (br (X. (ZZ) ({} (0))) (~~>) (0)) anbi1d (0) (if (/\ (br F (~~>) A) (br F (~~>) B)) B (0)) (X. (ZZ) ({} (0))) (~~>) breq2 (br (X. (ZZ) ({} (0))) (~~>) (if (/\ (br F (~~>) A) (br F (~~>) B)) A (0))) anbi2d (X. (ZZ) ({} (0))) (if (/\ (br F (~~>) A) (br F (~~>) B)) F (X. (ZZ) ({} (0)))) (~~>) (if (/\ (br F (~~>) A) (br F (~~>) B)) A (0)) breq1 (X. (ZZ) ({} (0))) (if (/\ (br F (~~>) A) (br F (~~>) B)) F (X. (ZZ) ({} (0)))) (~~>) (if (/\ (br F (~~>) A) (br F (~~>) B)) B (0)) breq1 anbi12d clim0 clim0 pm3.2i elimhyp3v climunii dedth2v)) thm (climeu ((A y) (x y) (F x) (F y)) ((climreu.1 (e. A (V)))) (-> (br F (~~>) A) (E! x (br F (~~>) (cv x)))) (climreu.1 (cv y) A F (~~>) breq2 cla4ev y visset x visset F climuni y x gen2 jctir (br F (~~>) (cv x)) y ax-17 (br F (~~>) (cv y)) x ax-17 (cv x) (cv y) F (~~>) breq2 cbveu (cv y) (cv x) F (~~>) breq2 eu4 bitr sylibr)) thm (climreu ((F x)) ((climreu.1 (e. A (V)))) (-> (br F (~~>) A) (E!e. x (CC) (br F (~~>) (cv x)))) (climreu.1 F x climeu x visset (cv x) (V) F climcl mpan pm4.71ri x eubii x (CC) (br F (~~>) (cv x)) df-reu bitr4 sylib)) thm (2climnn ((x y) (k x) (j x) (A x) (k y) (j y) (A y) (j k) (A k) (A j) (F j) (F k) (F x) (F y) (G x) (G y) (G k) (G j)) ((2climnn.1 (e. G (V)))) (-> (/\ (/\ (e. A (CC)) (A.e. k (NN) (/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC))))) (/\ (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` G (cv k)) (-) (` F (cv k)))) (<) (cv x))))))) (br F (~~>) A))) (br G (~~>) A)) ((cv y) rehalfclt (cv x) (opr (cv y) (/) (2)) (0) (<) breq2 (cv x) (opr (cv y) (/) (2)) (cv x) (opr (cv y) (/) (2)) (+) opreq12 anidms (` (abs) (opr (` G (cv k)) (-) A)) (<) breq2d (br (cv j) (<_) (cv k)) imbi2d j (NN) k (NN) rexralbidv imbi12d (RR) rcla4v syl A (CC) F (cv x) j k climcvgnn (A.e. k (NN) (/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)))) adantllr anassrs (` (abs) (opr (` G (cv k)) (-) (` F (cv k)))) (` (abs) (opr (` F (cv k)) (-) A)) (cv x) (cv x) lt2addt (` G (cv k)) (` F (cv k)) subclt ancoms (opr (` G (cv k)) (-) (` F (cv k))) absclt syl (e. A (CC)) adantl (` F (cv k)) A subclt ancoms (opr (` F (cv k)) (-) A) absclt syl (e. (` G (cv k)) (CC)) adantrr jca (e. (cv x) (RR)) adantlr (e. (cv x) (RR)) (e. (cv x) (RR)) pm3.2 pm2.43i (e. A (CC)) (/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC))) ad2antlr sylanc (` G (cv k)) (` F (cv k)) A npncant 3com13 3expb (abs) fveq2d (opr (` G (cv k)) (-) (` F (cv k))) (opr (` F (cv k)) (-) A) abstrit (` G (cv k)) (` F (cv k)) subclt ancoms (e. A (CC)) adantl (` F (cv k)) A subclt ancoms (e. (` G (cv k)) (CC)) adantrr sylanc eqbrtrrd (e. (cv x) (RR)) adantlr (` (abs) (opr (` G (cv k)) (-) A)) (opr (` (abs) (opr (` G (cv k)) (-) (` F (cv k)))) (+) (` (abs) (opr (` F (cv k)) (-) A))) (opr (cv x) (+) (cv x)) lelttrt (` G (cv k)) A subclt ancoms (opr (` G (cv k)) (-) A) absclt syl (e. (cv x) (RR)) adantlr (e. (` F (cv k)) (CC)) adantrl (` (abs) (opr (` G (cv k)) (-) (` F (cv k)))) (` (abs) (opr (` F (cv k)) (-) A)) axaddrcl (` G (cv k)) (` F (cv k)) subclt ancoms (opr (` G (cv k)) (-) (` F (cv k))) absclt syl (e. A (CC)) adantl (` F (cv k)) A subclt ancoms (opr (` F (cv k)) (-) A) absclt syl (e. (` G (cv k)) (CC)) adantrr sylanc (e. (cv x) (RR)) adantlr (cv x) (cv x) axaddrcl anidms (e. A (CC)) (/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC))) ad2antlr syl3anc mpand syld (br (cv j) (<_) (cv k)) imim2d ex k (NN) r19.20sdv imp an1rs k (NN) (-> (br (cv j) (<_) (cv k)) (/\ (br (` (abs) (opr (` G (cv k)) (-) (` F (cv k)))) (<) (cv x)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` G (cv k)) (-) A)) (<) (opr (cv x) (+) (cv x)))) r19.20 syl (br F (~~>) A) adantlr (br (0) (<) (cv x)) adantr j (NN) r19.22sdv j k (br (` (abs) (opr (` G (cv k)) (-) (` F (cv k)))) (<) (cv x)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)) cvgannn syl5ib mpan2d ex a2d r19.20dva ex com23 imp32 syl5 (cv y) halfpos2t bicomd (cv y) recnt (cv y) 2halvest syl (` (abs) (opr (` G (cv k)) (-) A)) (<) breq2d (br (cv j) (<_) (cv k)) imbi2d j (NN) k (NN) rexralbidv imbi12d sylibd com12 r19.21aiv 2climnn.1 A k y j climnn (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) pm3.27 k (NN) r19.20si sylan2 (/\ (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` G (cv k)) (-) (` F (cv k)))) (<) (cv x))))))) (br F (~~>) A)) adantr mpbird)) thm (climshft ((j t) (j m) (j n) (j x) (A j) (m t) (n t) (t x) (A t) (m n) (m x) (A m) (n x) (A n) (A x) (M j) (M t) (M m) (M n) (M x) (F j) (F t) (F m) (F n) (F x)) ((climshft.1 (e. F (V))) (climshft.2 (e. M (ZZ)))) (-> (e. A (CC)) (<-> (br (opr F (shift) M) (~~>) A) (br F (~~>) A))) (F (shift) M oprex A x j t clim1 climshft.2 zre (cv j) M (cv n) lesubaddt mp3an2 (cv j) zret (cv n) zret syl2an (A.e. t (ZZ) (-> (br (cv j) (<_) (cv t)) (/\ (e. (` (opr F (shift) M) (cv t)) (CC)) (br (` (abs) (opr (` (opr F (shift) M) (cv t)) (-) A)) (<) (cv x))))) adantr climshft.2 (cv n) M zaddclt mpan2 (cv t) (opr (cv n) (+) M) (cv j) (<_) breq2 (cv t) (opr (cv n) (+) M) (opr F (shift) M) fveq2 (CC) eleq1d (cv t) (opr (cv n) (+) M) (opr F (shift) M) fveq2 (-) A opreq1d (abs) fveq2d (<) (cv x) breq1d anbi12d imbi12d (ZZ) rcla4v syl imp (e. (cv j) (ZZ)) adantll sylbid imp climshft.2 (cv n) M zaddclt mpan2 (opr (cv n) (+) M) zcnt climshft.2 climshft.1 M (ZZ) (opr (cv n) (+) M) shftvalt mpan 3syl (cv n) zcnt climshft.2 zre recn (cv n) M pncant mpan2 syl F fveq2d eqtr2d (e. (cv j) (ZZ)) adantl (A.e. t (ZZ) (-> (br (cv j) (<_) (cv t)) (/\ (e. (` (opr F (shift) M) (cv t)) (CC)) (br (` (abs) (opr (` (opr F (shift) M) (cv t)) (-) A)) (<) (cv x))))) (br (opr (cv j) (-) M) (<_) (cv n)) ad2antrr (` F (cv n)) (` (opr F (shift) M) (opr (cv n) (+) M)) (CC) eleq1 (` F (cv n)) (` (opr F (shift) M) (opr (cv n) (+) M)) (-) A opreq1 (abs) fveq2d (<) (cv x) breq1d anbi12d syl mpbird exp41 com23 r19.21adv climshft.2 (cv j) M zsubclt mpan2 jctild (cv m) (opr (cv j) (-) M) (<_) (cv n) breq1 (/\ (e. (` F (cv n)) (CC)) (br (` (abs) (opr (` F (cv n)) (-) A)) (<) (cv x))) imbi1d n (ZZ) ralbidv (ZZ) rcla4ev syl6 r19.23aiv climshft.2 zre (cv m) M (cv t) leaddsubt mp3an2 (cv m) zret (cv t) zret syl2an (A.e. n (ZZ) (-> (br (cv m) (<_) (cv n)) (/\ (e. (` F (cv n)) (CC)) (br (` (abs) (opr (` F (cv n)) (-) A)) (<) (cv x))))) adantr climshft.2 (cv t) M zsubclt mpan2 (cv n) (opr (cv t) (-) M) (cv m) (<_) breq2 (cv n) (opr (cv t) (-) M) F fveq2 (CC) eleq1d (cv n) (opr (cv t) (-) M) F fveq2 (-) A opreq1d (abs) fveq2d (<) (cv x) breq1d anbi12d imbi12d (ZZ) rcla4v syl imp (e. (cv m) (ZZ)) adantll sylbid imp (cv t) zcnt climshft.2 climshft.1 M (ZZ) (cv t) shftvalt mpan syl (e. (cv m) (ZZ)) adantl (A.e. n (ZZ) (-> (br (cv m) (<_) (cv n)) (/\ (e. (` F (cv n)) (CC)) (br (` (abs) (opr (` F (cv n)) (-) A)) (<) (cv x))))) (br (opr (cv m) (+) M) (<_) (cv t)) ad2antrr (` (opr F (shift) M) (cv t)) (` F (opr (cv t) (-) M)) (CC) eleq1 (` (opr F (shift) M) (cv t)) (` F (opr (cv t) (-) M)) (-) A opreq1 (abs) fveq2d (<) (cv x) breq1d anbi12d syl mpbird exp41 com23 r19.21adv climshft.2 (cv m) M zaddclt mpan2 jctild (cv j) (opr (cv m) (+) M) (<_) (cv t) breq1 (/\ (e. (` (opr F (shift) M) (cv t)) (CC)) (br (` (abs) (opr (` (opr F (shift) M) (cv t)) (-) A)) (<) (cv x))) imbi1d t (ZZ) ralbidv (ZZ) rcla4ev syl6 r19.23aiv impbi (br (0) (<) (cv x)) imbi2i x (RR) ralbii syl6bb climshft.1 A x m n clim1 bitr4d)) thm (climres ((j t) (j x) (A j) (t x) (A t) (A x) (j k) (M j) (k t) (k x) (M k) (M t) (M x) (F j) (F t) (F x)) ((climshft.1 (e. F (V))) (climshft.2 (e. M (ZZ)))) (-> (e. A (CC)) (<-> (br (|` F (` (ZZ>) M)) (~~>) A) (br F (~~>) A))) (climshft.1 F (V) ({e.|} k (ZZ) (br M (<_) (cv k))) resexg ax-mp climshft.2 A x j t clim3b (cv k) (cv t) M (<_) breq2 (ZZ) elrab (cv t) ({e.|} k (ZZ) (br M (<_) (cv k))) F fvres sylbir (CC) eleq1d (cv k) (cv t) M (<_) breq2 (ZZ) elrab (cv t) ({e.|} k (ZZ) (br M (<_) (cv k))) F fvres sylbir (-) A opreq1d (abs) fveq2d (<) (cv x) breq1d anbi12d ex (br (cv j) (<_) (cv t)) adantrd pm5.74d ralbiia j (ZZ) rexbii (br (0) (<) (cv x)) imbi2i x (RR) ralbii syl6bb climshft.1 climshft.2 A x j t clim3b bitr4d climshft.2 M k uzvalt ax-mp (` (ZZ>) M) ({e.|} k (ZZ) (br M (<_) (cv k))) F reseq2 ax-mp (~~>) A breq1i syl5bb)) thm (climuz0 () ((climuz0.1 (e. M (ZZ)))) (br (X. (` (ZZ>) M) ({} (0))) (~~>) (0)) (clim0 0cn elisseti snnz ({} (0)) (` (ZZ>) M) dmxp ax-mp (dom (X. (` (ZZ>) M) ({} (0)))) (` (ZZ>) M) (X. (ZZ) ({} (0))) reseq2 ax-mp 0cn elisseti (ZZ) fconst (X. (ZZ) ({} (0))) (ZZ) ({} (0)) ffun ax-mp M uzssz ({} (0)) ssid (` (ZZ>) M) (ZZ) ({} (0)) ({} (0)) ssxp mp2an (X. (ZZ) ({} (0))) (X. (` (ZZ>) M) ({} (0))) funssres mp2an eqtr3 (~~>) (0) breq1i 0cn zex (0) snex xpex climuz0.1 (0) climres ax-mp bitr3 mpbir)) thm (climrecl ((j m) (A j) (A m) (j k) (F j) (k m) (F k) (F m) (M j) (M k) (M m)) ((climrecl.1 (e. A (V)))) (-> (/\/\ (e. M (ZZ)) (br F (~~>) A) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR)))) (e. A (RR))) (climrecl.1 A (V) F climcl mpan (e. M (ZZ)) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR))) 3ad2ant2 A imclt recnd (` (Im) A) absge0t syl (/\/\ (e. M (ZZ)) (br F (~~>) A) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR)))) adantl (cv m) (cv j) (cv j) (<_) breq2 (cv m) (cv j) F fveq2 (-) A opreq1d (abs) fveq2d (<) (` (abs) (` (Im) A)) breq1d negbid anbi12d (ZZ) rcla4ev (cv j) M eluzelz (e. (` F (cv j)) (RR)) (e. A (CC)) ad2antlr (cv j) M eluzelz (cv j) zret (cv j) leidt 3syl (e. (` F (cv j)) (RR)) (e. A (CC)) ad2antlr A imclt recnd (` (Im) A) absnegt syl (e. (` F (cv j)) (RR)) adantl (` F (cv j)) reim0t (-) (` (Im) A) opreq1d (` (Im) A) df-neg syl6reqr (e. A (CC)) adantr (` F (cv j)) A imsubt (` F (cv j)) recnt sylan eqtr4d (abs) fveq2d eqtr3d (` F (cv j)) A subclt (` F (cv j)) recnt sylan (opr (` F (cv j)) (-) A) absimlet syl eqbrtrd (` (abs) (` (Im) A)) (` (abs) (opr (` F (cv j)) (-) A)) lenltt A imclt recnd (` (Im) A) absclt syl (e. (` F (cv j)) (RR)) adantl (` F (cv j)) A subclt (` F (cv j)) recnt sylan (opr (` F (cv j)) (-) A) absclt syl sylanc mpbid (e. (cv j) (` (ZZ>) M)) adantlr jca sylanc m (ZZ) (br (cv j) (<_) (cv m)) (br (` (abs) (opr (` F (cv m)) (-) A)) (<) (` (abs) (` (Im) A))) rexanali sylib (cv k) (cv j) F fveq2 (RR) eleq1d (` (ZZ>) M) rcla4cva (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR))) (e. (cv j) (` (ZZ>) M)) pm3.27 jca sylan an1rs nrexdv (e. M (ZZ)) adantll (br F (~~>) A) 3adantl2 A imclt recnd (` (Im) A) absclt syl (e. M (ZZ)) (br F (~~>) A) ad2antrl M A (CC) F (` (abs) (` (Im) A)) j m climcvg2zbOLD anassrs anassrs ex mpdan anassrs an1rs (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR))) 3adantl3 mtod jca A imclt recnd (` (Im) A) absclt syl 0re (0) (` (abs) (` (Im) A)) eqleltt mpan syl (/\/\ (e. M (ZZ)) (br F (~~>) A) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR)))) adantl mpbird eqcomd A imclt recnd (` (Im) A) abs00t syl (/\/\ (e. M (ZZ)) (br F (~~>) A) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR)))) adantl mpbid A reim0bt (/\/\ (e. M (ZZ)) (br F (~~>) A) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR)))) adantl mpbird mpdan)) thm (climfnrcl ((F k)) ((climfnrcl.1 (e. A (V))) (climfnrcl.2 (:--> F (NN) (RR))) (climfnrcl.3 (br F (~~>) A))) (e. A (RR)) (1z climfnrcl.3 (cv k) elnnuz climfnrcl.2 (cv k) ffvrni sylbir rgen climfnrcl.1 (1) F k climrecl mp3an)) thm (climge0 ((j m) (A j) (A m) (j k) (F j) (k m) (F k) (F m) (M j) (M k) (M m)) ((climge0.1 (e. A (V)))) (-> (/\/\ (e. M (ZZ)) (br F (~~>) A) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k)))))) (br (0) (<_) A)) (climge0.1 M F k climrecl (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k))) pm3.26 k (` (ZZ>) M) r19.20si syl3an3 A lt0neg1t (/\ (e. M (ZZ)) (br F (~~>) A)) adantl M A (V) F (-u A) j m climcvg2zbOLD climge0.1 (br F (~~>) A) jctl sylanr1 anassrs anassrs ex A renegclt sylan2 sylbid (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k))))) 3adantl3 (cv m) (cv j) (cv j) (<_) breq2 (cv m) (cv j) F fveq2 (-) A opreq1d (abs) fveq2d (<) (-u A) breq1d negbid anbi12d (ZZ) rcla4ev (cv j) M eluzelz (/\ (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k))))) (/\ (e. A (RR)) (br A (<) (0)))) adantl (cv j) M eluzelz (cv j) zret (cv j) leidt 3syl (/\ (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k))))) (/\ (e. A (RR)) (br A (<) (0)))) adantl (-u A) (` F (cv j)) addge02t A renegclt sylan biimpa anasss ancoms (br A (<) (0)) adantrr (opr (` F (cv j)) (+) (-u A)) absidt (` F (cv j)) (-u A) axaddrcl A renegclt sylan2 (br (0) (<_) (` F (cv j))) (br A (<) (0)) ad2ant2r (` F (cv j)) (-u A) addge0t an4s A renegclt (br A (<) (0)) adantr 0re A (0) ltlet mpan2 A le0neg1t sylibd imp jca sylan2 sylanc (` F (cv j)) A negsubt (` F (cv j)) recnt A recnt syl2an (abs) fveq2d (br (0) (<_) (` F (cv j))) (br A (<) (0)) ad2ant2r eqtr3d breqtrd (-u A) (` (abs) (opr (` F (cv j)) (-) A)) lenltt A renegclt (e. (` F (cv j)) (RR)) adantl (` F (cv j)) A resubclt recnd (opr (` F (cv j)) (-) A) absclt syl sylanc (br (0) (<_) (` F (cv j))) (br A (<) (0)) ad2ant2r mpbid (cv k) (cv j) F fveq2 (RR) eleq1d (cv k) (cv j) F fveq2 (0) (<_) breq2d anbi12d (` (ZZ>) M) rcla4cva sylan an1rs jca sylanc m (ZZ) (br (cv j) (<_) (cv m)) (br (` (abs) (opr (` F (cv m)) (-) A)) (<) (-u A)) rexanali sylib nrexdv exp32 (e. M (ZZ)) (br F (~~>) A) 3ad2ant3 imp pm2.65d 0re (0) A lenltt mpan (/\/\ (e. M (ZZ)) (br F (~~>) A) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k)))))) adantl mpbird mpdan)) thm (climaddlem1 ((ph t) (A t) (B t) (k t) (F k) (F t) (G k) (G t) (H k) (H t) (M k) (M t)) ((climadd.1 (e. F (V))) (climadd.2 (e. G (V))) (climadd.3 (e. H (V))) (climadd.4 (e. A (V))) (climadd.5 (e. B (V))) (climaddlem.6 (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))))))) (-> (/\ (e. (cv t) (` (ZZ>) M)) ph) (/\/\ (e. (` F (cv t)) (CC)) (e. (` G (cv t)) (CC)) (= (` H (cv t)) (opr (` F (cv t)) (+) (` G (cv t)))))) ((cv k) (cv t) F fveq2 (CC) eleq1d (cv k) (cv t) G fveq2 (CC) eleq1d (cv k) (cv t) H fveq2 (cv k) (cv t) F fveq2 (cv k) (cv t) G fveq2 (+) opreq12d eqeq12d 3anbi123d (` (ZZ>) M) rcla4va (/\ (br F (~~>) A) (br G (~~>) B)) adantrl climaddlem.6 sylan2b)) thm (climaddlem2 ((F k) (G k) (H k) (M k)) ((climadd.1 (e. F (V))) (climadd.2 (e. G (V))) (climadd.3 (e. H (V))) (climadd.4 (e. A (V))) (climadd.5 (e. B (V))) (climaddlem.6 (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))))))) (-> ph (/\ (e. A (CC)) (e. B (CC)))) (climaddlem.6 climadd.4 A (V) F climcl mpan climadd.5 B (V) G climcl mpan anim12i (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))) adantr sylbi)) thm (climaddlem3 ((f t) (f u) (f v) (f ph) (t u) (t v) (ph t) (u v) (ph u) (ph v) (f g) (f h) (f t) (f u) (f v) (A f) (g h) (g t) (g u) (g v) (A g) (h t) (h u) (h v) (A h) (t u) (t v) (A t) (u v) (A u) (A v) (B f) (B g) (B h) (B t) (B u) (B v) (f k) (F f) (g k) (F g) (k t) (k u) (k v) (F k) (F t) (F u) (F v) (G f) (G g) (G k) (G t) (G u) (G v) (H f) (h k) (H h) (H k) (H t) (H u) (H v) (M f) (M g) (M h) (M k) (M t) (M u) (M v)) ((climadd.1 (e. F (V))) (climadd.2 (e. G (V))) (climadd.3 (e. H (V))) (climadd.4 (e. A (V))) (climadd.5 (e. B (V))) (climaddlem.6 (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))))))) (-> (/\ (e. M (ZZ)) ph) (br H (~~>) (opr A (+) B))) (climadd.4 0z (0) A (V) F (opr (cv v) (/) (2)) u t M climcvg4uzOLD mpan mpanl1 climadd.5 0z (0) B (V) G (opr (cv v) (/) (2)) f g M climcvg4uzOLD mpan mpanl1 anim12i anandirs (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))) adantlr climaddlem.6 sylanb (cv v) rehalfclt (br (0) (<) (cv v)) adantr 2re 2pos (cv v) (2) divgt0t mpanr2 mpanl2 jca sylan2 (/\ (e. (cv u) (NN0)) (e. (cv f) (NN0))) (e. (cv t) (` (ZZ>) M)) pm3.27 (cv u) (cv f) nn0addge1t (cv u) nn0ret sylan (e. (cv t) (RR)) adantr (cv u) (opr (cv u) (+) (cv f)) (cv t) letrt 3expa (e. (cv u) (RR)) (e. (cv f) (RR)) pm3.26 (cv u) (cv f) axaddrcl jca (cv u) nn0ret (cv f) nn0ret syl2an sylan mpand (cv t) M eluzelz (cv t) zret syl sylan2 (e. (cv g) (` (ZZ>) M)) adantr (cv f) (cv u) nn0addge2t ancoms (cv f) nn0ret sylan2 (e. (cv g) (RR)) adantr (cv f) (opr (cv u) (+) (cv f)) (cv g) letrt 3expa (e. (cv u) (RR)) (e. (cv f) (RR)) pm3.27 (cv u) (cv f) axaddrcl jca (cv u) nn0ret (cv f) nn0ret syl2an sylan mpand (cv g) M eluzelz (cv g) zret syl sylan2 (e. (cv t) (` (ZZ>) M)) adantlr anim12d (br (cv u) (<_) (cv t)) (br (` (abs) (opr (` F (cv t)) (-) A)) (<) (opr (cv v) (/) (2))) (br (cv f) (<_) (cv g)) (br (` (abs) (opr (` G (cv g)) (-) B)) (<) (opr (cv v) (/) (2))) prth syl9 exp4a r19.20dva (cv g) (cv t) (opr (cv u) (+) (cv f)) (<_) breq2 (cv g) (cv t) G fveq2 (-) B opreq1d (abs) fveq2d (<) (opr (cv v) (/) (2)) breq1d (br (` (abs) (opr (` F (cv t)) (-) A)) (<) (opr (cv v) (/) (2))) anbi2d imbi12d (` (ZZ>) M) rcla4v com23 a2d g (` (ZZ>) M) (br (opr (cv u) (+) (cv f)) (<_) (cv t)) (-> (br (opr (cv u) (+) (cv f)) (<_) (cv g)) (/\ (br (` (abs) (opr (` F (cv t)) (-) A)) (<) (opr (cv v) (/) (2))) (br (` (abs) (opr (` G (cv g)) (-) B)) (<) (opr (cv v) (/) (2))))) r19.21v syl5ib com12 syl6 mpid com23 (/\ ph (e. (cv v) (RR))) adantll (` (abs) (opr (` F (cv t)) (-) A)) (` (abs) (opr (` G (cv t)) (-) B)) (opr (cv v) (/) (2)) (opr (cv v) (/) (2)) lt2addt (` F (cv t)) A subclt climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp1d climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 climaddlem2 pm3.26d (e. (cv t) (` (ZZ>) M)) adantl sylanc (opr (` F (cv t)) (-) A) absclt syl (` G (cv t)) B subclt climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp2d climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 climaddlem2 pm3.27d (e. (cv t) (` (ZZ>) M)) adantl sylanc (opr (` G (cv t)) (-) B) absclt syl jca (cv v) rehalfclt (cv v) rehalfclt jca syl2an anasss (cv v) rehalfclt (opr (cv v) (/) (2)) recnt (opr (cv v) (/) (2)) 2timest 3syl (cv v) recnt 2cn 2re 2pos gt0ne0i (2) (cv v) divcan2t mp3an13 syl eqtr3d (e. (cv t) (` (ZZ>) M)) ph ad2antll (opr (` (abs) (opr (` F (cv t)) (-) A)) (+) (` (abs) (opr (` G (cv t)) (-) B))) (<) breq2d sylibd climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp3d (-) (opr A (+) B) opreq1d (` F (cv t)) (` G (cv t)) A B sub4t climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp1d climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp2d jca climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 climaddlem2 (e. (cv t) (` (ZZ>) M)) adantl sylanc eqtrd (abs) fveq2d (opr (` F (cv t)) (-) A) (opr (` G (cv t)) (-) B) abstrit (` F (cv t)) A subclt climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp1d climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 climaddlem2 pm3.26d (e. (cv t) (` (ZZ>) M)) adantl sylanc (` G (cv t)) B subclt climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp2d climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 climaddlem2 pm3.27d (e. (cv t) (` (ZZ>) M)) adantl sylanc sylanc eqbrtrd (e. (cv v) (RR)) adantrr (` (abs) (opr (` H (cv t)) (-) (opr A (+) B))) (opr (` (abs) (opr (` F (cv t)) (-) A)) (+) (` (abs) (opr (` G (cv t)) (-) B))) (cv v) lelttrt 3expa (` H (cv t)) (opr A (+) B) subclt climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp3d (` F (cv t)) (` G (cv t)) axaddcl climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp1d climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp2d sylanc eqeltrd A B axaddcl climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 climaddlem2 pm3.26d (e. (cv t) (` (ZZ>) M)) adantl climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 climaddlem2 pm3.27d (e. (cv t) (` (ZZ>) M)) adantl sylanc sylanc (opr (` H (cv t)) (-) (opr A (+) B)) absclt syl (` (abs) (opr (` F (cv t)) (-) A)) (` (abs) (opr (` G (cv t)) (-) B)) axaddrcl (` F (cv t)) A subclt climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp1d climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 climaddlem2 pm3.26d (e. (cv t) (` (ZZ>) M)) adantl sylanc (opr (` F (cv t)) (-) A) absclt syl (` G (cv t)) B subclt climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp2d climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 climaddlem2 pm3.27d (e. (cv t) (` (ZZ>) M)) adantl sylanc (opr (` G (cv t)) (-) B) absclt syl sylanc jca sylan anasss mpand syld ancoms (br (opr (cv u) (+) (cv f)) (<_) (cv t)) a1d (/\ (e. (cv u) (NN0)) (e. (cv f) (NN0))) adantlr syldd com23 r19.20dva (br (0) (<) (cv v)) adantlrr (cv h) (opr (cv u) (+) (cv f)) (<_) (cv t) breq1 (br (` (abs) (opr (` H (cv t)) (-) (opr A (+) B))) (<) (cv v)) imbi1d t (` (ZZ>) M) ralbidv (ZZ) rcla4ev (cv u) (cv f) nn0addclt (opr (cv u) (+) (cv f)) nn0zt syl sylan ex (/\ ph (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) adantl syld t (` (ZZ>) M) g (-> (br (cv u) (<_) (cv t)) (br (` (abs) (opr (` F (cv t)) (-) A)) (<) (opr (cv v) (/) (2)))) (-> (br (cv f) (<_) (cv g)) (br (` (abs) (opr (` G (cv g)) (-) B)) (<) (opr (cv v) (/) (2)))) raaan syl5ibr ex (cv u) elnn0uz (cv f) elnn0uz anbi12i syl5ibr r19.23advv u (` (ZZ>) (0)) f (` (ZZ>) (0)) (A.e. t (` (ZZ>) M) (-> (br (cv u) (<_) (cv t)) (br (` (abs) (opr (` F (cv t)) (-) A)) (<) (opr (cv v) (/) (2))))) (A.e. g (` (ZZ>) M) (-> (br (cv f) (<_) (cv g)) (br (` (abs) (opr (` G (cv g)) (-) B)) (<) (opr (cv v) (/) (2))))) reeanv syl5ibr mpd exp32 r19.21aiv (e. M (ZZ)) adantl climadd.3 M (opr A (+) B) t v h clim3zOLD climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 climaddlem2 A B axaddcl syl climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp3d (` F (cv t)) (` G (cv t)) axaddcl climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp1d climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 climaddlem.6 t climaddlem1 3simp2d sylanc eqeltrd expcom r19.21aiv jca sylan2 mpbird)) thm (climadd ((F k) (G k) (H k) (M k)) ((climadd.1 (e. F (V))) (climadd.2 (e. G (V))) (climadd.3 (e. H (V))) (climadd.4 (e. A (V))) (climadd.5 (e. B (V)))) (-> (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))))) (br H (~~>) (opr A (+) B))) (climadd.1 climadd.2 climadd.3 climadd.4 climadd.5 (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))) pm4.2 climaddlem3 an1s)) thm (climaddc1 ((A k) (C k) (F k) (G k) (M k)) ((climaddc1.1 (e. F (V))) (climaddc1.2 (e. G (V))) (climaddc1.3 (e. A (V))) (climaddc1.4 (e. C (V)))) (-> (/\ (/\ (br F (~~>) A) (e. C (CC))) (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (` F (cv k)) (+) C)))))) (br G (~~>) (opr A (+) C))) (climaddc1.1 zex C snex xpex climaddc1.2 climaddc1.3 climaddc1.4 M k climadd C climconst2 (br F (~~>) A) anim2i (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (` F (cv k)) (+) C))))) adantr (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (` F (cv k)) (+) C)))) pm3.26 (/\ (br F (~~>) A) (e. C (CC))) adantl (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (` F (cv k)) (+) C)) pm3.26 (/\ (e. C (CC)) (e. (cv k) (ZZ))) adantl C (CC) (cv k) (ZZ) fvconst2g (e. C (CC)) (e. (cv k) (ZZ)) pm3.26 eqeltrd (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (` F (cv k)) (+) C))) adantr (/\ (e. C (CC)) (e. (cv k) (ZZ))) (= (` G (cv k)) (opr (` F (cv k)) (+) C)) pm3.27 C (CC) (cv k) (ZZ) fvconst2g (` F (cv k)) (+) opreq2d (= (` G (cv k)) (opr (` F (cv k)) (+) C)) adantr eqtr4d (e. (` F (cv k)) (CC)) adantrl 3jca ex (cv k) M eluzelz sylan2 r19.20dva imp (br F (~~>) A) (e. M (ZZ)) ad2ant2l jca sylanc)) thm (climaddc2 ((A k) (C k) (F k) (G k) (M k)) ((climaddc1.1 (e. F (V))) (climaddc1.2 (e. G (V))) (climaddc1.3 (e. A (V))) (climaddc1.4 (e. C (V)))) (-> (/\ (/\ (br F (~~>) A) (e. C (CC))) (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr C (+) (` F (cv k)))))))) (br G (~~>) (opr C (+) A))) ((e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr C (+) (` F (cv k)))))) pm3.26 (e. C (CC)) adantl C (` F (cv k)) axaddcom (` G (cv k)) eqeq2d biimpd ex imdistand k (` (ZZ>) M) r19.20sdv imp (e. M (ZZ)) adantrl jca (br F (~~>) A) adantll climaddc1.1 climaddc1.2 climaddc1.3 climaddc1.4 M k climaddc1 A C axaddcom climaddc1.3 A (V) F climcl mpan sylan (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (` F (cv k)) (+) C))))) adantr breqtrd syldan)) thm (climmullem1 () () (-> (/\ (e. A (CC)) (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) (/\ (e. (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))) (RR)) (br (0) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A)))))) ((opr (cv v) (/) (2)) (opr (1) (+) (` (abs) A)) redivclt 3expb (cv v) rehalfclt A absclt ax1re (1) (` (abs) A) axaddrcl mpan syl (opr (1) (+) (` (abs) A)) gt0ne0t A absclt ax1re (1) (` (abs) A) axaddrcl mpan syl ax1re lt01 (1) (` (abs) A) addgtge0t mpanr1 mpanl1 A absclt A absge0t sylanc sylanc jca syl2an ancoms (br (0) (<) (cv v)) adantrr (opr (cv v) (/) (2)) (opr (1) (+) (` (abs) A)) divgt0t (cv v) rehalfclt (e. A (CC)) (br (0) (<) (cv v)) ad2antrl A absclt ax1re (1) (` (abs) A) axaddrcl mpan syl (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) adantr jca (cv v) halfpos2t biimpa (e. A (CC)) adantl ax1re lt01 (1) (` (abs) A) addgtge0t mpanr1 mpanl1 A absclt A absge0t sylanc (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) adantr jca sylanc jca)) thm (climmullem2 () () (-> (/\ (e. B (CC)) (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) (/\ (e. (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))))) (RR)) (br (0) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))))))) ((opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))) rerecclt (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) rerecclt B v climmullem1 pm3.26d B v climmullem1 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) gt0ne0t syl sylanc ax1re (1) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))) axaddrcl mpan syl (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))) gt0ne0t (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) rerecclt B v climmullem1 pm3.26d B v climmullem1 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) gt0ne0t syl sylanc ax1re (1) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))) axaddrcl mpan syl ax1re lt01 (1) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))) addgt0t mpanr1 mpanl1 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) rerecclt B v climmullem1 pm3.26d B v climmullem1 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) gt0ne0t syl sylanc B v climmullem1 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) recgt0t syl sylanc sylanc sylanc (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))) recgt0t (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) rerecclt B v climmullem1 pm3.26d B v climmullem1 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) gt0ne0t syl sylanc ax1re (1) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))) axaddrcl mpan syl ax1re lt01 (1) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))) addgt0t mpanr1 mpanl1 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) rerecclt B v climmullem1 pm3.26d B v climmullem1 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) gt0ne0t syl sylanc B v climmullem1 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) recgt0t syl sylanc sylanc jca)) thm (climmullem3 () () (-> (/\ (/\/\ (/\ (e. F (CC)) (e. G (CC))) (/\ (e. A (CC)) (e. B (CC))) (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) (/\ (br (` (abs) (opr F (-) A)) (<) (1)) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A)))))) (br (` (abs) (opr F (x.) (opr G (-) B))) (<) (opr (cv v) (/) (2)))) ((` (abs) F) (opr (1) (+) (` (abs) A)) (` (abs) (opr G (-) B)) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))) ltmul12it F absclt (e. G (CC)) (/\ (e. A (CC)) (e. B (CC))) ad2antrr (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (/\ (br (` (abs) (opr F (-) A)) (<) (1)) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) adantr A absclt ax1re (1) (` (abs) A) axaddrcl mpan syl (/\ (e. F (CC)) (e. G (CC))) (e. B (CC)) ad2antrl (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (/\ (br (` (abs) (opr F (-) A)) (<) (1)) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) adantr jca F absge0t (e. G (CC)) (/\ (e. A (CC)) (e. B (CC))) ad2antrr (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (/\ (br (` (abs) (opr F (-) A)) (<) (1)) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) adantr F A npcant (abs) fveq2d (opr F (-) A) A abstrit F A subclt (e. F (CC)) (e. A (CC)) pm3.27 sylanc eqbrtrrd (br (` (abs) (opr F (-) A)) (<) (1)) adantr (` (abs) (opr F (-) A)) (1) (` (abs) A) ltadd1t F A subclt (opr F (-) A) absclt syl ax1re (/\ (e. F (CC)) (e. A (CC))) a1i A absclt (e. F (CC)) adantl syl3anc biimpa (` (abs) F) (opr (` (abs) (opr F (-) A)) (+) (` (abs) A)) (opr (1) (+) (` (abs) A)) lelttrt F absclt (e. A (CC)) adantr (` (abs) (opr F (-) A)) (` (abs) A) axaddrcl F A subclt (opr F (-) A) absclt syl A absclt (e. F (CC)) adantl sylanc A absclt ax1re (1) (` (abs) A) axaddrcl mpan syl (e. F (CC)) adantl syl3anc (br (` (abs) (opr F (-) A)) (<) (1)) adantr mp2and (e. B (CC)) adantlrr (e. G (CC)) adantllr (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adantl3 (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A)))) adantrr jca jca G B subclt (opr G (-) B) absclt syl (e. F (CC)) (e. A (CC)) ad2ant2l (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (/\ (br (` (abs) (opr F (-) A)) (<) (1)) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) adantr A v climmullem1 pm3.26d (e. B (CC)) adantlr (/\ (e. F (CC)) (e. G (CC))) 3adant1 (/\ (br (` (abs) (opr F (-) A)) (<) (1)) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) adantr jca G B subclt (opr G (-) B) absge0t syl (e. F (CC)) (e. A (CC)) ad2ant2l (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (br (` (abs) (opr F (-) A)) (<) (1)) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A)))) pm3.27 anim12i jca sylanc F (opr G (-) B) absmult G B subclt sylan2 anassrs (e. A (CC)) adantrl (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (/\ (br (` (abs) (opr F (-) A)) (<) (1)) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) adantr eqcomd (opr (1) (+) (` (abs) A)) (opr (cv v) (/) (2)) divcan2t A absclt ax1re (1) (` (abs) A) axaddrcl mpan syl recnd (e. (cv v) (RR)) adantr (cv v) rehalfclt recnd (e. A (CC)) adantl (opr (1) (+) (` (abs) A)) gt0ne0t A absclt ax1re (1) (` (abs) A) axaddrcl mpan syl ax1re lt01 (1) (` (abs) A) addgtge0t mpanr1 mpanl1 A absclt A absge0t sylanc sylanc (e. (cv v) (RR)) adantr syl3anc (e. B (CC)) (br (0) (<) (cv v)) ad2ant2r (/\ (e. F (CC)) (e. G (CC))) 3adant1 (/\ (br (` (abs) (opr F (-) A)) (<) (1)) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) adantr 3brtr3d)) thm (climmullem4 () () (-> (/\ (/\/\ (/\ (e. F (CC)) (e. G (CC))) (/\ (e. A (CC)) (e. B (CC))) (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) (br (` (abs) (opr F (-) A)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))) (br (` (abs) (opr (opr F (-) A) (x.) B)) (<) (opr (cv v) (/) (2)))) ((opr F (-) A) B absmult F A subclt sylan anasss (e. G (CC)) adantlr (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (br (` (abs) (opr F (-) A)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))) adantr (` (abs) (opr F (-) A)) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) (` (abs) B) lemul1it F A subclt (opr F (-) A) absclt syl (e. G (CC)) (e. B (CC)) ad2ant2r (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 B v climmullem1 pm3.26d (e. A (CC)) adantll (/\ (e. F (CC)) (e. G (CC))) 3adant1 B absclt (/\ (e. F (CC)) (e. G (CC))) (e. A (CC)) ad2antll (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 3jca (br (` (abs) (opr F (-) A)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))) adantr B absge0t (/\ (e. F (CC)) (e. G (CC))) (e. A (CC)) ad2antll (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (br (` (abs) (opr F (-) A)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))) adantr (` (abs) (opr F (-) A)) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) ltlet F A subclt (opr F (-) A) absclt syl (e. G (CC)) (e. B (CC)) ad2ant2r (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 B v climmullem1 pm3.26d (e. A (CC)) adantll (/\ (e. F (CC)) (e. G (CC))) 3adant1 sylanc imp jca sylanc eqbrtrd (` (abs) B) recp1lt1 B absclt B absge0t sylanc (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) adantr ax1re (opr (` (abs) B) (/) (opr (1) (+) (` (abs) B))) (1) (opr (cv v) (/) (2)) ltmul1t mp3anl2 (` (abs) B) (opr (1) (+) (` (abs) B)) redivclt B absclt B absclt ax1re (1) (` (abs) B) axaddrcl mpan syl (opr (1) (+) (` (abs) B)) gt0ne0t B absclt ax1re (1) (` (abs) B) axaddrcl mpan syl ax1re lt01 (1) (` (abs) B) addgtge0t mpanr1 mpanl1 B absclt B absge0t sylanc sylanc syl3anc (cv v) rehalfclt (br (0) (<) (cv v)) adantr anim12i (cv v) halfpos2t biimpa (e. B (CC)) adantl sylanc mpbid (` (abs) B) (opr (1) (+) (` (abs) B)) (opr (cv v) (/) (2)) div13t B absclt (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) adantr recnd B absclt ax1re (1) (` (abs) B) axaddrcl mpan syl (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) adantr recnd (cv v) rehalfclt (e. B (CC)) (br (0) (<) (cv v)) ad2antrl recnd 3jca (opr (1) (+) (` (abs) B)) gt0ne0t B absclt ax1re (1) (` (abs) B) axaddrcl mpan syl ax1re lt01 (1) (` (abs) B) addgtge0t mpanr1 mpanl1 B absclt B absge0t sylanc sylanc (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) adantr sylanc (cv v) rehalfclt recnd (opr (cv v) (/) (2)) mulid2t syl (e. B (CC)) (br (0) (<) (cv v)) ad2antrl 3brtr3d (e. A (CC)) adantll (/\ (e. F (CC)) (e. G (CC))) 3adant1 (br (` (abs) (opr F (-) A)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))) adantr (` (abs) (opr (opr F (-) A) (x.) B)) (opr (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) (x.) (` (abs) B)) (opr (cv v) (/) (2)) lelttrt (opr F (-) A) B axmulcl F A subclt sylan anasss (opr (opr F (-) A) (x.) B) absclt syl (e. G (CC)) adantlr (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) (` (abs) B) axmulrcl B v climmullem1 pm3.26d B absclt (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) adantr sylanc (e. A (CC)) adantll (/\ (e. F (CC)) (e. G (CC))) 3adant1 (cv v) rehalfclt (/\ (e. A (CC)) (e. B (CC))) (br (0) (<) (cv v)) ad2antrl (/\ (e. F (CC)) (e. G (CC))) 3adant1 syl3anc (br (` (abs) (opr F (-) A)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))) adantr mp2and)) thm (climmullem5 () () (-> (/\/\ (/\ (e. F (CC)) (e. G (CC))) (/\ (e. A (CC)) (e. B (CC))) (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) (-> (/\ (br (` (abs) (opr F (-) A)) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))))) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) (br (` (abs) (opr (opr F (x.) G) (-) (opr A (x.) B))) (<) (cv v)))) (F G B subdit 3expa (e. A (CC)) adantrl F A B subdirt 3expb (e. G (CC)) adantlr (+) opreq12d (opr F (x.) G) (opr F (x.) B) (opr A (x.) B) npncant F G axmulcl (/\ (e. A (CC)) (e. B (CC))) adantr F B axmulcl (e. A (CC)) adantrl (e. G (CC)) adantlr A B axmulcl (/\ (e. F (CC)) (e. G (CC))) adantl syl3anc eqtr2d (abs) fveq2d (opr F (x.) (opr G (-) B)) (opr (opr F (-) A) (x.) B) abstrit F (opr G (-) B) axmulcl G B subclt sylan2 anassrs (e. A (CC)) adantrl (opr F (-) A) B axmulcl F A subclt sylan anasss (e. G (CC)) adantlr sylanc eqbrtrd (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (/\ (br (` (abs) (opr F (-) A)) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))))) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) adantr B v climmullem1 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) recrecltt pm3.26d syl (e. A (CC)) adantll (/\ (e. F (CC)) (e. G (CC))) 3adant1 ax1re (` (abs) (opr F (-) A)) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))))) (1) axlttrn mp3an3 F A subclt (opr F (-) A) absclt syl (e. G (CC)) (e. B (CC)) ad2ant2r (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 B v climmullem2 pm3.26d (e. A (CC)) adantll (/\ (e. F (CC)) (e. G (CC))) 3adant1 sylanc mpan2d (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A)))) anim1d imp F G A B v climmullem3 syldan B v climmullem1 (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) recrecltt pm3.27d syl (e. A (CC)) adantll (/\ (e. F (CC)) (e. G (CC))) 3adant1 (` (abs) (opr F (-) A)) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))))) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))) axlttrn F A subclt (opr F (-) A) absclt syl (e. G (CC)) (e. B (CC)) ad2ant2r (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 B v climmullem2 pm3.26d (e. A (CC)) adantll (/\ (e. F (CC)) (e. G (CC))) 3adant1 B v climmullem1 pm3.26d (e. A (CC)) adantll (/\ (e. F (CC)) (e. G (CC))) 3adant1 syl3anc mpan2d imp F G A B v climmullem4 syldan (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A)))) adantrr (` (abs) (opr F (x.) (opr G (-) B))) (` (abs) (opr (opr F (-) A) (x.) B)) (opr (cv v) (/) (2)) (opr (cv v) (/) (2)) lt2addt F (opr G (-) B) axmulcl G B subclt sylan2 anassrs (opr F (x.) (opr G (-) B)) absclt syl (e. A (CC)) adantrl (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (opr F (-) A) B axmulcl F A subclt sylan anasss (opr (opr F (-) A) (x.) B) absclt syl (e. G (CC)) adantlr (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 jca (cv v) rehalfclt (cv v) rehalfclt jca (/\ (e. A (CC)) (e. B (CC))) (br (0) (<) (cv v)) ad2antrl (/\ (e. F (CC)) (e. G (CC))) 3adant1 sylanc (/\ (br (` (abs) (opr F (-) A)) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))))) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) adantr mp2and (` (abs) (opr (opr F (x.) G) (-) (opr A (x.) B))) (opr (` (abs) (opr F (x.) (opr G (-) B))) (+) (` (abs) (opr (opr F (-) A) (x.) B))) (opr (opr (cv v) (/) (2)) (+) (opr (cv v) (/) (2))) lelttrt (opr F (x.) G) (opr A (x.) B) subclt F G axmulcl A B axmulcl syl2an (opr (opr F (x.) G) (-) (opr A (x.) B)) absclt syl (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (` (abs) (opr F (x.) (opr G (-) B))) (` (abs) (opr (opr F (-) A) (x.) B)) axaddrcl F (opr G (-) B) axmulcl G B subclt sylan2 anassrs (opr F (x.) (opr G (-) B)) absclt syl (e. A (CC)) adantrl (opr F (-) A) B axmulcl F A subclt sylan anasss (opr (opr F (-) A) (x.) B) absclt syl (e. G (CC)) adantlr sylanc (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) 3adant3 (opr (cv v) (/) (2)) (opr (cv v) (/) (2)) axaddrcl (cv v) rehalfclt (cv v) rehalfclt sylanc (/\ (e. A (CC)) (e. B (CC))) (br (0) (<) (cv v)) ad2antrl (/\ (e. F (CC)) (e. G (CC))) 3adant1 syl3anc (/\ (br (` (abs) (opr F (-) A)) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))))) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) adantr mp2and (cv v) recnt (cv v) 2halvest syl (/\ (e. A (CC)) (e. B (CC))) (br (0) (<) (cv v)) ad2antrl (/\ (e. F (CC)) (e. G (CC))) 3adant1 (/\ (br (` (abs) (opr F (-) A)) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))))) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) adantr breqtrd ex)) thm (climmullem6 ((ph t) (A t) (B t) (k t) (F k) (F t) (G k) (G t) (H k) (H t) (M k) (M t)) ((climmul.1 (e. F (V))) (climmul.2 (e. G (V))) (climmul.3 (e. H (V))) (climmul.4 (e. A (V))) (climmul.5 (e. B (V))) (climmullem.6 (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (x.) (` G (cv k)))))))))) (-> (/\ (e. (cv t) (` (ZZ>) M)) ph) (/\/\ (e. (` F (cv t)) (CC)) (e. (` G (cv t)) (CC)) (= (` H (cv t)) (opr (` F (cv t)) (x.) (` G (cv t)))))) ((cv k) (cv t) F fveq2 (CC) eleq1d (cv k) (cv t) G fveq2 (CC) eleq1d (cv k) (cv t) H fveq2 (cv k) (cv t) F fveq2 (cv k) (cv t) G fveq2 (x.) opreq12d eqeq12d 3anbi123d (` (ZZ>) M) rcla4va (/\ (br F (~~>) A) (br G (~~>) B)) adantrl climmullem.6 sylan2b)) thm (climmullem7 ((F k) (G k) (H k) (M k)) ((climmul.1 (e. F (V))) (climmul.2 (e. G (V))) (climmul.3 (e. H (V))) (climmul.4 (e. A (V))) (climmul.5 (e. B (V))) (climmullem.6 (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (x.) (` G (cv k)))))))))) (-> ph (/\ (e. A (CC)) (e. B (CC)))) (climmullem.6 climmul.4 A (V) F climcl mpan climmul.5 B (V) G climcl mpan anim12i (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (x.) (` G (cv k)))))) adantr sylbi)) thm (climmullem8 ((f t) (f u) (f v) (f ph) (t u) (t v) (ph t) (u v) (ph u) (ph v) (f g) (f h) (f t) (f u) (f v) (A f) (g h) (g t) (g u) (g v) (A g) (h t) (h u) (h v) (A h) (t u) (t v) (A t) (u v) (A u) (A v) (B f) (B g) (B h) (B t) (B u) (B v) (f k) (F f) (g k) (F g) (k t) (k u) (k v) (F k) (F t) (F u) (F v) (G f) (G g) (G k) (G t) (G u) (G v) (H f) (h k) (H h) (H k) (H t) (H u) (H v) (M f) (M g) (M h) (M k) (M t) (M u) (M v)) ((climmul.1 (e. F (V))) (climmul.2 (e. G (V))) (climmul.3 (e. H (V))) (climmul.4 (e. A (V))) (climmul.5 (e. B (V))) (climmullem.6 (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (x.) (` G (cv k)))))))))) (-> (/\ (e. M (ZZ)) ph) (br H (~~>) (opr A (x.) B))) (B v climmullem2 climmul.1 climmul.2 climmul.3 climmul.4 climmul.5 climmullem.6 climmullem7 pm3.27d sylan A v climmullem1 climmul.1 climmul.2 climmul.3 climmul.4 climmul.5 climmullem.6 climmullem7 pm3.26d sylan jca climmul.4 0z (0) A (V) F (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))))) u t M climcvg4uzOLD mpan mpanl1 climmul.5 0z (0) B (V) G (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))) f g M climcvg4uzOLD mpan mpanl1 anim12i an4s (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (x.) (` G (cv k)))))) adantlr climmullem.6 sylanb syldan (/\ (e. (cv u) (NN0)) (e. (cv f) (NN0))) (e. (cv t) (` (ZZ>) M)) pm3.27 (cv u) (cv f) nn0addge1t (cv u) nn0ret sylan (e. (cv t) (RR)) adantr (cv u) (opr (cv u) (+) (cv f)) (cv t) letrt 3expa (e. (cv u) (RR)) (e. (cv f) (RR)) pm3.26 (cv u) (cv f) axaddrcl jca (cv u) nn0ret (cv f) nn0ret syl2an sylan mpand (cv t) M eluzelz (cv t) zret syl sylan2 (e. (cv g) (` (ZZ>) M)) adantr (cv f) (cv u) nn0addge2t ancoms (cv f) nn0ret sylan2 (e. (cv g) (RR)) adantr (cv f) (opr (cv u) (+) (cv f)) (cv g) letrt 3expa (e. (cv u) (RR)) (e. (cv f) (RR)) pm3.27 (cv u) (cv f) axaddrcl jca (cv u) nn0ret (cv f) nn0ret syl2an sylan mpand (cv g) M eluzelz (cv g) zret syl sylan2 (e. (cv t) (` (ZZ>) M)) adantlr anim12d (br (cv u) (<_) (cv t)) (br (` (abs) (opr (` F (cv t)) (-) A)) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))))) (br (cv f) (<_) (cv g)) (br (` (abs) (opr (` G (cv g)) (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A)))) prth syl9 exp4a r19.20dva (cv g) (cv t) (opr (cv u) (+) (cv f)) (<_) breq2 (cv g) (cv t) G fveq2 (-) B opreq1d (abs) fveq2d (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))) breq1d (br (` (abs) (opr (` F (cv t)) (-) A)) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))))) anbi2d imbi12d (` (ZZ>) M) rcla4v com23 a2d g (` (ZZ>) M) (br (opr (cv u) (+) (cv f)) (<_) (cv t)) (-> (br (opr (cv u) (+) (cv f)) (<_) (cv g)) (/\ (br (` (abs) (opr (` F (cv t)) (-) A)) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))))) (br (` (abs) (opr (` G (cv g)) (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A)))))) r19.21v syl5ib com12 syl6 mpid com23 (/\ ph (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) adantll (` F (cv t)) (` G (cv t)) A B v climmullem5 3expa climmul.1 climmul.2 climmul.3 climmul.4 climmul.5 climmullem.6 t climmullem6 3simp1d climmul.1 climmul.2 climmul.3 climmul.4 climmul.5 climmullem.6 t climmullem6 3simp2d jca climmul.1 climmul.2 climmul.3 climmul.4 climmul.5 climmullem.6 climmullem7 (e. (cv t) (` (ZZ>) M)) adantl jca sylan anasss climmul.1 climmul.2 climmul.3 climmul.4 climmul.5 climmullem.6 t climmullem6 3simp3d (/\ (e. (cv v) (RR)) (br (0) (<) (cv v))) adantrr (-) (opr A (x.) B) opreq1d (abs) fveq2d (<) (cv v) breq1d sylibrd ancoms (br (opr (cv u) (+) (cv f)) (<_) (cv t)) a1d (/\ (e. (cv u) (NN0)) (e. (cv f) (NN0))) adantlr syldd com23 r19.20dva (cv h) (opr (cv u) (+) (cv f)) (<_) (cv t) breq1 (br (` (abs) (opr (` H (cv t)) (-) (opr A (x.) B))) (<) (cv v)) imbi1d t (` (ZZ>) M) ralbidv (ZZ) rcla4ev (cv u) (cv f) nn0addclt (opr (cv u) (+) (cv f)) nn0zt syl sylan ex (/\ ph (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) adantl syld t (` (ZZ>) M) g (-> (br (cv u) (<_) (cv t)) (br (` (abs) (opr (` F (cv t)) (-) A)) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))))))) (-> (br (cv f) (<_) (cv g)) (br (` (abs) (opr (` G (cv g)) (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) raaan syl5ibr ex (cv u) elnn0uz (cv f) elnn0uz anbi12i syl5ibr r19.23advv u (` (ZZ>) (0)) f (` (ZZ>) (0)) (A.e. t (` (ZZ>) M) (-> (br (cv u) (<_) (cv t)) (br (` (abs) (opr (` F (cv t)) (-) A)) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))))))) (A.e. g (` (ZZ>) M) (-> (br (cv f) (<_) (cv g)) (br (` (abs) (opr (` G (cv g)) (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A)))))) reeanv syl5ibr mpd exp32 r19.21aiv (e. M (ZZ)) adantl climmul.3 M (opr A (x.) B) t v h clim3zOLD climmul.1 climmul.2 climmul.3 climmul.4 climmul.5 climmullem.6 climmullem7 A B axmulcl syl climmul.1 climmul.2 climmul.3 climmul.4 climmul.5 climmullem.6 t climmullem6 3simp3d (` F (cv t)) (` G (cv t)) axmulcl climmul.1 climmul.2 climmul.3 climmul.4 climmul.5 climmullem.6 t climmullem6 3simp1d climmul.1 climmul.2 climmul.3 climmul.4 climmul.5 climmullem.6 t climmullem6 3simp2d sylanc eqeltrd expcom r19.21aiv jca sylan2 mpbird)) thm (climmul ((F k) (G k) (H k) (M k)) ((climmul.1 (e. F (V))) (climmul.2 (e. G (V))) (climmul.3 (e. H (V))) (climmul.4 (e. A (V))) (climmul.5 (e. B (V)))) (-> (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (x.) (` G (cv k)))))))) (br H (~~>) (opr A (x.) B))) (climmul.1 climmul.2 climmul.3 climmul.4 climmul.5 (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (x.) (` G (cv k))))))) pm4.2 climmullem8 an1s)) thm (climmulc2 ((A x) (k x) (C k) (C x) (F k) (F x) (G k) (G x) (M k) (M x)) ((climmulc2.1 (e. F (V))) (climmulc2.2 (e. G (V))) (climmulc2.3 (e. A (V)))) (-> (/\ (/\ (e. C (CC)) (br F (~~>) A)) (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr C (x.) (` F (cv k)))))))) (br G (~~>) (opr C (x.) A))) ((cv x) C (x.) (` F (cv k)) opreq1 (` G (cv k)) eqeq2d (e. (` F (cv k)) (CC)) anbi2d k (` (ZZ>) M) ralbidv (cv x) C (x.) A opreq1 G (~~>) breq2d imbi12d (e. M (ZZ)) imbi2d (br F (~~>) A) imbi2d zex (cv x) snex xpex climmulc2.1 climmulc2.2 x visset climmulc2.3 M k climmul (cv x) climconst2 (br F (~~>) A) anim1i (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (cv x) (x.) (` F (cv k))))))) adantr (cv x) (CC) (cv k) (ZZ) fvconst2g (cv k) M eluzelz sylan2 (e. (cv x) (CC)) (e. (cv k) (` (ZZ>) M)) pm3.26 eqeltrd (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (cv x) (x.) (` F (cv k))))) a1d (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (cv x) (x.) (` F (cv k)))) pm3.26 (/\ (e. (cv x) (CC)) (e. (cv k) (` (ZZ>) M))) a1i (cv x) (CC) (cv k) (ZZ) fvconst2g (cv k) M eluzelz sylan2 (x.) (` F (cv k)) opreq1d (` G (cv k)) eqeq2d biimprd (e. (` F (cv k)) (CC)) adantld 3jcad r19.20dva (e. M (ZZ)) anim2d imp (br F (~~>) A) adantlr sylanc exp43 vtoclga imp43)) thm (climaddc ((B k) (A k) (F k) (G k)) ((climaddc.1 (e. A (CC))) (climaddc.2 (e. B (V))) (climaddc.3 (br F (~~>) B)) (climaddc.4 (Fn G (NN))) (climaddc.5 (-> (e. (cv k) (NN)) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr A (+) (` F (cv k)))))))) (br G (~~>) (opr A (+) B)) (climaddc.3 climaddc.1 pm3.2i 1z (cv k) elnnuz climaddc.5 sylbir rgen pm3.2i climaddc.3 climrel F B brrelexi ax-mp climaddc.4 nnex G (NN) (V) fnex mp2an climaddc.2 climaddc.1 elisseti (1) k climaddc2 mp2an)) thm (climmulc ((A k) (F k) (G k)) ((climaddc.1 (e. A (CC))) (climaddc.2 (e. B (V))) (climaddc.3 (br F (~~>) B)) (climaddc.4 (Fn G (NN))) (climmulc.5 (-> (e. (cv k) (NN)) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr A (x.) (` F (cv k)))))))) (br G (~~>) (opr A (x.) B)) (climaddc.1 climaddc.3 pm3.2i 1z (cv k) elnnuz climmulc.5 sylbir rgen pm3.2i climaddc.3 climrel F B brrelexi ax-mp climaddc.4 nnex G (NN) (V) fnex mp2an climaddc.2 A (1) k climmulc2 mp2an)) thm (clim2serz ((A k) (j k) (F j) (F k) (M j) (M k) (N j) (N k)) ((clim2serz.1 (e. F (V))) (clim2serz.2 (e. A (V))) (clim2serz.3 (br (opr (<,> M (+)) (seq) F) (~~>) A)) (clim2serz.4 (:--> F (` (ZZ>) M) (CC)))) (-> (e. N (` (ZZ>) M)) (br (opr (<,> (opr N (+) (1)) (+)) (seq) F) (~~>) (opr A (-) (` (opr (<,> M (+)) (seq) F) N)))) (clim2serz.3 (<,> M (+)) (seq) F oprex (<,> (opr N (+) (1)) (+)) (seq) F oprex clim2serz.2 (` (opr (<,> M (+)) (seq) F) N) minusex (opr N (+) (1)) k climaddc1 mpanl1 (cv j) M N elfzuzt clim2serz.4 (cv j) ffvrni syl rgen clim2serz.1 N M j serzclt mpan2 (` (opr (<,> M (+)) (seq) F) N) negclt syl N M eluzelz peano2zd (cv k) (opr N (+) (1)) M uztrn N M peano2uz sylan2 ancoms (cv j) M (cv k) elfzuzt clim2serz.4 (cv j) ffvrni syl rgen clim2serz.1 (cv k) M j serzclt mpan2 syl (` (opr (<,> M (+)) (seq) F) (cv k)) (` (opr (<,> M (+)) (seq) F) N) negsubt (cv k) (opr N (+) (1)) M uztrn N M peano2uz sylan2 ancoms (cv j) M (cv k) elfzuzt clim2serz.4 (cv j) ffvrni syl rgen clim2serz.1 (cv k) M j serzclt mpan2 syl (cv j) M N elfzuzt clim2serz.4 (cv j) ffvrni syl rgen clim2serz.1 N M j serzclt mpan2 (e. (cv k) (` (ZZ>) (opr N (+) (1)))) adantr sylanc clim2serz.1 (cv k) N M j serzsplit (cv k) (opr N (+) (1)) eluzelz (e. N (` (ZZ>) M)) adantl N (cv k) eluzp1m1t N M eluzelz sylan N M (opr (cv k) (-) (1)) eluzfzt syldan (cv j) M (cv k) elfzuzt clim2serz.4 (cv j) ffvrni syl rgen (/\ (e. N (` (ZZ>) M)) (e. (cv k) (` (ZZ>) (opr N (+) (1))))) a1i syl3anc eqcomd (` (opr (<,> M (+)) (seq) F) (cv k)) (` (opr (<,> M (+)) (seq) F) N) (` (opr (<,> (opr N (+) (1)) (+)) (seq) F) (cv k)) subaddt (cv k) (opr N (+) (1)) M uztrn N M peano2uz sylan2 ancoms (cv j) M (cv k) elfzuzt clim2serz.4 (cv j) ffvrni syl rgen clim2serz.1 (cv k) M j serzclt mpan2 syl (cv j) M N elfzuzt clim2serz.4 (cv j) ffvrni syl rgen clim2serz.1 N M j serzclt mpan2 (e. (cv k) (` (ZZ>) (opr N (+) (1)))) adantr clim2serz.1 F (V) (` (ZZ>) (opr N (+) (1))) resexg ax-mp (cv k) (opr N (+) (1)) j serzclt (e. N (` (ZZ>) M)) (e. (cv k) (` (ZZ>) (opr N (+) (1)))) pm3.27 (cv j) (opr N (+) (1)) M uztrn (cv j) (opr N (+) (1)) (cv k) elfzuzt N M peano2uz syl2an clim2serz.4 (cv j) ffvrni syl ancoms (cv j) (opr N (+) (1)) (cv k) elfzuzt (cv j) (` (ZZ>) (opr N (+) (1))) F fvres syl (CC) eleq1d (e. N (` (ZZ>) M)) adantl mpbird r19.21aiva (e. (cv k) (` (ZZ>) (opr N (+) (1)))) adantr sylanc N M eluzelz peano2zd addex clim2serz.1 (opr N (+) (1)) seqzres syl (cv k) fveq1d (CC) eleq1d (e. (cv k) (` (ZZ>) (opr N (+) (1)))) adantr mpbid syl3anc mpbird eqtr2d jca r19.21aiva jca sylanc (cv j) M N elfzuzt clim2serz.4 (cv j) ffvrni syl rgen clim2serz.1 N M j serzclt mpan2 clim2serz.2 clim2serz.3 A (V) (opr (<,> M (+)) (seq) F) climcl mp2an A (` (opr (<,> M (+)) (seq) F) N) negsubt mpan syl breqtrd)) thm (climcj ((j k) (j x) (A j) (k x) (A k) (A x) (F j) (F k) (F x) (G j) (G k) (G x) (M j) (M k) (M x)) ((climcj.1 (e. A (V))) (climcj.4 (e. G (V))) (climcj.2 (e. M (ZZ))) (climcj.3 (br F (~~>) A)) (climcj.5 (-> (e. (cv k) (` (ZZ>) M)) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (` (*) (` F (cv k)))))))) (br G (~~>) (` (*) A)) (climcj.3 climcj.2 climcj.1 climcj.3 A (V) F climcl mp2an climcj.5 pm3.26d rgen climcj.3 climrel F A brrelexi ax-mp M A k x j clim3z mp3an mpbi climcj.5 pm3.27d (-) (` (*) A) opreq1d climcj.5 pm3.26d climcj.1 climcj.3 A (V) F climcl mp2an (` F (cv k)) A cjsubt mpan2 syl eqtr4d (abs) fveq2d climcj.5 pm3.26d climcj.1 climcj.3 A (V) F climcl mp2an (` F (cv k)) A subclt mpan2 (opr (` F (cv k)) (-) A) abscjt 3syl eqtrd (<) (cv x) breq1d (br (cv j) (<_) (cv k)) imbi2d ralbiia j (ZZ) rexbii (br (0) (<) (cv x)) imbi2i x (RR) ralbii mpbir climcj.2 climcj.1 climcj.3 A (V) F climcl mp2an cjcl climcj.5 pm3.27d climcj.5 pm3.26d (` F (cv k)) cjclt syl eqeltrd rgen climcj.4 M (` (*) A) k x j clim3z mp3an mpbir)) thm (climubi ((j m) (j w) (A j) (m w) (A m) (A w) (j k) (F j) (k m) (k w) (F k) (F m) (F w) (N j) (N m) (N w)) ((climub.1 (e. A (V))) (climub.2 (:--> F (NN) (RR))) (climub.3 (-> (e. (cv k) (NN)) (br (` F (cv k)) (<_) (` F (opr (cv k) (+) (1)))))) (climub.4 (br F (~~>) A)) (climubi.5 (e. N (NN)))) (br (` F N) (<_) A) ((cv j) (opr N (+) (cv m)) (cv m) (<_) breq2 (cv j) (opr N (+) (cv m)) F fveq2 (-) A opreq1d (abs) fveq2d (<) (opr (` F N) (-) A) breq1d imbi12d negbid (NN) rcla4ev climubi.5 N (cv m) nnaddclt mpan (cv m) nnret climubi.5 nnnn0 (cv m) N nn0addge2t mpan2 syl climub.2 climubi.5 F (NN) (RR) N ffvrn mp2an climub.1 climub.2 climub.4 climfnrcl resubcl (opr (` F N) (-) A) (opr (` F (opr N (+) (cv m))) (-) A) (` (abs) (opr (` F (opr N (+) (cv m))) (-) A)) letrt mp3an1 climubi.5 N (cv m) nnaddclt mpan climub.2 (opr N (+) (cv m)) ffvrni syl climub.1 climub.2 climub.4 climfnrcl (` F (opr N (+) (cv m))) A resubclt mpan2 syl climubi.5 N (cv m) nnaddclt mpan climub.2 (opr N (+) (cv m)) ffvrni syl climub.1 climub.2 climub.4 climfnrcl (` F (opr N (+) (cv m))) A resubclt mpan2 syl (opr (` F (opr N (+) (cv m))) (-) A) recnt syl (opr (` F (opr N (+) (cv m))) (-) A) absclt syl jca climubi.5 climub.2 climub.3 N (opr N (+) (cv m)) monoord mp3an1 climubi.5 N (cv m) nnaddclt mpan (cv m) nnnn0t climubi.5 nnre N (cv m) nn0addge1t mpan syl sylanc climubi.5 N (cv m) nnaddclt mpan climub.2 (opr N (+) (cv m)) ffvrni syl climub.2 climubi.5 F (NN) (RR) N ffvrn mp2an climub.1 climub.2 climub.4 climfnrcl (` F N) (` F (opr N (+) (cv m))) A lesub1t mp3an13 syl mpbid climubi.5 N (cv m) nnaddclt mpan climub.2 (opr N (+) (cv m)) ffvrni syl climub.1 climub.2 climub.4 climfnrcl (` F (opr N (+) (cv m))) A resubclt mpan2 syl (opr (` F (opr N (+) (cv m))) (-) A) leabst syl jca sylc climubi.5 N (cv m) nnaddclt mpan climub.2 (opr N (+) (cv m)) ffvrni syl climub.1 climub.2 climub.4 climfnrcl (` F (opr N (+) (cv m))) A resubclt mpan2 syl (opr (` F (opr N (+) (cv m))) (-) A) recnt syl (opr (` F (opr N (+) (cv m))) (-) A) absclt syl climub.2 climubi.5 F (NN) (RR) N ffvrn mp2an climub.1 climub.2 climub.4 climfnrcl resubcl (opr (` F N) (-) A) (` (abs) (opr (` F (opr N (+) (cv m))) (-) A)) lenltt mpan syl mpbid jca (br (cv m) (<_) (opr N (+) (cv m))) (br (` (abs) (opr (` F (opr N (+) (cv m))) (-) A)) (<) (opr (` F N) (-) A)) annim sylib sylanc j (NN) (-> (br (cv m) (<_) (cv j)) (br (` (abs) (opr (` F (cv j)) (-) A)) (<) (opr (` F N) (-) A))) rexnal sylib nrex climub.2 climubi.5 F (NN) (RR) N ffvrn mp2an climub.1 climub.2 climub.4 climfnrcl resubcl climub.4 climub.1 A (V) F (cv w) m j climcvgnn mpanl1 exp32 r19.21aiv ax-mp (cv w) (opr (` F N) (-) A) (0) (<) breq2 (cv w) (opr (` F N) (-) A) (` (abs) (opr (` F (cv j)) (-) A)) (<) breq2 (br (cv m) (<_) (cv j)) imbi2d m (NN) j (NN) rexralbidv imbi12d (RR) rcla4v mp2 mto climub.1 climub.2 climub.4 climfnrcl climub.2 climubi.5 F (NN) (RR) N ffvrn mp2an posdif mtbir climub.2 climubi.5 F (NN) (RR) N ffvrn mp2an climub.1 climub.2 climub.4 climfnrcl lenlt mpbir)) thm (climub ((F k)) ((climub.1 (e. A (V))) (climub.2 (:--> F (NN) (RR))) (climub.3 (-> (e. (cv k) (NN)) (br (` F (cv k)) (<_) (` F (opr (cv k) (+) (1)))))) (climub.4 (br F (~~>) A))) (-> (e. N (NN)) (br (` F N) (<_) A)) (N (if (e. N (NN)) N (1)) F fveq2 (<_) A breq1d climub.1 climub.2 climub.3 climub.4 1nn N elimel climubi dedth)) thm (climsup ((j k) (j x) (j y) (j z) (F j) (k x) (k y) (k z) (F k) (x y) (x z) (F x) (y z) (F y) (F z)) ((climsup.1 (:--> F (NN) (RR))) (climsup.2 (-> (e. (cv k) (NN)) (br (` F (cv k)) (<_) (` F (opr (cv k) (+) (1)))))) (climsup.3 (E.e. x (RR) (A.e. k (NN) (br (` F (cv k)) (<_) (cv x)))))) (br F (~~>) (sup (ran F) (RR) (<))) (climsup.1 axresscn F (NN) (RR) (CC) fss mp2an climsup.1 F (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp climsup.1 F (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir F dm0rn0 mtbi climsup.3 k (NN) (br (` F (cv k)) (<_) (cv x)) hbra1 (br (cv y) (<_) (cv x)) k ax-17 k (NN) (br (` F (cv k)) (<_) (cv x)) ra4 (` F (cv k)) (cv y) (<_) (cv x) breq1 biimpcd syl6 r19.23ad climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv y) k fvelrn ax-mp syl5ib r19.21aiv x (RR) r19.22si ax-mp 3pm3.2i suprcli recn F (sup (ran F) (RR) (<)) x j k climfnn mp2an climsup.1 F (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp climsup.1 F (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir F dm0rn0 mtbi climsup.3 k (NN) (br (` F (cv k)) (<_) (cv x)) hbra1 (br (cv y) (<_) (cv x)) k ax-17 k (NN) (br (` F (cv k)) (<_) (cv x)) ra4 (` F (cv k)) (cv y) (<_) (cv x) breq1 biimpcd syl6 r19.23ad climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv y) k fvelrn ax-mp syl5ib r19.21aiv x (RR) r19.22si ax-mp 3pm3.2i suprcli (cv x) (sup (ran F) (RR) (<)) ltsubpost mpan2 climsup.1 F (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp climsup.1 F (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir F dm0rn0 mtbi climsup.3 k (NN) (br (` F (cv k)) (<_) (cv x)) hbra1 (br (cv z) (<_) (cv x)) k ax-17 k (NN) (br (` F (cv k)) (<_) (cv x)) ra4 (` F (cv k)) (cv z) (<_) (cv x) breq1 biimpcd syl6 r19.23ad climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv z) k fvelrn ax-mp syl5ib r19.21aiv x (RR) r19.22si ax-mp 3pm3.2i (opr (sup (ran F) (RR) (<)) (-) (cv x)) y suprlubi climsup.1 F (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp climsup.1 F (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir F dm0rn0 mtbi climsup.3 k (NN) (br (` F (cv k)) (<_) (cv x)) hbra1 (br (cv y) (<_) (cv x)) k ax-17 k (NN) (br (` F (cv k)) (<_) (cv x)) ra4 (` F (cv k)) (cv y) (<_) (cv x) breq1 biimpcd syl6 r19.23ad climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv y) k fvelrn ax-mp syl5ib r19.21aiv x (RR) r19.22si ax-mp 3pm3.2i suprcli (sup (ran F) (RR) (<)) (cv x) resubclt mpan sylan ex sylbid climsup.1 F (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp climsup.1 F (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir F dm0rn0 mtbi climsup.3 k (NN) (br (` F (cv k)) (<_) (cv x)) hbra1 (br (cv y) (<_) (cv x)) k ax-17 k (NN) (br (` F (cv k)) (<_) (cv x)) ra4 (` F (cv k)) (cv y) (<_) (cv x) breq1 biimpcd syl6 r19.23ad climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv y) k fvelrn ax-mp syl5ib r19.21aiv x (RR) r19.22si ax-mp 3pm3.2i suprcli (sup (ran F) (RR) (<)) (cv x) resubclt mpan (br (opr (sup (ran F) (RR) (<)) (-) (cv x)) (<) (cv y)) adantr (/\ (e. (cv j) (NN)) (= (` F (cv j)) (cv y))) (/\ (e. (cv k) (NN)) (br (cv j) (<_) (cv k))) ad2antrr climsup.1 (cv j) ffvrni (= (` F (cv j)) (cv y)) adantr (/\ (e. (cv x) (RR)) (br (opr (sup (ran F) (RR) (<)) (-) (cv x)) (<) (cv y))) (/\ (e. (cv k) (NN)) (br (cv j) (<_) (cv k))) ad2antlr climsup.1 (cv k) ffvrni (/\ (/\ (e. (cv x) (RR)) (br (opr (sup (ran F) (RR) (<)) (-) (cv x)) (<) (cv y))) (/\ (e. (cv j) (NN)) (= (` F (cv j)) (cv y)))) (br (cv j) (<_) (cv k)) ad2antrl (` F (cv j)) (cv y) (opr (sup (ran F) (RR) (<)) (-) (cv x)) (<) breq2 biimparc (e. (cv x) (RR)) (e. (cv j) (NN)) ad2ant2l (/\ (e. (cv k) (NN)) (br (cv j) (<_) (cv k))) adantr climsup.1 climsup.2 (cv j) (cv k) monoord 3expb (= (` F (cv j)) (cv y)) adantlr (/\ (e. (cv x) (RR)) (br (opr (sup (ran F) (RR) (<)) (-) (cv x)) (<) (cv y))) adantll ltletrd climsup.1 (cv k) ffvrni (` F (cv k)) recnt climsup.1 F (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp climsup.1 F (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir F dm0rn0 mtbi climsup.3 k (NN) (br (` F (cv k)) (<_) (cv x)) hbra1 (br (cv y) (<_) (cv x)) k ax-17 k (NN) (br (` F (cv k)) (<_) (cv x)) ra4 (` F (cv k)) (cv y) (<_) (cv x) breq1 biimpcd syl6 r19.23ad climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv y) k fvelrn ax-mp syl5ib r19.21aiv x (RR) r19.22si ax-mp 3pm3.2i suprcli recn (` F (cv k)) (sup (ran F) (RR) (<)) abssubt mpan2 3syl (opr (sup (ran F) (RR) (<)) (-) (` F (cv k))) absidt climsup.1 (cv k) ffvrni climsup.1 F (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp climsup.1 F (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir F dm0rn0 mtbi climsup.3 k (NN) (br (` F (cv k)) (<_) (cv x)) hbra1 (br (cv y) (<_) (cv x)) k ax-17 k (NN) (br (` F (cv k)) (<_) (cv x)) ra4 (` F (cv k)) (cv y) (<_) (cv x) breq1 biimpcd syl6 r19.23ad climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv y) k fvelrn ax-mp syl5ib r19.21aiv x (RR) r19.22si ax-mp 3pm3.2i suprcli jctil (sup (ran F) (RR) (<)) (` F (cv k)) resubclt syl climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv k) fnfvrn mpan climsup.1 F (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp climsup.1 F (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir F dm0rn0 mtbi climsup.3 k (NN) (br (` F (cv k)) (<_) (cv x)) hbra1 (br (cv y) (<_) (cv x)) k ax-17 k (NN) (br (` F (cv k)) (<_) (cv x)) ra4 (` F (cv k)) (cv y) (<_) (cv x) breq1 biimpcd syl6 r19.23ad climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv y) k fvelrn ax-mp syl5ib r19.21aiv x (RR) r19.22si ax-mp 3pm3.2i (` F (cv k)) suprubi syl climsup.1 (cv k) ffvrni climsup.1 F (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp climsup.1 F (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir F dm0rn0 mtbi climsup.3 k (NN) (br (` F (cv k)) (<_) (cv x)) hbra1 (br (cv y) (<_) (cv x)) k ax-17 k (NN) (br (` F (cv k)) (<_) (cv x)) ra4 (` F (cv k)) (cv y) (<_) (cv x) breq1 biimpcd syl6 r19.23ad climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv y) k fvelrn ax-mp syl5ib r19.21aiv x (RR) r19.22si ax-mp 3pm3.2i suprcli (sup (ran F) (RR) (<)) (` F (cv k)) subge0t mpan syl mpbird sylanc eqtrd (<) (cv x) breq1d (e. (cv x) (RR)) adantl climsup.1 F (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp climsup.1 F (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir F dm0rn0 mtbi climsup.3 k (NN) (br (` F (cv k)) (<_) (cv x)) hbra1 (br (cv y) (<_) (cv x)) k ax-17 k (NN) (br (` F (cv k)) (<_) (cv x)) ra4 (` F (cv k)) (cv y) (<_) (cv x) breq1 biimpcd syl6 r19.23ad climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv y) k fvelrn ax-mp syl5ib r19.21aiv x (RR) r19.22si ax-mp 3pm3.2i suprcli (sup (ran F) (RR) (<)) (cv x) (` F (cv k)) ltsub23t mp3an1 climsup.1 (cv k) ffvrni sylan2 bitr4d (br (opr (sup (ran F) (RR) (<)) (-) (cv x)) (<) (cv y)) (br (cv j) (<_) (cv k)) ad2ant2r (/\ (e. (cv j) (NN)) (= (` F (cv j)) (cv y))) adantlr mpbird exp32 r19.21aiv exp32 r19.22dv climsup.1 F (NN) (RR) ffn ax-mp F (NN) (cv y) j fvelrn ax-mp syl5ib ex com23 r19.23adv syld mprgbir)) thm (climcau ((x y) (x z) (w x) (v x) (F x) (y z) (w y) (v y) (F y) (w z) (v z) (F z) (v w) (F w) (F v) (A y) (A z) (A w) (A v)) ((climcau.1 (e. A (V))) (climcau.2 (br F (~~>) A)) (climcau.3 (:--> F (NN) (CC)))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) (<) (cv x))))))) ((cv x) halfpos2t (cv x) rehalfclt (cv w) (opr (cv x) (/) (2)) (0) (<) breq2 (cv w) (opr (cv x) (/) (2)) (` (abs) (opr (` F (cv v)) (-) A)) (<) breq2 (br (cv y) (<_) (cv v)) imbi2d y (NN) v (NN) rexralbidv imbi12d climcau.2 climcau.1 A (V) F (cv w) y v climcvgnn mpanl1 exp32 r19.21aiv ax-mp vtoclri syl sylbid (cv y) (cv z) ltlet (cv y) nnret (cv z) nnret syl2an (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (` (abs) (opr (` F (cv v)) (-) A)) (<) (opr (cv x) (/) (2))))) adantrl (cv v) (cv z) (cv y) (<_) breq2 (cv v) (cv z) F fveq2 (-) A opreq1d (abs) fveq2d (<) (opr (cv x) (/) (2)) breq1d imbi12d (NN) rcla4cva (e. (cv y) (NN)) adantl syld imp climcau.3 (cv y) ffvrni climcau.1 climcau.2 A (V) F climcl mp2an A (` F (cv y)) abssubt mpan syl (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (` (abs) (opr (` F (cv v)) (-) A)) (<) (opr (cv x) (/) (2))))) adantr (cv y) nnret (cv y) leidt syl (cv v) (cv y) (cv y) (<_) breq2 (cv v) (cv y) F fveq2 (-) A opreq1d (abs) fveq2d (<) (opr (cv x) (/) (2)) breq1d imbi12d (NN) rcla4v mpid imp eqbrtrd (e. (cv z) (NN)) adantrr (br (cv y) (<) (cv z)) adantr jca (e. (cv x) (RR)) adantlll (` (abs) (opr (` F (cv z)) (-) A)) (` (abs) (opr A (-) (` F (cv y)))) (opr (cv x) (/) (2)) (opr (cv x) (/) (2)) lt2addt climcau.3 (cv z) ffvrni climcau.1 climcau.2 A (V) F climcl mp2an (` F (cv z)) A subclt mpan2 syl (opr (` F (cv z)) (-) A) absclt syl (/\ (e. (cv x) (RR)) (e. (cv y) (NN))) adantl climcau.3 (cv y) ffvrni climcau.1 climcau.2 A (V) F climcl mp2an A (` F (cv y)) subclt mpan syl (opr A (-) (` F (cv y))) absclt syl (e. (cv x) (RR)) (e. (cv z) (NN)) ad2antlr jca (cv x) rehalfclt (e. (cv y) (NN)) (e. (cv z) (NN)) ad2antrr (cv x) rehalfclt (e. (cv y) (NN)) (e. (cv z) (NN)) ad2antrr jca sylanc (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (` (abs) (opr (` F (cv v)) (-) A)) (<) (opr (cv x) (/) (2))))) adantrl (br (cv y) (<) (cv z)) adantr mpd (cv x) recnt (cv x) 2halvest syl (e. (cv y) (NN)) adantr (/\ (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (` (abs) (opr (` F (cv v)) (-) A)) (<) (opr (cv x) (/) (2))))) (e. (cv z) (NN))) (br (cv y) (<) (cv z)) ad2antrr breqtrd ex climcau.1 climcau.2 A (V) F climcl mp2an (` F (cv z)) A (` F (cv y)) npncant mp3an2 climcau.3 (cv z) ffvrni climcau.3 (cv y) ffvrni syl2an (abs) fveq2d (opr (` F (cv z)) (-) A) (opr A (-) (` F (cv y))) abstrit climcau.3 (cv z) ffvrni climcau.1 climcau.2 A (V) F climcl mp2an (` F (cv z)) A subclt mpan2 syl climcau.3 (cv y) ffvrni climcau.1 climcau.2 A (V) F climcl mp2an A (` F (cv y)) subclt mpan syl syl2an eqbrtrrd (e. (cv x) (RR)) adantll (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) (opr (` (abs) (opr (` F (cv z)) (-) A)) (+) (` (abs) (opr A (-) (` F (cv y))))) (cv x) lelttrt (` F (cv z)) (` F (cv y)) subclt climcau.3 (cv z) ffvrni climcau.3 (cv y) ffvrni syl2an (opr (` F (cv z)) (-) (` F (cv y))) absclt syl (e. (cv x) (RR)) adantll (` (abs) (opr (` F (cv z)) (-) A)) (` (abs) (opr A (-) (` F (cv y)))) axaddrcl climcau.3 (cv z) ffvrni climcau.1 climcau.2 A (V) F climcl mp2an (` F (cv z)) A subclt mpan2 syl (opr (` F (cv z)) (-) A) absclt syl climcau.3 (cv y) ffvrni climcau.1 climcau.2 A (V) F climcl mp2an A (` F (cv y)) subclt mpan syl (opr A (-) (` F (cv y))) absclt syl syl2an (e. (cv x) (RR)) adantll (e. (cv x) (RR)) (e. (cv z) (NN)) pm3.26 (e. (cv y) (NN)) adantr syl3anc mpand an1rs (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (` (abs) (opr (` F (cv v)) (-) A)) (<) (opr (cv x) (/) (2))))) adantrl syld exp32 r19.21adv r19.22dva syld rgen)) thm (caucvglem1 ((v y) (u y) (A y) (u v) (A v) (A u) (y z) (w y) (u y) (v y) (F y) (w z) (u z) (v z) (F z) (u w) (v w) (F w) (u v) (F u) (F v) (S z) (S w)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (<-> (e. A S) (/\ (e. A (RR)) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br A (<) (` F (cv y)))))))) (caucvg.3 A eleq2i (cv u) A (<) (` F (cv y)) breq1 (br (cv v) (<_) (cv y)) imbi2d v (NN) y (NN) rexralbidv (RR) elrab bitr)) thm (caucvglem2 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (/\/\ (C_ S (RR)) (-. (= S ({/}))) (E.e. x (RR) (A.e. f S (br (cv f) (<_) (cv x))))) (caucvg.3 u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))) ssrab2 eqsstr caucvg.1 axresscn F (NN) (RR) (CC) fss mp2an caucvg.2 x caubnd (` F (cv y)) (cv x) absltt (br (-u (cv x)) (<) (` F (cv y))) (br (` F (cv y)) (<) (cv x)) pm3.26 syl6bi caucvg.1 (cv y) ffvrni sylan ancoms (br (1) (<_) (cv y)) a1dd r19.20dva 1nn (e. (cv x) (RR)) a1i jctild (cv v) (1) (<_) (cv y) breq1 (br (-u (cv x)) (<) (` F (cv y))) imbi1d y (NN) ralbidv (NN) rcla4ev syl6 (cv x) renegclt jctild caucvg.1 caucvg.2 caucvg.3 (-u (cv x)) caucvglem1 syl6ibr (-u (cv x)) S n0i syl6 r19.23aiv ax-mp caucvg.1 axresscn F (NN) (RR) (CC) fss mp2an caucvg.2 x caubnd caucvg.1 caucvg.2 caucvg.3 (cv f) caucvglem1 (cv v) nnret (cv v) leidt syl (cv y) (cv v) (cv v) (<_) breq2 (cv y) (cv v) F fveq2 (cv f) (<) breq2d imbi12d (NN) rcla4v mpid (e. (cv x) (RR)) adantr (cv y) (cv v) F fveq2 (abs) fveq2d (<) (cv x) breq1d (NN) rcla4v (e. (cv x) (RR)) adantr (` F (cv v)) (cv x) absltt caucvg.1 (cv v) ffvrni sylan (br (-u (cv x)) (<) (` F (cv v))) (br (` F (cv v)) (<) (cv x)) pm3.27 syl6bi syld anim12d (e. (cv f) (RR)) 3adant1 (cv f) (` F (cv v)) (cv x) axlttrn caucvg.1 (cv v) ffvrni syl3an2 (cv f) (cv x) ltlet (e. (cv v) (NN)) 3adant2 syld syld exp3a 3exp com34 r19.23adv imp sylbi com3l r19.21adv r19.22i ax-mp 3pm3.2i)) thm (caucvglem3 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (e. (sup S (RR) (<)) (RR)) (caucvg.1 caucvg.2 caucvg.3 x f caucvglem2 suprcli)) thm (caucvglem4 ((v y) (u y) (t y) (A y) (u v) (t v) (A v) (t u) (A u) (A t) (x y) (x z) (w x) (u x) (v x) (t x) (f x) (F x) (y z) (w y) (u y) (v y) (t y) (f y) (F y) (w z) (u z) (v z) (t z) (f z) (F z) (u w) (v w) (t w) (f w) (F w) (u v) (t u) (f u) (F u) (t v) (f v) (F v) (f t) (F t) (F f) (S x) (S z) (S w) (S t) (S f)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (-> (e. A (RR)) (-> (br A (<) (sup S (RR) (<))) (e. A S))) (caucvg.1 caucvg.2 caucvg.3 x f caucvglem2 A t suprlubi ex A (cv t) (` F (cv y)) axlttrn caucvg.1 (cv y) ffvrni syl3an3 exp3a 3exp com3l com4t imp41 (br (cv v) (<_) (cv y)) imim2d r19.20dva v (NN) r19.22sdv ex imp3a (e. A (RR)) (br A (<) (cv t)) pm3.26 jctild caucvg.1 caucvg.2 caucvg.3 (cv t) caucvglem1 caucvg.1 caucvg.2 caucvg.3 A caucvglem1 3imtr4g ex com23 r19.23adv syld)) thm (caucvglem5 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (-> (/\ (e. (cv x) (RR)) (e. (cv w) (NN))) (-> (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2))))) (br (-u (opr (cv x) (/) (2))) (<_) (opr (sup S (RR) (<)) (-) (` F (cv w)))))) ((cv w) (cv y) nnltp1let biimprd (e. (opr (cv x) (/) (2)) (RR)) adantll (opr (` F (cv y)) (-) (` F (cv w))) (opr (cv x) (/) (2)) absltt (br (-u (opr (cv x) (/) (2))) (<) (opr (` F (cv y)) (-) (` F (cv w)))) (br (opr (` F (cv y)) (-) (` F (cv w))) (<) (opr (cv x) (/) (2))) pm3.26 syl6bi (` F (cv y)) (` F (cv w)) resubclt sylan 3impa 3com13 (-u (opr (cv x) (/) (2))) (` F (cv w)) (` F (cv y)) ltaddsubt (opr (cv x) (/) (2)) renegclt syl3an1 sylibrd caucvg.1 (cv w) ffvrni syl3an2 caucvg.1 (cv y) ffvrni syl3an3 3expa imim12d r19.20dva (cv w) peano2nn (e. (opr (cv x) (/) (2)) (RR)) adantl jctild (cv v) (opr (cv w) (+) (1)) (<_) (cv y) breq1 (br (opr (-u (opr (cv x) (/) (2))) (+) (` F (cv w))) (<) (` F (cv y))) imbi1d y (NN) ralbidv (NN) rcla4ev syl6 (-u (opr (cv x) (/) (2))) (` F (cv w)) axaddrcl (opr (cv x) (/) (2)) renegclt caucvg.1 (cv w) ffvrni syl2an jctild caucvg.1 caucvg.2 caucvg.3 (opr (-u (opr (cv x) (/) (2))) (+) (` F (cv w))) caucvglem1 caucvg.1 caucvg.2 caucvg.3 x f caucvglem2 (opr (-u (opr (cv x) (/) (2))) (+) (` F (cv w))) suprubi sylbir syl6 caucvg.1 caucvg.2 caucvg.3 caucvglem3 (-u (opr (cv x) (/) (2))) (` F (cv w)) (sup S (RR) (<)) leaddsubt mp3an3 (opr (cv x) (/) (2)) renegclt caucvg.1 (cv w) ffvrni syl2an sylibd (cv x) rehalfclt sylan)) thm (caucvglem6 ((x y) (x z) (w x) (u x) (v x) (F x) (y z) (w y) (u y) (v y) (F y) (w z) (u z) (v z) (F z) (u w) (v w) (F w) (u v) (F u) (F v) (S x) (S z) (S w)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (-> (/\ (e. (cv x) (RR)) (e. (cv w) (NN))) (-> (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2))))) (br (opr (sup S (RR) (<)) (-) (` F (cv w))) (<_) (opr (cv x) (/) (2))))) ((cv y) (opr (cv w) (+) (cv v)) (opr (cv w) (+) (cv v)) (<_) breq2 (NN) rcla4ev (cv w) (cv v) nnaddclt (cv w) (cv v) nnaddclt (opr (cv w) (+) (cv v)) nnret syl (opr (cv w) (+) (cv v)) leidt syl sylanc (e. (opr (cv x) (/) (2)) (RR)) adantll (/\ (e. (opr (cv x) (/) (2)) (RR)) (/\ (e. (cv w) (NN)) (e. (cv v) (NN)))) y ax-17 y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2)))) hbra1 y (NN) (-> (br (cv v) (<_) (cv y)) (br (opr (opr (cv x) (/) (2)) (+) (` F (cv w))) (<) (` F (cv y)))) hbra1 hbne hbim (cv v) nngt0t (e. (cv w) (NN)) adantl (cv v) (cv w) ltaddpost ancoms (cv w) nnret (cv v) nnret syl2an mpbid (e. (cv y) (RR)) adantl (cv w) (opr (cv w) (+) (cv v)) (cv y) ltletrt (cv w) nnret (e. (cv y) (RR)) (e. (cv v) (NN)) ad2antrl (cv w) (cv v) nnaddclt (opr (cv w) (+) (cv v)) nnret syl (e. (cv y) (RR)) adantl (e. (cv y) (RR)) (/\ (e. (cv w) (NN)) (e. (cv v) (NN))) pm3.26 syl3anc mpand (cv y) nnret sylan (e. (opr (cv x) (/) (2)) (RR)) adantrl (opr (` F (cv y)) (-) (` F (cv w))) (opr (cv x) (/) (2)) absltt (br (-u (opr (cv x) (/) (2))) (<) (opr (` F (cv y)) (-) (` F (cv w)))) (br (opr (` F (cv y)) (-) (` F (cv w))) (<) (opr (cv x) (/) (2))) pm3.27 syl6bi ancoms (` F (cv y)) (` F (cv w)) resubclt sylan2 an1s (` F (cv y)) (` F (cv w)) (opr (cv x) (/) (2)) ltsubaddt 3com23 3expb sylibd (` F (cv y)) (opr (opr (cv x) (/) (2)) (+) (` F (cv w))) ltnsymt (opr (cv x) (/) (2)) (` F (cv w)) axaddrcl sylan2 syld caucvg.1 (cv y) ffvrni sylan caucvg.1 (cv w) ffvrni sylanr2 (e. (cv v) (NN)) adantrrr imim12d ex com23 a2i imp4d com12 y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2)))) ra4 syl5 (cv w) nngt0t (e. (cv v) (NN)) adantr (cv w) (cv v) ltaddpos2t (cv w) nnret (cv v) nnret syl2an mpbid (e. (cv y) (RR)) adantr (cv v) (opr (cv w) (+) (cv v)) (cv y) ltletrt (cv v) nnret (e. (cv w) (NN)) (e. (cv y) (RR)) ad2antlr (cv w) (cv v) nnaddclt (opr (cv w) (+) (cv v)) nnret syl (e. (cv y) (RR)) adantr (/\ (e. (cv w) (NN)) (e. (cv v) (NN))) (e. (cv y) (RR)) pm3.27 syl3anc mpand (cv v) (cv y) ltlet (cv v) nnret sylan (e. (cv w) (NN)) adantll syld (cv y) nnret sylan2 ancoms (br (opr (opr (cv x) (/) (2)) (+) (` F (cv w))) (<) (` F (cv y))) imim1d ex com23 a2i imp4d com12 y (NN) (-> (br (cv v) (<_) (cv y)) (br (opr (opr (cv x) (/) (2)) (+) (` F (cv w))) (<) (` F (cv y)))) ra4 syl5 (e. (opr (cv x) (/) (2)) (RR)) adantrll nsyld exp32 com12 r19.23ad anassrs mpd imp an1rs nrexdv ex (opr (cv x) (/) (2)) (` F (cv w)) axaddrcl caucvg.1 (cv w) ffvrni sylan2 caucvg.1 caucvg.2 caucvg.3 (opr (opr (cv x) (/) (2)) (+) (` F (cv w))) caucvglem4 caucvg.1 caucvg.2 caucvg.3 (opr (opr (cv x) (/) (2)) (+) (` F (cv w))) caucvglem1 pm3.27bd syl6 con3d syl syld (opr (cv x) (/) (2)) (` F (cv w)) axaddrcl caucvg.1 (cv w) ffvrni sylan2 caucvg.1 caucvg.2 caucvg.3 caucvglem3 (sup S (RR) (<)) (opr (opr (cv x) (/) (2)) (+) (` F (cv w))) lenltt mpan syl sylibrd caucvg.1 caucvg.2 caucvg.3 caucvglem3 (sup S (RR) (<)) (` F (cv w)) (opr (cv x) (/) (2)) lesubaddt mp3an1 caucvg.1 (cv w) ffvrni sylan ancoms sylibrd (cv x) rehalfclt sylan)) thm (caucvg ((x y) (x z) (w x) (u x) (v x) (t x) (F x) (y z) (w y) (u y) (v y) (t y) (F y) (w z) (u z) (v z) (t z) (F z) (u w) (v w) (t w) (F w) (u v) (t u) (F u) (t v) (F v) (F t) (S x) (S z) (S w) (S t)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (br F (~~>) (sup S (RR) (<))) (caucvg.2 (cv z) (opr (cv x) (/) (2)) (0) (<) breq2 (cv z) (opr (cv x) (/) (2)) (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) breq2 (br (cv w) (<) (cv y)) imbi2d w (NN) y (NN) rexralbidv imbi12d (RR) rcla4cv (/\ (e. (opr (cv x) (/) (2)) (RR)) (br (0) (<) (opr (cv x) (/) (2)))) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2)))))) biimt (e. (opr (cv x) (/) (2)) (RR)) (br (0) (<) (opr (cv x) (/) (2))) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2)))))) impexp syl6bb (cv x) rehalfclt (br (0) (<) (cv x)) adantr (cv x) halfpos2t biimpa sylanc caucvg.1 caucvg.2 caucvg.3 x caucvglem5 caucvg.1 caucvg.2 caucvg.3 x caucvglem6 jcad (opr (sup S (RR) (<)) (-) (` F (cv w))) (opr (cv x) (/) (2)) abslet caucvg.1 (cv w) ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 (sup S (RR) (<)) (` F (cv w)) resubclt mpan syl (e. (cv x) (RR)) adantl (cv x) rehalfclt (e. (cv w) (NN)) adantr sylanc sylibrd (e. (cv t) (NN)) adantr (` (abs) (opr (sup S (RR) (<)) (-) (` F (cv w)))) (opr (cv x) (/) (2)) (` (abs) (opr (` F (cv t)) (-) (` F (cv w)))) leadd1t caucvg.1 (cv w) ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 (sup S (RR) (<)) (` F (cv w)) resubclt mpan syl recnd (opr (sup S (RR) (<)) (-) (` F (cv w))) absclt syl (e. (cv x) (RR)) (e. (cv t) (NN)) ad2antlr (cv x) rehalfclt (e. (cv w) (NN)) (e. (cv t) (NN)) ad2antrr (` F (cv t)) (` F (cv w)) resubclt caucvg.1 (cv t) ffvrni caucvg.1 (cv w) ffvrni syl2an ancoms recnd (opr (` F (cv t)) (-) (` F (cv w))) absclt syl (e. (cv x) (RR)) adantll syl3anc sylibd caucvg.1 caucvg.2 caucvg.3 caucvglem3 recn (sup S (RR) (<)) (` F (cv w)) (` F (cv t)) npncant mp3an1 caucvg.1 (cv w) ffvrni recnd caucvg.1 (cv t) ffvrni recnd syl2an (abs) fveq2d (opr (sup S (RR) (<)) (-) (` F (cv w))) (opr (` F (cv w)) (-) (` F (cv t))) abstrit caucvg.1 (cv w) ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 (sup S (RR) (<)) (` F (cv w)) resubclt mpan syl recnd (e. (cv t) (NN)) adantr (` F (cv w)) (` F (cv t)) resubclt caucvg.1 (cv w) ffvrni caucvg.1 (cv t) ffvrni syl2an recnd sylanc eqbrtrrd caucvg.1 (cv t) ffvrni recnd caucvg.1 caucvg.2 caucvg.3 caucvglem3 recn (sup S (RR) (<)) (` F (cv t)) abssubt mpan syl (e. (cv w) (NN)) adantl (` F (cv w)) (` F (cv t)) abssubt caucvg.1 (cv w) ffvrni recnd caucvg.1 (cv t) ffvrni recnd syl2an (` (abs) (opr (sup S (RR) (<)) (-) (` F (cv w)))) (+) opreq2d 3brtr3d (e. (cv x) (RR)) adantll (` (abs) (opr (` F (cv t)) (-) (sup S (RR) (<)))) (opr (` (abs) (opr (sup S (RR) (<)) (-) (` F (cv w)))) (+) (` (abs) (opr (` F (cv t)) (-) (` F (cv w))))) (opr (opr (cv x) (/) (2)) (+) (` (abs) (opr (` F (cv t)) (-) (` F (cv w))))) letrt caucvg.1 (cv t) ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 (` F (cv t)) (sup S (RR) (<)) resubclt mpan2 syl recnd (opr (` F (cv t)) (-) (sup S (RR) (<))) absclt syl (/\ (e. (cv x) (RR)) (e. (cv w) (NN))) adantl (` (abs) (opr (sup S (RR) (<)) (-) (` F (cv w)))) (` (abs) (opr (` F (cv t)) (-) (` F (cv w)))) axaddrcl caucvg.1 (cv w) ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 (sup S (RR) (<)) (` F (cv w)) resubclt mpan syl recnd (opr (sup S (RR) (<)) (-) (` F (cv w))) absclt syl (e. (cv t) (NN)) adantr (` F (cv t)) (` F (cv w)) resubclt caucvg.1 (cv t) ffvrni caucvg.1 (cv w) ffvrni syl2an ancoms recnd (opr (` F (cv t)) (-) (` F (cv w))) absclt syl sylanc (e. (cv x) (RR)) adantll (opr (cv x) (/) (2)) (` (abs) (opr (` F (cv t)) (-) (` F (cv w)))) axaddrcl (cv x) rehalfclt (` F (cv t)) (` F (cv w)) resubclt caucvg.1 (cv t) ffvrni caucvg.1 (cv w) ffvrni syl2an ancoms recnd (opr (` F (cv t)) (-) (` F (cv w))) absclt syl syl2an anassrs syl3anc mpand syld (br (cv w) (<) (cv t)) adantrl (cv y) (cv t) (cv w) (<) breq2 (cv y) (cv t) F fveq2 (-) (` F (cv w)) opreq1d (abs) fveq2d (<) (opr (cv x) (/) (2)) breq1d imbi12d (NN) rcla4v com3r imp (/\ (e. (cv x) (RR)) (e. (cv w) (NN))) adantl (` (abs) (opr (` F (cv t)) (-) (` F (cv w)))) (opr (cv x) (/) (2)) (opr (cv x) (/) (2)) ltadd2t (` F (cv t)) (` F (cv w)) resubclt caucvg.1 (cv t) ffvrni caucvg.1 (cv w) ffvrni syl2an ancoms recnd (opr (` F (cv t)) (-) (` F (cv w))) absclt syl (e. (cv x) (RR)) adantll (cv x) rehalfclt (e. (cv w) (NN)) (e. (cv t) (NN)) ad2antrr (cv x) rehalfclt (e. (cv w) (NN)) (e. (cv t) (NN)) ad2antrr syl3anc (br (cv w) (<) (cv t)) adantrl (cv x) recnt (cv x) 2halvest syl (e. (cv w) (NN)) (/\ (br (cv w) (<) (cv t)) (e. (cv t) (NN))) ad2antrr (opr (opr (cv x) (/) (2)) (+) (` (abs) (opr (` F (cv t)) (-) (` F (cv w))))) (<) breq2d bitrd sylibd jcad (` (abs) (opr (` F (cv t)) (-) (sup S (RR) (<)))) (opr (opr (cv x) (/) (2)) (+) (` (abs) (opr (` F (cv t)) (-) (` F (cv w))))) (cv x) lelttrt caucvg.1 (cv t) ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 (` F (cv t)) (sup S (RR) (<)) resubclt mpan2 syl recnd (opr (` F (cv t)) (-) (sup S (RR) (<))) absclt syl (/\ (e. (cv x) (RR)) (e. (cv w) (NN))) adantl (opr (cv x) (/) (2)) (` (abs) (opr (` F (cv t)) (-) (` F (cv w)))) axaddrcl (cv x) rehalfclt (` F (cv t)) (` F (cv w)) resubclt caucvg.1 (cv t) ffvrni caucvg.1 (cv w) ffvrni syl2an ancoms recnd (opr (` F (cv t)) (-) (` F (cv w))) absclt syl syl2an anassrs (e. (cv x) (RR)) (e. (cv w) (NN)) pm3.26 (e. (cv t) (NN)) adantr syl3anc (br (cv w) (<) (cv t)) adantrl syld exp32 com24 r19.21adv r19.22dva (br (0) (<) (cv x)) adantr sylbird com12 exp3a syl r19.21aiv ax-mp x (RR) (br (0) (<) (cv x)) w t (br (` (abs) (opr (` F (cv t)) (-) (sup S (RR) (<)))) (<) (cv x)) cvg1 mpbi caucvg.1 axresscn F (NN) (RR) (CC) fss mp2an caucvg.1 caucvg.2 caucvg.3 caucvglem3 recn F (sup S (RR) (<)) x w t climfnn mp2an mpbir)) thm (caucvg3a ((v x) (u x) (F x) (u v) (F v) (F u) (x y) (x z) (w x) (G x) (y z) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (G v) (G u) (H x) (H y) (H z) (H w) (H v) (H u) (D x) (D u) (D v) (R x) (R z) (R w) (S z) (S w)) ((caucvg3a.1 (:--> F (NN) (CC))) (caucvg3a.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg3a.3 (Fn G (NN))) (caucvg3a.4 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` F (cv x)))))) (caucvg3a.4a (= R ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` G (cv y))))))))) (caucvg3a.5 (Fn H (NN))) (caucvg3a.6 (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (Im) (` F (cv x)))))) (caucvg3a.6a (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` H (cv y))))))))) (caucvg3a.7 (Fn D (NN))) (caucvg3a.8 (-> (e. (cv x) (NN)) (= (` D (cv x)) (opr (i) (x.) (` H (cv x))))))) (br F (~~>) (opr (sup R (RR) (<)) (+) (opr (i) (x.) (sup S (RR) (<))))) (G (NN) (RR) x ffnfv caucvg3a.3 caucvg3a.4 caucvg3a.1 (cv x) ffvrni (` F (cv x)) reclt syl eqeltrd rgen mpbir2an caucvg3a.1 caucvg3a.2 caucvg3a.3 caucvg3a.4 caure caucvg3a.4a caucvg axicn ltso S supex H (NN) (RR) x ffnfv caucvg3a.5 caucvg3a.6 caucvg3a.1 (cv x) ffvrni (` F (cv x)) imclt syl eqeltrd rgen mpbir2an caucvg3a.1 caucvg3a.2 caucvg3a.5 caucvg3a.6 cauim caucvg3a.6a caucvg caucvg3a.7 caucvg3a.6 caucvg3a.1 (cv x) ffvrni (` F (cv x)) imclt syl eqeltrd recnd caucvg3a.8 jca climmulc pm3.2i 1z (cv x) elnnuz caucvg3a.4 caucvg3a.1 (cv x) ffvrni (` F (cv x)) reclt syl recnd eqeltrd caucvg3a.8 caucvg3a.6 caucvg3a.1 (cv x) ffvrni (` F (cv x)) imclt syl eqeltrd recnd axicn (i) (` H (cv x)) axmulcl mpan syl eqeltrd caucvg3a.1 (cv x) ffvrni (` F (cv x)) replimt syl caucvg3a.4 caucvg3a.8 caucvg3a.6 caucvg3a.1 (cv x) ffvrni (` F (cv x)) imclt syl eqeltrd recnd axicn (i) (` H (cv x)) axmulcom mpan syl caucvg3a.6 (x.) (i) opreq1d 3eqtrd (+) opreq12d eqtr4d 3jca sylbir rgen pm3.2i caucvg3a.3 nnex G (NN) (V) fnex mp2an caucvg3a.7 nnex D (NN) (V) fnex mp2an caucvg3a.1 nnex F (NN) (CC) (V) fex mp2an ltso R supex (i) (x.) (sup S (RR) (<)) oprex (1) x climadd mp2an)) thm (ser1f0 ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A k) (A m) (A n)) ((ser1f0.1 (:--> F (NN) (CC))) (ser1f0.2 (e. A (CC))) (ser1f0.3 (br (opr (+) (seq1) F) (~~>) A))) (br F (~~>) (0)) (0cn (cv k) zret ax1re 2re 1lt2 ltlei ax1re 2re (1) (2) (cv k) letrt mp3an12 mpani syl (cv k) elnnz1 ser1f0.1 (cv k) ffvrni sylbir ex syld rgen ser1f0.1 nnex F (NN) (CC) (V) fex mp2an 2z (0) k x j clim3 mp2an ser1f0.2 ser1f0.3 A (CC) (opr (+) (seq1) F) (opr (cv x) (/) (2)) m n climcvg1 mpanl12 (cv x) rehalfclt (br (0) (<) (cv x)) adantr (cv x) halfpos2t biimpa sylanc (cv j) (opr (cv m) (+) (1)) (<_) (cv k) breq1 (br (2) (<_) (cv k)) anbi2d (br (` (abs) (opr (` F (cv k)) (-) (0))) (<) (cv x)) imbi1d k (ZZ) ralbidv (ZZ) rcla4ev (cv m) peano2z (e. (cv x) (RR)) (A.e. n (ZZ) (-> (br (cv m) (<_) (cv n)) (br (` (abs) (opr (` (opr (+) (seq1) F) (cv n)) (-) A)) (<) (opr (cv x) (/) (2))))) ad2antrl (cv m) (cv k) p1let 3expa ex (cv m) zret (cv k) zret syl2an ancoms (A.e. n (ZZ) (-> (br (cv m) (<_) (cv n)) (br (` (abs) (opr (` (opr (+) (seq1) F) (cv n)) (-) A)) (<) (opr (cv x) (/) (2))))) adantrr (cv n) (cv k) (cv m) (<_) breq2 (cv n) (cv k) (opr (+) (seq1) F) fveq2 (-) A opreq1d (abs) fveq2d (<) (opr (cv x) (/) (2)) breq1d imbi12d (ZZ) rcla4va (e. (cv m) (ZZ)) adantrl syld ax1re (cv m) (1) (cv k) leaddsubt mp3an2 (cv m) zret (cv k) zret syl2an ancoms (A.e. n (ZZ) (-> (br (cv m) (<_) (cv n)) (br (` (abs) (opr (` (opr (+) (seq1) F) (cv n)) (-) A)) (<) (opr (cv x) (/) (2))))) adantrr (cv k) peano2zm (cv n) (opr (cv k) (-) (1)) (cv m) (<_) breq2 (cv n) (opr (cv k) (-) (1)) (opr (+) (seq1) F) fveq2 (-) A opreq1d (abs) fveq2d (<) (opr (cv x) (/) (2)) breq1d imbi12d (ZZ) rcla4v syl imp (e. (cv m) (ZZ)) adantrl sylbid jcad (br (2) (<_) (cv k)) adantlr (e. (cv x) (RR)) adantrl (` (abs) (opr (` (opr (+) (seq1) F) (cv k)) (-) A)) (` (abs) (opr (` (opr (+) (seq1) F) (opr (cv k) (-) (1))) (-) A)) (opr (cv x) (/) (2)) (opr (cv x) (/) (2)) lt2addt ser1f0.1 (cv k) ser1cl ser1f0.2 (` (opr (+) (seq1) F) (cv k)) A subclt mpan2 syl (opr (` (opr (+) (seq1) F) (cv k)) (-) A) absclt syl (br (1) (<) (cv k)) adantr 1nn (1) (cv k) nnsubt mpan biimpa ser1f0.1 (opr (cv k) (-) (1)) ser1cl syl ser1f0.2 (` (opr (+) (seq1) F) (opr (cv k) (-) (1))) A subclt mpan2 syl (opr (` (opr (+) (seq1) F) (opr (cv k) (-) (1))) (-) A) absclt syl jca (cv x) rehalfclt (cv x) rehalfclt jca syl2an 1nn (1) (cv k) nnsubt mpan biimpa ser1f0.1 (opr (cv k) (-) (1)) ser1cl syl ser1f0.2 (` (opr (+) (seq1) F) (opr (cv k) (-) (1))) A abssubt mpan2 syl (` (abs) (opr (` (opr (+) (seq1) F) (cv k)) (-) A)) (+) opreq2d (cv x) recnt (cv x) 2halvest syl (<) breqan12d sylibd (opr (` (opr (+) (seq1) F) (cv k)) (-) A) (opr A (-) (` (opr (+) (seq1) F) (opr (cv k) (-) (1)))) abstrit ser1f0.1 (cv k) ser1cl ser1f0.2 (` (opr (+) (seq1) F) (cv k)) A subclt mpan2 syl (br (1) (<) (cv k)) adantr 1nn (1) (cv k) nnsubt mpan biimpa ser1f0.1 (opr (cv k) (-) (1)) ser1cl syl ser1f0.2 A (` (opr (+) (seq1) F) (opr (cv k) (-) (1))) subclt mpan syl sylanc (e. (cv x) (RR)) adantr (` (abs) (opr (opr (` (opr (+) (seq1) F) (cv k)) (-) A) (+) (opr A (-) (` (opr (+) (seq1) F) (opr (cv k) (-) (1)))))) (opr (` (abs) (opr (` (opr (+) (seq1) F) (cv k)) (-) A)) (+) (` (abs) (opr A (-) (` (opr (+) (seq1) F) (opr (cv k) (-) (1)))))) (cv x) lelttrt (opr (` (opr (+) (seq1) F) (cv k)) (-) A) (opr A (-) (` (opr (+) (seq1) F) (opr (cv k) (-) (1)))) axaddcl ser1f0.1 (cv k) ser1cl ser1f0.2 (` (opr (+) (seq1) F) (cv k)) A subclt mpan2 syl (br (1) (<) (cv k)) adantr 1nn (1) (cv k) nnsubt mpan biimpa ser1f0.1 (opr (cv k) (-) (1)) ser1cl syl ser1f0.2 A (` (opr (+) (seq1) F) (opr (cv k) (-) (1))) subclt mpan syl sylanc (opr (opr (` (opr (+) (seq1) F) (cv k)) (-) A) (+) (opr A (-) (` (opr (+) (seq1) F) (opr (cv k) (-) (1))))) absclt syl (e. (cv x) (RR)) adantr (` (abs) (opr (` (opr (+) (seq1) F) (cv k)) (-) A)) (` (abs) (opr A (-) (` (opr (+) (seq1) F) (opr (cv k) (-) (1))))) axaddrcl ser1f0.1 (cv k) ser1cl ser1f0.2 (` (opr (+) (seq1) F) (cv k)) A subclt mpan2 syl (opr (` (opr (+) (seq1) F) (cv k)) (-) A) absclt syl (br (1) (<) (cv k)) adantr 1nn (1) (cv k) nnsubt mpan biimpa ser1f0.1 (opr (cv k) (-) (1)) ser1cl syl ser1f0.2 A (` (opr (+) (seq1) F) (opr (cv k) (-) (1))) subclt mpan syl (opr A (-) (` (opr (+) (seq1) F) (opr (cv k) (-) (1)))) absclt syl sylanc (e. (cv x) (RR)) adantr (/\ (e. (cv k) (NN)) (br (1) (<) (cv k))) (e. (cv x) (RR)) pm3.27 syl3anc mpand addex ser1f0.1 nnex F (NN) (CC) (V) fex mp2an (cv k) seq1m1 eqcomd (` (opr (+) (seq1) F) (cv k)) (` (opr (+) (seq1) F) (opr (cv k) (-) (1))) (` F (cv k)) subaddt ser1f0.1 (cv k) ser1cl (br (1) (<) (cv k)) adantr 1nn (1) (cv k) nnsubt mpan biimpa ser1f0.1 (opr (cv k) (-) (1)) ser1cl syl ser1f0.1 (cv k) ffvrni (br (1) (<) (cv k)) adantr syl3anc mpbird ser1f0.2 (` (opr (+) (seq1) F) (cv k)) A (` (opr (+) (seq1) F) (opr (cv k) (-) (1))) npncant mp3an2 ser1f0.1 (cv k) ser1cl (br (1) (<) (cv k)) adantr 1nn (1) (cv k) nnsubt mpan biimpa ser1f0.1 (opr (cv k) (-) (1)) ser1cl syl sylanc ser1f0.1 (cv k) ffvrni (` F (cv k)) subid1t syl (br (1) (<) (cv k)) adantr 3eqtr4d (abs) fveq2d x equid (e. (cv x) (RR)) a1i (<) breqan12d sylibd syld 1z (1) (cv k) zltp1let mpan df-2 (<_) (cv k) breq1i syl6bbr (cv k) zret ax1re (1) (cv k) ltlet mpan syl sylbird imdistani (cv k) elnnz1 sylibr 1z (1) (cv k) zltp1let mpan df-2 (<_) (cv k) breq1i syl6bbr biimpar jca sylan (/\ (e. (cv m) (ZZ)) (A.e. n (ZZ) (-> (br (cv m) (<_) (cv n)) (br (` (abs) (opr (` (opr (+) (seq1) F) (cv n)) (-) A)) (<) (opr (cv x) (/) (2)))))) adantrr syld exp31 com3r imp4a r19.21aiv sylanc exp32 r19.23adv (br (0) (<) (cv x)) adantr mpd ex mprgbir)) thm (ser1const ((j k) (A j) (A k) (N j)) ((ser1const.1 (e. A (CC)))) (-> (e. N (NN)) (= (` (opr (+) (seq1) (X. (NN) ({} A))) N) (opr N (x.) A))) ((cv j) (1) (opr (+) (seq1) (X. (NN) ({} A))) fveq2 (cv j) (1) (x.) A opreq1 eqeq12d (cv j) (cv k) (opr (+) (seq1) (X. (NN) ({} A))) fveq2 (cv j) (cv k) (x.) A opreq1 eqeq12d (cv j) (opr (cv k) (+) (1)) (opr (+) (seq1) (X. (NN) ({} A))) fveq2 (cv j) (opr (cv k) (+) (1)) (x.) A opreq1 eqeq12d (cv j) N (opr (+) (seq1) (X. (NN) ({} A))) fveq2 (cv j) N (x.) A opreq1 eqeq12d 1nn ser1const.1 elisseti (1) (NN) fvconst2 ax-mp addex nnex A snex xpex seq11 ser1const.1 mulid2 3eqtr4 addex nnex A snex xpex (cv k) seq1p1 (cv k) peano2nn ser1const.1 elisseti (opr (cv k) (+) (1)) (NN) fvconst2 syl (` (opr (+) (seq1) (X. (NN) ({} A))) (cv k)) (+) opreq2d eqtrd (= (` (opr (+) (seq1) (X. (NN) ({} A))) (cv k)) (opr (cv k) (x.) A)) adantr (` (opr (+) (seq1) (X. (NN) ({} A))) (cv k)) (opr (cv k) (x.) A) (+) A opreq1 (cv k) nncnt 1cn ser1const.1 (cv k) (1) A adddirt mp3an23 syl ser1const.1 mulid2 (opr (cv k) (x.) A) (+) opreq2i syl6req sylan9eqr eqtrd ex nnind)) thm (ser10 () () (-> (e. N (NN)) (= (` (opr (+) (seq1) (X. (NN) ({} (0)))) N) (0))) (0cn N ser1const N nncnt N mul01t syl eqtrd)) thm (ser1clim0 ((j k) (j x) (k x)) () (br (opr (+) (seq1) (X. (NN) ({} (0)))) (~~>) (0)) (0cn elisseti (NN) fconst 0cn (0) (CC) snssi ax-mp (X. (NN) ({} (0))) (NN) ({} (0)) (CC) fss mp2an ser1f 0cn (opr (+) (seq1) (X. (NN) ({} (0)))) (0) x j k climfnn mp2an (cv k) ser10 (-) (0) opreq1d 0cn subid syl6eq (abs) fveq2d abs0 syl6eq (<) (cv x) breq1d biimprcd (br (1) (<_) (cv k)) a1dd r19.21aiv 1nn jctil (cv j) (1) (<_) (cv k) breq1 (br (` (abs) (opr (` (opr (+) (seq1) (X. (NN) ({} (0)))) (cv k)) (-) (0))) (<) (cv x)) imbi1d k (NN) ralbidv (NN) rcla4ev syl (e. (cv x) (RR)) a1i mprgbir)) thm (ser1cmp ((A y) (x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z)) ((ser1cmp.1 (:--> F (NN) (RR))) (ser1cmp.2 (:--> G (NN) (RR))) (ser1cmp.3 (-> (e. (cv x) (NN)) (br (` G (cv x)) (<_) (` F (cv x)))))) (-> (e. A (NN)) (br (` (opr (+) (seq1) G) A) (<_) (` (opr (+) (seq1) F) A))) ((cv y) (1) (opr (+) (seq1) G) fveq2 (cv y) (1) (opr (+) (seq1) F) fveq2 (<_) breq12d (cv y) (cv z) (opr (+) (seq1) G) fveq2 (cv y) (cv z) (opr (+) (seq1) F) fveq2 (<_) breq12d (cv y) (opr (cv z) (+) (1)) (opr (+) (seq1) G) fveq2 (cv y) (opr (cv z) (+) (1)) (opr (+) (seq1) F) fveq2 (<_) breq12d (cv y) A (opr (+) (seq1) G) fveq2 (cv y) A (opr (+) (seq1) F) fveq2 (<_) breq12d 1nn (cv x) (1) G fveq2 (cv x) (1) F fveq2 (<_) breq12d ser1cmp.3 vtoclga ax-mp addex ser1cmp.2 nnex G (NN) (RR) (V) fex mp2an seq11 addex ser1cmp.1 nnex F (NN) (RR) (V) fex mp2an seq11 3brtr4 (` (opr (+) (seq1) G) (cv z)) (` G (opr (cv z) (+) (1))) axaddrcl ser1cmp.2 (cv z) ser1recl (cv z) peano2nn ser1cmp.2 (opr (cv z) (+) (1)) ffvrni syl sylanc (br (` (opr (+) (seq1) G) (cv z)) (<_) (` (opr (+) (seq1) F) (cv z))) adantr (` (opr (+) (seq1) F) (cv z)) (` G (opr (cv z) (+) (1))) axaddrcl ser1cmp.1 (cv z) ser1recl (cv z) peano2nn ser1cmp.2 (opr (cv z) (+) (1)) ffvrni syl sylanc (br (` (opr (+) (seq1) G) (cv z)) (<_) (` (opr (+) (seq1) F) (cv z))) adantr (` (opr (+) (seq1) F) (cv z)) (` F (opr (cv z) (+) (1))) axaddrcl ser1cmp.1 (cv z) ser1recl (cv z) peano2nn ser1cmp.1 (opr (cv z) (+) (1)) ffvrni syl sylanc (br (` (opr (+) (seq1) G) (cv z)) (<_) (` (opr (+) (seq1) F) (cv z))) adantr (` (opr (+) (seq1) G) (cv z)) (` (opr (+) (seq1) F) (cv z)) (` G (opr (cv z) (+) (1))) leadd1t ser1cmp.2 (cv z) ser1recl ser1cmp.1 (cv z) ser1recl (cv z) peano2nn ser1cmp.2 (opr (cv z) (+) (1)) ffvrni syl syl3anc biimpa (cv z) peano2nn (cv x) (opr (cv z) (+) (1)) G fveq2 (cv x) (opr (cv z) (+) (1)) F fveq2 (<_) breq12d ser1cmp.3 vtoclga syl (` G (opr (cv z) (+) (1))) (` F (opr (cv z) (+) (1))) (` (opr (+) (seq1) F) (cv z)) leadd2t (cv z) peano2nn ser1cmp.2 (opr (cv z) (+) (1)) ffvrni syl (cv z) peano2nn ser1cmp.1 (opr (cv z) (+) (1)) ffvrni syl ser1cmp.1 (cv z) ser1recl syl3anc mpbid (br (` (opr (+) (seq1) G) (cv z)) (<_) (` (opr (+) (seq1) F) (cv z))) adantr letrd addex ser1cmp.2 nnex G (NN) (RR) (V) fex mp2an (cv z) seq1p1 (br (` (opr (+) (seq1) G) (cv z)) (<_) (` (opr (+) (seq1) F) (cv z))) adantr addex ser1cmp.1 nnex F (NN) (RR) (V) fex mp2an (cv z) seq1p1 (br (` (opr (+) (seq1) G) (cv z)) (<_) (` (opr (+) (seq1) F) (cv z))) adantr 3brtr4d ex nnind)) thm (ser1cmp0 ((F x)) ((ser1cmp0.1 (:--> F (NN) (RR))) (ser1cmp0.2 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))))) (-> (e. A (NN)) (br (0) (<_) (` (opr (+) (seq1) F) A))) (A ser10 ser1cmp0.1 0re elisseti (NN) fconst 0re (0) (RR) snssi ax-mp (X. (NN) ({} (0))) (NN) ({} (0)) (RR) fss mp2an 0re elisseti (cv x) (NN) fvconst2 ser1cmp0.2 eqbrtrd A ser1cmp eqbrtrrd)) thm (ser1cmp2lem ((A z) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F z) (G x) (G z) (S z)) ((ser1cmp2.1 (:--> F (NN) (RR))) (ser1cmp2.2 (:--> G (NN) (RR))) (ser1cmp2.3 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (ser1cmp2.4 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (ser1cmp2.5 (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<))))) (-> (/\ (e. A (NN)) (e. B (NN))) (-> (br A (<_) B) (br (` (opr (+) (seq1) G) A) (<_) (opr S (+) (` (opr (+) (seq1) F) A))))) (ser1cmp2.2 ser1re A B y seq1ub ser1cmp2.5 syl6breqr 3expa ex S (` (opr (+) (seq1) F) A) addge01t biimp3a ser1cmp2.2 ser1re B y x z seq1ublem (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) x z suprcl syl ser1cmp2.5 syl5eqel (e. A (NN)) adantl ser1cmp2.1 A ser1recl (e. B (NN)) adantr ser1cmp2.1 ser1cmp2.3 A ser1cmp0 (e. B (NN)) adantr syl3anc (` (opr (+) (seq1) G) A) S (opr S (+) (` (opr (+) (seq1) F) A)) letrt ser1cmp2.2 A ser1recl (e. B (NN)) adantr ser1cmp2.2 ser1re B y x z seq1ublem (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) x z suprcl syl ser1cmp2.5 syl5eqel (e. A (NN)) adantl S (` (opr (+) (seq1) F) A) axaddrcl ser1cmp2.2 ser1re B y x z seq1ublem (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) x z suprcl syl ser1cmp2.5 syl5eqel ser1cmp2.1 A ser1recl syl2an ancoms syl3anc mpan2d syld)) thm (ser1cmp2 ((A z) (x y) (w x) (x z) (B x) (w y) (y z) (B y) (w z) (B w) (B z) (F x) (F w) (F z) (G x) (G w) (G z) (S w) (S z)) ((ser1cmp2.1 (:--> F (NN) (RR))) (ser1cmp2.2 (:--> G (NN) (RR))) (ser1cmp2.3 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (ser1cmp2.4 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (ser1cmp2.5 (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<))))) (-> (/\ (e. A (NN)) (e. B (NN))) (br (` (opr (+) (seq1) G) A) (<_) (opr S (+) (` (opr (+) (seq1) F) A)))) ((cv z) (1) (opr (+) (seq1) G) fveq2 (cv z) (1) (opr (+) (seq1) F) fveq2 S (+) opreq2d (<_) breq12d (e. B (NN)) imbi2d (cv z) (cv w) (opr (+) (seq1) G) fveq2 (cv z) (cv w) (opr (+) (seq1) F) fveq2 S (+) opreq2d (<_) breq12d (e. B (NN)) imbi2d (cv z) (opr (cv w) (+) (1)) (opr (+) (seq1) G) fveq2 (cv z) (opr (cv w) (+) (1)) (opr (+) (seq1) F) fveq2 S (+) opreq2d (<_) breq12d (e. B (NN)) imbi2d (cv z) A (opr (+) (seq1) G) fveq2 (cv z) A (opr (+) (seq1) F) fveq2 S (+) opreq2d (<_) breq12d (e. B (NN)) imbi2d B nnge1t 1nn ser1cmp2.1 ser1cmp2.2 ser1cmp2.3 ser1cmp2.4 ser1cmp2.5 (1) ser1cmp2lem mpan mpd (opr (cv w) (+) (1)) B lelttrit (cv w) peano2nn (opr (cv w) (+) (1)) nnret syl B nnret syl2an ser1cmp2.1 ser1cmp2.2 ser1cmp2.3 ser1cmp2.4 ser1cmp2.5 (opr (cv w) (+) (1)) ser1cmp2lem (cv w) peano2nn sylan (br (` (opr (+) (seq1) G) (cv w)) (<_) (opr S (+) (` (opr (+) (seq1) F) (cv w)))) a1dd (` (opr (+) (seq1) G) (cv w)) (` G (opr (cv w) (+) (1))) axaddrcl ser1cmp2.2 (cv w) ser1recl (cv w) peano2nn ser1cmp2.2 (opr (cv w) (+) (1)) ffvrni syl sylanc (/\ (e. B (NN)) (br B (<) (opr (cv w) (+) (1)))) (br (` (opr (+) (seq1) G) (cv w)) (<_) (opr S (+) (` (opr (+) (seq1) F) (cv w)))) ad2antrr (opr S (+) (` (opr (+) (seq1) F) (cv w))) (` G (opr (cv w) (+) (1))) axaddrcl S (` (opr (+) (seq1) F) (cv w)) axaddrcl ser1cmp2.2 ser1re B y z w seq1ublem (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) z w suprcl syl ser1cmp2.5 syl5eqel ser1cmp2.1 (cv w) ser1recl syl2an ancoms (cv w) peano2nn ser1cmp2.2 (opr (cv w) (+) (1)) ffvrni syl (e. B (NN)) adantr sylanc (br B (<) (opr (cv w) (+) (1))) adantrr (br (` (opr (+) (seq1) G) (cv w)) (<_) (opr S (+) (` (opr (+) (seq1) F) (cv w)))) adantr (opr S (+) (` (opr (+) (seq1) F) (cv w))) (` F (opr (cv w) (+) (1))) axaddrcl S (` (opr (+) (seq1) F) (cv w)) axaddrcl ser1cmp2.2 ser1re B y z w seq1ublem (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) z w suprcl syl ser1cmp2.5 syl5eqel ser1cmp2.1 (cv w) ser1recl syl2an ancoms (cv w) peano2nn ser1cmp2.1 (opr (cv w) (+) (1)) ffvrni syl (e. B (NN)) adantr sylanc (br B (<) (opr (cv w) (+) (1))) adantrr (br (` (opr (+) (seq1) G) (cv w)) (<_) (opr S (+) (` (opr (+) (seq1) F) (cv w)))) adantr (` (opr (+) (seq1) G) (cv w)) (opr S (+) (` (opr (+) (seq1) F) (cv w))) (` G (opr (cv w) (+) (1))) leadd1t ser1cmp2.2 (cv w) ser1recl (e. B (NN)) adantr S (` (opr (+) (seq1) F) (cv w)) axaddrcl ser1cmp2.2 ser1re B y z w seq1ublem (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) z w suprcl syl ser1cmp2.5 syl5eqel ser1cmp2.1 (cv w) ser1recl syl2an ancoms (cv w) peano2nn ser1cmp2.2 (opr (cv w) (+) (1)) ffvrni syl (e. B (NN)) adantr syl3anc biimpa (br B (<) (opr (cv w) (+) (1))) adantlrr (cv x) (opr (cv w) (+) (1)) B (<) breq2 (cv x) (opr (cv w) (+) (1)) G fveq2 (cv x) (opr (cv w) (+) (1)) F fveq2 (<_) breq12d imbi12d ser1cmp2.4 ex vtoclga imp (cv w) peano2nn sylan (e. B (NN)) adantrl (` G (opr (cv w) (+) (1))) (` F (opr (cv w) (+) (1))) (opr S (+) (` (opr (+) (seq1) F) (cv w))) leadd2t (cv w) peano2nn ser1cmp2.2 (opr (cv w) (+) (1)) ffvrni syl (e. B (NN)) adantr (cv w) peano2nn ser1cmp2.1 (opr (cv w) (+) (1)) ffvrni syl (e. B (NN)) adantr S (` (opr (+) (seq1) F) (cv w)) axaddrcl ser1cmp2.2 ser1re B y z w seq1ublem (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) z w suprcl syl ser1cmp2.5 syl5eqel ser1cmp2.1 (cv w) ser1recl syl2an ancoms syl3anc (br B (<) (opr (cv w) (+) (1))) adantrr mpbid (br (` (opr (+) (seq1) G) (cv w)) (<_) (opr S (+) (` (opr (+) (seq1) F) (cv w)))) adantr letrd addex ser1cmp2.2 nnex G (NN) (RR) (V) fex mp2an (cv w) seq1p1 (/\ (e. B (NN)) (br B (<) (opr (cv w) (+) (1)))) (br (` (opr (+) (seq1) G) (cv w)) (<_) (opr S (+) (` (opr (+) (seq1) F) (cv w)))) ad2antrr addex ser1cmp2.1 nnex F (NN) (RR) (V) fex mp2an (cv w) seq1p1 S (+) opreq2d (e. B (NN)) adantr S (` (opr (+) (seq1) F) (cv w)) (` F (opr (cv w) (+) (1))) axaddass ser1cmp2.2 ser1re B y z w seq1ublem (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) z w suprcl syl ser1cmp2.5 syl5eqel recnd (e. (cv w) (NN)) adantl ser1cmp2.1 (cv w) ser1recl recnd (e. B (NN)) adantr (cv w) peano2nn ser1cmp2.1 (opr (cv w) (+) (1)) ffvrni syl recnd (e. B (NN)) adantr syl3anc eqtr4d (br B (<) (opr (cv w) (+) (1))) adantrr (br (` (opr (+) (seq1) G) (cv w)) (<_) (opr S (+) (` (opr (+) (seq1) F) (cv w)))) adantr 3brtr4d ex anassrs ex jaod mpd ex a2d nnind imp)) thm (cvgcmp2lem ((x z) (A x) (A z) (x y) (w x) (B x) (y z) (w y) (B y) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w) (S x) (S z) (S w) (H x) (H z)) ((cvgcmp2.1 (e. A (V))) (cvgcmp2.2 (:--> F (NN) (RR))) (cvgcmp2.3 (:--> G (NN) (RR))) (cvgcmp2.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp2.5 (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x))))) (cvgcmp2.6 (br (opr (+) (seq1) F) (~~>) A)) (cvgcmp2.7 (e. B (NN))) (cvgcmp2.8 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (cvgcmp2lem.9 (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)))) (cvgcmp2lem.10 (Fn H (NN))) (cvgcmp2lem.11 (-> (e. (cv x) (NN)) (= (` H (cv x)) (opr S (+) (` (opr (+) (seq1) F) (cv x))))))) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (cvgcmp2.3 ser1re cvgcmp2.3 cvgcmp2.5 (cv x) ser1mono cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr cvgcmp2.1 cvgcmp2.2 ser1re cvgcmp2.6 climfnrcl readdcl cvgcmp2.3 (cv x) ser1recl cvgcmp2lem.11 cvgcmp2.2 (cv x) ser1recl cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr S (` (opr (+) (seq1) F) (cv x)) axaddrcl mpan syl eqeltrd cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr cvgcmp2.1 cvgcmp2.2 ser1re cvgcmp2.6 climfnrcl readdcl (e. (cv x) (NN)) a1i cvgcmp2.7 cvgcmp2.2 cvgcmp2.3 cvgcmp2.4 cvgcmp2.8 cvgcmp2lem.9 (cv x) ser1cmp2 mpan2 cvgcmp2lem.11 breqtrrd S (+) A oprex H (NN) (RR) x ffnfv cvgcmp2lem.10 cvgcmp2lem.11 cvgcmp2.2 (cv x) ser1recl cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr S (` (opr (+) (seq1) F) (cv x)) axaddrcl mpan syl eqeltrd rgen mpbir2an cvgcmp2.2 cvgcmp2.4 (cv z) ser1mono cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr (` (opr (+) (seq1) F) (cv z)) (` (opr (+) (seq1) F) (opr (cv z) (+) (1))) S leadd2t mp3an3 cvgcmp2.2 (cv z) ser1recl (cv z) peano2nn cvgcmp2.2 (opr (cv z) (+) (1)) ser1recl syl sylanc mpbid (cv x) (cv z) H fveq2 (cv x) (cv z) (opr (+) (seq1) F) fveq2 S (+) opreq2d eqeq12d cvgcmp2lem.11 vtoclga (cv z) peano2nn (cv x) (opr (cv z) (+) (1)) H fveq2 (cv x) (opr (cv z) (+) (1)) (opr (+) (seq1) F) fveq2 S (+) opreq2d eqeq12d cvgcmp2lem.11 vtoclga syl 3brtr4d cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr recn cvgcmp2.1 cvgcmp2.6 cvgcmp2lem.10 cvgcmp2.2 (cv x) ser1recl recnd cvgcmp2lem.11 jca climaddc (cv x) climub letrd rgen (cv z) (opr S (+) A) (` (opr (+) (seq1) G) (cv x)) (<_) breq2 x (NN) ralbidv (RR) rcla4ev mp2an climsup)) thm (cvgcmp2 ((x z) (A x) (A z) (x y) (w x) (B x) (y z) (w y) (B y) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w)) ((cvgcmp2.1 (e. A (V))) (cvgcmp2.2 (:--> F (NN) (RR))) (cvgcmp2.3 (:--> G (NN) (RR))) (cvgcmp2.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp2.5 (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x))))) (cvgcmp2.6 (br (opr (+) (seq1) F) (~~>) A)) (cvgcmp2.7 (e. B (NN))) (cvgcmp2.8 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))))) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (cvgcmp2.1 cvgcmp2.2 cvgcmp2.3 cvgcmp2.4 cvgcmp2.5 cvgcmp2.6 cvgcmp2.7 cvgcmp2.8 (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)) eqid (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)) (+) (` (opr (+) (seq1) F) (cv z)) oprex ({<,>|} z w (/\ (e. (cv z) (NN)) (= (cv w) (opr (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)) (+) (` (opr (+) (seq1) F) (cv z)))))) eqid fnopab2 (cv z) (cv x) (opr (+) (seq1) F) fveq2 (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)) (+) opreq2d ({<,>|} z w (/\ (e. (cv z) (NN)) (= (cv w) (opr (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)) (+) (` (opr (+) (seq1) F) (cv z)))))) eqid (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)) (+) (` (opr (+) (seq1) F) (cv x)) oprex fvopab4 cvgcmp2lem)) thm (cvgcmp2clem ((A x) (B x) (F x) (G x) (S x) (C x)) ((cvgcmp2.1 (e. A (V))) (cvgcmp2.2 (:--> F (NN) (RR))) (cvgcmp2.3 (:--> G (NN) (RR))) (cvgcmp2.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp2.5 (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x))))) (cvgcmp2.6 (br (opr (+) (seq1) F) (~~>) A)) (cvgcmp2.7 (e. B (NN))) (cvgcmp2c.9 (e. C (RR))) (cvgcmp2c.10 (br (0) (<_) C)) (cvgcmp2c.11 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (opr C (x.) (` F (cv x)))))) (cvgcmp2clem.12 (Fn S (NN))) (cvgcmp2clem.13 (-> (e. (cv x) (NN)) (= (` S (cv x)) (opr C (x.) (` F (cv x))))))) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (C (x.) A oprex S (NN) (RR) x ffnfv cvgcmp2clem.12 cvgcmp2clem.13 cvgcmp2.2 (cv x) ffvrni cvgcmp2c.9 C (` F (cv x)) axmulrcl mpan syl eqeltrd rgen mpbir2an cvgcmp2.3 cvgcmp2c.9 C (` F (cv x)) mulge0tOLD mpan cvgcmp2.2 (cv x) ffvrni cvgcmp2.4 cvgcmp2c.10 jctil sylc cvgcmp2clem.13 breqtrrd cvgcmp2.5 cvgcmp2c.9 recn cvgcmp2.1 cvgcmp2.6 addex cvgcmp2clem.12 nnex S (NN) (V) fnex mp2an seq1fn cvgcmp2.2 axresscn F (NN) (RR) (CC) fss mp2an (cv x) ser1cl cvgcmp2c.9 recn cvgcmp2.2 axresscn F (NN) (RR) (CC) fss mp2an cvgcmp2clem.12 nnex S (NN) (V) fnex mp2an cvgcmp2clem.13 (cv x) ser1mulc jca climmulc cvgcmp2.7 cvgcmp2c.11 cvgcmp2clem.13 (br B (<) (cv x)) adantr breqtrrd cvgcmp2)) thm (cvgcmp2c ((x z) (A x) (A z) (w x) (B x) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w) (C x) (C z) (C w)) ((cvgcmp2.1 (e. A (V))) (cvgcmp2.2 (:--> F (NN) (RR))) (cvgcmp2.3 (:--> G (NN) (RR))) (cvgcmp2.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp2.5 (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x))))) (cvgcmp2.6 (br (opr (+) (seq1) F) (~~>) A)) (cvgcmp2.7 (e. B (NN))) (cvgcmp2c.9 (e. C (RR))) (cvgcmp2c.10 (br (0) (<_) C)) (cvgcmp2c.11 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (opr C (x.) (` F (cv x))))))) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (cvgcmp2.1 cvgcmp2.2 cvgcmp2.3 cvgcmp2.4 cvgcmp2.5 cvgcmp2.6 cvgcmp2.7 cvgcmp2c.9 cvgcmp2c.10 cvgcmp2c.11 C (x.) (` F (cv z)) oprex ({<,>|} z w (/\ (e. (cv z) (NN)) (= (cv w) (opr C (x.) (` F (cv z)))))) eqid fnopab2 (cv z) (cv x) F fveq2 C (x.) opreq2d ({<,>|} z w (/\ (e. (cv z) (NN)) (= (cv w) (opr C (x.) (` F (cv z)))))) eqid C (x.) (` F (cv x)) oprex fvopab4 cvgcmp2clem)) thm (cvgcmp ((A x) (F x) (G x)) ((cvgcmp.1 (e. A (V))) (cvgcmp.2 (:--> F (NN) (RR))) (cvgcmp.3 (:--> G (NN) (RR))) (cvgcmp.4 (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))))) (cvgcmp.5 (br (opr (+) (seq1) F) (~~>) A))) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (cvgcmp.1 cvgcmp.2 cvgcmp.3 0re (0) (` G (cv x)) (` F (cv x)) letrt mp3an1 cvgcmp.3 (cv x) ffvrni cvgcmp.2 (cv x) ffvrni jca cvgcmp.4 sylc cvgcmp.4 pm3.26d cvgcmp.5 1nn cvgcmp.4 pm3.27d (br (1) (<) (cv x)) adantr cvgcmp2)) thm (cvgcmpub ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (G x) (G y) (G z) (G w)) ((cvgcmp.1 (e. A (V))) (cvgcmp.2 (:--> F (NN) (RR))) (cvgcmp.3 (:--> G (NN) (RR))) (cvgcmp.4 (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))))) (cvgcmp.5 (br (opr (+) (seq1) F) (~~>) A)) (cvgcmpub.6 (e. B (V))) (cvgcmpub.7 (br (opr (+) (seq1) G) (~~>) B))) (br B (<_) A) (cvgcmpub.6 cvgcmp.3 ser1re (opr (+) (seq1) G) (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp cvgcmp.3 ser1re (opr (+) (seq1) G) (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir (opr (+) (seq1) G) dm0rn0 mtbi cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climfnrcl cvgcmp.3 ser1re (opr (+) (seq1) G) (NN) (RR) ffn ax-mp (opr (+) (seq1) G) (NN) (cv z) x fvelrn ax-mp (` (opr (+) (seq1) G) (cv x)) (cv z) (<_) A breq1 cvgcmp.3 (cv x) ser1recl cvgcmp.2 (cv x) ser1recl cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climfnrcl (e. (cv x) (NN)) a1i cvgcmp.2 cvgcmp.3 cvgcmp.4 pm3.27d (cv x) ser1cmp cvgcmp.1 cvgcmp.2 ser1re cvgcmp.2 0re (0) (` G (cv x)) (` F (cv x)) letrt mp3an1 cvgcmp.3 (cv x) ffvrni cvgcmp.2 (cv x) ffvrni jca cvgcmp.4 sylc (cv x) ser1mono cvgcmp.5 (cv x) climub letrd syl5bi com12 r19.23aiv sylbi rgen (cv y) A (cv z) (<_) breq2 z (ran (opr (+) (seq1) G)) ralbidv (RR) rcla4ev mp2an 3pm3.2i suprcli elisseti cvgcmpub.7 cvgcmp.1 cvgcmp.2 cvgcmp.3 cvgcmp.4 cvgcmp.5 cvgcmp pm3.2i climunii cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climfnrcl cvgcmp.3 ser1re (opr (+) (seq1) G) (NN) (RR) ffn ax-mp (opr (+) (seq1) G) (NN) (cv w) x fvelrn ax-mp (` (opr (+) (seq1) G) (cv x)) (cv w) (<_) A breq1 cvgcmp.3 (cv x) ser1recl cvgcmp.2 (cv x) ser1recl cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climfnrcl (e. (cv x) (NN)) a1i cvgcmp.2 cvgcmp.3 cvgcmp.4 pm3.27d (cv x) ser1cmp cvgcmp.1 cvgcmp.2 ser1re cvgcmp.2 0re (0) (` G (cv x)) (` F (cv x)) letrt mp3an1 cvgcmp.3 (cv x) ffvrni cvgcmp.2 (cv x) ffvrni jca cvgcmp.4 sylc (cv x) ser1mono cvgcmp.5 (cv x) climub letrd syl5bi com12 r19.23aiv sylbi rgen cvgcmp.3 ser1re (opr (+) (seq1) G) (NN) (RR) frn ax-mp 1nn (1) (NN) n0i ax-mp cvgcmp.3 ser1re (opr (+) (seq1) G) (NN) (RR) fdm ax-mp ({/}) eqeq1i mtbir (opr (+) (seq1) G) dm0rn0 mtbi cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climfnrcl cvgcmp.3 ser1re (opr (+) (seq1) G) (NN) (RR) ffn ax-mp (opr (+) (seq1) G) (NN) (cv z) x fvelrn ax-mp (` (opr (+) (seq1) G) (cv x)) (cv z) (<_) A breq1 cvgcmp.3 (cv x) ser1recl cvgcmp.2 (cv x) ser1recl cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climfnrcl (e. (cv x) (NN)) a1i cvgcmp.2 cvgcmp.3 cvgcmp.4 pm3.27d (cv x) ser1cmp cvgcmp.1 cvgcmp.2 ser1re cvgcmp.2 0re (0) (` G (cv x)) (` F (cv x)) letrt mp3an1 cvgcmp.3 (cv x) ffvrni cvgcmp.2 (cv x) ffvrni jca cvgcmp.4 sylc (cv x) ser1mono cvgcmp.5 (cv x) climub letrd syl5bi com12 r19.23aiv sylbi rgen (cv y) A (cv z) (<_) breq2 z (ran (opr (+) (seq1) G)) ralbidv (RR) rcla4ev mp2an 3pm3.2i A w suprleubi mp2an eqbrtr)) thm (cvgcmp3c ((Q x) (A x) (B x) (F x) (x y) (x z) (w x) (v x) (u x) (G x) (y z) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (u v) (G v) (G u) (H x) (H z) (H w) (H v) (H u) (T x) (T y) (T u) (R x) (R z) (R w) (S z) (S w) (D x) (D y) (D u) (C x) (C y) (C z) (C w) (C v) (C u)) ((cvgcmp3.1 (:--> F (NN) (RR))) (cvgcmp3.2 (:--> T (NN) (CC))) (cvgcmp3.3 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp3.6 (e. A (V))) (cvgcmp3.7 (br (opr (+) (seq1) F) (~~>) A)) (cvgcmp3.8 (Fn H (NN))) (cvgcmp3.9 (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (abs) (` T (cv x)))))) (cvgcmp3.10 (Fn G (NN))) (cvgcmp3.11 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` (opr (+) (seq1) T) (cv x)))))) (cvgcmp3.12 (= R ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` G (cv v))))))))) (cvgcmp3.13 (Fn C (NN))) (cvgcmp3.14 (-> (e. (cv x) (NN)) (= (` C (cv x)) (` (Im) (` (opr (+) (seq1) T) (cv x)))))) (cvgcmp3.15 (= S ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` C (cv v))))))))) (cvgcmp3.16 (Fn D (NN))) (cvgcmp3.17 (-> (e. (cv x) (NN)) (= (` D (cv x)) (opr (i) (x.) (` C (cv x)))))) (cvgcmp3.4 (e. B (NN))) (cvgcmp3c.18 (e. Q (RR))) (cvgcmp3c.19 (br (0) (<_) Q)) (cvgcmp3c.5 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` (abs) (` T (cv x))) (<_) (opr Q (x.) (` F (cv x))))))) (br (opr (+) (seq1) T) (~~>) (opr (sup R (RR) (<)) (+) (opr (i) (x.) (sup S (RR) (<))))) (cvgcmp3.2 ser1f ltso (ran (opr (+) (seq1) H)) supex cvgcmp3.6 cvgcmp3.1 H (NN) (RR) x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 (cv x) ffvrni (` T (cv x)) absclt syl eqeltrd rgen mpbir2an cvgcmp3.3 cvgcmp3.2 (cv x) ffvrni (` T (cv x)) absge0t syl cvgcmp3.9 breqtrrd cvgcmp3.7 cvgcmp3.4 cvgcmp3c.18 cvgcmp3c.19 cvgcmp3.9 (br B (<) (cv x)) adantr cvgcmp3c.5 eqbrtrd cvgcmp2c H (NN) (RR) x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 (cv x) ffvrni (` T (cv x)) absclt syl eqeltrd rgen mpbir2an axresscn H (NN) (RR) (CC) fss mp2an ser1f z w v climcau (` (opr (+) (seq1) T) (cv v)) (` (opr (+) (seq1) T) (cv w)) subclt cvgcmp3.2 (cv v) ser1cl cvgcmp3.2 (cv w) ser1cl syl2an (opr (` (opr (+) (seq1) T) (cv v)) (-) (` (opr (+) (seq1) T) (cv w))) absclt syl (br (cv w) (<) (cv v)) adantr (` (opr (+) (seq1) H) (cv v)) (` (opr (+) (seq1) H) (cv w)) resubclt H (NN) (RR) x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 (cv x) ffvrni (` T (cv x)) absclt syl eqeltrd rgen mpbir2an (cv v) ser1recl H (NN) (RR) x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 (cv x) ffvrni (` T (cv x)) absclt syl eqeltrd rgen mpbir2an (cv w) ser1recl syl2an (br (cv w) (<) (cv v)) adantr (` (opr (+) (seq1) H) (cv v)) (` (opr (+) (seq1) H) (cv w)) resubclt H (NN) (RR) x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 (cv x) ffvrni (` T (cv x)) absclt syl eqeltrd rgen mpbir2an (cv v) ser1recl H (NN) (RR) x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 (cv x) ffvrni (` T (cv x)) absclt syl eqeltrd rgen mpbir2an (cv w) ser1recl syl2an recnd (opr (` (opr (+) (seq1) H) (cv v)) (-) (` (opr (+) (seq1) H) (cv w))) absclt syl (br (cv w) (<) (cv v)) adantr cvgcmp3.2 cvgcmp3.8 cvgcmp3.9 (cv w) (cv v) ser1absdif 3com12 3expa (` (opr (+) (seq1) H) (cv v)) (` (opr (+) (seq1) H) (cv w)) resubclt H (NN) (RR) x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 (cv x) ffvrni (` T (cv x)) absclt syl eqeltrd rgen mpbir2an (cv v) ser1recl H (NN) (RR) x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 (cv x) ffvrni (` T (cv x)) absclt syl eqeltrd rgen mpbir2an (cv w) ser1recl syl2an (opr (` (opr (+) (seq1) H) (cv v)) (-) (` (opr (+) (seq1) H) (cv w))) leabst syl (br (cv w) (<) (cv v)) adantr letrd (e. (cv z) (RR)) adantlll (` (abs) (opr (` (opr (+) (seq1) T) (cv v)) (-) (` (opr (+) (seq1) T) (cv w)))) (` (abs) (opr (` (opr (+) (seq1) H) (cv v)) (-) (` (opr (+) (seq1) H) (cv w)))) (cv z) lelttrt (` (opr (+) (seq1) T) (cv v)) (` (opr (+) (seq1) T) (cv w)) subclt cvgcmp3.2 (cv v) ser1cl cvgcmp3.2 (cv w) ser1cl syl2an (opr (` (opr (+) (seq1) T) (cv v)) (-) (` (opr (+) (seq1) T) (cv w))) absclt syl (e. (cv z) (RR)) adantll (` (opr (+) (seq1) H) (cv v)) (` (opr (+) (seq1) H) (cv w)) resubclt H (NN) (RR) x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 (cv x) ffvrni (` T (cv x)) absclt syl eqeltrd rgen mpbir2an (cv v) ser1recl H (NN) (RR) x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 (cv x) ffvrni (` T (cv x)) absclt syl eqeltrd rgen mpbir2an (cv w) ser1recl syl2an recnd (opr (` (opr (+) (seq1) H) (cv v)) (-) (` (opr (+) (seq1) H) (cv w))) absclt syl (e. (cv z) (RR)) adantll (e. (cv z) (RR)) (e. (cv v) (NN)) pm3.26 (e. (cv w) (NN)) adantr syl3anc (br (cv w) (<) (cv v)) adantr mpand ex an1rs a2d r19.20dva r19.22dva (br (0) (<) (cv z)) imim2d r19.20i ax-mp cvgcmp3.10 cvgcmp3.11 cvgcmp3.12 cvgcmp3.13 cvgcmp3.14 cvgcmp3.15 cvgcmp3.16 cvgcmp3.17 caucvg3a)) thm (cvgcmp3ce ((A x) (B x) (F x) (x y) (x z) (w x) (v x) (u x) (t x) (h x) (g x) (G x) (y z) (w y) (v y) (u y) (t y) (h y) (g y) (G y) (w z) (v z) (u z) (t z) (h z) (g z) (G z) (v w) (u w) (t w) (h w) (g w) (G w) (u v) (t v) (h v) (g v) (G v) (t u) (h u) (g u) (G u) (h t) (g t) (G t) (g h) (G h) (G g) (C x)) ((cvgcmp3ce.1 (e. A (V))) (cvgcmp3ce.2 (e. B (NN))) (cvgcmp3ce.3 (:--> F (NN) (RR))) (cvgcmp3ce.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp3ce.5 (br (opr (+) (seq1) F) (~~>) A)) (cvgcmp3ce.6 (e. C (RR))) (cvgcmp3ce.7 (br (0) (<_) C)) (cvgcmp3ce.8 (:--> G (NN) (CC))) (cvgcmp3ce.9 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x))))))) (E. y (br (opr (+) (seq1) G) (~~>) (cv y))) (cvgcmp3ce.3 cvgcmp3ce.8 cvgcmp3ce.4 cvgcmp3ce.1 cvgcmp3ce.5 (abs) (` G (cv u)) fvex ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (abs) (` G (cv u)))))) eqid fnopab2 (cv u) (cv x) G fveq2 (abs) fveq2d ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (abs) (` G (cv u)))))) eqid (abs) (` G (cv x)) fvex fvopab4 (Re) (` (opr (+) (seq1) G) (cv u)) fvex ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Re) (` (opr (+) (seq1) G) (cv u)))))) eqid fnopab2 (cv u) (cv x) (opr (+) (seq1) G) fveq2 (Re) fveq2d ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Re) (` (opr (+) (seq1) G) (cv u)))))) eqid (Re) (` (opr (+) (seq1) G) (cv x)) fvex fvopab4 ({e.|} z (RR) (E.e. w (NN) (A.e. v (NN) (-> (br (cv w) (<_) (cv v)) (br (cv z) (<) (` ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Re) (` (opr (+) (seq1) G) (cv u)))))) (cv v))))))) eqid (Im) (` (opr (+) (seq1) G) (cv u)) fvex ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Im) (` (opr (+) (seq1) G) (cv u)))))) eqid fnopab2 (cv u) (cv x) (opr (+) (seq1) G) fveq2 (Im) fveq2d ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Im) (` (opr (+) (seq1) G) (cv u)))))) eqid (Im) (` (opr (+) (seq1) G) (cv x)) fvex fvopab4 ({e.|} z (RR) (E.e. w (NN) (A.e. v (NN) (-> (br (cv w) (<_) (cv v)) (br (cv z) (<) (` ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Im) (` (opr (+) (seq1) G) (cv u)))))) (cv v))))))) eqid (i) (x.) (` ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Im) (` (opr (+) (seq1) G) (cv u)))))) (cv h)) oprex ({<,>|} h g (/\ (e. (cv h) (NN)) (= (cv g) (opr (i) (x.) (` ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Im) (` (opr (+) (seq1) G) (cv u)))))) (cv h)))))) eqid fnopab2 (cv h) (cv x) ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Im) (` (opr (+) (seq1) G) (cv u)))))) fveq2 (i) (x.) opreq2d ({<,>|} h g (/\ (e. (cv h) (NN)) (= (cv g) (opr (i) (x.) (` ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Im) (` (opr (+) (seq1) G) (cv u)))))) (cv h)))))) eqid (i) (x.) (` ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Im) (` (opr (+) (seq1) G) (cv u)))))) (cv x)) oprex fvopab4 cvgcmp3ce.2 cvgcmp3ce.6 cvgcmp3ce.7 cvgcmp3ce.9 cvgcmp3c (sup ({e.|} z (RR) (E.e. w (NN) (A.e. v (NN) (-> (br (cv w) (<_) (cv v)) (br (cv z) (<) (` ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Re) (` (opr (+) (seq1) G) (cv u)))))) (cv v))))))) (RR) (<)) (+) (opr (i) (x.) (sup ({e.|} z (RR) (E.e. w (NN) (A.e. v (NN) (-> (br (cv w) (<_) (cv v)) (br (cv z) (<) (` ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Im) (` (opr (+) (seq1) G) (cv u)))))) (cv v))))))) (RR) (<))) oprex (cv y) (opr (sup ({e.|} z (RR) (E.e. w (NN) (A.e. v (NN) (-> (br (cv w) (<_) (cv v)) (br (cv z) (<) (` ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Re) (` (opr (+) (seq1) G) (cv u)))))) (cv v))))))) (RR) (<)) (+) (opr (i) (x.) (sup ({e.|} z (RR) (E.e. w (NN) (A.e. v (NN) (-> (br (cv w) (<_) (cv v)) (br (cv z) (<) (` ({<,>|} u t (/\ (e. (cv u) (NN)) (= (cv t) (` (Im) (` (opr (+) (seq1) G) (cv u)))))) (cv v))))))) (RR) (<)))) (opr (+) (seq1) G) (~~>) breq2 cla4ev ax-mp)) thm (cvgcmp3cetlem1 ((A z) (x z) (B x) (B z) (F x) (F z) (x y) (G x) (y z) (G y) (G z) (C x) (C z) (ph y) (ph z)) ((cvgcmp3ce.1 (e. A (V))) (cvgcmp3ce.2 (e. B (NN))) (cvgcmp3ce.3 (:--> F (NN) (RR))) (cvgcmp3ce.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp3ce.5 (br (opr (+) (seq1) F) (~~>) A)) (cvgcmp3cetlem1.6 (<-> ph (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x))))))))))) (-> ph (E. y (br (opr (+) (seq1) G) (~~>) (cv y)))) (G (if ph G (X. (NN) ({} (0)))) (+) (seq1) opreq2 (~~>) (cv y) breq1d y exbidv cvgcmp3ce.1 cvgcmp3ce.2 cvgcmp3ce.3 (cv x) (cv z) F fveq2 (0) (<_) breq2d cvgcmp3ce.4 vtoclga cvgcmp3ce.5 G (if ph G (X. (NN) ({} (0)))) (NN) (CC) feq1 G (if ph G (X. (NN) ({} (0)))) (cv z) fveq1 (abs) fveq2d (<_) (opr C (x.) (` F (cv z))) breq1d (br B (<) (cv z)) imbi2d z (NN) ralbidv anbi12d (/\ (e. C (RR)) (br (0) (<_) C)) anbi2d cvgcmp3cetlem1.6 (cv x) (cv z) B (<) breq2 (cv x) (cv z) G fveq2 (abs) fveq2d (cv x) (cv z) F fveq2 C (x.) opreq2d (<_) breq12d imbi12d (NN) cbvralv (:--> G (NN) (CC)) anbi2i (/\ (e. C (RR)) (br (0) (<_) C)) anbi2i bitr syl5bb C (if ph C (0)) (RR) eleq1 C (if ph C (0)) (0) (<_) breq2 anbi12d C (if ph C (0)) (x.) (` F (cv z)) opreq1 (` (abs) (` (if ph G (X. (NN) ({} (0)))) (cv z))) (<_) breq2d (br B (<) (cv z)) imbi2d z (NN) ralbidv (:--> (if ph G (X. (NN) ({} (0)))) (NN) (CC)) anbi2d anbi12d (X. (NN) ({} (0))) (if ph G (X. (NN) ({} (0)))) (NN) (CC) feq1 (X. (NN) ({} (0))) (if ph G (X. (NN) ({} (0)))) (cv z) fveq1 (abs) fveq2d (<_) (opr (0) (x.) (` F (cv z))) breq1d (br B (<) (cv z)) imbi2d z (NN) ralbidv anbi12d (/\ (e. (0) (RR)) (br (0) (<_) (0))) anbi2d (0) (if ph C (0)) (RR) eleq1 (0) (if ph C (0)) (0) (<_) breq2 anbi12d (0) (if ph C (0)) (x.) (` F (cv z)) opreq1 (` (abs) (` (if ph G (X. (NN) ({} (0)))) (cv z))) (<_) breq2d (br B (<) (cv z)) imbi2d z (NN) ralbidv (:--> (if ph G (X. (NN) ({} (0)))) (NN) (CC)) anbi2d anbi12d 0re 0re leid pm3.2i 0cn elisseti (NN) fconst 0cn (0) (CC) snssi ax-mp (X. (NN) ({} (0))) (NN) ({} (0)) (CC) fss mp2an 0cn elisseti (cv z) (NN) fvconst2 (abs) fveq2d abs0 syl6eq cvgcmp3ce.3 (cv z) ffvrni recnd (` F (cv z)) mul02t syl 0re leid syl5breqr eqbrtrd (br B (<) (cv z)) a1d rgen pm3.2i pm3.2i elimhyp2v pm3.26i pm3.26i G (if ph G (X. (NN) ({} (0)))) (NN) (CC) feq1 G (if ph G (X. (NN) ({} (0)))) (cv z) fveq1 (abs) fveq2d (<_) (opr C (x.) (` F (cv z))) breq1d (br B (<) (cv z)) imbi2d z (NN) ralbidv anbi12d (/\ (e. C (RR)) (br (0) (<_) C)) anbi2d cvgcmp3cetlem1.6 (cv x) (cv z) B (<) breq2 (cv x) (cv z) G fveq2 (abs) fveq2d (cv x) (cv z) F fveq2 C (x.) opreq2d (<_) breq12d imbi12d (NN) cbvralv (:--> G (NN) (CC)) anbi2i (/\ (e. C (RR)) (br (0) (<_) C)) anbi2i bitr syl5bb C (if ph C (0)) (RR) eleq1 C (if ph C (0)) (0) (<_) breq2 anbi12d C (if ph C (0)) (x.) (` F (cv z)) opreq1 (` (abs) (` (if ph G (X. (NN) ({} (0)))) (cv z))) (<_) breq2d (br B (<) (cv z)) imbi2d z (NN) ralbidv (:--> (if ph G (X. (NN) ({} (0)))) (NN) (CC)) anbi2d anbi12d (X. (NN) ({} (0))) (if ph G (X. (NN) ({} (0)))) (NN) (CC) feq1 (X. (NN) ({} (0))) (if ph G (X. (NN) ({} (0)))) (cv z) fveq1 (abs) fveq2d (<_) (opr (0) (x.) (` F (cv z))) breq1d (br B (<) (cv z)) imbi2d z (NN) ralbidv anbi12d (/\ (e. (0) (RR)) (br (0) (<_) (0))) anbi2d (0) (if ph C (0)) (RR) eleq1 (0) (if ph C (0)) (0) (<_) breq2 anbi12d (0) (if ph C (0)) (x.) (` F (cv z)) opreq1 (` (abs) (` (if ph G (X. (NN) ({} (0)))) (cv z))) (<_) breq2d (br B (<) (cv z)) imbi2d z (NN) ralbidv (:--> (if ph G (X. (NN) ({} (0)))) (NN) (CC)) anbi2d anbi12d 0re 0re leid pm3.2i 0cn elisseti (NN) fconst 0cn (0) (CC) snssi ax-mp (X. (NN) ({} (0))) (NN) ({} (0)) (CC) fss mp2an 0cn elisseti (cv z) (NN) fvconst2 (abs) fveq2d abs0 syl6eq cvgcmp3ce.3 (cv z) ffvrni recnd (` F (cv z)) mul02t syl 0re leid syl5breqr eqbrtrd (br B (<) (cv z)) a1d rgen pm3.2i pm3.2i elimhyp2v pm3.26i pm3.27i G (if ph G (X. (NN) ({} (0)))) (NN) (CC) feq1 G (if ph G (X. (NN) ({} (0)))) (cv z) fveq1 (abs) fveq2d (<_) (opr C (x.) (` F (cv z))) breq1d (br B (<) (cv z)) imbi2d z (NN) ralbidv anbi12d (/\ (e. C (RR)) (br (0) (<_) C)) anbi2d cvgcmp3cetlem1.6 (cv x) (cv z) B (<) breq2 (cv x) (cv z) G fveq2 (abs) fveq2d (cv x) (cv z) F fveq2 C (x.) opreq2d (<_) breq12d imbi12d (NN) cbvralv (:--> G (NN) (CC)) anbi2i (/\ (e. C (RR)) (br (0) (<_) C)) anbi2i bitr syl5bb C (if ph C (0)) (RR) eleq1 C (if ph C (0)) (0) (<_) breq2 anbi12d C (if ph C (0)) (x.) (` F (cv z)) opreq1 (` (abs) (` (if ph G (X. (NN) ({} (0)))) (cv z))) (<_) breq2d (br B (<) (cv z)) imbi2d z (NN) ralbidv (:--> (if ph G (X. (NN) ({} (0)))) (NN) (CC)) anbi2d anbi12d (X. (NN) ({} (0))) (if ph G (X. (NN) ({} (0)))) (NN) (CC) feq1 (X. (NN) ({} (0))) (if ph G (X. (NN) ({} (0)))) (cv z) fveq1 (abs) fveq2d (<_) (opr (0) (x.) (` F (cv z))) breq1d (br B (<) (cv z)) imbi2d z (NN) ralbidv anbi12d (/\ (e. (0) (RR)) (br (0) (<_) (0))) anbi2d (0) (if ph C (0)) (RR) eleq1 (0) (if ph C (0)) (0) (<_) breq2 anbi12d (0) (if ph C (0)) (x.) (` F (cv z)) opreq1 (` (abs) (` (if ph G (X. (NN) ({} (0)))) (cv z))) (<_) breq2d (br B (<) (cv z)) imbi2d z (NN) ralbidv (:--> (if ph G (X. (NN) ({} (0)))) (NN) (CC)) anbi2d anbi12d 0re 0re leid pm3.2i 0cn elisseti (NN) fconst 0cn (0) (CC) snssi ax-mp (X. (NN) ({} (0))) (NN) ({} (0)) (CC) fss mp2an 0cn elisseti (cv z) (NN) fvconst2 (abs) fveq2d abs0 syl6eq cvgcmp3ce.3 (cv z) ffvrni recnd (` F (cv z)) mul02t syl 0re leid syl5breqr eqbrtrd (br B (<) (cv z)) a1d rgen pm3.2i pm3.2i elimhyp2v pm3.27i pm3.26i G (if ph G (X. (NN) ({} (0)))) (NN) (CC) feq1 G (if ph G (X. (NN) ({} (0)))) (cv z) fveq1 (abs) fveq2d (<_) (opr C (x.) (` F (cv z))) breq1d (br B (<) (cv z)) imbi2d z (NN) ralbidv anbi12d (/\ (e. C (RR)) (br (0) (<_) C)) anbi2d cvgcmp3cetlem1.6 (cv x) (cv z) B (<) breq2 (cv x) (cv z) G fveq2 (abs) fveq2d (cv x) (cv z) F fveq2 C (x.) opreq2d (<_) breq12d imbi12d (NN) cbvralv (:--> G (NN) (CC)) anbi2i (/\ (e. C (RR)) (br (0) (<_) C)) anbi2i bitr syl5bb C (if ph C (0)) (RR) eleq1 C (if ph C (0)) (0) (<_) breq2 anbi12d C (if ph C (0)) (x.) (` F (cv z)) opreq1 (` (abs) (` (if ph G (X. (NN) ({} (0)))) (cv z))) (<_) breq2d (br B (<) (cv z)) imbi2d z (NN) ralbidv (:--> (if ph G (X. (NN) ({} (0)))) (NN) (CC)) anbi2d anbi12d (X. (NN) ({} (0))) (if ph G (X. (NN) ({} (0)))) (NN) (CC) feq1 (X. (NN) ({} (0))) (if ph G (X. (NN) ({} (0)))) (cv z) fveq1 (abs) fveq2d (<_) (opr (0) (x.) (` F (cv z))) breq1d (br B (<) (cv z)) imbi2d z (NN) ralbidv anbi12d (/\ (e. (0) (RR)) (br (0) (<_) (0))) anbi2d (0) (if ph C (0)) (RR) eleq1 (0) (if ph C (0)) (0) (<_) breq2 anbi12d (0) (if ph C (0)) (x.) (` F (cv z)) opreq1 (` (abs) (` (if ph G (X. (NN) ({} (0)))) (cv z))) (<_) breq2d (br B (<) (cv z)) imbi2d z (NN) ralbidv (:--> (if ph G (X. (NN) ({} (0)))) (NN) (CC)) anbi2d anbi12d 0re 0re leid pm3.2i 0cn elisseti (NN) fconst 0cn (0) (CC) snssi ax-mp (X. (NN) ({} (0))) (NN) ({} (0)) (CC) fss mp2an 0cn elisseti (cv z) (NN) fvconst2 (abs) fveq2d abs0 syl6eq cvgcmp3ce.3 (cv z) ffvrni recnd (` F (cv z)) mul02t syl 0re leid syl5breqr eqbrtrd (br B (<) (cv z)) a1d rgen pm3.2i pm3.2i elimhyp2v pm3.27i pm3.27i rspec imp y cvgcmp3ce dedth)) thm (cvgcmp3cetlem2 ((x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (G x) (G y) (G z) (C x) (C y) (C z) (ph y) (ph z)) ((cvgcmp3cetlem2.1 (e. A (V))) (cvgcmp3cetlem2.2 (e. B (NN))) (cvgcmp3cetlem2.3 (<-> ph (/\ (/\ (:--> F (NN) (RR)) (A.e. x (NN) (br (0) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>) A))))) (-> (/\ ph (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) G) (~~>) (cv y)))) (F (if ph F (X. (NN) ({} (0)))) (cv z) fveq1 C (x.) opreq2d (` (abs) (` G (cv z))) (<_) breq2d (br B (<) (cv z)) imbi2d z (NN) ralbidv (:--> G (NN) (CC)) anbi2d (/\ (e. C (RR)) (br (0) (<_) C)) anbi2d (E. y (br (opr (+) (seq1) G) (~~>) (cv y))) imbi1d cvgcmp3cetlem2.1 0re elisseti ph ifex cvgcmp3cetlem2.2 F (if ph F (X. (NN) ({} (0)))) (NN) (RR) feq1 F (if ph F (X. (NN) ({} (0)))) (cv z) fveq1 (0) (<_) breq2d z (NN) ralbidv anbi12d F (if ph F (X. (NN) ({} (0)))) (+) (seq1) opreq2 (~~>) A breq1d anbi12d cvgcmp3cetlem2.3 (cv x) (cv z) F fveq2 (0) (<_) breq2d (NN) cbvralv (:--> F (NN) (RR)) anbi2i (br (opr (+) (seq1) F) (~~>) A) anbi1i bitr syl5bb A (if ph A (0)) (opr (+) (seq1) (if ph F (X. (NN) ({} (0))))) (~~>) breq2 (/\ (:--> (if ph F (X. (NN) ({} (0)))) (NN) (RR)) (A.e. z (NN) (br (0) (<_) (` (if ph F (X. (NN) ({} (0)))) (cv z))))) anbi2d (X. (NN) ({} (0))) (if ph F (X. (NN) ({} (0)))) (NN) (RR) feq1 (X. (NN) ({} (0))) (if ph F (X. (NN) ({} (0)))) (cv z) fveq1 (0) (<_) breq2d z (NN) ralbidv anbi12d (X. (NN) ({} (0))) (if ph F (X. (NN) ({} (0)))) (+) (seq1) opreq2 (~~>) (0) breq1d anbi12d (0) (if ph A (0)) (opr (+) (seq1) (if ph F (X. (NN) ({} (0))))) (~~>) breq2 (/\ (:--> (if ph F (X. (NN) ({} (0)))) (NN) (RR)) (A.e. z (NN) (br (0) (<_) (` (if ph F (X. (NN) ({} (0)))) (cv z))))) anbi2d 0re elisseti (NN) fconst 0re (0) (RR) snssi ax-mp (X. (NN) ({} (0))) (NN) ({} (0)) (RR) fss mp2an 0re elisseti (cv z) (NN) fvconst2 0re leid syl5breqr rgen pm3.2i ser1clim0 pm3.2i elimhyp2v pm3.26i pm3.26i F (if ph F (X. (NN) ({} (0)))) (NN) (RR) feq1 F (if ph F (X. (NN) ({} (0)))) (cv z) fveq1 (0) (<_) breq2d z (NN) ralbidv anbi12d F (if ph F (X. (NN) ({} (0)))) (+) (seq1) opreq2 (~~>) A breq1d anbi12d cvgcmp3cetlem2.3 (cv x) (cv z) F fveq2 (0) (<_) breq2d (NN) cbvralv (:--> F (NN) (RR)) anbi2i (br (opr (+) (seq1) F) (~~>) A) anbi1i bitr syl5bb A (if ph A (0)) (opr (+) (seq1) (if ph F (X. (NN) ({} (0))))) (~~>) breq2 (/\ (:--> (if ph F (X. (NN) ({} (0)))) (NN) (RR)) (A.e. z (NN) (br (0) (<_) (` (if ph F (X. (NN) ({} (0)))) (cv z))))) anbi2d (X. (NN) ({} (0))) (if ph F (X. (NN) ({} (0)))) (NN) (RR) feq1 (X. (NN) ({} (0))) (if ph F (X. (NN) ({} (0)))) (cv z) fveq1 (0) (<_) breq2d z (NN) ralbidv anbi12d (X. (NN) ({} (0))) (if ph F (X. (NN) ({} (0)))) (+) (seq1) opreq2 (~~>) (0) breq1d anbi12d (0) (if ph A (0)) (opr (+) (seq1) (if ph F (X. (NN) ({} (0))))) (~~>) breq2 (/\ (:--> (if ph F (X. (NN) ({} (0)))) (NN) (RR)) (A.e. z (NN) (br (0) (<_) (` (if ph F (X. (NN) ({} (0)))) (cv z))))) anbi2d 0re elisseti (NN) fconst 0re (0) (RR) snssi ax-mp (X. (NN) ({} (0))) (NN) ({} (0)) (RR) fss mp2an 0re elisseti (cv z) (NN) fvconst2 0re leid syl5breqr rgen pm3.2i ser1clim0 pm3.2i elimhyp2v pm3.26i pm3.27i rspec F (if ph F (X. (NN) ({} (0)))) (NN) (RR) feq1 F (if ph F (X. (NN) ({} (0)))) (cv z) fveq1 (0) (<_) breq2d z (NN) ralbidv anbi12d F (if ph F (X. (NN) ({} (0)))) (+) (seq1) opreq2 (~~>) A breq1d anbi12d cvgcmp3cetlem2.3 (cv x) (cv z) F fveq2 (0) (<_) breq2d (NN) cbvralv (:--> F (NN) (RR)) anbi2i (br (opr (+) (seq1) F) (~~>) A) anbi1i bitr syl5bb A (if ph A (0)) (opr (+) (seq1) (if ph F (X. (NN) ({} (0))))) (~~>) breq2 (/\ (:--> (if ph F (X. (NN) ({} (0)))) (NN) (RR)) (A.e. z (NN) (br (0) (<_) (` (if ph F (X. (NN) ({} (0)))) (cv z))))) anbi2d (X. (NN) ({} (0))) (if ph F (X. (NN) ({} (0)))) (NN) (RR) feq1 (X. (NN) ({} (0))) (if ph F (X. (NN) ({} (0)))) (cv z) fveq1 (0) (<_) breq2d z (NN) ralbidv anbi12d (X. (NN) ({} (0))) (if ph F (X. (NN) ({} (0)))) (+) (seq1) opreq2 (~~>) (0) breq1d anbi12d (0) (if ph A (0)) (opr (+) (seq1) (if ph F (X. (NN) ({} (0))))) (~~>) breq2 (/\ (:--> (if ph F (X. (NN) ({} (0)))) (NN) (RR)) (A.e. z (NN) (br (0) (<_) (` (if ph F (X. (NN) ({} (0)))) (cv z))))) anbi2d 0re elisseti (NN) fconst 0re (0) (RR) snssi ax-mp (X. (NN) ({} (0))) (NN) ({} (0)) (RR) fss mp2an 0re elisseti (cv z) (NN) fvconst2 0re leid syl5breqr rgen pm3.2i ser1clim0 pm3.2i elimhyp2v pm3.27i (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. z (NN) (-> (br B (<) (cv z)) (br (` (abs) (` G (cv z))) (<_) (opr C (x.) (` (if ph F (X. (NN) ({} (0)))) (cv z)))))))) pm4.2 y cvgcmp3cetlem1 dedth (cv x) (cv z) B (<) breq2 (cv x) (cv z) G fveq2 (abs) fveq2d (cv x) (cv z) F fveq2 C (x.) opreq2d (<_) breq12d imbi12d (NN) cbvralv (:--> G (NN) (CC)) anbi2i (/\ (e. C (RR)) (br (0) (<_) C)) anbi2i syl5ib imp)) thm (cvgcmp3cet ((A y) (x y) (B x) (B y) (F x) (F y) (G x) (G y) (C x) (C y)) ((cvgcmp3cet.6 (e. A (V)))) (-> (/\ (e. B (NN)) (/\ (/\ (/\ (:--> F (NN) (RR)) (A.e. x (NN) (br (0) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>) A)) (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x)))))))))) (E. y (br (opr (+) (seq1) G) (~~>) (cv y)))) (B (if (e. B (NN)) B (1)) (<) (cv x) breq1 (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x)))) imbi1d x (NN) ralbidv (:--> G (NN) (CC)) anbi2d (/\ (e. C (RR)) (br (0) (<_) C)) anbi2d (/\ (/\ (:--> F (NN) (RR)) (A.e. x (NN) (br (0) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>) A)) anbi2d (E. y (br (opr (+) (seq1) G) (~~>) (cv y))) imbi1d cvgcmp3cet.6 1nn B elimel (/\ (/\ (:--> F (NN) (RR)) (A.e. x (NN) (br (0) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>) A)) pm4.2 C G y cvgcmp3cetlem2 dedth imp)) thm (dfisum ((m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y) (M m) (M n) (M x) (M y) (k m) (k n) (k x) (k y)) () (-> (e. M (ZZ)) (= (sum_ k (` (ZZ>) M) A) (U. ({|} x (br (opr (<,> M (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (~~>) (cv x)))))) ((cv n) (cv m) M fzneuzt (opr (cv m) (...) (cv n)) (` (ZZ>) M) eqcom negbii sylib ancoms (e. (cv x) (` (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (cv n))) intnanrd nrexdv m nexdv x nexdv x (E. m (E.e. n (` (ZZ>) (cv m)) (/\ (= (` (ZZ>) M) (opr (cv m) (...) (cv n))) (e. (cv x) (` (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (cv n)))))) abn0 con1bii sylib M (cv m) uz11t M (cv m) eqcom syl6bb (br (opr (<,> (cv m) (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (~~>) (cv x)) anbi1d m (ZZ) rexbidv (cv m) M (+) opeq1 (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ)) opreq1d (~~>) (cv x) breq1d (ZZ) ceqsrexv bitrd x abbidv unieqd uneq12d ({/}) (U. ({|} x (br (opr (<,> M (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (~~>) (cv x)))) uncom (U. ({|} x (br (opr (<,> M (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (~~>) (cv x)))) un0 eqtr syl6eq k (` (ZZ>) M) A x m n y df-sum syl5eq)) thm (isumvalt ((k x) (k y) (F k) (x y) (F x) (F y) (M x) (M y)) ((isumvalt.1 (e. F (V)))) (-> (e. M (ZZ)) (= (sum_ k (` (ZZ>) M) (` F (cv k))) (U. ({|} x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))))) (M k (` F (cv k)) x y dfisum addex isumvalt.1 M k y seqzres2 (~~>) (cv x) breq1d x abbidv unieqd eqtrd)) thm (isumvaltf ((j x) (j y) (j z) (F j) (x y) (x z) (F x) (y z) (F y) (F z) (M x) (M z) (j k) (k y) (k z)) ((isumvaltf.1 (e. F (V))) (isumvaltf.2 (-> (e. (cv y) F) (A. k (e. (cv y) F))))) (-> (e. M (ZZ)) (= (sum_ k (` (ZZ>) M) (` F (cv k))) (U. ({|} x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))))) (isumvaltf.1 M j x isumvalt (e. (cv z) (` F (cv k))) j ax-17 isumvaltf.2 z hblem (e. (cv z) (cv j)) k ax-17 hbfv (cv k) (cv j) F fveq2 (` (ZZ>) M) cbvsum syl5eq)) thm (isumclimtf ((A x) (x y) (F x) (F y) (M x) (k y)) ((isumclimtf.1 (-> (e. (cv y) F) (A. k (e. (cv y) F)))) (isumclimtf.2 (e. F (V))) (isumclimtf.3 (e. A (V)))) (-> (/\ (e. M (ZZ)) (br (opr (<,> M (+)) (seq) F) (~~>) A)) (= (sum_ k (` (ZZ>) M) (` F (cv k))) A)) (isumclimtf.2 isumclimtf.1 M x isumvaltf x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)) rabab unieqi syl6eqr isumclimtf.3 (opr (<,> M (+)) (seq) F) x climeu x (V) (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)) df-reu x visset (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)) biantrur x eubii bitr4 sylibr isumclimtf.3 jctil (cv x) A (opr (<,> M (+)) (seq) F) (~~>) breq2 (V) reuuni2 syl ibi sylan9eq)) thm (isumclimt ((k x) (F x) (F k)) ((isumclimt.1 (e. F (V))) (isumclimt.2 (e. A (V)))) (-> (/\ (e. M (ZZ)) (br (opr (<,> M (+)) (seq) F) (~~>) A)) (= (sum_ k (` (ZZ>) M) (` F (cv k))) A)) ((e. (cv x) F) k ax-17 isumclimt.1 isumclimt.2 M isumclimtf)) thm (isumclim2tf ((x y) (F x) (F y) (M x) (k x) (k y)) ((isumclim2tf.1 (-> (e. (cv y) F) (A. k (e. (cv y) F)))) (isumclim2tf.2 (e. F (V)))) (-> (/\ (e. M (ZZ)) (E. x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))) (br (opr (<,> M (+)) (seq) F) (~~>) (sum_ k (` (ZZ>) M) (` F (cv k))))) ((e. M (ZZ)) (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)) pm3.27 isumclim2tf.1 isumclim2tf.2 x visset M isumclimtf breqtrrd ex x 19.23adv imp)) thm (isumclim2t ((k x) (k y) (F k) (x y) (F x) (F y) (M x)) ((isumclim2t.1 (e. F (V)))) (-> (/\ (e. M (ZZ)) (E. x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))) (br (opr (<,> M (+)) (seq) F) (~~>) (sum_ k (` (ZZ>) M) (` F (cv k))))) ((e. (cv y) F) k ax-17 isumclim2t.1 M x isumclim2tf)) thm (isumclim3t ((x y) (x z) (A x) (y z) (A y) (A z) (k x) (k y) (k z) (M k) (M x) (M y) (M z) (F z)) ((isumclim3t.1 (e. A (V))) (isumclim3t.2 (= F ({<,>|} k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A)))))) (-> (/\ (e. M (ZZ)) (E. x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))) (br (opr (<,> M (+)) (seq) F) (~~>) (sum_ k (` (ZZ>) M) A))) (isumclim3t.2 (<,> M (+)) (seq) opreq2i (e. M (ZZ)) a1i eqcomd (~~>) (cv x) breq1d x exbidv biimpar z k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A)) hbopab1 (ZZ>) M fvex k y A funopabex2 M x isumclim2tf syldan isumclim3t.1 k (` (ZZ>) M) A (V) y fvopab2 mpan2 sumeq2i syl6breq isumclim3t.2 (<,> M (+)) (seq) opreq2i syl5eqbr)) thm (isumclim4t ((y z) (A y) (A z) (k y) (k z) (M k) (M y) (M z) (F z)) ((isumclim3t.1 (e. A (V))) (isumclim3t.2 (= F ({<,>|} k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A))))) (isumclim4t.3 (e. B (V)))) (-> (/\ (e. M (ZZ)) (br (opr (<,> M (+)) (seq) F) (~~>) B)) (= (sum_ k (` (ZZ>) M) A) B)) (z k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A)) hbopab1 isumclim3t.2 (cv z) eleq2i isumclim3t.2 (cv z) eleq2i k albii 3imtr4 isumclim3t.2 (ZZ>) M fvex k y A funopabex2 eqeltr isumclim4t.3 M isumclimtf (` (ZZ>) M) ssid isumclim3t.1 isumclim3t.2 (` (ZZ>) M) sumeqfv ax-mp syl5eqr)) thm (isumclim5t ((j x) (j y) (j z) (A j) (x y) (x z) (A x) (y z) (A y) (A z) (j k) (M j) (k x) (k y) (k z) (M k) (M x) (M y) (M z)) ((isumclim5t.1 (e. A (V))) (isumclim5t.2 (= F ({<,>|} j y (/\ (e. (cv j) (` (ZZ>) M)) (= (cv y) (sum_ k (opr M (...) (cv j)) A))))))) (-> (/\ (e. M (ZZ)) (E. x (br F (~~>) (cv x)))) (br F (~~>) (sum_ k (` (ZZ>) M) A))) (isumclim5t.1 ({<,>|} k z (/\ (e. (cv k) (` (ZZ>) M)) (= (cv z) A))) eqid isumclim5t.2 serzfsum (E. x (br F (~~>) (cv x))) adantr isumclim5t.1 ({<,>|} k z (/\ (e. (cv k) (` (ZZ>) M)) (= (cv z) A))) eqid isumclim5t.2 serzfsum (~~>) (cv x) breq1d x exbidv biimpar isumclim5t.1 ({<,>|} k z (/\ (e. (cv k) (` (ZZ>) M)) (= (cv z) A))) eqid x isumclim3t syldan eqbrtrrd)) thm (isum1p ((j k) (j n) (j x) (F j) (k n) (k x) (F k) (n x) (F n) (F x) (M j) (M k) (M n) (M x)) ((isum1p.1 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (CC))) (E. x (br (opr (<,> (opr M (+) (1)) (+)) (seq) F) (~~>) (cv x)))) (= (sum_ k (` (ZZ>) M) (` F (cv k))) (opr (` F M) (+) (sum_ k (` (ZZ>) (opr M (+) (1))) (` F (cv k)))))) (isum1p.1 (opr M (+) (1)) x k isumclim2t M peano2z sylan (A.e. j (` (ZZ>) M) (e. (` F (cv j)) (CC))) adantlr isum1p.1 (` F M) (+) (sum_ k (` (ZZ>) (opr M (+) (1))) (` F (cv k))) oprex M k isumclimt (e. M (ZZ)) (A.e. j (` (ZZ>) M) (e. (` F (cv j)) (CC))) pm3.26 (br (opr (<,> (opr M (+) (1)) (+)) (seq) F) (~~>) (sum_ k (` (ZZ>) (opr M (+) (1))) (` F (cv k)))) adantr (<,> (opr M (+) (1)) (+)) (seq) F oprex (<,> M (+)) (seq) F oprex k (` (ZZ>) (opr M (+) (1))) (` F (cv k)) sumex F M fvex (opr M (+) (1)) n climaddc2 (/\ (e. M (ZZ)) (A.e. j (` (ZZ>) M) (e. (` F (cv j)) (CC)))) (br (opr (<,> (opr M (+) (1)) (+)) (seq) F) (~~>) (sum_ k (` (ZZ>) (opr M (+) (1))) (` F (cv k)))) pm3.27 (cv j) M F fveq2 (CC) eleq1d (` (ZZ>) M) rcla4va M uzidt sylan (br (opr (<,> (opr M (+) (1)) (+)) (seq) F) (~~>) (sum_ k (` (ZZ>) (opr M (+) (1))) (` F (cv k)))) adantr jca M peano2z (A.e. j (` (ZZ>) M) (e. (` F (cv j)) (CC))) (br (opr (<,> (opr M (+) (1)) (+)) (seq) F) (~~>) (sum_ k (` (ZZ>) (opr M (+) (1))) (` F (cv k)))) ad2antrr isum1p.1 (cv n) (opr M (+) (1)) j serzclt (/\ (e. M (ZZ)) (A.e. j (` (ZZ>) M) (e. (` F (cv j)) (CC)))) (e. (cv n) (` (ZZ>) (opr M (+) (1)))) pm3.27 M (cv j) peano2uzr ex (cv j) (opr M (+) (1)) (cv n) elfzuzt syl5 (e. (` F (cv j)) (CC)) imim1d r19.20dv2 (e. (cv n) (` (ZZ>) (opr M (+) (1)))) a1dd imp31 sylanc isum1p.1 (cv n) M j serz1p 3expa M (cv n) peano2uzr M (cv n) eluzp1lt jca sylan an1rs (cv j) M (cv n) elfzuzt (e. (` F (cv j)) (CC)) imim1i r19.20i2 sylanl2 jca r19.21aiva (br (opr (<,> (opr M (+) (1)) (+)) (seq) F) (~~>) (sum_ k (` (ZZ>) (opr M (+) (1))) (` F (cv k)))) adantr jca sylanc sylanc syldan 3impa (cv k) (cv j) F fveq2 (CC) eleq1d (` (ZZ>) M) cbvralv syl3an2b)) thm (isumnn0nn ((k x) (F x) (F k)) ((isumnn0nn.1 (e. F (V)))) (-> (/\ (A.e. k (NN0) (e. (` F (cv k)) (CC))) (E. x (br (opr (+) (seq1) F) (~~>) (cv x)))) (= (sum_ k (NN0) (` F (cv k))) (opr (` F (0)) (+) (sum_ k (NN) (` F (cv k)))))) (0z isumnn0nn.1 (0) k x isum1p mp3an1 nn0uz (NN0) (` (ZZ>) (0)) k (e. (` F (cv k)) (CC)) raleq1 ax-mp addex isumnn0nn.1 seq1seqz 1cn addid2 (opr (0) (+) (1)) (1) (+) opeq1 ax-mp (seq) F opreq1i eqtr4 (~~>) (cv x) breq1i x exbii syl2anb nn0uz k (` F (cv k)) sumeq1i nnuz 1cn addid2 (ZZ>) fveq2i eqtr4 k (` F (cv k)) sumeq1i (` F (0)) (+) opreq2i 3eqtr4g)) thm (isumclt ((k x) (F k) (F x) (M x)) ((isumclt.1 (e. F (V)))) (-> (/\ (e. M (ZZ)) (E. x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))) (e. (sum_ k (` (ZZ>) M) (` F (cv k))) (CC))) (isumclt.1 M k x isumvalt (CC) eleq1d x (CC) (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)) hbreu1 x visset (opr (<,> M (+)) (seq) F) x climreu 19.23ai x (CC) (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)) reucl syl x visset (cv x) (V) (opr (<,> M (+)) (seq) F) climcl mpan pm4.71ri x abbii x (CC) (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)) df-rab eqtr4 unieqi syl5eqel syl5bir imp)) thm (isumreclt ((j k) (j x) (F j) (k x) (F k) (F x) (M j) (M k) (M x)) ((isumreclt.1 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR))) (E. x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))) (e. (sum_ k (` (ZZ>) M) (` F (cv k))) (RR))) (isumreclt.1 x visset M k isumclimt (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR))) adantlr isumreclt.1 (cv j) M k serzreclt (cv k) M (cv j) elfzuzt (e. (` F (cv k)) (RR)) imim1i r19.20i2 sylan2 expcom (e. M (ZZ)) adantl r19.21aiv (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)) adantlr x visset M (opr (<,> M (+)) (seq) F) j climrecl 3expa syldan an1rs eqeltrd ex x 19.23adv 3impia)) thm (expcnvlem1 ((ph x) (x y) (A x) (A y)) ((expcnvlem1.1 (e. A (RR))) (expcnvlem1.2 (-> (e. (cv y) (NN)) (-> (br A (<) (cv y)) ph)))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) ph))) (expcnvlem1.1 A flclt ax-mp (` (floor) A) nn0absclt ax-mp (` (abs) (` (floor) A)) nn0p1nnt ax-mp (cv y) nnret expcnvlem1.1 A flltp1t ax-mp expcnvlem1.1 A flclt ax-mp zre leabs expcnvlem1.1 A flclt ax-mp zre expcnvlem1.1 A flclt ax-mp (` (floor) A) zcnt ax-mp abscl ax1re leadd1 mpbi expcnvlem1.1 expcnvlem1.1 A flclt ax-mp zre ax1re readdcl expcnvlem1.1 A flclt ax-mp (` (floor) A) zcnt ax-mp abscl ax1re readdcl ltletr mp2an expcnvlem1.1 expcnvlem1.1 A flclt ax-mp (` (floor) A) zcnt ax-mp abscl ax1re readdcl A (opr (` (abs) (` (floor) A)) (+) (1)) (cv y) ltletrt mp3an12 mpani syl ph imim1d r19.20i expcnvlem1.2 mprg (cv x) (opr (` (abs) (` (floor) A)) (+) (1)) (<_) (cv y) breq1 ph imbi1d y (NN) ralbidv (NN) rcla4ev mp2an)) thm (expcnvlem2 () ((expcnvlem.1 (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1)))) (expcnvlem.2 (/\ (e. B (RR)) (br (0) (<) B))) (expcnvlem.3 (e. C (NN)))) (-> (br (opr (opr (opr (1) (/) B) (-) (1)) (/) (opr (opr (1) (/) A) (-) (1))) (<) C) (br (opr A (^) C) (<) B)) (expcnvlem.2 pm3.26i expcnvlem.2 pm3.26i expcnvlem.2 pm3.27i gt0ne0i rereccl ax1re resubcl expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre expcnvlem.1 3simp3i expcnvlem.1 3simp1i expcnvlem.1 3simp2i A reclt1t mp2an mpbi ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl posdif mpbi (opr (opr (1) (/) B) (-) (1)) (opr (opr (1) (/) A) (-) (1)) C ltdivmult mpan2 mp3an expcnvlem.2 pm3.26i expcnvlem.2 pm3.26i expcnvlem.2 pm3.27i gt0ne0i rereccl ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl ltsubadd2 ax1re expcnvlem.2 pm3.26i ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl readdcl expcnvlem.2 pm3.27i ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl lt01 expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre expcnvlem.1 3simp3i expcnvlem.1 3simp1i expcnvlem.1 3simp2i A reclt1t mp2an mpbi ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl posdif mpbi expcnvlem.3 nngt0 mulgt0i addgt0i ltdiv23i 3bitr expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnnn0 0re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl expcnvlem.1 3simp1i expcnvlem.1 3simp2i recgt0i ltlei 1cn negid 1cn expcnvlem.1 3simp1i recn expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i reccl pncan3 3brtr4 ax1re renegcl expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl ax1re leadd2 mpbir (opr (opr (1) (/) A) (-) (1)) C bernneq mp3an 1cn expcnvlem.1 3simp1i recn expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i reccl pncan3 (^) C opreq1i expcnvlem.1 3simp1i recn expcnvlem.3 expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i A C recexpt C nnnn0t syl3an2 mp3an eqtr breqtr ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl readdcl expcnvlem.1 3simp1i expcnvlem.3 nnnn0 A C reexpclt mp2an pm3.2i ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl lt01 expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre expcnvlem.1 3simp3i expcnvlem.1 3simp1i expcnvlem.1 3simp2i A reclt1t mp2an mpbi ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl posdif mpbi expcnvlem.3 nngt0 mulgt0i addgt0i expcnvlem.1 3simp1i expcnvlem.3 nnnn0 expcnvlem.1 3simp2i A C expgt0t mp3an pm3.2i (opr (1) (+) (opr (opr (opr (1) (/) A) (-) (1)) (x.) C)) (opr A (^) C) lerec2t mp2an mpbi expcnvlem.1 3simp1i expcnvlem.3 nnnn0 A C reexpclt mp2an ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl readdcl ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl readdcl ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl lt01 expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre expcnvlem.1 3simp3i expcnvlem.1 3simp1i expcnvlem.1 3simp2i A reclt1t mp2an mpbi ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl posdif mpbi expcnvlem.3 nngt0 mulgt0i addgt0i gt0ne0i rereccl expcnvlem.2 pm3.26i lelttr mpan sylbi)) thm (expcnvlem3 () ((expcnvlem.1 (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1)))) (expcnvlem.2 (/\ (e. B (RR)) (br (0) (<) B)))) (-> (e. C (NN)) (-> (br (opr (opr (opr (1) (/) B) (-) (1)) (/) (opr (opr (1) (/) A) (-) (1))) (<) C) (br (opr A (^) C) (<) B))) (C (if (e. C (NN)) C (1)) (opr (opr (opr (1) (/) B) (-) (1)) (/) (opr (opr (1) (/) A) (-) (1))) (<) breq2 C (if (e. C (NN)) C (1)) A (^) opreq2 (<) B breq1d imbi12d expcnvlem.1 expcnvlem.2 1nn C elimel expcnvlem2 dedth)) thm (expcnvlem4 ((x y) (A x) (A y) (B x) (B y)) ((expcnvlem.1 (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1)))) (expcnvlem.2 (/\ (e. B (RR)) (br (0) (<) B)))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B)))) (expcnvlem.2 pm3.26i expcnvlem.2 pm3.26i expcnvlem.2 pm3.27i gt0ne0i rereccl ax1re resubcl expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.1 3simp3i expcnvlem.1 3simp1i expcnvlem.1 3simp2i A reclt1t mp2an mpbi ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl posdif mpbi gt0ne0i redivcl expcnvlem.1 expcnvlem.2 (cv y) expcnvlem3 x expcnvlem1)) thm (expcnvlem5 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B))))) (A (if (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) A (opr (1) (/) (2))) (^) (cv y) opreq1 (<) B breq1d (br (cv x) (<_) (cv y)) imbi2d x (NN) y (NN) rexralbidv B (if (/\ (e. B (RR)) (br (0) (<) B)) B (1)) (opr (if (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) A (opr (1) (/) (2))) (^) (cv y)) (<) breq2 (br (cv x) (<_) (cv y)) imbi2d x (NN) y (NN) rexralbidv A (if (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) A (opr (1) (/) (2))) (RR) eleq1 A (if (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) A (opr (1) (/) (2))) (0) (<) breq2 A (if (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) A (opr (1) (/) (2))) (<) (1) breq1 3anbi123d (opr (1) (/) (2)) (if (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) A (opr (1) (/) (2))) (RR) eleq1 (opr (1) (/) (2)) (if (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) A (opr (1) (/) (2))) (0) (<) breq2 (opr (1) (/) (2)) (if (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) A (opr (1) (/) (2))) (<) (1) breq1 3anbi123d 2re 2re 2pos gt0ne0i rereccl halfgt0 halflt1 3pm3.2i elimhyp B (if (/\ (e. B (RR)) (br (0) (<) B)) B (1)) (RR) eleq1 B (if (/\ (e. B (RR)) (br (0) (<) B)) B (1)) (0) (<) breq2 anbi12d (1) (if (/\ (e. B (RR)) (br (0) (<) B)) B (1)) (RR) eleq1 (1) (if (/\ (e. B (RR)) (br (0) (<) B)) B (1)) (0) (<) breq2 anbi12d ax1re lt01 pm3.2i elimhyp x y expcnvlem4 dedth2h)) thm (expcnvlem6 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\/\ (e. A (RR)) (br (0) (<_) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B))))) (0re (0) A leloet mpan A B x y expcnvlem5 3exp1 1nn (br (0) (<) B) a1i (= (0) A) a1i (cv y) 0expt (<) B breq1d (= (0) A) adantl (0) A (^) (cv y) opreq1 (<) B breq1d (e. (cv y) (NN)) adantr bitr3d biimpd ex (br (1) (<_) (cv y)) a1d com24 r19.21adv jcad (cv x) (1) (<_) (cv y) breq1 (br (opr A (^) (cv y)) (<) B) imbi1d y (NN) ralbidv (NN) rcla4ev syl6 (e. B (RR)) adantld (br A (<) (1)) a1d (e. A (RR)) a1i jaod sylbid 3imp1)) thm (expcnv ((j k) (j x) (A j) (k x) (A k) (A x) (F j) (F x) (F k)) ((expcnv.1 (e. F (V)))) (-> (/\/\ (e. A (CC)) (A.e. k (NN) (= (` F (cv k)) (opr A (^) (cv k)))) (br (` (abs) A) (<) (1))) (br F (~~>) (0))) ((` (abs) A) (cv x) j k expcnvlem6 A absclt (br (` (abs) A) (<) (1)) adantr A absge0t (br (` (abs) A) (<) (1)) adantr (e. A (CC)) (br (` (abs) A) (<) (1)) pm3.27 3jca sylan (A.e. k (NN) (= (` F (cv k)) (opr A (^) (cv k)))) adantllr (` F (cv k)) (opr A (^) (cv k)) (abs) fveq2 A (cv k) absexpt (cv k) nnnn0t sylan2 sylan9eqr (<) (cv x) breq1d (br (cv j) (<_) (cv k)) imbi2d ex r19.20dva imp k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (` F (cv k))) (<) (cv x))) (-> (br (cv j) (<_) (cv k)) (br (opr (` (abs) A) (^) (cv k)) (<) (cv x))) r19.15 syl j (NN) rexbidv (br (` (abs) A) (<) (1)) (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) ad2antrr mpbird exp32 3impa r19.21aiv (` F (cv k)) (opr A (^) (cv k)) (CC) eleq1 A (cv k) expclt (cv k) nnnn0t sylan2 syl5bir com12 r19.20dva imp expcnv.1 k x j clim0nn syl (br (` (abs) A) (<) (1)) 3adant3 mpbird)) thm (geoser ((j k) (j m) (j y) (A j) (k m) (k y) (A k) (m y) (A m) (A y) (F j) (F m) (N j)) ((geoser.1 (= F ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k))))))) (geoser.2 (e. A (CC)))) (-> (/\ (e. N (NN0)) (=/= A (1))) (= (` (opr (+) (seq0) F) N) (opr (opr (1) (-) (opr A (^) (opr N (+) (1)))) (/) (opr (1) (-) A)))) (1cn geoser.2 subcl (opr (1) (-) A) (` (opr (+) (seq0) F) N) divcan4t mp3an1 geoser.1 geoser.2 A (cv k) expclt mpan fopab N ser0cl geoser.2 subid (opr (1) (-) A) eqeq1i geoser.2 1cn geoser.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi biimp syl2an geoser.1 A (^) (opr (cv m) (+) (1)) oprex (cv k) (opr (cv m) (+) (1)) A (^) opreq2 ser0p1 (x.) (opr (1) (-) A) opreq1d 1cn geoser.2 subcl (` (opr (+) (seq0) F) (cv m)) (opr A (^) (opr (cv m) (+) (1))) (opr (1) (-) A) adddirt mp3an3 geoser.1 geoser.2 A (cv k) expclt mpan fopab (cv m) ser0cl (cv m) peano2nn0 geoser.2 A (opr (cv m) (+) (1)) expclt mpan syl sylanc eqtrd (= (opr (` (opr (+) (seq0) F) (cv m)) (x.) (opr (1) (-) A)) (opr (1) (-) (opr A (^) (opr (cv m) (+) (1))))) adantr (opr (` (opr (+) (seq0) F) (cv m)) (x.) (opr (1) (-) A)) (opr (1) (-) (opr A (^) (opr (cv m) (+) (1)))) (+) (opr (opr A (^) (opr (cv m) (+) (1))) (x.) (opr (1) (-) A)) opreq1 (cv m) peano2nn0 geoser.2 A (opr (cv m) (+) (1)) expp1t mpan syl (1) (-) opreq2d (cv m) peano2nn0 geoser.2 A (opr (cv m) (+) (1)) expclt mpan syl (opr A (^) (opr (cv m) (+) (1))) ax1id syl (-) (opr (opr A (^) (opr (cv m) (+) (1))) (x.) (opr (1) (-) A)) opreq1d (cv m) peano2nn0 geoser.2 A (opr (cv m) (+) (1)) expclt mpan syl 1cn 1cn geoser.2 subcl (opr A (^) (opr (cv m) (+) (1))) (1) (opr (1) (-) A) subdit mp3an23 syl 1cn geoser.2 (1) A nncant mp2an (opr A (^) (opr (cv m) (+) (1))) (x.) opreq2i syl5reqr eqtr3d (1) (-) opreq2d 1cn (1) (opr A (^) (opr (cv m) (+) (1))) (opr (opr A (^) (opr (cv m) (+) (1))) (x.) (opr (1) (-) A)) subsubt mp3an1 (cv m) peano2nn0 geoser.2 A (opr (cv m) (+) (1)) expclt mpan syl (cv m) peano2nn0 geoser.2 A (opr (cv m) (+) (1)) expclt mpan syl 1cn geoser.2 subcl (opr A (^) (opr (cv m) (+) (1))) (opr (1) (-) A) axmulcl mpan2 syl sylanc 3eqtr2rd sylan9eqr eqtrd ex geoser.2 A exp0t ax-mp (x.) (opr (1) (-) A) opreq1i 1cn geoser.2 subcl mulid2 eqtr geoser.1 A (^) (0) oprex (cv k) (0) A (^) opreq2 ser00 (x.) (opr (1) (-) A) opreq1i 1cn addid2 A (^) opreq2i geoser.2 A exp1t ax-mp eqtr (1) (-) opreq2i 3eqtr4 (cv j) (0) (opr (+) (seq0) F) fveq2 (x.) (opr (1) (-) A) opreq1d (cv j) (0) (+) (1) opreq1 A (^) opreq2d (1) (-) opreq2d eqeq12d (cv j) (cv m) (opr (+) (seq0) F) fveq2 (x.) (opr (1) (-) A) opreq1d (cv j) (cv m) (+) (1) opreq1 A (^) opreq2d (1) (-) opreq2d eqeq12d (cv j) (opr (cv m) (+) (1)) (opr (+) (seq0) F) fveq2 (x.) (opr (1) (-) A) opreq1d (cv j) (opr (cv m) (+) (1)) (+) (1) opreq1 A (^) opreq2d (1) (-) opreq2d eqeq12d (cv j) N (opr (+) (seq0) F) fveq2 (x.) (opr (1) (-) A) opreq1d (cv j) N (+) (1) opreq1 A (^) opreq2d (1) (-) opreq2d eqeq12d nn0indALT (/) (opr (1) (-) A) opreq1d (=/= A (1)) adantr eqtr3d)) thm (geolimilem ((k n) (k y) (A k) (n y) (A n) (A y) (F n) (G n)) ((geoser.1 (= F ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k))))))) (geoser.2 (e. A (CC))) (geolimilem.3 (br (` (abs) A) (<) (1))) (geolimilem.4 (Fn G (NN0))) (geolimilem.5 (-> (e. (cv n) (NN0)) (= (` G (cv n)) (opr (-u (opr A (x.) (` F (cv n)))) (/) (opr (1) (-) A)))))) (br (opr (+) (seq0) F) (~~>) (opr (1) (/) (opr (1) (-) A))) (1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geoser.2 subid (opr (1) (-) A) eqeq1i geoser.2 1cn geoser.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi mpbi reccl negcl geoser.2 mulcl geoser.2 (cv n) nnnn0t (cv k) (cv n) A (^) opreq2 geoser.1 A (^) (cv n) oprex fvopab4 syl rgen geolimilem.3 geoser.1 nn0ex k y (opr A (^) (cv k)) funopabex2 eqeltr A n expcnv mp3an pm3.2i 0z (cv n) elnn0uz geoser.1 geoser.2 A (cv k) expclt mpan fopab (cv n) ffvrni geolimilem.5 geoser.1 geoser.2 A (cv k) expclt mpan fopab (cv n) ffvrni geoser.2 negcl 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geoser.2 subid (opr (1) (-) A) eqeq1i geoser.2 1cn geoser.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi mpbi (-u A) (` F (cv n)) (opr (1) (-) A) div23t mpan2 mp3an13 syl geoser.1 geoser.2 A (cv k) expclt mpan fopab (cv n) ffvrni geoser.2 A (` F (cv n)) mulneg1t mpan syl (/) (opr (1) (-) A) opreq1d 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geoser.2 subid (opr (1) (-) A) eqeq1i geoser.2 1cn geoser.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi mpbi reccl geoser.2 (opr (1) (/) (opr (1) (-) A)) A mulneg12t mp2an geoser.2 negcl 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geoser.2 subid (opr (1) (-) A) eqeq1i geoser.2 1cn geoser.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi mpbi (-u A) (opr (1) (-) A) divrec2t mp3an eqtr4 (x.) (` F (cv n)) opreq1i eqcomi (e. (cv n) (NN0)) a1i 3eqtr3d eqtrd jca sylbir rgen pm3.2i geoser.1 nn0ex k y (opr A (^) (cv k)) funopabex2 eqeltr geolimilem.4 nn0ex G (NN0) (V) fnex mp2an 0z elisseti (opr (-u (opr (1) (/) (opr (1) (-) A))) (x.) A) (0) n climmulc2 mp2an 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geoser.2 subid (opr (1) (-) A) eqeq1i geoser.2 1cn geoser.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi mpbi reccl negcl geoser.2 mulcl mul01 breqtr 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geoser.2 subid (opr (1) (-) A) eqeq1i geoser.2 1cn geoser.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi mpbi reccl pm3.2i 0z (cv n) elnn0uz geolimilem.5 geoser.1 geoser.2 A (cv k) expclt mpan fopab (cv n) ffvrni geoser.2 A (` F (cv n)) axmulcl mpan syl (opr A (x.) (` F (cv n))) negclt syl 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geoser.2 subid (opr (1) (-) A) eqeq1i geoser.2 1cn geoser.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi mpbi (-u (opr A (x.) (` F (cv n)))) (opr (1) (-) A) divclt mp3an23 syl eqeltrd geoser.2 A (cv n) expclt mpan geoser.2 A (opr A (^) (cv n)) axmulcom mpan syl (cv k) (cv n) A (^) opreq2 geoser.1 A (^) (cv n) oprex fvopab4 A (x.) opreq2d geoser.2 A (cv n) expp1t mpan 3eqtr4rd (1) (-) opreq2d geoser.1 geoser.2 A (cv k) expclt mpan fopab (cv n) ffvrni geoser.2 A (` F (cv n)) axmulcl mpan syl 1cn (1) (opr A (x.) (` F (cv n))) negsubt mpan syl eqtr4d (/) (opr (1) (-) A) opreq1d geoser.1 geoser.2 A (cv k) expclt mpan fopab (cv n) ffvrni geoser.2 A (` F (cv n)) axmulcl mpan syl (opr A (x.) (` F (cv n))) negclt syl 1cn 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geoser.2 subid (opr (1) (-) A) eqeq1i geoser.2 1cn geoser.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi mpbi (1) (-u (opr A (x.) (` F (cv n)))) (opr (1) (-) A) divdirt mpan2 mp3an13 syl eqtrd geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geoser.1 geoser.2 (cv n) geoser mpan2 geolimilem.5 (opr (1) (/) (opr (1) (-) A)) (+) opreq2d 3eqtr4d jca sylbir rgen pm3.2i geolimilem.4 nn0ex G (NN0) (V) fnex mp2an (+) (seq0) F oprex 0z elisseti (1) (/) (opr (1) (-) A) oprex (0) n climaddc2 mp2an 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geoser.2 subid (opr (1) (-) A) eqeq1i geoser.2 1cn geoser.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi mpbi reccl addid1 breqtr)) thm (geolimi ((j k) (j n) (j y) (j z) (A j) (k n) (k y) (k z) (A k) (n y) (n z) (A n) (y z) (A y) (A z) (F j) (F n) (F z)) ((geoser.1 (= F ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k))))))) (geoser.2 (e. A (CC))) (geolimi.3 (br (` (abs) A) (<) (1)))) (br (opr (+) (seq0) F) (~~>) (opr (1) (/) (opr (1) (-) A))) (geoser.1 geoser.2 geolimi.3 (-u (opr A (x.) (` F (cv j)))) (/) (opr (1) (-) A) oprex ({<,>|} j z (/\ (e. (cv j) (NN0)) (= (cv z) (opr (-u (opr A (x.) (` F (cv j)))) (/) (opr (1) (-) A))))) eqid fnopab2 (cv j) (cv n) F fveq2 A (x.) opreq2d negeqd (/) (opr (1) (-) A) opreq1d ({<,>|} j z (/\ (e. (cv j) (NN0)) (= (cv z) (opr (-u (opr A (x.) (` F (cv j)))) (/) (opr (1) (-) A))))) eqid (-u (opr A (x.) (` F (cv n)))) (/) (opr (1) (-) A) oprex fvopab4 geolimilem)) thm (geolim1i ((k y) (A k) (A y)) ((geolim1i.1 (= F ({<,>|} k y (/\ (e. (cv k) (NN)) (= (cv y) (opr A (^) (cv k))))))) (geolim1i.2 (e. A (CC))) (geolim1i.3 (br (` (abs) A) (<) (1)))) (br (opr (+) (seq1) F) (~~>) (opr A (/) (opr (1) (-) A))) (0z (0) uzidt ax-mp nn0ex k y (opr A (^) (cv k)) funopabex2 (1) (/) (opr (1) (-) A) oprex addex nn0ex k y (opr A (^) (cv k)) funopabex2 seq0seqz ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k))))) eqid geolim1i.2 geolim1i.3 geolimi eqbrtrr ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k))))) eqid geolim1i.2 A (cv k) expclt mpan fopab nn0uz (NN0) (` (ZZ>) (0)) ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k))))) (CC) feq2 ax-mp mpbi (0) clim2serz ax-mp (cv k) nnnn0t pm4.71i (= (cv y) (opr A (^) (cv k))) anbi1i (e. (cv k) (NN)) (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k))) anass bitr k y opabbii geolim1i.1 k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k)))) (NN) resopab 3eqtr4 nnuz (NN) (` (ZZ>) (1)) ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k))))) reseq2 ax-mp eqtr (<,> (1) (+)) (seq) opreq2i 1z addex nn0ex k y (opr A (^) (cv k)) funopabex2 (1) seqzres ax-mp eqtr addex geolim1i.1 nnex k y (opr A (^) (cv k)) funopabex2 eqeltr seq1seqz 1cn addid2 (opr (0) (+) (1)) (1) (+) opeq1 ax-mp (seq) ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k))))) opreq1i 3eqtr4 1cn 1cn geolim1i.2 subcl 1cn geolim1i.2 subcl 3pm3.2i geolim1i.3 geolim1i.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geolim1i.2 subid (opr (1) (-) A) eqeq1i geolim1i.2 1cn geolim1i.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi mpbi (1) (opr (1) (-) A) (opr (1) (-) A) divsubdirt mp2an 1cn geolim1i.2 (1) A nncant mp2an (/) (opr (1) (-) A) opreq1i 1cn geolim1i.2 subcl geolim1i.3 geolim1i.2 abscl ax1re ltne ax-mp A (1) (abs) fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A (1) df-ne mpbir geolim1i.2 subid (opr (1) (-) A) eqeq1i geolim1i.2 1cn geolim1i.2 subcan2 (0) (opr (1) (-) A) eqcom 3bitr3 eqneqi mpbi divid (opr (1) (/) (opr (1) (-) A)) (-) opreq2i 3eqtr3 0z addex nn0ex k y (opr A (^) (cv k)) funopabex2 (0) seqz1 ax-mp 0nn0 (cv k) (0) A (^) opreq2 ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k))))) eqid A (^) (0) oprex fvopab4 ax-mp geolim1i.2 A exp0t ax-mp 3eqtr (opr (1) (/) (opr (1) (-) A)) (-) opreq2i eqtr4 3brtr4)) thm (geolim1 ((k y) (A k) (A y)) ((geolim1.1 (= F ({<,>|} k y (/\ (e. (cv k) (NN)) (= (cv y) (opr A (^) (cv k)))))))) (-> (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) (br (opr (+) (seq1) F) (~~>) (opr A (/) (opr (1) (-) A)))) (A (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0)) (^) (cv k) opreq1 (cv y) eqeq2d (e. (cv k) (NN)) anbi2d k y opabbidv (+) (seq1) opreq2d (= A (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0))) id A (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0)) (1) (-) opreq2 (/) opreq12d (~~>) breq12d ({<,>|} k y (/\ (e. (cv k) (NN)) (= (cv y) (opr (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0)) (^) (cv k))))) eqid A (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0)) (CC) eleq1 A (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0)) (abs) fveq2 (<) (1) breq1d anbi12d (0) (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0)) (CC) eleq1 (0) (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0)) (abs) fveq2 (<) (1) breq1d anbi12d 0cn abs0 lt01 eqbrtr pm3.2i elimhyp pm3.26i A (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0)) (CC) eleq1 A (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0)) (abs) fveq2 (<) (1) breq1d anbi12d (0) (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0)) (CC) eleq1 (0) (if (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) A (0)) (abs) fveq2 (<) (1) breq1d anbi12d 0cn abs0 lt01 eqbrtr pm3.2i elimhyp pm3.27i geolim1i dedth geolim1.1 (+) (seq1) opreq2i syl5eqbr)) thm (geosum ((k y) (A k) (A y) (N y) (N k)) ((geosum.1 (e. A (CC)))) (-> (/\ (e. N (NN0)) (=/= A (1))) (= (sum_ k (opr (0) (...) N) (opr A (^) (cv k))) (opr (opr (1) (-) (opr A (^) (opr N (+) (1)))) (/) (opr (1) (-) A)))) (N elnn0uz A (^) (cv k) oprex (cv k) elnn0uz (= (cv y) (opr A (^) (cv k))) anbi1i k y opabbii N fsumserz2 sylbi addex nn0ex k y (opr A (^) (cv k)) funopabex2 seq0seqz N fveq1i syl6eqr (=/= A (1)) adantr ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k))))) eqid geosum.1 N geoser eqtrd)) thm (cvgratlem1ALT ((x y) (x z) (y z) (A x) (A y) (A z) (B x) (B y) (B z) (C y) (F x) (F y) (F z)) ((cvgratlem1ALT.1 (:--> F (NN) (RR))) (cvgratlem1ALT.2 (e. A (RR))) (cvgratlem1ALT.3 (e. B (NN))) (cvgratlem1ALT.4 (e. C (NN)))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) (br (` F (opr B (+) C)) (<_) (opr (opr A (^) C) (x.) (` F B)))) (cvgratlem1ALT.4 (cv y) (1) B (+) opreq2 F fveq2d (cv y) (1) A (^) opreq2 (x.) (` F B) opreq1d (<) breq12d (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) imbi2d (cv y) (cv z) B (+) opreq2 F fveq2d (cv y) (cv z) A (^) opreq2 (x.) (` F B) opreq1d (<) breq12d (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) imbi2d (cv y) (opr (cv z) (+) (1)) B (+) opreq2 F fveq2d (cv y) (opr (cv z) (+) (1)) A (^) opreq2 (x.) (` F B) opreq1d (<) breq12d (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) imbi2d (cv y) C B (+) opreq2 F fveq2d (cv y) C A (^) opreq2 (x.) (` F B) opreq1d (<) breq12d (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) imbi2d cvgratlem1ALT.3 nnre leid cvgratlem1ALT.3 (cv x) B B (<_) breq2 (cv x) B (+) (1) opreq1 F fveq2d (cv x) B F fveq2 A (x.) opreq2d (<) breq12d imbi12d (NN) rcla4v ax-mp mpi cvgratlem1ALT.2 recn A exp1t ax-mp (x.) (` F B) opreq1i syl6breqr (br (0) (<) A) adantl (` F (opr (opr B (+) (cv z)) (+) (1))) (opr A (x.) (` F (opr B (+) (cv z)))) (opr A (x.) (opr (opr A (^) (cv z)) (x.) (` F B))) axlttrn cvgratlem1ALT.3 B (cv z) nnaddclt mpan (opr B (+) (cv z)) peano2nn syl cvgratlem1ALT.1 (opr (opr B (+) (cv z)) (+) (1)) ffvrni syl cvgratlem1ALT.3 B (cv z) nnaddclt mpan cvgratlem1ALT.1 (opr B (+) (cv z)) ffvrni syl cvgratlem1ALT.2 A (` F (opr B (+) (cv z))) axmulrcl mpan syl cvgratlem1ALT.2 A (cv z) reexpclt (cv z) nnnn0t sylan2 mpan cvgratlem1ALT.3 cvgratlem1ALT.1 B ffvrni ax-mp (opr A (^) (cv z)) (` F B) axmulrcl mpan2 syl cvgratlem1ALT.2 A (opr (opr A (^) (cv z)) (x.) (` F B)) axmulrcl mpan syl syl3anc (cv z) nnnn0t cvgratlem1ALT.3 nnre B (cv z) nn0addge1t mpan syl cvgratlem1ALT.3 B (cv z) nnaddclt mpan (cv x) (opr B (+) (cv z)) B (<_) breq2 (cv x) (opr B (+) (cv z)) (+) (1) opreq1 F fveq2d (cv x) (opr B (+) (cv z)) F fveq2 A (x.) opreq2d (<) breq12d imbi12d (NN) rcla4v syl mpid (` F (opr B (+) (cv z))) (opr (opr A (^) (cv z)) (x.) (` F B)) A ltmul2t biimpd ex cvgratlem1ALT.3 B (cv z) nnaddclt mpan cvgratlem1ALT.1 (opr B (+) (cv z)) ffvrni syl cvgratlem1ALT.2 A (cv z) reexpclt (cv z) nnnn0t sylan2 mpan cvgratlem1ALT.3 cvgratlem1ALT.1 B ffvrni ax-mp (opr A (^) (cv z)) (` F B) axmulrcl mpan2 syl cvgratlem1ALT.2 (e. (cv z) (NN)) a1i syl3anc imp3a syl2and exp4d com23 imp32 (cv z) nncnt cvgratlem1ALT.3 nncn 1cn B (cv z) (1) axaddass mp3an13 syl F fveq2d cvgratlem1ALT.2 recn A (cv z) expp1t (cv z) nnnn0t sylan2 mpan cvgratlem1ALT.2 recn A (cv z) expclt (cv z) nnnn0t sylan2 mpan cvgratlem1ALT.2 recn (opr A (^) (cv z)) A axmulcom mpan2 syl eqtrd (x.) (` F B) opreq1d cvgratlem1ALT.2 recn A (cv z) expclt (cv z) nnnn0t sylan2 mpan cvgratlem1ALT.2 recn cvgratlem1ALT.3 cvgratlem1ALT.1 B ffvrni ax-mp recn A (opr A (^) (cv z)) (` F B) axmulass mp3an13 syl eqtr2d (<) breq12d (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) adantr sylibd ex a2d nnind ax-mp cvgratlem1ALT.3 cvgratlem1ALT.4 B C nnaddclt mp2an cvgratlem1ALT.1 (opr B (+) C) ffvrni ax-mp cvgratlem1ALT.2 cvgratlem1ALT.4 A C reexpclt C nnnn0t sylan2 mp2an cvgratlem1ALT.3 cvgratlem1ALT.1 B ffvrni ax-mp remulcl ltle syl)) thm (cvgratlem2ALT ((A x) (B x) (F x)) ((cvgratlem1ALT.1 (:--> F (NN) (RR))) (cvgratlem1ALT.2 (e. A (RR))) (cvgratlem1ALT.3 (e. B (NN))) (cvgratlem1ALT.4 (e. C (NN)))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) (-> (br B (<) C) (br (` F C) (<_) (opr (opr (` F B) (/) (opr A (^) B)) (x.) (opr A (^) C))))) (cvgratlem1ALT.3 cvgratlem1ALT.4 nnsub (opr C (-) B) (if (e. (opr C (-) B) (NN)) (opr C (-) B) (1)) B (+) opreq2 F fveq2d (opr C (-) B) (if (e. (opr C (-) B) (NN)) (opr C (-) B) (1)) A (^) opreq2 (x.) (` F B) opreq1d (<_) breq12d (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) imbi2d cvgratlem1ALT.1 cvgratlem1ALT.2 cvgratlem1ALT.3 1nn (opr C (-) B) elimel x cvgratlem1ALT dedth cvgratlem1ALT.3 nncn cvgratlem1ALT.4 nncn pncan3 F fveq2i (<_) (opr (opr A (^) (opr C (-) B)) (x.) (` F B)) breq1i syl6ib sylbi impcom cvgratlem1ALT.2 recn cvgratlem1ALT.4 cvgratlem1ALT.3 3pm3.2i A C B expsubt (e. A (CC)) id C nnnn0t B nnnn0t 3anim123i (/\ (=/= A (0)) (br B (<) C)) adantr B C ltlet B nnret C nnret syl2an ancoms (e. A (CC)) 3adant1 (=/= A (0)) anim2d imp sylanc mpan cvgratlem1ALT.2 gt0ne0 sylan (x.) (` F B) opreq1d cvgratlem1ALT.2 cvgratlem1ALT.3 A B expgt0t B nnnn0t syl3an2 mp3an12 cvgratlem1ALT.2 cvgratlem1ALT.3 A B reexpclt B nnnn0t sylan2 mp2an gt0ne0 cvgratlem1ALT.3 cvgratlem1ALT.1 B ffvrni ax-mp recn cvgratlem1ALT.2 cvgratlem1ALT.3 A B reexpclt B nnnn0t sylan2 mp2an recn cvgratlem1ALT.2 recn cvgratlem1ALT.4 A C expclt C nnnn0t sylan2 mp2an 3pm3.2i (` F B) (opr A (^) B) (opr A (^) C) div13t mpan 3syl (br B (<) C) adantr eqtr4d (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))) adantlr breqtrd ex)) thm (cvgratlem3ALT ((A x) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (C y) (C z) (G x)) ((cvgratlem3ALT.1 (:--> F (NN) (CC))) (cvgratlem3ALT.2 (= G ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (` (abs) (` F (cv y)))))))) (cvgratlem3ALT.3 (e. A (RR))) (cvgratlem3ALT.4 (e. B (NN))) (cvgratlem3ALT.5 (e. C (NN)))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x)))))))) (-> (br B (<) C) (br (` (abs) (` F C)) (<_) (opr (opr (` (abs) (` F B)) (/) (opr A (^) B)) (x.) (opr A (^) C))))) (cvgratlem3ALT.2 cvgratlem3ALT.1 (cv y) ffvrni (` F (cv y)) absclt syl fopab cvgratlem3ALT.3 cvgratlem3ALT.4 cvgratlem3ALT.5 x cvgratlem2ALT imp cvgratlem3ALT.5 (cv y) C F fveq2 (abs) fveq2d cvgratlem3ALT.2 (abs) (` F C) fvex fvopab4 ax-mp cvgratlem3ALT.4 (cv y) B F fveq2 (abs) fveq2d cvgratlem3ALT.2 (abs) (` F B) fvex fvopab4 ax-mp (/) (opr A (^) B) opreq1i (x.) (opr A (^) C) opreq1i 3brtr3g ex (cv x) peano2nn (cv y) (opr (cv x) (+) (1)) F fveq2 (abs) fveq2d cvgratlem3ALT.2 (abs) (` F (opr (cv x) (+) (1))) fvex fvopab4 syl (cv y) (cv x) F fveq2 (abs) fveq2d cvgratlem3ALT.2 (abs) (` F (cv x)) fvex fvopab4 A (x.) opreq2d (<) breq12d (br B (<_) (cv x)) imbi2d ralbiia sylan2br)) thm (cvgratlem1 ((x y) (x z) (y z) (A x) (A y) (A z) (B x) (B y) (B z) (C y) (F x) (F y) (F z)) ((cvgratlem1.1 (:--> F (NN) (RR)))) (-> (/\ (e. C (NN)) (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))))) (br (` F (opr B (+) C)) (<_) (opr (opr A (^) C) (x.) (` F B)))) ((cv y) (1) B (+) opreq2 F fveq2d (cv y) (1) A (^) opreq2 (x.) (` F B) opreq1d (<) breq12d (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))))) imbi2d (cv y) (cv z) B (+) opreq2 F fveq2d (cv y) (cv z) A (^) opreq2 (x.) (` F B) opreq1d (<) breq12d (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))))) imbi2d (cv y) (opr (cv z) (+) (1)) B (+) opreq2 F fveq2d (cv y) (opr (cv z) (+) (1)) A (^) opreq2 (x.) (` F B) opreq1d (<) breq12d (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))))) imbi2d (cv y) C B (+) opreq2 F fveq2d (cv y) C A (^) opreq2 (x.) (` F B) opreq1d (<) breq12d (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))))) imbi2d B nnret B leidt syl (cv x) B B (<_) breq2 (cv x) B (+) (1) opreq1 F fveq2d (cv x) B F fveq2 A (x.) opreq2d (<) breq12d imbi12d (NN) rcla4v mpid imp (/\ (e. A (RR)) (br (0) (<) A)) adantl A recnt A exp1t syl (br (0) (<) A) adantr (x.) (` F B) opreq1d (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) adantr breqtrrd (` F (opr (opr B (+) (cv z)) (+) (1))) (opr A (x.) (` F (opr B (+) (cv z)))) (opr A (x.) (opr (opr A (^) (cv z)) (x.) (` F B))) axlttrn B (cv z) nnaddclt ancoms (opr B (+) (cv z)) peano2nn syl cvgratlem1.1 (opr (opr B (+) (cv z)) (+) (1)) ffvrni syl (e. A (RR)) adantll A (` F (opr B (+) (cv z))) axmulrcl B (cv z) nnaddclt ancoms cvgratlem1.1 (opr B (+) (cv z)) ffvrni syl sylan2 anassrs A (opr (opr A (^) (cv z)) (x.) (` F B)) axmulrcl (e. A (RR)) (e. (cv z) (NN)) pm3.26 (e. B (NN)) adantr (opr A (^) (cv z)) (` F B) axmulrcl A (cv z) reexpclt (cv z) nnnn0t sylan2 cvgratlem1.1 B ffvrni syl2an sylanc syl3anc B (cv z) nn0addge1t B nnret (cv z) nnnn0t syl2an B (cv z) nnaddclt (cv x) (opr B (+) (cv z)) B (<_) breq2 (cv x) (opr B (+) (cv z)) (+) (1) opreq1 F fveq2d (cv x) (opr B (+) (cv z)) F fveq2 A (x.) opreq2d (<) breq12d imbi12d (NN) rcla4v syl mpid ancoms (e. A (RR)) adantll (` F (opr B (+) (cv z))) (opr (opr A (^) (cv z)) (x.) (` F B)) A ltmul2t biimpd ex B (cv z) nnaddclt ancoms (e. A (RR)) adantll cvgratlem1.1 (opr B (+) (cv z)) ffvrni syl (opr A (^) (cv z)) (` F B) axmulrcl A (cv z) reexpclt (cv z) nnnn0t sylan2 cvgratlem1.1 B ffvrni syl2an (e. A (RR)) (e. (cv z) (NN)) pm3.26 (e. B (NN)) adantr syl3anc imp3a syl2and exp4d exp31 imp4a com12 com34 imp44 1cn B (cv z) (1) axaddass mp3an3 B nncnt (cv z) nncnt syl2an F fveq2d (e. A (CC)) adantll A (cv z) expp1t (cv z) nnnn0t sylan2 (opr A (^) (cv z)) A axmulcom A (cv z) expclt (cv z) nnnn0t sylan2 (e. A (CC)) (e. (cv z) (NN)) pm3.26 sylanc eqtrd (e. B (NN)) adantlr (x.) (` F B) opreq1d A (opr A (^) (cv z)) (` F B) axmulass 3expa (e. A (CC)) (e. (cv z) (NN)) pm3.26 A (cv z) expclt (cv z) nnnn0t sylan2 jca cvgratlem1.1 B ffvrni recnd syl2an an1rs eqtr2d (<) breq12d A recnt sylanl1 ancoms (br (0) (<) A) adantrlr (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))) adantrrr sylibd ex a2d nnind imp (` F (opr B (+) C)) (opr (opr A (^) C) (x.) (` F B)) ltlet B C nnaddclt ancoms cvgratlem1.1 (opr B (+) C) ffvrni syl (e. A (RR)) adantrl (opr A (^) C) (` F B) axmulrcl A C reexpclt C nnnn0t sylan2 ancoms cvgratlem1.1 B ffvrni syl2an anasss sylanc (br (0) (<) A) adantrlr (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))) adantrrr mpd)) thm (cvgratlem2 ((A x) (B x) (F x)) ((cvgratlem1.1 (:--> F (NN) (RR)))) (-> (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))))) (-> (/\ (e. C (NN)) (br B (<) C)) (br (` F C) (<_) (opr (opr (` F B) (/) (opr A (^) B)) (x.) (opr A (^) C))))) (B C nnsubt cvgratlem1.1 (opr C (-) B) A B x cvgratlem1 exp45 com3r imp4d (e. C (NN)) adantr B C pncan3t B nncnt C nncnt syl2an F fveq2d (<_) (opr (opr A (^) (opr C (-) B)) (x.) (` F B)) breq1d sylibd exp3a sylbid ex imp3a exp4a com12 com4l com12 imp42 A C B expsubt (e. A (CC)) id C nnnn0t B nnnn0t 3anim123i (/\ (=/= A (0)) (br B (<) C)) adantr B C ltlet ancoms C nnret B nnret syl2an (e. A (CC)) 3adant1 (=/= A (0)) anim2d imp sylanc A recnt (br (0) (<) A) adantr (e. B (NN)) (/\ (e. C (NN)) (br B (<) C)) ad2antrr (e. C (NN)) (br B (<) C) pm3.26 (/\ (/\ (e. A (RR)) (br (0) (<) A)) (e. B (NN))) adantl (/\ (e. A (RR)) (br (0) (<) A)) (e. B (NN)) pm3.27 (/\ (e. C (NN)) (br B (<) C)) adantr 3jca A gt0ne0t (e. B (NN)) adantr (e. C (NN)) (br B (<) C) pm3.27 anim12i sylanc (x.) (` F B) opreq1d (` F B) (opr A (^) B) (opr A (^) C) div13t cvgratlem1.1 B ffvrni recnd (/\ (e. A (RR)) (br (0) (<) A)) (e. C (NN)) ad2antlr A B reexpclt B nnnn0t sylan2 recnd (br (0) (<) A) adantlr (e. C (NN)) adantr A C reexpclt C nnnn0t sylan2 recnd (br (0) (<) A) adantlr (e. B (NN)) adantlr 3jca (opr A (^) B) gt0ne0t A B reexpclt B nnnn0t sylan2 (br (0) (<) A) adantlr A B expgt0t B nnnn0t syl3an2 3expa an1rs sylanc (e. C (NN)) adantr sylanc (br B (<) C) adantrr eqtr4d (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))) adantlrr breqtrd ex)) thm (cvgratlem3 ((A x) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (C y) (C z) (G x)) ((cvgratlem3.1 (:--> F (NN) (CC))) (cvgratlem3.2 (= G ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (` (abs) (` F (cv y))))))))) (-> (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (-> (/\ (e. C (NN)) (br B (<) C)) (br (` (abs) (` F C)) (<_) (opr (opr (` (abs) (` F B)) (/) (opr A (^) B)) (x.) (opr A (^) C))))) (cvgratlem3.2 cvgratlem3.1 (cv y) ffvrni (` F (cv y)) absclt syl fopab A B x C cvgratlem2 imp (cv y) C F fveq2 (abs) fveq2d cvgratlem3.2 (abs) (` F C) fvex fvopab4 (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` G (opr (cv x) (+) (1))) (<) (opr A (x.) (` G (cv x)))))))) (br B (<) C) ad2antrl (cv y) B F fveq2 (abs) fveq2d cvgratlem3.2 (abs) (` F B) fvex fvopab4 (/) (opr A (^) B) opreq1d (x.) (opr A (^) C) opreq1d (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` G (opr (cv x) (+) (1))) (<) (opr A (x.) (` G (cv x)))))) adantr (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. C (NN)) (br B (<) C)) ad2antlr 3brtr3d ex (cv x) peano2nn (cv y) (opr (cv x) (+) (1)) F fveq2 (abs) fveq2d cvgratlem3.2 (abs) (` F (opr (cv x) (+) (1))) fvex fvopab4 syl (cv y) (cv x) F fveq2 (abs) fveq2d cvgratlem3.2 (abs) (` F (cv x)) fvex fvopab4 A (x.) opreq2d (<) breq12d (br B (<_) (cv x)) imbi2d ralbiia (e. B (NN)) anbi2i sylan2br)) thm (cvgratlem4 ((A x) (B x) (F x)) ((cvgrat.1 (:--> F (NN) (CC)))) (-> (/\ (e. A (RR)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (br (0) (<) A)) (B nnret B leidt syl (cv x) B B (<_) breq2 (cv x) B (+) (1) opreq1 F fveq2d (abs) fveq2d (cv x) B F fveq2 (abs) fveq2d A (x.) opreq2d (<) breq12d imbi12d (NN) rcla4v mpid (e. A (RR)) adantl B peano2nn cvgrat.1 (opr B (+) (1)) ffvrni (` F (opr B (+) (1))) absge0t 3syl (e. A (RR)) adantl 0re (0) (` (abs) (` F (opr B (+) (1)))) (opr A (x.) (` (abs) (` F B))) lelttrt mp3an1 B peano2nn cvgrat.1 (opr B (+) (1)) ffvrni (` F (opr B (+) (1))) absclt 3syl (e. A (RR)) adantl A (` (abs) (` F B)) axmulrcl cvgrat.1 B ffvrni (` F B) absclt syl sylan2 sylanc mpand cvgrat.1 B ffvrni (` F B) absge0t syl (e. A (RR)) adantl jctild A (` (abs) (` F B)) prodgt02t ex cvgrat.1 B ffvrni (` F B) absclt syl sylan2 syld syld ex imp32)) thm (cvgratlem5 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (u x) (B x) (u y) (B y) (u v) (B v) (B u) (F x) (F y) (F v) (F u) (G y) (G v)) ((cvgrat.1 (:--> F (NN) (CC))) (cvgratlem5.2 (= G ({<,>|} z w (/\ (e. (cv z) (NN)) (= (cv w) (opr A (^) (cv z)))))))) (-> (/\ (/\ (e. A (RR)) (br A (<) (1))) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) F) (~~>) (cv y)))) (A (/) (opr (1) (-) A) oprex B G v (opr (` (abs) (` F B)) (/) (opr A (^) B)) F y cvgcmp3cet (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))) pm3.26 (/\ (e. A (RR)) (br A (<) (1))) adantl cvgrat.1 A B x cvgratlem4 (br A (<) (1)) adantlr 0re (0) A ltlet mpan imp (br A (<) (1)) adantlr A (cv z) reexpclt (cv z) nnnn0t sylan2 r19.21aiva cvgratlem5.2 (RR) fopab2 sylib (br (0) (<_) A) adantr A (cv v) expge0t (cv v) nnnn0t syl3an2 3expa an1rs (cv z) (cv v) A (^) opreq2 cvgratlem5.2 A (^) (cv v) oprex fvopab4 (/\ (e. A (RR)) (br (0) (<_) A)) adantl breqtrrd r19.21aiva jca (br A (<) (1)) adantlr cvgratlem5.2 geolim1 A recnt (br A (<) (1)) (br (0) (<_) A) ad2antrr A absidt (br A (<) (1)) adantlr (e. A (RR)) (br A (<) (1)) pm3.27 (br (0) (<_) A) adantr eqbrtrd sylanc jca syldan (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x)))))))) adantr (` (abs) (` F B)) (opr A (^) B) redivclt cvgrat.1 B ffvrni (` F B) absclt syl (e. A (RR)) (br (0) (<) A) 3ad2ant2 A B reexpclt B nnnn0t sylan2 (br (0) (<) A) 3adant3 A gt0ne0t (e. B (NN)) 3adant2 A B expne0t A recnt sylan (br (0) (<) A) 3adant3 mpbid syl3anc (` (abs) (` F B)) (opr A (^) B) divge0t cvgrat.1 B ffvrni (` F B) absclt syl (e. A (RR)) (br (0) (<) A) 3ad2ant2 A B reexpclt B nnnn0t sylan2 (br (0) (<) A) 3adant3 jca cvgrat.1 B ffvrni (` F B) absge0t syl (e. A (RR)) (br (0) (<) A) 3ad2ant2 A B expgt0t B nnnn0t syl3an2 jca sylanc jca 3expa an1rs (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))) adantrr cvgrat.1 ({<,>|} y u (/\ (e. (cv y) (NN)) (= (cv u) (` (abs) (` F (cv y)))))) eqid A B x (cv v) cvgratlem3 imp (cv z) (cv v) A (^) opreq2 cvgratlem5.2 A (^) (cv v) oprex fvopab4 (opr (` (abs) (` F B)) (/) (opr A (^) B)) (x.) opreq2d (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (br B (<) (cv v)) ad2antrl breqtrrd exp32 r19.21aiv cvgrat.1 jctil jca (br A (<) (1)) adantllr jca exp31 com23 imp mpd sylanc)) thm (cvgrat ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (F x) (F y)) ((cvgrat.1 (:--> F (NN) (CC)))) (-> (/\ (/\ (e. A (RR)) (br A (<) (1))) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) F) (~~>) (cv y)))) (cvgrat.1 ({<,>|} z w (/\ (e. (cv z) (NN)) (= (cv w) (opr A (^) (cv z))))) eqid B x y cvgratlem5)) thm (fsum0diaglem1 () () (-> (/\ (e. N A) (e. K (opr (0) (...) N))) (C_ (opr (0) (...) (opr N (-) K)) (opr (0) (...) N))) (K (0) N elfzle1 (e. N A) adantl N K subge02t N A K (0) elfzel2g N zret syl K (0) N elfzelz (e. N A) adantl K zret syl sylanc mpbid (opr N (-) K) N eluzt N K zsubclt N A K (0) elfzel2g K (0) N elfzelz (e. N A) adantl sylanc N A K (0) elfzel2g sylanc mpbird 0z N (opr N (-) K) (0) fzss2t mpan2 syl)) thm (fsum0diaglem2 ((k m) (k n) (A k) (m n) (A m) (A n) (j m) (j n) (B j) (B m) (B n) (j k)) () (-> (e. (cv n) (NN0)) (-> (-> (/\ (A.e. j (opr (0) (...) (cv n)) (e. A (CC))) (A.e. k (opr (0) (...) (cv n)) (e. B (CC)))) (= (sum_ j (opr (0) (...) (cv n)) (sum_ k (opr (0) (...) (opr (cv n) (-) (cv j))) (opr A (x.) B))) (sum_ k (opr (0) (...) (cv n)) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B)))))) (-> (/\ (A.e. j (opr (0) (...) (opr (cv n) (+) (1))) (e. A (CC))) (A.e. k (opr (0) (...) (opr (cv n) (+) (1))) (e. B (CC)))) (= (sum_ j (opr (0) (...) (opr (cv n) (+) (1))) (sum_ k (opr (0) (...) (opr (opr (cv n) (+) (1)) (-) (cv j))) (opr A (x.) B))) (sum_ k (opr (0) (...) (opr (cv n) (+) (1))) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B)))))))) ((cv n) elnn0uz (opr (cv n) (-) (cv j)) (0) k (opr A (x.) B) fsumclt (cv n) (` (ZZ>) (0)) (cv j) (0) fznn0subt nn0uz syl6eleq (A.e. k (opr (0) (...) (opr (cv n) (+) (1))) (e. B (CC))) (e. A (CC)) ad2ant2r (cv n) (` (ZZ>) (0)) (cv j) fsum0diaglem1 (cv k) sseld n visset (cv k) (0) fzelp1 syl6 (e. A (CC)) adantrr A B axmulcl ex (e. (cv n) (` (ZZ>) (0))) (e. (cv j) (opr (0) (...) (cv n))) ad2antll imim12d r19.20dv2 imp an1rs sylanc A ([_/]_ (opr (opr (cv n) (-) (cv j)) (+) (1)) k B) axmulcl (e. (cv j) (opr (0) (...) (cv n))) (e. A (CC)) pm3.27 (/\ (e. (cv n) (` (ZZ>) (0))) (A.e. k (opr (0) (...) (opr (cv n) (+) (1))) (e. B (CC)))) adantl (opr (opr (cv n) (-) (cv j)) (+) (1)) (opr (0) (...) (opr (cv n) (+) (1))) k B (CC) ra4csbela 1cn (cv n) (1) (cv j) addsubt mp3an2 (cv n) (0) eluzelz (cv n) zcnt syl (cv j) (0) (cv n) elfzelz (cv j) zcnt syl syl2an (opr (cv n) (+) (1)) (` (ZZ>) (0)) (cv j) fznn0sub2t (cv n) (0) peano2uz n visset (cv j) (0) fzelp1 syl2an eqeltrrd sylan an1rs (e. A (CC)) adantrr sylanc jca exp32 a2d n visset (cv j) (0) fzelp1 (e. A (CC)) imim1i syl5 r19.20dv2 impcom an1s (cv n) (0) j (sum_ k (opr (0) (...) (opr (cv n) (-) (cv j))) (opr A (x.) B)) (opr A (x.) ([_/]_ (opr (opr (cv n) (-) (cv j)) (+) (1)) k B)) fsumadd syldan (+) (opr ([_/]_ (opr (cv n) (+) (1)) j A) (x.) ([_/]_ (0) k B)) opreq1d (sum_ j (opr (0) (...) (cv n)) (sum_ k (opr (0) (...) (opr (cv n) (-) (cv j))) (opr A (x.) B))) (sum_ j (opr (0) (...) (cv n)) (opr A (x.) ([_/]_ (opr (opr (cv n) (-) (cv j)) (+) (1)) k B))) (opr ([_/]_ (opr (cv n) (+) (1)) j A) (x.) ([_/]_ (0) k B)) axaddass (cv n) (0) j (sum_ k (opr (0) (...) (opr (cv n) (-) (cv j))) (opr A (x.) B)) fsumclt (opr (cv n) (-) (cv j)) (0) k (opr A (x.) B) fsumclt n visset (cv n) (V) (cv j) (0) fznn0subt mpan nn0uz syl6eleq (A.e. k (opr (0) (...) (cv n)) (e. B (CC))) (e. A (CC)) ad2antlr n visset (cv n) (V) (cv j) fsum0diaglem1 mpan (cv k) sseld (e. B (CC)) imim1d (e. A (CC)) adantr A B axmulcl ex (e. (cv j) (opr (0) (...) (cv n))) adantl syl6d r19.20dv2 impcom anassrs sylanc ex r19.20dva impcom n visset (cv j) (0) fzelp1 (e. A (CC)) imim1i r19.20i2 n visset (cv k) (0) fzelp1 (e. B (CC)) imim1i r19.20i2 syl2an sylan2 A ([_/]_ (opr (opr (cv n) (-) (cv j)) (+) (1)) k B) axmulcl (e. (cv j) (opr (0) (...) (cv n))) (e. A (CC)) pm3.27 (/\ (e. (cv n) (` (ZZ>) (0))) (A.e. k (opr (0) (...) (opr (cv n) (+) (1))) (e. B (CC)))) adantl (opr (opr (cv n) (-) (cv j)) (+) (1)) (opr (0) (...) (opr (cv n) (+) (1))) k B (CC) ra4csbela 1cn (cv n) (1) (cv j) addsubt mp3an2 (cv n) (0) eluzelz (cv n) zcnt syl (cv j) (0) (cv n) elfzelz (cv j) zcnt syl syl2an (opr (cv n) (+) (1)) (` (ZZ>) (0)) (cv j) fznn0sub2t (cv n) (0) peano2uz n visset (cv j) (0) fzelp1 syl2an eqeltrrd sylan an1rs (e. A (CC)) adantrr sylanc exp32 a2d n visset (cv j) (0) fzelp1 (e. A (CC)) imim1i syl5 r19.20dv2 impcom an1s (cv n) (0) j (opr A (x.) ([_/]_ (opr (opr (cv n) (-) (cv j)) (+) (1)) k B)) fsumclt syldan (opr (cv n) (+) (1)) (opr (0) (...) (opr (cv n) (+) (1))) j A (CC) ra4csbela (opr (cv n) (+) (1)) (0) eluzfz2t sylan (0) (opr (0) (...) (opr (cv n) (+) (1))) k B (CC) ra4csbela (opr (cv n) (+) (1)) (0) eluzfz1t sylan anim12i anandis (cv n) (0) peano2uz sylan ([_/]_ (opr (cv n) (+) (1)) j A) ([_/]_ (0) k B) axmulcl syl syl3anc eqtrd (sum_ j (opr (0) (...) (cv n)) (sum_ k (opr (0) (...) (opr (cv n) (-) (cv j))) (opr A (x.) B))) (sum_ k (opr (0) (...) (cv n)) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B)))) (+) (opr (sum_ j (opr (0) (...) (cv n)) (opr A (x.) ([_/]_ (opr (opr (cv n) (-) (cv j)) (+) (1)) k B))) (+) (opr ([_/]_ (opr (cv n) (+) (1)) j A) (x.) ([_/]_ (0) k B))) opreq1 sylan9eq k (opr (0) (...) (opr (opr (cv n) (+) (1)) (-) (cv j))) (opr A (x.) B) sumex k (opr (0) (...) (opr (opr (cv n) (+) (1)) (-) (opr (cv n) (+) (1)))) (opr ([_/]_ (opr (cv n) (+) (1)) j A) (x.) B) sumex (e. (cv m) (opr (0) (...) (opr (opr (cv n) (+) (1)) (-) (opr (cv n) (+) (1))))) j ax-17 (cv n) (+) (1) oprex (e. (cv m) (opr (cv n) (+) (1))) j ax-17 A hbcsb1 (e. (cv m) (x.)) j ax-17 (e. (cv m) B) j ax-17 hbopr k hbsum (cv j) (opr (cv n) (+) (1)) (opr (cv n) (+) (1)) (-) opreq2 (0) (...) opreq2d j (opr (cv n) (+) (1)) A csbeq1a (x.) B opreq1d (e. (cv k) (opr (0) (...) (opr (opr (cv n) (+) (1)) (-) (opr (cv n) (+) (1))))) a1d r19.21aiv sumeq12rd (0) fsump1f 1cn (cv n) (1) (cv j) addsubt mp3an2 (0) (...) opreq2d k (opr A (x.) B) sumeq1d (cv n) (0) eluzelz (cv n) zcnt syl (cv j) (0) (cv n) elfzelz (cv j) zcnt syl syl2an (cv n) (` (ZZ>) (0)) (cv j) (0) fznn0subt nn0uz syl6eleq A (x.) B oprex A (x.) ([_/]_ (opr (opr (cv n) (-) (cv j)) (+) (1)) k B) oprex (e. (cv m) A) k ax-17 (e. (cv m) (x.)) k ax-17 (opr (cv n) (-) (cv j)) (+) (1) oprex (e. (cv m) (opr (opr (cv n) (-) (cv j)) (+) (1))) k ax-17 B hbcsb1 hbopr k (opr (opr (cv n) (-) (cv j)) (+) (1)) B csbeq1a A (x.) opreq2d (0) fsump1f syl eqtrd sumeq2dv (cv n) (0) eluzelz (cv n) zcnt syl (cv n) peano2cn syl (opr (cv n) (+) (1)) subidt syl (0) (...) opreq2d k (opr ([_/]_ (opr (cv n) (+) (1)) j A) (x.) B) sumeq1d (cv n) (0) eluzel2 ([_/]_ (opr (cv n) (+) (1)) j A) (x.) ([_/]_ (0) k B) oprex (e. (cv m) ([_/]_ (opr (cv n) (+) (1)) j A)) k ax-17 (e. (cv m) (x.)) k ax-17 0z (e. (cv m) (0)) k ax-17 (ZZ) B hbcsb1g ax-mp hbopr k (0) B csbeq1a ([_/]_ (opr (cv n) (+) (1)) j A) (x.) opreq2d (V) fsum1f mpan syl eqtrd (+) opreq12d eqtrd (/\ (A.e. j (opr (0) (...) (opr (cv n) (+) (1))) (e. A (CC))) (A.e. k (opr (0) (...) (opr (cv n) (+) (1))) (e. B (CC)))) (= (sum_ j (opr (0) (...) (cv n)) (sum_ k (opr (0) (...) (opr (cv n) (-) (cv j))) (opr A (x.) B))) (sum_ k (opr (0) (...) (cv n)) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B))))) ad2antrr j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B)) sumex j (opr (0) (...) (opr (cv n) (+) (1))) (opr A (x.) ([_/]_ (opr (opr (cv n) (+) (1)) (-) (cv j)) k B)) sumex (e. (cv m) (opr (0) (...) (opr (cv n) (+) (1)))) k ax-17 (e. (cv m) A) k ax-17 (e. (cv m) (x.)) k ax-17 (opr (cv n) (+) (1)) (-) (cv j) oprex (e. (cv m) (opr (opr (cv n) (+) (1)) (-) (cv j))) k ax-17 B hbcsb1 hbopr j hbsum (cv k) (opr (cv n) (+) (1)) (0) (...) opreq2 (cv k) (opr (cv n) (+) (1)) (-) (cv j) opreq1 k B csbeq1d A (x.) opreq2d (e. (cv j) (opr (0) (...) (opr (cv n) (+) (1)))) a1d r19.21aiv sumeq12rd (0) fsump1f A (x.) ([_/]_ (opr (opr (cv n) (+) (1)) (-) (cv j)) k B) oprex ([_/]_ (opr (cv n) (+) (1)) j A) (x.) ([_/]_ (opr (opr (cv n) (+) (1)) (-) (opr (cv n) (+) (1))) k B) oprex (cv n) (+) (1) oprex (e. (cv m) (opr (cv n) (+) (1))) j ax-17 A hbcsb1 (e. (cv m) (x.)) j ax-17 (e. (cv m) ([_/]_ (opr (opr (cv n) (+) (1)) (-) (opr (cv n) (+) (1))) k B)) j ax-17 hbopr j (opr (cv n) (+) (1)) A csbeq1a (cv j) (opr (cv n) (+) (1)) (opr (cv n) (+) (1)) (-) opreq2 k B csbeq1d (x.) opreq12d (0) fsump1f 1cn (cv n) (1) (cv j) addsubt mp3an2 k B csbeq1d A (x.) opreq2d (cv n) (0) eluzelz (cv n) zcnt syl (cv j) (0) (cv n) elfzelz (cv j) zcnt syl syl2an sumeq2dv (cv n) (0) eluzelz (cv n) zcnt syl (cv n) peano2cn syl (opr (cv n) (+) (1)) subidt syl k B csbeq1d ([_/]_ (opr (cv n) (+) (1)) j A) (x.) opreq2d (+) opreq12d eqtrd (sum_ k (opr (0) (...) (cv n)) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B)))) (+) opreq2d eqtrd (/\ (A.e. j (opr (0) (...) (opr (cv n) (+) (1))) (e. A (CC))) (A.e. k (opr (0) (...) (opr (cv n) (+) (1))) (e. B (CC)))) (= (sum_ j (opr (0) (...) (cv n)) (sum_ k (opr (0) (...) (opr (cv n) (-) (cv j))) (opr A (x.) B))) (sum_ k (opr (0) (...) (cv n)) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B))))) ad2antrr 3eqtr4d exp31 sylbi a2d n visset (cv j) (0) fzelp1 (e. A (CC)) imim1i r19.20i2 n visset (cv k) (0) fzelp1 (e. B (CC)) imim1i r19.20i2 anim12i (= (sum_ j (opr (0) (...) (cv n)) (sum_ k (opr (0) (...) (opr (cv n) (-) (cv j))) (opr A (x.) B))) (sum_ k (opr (0) (...) (cv n)) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B))))) imim1i syl5)) thm (fsum0diag ((k m) (k n) (A k) (m n) (A m) (A n) (j m) (j n) (B j) (B m) (B n) (j k) (N j) (N k) (N m) (N n)) () (-> (/\/\ (e. N (NN0)) (A.e. j (opr (0) (...) N) (e. A (CC))) (A.e. k (opr (0) (...) N) (e. B (CC)))) (= (sum_ j (opr (0) (...) N) (sum_ k (opr (0) (...) (opr N (-) (cv j))) (opr A (x.) B))) (sum_ k (opr (0) (...) N) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B)))))) (n j A k B fsum0diaglem2 ([_/]_ (0) j A) (x.) ([_/]_ (0) k B) oprex 0z 0z (e. (cv m) (0)) j ax-17 (ZZ) A hbcsb1g ax-mp (e. (cv m) (x.)) j ax-17 (e. (cv m) ([_/]_ (0) k B)) j ax-17 hbopr j (0) A csbeq1a (x.) ([_/]_ (0) k B) opreq1d (V) fsum1f mp2an j (opr (0) (...) (0)) (opr A (x.) ([_/]_ (0) k B)) sumex 0z (e. (cv m) (opr (0) (...) (0))) k ax-17 (e. (cv m) A) k ax-17 (e. (cv m) (x.)) k ax-17 0z (e. (cv m) (0)) k ax-17 (ZZ) B hbcsb1g ax-mp hbopr j hbsum (cv k) (0) (0) (...) opreq2 (cv k) (0) (cv j) (0) (-) opreq12 0cn subid syl6eq k B csbeq1d A (x.) opreq2d ex (cv j) (0) elfz1eqt syl5 r19.21aiv sumeq12rd (V) fsum1f mp2an k (opr (0) (...) (0)) (opr ([_/]_ (0) j A) (x.) B) sumex 0z (e. (cv m) (opr (0) (...) (0))) j ax-17 0z (e. (cv m) (0)) j ax-17 (ZZ) A hbcsb1g ax-mp (e. (cv m) (x.)) j ax-17 (e. (cv m) B) j ax-17 hbopr k hbsum (cv j) (0) (0) (-) opreq2 0cn subid syl6eq (0) (...) opreq2d j (0) A csbeq1a (x.) B opreq1d (e. (cv k) (opr (0) (...) (0))) a1d r19.21aiv sumeq12rd (V) fsum1f mp2an ([_/]_ (0) j A) (x.) ([_/]_ (0) k B) oprex 0z (e. (cv m) ([_/]_ (0) j A)) k ax-17 (e. (cv m) (x.)) k ax-17 0z (e. (cv m) (0)) k ax-17 (ZZ) B hbcsb1g ax-mp hbopr k (0) B csbeq1a ([_/]_ (0) j A) (x.) opreq2d (V) fsum1f mp2an eqtr 3eqtr4r (/\ (A.e. j (opr (0) (...) (0)) (e. A (CC))) (A.e. k (opr (0) (...) (0)) (e. B (CC)))) a1i (cv m) (0) (0) (...) opreq2 j (e. A (CC)) raleq1d (cv m) (0) (0) (...) opreq2 k (e. B (CC)) raleq1d anbi12d (cv m) (0) (0) (...) opreq2 (cv m) (0) (-) (cv j) opreq1 (0) (...) opreq2d k (opr A (x.) B) sumeq1d (e. (cv j) (opr (0) (...) (cv m))) a1d r19.21aiv sumeq12d (cv m) (0) (0) (...) opreq2 k (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B))) sumeq1d eqeq12d imbi12d (cv m) (cv n) (0) (...) opreq2 j (e. A (CC)) raleq1d (cv m) (cv n) (0) (...) opreq2 k (e. B (CC)) raleq1d anbi12d (cv m) (cv n) (0) (...) opreq2 (cv m) (cv n) (-) (cv j) opreq1 (0) (...) opreq2d k (opr A (x.) B) sumeq1d (e. (cv j) (opr (0) (...) (cv m))) a1d r19.21aiv sumeq12d (cv m) (cv n) (0) (...) opreq2 k (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B))) sumeq1d eqeq12d imbi12d (cv m) (opr (cv n) (+) (1)) (0) (...) opreq2 j (e. A (CC)) raleq1d (cv m) (opr (cv n) (+) (1)) (0) (...) opreq2 k (e. B (CC)) raleq1d anbi12d (cv m) (opr (cv n) (+) (1)) (0) (...) opreq2 (cv m) (opr (cv n) (+) (1)) (-) (cv j) opreq1 (0) (...) opreq2d k (opr A (x.) B) sumeq1d (e. (cv j) (opr (0) (...) (cv m))) a1d r19.21aiv sumeq12d (cv m) (opr (cv n) (+) (1)) (0) (...) opreq2 k (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B))) sumeq1d eqeq12d imbi12d (cv m) N (0) (...) opreq2 j (e. A (CC)) raleq1d (cv m) N (0) (...) opreq2 k (e. B (CC)) raleq1d anbi12d (cv m) N (0) (...) opreq2 (cv m) N (-) (cv j) opreq1 (0) (...) opreq2d k (opr A (x.) B) sumeq1d (e. (cv j) (opr (0) (...) (cv m))) a1d r19.21aiv sumeq12d (cv m) N (0) (...) opreq2 k (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B))) sumeq1d eqeq12d imbi12d nn0indALT 3impib)) thm (fsum0diag2 ((k m) (k n) (k x) (A k) (m n) (m x) (A m) (n x) (A n) (A x) (j m) (j n) (j x) (B j) (B m) (B n) (B x) (j k) (N j) (N k) (N m) (N n) (N x)) () (-> (/\/\ (e. N (NN0)) (A.e. j (opr (0) (...) N) (e. A (CC))) (A.e. k (opr (0) (...) N) (e. B (CC)))) (= (sum_ j (opr (0) (...) N) (sum_ k (opr (0) (...) (opr N (-) (cv j))) (opr A (x.) B))) (sum_ j (opr (0) (...) N) (sum_ k (opr (0) (...) (cv j)) (opr ([_/]_ (opr (cv j) (-) (cv k)) j A) (x.) B))))) (N j A k B fsum0diag (cv m) (0) (cv m) j (opr A (x.) ([_/]_ (opr (cv m) (-) (cv j)) k B)) n fsumrev (cv m) (0) N elfzuzt (/\ (e. N (NN0)) (/\ (A.e. j (opr (0) (...) N) (e. A (CC))) (A.e. k (opr (0) (...) N) (e. B (CC))))) adantl (cv m) (0) N elfzelz (/\ (e. N (NN0)) (/\ (A.e. j (opr (0) (...) N) (e. A (CC))) (A.e. k (opr (0) (...) N) (e. B (CC))))) adantl N (NN0) (cv m) (0) elfzuz3t 0z N (cv m) (0) fzss2t mpan2 syl (cv j) sseld (e. A (CC)) imim1d (A.e. k (opr (0) (...) N) (e. B (CC))) adantr A ([_/]_ (opr (cv m) (-) (cv j)) k B) axmulcl (opr (cv m) (-) (cv j)) (opr (0) (...) N) k B (CC) ra4csbela N (NN0) (cv m) (0) elfzuz3t 0z N (cv m) (0) fzss2t mpan2 syl (e. (cv j) (opr (0) (...) (cv m))) adantr m visset (cv m) (V) (cv j) fznn0sub2t mpan (/\ (e. N (NN0)) (e. (cv m) (opr (0) (...) N))) adantl sseldd sylan an1rs sylan2 exp32 com3l a2d syld r19.20dv2 exp31 com24 imp42 syl3anc (cv m) (0) N elfzelz (cv m) zcnt syl (cv m) subidt (cv m) subid1t (...) opreq12d syl (e. N (NN0)) adantl (cv m) (cv n) nncant m visset (cv n) (0) elfzel2 (cv m) zcnt syl (cv n) (0) (cv m) elfzelz (cv n) zcnt syl sylanc k B csbeq1d ([_/]_ (opr (cv m) (-) (cv n)) j A) (x.) opreq2d (/\ (e. N (NN0)) (e. (cv m) (opr (0) (...) N))) adantl (cv m) (-) (cv n) oprex (opr (cv m) (-) (cv n)) (V) j A (x.) ([_/]_ (opr (cv m) (-) (cv j)) k B) csbopr12g ax-mp (cv m) (-) (cv n) oprex (cv m) (-) (cv j) oprex j ax-gen (cv j) (opr (cv m) (-) (cv n)) (cv m) (-) opreq2 (V) (V) k B csbco3g mp2an ([_/]_ (opr (cv m) (-) (cv n)) j A) (x.) opreq2i eqtr syl5eq r19.21aiva sumeq12rd (/\ (A.e. j (opr (0) (...) N) (e. A (CC))) (A.e. k (opr (0) (...) N) (e. B (CC)))) adantlr eqtrd (e. (cv x) (opr ([_/]_ (opr (cv m) (-) (cv k)) j A) (x.) B)) n ax-17 (e. (cv x) ([_/]_ (opr (cv m) (-) (cv n)) j A)) k ax-17 (e. (cv x) (x.)) k ax-17 n visset (e. (cv x) (cv n)) k ax-17 B hbcsb1 hbopr (cv k) (cv n) (cv m) (-) opreq2 j A csbeq1d k (cv n) B csbeq1a (x.) opreq12d (opr (0) (...) (cv m)) cbvsum syl6eqr ex 3impb r19.21aiv sumeq2d (e. (cv x) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B)))) m ax-17 (e. (cv x) (opr (0) (...) (cv m))) k ax-17 (e. (cv x) A) k ax-17 (e. (cv x) (x.)) k ax-17 (cv m) (-) (cv j) oprex (e. (cv x) (opr (cv m) (-) (cv j))) k ax-17 B hbcsb1 hbopr j hbsum (cv k) (cv m) (0) (...) opreq2 (cv k) (cv m) (-) (cv j) opreq1 k B csbeq1d A (x.) opreq2d (e. (cv j) (opr (0) (...) (cv m))) a1d r19.21aiv sumeq12rd (opr (0) (...) N) cbvsum (e. (cv x) (sum_ k (opr (0) (...) (cv j)) (opr ([_/]_ (opr (cv j) (-) (cv k)) j A) (x.) B))) m ax-17 (e. (cv x) (opr (0) (...) (cv m))) j ax-17 (cv m) (-) (cv k) oprex (e. (cv x) (opr (cv m) (-) (cv k))) j ax-17 A hbcsb1 (e. (cv x) (x.)) j ax-17 (e. (cv x) B) j ax-17 hbopr k hbsum (cv j) (cv m) (0) (...) opreq2 (cv j) (cv m) (-) (cv k) opreq1 j A csbeq1d (x.) B opreq1d (e. (cv k) (opr (0) (...) (cv m))) a1d r19.21aiv sumeq12rd (opr (0) (...) N) cbvsum 3eqtr4g eqtrd)) thm (efcltlem1 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (F w)) ((efcltlem.1 (= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y)))))))) (efcltlem1.2 (e. A (CC))) (efcltlem1.3 (=/= A (0)))) (E. x (br (opr (+) (seq1) F) (~~>) (cv x))) (efcltlem1.2 abscl efcltlem1.2 abscl (` (abs) A) flclt ax-mp (` (floor) (` (abs) A)) peano2z ax-mp zre efcltlem1.2 abscl (` (abs) A) flclt ax-mp (` (floor) (` (abs) A)) peano2z ax-mp zre efcltlem1.2 absge0 efcltlem1.2 abscl 0z (` (abs) A) (0) flget mp2an mpbi lt01 efcltlem1.2 abscl (` (abs) A) flclt ax-mp zre ax1re addgegt0 mp2an gt0ne0i redivcl efcltlem1.2 abscl recn div1 efcltlem1.2 abscl (` (abs) A) flleltt ax-mp pm3.27i eqbrtr efcltlem1.2 abscl ax1re efcltlem1.2 abscl (` (abs) A) flclt ax-mp (` (floor) (` (abs) A)) peano2z ax-mp zre lt01 efcltlem1.2 absge0 efcltlem1.2 abscl 0z (` (abs) A) (0) flget mp2an mpbi lt01 efcltlem1.2 abscl (` (abs) A) flclt ax-mp zre ax1re addgegt0 mp2an ltdiv23i mpbi pm3.2i efcltlem1.2 abscl efcltlem1.2 absge0 (` (abs) A) flge0nn0t mp2an (` (floor) (` (abs) A)) nn0p1nnt ax-mp efcltlem1.2 abscl efcltlem1.2 absge0 (` (abs) A) flge0nn0t mp2an (` (floor) (` (abs) A)) nn0p1nnt ax-mp (opr (` (floor) (` (abs) A)) (+) (1)) (cv w) nnleltp1t mpan efcltlem1.2 abscl (` (abs) A) flclt ax-mp (` (floor) (` (abs) A)) peano2z ax-mp zre efcltlem1.2 absge0 efcltlem1.2 abscl 0z (` (abs) A) (0) flget mp2an mpbi lt01 efcltlem1.2 abscl (` (abs) A) flclt ax-mp zre ax1re addgegt0 mp2an efcltlem1.2 abscl efcltlem1.3 efcltlem1.2 absgt0 mpbi pm3.2i (e. (opr (` (floor) (` (abs) A)) (+) (1)) (RR)) (e. (` (abs) (opr (cv w) (+) (1))) (RR)) (e. (` (abs) A) (RR)) (br (0) (<) (opr (` (floor) (` (abs) A)) (+) (1))) (br (0) (<) (` (abs) (opr (cv w) (+) (1)))) (br (0) (<) (` (abs) A)) an6 (opr (` (floor) (` (abs) A)) (+) (1)) (` (abs) (opr (cv w) (+) (1))) (` (abs) A) ltdiv2t sylbir mp3an3 mpanl12 (cv w) peano2nn (opr (cv w) (+) (1)) nncnt (opr (cv w) (+) (1)) absclt 3syl (cv w) peano2nn (opr (cv w) (+) (1)) nngt0t (opr (cv w) (+) (1)) absidt (opr (cv w) (+) (1)) nnret (opr (cv w) (+) (1)) nnnn0t (opr (cv w) (+) (1)) nn0ge0t syl sylanc breqtrrd syl sylanc (opr (cv w) (+) (1)) absidt (cv w) peano2nn (opr (cv w) (+) (1)) nnret syl (cv w) nnnn0t (cv w) peano2nn0 (opr (cv w) (+) (1)) nn0ge0t 3syl sylanc (opr (` (floor) (` (abs) A)) (+) (1)) (<) breq2d efcltlem1.2 abscl efcltlem1.2 abscl (` (abs) A) flclt ax-mp (` (floor) (` (abs) A)) peano2z ax-mp zre efcltlem1.2 abscl (` (abs) A) flclt ax-mp (` (floor) (` (abs) A)) peano2z ax-mp zre efcltlem1.2 absge0 efcltlem1.2 abscl 0z (` (abs) A) (0) flget mp2an mpbi lt01 efcltlem1.2 abscl (` (abs) A) flclt ax-mp zre ax1re addgegt0 mp2an gt0ne0i redivcl (opr (` (abs) A) (/) (` (abs) (opr (cv w) (+) (1)))) (opr (` (abs) A) (/) (opr (` (floor) (` (abs) A)) (+) (1))) (` (abs) (opr (opr A (^) (cv w)) (/) (` (!) (cv w)))) ltmul1t mp3anl2 efcltlem1.2 abscl (` (abs) A) (` (abs) (opr (cv w) (+) (1))) redivclt mp3an1 (cv w) peano2nn (opr (cv w) (+) (1)) nncnt (opr (cv w) (+) (1)) absclt 3syl (` (abs) (opr (cv w) (+) (1))) gt0ne0t (cv w) peano2nn (opr (cv w) (+) (1)) nncnt (opr (cv w) (+) (1)) absclt 3syl (cv w) peano2nn (opr (cv w) (+) (1)) nngt0t (opr (cv w) (+) (1)) absidt (opr (cv w) (+) (1)) nnret (opr (cv w) (+) (1)) nnnn0t (opr (cv w) (+) (1)) nn0ge0t syl sylanc breqtrrd syl sylanc sylanc (opr A (^) (cv w)) (` (!) (cv w)) divclt efcltlem1.2 A (cv w) expclt (cv w) nnnn0t sylan2 mpan (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nncnt 3syl (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nnne0t 3syl syl3anc (opr (opr A (^) (cv w)) (/) (` (!) (cv w))) absclt syl jca (opr (opr A (^) (cv w)) (/) (` (!) (cv w))) absgt0t biimpa (opr A (^) (cv w)) (` (!) (cv w)) divclt efcltlem1.2 A (cv w) expclt (cv w) nnnn0t sylan2 mpan (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nncnt 3syl (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nnne0t 3syl syl3anc (opr A (^) (cv w)) (` (!) (cv w)) divne0bt biimpa efcltlem1.2 A (cv w) expclt (cv w) nnnn0t sylan2 mpan (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nncnt 3syl (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nnne0t 3syl 3jca efcltlem1.3 efcltlem1.2 A (cv w) expne0t mpan mpbii sylanc sylanc sylanc (cv w) peano2nn (cv y) (opr (cv w) (+) (1)) A (^) opreq2 (cv y) (opr (cv w) (+) (1)) (!) fveq2 (/) opreq12d efcltlem.1 (opr A (^) (opr (cv w) (+) (1))) (/) (` (!) (opr (cv w) (+) (1))) oprex fvopab4 syl efcltlem1.2 A (cv w) expp1t (cv w) nnnn0t sylan2 mpan (cv w) nnnn0t (cv w) facp1t syl (/) opreq12d (opr A (^) (cv w)) (` (!) (cv w)) A (opr (cv w) (+) (1)) divmuldivt efcltlem1.2 A (cv w) expclt (cv w) nnnn0t sylan2 mpan (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nncnt 3syl jca (cv w) peano2nn (opr (cv w) (+) (1)) nncnt syl efcltlem1.2 jctil jca (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nnne0t 3syl (cv w) peano2nn (opr (cv w) (+) (1)) nnne0t syl jca sylanc eqtr4d (opr (opr A (^) (cv w)) (/) (` (!) (cv w))) (opr A (/) (opr (cv w) (+) (1))) axmulcom (opr A (^) (cv w)) (` (!) (cv w)) divclt efcltlem1.2 A (cv w) expclt (cv w) nnnn0t sylan2 mpan (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nncnt 3syl (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nnne0t 3syl syl3anc efcltlem1.2 A (opr (cv w) (+) (1)) divclt mp3an1 (cv w) peano2nn (opr (cv w) (+) (1)) nncnt syl (cv w) peano2nn (opr (cv w) (+) (1)) nnne0t syl sylanc sylanc 3eqtrd (abs) fveq2d (opr A (/) (opr (cv w) (+) (1))) (opr (opr A (^) (cv w)) (/) (` (!) (cv w))) absmult efcltlem1.2 A (opr (cv w) (+) (1)) divclt mp3an1 (cv w) peano2nn (opr (cv w) (+) (1)) nncnt syl (cv w) peano2nn (opr (cv w) (+) (1)) nnne0t syl sylanc (opr A (^) (cv w)) (` (!) (cv w)) divclt efcltlem1.2 A (cv w) expclt (cv w) nnnn0t sylan2 mpan (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nncnt 3syl (cv w) nnnn0t (cv w) facclt (` (!) (cv w)) nnne0t 3syl syl3anc sylanc efcltlem1.2 A (opr (cv w) (+) (1)) absdivt mp3an1 (cv w) peano2nn (opr (cv w) (+) (1)) nncnt syl (cv w) peano2nn (opr (cv w) (+) (1)) nnne0t syl sylanc (x.) (` (abs) (opr (opr A (^) (cv w)) (/) (` (!) (cv w)))) opreq1d 3eqtrd (cv y) (cv w) A (^) opreq2 (cv y) (cv w) (!) fveq2 (/) opreq12d efcltlem.1 (opr A (^) (cv w)) (/) (` (!) (cv w)) oprex fvopab4 (abs) fveq2d (opr (` (abs) A) (/) (opr (` (floor) (` (abs) A)) (+) (1))) (x.) opreq2d (<) breq12d bitr4d 3bitr3d bitrd biimpd rgen pm3.2i efcltlem.1 (opr A (^) (cv y)) (` (!) (cv y)) divclt efcltlem1.2 A (cv y) expclt (cv y) nnnn0t sylan2 mpan (cv y) nnnn0t (cv y) facclt (` (!) (cv y)) nncnt 3syl (cv y) nnnn0t (cv y) facne0t syl syl3anc fopab (opr (` (abs) A) (/) (opr (` (floor) (` (abs) A)) (+) (1))) (opr (` (floor) (` (abs) A)) (+) (1)) w x cvgrat mp2an)) thm (efcltlem2 ((y z) (A y) (A z)) ((efcltlem.1 (= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y))))))))) (-> (= A (0)) (br (opr (+) (seq1) F) (~~>) (0))) (A (0) (^) (cv y) opreq1 (/) (` (!) (cv y)) opreq1d (0) eqeq1d (cv y) 0expt (/) (` (!) (cv y)) opreq1d (cv y) nnnn0t (` (!) (cv y)) div0t (cv y) facclt (` (!) (cv y)) nncnt syl (cv y) facne0t sylanc syl eqtrd syl5bir (opr (opr A (^) (cv y)) (/) (` (!) (cv y))) (0) (cv z) eqeq2 syl6 pm5.32d y z opabbidv efcltlem.1 (NN) (0) y z fconstopab 3eqtr4g (+) (seq1) opreq2d ser1clim0 syl6eqbr)) thm (efcltlem3 ((x y) (x z) (A x) (y z) (A y) (A z) (F x)) ((efcltlem.1 (= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y))))))))) (-> (e. A (CC)) (E. x (br (opr (+) (seq1) F) (~~>) (cv x)))) (A (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (^) (cv y) opreq1 (/) (` (!) (cv y)) opreq1d (cv z) eqeq2d (e. (cv y) (NN)) anbi2d y z opabbidv (+) (seq1) opreq2d (~~>) (cv x) breq1d x exbidv ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (^) (cv y)) (/) (` (!) (cv y)))))) eqid A (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (CC) eleq1 A (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (0) neeq1 anbi12d (1) (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (CC) eleq1 (1) (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (0) neeq1 anbi12d 1cn ax1ne0 pm3.2i elimhyp pm3.26i A (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (CC) eleq1 A (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (0) neeq1 anbi12d (1) (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (CC) eleq1 (1) (if (/\ (e. A (CC)) (=/= A (0))) A (1)) (0) neeq1 anbi12d 1cn ax1ne0 pm3.2i elimhyp pm3.27i x efcltlem1 dedth efcltlem.1 (+) (seq1) opreq2i (~~>) (cv x) breq1i x exbii sylibr ex A (0) df-ne syl5ibr efcltlem.1 efcltlem2 0cn elisseti (cv x) (0) (opr (+) (seq1) F) (~~>) breq2 cla4ev syl pm2.61d2)) thm (efcltlem2a ((j k) (j m) (j y) (A j) (k m) (k y) (A k) (m y) (A m) (A y) (F k) (F m)) ((efcltlem4.1 (= F ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j))))))))) (-> (= A (0)) (br (opr (+) (seq0) F) (~~>) (1))) (efcltlem4.1 (opr A (^) (opr (cv m) (+) (1))) (/) (` (!) (opr (cv m) (+) (1))) oprex (cv j) (opr (cv m) (+) (1)) A (^) opreq2 (cv j) (opr (cv m) (+) (1)) (!) fveq2 (/) opreq12d ser0p1 (= A (0)) (= (` (opr (+) (seq0) F) (cv m)) (1)) ad2antrr (` (opr (+) (seq0) F) (cv m)) (1) (+) (opr (opr A (^) (opr (cv m) (+) (1))) (/) (` (!) (opr (cv m) (+) (1)))) opreq1 A (0) (^) (opr (cv m) (+) (1)) opreq1 (cv m) nn0p1nnt (opr (cv m) (+) (1)) 0expt syl sylan9eqr (/) (` (!) (opr (cv m) (+) (1))) opreq1d (` (!) (opr (cv m) (+) (1))) div0t (cv m) peano2nn0 (opr (cv m) (+) (1)) facclt (` (!) (opr (cv m) (+) (1))) nncnt syl syl (cv m) peano2nn0 (opr (cv m) (+) (1)) facne0t syl sylanc (= A (0)) adantr eqtrd (1) (+) opreq2d 1cn addid1 syl6eq sylan9eqr eqtrd exp31 a2d addex efcltlem4.1 nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 eqeltr seq00 (= A (0)) a1i 0nn0 (= A (0)) a1i (cv j) (0) A (^) opreq2 (cv j) (0) (!) fveq2 (/) opreq12d efcltlem4.1 (opr A (^) (0)) (/) (` (!) (0)) oprex fvopab4 syl A (0) (^) (0) opreq1 0cn (0) exp0t ax-mp syl6eq (/) (` (!) (0)) opreq1d fac0 (1) (/) opreq2i 1cn div1 eqtr syl6eq 3eqtrd (cv k) (0) (opr (+) (seq0) F) fveq2 (1) eqeq1d (= A (0)) imbi2d (cv k) (cv m) (opr (+) (seq0) F) fveq2 (1) eqeq1d (= A (0)) imbi2d (cv k) (opr (cv m) (+) (1)) (opr (+) (seq0) F) fveq2 (1) eqeq1d (= A (0)) imbi2d (cv k) (cv m) (opr (+) (seq0) F) fveq2 (1) eqeq1d (= A (0)) imbi2d nn0indALT impcom (cv m) elnn0uz sylan2br r19.21aiva 0z 1cn (+) (seq0) F oprex (0) (1) m climconst mp3an12 syl)) thm (efcltlem4 ((j k) (j m) (j w) (j x) (j y) (A j) (k m) (k w) (k x) (k y) (A k) (m w) (m x) (m y) (A m) (w x) (w y) (A w) (x y) (A x) (A y) (F k) (F m) (F w) (F x)) ((efcltlem4.1 (= F ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j))))))))) (-> (e. A (CC)) (E. x (br (opr (+) (seq0) F) (~~>) (cv x)))) (j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j))))) (NN) resopab efcltlem4.1 F ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) (NN) reseq1 ax-mp (cv j) nnnn0t pm4.71i (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) anbi1i (e. (cv j) (NN)) (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) anass bitr j y opabbii 3eqtr4 w efcltlem3 addex efcltlem4.1 nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 eqeltr seq1res (~~>) (cv w) breq1i w exbii sylib (<,> (1) (+)) (seq) F oprex (<,> (0) (+)) (seq) F oprex w visset F (0) fvex (1) k climaddc2 (e. A (CC)) (br (opr (<,> (1) (+)) (seq) F) (~~>) (cv w)) pm3.27 (opr A (^) (cv j)) (` (!) (cv j)) divclt A (cv j) expclt (cv j) facclt (` (!) (cv j)) nncnt syl (e. A (CC)) adantl (cv j) facne0t (e. A (CC)) adantl syl3anc r19.21aiva efcltlem4.1 (CC) fopab2 sylib 0nn0 F (NN0) (CC) (0) ffvrn mpan2 syl (br (opr (<,> (1) (+)) (seq) F) (~~>) (cv w)) adantr jca efcltlem4.1 nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 eqeltr (cv k) (1) m serzclt (e. A (CC)) (e. (cv k) (` (ZZ>) (1))) pm3.27 F (NN0) (CC) (cv m) ffvrn (opr A (^) (cv j)) (` (!) (cv j)) divclt A (cv j) expclt (cv j) facclt (` (!) (cv j)) nncnt syl (e. A (CC)) adantl (cv j) facne0t (e. A (CC)) adantl syl3anc r19.21aiva efcltlem4.1 (CC) fopab2 sylib (cv m) (cv k) elfznnt (cv m) nnnn0t syl syl2an r19.21aiva (e. (cv k) (` (ZZ>) (1))) adantr sylanc efcltlem4.1 nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 eqeltr (cv k) (0) m serz1p (cv k) nnnn0t (cv k) elnnuz (cv k) elnn0uz 3imtr3 (e. A (CC)) adantl (cv k) elnnuz (cv k) nngt0t sylbir (e. A (CC)) adantl F (NN0) (CC) (cv m) ffvrn (opr A (^) (cv j)) (` (!) (cv j)) divclt A (cv j) expclt (cv j) facclt (` (!) (cv j)) nncnt syl (e. A (CC)) adantl (cv j) facne0t (e. A (CC)) adantl syl3anc r19.21aiva efcltlem4.1 (CC) fopab2 sylib (cv m) (cv k) elfznn0t syl2an r19.21aiva (e. (cv k) (` (ZZ>) (1))) adantr syl3anc 1cn addid2 (opr (0) (+) (1)) (1) (+) opeq1 ax-mp (seq) F opreq1i (cv k) fveq1i (` F (0)) (+) opreq2i syl6eq jca r19.21aiva (br (opr (<,> (1) (+)) (seq) F) (~~>) (cv w)) adantr 1z jctil sylanc ex addex efcltlem4.1 nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 eqeltr seq1seqz (~~>) (cv w) breq1i addex efcltlem4.1 nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 eqeltr seq0seqz (~~>) (opr (` F (0)) (+) (cv w)) breq1i 3imtr4g (` F (0)) (+) (cv w) oprex (cv x) (opr (` F (0)) (+) (cv w)) (opr (+) (seq0) F) (~~>) breq2 cla4ev syl6 w 19.23adv mpd)) thm (dfef2lem ((j k) (j x) (j y) (A j) (k x) (k y) (A k) (x y) (A x) (A y)) ((dfef2lem.1 (e. A (CC)))) (= (sum_ k (NN0) (opr (opr A (^) (cv k)) (/) (` (!) (cv k)))) (opr (1) (+) (sum_ k (NN) (` ({<,>|} j y (/\ (e. (cv j) (NN)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) (cv k))))) ((cv j) (cv k) A (^) opreq2 (cv j) (cv k) (!) fveq2 (/) opreq12d ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid (opr A (^) (cv k)) (/) (` (!) (cv k)) oprex fvopab4 (opr A (^) (cv k)) (` (!) (cv k)) divclt dfef2lem.1 A (cv k) expclt mpan (cv k) facclt (` (!) (cv k)) nncnt syl (cv k) facne0t syl3anc eqeltrd rgen dfef2lem.1 ({<,>|} j y (/\ (e. (cv j) (NN)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid x efcltlem3 ax-mp (opr A (^) (cv j)) (/) (` (!) (cv j)) oprex ({<,>|} j y (/\ (e. (cv j) (NN)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid dmopab2 (dom ({<,>|} j y (/\ (e. (cv j) (NN)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j))))))) (NN) ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) reseq2 ax-mp ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid (opr A (^) (cv j)) (` (!) (cv j)) divclt dfef2lem.1 A (cv j) expclt mpan (cv j) facclt (` (!) (cv j)) nncnt syl (cv j) facne0t syl3anc fopab ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) (NN0) (CC) ffun ax-mp (cv j) nnnn0t (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) anim1i j y ssopab2i ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) ({<,>|} j y (/\ (e. (cv j) (NN)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) funssres mp2an eqtr3 (+) (seq1) opreq2i addex nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 seq1res eqtr3 (~~>) (cv x) breq1i x exbii mpbi nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 k x isumnn0nn mp2an (cv j) (cv k) A (^) opreq2 (cv j) (cv k) (!) fveq2 (/) opreq12d ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid (opr A (^) (cv k)) (/) (` (!) (cv k)) oprex fvopab4 sumeq2i 0nn0 (cv j) (0) A (^) opreq2 dfef2lem.1 A exp0t ax-mp syl6eq (cv j) (0) (!) fveq2 fac0 syl6eq (/) opreq12d 1cn div1 syl6eq ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid 1cn elisseti fvopab4 ax-mp (cv k) nnnn0t (cv j) (cv k) A (^) opreq2 (cv j) (cv k) (!) fveq2 (/) opreq12d ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid (opr A (^) (cv k)) (/) (` (!) (cv k)) oprex fvopab4 syl (cv j) (cv k) A (^) opreq2 (cv j) (cv k) (!) fveq2 (/) opreq12d ({<,>|} j y (/\ (e. (cv j) (NN)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid (opr A (^) (cv k)) (/) (` (!) (cv k)) oprex fvopab4 eqtr4d sumeq2i (+) opreq12i 3eqtr3)) thm (efvalt ((x y) (k x) (A x) (k y) (A y) (A k)) () (-> (e. A (CC)) (= (` (exp) A) (sum_ k (NN0) (opr (opr A (^) (cv k)) (/) (` (!) (cv k)))))) ((cv x) A (^) (cv k) opreq1 (/) (` (!) (cv k)) opreq1d k (NN0) sumeq2sdv x y k df-ef k (NN0) (opr (opr A (^) (cv k)) (/) (` (!) (cv k))) sumex fvopab4)) thm (eval () () (= (e) (sum_ k (NN0) (opr (opr (1) (^) (cv k)) (/) (` (!) (cv k))))) (df-e 1cn (1) k efvalt ax-mp eqtr)) thm (efclt ((j k) (j x) (j y) (A j) (k x) (k y) (A k) (x y) (A x) (A y)) () (-> (e. A (CC)) (e. (` (exp) A) (CC))) (A k efvalt nn0uz eqcomi (cv j) eleq2i (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) anbi1i j y opabbii x efcltlem4 addex (ZZ>) (0) fvex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 seq0seqz (~~>) (cv x) breq1i x exbii sylib 0z (ZZ>) (0) fvex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 (0) x k isumclt mpan syl nn0uz k (opr (opr A (^) (cv k)) (/) (` (!) (cv k))) sumeq1i (cv j) (cv k) A (^) opreq2 (cv j) (cv k) (!) fveq2 (/) opreq12d ({<,>|} j y (/\ (e. (cv j) (` (ZZ>) (0))) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid (opr A (^) (cv k)) (/) (` (!) (cv k)) oprex fvopab4 sumeq2i eqtr4 syl5eqel eqeltrd)) thm (efcvg ((A j) (A k) (A x) (A y) (j k) (j x) (j y) (k x) (k y) (x y) (F k) (F x)) ((efcvg.1 (= F ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j))))))))) (-> (e. A (CC)) (br (opr (+) (seq0) F) (~~>) (` (exp) A))) (efcvg.1 x efcltlem4 addex efcvg.1 nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 eqeltr seq0seqz (~~>) (cv x) breq1i x exbii sylib 0z efcvg.1 nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 eqeltr (0) x k isumclim2t mpan syl nn0uz k (` F (cv k)) sumeq1i (cv j) (cv k) A (^) opreq2 (cv j) (cv k) (!) fveq2 (/) opreq12d efcvg.1 (opr A (^) (cv k)) (/) (` (!) (cv k)) oprex fvopab4 sumeq2i eqtr3 syl6breq addex efcvg.1 nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 eqeltr seq0seqz syl5eqbr A k efvalt breqtrrd)) thm (efcvgfsum ((k n) (k x) (k y) (k z) (A k) (n x) (n y) (n z) (A n) (x y) (x z) (A x) (y z) (A y) (A z)) ((efcvgfsum.1 (= F ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (sum_ k (opr (0) (...) (cv n)) (opr (opr A (^) (cv k)) (/) (` (!) (cv k)))))))))) (-> (e. A (CC)) (br F (~~>) (` (exp) A))) (({<,>|} k z (/\ (e. (cv k) (NN0)) (= (cv z) (opr (opr A (^) (cv k)) (/) (` (!) (cv k)))))) eqid x efcltlem4 addex nn0ex k z (opr (opr A (^) (cv k)) (/) (` (!) (cv k))) funopabex2 seq0seqz 0z (opr A (^) (cv k)) (/) (` (!) (cv k)) oprex (cv k) elnn0uz (= (cv z) (opr (opr A (^) (cv k)) (/) (` (!) (cv k)))) anbi1i k z opabbii efcvgfsum.1 (cv n) elnn0uz (= (cv y) (sum_ k (opr (0) (...) (cv n)) (opr (opr A (^) (cv k)) (/) (` (!) (cv k))))) anbi1i n y opabbii eqtr serzfsum ax-mp eqtr (~~>) (cv x) breq1i x exbii sylib 0z (opr A (^) (cv k)) (/) (` (!) (cv k)) oprex efcvgfsum.1 (cv n) elnn0uz (= (cv y) (sum_ k (opr (0) (...) (cv n)) (opr (opr A (^) (cv k)) (/) (` (!) (cv k))))) anbi1i n y opabbii eqtr x isumclim5t mpan syl A k efvalt nn0uz k (opr (opr A (^) (cv k)) (/) (` (!) (cv k))) sumeq1i syl6req breqtrd)) thm (reefcl ((j k) (j x) (j y) (A j) (k x) (k y) (A k) (x y) (A x) (A y)) ((reefcl.1 (e. A (RR)))) (e. (` (exp) A) (RR)) (reefcl.1 recn A k efvalt ax-mp (cv j) (cv k) A (^) opreq2 (cv j) (cv k) (!) fveq2 (/) opreq12d ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid (opr A (^) (cv k)) (/) (` (!) (cv k)) oprex fvopab4 sumeq2i nn0uz k (` ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) (cv k)) sumeq1i 3eqtr2 0z (cv k) elnn0uz (cv j) (cv k) A (^) opreq2 (cv j) (cv k) (!) fveq2 (/) opreq12d ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid (opr A (^) (cv k)) (/) (` (!) (cv k)) oprex fvopab4 (opr A (^) (cv k)) (` (!) (cv k)) redivclt reefcl.1 A (cv k) reexpclt mpan (cv k) facclt (` (!) (cv k)) nnret syl (cv k) facne0t syl3anc eqeltrd sylbir rgen reefcl.1 recn ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) eqid x efcltlem4 ax-mp addex nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 seq0seqz (~~>) (cv x) breq1i x exbii mpbi nn0ex j y (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) funopabex2 (0) k x isumreclt mp3an eqeltr)) thm (erelem1 ((x y)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (/\ (:--> F (NN) (RR)) (:--> G (NN) (RR))) (erelem1.1 2re 2re 2pos gt0ne0i rereccl (opr (1) (/) (2)) (cv x) reexpclt (cv x) nnnn0t sylan2 mpan 2re (2) (opr (opr (1) (/) (2)) (^) (cv x)) axmulrcl mpan syl fopab erelem1.2 (` (!) (cv x)) rerecclt (cv x) nnnn0t (cv x) facclt syl (` (!) (cv x)) nnret syl (cv x) nnnn0t (cv x) facclt syl (` (!) (cv x)) nnne0t syl sylanc fopab pm3.2i)) thm (erelem2 ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (br (opr (+) (seq1) F) (~~>) (2)) (2cn 1cn elisseti ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (opr (1) (/) (2)) (^) (cv x))))) eqid 2cn 2re 2pos gt0ne0i reccl 0re 2re 2re 2pos gt0ne0i rereccl halfgt0 ltlei 2re 2re 2pos gt0ne0i rereccl absid ax-mp halflt1 eqbrtr geolim1i 1cn (1) 2halvest ax-mp 1cn 2cn 2re 2pos gt0ne0i reccl 2cn 2re 2pos gt0ne0i reccl subadd mpbir (opr (1) (/) (2)) (/) opreq2i 2cn 2re 2pos gt0ne0i reccl 1cn 2cn ax1ne0 2re 2pos gt0ne0i divne0 divid eqtr breqtr addex erelem1.1 nnex x y (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))) funopabex2 eqeltr seq1fn ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (opr (1) (/) (2)) (^) (cv x))))) eqid (cv x) nnnn0t 2cn 2re 2pos gt0ne0i reccl (opr (1) (/) (2)) (cv x) expclt mpan syl rgen (cv z) ser1cl2 2cn ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (opr (1) (/) (2)) (^) (cv x))))) eqid (cv x) nnnn0t 2cn 2re 2pos gt0ne0i reccl (opr (1) (/) (2)) (cv x) expclt mpan syl fopab erelem1.1 nnex x y (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))) funopabex2 eqeltr (cv x) (cv z) (opr (1) (/) (2)) (^) opreq2 (2) (x.) opreq2d erelem1.1 (2) (x.) (opr (opr (1) (/) (2)) (^) (cv z)) oprex fvopab4 (cv x) (cv z) (opr (1) (/) (2)) (^) opreq2 ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (opr (1) (/) (2)) (^) (cv x))))) eqid (opr (1) (/) (2)) (^) (cv z) oprex fvopab4 (2) (x.) opreq2d eqtr4d (cv z) ser1mulc jca climmulc 2cn mulid1 breqtr)) thm (erelem3 ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (-> (e. (cv z) (NN)) (/\ (br (0) (<_) (` G (cv z))) (br (` G (cv z)) (<_) (` F (cv z))))) (0re (0) (opr (1) (/) (` (!) (cv z))) ltlet mpan (` (!) (cv z)) rerecclt (cv z) nnnn0t (cv z) facclt syl (` (!) (cv z)) nnret syl (cv z) nnnn0t (cv z) facclt syl (` (!) (cv z)) nnne0t syl sylanc (` (!) (cv z)) recgt0t (cv z) nnnn0t (cv z) facclt syl (` (!) (cv z)) nnret syl (cv z) nnnn0t (cv z) facclt syl (` (!) (cv z)) nngt0t syl sylanc sylc (cv x) (cv z) (!) fveq2 (1) (/) opreq2d erelem1.2 (1) (/) (` (!) (cv z)) oprex fvopab4 breqtrrd (cv z) nnnn0t (cv z) faclbnd2 syl (opr (opr (2) (^) (cv z)) (/) (2)) (` (!) (cv z)) lerect an4s 2re (2) (cv z) reexpclt (cv z) nnnn0t sylan2 mpan (opr (2) (^) (cv z)) rehalfclt syl (cv z) nnnn0t (cv z) facclt syl (` (!) (cv z)) nnret syl jca 2pos 2re (opr (2) (^) (cv z)) (2) divgt0t mpanl2 mpanr2 2re (2) (cv z) reexpclt (cv z) nnnn0t sylan2 mpan 2re 2pos (2) (cv z) expgt0t (cv z) nnnn0t syl3an2 mp3an13 sylanc (cv z) nnnn0t (cv z) facclt syl (` (!) (cv z)) nngt0t syl jca sylanc mpbid 2cn (2) (opr (2) (^) (cv z)) divrect mp3an1 2re (2) (cv z) reexpclt (cv z) nnnn0t sylan2 mpan recnd (opr (2) (^) (cv z)) gt0ne0t 2re (2) (cv z) reexpclt (cv z) nnnn0t sylan2 mpan 2re 2pos (2) (cv z) expgt0t (cv z) nnnn0t syl3an2 mp3an13 sylanc sylanc 2re 2pos gt0ne0i 2cn (opr (2) (^) (cv z)) (2) recdivt mpanl2 mpanr2 2re (2) (cv z) reexpclt (cv z) nnnn0t sylan2 mpan recnd (opr (2) (^) (cv z)) gt0ne0t 2re (2) (cv z) reexpclt (cv z) nnnn0t sylan2 mpan 2re 2pos (2) (cv z) expgt0t (cv z) nnnn0t syl3an2 mp3an13 sylanc sylanc 2cn 2re 2pos gt0ne0i (2) (cv z) recexpt (cv z) nnnn0t syl3an2 mp3an13 (2) (x.) opreq2d 3eqtr4d breqtrd (cv x) (cv z) (!) fveq2 (1) (/) opreq2d erelem1.2 (1) (/) (` (!) (cv z)) oprex fvopab4 (cv x) (cv z) (opr (1) (/) (2)) (^) opreq2 (2) (x.) opreq2d erelem1.1 (2) (x.) (opr (opr (1) (/) (2)) (^) (cv z)) oprex fvopab4 3brtr4d jca)) thm (erelem4 ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (2re elisseti erelem1.1 erelem1.2 erelem1 pm3.26i erelem1.1 erelem1.2 erelem1 pm3.27i erelem1.1 erelem1.2 z erelem3 erelem1.1 erelem1.2 erelem2 cvgcmp)) thm (erelem5 ((x y)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (e. (sup (ran (opr (+) (seq1) G)) (RR) (<)) (RR)) (ltso (ran (opr (+) (seq1) G)) supex erelem1.1 erelem1.2 erelem1 pm3.27i ser1re erelem1.1 erelem1.2 erelem4 climfnrcl)) thm (erelem6 ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (= (e) (opr (1) (+) (sup (ran (opr (+) (seq1) G)) (RR) (<)))) (x eval 1cn x z y dfef2lem (cv z) (cv x) (1) (^) opreq2 (cv z) (cv x) (!) fveq2 (/) opreq12d ({<,>|} z y (/\ (e. (cv z) (NN)) (= (cv y) (opr (opr (1) (^) (cv z)) (/) (` (!) (cv z)))))) eqid (opr (1) (^) (cv x)) (/) (` (!) (cv x)) oprex fvopab4 (cv x) nnnn0t (cv x) 1expt syl (/) (` (!) (cv x)) opreq1d eqtrd sumeq2i nnuz x (opr (1) (/) (` (!) (cv x))) sumeq1i eqtr (1) (+) opreq2i eqtr 1z addex erelem1.2 nnex x y (opr (1) (/) (` (!) (cv x))) funopabex2 eqeltr seq1seqz erelem1.1 erelem1.2 erelem4 eqbrtrr (1) (/) (` (!) (cv x)) oprex erelem1.2 (cv x) elnnuz (= (cv y) (opr (1) (/) (` (!) (cv x)))) anbi1i x y opabbii eqtr ltso (ran (opr (+) (seq1) G)) supex isumclim4t mp2an (1) (+) opreq2i 3eqtr)) thm (erelem7 ((x y)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (e. (e) (RR)) (erelem1.1 erelem1.2 erelem6 ax1re erelem1.1 erelem1.2 erelem5 readdcl eqeltr)) thm (ele3lem ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (br (e) (<_) (3)) (2re elisseti erelem1.1 erelem1.2 erelem1 pm3.26i erelem1.1 erelem1.2 erelem1 pm3.27i erelem1.1 erelem1.2 z erelem3 erelem1.1 erelem1.2 erelem2 ltso (ran (opr (+) (seq1) G)) supex erelem1.1 erelem1.2 erelem4 cvgcmpub erelem1.1 erelem1.2 erelem5 2re ax1re leadd2 mpbi erelem1.1 erelem1.2 erelem6 df-3 2cn 1cn addcom eqtr 3brtr4)) thm (ege2le3lem1 ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (br (` (opr (+) (seq1) G) (1)) (<_) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (ltso (ran (opr (+) (seq1) G)) supex erelem1.1 erelem1.2 erelem1 pm3.27i ser1re erelem1.1 erelem1.2 erelem1 pm3.27i erelem1.1 erelem1.2 z erelem3 pm3.26d (cv z) ser1mono erelem1.1 erelem1.2 erelem4 1nn climubi)) thm (ege2lem2 ((x y)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (br (2) (<_) (e)) (addex erelem1.2 nnex x y (opr (1) (/) (` (!) (cv x))) funopabex2 eqeltr seq11 1nn (cv x) (1) (!) fveq2 (1) (/) opreq2d erelem1.2 (1) (/) (` (!) (1)) oprex fvopab4 ax-mp fac1 (1) (/) opreq2i 1cn div1 3eqtr eqtr2 erelem1.1 erelem1.2 ege2le3lem1 eqbrtr ax1re erelem1.1 erelem1.2 erelem5 ax1re leadd2 mpbi df-2 erelem1.1 erelem1.2 erelem6 3brtr4)) thm (ege2le3lem2 ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (/\ (br (2) (<_) (e)) (br (e) (<_) (3))) (addex erelem1.2 nnex x y (opr (1) (/) (` (!) (cv x))) funopabex2 eqeltr seq11 1nn (cv x) (1) (!) fveq2 (1) (/) opreq2d erelem1.2 (1) (/) (` (!) (1)) oprex fvopab4 ax-mp fac1 (1) (/) opreq2i 1cn div1 3eqtr eqtr2 erelem1.1 erelem1.2 ege2le3lem1 eqbrtr ax1re erelem1.1 erelem1.2 erelem5 ax1re leadd2 mpbi df-2 erelem1.1 erelem1.2 erelem6 3brtr4 2re elisseti erelem1.1 erelem1.2 erelem1 pm3.26i erelem1.1 erelem1.2 erelem1 pm3.27i erelem1.1 erelem1.2 z erelem3 erelem1.1 erelem1.2 erelem2 ltso (ran (opr (+) (seq1) G)) supex erelem1.1 erelem1.2 erelem4 cvgcmpub erelem1.1 erelem1.2 erelem5 2re ax1re leadd2 mpbi erelem1.1 erelem1.2 erelem6 df-3 2cn 1cn addcom eqtr 3brtr4 pm3.2i)) thm (ere () () (e. (e) (RR)) (df-e ax1re reefcl eqeltr)) thm (ereALT ((x y)) () (e. (e) (RR)) (({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))) eqid ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))) eqid erelem7)) thm (ege2 ((x y)) () (br (2) (<_) (e)) (({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))) eqid ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))) eqid ege2lem2)) thm (ele3 ((x y)) () (br (e) (<_) (3)) (({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))) eqid ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))) eqid ele3lem)) thm (ege2le3 ((x y)) () (/\ (br (2) (<_) (e)) (br (e) (<_) (3))) (({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))) eqid ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))) eqid ege2le3lem2)) thm (ef0 ((j y)) () (= (` (exp) (0)) (1)) ((exp) (0) fvex 1cn elisseti 0cn ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr (0) (^) (cv j)) (/) (` (!) (cv j)))))) eqid efcvg ax-mp (0) eqid ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr (0) (^) (cv j)) (/) (` (!) (cv j)))))) eqid efcltlem2a ax-mp pm3.2i climunii)) thm (efcj ((j k) (j m) (j y) (A j) (k m) (k y) (A k) (m y) (A m) (A y)) ((efcj.1 (e. A (CC)))) (= (` (exp) (` (*) A)) (` (*) (` (exp) A))) (efcj.1 cjcl (` (*) A) k efvalt ax-mp nn0uz k (opr (opr (` (*) A) (^) (cv k)) (/) (` (!) (cv k))) sumeq1i 0z addex nn0ex k y (opr (opr (` (*) A) (^) (cv k)) (/) (` (!) (cv k))) funopabex2 seq0seqz (exp) A fvex (+) (seq0) ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr (opr (` (*) A) (^) (cv k)) (/) (` (!) (cv k)))))) oprex 0z efcj.1 ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr (opr A (^) (cv k)) (/) (` (!) (cv k)))))) eqid efcvg ax-mp (cv j) elnn0uz ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr (opr A (^) (cv k)) (/) (` (!) (cv k)))))) eqid (opr A (^) (cv k)) (` (!) (cv k)) divclt efcj.1 A (cv k) expclt mpan (cv k) facclt (` (!) (cv k)) nncnt syl (cv k) facne0t syl3anc fopab (cv j) ser0cl ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr (opr A (^) (cv k)) (/) (` (!) (cv k)))))) eqid (opr A (^) (cv k)) (` (!) (cv k)) divclt efcj.1 A (cv k) expclt mpan (cv k) facclt (` (!) (cv k)) nncnt syl (cv k) facne0t syl3anc fopab nn0ex k y (opr (opr (` (*) A) (^) (cv k)) (/) (` (!) (cv k))) funopabex2 (opr A (^) (cv m)) (` (!) (cv m)) cjdivt efcj.1 A (cv m) expclt mpan (cv m) facclt (` (!) (cv m)) nnret syl recnd (cv m) facne0t syl3anc efcj.1 A (cv m) cjexpt mpan (cv m) facclt (` (!) (cv m)) nnret syl (` (!) (cv m)) cjret syl (/) opreq12d eqtr2d (cv k) (cv m) (` (*) A) (^) opreq2 (cv k) (cv m) (!) fveq2 (/) opreq12d ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr (opr (` (*) A) (^) (cv k)) (/) (` (!) (cv k)))))) eqid (opr (` (*) A) (^) (cv m)) (/) (` (!) (cv m)) oprex fvopab4 (cv k) (cv m) A (^) opreq2 (cv k) (cv m) (!) fveq2 (/) opreq12d ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr (opr A (^) (cv k)) (/) (` (!) (cv k)))))) eqid (opr A (^) (cv m)) (/) (` (!) (cv m)) oprex fvopab4 (*) fveq2d 3eqtr4d (cv j) ser0cj jca sylbir climcj eqbrtrr (opr (` (*) A) (^) (cv k)) (/) (` (!) (cv k)) oprex (cv k) elnn0uz (= (cv y) (opr (opr (` (*) A) (^) (cv k)) (/) (` (!) (cv k)))) anbi1i k y opabbii (*) (` (exp) A) fvex isumclim4t mp2an 3eqtr)) thm (efcjt () () (-> (e. A (CC)) (= (` (exp) (` (*) A)) (` (*) (` (exp) A)))) (A (if (e. A (CC)) A (0)) (*) fveq2 (exp) fveq2d A (if (e. A (CC)) A (0)) (exp) fveq2 (*) fveq2d eqeq12d 0cn A elimel efcj dedth)) thm (efaddlem1 () ((efaddlem1.1 (e. N (NN)))) (-> (e. (cv j) (opr (1) (...) N)) (e. N (` (ZZ>) (opr (opr N (-) (cv j)) (+) (1))))) ((cv j) (1) N elfzle1 (cv j) (1) N elfzelz (cv j) zret syl ax1re efaddlem1.1 nnre (1) (cv j) N lesub2t mp3an13 syl mpbid (cv j) (1) N elfzelz (cv j) zret syl efaddlem1.1 nnre N (cv j) resubclt mpan syl ax1re efaddlem1.1 nnre (opr N (-) (cv j)) (1) N leaddsubt mp3an23 syl mpbird (cv j) (1) N elfzelz efaddlem1.1 N nnzt ax-mp N (cv j) zsubclt mpan syl peano2zd efaddlem1.1 N nnzt ax-mp (opr (opr N (-) (cv j)) (+) (1)) N eluzt mpan2 syl mpbird)) thm (efaddlem2 () ((efaddlem1.1 (e. N (NN)))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (e. (cv k) (NN))) ((cv k) (opr (opr N (-) (cv j)) (+) (1)) N elfzelz (e. (cv j) (opr (1) (...) N)) adantl ax1re (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) a1i (cv k) (opr (opr N (-) (cv j)) (+) (1)) N elfzel1 (opr (opr N (-) (cv j)) (+) (1)) zret syl (e. (cv j) (opr (1) (...) N)) adantl (cv k) (opr (opr N (-) (cv j)) (+) (1)) N elfzelz (cv k) zret syl (e. (cv j) (opr (1) (...) N)) adantl efaddlem1.1 N (NN) (cv j) (1) fznn0subt mpan (opr N (-) (cv j)) nn0p1nnt (opr (opr N (-) (cv j)) (+) (1)) nnge1t 3syl (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv k) (opr (opr N (-) (cv j)) (+) (1)) N elfzle1 (e. (cv j) (opr (1) (...) N)) adantl letrd jca (cv k) elnnz1 sylibr)) thm (efaddlem3 ((j k) (N j) (N k)) ((efaddlem3.1 (e. N (NN))) (efaddlem3.2 (e. A (CC))) (efaddlem3.3 (e. B (CC)))) (e. (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))) (CC)) (efaddlem3.1 nnuz eleqtr N (opr (opr N (-) (cv j)) (+) (1)) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt efaddlem3.1 j efaddlem1 (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem3.2 A (cv j) expclt mpan efaddlem3.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nncnt syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnne0t syl syl3anc (cv j) nnnn0t (cv k) nnnn0t syl2an (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr efaddlem3.1 j k efaddlem2 sylanc r19.21aiva sylanc rgen N (1) j (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) fsumclt mp2an)) thm (efaddlem4 ((j k) (N j) (N k)) ((efaddlem3.1 (e. N (NN))) (efaddlem3.2 (e. A (CC))) (efaddlem3.3 (e. B (CC)))) (e. (` (abs) (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))))) (RR)) (efaddlem3.1 efaddlem3.2 efaddlem3.3 j k efaddlem3 abscl)) thm (efaddlem5 ((j k) (A j) (A k) (B j) (B k) (N j) (N k)) ((efaddlem5.1 (e. N (NN))) (efaddlem5.2 (e. A (CC))) (efaddlem5.3 (e. B (CC)))) (= (sum_ j (opr (0) (...) N) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j)))) (sum_ j (opr (0) (...) N) (sum_ k (opr (0) (...) (opr N (-) (cv j))) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))))) ((cv j) N elfznn0t efaddlem5.2 efaddlem5.3 (cv j) k binom (/) (` (!) (cv j)) opreq1d (cv j) (0) (` (!) (cv j)) k (opr (opr (cv j) (C.) (cv k)) (x.) (opr (opr A (^) (opr (cv j) (-) (cv k))) (x.) (opr B (^) (cv k)))) fsumdivc (cv j) elnn0uz biimp (cv j) facclt (` (!) (cv j)) nncnt syl jca (cv j) facne0t (opr (cv j) (C.) (cv k)) (opr (opr A (^) (opr (cv j) (-) (cv k))) (x.) (opr B (^) (cv k))) axmulcl (cv j) (cv k) bcclt (cv k) (0) (cv j) elfzelz sylan2 (opr (cv j) (C.) (cv k)) nn0cnt syl (opr A (^) (opr (cv j) (-) (cv k))) (opr B (^) (cv k)) axmulcl (cv j) (NN0) (cv k) (0) fznn0subt efaddlem5.2 A (opr (cv j) (-) (cv k)) expclt mpan syl (cv k) (cv j) elfznn0t efaddlem5.3 B (cv k) expclt mpan syl (e. (cv j) (NN0)) adantl sylanc sylanc r19.21aiva jca sylanc (opr (cv j) (C.) (cv k)) (opr (opr A (^) (opr (cv j) (-) (cv k))) (x.) (opr B (^) (cv k))) (` (!) (cv j)) div23t (cv j) (cv k) bcclt (cv k) (0) (cv j) elfzelz sylan2 (opr (cv j) (C.) (cv k)) nn0cnt syl (opr A (^) (opr (cv j) (-) (cv k))) (opr B (^) (cv k)) axmulcl (cv j) (NN0) (cv k) (0) fznn0subt efaddlem5.2 A (opr (cv j) (-) (cv k)) expclt mpan syl (cv k) (cv j) elfznn0t efaddlem5.3 B (cv k) expclt mpan syl (e. (cv j) (NN0)) adantl sylanc (cv j) facclt (` (!) (cv j)) nncnt syl (e. (cv k) (opr (0) (...) (cv j))) adantr 3jca (cv j) facne0t (e. (cv k) (opr (0) (...) (cv j))) adantr sylanc (cv j) (cv k) bcval3t (/) (` (!) (cv j)) opreq1d (` (!) (cv j)) (opr (` (!) (opr (cv j) (-) (cv k))) (x.) (` (!) (cv k))) (` (!) (cv j)) divdiv23t (cv j) facclt (` (!) (cv j)) nncnt syl (e. (cv k) (opr (0) (...) (cv j))) adantr (` (!) (opr (cv j) (-) (cv k))) (` (!) (cv k)) nnmulclt (cv j) (NN0) (cv k) (0) fznn0subt (opr (cv j) (-) (cv k)) facclt syl (cv k) (cv j) elfznn0t (cv k) facclt syl (e. (cv j) (NN0)) adantl sylanc (opr (` (!) (opr (cv j) (-) (cv k))) (x.) (` (!) (cv k))) nncnt syl (cv j) facclt (` (!) (cv j)) nncnt syl (e. (cv k) (opr (0) (...) (cv j))) adantr 3jca (` (!) (opr (cv j) (-) (cv k))) (` (!) (cv k)) nnmulclt (cv j) (NN0) (cv k) (0) fznn0subt (opr (cv j) (-) (cv k)) facclt syl (cv k) (cv j) elfznn0t (cv k) facclt syl (e. (cv j) (NN0)) adantl sylanc (opr (` (!) (opr (cv j) (-) (cv k))) (x.) (` (!) (cv k))) nnne0t syl (cv j) facne0t (e. (cv k) (opr (0) (...) (cv j))) adantr jca sylanc (` (!) (cv j)) dividt (cv j) facclt (` (!) (cv j)) nncnt syl (cv j) facne0t sylanc (e. (cv k) (opr (0) (...) (cv j))) adantr (/) (opr (` (!) (opr (cv j) (-) (cv k))) (x.) (` (!) (cv k))) opreq1d 3eqtrd (x.) (opr (opr A (^) (opr (cv j) (-) (cv k))) (x.) (opr B (^) (cv k))) opreq1d (opr A (^) (opr (cv j) (-) (cv k))) (opr B (^) (cv k)) axmulcl (cv j) (NN0) (cv k) (0) fznn0subt efaddlem5.2 A (opr (cv j) (-) (cv k)) expclt mpan syl (cv k) (cv j) elfznn0t efaddlem5.3 B (cv k) expclt mpan syl (e. (cv j) (NN0)) adantl sylanc (opr (opr A (^) (opr (cv j) (-) (cv k))) (x.) (opr B (^) (cv k))) mulid2t syl (/) (opr (` (!) (opr (cv j) (-) (cv k))) (x.) (` (!) (cv k))) opreq1d 1cn (1) (opr (opr A (^) (opr (cv j) (-) (cv k))) (x.) (opr B (^) (cv k))) (opr (` (!) (opr (cv j) (-) (cv k))) (x.) (` (!) (cv k))) div23t mp3anl1 (opr A (^) (opr (cv j) (-) (cv k))) (opr B (^) (cv k)) axmulcl (cv j) (NN0) (cv k) (0) fznn0subt efaddlem5.2 A (opr (cv j) (-) (cv k)) expclt mpan syl (cv k) (cv j) elfznn0t efaddlem5.3 B (cv k) expclt mpan syl (e. (cv j) (NN0)) adantl sylanc (` (!) (opr (cv j) (-) (cv k))) (` (!) (cv k)) nnmulclt (cv j) (NN0) (cv k) (0) fznn0subt (opr (cv j) (-) (cv k)) facclt syl (cv k) (cv j) elfznn0t (cv k) facclt syl (e. (cv j) (NN0)) adantl sylanc (opr (` (!) (opr (cv j) (-) (cv k))) (x.) (` (!) (cv k))) nncnt syl jca (` (!) (opr (cv j) (-) (cv k))) (` (!) (cv k)) nnmulclt (cv j) (NN0) (cv k) (0) fznn0subt (opr (cv j) (-) (cv k)) facclt syl (cv k) (cv j) elfznn0t (cv k) facclt syl (e. (cv j) (NN0)) adantl sylanc (opr (` (!) (opr (cv j) (-) (cv k))) (x.) (` (!) (cv k))) nnne0t syl sylanc (opr A (^) (opr (cv j) (-) (cv k))) (` (!) (opr (cv j) (-) (cv k))) (opr B (^) (cv k)) (` (!) (cv k)) divmuldivt (cv j) (NN0) (cv k) (0) fznn0subt efaddlem5.2 A (opr (cv j) (-) (cv k)) expclt mpan syl (cv j) (NN0) (cv k) (0) fznn0subt (opr (cv j) (-) (cv k)) facclt (` (!) (opr (cv j) (-) (cv k))) nncnt 3syl jca (cv k) (cv j) elfznn0t efaddlem5.3 B (cv k) expclt mpan syl (e. (cv j) (NN0)) adantl (cv k) (cv j) elfznn0t (cv k) facclt (` (!) (cv k)) nncnt 3syl (e. (cv j) (NN0)) adantl jca jca (cv j) (NN0) (cv k) (0) fznn0subt (opr (cv j) (-) (cv k)) facne0t syl (cv k) (cv j) elfznn0t (cv k) facne0t syl (e. (cv j) (NN0)) adantl jca sylanc (cv j) (-) (cv k) oprex (opr (cv j) (-) (cv k)) (V) j (opr A (^) (cv j)) (/) (` (!) (cv j)) csbopr12g ax-mp (cv j) (-) (cv k) oprex (opr (cv j) (-) (cv k)) (V) j A (^) (cv j) csbopr2g ax-mp (cv j) (-) (cv k) oprex (opr (cv j) (-) (cv k)) (V) j csbvarg ax-mp A (^) opreq2i eqtr (cv j) (-) (cv k) oprex (opr (cv j) (-) (cv k)) (V) j (!) (cv j) csbfv2g ax-mp (cv j) (-) (cv k) oprex (opr (cv j) (-) (cv k)) (V) j csbvarg ax-mp (!) fveq2i eqtr (/) opreq12i eqtr (x.) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) opreq1i syl5req 3eqtr3d 3eqtrd sumeq2dv 3eqtrd syl sumeq2i efaddlem5.1 nnnn0 (cv j) N elfznn0t (opr A (^) (cv j)) (` (!) (cv j)) divclt efaddlem5.2 A (cv j) expclt mpan (cv j) facclt (` (!) (cv j)) nncnt syl (cv j) facne0t syl3anc syl rgen (cv k) N elfznn0t (opr B (^) (cv k)) (` (!) (cv k)) divclt efaddlem5.3 B (cv k) expclt mpan (cv k) facclt (` (!) (cv k)) nncnt syl (cv k) facne0t syl3anc syl rgen N j (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) k (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) fsum0diag2 mp3an (opr A (^) (cv j)) (` (!) (cv j)) (opr B (^) (cv k)) (` (!) (cv k)) divmuldivt efaddlem5.2 A (cv j) expclt mpan (cv j) facclt (` (!) (cv j)) nncnt syl jca efaddlem5.3 B (cv k) expclt mpan (cv k) facclt (` (!) (cv k)) nncnt syl jca anim12i (cv j) facne0t (cv k) facne0t anim12i sylanc (cv j) N elfznn0t (cv k) (opr N (-) (cv j)) elfznn0t syl2an sumeq2dv sumeq2i 3eqtr2)) thm (efaddlem6 ((j k) (A j) (A k) (B j) (B k) (N j) (N k)) ((efaddlem5.1 (e. N (NN))) (efaddlem5.2 (e. A (CC))) (efaddlem5.3 (e. B (CC)))) (= (opr (opr (sum_ j (opr (0) (...) N) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) N) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))) (-) (sum_ j (opr (0) (...) N) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j))))) (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))))) (efaddlem5.1 nnnn0 nn0uz eleqtr (cv j) N elfznn0t (opr A (^) (cv j)) (` (!) (cv j)) divclt efaddlem5.2 A (cv j) expclt mpan (cv j) facclt (` (!) (cv j)) nncnt syl (cv j) facne0t syl3anc syl rgen pm3.2i efaddlem5.1 nnnn0 nn0uz eleqtr (cv k) N elfznn0t (opr B (^) (cv k)) (` (!) (cv k)) divclt efaddlem5.3 B (cv k) expclt mpan (cv k) facclt (` (!) (cv k)) nncnt syl (cv k) facne0t syl3anc syl rgen pm3.2i N (0) j (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) N (0) k (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) fsum2mul mp2an (opr A (^) (cv j)) (` (!) (cv j)) (opr B (^) (cv k)) (` (!) (cv k)) divmuldivt efaddlem5.2 A (cv j) expclt mpan (cv j) facclt (` (!) (cv j)) nncnt syl jca efaddlem5.3 B (cv k) expclt mpan (cv k) facclt (` (!) (cv k)) nncnt syl jca anim12i (cv j) facne0t (cv k) facne0t anim12i sylanc (cv j) N elfznn0t (cv k) N elfznn0t syl2an sumeq2dv sumeq2i efaddlem5.1 nnnn0 nn0uz eleqtr efaddlem5.1 nngt0 (cv j) N elfznn0t (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem5.2 A (cv j) expclt mpan efaddlem5.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) axmulcl (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an (cv j) facne0t (cv k) facne0t anim12i (` (!) (cv j)) (` (!) (cv k)) muln0bt (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an mpbid syl3anc (cv k) N elfznn0t sylan2 r19.21aiva efaddlem5.1 nnnn0 nn0uz eleqtr N (0) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt mpan 3syl rgen (cv j) (0) A (^) opreq2 efaddlem5.2 A exp0t ax-mp syl6eq (x.) (opr B (^) (cv k)) opreq1d (cv j) (0) (!) fveq2 fac0 syl6eq (x.) (` (!) (cv k)) opreq1d (/) opreq12d k (opr (0) (...) N) sumeq2sdv N fsum1p mp3an 1cn addid2 (...) N opreq1i j (sum_ k (opr (0) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) sumeq1i (sum_ k (opr (0) (...) N) (opr (opr (1) (x.) (opr B (^) (cv k))) (/) (opr (1) (x.) (` (!) (cv k))))) (+) opreq2i 3eqtr eqtr3 efaddlem5.1 efaddlem5.2 efaddlem5.3 j k efaddlem5 efaddlem5.1 nnnn0 nn0uz eleqtr efaddlem5.1 nngt0 (opr N (-) (cv j)) (0) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt efaddlem5.1 N (NN) (cv j) (0) fznn0subt mpan nn0uz syl6eleq (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem5.2 A (cv j) expclt mpan efaddlem5.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) axmulcl (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an (cv j) facne0t (cv k) facne0t anim12i (` (!) (cv j)) (` (!) (cv k)) muln0bt (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an mpbid syl3anc (cv j) N elfznn0t (cv k) (opr N (-) (cv j)) elfznn0t syl2an r19.21aiva sylanc rgen (cv j) (0) N (-) opreq2 (0) (...) opreq2d (cv j) (0) A (^) opreq2 efaddlem5.2 A exp0t ax-mp syl6eq (x.) (opr B (^) (cv k)) opreq1d (cv j) (0) (!) fveq2 fac0 syl6eq (x.) (` (!) (cv k)) opreq1d (/) opreq12d (e. (cv k) (opr (0) (...) (opr N (-) (cv j)))) adantr r19.21aiva sumeq12d N fsum1p mp3an efaddlem5.1 nncn subid1 (0) (...) opreq2i k (opr (opr (1) (x.) (opr B (^) (cv k))) (/) (opr (1) (x.) (` (!) (cv k)))) sumeq1i 1cn addid2 (...) N opreq1i j (sum_ k (opr (0) (...) (opr N (-) (cv j))) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) sumeq1i (+) opreq12i 3eqtr (-) opreq12i efaddlem5.1 nnnn0 nn0uz eleqtr (cv k) N elfznn0t (cv k) facclt (` (!) (cv k)) nncnt (` (!) (cv k)) mulid2t 3syl (opr (1) (x.) (opr B (^) (cv k))) (/) opreq2d (opr (1) (x.) (opr B (^) (cv k))) (` (!) (cv k)) divclt efaddlem5.3 B (cv k) expclt mpan 1cn (1) (opr B (^) (cv k)) axmulcl mpan syl (cv k) facclt (` (!) (cv k)) nncnt syl (cv k) facne0t syl3anc eqeltrd syl rgen N (0) k (opr (opr (1) (x.) (opr B (^) (cv k))) (/) (opr (1) (x.) (` (!) (cv k)))) fsumclt mp2an efaddlem5.1 nnuz eleqtr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem5.2 A (cv j) expclt mpan efaddlem5.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) axmulcl (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an (cv j) facne0t (cv k) facne0t anim12i (` (!) (cv j)) (` (!) (cv k)) muln0bt (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an mpbid syl3anc (cv j) N elfznnt (cv j) nnnn0t syl (cv k) N elfznn0t syl2an r19.21aiva efaddlem5.1 nnnn0 nn0uz eleqtr N (0) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt mpan syl rgen N (1) j (sum_ k (opr (0) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) fsumclt mp2an efaddlem5.1 nnuz eleqtr (opr N (-) (cv j)) (0) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt efaddlem5.1 N (NN) (cv j) (1) fznn0subt mpan nn0uz syl6eleq (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem5.2 A (cv j) expclt mpan efaddlem5.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) axmulcl (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an (cv j) facne0t (cv k) facne0t anim12i (` (!) (cv j)) (` (!) (cv k)) muln0bt (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an mpbid syl3anc (cv j) N elfznnt (cv j) nnnn0t syl (cv k) (opr N (-) (cv j)) elfznn0t syl2an r19.21aiva sylanc rgen N (1) j (sum_ k (opr (0) (...) (opr N (-) (cv j))) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) fsumclt mp2an (sum_ k (opr (0) (...) N) (opr (opr (1) (x.) (opr B (^) (cv k))) (/) (opr (1) (x.) (` (!) (cv k))))) (sum_ j (opr (1) (...) N) (sum_ k (opr (0) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))) (sum_ j (opr (1) (...) N) (sum_ k (opr (0) (...) (opr N (-) (cv j))) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))) pnpcant mp3an (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem5.2 A (cv j) expclt mpan efaddlem5.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) axmulcl (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an (cv j) facne0t (cv k) facne0t anim12i (` (!) (cv j)) (` (!) (cv k)) muln0bt (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an mpbid syl3anc (cv j) N elfznnt (cv j) nnnn0t syl (cv k) N elfznn0t syl2an r19.21aiva efaddlem5.1 N nnzt ax-mp N (cv j) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsum0split mp3an1 mpdan sumeq2i efaddlem5.1 nnuz eleqtr (opr N (-) (cv j)) (0) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt efaddlem5.1 N (NN) (cv j) (1) fznn0subt mpan nn0uz syl6eleq (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem5.2 A (cv j) expclt mpan efaddlem5.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) axmulcl (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an (cv j) facne0t (cv k) facne0t anim12i (` (!) (cv j)) (` (!) (cv k)) muln0bt (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an mpbid syl3anc (cv j) N elfznnt (cv j) nnnn0t syl (cv k) (opr N (-) (cv j)) elfznn0t syl2an r19.21aiva sylanc N (opr (opr N (-) (cv j)) (+) (1)) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt efaddlem5.1 j efaddlem1 efaddlem5.1 j k efaddlem2 (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem5.2 A (cv j) expclt mpan efaddlem5.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) axmulcl (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an (cv j) facne0t (cv k) facne0t anim12i (` (!) (cv j)) (` (!) (cv k)) muln0bt (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an mpbid syl3anc (cv j) N elfznnt (cv j) nnnn0t syl (cv k) nnnn0t syl2an syldan r19.21aiva sylanc jca rgen N (1) j (sum_ k (opr (0) (...) (opr N (-) (cv j))) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) fsumadd mp2an eqtr2 efaddlem5.1 nnuz eleqtr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem5.2 A (cv j) expclt mpan efaddlem5.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) axmulcl (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an (cv j) facne0t (cv k) facne0t anim12i (` (!) (cv j)) (` (!) (cv k)) muln0bt (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an mpbid syl3anc (cv j) N elfznnt (cv j) nnnn0t syl (cv k) N elfznn0t syl2an r19.21aiva efaddlem5.1 nnnn0 nn0uz eleqtr N (0) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt mpan syl rgen N (1) j (sum_ k (opr (0) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) fsumclt mp2an efaddlem5.1 nnuz eleqtr (opr N (-) (cv j)) (0) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt efaddlem5.1 N (NN) (cv j) (1) fznn0subt mpan nn0uz syl6eleq (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem5.2 A (cv j) expclt mpan efaddlem5.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) axmulcl (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an (cv j) facne0t (cv k) facne0t anim12i (` (!) (cv j)) (` (!) (cv k)) muln0bt (cv j) facclt (` (!) (cv j)) nncnt syl (cv k) facclt (` (!) (cv k)) nncnt syl syl2an mpbid syl3anc (cv j) N elfznnt (cv j) nnnn0t syl (cv k) (opr N (-) (cv j)) elfznn0t syl2an r19.21aiva sylanc rgen N (1) j (sum_ k (opr (0) (...) (opr N (-) (cv j))) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) fsumclt mp2an efaddlem5.1 efaddlem5.2 efaddlem5.3 j k efaddlem3 subadd mpbir 3eqtr)) thm (efaddlem7 () ((efaddlem7.1 (e. A (CC))) (efaddlem7.2 (e. B (CC))) (efaddlem7.3 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))))) (e. T (NN)) (efaddlem7.3 efaddlem7.1 abscl ax1re readdcl efaddlem7.1 absge0 ax1re efaddlem7.1 abscl (1) (` (abs) A) addge02t mp2an mpbi (opr (` (abs) A) (+) (1)) flge1nnt mp2an efaddlem7.2 abscl ax1re readdcl efaddlem7.2 absge0 ax1re efaddlem7.2 abscl (1) (` (abs) B) addge02t mp2an mpbi (opr (` (abs) B) (+) (1)) flge1nnt mp2an (` (floor) (opr (` (abs) A) (+) (1))) (` (floor) (opr (` (abs) B) (+) (1))) nnmulclt mp2an nnsqcl eqeltr)) thm (efaddlem8 () ((efaddlem8.1 (e. N (NN))) (efaddlem8.2 (= S (` (floor) (opr (opr N (+) (1)) (/) (2)))))) (e. S (NN)) (efaddlem8.2 efaddlem8.1 nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl efaddlem8.1 N nnge1t ax-mp ax1re efaddlem8.1 nnre ax1re leadd1 mpbi 2cn mulid2 df-2 eqtr 2cn efaddlem8.1 nnre ax1re readdcl recn 2re 2pos gt0ne0i divcan1 3brtr4 2pos ax1re efaddlem8.1 nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl 2re lemul1 ax-mp mpbir (opr (opr N (+) (1)) (/) (2)) flge1nnt mp2an eqeltr)) thm (efaddlem9 () ((efaddlem9.1 (e. N (NN)))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (/\ (/\ (e. (cv j) (NN0)) (br (cv j) (<_) N)) (/\ (e. (cv k) (NN0)) (br (cv k) (<_) N)))) ((cv j) N elfznnt (cv j) nnnn0t syl (cv j) (1) N elfzle2 jca (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr efaddlem9.1 j k efaddlem2 (cv k) nnnn0t syl (cv k) (opr (opr N (-) (cv j)) (+) (1)) N elfzle2 (e. (cv j) (opr (1) (...) N)) adantl jca jca)) thm (efaddlem10 () ((efaddlem10.1 (e. N (NN))) (efaddlem10.2 (e. A (CC)))) (-> (/\ (e. (cv j) (NN0)) (br (cv j) (<_) N)) (/\ (/\ (e. (` (abs) (opr A (^) (cv j))) (RR)) (e. (opr (` (floor) (opr (` (abs) A) (+) (1))) (^) N) (RR))) (/\ (br (0) (<_) (` (abs) (opr A (^) (cv j)))) (br (` (abs) (opr A (^) (cv j))) (<_) (opr (` (floor) (opr (` (abs) A) (+) (1))) (^) N))))) (efaddlem10.2 A (cv j) expclt mpan (opr A (^) (cv j)) absclt syl efaddlem10.2 abscl ax1re readdcl (opr (` (abs) A) (+) (1)) flclt ax-mp zre efaddlem10.1 nnnn0 (` (floor) (opr (` (abs) A) (+) (1))) N reexpclt mp2an jctir (br (cv j) (<_) N) adantr efaddlem10.2 A (cv j) expclt mpan (opr A (^) (cv j)) absge0t syl (br (cv j) (<_) N) adantr efaddlem10.2 A (cv j) absexpt mpan (br (cv j) (<_) N) adantr efaddlem10.2 abscl (` (abs) A) (cv j) reexpclt mpan (br (cv j) (<_) N) adantr efaddlem10.2 abscl ax1re readdcl (opr (` (abs) A) (+) (1)) flclt ax-mp zre (` (floor) (opr (` (abs) A) (+) (1))) (cv j) reexpclt mpan (br (cv j) (<_) N) adantr efaddlem10.2 abscl ax1re readdcl (opr (` (abs) A) (+) (1)) flclt ax-mp zre efaddlem10.1 nnnn0 (` (floor) (opr (` (abs) A) (+) (1))) N reexpclt mp2an (/\ (e. (cv j) (NN0)) (br (cv j) (<_) N)) a1i (` (abs) A) (` (floor) (opr (` (abs) A) (+) (1))) (cv j) expmwordit efaddlem10.2 abscl (e. (cv j) (NN0)) a1i efaddlem10.2 abscl ax1re readdcl (opr (` (abs) A) (+) (1)) flclt ax-mp zre (e. (cv j) (NN0)) a1i (e. (cv j) (NN0)) id 3jca efaddlem10.2 absge0 efaddlem10.2 abscl efaddlem10.2 abscl (` (abs) A) flclt ax-mp zre ax1re readdcl efaddlem10.2 abscl (` (abs) A) flltp1t ax-mp ltlei efaddlem10.2 abscl 1z (` (abs) A) (1) fladdzt mp2an breqtrr pm3.2i (e. (cv j) (NN0)) a1i sylanc (br (cv j) (<_) N) adantr efaddlem10.1 nnnn0 (` (floor) (opr (` (abs) A) (+) (1))) (cv j) N expwordit mp3anl3 (e. (cv j) (NN0)) (br (cv j) (<_) N) pm3.26 efaddlem10.2 abscl ax1re readdcl (opr (` (abs) A) (+) (1)) flclt ax-mp zre jctil (e. (cv j) (NN0)) (br (cv j) (<_) N) pm3.27 efaddlem10.2 absge0 ax1re efaddlem10.2 abscl (1) (` (abs) A) addge02t mp2an mpbi efaddlem10.2 abscl ax1re readdcl 1z (opr (` (abs) A) (+) (1)) (1) flget mp2an mpbi jctil sylanc letrd eqbrtrd jca jca)) thm (efaddlem11 () ((efaddlem11.1 (e. N (NN))) (efaddlem11.2 (e. A (CC))) (efaddlem11.3 (e. B (CC)))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (br (` (abs) (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k)))) (<_) (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (^) N) (x.) (opr (` (floor) (opr (` (abs) B) (+) (1))) (^) N)))) (efaddlem11.1 j k efaddlem9 (opr A (^) (cv j)) (opr B (^) (cv k)) absmult efaddlem11.2 A (cv j) expclt mpan efaddlem11.3 B (cv k) expclt mpan syl2an (br (cv j) (<_) N) (br (cv k) (<_) N) ad2ant2r (` (abs) (opr A (^) (cv j))) (opr (` (floor) (opr (` (abs) A) (+) (1))) (^) N) (` (abs) (opr B (^) (cv k))) (opr (` (floor) (opr (` (abs) B) (+) (1))) (^) N) lemul12it efaddlem11.1 efaddlem11.2 j efaddlem10 efaddlem11.1 efaddlem11.3 k efaddlem10 syl2an eqbrtrd syl)) thm (efaddlem12 () ((efaddlem12.1 (e. N (NN))) (efaddlem12.2 (e. A (CC))) (efaddlem12.3 (e. B (CC))) (efaddlem12.4 (= S (` (floor) (opr (opr N (+) (1)) (/) (2))))) (efaddlem12.5 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))))) (br (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (^) N) (x.) (opr (` (floor) (opr (` (abs) B) (+) (1))) (^) N)) (<_) (opr T (^) S)) (efaddlem12.2 abscl ax1re readdcl (opr (` (abs) A) (+) (1)) flclt ax-mp zre efaddlem12.3 abscl ax1re readdcl (opr (` (abs) B) (+) (1)) flclt ax-mp zre remulcl efaddlem12.1 nnnn0 2nn0 efaddlem12.1 nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl 0re efaddlem12.1 nnre efaddlem12.1 nnnn0 nn0ge0 (0) N letrp1t mp3an 2cn mul02 2cn efaddlem12.1 nnre ax1re readdcl recn 2re 2pos gt0ne0i divcan1 3brtr4 2pos 0re efaddlem12.1 nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl 2re lemul1 ax-mp mpbir (opr (opr N (+) (1)) (/) (2)) flge0nn0t mp2an nn0mulcl 3pm3.2i (` (floor) (opr (` (abs) A) (+) (1))) elnnz1 efaddlem12.2 abscl ax1re readdcl (opr (` (abs) A) (+) (1)) flclt ax-mp efaddlem12.2 absge0 ax1re efaddlem12.2 abscl (1) (` (abs) A) addge02t mp2an mpbi efaddlem12.2 abscl ax1re readdcl 1z (opr (` (abs) A) (+) (1)) (1) flget mp2an mpbi mpbir2an (` (floor) (opr (` (abs) B) (+) (1))) elnnz1 efaddlem12.3 abscl ax1re readdcl (opr (` (abs) B) (+) (1)) flclt ax-mp efaddlem12.3 absge0 ax1re efaddlem12.3 abscl (1) (` (abs) B) addge02t mp2an mpbi efaddlem12.3 abscl ax1re readdcl 1z (opr (` (abs) B) (+) (1)) (1) flget mp2an mpbi mpbir2an (` (floor) (opr (` (abs) A) (+) (1))) (` (floor) (opr (` (abs) B) (+) (1))) nnmulclt mp2an (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) nnge1t ax-mp efaddlem12.1 N nnzt ax-mp N flhalft ax-mp pm3.2i (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) N (opr (2) (x.) (` (floor) (opr (opr N (+) (1)) (/) (2)))) expwordit mp2an efaddlem12.2 abscl ax1re readdcl (opr (` (abs) A) (+) (1)) flclt ax-mp (` (floor) (opr (` (abs) A) (+) (1))) zcnt ax-mp efaddlem12.3 abscl ax1re readdcl (opr (` (abs) B) (+) (1)) flclt ax-mp (` (floor) (opr (` (abs) B) (+) (1))) zcnt ax-mp efaddlem12.1 nnnn0 (` (floor) (opr (` (abs) A) (+) (1))) (` (floor) (opr (` (abs) B) (+) (1))) N mulexpt mp3an efaddlem12.2 abscl ax1re readdcl (opr (` (abs) A) (+) (1)) flclt ax-mp (` (floor) (opr (` (abs) A) (+) (1))) zcnt ax-mp efaddlem12.3 abscl ax1re readdcl (opr (` (abs) B) (+) (1)) flclt ax-mp (` (floor) (opr (` (abs) B) (+) (1))) zcnt ax-mp mulcl 2nn0 efaddlem12.1 nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl 0re efaddlem12.1 nnre efaddlem12.1 nnnn0 nn0ge0 (0) N letrp1t mp3an 2cn mul02 2cn efaddlem12.1 nnre ax1re readdcl recn 2re 2pos gt0ne0i divcan1 3brtr4 2pos 0re efaddlem12.1 nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl 2re lemul1 ax-mp mpbir (opr (opr N (+) (1)) (/) (2)) flge0nn0t mp2an (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (2) (` (floor) (opr (opr N (+) (1)) (/) (2))) expmult mp3an efaddlem12.5 efaddlem12.4 (^) opreq12i eqtr4 3brtr3)) thm (efaddlem13 () ((efaddlem12.1 (e. N (NN))) (efaddlem12.2 (e. A (CC))) (efaddlem12.3 (e. B (CC))) (efaddlem12.4 (= S (` (floor) (opr (opr N (+) (1)) (/) (2))))) (efaddlem12.5 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (br (` (abs) (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k)))) (<_) (opr T (^) S))) (efaddlem12.1 j k efaddlem9 (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem12.2 A (cv j) expclt mpan efaddlem12.3 B (cv k) expclt mpan syl2an (br (cv j) (<_) N) (br (cv k) (<_) N) ad2ant2r (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) absclt 3syl efaddlem12.2 abscl ax1re readdcl (opr (` (abs) A) (+) (1)) flclt ax-mp zre efaddlem12.1 nnnn0 (` (floor) (opr (` (abs) A) (+) (1))) N reexpclt mp2an efaddlem12.3 abscl ax1re readdcl (opr (` (abs) B) (+) (1)) flclt ax-mp zre efaddlem12.1 nnnn0 (` (floor) (opr (` (abs) B) (+) (1))) N reexpclt mp2an remulcl (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) a1i efaddlem12.2 efaddlem12.3 efaddlem12.5 efaddlem7 nnre efaddlem12.1 efaddlem12.4 efaddlem8 nnnn0 T S reexpclt mp2an (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) a1i efaddlem12.1 efaddlem12.2 efaddlem12.3 j k efaddlem11 efaddlem12.1 efaddlem12.2 efaddlem12.3 efaddlem12.4 efaddlem12.5 efaddlem12 (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) a1i letrd)) thm (efaddlem14 () ((efaddlem14.1 (e. N (NN)))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (br (opr N (+) (1)) (<_) (opr (cv j) (+) (cv k)))) ((cv k) (opr (opr N (-) (cv j)) (+) (1)) N elfzle1 (e. (cv j) (opr (1) (...) N)) adantl (opr (opr N (-) (cv j)) (+) (1)) (cv k) (cv j) leadd1t (opr (opr N (-) (cv j)) (+) (1)) zret (cv k) zret (cv j) zret syl3an (cv k) (opr (opr N (-) (cv j)) (+) (1)) N elfzel1 (e. (cv j) (opr (1) (...) N)) adantl (cv k) (opr (opr N (-) (cv j)) (+) (1)) N elfzelz (e. (cv j) (opr (1) (...) N)) adantl (cv j) (1) N elfzelz (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr syl3anc mpbid (cv j) (1) N elfzelz (cv j) zcnt efaddlem14.1 nncn 1cn N (cv j) (1) nppcant mp3an13 3syl eqcomd (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) (cv k) axaddcom (cv j) zcnt (cv k) zcnt syl2an (cv j) (1) N elfzelz (cv k) (opr (opr N (-) (cv j)) (+) (1)) N elfzelz syl2an 3brtr4d)) thm (efaddlem15 () ((efaddlem15.1 (e. N (NN))) (efaddlem15.2 (= S (` (floor) (opr (opr N (+) (1)) (/) (2)))))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (br (` (!) S) (<_) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) (efaddlem15.1 efaddlem15.2 efaddlem8 nnnn0 S facclt ax-mp nnre (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) a1i (cv j) (cv k) nn0addclt (cv j) N elfznnt (cv j) nnnn0t syl (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr efaddlem15.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (opr (cv j) (+) (cv k)) (/) (2)) flge0nn0t (opr (cv j) (+) (cv k)) nn0ret (opr (cv j) (+) (cv k)) rehalfclt syl (opr (cv j) (+) (cv k)) nn0ge0t (opr (cv j) (+) (cv k)) nn0ret (opr (cv j) (+) (cv k)) halfnneg2t syl mpbid sylanc (` (floor) (opr (opr (cv j) (+) (cv k)) (/) (2))) facclt syl (` (!) (` (floor) (opr (opr (cv j) (+) (cv k)) (/) (2)))) nnret syl syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) N elfznnt (cv j) nnnn0t syl (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) facclt syl efaddlem15.1 j k efaddlem2 (cv k) nnnn0t syl (cv k) facclt syl sylanc (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnret syl efaddlem15.1 efaddlem15.2 efaddlem8 nnnn0 S (` (floor) (opr (opr (cv j) (+) (cv k)) (/) (2))) facwordit mp3an1 (opr (opr (cv j) (+) (cv k)) (/) (2)) flge0nn0t (cv j) (cv k) nn0addclt (cv j) N elfznnt (cv j) nnnn0t syl (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr efaddlem15.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (cv j) (+) (cv k)) nn0ret syl (opr (cv j) (+) (cv k)) rehalfclt syl (cv j) (cv k) nn0addclt (cv j) N elfznnt (cv j) nnnn0t syl (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr efaddlem15.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (cv j) (+) (cv k)) nn0ge0t syl (cv j) (cv k) nn0addclt (cv j) N elfznnt (cv j) nnnn0t syl (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr efaddlem15.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (cv j) (+) (cv k)) nn0ret syl (opr (cv j) (+) (cv k)) halfnneg2t syl mpbid sylanc efaddlem15.1 nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl (opr (opr N (+) (1)) (/) (2)) (opr (opr (cv j) (+) (cv k)) (/) (2)) flwordit mp3an1 (cv j) (cv k) nn0addclt (cv j) N elfznnt (cv j) nnnn0t syl (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr efaddlem15.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (cv j) (+) (cv k)) nn0ret syl (opr (cv j) (+) (cv k)) rehalfclt syl efaddlem15.1 j k efaddlem14 (cv j) (cv k) nn0addclt (cv j) N elfznnt (cv j) nnnn0t syl (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr efaddlem15.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (cv j) (+) (cv k)) nn0ret syl efaddlem15.1 nnre ax1re readdcl 2re 2pos (opr N (+) (1)) (opr (cv j) (+) (cv k)) (2) lediv1t mpan2 mp3an13 syl mpbid sylanc efaddlem15.2 syl5eqbr sylanc (cv j) (cv k) facavgt (cv j) N elfznnt (cv j) nnnn0t syl (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr efaddlem15.1 j k efaddlem2 (cv k) nnnn0t syl sylanc letrd)) thm (efaddlem16 ((j k) (N j) (N k) (C j) (C k)) ((efaddlem16.1 (e. N (NN))) (efaddlem16.2 (e. C (RR))) (efaddlem16.3 (br (0) (<_) C))) (br (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) C)) (<_) (opr (opr N (^) (2)) (x.) C)) (efaddlem16.1 nnuz eleqtr N (opr (opr N (-) (cv j)) (+) (1)) k C fsumreclt efaddlem16.1 j efaddlem1 efaddlem16.2 (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) a1i r19.21aiva sylanc efaddlem16.1 nnre efaddlem16.2 remulcl (e. (cv j) (opr (1) (...) N)) a1i efaddlem16.1 j efaddlem1 efaddlem16.2 recn N (opr (opr N (-) (cv j)) (+) (1)) C k fsumconst mpan2 syl (cv j) (1) N elfzelz (cv j) zcnt efaddlem16.1 nncn N (cv j) subclt mpan efaddlem16.1 nncn 1cn N (opr N (-) (cv j)) (1) nppcan2t mp3an13 syl efaddlem16.1 nncn N (cv j) nncant mpan eqtrd 3syl (x.) C opreq1d eqtrd efaddlem16.3 efaddlem16.1 nnre efaddlem16.2 (cv j) N C lemul1it ex mp3an23 mpani (cv j) (1) N elfzelz (cv j) zret syl (cv j) (1) N elfzle2 sylc eqbrtrd 3jca rgen N (1) j (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) C) (opr N (x.) C) fsumcmp mp2an efaddlem16.1 nnuz eleqtr efaddlem16.1 nncn efaddlem16.2 recn mulcl N (1) (opr N (x.) C) j fsumconst mp2an efaddlem16.1 nncn 1cn N (1) npcant mp2an (x.) (opr N (x.) C) opreq1i efaddlem16.1 nncn sqval (x.) C opreq1i efaddlem16.1 nncn efaddlem16.1 nncn efaddlem16.2 recn mulass eqtr2 3eqtr breqtr)) thm (efaddlem17 ((j k) (N j) (N k)) ((efaddlem17.1 (e. N (NN))) (efaddlem17.2 (e. A (CC))) (efaddlem17.3 (e. B (CC))) (efaddlem17.4 (= S (` (floor) (opr (opr N (+) (1)) (/) (2))))) (efaddlem17.5 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (br (` (abs) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) (<_) (opr (opr T (^) S) (/) (` (!) S)))) ((opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) absdivt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nncnt syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnne0t syl syl3anc (opr (` (!) (cv j)) (x.) (` (!) (cv k))) absidt (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnret syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnnn0t (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nn0ge0t 3syl sylanc (` (abs) (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k)))) (/) opreq2d eqtrd (` (abs) (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k)))) (opr T (^) S) (` (!) S) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) lediv12it (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) absclt syl efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an jctir (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) absge0t syl efaddlem17.1 efaddlem17.2 efaddlem17.3 efaddlem17.4 efaddlem17.5 j k efaddlem13 jca jca (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnret syl efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nnre jctil efaddlem17.1 efaddlem17.4 j k efaddlem15 efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nngt0 jctil jca sylanc eqbrtrd)) thm (efaddlem18 ((j k) (N j) (N k)) ((efaddlem17.1 (e. N (NN))) (efaddlem17.2 (e. A (CC))) (efaddlem17.3 (e. B (CC))) (efaddlem17.4 (= S (` (floor) (opr (opr N (+) (1)) (/) (2))))) (efaddlem17.5 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))))) (e. (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr T (^) S) (/) (` (!) S)))) (RR)) (efaddlem17.1 nnuz eleqtr N (opr (opr N (-) (cv j)) (+) (1)) k (opr (opr T (^) S) (/) (` (!) S)) fsumreclt efaddlem17.1 j efaddlem1 efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facne0t ax-mp redivcl (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) a1i r19.21aiva sylanc rgen N (1) j (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr T (^) S) (/) (` (!) S))) fsumreclt mp2an)) thm (efaddlem19 ((j k) (N j) (N k)) ((efaddlem17.1 (e. N (NN))) (efaddlem17.2 (e. A (CC))) (efaddlem17.3 (e. B (CC))) (efaddlem17.4 (= S (` (floor) (opr (opr N (+) (1)) (/) (2))))) (efaddlem17.5 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))))) (br (` (abs) (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))))) (<_) (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr T (^) S) (/) (` (!) S))))) (efaddlem17.1 nnuz eleqtr N (opr (opr N (-) (cv j)) (+) (1)) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt efaddlem17.1 j efaddlem1 (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nncnt syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnne0t syl syl3anc (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc r19.21aiva sylanc rgen N (1) j (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) fsumabs mp2an efaddlem17.1 nnuz eleqtr N (opr (opr N (-) (cv j)) (+) (1)) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt efaddlem17.1 j efaddlem1 (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nncnt syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnne0t syl syl3anc (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc r19.21aiva sylanc (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) absclt syl N (opr (opr N (-) (cv j)) (+) (1)) k (` (abs) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) fsumreclt efaddlem17.1 j efaddlem1 (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nncnt syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnne0t syl syl3anc (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) absclt syl r19.21aiva sylanc N (opr (opr N (-) (cv j)) (+) (1)) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumabs efaddlem17.1 j efaddlem1 (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nncnt syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnne0t syl syl3anc (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc r19.21aiva sylanc 3jca rgen N (1) j (` (abs) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (` (abs) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))) fsumcmp mp2an efaddlem17.1 efaddlem17.2 efaddlem17.3 j k efaddlem4 efaddlem17.1 nnuz eleqtr N (opr (opr N (-) (cv j)) (+) (1)) k (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) fsumclt efaddlem17.1 j efaddlem1 (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nncnt syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnne0t syl syl3anc (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc r19.21aiva sylanc (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) absclt syl rgen N (1) j (` (abs) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))) fsumreclt mp2an efaddlem17.1 nnuz eleqtr N (opr (opr N (-) (cv j)) (+) (1)) k (` (abs) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) fsumreclt efaddlem17.1 j efaddlem1 (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nncnt syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnne0t syl syl3anc (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) absclt syl r19.21aiva sylanc rgen N (1) j (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (` (abs) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))) fsumreclt mp2an letr mp2an efaddlem17.1 nnuz eleqtr N (opr (opr N (-) (cv j)) (+) (1)) k (` (abs) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) fsumreclt efaddlem17.1 j efaddlem1 (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nncnt syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnne0t syl syl3anc (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) absclt syl r19.21aiva sylanc N (opr (opr N (-) (cv j)) (+) (1)) k (opr (opr T (^) S) (/) (` (!) S)) fsumreclt efaddlem17.1 j efaddlem1 efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facne0t ax-mp redivcl (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) a1i r19.21aiva sylanc N (opr (opr N (-) (cv j)) (+) (1)) k (` (abs) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) (opr (opr T (^) S) (/) (` (!) S)) fsumcmp efaddlem17.1 j efaddlem1 (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nncnt syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnne0t syl syl3anc (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) absclt syl efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facne0t ax-mp redivcl (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) a1i efaddlem17.1 efaddlem17.2 efaddlem17.3 efaddlem17.4 efaddlem17.5 j k efaddlem17 3jca r19.21aiva sylanc 3jca rgen N (1) j (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (` (abs) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr T (^) S) (/) (` (!) S))) fsumcmp mp2an efaddlem17.1 efaddlem17.2 efaddlem17.3 j k efaddlem4 efaddlem17.1 nnuz eleqtr N (opr (opr N (-) (cv j)) (+) (1)) k (` (abs) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) fsumreclt efaddlem17.1 j efaddlem1 (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (opr (` (!) (cv j)) (x.) (` (!) (cv k))) divclt (opr A (^) (cv j)) (opr B (^) (cv k)) axmulcl efaddlem17.2 A (cv j) expclt mpan efaddlem17.3 B (cv k) expclt mpan syl2an (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nncnt syl (` (!) (cv j)) (` (!) (cv k)) nnmulclt (cv j) facclt (cv k) facclt syl2an (opr (` (!) (cv j)) (x.) (` (!) (cv k))) nnne0t syl syl3anc (cv j) N elfznnt (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N)) adantr (cv j) nnnn0t syl efaddlem17.1 j k efaddlem2 (cv k) nnnn0t syl sylanc (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))) absclt syl r19.21aiva sylanc rgen N (1) j (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (` (abs) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))) fsumreclt mp2an efaddlem17.1 efaddlem17.2 efaddlem17.3 efaddlem17.4 efaddlem17.5 j k efaddlem18 letr mp2an)) thm (efaddlem20 () ((efaddlem17.1 (e. N (NN))) (efaddlem17.2 (e. A (CC))) (efaddlem17.3 (e. B (CC))) (efaddlem17.4 (= S (` (floor) (opr (opr N (+) (1)) (/) (2))))) (efaddlem17.5 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))))) (br (opr (opr N (^) (2)) (x.) (opr (opr T (^) S) (/) (` (!) S))) (<_) (opr (opr (opr S (^) (2)) (x.) (opr (opr (4) (x.) T) (^) S)) (/) (` (!) S))) (efaddlem17.1 nnsqcl nnre 2re efaddlem17.1 efaddlem17.4 efaddlem8 nnre remulcl resqcl efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facne0t ax-mp redivcl 3pm3.2i efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nnre pm3.2i efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnnn0 efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S nn0expclt mp2an nn0ge0 efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nngt0 pm3.2i (opr T (^) S) (` (!) S) divge0t mp2an efaddlem17.1 N nnzt ax-mp N flhalft ax-mp efaddlem17.4 (2) (x.) opreq2i breqtrr efaddlem17.1 nnre 0re efaddlem17.1 nnre efaddlem17.1 nngt0 ltlei pm3.2i 2re efaddlem17.1 efaddlem17.4 efaddlem8 nnre remulcl 2nn0 efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 nn0mulcl nn0ge0 pm3.2i N (opr (2) (x.) S) le2sqt mp2an mpbi pm3.2i (opr N (^) (2)) (opr (opr (2) (x.) S) (^) (2)) (opr (opr T (^) S) (/) (` (!) S)) lemul1it mp2an 2re efaddlem17.1 efaddlem17.4 efaddlem8 nnre remulcl resqcl recn efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an recn efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nncn efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facne0t ax-mp divass breqtrr 2cn efaddlem17.1 efaddlem17.4 efaddlem8 nncn sqmul 2cn sqcl efaddlem17.1 efaddlem17.4 efaddlem8 nnsqcl nncn mulcom eqtr (x.) (opr T (^) S) opreq1i efaddlem17.1 efaddlem17.4 efaddlem8 nnsqcl nncn 2cn sqcl efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an recn mulass eqtr 2re resqcl efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an remulcl 4re efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre remulcl efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 (opr (4) (x.) T) S reexpclt mp2an efaddlem17.1 efaddlem17.4 efaddlem8 nnsqcl nnre 3pm3.2i 0re efaddlem17.1 efaddlem17.4 efaddlem8 nnsqcl nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnsqcl nngt0 ltlei 2re resqcl 4re efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 (4) S reexpclt mp2an efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an 3pm3.2i efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnnn0 efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S nn0expclt mp2an nn0ge0 sq2 4re recn (4) exp1t ax-mp eqtr4 4re 1nn0 efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 3pm3.2i 0re 3re 3pos ltlei ax1re 3re (1) (3) addge02t mp2an mpbi df-4 breqtrr efaddlem17.1 efaddlem17.4 efaddlem8 S nnge1t ax-mp pm3.2i (4) (1) S expwordit mp2an eqbrtr pm3.2i (opr (2) (^) (2)) (opr (4) (^) S) (opr T (^) S) lemul1it mp2an 4re recn efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nncn efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 (4) T S mulexpt mp3an breqtrr pm3.2i (opr (opr (2) (^) (2)) (x.) (opr T (^) S)) (opr (opr (4) (x.) T) (^) S) (opr S (^) (2)) lemul2it mp2an eqbrtr 2re efaddlem17.1 efaddlem17.4 efaddlem8 nnre remulcl resqcl efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an remulcl efaddlem17.1 efaddlem17.4 efaddlem8 nnsqcl nnre 4re efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre remulcl efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 (opr (4) (x.) T) S reexpclt mp2an remulcl efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nngt0 (opr (opr (opr (2) (x.) S) (^) (2)) (x.) (opr T (^) S)) (opr (opr S (^) (2)) (x.) (opr (opr (4) (x.) T) (^) S)) (` (!) S) lediv1t mpan2 mp3an mpbi efaddlem17.1 nnsqcl nnre efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facne0t ax-mp redivcl remulcl 2re efaddlem17.1 efaddlem17.4 efaddlem8 nnre remulcl resqcl efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 T S reexpclt mp2an remulcl efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facne0t ax-mp redivcl efaddlem17.1 efaddlem17.4 efaddlem8 nnsqcl nnre 4re efaddlem17.2 efaddlem17.3 efaddlem17.5 efaddlem7 nnre remulcl efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 (opr (4) (x.) T) S reexpclt mp2an remulcl efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem17.1 efaddlem17.4 efaddlem8 nnnn0 S facne0t ax-mp redivcl letr mp2an)) thm (efaddlem21 () ((efaddlem21.1 (e. A (CC))) (efaddlem21.2 (e. B (CC))) (efaddlem21.3 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (efaddlem21.4 (= R (opr (2) (x.) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3)))))))) (e. R (NN)) (efaddlem21.4 2nn 2nn 3nn nnsqcl nnnn0 (2) (opr (3) (^) (2)) nnexpclt mp2an 4nn efaddlem21.1 efaddlem21.2 efaddlem21.3 efaddlem7 (4) T nnmulclt mp2an 4nn efaddlem21.1 efaddlem21.2 efaddlem21.3 efaddlem7 (4) T nnmulclt mp2an nnnn0 3nn nnnn0 nn0addcl (opr (4) (x.) T) (opr (opr (4) (x.) T) (+) (3)) nnexpclt mp2an (opr (2) (^) (opr (3) (^) (2))) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3))) nnmulclt mp2an (2) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3)))) nnmulclt mp2an eqeltr)) thm (efaddlem22 ((j k) (N j) (N k) (S j) (S k) (T j) (T k)) ((efaddlem22.1 (e. N (NN))) (efaddlem22.2 (e. A (CC))) (efaddlem22.3 (e. B (CC))) (efaddlem22.4 (= S (` (floor) (opr (opr N (+) (1)) (/) (2))))) (efaddlem22.5 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (efaddlem22.6 (= R (opr (2) (x.) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3)))))))) (br (` (abs) (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))))) (<) (opr (opr (2) (x.) R) (/) N)) (efaddlem22.1 efaddlem22.2 efaddlem22.3 efaddlem22.4 efaddlem22.5 j k efaddlem19 efaddlem22.1 efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem7 nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 T S reexpclt mp2an efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnne0 redivcl efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem7 nnnn0 efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 T S nn0expclt mp2an nn0ge0 efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nngt0 efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem7 nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 T S reexpclt mp2an efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnre divge0 mp2an j k efaddlem16 efaddlem22.1 efaddlem22.2 efaddlem22.3 j k efaddlem4 efaddlem22.1 efaddlem22.2 efaddlem22.3 efaddlem22.4 efaddlem22.5 j k efaddlem18 efaddlem22.1 nnsqcl nnre efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem7 nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 T S reexpclt mp2an efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnne0 redivcl remulcl letr mp2an efaddlem22.1 efaddlem22.2 efaddlem22.3 efaddlem22.4 efaddlem22.5 efaddlem20 efaddlem22.1 efaddlem22.2 efaddlem22.3 j k efaddlem4 efaddlem22.1 nnsqcl nnre efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem7 nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 T S reexpclt mp2an efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnne0 redivcl remulcl efaddlem22.1 efaddlem22.4 efaddlem8 nnre resqcl 4re efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem7 nnre remulcl efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 (opr (4) (x.) T) S reexpclt mp2an remulcl efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnne0 redivcl letr mp2an efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 3nn nnnn0 4nn efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem7 (4) T nnmulclt mp2an S (3) (opr (4) (x.) T) faclbnd5 mp3an efaddlem22.1 efaddlem22.4 efaddlem8 nnre resqcl recn 4re efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem7 nnre remulcl efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 (opr (4) (x.) T) S reexpclt mp2an recn efaddlem22.1 efaddlem22.4 efaddlem8 nncn mul23 df-3 S (^) opreq2i efaddlem22.1 efaddlem22.4 efaddlem8 nncn 2nn0 S (2) expp1t mp2an eqtr (x.) (opr (opr (4) (x.) T) (^) S) opreq1i eqtr4 efaddlem22.6 (x.) (` (!) S) opreq1i 3brtr4 efaddlem22.1 efaddlem22.4 efaddlem8 nnre resqcl 4re efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem7 nnre remulcl efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 (opr (4) (x.) T) S reexpclt mp2an remulcl efaddlem22.1 efaddlem22.4 efaddlem8 nnre pm3.2i efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem22.6 efaddlem21 nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnre pm3.2i efaddlem22.1 efaddlem22.4 efaddlem8 nngt0 efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nngt0 pm3.2i (opr (opr S (^) (2)) (x.) (opr (opr (4) (x.) T) (^) S)) S R (` (!) S) lt2mul2divt mp3an mpbi efaddlem22.1 efaddlem22.2 efaddlem22.3 j k efaddlem4 efaddlem22.1 efaddlem22.4 efaddlem8 nnre resqcl 4re efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem7 nnre remulcl efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 (opr (4) (x.) T) S reexpclt mp2an remulcl efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnnn0 S facclt ax-mp nnne0 redivcl efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem22.6 efaddlem21 nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnne0 redivcl lelttr mp2an efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem22.6 efaddlem21 nncn efaddlem22.1 efaddlem22.4 efaddlem8 nncn efaddlem22.1 efaddlem22.4 efaddlem8 nnne0 pm3.2i 2cn 2re 2pos gt0ne0i pm3.2i R S (2) divcan5t mp3an efaddlem22.1 N nnzt ax-mp N flhalft ax-mp efaddlem22.4 (2) (x.) opreq2i breqtrr efaddlem22.1 nnre efaddlem22.1 nngt0 pm3.2i 2re efaddlem22.1 efaddlem22.4 efaddlem8 nnre remulcl 2nn efaddlem22.1 efaddlem22.4 efaddlem8 (2) S nnmulclt mp2an nngt0 pm3.2i 2re efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem22.6 efaddlem21 nnre remulcl 2nn efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem22.6 efaddlem21 (2) R nnmulclt mp2an nngt0 pm3.2i N (opr (2) (x.) S) (opr (2) (x.) R) lediv2t mp3an mpbi eqbrtrr efaddlem22.1 efaddlem22.2 efaddlem22.3 j k efaddlem4 efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem22.6 efaddlem21 nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnre efaddlem22.1 efaddlem22.4 efaddlem8 nnne0 redivcl 2re efaddlem22.2 efaddlem22.3 efaddlem22.5 efaddlem22.6 efaddlem21 nnre remulcl efaddlem22.1 nnre efaddlem22.1 nnne0 redivcl ltletr mp2an)) thm (efaddlem23 ((j k) (A j) (A k) (B j) (B k) (N j) (N k) (T j) (T k)) ((efaddlem23.1 (e. N (NN))) (efaddlem23.2 (e. A (CC))) (efaddlem23.3 (e. B (CC))) (efaddlem23.4 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (efaddlem23.5 (= R (opr (2) (x.) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3)))))))) (-> (/\ (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) (br (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) (<_) N)) (br (` (abs) (opr (opr (sum_ j (opr (0) (...) N) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) N) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))) (-) (sum_ j (opr (0) (...) N) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j)))))) (<) (cv x))) (efaddlem23.1 efaddlem23.2 efaddlem23.3 j k efaddlem4 (/\ (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) (br (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) (<_) N)) a1i 2re efaddlem23.2 efaddlem23.3 efaddlem23.4 efaddlem23.5 efaddlem21 nnre remulcl efaddlem23.1 nnre efaddlem23.1 nnne0 redivcl (/\ (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) (br (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) (<_) N)) a1i (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (br (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) (<_) N) adantr efaddlem23.1 efaddlem23.2 efaddlem23.3 (` (floor) (opr (opr N (+) (1)) (/) (2))) eqid efaddlem23.4 efaddlem23.5 j k efaddlem22 (/\ (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) (br (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) (<_) N)) a1i (opr (2) (x.) R) (cv x) redivclt 2re efaddlem23.2 efaddlem23.3 efaddlem23.4 efaddlem23.5 efaddlem21 nnre remulcl (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) a1i (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (cv x) gt0ne0t syl3anc (opr (opr (2) (x.) R) (/) (cv x)) flltp1t syl (opr (opr (2) (x.) R) (/) (cv x)) (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) N ltletrt (opr (2) (x.) R) (cv x) redivclt 2re efaddlem23.2 efaddlem23.3 efaddlem23.4 efaddlem23.5 efaddlem21 nnre remulcl (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) a1i (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (cv x) gt0ne0t syl3anc (opr (2) (x.) R) (cv x) redivclt 2re efaddlem23.2 efaddlem23.3 efaddlem23.4 efaddlem23.5 efaddlem21 nnre remulcl (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) a1i (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (cv x) gt0ne0t syl3anc (opr (opr (2) (x.) R) (/) (cv x)) flclt syl (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) zret syl (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) peano2re syl efaddlem23.1 nnre (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) a1i syl3anc mpand imp (opr (2) (x.) R) (cv x) N ltdiv23t 2re efaddlem23.2 efaddlem23.3 efaddlem23.4 efaddlem23.5 efaddlem21 nnre remulcl (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) a1i (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 efaddlem23.1 nnre (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) a1i 3jca (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.27 efaddlem23.1 nngt0 jctir sylanc (br (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) (<_) N) adantr mpbid lttrd efaddlem23.1 efaddlem23.2 efaddlem23.3 j k efaddlem6 (abs) fveq2i syl5eqbr)) thm (efaddlem24 ((j k) (A j) (A k) (B j) (B k) (N j) (N k) (T j) (T k) (j x) (k x)) ((efaddlem24.1 (e. A (CC))) (efaddlem24.2 (e. B (CC))) (efaddlem24.3 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (efaddlem24.4 (= R (opr (2) (x.) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3)))))))) (-> (/\/\ (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) (e. N (NN)) (br (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) (<_) N)) (br (` (abs) (opr (opr (sum_ j (opr (0) (...) N) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) N) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))) (-) (sum_ j (opr (0) (...) N) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j)))))) (<) (cv x))) (N (if (e. N (NN)) N (1)) (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) (<_) breq2 (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) anbi2d N (if (e. N (NN)) N (1)) (0) (...) opreq2 j (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) sumeq1d N (if (e. N (NN)) N (1)) (0) (...) opreq2 k (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) sumeq1d (x.) opreq12d N (if (e. N (NN)) N (1)) (0) (...) opreq2 j (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j))) sumeq1d (-) opreq12d (abs) fveq2d (<) (cv x) breq1d imbi12d 1nn N elimel efaddlem24.1 efaddlem24.2 efaddlem24.3 efaddlem24.4 x j k efaddlem23 dedth 3impib 3com12)) thm (efaddlem25 ((j k) (j m) (A j) (k m) (A k) (A m) (B j) (B k) (B m) (m n) (R m) (R n) (T j) (T k) (m x) (j n) (k n) (n x) (j x) (k x)) ((efaddlem24.1 (e. A (CC))) (efaddlem24.2 (e. B (CC))) (efaddlem24.3 (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (efaddlem24.4 (= R (opr (2) (x.) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3)))))))) (-> (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) (E.e. m (NN) (A.e. n (NN) (-> (br (cv m) (<_) (cv n)) (br (` (abs) (opr (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))) (-) (sum_ j (opr (0) (...) (cv n)) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j)))))) (<) (cv x)))))) ((cv m) (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) (<_) (cv n) breq1 (br (` (abs) (opr (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))) (-) (sum_ j (opr (0) (...) (cv n)) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j)))))) (<) (cv x)) imbi1d n (NN) ralbidv (NN) rcla4ev (opr (opr (2) (x.) R) (/) (cv x)) flge0nn0t (opr (2) (x.) R) (cv x) redivclt 2re efaddlem24.1 efaddlem24.2 efaddlem24.3 efaddlem24.4 efaddlem21 nnre remulcl (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) a1i (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (cv x) gt0ne0t syl3anc 0re (0) (opr (opr (2) (x.) R) (/) (cv x)) ltlet mpan (opr (2) (x.) R) (cv x) redivclt 2re efaddlem24.1 efaddlem24.2 efaddlem24.3 efaddlem24.4 efaddlem21 nnre remulcl (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) a1i (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (cv x) gt0ne0t syl3anc (opr (2) (x.) R) (cv x) divgt0t 2nn efaddlem24.1 efaddlem24.2 efaddlem24.3 efaddlem24.4 efaddlem21 (2) R nnmulclt mp2an nnre (e. (cv x) (RR)) jctl 2nn efaddlem24.1 efaddlem24.2 efaddlem24.3 efaddlem24.4 efaddlem21 (2) R nnmulclt mp2an nngt0 (br (0) (<) (cv x)) jctl syl2an sylc sylanc (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) nn0p1nnt syl efaddlem24.1 efaddlem24.2 efaddlem24.3 efaddlem24.4 x (cv n) j k efaddlem24 3expa ex r19.21aiva sylanc)) thm (efaddlem26 ((j n) (j y) (A j) (n y) (A n) (A y) (k n) (k y) (B k) (B n) (B y) (F m) (G m) (H m) (j k) (j m) (k m) (m n) (m y)) ((efaddlem26.1 (e. A (CC))) (efaddlem26.2 (e. B (CC))) (efaddlem26.3 (= F ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j))))))))) (efaddlem26.4 (= G ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))))))) (efaddlem26.5 (= H ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))))))))) (br H (~~>) (opr (` (exp) A) (x.) (` (exp) B))) (efaddlem26.1 efaddlem26.3 efcvgfsum ax-mp efaddlem26.2 efaddlem26.4 efcvgfsum ax-mp pm3.2i 0z (cv m) elnn0uz (cv n) (cv m) (0) (...) opreq2 j (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) sumeq1d efaddlem26.3 j (opr (0) (...) (cv m)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) sumex fvopab4 (cv m) elnn0uz (cv j) (cv m) elfznn0t (opr A (^) (cv j)) (` (!) (cv j)) divclt efaddlem26.1 A (cv j) expclt mpan (cv j) facclt (` (!) (cv j)) nncnt syl (cv j) facne0t syl3anc syl rgen (cv m) (0) j (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) fsumclt mpan2 sylbi eqeltrd (cv n) (cv m) (0) (...) opreq2 k (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) sumeq1d efaddlem26.4 k (opr (0) (...) (cv m)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) sumex fvopab4 (cv m) elnn0uz (cv k) (cv m) elfznn0t (opr B (^) (cv k)) (` (!) (cv k)) divclt efaddlem26.2 B (cv k) expclt mpan (cv k) facclt (` (!) (cv k)) nncnt syl (cv k) facne0t syl3anc syl rgen (cv m) (0) k (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) fsumclt mpan2 sylbi eqeltrd (cv n) (cv m) (0) (...) opreq2 j (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) sumeq1d (cv n) (cv m) (0) (...) opreq2 k (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) sumeq1d (x.) opreq12d efaddlem26.5 (sum_ j (opr (0) (...) (cv m)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv m)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k)))) oprex fvopab4 (cv n) (cv m) (0) (...) opreq2 j (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) sumeq1d efaddlem26.3 j (opr (0) (...) (cv m)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) sumex fvopab4 (cv n) (cv m) (0) (...) opreq2 k (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) sumeq1d efaddlem26.4 k (opr (0) (...) (cv m)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) sumex fvopab4 (x.) opreq12d eqtr4d 3jca sylbir rgen pm3.2i efaddlem26.3 nn0ex n y (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) funopabex2 eqeltr efaddlem26.4 nn0ex n y (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k)))) funopabex2 eqeltr efaddlem26.5 nn0ex n y (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))) funopabex2 eqeltr (exp) A fvex (exp) B fvex (0) m climmul mp2an)) thm (efaddlem27 ((f j) (f k) (f m) (f n) (f x) (f y) (A f) (j k) (j m) (j n) (j x) (j y) (A j) (k m) (k n) (k x) (k y) (A k) (m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y) (B f) (B j) (B k) (B m) (B n) (B x) (B y) (F f) (F m) (F x) (H f) (H m) (H x)) ((efaddlem27.1 (e. A (CC))) (efaddlem27.2 (e. B (CC))) (efaddlem27.3 (= F ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (sum_ j (opr (0) (...) (cv n)) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j))))))))) (efaddlem27.4 (= H ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))))))))) (br H (~~>) (` (exp) (opr A (+) B))) (efaddlem27.1 efaddlem27.2 addcl (opr A (+) B) efclt ax-mp (cv m) nnnn0t (cv n) (cv m) (0) (...) opreq2 j (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j))) sumeq1d efaddlem27.3 j (opr (0) (...) (cv m)) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j))) sumex fvopab4 (cv j) (cv m) elfznn0t (opr (opr A (+) B) (^) (cv j)) (` (!) (cv j)) divclt efaddlem27.1 efaddlem27.2 addcl (opr A (+) B) (cv j) expclt mpan (cv j) facclt (` (!) (cv j)) nncnt syl (cv j) facne0t syl3anc syl rgen (cv m) j (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j))) fsum0clt mpan2 eqeltrd (cv n) (cv m) (0) (...) opreq2 j (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) sumeq1d (cv n) (cv m) (0) (...) opreq2 k (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) sumeq1d (x.) opreq12d efaddlem27.4 (sum_ j (opr (0) (...) (cv m)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv m)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k)))) oprex fvopab4 (sum_ j (opr (0) (...) (cv m)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (sum_ k (opr (0) (...) (cv m)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k)))) axmulcl (cv j) (cv m) elfznn0t (opr A (^) (cv j)) (` (!) (cv j)) divclt efaddlem27.1 A (cv j) expclt mpan (cv j) facclt (` (!) (cv j)) nncnt syl (cv j) facne0t syl3anc syl rgen (cv m) j (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) fsum0clt mpan2 (cv k) (cv m) elfznn0t (opr B (^) (cv k)) (` (!) (cv k)) divclt efaddlem27.2 B (cv k) expclt mpan (cv k) facclt (` (!) (cv k)) nncnt syl (cv k) facne0t syl3anc syl rgen (cv m) k (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) fsum0clt mpan2 sylanc eqeltrd jca syl rgen pm3.2i efaddlem27.1 efaddlem27.2 (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)) eqid (opr (2) (x.) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))) (^) (opr (opr (4) (x.) (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))) (+) (3))))) eqid x f m j k efaddlem25 (cv m) nnnn0t (cv n) (cv m) (0) (...) opreq2 j (opr (opr A (^) (cv j)) (/) (` (!) (cv j))) sumeq1d (cv n) (cv m) (0) (...) opreq2 k (opr (opr B (^) (cv k)) (/) (` (!) (cv k))) sumeq1d (x.) opreq12d efaddlem27.4 (sum_ j (opr (0) (...) (cv m)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv m)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k)))) oprex fvopab4 (cv n) (cv m) (0) (...) opreq2 j (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j))) sumeq1d efaddlem27.3 j (opr (0) (...) (cv m)) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j))) sumex fvopab4 (-) opreq12d syl (abs) fveq2d (<) (cv x) breq1d (br (cv f) (<_) (cv m)) imbi2d ralbiia f (NN) rexbii sylibr ex rgen efaddlem27.1 efaddlem27.2 addcl efaddlem27.3 efcvgfsum ax-mp pm3.2i efaddlem27.4 nn0ex n y (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))) funopabex2 eqeltr (` (exp) (opr A (+) B)) m F x f 2climnn mp2an)) thm (efaddlem28 ((j k) (j n) (j y) (A j) (k n) (k y) (A k) (n y) (A n) (A y) (B j) (B k) (B n) (B y)) ((efaddlem28.1 (e. A (CC))) (efaddlem28.2 (e. B (CC))) (efaddlem28.4 (= H ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))))))))) (= (` (exp) (opr A (+) B)) (opr (` (exp) A) (x.) (` (exp) B))) ((exp) (opr A (+) B) fvex (` (exp) A) (x.) (` (exp) B) oprex efaddlem28.1 efaddlem28.2 ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (sum_ j (opr (0) (...) (cv n)) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j))))))) eqid efaddlem28.4 efaddlem27 efaddlem28.1 efaddlem28.2 ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j))))))) eqid ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))))) eqid efaddlem28.4 efaddlem26 pm3.2i climunii)) thm (efadd ((j k) (j n) (j y) (A j) (k n) (k y) (A k) (n y) (A n) (A y) (B j) (B k) (B n) (B y)) ((efadd.1 (e. A (CC))) (efadd.2 (e. B (CC)))) (= (` (exp) (opr A (+) B)) (opr (` (exp) A) (x.) (` (exp) B))) (efadd.1 efadd.2 ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k)))))))) eqid efaddlem28)) thm (efaddt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (exp) (opr A (+) B)) (opr (` (exp) A) (x.) (` (exp) B)))) (A (if (e. A (CC)) A (0)) (+) B opreq1 (exp) fveq2d A (if (e. A (CC)) A (0)) (exp) fveq2 (x.) (` (exp) B) opreq1d eqeq12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (+) opreq2 (exp) fveq2d B (if (e. B (CC)) B (0)) (exp) fveq2 (` (exp) (if (e. A (CC)) A (0))) (x.) opreq2d eqeq12d 0cn A elimel 0cn B elimel efadd dedth2h)) thm (efcant () () (-> (e. A (CC)) (= (opr (` (exp) A) (x.) (` (exp) (-u A))) (1))) (A negclt A (-u A) efaddt mpdan A negidt (exp) fveq2d ef0 syl6eq eqtr3d)) thm (efne0t () () (-> (e. A (CC)) (=/= (` (exp) A) (0))) (ax1ne0 (1) (0) df-ne mpbi (` (exp) A) (0) (x.) (` (exp) (-u A)) opreq1 (e. A (CC)) adantl A efcant (= (` (exp) A) (0)) adantr A negclt (-u A) efclt (` (exp) (-u A)) mul02t 3syl (= (` (exp) A) (0)) adantr 3eqtr3d ex mtoi (` (exp) A) (0) df-ne sylibr)) thm (efnn0valtlem ((x y) (N x) (N y)) () (-> (e. N (NN)) (= (` (exp) N) (` (opr (x.) (seq1) (X. (NN) ({} (e)))) N))) ((cv x) (1) (exp) fveq2 (cv x) (1) (opr (x.) (seq1) (X. (NN) ({} (e)))) fveq2 eqeq12d (cv x) (cv y) (exp) fveq2 (cv x) (cv y) (opr (x.) (seq1) (X. (NN) ({} (e)))) fveq2 eqeq12d (cv x) (opr (cv y) (+) (1)) (exp) fveq2 (cv x) (opr (cv y) (+) (1)) (opr (x.) (seq1) (X. (NN) ({} (e)))) fveq2 eqeq12d (cv x) N (exp) fveq2 (cv x) N (opr (x.) (seq1) (X. (NN) ({} (e)))) fveq2 eqeq12d mulex nnex (e) snex xpex seq11 1nn ere recn (X. (NN) ({} (e))) (NN) (e) (1) fvconst (e) (CC) (NN) fconstg sylan mpan ax-mp eqtr df-e eqtr2 (cv y) nncnt 1cn (cv y) (1) efaddt mpan2 syl df-e (` (exp) (cv y)) (x.) opreq2i syl6eqr ere recn 1cn (1) efne0t df-e (0) neeq1i sylibr ax-mp (e) (` (exp) (cv y)) divcan4t (cv y) nncnt (cv y) efclt syl syl3an2 mp3an13 (opr (opr (` (exp) (cv y)) (x.) (e)) (/) (e)) (` (exp) (cv y)) (opr (` (exp) (opr (cv y) (+) (1))) (/) (e)) eqeq2 syl ere recn 1cn (1) efne0t df-e (0) neeq1i sylibr ax-mp (` (exp) (opr (cv y) (+) (1))) (opr (` (exp) (cv y)) (x.) (e)) (e) div11t (cv y) peano2nn (opr (cv y) (+) (1)) nncnt (opr (cv y) (+) (1)) efclt 3syl syl3an1 (cv y) nncnt (cv y) efclt syl ere recn (` (exp) (cv y)) (e) axmulcl mpan2 syl syl3an2 3expa anabsan 3impb mp3an23 bitr3d mpbird (opr (` (exp) (opr (cv y) (+) (1))) (/) (e)) (` (exp) (cv y)) (` (opr (x.) (seq1) (X. (NN) ({} (e)))) (cv y)) eqeq1 syl ere recn (e) (CC) (NN) fconstg ax-mp ere recn (e) (CC) snssi ax-mp (X. (NN) ({} (e))) (NN) ({} (e)) (CC) fss mp2an axmulopr mulex nnex (e) snex xpex (cv y) (CC) seq1cl mp3an23 ere recn 1cn (1) efne0t df-e (0) neeq1i sylibr ax-mp (e) (` (opr (x.) (seq1) (X. (NN) ({} (e)))) (cv y)) divcan4t mp3an13 syl (opr (opr (` (opr (x.) (seq1) (X. (NN) ({} (e)))) (cv y)) (x.) (e)) (/) (e)) (` (opr (x.) (seq1) (X. (NN) ({} (e)))) (cv y)) (opr (` (exp) (opr (cv y) (+) (1))) (/) (e)) eqeq2 syl ere recn (e) (CC) (NN) fconstg ax-mp ere recn (e) (CC) snssi ax-mp (X. (NN) ({} (e))) (NN) ({} (e)) (CC) fss mp2an axmulopr mulex nnex (e) snex xpex (cv y) (CC) seq1cl mp3an23 ere recn (` (opr (x.) (seq1) (X. (NN) ({} (e)))) (cv y)) (e) axmulcl mpan2 syl ere recn 1cn (1) efne0t df-e (0) neeq1i sylibr ax-mp pm3.2i (` (exp) (opr (cv y) (+) (1))) (opr (` (opr (x.) (seq1) (X. (NN) ({} (e)))) (cv y)) (x.) (e)) (e) div11t mp3an3 (cv y) peano2nn (opr (cv y) (+) (1)) nncnt (opr (cv y) (+) (1)) efclt 3syl sylan mpdan bitr3d bitr3d (cv y) peano2nn (cv y) nngt0t (cv y) nnret 0re ax1re (0) (cv y) (1) ltadd1t mp3an13 syl mpbid 1cn (1) addid2t ax-mp (<) (opr (cv y) (+) (1)) breq1i (e. (cv y) (NN)) a1i mpbid jca mulex nnex (e) snex xpex (opr (cv y) (+) (1)) seq1m1 ere recn (X. (NN) ({} (e))) (NN) (e) (opr (cv y) (+) (1)) fvconst (e) (CC) (NN) fconstg sylan mpan (br (1) (<) (opr (cv y) (+) (1))) adantr (` (X. (NN) ({} (e))) (opr (cv y) (+) (1))) (e) (` (opr (x.) (seq1) (X. (NN) ({} (e)))) (opr (opr (cv y) (+) (1)) (-) (1))) (x.) opreq2 syl eqtrd syl (cv y) nncnt 1cn (cv y) (1) pncant mpan2 syl (opr (x.) (seq1) (X. (NN) ({} (e)))) fveq2d (x.) (e) opreq1d eqtrd (` (exp) (opr (cv y) (+) (1))) eqeq2d bitr4d biimpd nnind)) thm (efnn0valt () () (-> (e. N (NN0)) (= (` (exp) N) (opr (e) (^) N))) (N elnn0 N efnn0valtlem ere recn (e) N expnnvalt mpan eqtr4d ef0 ere recn (e) exp0t ax-mp eqtr4 N (0) (exp) fveq2 N (0) (e) (^) opreq2 eqeq12d mpbiri jaoi sylbi)) thm (sinvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (sin) A) (opr (opr (` (exp) (opr (i) (x.) A)) (-) (` (exp) (opr (-u (i)) (x.) A))) (/) (opr (2) (x.) (i))))) ((cv x) A (i) (x.) opreq2 (exp) fveq2d (cv x) A (-u (i)) (x.) opreq2 (exp) fveq2d (-) opreq12d (/) (opr (2) (x.) (i)) opreq1d x y df-sin (opr (` (exp) (opr (i) (x.) A)) (-) (` (exp) (opr (-u (i)) (x.) A))) (/) (opr (2) (x.) (i)) oprex fvopab4)) thm (cosvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (cos) A) (opr (opr (` (exp) (opr (i) (x.) A)) (+) (` (exp) (opr (-u (i)) (x.) A))) (/) (2)))) ((cv x) A (i) (x.) opreq2 (exp) fveq2d (cv x) A (-u (i)) (x.) opreq2 (exp) fveq2d (+) opreq12d (/) (2) opreq1d x y df-cos (opr (` (exp) (opr (i) (x.) A)) (+) (` (exp) (opr (-u (i)) (x.) A))) (/) (2) oprex fvopab4)) thm (sinclt () () (-> (e. A (CC)) (e. (` (sin) A) (CC))) (A sinvalt axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl jca (` (exp) (opr (i) (x.) A)) (` (exp) (opr (-u (i)) (x.) A)) subclt 2cn axicn mulcl 2cn axicn 2re 2pos gt0ne0i ine0 muln0 (opr (` (exp) (opr (i) (x.) A)) (-) (` (exp) (opr (-u (i)) (x.) A))) (opr (2) (x.) (i)) divclt mp3an23 3syl eqeltrd)) thm (cosclt () () (-> (e. A (CC)) (e. (` (cos) A) (CC))) (A cosvalt axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl jca (` (exp) (opr (i) (x.) A)) (` (exp) (opr (-u (i)) (x.) A)) axaddcl (opr (` (exp) (opr (i) (x.) A)) (+) (` (exp) (opr (-u (i)) (x.) A))) halfclt 3syl eqeltrd)) thm (resinvalt () () (-> (e. A (RR)) (= (` (sin) A) (` (Im) (` (exp) (opr (i) (x.) A))))) (A recnt axicn (i) A cjmult mpan syl A cjret (-u (i)) (x.) opreq2d cji (x.) (` (*) A) opreq1i syl5eq eqtrd (exp) fveq2d A recnt axicn (i) A axmulcl mpan syl (opr (i) (x.) A) efcjt syl eqtr3d (` (exp) (opr (i) (x.) A)) (-) opreq2d (/) (opr (2) (x.) (i)) opreq1d A recnt A sinvalt syl A recnt axicn (i) A axmulcl mpan syl (opr (i) (x.) A) efclt syl (` (exp) (opr (i) (x.) A)) imcjt syl 3eqtr4d)) thm (recosvalt () () (-> (e. A (RR)) (= (` (cos) A) (` (Re) (` (exp) (opr (i) (x.) A))))) (A recnt axicn (i) A cjmult mpan syl A cjret (-u (i)) (x.) opreq2d cji (x.) (` (*) A) opreq1i syl5eq eqtrd (exp) fveq2d A recnt axicn (i) A axmulcl mpan syl (opr (i) (x.) A) efcjt syl eqtr3d (` (exp) (opr (i) (x.) A)) (+) opreq2d (/) (2) opreq1d A recnt A cosvalt syl A recnt axicn (i) A axmulcl mpan syl (opr (i) (x.) A) efclt syl (` (exp) (opr (i) (x.) A)) recjt syl 3eqtr4d)) thm (resinclt () () (-> (e. A (RR)) (e. (` (sin) A) (RR))) (A resinvalt A recnt axicn (i) A axmulcl mpan syl (opr (i) (x.) A) efclt syl (` (exp) (opr (i) (x.) A)) imclt syl eqeltrd)) thm (recosclt () () (-> (e. A (RR)) (e. (` (cos) A) (RR))) (A recosvalt A recnt axicn (i) A axmulcl mpan syl (opr (i) (x.) A) efclt syl (` (exp) (opr (i) (x.) A)) reclt syl eqeltrd)) thm (sinnegt () () (-> (e. A (CC)) (= (` (sin) (-u A)) (-u (` (sin) A)))) (axicn (i) A mulneg12t mpan eqcomd (exp) fveq2d axicn (i) A mul2negt mpan (exp) fveq2d (-) opreq12d (` (exp) (opr (i) (x.) A)) (` (exp) (opr (-u (i)) (x.) A)) negsubdi2t axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl sylanc eqtr4d (/) (opr (2) (x.) (i)) opreq1d A negclt (-u A) sinvalt syl A sinvalt negeqd (` (exp) (opr (i) (x.) A)) (` (exp) (opr (-u (i)) (x.) A)) subclt axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl sylanc 2cn axicn mulcl 2cn axicn 2re 2pos gt0ne0i ine0 muln0 (opr (` (exp) (opr (i) (x.) A)) (-) (` (exp) (opr (-u (i)) (x.) A))) (opr (2) (x.) (i)) divnegt mp3an23 syl eqtrd 3eqtr4d)) thm (cosnegt () () (-> (e. A (CC)) (= (` (cos) (-u A)) (` (cos) A))) (axicn (i) A mulneg12t mpan eqcomd (exp) fveq2d axicn (i) A mul2negt mpan (exp) fveq2d (+) opreq12d (` (exp) (opr (-u (i)) (x.) A)) (` (exp) (opr (i) (x.) A)) axaddcom axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl sylanc eqtrd (/) (2) opreq1d A negclt (-u A) cosvalt syl A cosvalt 3eqtr4d)) thm (sin0 () () (= (` (sin) (0)) (0)) (0re (0) resinvalt ax-mp axicn mul01 (exp) fveq2i ef0 eqtr (Im) fveq2i im1 3eqtr)) thm (cos0 () () (= (` (cos) (0)) (1)) (0re (0) recosvalt ax-mp axicn mul01 (exp) fveq2i ef0 eqtr (Re) fveq2i re1 3eqtr)) thm (efivalt () () (-> (e. A (CC)) (= (` (exp) (opr (i) (x.) A)) (opr (` (cos) A) (+) (opr (i) (x.) (` (sin) A))))) (2cn 2re 2pos gt0ne0i (opr (` (exp) (opr (i) (x.) A)) (+) (` (exp) (opr (-u (i)) (x.) A))) (opr (` (exp) (opr (i) (x.) A)) (-) (` (exp) (opr (-u (i)) (x.) A))) (2) divdirt mpan2 mp3an3 (` (exp) (opr (i) (x.) A)) (` (exp) (opr (-u (i)) (x.) A)) axaddcl axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl sylanc (` (exp) (opr (i) (x.) A)) (` (exp) (opr (-u (i)) (x.) A)) subclt axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl sylanc sylanc (` (exp) (opr (-u (i)) (x.) A)) (` (exp) (opr (i) (x.) A)) pncan3t axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl sylanc (` (exp) (opr (i) (x.) A)) (+) opreq2d (` (exp) (opr (i) (x.) A)) (` (exp) (opr (-u (i)) (x.) A)) (opr (` (exp) (opr (i) (x.) A)) (-) (` (exp) (opr (-u (i)) (x.) A))) axaddass axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl (` (exp) (opr (i) (x.) A)) (` (exp) (opr (-u (i)) (x.) A)) subclt axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl sylanc syl3anc axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl (` (exp) (opr (i) (x.) A)) 2timest syl 3eqtr4d (/) (2) opreq1d axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl 2cn 2re 2pos gt0ne0i (2) (` (exp) (opr (i) (x.) A)) divcan3t mp3an13 syl eqtr2d A cosvalt (` (exp) (opr (i) (x.) A)) (` (exp) (opr (-u (i)) (x.) A)) subclt axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl sylanc axicn 2cn axicn mulcl 2cn axicn 2re 2pos gt0ne0i ine0 muln0 (i) (opr (` (exp) (opr (i) (x.) A)) (-) (` (exp) (opr (-u (i)) (x.) A))) (opr (2) (x.) (i)) div12t mpan2 mp3an13 syl A sinvalt (i) (x.) opreq2d (` (exp) (opr (i) (x.) A)) (` (exp) (opr (-u (i)) (x.) A)) subclt axicn (i) A axmulcl mpan (opr (i) (x.) A) efclt syl axicn negcl (-u (i)) A axmulcl mpan (opr (-u (i)) (x.) A) efclt syl sylanc 2cn 2re 2pos gt0ne0i (opr (` (exp) (opr (i) (x.) A)) (-) (` (exp) (opr (-u (i)) (x.) A))) (2) divrect mp3an23 syl axicn mulid2 (/) (opr (2) (x.) (i)) opreq1i axicn ine0 divid (opr (1) (/) (2)) (x.) opreq2i 1cn 2cn axicn axicn 2re 2pos gt0ne0i ine0 divmuldiv 2cn 2re 2pos gt0ne0i reccl mulid1 3eqtr3 eqtr3 (opr (` (exp) (opr (i) (x.) A)) (-) (` (exp) (opr (-u (i)) (x.) A))) (x.) opreq2i syl6eqr 3eqtr4d (+) opreq12d 3eqtr4d)) thm (efmivalt () () (-> (e. A (CC)) (= (` (exp) (opr (-u (i)) (x.) A)) (opr (` (cos) A) (-) (opr (i) (x.) (` (sin) A))))) (axicn (i) A mulneg12t mpan (exp) fveq2d A negclt (-u A) efivalt syl A cosnegt A sinnegt (i) (x.) opreq2d A sinclt axicn (i) (` (sin) A) mulneg2t mpan syl eqtrd (+) opreq12d (` (cos) A) (opr (i) (x.) (` (sin) A)) negsubt A cosclt A sinclt axicn (i) (` (sin) A) axmulcl mpan syl sylanc eqtrd 3eqtrd)) thm (efeult () () (-> (e. A (CC)) (= (` (exp) A) (opr (` (exp) (` (Re) A)) (x.) (opr (` (cos) (` (Im) A)) (+) (opr (i) (x.) (` (sin) (` (Im) A))))))) (A replimt A imclt recnd axicn (i) (` (Im) A) axmulcom mpan syl (` (Re) A) (+) opreq2d eqtr4d (exp) fveq2d (` (Re) A) (opr (i) (x.) (` (Im) A)) efaddt A reclt recnd A imclt recnd axicn (i) (` (Im) A) axmulcl mpan syl sylanc A imclt recnd (` (Im) A) efivalt syl (` (exp) (` (Re) A)) (x.) opreq2d 3eqtrd)) thm (sinadd () ((sinadd.1 (e. A (CC))) (sinadd.2 (e. B (CC)))) (= (` (sin) (opr A (+) B)) (opr (opr (` (sin) A) (x.) (` (cos) B)) (+) (opr (` (cos) A) (x.) (` (sin) B)))) (sinadd.1 sinadd.2 addcl (opr A (+) B) sinvalt ax-mp sinadd.1 A cosclt ax-mp sinadd.2 B cosclt ax-mp mulcl axicn sinadd.2 B sinclt ax-mp mulcl axicn sinadd.1 A sinclt ax-mp mulcl mulcl addcl sinadd.1 A cosclt ax-mp axicn sinadd.2 B sinclt ax-mp mulcl mulcl sinadd.2 B cosclt ax-mp axicn sinadd.1 A sinclt ax-mp mulcl mulcl addcl sinadd.1 A cosclt ax-mp axicn sinadd.2 B sinclt ax-mp mulcl mulcl sinadd.2 B cosclt ax-mp axicn sinadd.1 A sinclt ax-mp mulcl mulcl addcl pnncan axicn sinadd.1 sinadd.2 adddi (exp) fveq2i axicn sinadd.1 mulcl axicn sinadd.2 mulcl efadd sinadd.1 A efivalt ax-mp sinadd.2 B efivalt ax-mp (x.) opreq12i sinadd.1 A cosclt ax-mp axicn sinadd.1 A sinclt ax-mp mulcl sinadd.2 B cosclt ax-mp axicn sinadd.2 B sinclt ax-mp mulcl muladd eqtr 3eqtr axicn negcl sinadd.1 sinadd.2 adddi (exp) fveq2i axicn negcl sinadd.1 mulcl axicn negcl sinadd.2 mulcl efadd sinadd.1 A efmivalt ax-mp sinadd.2 B efmivalt ax-mp (x.) opreq12i sinadd.1 A cosclt ax-mp axicn sinadd.1 A sinclt ax-mp mulcl pm3.2i sinadd.2 B cosclt ax-mp axicn sinadd.2 B sinclt ax-mp mulcl pm3.2i (` (cos) A) (opr (i) (x.) (` (sin) A)) (` (cos) B) (opr (i) (x.) (` (sin) B)) mulsubt mp2an eqtr 3eqtr (-) opreq12i sinadd.1 A cosclt ax-mp axicn sinadd.2 B sinclt ax-mp mulcl mulcl sinadd.2 B cosclt ax-mp axicn sinadd.1 A sinclt ax-mp mulcl mulcl addcl 2times 3eqtr4 (/) (opr (2) (x.) (i)) opreq1i 2cn axicn sinadd.1 A cosclt ax-mp sinadd.2 B sinclt ax-mp mulcl sinadd.1 A sinclt ax-mp sinadd.2 B cosclt ax-mp mulcl addcl mulass axicn sinadd.1 A cosclt ax-mp sinadd.2 B sinclt ax-mp mulcl sinadd.1 A sinclt ax-mp sinadd.2 B cosclt ax-mp mulcl adddi axicn sinadd.1 A cosclt ax-mp sinadd.2 B sinclt ax-mp mul12 sinadd.1 A sinclt ax-mp sinadd.2 B cosclt ax-mp mulcom (i) (x.) opreq2i axicn sinadd.2 B cosclt ax-mp sinadd.1 A sinclt ax-mp mul12 eqtr (+) opreq12i eqtr (2) (x.) opreq2i eqtr 2cn sinadd.1 A cosclt ax-mp axicn sinadd.2 B sinclt ax-mp mulcl mulcl sinadd.2 B cosclt ax-mp axicn sinadd.1 A sinclt ax-mp mulcl mulcl addcl mulcl 2cn axicn mulcl sinadd.1 A cosclt ax-mp sinadd.2 B sinclt ax-mp mulcl sinadd.1 A sinclt ax-mp sinadd.2 B cosclt ax-mp mulcl addcl 2cn axicn 2re 2pos gt0ne0i ine0 muln0 divmul mpbir sinadd.1 A sinclt ax-mp sinadd.2 B cosclt ax-mp mulcl sinadd.1 A cosclt ax-mp sinadd.2 B sinclt ax-mp mulcl addcom eqtr4 3eqtr)) thm (cosadd () ((sinadd.1 (e. A (CC))) (sinadd.2 (e. B (CC)))) (= (` (cos) (opr A (+) B)) (opr (opr (` (cos) A) (x.) (` (cos) B)) (-) (opr (` (sin) A) (x.) (` (sin) B)))) (sinadd.1 sinadd.2 addcl (opr A (+) B) cosvalt ax-mp sinadd.1 A cosclt ax-mp sinadd.2 B cosclt ax-mp mulcl axicn sinadd.2 B sinclt ax-mp mulcl axicn sinadd.1 A sinclt ax-mp mulcl mulcl addcl sinadd.1 A cosclt ax-mp axicn sinadd.2 B sinclt ax-mp mulcl mulcl sinadd.2 B cosclt ax-mp axicn sinadd.1 A sinclt ax-mp mulcl mulcl addcl sinadd.1 A cosclt ax-mp sinadd.2 B cosclt ax-mp mulcl axicn sinadd.2 B sinclt ax-mp mulcl axicn sinadd.1 A sinclt ax-mp mulcl mulcl addcl (opr (opr (` (cos) A) (x.) (` (cos) B)) (+) (opr (opr (i) (x.) (` (sin) B)) (x.) (opr (i) (x.) (` (sin) A)))) (opr (opr (` (cos) A) (x.) (opr (i) (x.) (` (sin) B))) (+) (opr (` (cos) B) (x.) (opr (i) (x.) (` (sin) A)))) (opr (opr (` (cos) A) (x.) (` (cos) B)) (+) (opr (opr (i) (x.) (` (sin) B)) (x.) (opr (i) (x.) (` (sin) A)))) ppncant mp3an axicn sinadd.1 sinadd.2 adddi (exp) fveq2i axicn sinadd.1 mulcl axicn sinadd.2 mulcl efadd sinadd.1 A efivalt ax-mp sinadd.2 B efivalt ax-mp (x.) opreq12i sinadd.1 A cosclt ax-mp axicn sinadd.1 A sinclt ax-mp mulcl sinadd.2 B cosclt ax-mp axicn sinadd.2 B sinclt ax-mp mulcl muladd eqtr 3eqtr axicn negcl sinadd.1 sinadd.2 adddi (exp) fveq2i axicn negcl sinadd.1 mulcl axicn negcl sinadd.2 mulcl efadd sinadd.1 A efmivalt ax-mp sinadd.2 B efmivalt ax-mp (x.) opreq12i sinadd.1 A cosclt ax-mp axicn sinadd.1 A sinclt ax-mp mulcl pm3.2i sinadd.2 B cosclt ax-mp axicn sinadd.2 B sinclt ax-mp mulcl pm3.2i (` (cos) A) (opr (i) (x.) (` (sin) A)) (` (cos) B) (opr (i) (x.) (` (sin) B)) mulsubt mp2an eqtr 3eqtr (+) opreq12i sinadd.1 A cosclt ax-mp sinadd.2 B cosclt ax-mp mulcl axicn sinadd.2 B sinclt ax-mp mulcl axicn sinadd.1 A sinclt ax-mp mulcl mulcl addcl 2times 3eqtr4 (/) (2) opreq1i 2cn sinadd.1 A cosclt ax-mp sinadd.2 B cosclt ax-mp mulcl axicn sinadd.2 B sinclt ax-mp mulcl axicn sinadd.1 A sinclt ax-mp mulcl mulcl addcl 2re 2pos gt0ne0i divcan3 axicn sinadd.2 B sinclt ax-mp axicn sinadd.1 A sinclt ax-mp mul4 itimesi sinadd.2 B sinclt ax-mp sinadd.1 A sinclt ax-mp mulcom (x.) opreq12i sinadd.1 A sinclt ax-mp sinadd.2 B sinclt ax-mp mulcl mulm1 3eqtr (opr (` (cos) A) (x.) (` (cos) B)) (+) opreq2i sinadd.1 A cosclt ax-mp sinadd.2 B cosclt ax-mp mulcl sinadd.1 A sinclt ax-mp sinadd.2 B sinclt ax-mp mulcl negsub 3eqtr 3eqtr)) thm (cosaddt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (cos) (opr A (+) B)) (opr (opr (` (cos) A) (x.) (` (cos) B)) (-) (opr (` (sin) A) (x.) (` (sin) B))))) (A (if (e. A (CC)) A (0)) (+) B opreq1 (cos) fveq2d A (if (e. A (CC)) A (0)) (cos) fveq2 (x.) (` (cos) B) opreq1d A (if (e. A (CC)) A (0)) (sin) fveq2 (x.) (` (sin) B) opreq1d (-) opreq12d eqeq12d B (if (e. B (CC)) B (0)) (if (e. A (CC)) A (0)) (+) opreq2 (cos) fveq2d B (if (e. B (CC)) B (0)) (cos) fveq2 (` (cos) (if (e. A (CC)) A (0))) (x.) opreq2d B (if (e. B (CC)) B (0)) (sin) fveq2 (` (sin) (if (e. A (CC)) A (0))) (x.) opreq2d (-) opreq12d eqeq12d 0cn A elimel 0cn B elimel cosadd dedth2h)) thm (sincossqt () () (-> (e. A (CC)) (= (opr (opr (` (sin) A) (^) (2)) (+) (opr (` (cos) A) (^) (2))) (1))) (A negclt A (-u A) cosaddt mpdan A negidt (cos) fveq2d cos0 syl6eq (opr (` (sin) A) (^) (2)) (opr (` (cos) A) (^) (2)) axaddcom A sinclt (` (sin) A) sqclt syl A cosclt (` (cos) A) sqclt syl sylanc A cosclt (` (cos) A) sqvalt syl A cosnegt (` (cos) A) (x.) opreq2d eqtr4d A sinclt (` (sin) A) sqvalt syl A sinnegt negeqd A sinclt (` (sin) A) negnegt syl eqtrd (` (sin) A) (x.) opreq2d (` (sin) A) (` (sin) (-u A)) mulneg2t A sinclt A negclt (-u A) sinclt syl sylanc 3eqtr2d (+) opreq12d (opr (` (cos) A) (x.) (` (cos) (-u A))) (opr (` (sin) A) (x.) (` (sin) (-u A))) negsubt (` (cos) A) (` (cos) (-u A)) axmulcl A cosclt A negclt (-u A) cosclt syl sylanc (` (sin) A) (` (sin) (-u A)) axmulcl A sinclt A negclt (-u A) sinclt syl sylanc sylanc 3eqtrrd 3eqtr3rd)) thm (sinbndt () () (-> (e. A (RR)) (/\ (br (-u (1)) (<_) (` (sin) A)) (br (` (sin) A) (<_) (1)))) (A recosclt (` (cos) A) sqge0t syl (opr (` (sin) A) (^) (2)) (opr (` (cos) A) (^) (2)) addge01t A resinclt (` (sin) A) resqclt syl A recosclt (` (cos) A) resqclt syl sylanc mpbid A recnt A sincossqt syl sq1 syl6eqr breqtrd A resinclt ax1re 0re ax1re lt01 ltlei (` (sin) A) (1) lenegsqt mp3an23 ax1re (` (sin) A) (1) lenegcon1t mpan2 (br (` (sin) A) (<_) (1)) anbi2d bitr3d syl mpbid (br (` (sin) A) (<_) (1)) (br (-u (1)) (<_) (` (sin) A)) ancom sylib)) thm (nn0ennn ((x y)) () (br (NN0) (~~) (NN)) (nn0ex (cv x) nn0p1nnt (cv y) elnnnn0 pm3.27bd 1cn (cv y) (1) (cv x) subaddt mp3an2 (cv x) (opr (cv y) (-) (1)) eqcom (cv y) (opr (1) (+) (cv x)) eqcom 3bitr4g 1cn (1) (cv x) axaddcom mpan (cv y) eqeq2d (e. (cv y) (CC)) adantl bitrd (cv y) nncnt (cv x) nn0cnt syl2an ancoms en3)) thm (nnenom ((x y) (x z) (y z)) () (br (NN) (~~) (om)) (nnex 0z (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) (0)) (om)) eqid z om2uzf1o z nn0zrab (NN0) ({e.|} z (ZZ) (br (0) (<_) (cv z))) (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) (0)) (om)) (om) f1oeq3 ax-mp mpbir omex (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) (0)) (om)) (NN0) f1oen ax-mp nn0ennn entr2)) thm (xpnnen ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (br (X. (NN) (NN)) (~~) (NN)) (nnex nnex xpex (cv x) (NN) (NN) elxp5 (opr (opr (|^| (|^| (cv x))) (+) (U. (ran ({} (cv x))))) (^) (2)) (U. (ran ({} (cv x)))) nnaddclt (|^| (|^| (cv x))) (U. (ran ({} (cv x)))) nnaddclt 2nn0 (opr (|^| (|^| (cv x))) (+) (U. (ran ({} (cv x))))) (2) nnexpclt mpan2 syl (e. (|^| (|^| (cv x))) (NN)) (e. (U. (ran ({} (cv x)))) (NN)) pm3.27 sylanc (= (cv x) (<,> (|^| (|^| (cv x))) (U. (ran ({} (cv x)))))) adantl sylbi (cv x) (<,> (cv z) (cv w)) inteq inteqd z visset (cv w) op1stb syl6eq (cv x) (<,> (cv z) (cv w)) sneq rneqd unieqd z visset w visset op2nda syl6eq (+) opreq12d (^) (2) opreq1d (cv x) (<,> (cv z) (cv w)) sneq rneqd unieqd z visset w visset op2nda syl6eq (+) opreq12d (cv y) (<,> (cv v) (cv u)) inteq inteqd v visset (cv u) op1stb syl6eq (cv y) (<,> (cv v) (cv u)) sneq rneqd unieqd v visset u visset op2nda syl6eq (+) opreq12d (^) (2) opreq1d (cv y) (<,> (cv v) (cv u)) sneq rneqd unieqd v visset u visset op2nda syl6eq (+) opreq12d eqeqan12d (cv z) (cv w) (cv v) (cv u) nn0opth2t (cv z) nnnn0t (cv w) nnnn0t anim12i (cv v) nnnn0t (cv u) nnnn0t anim12i syl2an sylan9bbr (cv x) (<,> (cv z) (cv w)) (cv y) (<,> (cv v) (cv u)) eqeq12 z visset w visset u visset (cv v) opth syl6bb (/\ (/\ (e. (cv z) (NN)) (e. (cv w) (NN))) (/\ (e. (cv v) (NN)) (e. (cv u) (NN)))) adantl bitr4d exp43 com23 r19.23aivv r19.23advv imp (cv x) (NN) (NN) z w elxp2 (cv y) (NN) (NN) v u elxp2 syl2anb (V) dom2 ax-mp nnex nnex 1nn elisseti xpsnen ensymi nnex nnex xpex (NN) ssid 1nn (1) (NN) snssi ax-mp (NN) (NN) ({} (1)) (NN) ssxp mp2an (X. (NN) (NN)) (V) (X. (NN) ({} (1))) ssdom2g mp2 (NN) (X. (NN) ({} (1))) (X. (NN) (NN)) endomtr mp2an (X. (NN) (NN)) (NN) sbth mp2an)) thm (xpomen () () (br (X. (om) (om)) (~~) (om)) (omex omex xpex nnenom nnenom nnex omex nnex omex xpen mp2an xpnnen entr3 nnenom entr)) thm (znnenlem () () (-> (/\ (/\ (br (0) (<_) (cv x)) (-. (br (0) (<_) (cv y)))) (/\ (e. (cv x) (ZZ)) (e. (cv y) (ZZ)))) (<-> (= (cv y) (cv x)) (= (opr (2) (x.) (cv x)) (opr (opr (-u (2)) (x.) (cv y)) (+) (1))))) ((= (cv y) (cv x)) (= (opr (2) (x.) (cv x)) (opr (opr (-u (2)) (x.) (cv y)) (+) (1))) pm5.21 0re (cv y) (0) ltnlet mpan2 (e. (cv x) (RR)) adantr 0re (cv y) (0) (cv x) ltletrt mp3an2 exp3a sylbird com23 imp3a (cv y) (cv x) ltnet syld ancoms (cv x) zret (cv y) zret syl2an impcom (cv x) (-u (cv y)) zneo (cv y) znegclt sylan2 (cv y) zcnt 2cn (2) (cv y) mulneg12t mpan syl (e. (cv x) (ZZ)) adantl (+) (1) opreq1d (opr (2) (x.) (cv x)) eqeq2d mtbird (/\ (br (0) (<_) (cv x)) (-. (br (0) (<_) (cv y)))) adantl sylanc)) thm (znnen ((x y)) () (br (ZZ) (~~) (NN)) (zex (cv x) elnn0z 2nn0 (2) (cv x) nn0mulclt mpan sylbir (br (0) (<_) (cv x)) (opr (2) (x.) (cv x)) (opr (opr (-u (2)) (x.) (cv x)) (+) (1)) iftrue (NN0) eleq1d (e. (cv x) (ZZ)) adantl mpbird 2z (2) znegclt ax-mp (-u (2)) (cv x) zmulclt mpan (-. (br (0) (<_) (cv x))) adantr (cv x) zret (cv x) renegclt 2pos 2re (2) (-u (cv x)) axmulgt0 mpan mpani syl 0re (cv x) (0) ltnlet mpan2 (cv x) lt0neg1t bitr3d (cv x) recnt 2cn (2) (cv x) mulneg12t mpan syl (0) (<) breq2d 3imtr4d 2re renegcl (-u (2)) (cv x) axmulrcl mpan 0re (0) (opr (-u (2)) (x.) (cv x)) ltlet mpan syl syld syl imp jca (opr (-u (2)) (x.) (cv x)) elnn0z biimpr (opr (-u (2)) (x.) (cv x)) peano2nn0 3syl (br (0) (<_) (cv x)) (opr (2) (x.) (cv x)) (opr (opr (-u (2)) (x.) (cv x)) (+) (1)) iffalse (NN0) eleq1d (e. (cv x) (ZZ)) adantl mpbird pm2.61dan (br (0) (<_) (cv x)) (opr (2) (x.) (cv x)) (opr (opr (-u (2)) (x.) (cv x)) (+) (1)) iftrue (br (0) (<_) (cv y)) (opr (2) (x.) (cv y)) (opr (opr (-u (2)) (x.) (cv y)) (+) (1)) iftrue eqeqan12d 2cn 2re 2pos gt0ne0i (cv x) (cv y) mulcant2 mp3an1 (cv x) zcnt (cv y) zcnt syl2an sylan9bb bicomd ex x y znnenlem x y equcom syl5bb (br (0) (<_) (cv x)) (opr (2) (x.) (cv x)) (opr (opr (-u (2)) (x.) (cv x)) (+) (1)) iftrue (br (0) (<_) (cv y)) (opr (2) (x.) (cv y)) (opr (opr (-u (2)) (x.) (cv y)) (+) (1)) iffalse eqeqan12d (/\ (e. (cv x) (ZZ)) (e. (cv y) (ZZ))) adantr bitr4d ex y x znnenlem (-. (br (0) (<_) (cv x))) (br (0) (<_) (cv y)) ancom (e. (cv x) (ZZ)) (e. (cv y) (ZZ)) ancom syl2anb (opr (opr (-u (2)) (x.) (cv x)) (+) (1)) (opr (2) (x.) (cv y)) eqcom syl6bbr (br (0) (<_) (cv x)) (opr (2) (x.) (cv x)) (opr (opr (-u (2)) (x.) (cv x)) (+) (1)) iffalse (br (0) (<_) (cv y)) (opr (2) (x.) (cv y)) (opr (opr (-u (2)) (x.) (cv y)) (+) (1)) iftrue eqeqan12d (/\ (e. (cv x) (ZZ)) (e. (cv y) (ZZ))) adantr bitr4d ex 1cn (opr (-u (2)) (x.) (cv x)) (opr (-u (2)) (x.) (cv y)) (1) addcan2t mp3an3 2cn negcl (-u (2)) (cv x) axmulcl mpan 2cn negcl (-u (2)) (cv y) axmulcl mpan syl2an 2cn negcl 2cn 2re 2pos gt0ne0i negn0 (cv x) (cv y) mulcant2 mp3an1 bitr2d (cv x) zcnt (cv y) zcnt syl2an (br (0) (<_) (cv x)) (opr (2) (x.) (cv x)) (opr (opr (-u (2)) (x.) (cv x)) (+) (1)) iffalse (br (0) (<_) (cv y)) (opr (2) (x.) (cv y)) (opr (opr (-u (2)) (x.) (cv y)) (+) (1)) iffalse eqeqan12d bicomd sylan9bbr ex 4cases bicomd (V) dom2 ax-mp nn0ex nn0ssz (NN0) (V) (ZZ) ssdomg mp2 (ZZ) (NN0) sbth mp2an nn0ennn entr)) thm (qnnen ((x y) (x z) (w x) (y z) (w y) (w z)) () (br (QQ) (~~) (NN)) (({<<,>,>|} x y z (/\ (/\ (e. (cv x) (ZZ)) (e. (cv y) (NN))) (= (cv z) (opr (cv x) (/) (cv y))))) (X. (ZZ) (NN)) (QQ) df-fo (cv x) (/) (cv y) oprex ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (ZZ)) (e. (cv y) (NN))) (= (cv z) (opr (cv x) (/) (cv y))))) eqid fnoprab2 (cv x) (/) (cv y) oprex ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (ZZ)) (e. (cv y) (NN))) (= (cv z) (opr (cv x) (/) (cv y))))) eqid (cv w) elrnoprab (cv w) x y elq bitr4 eqriv mpbir2an zex nnex xpex ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (ZZ)) (e. (cv y) (NN))) (= (cv z) (opr (cv x) (/) (cv y))))) (QQ) fodom ax-mp znnen nnex enref zex nnex nnex nnex xpen mp2an xpnnen entr (QQ) (X. (ZZ) (NN)) (NN) domentr mp2an nnex nnssq (NN) (V) (QQ) ssdomg mp2 (QQ) (NN) sbth mp2an)) thm (unbenlem ((m n) (m z) (m w) (m v) (m u) (m t) (A m) (n z) (n w) (n v) (n u) (n t) (A n) (w z) (v z) (u z) (t z) (A z) (v w) (u w) (t w) (A w) (u v) (t v) (A v) (t u) (A u) (A t) (G m) (G n) (G v) (G u) (G t)) ((unbenlem.1 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))))) (-> (/\ (C_ A (NN)) (A.e. m (NN) (E.e. n A (br (cv m) (<) (cv n))))) (br A (~~) (om))) (A (" (`' G) A) (om) entrt A (V) (|` (`' G) A) (" (`' G) A) f1oeng nnex A ssex 1z unbenlem.1 t om2uzf1o t nnzrab (NN) ({e.|} t (ZZ) (br (1) (<_) (cv t))) G (om) f1oeq3 ax-mp mpbir G (om) (NN) f1ocnv ax-mp (`' G) (NN) (om) f1of1 ax-mp (`' G) (NN) (om) A f1ores mpan sylc (A.e. m (NN) (E.e. n A (br (cv m) (<) (cv n)))) adantr 1z unbenlem.1 (cv v) t om2uzuz t nnzrab syl6eleqr (cv m) (` G (cv v)) (<) (cv n) breq1 n A rexbidv (NN) rcla4v syl (C_ A (NN)) adantr (cv u) (" (`' G) A) G fvres (cv n) eqeq1d biimpa (e. (cv v) (om)) adantll 1z unbenlem.1 (cv v) (cv u) om2uzlt2 (`' G) A imassrn G dfdm4 1z unbenlem.1 t om2uzf1o t nnzrab (NN) ({e.|} t (ZZ) (br (1) (<_) (cv t))) G (om) f1oeq3 ax-mp mpbir G (om) (NN) f1of ax-mp G (om) (NN) fdm ax-mp eqtr3 sseqtr (cv u) sseli sylan2 (` G (cv u)) (cv n) (` G (cv v)) (<) breq2 sylan9bb syldan biimparc exp44 imp31 r19.22dva 1z unbenlem.1 t om2uzf1o t nnzrab (NN) ({e.|} t (ZZ) (br (1) (<_) (cv t))) G (om) f1oeq3 ax-mp mpbir G (om) (NN) f1ocnv ax-mp (`' G) (NN) (om) f1of1 ax-mp (`' G) (NN) (om) A f1ores mpan (|` (`' G) A) A (" (`' G) A) f1ocnv syl 1z unbenlem.1 t om2uzf1o t nnzrab (NN) ({e.|} t (ZZ) (br (1) (<_) (cv t))) G (om) f1oeq3 ax-mp mpbir G (om) (NN) f1ofun ax-mp G A funcnvres2 ax-mp (`' (|` (`' G) A)) (|` G (" (`' G) A)) (" (`' G) A) A f1oeq1 ax-mp sylib (|` G (" (`' G) A)) (" (`' G) A) A f1ofo (|` G (" (`' G) A)) (" (`' G) A) A forn syl (cv n) eleq2d (|` G (" (`' G) A)) (" (`' G) A) A f1ofn (|` G (" (`' G) A)) (" (`' G) A) (cv n) u fvelrn syl bitr3d syl biimpa syl5 exp4b com4l imp r19.23adv syld ex com3l imp r19.21aiv (`' G) A imassrn G dfdm4 1z unbenlem.1 t om2uzf1o t nnzrab (NN) ({e.|} t (ZZ) (br (1) (<_) (cv t))) G (om) f1oeq3 ax-mp mpbir G (om) (NN) f1of ax-mp G (om) (NN) fdm ax-mp eqtr3 sseqtr jctil (" (`' G) A) v u unbnnt syl sylanc)) thm (unben ((m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y)) () (-> (/\ (C_ A (NN)) (A.e. m (NN) (E.e. n A (br (cv m) (<) (cv n))))) (br A (~~) (NN))) ((|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) (1)) (om)) eqid A m n unbenlem omex nnenom ensymi A (om) (NN) entrt mpan2 syl)) thm (infpnlem1 ((N j) (M j) (K j)) ((infpnlem.1 (= K (opr (` (!) N) (+) (1))))) (-> (/\ (e. N (NN)) (e. M (NN))) (-> (/\ (/\ (br (1) (<) M) (e. (opr K (/) M) (NN))) (A.e. j (NN) (-> (/\ (br (1) (<) (cv j)) (e. (opr K (/) (cv j)) (NN))) (br M (<_) (cv j))))) (/\ (br N (<) M) (A.e. j (NN) (-> (e. (opr M (/) (cv j)) (NN)) (\/ (= (cv j) (1)) (= (cv j) M))))))) (M N lenltt M nnret N nnret syl2an ancoms (br (1) (<) M) adantr N M facndivt (opr K (/) M) nnzt infpnlem.1 (/) M opreq1i syl5eqelr nsyl N nnnn0t sylanl1 exp32 imp sylbird a3d ex imp3a (A.e. j (NN) (-> (/\ (br (1) (<) (cv j)) (e. (opr K (/) (cv j)) (NN))) (br M (<_) (cv j)))) adantrd (cv j) M letri3t (cv j) nnret M nnret syl2an biimprd exp4b com3l imp32 (e. N (NN)) adantll (/\ (br (1) (<) (cv j)) (e. (opr K (/) (cv j)) (NN))) imim2d com23 (cv j) M K nndivtrt ex 3com13 3expa N nnnn0t N facclt syl (` (!) N) peano2nn syl infpnlem.1 syl5eqel K nncnt syl sylanl1 (br (cv j) (<_) M) adantrl sylan2d exp4d com24 exp32 com24 imp31 com14 3imp com3l r19.20dva ex (br (1) (<) M) adantld imp3a M j primet (e. N (NN)) adantl sylibrd jcad)) thm (infpnlem2 ((j k) (N j) (N k) (K j) (K k)) ((infpnlem.1 (= K (opr (` (!) N) (+) (1))))) (-> (e. N (NN)) (E.e. j (NN) (/\ (br N (<) (cv j)) (A.e. k (NN) (-> (e. (opr (cv j) (/) (cv k)) (NN)) (\/ (= (cv k) (1)) (= (cv k) (cv j)))))))) ((cv j) K (1) (<) breq2 (cv j) K K (/) opreq2 (NN) eleq1d anbi12d (NN) rcla4ev N nnnn0t N facclt syl (` (!) N) peano2nn syl infpnlem.1 syl5eqel N nnnn0t N facclt syl (` (!) N) nnge1t syl N nnnn0t N facclt syl 1nn jctil (1) (` (!) N) nnleltp1t syl mpbid infpnlem.1 syl6breqr N nnnn0t N facclt syl (` (!) N) peano2nn syl infpnlem.1 syl5eqel K nncnt K nnne0t jca syl K dividt syl 1nn syl6eqel jca sylanc (cv j) (cv k) (1) (<) breq2 (cv j) (cv k) K (/) opreq2 (NN) eleq1d anbi12d nnwos syl infpnlem.1 (cv j) k infpnlem1 r19.22dva mpd)) thm (infpn ((j k) (N j) (N k)) () (-> (e. N (NN)) (E.e. j (NN) (/\ (br N (<) (cv j)) (A.e. k (NN) (-> (e. (opr (cv j) (/) (cv k)) (NN)) (\/ (= (cv k) (1)) (= (cv k) (cv j)))))))) ((opr (` (!) N) (+) (1)) eqid j k infpnlem2)) thm (infpn2 ((j k) (j n) (j m) (k n) (k m) (m n) (S j) (S k)) ((infpn2.1 (= S ({e.|} n (NN) (/\ (br (1) (<) (cv n)) (A.e. m (NN) (-> (e. (opr (cv n) (/) (cv m)) (NN)) (\/ (= (cv m) (1)) (= (cv m) (cv n)))))))))) (br S (~~) (NN)) (infpn2.1 n (NN) (/\ (br (1) (<) (cv n)) (A.e. m (NN) (-> (e. (opr (cv n) (/) (cv m)) (NN)) (\/ (= (cv m) (1)) (= (cv m) (cv n)))))) ssrab2 eqsstr (cv j) k m infpn (cv j) nnge1t (e. (cv k) (NN)) adantr ax1re (1) (cv j) (cv k) lelttrt mp3an1 (cv j) nnret (cv k) nnret syl2an mpand ancld (A.e. m (NN) (-> (e. (opr (cv k) (/) (cv m)) (NN)) (\/ (= (cv m) (1)) (= (cv m) (cv k))))) anim1d (br (cv j) (<) (cv k)) (br (1) (<) (cv k)) (A.e. m (NN) (-> (e. (opr (cv k) (/) (cv m)) (NN)) (\/ (= (cv m) (1)) (= (cv m) (cv k))))) anass syl6ib r19.22dva mpd infpn2.1 (cv k) eleq2i (cv n) (cv k) (1) (<) breq2 (cv n) (cv k) (/) (cv m) opreq1 (NN) eleq1d (cv n) (cv k) (cv m) eqeq2 (= (cv m) (1)) orbi2d imbi12d m (NN) ralbidv anbi12d (NN) elrab bitr (br (cv j) (<) (cv k)) anbi1i (e. (cv k) (NN)) (/\ (br (1) (<) (cv k)) (A.e. m (NN) (-> (e. (opr (cv k) (/) (cv m)) (NN)) (\/ (= (cv m) (1)) (= (cv m) (cv k)))))) (br (cv j) (<) (cv k)) anass (/\ (br (1) (<) (cv k)) (A.e. m (NN) (-> (e. (opr (cv k) (/) (cv m)) (NN)) (\/ (= (cv m) (1)) (= (cv m) (cv k)))))) (br (cv j) (<) (cv k)) ancom (e. (cv k) (NN)) anbi2i 3bitr rexbii2 sylibr rgen S j k unben mp2an)) thm (ruclem1 () ((ruclem1.a (e. A (RR))) (ruclem1.b (e. B (RR)))) (<-> (br A (<) B) (br A (<) (opr (opr (opr (2) (x.) A) (+) B) (/) (3)))) (ruclem1.a ruclem1.b 2re ruclem1.a remulcl ltadd2 df-3 (x.) A opreq1i 2cn 1cn ruclem1.a recn adddir ruclem1.a recn mulid2 (opr (2) (x.) A) (+) opreq2i 3eqtrr (<) (opr (opr (2) (x.) A) (+) B) breq1i bitr 3re ruclem1.a remulcl 2re ruclem1.a remulcl ruclem1.b readdcl 3re 3pos ltdiv1i 3re recn ruclem1.a recn 3re 3pos gt0ne0i divcan3 (<) (opr (opr (opr (2) (x.) A) (+) B) (/) (3)) breq1i 3bitr)) thm (ruclem2 () ((ruclem1.a (e. A (RR))) (ruclem1.b (e. B (RR)))) (<-> (br A (<) B) (br (opr (opr (opr (2) (x.) A) (+) B) (/) (3)) (<) (opr (opr A (+) (opr (2) (x.) B)) (/) (3)))) (ruclem1.a ruclem1.b ruclem1.a ruclem1.b readdcl ltadd1 ruclem1.a recn 2times (+) B opreq1i ruclem1.a recn ruclem1.a recn ruclem1.b recn addass eqtr ruclem1.b recn 2times A (+) opreq2i ruclem1.a recn ruclem1.b recn ruclem1.b recn add12 eqtr (<) breq12i 2re ruclem1.a remulcl ruclem1.b readdcl ruclem1.a 2re ruclem1.b remulcl readdcl 3re 3pos ltdiv1i 3bitr2)) thm (ruclem3 () ((ruclem1.a (e. A (RR))) (ruclem1.b (e. B (RR)))) (<-> (br A (<) B) (br (opr (opr A (+) (opr (2) (x.) B)) (/) (3)) (<) B)) (ruclem1.a ruclem1.b 2re ruclem1.b remulcl ltadd1 df-3 2cn 1cn addcom eqtr (x.) B opreq1i 1cn 2cn ruclem1.b recn adddir ruclem1.b recn mulid2 (+) (opr (2) (x.) B) opreq1i 3eqtrr (opr A (+) (opr (2) (x.) B)) (<) breq2i bitr ruclem1.a 2re ruclem1.b remulcl readdcl 3re ruclem1.b remulcl 3re 3pos ltdiv1i 3re recn ruclem1.b recn 3re 3pos gt0ne0i divcan3 (opr (opr A (+) (opr (2) (x.) B)) (/) (3)) (<) breq2i 3bitr)) thm (ruclem4 () () (-> (/\ (= A C) (= B D)) (= (if (/\ (br (` (1st) A) (<) B) (br B (<) (` (2nd) A))) (<,> (opr (opr (opr (2) (x.) B) (+) (` (2nd) A)) (/) (3)) (opr (opr B (+) (opr (2) (x.) (` (2nd) A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) A)) (+) (` (2nd) A)) (/) (3)) (opr (opr (` (1st) A) (+) (opr (2) (x.) (` (2nd) A))) (/) (3)))) (if (/\ (br (` (1st) C) (<) D) (br D (<) (` (2nd) C))) (<,> (opr (opr (opr (2) (x.) D) (+) (` (2nd) C)) (/) (3)) (opr (opr D (+) (opr (2) (x.) (` (2nd) C))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) C)) (+) (` (2nd) C)) (/) (3)) (opr (opr (` (1st) C) (+) (opr (2) (x.) (` (2nd) C))) (/) (3)))))) (A C (1st) fveq2 (<) B breq1d A C (2nd) fveq2 B (<) breq2d anbi12d (<,> (opr (opr (opr (2) (x.) B) (+) (` (2nd) A)) (/) (3)) (opr (opr B (+) (opr (2) (x.) (` (2nd) A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) A)) (+) (` (2nd) A)) (/) (3)) (opr (opr (` (1st) A) (+) (opr (2) (x.) (` (2nd) A))) (/) (3))) ifbid (<,> (opr (opr (opr (2) (x.) B) (+) (` (2nd) A)) (/) (3)) (opr (opr B (+) (opr (2) (x.) (` (2nd) A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) B) (+) (` (2nd) C)) (/) (3)) (opr (opr B (+) (opr (2) (x.) (` (2nd) C))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) A)) (+) (` (2nd) A)) (/) (3)) (opr (opr (` (1st) A) (+) (opr (2) (x.) (` (2nd) A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) C)) (+) (` (2nd) C)) (/) (3)) (opr (opr (` (1st) C) (+) (opr (2) (x.) (` (2nd) C))) (/) (3))) (/\ (br (` (1st) C) (<) B) (br B (<) (` (2nd) C))) ifeq12 (opr (opr (opr (2) (x.) B) (+) (` (2nd) A)) (/) (3)) (opr (opr (opr (2) (x.) B) (+) (` (2nd) C)) (/) (3)) (opr (opr B (+) (opr (2) (x.) (` (2nd) A))) (/) (3)) (opr (opr B (+) (opr (2) (x.) (` (2nd) C))) (/) (3)) opeq12 A C (2nd) fveq2 (opr (2) (x.) B) (+) opreq2d (/) (3) opreq1d A C (2nd) fveq2 (2) (x.) opreq2d B (+) opreq2d (/) (3) opreq1d sylanc (opr (opr (opr (2) (x.) (` (1st) A)) (+) (` (2nd) A)) (/) (3)) (opr (opr (opr (2) (x.) (` (1st) C)) (+) (` (2nd) C)) (/) (3)) (opr (opr (` (1st) A) (+) (opr (2) (x.) (` (2nd) A))) (/) (3)) (opr (opr (` (1st) C) (+) (opr (2) (x.) (` (2nd) C))) (/) (3)) opeq12 A C (1st) fveq2 (2) (x.) opreq2d A C (2nd) fveq2 (+) opreq12d (/) (3) opreq1d A C (1st) fveq2 A C (2nd) fveq2 (2) (x.) opreq2d (+) opreq12d (/) (3) opreq1d sylanc sylanc eqtrd B D (` (1st) C) (<) breq2 B D (<) (` (2nd) C) breq1 anbi12d (<,> (opr (opr (opr (2) (x.) B) (+) (` (2nd) C)) (/) (3)) (opr (opr B (+) (opr (2) (x.) (` (2nd) C))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) C)) (+) (` (2nd) C)) (/) (3)) (opr (opr (` (1st) C) (+) (opr (2) (x.) (` (2nd) C))) (/) (3))) ifbid (opr (opr (opr (2) (x.) B) (+) (` (2nd) C)) (/) (3)) (opr (opr (opr (2) (x.) D) (+) (` (2nd) C)) (/) (3)) (opr (opr B (+) (opr (2) (x.) (` (2nd) C))) (/) (3)) (opr (opr D (+) (opr (2) (x.) (` (2nd) C))) (/) (3)) opeq12 B D (2) (x.) opreq2 (+) (` (2nd) C) opreq1d (/) (3) opreq1d B D (+) (opr (2) (x.) (` (2nd) C)) opreq1 (/) (3) opreq1d sylanc (/\ (br (` (1st) C) (<) D) (br D (<) (` (2nd) C))) (<,> (opr (opr (opr (2) (x.) (` (1st) C)) (+) (` (2nd) C)) (/) (3)) (opr (opr (` (1st) C) (+) (opr (2) (x.) (` (2nd) C))) (/) (3))) ifeq1d eqtrd sylan9eq)) thm (ruclem5 () ((ruclem5.0 (:--> F (NN) (RR))) (ruclem5.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))))) (e. C (V)) (ruclem5.1 (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2)))) snex ruclem5.0 nnex F (NN) (RR) (V) fex mp2an F (V) (\ (NN) ({} (1))) resexg ax-mp unex eqeltr)) thm (ruclem6 () ((ruclem5.0 (:--> F (NN) (RR))) (ruclem5.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))))) (= (|` C (\ (NN) ({} (1)))) (|` F (\ (NN) ({} (1))))) (ruclem5.1 C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))) (\ (NN) ({} (1))) reseq1 ax-mp ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) resundir ({} (1)) (NN) difdisj 1nn elisseti (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2)) opex f1osn ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) ({} (1)) ({} (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2)))) f1ofn ax-mp ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) ({} (1)) (\ (NN) ({} (1))) fnresdisj ax-mp mpbi F (\ (NN) ({} (1))) residm uneq12i ({/}) (|` F (\ (NN) ({} (1)))) uncom (|` F (\ (NN) ({} (1)))) un0 3eqtr 3eqtr)) thm (ruclem7 () ((ruclem5.0 (:--> F (NN) (RR))) (ruclem5.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))))) (= (` C (1)) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2)))) (1nn elisseti snid (1) ({} (1)) C fvres ax-mp ruclem5.1 C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))) ({} (1)) reseq1 ax-mp ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))) ({} (1)) resundir (\ (NN) ({} (1))) ({} (1)) incom ({} (1)) (NN) difdisj eqtr (\ (NN) ({} (1))) ({} (1)) F resdisj ax-mp (|` ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) ({} (1))) uneq2i (|` ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) ({} (1))) un0 eqtr 3eqtr (1) fveq1i 1nn elisseti snid (1) ({} (1)) ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) fvres ax-mp 1nn elisseti (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2)) opex fvsn 3eqtr eqtr3)) thm (ruclem8 () ((ruclem5.0 (:--> F (NN) (RR))) (ruclem5.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem8.a (e. A (NN)))) (= (` C (opr A (+) (1))) (` F (opr A (+) (1)))) (ruclem8.a nngt0 0re ruclem8.a nnre ax1re ltadd1 1cn addid2 (<) (opr A (+) (1)) breq1i bitr mpbi ruclem8.a A peano2nn ax-mp (opr A (+) (1)) nngt1ne1t ax-mp A (+) (1) oprex (1) elsnc negbii bitr4 biimp ruclem8.a A peano2nn ax-mp jctil (opr A (+) (1)) (NN) ({} (1)) eldif sylibr (opr A (+) (1)) (\ (NN) ({} (1))) C fvres (opr A (+) (1)) (\ (NN) ({} (1))) F fvres ruclem5.0 ruclem5.1 ruclem6 (opr A (+) (1)) fveq1i syl5eq eqtr3d syl ax-mp)) thm (ruclem9 ((x y) (x z) (y z) (A z)) ((ruclem9.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) A)))))) (e. D (V)) (reex reex xpex reex ruclem9.2 oprabex2)) thm (ruclem10 () ((ruclem10.a (e. A (NN))) (ruclem.d (e. D (V))) (ruclem.c (e. C (V))) (ruclem10.3 (= G (o. (1st) (opr D (seq1) C))))) (= (` (1st) (` (opr D (seq1) C) A)) (` G A)) (ruclem10.3 A fveq1i fo1st (1st) (V) (V) fof ax-mp (1st) (V) (V) ffun ax-mp ruclem.d ruclem.c seq1fn ruclem10.a (1st) (opr D (seq1) C) (NN) A fvco2 mp3an eqtr2)) thm (ruclem11 () ((ruclem11.a (e. A (NN))) (ruclem11.d (e. D (V))) (ruclem11.c (e. C (V))) (ruclem11.4 (= H (o. (2nd) (opr D (seq1) C))))) (= (` (2nd) (` (opr D (seq1) C) A)) (` H A)) (ruclem11.4 A fveq1i fo2nd (2nd) (V) (V) fof ax-mp (2nd) (V) (V) ffun ax-mp ruclem11.d ruclem11.c seq1fn ruclem11.a (2nd) (opr D (seq1) C) (NN) A fvco2 mp3an eqtr2)) thm (ruclem12 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) ((ruclem12.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))))) (= D ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (X. (RR) (RR))) (e. (cv v) (RR))) (= (cv u) (if (/\ (br (` (1st) (cv w)) (<) (cv v)) (br (cv v) (<) (` (2nd) (cv w)))) (<,> (opr (opr (opr (2) (x.) (cv v)) (+) (` (2nd) (cv w))) (/) (3)) (opr (opr (cv v) (+) (opr (2) (x.) (` (2nd) (cv w)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv w))) (+) (` (2nd) (cv w))) (/) (3)) (opr (opr (` (1st) (cv w)) (+) (opr (2) (x.) (` (2nd) (cv w)))) (/) (3)))))))) (ruclem12.2 (cv x) (cv w) (X. (RR) (RR)) eleq1 (cv y) (cv v) (RR) eleq1 bi2anan9 (cv x) (cv w) (cv y) (cv v) ruclem4 (cv z) eqeq2d anbi12d z cbvoprab12v (cv z) (cv u) (if (/\ (br (` (1st) (cv w)) (<) (cv v)) (br (cv v) (<) (` (2nd) (cv w)))) (<,> (opr (opr (opr (2) (x.) (cv v)) (+) (` (2nd) (cv w))) (/) (3)) (opr (opr (cv v) (+) (opr (2) (x.) (` (2nd) (cv w)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv w))) (+) (` (2nd) (cv w))) (/) (3)) (opr (opr (` (1st) (cv w)) (+) (opr (2) (x.) (` (2nd) (cv w)))) (/) (3)))) eqeq1 (/\ (e. (cv w) (X. (RR) (RR))) (e. (cv v) (RR))) anbi2d w v cbvoprab3v 3eqtr)) thm (ruclem13 ((x y) (x z) (y z)) ((ruclem13.0 (:--> F (NN) (RR))) (ruclem13.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem13.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))))) (:--> (opr D (seq1) C) (NN) (X. (RR) (RR))) ((opr D (seq1) C) (NN) (X. (RR) (RR)) df-f ruclem13.2 ruclem9 ruclem13.0 ruclem13.1 ruclem5 seq1fn ruclem13.0 ruclem13.1 ruclem7 (` F (1)) (+) (2) oprex (opr (` F (1)) (+) (1)) (RR) (RR) opelxp ruclem13.0 1nn F (NN) (RR) (1) ffvrn mp2an ax1re readdcl ruclem13.0 1nn F (NN) (RR) (1) ffvrn mp2an 2re readdcl mpbir2an eqeltr ruclem13.0 (NN) ({} (1)) difss F (NN) (RR) (\ (NN) ({} (1))) fssres mp2an ruclem13.0 ruclem13.1 ruclem6 (|` C (\ (NN) ({} (1)))) (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) (RR) feq1 ax-mp mpbir (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) iftrue (X. (RR) (RR)) eleq1d (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (RR) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)) (RR) opelxpi (opr (2) (x.) (cv y)) (` (2nd) (cv x)) axaddrcl 2re (2) (cv y) axmulrcl mpan sylan 3re 3re 3pos gt0ne0i (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (3) redivclt mp3an23 syl (cv y) (opr (2) (x.) (` (2nd) (cv x))) axaddrcl 2re (2) (` (2nd) (cv x)) axmulrcl mpan sylan2 3re 3re 3pos gt0ne0i (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (3) redivclt mp3an23 syl sylanc ancoms (e. (` (1st) (cv x)) (RR)) adantll syl5bir (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) iffalse (X. (RR) (RR)) eleq1d (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (RR) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)) (RR) opelxpi (opr (2) (x.) (` (1st) (cv x))) (` (2nd) (cv x)) axaddrcl 2re (2) (` (1st) (cv x)) axmulrcl mpan sylan 3re 3re 3pos gt0ne0i (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (3) redivclt mp3an23 syl (` (1st) (cv x)) (opr (2) (x.) (` (2nd) (cv x))) axaddrcl 2re (2) (` (2nd) (cv x)) axmulrcl mpan sylan2 3re 3re 3pos gt0ne0i (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (3) redivclt mp3an23 syl sylanc syl5bir (e. (cv y) (RR)) adantrd pm2.61i (= (cv x) (<,> (` (1st) (cv x)) (` (2nd) (cv x)))) adantll (cv x) (RR) (RR) elxp6 sylanb rgen2a ruclem13.2 (X. (RR) (RR)) foprab2 mpbi ruclem13.2 ruclem9 ruclem13.0 ruclem13.1 ruclem5 (X. (RR) (RR)) (RR) seq1rn mp3an mpbir2an)) thm (ruclem14 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C))))) (= (` (opr D (seq1) C) (1)) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2)))) (ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 seq11 ruclem.0 ruclem.1 ruclem7 eqtr)) thm (ruclem15 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A w) (A v) (A u) (C w) (C v) (C u) (D w) (D v) (D u) (G w) (G v) (G u) (H w) (H v) (H u) (F z) (F w) (F v) (F u)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem15.a (e. A (NN)))) (= (` (opr D (seq1) C) (opr A (+) (1))) (if (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (<,> (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3))))) (ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 A seq1p1 ax-mp ruclem.0 ruclem.1 ruclem.2 ruclem13 ruclem15.a (opr D (seq1) C) (NN) (X. (RR) (RR)) A ffvrn mp2an ruclem.0 ruclem.1 ruclem15.a ruclem8 ruclem.0 ruclem15.a A peano2nn ax-mp F (NN) (RR) (opr A (+) (1)) ffvrn mp2an eqeltr (opr (opr (opr (2) (x.) (` C (opr A (+) (1)))) (+) (` (2nd) (` (opr D (seq1) C) A))) (/) (3)) (opr (opr (` C (opr A (+) (1))) (+) (opr (2) (x.) (` (2nd) (` (opr D (seq1) C) A)))) (/) (3)) opex (opr (opr (opr (2) (x.) (` (1st) (` (opr D (seq1) C) A))) (+) (` (2nd) (` (opr D (seq1) C) A))) (/) (3)) (opr (opr (` (1st) (` (opr D (seq1) C) A)) (+) (opr (2) (x.) (` (2nd) (` (opr D (seq1) C) A)))) (/) (3)) opex (/\ (br (` (1st) (` (opr D (seq1) C) A)) (<) (` C (opr A (+) (1)))) (br (` C (opr A (+) (1))) (<) (` (2nd) (` (opr D (seq1) C) A)))) ifex (cv v) eqid (cv w) (` (opr D (seq1) C) A) (cv v) (cv v) ruclem4 mpan2 (` (opr D (seq1) C) A) eqid (` (opr D (seq1) C) A) (` (opr D (seq1) C) A) (cv v) (` C (opr A (+) (1))) ruclem4 mpan ruclem.2 w v u ruclem12 oprabval2 mp2an ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 ruclem.0 ruclem.1 ruclem15.a ruclem8 (<) breq12i ruclem.0 ruclem.1 ruclem15.a ruclem8 ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 (<) breq12i anbi12i (/\ (br (` (1st) (` (opr D (seq1) C) A)) (<) (` C (opr A (+) (1)))) (br (` C (opr A (+) (1))) (<) (` (2nd) (` (opr D (seq1) C) A)))) (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (<,> (opr (opr (opr (2) (x.) (` C (opr A (+) (1)))) (+) (` (2nd) (` (opr D (seq1) C) A))) (/) (3)) (opr (opr (` C (opr A (+) (1))) (+) (opr (2) (x.) (` (2nd) (` (opr D (seq1) C) A)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (` (opr D (seq1) C) A))) (+) (` (2nd) (` (opr D (seq1) C) A))) (/) (3)) (opr (opr (` (1st) (` (opr D (seq1) C) A)) (+) (opr (2) (x.) (` (2nd) (` (opr D (seq1) C) A)))) (/) (3))) ifbi ax-mp ruclem.0 ruclem.1 ruclem15.a ruclem8 (2) (x.) opreq2i ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 (+) opreq12i (/) (3) opreq1i ruclem.0 ruclem.1 ruclem15.a ruclem8 ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 (2) (x.) opreq2i (+) opreq12i (/) (3) opreq1i (opr (opr (opr (2) (x.) (` C (opr A (+) (1)))) (+) (` (2nd) (` (opr D (seq1) C) A))) (/) (3)) (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)) (opr (opr (` C (opr A (+) (1))) (+) (opr (2) (x.) (` (2nd) (` (opr D (seq1) C) A)))) (/) (3)) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3)) opeq12 mp2an ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 (2) (x.) opreq2i ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 (+) opreq12i (/) (3) opreq1i ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 (2) (x.) opreq2i (+) opreq12i (/) (3) opreq1i (opr (opr (opr (2) (x.) (` (1st) (` (opr D (seq1) C) A))) (+) (` (2nd) (` (opr D (seq1) C) A))) (/) (3)) (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)) (opr (opr (` (1st) (` (opr D (seq1) C) A)) (+) (opr (2) (x.) (` (2nd) (` (opr D (seq1) C) A)))) (/) (3)) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3)) opeq12 mp2an (<,> (opr (opr (opr (2) (x.) (` C (opr A (+) (1)))) (+) (` (2nd) (` (opr D (seq1) C) A))) (/) (3)) (opr (opr (` C (opr A (+) (1))) (+) (opr (2) (x.) (` (2nd) (` (opr D (seq1) C) A)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (` (opr D (seq1) C) A))) (+) (` (2nd) (` (opr D (seq1) C) A))) (/) (3)) (opr (opr (` (1st) (` (opr D (seq1) C) A)) (+) (opr (2) (x.) (` (2nd) (` (opr D (seq1) C) A)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3))) (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) ifeq12 mp2an eqtr 3eqtr)) thm (ruclem16 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C))))) (= (` G (1)) (opr (` F (1)) (+) (1))) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem14 (1st) fveq2i 1nn ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 (` F (1)) (+) (1) oprex (opr (` F (1)) (+) (2)) op1st 3eqtr3)) thm (ruclem17 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C))))) (:--> G (NN) (RR)) ((RR) (RR) f1stres ruclem.0 ruclem.1 ruclem.2 ruclem13 (|` (1st) (X. (RR) (RR))) (X. (RR) (RR)) (RR) (opr D (seq1) C) (NN) fco mp2an ruclem.0 ruclem.1 ruclem.2 ruclem13 (opr D (seq1) C) (NN) (X. (RR) (RR)) frn ax-mp (opr D (seq1) C) (X. (RR) (RR)) (1st) cores ax-mp ruclem.3 eqtr4 (o. (|` (1st) (X. (RR) (RR))) (opr D (seq1) C)) G (NN) (RR) feq1 ax-mp mpbi)) thm (ruclem18 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (-> (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (= (` G (opr A (+) (1))) (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)))) ((/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (<,> (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3))) iftrue ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem15 syl5eq (1st) fveq2d ruclem18.a A peano2nn ax-mp ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3) oprex (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3)) op1st 3eqtr3g)) thm (ruclem19 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (-> (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (= (` H (opr A (+) (1))) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3)))) ((/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (<,> (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3))) iftrue ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem15 syl5eq (2nd) fveq2d ruclem18.a A peano2nn ax-mp ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3) oprex (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3) oprex op2nd 3eqtr3g)) thm (ruclem20 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (-> (-. (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A)))) (= (` G (opr A (+) (1))) (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)))) ((/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (<,> (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3))) iffalse ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem15 syl5eq (1st) fveq2d ruclem18.a A peano2nn ax-mp ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3) oprex (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3)) op1st 3eqtr3g)) thm (ruclem21 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (-> (-. (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A)))) (= (` H (opr A (+) (1))) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3)))) ((/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (<,> (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3))) iffalse ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem15 syl5eq (2nd) fveq2d ruclem18.a A peano2nn ax-mp ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3) oprex (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3) oprex op2nd 3eqtr3g)) thm (ruclem22 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (e. (` G A) (RR)) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 ruclem18.a G (NN) (RR) A ffvrn mp2an)) thm (ruclem23 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (e. (` H A) (RR)) (ruclem18.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 ruclem.0 ruclem.1 ruclem.2 ruclem13 ruclem18.a (opr D (seq1) C) (NN) (X. (RR) (RR)) A ffvrn mp2an (` (opr D (seq1) C) A) (RR) (RR) elxp6 pm3.27bd ax-mp pm3.27i eqeltrr)) thm (ruclem24 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (-> (br (` G A) (<) (` H A)) (br (` G (opr A (+) (1))) (<) (` H (opr A (+) (1))))) (ruclem.0 ruclem18.a A peano2nn ax-mp F (NN) (RR) (opr A (+) (1)) ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem2 biimp (br (` G A) (<) (` F (opr A (+) (1)))) adantl ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem18 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem19 3brtr4d (br (` G A) (<) (` H A)) a1d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem20 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem21 (<) breq12d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem2 syl6rbbr biimpd pm2.61i)) thm (ruclem25 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (br (` G A) (<) (` H A)) (ruclem18.a (cv w) (1) G fveq2 (cv w) (1) H fveq2 (<) breq12d (cv w) (cv v) G fveq2 (cv w) (cv v) H fveq2 (<) breq12d (cv w) (opr (cv v) (+) (1)) G fveq2 (cv w) (opr (cv v) (+) (1)) H fveq2 (<) breq12d (cv w) A G fveq2 (cv w) A H fveq2 (<) breq12d 1lt2 ax1re 2re ruclem.0 1nn F (NN) (RR) (1) ffvrn mp2an ltadd2 mpbi ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem16 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem14 (2nd) fveq2i 1nn ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 (` F (1)) (+) (1) oprex (` F (1)) (+) (2) oprex op2nd 3eqtr3 3brtr4 (cv v) (if (e. (cv v) (NN)) (cv v) (1)) G fveq2 (cv v) (if (e. (cv v) (NN)) (cv v) (1)) H fveq2 (<) breq12d (cv v) (if (e. (cv v) (NN)) (cv v) (1)) (+) (1) opreq1 G fveq2d (cv v) (if (e. (cv v) (NN)) (cv v) (1)) (+) (1) opreq1 H fveq2d (<) breq12d imbi12d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn (cv v) elimel ruclem24 dedth nnind ax-mp)) thm (ruclem26 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (br (` G A) (<) (` G (opr A (+) (1)))) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem22 ruclem.0 ruclem18.a A peano2nn ax-mp F (NN) (RR) (opr A (+) (1)) ffvrn mp2an 2re ruclem.0 ruclem18.a A peano2nn ax-mp F (NN) (RR) (opr A (+) (1)) ffvrn mp2an remulcl ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 readdcl 3re 3re 3pos gt0ne0i redivcl lttr ruclem.0 ruclem18.a A peano2nn ax-mp F (NN) (RR) (opr A (+) (1)) ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem1 sylan2b ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem18 breqtrrd ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem20 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem25 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem1 mpbi syl5breqr pm2.61i)) thm (ruclem27 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (br (` H (opr A (+) (1))) (<) (` H A)) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem19 ruclem.0 ruclem18.a A peano2nn ax-mp F (NN) (RR) (opr A (+) (1)) ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem3 biimp (br (` G A) (<) (` F (opr A (+) (1)))) adantl eqbrtrd ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem21 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem25 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem3 mpbi syl6eqbr pm2.61i)) thm (ruclem28 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem28.a (e. A (NN)))) (-. (/\ (br (` G (opr A (+) (1))) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H (opr A (+) (1)))))) (ruclem.0 ruclem28.a A peano2nn ax-mp F (NN) (RR) (opr A (+) (1)) ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem23 ruclem1 biimp (br (` G A) (<) (` F (opr A (+) (1)))) adantl ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem18 breqtrrd ruclem.0 ruclem28.a A peano2nn ax-mp F (NN) (RR) (opr A (+) (1)) ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a A peano2nn ax-mp ruclem22 ltnsym syl (br (` F (opr A (+) (1))) (<) (` H (opr A (+) (1)))) intnanrd ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem26 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a A peano2nn ax-mp ruclem22 ruclem.0 ruclem28.a A peano2nn ax-mp F (NN) (RR) (opr A (+) (1)) ffvrn mp2an lttr mpan ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem27 ruclem.0 ruclem28.a A peano2nn ax-mp F (NN) (RR) (opr A (+) (1)) ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a A peano2nn ax-mp ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem23 lttr mpan2 anim12i con3i pm2.61i)) thm (ruclem29 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem28.a (e. A (NN)))) (-. (/\ (br (` G A) (<) (` F A)) (br (` F A) (<) (` H A)))) (ruclem28.a (cv w) (1) G fveq2 (cv w) (1) F fveq2 (<) breq12d (cv w) (1) F fveq2 (cv w) (1) H fveq2 (<) breq12d anbi12d negbid (cv w) (opr (cv v) (+) (1)) G fveq2 (cv w) (opr (cv v) (+) (1)) F fveq2 (<) breq12d (cv w) (opr (cv v) (+) (1)) F fveq2 (cv w) (opr (cv v) (+) (1)) H fveq2 (<) breq12d anbi12d negbid (cv w) A G fveq2 (cv w) A F fveq2 (<) breq12d (cv w) A F fveq2 (cv w) A H fveq2 (<) breq12d anbi12d negbid ruclem.0 1nn F (NN) (RR) (1) ffvrn mp2an ltp1 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem16 breqtrr ruclem.0 1nn F (NN) (RR) (1) ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn ruclem22 ltnsym ax-mp (br (` F (1)) (<) (` H (1))) intnanr (cv v) (if (e. (cv v) (NN)) (cv v) (1)) (+) (1) opreq1 G fveq2d (cv v) (if (e. (cv v) (NN)) (cv v) (1)) (+) (1) opreq1 F fveq2d (<) breq12d (cv v) (if (e. (cv v) (NN)) (cv v) (1)) (+) (1) opreq1 F fveq2d (cv v) (if (e. (cv v) (NN)) (cv v) (1)) (+) (1) opreq1 H fveq2d (<) breq12d anbi12d negbid ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn (cv v) elimel ruclem28 dedth nn1suc ax-mp)) thm (ruclem30 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem28.a (e. A (NN))) (ruclem.b (e. B (NN)))) (-> (br (` G A) (<) (` G (opr A (+) B))) (br (` G A) (<) (` G (opr A (+) (opr B (+) (1)))))) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem26 ruclem28.a nncn ruclem.b nncn 1cn addass G fveq2i breqtr ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b B peano2nn ax-mp A (opr B (+) (1)) nnaddclt mp2an ruclem22 lttr mpan2)) thm (ruclem31 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem28.a (e. A (NN))) (ruclem.b (e. B (NN)))) (-> (br (` H (opr A (+) B)) (<) (` H B)) (br (` H (opr (opr A (+) (1)) (+) B)) (<) (` H B))) (ruclem28.a nncn 1cn ruclem.b nncn add23 H fveq2i ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem27 eqbrtr ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a A peano2nn ax-mp ruclem.b (opr A (+) (1)) B nnaddclt mp2an ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.b ruclem23 lttr mpan)) thm (ruclem32 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v) (B w) (B v)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem28.a (e. A (NN))) (ruclem.b (e. B (NN)))) (br (` G A) (<) (` H B)) (ruclem.b (cv w) (1) A (+) opreq2 G fveq2d (` G A) (<) breq2d (cv w) (cv v) A (+) opreq2 G fveq2d (` G A) (<) breq2d (cv w) (opr (cv v) (+) (1)) A (+) opreq2 G fveq2d (` G A) (<) breq2d (cv w) B A (+) opreq2 G fveq2d (` G A) (<) breq2d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem26 (cv v) (if (e. (cv v) (NN)) (cv v) (1)) A (+) opreq2 G fveq2d (` G A) (<) breq2d (cv v) (if (e. (cv v) (NN)) (cv v) (1)) (+) (1) opreq1 A (+) opreq2d G fveq2d (` G A) (<) breq2d imbi12d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a 1nn (cv v) elimel ruclem30 dedth nnind ax-mp ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem25 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem23 lttr mp2an ruclem28.a (cv w) (1) (+) B opreq1 H fveq2d (<) (` H B) breq1d (cv w) (cv v) (+) B opreq1 H fveq2d (<) (` H B) breq1d (cv w) (opr (cv v) (+) (1)) (+) B opreq1 H fveq2d (<) (` H B) breq1d (cv w) A (+) B opreq1 H fveq2d (<) (` H B) breq1d 1cn ruclem.b nncn addcom H fveq2i ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.b ruclem27 eqbrtr (cv v) (if (e. (cv v) (NN)) (cv v) (1)) (+) B opreq1 H fveq2d (<) (` H B) breq1d (cv v) (if (e. (cv v) (NN)) (cv v) (1)) (+) (1) opreq1 (+) B opreq1d H fveq2d (<) (` H B) breq1d imbi12d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn (cv v) elimel ruclem.b ruclem31 dedth nnind ax-mp ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.b ruclem23 lttr mp2an)) thm (ruclem33 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C))))) (/\/\ (C_ (ran G) (RR)) (-. (= (ran G) ({/}))) (E.e. w (RR) (A.e. v (ran G) (br (cv v) (<_) (cv w))))) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 G (NN) (RR) frn ax-mp ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 G (NN) (RR) ffn ax-mp 1nn G (NN) (1) fnfvrn mp2an (` G (1)) (ran G) n0i ax-mp ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 G (NN) (RR) ffn ax-mp G (NN) (cv v) w fvelrn ax-mp (` G (cv w)) (cv v) (<_) (` H (1)) breq1 (` G (cv w)) (` H (1)) ltlet (cv w) (if (e. (cv w) (NN)) (cv w) (1)) G fveq2 (RR) eleq1d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn (cv w) elimel ruclem22 dedth ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn ruclem23 jctir (cv w) (if (e. (cv w) (NN)) (cv w) (1)) G fveq2 (<) (` H (1)) breq1d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn (cv w) elimel 1nn ruclem32 dedth sylc syl5bi com12 r19.23aiv sylbi rgen (cv w) (` H (1)) (cv v) (<_) breq2 v (ran G) ralbidv (RR) rcla4ev mp2an 3pm3.2i)) thm (ruclem34 ((S w) (x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem.5 (= S (sup (ran G) (RR) (<))))) (e. S (RR)) (ruclem.5 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 w v ruclem33 suprcli eqeltr)) thm (ruclem35 ((S w) (x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A w) (A v) (A u) (C w) (C v) (C u) (D w) (D v) (D u) (G w) (G v) (G u) (H w) (H v) (H u) (F z) (F w) (F v) (F u)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem.5 (= S (sup (ran G) (RR) (<)))) (ruclem.a (e. A (NN)))) (/\ (br (` G A) (<) S) (br S (<) (` H A))) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a ruclem26 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 G (NN) (RR) ffn ax-mp ruclem.a A peano2nn ax-mp G (NN) (opr A (+) (1)) fnfvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 w v ruclem33 (` G (opr A (+) (1))) suprubi ax-mp ruclem.5 breqtrr ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a A peano2nn ax-mp ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 ruclem34 ltletr mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a A peano2nn ax-mp ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 G (NN) (RR) ffn ax-mp G (NN) (cv u) w fvelrn ax-mp (` G (cv w)) (cv u) (` H (opr A (+) (1))) (<) breq2 negbid (` G (cv w)) (` H (opr A (+) (1))) ltnsymt (cv w) (if (e. (cv w) (NN)) (cv w) (1)) G fveq2 (RR) eleq1d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn (cv w) elimel ruclem22 dedth ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a A peano2nn ax-mp ruclem23 jctir (cv w) (if (e. (cv w) (NN)) (cv w) (1)) G fveq2 (<) (` H (opr A (+) (1))) breq1d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn (cv w) elimel ruclem.a A peano2nn ax-mp ruclem32 dedth sylc syl5bi com12 r19.23aiv sylbi rgen ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 w v ruclem33 (` H (opr A (+) (1))) u suprnubi mp2an ruclem.5 (` H (opr A (+) (1))) (<) breq2i mtbir ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 ruclem34 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a A peano2nn ax-mp ruclem23 lenlt mpbir ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a ruclem27 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 ruclem34 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a A peano2nn ax-mp ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a ruclem23 lelttr mp2an pm3.2i)) thm (ruclem36 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem.5 (= S (sup (ran G) (RR) (<)))) (ruclem.a (e. A (NN)))) (-. (= (` F A) S)) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a ruclem29 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 ruclem.a ruclem35 (` F A) S (` G A) (<) breq2 (` F A) S (<) (` H A) breq1 anbi12d mpbiri mto)) thm (ruclem37 ((S w) (x y) (x z) (w x) (y z) (w y) (w z) (C w) (D w) (G w) (H w) (F z) (F w)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem.5 (= S (sup (ran G) (RR) (<))))) (-. (:-onto-> F (NN) (RR))) ((cv w) (if (e. (cv w) (NN)) (cv w) (1)) F fveq2 S eqeq1d negbid ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 1nn (cv w) elimel ruclem36 dedth nrex ruclem.0 F (NN) (RR) ffn ax-mp F (NN) S w fvelrn ax-mp mtbir F (NN) (RR) forn ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 ruclem34 syl5eleqr mto)) thm (ruclem38 ((F z) (x y) (x z) (y z)) ((ruclem38.0 (:--> F (NN) (RR)))) (-. (:-onto-> F (NN) (RR))) (ruclem38.0 (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))) eqid ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))) eqid (o. (1st) (opr ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))) (seq1) (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) eqid (o. (2nd) (opr ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))) (seq1) (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) eqid (sup (ran (o. (1st) (opr ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))) (seq1) (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))))) (RR) (<)) eqid ruclem37)) thm (ruclem39 () () (-. (:-onto-> F (NN) (RR))) (F (NN) (RR) fof F (if (:--> F (NN) (RR)) F (X. (NN) ({} (0)))) (NN) (RR) foeq1 negbid F (if (:--> F (NN) (RR)) F (X. (NN) ({} (0)))) (NN) (RR) feq1 (X. (NN) ({} (0))) (if (:--> F (NN) (RR)) F (X. (NN) ({} (0)))) (NN) (RR) feq1 0re elisseti (NN) fconst 0re (0) (RR) snssi ax-mp (X. (NN) ({} (0))) (NN) ({} (0)) (RR) fss mp2an elimhyp ruclem38 dedth syl (:-onto-> F (NN) (RR)) pm2.01 ax-mp)) thm (ruc () () (br (NN) (~<) (RR)) ((NN) (RR) brsdom reex nnssre (RR) (V) (NN) ssdom2g mp2 (cv f) ruclem39 (cv f) (NN) (RR) f1ofo mto f nex reex (NN) f bren mtbir mpbir2an)) thm (resdomq () () (br (QQ) (~<) (RR)) (qnnen ruc (QQ) (NN) (RR) ensdomtr mp2an)) thm (aleph1re () () (br (` (aleph) (1o)) (~<_) (RR)) (aleph0 omex nnenom ensymi eqbrtr ruc (` (aleph) ({/})) (NN) (RR) ensdomtr mp2an ({/}) (RR) alephnbtwn2 (br (` (aleph) ({/})) (~<) (RR)) (br (RR) (~<) (` (aleph) (suc ({/})))) imnan mpbir ax-mp df-1o (aleph) fveq2i (RR) (~<) breq2i mtbir (aleph) (1o) fvex reex (` (aleph) (1o)) (V) (RR) (V) domtri mp2an mpbir)) thm (infxpidmlem1 () ((infxpidmlem1.1 (e. A (V))) (infxpidmlem1.2 (e. B (V)))) (-> (/\ (br (om) (~<_) (cv x)) (br (cv x) (~~) (X. (cv x) (cv x)))) (-> (/\ (br (cv x) (~~) A) (br (cv x) (~~) B)) (br (cv x) (~~) (u. A B)))) ((cv x) (u. A B) sbth infxpidmlem1.1 A B ssun1 A (V) (u. A B) ssdomg mp2 (cv x) A (u. A B) endomtr mpan2 (/\ (br (1o) (~<) (cv x)) (br (cv x) (~~) (X. (cv x) (cv x)))) (br (cv x) (~~) B) ad2antrl (u. A B) (X. A B) (cv x) domentr A B unxpdom infxpidmlem1.1 A (V) (1o) (cv x) sdomentr ax-mp infxpidmlem1.2 B (V) (1o) (cv x) sdomentr ax-mp syl2an anandis (br (cv x) (~~) (X. (cv x) (cv x))) adantlr (cv x) (X. (cv x) (cv x)) (X. A B) entrt infxpidmlem1.1 infxpidmlem1.2 xpex (cv x) ensym syl x visset infxpidmlem1.1 x visset infxpidmlem1.2 xpen sylan2 (br (1o) (~<) (cv x)) adantll sylanc sylanc ex 1onn x visset (1o) infsdomnn mpan2 sylan)) thm (infxpidmlem2 ((f x) (t x) (A x) (f t) (A f) (A t) (B x) (B f) (B t) (H x)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem2.2 (e. B (V)))) (<-> (e. B H) (\/ (= B ({/})) (E. x (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> B (X. (cv x) (cv x)) (cv x)))))) (infxpidmlem2.2 (cv f) B ({/}) eqeq1 (cv f) B (X. (cv t) (cv t)) (cv t) f1oeq1 (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) anbi2d t exbidv orbi12d infxpidmlem.1 elab2 (cv x) (cv t) (om) (~<_) breq2 (cv x) (cv t) A sseq1 anbi12d (cv x) (cv t) (cv x) xpeq1 (cv x) (cv t) (cv t) xpeq2 eqtrd (X. (cv x) (cv x)) (X. (cv t) (cv t)) B (cv x) f1oeq2 syl (cv x) (cv t) B (X. (cv t) (cv t)) f1oeq3 bitrd anbi12d cbvexv (= B ({/})) orbi2i bitr4)) thm (infxpidmlem3 ((D x) (f x) (t x) (A x) (f t) (A f) (A t) (B x) (B f) (B t) (H x)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem2.2 (e. B (V))) (infxpidmlem3.3 (e. D (V)))) (-> (/\ (/\ (br (om) (~<_) D) (C_ D A)) (:-1-1-onto-> B (X. D D) D)) (e. B H)) (infxpidmlem3.3 (cv x) D (om) (~<_) breq2 (cv x) D A sseq1 anbi12d (cv x) D (cv x) xpeq1 (cv x) D D xpeq2 eqtrd (X. (cv x) (cv x)) (X. D D) B (cv x) f1oeq2 syl (cv x) D B (X. D D) f1oeq3 bitrd anbi12d cla4ev (E. x (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> B (X. (cv x) (cv x)) (cv x)))) (= B ({/})) olc infxpidmlem.1 infxpidmlem2.2 x infxpidmlem2 sylibr syl)) thm (infxpidmlem4 ((f x) (g x) (t x) (A x) (f g) (f t) (A f) (g t) (A g) (A t) (H x) (H g)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))))) (-> (e. (cv g) H) (= (dom (cv g)) (X. (ran (cv g)) (ran (cv g))))) (infxpidmlem.1 g visset x infxpidmlem2 (cv g) ({/}) dmeq dm0 syl6eq (cv g) ({/}) rneq rn0 syl6eq (ran (cv g)) ({/}) (ran (cv g)) xpeq2 syl (ran (cv g)) xp0 syl6eq eqtr4d (cv g) (X. (cv x) (cv x)) (cv x) f1o2 (cv g) (X. (cv x) (cv x)) fndm (ran (cv g)) (cv x) (ran (cv g)) xpeq1 (ran (cv g)) (cv x) (cv x) xpeq2 eqtr2d sylan9eq (Fun (`' (cv g))) 3adant2 sylbi (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) adantl x 19.23aiv jaoi sylbi)) thm (infxpidmlem5 ((y z) (f y) (g y) (t y) (A y) (f z) (g z) (t z) (A z) (f g) (f t) (A f) (g t) (A g) (A t) (C y) (C z) (C f) (C g) (C t) (H y) (H z) (H g)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))))) (-> (/\ (C_ C H) (e. (cv g) C)) (-> (/\ (e. (cv y) (ran (cv g))) (e. (cv z) (ran (cv g)))) (e. (<,> (cv y) (cv z)) (dom (U. C))))) (C H (cv g) ssel2 infxpidmlem.1 g infxpidmlem4 syl (<,> (cv y) (cv z)) eleq2d z visset (cv y) (ran (cv g)) (ran (cv g)) opelxp syl6bb g C (dom (cv g)) ssiun2 C g dmuni syl6ssr (<,> (cv y) (cv z)) sseld (C_ C H) adantl sylbird)) thm (infxpidmlem6 ((f y) (g y) (t y) (A y) (f g) (f t) (A f) (g t) (A g) (A t) (B y) (B f) (B g) (B t) (C y) (C f) (C g) (C t) (H y) (H g)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem6.2 (= B (ran (U. C))))) (<-> (e. (cv y) B) (E.e. g C (e. (cv y) (ran (cv g))))) (infxpidmlem6.2 C g rnuni eqtr (cv y) eleq2i (cv y) g C (ran (cv g)) eliun bitr)) thm (infxpidmlem7 ((x y) (x z) (w x) (f x) (g x) (h x) (v x) (t x) (A x) (y z) (w y) (f y) (g y) (h y) (v y) (t y) (A y) (w z) (f z) (g z) (h z) (v z) (t z) (A z) (f w) (g w) (h w) (v w) (t w) (A w) (f g) (f h) (f v) (f t) (A f) (g h) (g v) (g t) (A g) (h v) (h t) (A h) (t v) (A v) (A t) (B x) (B y) (B z) (B w) (B v) (B f) (B g) (B h) (B t) (C x) (C y) (C z) (C w) (C v) (C f) (C g) (C h) (C t) (H x) (H y) (H z) (H w) (H v) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem6.2 (= B (ran (U. C))))) (-> (/\ (C_ C H) (A.e. g C (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g)))))) (:-1-1-onto-> (U. C) (X. B B) B)) (g C (/\ (Fun (cv g)) (Fun (`' (cv g)))) (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g)))) r19.26 g C h fun11uni sylbir C H (cv g) ssel infxpidmlem.1 g visset x infxpidmlem2 ({/}) f10 (cv g) ({/}) ({/}) ({/}) f1eq1 mpbiri (cv g) ({/}) ({/}) df-f1 (cv g) ({/}) ({/}) ffun (Fun (`' (cv g))) anim1i sylbi syl (cv g) (X. (cv x) (cv x)) (cv x) f1of1 (cv g) (X. (cv x) (cv x)) (cv x) df-f1 (cv g) (X. (cv x) (cv x)) (cv x) ffun (Fun (`' (cv g))) anim1i sylbi syl (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) adantl x 19.23aiv jaoi sylbi syl6 r19.21aiv sylan C H (cv g) ssel2 infxpidmlem.1 g infxpidmlem4 syl g C (e. (cv y) (ran (cv g))) ra4e infxpidmlem.1 infxpidmlem6.2 y g infxpidmlem6 sylibr ex ssrdv (ran (cv g)) B (ran (cv g)) B ssxp anidms syl (C_ C H) adantl eqsstrd (cv y) sseld ex r19.23adv C g dmuni (cv y) eleq2i (cv y) g C (dom (cv g)) eliun bitr syl5ib ssrdv (A.e. g C (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g))))) adantr B B relxp (/\ (C_ C H) (A.e. g C (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g)))))) a1i (cv g) (cv w) (cv h) sseq1 (cv g) (cv w) (cv h) sseq2 orbi12d (cv h) (cv v) (cv w) sseq2 (cv h) (cv v) (cv w) sseq1 orbi12d C C rcla42v (cv w) (cv v) rnss (cv y) sseld (e. (cv z) (ran (cv v))) anim1d infxpidmlem.1 C v y z infxpidmlem5 syl9r (e. (cv w) C) adantrl (cv v) (cv w) rnss (cv z) sseld (e. (cv y) (ran (cv w))) anim2d infxpidmlem.1 C w y z infxpidmlem5 syl9r (e. (cv v) C) adantrr jaod ex com3l syld com13 imp r19.23advv infxpidmlem.1 infxpidmlem6.2 y w infxpidmlem6 infxpidmlem.1 infxpidmlem6.2 z v infxpidmlem6 anbi12i z visset (cv y) B B opelxp w C v C (e. (cv y) (ran (cv w))) (e. (cv z) (ran (cv v))) reeanv 3bitr4 syl5ib relssdv eqssd jca (Fun (U. C)) (Fun (`' (U. C))) (= (dom (U. C)) (X. B B)) an23 (U. C) (X. B B) df-fn (Fun (`' (U. C))) anbi1i bitr4 sylib infxpidmlem6.2 eqcomi jctir (U. C) (X. B B) B f1o2 (Fn (U. C) (X. B B)) (Fun (`' (U. C))) (= (ran (U. C)) B) df-3an bitr sylibr)) thm (infxpidmlem8 ((x y) (f x) (g x) (h x) (t x) (A x) (f y) (g y) (h y) (t y) (A y) (f g) (f h) (f t) (A f) (g h) (g t) (A g) (h t) (A h) (A t) (B x) (B y) (B f) (B g) (B h) (B t) (C x) (C y) (C f) (C g) (C h) (C t) (H x) (H y) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem6.2 (= B (ran (U. C)))) (infxpidmlem8.3 (e. C (V)))) (-> (/\ (C_ C H) (A.e. g C (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g)))))) (e. (U. C) H)) (C H (cv g) ssel2 infxpidmlem.1 g visset x infxpidmlem2 biimp ord (cv g) (X. (cv x) (cv x)) (cv x) f1ofo (cv g) (X. (cv x) (cv x)) (cv x) forn syl eqcomd (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) anim1i ancoms x 19.22i g visset (cv g) (V) rnexg ax-mp (cv x) (ran (cv g)) (om) (~<_) breq2 (cv x) (ran (cv g)) A sseq1 anbi12d ceqsexv sylib syl6 syl (om) (ran (cv g)) B domtr g C (e. (cv y) (ran (cv g))) ra4e infxpidmlem.1 infxpidmlem6.2 y g infxpidmlem6 sylibr ex ssrdv g visset (cv g) (V) rnexg ax-mp (ran (cv g)) (V) B ssdomg ax-mp syl sylan2 expcom (C_ C H) adantl (C_ (ran (cv g)) A) adantrd syld ex r19.23adv C uni0b C ({} ({/})) g dfss3 g ({/}) elsn g C ralbii 3bitr negbii g C (= (cv g) ({/})) rexnal bitr4 syl5ib C H (cv g) ssel2 infxpidmlem.1 g visset x infxpidmlem2 biimp ord (cv g) (X. (cv x) (cv x)) (cv x) f1ofo (cv g) (X. (cv x) (cv x)) (cv x) forn syl eqcomd (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) anim1i ancoms x 19.22i g visset (cv g) (V) rnexg ax-mp (cv x) (ran (cv g)) (om) (~<_) breq2 (cv x) (ran (cv g)) A sseq1 anbi12d ceqsexv sylib syl6 syl (br (om) (~<_) (ran (cv g))) (C_ (ran (cv g)) A) pm3.27 syl6 (cv g) ({/}) rneq rn0 syl6eq A 0ss (= (cv g) ({/})) a1i eqsstrd pm2.61d2 (cv y) sseld ex r19.23adv infxpidmlem.1 infxpidmlem6.2 y g infxpidmlem6 syl5ib ssrdv jctird (A.e. g C (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g))))) adantr infxpidmlem.1 infxpidmlem6.2 g h infxpidmlem7 jctird infxpidmlem.1 infxpidmlem8.3 uniex infxpidmlem6.2 infxpidmlem8.3 uniex (U. C) (V) rnexg ax-mp eqeltr infxpidmlem3 syl6 (= (U. C) ({/})) (E. x (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> (U. C) (X. (cv x) (cv x)) (cv x)))) orc infxpidmlem.1 infxpidmlem8.3 uniex x infxpidmlem2 sylibr pm2.61d2)) thm (infxpidmlem9 ((f z) (g z) (h z) (t z) (A z) (f g) (f h) (f t) (A f) (g h) (g t) (A g) (h t) (A h) (A t) (H z) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem.2 (e. A (V)))) (E.e. g H (A.e. h H (-. (C: (cv g) (cv h))))) (infxpidmlem.1 f (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))) unab eqtr4 ({/}) f df-sn p0ex eqeltrr t (P~ A) (/\ (br (om) (~<_) (cv t)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))) df-rex t visset A elpw (/\ (br (om) (~<_) (cv t)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))) anbi1i (C_ (cv t) A) (/\ (br (om) (~<_) (cv t)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))) ancom (br (om) (~<_) (cv t)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)) (C_ (cv t) A) an23 3bitr t exbii bitr f abbii infxpidmlem.2 pwex t visset t visset xpex t visset (X. (cv t) (cv t)) (V) (cv t) (V) f mapex mp2an (cv f) (X. (cv t) (cv t)) (cv t) f1of (br (om) (~<_) (cv t)) adantl f ss2abi ssexi t abrexex2 eqeltrr unex eqeltr z g h zorn2 infxpidmlem.1 (ran (U. (cv z))) eqid z visset g h infxpidmlem8 mpg)) thm (infxpidmlem10 ((f y) (g y) (h y) (v y) (t y) (A y) (f g) (f h) (f v) (f t) (A f) (g h) (g v) (g t) (A g) (h v) (h t) (A h) (t v) (A v) (A t) (H y) (H v) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem.2 (e. A (V)))) (-> (A.e. h H (-. (C: (cv g) (cv h)))) (-> (br (om) (~<_) A) (-. (= (cv g) ({/}))))) ((cv g) ({/}) (cv v) psseq1 (cv v) 0pss syl6bb con2bid (cv h) (cv v) (cv g) psseq2 negbid H rcla4cva syl5bir com12 con3d ex r19.23adv infxpidmlem.1 v visset y visset infxpidmlem3 ex (cv v) (X. (cv y) (cv y)) (cv y) f1ofo (cv v) (X. (cv y) (cv y)) (cv y) forn syl eqcomd (cv v) ({/}) rneq rn0 syl6eq sylan9eq ex y visset infn0 nsyli com12 (C_ (cv y) A) adantr jcad v 19.22dv imp an1rs (om) (cv y) endom omex y visset omex y visset xpen anidms xpomen y visset y visset xpex (X. (cv y) (cv y)) (V) (X. (om) (om)) (om) enen1 mpan mpbii syl y visset (cv y) (V) (om) (X. (cv y) (cv y)) enen2 mpan mpbid y visset (X. (cv y) (cv y)) v bren sylib jca sylan y 19.23aiv infxpidmlem.2 (om) y domen v H (-. (= (cv v) ({/}))) df-rex 3imtr4 syl5)) thm (infxpidmlem11 ((x y) (f x) (g x) (h x) (u x) (t x) (A x) (f y) (g y) (h y) (u y) (t y) (A y) (f g) (f h) (f u) (f t) (A f) (g h) (g u) (g t) (A g) (h u) (h t) (A h) (t u) (A u) (A t) (H x) (H y) (H u) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem.2 (e. A (V)))) (-> (/\ (/\ (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> (cv g) (X. (cv x) (cv x)) (cv x))) (/\ (br (cv x) (~~) (cv y)) (C_ (cv y) (\ A (cv x))))) (:-1-1-onto-> (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y))) (E.e. h H (C: (cv g) (cv h)))) ((cv h) (u. (cv g) (cv u)) (cv g) psseq2 H rcla4ev (cv g) (X. (cv x) (cv x)) (cv x) (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y) f1oun (cv x) (cv y) (cv x) (cv x) xpdisj2 (cv x) (cv y) (cv x) (cv x) xpdisj1 (cv x) (cv y) (cv x) (cv y) xpdisj1 jca (X. (cv x) (cv x)) (X. (cv y) (cv x)) (X. (cv y) (cv y)) undisj2 sylib jca (X. (cv x) (cv x)) (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y))) undisj2 sylib ancri sylan2 (cv x) (cv y) (cv x) (cv y) xpun (X. (cv x) (cv x)) (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y))) unass eqtr (X. (u. (cv x) (cv y)) (u. (cv x) (cv y))) (u. (X. (cv x) (cv x)) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y))))) (u. (cv g) (cv u)) (u. (cv x) (cv y)) f1oeq2 ax-mp sylibr (cv y) (\ A (cv x)) (cv x) sslin (cv x) A difdisj syl6ss (i^i (cv x) (cv y)) ss0b sylib sylan2 (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) adantll infxpidmlem.1 g visset u visset unex x visset y visset unex infxpidmlem3 x visset (cv x) (cv y) ssun1 (cv x) (V) (u. (cv x) (cv y)) ssdomg mp2 (om) (cv x) (u. (cv x) (cv y)) domtr mpan2 A (cv x) difss (cv y) (\ A (cv x)) A sstr mpan2 (C_ (cv x) A) anim2i (cv x) A (cv y) unss sylib anim12i anassrs sylan ex (/\ (:-1-1-onto-> (cv g) (X. (cv x) (cv x)) (cv x)) (:-1-1-onto-> (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y))) adantlr mpd (br (cv x) (~~) (cv y)) adantrl (cv g) (cv u) disjpss (ran (cv g)) (cv x) (ran (cv u)) (cv y) ineq12 ({/}) eqeq1d (cv g) (X. (cv x) (cv x)) (cv x) f1ofo (cv g) (X. (cv x) (cv x)) (cv x) forn syl (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y) f1ofo (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y) forn syl syl2an (cv g) (X. (cv x) (cv x)) (cv x) f1orel (cv g) (cv u) relin1 (i^i (cv g) (cv u)) relrn0 (cv g) (cv u) rnin (i^i (ran (cv g)) (ran (cv u))) ({/}) (ran (i^i (cv g) (cv u))) sseq2 mpbii (ran (i^i (cv g) (cv u))) ss0 syl syl5bir 3syl (:-1-1-onto-> (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y)) adantr sylbird imp (/\ (br (om) (~<_) (cv x)) (br (cv x) (~~) (cv y))) adantr (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y) f1orel (cv u) relrn0 syl (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y) f1ofo (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y) forn syl ({/}) eqeq1d bitrd negbid y visset infn0 syl5bir (om) (cv x) (cv y) domentr syl5 imp (:-1-1-onto-> (cv g) (X. (cv x) (cv x)) (cv x)) adantll (= (i^i (cv x) (cv y)) ({/})) adantlr sylanc exp43 (cv y) (\ A (cv x)) (cv x) sslin (cv x) A difdisj syl6ss (i^i (cv x) (cv y)) ss0b sylib syl5 com4t com23 (C_ (cv x) A) adantr imp43 sylanc exp42 com34 imp41)) thm (infxpidmlem12 ((x y) (f x) (g x) (h x) (u x) (t x) (A x) (f y) (g y) (h y) (u y) (t y) (A y) (f g) (f h) (f u) (f t) (A f) (g h) (g u) (g t) (A g) (h u) (h t) (A h) (t u) (A u) (A t) (H x) (H y) (H u) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem.2 (e. A (V)))) (-> (br (om) (~<_) A) (br (X. A A) (~~) A)) (infxpidmlem.1 infxpidmlem.2 g h infxpidmlem9 infxpidmlem.1 infxpidmlem.2 h g infxpidmlem10 infxpidmlem.1 g visset x infxpidmlem2 (= (cv g) ({/})) (E. x (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> (cv g) (X. (cv x) (cv x)) (cv x)))) df-or bitr (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv x) (cv y) entrt (cv x) (X. (cv x) (cv x)) (X. (cv x) (cv y)) entrt x visset enref x visset x visset x visset y visset xpen mpan sylan2 (br (om) (~<_) (cv x)) adantll (cv x) (X. (cv x) (cv x)) (X. (cv y) (cv x)) entrt x visset enref x visset y visset x visset x visset xpen mpan2 sylan2 (cv x) (X. (cv x) (cv x)) (X. (cv y) (cv y)) entrt x visset y visset x visset y visset xpen anidms sylan2 jca (br (om) (~<_) (cv x)) adantll y visset x visset xpex y visset y visset xpex x infxpidmlem1 (br (cv x) (~~) (cv y)) adantr mpd x visset y visset xpex y visset x visset xpex y visset y visset xpex unex x infxpidmlem1 (br (cv x) (~~) (cv y)) adantr mp2and x visset y visset xpex y visset x visset xpex y visset y visset xpex unex unex (cv x) ensym syl (/\ (br (om) (~<_) (cv x)) (br (cv x) (~~) (X. (cv x) (cv x)))) (br (cv x) (~~) (cv y)) pm3.27 sylanc x visset (X. (cv x) (cv x)) ensym sylanl2 y visset (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) u bren sylib x visset x visset xpex (cv g) (cv x) f1oen (br (om) (~<_) (cv x)) anim2i (C_ (cv x) A) adantlr sylan (C_ (cv y) (\ A (cv x))) adantrr infxpidmlem.1 infxpidmlem.2 x g y u h infxpidmlem11 ex u 19.23adv mpd ex y 19.23adv infxpidmlem.2 A (V) (cv x) difexg ax-mp (cv x) y domen syl5ib x visset infxpidmlem.2 A (V) (cv x) difexg ax-mp (cv x) (V) (\ A (cv x)) (V) domtri mp2an h H (C: (cv g) (cv h)) dfrex2 3imtr3g a3d A (cv x) sbth A (X. (cv x) (cv x)) (cv x) domentr x visset infxpidmlem.2 A (V) (cv x) difexg ax-mp unxpdom2 infxpidmlem.2 A (cv x) ssun2 A (V) (u. (cv x) A) ssdomg mp2 (cv x) A undif2 breqtrr A (u. (cv x) (\ A (cv x))) (X. (cv x) (cv x)) domtr mpan syl 1onn x visset (1o) infsdomnn mpan2 (\ A (cv x)) (cv x) sdomdom syl2an sylan x visset (cv x) (V) A ssdomg ax-mp syl2an (X. A A) (cv x) A entrt (X. A A) (X. (cv x) (cv x)) (cv x) entrt infxpidmlem.2 x visset infxpidmlem.2 x visset xpen anidms sylan x visset A ensym (br (X. (cv x) (cv x)) (~~) (cv x)) adantr sylanc expcom (/\ (br (om) (~<_) (cv x)) (br (\ A (cv x)) (~<) (cv x))) (C_ (cv x) A) ad2antlr mpd exp41 com24 imp31 x visset x visset xpex (cv g) (cv x) f1oen sylan2 syld x 19.23aiv (-. (= (cv g) ({/}))) imim2i sylbi com13 syld com3r r19.23aiv ax-mp)) thm (infxpidm ((f x) (A f) (A x)) ((infxpidm.1 (e. A (V)))) (-> (br (om) (~<_) A) (br (X. A A) (~~) A)) (({|} f (\/ (= (cv f) ({/})) (E. x (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> (cv f) (X. (cv x) (cv x)) (cv x)))))) eqid infxpidm.1 infxpidmlem12)) thm (infunabs () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (u. A B) (~~) A)) ((opr A (+c) B) (X. A A) A domentr (opr A (+c) B) (X. A (2o)) (X. A A) domtr infunabs.2 infunabs.1 infunabs.1 cdadom2 infunabs.1 xp2cda syl6breqr 2onn (2o) omsdomnn ax-mp pm3.26i (2o) (om) A domtr mpan infunabs.1 infunabs.1 (2o) xpdom2 syl syl2an infunabs.1 infxpidm (br B (~<_) A) adantl sylanc infunabs.1 infunabs.2 cdadom3 jctir (opr A (+c) B) A sbth syl ancoms infunabs.1 infunabs.2 uncdadom (u. A B) (opr A (+c) B) A domentr mpan syl infunabs.1 A B ssun1 A (V) (u. A B) ssdomg mp2 jctir (u. A B) A sbth syl)) thm (infcdaabs () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (opr A (+c) B) (~~) A)) ((opr A (+c) B) (X. A A) A domentr (opr A (+c) B) (X. A (2o)) (X. A A) domtr infunabs.2 infunabs.1 infunabs.1 cdadom2 infunabs.1 xp2cda syl6breqr 2onn (2o) omsdomnn ax-mp pm3.26i (2o) (om) A domtr mpan infunabs.1 infunabs.1 (2o) xpdom2 syl syl2an infunabs.1 infxpidm (br B (~<_) A) adantl sylanc infunabs.1 infunabs.2 cdadom3 jctir (opr A (+c) B) A sbth syl ancoms)) thm (infcda () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (br (om) (~<_) A) (br (opr A (+c) B) (~~) (u. A B))) (infunabs.2 infunabs.1 B (V) A (V) entri3 mp2an (opr A (+c) B) A (u. A B) entrt infunabs.1 infunabs.2 infcdaabs infunabs.1 infunabs.2 infunabs infunabs.1 (u. A B) ensym syl sylanc ex (opr B (+c) A) B (u. B A) entrt infunabs.2 infunabs.1 infcdaabs infunabs.2 infunabs.1 infunabs infunabs.2 (u. B A) ensym syl sylanc infunabs.1 infunabs.2 cdacomen (opr A (+c) B) (opr B (+c) A) (u. B A) entrt mpan syl B A uncom syl6breq (om) A B domtr sylan exp31 pm2.43d jaod mpi)) thm (infdif () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (/\ (br (om) (~<_) A) (br B (~<) A)) (br (\ A B) (~~) A)) (A (u. A B) (opr (\ A B) (+c) (\ A B)) endomtr infunabs.1 infunabs.2 infunabs infunabs.1 (u. A B) ensym syl B A sdomdom sylan2 omex infunabs.2 (om) (V) B (V) entri2 mp2an infunabs.1 A B ssun1 A (V) (u. A B) ssdomg mp2 infunabs.1 infunabs.1 infunabs.2 unex A (V) (u. A B) (V) domtri mp2an mpbi (u. A B) (opr B (+c) B) A domsdomtr infunabs.1 A (V) B difexg ax-mp infunabs.2 infunabs.2 cdadom1 infunabs.1 A (V) B difexg ax-mp infunabs.2 uncdadom (u. (\ A B) B) (opr (\ A B) (+c) B) (opr B (+c) B) domtr mpan syl A B undif1 syl5eqbrr (opr B (+c) B) B A ensdomtr infunabs.2 B (V) domrefg ax-mp infunabs.2 infunabs.2 infcdaabs mpan2 sylan ancoms syl2an expcom mtoi ex (br (om) (~<_) A) adantl (br (om) (~<_) A) (br B (~<) A) pm3.26 (br B (~<) (om)) adantr A (opr B (+c) B) (om) domsdomtr A (u. A B) (opr B (+c) B) endomtr infunabs.1 infunabs.2 infunabs infunabs.1 (u. A B) ensym syl B A sdomdom sylan2 infunabs.1 A (V) B difexg ax-mp infunabs.2 infunabs.2 cdadom1 infunabs.1 A (V) B difexg ax-mp infunabs.2 uncdadom (u. (\ A B) B) (opr (\ A B) (+c) B) (opr B (+c) B) domtr mpan syl A B undif1 syl5eqbrr syl2an B B cdafi anidms syl2an an1rs ex con3d omex infunabs.1 (om) (V) A (V) domtri mp2an syl5ib mpd ex jaod mpi infunabs.1 A (V) B difexg ax-mp infunabs.2 (\ A B) (V) B (V) domtri mp2an con2bii sylibr B (\ A B) sdomdom infunabs.2 infunabs.1 A (V) B difexg ax-mp infunabs.1 A (V) B difexg ax-mp cdadom2 A B undif1 infunabs.1 A (V) B difexg ax-mp infunabs.2 uncdadom eqbrtrr (u. A B) (opr (\ A B) (+c) B) (opr (\ A B) (+c) (\ A B)) domtr mpan syl 3syl sylanc (om) A (opr (\ A B) (+c) (\ A B)) domtr infunabs.1 A (V) B difexg ax-mp cdainf sylibr infunabs.1 A (V) B difexg ax-mp (\ A B) (V) domrefg ax-mp infunabs.1 A (V) B difexg ax-mp infunabs.1 A (V) B difexg ax-mp infcdaabs mpan2 syl A (opr (\ A B) (+c) (\ A B)) (\ A B) domentr ex (br (om) (~<_) A) adantl mpd syldan infunabs.1 A B difss A (V) (\ A B) ssdom2g mp2 jctil (\ A B) A sbth syl)) thm (infxpabs () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (/\ (/\ (br (om) (~<_) A) (-. (= B ({/})))) (br B (~<_) A)) (br (X. A B) (~~) A)) ((X. A B) A sbth (X. A B) (X. A A) A domentr infunabs.1 infunabs.1 B xpdom2 infunabs.1 infxpidm syl2an ancoms infunabs.1 B xpdom3 syl2an an1rs)) thm (infxpdom () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (X. A B) (~<_) A)) (B ({/}) A xpeq2 A xp0 syl6eq A 0dom syl6eqbr (/\ (br (om) (~<_) A) (br B (~<_) A)) adantl infunabs.1 infunabs.2 infxpabs (X. A B) A endom syl an1rs pm2.61dan)) thm (infmap1 () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (/\ (/\ (br (2o) (~<_) A) (br (om) (~<_) B)) (br A (~<_) B)) (br (opr A (^m) B) (~~) (opr (2o) (^m) B))) ((opr A (^m) B) (opr (2o) (^m) B) sbth (opr A (^m) B) (opr B (^m) B) (opr (2o) (^m) B) domtr infunabs.1 infunabs.2 infunabs.2 mapdom1 infunabs.2 infxpidm 2on elisseti enref 2on elisseti 2on elisseti infunabs.2 infunabs.2 xpex infunabs.2 mapen mpan infunabs.2 canth2 B (P~ B) sdomdom ax-mp infunabs.2 pw2en B (P~ B) (opr (2o) (^m) B) domentr mp2an infunabs.2 (2o) (^m) B oprex infunabs.2 mapdom1 ax-mp 2on elisseti infunabs.2 infunabs.2 mapxpen (opr B (^m) B) (opr (opr (2o) (^m) B) (^m) B) (opr (2o) (^m) (X. B B)) domentr mp2an (opr B (^m) B) (opr (2o) (^m) (X. B B)) (opr (2o) (^m) B) domentr mpan 3syl syl2an 2on elisseti infunabs.1 infunabs.2 mapdom1 syl2an ancom31s)) thm (iunctb ((A x)) ((iunctb.1 (e. A (V))) (iunctb.2 (e. B (V)))) (-> (/\ (br A (~<_) (om)) (A.e. x A (br B (~<_) (om)))) (br (U_ x A B) (~<_) (om))) ((U_ x A B) (X. A (om)) (om) domtr iunctb.1 omex iunctb.2 x iundom omex (om) (V) domrefg ax-mp omex iunctb.1 infxpdom mpan iunctb.1 omex xpex omex iunctb.1 xpcomen (X. A (om)) (V) (X. (om) A) (om) domen1 mp2an sylib syl2an ancoms)) thm (unictb ((A x)) ((unictb.1 (e. A (V)))) (-> (/\ (br A (~<_) (om)) (A.e. x A (br (cv x) (~<_) (om)))) (br (U. A) (~<_) (om))) (unictb.1 x visset x iunctb A x uniiun syl5eqbr)) thm (unctb ((A x) (B x)) () (-> (/\ (br A (~<_) (om)) (br B (~<_) (om))) (br (u. A B) (~<_) (om))) (A (V) B (V) uniprg reldom A (om) brrelexi reldom B (om) brrelexi syl2an (cv x) A (~<_) (om) breq1 biimprcd (cv x) B (~<_) (om) breq1 biimprcd jaao x visset A B elpr syl5ib r19.21aiv x A B prfi x ({,} A B) isfinite1 ax-mp pm3.26i A B prex x unictb mpan syl eqbrtrrd)) thm (aleph1irr () () (br (` (aleph) (1o)) (~<_) (\ (RR) (QQ))) (aleph1re reex omex nnenom ensymi ruc (om) (NN) (RR) ensdomtr mp2an (om) (RR) sdomdom ax-mp resdomq reex qex infdif mp2an ensymi (` (aleph) (1o)) (RR) (\ (RR) (QQ)) domentr mp2an)) thm (infmap2lem1 ((x z) (w x) (v x) (f x) (A x) (w z) (v z) (f z) (A z) (v w) (f w) (A w) (f v) (A v) (A f) (B x) (B z) (B w) (B f) (B v) (R v) (R f)) ((infmap2lem.1 (e. A (V))) (infmap2lem.2 (e. B (V))) (infmap2lem.3 (= R ({<,>|} z w (/\ (/\ (C_ (cv z) A) (br (cv z) (~~) B)) (:-onto-> (cv w) B (cv z))))))) (-> (/\ (C_ (cv f) R) (Fn (cv f) (dom R))) (-> (e. (cv v) ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B)))) (/\ (C_ (cv v) A) (:-onto-> (` (cv f) (cv v)) B (cv v))))) ((cv f) R (<,> (cv v) (` (cv f) (cv v))) ssel infmap2lem.3 (<,> (cv v) (` (cv f) (cv v))) eleq2i v visset (cv f) (cv v) fvex (cv z) (cv v) A sseq1 (cv z) (cv v) (~~) B breq1 anbi12d (cv z) (cv v) (cv w) B foeq3 anbi12d (cv w) (` (cv f) (cv v)) B (cv v) foeq1 (/\ (C_ (cv v) A) (br (cv v) (~~) B)) anbi2d opelopab bitr (C_ (cv v) A) (br (cv v) (~~) B) pm3.26 (:-onto-> (` (cv f) (cv v)) B (cv v)) anim1i sylbi syl6 (cv f) (dom R) (cv v) fnopfv syl5 exp3a imp infmap2lem.3 dmeqi z w (/\ (/\ (C_ (cv z) A) (br (cv z) (~~) B)) (:-onto-> (cv w) B (cv z))) dmopab (C_ (cv z) A) (br (cv z) (~~) B) (E. w (:-onto-> (cv w) B (cv z))) anass w (/\ (C_ (cv z) A) (br (cv z) (~~) B)) (:-onto-> (cv w) B (cv z)) 19.42v infmap2lem.2 (cv z) ensym z visset B w bren (cv w) B (cv z) f1ofo w 19.22i sylbi syl pm4.71i (C_ (cv z) A) anbi2i 3bitr4 z abbii 3eqtr (cv z) (cv x) A sseq1 (cv z) (cv x) (~~) B breq1 anbi12d cbvabv eqtr (cv v) eleq2i syl5ibr)) thm (infmap2lem2 ((x z) (w x) (v x) (u x) (f x) (A x) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B z) (B w) (B f) (B v) (B u) (R u) (R v) (R f)) ((infmap2lem.1 (e. A (V))) (infmap2lem.2 (e. B (V))) (infmap2lem.3 (= R ({<,>|} z w (/\ (/\ (C_ (cv z) A) (br (cv z) (~~) B)) (:-onto-> (cv w) B (cv z))))))) (br ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B))) (~<_) (opr A (^m) B)) (infmap2lem.3 (P~ A) (opr A (^m) B) z w df-xp infmap2lem.1 pwex A (^m) B oprex xpex eqeltrr (C_ (cv z) A) (:-onto-> (cv w) B (cv z)) pm3.26 z visset A elpw sylibr (cv w) B (cv z) fof (cv w) B (cv z) ffn syl (C_ (cv z) A) adantl (cv w) B (cv z) forn A sseq1d biimparc jca infmap2lem.1 infmap2lem.2 (cv w) elmap (cv w) B A df-f bitr sylibr jca (br (cv z) (~~) B) adantlr z w ssopab2i ssexi eqeltr R (V) f ac7g ax-mp A x df-pw infmap2lem.1 pwex eqeltrr (C_ (cv x) A) (br (cv x) (~~) B) pm3.26 x ss2abi ssexi infmap2lem.1 infmap2lem.2 infmap2lem.3 f v x infmap2lem1 (` (cv f) (cv v)) B (cv v) A fss (` (cv f) (cv v)) B (cv v) fof sylan ancoms infmap2lem.1 infmap2lem.2 (` (cv f) (cv v)) elmap sylibr syl6 infmap2lem.1 infmap2lem.2 infmap2lem.3 f v x infmap2lem1 (C_ (cv v) A) (:-onto-> (` (cv f) (cv v)) B (cv v)) pm3.27 syl6 infmap2lem.1 infmap2lem.2 infmap2lem.3 f u x infmap2lem1 (C_ (cv u) A) (:-onto-> (` (cv f) (cv u)) B (cv u)) pm3.27 syl6 anim12d (` (cv f) (cv v)) B (cv v) forn (` (cv f) (cv u)) B (cv u) forn eqeqan12d (` (cv f) (cv v)) (` (cv f) (cv u)) rneq syl5bi (cv v) (cv u) (cv f) fveq2 (/\ (:-onto-> (` (cv f) (cv v)) B (cv v)) (:-onto-> (` (cv f) (cv u)) B (cv u))) a1i impbid syl6 (V) dom2d mpi f 19.23aiv ax-mp)) thm (infmap2 ((x z) (w x) (A x) (w z) (A z) (A w) (B x) (B z) (B w)) ((infmap2.1 (e. A (V))) (infmap2.2 (e. B (V)))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (opr A (^m) B) (~~) ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B))))) (infmap2.1 infmap2.2 infxpabs infmap2.2 infmap2.1 xpcomen (X. B A) (X. A B) A entrt mpan syl infmap2.2 infmap2.1 xpex infmap2.1 infmap2.2 x ssenen A (^m) B oprex x (opr A (^m) B) abid2 infmap2.1 infmap2.2 (cv x) elmap (cv x) B A fssxp (cv x) B A ffun x visset fundmen x visset (dom (cv x)) ensym 3syl (cv x) B A fdm breqtrd jca sylbi x ss2abi eqsstr3 (opr A (^m) B) (V) ({|} x (/\ (C_ (cv x) (X. B A)) (br (cv x) (~~) B))) ssdomg mp2 (opr A (^m) B) ({|} x (/\ (C_ (cv x) (X. B A)) (br (cv x) (~~) B))) ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B))) domentr mpan 3syl infmap2.1 infmap2.2 ({<,>|} z w (/\ (/\ (C_ (cv z) A) (br (cv z) (~~) B)) (:-onto-> (cv w) B (cv z)))) eqid x infmap2lem2 (opr A (^m) B) ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B))) sbth mpan2 syl exp31 B ({/}) A (^m) opreq2 infmap2.1 map0e syl6eq B ({/}) (cv x) (~~) breq2 (C_ (cv x) A) anbi2d x abbidv ({/}) x df-sn df1o2 (cv x) en0 (C_ (cv x) A) anbi2i A 0ss (cv x) ({/}) A sseq1 mpbiri pm4.71ri bitr4 x abbii 3eqtr4r syl6eq eqtr4d A (^m) B oprex (opr A (^m) B) (V) ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B))) eqeng ax-mp syl (br B (~<_) A) a1d pm2.61d2 imp)) thm (alephadd () () (br (opr (` (aleph) A) (+c) (` (aleph) B)) (~~) (u. (` (aleph) A) (` (aleph) B))) (A (aleph) ndmfv B (aleph) ndmfv (+c) opreqan12d 0ex 0ex cdaval ({/}) ({} ({/})) ({} (1o)) xpundi (u. ({} ({/})) ({} (1o))) xp0r 3eqtr2 syl6eq A (aleph) ndmfv (-. (e. B (dom (aleph)))) adantr B (aleph) ndmfv (-. (e. A (dom (aleph)))) adantl uneq12d ({/}) un0 syl6eq eqtr4d alephfnon (aleph) (On) fndm ax-mp A eleq2i negbii alephfnon (aleph) (On) fndm ax-mp B eleq2i negbii syl2anbr (` (aleph) A) (+c) (` (aleph) B) oprex (opr (` (aleph) A) (+c) (` (aleph) B)) (V) (u. (` (aleph) A) (` (aleph) B)) eqeng ax-mp syl ex A alephgeom (aleph) A fvex (` (aleph) A) (V) (om) ssdom2g ax-mp (aleph) A fvex (aleph) B fvex infcda syl sylbi B alephgeom (aleph) B fvex (` (aleph) B) (V) (om) ssdom2g ax-mp (aleph) B fvex (aleph) A fvex infcda (aleph) A fvex (aleph) B fvex cdacomen (opr (` (aleph) A) (+c) (` (aleph) B)) (opr (` (aleph) B) (+c) (` (aleph) A)) (u. (` (aleph) B) (` (aleph) A)) entrt mpan syl (` (aleph) B) (` (aleph) A) uncom syl6breq syl sylbi pm2.61ii)) thm (alephexp1 () () (-> (/\ (/\ (e. A (On)) (e. B (On))) (C_ A B)) (br (opr (` (aleph) A) (^m) (` (aleph) B)) (~~) (opr (2o) (^m) (` (aleph) B)))) ((aleph) A fvex (aleph) B fvex infmap1 A alephgeom (aleph) A fvex (` (aleph) A) (V) (om) ssdom2g ax-mp sylbi 2onn (2o) nnsdom ax-mp (2o) (om) sdomdom ax-mp (2o) (om) (` (aleph) A) domtr mpan syl B alephgeom (aleph) B fvex (` (aleph) B) (V) (om) ssdom2g ax-mp sylbi anim12i (C_ A B) adantr A B alephord3 (aleph) A fvex (` (aleph) A) (V) (` (aleph) B) ssdomg ax-mp syl6bi imp sylanc)) thm (alephsuc3 ((A x)) () (-> (e. A (On)) (br (` (aleph) (suc A)) (~~) ({e.|} x (On) (br (cv x) (~~) (` (aleph) A))))) (A x alephsuc2 (` (aleph) A) difeq1d A alephcard (` (aleph) A) x cardval2 eqtr3 (e. A (On)) a1i ({e.|} x (On) (br (cv x) (~<_) (` (aleph) A))) difeq2d eqtrd x (On) (br (cv x) (~<_) (` (aleph) A)) (br (cv x) (~<) (` (aleph) A)) difrab (cv x) (` (aleph) A) bren2 (e. (cv x) (On)) a1i rabbii eqtr4 syl6req (aleph) (suc A) fvex (aleph) A fvex infdif A sucelon (suc A) alephgeom bitr (aleph) (suc A) fvex (` (aleph) (suc A)) (V) (om) ssdom2g ax-mp sylbi A alephordlem1 sylanc eqbrtrd (aleph) (suc A) fvex ({e.|} x (On) (br (cv x) (~~) (` (aleph) A))) ensym syl)) thm (alephexp2 ((A x)) () (-> (e. A (On)) (br (opr (2o) (^m) (` (aleph) A)) (~~) ({|} x (/\ (C_ (cv x) (` (aleph) A)) (br (cv x) (~~) (` (aleph) A)))))) (A alephgeom (aleph) A fvex (` (aleph) A) (V) (om) ssdom2g ax-mp sylbi (aleph) A fvex (` (aleph) A) (V) domrefg ax-mp jctir (aleph) A fvex (aleph) A fvex x infmap2 syl (e. A (On)) (e. A (On)) pm3.2 pm2.43i A ssid jctir A A alephexp1 (2o) (^m) (` (aleph) A) oprex (opr (2o) (^m) (` (aleph) A)) (V) (opr (` (aleph) A) (^m) (` (aleph) A)) ({|} x (/\ (C_ (cv x) (` (aleph) A)) (br (cv x) (~~) (` (aleph) A)))) enen1 mpan 3syl mpbid)) thm (gch-kn ((A x)) () (-> (e. A (On)) (<-> (br (` (aleph) (suc A)) (~~) ({|} x (/\ (C_ (cv x) (` (aleph) A)) (br (cv x) (~~) (` (aleph) A))))) (br (` (aleph) (suc A)) (~~) (opr (2o) (^m) (` (aleph) A))))) (A x alephexp2 (` (aleph) A) x df-pw (aleph) A fvex pwex eqeltrr (C_ (cv x) (` (aleph) A)) (br (cv x) (~~) (` (aleph) A)) pm3.26 x ss2abi ssexi ({|} x (/\ (C_ (cv x) (` (aleph) A)) (br (cv x) (~~) (` (aleph) A)))) (V) (opr (2o) (^m) (` (aleph) A)) (` (aleph) (suc A)) enen2 mpan syl bicomd)) thm (istopg ((x y) (x z) (J x) (y z) (J y) (J z)) () (-> (e. J A) (<-> (e. J (Top)) (/\ (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) (A.e. x J (A.e. y J (e. (i^i (cv x) (cv y)) J)))))) ((cv z) J (cv x) sseq2 (cv z) J (U. (cv x)) eleq2 imbi12d x albidv (cv z) J (i^i (cv x) (cv y)) eleq2 y raleqd x raleqd anbi12d z x y df-top A elab2g)) thm (istop2g ((x y) (J x) (J y)) () (-> (e. J A) (<-> (e. J (Top)) (/\ (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) (A. x (-> (/\/\ (C_ (cv x) J) (-. (= (cv x) ({/}))) (E.e. y (om) (br (cv x) (~~) (cv y)))) (e. (|^| (cv x)) J)))))) (J A x y istopg x J y fiint (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) anbi2i syl6bb)) thm (uniopnt ((x y) (A x) (A y) (J x) (J y)) () (-> (/\ (e. J (Top)) (C_ A J)) (e. (U. A) J)) (J (Top) x y istopg ibi pm3.26d J (Top) A elpw2g biimpar (cv x) A J sseq1 (cv x) A unieq J eleq1d imbi12d (P~ J) cla4gv syl com23 ex pm2.43d mpid imp)) thm (iunopnt ((x y) (A x) (A y) (B y) (J x) (J y)) () (-> (/\ (e. J (Top)) (A.e. x A (e. B J))) (e. (U_ x A B) J)) (x A B J y dfiun2g (e. J (Top)) adantl J ({|} y (E.e. x A (= (cv y) B))) uniopnt x A B J J y uniiunlem ibi sylan2 eqeltrd)) thm (inopnt ((x y) (A x) (A y) (B y) (J x) (J y)) () (-> (/\/\ (e. J (Top)) (e. A J) (e. B J)) (e. (i^i A B) J)) ((cv x) A (cv y) ineq1 J eleq1d (cv y) B A ineq2 J eleq1d J J rcla42v J (Top) x y istopg ibi pm3.27d syl5com 3impib)) thm (0opnt () () (-> (e. J (Top)) (e. ({/}) J)) (J 0ss J ({/}) uniopnt mpan2 uni0 syl5eqelr)) thm (1open () ((1open.1 (= X (U. J)))) (-> (e. J (Top)) (e. X J)) (J ssid J J uniopnt mpan2 1open.1 syl5eqel)) thm (eltopss () ((1open.1 (= X (U. J)))) (-> (/\ (e. J (Top)) (e. A J)) (C_ A X)) (A J elssuni 1open.1 syl6ssr (e. J (Top)) adantl)) thm (unitopt () () (-> (e. J (Top)) (e. (U. J) J)) (J ssid J J uniopnt mpan2)) thm (eltopsp ((x y) (J x) (J y)) () (<-> (e. (<,> (U. J) J) (TopSp)) (e. J (Top))) ((U. J) (TopSp) J df-br x y (/\ (e. (cv y) (Top)) (= (cv x) (U. (cv y)))) relopab x y df-topsp (TopSp) ({<,>|} x y (/\ (e. (cv y) (Top)) (= (cv x) (U. (cv y))))) releq ax-mp mpbir (U. J) J brrelexi sylbir (U. J) (TopSp) J df-br x y (/\ (e. (cv y) (Top)) (= (cv x) (U. (cv y)))) relopab x y df-topsp (TopSp) ({<,>|} x y (/\ (e. (cv y) (Top)) (= (cv x) (U. (cv y))))) releq ax-mp mpbir (U. J) J brrelexi sylbir J uniexb sylibr jca J (Top) uniexg J (Top) elisset jca (cv x) (U. J) (U. (cv y)) eqeq1 (e. (cv y) (Top)) anbi2d (cv y) J (Top) eleq1 (cv y) J unieq (U. J) eqeq2d anbi12d (V) (V) opelopabg x y df-topsp (<,> (U. J) J) eleq2i (U. J) eqid (e. J (Top)) biantru 3bitr4g pm5.21nii)) thm (tpsex ((x y) (A x) (A y) (J x) (J y)) () (-> (e. (<,> A J) (TopSp)) (/\ (e. A (V)) (e. J (V)))) (A (TopSp) J df-br x y (/\ (e. (cv y) (Top)) (= (cv x) (U. (cv y)))) relopab x y df-topsp (TopSp) ({<,>|} x y (/\ (e. (cv y) (Top)) (= (cv x) (U. (cv y))))) releq ax-mp mpbir A J brrelexi sylbir A eirr (e. A (Top)) (= A (U. A)) pm3.27 A unitopt (= A (U. A)) adantr eqeltrd mto A (TopSp) A df-br x y (/\ (e. (cv y) (Top)) (= (cv x) (U. (cv y)))) relopab x y df-topsp (TopSp) ({<,>|} x y (/\ (e. (cv y) (Top)) (= (cv x) (U. (cv y))))) releq ax-mp mpbir A A brrelexi sylbir A (TopSp) A df-br x y (/\ (e. (cv y) (Top)) (= (cv x) (U. (cv y)))) relopab x y df-topsp (TopSp) ({<,>|} x y (/\ (e. (cv y) (Top)) (= (cv x) (U. (cv y))))) releq ax-mp mpbir A A brrelexi sylbir jca A (Top) elisset A (Top) elisset jca (= A (U. A)) adantr (cv x) A (U. (cv y)) eqeq1 (e. (cv y) (Top)) anbi2d (cv y) A (Top) eleq1 (cv y) A unieq A eqeq2d anbi12d (V) (V) opelopabg x y df-topsp (<,> A A) eleq2i syl5bb pm5.21nii mtbir J A opprc2 (TopSp) eleq1d mtbiri a3i jca)) thm (istps ((x y) (A x) (A y) (J x) (J y)) () (<-> (e. (<,> A J) (TopSp)) (/\ (e. J (Top)) (= A (U. J)))) (A J tpsex (e. J (Top)) (= A (U. J)) pm3.27 J (Top) uniexg (= A (U. J)) adantr eqeltrd J (Top) elisset (= A (U. J)) adantr jca (cv x) A (U. (cv y)) eqeq1 (e. (cv y) (Top)) anbi2d (cv y) J (Top) eleq1 (cv y) J unieq A eqeq2d anbi12d (V) (V) opelopabg x y df-topsp (<,> A J) eleq2i syl5bb pm5.21nii)) thm (istps2 () () (<-> (e. (<,> A J) (TopSp)) (/\ (/\ (e. J (Top)) (C_ J (P~ A))) (/\ (e. ({/}) J) (e. A J)))) ((U. J) A eqimss (e. J (Top)) adantl J 0opnt (= (U. J) A) adantr (e. J (Top)) (= (U. J) A) pm3.27 J unitopt (= (U. J) A) adantr eqeltrrd jca jca ex J A unissel (e. ({/}) J) adantrl (e. J (Top)) a1i impbid A (U. J) eqcom J A sspwuni (/\ (e. ({/}) J) (e. A J)) anbi1i 3bitr4g pm5.32i A J istps (e. J (Top)) (C_ J (P~ A)) (/\ (e. ({/}) J) (e. A J)) anass 3bitr4)) thm (istps3 ((x y) (A x) (A y) (J x) (J y)) () (<-> (e. (<,> A J) (TopSp)) (/\ (/\/\ (C_ J (P~ A)) (e. ({/}) J) (e. A J)) (/\ (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) (A.e. x J (A.e. y J (e. (i^i (cv x) (cv y)) J)))))) (A J istps2 (e. J (Top)) (C_ J (P~ A)) (/\ (e. ({/}) J) (e. A J)) anass (e. J (Top)) (/\ (C_ J (P~ A)) (/\ (e. ({/}) J) (e. A J))) ancom (C_ J (P~ A)) (e. ({/}) J) (e. A J) 3anass (e. J (Top)) anbi1i J (P~ A) (V) ssexg A J pwexg sylan2 (e. ({/}) J) 3adant2 J (V) x y istopg syl pm5.32i 3bitr2 3bitr)) thm (istps4 ((x y) (A x) (A y) (J x) (J y)) () (<-> (e. (<,> A J) (TopSp)) (/\ (/\/\ (C_ J (P~ A)) (e. ({/}) J) (e. A J)) (/\ (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) (A. x (-> (/\/\ (C_ (cv x) J) (-. (= (cv x) ({/}))) (E.e. y (om) (br (cv x) (~~) (cv y)))) (e. (|^| (cv x)) J)))))) (A J x y istps3 x J y fiint (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) anbi2i (/\/\ (C_ J (P~ A)) (e. ({/}) J) (e. A J)) anbi2i bitr)) thm (istps5 ((x y) (A x) (A y) (J x) (J y)) () (<-> (e. (<,> A J) (TopSp)) (/\ (/\/\ (A.e. x J (C_ (cv x) A)) (e. ({/}) J) (e. A J)) (/\ (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) (A. x (-> (/\/\ (C_ (cv x) J) (-. (= (cv x) ({/}))) (E.e. y (om) (br (cv x) (~~) (cv y)))) (e. (|^| (cv x)) J)))))) (A J x y istps4 J A sspwuni J A x unissb bitr (e. ({/}) J) (e. A J) 3anbi1i (/\ (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) (A. x (-> (/\/\ (C_ (cv x) J) (-. (= (cv x) ({/}))) (E.e. y (om) (br (cv x) (~~) (cv y)))) (e. (|^| (cv x)) J)))) anbi1i bitr)) thm (isbasisg ((x y) (x z) (B x) (y z) (B y) (B z)) () (-> (e. B C) (<-> (e. B (Bases)) (A.e. x B (A.e. y B (C_ (i^i (cv x) (cv y)) (U. (i^i B (P~ (i^i (cv x) (cv y)))))))))) ((cv z) B (P~ (i^i (cv x) (cv y))) ineq1 unieqd (i^i (cv x) (cv y)) sseq2d y raleqd x raleqd z x y df-bases C elab2g)) thm (isbasis2g ((w x) (w y) (w z) (B w) (x y) (x z) (B x) (y z) (B y) (B z)) () (-> (e. B C) (<-> (e. B (Bases)) (A.e. x B (A.e. y B (A.e. z (i^i (cv x) (cv y)) (E.e. w B (/\ (e. (cv z) (cv w)) (C_ (cv w) (i^i (cv x) (cv y)))))))))) (B C x y isbasisg (i^i (cv x) (cv y)) (U. (i^i B (P~ (i^i (cv x) (cv y))))) z dfss3 (cv w) B (P~ (i^i (cv x) (cv y))) elin (i^i (cv x) (cv y)) w df-pw abeq2i (e. (cv w) B) anbi2i bitr (e. (cv z) (cv w)) anbi2i (e. (cv z) (cv w)) (e. (cv w) B) (C_ (cv w) (i^i (cv x) (cv y))) an12 bitr w exbii (cv z) (i^i B (P~ (i^i (cv x) (cv y)))) w eluni w B (/\ (e. (cv z) (cv w)) (C_ (cv w) (i^i (cv x) (cv y)))) df-rex 3bitr4 z (i^i (cv x) (cv y)) ralbii bitr x B y B 2ralbii syl6bb)) thm (isbasis3g ((w x) (w y) (w z) (B w) (x y) (x z) (B x) (y z) (B y) (B z)) () (-> (e. B C) (<-> (e. B (Bases)) (/\/\ (A.e. x B (C_ (cv x) (U. B))) (A.e. x (U. B) (E.e. y B (e. (cv x) (cv y)))) (A.e. x B (A.e. y B (A.e. z (i^i (cv x) (cv y)) (E.e. w B (/\ (e. (cv z) (cv w)) (C_ (cv w) (i^i (cv x) (cv y))))))))))) (B C x y z w isbasis2g (cv x) B elssuni rgen (cv x) B y eluni2 biimp rgen pm3.2i (A.e. x B (A.e. y B (A.e. z (i^i (cv x) (cv y)) (E.e. w B (/\ (e. (cv z) (cv w)) (C_ (cv w) (i^i (cv x) (cv y)))))))) biantrur (A.e. x B (C_ (cv x) (U. B))) (A.e. x (U. B) (E.e. y B (e. (cv x) (cv y)))) (A.e. x B (A.e. y B (A.e. z (i^i (cv x) (cv y)) (E.e. w B (/\ (e. (cv z) (cv w)) (C_ (cv w) (i^i (cv x) (cv y)))))))) df-3an bitr4 syl6bb)) thm (basis1t ((x y) (B x) (B y) (C x) (C y) (D y) (D x)) () (-> (/\/\ (e. B (Bases)) (e. C B) (e. D B)) (C_ (i^i C D) (U. (i^i B (P~ (i^i C D)))))) ((cv x) C (cv y) ineq1 (cv x) C (cv y) ineq1 (i^i (cv x) (cv y)) (i^i C (cv y)) pweq syl B ineq2d unieqd sseq12d (cv y) D C ineq2 (cv y) D C ineq2 (i^i C (cv y)) (i^i C D) pweq syl B ineq2d unieqd sseq12d B B rcla42v B (Bases) x y isbasisg ibi syl5com 3impib)) thm (basis2t ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (B w) (B x) (B y) (B z) (C w) (C x) (C y) (C z) (D w) (D y) (D x) (D z)) () (-> (/\ (/\ (e. B (Bases)) (e. C B)) (/\ (e. D B) (e. A (i^i C D)))) (E.e. x B (/\ (e. A (cv x)) (C_ (cv x) (i^i C D))))) (B (Bases) y z w x isbasis2g ibi (cv y) C (cv z) ineq1 (i^i (cv y) (cv z)) (i^i C (cv z)) (cv x) sseq2 (e. (cv w) (cv x)) anbi2d x B rexbidv w raleqd syl (cv z) D C ineq2 (i^i C (cv z)) (i^i C D) (cv x) sseq2 (e. (cv w) (cv x)) anbi2d x B rexbidv w raleqd syl B B rcla42v (cv w) A (cv x) eleq1 (C_ (cv x) (i^i C D)) anbi1d x B rexbidv (i^i C D) rcla4cv syl6com syl exp3a imp43)) thm (elbast () () (-> (/\ (e. B (Bases)) (e. A B)) (C_ A (U. B))) (B A A basis1t 3expb anabsan2 A inidm syl5ssr B (P~ (i^i A A)) inss1 (i^i B (P~ (i^i A A))) B uniss ax-mp (/\ (e. B (Bases)) (e. A B)) a1i sstrd)) thm (tgvalt ((x y) (x z) (y z) (B x) (B y) (B z)) () (-> (e. B (Bases)) (= (` (topGen) B) ({|} x (C_ (cv x) (U. (i^i B (P~ (cv x)))))))) (B (Bases) uniexg (U. B) (V) x (C_ (cv x) (U. (P~ (cv x)))) abssexg B (P~ (cv x)) uniin (cv x) (U. (i^i B (P~ (cv x)))) (i^i (U. B) (U. (P~ (cv x)))) sstr mpan2 (cv x) (U. B) (U. (P~ (cv x))) ssin sylibr x ss2abi ({|} x (C_ (cv x) (U. (i^i B (P~ (cv x)))))) ({|} x (/\ (C_ (cv x) (U. B)) (C_ (cv x) (U. (P~ (cv x)))))) (V) ssexg mpan 3syl (cv y) B (P~ (cv x)) ineq1 unieqd (cv x) sseq2d x abbidv y z x df-topgen (V) fvopab4g mpdan)) thm (tgval2t ((x y) (x z) (y z) (B x) (B y) (B z)) () (-> (e. B (Bases)) (= (` (topGen) B) ({|} x (/\ (C_ (cv x) (U. B)) (A.e. y (cv x) (E.e. z B (/\ (e. (cv y) (cv z)) (C_ (cv z) (cv x))))))))) (B x tgvalt (cv x) (U. (i^i B (P~ (cv x)))) y dfss3 B (P~ (cv x)) inss1 (i^i B (P~ (cv x))) B uniss ax-mp (cv y) sseli pm4.71ri y (cv x) ralbii y (cv x) (e. (cv y) (U. B)) (e. (cv y) (U. (i^i B (P~ (cv x))))) r19.26 3bitr (cv x) (U. B) y dfss3 (cv z) B (P~ (cv x)) elin (e. (cv y) (cv z)) anbi2i (e. (cv y) (cv z)) (e. (cv z) B) (e. (cv z) (P~ (cv x))) an12 bitr z exbii (cv y) (i^i B (P~ (cv x))) z eluni z B (/\ (e. (cv y) (cv z)) (e. (cv z) (P~ (cv x)))) df-rex 3bitr4 z visset (cv x) elpw (e. (cv y) (cv z)) anbi2i z B rexbii bitr2 y (cv x) ralbii anbi12i bitr4 x abbii syl6eq)) thm (eltgt ((A x) (B x)) () (-> (e. B (Bases)) (<-> (e. A (` (topGen) B)) (C_ A (U. (i^i B (P~ A)))))) (B x tgvalt A eleq2d A ({|} x (C_ (cv x) (U. (i^i B (P~ (cv x)))))) elisset (e. B (Bases)) adantl A (U. (i^i B (P~ A))) (V) ssexg ancoms B (Bases) (P~ A) inex1g (i^i B (P~ A)) (V) uniexg syl sylan (= (cv x) A) id (cv x) A pweq B ineq2d unieqd sseq12d (V) elabg pm5.21nd bitrd)) thm (eltg2t ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (e. B (Bases)) (<-> (e. A (` (topGen) B)) (/\ (C_ A (U. B)) (A.e. x A (E.e. y B (/\ (e. (cv x) (cv y)) (C_ (cv y) A))))))) (B z x y tgval2t A eleq2d A ({|} z (/\ (C_ (cv z) (U. B)) (A.e. x (cv z) (E.e. y B (/\ (e. (cv x) (cv y)) (C_ (cv y) (cv z))))))) elisset (e. B (Bases)) adantl A (U. B) (V) ssexg B (Bases) uniexg sylan2 ancoms (A.e. x A (E.e. y B (/\ (e. (cv x) (cv y)) (C_ (cv y) A)))) adantrr (cv z) A (U. B) sseq1 (cv z) A (cv y) sseq2 (e. (cv x) (cv y)) anbi2d y B rexbidv x raleqd anbi12d (V) elabg pm5.21nd bitrd)) thm (tg1t ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. B (Bases)) (e. A (` (topGen) B))) (C_ A (U. B))) (B A x y eltg2t (C_ A (U. B)) (A.e. x A (E.e. y B (/\ (e. (cv x) (cv y)) (C_ (cv y) A)))) pm3.26 syl6bi imp)) thm (tg2t ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\/\ (e. B (Bases)) (e. A (` (topGen) B)) (e. C A)) (E.e. x B (/\ (e. C (cv x)) (C_ (cv x) A)))) (B A y x eltg2t (cv y) C (cv x) eleq1 (C_ (cv x) A) anbi1d x B rexbidv A rcla4cv (C_ A (U. B)) adantl syl6bi 3imp)) thm (bastgt ((B x)) () (-> (e. B (Bases)) (C_ B (` (topGen) B))) (B (cv x) (cv x) basis1t (e. B (Bases)) (e. (cv x) B) pm3.26 (e. B (Bases)) (e. (cv x) B) pm3.27 (e. B (Bases)) (e. (cv x) B) pm3.27 syl3anc (cv x) inidm (cv x) inidm (i^i (cv x) (cv x)) (cv x) pweq ax-mp B ineq2i unieqi 3sstr3g ex B (cv x) eltgt sylibrd ssrdv)) thm (unitgt ((x y) (B x) (B y)) () (-> (e. B (Bases)) (= (U. (` (topGen) B)) (U. B))) (B (cv y) eltgt B (P~ (cv y)) inss1 (i^i B (P~ (cv y))) B uniss ax-mp (cv y) (U. (i^i B (P~ (cv y)))) (U. B) sstr mpan2 (cv x) sseld syl6bi r19.23adv (cv x) (` (topGen) B) y eluni2 syl5ib B bastgt B (` (topGen) B) uniss syl (cv x) sseld impbid eqrdv)) thm (0eltgt () () (-> (e. B (Bases)) (e. ({/}) (` (topGen) B))) ((U. (i^i B (P~ ({/})))) 0ss B ({/}) eltgt mpbiri)) thm (tgclt ((v w) (v x) (v y) (v z) (w x) (w y) (w z) (x y) (x z) (y z) (t u) (t v) (t w) (t x) (t y) (t z) (B t) (u v) (u w) (u x) (u y) (u z) (B u) (B v) (B w) (B x) (B y) (B z)) () (-> (e. B (Bases)) (e. (` (topGen) B) (Top))) ((cv u) (` (topGen) B) uniss (e. B (Bases)) adantl B unitgt (C_ (cv u) (` (topGen) B)) adantr sseqtrd B (cv t) x y eltg2t (C_ (cv t) (U. B)) (A.e. x (cv t) (E.e. y B (/\ (e. (cv x) (cv y)) (C_ (cv y) (cv t))))) pm3.27 syl6bi x (cv t) (E.e. y B (/\ (e. (cv x) (cv y)) (C_ (cv y) (cv t)))) ra4 syl6 imp31 an1rs (cv u) (` (topGen) B) (cv t) ssel2 sylan2 an42s (cv y) (cv t) (U. (cv u)) sstr2 (cv t) (cv u) elssuni syl5com (e. (cv x) (cv y)) anim2d y B r19.22sdv (/\ (e. B (Bases)) (C_ (cv u) (` (topGen) B))) (e. (cv x) (cv t)) ad2antrl mpd exp32 r19.23adv (cv x) (cv u) t eluni2 syl5ib r19.21aiv jca ex B (U. (cv u)) x y eltg2t sylibrd u 19.21aiv B (cv u) tg1t (cv u) (U. B) (cv v) ssinss1 syl (e. (cv v) (` (topGen) B)) adantrr B (cv u) x z eltg2t biimpa pm3.27d x (cv u) (E.e. z B (/\ (e. (cv x) (cv z)) (C_ (cv z) (cv u)))) ra4 syl B (cv v) x w eltg2t biimpa pm3.27d x (cv v) (E.e. w B (/\ (e. (cv x) (cv w)) (C_ (cv w) (cv v)))) ra4 syl im2anan9 (cv x) (cv u) (cv v) elin z B w B (/\ (e. (cv x) (cv z)) (C_ (cv z) (cv u))) (/\ (e. (cv x) (cv w)) (C_ (cv w) (cv v))) reeanv 3imtr4g anandis B (cv z) (cv w) (cv x) t basis2t (e. (cv x) (i^i (cv u) (cv v))) adantllr (C_ (i^i (cv z) (cv w)) (i^i (cv u) (cv v))) adantrrr (cv t) (i^i (cv z) (cv w)) (i^i (cv u) (cv v)) sstr2 com12 (e. (cv x) (cv t)) anim2d t B r19.22sdv (e. (cv x) (i^i (cv z) (cv w))) adantl (/\ (/\ (e. B (Bases)) (e. (cv x) (i^i (cv u) (cv v)))) (e. (cv z) B)) (e. (cv w) B) ad2antll mpd (cv x) (cv z) (cv w) elin biimpr (cv z) (cv u) (cv w) (cv v) ss2in anim12i an4s sylanr2 exp32 r19.23adv ex r19.23adv ex a2d imp syldan r19.21aiv jca ex B (i^i (cv u) (cv v)) x t eltg2t sylibrd r19.21aivv jca (topGen) B fvex (` (topGen) B) (V) u v istopg ax-mp sylibr)) thm (tgval3t ((w x) (w y) (w z) (x y) (x z) (y z) (B w) (B x) (B y) (B z)) () (-> (e. B (Bases)) (= (` (topGen) B) ({|} x (E. y (/\ (C_ (cv y) B) (= (cv x) (U. (cv y)))))))) (B (cv x) w y eltg2t biimpa pm3.27d w z y elequ1 (C_ (cv y) (cv x)) anbi1d y B rexbidv (cv x) rcla4cv y B (/\ (e. (cv z) (cv y)) (C_ (cv y) (cv x))) df-rex (e. (cv y) B) (e. (cv z) (cv y)) (C_ (cv y) (cv x)) an12 (e. (cv y) B) (C_ (cv y) (cv x)) ancom (e. (cv z) (cv y)) anbi2i bitr y exbii bitr syl6ib syl (cv y) (cv x) (cv z) ssel impcom (e. (cv y) B) adantrr y 19.23aiv (/\ (e. B (Bases)) (e. (cv x) (` (topGen) B))) a1i impbid (cv z) y (/\ (C_ (cv y) (cv x)) (e. (cv y) B)) eluniab syl6bbr eqrdv (C_ (cv y) (cv x)) (e. (cv y) B) pm3.27 abssi jctil x visset (cv x) (V) y (e. (cv y) B) abssexg ax-mp z y (/\ (C_ (cv y) (cv x)) (e. (cv y) B)) hbab1 z y (/\ (C_ (cv y) (cv x)) (e. (cv y) B)) hbab1 (e. (cv z) B) y ax-17 hbss (e. (cv z) (cv x)) y ax-17 z y (/\ (C_ (cv y) (cv x)) (e. (cv y) B)) hbab1 hbuni hbeq hban (cv y) ({|} y (/\ (C_ (cv y) (cv x)) (e. (cv y) B))) B sseq1 (cv y) ({|} y (/\ (C_ (cv y) (cv x)) (e. (cv y) B))) unieq (cv x) eqeq2d anbi12d (V) cla4egf ax-mp syl ex (C_ (cv y) B) (= (cv x) (U. (cv y))) pm3.27 (e. B (Bases)) adantl (` (topGen) B) (cv y) uniopnt B tgclt (C_ (cv y) B) adantr (cv y) B (` (topGen) B) sstr B bastgt sylan2 ancoms sylanc (= (cv x) (U. (cv y))) adantrr eqeltrd ex y 19.23adv impbid abbi2dv)) thm (eltg3t ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B (Bases)) (<-> (e. A (` (topGen) B)) (E. x (/\ (C_ (cv x) B) (= A (U. (cv x))))))) (B y x tgval3t A eleq2d x visset uniex A (U. (cv x)) (V) eleq1 mpbiri (C_ (cv x) B) adantl x 19.23aiv (cv y) A (U. (cv x)) eqeq1 (C_ (cv x) B) anbi2d x exbidv elab3 syl6bb)) thm (topbast ((w x) (w y) (w z) (x y) (x z) (y z) (J w) (J x) (J y) (J z)) () (-> (e. J (Top)) (e. J (Bases))) ((cv w) (i^i (cv x) (cv y)) (cv z) eleq2 (cv w) (i^i (cv x) (cv y)) (i^i (cv x) (cv y)) sseq1 anbi12d J rcla4ev J (cv x) (cv y) inopnt 3expb (e. (cv z) (i^i (cv x) (cv y))) adantr (/\ (e. J (Top)) (/\ (e. (cv x) J) (e. (cv y) J))) (e. (cv z) (i^i (cv x) (cv y))) pm3.27 (i^i (cv x) (cv y)) ssid jctir sylanc exp31 r19.21adv r19.21aivv J (Top) x y z w isbasis2g mpbird)) thm (tgtopt ((x y) (J x) (J y)) () (-> (e. J (Top)) (= (` (topGen) J) J)) (J topbast J (cv x) y eltg3t syl (/\ (e. J (Top)) (C_ (cv y) J)) (= (cv x) (U. (cv y))) pm3.27 J (cv y) uniopnt (= (cv x) (U. (cv y))) adantr eqeltrd anasss ex y 19.23adv sylbid ssrdv J topbast J bastgt syl eqssd)) thm (topss2t ((u v) (u w) (u x) (u y) (u z) (B u) (v w) (v x) (v y) (v z) (B v) (w x) (w y) (w z) (B w) (x y) (x z) (B x) (y z) (B y) (B z) (C u) (C v) (C w) (C x) (C y) (C z)) () (-> (/\/\ (e. B (Bases)) (e. C (Bases)) (= (U. B) (U. C))) (<-> (C_ (` (topGen) B) (` (topGen) C)) (A.e. x (U. B) (A.e. y B (-> (e. (cv x) (cv y)) (E.e. z C (/\ (e. (cv x) (cv z)) (C_ (cv z) (cv y))))))))) (C (cv y) (cv x) z tg2t (e. B (Bases)) (e. C (Bases)) pm3.27 (C_ (` (topGen) B) (` (topGen) C)) (/\ (/\ (e. (cv x) (U. B)) (e. (cv y) B)) (e. (cv x) (cv y))) ad2antrr B (` (topGen) B) (` (topGen) C) sstr B bastgt sylan (cv y) sseld imp (e. C (Bases)) adantllr (e. (cv x) (U. B)) adantrl (e. (cv x) (cv y)) adantrr (/\ (e. (cv x) (U. B)) (e. (cv y) B)) (e. (cv x) (cv y)) pm3.27 (/\ (/\ (e. B (Bases)) (e. C (Bases))) (C_ (` (topGen) B) (` (topGen) C))) adantl syl3anc exp43 r19.21advv (= (U. B) (U. C)) 3adant3 C (cv u) w z eltg2t B (cv u) tg1t (= (U. B) (U. C)) adantll (= (U. B) (U. C)) (e. B (Bases)) pm3.26 (e. (cv u) (` (topGen) B)) adantr sseqtrd (A.e. x (U. B) (A.e. y B (-> (e. (cv x) (cv y)) (E.e. z C (/\ (e. (cv x) (cv z)) (C_ (cv z) (cv y))))))) adantrl (cv v) B elssuni (cv w) sseld imp (C_ (cv v) (cv u)) adantrr (A.e. x (U. B) (A.e. y B (-> (e. (cv x) (cv y)) (E.e. z C (/\ (e. (cv x) (cv z)) (C_ (cv z) (cv y))))))) adantrl x w y elequ1 x w z elequ1 (C_ (cv z) (cv y)) anbi1d z C rexbidv imbi12d y v w elequ2 (cv y) (cv v) (cv z) sseq2 (e. (cv w) (cv z)) anbi2d z C rexbidv imbi12d (U. B) B rcla42v ex com4l imp32 (C_ (cv v) (cv u)) adantrrr mpd an1s (cv z) (cv v) (cv u) sstr2 com12 (e. (cv w) (cv z)) anim2d z C r19.22sdv (e. (cv w) (cv v)) adantl (A.e. x (U. B) (A.e. y B (-> (e. (cv x) (cv y)) (E.e. z C (/\ (e. (cv x) (cv z)) (C_ (cv z) (cv y))))))) (e. (cv v) B) ad2antll mpd exp32 r19.23adv B (cv u) (cv w) v tg2t 3expa syl5 exp4c com12 imp42 r19.21aiva (= (U. B) (U. C)) adantll jca syl5bir exp4c 3imp 3comr exp3a imp ssrdv ex impbid)) thm (basgen2t ((v w) (v x) (v y) (v z) (B v) (w x) (w y) (w z) (B w) (x y) (x z) (B x) (y z) (B y) (B z) (J v) (J w) (J x) (J y) (J z)) () (-> (/\/\ (e. J (Top)) (C_ B J) (A.e. x J (A.e. y (cv x) (E.e. z B (/\ (e. (cv y) (cv z)) (C_ (cv z) (cv x))))))) (/\ (e. B (Bases)) (= (` (topGen) B) J))) (J (cv w) (cv v) inopnt 3expb B J (cv w) ssel B J (cv v) ssel anim12d imp sylan2 anassrs (cv x) (i^i (cv w) (cv v)) (cv z) sseq2 (e. (cv y) (cv z)) anbi2d z B rexbidv y raleqd J rcla4v syl ex com23 r19.21advv B J (Top) ssexg ancoms B (V) w v y z isbasis2g syl sylibrd imp J (cv w) (cv v) inopnt 3expb B J (cv w) ssel B J (cv v) ssel anim12d imp sylan2 anassrs (cv x) (i^i (cv w) (cv v)) (cv z) sseq2 (e. (cv y) (cv z)) anbi2d z B rexbidv y raleqd J rcla4v syl ex com23 r19.21advv B J (Top) ssexg ancoms B (V) w v y z isbasis2g syl sylibrd imp B (cv x) y eltg3t (/\ (e. J (Top)) (C_ B J)) adantl (/\ (e. J (Top)) (/\ (C_ (cv y) B) (C_ B J))) (= (cv x) (U. (cv y))) pm3.27 J (cv y) uniopnt (cv y) B J sstr sylan2 (= (cv x) (U. (cv y))) adantr eqeltrd exp42 com23 imp4b y 19.23adv (e. B (Bases)) adantr sylbid ssrdv syldan J tgtopt (C_ B J) (A.e. x J (A.e. y (cv x) (E.e. z B (/\ (e. (cv y) (cv z)) (C_ (cv z) (cv x)))))) ad2antrr J B y x z topss2t 3expa biimpar J topbast (C_ B J) (A.e. x J (A.e. y (cv x) (E.e. z B (/\ (e. (cv y) (cv z)) (C_ (cv z) (cv x)))))) ad2antrr J (cv w) (cv v) inopnt 3expb B J (cv w) ssel B J (cv v) ssel anim12d imp sylan2 anassrs (cv x) (i^i (cv w) (cv v)) (cv z) sseq2 (e. (cv y) (cv z)) anbi2d z B rexbidv y raleqd J rcla4v syl ex com23 r19.21advv B J (Top) ssexg ancoms B (V) w v y z isbasis2g syl sylibrd imp jca J unitopt (cv x) (U. J) y (E.e. z B (e. (cv y) (cv z))) raleq1 J rcla4v y w z elequ1 z B rexbidv (U. J) rcla4cv (cv w) B z eluni2 syl6ibr syl6 (e. (cv y) (cv z)) (C_ (cv z) (cv x)) pm3.26 z B r19.22si y (cv x) r19.20si x J r19.20si syl5 syl imp ssrdv (C_ B J) adantlr B J uniss (e. J (Top)) (A.e. x J (A.e. y (cv x) (E.e. z B (/\ (e. (cv y) (cv z)) (C_ (cv z) (cv x)))))) ad2antlr eqssd jca (-> (e. (cv y) (cv x)) (E.e. z B (/\ (e. (cv y) (cv z)) (C_ (cv z) (cv x))))) (e. (cv y) (U. J)) ax-1 r19.20i2 x J r19.20si x J y (U. J) (-> (e. (cv y) (cv x)) (E.e. z B (/\ (e. (cv y) (cv z)) (C_ (cv z) (cv x))))) ralcom sylib (/\ (e. J (Top)) (C_ B J)) adantl sylanc eqsstr3d eqssd jca 3impa)) thm (subbas ((f g) (f x) (f y) (A f) (g x) (g y) (A g) (x y) (A x) (A y) (f t) (f u) (f v) (f w) (B f) (g t) (g u) (g v) (g w) (B g) (t u) (t v) (t w) (B t) (u v) (u w) (B u) (v w) (B v) (B w) (f z) (g z) (v x) (v y) (v z) (w x) (w y) (w z) (x z) (y z)) ((subbas.1 (e. A (V))) (subbas.2 (= B ({|} x (E. y (/\/\ (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) (= (cv x) (|^| (cv y))))))))) (e. B (Bases)) (f g (/\/\ (C_ (cv f) A) (E.e. z (om) (br (cv f) (~~) (cv z))) (= (cv w) (|^| (cv f)))) (/\/\ (C_ (cv g) A) (E.e. z (om) (br (cv g) (~~) (cv z))) (= (cv v) (|^| (cv g)))) eeanv (C_ (cv f) A) (E.e. z (om) (br (cv f) (~~) (cv z))) (= (cv w) (|^| (cv f))) (C_ (cv g) A) (E.e. z (om) (br (cv g) (~~) (cv z))) (= (cv v) (|^| (cv g))) an6 (cv t) (i^i (cv w) (cv v)) (cv u) eleq2 (cv t) (i^i (cv w) (cv v)) (i^i (cv w) (cv v)) sseq1 anbi12d B rcla4ev f visset g visset unex (cv y) (u. (cv f) (cv g)) A sseq1 (cv y) (u. (cv f) (cv g)) (~~) (cv z) breq1 z (om) rexbidv (cv y) (u. (cv f) (cv g)) inteq (i^i (cv w) (cv v)) eqeq2d 3anbi123d cla4ev (cv f) A (cv g) unss biimp z (cv f) (cv g) unfi (cv w) (|^| (cv f)) (cv v) (|^| (cv g)) ineq12 (cv f) (cv g) intun syl6eqr syl3an subbas.2 (i^i (cv w) (cv v)) eleq2i w visset (cv v) inex1 (cv x) (i^i (cv w) (cv v)) (|^| (cv y)) eqeq1 (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) 3anbi3d y exbidv elab bitr2 sylib (i^i (cv w) (cv v)) ssid (e. (cv u) (i^i (cv w) (cv v))) jctr syl2an ex sylbi f g 19.23aivv sylbir subbas.2 (cv w) eleq2i w visset (cv x) (cv w) (|^| (cv y)) eqeq1 (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) 3anbi3d y exbidv elab (cv y) (cv f) A sseq1 (cv y) (cv f) (~~) (cv z) breq1 z (om) rexbidv (cv y) (cv f) inteq (cv w) eqeq2d 3anbi123d cbvexv 3bitr subbas.2 (cv v) eleq2i v visset (cv x) (cv v) (|^| (cv y)) eqeq1 (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) 3anbi3d y exbidv elab (cv y) (cv g) A sseq1 (cv y) (cv g) (~~) (cv z) breq1 z (om) rexbidv (cv y) (cv g) inteq (cv v) eqeq2d 3anbi123d cbvexv 3bitr syl2anb r19.21aiv rgen2 subbas.1 (|^| (cv y)) x df-sn (|^| (cv y)) snex eqeltrr y abexssex subbas.2 (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) (= (cv x) (|^| (cv y))) 3simpb y 19.22i x ss2abi eqsstr ssexi B (V) w v u t isbasis2g ax-mp mpbir)) thm (subbas2 ((w x) (w y) (A w) (x y) (A x) (A y) (w z) (x z) (y z)) ((subbas2.1 (= B ({|} x (E. y (/\/\ (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) (= (cv x) (|^| (cv y))))))))) (= (U. B) (U. A)) (subbas2.1 unieqi (cv w) x (E. y (/\/\ (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) (= (cv x) (|^| (cv y))))) eluniab (cv x) (U. A) (cv w) ssel com12 (C_ (cv y) A) (= (cv x) (|^| (cv y))) pm3.27 (cv y) A intssuni2 x visset (cv x) (|^| (cv y)) (V) eleq1 mpbii (cv y) intex sylibr sylan2 eqsstrd (E.e. z (om) (br (cv y) (~~) (cv z))) 3adant2 syl5 y 19.23adv imp x 19.23aiv sylbi ssriv (cv w) snex (cv y) ({} (cv w)) A sseq1 (cv y) ({} (cv w)) (~~) (cv z) breq1 z (om) rexbidv (cv y) ({} (cv w)) inteq (cv w) eqeq2d 3anbi123d cla4ev (cv w) A snssi z (cv w) snfi (e. (cv w) A) a1i w visset intsn eqcomi (e. (cv w) A) a1i syl3anc w visset (cv x) (cv w) (|^| (cv y)) eqeq1 (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) 3anbi3d y exbidv elab sylibr ssriv A ({|} x (E. y (/\/\ (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) (= (cv x) (|^| (cv y)))))) uniss ax-mp eqssi eqtr)) thm (sn0top ((x y)) () (e. ({} ({/})) (Top)) (p0ex ({} ({/})) (V) x y istopg ax-mp (cv x) ({/}) sssn (cv x) ({/}) unieq uni0 0ex (U. ({/})) elsnc2 mpbir syl6eqel (cv x) ({} ({/})) unieq 0ex unisn (U. (cv x)) (U. ({} ({/}))) ({/}) eqtrt mpan2 x visset uniex ({/}) elsnc sylibr syl jaoi sylbi x ax-gen y ({/}) elsn (cv y) ({/}) (cv x) ineq2 (cv x) in0 (i^i (cv x) (cv y)) eqeq2i biimp syl x visset (cv y) inex1 ({/}) elsnc biimpr syl sylbi (e. (cv x) ({} ({/}))) adantl rgen2 mpbir2an)) thm (indistop ((x y) (A x) (A y)) ((indistop.1 (e. A (V)))) (e. ({,} ({/}) A) (Top)) (({/}) A prex ({,} ({/}) A) (V) x y istopg ax-mp (cv x) ({/}) A sspr (cv x) ({/}) unieq uni0 0ex A pri1 eqeltr syl6eqel (cv x) ({} ({/})) unieq 0ex A pri1 (= (cv x) ({} ({/}))) a1i 0ex unisn syl5eqel eqeltrd jaoi (cv x) ({} A) unieq indistop.1 ({/}) pri2 (= (cv x) ({} A)) a1i indistop.1 unisn syl5eqel eqeltrd (cv x) ({,} ({/}) A) unieq A ({/}) uncom A un0 indistop.1 ({/}) pri2 eqeltr eqeltrr (= (cv x) ({,} ({/}) A)) a1i 0ex indistop.1 unipr syl5eqel eqeltrd jaoi jaoi sylbi x ax-gen (= (cv y) ({/})) (= (cv x) ({/})) pm3.26 (cv x) ineq2d (cv x) in0 syl6eq 0ex A pri1 syl6eqel ex (= (cv y) A) (= (cv x) ({/})) pm3.27 (cv y) ineq1d ({/}) (cv y) incom (cv y) in0 eqtr syl6eq 0ex A pri1 syl6eqel ex jaoi com12 (cv x) A (cv y) ({/}) ineq12 ancoms A in0 syl6eq 0ex A pri1 syl6eqel ex (cv x) A (cv y) A ineq12 ancoms A inidm syl6eq indistop.1 ({/}) pri2 syl6eqel ex jaoi com12 jaoi y visset ({/}) A elpr syl5ib y 19.21aiv x visset ({/}) A elpr y ({,} ({/}) A) (e. (i^i (cv x) (cv y)) ({,} ({/}) A)) df-ral 3imtr4 rgen mpbir2an)) thm (distop ((x y) (A x) (A y)) ((indistop.1 (e. A (V)))) (e. (P~ A) (Top)) (indistop.1 pwex (P~ A) (V) x y istopg ax-mp (cv x) (P~ A) uniss A unipw syl6ss x visset uniex A elpw sylibr x ax-gen x visset A elpw y visset A elpw (cv x) A (cv y) ssinss1 (C_ (cv y) A) a1i y visset (cv x) inex2 A elpw syl6ibr sylbi com12 sylbi r19.21aiv rgen mpbir2an)) thm (fctop ((n x) (n y) (n z) (A n) (x y) (x z) (A x) (y z) (A y) (A z)) ((indistop.1 (e. A (V)))) (e. ({|} x (/\ (C_ (cv x) A) (\/ (E.e. n (om) (br (\ A (cv x)) (~~) (cv n))) (= (cv x) ({/}))))) (Top)) (indistop.1 A (V) x (\/ (E.e. n (om) (br (\ A (cv x)) (~~) (cv n))) (= (cv x) ({/}))) abssexg ax-mp ({|} x (/\ (C_ (cv x) A) (\/ (E.e. n (om) (br (\ A (cv x)) (~~) (cv n))) (= (cv x) ({/}))))) (V) y z istopg ax-mp (cv y) ({|} x (/\ (C_ (cv x) A) (\/ (E.e. n (om) (br (\ A (cv x)) (~~) (cv n))) (= (cv x) ({/}))))) uniss x (/\ (C_ (cv x) A) (\/ (E.e. n (om) (br (\ A (cv x)) (~~) (cv n))) (= (cv x) ({/})))) rabab unieqi (C_ (cv x) A) (\/ (E.e. n (om) (br (\ A (cv x)) (~~) (cv n))) (= (cv x) ({/}))) pm3.26 (e. (cv x) (V)) a1i ss2rabi ({e.|} x (V) (/\ (C_ (cv x) A) (\/ (E.e. n (om) (br (\ A (cv x)) (~~) (cv n))) (= (cv x) ({/}))))) ({e.|} x (V) (C_ (cv x) A)) uniss ax-mp indistop.1 A (V) x unimax ax-mp sseqtr eqsstr3 (U. (cv y)) (U. ({|} x (/\ (C_ (cv x) A) (\/ (E.e. n (om) (br (\ A (cv x)) (~~) (cv n))) (= (cv x) ({/})))))) A sstr mpan2 syl (cv y) ({|} x (/\ (C_ (cv x) A) (\/ (E.e. n (om) (br (\ A (cv x)) (~~) (cv n))) (= (cv x) ({/}))))) (cv z) ssel2 z visset (cv x) (cv z) A sseq1 (cv x) (cv z) A difeq2 (~~) (cv n) breq1d n (om) rexbidv (cv x) (cv z) ({/}) eqeq1 orbi12d anbi12d elab sylib pm3.27d ord con1d imp n (\ A (cv z)) (\ A (U. (cv y))) ssfi (cv z) (cv y) elssuni (cv z) (U. (cv y)) A sscon syl sylan2 expcom (C_ (cv y) ({|} x (/\ (C_ (cv x) A) (\/ (E.e. n (om) (br (\ A (cv x)) (~~) (cv n))) (= (cv x) ({/})))))) (-. (= (cv z) ({/}))) ad2antlr mpd exp31 r19.23adv (cv y) uni0b (cv y) ({} ({/})) z dfss3 z ({/}) elsn z (cv y) ralbii 3bitr negbii z (cv y) (= (cv z) ({/})) rexnal bitr4 syl5ib con1d orrd jca y visset uniex (cv x) (U. (cv y)) A sseq1 (cv x) (U. (cv y)) A difeq2 (~~) (cv n) breq1d n (om) rexbidv (cv x) (U. (cv y)) ({/}) eqeq1 orbi12d anbi12d elab sylibr y ax-gen (cv y) A (cv z) ssinss1 (\/ (E.e. n (om) (br (\ A (cv y)) (~~) (cv n))) (= (cv y) ({/}))) (/\ (C_ (cv z) A) (\/ (E.e. n (om) (br (\ A (cv z)) (~~) (cv n))) (= (cv z) ({/})))) ad2antrr n (\ A (cv y)) (\ A (cv z)) unfi A (cv y) (cv z) difindi (~~) (cv n) breq1i n (om) rexbii sylibr (E.e. n (om) (br (\ A (i^i (cv y) (cv z))) (~~) (cv n))) (= (i^i (cv y) (cv z)) ({/})) orc syl (cv y) ({/}) (cv z) ineq1 ({/}) (cv z) incom (cv z) in0 eqtr syl6eq (= (i^i (cv y) (cv z)) ({/})) (E.e. n (om) (br (\ A (i^i (cv y) (cv z))) (~~) (cv n))) olc syl (cv z) ({/}) (cv y) ineq2 (cv y) in0 syl6eq (= (i^i (cv y) (cv z)) ({/})) (E.e. n (om) (br (\ A (i^i (cv y) (cv z))) (~~) (cv n))) olc syl ccase2 (C_ (cv y) A) (C_ (cv z) A) ad2ant2l jca y visset (cv x) (cv y) A sseq1 (cv x) (cv y) A difeq2 (~~) (cv n) breq1d n (om) rexbidv (cv x) (cv y) ({/}) eqeq1 orbi12d anbi12d elab z visset (cv x) (cv z) A sseq1 (cv x) (cv z) A difeq2 (~~) (cv n) breq1d n (om) rexbidv (cv x) (cv z) ({/}) eqeq1 orbi12d anbi12d elab anbi12i y visset (cv z) inex1 (cv x) (i^i (cv y) (cv z)) A sseq1 (cv x) (i^i (cv y) (cv z)) A difeq2 (\ A (cv x)) (\ A (i^i (cv y) (cv z))) (~~) (cv n) breq1 n (om) rexbidv syl (cv x) (i^i (cv y) (cv z)) ({/}) eqeq1 orbi12d anbi12d elab 3imtr4 rgen2 mpbir2an)) thm (fctop2 ((n x) (A n) (A x)) ((indistop.1 (e. A (V)))) (e. ({|} x (/\ (C_ (cv x) A) (\/ (br (\ A (cv x)) (~<) (om)) (= (cv x) ({/}))))) (Top)) ((\ A (cv x)) n isfinite (= (cv x) ({/})) orbi1i (C_ (cv x) A) anbi2i x abbii indistop.1 x n fctop eqeltr)) thm (cctop ((x y) (x z) (A x) (y z) (A y) (A z)) ((indistop.1 (e. A (V)))) (e. ({|} x (/\ (C_ (cv x) A) (\/ (br (\ A (cv x)) (~<_) (om)) (= (cv x) ({/}))))) (Top)) (indistop.1 A (V) x (\/ (br (\ A (cv x)) (~<_) (om)) (= (cv x) ({/}))) abssexg ax-mp ({|} x (/\ (C_ (cv x) A) (\/ (br (\ A (cv x)) (~<_) (om)) (= (cv x) ({/}))))) (V) y z istopg ax-mp (cv y) ({|} x (/\ (C_ (cv x) A) (\/ (br (\ A (cv x)) (~<_) (om)) (= (cv x) ({/}))))) uniss x (/\ (C_ (cv x) A) (\/ (br (\ A (cv x)) (~<_) (om)) (= (cv x) ({/})))) rabab unieqi (C_ (cv x) A) (\/ (br (\ A (cv x)) (~<_) (om)) (= (cv x) ({/}))) pm3.26 (e. (cv x) (V)) a1i ss2rabi ({e.|} x (V) (/\ (C_ (cv x) A) (\/ (br (\ A (cv x)) (~<_) (om)) (= (cv x) ({/}))))) ({e.|} x (V) (C_ (cv x) A)) uniss ax-mp indistop.1 A (V) x unimax ax-mp sseqtr eqsstr3 (U. (cv y)) (U. ({|} x (/\ (C_ (cv x) A) (\/ (br (\ A (cv x)) (~<_) (om)) (= (cv x) ({/})))))) A sstr mpan2 syl (cv y) ({|} x (/\ (C_ (cv x) A) (\/ (br (\ A (cv x)) (~<_) (om)) (= (cv x) ({/}))))) (cv z) ssel2 z visset (cv x) (cv z) A sseq1 (cv x) (cv z) A difeq2 (~<_) (om) breq1d (cv x) (cv z) ({/}) eqeq1 orbi12d anbi12d elab sylib pm3.27d ord con1d imp (\ A (U. (cv y))) (\ A (cv z)) (om) domtr (C_ (cv y) ({|} x (/\ (C_ (cv x) A) (\/ (br (\ A (cv x)) (~<_) (om)) (= (cv x) ({/})))))) (e. (cv z) (cv y)) pm3.27 (-. (= (cv z) ({/}))) (br (\ A (cv z)) (~<_) (om)) ad2antrr (cv z) (cv y) elssuni syl (cv z) (U. (cv y)) A sscon indistop.1 A (V) (cv z) difexg ax-mp (\ A (cv z)) (V) (\ A (U. (cv y))) ssdom2g ax-mp 3syl (/\ (/\ (C_ (cv y) ({|} x (/\ (C_ (cv x) A) (\/ (br (\ A (cv x)) (~<_) (om)) (= (cv x) ({/})))))) (e. (cv z) (cv y))) (-. (= (cv z) ({/})))) (br (\ A (cv z)) (~<_) (om)) pm3.27 sylanc mpdan exp31 r19.23adv (cv y) uni0b (cv y) ({} ({/})) z dfss3 z ({/}) elsn z (cv y) ralbii 3bitr negbii z (cv y) (= (cv z) ({/})) rexnal bitr4 syl5ib con1d orrd jca y visset uniex (cv x) (U. (cv y)) A sseq1 (cv x) (U. (cv y)) A difeq2 (~<_) (om) breq1d (cv x) (U. (cv y)) ({/}) eqeq1 orbi12d anbi12d elab sylibr y ax-gen (cv y) A (cv z) ssinss1 (\/ (br (\ A (cv y)) (~<_) (om)) (= (cv y) ({/}))) (/\ (C_ (cv z) A) (\/ (br (\ A (cv z)) (~<_) (om)) (= (cv z) ({/})))) ad2antrr (\ A (cv y)) (\ A (cv z)) unctb A (cv y) (cv z) difindi syl5eqbr (br (\ A (i^i (cv y) (cv z))) (~<_) (om)) (= (i^i (cv y) (cv z)) ({/})) orc syl (cv y) ({/}) (cv z) ineq1 ({/}) (cv z) incom (cv z) in0 eqtr syl6eq (= (i^i (cv y) (cv z)) ({/})) (br (\ A (i^i (cv y) (cv z))) (~<_) (om)) olc syl (cv z) ({/}) (cv y) ineq2 (cv y) in0 syl6eq (= (i^i (cv y) (cv z)) ({/})) (br (\ A (i^i (cv y) (cv z))) (~<_) (om)) olc syl ccase2 (C_ (cv y) A) (C_ (cv z) A) ad2ant2l jca y visset (cv x) (cv y) A sseq1 (cv x) (cv y) A difeq2 (~<_) (om) breq1d (cv x) (cv y) ({/}) eqeq1 orbi12d anbi12d elab z visset (cv x) (cv z) A sseq1 (cv x) (cv z) A difeq2 (~<_) (om) breq1d (cv x) (cv z) ({/}) eqeq1 orbi12d anbi12d elab anbi12i y visset (cv z) inex1 (cv x) (i^i (cv y) (cv z)) A sseq1 (cv x) (i^i (cv y) (cv z)) A difeq2 (~<_) (om) breq1d (cv x) (i^i (cv y) (cv z)) ({/}) eqeq1 orbi12d anbi12d elab 3imtr4 rgen2 mpbir2an)) thm (indistps () ((indistop.1 (e. A (V)))) (e. (<,> A ({,} ({/}) A)) (TopSp)) (0ex indistop.1 unipr ({/}) A uncom A un0 3eqtr ({,} ({/}) A) eqid (U. ({,} ({/}) A)) A ({,} ({/}) A) ({,} ({/}) A) opeq12 mp2an indistop.1 indistop ({,} ({/}) A) eltopsp mpbir eqeltrr)) thm (distps () ((indistop.1 (e. A (V)))) (e. (<,> A (P~ A)) (TopSp)) (A unipw (P~ A) eqid (U. (P~ A)) A (P~ A) (P~ A) opeq12 mp2an indistop.1 distop (P~ A) eltopsp mpbir eqeltrr)) thm (clsdval ((w x) (w z) (J w) (x z) (J x) (J z) (X w) (X x) (X z)) ((clsdval.1 (= X (U. J)))) (-> (e. J (Top)) (= (` (Cls) J) ({|} x (/\ (C_ (cv x) X) (e. (\ X (cv x)) J))))) (J (Top) uniexg clsdval.1 syl5eqel X (V) x (e. (\ X (cv x)) J) abssexg syl (cv z) J unieq clsdval.1 syl6eqr (cv x) sseq2d (cv z) J unieq clsdval.1 syl6eqr (cv x) difeq1d (= (cv z) J) id eleq12d anbi12d x abbidv z w x df-clsd (V) fvopab4g mpdan)) thm (ntrfval ((v w) (v x) (v y) (v z) (J v) (w x) (w y) (w z) (J w) (x y) (x z) (J x) (y z) (J y) (J z) (X w) (X x) (X y) (X z)) ((clsdval.1 (= X (U. J)))) (-> (e. J (Top)) (= (` (int) J) ({<,>|} x y (/\ (C_ (cv x) X) (= (cv y) (U. ({e.|} v J (C_ (cv v) (cv x))))))))) (J (Top) uniexg clsdval.1 syl5eqel X (V) pwexg (P~ X) (V) x y (U. ({e.|} v J (C_ (cv v) (cv x)))) funopabex2g 3syl x visset X elpw (= (cv y) (U. ({e.|} v J (C_ (cv v) (cv x))))) anbi1i x y opabbii syl5eqelr (cv z) J unieq clsdval.1 syl6eqr (cv x) sseq2d (cv z) J v (C_ (cv v) (cv x)) rabeq unieqd (cv y) eqeq2d anbi12d x y opabbidv z w x y v df-ntr (V) fvopab4g mpdan)) thm (clsfval ((v w) (v x) (v y) (v z) (J v) (w x) (w y) (w z) (J w) (x y) (x z) (J x) (y z) (J y) (J z) (X w) (X x) (X y) (X z)) ((clsdval.1 (= X (U. J)))) (-> (e. J (Top)) (= (` (cls) J) ({<,>|} x y (/\ (C_ (cv x) X) (= (cv y) (|^| ({e.|} v (` (Cls) J) (C_ (cv x) (cv v))))))))) (J (Top) uniexg clsdval.1 syl5eqel X (V) pwexg (P~ X) (V) x y (|^| ({e.|} v (` (Cls) J) (C_ (cv x) (cv v)))) funopabex2g 3syl x visset X elpw (= (cv y) (|^| ({e.|} v (` (Cls) J) (C_ (cv x) (cv v))))) anbi1i x y opabbii syl5eqelr (cv z) J unieq clsdval.1 syl6eqr (cv x) sseq2d (cv z) J (Cls) fveq2 (` (Cls) (cv z)) (` (Cls) J) v (C_ (cv x) (cv v)) rabeq syl inteqd (cv y) eqeq2d anbi12d x y opabbidv z w x y v df-cls (V) fvopab4g mpdan)) thm (isclsd ((J x) (S x) (X x)) ((isclsd.1 (= X (U. J)))) (-> (e. J (Top)) (<-> (e. S (` (Cls) J)) (/\ (C_ S X) (e. (\ X S) J)))) (isclsd.1 x clsdval S eleq2d S ({|} x (/\ (C_ (cv x) X) (e. (\ X (cv x)) J))) elisset (e. J (Top)) adantl S X (V) ssexg ancoms J (Top) uniexg isclsd.1 syl5eqel sylan (e. (\ X S) J) adantrr (cv x) S X sseq1 (cv x) S X difeq2 J eleq1d anbi12d (V) elabg pm5.21nd bitrd)) thm (isclsd2 () ((isclsd.1 (= X (U. J)))) (-> (/\ (e. J (Top)) (C_ S X)) (<-> (e. S (` (Cls) J)) (e. (\ X S) J))) (isclsd.1 S isclsd (C_ S X) (e. (\ X S) J) ibar bicomd sylan9bb)) thm (clsdss () ((isclsd.1 (= X (U. J)))) (-> (/\ (e. J (Top)) (e. S (` (Cls) J))) (C_ S X)) (isclsd.1 S isclsd (C_ S X) (e. (\ X S) J) pm3.26 syl6bi imp)) thm (clsdopen () ((isclsd.1 (= X (U. J)))) (-> (/\ (e. J (Top)) (e. S (` (Cls) J))) (e. (\ X S) J)) (isclsd.1 S isclsd (C_ S X) (e. (\ X S) J) pm3.27 syl6bi imp)) thm (isopen2 () ((isclsd.1 (= X (U. J)))) (-> (/\ (e. J (Top)) (C_ S X)) (<-> (e. S J) (e. (\ X S) (` (Cls) J)))) (X S difss isclsd.1 (\ X S) isclsd2 mpan2 S X dfss4 biimp J eleq1d sylan9bb bicomd)) thm (openclsd () ((isclsd.1 (= X (U. J)))) (-> (/\ (e. J (Top)) (e. S J)) (e. (\ X S) (` (Cls) J))) ((e. J (Top)) (e. S J) pm3.27 isclsd.1 S eltopss isclsd.1 S isopen2 syldan mpbid)) thm (1clsd () ((isclsd.1 (= X (U. J)))) (-> (e. J (Top)) (e. X (` (Cls) J))) (J 0opnt X difid syl5eqel X ssid jctil isclsd.1 X isclsd mpbird)) thm (ntrval ((x y) (x z) (J x) (y z) (J y) (J z) (S x) (S y) (S z) (X x) (X y) (X z)) ((isclsd.1 (= X (U. J)))) (-> (/\ (e. J (Top)) (C_ S X)) (= (` (` (int) J) S) (U. ({e.|} x J (C_ (cv x) S))))) (isclsd.1 y z x ntrfval (C_ S X) adantr y visset X elpw (= (cv z) (U. ({e.|} x J (C_ (cv x) (cv y))))) anbi1i y z opabbii syl6eqr S fveq1d (cv y) S (cv x) sseq2 x J rabbisdv unieqd ({<,>|} y z (/\ (e. (cv y) (P~ X)) (= (cv z) (U. ({e.|} x J (C_ (cv x) (cv y))))))) eqid (V) fvopab4g J (Top) uniexg isclsd.1 syl5eqel X (V) S elpw2g biimprd syl imp J (Top) x (C_ (cv x) S) rabexg ({e.|} x J (C_ (cv x) S)) (V) uniexg syl (C_ S X) adantr sylanc eqtrd)) thm (clsval ((x y) (x z) (J x) (y z) (J y) (J z) (S x) (S y) (S z) (X x) (X y) (X z)) ((isclsd.1 (= X (U. J)))) (-> (/\ (e. J (Top)) (C_ S X)) (= (` (` (cls) J) S) (|^| ({e.|} x (` (Cls) J) (C_ S (cv x)))))) (isclsd.1 y z x clsfval (C_ S X) adantr y visset X elpw (= (cv z) (|^| ({e.|} x (` (Cls) J) (C_ (cv y) (cv x))))) anbi1i y z opabbii syl6eqr S fveq1d (cv y) S (cv x) sseq1 x (` (Cls) J) rabbisdv inteqd ({<,>|} y z (/\ (e. (cv y) (P~ X)) (= (cv z) (|^| ({e.|} x (` (Cls) J) (C_ (cv y) (cv x))))))) eqid (V) fvopab4g X (V) S elpw2g biimpar J (Top) uniexg isclsd.1 syl5eqel sylan (cv x) X S sseq2 (` (Cls) J) rcla4ev isclsd.1 1clsd sylan x (` (Cls) J) (C_ S (cv x)) intexrab sylib sylanc eqtrd)) thm (clsval2 ((x y) (x z) (J x) (y z) (J y) (J z) (S x) (S y) (S z) (X x) (X y) (X z)) ((isclsd.1 (= X (U. J)))) (-> (/\ (e. J (Top)) (C_ S X)) (= (` (` (cls) J) S) (\ X (` (` (int) J) (\ X S))))) (isclsd.1 (cv z) clsdopen (cv y) (\ X (cv z)) (\ X S) sseq1 (cv y) (\ X (cv z)) (cv x) eleq2 negbid imbi12d J rcla4v syl (e. (cv x) X) adantrl S (cv z) X sscon (e. (cv x) X) a1i (cv x) X (cv z) neldif ex imim12d (e. J (Top)) (e. (cv z) (` (Cls) J)) ad2antrl syld exp32 com34 imp3a r19.21adv (C_ S X) adantr isclsd.1 1clsd (cv z) X S sseq2 (cv z) X (cv x) eleq2 imbi12d (` (Cls) J) rcla4v syl com23 imp isclsd.1 (cv y) openclsd (cv z) (\ X (cv y)) S sseq2 (cv z) (\ X (cv y)) (cv x) eleq2 imbi12d (` (Cls) J) rcla4v syl (C_ S X) adantlr (cv y) X S ssconb biimpd isclsd.1 (cv y) eltopss sylan an1rs (cv x) X (cv y) eldifn (/\ (/\ (e. J (Top)) (C_ S X)) (e. (cv y) J)) a1i imim12d syld ex com23 r19.21adv jcad impbid (cv x) X (U. ({e.|} y J (C_ (cv y) (\ X S)))) eldif y J (/\ (e. (cv x) (cv y)) (C_ (cv y) (\ X S))) ralnex (C_ (cv y) (\ X S)) (e. (cv x) (cv y)) imnan (C_ (cv y) (\ X S)) (e. (cv x) (cv y)) ancom negbii bitr y J ralbii (cv x) y J (C_ (cv y) (\ X S)) elunirab negbii 3bitr4r (e. (cv x) X) anbi2i bitr x visset z (` (Cls) J) (C_ S (cv z)) elintrab 3bitr4g eqrdv isclsd.1 (\ X S) y ntrval X S difss (C_ S X) a1i sylan2 X difeq2d isclsd.1 S z clsval 3eqtr4rd)) thm (ntropn ((J x) (S x) (X x)) ((ntropn.1 (= X (U. J)))) (-> (/\ (e. J (Top)) (C_ S X)) (e. (` (` (int) J) S) J)) (ntropn.1 S x ntrval x J (C_ (cv x) S) ssrab2 J ({e.|} x J (C_ (cv x) S)) uniopnt mpan2 (C_ S X) adantr eqeltrd)) thm (iinclsd ((A x) (J x)) () (-> (/\/\ (e. J (Top)) (=/= A ({/})) (A.e. x A (e. B (` (Cls) J)))) (e. (|^|_ x A B) (` (Cls) J))) ((U. J) eqid B clsdss ex B (U. J) dfss4 syl6ib x A r19.20sdv imp x A (\ (U. J) (\ (U. J) B)) B iineq2 syl (=/= A ({/})) 3adant2 A x (U. J) (\ (U. J) B) iindif2 (e. J (Top)) (A.e. x A (e. B (` (Cls) J))) 3ad2ant2 eqtr3d (U. J) eqid B isclsd (C_ B (U. J)) (e. (\ (U. J) B) J) pm3.27 syl6bi x A r19.20sdv imp J x A (\ (U. J) B) iunopnt syldan x A (\ (U. J) B) (U. J) iunss (U. J) B difss (e. (cv x) A) a1i mprgbir (U. J) eqid (U_ x A (\ (U. J) B)) isopen2 mpan2 (A.e. x A (e. B (` (Cls) J))) adantr mpbid (=/= A ({/})) 3adant2 eqeltrd)) thm (intclsd ((A x) (J x)) () (-> (/\/\ (e. J (Top)) (=/= A ({/})) (C_ A (` (Cls) J))) (e. (|^| A) (` (Cls) J))) (J A x (cv x) iinclsd A (` (Cls) J) x dfss3 syl3an3b A x intiin syl5eqel)) thm (isopen3 ((J x) (S x) (X x)) ((isopen3.1 (= X (U. J)))) (-> (/\ (e. J (Top)) (C_ S X)) (<-> (e. S J) (= (` (` (int) J) S) S))) (isopen3.1 S x ntrval S J x unimax sylan9eq ex (` (` (int) J) S) S J eleq1 isopen3.1 S ntropn syl5bi com12 impbid)) thm (ntrtop () ((isopen3.1 (= X (U. J)))) (-> (e. J (Top)) (= (` (` (int) J) X) X)) (isopen3.1 1open X ssid isopen3.1 X isopen3 mpan2 mpbid)) thm (isnei ((J u) (J w) (J j) (J z) (J g) (J y) (u w) (j u) (u z) (g u) (u y) (j w) (w z) (g w) (w y) (j z) (g j) (j y) (g z) (y z) (g y) (X u) (X w) (X j) (X z) (X g) (X y) (A g) (A w) (A z) (B g) (B w) (B z)) ((isnei.1 (= X (U. J))) (isnei.2 (e. A (V))) (isnei.3 (e. B (V))) (isnei.4 (e. J (Top)))) (<-> (br A (` (nei) J) B) (/\ (C_ A X) (E.e. g J (/\ (C_ B (cv g)) (C_ (cv g) A))))) (isnei.2 isnei.3 (cv w) A X sseq1 (cv w) A (cv g) sseq2 (C_ (cv z) (cv g)) anbi2d g J rexbidv anbi12d (cv z) B (cv g) sseq1 (C_ (cv g) A) anbi1d g J rexbidv (C_ A X) anbi2d isnei.4 w z (/\ (C_ (cv w) X) (E.e. g J (/\ (C_ (cv z) (cv g)) (C_ (cv g) (cv w))))) u df-opab (e. (cv u) (X. (P~ X) (P~ X))) (E. w (E. z (/\ (= (cv u) (<,> (cv w) (cv z))) (/\ (C_ (cv w) X) (E.e. g J (/\ (C_ (cv z) (cv g)) (C_ (cv g) (cv w)))))))) pm3.27 w visset X elpw biimpr g J (/\ (C_ (cv z) (cv g)) (C_ (cv g) (cv w))) df-rex (C_ (cv z) (cv g)) (C_ (cv g) (cv w)) pm3.26 (e. (cv g) J) anim2i g 19.22i (cv z) (cv g) J ssuni isnei.1 syl6ssr z visset X elpw sylibr ancoms g 19.23aiv syl sylbi anim12i (= (cv u) (<,> (cv w) (cv z))) anim2i w z 19.22i2 (cv u) (P~ X) (P~ X) w z elxp sylibr ancri impbi u abbii isnei.1 isnei.4 elisseti uniex eqeltr pwex isnei.1 isnei.4 elisseti uniex eqeltr pwex xpex u (E. w (E. z (/\ (= (cv u) (<,> (cv w) (cv z))) (/\ (C_ (cv w) X) (E.e. g J (/\ (C_ (cv z) (cv g)) (C_ (cv g) (cv w)))))))) zfausab eqeltrr eqeltr (cv j) J unieq isnei.1 syl6eqr (cv w) sseq2d (cv j) J g (/\ (C_ (cv z) (cv g)) (C_ (cv g) (cv w))) rexeq1 anbi12d w z opabbidv j y w z g df-nei (V) fvopab4g mp2an brab)) thm (scnei ((A x) (B x) (C x) (J x)) ((scnei.1 (e. A (V))) (scnei.2 (e. B (V))) (scnei.3 (e. C (V))) (scnei.4 (e. J (Top)))) (-> (/\ (br A (` (nei) J) B) (C_ C B)) (br A (` (nei) J) C)) ((U. J) eqid scnei.1 scnei.2 scnei.4 x isnei C B (cv x) sstr2 (e. (cv x) J) adantr (C_ (cv x) A) anim1d r19.22dva impcom (C_ A (U. J)) anim2i exp32 imp (U. J) eqid scnei.1 scnei.3 scnei.4 x isnei syl6ibr sylbi imp)) thm (neibel ((A g) (B g) (J g)) ((neibel.1 (e. J (Top))) (neibel.2 (e. A (V))) (neibel.3 (e. B (V)))) (-> (br A (` (nei) J) ({} B)) (e. B A)) ((U. J) eqid neibel.2 B snex neibel.1 g isnei g J (/\ (C_ ({} B) (cv g)) (C_ (cv g) A)) df-rex (cv g) A B ssel neibel.3 (cv g) snss biimpr syl5com imp (C_ A (U. J)) a1d (e. (cv g) J) adantl g 19.23aiv sylbi impcom sylbi)) thm (intnei ((J g) (J j) (J k) (g j) (g k) (j k) (A g) (A j) (A k) (B g) (B j) (B k) (C g) (C j) (C k)) ((intnei.1 (e. J (Top))) (intnei.2 (e. A (V))) (intnei.3 (e. B (V))) (intnei.4 (e. C (V)))) (-> (/\ (br A (` (nei) J) B) (br C (` (nei) J) B)) (br (i^i A C) (` (nei) J) B)) ((U. J) eqid intnei.2 intnei.3 intnei.1 g isnei A (U. J) C ssinss1 (C_ C (U. J)) adantl (cv g) (cv j) B sseq2 (cv g) (cv j) A sseq1 anbi12d J cbvrexv intnei.1 J (cv j) (cv k) inopnt mp3an1 B (cv j) (cv k) ssin biimp (cv j) A (cv k) C ss2in anim12i an4s anim12i exp43 com23 imp41 (cv g) (i^i (cv j) (cv k)) B sseq2 (cv g) (i^i (cv j) (cv k)) (i^i A C) sseq1 anbi12d J rcla4ev syl exp41 r19.23aiv r19.23adv (cv g) (cv k) B sseq2 (cv g) (cv k) C sseq1 anbi12d J cbvrexv syl5ib sylbi imp anim12i exp43 com24 imp com13 imp (U. J) eqid intnei.4 intnei.3 intnei.1 g isnei (U. J) eqid intnei.2 C inex1 intnei.3 intnei.1 g isnei 3imtr4g sylbi imp)) thm (opneit ((J g) (A g)) ((opneit.1 (e. J (Top))) (opneit.2 (e. A (V)))) (<-> (br A (` (nei) J) A) (e. A J)) ((U. J) eqid opneit.2 opneit.2 opneit.1 g isnei (cv g) A eqcom A (cv g) eqss bitr biimpr g J r19.22si (C_ A (U. J)) adantl sylbi A (cv g) J eleq1 eqcoms A J elssuni syl6bir com12 r19.23aiv (cv g) A eqcom A (cv g) eqss bitr biimp g J r19.22si jca (U. J) eqid opneit.2 opneit.2 opneit.1 g isnei sylibr impbi g J (= (cv g) A) df-rex bitr g (e. (cv g) J) (= (cv g) A) exancom opneit.2 (cv g) A J eleq1 ceqsexv 3bitr)) thm (onp ((A g) (B g) (J g)) ((onp.1 (e. J (Top))) (onp.2 (e. A J)) (onp.3 (e. B (V)))) (-> (C_ B A) (br A (` (nei) J) B)) (onp.2 elisseti (cv g) A B sseq2 biimpcd (e. (cv g) J) (C_ (cv g) A) ad2antrl com12 onp.2 A J (cv g) eleq1a ax-mp (C_ B A) adantr A (cv g) B sseq2 biimpd eqcoms imp (cv g) A eqimss (C_ B A) adantr jca jca ex impbid cla4ev g J (/\ (C_ B (cv g)) (C_ (cv g) A)) df-rex sylibr onp.2 A J elssuni ax-mp jctil (U. J) eqid onp.2 elisseti onp.3 onp.1 g isnei sylibr)) thm (ssneip ((A g) (B g) (C g) (J g) (X g)) ((ssneip.1 (= X (U. J))) (ssneip.2 (e. A (V))) (ssneip.3 (e. B (V))) (ssneip.4 (e. C (V))) (ssneip.5 (e. J (Top)))) (-> (/\/\ (C_ A B) (C_ B X) (br A (` (nei) J) C)) (br B (` (nei) J) C)) ((C_ B X) (C_ A X) ax-1 (C_ A B) adantr (cv g) A B sstr2 com12 (C_ B X) adantl (C_ C (cv g)) anim2d g J r19.22sdv anim12d ancoms ssneip.1 ssneip.2 ssneip.4 ssneip.5 g isnei ssneip.1 ssneip.3 ssneip.4 ssneip.5 g isnei 3imtr4g 3impia)) thm (enonei ((J x) (J y) (x y) (A x) (A y) (B x) (B y)) ((enonei.1 (e. J (Top))) (enonei.2 (e. A (V))) (enonei.3 (e. B (V)))) (-> (br B (` (nei) J) A) (E. x (/\ (br (cv x) (` (nei) J) A) (A. y (-> (C_ (cv y) (cv x)) (br B (` (nei) J) (cv y))))))) ((U. J) eqid enonei.3 enonei.2 enonei.1 x isnei enonei.1 x visset opneit x visset x visset enonei.2 enonei.1 scnei ex sylbir com12 (C_ (cv x) B) adantr impcom (C_ B (U. J)) adantl (U. J) eqid x visset enonei.3 y visset enonei.1 ssneip 3com12 3exp imp x visset x visset y visset enonei.1 scnei enonei.1 x visset opneit sylanbr syl5 exp4b com23 imp32 y 19.21aiv (C_ A (cv x)) adantrrl jca ex x 19.22dv imp x J (/\ (C_ A (cv x)) (C_ (cv x) B)) df-rex sylan2b sylbi)) thm (ucnp () ((uncp.1 (= X (U. J))) (ucnp.2 (e. J (Top))) (ucnp.3 (e. A (V)))) (-> (C_ A X) (br X (` (nei) J) A)) (ucnp.2 uncp.1 ucnp.2 J unitopt ax-mp eqeltr ucnp.3 onp)) thm (cnfval ((f j) (f k) (f y) (f z) (J f) (j k) (j y) (j z) (J j) (k y) (k z) (J k) (y z) (J y) (J z) (K f) (K j) (K k) (K y) (K z) (X f) (X j) (X k) (X y) (X z) (Y f) (Y k) (Y y) (Y z)) ((cnfval.1 (= X (U. J))) (cnfval.2 (= Y (U. K)))) (-> (/\ (e. J (Top)) (e. K (Top))) (= (opr J (Cn) K) ({e.|} f (opr Y (^m) X) (A.e. y K (e. (" (`' (cv f)) (cv y)) J))))) (Y (^m) X oprex f (A.e. y K (e. (" (`' (cv f)) (cv y)) J)) rabex (cv j) J unieq cnfval.1 syl6eqr (U. (cv k)) (^m) opreq2d (opr (U. (cv k)) (^m) (U. (cv j))) (opr (U. (cv k)) (^m) X) f (A.e. y (cv k) (e. (" (`' (cv f)) (cv y)) (cv j))) rabeq syl (cv j) J (" (`' (cv f)) (cv y)) eleq2 y (cv k) ralbidv f (opr (U. (cv k)) (^m) X) rabbisdv eqtrd (cv k) K unieq cnfval.2 syl6eqr (^m) X opreq1d (opr (U. (cv k)) (^m) X) (opr Y (^m) X) f (A.e. y (cv k) (e. (" (`' (cv f)) (cv y)) J)) rabeq syl (cv k) K y (e. (" (`' (cv f)) (cv y)) J) raleq1 f (opr Y (^m) X) rabbisdv eqtrd j k z f y df-cn oprabval2)) thm (cnpfval ((f j) (f k) (f v) (f w) (f x) (f y) (f z) (J f) (j k) (j v) (j w) (j x) (j y) (j z) (J j) (k v) (k w) (k x) (k y) (k z) (J k) (v w) (v x) (v y) (v z) (J v) (w x) (w y) (w z) (J w) (x y) (x z) (J x) (y z) (J y) (J z) (K f) (K j) (K k) (K w) (K x) (K y) (K z) (X f) (X j) (X k) (X x) (X y) (X z) (Y f) (Y k) (Y y) (Y z)) ((cnfval.1 (= X (U. J))) (cnfval.2 (= Y (U. K)))) (-> (/\ (e. J (Top)) (e. K (Top))) (= (opr J (CnP) K) ({<,>|} x y (/\ (e. (cv x) X) (= (cv y) ({e.|} f (opr Y (^m) X) (A.e. w K (-> (e. (` (cv f) (cv x)) (cv w)) (E.e. v J (/\ (e. (cv x) (cv v)) (C_ (" (cv f) (cv v)) (cv w)))))))))))) ((cv j) J unieq cnfval.1 syl6eqr (cv x) eleq2d (cv j) J unieq cnfval.1 syl6eqr (U. (cv k)) (^m) opreq2d (opr (U. (cv k)) (^m) (U. (cv j))) (opr (U. (cv k)) (^m) X) f (A.e. w (cv k) (-> (e. (` (cv f) (cv x)) (cv w)) (E.e. v (cv j) (/\ (e. (cv x) (cv v)) (C_ (" (cv f) (cv v)) (cv w)))))) rabeq syl (cv j) J v (/\ (e. (cv x) (cv v)) (C_ (" (cv f) (cv v)) (cv w))) rexeq1 (e. (` (cv f) (cv x)) (cv w)) imbi2d w (cv k) ralbidv f (opr (U. (cv k)) (^m) X) rabbisdv eqtrd (cv y) eqeq2d anbi12d x y opabbidv (cv k) K unieq cnfval.2 syl6eqr (^m) X opreq1d (opr (U. (cv k)) (^m) X) (opr Y (^m) X) f (A.e. w (cv k) (-> (e. (` (cv f) (cv x)) (cv w)) (E.e. v J (/\ (e. (cv x) (cv v)) (C_ (" (cv f) (cv v)) (cv w)))))) rabeq syl (cv k) K w (-> (e. (` (cv f) (cv x)) (cv w)) (E.e. v J (/\ (e. (cv x) (cv v)) (C_ (" (cv f) (cv v)) (cv w))))) raleq1 f (opr Y (^m) X) rabbisdv eqtrd (cv y) eqeq2d (e. (cv x) X) anbi2d x y opabbidv j k z x y f w v df-cnp (V) oprabval2g (e. J (Top)) (e. K (Top)) pm3.26 (e. J (Top)) (e. K (Top)) pm3.27 J (Top) uniexg cnfval.1 (V) eleq1i biimpr X (V) x y ({e.|} f (opr Y (^m) X) (A.e. w K (-> (e. (` (cv f) (cv x)) (cv w)) (E.e. v J (/\ (e. (cv x) (cv v)) (C_ (" (cv f) (cv v)) (cv w))))))) funopabex2g 3syl (e. K (Top)) adantr syl3anc)) thm (iscn ((f y) (F f) (F y) (J f) (J y) (K y) (K f) (X f) (X y) (Y f) (Y y)) ((iscn.1 (= X (U. J))) (iscn.2 (= Y (U. K)))) (-> (/\ (e. J (Top)) (e. K (Top))) (<-> (e. F (opr J (Cn) K)) (/\ (:--> F X Y) (A.e. y K (e. (" (`' F) (cv y)) J))))) (iscn.1 iscn.2 f y cnfval F eleq2d Y (V) X (V) F elmapg K (Top) uniexg iscn.2 syl5eqel J (Top) uniexg iscn.1 syl5eqel syl2an ancoms (A.e. y K (e. (" (`' F) (cv y)) J)) anbi1d (cv f) F cnveq (`' (cv f)) (`' F) (cv y) imaeq1 syl J eleq1d y K ralbidv (opr Y (^m) X) elrab syl5bb bitrd)) thm (cnpval ((f x) (f y) (x y) (f v) (f w) (J f) (v w) (v x) (v y) (J v) (w x) (w y) (J w) (J x) (J y) (K y) (K w) (K v) (K f) (P f) (P v) (P w) (P x) (P y) (X f) (X v) (X w) (X y) (Y f) (Y v) (Y w) (Y y)) ((iscn.1 (= X (U. J))) (iscn.2 (= Y (U. K)))) (-> (/\/\ (e. J (Top)) (e. K (Top)) (e. P X)) (= (` (opr J (CnP) K) P) ({e.|} f (opr Y (^m) X) (A.e. y K (-> (e. (` (cv f) P) (cv y)) (E.e. x J (/\ (e. P (cv x)) (C_ (" (cv f) (cv x)) (cv y))))))))) (iscn.1 iscn.2 w v f y x cnpfval P fveq1d (e. P X) 3adant3 (cv w) P (cv f) fveq2 (cv y) eleq1d (cv w) P (cv x) eleq1 (C_ (" (cv f) (cv x)) (cv y)) anbi1d x J rexbidv imbi12d y K ralbidv f (opr Y (^m) X) rabbisdv ({<,>|} w v (/\ (e. (cv w) X) (= (cv v) ({e.|} f (opr Y (^m) X) (A.e. y K (-> (e. (` (cv f) (cv w)) (cv y)) (E.e. x J (/\ (e. (cv w) (cv x)) (C_ (" (cv f) (cv x)) (cv y)))))))))) eqid Y (^m) X oprex f (A.e. y K (-> (e. (` (cv f) P) (cv y)) (E.e. x J (/\ (e. P (cv x)) (C_ (" (cv f) (cv x)) (cv y)))))) rabex fvopab4 (e. J (Top)) (e. K (Top)) 3ad2ant3 eqtrd)) thm (iscnp ((f x) (f y) (F f) (x y) (F x) (F y) (J f) (J x) (J y) (K y) (K f) (P f) (P x) (P y) (X f) (X y) (Y f) (Y y)) ((iscn.1 (= X (U. J))) (iscn.2 (= Y (U. K)))) (-> (/\/\ (e. J (Top)) (e. K (Top)) (e. P X)) (<-> (e. F (` (opr J (CnP) K) P)) (/\ (:--> F X Y) (A.e. y K (-> (e. (` F P) (cv y)) (E.e. x J (/\ (e. P (cv x)) (C_ (" F (cv x)) (cv y))))))))) (iscn.1 iscn.2 P f y x cnpval F eleq2d Y (V) X (V) F elmapg K (Top) uniexg iscn.2 syl5eqel J (Top) uniexg iscn.1 syl5eqel syl2an ancoms (A.e. y K (-> (e. (` F P) (cv y)) (E.e. x J (/\ (e. P (cv x)) (C_ (" F (cv x)) (cv y)))))) anbi1d (cv f) F P fveq1 (cv y) eleq1d (cv f) F (cv x) imaeq1 (cv y) sseq1d (e. P (cv x)) anbi2d x J rexbidv imbi12d y K ralbidv (opr Y (^m) X) elrab syl5bb (e. P X) 3adant3 bitrd)) thm (msrel ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (Rel (Met)) (x y (/\ (:--> (cv y) (X. (cv x) (cv x)) (RR)) (A.e. z (cv x) (A.e. w (cv x) (/\ (<-> (= (opr (cv z) (cv y) (cv w)) (0)) (= (cv z) (cv w))) (A.e. v (cv x) (br (opr (cv z) (cv y) (cv w)) (<_) (opr (opr (cv v) (cv y) (cv z)) (+) (opr (cv v) (cv y) (cv w))))))))) relopab x y z w v df-ms (Met) ({<,>|} x y (/\ (:--> (cv y) (X. (cv x) (cv x)) (RR)) (A.e. z (cv x) (A.e. w (cv x) (/\ (<-> (= (opr (cv z) (cv y) (cv w)) (0)) (= (cv z) (cv w))) (A.e. v (cv x) (br (opr (cv z) (cv y) (cv w)) (<_) (opr (opr (cv v) (cv y) (cv z)) (+) (opr (cv v) (cv y) (cv w)))))))))) releq ax-mp mpbir)) thm (ismsg ((v w) (v x) (v y) (v z) (X v) (w x) (w y) (w z) (X w) (x y) (x z) (X x) (y z) (X y) (X z) (D v) (D w) (D x) (D y) (D z)) () (-> (e. D A) (<-> (e. (<,> X D) (Met)) (/\ (:--> D (X. X X) (RR)) (A.e. x X (A.e. y X (/\ (<-> (= (opr (cv x) D (cv y)) (0)) (= (cv x) (cv y))) (A.e. z X (br (opr (cv x) D (cv y)) (<_) (opr (opr (cv z) D (cv x)) (+) (opr (cv z) D (cv y))))))))))) (X (Met) D df-br msrel X D brrelexi sylbir (e. D A) anim1i ancoms D A (X. X X) (RR) dmfex X X (V) xpexr (e. X (V)) oridm sylib syl (e. D A) (:--> D (X. X X) (RR)) pm3.26 jca (A.e. x X (A.e. y X (/\ (<-> (= (opr (cv x) D (cv y)) (0)) (= (cv x) (cv y))) (A.e. z X (br (opr (cv x) D (cv y)) (<_) (opr (opr (cv z) D (cv x)) (+) (opr (cv z) D (cv y)))))))) adantrr (cv w) X (cv w) xpeq1 (cv w) X X xpeq2 eqtrd (X. (cv w) (cv w)) (X. X X) (cv v) (RR) feq2 syl (cv w) X z (br (opr (cv x) (cv v) (cv y)) (<_) (opr (opr (cv z) (cv v) (cv x)) (+) (opr (cv z) (cv v) (cv y)))) raleq1 (<-> (= (opr (cv x) (cv v) (cv y)) (0)) (= (cv x) (cv y))) anbi2d y raleqd x raleqd anbi12d (cv v) D (X. X X) (RR) feq1 (cv v) D (cv x) (cv y) opreq (0) eqeq1d (= (cv x) (cv y)) bibi1d (cv v) D (cv x) (cv y) opreq (cv v) D (cv z) (cv x) opreq (cv v) D (cv z) (cv y) opreq (+) opreq12d (<_) breq12d z X ralbidv anbi12d x X y X 2ralbidv anbi12d (V) A opelopabg w v x y z df-ms (<,> X D) eleq2i syl5bb pm5.21nd)) thm (msflem ((x y) (x z) (y z) (v w) (v x) (v y) (v z) (D v) (w x) (w y) (w z) (D w) (D x) (D y) (D z) (M v) (M w) (X v) (X w) (X x) (X y) (X z)) ((msf.1 (= X (` (1st) M))) (msf.2 (= D (` (2nd) M)))) (-> (e. M (Met)) (/\ (:--> D (X. X X) (RR)) (A.e. x X (A.e. y X (/\ (<-> (= (opr (cv x) D (cv y)) (0)) (= (cv x) (cv y))) (A.e. z X (br (opr (cv x) D (cv y)) (<_) (opr (opr (cv z) D (cv x)) (+) (opr (cv z) D (cv y)))))))))) (w v x y z df-ms M eleq2i msf.1 (cv w) eqeq2i (cv w) X (cv w) xpeq1 (cv w) X X xpeq2 eqtrd (X. (cv w) (cv w)) (X. X X) (cv v) (RR) feq2 syl (cv w) X z (br (opr (cv x) (cv v) (cv y)) (<_) (opr (opr (cv z) (cv v) (cv x)) (+) (opr (cv z) (cv v) (cv y)))) raleq1 (<-> (= (opr (cv x) (cv v) (cv y)) (0)) (= (cv x) (cv y))) anbi2d y raleqd x raleqd anbi12d sylbir msf.2 (cv v) eqeq2i (cv v) D (X. X X) (RR) feq1 (cv v) D (cv x) (cv y) opreq (0) eqeq1d (= (cv x) (cv y)) bibi1d (cv v) D (cv x) (cv y) opreq (cv v) D (cv z) (cv x) opreq (cv v) D (cv z) (cv y) opreq (+) opreq12d (<_) breq12d z X ralbidv anbi12d x X y X 2ralbidv anbi12d sylbir elopabi sylbi)) thm (msf ((x y) (x z) (y z) (D x) (D y) (D z) (X x) (X y) (X z)) ((msf.1 (= X (` (1st) M))) (msf.2 (= D (` (2nd) M)))) (-> (e. M (Met)) (:--> D (X. X X) (RR))) (msf.1 msf.2 x y z msflem pm3.26d)) thm (mscl () ((msf.1 (= X (` (1st) M))) (msf.2 (= D (` (2nd) M)))) (-> (/\/\ (e. M (Met)) (e. A X) (e. B X)) (e. (opr A D B) (RR))) (D X X (RR) A B foprrn msf.1 msf.2 msf syl3an1)) thm (mseq0 ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z) (D x) (D y) (D z) (X x) (X y) (X z)) ((msf.1 (= X (` (1st) M))) (msf.2 (= D (` (2nd) M)))) (-> (/\/\ (e. M (Met)) (e. A X) (e. B X)) (<-> (= (opr A D B) (0)) (= A B))) (msf.1 msf.2 x y z msflem pm3.27d (<-> (= (opr (cv x) D (cv y)) (0)) (= (cv x) (cv y))) (A.e. z X (br (opr (cv x) D (cv y)) (<_) (opr (opr (cv z) D (cv x)) (+) (opr (cv z) D (cv y))))) pm3.26 y X r19.20si x X r19.20si (cv x) A D (cv y) opreq1 (0) eqeq1d (cv x) A (cv y) eqeq1 bibi12d (cv y) B A D opreq2 (0) eqeq1d (cv y) B A eqeq2 bibi12d X X rcla42v com12 3syl 3impib)) thm (mstri2 ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z) (C z) (D x) (D y) (D z) (X x) (X y) (X z)) ((msf.1 (= X (` (1st) M))) (msf.2 (= D (` (2nd) M)))) (-> (/\ (/\ (e. M (Met)) (e. A X)) (/\ (e. B X) (e. C X))) (br (opr A D B) (<_) (opr (opr C D A) (+) (opr C D B)))) ((cv x) A D (cv y) opreq1 (cv x) A (cv z) D opreq2 (+) (opr (cv z) D (cv y)) opreq1d (<_) breq12d (cv y) B A D opreq2 (cv y) B (cv z) D opreq2 (opr (cv z) D A) (+) opreq2d (<_) breq12d (cv z) C D A opreq1 (cv z) C D B opreq1 (+) opreq12d (opr A D B) (<_) breq2d X X X rcla43v msf.1 msf.2 x y z msflem pm3.27d (<-> (= (opr (cv x) D (cv y)) (0)) (= (cv x) (cv y))) (A.e. z X (br (opr (cv x) D (cv y)) (<_) (opr (opr (cv z) D (cv x)) (+) (opr (cv z) D (cv y))))) pm3.27 y X r19.20si x X r19.20si syl syl5 3expb ex com3r imp31)) thm (ms0 () ((msf.1 (= X (` (1st) M))) (msf.2 (= D (` (2nd) M)))) (-> (/\ (e. M (Met)) (e. A X)) (= (opr A D A) (0))) (A eqid msf.1 msf.2 A A mseq0 3expb anabsan2 mpbiri)) thm (mssym () ((msf.1 (= X (` (1st) M))) (msf.2 (= D (` (2nd) M)))) (-> (/\/\ (e. M (Met)) (e. A X) (e. B X)) (= (opr A D B) (opr B D A))) (msf.1 msf.2 A B B mstri2 anabsan2 3impa msf.1 msf.2 B ms0 (e. A X) 3adant2 (opr B D A) (+) opreq2d msf.1 msf.2 B A mscl recnd (opr B D A) ax0id syl 3com23 eqtrd breqtrd msf.1 msf.2 B A A mstri2 anabsan2 3impa 3com23 msf.1 msf.2 A ms0 (e. B X) 3adant3 (opr A D B) (+) opreq2d msf.1 msf.2 A B mscl recnd (opr A D B) ax0id syl eqtrd breqtrd jca (opr A D B) (opr B D A) letri3t msf.1 msf.2 A B mscl msf.1 msf.2 B A mscl 3com23 sylanc mpbird)) thm (mstri () ((msf.1 (= X (` (1st) M))) (msf.2 (= D (` (2nd) M)))) (-> (/\ (/\ (e. M (Met)) (e. A X)) (/\ (e. B X) (e. C X))) (br (opr A D B) (<_) (opr (opr A D C) (+) (opr C D B)))) (msf.1 msf.2 A B C mstri2 msf.1 msf.2 C A mssym 3com23 3expa (e. B X) adantrl (+) (opr C D B) opreq1d breqtrd)) thm (msge0 () ((msf.1 (= X (` (1st) M))) (msf.2 (= D (` (2nd) M)))) (-> (/\/\ (e. M (Met)) (e. A X) (e. B X)) (br (0) (<_) (opr A D B))) (2re 2pos (2) (opr A D B) prodge0t mpanr1 mpanl1 msf.1 msf.2 A B mscl msf.1 msf.2 B B A mstri2 exp43 pm2.43d com23 3imp msf.1 msf.2 B ms0 (e. A X) 3adant2 msf.1 msf.2 A B mscl recnd (opr A D B) 2timest syl eqcomd 3brtr3d sylanc)) thm (msgt0 () ((msf.1 (= X (` (1st) M))) (msf.2 (= D (` (2nd) M)))) (-> (/\/\ (e. M (Met)) (e. A X) (e. B X)) (<-> (=/= A B) (br (0) (<) (opr A D B)))) (msf.1 msf.2 A B mseq0 (0) (opr A D B) eqcom syl5bb eqneqd msf.1 msf.2 A B msge0 (-. (= (0) (opr A D B))) biantrurd msf.1 msf.2 A B mscl 0re (0) (opr A D B) ltlent mpan syl bitr4d (0) (opr A D B) df-ne syl5bb bitr3d)) thm (blfval ((t u) (t z) (D t) (u z) (D u) (D z) (t w) (t x) (t y) (M t) (u w) (u x) (u y) (M u) (w x) (w y) (w z) (M w) (x y) (x z) (M x) (y z) (M y) (M z) (X t) (X u) (X w) (X x) (X y) (X z)) ((blfval.1 (= X (` (1st) M))) (blfval.2 (= D (` (2nd) M)))) (-> (e. M (Met)) (= (` (ball) M) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) X) (e. (cv y) (RR))) (/\ (br (0) (<) (cv y)) (= (cv z) ({e.|} w X (br (opr (cv x) D (cv w)) (<) (cv y))))))))) ((cv t) M (1st) fveq2 blfval.1 syl6eqr (cv x) eleq2d (e. (cv y) (RR)) anbi1d (cv t) M (1st) fveq2 blfval.1 syl6eqr (` (1st) (cv t)) X w (br (opr (cv x) (` (2nd) (cv t)) (cv w)) (<) (cv y)) rabeq syl (cv t) M (2nd) fveq2 blfval.2 syl6eqr (cv x) (cv w) opreqd (<) (cv y) breq1d w X rabbisdv eqtrd (cv z) eqeq2d (br (0) (<) (cv y)) anbi2d anbi12d x y z oprabbidv t u x y z w df-bl blfval.1 (1st) M fvex eqeltr reex ({<<,>,>|} x y z (/\ (/\ (e. (cv x) X) (e. (cv y) (RR))) (= (cv z) ({e.|} w X (br (opr (cv x) D (cv w)) (<) (cv y)))))) eqid oprabex2 (br (0) (<) (cv y)) (= (cv z) ({e.|} w X (br (opr (cv x) D (cv w)) (<) (cv y)))) pm3.27 (/\ (e. (cv x) X) (e. (cv y) (RR))) anim2i x y z ssoprab2i ssexi fvopab4)) thm (blfval2 ((D z) (w x) (w y) (w z) (M w) (x y) (x z) (M x) (y z) (M y) (M z) (X w) (X x) (X y) (X z) (v y)) ((blfval.1 (= X (` (1st) M))) (blfval.2 (= D (` (2nd) M)))) (-> (e. M (Met)) (= (` (ball) M) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) X) (e. (cv y) ({e.|} v (RR) (br (0) (<) (cv v))))) (= (cv z) ({e.|} w X (br (opr (cv x) D (cv w)) (<) (cv y)))))))) (blfval.1 blfval.2 x y z w blfval (/\ (e. (cv x) X) (e. (cv y) (RR))) (br (0) (<) (cv y)) (= (cv z) ({e.|} w X (br (opr (cv x) D (cv w)) (<) (cv y)))) anass (e. (cv x) X) (e. (cv y) (RR)) (br (0) (<) (cv y)) anass (cv v) (cv y) (0) (<) breq2 (RR) elrab (e. (cv x) X) anbi2i bitr4 (= (cv z) ({e.|} w X (br (opr (cv x) D (cv w)) (<) (cv y)))) anbi1i bitr3 x y z oprabbii syl6eq)) thm (blval ((x y) (v w) (v x) (v y) (v z) (D v) (w x) (w y) (w z) (D w) (x z) (D x) (y z) (D y) (D z) (M v) (M w) (M x) (M y) (M z) (P w) (P x) (P y) (P z) (R v) (R w) (R x) (R y) (R z) (X v) (X w) (X x) (X y) (X z)) ((blval.1 (= X (` (1st) M))) (blval.2 (= D (` (2nd) M)))) (-> (/\ (/\ (e. M (Met)) (e. P X)) (/\ (e. R (RR)) (br (0) (<) R))) (= (opr P (` (ball) M) R) ({e.|} x X (br (opr P D (cv x)) (<) R)))) (blval.1 blval.2 y z w v x blfval2 P R opreqd (e. P X) (/\ (e. R (RR)) (br (0) (<) R)) ad2antrr blval.1 (1st) M fvex eqeltr x (br (opr P D (cv x)) (<) R) rabex (cv y) P D (cv x) opreq1 (<) (cv z) breq1d x X rabbisdv (cv z) R (opr P D (cv x)) (<) breq2 x X rabbisdv ({<<,>,>|} y z w (/\ (/\ (e. (cv y) X) (e. (cv z) ({e.|} v (RR) (br (0) (<) (cv v))))) (= (cv w) ({e.|} x X (br (opr (cv y) D (cv x)) (<) (cv z)))))) eqid oprabval2 (cv v) R (0) (<) breq2 (RR) elrab sylan2br (e. M (Met)) adantll eqtrd)) thm (elbl ((A x) (D x) (M x) (P x) (R x) (X x)) ((blval.1 (= X (` (1st) M))) (blval.2 (= D (` (2nd) M)))) (-> (/\ (/\ (e. M (Met)) (e. P X)) (/\ (e. R (RR)) (br (0) (<) R))) (<-> (e. A (opr P (` (ball) M) R)) (/\ (e. A X) (br (opr P D A) (<) R)))) (blval.1 blval.2 P R x blval A eleq2d (cv x) A P D opreq2 (<) R breq1d X elrab syl6bb)) thm (blcntr () ((blval.1 (= X (` (1st) M))) (blval.2 (= D (` (2nd) M)))) (-> (/\ (/\ (e. M (Met)) (e. P X)) (/\ (e. R (RR)) (br (0) (<) R))) (e. P (opr P (` (ball) M) R))) ((e. M (Met)) (e. P X) pm3.27 (/\ (e. R (RR)) (br (0) (<) R)) adantr blval.1 blval.2 P ms0 (/\ (e. R (RR)) (br (0) (<) R)) adantr (e. R (RR)) (br (0) (<) R) pm3.27 (/\ (e. M (Met)) (e. P X)) adantl eqbrtrd jca blval.1 blval.2 P R P elbl mpbird)) thm (blrn ((x y) (B x) (B y) (v w) (v x) (v y) (v z) (D v) (w x) (w y) (w z) (D w) (x z) (D x) (y z) (D y) (D z) (M v) (M w) (M x) (M y) (M z) (X v) (X w) (X x) (X y) (X z)) ((blval.1 (= X (` (1st) M))) (blval.2 (= D (` (2nd) M)))) (-> (e. M (Met)) (<-> (e. B (ran (` (ball) M))) (E.e. x X (E.e. y ({e.|} w (RR) (br (0) (<) (cv w))) (= B ({e.|} z X (br (opr (cv x) D (cv z)) (<) (cv y)))))))) (blval.1 blval.2 x y v w z blfval2 rneqd B eleq2d blval.1 (1st) M fvex eqeltr z (br (opr (cv x) D (cv z)) (<) (cv y)) rabex ({<<,>,>|} x y v (/\ (/\ (e. (cv x) X) (e. (cv y) ({e.|} w (RR) (br (0) (<) (cv w))))) (= (cv v) ({e.|} z X (br (opr (cv x) D (cv z)) (<) (cv y)))))) eqid B elrnoprab syl6bb)) thm (blrn2 ((x y) (B x) (B y) (w x) (w y) (w z) (D w) (x z) (D x) (y z) (D y) (D z) (M w) (M x) (M y) (M z) (X w) (X x) (X y) (X z)) ((blval.1 (= X (` (1st) M))) (blval.2 (= D (` (2nd) M)))) (-> (e. M (Met)) (<-> (e. B (ran (` (ball) M))) (E.e. x X (E.e. y (RR) (/\ (br (0) (<) (cv y)) (= B ({e.|} z X (br (opr (cv x) D (cv z)) (<) (cv y))))))))) (blval.1 blval.2 B x y w z blrn (cv w) (cv y) (0) (<) breq2 (RR) elrab (= B ({e.|} z X (br (opr (cv x) D (cv z)) (<) (cv y)))) anbi1i (e. (cv y) (RR)) (br (0) (<) (cv y)) (= B ({e.|} z X (br (opr (cv x) D (cv z)) (<) (cv y)))) anass bitr rexbii2 x X rexbii syl6bb)) thm (blrn3 ((x y) (x z) (A x) (y z) (A y) (A z) (M x) (M y) (M z) (X x) (X y) (X z)) ((blrn3.1 (= X (` (1st) M)))) (-> (e. M (Met)) (<-> (e. A (ran (` (ball) M))) (E.e. x X (E.e. y (RR) (/\ (br (0) (<) (cv y)) (= A (opr (cv x) (` (ball) M) (cv y)))))))) (blrn3.1 (` (2nd) M) eqid A x y z blrn2 blrn3.1 (` (2nd) M) eqid (cv x) (cv y) z blval anassrs A eqeq2d ex pm5.32d rexbidva rexbidva bitr4d)) thm (blelrn ((x y) (M x) (M y) (X x) (X y) (P x) (P y) (R x) (R y)) ((blrn3.1 (= X (` (1st) M)))) (-> (/\ (/\ (e. M (Met)) (e. P X)) (/\ (e. R (RR)) (br (0) (<) R))) (e. (opr P (` (ball) M) R) (ran (` (ball) M)))) ((cv x) P (` (ball) M) (cv y) opreq1 (opr P (` (ball) M) R) eqeq2d (br (0) (<) (cv y)) anbi2d (cv y) R (0) (<) breq2 (cv y) R P (` (ball) M) opreq2 (opr P (` (ball) M) R) eqeq2d anbi12d X (RR) rcla42ev (e. M (Met)) (e. P X) pm3.27 (e. R (RR)) (br (0) (<) R) pm3.26 anim12i (e. R (RR)) (br (0) (<) R) pm3.27 (/\ (e. M (Met)) (e. P X)) adantl (opr P (` (ball) M) R) eqid jctir sylanc blrn3.1 (opr P (` (ball) M) R) x y blrn3 (e. P X) (/\ (e. R (RR)) (br (0) (<) R)) ad2antrr mpbird)) thm (blex ((M x) (X x) (P x)) ((blrn3.1 (= X (` (1st) M)))) (-> (/\ (e. M (Met)) (e. P X)) (E.e. x (ran (` (ball) M)) (e. P (cv x)))) (ax1re lt01 pm3.2i (cv x) (opr P (` (ball) M) (1)) P eleq2 (ran (` (ball) M)) rcla4ev blrn3.1 P (1) blelrn blrn3.1 (` (2nd) M) eqid P (1) blcntr sylanc mpan2)) thm (blssm ((M x) (X x) (P x) (R x)) ((blrn3.1 (= X (` (1st) M)))) (-> (/\ (/\ (e. M (Met)) (e. P X)) (/\ (e. R (RR)) (br (0) (<) R))) (C_ (opr P (` (ball) M) R) X)) (blrn3.1 (` (2nd) M) eqid P R x blval x X (br (opr P (` (2nd) M) (cv x)) (<) R) ssrab2 (/\ (/\ (e. M (Met)) (e. P X)) (/\ (e. R (RR)) (br (0) (<) R))) a1i eqsstrd)) thm (rnblssm ((x y) (B x) (B y) (M x) (M y) (X x) (X y)) ((blrn3.1 (= X (` (1st) M)))) (-> (/\ (e. M (Met)) (e. B (ran (` (ball) M)))) (C_ B X)) (blrn3.1 B x y blrn3 (/\ (/\ (e. M (Met)) (e. (cv x) X)) (/\ (e. (cv y) (RR)) (br (0) (<) (cv y)))) (= B (opr (cv x) (` (ball) M) (cv y))) pm3.27 blrn3.1 (cv x) (cv y) blssm (= B (opr (cv x) (` (ball) M) (cv y))) adantr eqsstrd ex exp43 imp3a imp4a r19.23advv sylbid imp)) thm (blin ((S x) (M x) (X x) (P x) (R x)) ((blrn3.1 (= X (` (1st) M)))) (-> (/\/\ (/\ (e. M (Met)) (e. P X)) (/\ (e. R (RR)) (br (0) (<) R)) (/\ (e. S (RR)) (br (0) (<) S))) (= (i^i (opr P (` (ball) M) R) (opr P (` (ball) M) S)) (opr P (` (ball) M) (if (br R (<_) S) R S)))) (blrn3.1 (` (2nd) M) eqid P R x blval (/\ (e. S (RR)) (br (0) (<) S)) 3adant3 blrn3.1 (` (2nd) M) eqid P S x blval (/\ (e. R (RR)) (br (0) (<) R)) 3adant2 ineq12d blrn3.1 (` (2nd) M) eqid P (if (br R (<_) S) R S) x blval R (if (br R (<_) S) R S) (RR) eleq1 R (if (br R (<_) S) R S) (0) (<) breq2 anbi12d S (if (br R (<_) S) R S) (RR) eleq1 S (if (br R (<_) S) R S) (0) (<) breq2 anbi12d ifboth sylan2 3impb (opr P (` (2nd) M) (cv x)) R S ltmint blrn3.1 (` (2nd) M) eqid P (cv x) mscl 3expa (e. R (RR)) (br (0) (<) R) pm3.26 (e. S (RR)) (br (0) (<) S) pm3.26 syl3an 3expb an1rs ex rabbidv 3impb eqtr2d x X (br (opr P (` (2nd) M) (cv x)) (<) R) (br (opr P (` (2nd) M) (cv x)) (<) S) inrab syl5eq eqtrd)) thm (blss ((v w) (v x) (v z) (B v) (w x) (w z) (B w) (x z) (B x) (B z) (M v) (M w) (M x) (M z) (P v) (P w) (P x) (P z)) () (-> (/\/\ (e. M (Met)) (e. B (ran (` (ball) M))) (e. P B)) (E.e. x (RR) (/\ (br (0) (<) (cv x)) (C_ (opr P (` (ball) M) (cv x)) B)))) ((` (1st) M) eqid (` (2nd) M) eqid B z w v blrn2 (cv x) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)) (0) (<) breq2 (cv x) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)) P (` (ball) M) opreq2 B sseq1d anbi12d (RR) rcla4ev (cv w) (opr (cv z) (` (2nd) M) P) resubclt (e. (cv z) (` (1st) M)) (e. (cv w) (RR)) pm3.27 (e. M (Met)) (e. P (` (1st) M)) ad2antlr (` (1st) M) eqid (` (2nd) M) eqid (cv z) P mscl 3expa (e. (cv w) (RR)) adantlrr sylanc (br (opr (cv z) (` (2nd) M) P) (<) (cv w)) adantrr B ({e.|} v (` (1st) M) (br (opr (cv z) (` (2nd) M) (cv v)) (<) (cv w))) P eleq2 (cv v) P (cv z) (` (2nd) M) opreq2 (<) (cv w) breq1d (` (1st) M) elrab syl6bb biimpa sylan2 (opr (cv z) (` (2nd) M) P) (cv w) posdift (` (1st) M) eqid (` (2nd) M) eqid (cv z) P mscl 3expa (e. (cv w) (RR)) adantlrr (e. (cv z) (` (1st) M)) (e. (cv w) (RR)) pm3.27 (e. M (Met)) (e. P (` (1st) M)) ad2antlr sylanc biimpa anasss B ({e.|} v (` (1st) M) (br (opr (cv z) (` (2nd) M) (cv v)) (<) (cv w))) P eleq2 (cv v) P (cv z) (` (2nd) M) opreq2 (<) (cv w) breq1d (` (1st) M) elrab syl6bb biimpa sylan2 (` (1st) M) eqid P (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)) blssm (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR))) pm3.26 (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w)) pm3.26 anim12i (cv w) (opr (cv z) (` (2nd) M) P) resubclt (e. (cv z) (` (1st) M)) (e. (cv w) (RR)) pm3.27 (e. M (Met)) (e. P (` (1st) M)) ad2antlr (` (1st) M) eqid (` (2nd) M) eqid (cv z) P mscl 3expa (e. (cv w) (RR)) adantlrr sylanc (br (opr (cv z) (` (2nd) M) P) (<) (cv w)) adantrr (opr (cv z) (` (2nd) M) P) (cv w) posdift (` (1st) M) eqid (` (2nd) M) eqid (cv z) P mscl 3expa (e. (cv w) (RR)) adantlrr (e. (cv z) (` (1st) M)) (e. (cv w) (RR)) pm3.27 (e. M (Met)) (e. P (` (1st) M)) ad2antlr sylanc biimpa anasss jca sylanc (` (1st) M) eqid (` (2nd) M) eqid P (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)) (cv v) elbl (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR))) pm3.26 (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w)) pm3.26 anim12i (cv w) (opr (cv z) (` (2nd) M) P) resubclt (e. (cv z) (` (1st) M)) (e. (cv w) (RR)) pm3.27 (e. M (Met)) (e. P (` (1st) M)) ad2antlr (` (1st) M) eqid (` (2nd) M) eqid (cv z) P mscl 3expa (e. (cv w) (RR)) adantlrr sylanc (br (opr (cv z) (` (2nd) M) P) (<) (cv w)) adantrr (opr (cv z) (` (2nd) M) P) (cv w) posdift (` (1st) M) eqid (` (2nd) M) eqid (cv z) P mscl 3expa (e. (cv w) (RR)) adantlrr (e. (cv z) (` (1st) M)) (e. (cv w) (RR)) pm3.27 (e. M (Met)) (e. P (` (1st) M)) ad2antlr sylanc biimpa anasss jca sylanc (` (1st) M) eqid (` (2nd) M) eqid (cv z) (cv v) mscl (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR))) pm3.26 (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w))) (/\ (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)))) ad2antrr (e. (cv z) (` (1st) M)) (e. (cv w) (RR)) pm3.26 (e. M (Met)) (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w))) ad2antlr (/\ (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)))) adantr (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P))) pm3.26 (/\ (/\ (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR)))) (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w)))) adantl syl3anc (opr (cv z) (` (2nd) M) P) (opr P (` (2nd) M) (cv v)) axaddrcl (` (1st) M) eqid (` (2nd) M) eqid (cv z) P mscl (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR))) pm3.26 (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w))) (/\ (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)))) ad2antrr (e. (cv z) (` (1st) M)) (e. (cv w) (RR)) pm3.26 (e. M (Met)) (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w))) ad2antlr (/\ (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)))) adantr (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w)) pm3.26 (/\ (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR)))) (/\ (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)))) ad2antlr syl3anc (` (1st) M) eqid (` (2nd) M) eqid P (cv v) mscl (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR))) pm3.26 (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w))) (/\ (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)))) ad2antrr (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w)) pm3.26 (/\ (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR)))) (/\ (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)))) ad2antlr (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P))) pm3.26 (/\ (/\ (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR)))) (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w)))) adantl syl3anc sylanc (e. (cv z) (` (1st) M)) (e. (cv w) (RR)) pm3.27 (e. M (Met)) adantl (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w))) (/\ (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)))) ad2antrr (` (1st) M) eqid (` (2nd) M) eqid (cv z) (cv v) P mstri (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR))) pm3.26 (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w))) (/\ (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)))) ad2antrr (e. (cv z) (` (1st) M)) (e. (cv w) (RR)) pm3.26 (e. M (Met)) (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w))) ad2antlr (/\ (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)))) adantr jca (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P))) pm3.26 (/\ (/\ (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR)))) (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w)))) adantl (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w)) pm3.26 (/\ (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR)))) (/\ (e. (cv v) (` (1st) M)) (br (opr P (` (2nd) M) (cv v)) (<) (opr (cv w) (-) (opr (cv z) (` (2nd) M) P)))) ad2antlr jca sylanc (opr (cv z) (` (2nd) M) P) (opr P (` (2nd) M) (cv v)) (cv w) ltaddsub2t (` (1st) M) eqid (` (2nd) M) eqid (cv z) P mscl 3expa (e. (cv w) (RR)) adantlrr (br (opr (cv z) (` (2nd) M) P) (<) (cv w)) adantrr (e. (cv v) (` (1st) M)) adantr (` (1st) M) eqid (` (2nd) M) eqid P (cv v) mscl 3expa (br (opr (cv z) (` (2nd) M) P) (<) (cv w)) adantlrr (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR))) adantllr (e. (cv z) (` (1st) M)) (e. (cv w) (RR)) pm3.27 (e. M (Met)) adantl (/\ (e. P (` (1st) M)) (br (opr (cv z) (` (2nd) M) P) (<) (cv w))) (e. (cv v) (` (1st) M)) ad2antrr syl3anc biimpar anasss lelttrd ex sylbid imp ssrabdv B ({e.|} v (` (1st) M) (br (opr (cv z) (` (2nd) M) (cv v)) (<) (cv w))) P eleq2 (cv v) P (cv z) (` (2nd) M) opreq2 (<) (cv w) breq1d (` (1st) M) elrab syl6bb biimpa sylan2 (= B ({e.|} v (` (1st) M) (br (opr (cv z) (` (2nd) M) (cv v)) (<) (cv w)))) (e. P B) pm3.26 (/\ (e. M (Met)) (/\ (e. (cv z) (` (1st) M)) (e. (cv w) (RR)))) adantl sseqtr4d jca sylanc (br (0) (<) (cv w)) adantrll exp43 r19.23advv sylbid 3imp)) thm (opnfval ((x y) (x z) (y z) (v w) (v x) (v y) (v z) (M v) (w x) (w y) (w z) (M w) (M x) (M y) (M z) (X v) (X w) (X x) (X y) (X z)) ((opnfval.1 (= X (` (1st) M)))) (-> (e. M (Met)) (= (` (open) M) ({|} x (/\ (C_ (cv x) X) (A.e. y (cv x) (E.e. z (ran (` (ball) M)) (/\ (e. (cv y) (cv z)) (C_ (cv z) (cv x))))))))) ((cv w) M (1st) fveq2 opnfval.1 syl6eqr (cv x) sseq2d (cv w) M (ball) fveq2 rneqd (ran (` (ball) (cv w))) (ran (` (ball) M)) z (/\ (e. (cv y) (cv z)) (C_ (cv z) (cv x))) rexeq1 syl y (cv x) ralbidv anbi12d x abbidv w v x y z df-opn opnfval.1 (1st) M fvex eqeltr X (V) x (A.e. y (cv x) (E.e. z (ran (` (ball) M)) (/\ (e. (cv y) (cv z)) (C_ (cv z) (cv x))))) abssexg ax-mp fvopab4)) thm (opnfss ((x y) (x z) (y z) (M x) (M y) (M z) (X x) (X y) (X z)) ((opnfval.1 (= X (` (1st) M)))) (-> (e. M (Met)) (C_ (` (open) M) (P~ X))) (opnfval.1 x y z opnfval (C_ (cv x) X) (A.e. y (cv x) (E.e. z (ran (` (ball) M)) (/\ (e. (cv y) (cv z)) (C_ (cv z) (cv x))))) pm3.26 x ss2abi X x df-pw sseqtr4 (e. M (Met)) a1i eqsstrd)) thm (isopn ((x y) (x z) (A x) (y z) (A y) (A z) (M x) (M y) (M z) (X x) (X y) (X z)) ((opnfval.1 (= X (` (1st) M)))) (-> (e. M (Met)) (<-> (e. A (` (open) M)) (/\ (C_ A X) (A.e. x A (E.e. y (ran (` (ball) M)) (/\ (e. (cv x) (cv y)) (C_ (cv y) A))))))) (opnfval.1 z x y opnfval A eleq2d opnfval.1 (1st) M fvex eqeltr (cv z) A (cv y) sseq2 (e. (cv x) (cv y)) anbi2d y (ran (` (ball) M)) rexbidv x raleqd elssab syl6bb)) thm (opnm ((x y) (M x) (M y) (X x) (X y)) ((opnfval.1 (= X (` (1st) M)))) (-> (e. M (Met)) (e. X (` (open) M))) (ax1re lt01 pm3.2i (cv y) (opr (cv x) (` (ball) M) (1)) (cv x) eleq2 (cv y) (opr (cv x) (` (ball) M) (1)) X sseq1 anbi12d (ran (` (ball) M)) rcla4ev opnfval.1 (cv x) (1) blelrn opnfval.1 (` (2nd) M) eqid (cv x) (1) blcntr opnfval.1 (cv x) (1) blssm jca sylanc mpan2 r19.21aiva X ssid jctil opnfval.1 X x y isopn mpbird)) thm (opnfuni () ((opnfval.1 (= X (` (1st) M)))) (-> (e. M (Met)) (= (U. (` (open) M)) X)) (opnfval.1 opnfss (` (open) M) (P~ X) uniss X unipw syl6ss syl opnfval.1 opnm X (` (open) M) elssuni syl eqssd)) thm (opni ((x y) (A x) (A y) (M x) (M y) (P x) (P y)) () (-> (/\/\ (e. M (Met)) (e. A (` (open) M)) (e. P A)) (E.e. x (ran (` (ball) M)) (/\ (e. P (cv x)) (C_ (cv x) A)))) ((` (1st) M) eqid A y x isopn (cv y) P (cv x) eleq1 (C_ (cv x) A) anbi1d x (ran (` (ball) M)) rexbidv A rcla4cv (C_ A (` (1st) M)) adantl syl6bi 3imp)) thm (blssopn ((x y) (x z) (y z) (M x) (M y) (M z)) () (-> (e. M (Met)) (C_ (ran (` (ball) M)) (` (open) M))) ((` (1st) M) eqid (cv x) rnblssm (cv x) ssid z x y elequ2 (cv z) (cv x) (cv x) sseq1 anbi12d (ran (` (ball) M)) rcla4ev mpanr2 r19.21aiva (e. M (Met)) adantl jca ex (` (1st) M) eqid (cv x) y z isopn sylibrd ssrdv)) thm (opnuni ((x y) (x z) (A x) (y z) (A y) (A z) (M x) (M y) (M z)) () (-> (/\ (e. M (Met)) (C_ A (` (open) M))) (e. (U. A) (` (open) M))) (A (` (open) M) (P~ (` (1st) M)) sstr2 (` (1st) M) eqid opnfss syl5com A (P~ (` (1st) M)) uniss (` (1st) M) unipw syl6ss syl6 M (cv z) (cv x) y opni (e. M (Met)) (C_ A (` (open) M)) pm3.26 (/\ (e. (cv z) A) (e. (cv x) (cv z))) adantr A (` (open) M) (cv z) ssel2 (e. (cv x) (cv z)) adantrr (e. M (Met)) adantll (e. (cv z) A) (e. (cv x) (cv z)) pm3.27 (/\ (e. M (Met)) (C_ A (` (open) M))) adantl syl3anc (cv y) (cv z) A ssuni expcom (e. (cv x) (cv y)) anim2d y (ran (` (ball) M)) r19.22sdv (/\ (e. M (Met)) (C_ A (` (open) M))) (e. (cv x) (cv z)) ad2antrl mpd exp32 r19.23adv (cv x) A z eluni2 syl5ib ex r19.21adv jcad (` (1st) M) eqid (U. A) x y isopn sylibrd imp)) thm (opnin ((f g) (f u) (f v) (f x) (f y) (f z) (A f) (g u) (g v) (g x) (g y) (g z) (A g) (u v) (u x) (u y) (u z) (A u) (v x) (v y) (v z) (A v) (x y) (x z) (A x) (y z) (A y) (A z) (f w) (B f) (g w) (B g) (u w) (B u) (v w) (B v) (w x) (w y) (B w) (B x) (B y) (M f) (M g) (M u) (M v) (w z) (M w) (M x) (M y) (M z)) () (-> (/\/\ (e. M (Met)) (e. A (` (open) M)) (e. B (` (open) M))) (e. (i^i A B) (` (open) M))) ((e. M (Met)) (C_ (i^i A B) (` (1st) M)) pm3.27 (/\ (A.e. z A (E.e. v (ran (` (ball) M)) (/\ (e. (cv z) (cv v)) (C_ (cv v) A)))) (A.e. w B (E.e. u (ran (` (ball) M)) (/\ (e. (cv w) (cv u)) (C_ (cv u) B))))) a1i v (ran (` (ball) M)) u (ran (` (ball) M)) (/\ (e. (cv x) (cv v)) (C_ (cv v) A)) (/\ (e. (cv x) (cv u)) (C_ (cv u) B)) reeanv M (cv v) (cv x) f blss 3expb ex M (cv u) (cv x) g blss 3expb ex anim12d (e. (cv x) (` (1st) M)) adantr (` (1st) M) eqid (cv x) (cv f) (cv g) blin 3expb (i^i A B) sseq1d (cv y) (opr (cv x) (` (ball) M) (if (br (cv f) (<_) (cv g)) (cv f) (cv g))) (cv x) eleq2 (cv y) (opr (cv x) (` (ball) M) (if (br (cv f) (<_) (cv g)) (cv f) (cv g))) (i^i A B) sseq1 anbi12d (ran (` (ball) M)) rcla4ev (` (1st) M) eqid (cv x) (if (br (cv f) (<_) (cv g)) (cv f) (cv g)) blelrn (cv f) (if (br (cv f) (<_) (cv g)) (cv f) (cv g)) (RR) eleq1 (cv f) (if (br (cv f) (<_) (cv g)) (cv f) (cv g)) (0) (<) breq2 anbi12d (cv g) (if (br (cv f) (<_) (cv g)) (cv f) (cv g)) (RR) eleq1 (cv g) (if (br (cv f) (<_) (cv g)) (cv f) (cv g)) (0) (<) breq2 anbi12d ifboth sylan2 (C_ (opr (cv x) (` (ball) M) (if (br (cv f) (<_) (cv g)) (cv f) (cv g))) (i^i A B)) adantr (` (1st) M) eqid (` (2nd) M) eqid (cv x) (if (br (cv f) (<_) (cv g)) (cv f) (cv g)) blcntr (cv f) (if (br (cv f) (<_) (cv g)) (cv f) (cv g)) (RR) eleq1 (cv f) (if (br (cv f) (<_) (cv g)) (cv f) (cv g)) (0) (<) breq2 anbi12d (cv g) (if (br (cv f) (<_) (cv g)) (cv f) (cv g)) (RR) eleq1 (cv g) (if (br (cv f) (<_) (cv g)) (cv f) (cv g)) (0) (<) breq2 anbi12d ifboth sylan2 (C_ (opr (cv x) (` (ball) M) (if (br (cv f) (<_) (cv g)) (cv f) (cv g))) (i^i A B)) anim1i sylanc ex sylbid ex com3l an4s ex com4l (opr (cv x) (` (ball) M) (cv f)) A (opr (cv x) (` (ball) M) (cv g)) B ss2in (opr (cv x) (` (ball) M) (cv f)) (cv v) A sstr (opr (cv x) (` (ball) M) (cv g)) (cv u) B sstr syl2an an4s syl5 exp3a imp an4s com4t r19.23advv f (RR) g (RR) (/\ (br (0) (<) (cv f)) (C_ (opr (cv x) (` (ball) M) (cv f)) (cv v))) (/\ (br (0) (<) (cv g)) (C_ (opr (cv x) (` (ball) M) (cv g)) (cv u))) reeanv syl5ibr syld com3l imp an4s (e. (cv v) (ran (` (ball) M))) (e. (cv x) (cv v)) (C_ (cv v) A) anass (e. (cv u) (ran (` (ball) M))) (e. (cv x) (cv u)) (C_ (cv u) B) anass syl2anbr an4s ex r19.23aivv sylbir (cv z) (cv x) (cv v) eleq1 (C_ (cv v) A) anbi1d v (ran (` (ball) M)) rexbidv A rcla4va (cv w) (cv x) (cv u) eleq1 (C_ (cv u) B) anbi1d u (ran (` (ball) M)) rexbidv B rcla4va syl2an an4s (cv x) A B elin sylanb ex com23 (i^i A B) (` (1st) M) (cv x) ssel com12 sylan2d com13 r19.21adv jcad exp3a com3r imp A (` (1st) M) B ssinss1 (C_ B (` (1st) M)) adantr sylan an4s com12 (` (1st) M) eqid A z v isopn (` (1st) M) eqid B w u isopn anbi12d (` (1st) M) eqid (i^i A B) x y isopn 3imtr4d 3impib)) thm (opntop ((x y) (M x) (M y)) () (-> (e. M (Met)) (e. (` (open) M) (Top))) (M (cv x) opnuni ex x 19.21aiv M (cv x) (cv y) opnin 3expb ex r19.21aivv jca (open) M fvex (` (open) M) (V) x y istopg ax-mp sylibr)) thm (0ms ((x y) (x z) (y z)) () (e. (<,> ({/}) ({/})) (Met)) ((RR) f0 ({/}) xp0r (X. ({/}) ({/})) ({/}) ({/}) (RR) feq2 ax-mp mpbir x (A.e. y ({/}) (/\ (<-> (= (opr (cv x) ({/}) (cv y)) (0)) (= (cv x) (cv y))) (A.e. z ({/}) (br (opr (cv x) ({/}) (cv y)) (<_) (opr (opr (cv z) ({/}) (cv x)) (+) (opr (cv z) ({/}) (cv y))))))) ral0 pm3.2i 0ex ({/}) (V) ({/}) x y z ismsg ax-mp mpbir)) thm (ccms ((D u) (D v) (D w) (u v) (u w) (v w) (x y) (x z) (u x) (v x) (w x) (y z) (u y) (v y) (w y) (u z) (v z) (w z)) ((ccms.1 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (CC))) (= (cv z) (` (abs) (opr (cv x) (-) (cv y))))))))) (e. (<,> (CC) D) (Met)) (axcnex axcnex ccms.1 oprabex2 D (V) (CC) w v u ismsg ax-mp D (X. (CC) (CC)) (RR) df-f (abs) (opr (cv x) (-) (cv y)) fvex ccms.1 fnoprab2 ccms.1 rneqi x y z (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (CC))) (= (cv z) (` (abs) (opr (cv x) (-) (cv y))))) rnoprab (/\ (e. (cv x) (CC)) (e. (cv y) (CC))) (= (cv z) (` (abs) (opr (cv x) (-) (cv y)))) pm3.27 (cv x) (cv y) subclt (= (cv z) (` (abs) (opr (cv x) (-) (cv y)))) adantr (opr (cv x) (-) (cv y)) absclt syl eqeltrd x y 19.23aivv abssi eqsstr eqsstr mpbir2an (cv w) (cv v) subclt (opr (cv w) (-) (cv v)) abs00t syl (cv w) (cv v) subclt (opr (cv w) (-) (cv v)) absclt syl (cv x) (cv w) (-) (cv y) opreq1 (abs) fveq2d (cv y) (cv v) (cv w) (-) opreq2 (abs) fveq2d ccms.1 (RR) oprabval2g 3expa mpdan eqcomd (0) eqeq1d (cv w) (cv v) subeq0t 3bitr3d (cv w) (cv v) (cv u) abs3dift 3expa (cv w) (cv u) abssubt (+) (` (abs) (opr (cv u) (-) (cv v))) opreq1d (e. (cv v) (CC)) adantlr breqtrd (cv w) (cv v) subclt (opr (cv w) (-) (cv v)) absclt syl (cv x) (cv w) (-) (cv y) opreq1 (abs) fveq2d (cv y) (cv v) (cv w) (-) opreq2 (abs) fveq2d ccms.1 (RR) oprabval2g 3expa mpdan (e. (cv u) (CC)) adantr (cv u) (cv w) subclt ancoms (opr (cv u) (-) (cv w)) absclt syl (cv x) (cv u) (-) (cv y) opreq1 (abs) fveq2d (cv y) (cv w) (cv u) (-) opreq2 (abs) fveq2d ccms.1 (RR) oprabval2g 3com12 3expa mpdan (e. (cv v) (CC)) adantlr (cv u) (cv v) subclt ancoms (opr (cv u) (-) (cv v)) absclt syl (cv x) (cv u) (-) (cv y) opreq1 (abs) fveq2d (cv y) (cv v) (cv u) (-) opreq2 (abs) fveq2d ccms.1 (RR) oprabval2g 3com12 3expa mpdan (e. (cv w) (CC)) adantll (+) opreq12d 3brtr4d r19.21aiva jca rgen2 mpbir2an)) thm (dscms ((X x) (X y) (X z) (x y) (x z) (y z) (X u) (X v) (X w) (u v) (u w) (v w) (D u) (D v) (D w) (u x) (u y) (u z) (v x) (v y) (v z) (w x) (w y) (w z)) ((dscms.1 (e. X (V))) (dscms.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) X) (e. (cv y) X)) (= (cv z) (if (= (cv x) (cv y)) (0) (1)))))))) (e. (<,> X D) (Met)) (dscms.1 dscms.1 dscms.2 oprabex2 D (V) X w v u ismsg ax-mp D (X. X X) (RR) df-f 0re elisseti ax1re elisseti (= (cv x) (cv y)) ifex dscms.2 fnoprab2 dscms.2 rneqi x y z (/\ (/\ (e. (cv x) X) (e. (cv y) X)) (= (cv z) (if (= (cv x) (cv y)) (0) (1)))) rnoprab (cv z) (= (cv x) (cv y)) (0) (1) eqif (= (cv z) (0)) id 0re syl6eqel (= (cv x) (cv y)) adantl (= (cv z) (1)) id ax1re syl6eqel (-. (= (cv x) (cv y))) adantl jaoi sylbi (/\ (e. (cv x) X) (e. (cv y) X)) adantl x y 19.23aivv abssi eqsstr eqsstr mpbir2an 0re ax1re (= (cv w) (cv v)) keepel x w y equequ1 (0) (1) ifbid y v w equequ2 (0) (1) ifbid dscms.2 (RR) oprabval2g mp3an3 eqcomd (0) eqeq1d (0) (= (cv w) (cv v)) (0) (1) eqif (= (cv w) (cv v)) (= (0) (0)) pm3.26 ax1ne0 (1) (0) necom mpbi (0) (1) df-ne mpbi (\/ (= (cv w) (cv v)) (= (cv w) (cv v))) pm2.21i orcanai ancoms jaoi sylbi eqcoms (/\ (e. (cv w) X) (e. (cv v) X)) a1i (= (cv w) (cv v)) (0) (1) iftrue (/\ (e. (cv w) X) (e. (cv v) X)) a1i impbid bitr3d (= (cv w) (cv v)) (0) (1) iftrue u w v equtr imdistani (= (cv u) (cv w)) (0) (1) iftrue (= (cv u) (cv v)) (0) (1) iftrue (+) opreqan12d 0re leid 0re 0re (0) (0) addge01t mp2an mpbi syl5breqr syl w v u equequ2 negbid biimpcd imdistani (= (cv u) (cv w)) (0) (1) iffalse (= (cv u) (cv v)) (0) (1) iffalse (+) opreqan12d 0re ax1re lt01 ltlei 0re ax1re lt01 ltlei ax1re ax1re addge0 mp2an syl5breqr syl pm2.61ian eqbrtrd (= (cv w) (cv v)) (0) (1) iffalse (cv u) (cv w) (cv v) neeq1 biimprd (cv w) (cv v) df-ne (cv u) (cv v) df-ne 3imtr3g imdistani (= (cv u) (cv w)) (0) (1) iftrue (= (cv u) (cv v)) (0) (1) iffalse (+) opreqan12d 1cn addid2 syl6eq ax1re leid syl5breqr syl ex (= (cv u) (cv w)) (0) (1) iffalse (= (cv u) (cv v)) (0) (1) iftrue (+) opreqan12d 1cn addid1 syl6eq ax1re leid syl5breqr ex (-. (= (cv w) (cv v))) a1i com13 (= (cv u) (cv w)) (0) (1) iffalse (= (cv u) (cv v)) (0) (1) iffalse (+) opreqan12d df-2 syl6eqr ax1re 2re 1lt2 ltlei syl5breqr ex (-. (= (cv w) (cv v))) a1i com13 pm2.61i pm2.61i eqbrtrd pm2.61i (/\ (/\ (e. (cv w) X) (e. (cv v) X)) (e. (cv u) X)) a1i 0re ax1re (= (cv w) (cv v)) keepel x w y equequ1 (0) (1) ifbid y v w equequ2 (0) (1) ifbid dscms.2 (RR) oprabval2g mp3an3 (e. (cv u) X) adantr 0re ax1re (= (cv u) (cv w)) keepel x u y equequ1 (0) (1) ifbid y w u equequ2 (0) (1) ifbid dscms.2 (RR) oprabval2g 3com12 mp3an3 (e. (cv v) X) adantlr 0re ax1re (= (cv u) (cv v)) keepel x u y equequ1 (0) (1) ifbid y v u equequ2 (0) (1) ifbid dscms.2 (RR) oprabval2g 3com12 mp3an3 (e. (cv w) X) adantll (+) opreq12d 3brtr4d r19.21aiva jca rgen2 mpbir2an)) thm (lmfval ((w z) (D w) (D z) (f j) (f k) (f w) (f x) (f y) (f z) (M f) (j k) (j w) (j x) (j y) (j z) (M j) (k w) (k x) (k y) (k z) (M k) (w x) (w y) (M w) (x y) (x z) (M x) (y z) (M y) (M z) (X f) (X w) (X y) (X z)) ((lmfval.1 (= X (` (1st) M))) (lmfval.2 (= D (` (2nd) M)))) (-> (e. M (Met)) (= (` (~~>m) M) ({<,>|} f y (/\/\ (C_ (cv f) (X. (CC) X)) (e. (cv y) X) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` (cv f) (cv k)) X) (br (opr (` (cv f) (cv k)) D (cv y)) (<) (cv x)))))))))))) ((cv z) M (1st) fveq2 lmfval.1 syl6eqr (` (1st) (cv z)) X (CC) xpeq2 syl (cv f) sseq2d (cv z) M (1st) fveq2 lmfval.1 syl6eqr (cv y) eleq2d (cv z) M (1st) fveq2 lmfval.1 syl6eqr (` (cv f) (cv k)) eleq2d (cv z) M (2nd) fveq2 lmfval.2 syl6eqr (` (cv f) (cv k)) (cv y) opreqd (<) (cv x) breq1d anbi12d (br (cv j) (<_) (cv k)) imbi2d j (ZZ) k (ZZ) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv 3anbi123d f y opabbidv z w f y x j k df-lm axcnex lmfval.1 (1st) M fvex eqeltr xpex pwex lmfval.1 (1st) M fvex eqeltr xpex (e. (cv f) (P~ (X. (CC) X))) (e. (cv y) X) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` (cv f) (cv k)) X) (br (opr (` (cv f) (cv k)) D (cv y)) (<) (cv x)))))))) df-3an f visset (X. (CC) X) elpw (e. (cv y) X) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` (cv f) (cv k)) X) (br (opr (` (cv f) (cv k)) D (cv y)) (<) (cv x)))))))) 3anbi1i bitr3 f y opabbii f y (P~ (X. (CC) X)) X (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` (cv f) (cv k)) X) (br (opr (` (cv f) (cv k)) D (cv y)) (<) (cv x)))))))) opabssxp eqsstr3 ssexi fvopab4)) thm (caufval ((w z) (D w) (D z) (f j) (f k) (f m) (f w) (f x) (f z) (M f) (j k) (j m) (j w) (j x) (j z) (M j) (k m) (k w) (k x) (k z) (M k) (m w) (m x) (m z) (M m) (w x) (M w) (x z) (M x) (M z) (X f) (X w) (X z)) ((lmfval.1 (= X (` (1st) M))) (lmfval.2 (= D (` (2nd) M)))) (-> (e. M (Met)) (= (` (cau) M) ({|} f (/\ (C_ (cv f) (X. (CC) X)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (A.e. m (ZZ) (-> (/\ (br (cv j) (<_) (cv k)) (br (cv j) (<_) (cv m))) (/\/\ (e. (` (cv f) (cv k)) X) (e. (` (cv f) (cv m)) X) (br (opr (` (cv f) (cv k)) D (` (cv f) (cv m))) (<) (cv x))))))))))))) ((cv z) M (1st) fveq2 lmfval.1 syl6eqr (` (1st) (cv z)) X (CC) xpeq2 syl (cv f) sseq2d (cv z) M (1st) fveq2 lmfval.1 syl6eqr (` (cv f) (cv k)) eleq2d (cv z) M (1st) fveq2 lmfval.1 syl6eqr (` (cv f) (cv m)) eleq2d (cv z) M (2nd) fveq2 lmfval.2 syl6eqr (` (cv f) (cv k)) (` (cv f) (cv m)) opreqd (<) (cv x) breq1d 3anbi123d (/\ (br (cv j) (<_) (cv k)) (br (cv j) (<_) (cv m))) imbi2d m (ZZ) ralbidv j (ZZ) k (ZZ) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv anbi12d f abbidv z w f x j k m df-cau axcnex lmfval.1 (1st) M fvex eqeltr xpex (X. (CC) X) (V) f (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (A.e. m (ZZ) (-> (/\ (br (cv j) (<_) (cv k)) (br (cv j) (<_) (cv m))) (/\/\ (e. (` (cv f) (cv k)) X) (e. (` (cv f) (cv m)) X) (br (opr (` (cv f) (cv k)) D (` (cv f) (cv m))) (<) (cv x))))))))) abssexg ax-mp fvopab4)) thm (lmbr ((w z) (D w) (D z) (j k) (j w) (j x) (j z) (F j) (k w) (k x) (k z) (F k) (w x) (F w) (x z) (F x) (F z) (M j) (M k) (M w) (M x) (M z) (P j) (P k) (P w) (P x) (P z) (X w) (X z)) ((lmbr.1 (= X (` (1st) M))) (lmbr.2 (= D (` (2nd) M)))) (-> (/\/\ (e. M (Met)) (C_ F (X. (CC) X)) (e. P X)) (<-> (br F (` (~~>m) M) P) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) X) (br (opr (` F (cv k)) D P) (<) (cv x)))))))))) (lmbr.1 lmbr.2 z w x j k lmfval (` (~~>m) M) ({<,>|} z w (/\/\ (C_ (cv z) (X. (CC) X)) (e. (cv w) X) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` (cv z) (cv k)) X) (br (opr (` (cv z) (cv k)) D (cv w)) (<) (cv x)))))))))) F P breq syl (C_ F (X. (CC) X)) (e. P X) 3ad2ant1 (cv z) F (X. (CC) X) sseq1 (cv z) F (cv k) fveq1 X eleq1d (cv z) F (cv k) fveq1 D (cv w) opreq1d (<) (cv x) breq1d anbi12d (br (cv j) (<_) (cv k)) imbi2d j (ZZ) k (ZZ) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv (e. (cv w) X) 3anbi13d (cv w) P X eleq1 (cv w) P (` F (cv k)) D opreq2 (<) (cv x) breq1d (e. (` F (cv k)) X) anbi2d (br (cv j) (<_) (cv k)) imbi2d j (ZZ) k (ZZ) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv (C_ F (X. (CC) X)) 3anbi23d ({<,>|} z w (/\/\ (C_ (cv z) (X. (CC) X)) (e. (cv w) X) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` (cv z) (cv k)) X) (br (opr (` (cv z) (cv k)) D (cv w)) (<) (cv x)))))))))) eqid (P~ (X. (CC) X)) X brabg axcnex lmbr.1 (1st) M fvex eqeltr xpex (X. (CC) X) (V) F elpw2g ax-mp sylanbr (C_ F (X. (CC) X)) (e. P X) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) X) (br (opr (` F (cv k)) D P) (<) (cv x)))))))) df-3an baibr bitr4d (e. M (Met)) 3adant1 bitrd)) thm (iscau ((D f) (f j) (f k) (f m) (f x) (F f) (j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (M f) (M j) (M k) (M m) (M x) (X f)) ((lmbr.1 (= X (` (1st) M))) (lmbr.2 (= D (` (2nd) M)))) (-> (/\ (e. M (Met)) (C_ F (X. (CC) X))) (<-> (e. F (` (cau) M)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (A.e. m (ZZ) (-> (/\ (br (cv j) (<_) (cv k)) (br (cv j) (<_) (cv m))) (/\/\ (e. (` F (cv k)) X) (e. (` F (cv m)) X) (br (opr (` F (cv k)) D (` F (cv m))) (<) (cv x))))))))))) (lmbr.1 lmbr.2 f x j k m caufval F eleq2d axcnex lmbr.1 (1st) M fvex eqeltr xpex (X. (CC) X) (V) F elpw2g ax-mp (cv f) F (X. (CC) X) sseq1 (cv f) F (cv k) fveq1 X eleq1d (cv f) F (cv m) fveq1 X eleq1d (cv f) F (cv k) fveq1 (cv f) F (cv m) fveq1 D opreq12d (<) (cv x) breq1d 3anbi123d (/\ (br (cv j) (<_) (cv k)) (br (cv j) (<_) (cv m))) imbi2d m (ZZ) ralbidv j (ZZ) k (ZZ) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv anbi12d (P~ (X. (CC) X)) elabg sylbir (C_ F (X. (CC) X)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (A.e. m (ZZ) (-> (/\ (br (cv j) (<_) (cv k)) (br (cv j) (<_) (cv m))) (/\/\ (e. (` F (cv k)) X) (e. (` F (cv m)) X) (br (opr (` F (cv k)) D (` F (cv m))) (<) (cv x))))))))) ibar bitr4d sylan9bb)) thm (iscms ((f x) (f y) (M f) (x y) (M x) (M y) (X x) (X y)) ((iscms.1 (= X (` (1st) M)))) (<-> (e. M (Cm)) (/\ (e. M (Met)) (A.e. f (` (cau) M) (E.e. x X (br (cv f) (` (~~>m) M) (cv x)))))) (y f x df-cms M eleq2i (cv y) M (cau) fveq2 f (E.e. x (` (1st) (cv y)) (br (cv f) (` (~~>m) (cv y)) (cv x))) raleq1d (cv y) M (1st) fveq2 iscms.1 syl6eqr (` (1st) (cv y)) X x (br (cv f) (` (~~>m) (cv y)) (cv x)) rexeq1 syl (cv y) M (~~>m) fveq2 (` (~~>m) (cv y)) (` (~~>m) M) (cv f) (cv x) breq syl x X rexbidv bitrd f (` (cau) M) ralbidv bitrd (Met) elrab bitr)) thm (avril1 ((F x)) () (-. (/\ (br A (P~ (RR)) (` (i) (1))) (br F ({/}) (opr (0) (x.) (1))))) (x equid x dfnul2 abeq2i con2bii mpbi (cv x) (<,> F (0)) ({/}) eleq1 mtbii (V) vtocleg (<,> F (0)) ({/}) elisset con3i pm2.61i F ({/}) (opr (0) (x.) (1)) df-br 0cn mulid1 (opr (0) (x.) (1)) (0) F opeq2 ax-mp ({/}) eleq1i bitr mtbir (br A (P~ (X. (R.) ({} (0R)))) (U. ({|} y (= (" (<,> (0R) (1R)) ({} (1))) ({} (cv y)))))) intnan df-i (1) fveq1i (<,> (0R) (1R)) (1) y df-fv eqtr A (P~ (RR)) breq2i df-r (RR) (X. (R.) ({} (0R))) (cv z) sseq2 z abbidv (RR) z df-pw (X. (R.) ({} (0R))) z df-pw 3eqtr4g ax-mp A (U. ({|} y (= (" (<,> (0R) (1R)) ({} (1))) ({} (cv y))))) breqi bitr (br F ({/}) (opr (0) (x.) (1))) anbi1i negbii mpbir)) thm (hvmulcl () ((hvmulcl.1 (e. A (CC))) (hvmulcl.2 (e. B (H~)))) (e. (opr A (.s) B) (H~)) (hvmulcl.1 hvmulcl.2 A B ax-hvmulcl mp2an)) thm (hvsubvalt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr A (-v) B) (opr A (+v) (opr (-u (1)) (.s) B)))) (A (+v) (opr (-u (1)) (.s) B) oprex (cv x) A (+v) (opr (-u (1)) (.s) (cv y)) opreq1 (cv y) B (-u (1)) (.s) opreq2 A (+v) opreq2d x y z df-hvsub oprabval2)) thm (hvsubclt () () (-> (/\ (e. A (H~)) (e. B (H~))) (e. (opr A (-v) B) (H~))) (A B hvsubvalt A (opr (-u (1)) (.s) B) ax-hvaddcl 1cn negcl (-u (1)) B ax-hvmulcl mpan sylan2 eqeltrd)) thm (hvaddcl () ((hvaddcl.1 (e. A (H~))) (hvaddcl.2 (e. B (H~)))) (e. (opr A (+v) B) (H~)) (hvaddcl.1 hvaddcl.2 A B ax-hvaddcl mp2an)) thm (hvcom () ((hvaddcl.1 (e. A (H~))) (hvaddcl.2 (e. B (H~)))) (= (opr A (+v) B) (opr B (+v) A)) (hvaddcl.1 hvaddcl.2 A B ax-hvcom mp2an)) thm (hvsubval () ((hvaddcl.1 (e. A (H~))) (hvaddcl.2 (e. B (H~)))) (= (opr A (-v) B) (opr A (+v) (opr (-u (1)) (.s) B))) (hvaddcl.1 hvaddcl.2 A B hvsubvalt mp2an)) thm (hvsubcl () ((hvaddcl.1 (e. A (H~))) (hvaddcl.2 (e. B (H~)))) (e. (opr A (-v) B) (H~)) (hvaddcl.1 hvaddcl.2 A B hvsubclt mp2an)) thm (hvaddid2t () () (-> (e. A (H~)) (= (opr (0v) (+v) A) A)) (ax-hv0cl A (0v) ax-hvcom mpan2 A ax-hvaddid eqtr3d)) thm (hvmul0t () () (-> (e. A (CC)) (= (opr A (.s) (0v)) (0v))) (A mul01t (.s) (0v) opreq1d ax-hv0cl (0v) ax-hvmul0 ax-mp syl6eq 0cn ax-hv0cl A (0) (0v) ax-hvmulass mp3an23 eqtr3d ax-hv0cl (0v) ax-hvmul0 ax-mp A (.s) opreq2i syl6req)) thm (hvmul0ort () () (-> (/\ (e. A (CC)) (e. B (H~))) (<-> (= (opr A (.s) B) (0v)) (\/ (= A (0)) (= B (0v))))) ((opr A (.s) B) (0v) (opr (1) (/) A) (.s) opreq2 (/\ (e. A (CC)) (e. B (H~))) (=/= A (0)) ad2antlr A recid2t (.s) B opreq1d (e. B (H~)) adantlr (opr (1) (/) A) A B ax-hvmulass A recclt (e. B (H~)) adantlr (e. A (CC)) (e. B (H~)) pm3.26 (=/= A (0)) adantr (e. A (CC)) (e. B (H~)) pm3.27 (=/= A (0)) adantr syl3anc B ax-hvmulid (e. A (CC)) (=/= A (0)) ad2antlr 3eqtr3d (= (opr A (.s) B) (0v)) adantlr A recclt (opr (1) (/) A) hvmul0t syl (e. B (H~)) adantlr (= (opr A (.s) B) (0v)) adantlr 3eqtr3d ex A (0) df-ne syl5ibr orrd ex A (0) (.s) B opreq1 (0v) eqeq1d B ax-hvmul0 syl5bir com12 (e. A (CC)) adantl B (0v) A (.s) opreq2 (0v) eqeq1d A hvmul0t syl5bir com12 (e. B (H~)) adantr jaod impbid)) thm (hvsubidt () () (-> (e. A (H~)) (= (opr A (-v) A) (0v))) (A ax-hvmulid (+v) (opr (-u (1)) (.s) A) opreq1d 1cn 1cn negcl (1) (-u (1)) A ax-hvdistr2 mp3an12 A A hvsubvalt anidms 3eqtr4rd 1cn negid (.s) A opreq1i syl6eq A ax-hvmul0 eqtrd)) thm (hvnegidt () () (-> (e. A (H~)) (= (opr A (+v) (opr (-u (1)) (.s) A)) (0v))) (A A hvsubvalt anidms A hvsubidt eqtr3d)) thm (hv2negt () () (-> (e. A (H~)) (= (opr (0v) (-v) A) (opr (-u (1)) (.s) A))) (ax-hv0cl (0v) A hvsubvalt mpan 1cn negcl (-u (1)) A ax-hvmulcl mpan (opr (-u (1)) (.s) A) hvaddid2t syl eqtrd)) thm (hvaddid2 () ((hvaddid2.1 (e. A (H~)))) (= (opr (0v) (+v) A) A) (hvaddid2.1 A hvaddid2t ax-mp)) thm (hvnegid () ((hvaddid2.1 (e. A (H~)))) (= (opr A (+v) (opr (-u (1)) (.s) A)) (0v)) (hvaddid2.1 A hvnegidt ax-mp)) thm (hv2neg () ((hvaddid2.1 (e. A (H~)))) (= (opr (0v) (-v) A) (opr (-u (1)) (.s) A)) (hvaddid2.1 A hv2negt ax-mp)) thm (hvm1negt () () (-> (/\ (e. A (CC)) (e. B (H~))) (= (opr (-u (1)) (.s) (opr A (.s) B)) (opr (-u A) (.s) B))) (1cn negcl (-u (1)) A B ax-hvmulass mp3an1 A mulm1t (e. B (H~)) adantr (.s) B opreq1d eqtr3d)) thm (hvaddsubvalt () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr A (+v) B) (opr A (-v) (opr (-u (1)) (.s) B)))) (A (opr (-u (1)) (.s) B) hvsubvalt 1cn negcl (-u (1)) B ax-hvmulcl mpan sylan2 1cn negcl (-u (1)) B hvm1negt mpan 1cn negneg (.s) B opreq1i syl6eq B ax-hvmulid eqtrd (e. A (H~)) adantl A (+v) opreq2d eqtr2d)) thm (hvadd23t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (+v) B) (+v) C) (opr (opr A (+v) C) (+v) B))) (B C ax-hvcom A (+v) opreq2d (e. A (H~)) 3adant1 A B C ax-hvass A C B ax-hvass 3com23 3eqtr4d)) thm (hvadd12t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr A (+v) (opr B (+v) C)) (opr B (+v) (opr A (+v) C)))) (A B ax-hvcom (+v) C opreq1d (e. C (H~)) 3adant3 A B C ax-hvass B A C ax-hvass 3com12 3eqtr3d)) thm (hvadd4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr A (+v) B) (+v) (opr C (+v) D)) (opr (opr A (+v) C) (+v) (opr B (+v) D)))) (A B C hvadd23t (+v) D opreq1d 3expa (e. D (H~)) adantrr (opr A (+v) B) C D ax-hvass 3expb A B ax-hvaddcl sylan (opr A (+v) C) B D ax-hvass 3expb A C ax-hvaddcl sylan an4s 3eqtr3d)) thm (hvsub4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr A (+v) B) (-v) (opr C (+v) D)) (opr (opr A (-v) C) (+v) (opr B (-v) D)))) (1cn negcl (-u (1)) C D ax-hvdistr1 mp3an1 (/\ (e. A (H~)) (e. B (H~))) adantl (opr A (+v) B) (+v) opreq2d 1cn negcl (-u (1)) C ax-hvmulcl mpan (e. A (H~)) anim2i 1cn negcl (-u (1)) D ax-hvmulcl mpan (e. B (H~)) anim2i anim12i an4s A (opr (-u (1)) (.s) C) B (opr (-u (1)) (.s) D) hvadd4t syl eqtr4d (opr A (+v) B) (opr C (+v) D) hvsubvalt A B ax-hvaddcl C D ax-hvaddcl syl2an A C hvsubvalt (e. B (H~)) (e. D (H~)) ad2ant2r B D hvsubvalt (e. A (H~)) (e. C (H~)) ad2ant2l (+v) opreq12d 3eqtr4d)) thm (hvaddsub12t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr A (+v) (opr B (-v) C)) (opr B (+v) (opr A (-v) C)))) (A B (opr (-u (1)) (.s) C) hvadd12t 1cn negcl (-u (1)) C ax-hvmulcl mpan syl3an3 B C hvsubvalt A (+v) opreq2d (e. A (H~)) 3adant1 A C hvsubvalt B (+v) opreq2d (e. B (H~)) 3adant2 3eqtr4d)) thm (hvpncant () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (opr A (+v) B) (-v) B) A)) ((opr A (+v) B) B hvsubvalt A B ax-hvaddcl (e. A (H~)) (e. B (H~)) pm3.27 sylanc A B (opr (-u (1)) (.s) B) ax-hvass 3expb 1cn negcl (-u (1)) B ax-hvmulcl mpan ancli sylan2 eqtrd B hvnegidt A (+v) opreq2d (e. A (H~)) adantl A ax-hvaddid (e. B (H~)) adantr 3eqtrd)) thm (hvpncan2t () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (opr A (+v) B) (-v) A) B)) (B A ax-hvcom (-v) A opreq1d B A hvpncant eqtr3d ancoms)) thm (hvaddsubasst () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (+v) B) (-v) C) (opr A (+v) (opr B (-v) C)))) (A B (opr (-u (1)) (.s) C) ax-hvass 1cn negcl (-u (1)) C ax-hvmulcl mpan syl3an3 (opr A (+v) B) C hvsubvalt A B ax-hvaddcl sylan 3impa B C hvsubvalt (e. A (H~)) 3adant1 A (+v) opreq2d 3eqtr4d)) thm (hvpncan3t () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr A (+v) (opr B (-v) A)) B)) (A B A hvaddsubasst (e. A (H~)) (e. B (H~)) pm3.26 (e. A (H~)) (e. B (H~)) pm3.27 (e. A (H~)) (e. B (H~)) pm3.26 syl3anc A B hvpncan2t eqtr3d)) thm (hvmulcomt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) (= (opr A (.s) (opr B (.s) C)) (opr B (.s) (opr A (.s) C)))) (A B axmulcom (.s) C opreq1d (e. C (H~)) 3adant3 A B C ax-hvmulass B A C ax-hvmulass 3com12 3eqtr3d)) thm (hvmulass () ((hvmulcom.1 (e. A (CC))) (hvmulcom.2 (e. B (CC))) (hvmulcom.3 (e. C (H~)))) (= (opr (opr A (x.) B) (.s) C) (opr A (.s) (opr B (.s) C))) (hvmulcom.1 hvmulcom.2 hvmulcom.3 A B C ax-hvmulass mp3an)) thm (hvmulcom () ((hvmulcom.1 (e. A (CC))) (hvmulcom.2 (e. B (CC))) (hvmulcom.3 (e. C (H~)))) (= (opr A (.s) (opr B (.s) C)) (opr B (.s) (opr A (.s) C))) (hvmulcom.1 hvmulcom.2 hvmulcom.3 A B C hvmulcomt mp3an)) thm (hvmul2neg () ((hvmulcom.1 (e. A (CC))) (hvmulcom.2 (e. B (CC))) (hvmulcom.3 (e. C (H~)))) (= (opr (-u A) (.s) (opr (-u B) (.s) C)) (opr A (.s) (opr B (.s) C))) (hvmulcom.1 hvmulcom.2 mul2neg (.s) C opreq1i hvmulcom.1 negcl hvmulcom.2 negcl hvmulcom.3 hvmulass hvmulcom.1 hvmulcom.2 hvmulcom.3 hvmulass 3eqtr3)) thm (hvsubdistr1t () () (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr A (.s) (opr B (-v) C)) (opr (opr A (.s) B) (-v) (opr A (.s) C)))) (A B (opr (-u (1)) (.s) C) ax-hvdistr1 1cn negcl (-u (1)) C ax-hvmulcl mpan syl3an3 1cn negcl A (-u (1)) C hvmulcomt mp3an2 (opr A (.s) B) (+v) opreq2d (e. B (H~)) 3adant2 eqtrd B C hvsubvalt (e. A (CC)) 3adant1 A (.s) opreq2d (opr A (.s) B) (opr A (.s) C) hvsubvalt A B ax-hvmulcl (e. C (H~)) 3adant3 A C ax-hvmulcl (e. B (H~)) 3adant2 sylanc 3eqtr4d)) thm (hvsubdistr2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) (= (opr (opr A (-) B) (.s) C) (opr (opr A (.s) C) (-v) (opr B (.s) C)))) (A (-u B) C ax-hvdistr2 B negclt syl3an2 A B negsubt (e. C (H~)) 3adant3 (.s) C opreq1d B mulm1t (.s) C opreq1d (e. C (H~)) adantr 1cn negcl (-u (1)) B C ax-hvmulass mp3an1 eqtr3d (e. A (CC)) 3adant1 (opr A (.s) C) (+v) opreq2d 3eqtr3d (opr A (.s) C) (opr B (.s) C) hvsubvalt A C ax-hvmulcl (e. B (CC)) 3adant2 B C ax-hvmulcl (e. A (CC)) 3adant1 sylanc eqtr4d)) thm (hvdistr1 () ((hvdistr1.1 (e. A (CC))) (hvdistr1.2 (e. B (H~))) (hvdistr1.3 (e. C (H~)))) (= (opr A (.s) (opr B (+v) C)) (opr (opr A (.s) B) (+v) (opr A (.s) C))) (hvdistr1.1 hvdistr1.2 hvdistr1.3 A B C ax-hvdistr1 mp3an)) thm (hvsubdistr1 () ((hvdistr1.1 (e. A (CC))) (hvdistr1.2 (e. B (H~))) (hvdistr1.3 (e. C (H~)))) (= (opr A (.s) (opr B (-v) C)) (opr (opr A (.s) B) (-v) (opr A (.s) C))) (hvdistr1.1 hvdistr1.2 hvdistr1.3 A B C hvsubdistr1t mp3an)) thm (hvass () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~)))) (= (opr (opr A (+v) B) (+v) C) (opr A (+v) (opr B (+v) C))) (hvass.1 hvass.2 hvass.3 A B C ax-hvass mp3an)) thm (hvadd23 () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~)))) (= (opr (opr A (+v) B) (+v) C) (opr (opr A (+v) C) (+v) B)) (hvass.1 hvass.2 hvass.3 A B C hvadd23t mp3an)) thm (hvsubass () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~)))) (= (opr (opr A (-v) B) (-v) C) (opr A (-v) (opr B (+v) C))) (hvass.1 hvass.2 hvsubcl hvass.3 hvsubval hvass.1 1cn negcl hvass.2 hvmulcl 1cn negcl hvass.3 hvmulcl hvass 1cn negcl hvass.2 hvass.3 hvdistr1 A (+v) opreq2i eqtr4 hvass.1 hvass.2 hvsubval (+v) (opr (-u (1)) (.s) C) opreq1i hvass.1 hvass.2 hvass.3 hvaddcl hvsubval 3eqtr4 eqtr)) thm (hvsub23 () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~)))) (= (opr (opr A (-v) B) (-v) C) (opr (opr A (-v) C) (-v) B)) (hvass.2 hvass.3 hvcom A (-v) opreq2i hvass.1 hvass.2 hvass.3 hvsubass hvass.1 hvass.3 hvass.2 hvsubass 3eqtr4)) thm (hvadd12 () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~)))) (= (opr A (+v) (opr B (+v) C)) (opr B (+v) (opr A (+v) C))) (hvass.1 hvass.2 hvcom (+v) C opreq1i hvass.1 hvass.2 hvass.3 hvass hvass.2 hvass.1 hvass.3 hvass 3eqtr3)) thm (hvadd4 () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~))) (hvadd4.4 (e. D (H~)))) (= (opr (opr A (+v) B) (+v) (opr C (+v) D)) (opr (opr A (+v) C) (+v) (opr B (+v) D))) (hvass.1 hvass.2 pm3.2i hvass.3 hvadd4.4 pm3.2i A B C D hvadd4t mp2an)) thm (hvsubsub4 () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~))) (hvadd4.4 (e. D (H~)))) (= (opr (opr A (-v) B) (-v) (opr C (-v) D)) (opr (opr A (-v) C) (-v) (opr B (-v) D))) (hvass.1 1cn negcl hvass.2 hvmulcl 1cn negcl hvass.3 hvmulcl 1cn negcl 1cn negcl hvadd4.4 hvmulcl hvmulcl hvadd4 1cn negcl hvass.3 1cn negcl hvadd4.4 hvmulcl hvdistr1 (opr A (+v) (opr (-u (1)) (.s) B)) (+v) opreq2i 1cn negcl hvass.2 1cn negcl hvadd4.4 hvmulcl hvdistr1 (opr A (+v) (opr (-u (1)) (.s) C)) (+v) opreq2i 3eqtr4 hvass.1 1cn negcl hvass.2 hvmulcl hvaddcl hvass.3 1cn negcl hvadd4.4 hvmulcl hvaddcl hvsubval hvass.1 1cn negcl hvass.3 hvmulcl hvaddcl hvass.2 1cn negcl hvadd4.4 hvmulcl hvaddcl hvsubval 3eqtr4 hvass.1 hvass.2 hvsubval hvass.3 hvadd4.4 hvsubval (-v) opreq12i hvass.1 hvass.3 hvsubval hvass.2 hvadd4.4 hvsubval (-v) opreq12i 3eqtr4)) thm (hvsubsub4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr A (-v) B) (-v) (opr C (-v) D)) (opr (opr A (-v) C) (-v) (opr B (-v) D)))) (A (if (e. A (H~)) A (0v)) (-v) B opreq1 (-v) (opr C (-v) D) opreq1d A (if (e. A (H~)) A (0v)) (-v) C opreq1 (-v) (opr B (-v) D) opreq1d eqeq12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (-v) (opr C (-v) D) opreq1d B (if (e. B (H~)) B (0v)) (-v) D opreq1 (opr (if (e. A (H~)) A (0v)) (-v) C) (-v) opreq2d eqeq12d C (if (e. C (H~)) C (0v)) (-v) D opreq1 (opr (if (e. A (H~)) A (0v)) (-v) (if (e. B (H~)) B (0v))) (-v) opreq2d C (if (e. C (H~)) C (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (-v) (opr (if (e. B (H~)) B (0v)) (-v) D) opreq1d eqeq12d D (if (e. D (H~)) D (0v)) (if (e. C (H~)) C (0v)) (-v) opreq2 (opr (if (e. A (H~)) A (0v)) (-v) (if (e. B (H~)) B (0v))) (-v) opreq2d D (if (e. D (H~)) D (0v)) (if (e. B (H~)) B (0v)) (-v) opreq2 (opr (if (e. A (H~)) A (0v)) (-v) (if (e. C (H~)) C (0v))) (-v) opreq2d eqeq12d ax-hv0cl A elimel ax-hv0cl B elimel ax-hv0cl C elimel ax-hv0cl D elimel hvsubsub4 dedth4h)) thm (hv2timest () () (-> (e. A (H~)) (= (opr (2) (.s) A) (opr A (+v) A))) (1cn 1cn (1) (1) A ax-hvdistr2 mp3an12 df-2 (.s) A opreq1i syl5eq 1cn (1) A A ax-hvdistr1 mp3an1 anidms A A ax-hvaddcl anidms (opr A (+v) A) ax-hvmulid syl 3eqtr2d)) thm (hvnegdi () ((hvnegdi.1 (e. A (H~))) (hvnegdi.2 (e. B (H~)))) (= (opr (-u (1)) (.s) (opr A (-v) B)) (opr B (-v) A)) (hvnegdi.1 hvnegdi.2 hvsubval (-u (1)) (.s) opreq2i 1cn negcl hvnegdi.1 1cn negcl hvnegdi.2 hvmulcl hvdistr1 1cn 1cn mul2neg 1cn mulid1 eqtr (.s) B opreq1i 1cn negcl 1cn negcl hvnegdi.2 hvmulass hvnegdi.2 B ax-hvmulid ax-mp 3eqtr3 (+v) (opr (-u (1)) (.s) A) opreq1i 1cn negcl hvnegdi.1 hvmulcl 1cn negcl 1cn negcl hvnegdi.2 hvmulcl hvmulcl hvcom hvnegdi.2 hvnegdi.1 hvsubval 3eqtr4 3eqtr)) thm (hvsubeq0 () ((hvnegdi.1 (e. A (H~))) (hvnegdi.2 (e. B (H~)))) (<-> (= (opr A (-v) B) (0v)) (= A B)) (hvnegdi.1 hvnegdi.2 hvsubval (0v) eqeq1i (opr A (+v) (opr (-u (1)) (.s) B)) (0v) (+v) B opreq1 sylbi hvnegdi.1 1cn negcl hvnegdi.2 hvmulcl hvnegdi.2 hvadd23 hvnegdi.1 hvnegdi.2 1cn negcl hvnegdi.2 hvmulcl hvass hvnegdi.2 hvnegid A (+v) opreq2i hvnegdi.1 A ax-hvaddid ax-mp eqtr 3eqtr hvnegdi.2 hvaddid2 3eqtr3g A B (-v) B opreq1 hvnegdi.2 B hvsubidt ax-mp syl6eq impbi)) thm (hvsubcan2 () ((hvnegdi.1 (e. A (H~))) (hvnegdi.2 (e. B (H~)))) (= (opr (opr A (+v) B) (+v) (opr A (-v) B)) (opr (2) (.s) A)) (hvnegdi.1 hvnegdi.2 hvsubval (opr A (+v) B) (+v) opreq2i hvnegdi.1 hvnegdi.2 hvnegdi.1 1cn negcl hvnegdi.2 hvmulcl hvadd4 hvnegdi.1 A hv2timest ax-mp eqcomi hvnegdi.2 hvnegid (+v) opreq12i eqtr 2cn hvnegdi.1 hvmulcl (opr (2) (.s) A) ax-hvaddid ax-mp 3eqtr)) thm (hvaddcan () ((hvnegdi.1 (e. A (H~))) (hvnegdi.2 (e. B (H~))) (hvaddcan.3 (e. C (H~)))) (<-> (= (opr A (+v) B) (opr A (+v) C)) (= B C)) ((opr A (+v) B) (opr A (+v) C) (+v) (opr (-u (1)) (.s) A) opreq1 hvnegdi.1 hvnegdi.2 1cn negcl hvnegdi.1 hvmulcl hvadd23 hvnegdi.1 hvnegid (+v) B opreq1i hvnegdi.2 hvaddid2 3eqtr hvnegdi.1 hvaddcan.3 1cn negcl hvnegdi.1 hvmulcl hvadd23 hvnegdi.1 hvnegid (+v) C opreq1i hvaddcan.3 hvaddid2 3eqtr 3eqtr3g B C A (+v) opreq2 impbi)) thm (hvsubadd () ((hvnegdi.1 (e. A (H~))) (hvnegdi.2 (e. B (H~))) (hvaddcan.3 (e. C (H~)))) (<-> (= (opr A (-v) B) C) (= (opr B (+v) C) A)) (hvnegdi.1 hvnegdi.2 hvsubval C eqeq1i hvnegdi.2 hvnegdi.1 1cn negcl hvnegdi.2 hvmulcl hvadd12 hvnegdi.2 hvnegid A (+v) opreq2i hvnegdi.1 A ax-hvaddid ax-mp 3eqtr (opr B (+v) C) eqeq1i hvnegdi.2 hvnegdi.1 1cn negcl hvnegdi.2 hvmulcl hvaddcl hvaddcan.3 hvaddcan A (opr B (+v) C) eqcom 3bitr3 bitr)) thm (hvnegdit () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (-u (1)) (.s) (opr A (-v) B)) (opr B (-v) A))) (A (if (e. A (H~)) A (0v)) (-v) B opreq1 (-u (1)) (.s) opreq2d A (if (e. A (H~)) A (0v)) B (-v) opreq2 eqeq12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (-u (1)) (.s) opreq2d B (if (e. B (H~)) B (0v)) (-v) (if (e. A (H~)) A (0v)) opreq1 eqeq12d ax-hv0cl A elimel ax-hv0cl B elimel hvnegdi dedth2h)) thm (hvsubeq0t () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (= (opr A (-v) B) (0v)) (= A B))) (A (if (e. A (H~)) A (0v)) (-v) B opreq1 (0v) eqeq1d A (if (e. A (H~)) A (0v)) B eqeq1 bibi12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (0v) eqeq1d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) eqeq2 bibi12d ax-hv0cl A elimel ax-hv0cl B elimel hvsubeq0 dedth2h)) thm (hvaddeq0t () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (= (opr A (+v) B) (0v)) (= A (opr (-u (1)) (.s) B)))) (A B hvaddsubvalt (0v) eqeq1d A (opr (-u (1)) (.s) B) hvsubeq0t 1cn negcl (-u (1)) B ax-hvmulcl mpan sylan2 bitrd)) thm (hvaddcant () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (+v) B) (opr A (+v) C)) (= B C))) (A (if (e. A (H~)) A (0v)) (+v) B opreq1 A (if (e. A (H~)) A (0v)) (+v) C opreq1 eqeq12d (= B C) bibi1d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (+v) opreq2 (opr (if (e. A (H~)) A (0v)) (+v) C) eqeq1d B (if (e. B (H~)) B (0v)) C eqeq1 bibi12d C (if (e. C (H~)) C (0v)) (if (e. A (H~)) A (0v)) (+v) opreq2 (opr (if (e. A (H~)) A (0v)) (+v) (if (e. B (H~)) B (0v))) eqeq2d C (if (e. C (H~)) C (0v)) (if (e. B (H~)) B (0v)) eqeq2 bibi12d ax-hv0cl A elimel ax-hv0cl B elimel ax-hv0cl C elimel hvaddcan dedth3h)) thm (hvaddcan2t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (+v) C) (opr B (+v) C)) (= A B))) (C A ax-hvcom (e. B (H~)) 3adant3 C B ax-hvcom (e. A (H~)) 3adant2 eqeq12d C A B hvaddcant bitr3d 3coml)) thm (hvmulcant () () (-> (/\ (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (=/= A (0))) (<-> (= (opr A (.s) B) (opr A (.s) C)) (= B C))) (A (0) df-ne (= A (0)) (= (opr B (-v) C) (0v)) biorf sylbi (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) adantl A (opr B (-v) C) hvmul0ort B C hvsubclt sylan2 3impb (=/= A (0)) adantr bitr4d A B C hvsubdistr1t (=/= A (0)) adantr (0v) eqeq1d (opr A (.s) B) (opr A (.s) C) hvsubeq0t A B ax-hvmulcl (e. C (H~)) 3adant3 A C ax-hvmulcl (e. B (H~)) 3adant2 sylanc (=/= A (0)) adantr 3bitrd B C hvsubeq0t (e. A (CC)) 3adant1 (=/= A (0)) adantr bitr3d)) thm (hvmulcan2t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) (=/= C (0v))) (<-> (= (opr A (.s) C) (opr B (.s) C)) (= A B))) (C (0v) df-ne (= C (0v)) (= (opr A (-) B) (0)) biorf (= C (0v)) (= (opr A (-) B) (0)) orcom syl6bb sylbi (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) adantl (opr A (-) B) C hvmul0ort A B subclt sylan 3impa (=/= C (0v)) adantr A B C hvsubdistr2t (=/= C (0v)) adantr (0v) eqeq1d 3bitr2rd (opr A (.s) C) (opr B (.s) C) hvsubeq0t A C ax-hvmulcl (e. B (CC)) 3adant2 B C ax-hvmulcl (e. A (CC)) 3adant1 sylanc (=/= C (0v)) adantr A B subeq0t (e. C (H~)) 3adant3 (=/= C (0v)) adantr 3bitr3d)) thm (hvsubcant () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (-v) B) (opr A (-v) C)) (= B C))) (A B hvsubvalt (e. C (H~)) 3adant3 A C hvsubvalt (e. B (H~)) 3adant2 eqeq12d A (opr (-u (1)) (.s) B) (opr (-u (1)) (.s) C) hvaddcant 1cn negcl (-u (1)) C ax-hvmulcl mpan syl3an3 1cn negcl (-u (1)) B ax-hvmulcl mpan syl3an2 1cn negcl 1cn ax1ne0 negn0 (-u (1)) B C hvmulcant mpan2 mp3an1 (e. A (H~)) 3adant1 3bitrd)) thm (hvsubcan2t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (-v) C) (opr B (-v) C)) (= A B))) (1cn negcl 1cn ax1ne0 negn0 (-u (1)) (opr C (-v) A) (opr C (-v) B) hvmulcant mpan2 mp3an1 C A hvsubclt (e. B (H~)) 3adant3 C B hvsubclt (e. A (H~)) 3adant2 sylanc C A hvnegdit (e. B (H~)) 3adant3 C B hvnegdit (e. A (H~)) 3adant2 eqeq12d C A B hvsubcant 3bitr3d 3coml)) thm (hvsub0t () () (-> (e. A (H~)) (= (opr A (-v) (0v)) A)) (ax-hv0cl A (0v) hvsubvalt mpan2 1cn negcl (-u (1)) hvmul0t ax-mp A (+v) opreq2i syl6eq A ax-hvaddid eqtrd)) thm (hvsubaddt () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (-v) B) C) (= (opr B (+v) C) A))) (A (if (e. A (H~)) A (0v)) (-v) B opreq1 C eqeq1d A (if (e. A (H~)) A (0v)) (opr B (+v) C) eqeq2 bibi12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 C eqeq1d B (if (e. B (H~)) B (0v)) (+v) C opreq1 (if (e. A (H~)) A (0v)) eqeq1d bibi12d C (if (e. C (H~)) C (0v)) (opr (if (e. A (H~)) A (0v)) (-v) (if (e. B (H~)) B (0v))) eqeq2 C (if (e. C (H~)) C (0v)) (if (e. B (H~)) B (0v)) (+v) opreq2 (if (e. A (H~)) A (0v)) eqeq1d bibi12d ax-hv0cl A elimel ax-hv0cl B elimel ax-hv0cl C elimel hvsubadd dedth3h)) thm (hvaddsub4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (<-> (= (opr A (+v) B) (opr C (+v) D)) (= (opr A (-v) C) (opr D (-v) B)))) ((opr A (+v) B) (opr C (+v) D) (opr C (+v) B) hvsubcan2t A B ax-hvaddcl (/\ (e. C (H~)) (e. D (H~))) adantr C D ax-hvaddcl (/\ (e. A (H~)) (e. B (H~))) adantl C B ax-hvaddcl ancoms (e. D (H~)) adantrr (e. A (H~)) adantll syl3anc (e. A (H~)) (e. B (H~)) pm3.27 (e. C (H~)) anim2i ancoms A B C B hvsub4t syldan B hvsubidt (e. A (H~)) (e. C (H~)) ad2antlr (opr A (-v) C) (+v) opreq2d A C hvsubclt (opr A (-v) C) ax-hvaddid syl (e. B (H~)) adantlr 3eqtrd (e. D (H~)) adantrr (e. C (H~)) (e. D (H~)) pm3.26 (e. B (H~)) anim1i C D C B hvsub4t syldan C hvsubidt (e. D (H~)) (e. B (H~)) ad2antrr (+v) (opr D (-v) B) opreq1d D B hvsubclt (opr D (-v) B) hvaddid2t syl (e. C (H~)) adantll 3eqtrd ancoms (e. A (H~)) adantll eqeq12d bitr3d)) thm (hicl () ((hicl.1 (e. A (H~))) (hicl.2 (e. B (H~)))) (e. (opr A (.i) B) (CC)) (hicl.1 hicl.2 A B ax-hicl mp2an)) thm (ax-his4OLD () () (-> (/\ (e. A (H~)) (-. (= A (0v)))) (br (0) (<) (opr A (.i) A))) (A ax-his4 A (0v) df-ne sylan2br)) thm (his5t () () (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr B (.i) (opr A (.s) C)) (opr (` (*) A) (x.) (opr B (.i) C)))) (B (opr A (.s) C) ax-his1 A C ax-hvmulcl sylan2 3impb 3com12 A C B ax-his3 3com23 (*) fveq2d A (opr C (.i) B) cjmult C B ax-hicl sylan2 3impb 3com23 B C ax-his1 (e. A (CC)) 3adant1 (` (*) A) (x.) opreq2d eqtr4d 3eqtrd)) thm (his52t () () (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr B (.i) (opr (` (*) A) (.s) C)) (opr A (x.) (opr B (.i) C)))) ((` (*) A) B C his5t A cjclt syl3an1 A cjcjt (x.) (opr B (.i) C) opreq1d (e. B (H~)) (e. C (H~)) 3ad2ant1 eqtrd)) thm (his35 () ((his35.1 (e. A (CC))) (his35.2 (e. B (CC))) (his35.3 (e. C (H~))) (his35.4 (e. D (H~)))) (= (opr (opr A (.s) C) (.i) (opr B (.s) D)) (opr (opr A (x.) (` (*) B)) (x.) (opr C (.i) D))) (his35.2 his35.3 his35.4 B C D his5t mp3an A (x.) opreq2i his35.1 his35.3 his35.2 his35.4 hvmulcl A C (opr B (.s) D) ax-his3 mp3an his35.1 his35.2 cjcl his35.3 his35.4 hicl mulass 3eqtr4)) thm (his7t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr A (.i) (opr B (+v) C)) (opr (opr A (.i) B) (+) (opr A (.i) C)))) (B C A ax-his2 (*) fveq2d (opr B (.i) A) (opr C (.i) A) cjaddt B A ax-hicl C A ax-hicl syl2an 3impdir eqtrd 3comr A (opr B (+v) C) ax-his1 B C ax-hvaddcl sylan2 3impb A B ax-his1 (e. C (H~)) 3adant3 A C ax-his1 (e. B (H~)) 3adant2 (+) opreq12d 3eqtr4d)) thm (hiassdit () () (-> (/\ (/\ (e. A (CC)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr (opr A (.s) B) (+v) C) (.i) D) (opr (opr A (x.) (opr B (.i) D)) (+) (opr C (.i) D)))) ((opr A (.s) B) C D ax-his2 3expb A B ax-hvmulcl sylan A B D ax-his3 3expa (e. C (H~)) adantrl (+) (opr C (.i) D) opreq1d eqtrd)) thm (his2subt () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (-v) B) (.i) C) (opr (opr A (.i) C) (-) (opr B (.i) C)))) (A B hvsubvalt (.i) C opreq1d (e. C (H~)) 3adant3 A (opr (-u (1)) (.s) B) C ax-his2 1cn negcl (-u (1)) B ax-hvmulcl mpan syl3an2 1cn negcl (-u (1)) B C ax-his3 mp3an1 B C ax-hicl (opr B (.i) C) mulm1t syl eqtrd (opr A (.i) C) (+) opreq2d (e. A (H~)) 3adant1 eqtrd (opr A (.i) C) (opr B (.i) C) negsubt A C ax-hicl (e. B (H~)) 3adant2 B C ax-hicl (e. A (H~)) 3adant1 sylanc 3eqtrd)) thm (his2sub2t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr A (.i) (opr B (-v) C)) (opr (opr A (.i) B) (-) (opr A (.i) C)))) (B C A his2subt (*) fveq2d (opr B (.i) A) (opr C (.i) A) cjsubt B A ax-hicl C A ax-hicl syl2an 3impdir eqtrd 3comr A (opr B (-v) C) ax-his1 B C hvsubclt sylan2 3impb A B ax-his1 (e. C (H~)) 3adant3 A C ax-his1 (e. B (H~)) 3adant2 (-) opreq12d 3eqtr4d)) thm (hiret () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (e. (opr A (.i) B) (RR)) (= (opr A (.i) B) (opr B (.i) A)))) (A B ax-hicl (opr A (.i) B) cjrebt syl (` (*) (opr A (.i) B)) (opr A (.i) B) eqcom syl6bb B A ax-his1 ancoms (opr A (.i) B) eqeq2d bitr4d)) thm (hiidrclt () () (-> (e. A (H~)) (e. (opr A (.i) A) (RR))) ((opr A (.i) A) eqid A A hiret mpbiri anidms)) thm (hi01t () () (-> (e. A (H~)) (= (opr (0v) (.i) A) (0))) (0cn ax-hv0cl (0) (0v) A ax-his3 mp3an12 ax-hv0cl (0v) ax-hvmul0 ax-mp (.i) A opreq1i syl5eqr ax-hv0cl (0v) A ax-hicl mpan (opr (0v) (.i) A) mul02t syl eqtrd)) thm (hi02t () () (-> (e. A (H~)) (= (opr A (.i) (0v)) (0))) (ax-hv0cl A (0v) ax-his1 mpan2 A hi01t (*) fveq2d cj0 syl6eq eqtrd)) thm (hiidge0t () () (-> (e. A (H~)) (br (0) (<_) (opr A (.i) A))) ((= A (0v)) pm2.1 A ax-his4OLD ex A (0v) (.i) A opreq1 A hi01t sylan9eqr eqcomd ex orim12d mpi A hiidrclt 0re jctil (0) (opr A (.i) A) leloet syl mpbird)) thm (his6t () () (-> (e. A (H~)) (<-> (= (opr A (.i) A) (0)) (= A (0v)))) (A ax-his4OLD 0re (0) (opr A (.i) A) leltnet mp3an1 A hiidrclt A hiidge0t sylanc (-. (= A (0v))) adantr mpbid ex a3d (opr A (.i) A) (0) eqcom syl5ib A (0v) (.i) A opreq1 (0) eqeq1d A hi01t syl5bir com12 impbid)) thm (his1 () ((his1.1 (e. A (H~))) (his1.2 (e. B (H~)))) (= (opr A (.i) B) (` (*) (opr B (.i) A))) (his1.1 his1.2 A B ax-his1 mp2an)) thm (abshicomt () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (` (abs) (opr A (.i) B)) (` (abs) (opr B (.i) A)))) (A B ax-his1 (abs) fveq2d B A ax-hicl ancoms (opr B (.i) A) abscjt syl eqtrd)) thm (hial0 ((A x)) () (-> (e. A (H~)) (<-> (A.e. x (H~) (= (opr A (.i) (cv x)) (0))) (= A (0v)))) ((cv x) A A (.i) opreq2 (0) eqeq1d (H~) rcla4v A his6t sylibd A (0v) (.i) (cv x) opreq1 (cv x) hi01t sylan9eq ex (e. A (H~)) a1i r19.21adv impbid)) thm (hial02 ((A x)) () (-> (e. A (H~)) (<-> (A.e. x (H~) (= (opr (cv x) (.i) A) (0))) (= A (0v)))) ((cv x) A (.i) A opreq1 (0) eqeq1d (H~) rcla4v A his6t sylibd A (0v) (cv x) (.i) opreq2 (cv x) hi02t sylan9eq ex (e. A (H~)) a1i r19.21adv impbid)) thm (hisubcom () ((hisubcom.1 (e. A (H~))) (hisubcom.2 (e. B (H~))) (hisubcom.3 (e. C (H~))) (hisubcom.4 (e. D (H~)))) (= (opr (opr A (-v) B) (.i) (opr C (-v) D)) (opr (opr B (-v) A) (.i) (opr D (-v) C))) (hisubcom.2 hisubcom.1 hvnegdi hisubcom.4 hisubcom.3 hvnegdi (.i) opreq12i 1cn negcl 1cn negcl hisubcom.2 hisubcom.1 hvsubcl hisubcom.4 hisubcom.3 hvsubcl his35 ax1re renegcl (-u (1)) cjret ax-mp (-u (1)) (x.) opreq2i 1cn 1cn mul2neg 1cn mulid1 3eqtr (x.) (opr (opr B (-v) A) (.i) (opr D (-v) C)) opreq1i hisubcom.2 hisubcom.1 hvsubcl hisubcom.4 hisubcom.3 hvsubcl hicl mulid2 3eqtr eqtr3)) thm (hi2eqt () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (= (opr A (.i) (opr A (-v) B)) (opr B (.i) (opr A (-v) B))) (= A B))) (A B (opr A (-v) B) his2subt (e. A (H~)) (e. B (H~)) pm3.26 (e. A (H~)) (e. B (H~)) pm3.27 A B hvsubclt syl3anc (0) eqeq1d A B hvsubclt (opr A (-v) B) his6t syl bitr3d (opr A (.i) (opr A (-v) B)) (opr B (.i) (opr A (-v) B)) subeq0t A B hvsubclt A (opr A (-v) B) ax-hicl syldan B (opr A (-v) B) ax-hicl (e. A (H~)) (e. B (H~)) pm3.27 A B hvsubclt sylanc sylanc A B hvsubeq0t 3bitr3d)) thm (hial2eqt ((A x) (B x)) () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (A.e. x (H~) (= (opr A (.i) (cv x)) (opr B (.i) (cv x)))) (= A B))) (A B hvsubclt (cv x) (opr A (-v) B) A (.i) opreq2 (cv x) (opr A (-v) B) B (.i) opreq2 eqeq12d (H~) rcla4v syl A B hi2eqt sylibd A B (.i) (cv x) opreq1 (e. (cv x) (H~)) a1d r19.21aiv (/\ (e. A (H~)) (e. B (H~))) a1i impbid)) thm (hial2eq2t ((A x) (B x)) () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (A.e. x (H~) (= (opr (cv x) (.i) A) (opr (cv x) (.i) B))) (= A B))) (A (cv x) ax-his1 B (cv x) ax-his1 eqeqan12d (opr (cv x) (.i) A) (opr (cv x) (.i) B) cj11t (cv x) A ax-hicl ancoms (cv x) B ax-hicl ancoms syl2an bitr2d anandirs ralbidva A B x hial2eqt bitrd)) thm (orthcom () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (= (opr A (.i) B) (0)) (= (opr B (.i) A) (0)))) (B A ax-his1 ancoms (0) eqeq1d (opr A (.i) B) (0) (*) fveq2 cj0 syl6eq syl5bir A B ax-his1 (0) eqeq1d (opr B (.i) A) (0) (*) fveq2 cj0 syl6eq syl5bir impbid)) thm (normlem0 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~)))) (= (opr (opr F (-v) (opr S (.s) G)) (.i) (opr F (-v) (opr S (.s) G))) (opr (opr (opr F (.i) F) (+) (opr (-u (` (*) S)) (x.) (opr F (.i) G))) (+) (opr (opr (-u S) (x.) (opr G (.i) F)) (+) (opr (opr S (x.) (` (*) S)) (x.) (opr G (.i) G))))) (normlem1.2 normlem1.1 normlem1.3 hvmulcl hvsubval normlem1.1 mulm1 (.s) G opreq1i 1cn negcl normlem1.1 normlem1.3 hvmulass eqtr3 F (+v) opreq2i eqtr4 normlem1.2 normlem1.1 normlem1.3 hvmulcl hvsubval normlem1.1 mulm1 (.s) G opreq1i 1cn negcl normlem1.1 normlem1.3 hvmulass eqtr3 F (+v) opreq2i eqtr4 (.i) opreq12i normlem1.2 normlem1.1 negcl normlem1.3 hvmulcl normlem1.2 normlem1.1 negcl normlem1.3 hvmulcl hvaddcl F (opr (-u S) (.s) G) (opr F (+v) (opr (-u S) (.s) G)) ax-his2 mp3an normlem1.2 normlem1.2 normlem1.1 negcl normlem1.3 hvmulcl F F (opr (-u S) (.s) G) his7t mp3an normlem1.1 negcl normlem1.2 normlem1.3 (-u S) F G his5t mp3an normlem1.1 cjneg (x.) (opr F (.i) G) opreq1i eqtr (opr F (.i) F) (+) opreq2i eqtr normlem1.1 negcl normlem1.3 normlem1.2 normlem1.1 negcl normlem1.3 hvmulcl hvaddcl (-u S) G (opr F (+v) (opr (-u S) (.s) G)) ax-his3 mp3an normlem1.3 normlem1.2 normlem1.1 negcl normlem1.3 hvmulcl G F (opr (-u S) (.s) G) his7t mp3an normlem1.1 negcl normlem1.3 normlem1.3 (-u S) G G his5t mp3an (opr G (.i) F) (+) opreq2i eqtr (-u S) (x.) opreq2i normlem1.1 negcl normlem1.3 normlem1.2 hicl normlem1.1 negcl cjcl normlem1.3 normlem1.3 hicl mulcl adddi normlem1.1 negcl normlem1.1 negcl cjcl normlem1.3 normlem1.3 hicl mulass normlem1.1 cjneg (-u S) (x.) opreq2i normlem1.1 normlem1.1 cjcl mul2neg eqtr (x.) (opr G (.i) G) opreq1i eqtr3 (opr (-u S) (x.) (opr G (.i) F)) (+) opreq2i eqtr 3eqtr (+) opreq12i 3eqtr)) thm (normlem1 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem1.4 (e. R (RR))) (normlem1.5 (= (` (abs) S) (1)))) (= (opr (opr F (-v) (opr (opr S (x.) R) (.s) G)) (.i) (opr F (-v) (opr (opr S (x.) R) (.s) G))) (opr (opr (opr F (.i) F) (+) (opr (opr (` (*) S) (x.) (-u R)) (x.) (opr F (.i) G))) (+) (opr (opr (opr S (x.) (-u R)) (x.) (opr G (.i) F)) (+) (opr (opr R (^) (2)) (x.) (opr G (.i) G))))) (normlem1.1 normlem1.4 recn mulcl normlem1.2 normlem1.3 normlem0 normlem1.1 normlem1.4 recn cjmul normlem1.4 normlem1.4 recn cjreb mpbi (` (*) S) (x.) opreq2i eqtr negeqi normlem1.1 cjcl normlem1.4 recn mulneg2 eqtr4 (x.) (opr F (.i) G) opreq1i (opr F (.i) F) (+) opreq2i normlem1.1 normlem1.4 recn mulneg2 eqcomi (x.) (opr G (.i) F) opreq1i normlem1.1 normlem1.4 recn cjmul (opr S (x.) R) (x.) opreq2i normlem1.1 normlem1.4 recn normlem1.1 cjcl normlem1.4 recn cjcl mul4 normlem1.5 (^) (2) opreq1i normlem1.1 absvalsq sq1 3eqtr3 normlem1.4 normlem1.4 recn cjreb mpbi R (x.) opreq2i (x.) opreq12i normlem1.4 recn normlem1.4 recn mulcl mulid2 eqtr 3eqtr normlem1.4 recn sqval eqtr4 (x.) (opr G (.i) G) opreq1i (+) opreq12i (+) opreq12i eqtr)) thm (normlem2 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem2.4 (= B (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F))))))) (e. B (RR)) (normlem2.4 normlem1.1 cjcj eqcomi normlem1.3 normlem1.2 his1 (x.) opreq12i normlem1.1 cjcl normlem1.2 normlem1.3 hicl cjmul eqtr4 normlem1.2 normlem1.3 his1 (` (*) S) (x.) opreq2i normlem1.1 normlem1.3 normlem1.2 hicl cjmul eqtr4 (+) opreq12i normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcom normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl cjadd 3eqtr4r normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl cjreb mpbir renegcl eqeltr)) thm (normlem3 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem2.4 (= B (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F)))))) (normlem3.5 (= A (opr G (.i) G))) (normlem3.6 (= C (opr F (.i) F))) (normlem3.7 (e. R (RR)))) (= (opr (opr (opr A (x.) (opr R (^) (2))) (+) (opr B (x.) R)) (+) C) (opr (opr (opr F (.i) F) (+) (opr (opr (` (*) S) (x.) (-u R)) (x.) (opr F (.i) G))) (+) (opr (opr (opr S (x.) (-u R)) (x.) (opr G (.i) F)) (+) (opr (opr R (^) (2)) (x.) (opr G (.i) G))))) (normlem3.6 normlem3.5 normlem1.3 normlem1.3 hicl eqeltr normlem3.7 recn sqcl mulcl normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 recn normlem3.7 recn mulcl addcom normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl normlem3.7 recn mulneg1 normlem2.4 (x.) R opreq1i normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl normlem3.7 recn mulneg2 3eqtr4 normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl normlem3.7 recn negcl adddir normlem1.1 cjcl normlem1.2 normlem1.3 hicl normlem3.7 recn negcl mul23 normlem1.1 normlem1.3 normlem1.2 hicl normlem3.7 recn negcl mul23 (+) opreq12i 3eqtr normlem3.7 recn sqcl normlem3.5 normlem1.3 normlem1.3 hicl eqeltr mulcom normlem3.5 (opr R (^) (2)) (x.) opreq2i eqtr3 (+) opreq12i normlem1.1 cjcl normlem3.7 recn negcl mulcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem3.7 recn negcl mulcl normlem1.3 normlem1.2 hicl mulcl normlem3.7 recn sqcl normlem1.3 normlem1.3 hicl mulcl addass 3eqtr (+) opreq12i normlem3.5 normlem1.3 normlem1.3 hicl eqeltr normlem3.7 recn sqcl mulcl normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 recn normlem3.7 recn mulcl addcl normlem3.6 normlem1.2 normlem1.2 hicl eqeltr addcom normlem1.2 normlem1.2 hicl normlem1.1 cjcl normlem3.7 recn negcl mulcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem3.7 recn negcl mulcl normlem1.3 normlem1.2 hicl mulcl normlem3.7 recn sqcl normlem1.3 normlem1.3 hicl mulcl addcl addass 3eqtr4)) thm (normlem4 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem2.4 (= B (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F)))))) (normlem3.5 (= A (opr G (.i) G))) (normlem3.6 (= C (opr F (.i) F))) (normlem4.7 (e. R (RR))) (normlem4.8 (= (` (abs) S) (1)))) (= (opr (opr F (-v) (opr (opr S (x.) R) (.s) G)) (.i) (opr F (-v) (opr (opr S (x.) R) (.s) G))) (opr (opr (opr A (x.) (opr R (^) (2))) (+) (opr B (x.) R)) (+) C)) (normlem1.1 normlem1.2 normlem1.3 normlem4.7 normlem4.8 normlem1 normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem3.5 normlem3.6 normlem4.7 normlem3 eqtr4)) thm (normlem5 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem2.4 (= B (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F)))))) (normlem3.5 (= A (opr G (.i) G))) (normlem3.6 (= C (opr F (.i) F))) (normlem4.7 (e. R (RR))) (normlem4.8 (= (` (abs) S) (1)))) (br (0) (<_) (opr (opr (opr A (x.) (opr R (^) (2))) (+) (opr B (x.) R)) (+) C)) (normlem1.2 normlem1.1 normlem4.7 recn mulcl normlem1.3 hvmulcl hvsubcl (opr F (-v) (opr (opr S (x.) R) (.s) G)) hiidge0t ax-mp normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem3.5 normlem3.6 normlem4.7 normlem4.8 normlem4 breqtr)) thm (normlem6 ((A x) (B x) (C x)) ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem2.4 (= B (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F)))))) (normlem3.5 (= A (opr G (.i) G))) (normlem3.6 (= C (opr F (.i) F))) (normlem6.7 (= (` (abs) S) (1)))) (br (` (abs) B) (<_) (opr (2) (x.) (opr (` (sqr) A) (x.) (` (sqr) C)))) (normlem1.3 G hiidge0t ax-mp normlem3.5 breqtrr normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr (cv x) (if (e. (cv x) (RR)) (cv x) (0)) (^) (2) opreq1 A (x.) opreq2d (cv x) (if (e. (cv x) (RR)) (cv x) (0)) B (x.) opreq2 (+) opreq12d (+) C opreq1d (0) (<_) breq2d normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem3.5 normlem3.6 0re (cv x) elimel normlem6.7 normlem5 dedth rgen discrlem ax-mp normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 resqcl 4re normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr remulcl remulcl 0re lesubadd2 mpbi 4re normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr remulcl remulcl recn addid1 breqtr normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 sqge0 0re 4re 4pos ltlei normlem1.3 G hiidge0t ax-mp normlem3.5 breqtrr normlem1.2 F hiidge0t ax-mp normlem3.6 breqtrr normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr mulge0 mp2an 4re normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr remulcl mulge0 mp2an normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 resqcl 4re normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr remulcl remulcl sqrle mp2an mpbi normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 absre 4re normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr remulcl 0re 4re 4pos ltlei normlem1.3 G hiidge0t ax-mp normlem3.5 breqtrr normlem1.2 F hiidge0t ax-mp normlem3.6 breqtrr normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr mulge0 mp2an sqrmuli sqr4 normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr normlem1.3 G hiidge0t ax-mp normlem3.5 breqtrr normlem1.2 F hiidge0t ax-mp normlem3.6 breqtrr sqrmuli (x.) opreq12i eqtr2 3brtr4)) thm (normlem7 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem7.4 (= (` (abs) S) (1)))) (br (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F))) (<_) (opr (2) (x.) (opr (` (sqr) (opr G (.i) G)) (x.) (` (sqr) (opr F (.i) F))))) (normlem1.1 normlem1.2 normlem1.3 (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F)))) eqid normlem2 normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl negreb mpbi leabs normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl absneg breqtrr normlem1.1 normlem1.2 normlem1.3 (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F)))) eqid (opr G (.i) G) eqid (opr F (.i) F) eqid normlem7.4 normlem6 normlem1.1 normlem1.2 normlem1.3 (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F)))) eqid normlem2 normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl negreb mpbi normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl negcl abscl 2re normlem1.3 G hiidge0t ax-mp normlem1.3 G hiidrclt ax-mp sqrcl ax-mp normlem1.2 F hiidge0t ax-mp normlem1.2 F hiidrclt ax-mp sqrcl ax-mp remulcl remulcl letr mp2an)) thm (normlem8 () ((normlem8.1 (e. A (H~))) (normlem8.2 (e. B (H~))) (normlem8.3 (e. C (H~))) (normlem8.4 (e. D (H~)))) (= (opr (opr A (+v) B) (.i) (opr C (+v) D)) (opr (opr (opr A (.i) C) (+) (opr B (.i) D)) (+) (opr (opr A (.i) D) (+) (opr B (.i) C)))) (normlem8.1 normlem8.3 normlem8.4 A C D his7t mp3an normlem8.2 normlem8.3 normlem8.4 B C D his7t mp3an (+) opreq12i normlem8.1 normlem8.2 normlem8.3 normlem8.4 hvaddcl A B (opr C (+v) D) ax-his2 mp3an normlem8.1 normlem8.3 hicl normlem8.2 normlem8.4 hicl normlem8.1 normlem8.4 hicl normlem8.2 normlem8.3 hicl add42 3eqtr4)) thm (normlem9 () ((normlem8.1 (e. A (H~))) (normlem8.2 (e. B (H~))) (normlem8.3 (e. C (H~))) (normlem8.4 (e. D (H~)))) (= (opr (opr A (-v) B) (.i) (opr C (-v) D)) (opr (opr (opr A (.i) C) (+) (opr B (.i) D)) (-) (opr (opr A (.i) D) (+) (opr B (.i) C)))) (normlem8.1 normlem8.2 hvsubval normlem8.3 normlem8.4 hvsubval (.i) opreq12i normlem8.1 1cn negcl normlem8.2 hvmulcl normlem8.3 1cn negcl normlem8.4 hvmulcl normlem8 1cn negcl normlem8.2 1cn negcl normlem8.4 hvmulcl (-u (1)) B (opr (-u (1)) (.s) D) ax-his3 mp3an 1cn negcl normlem8.2 normlem8.4 (-u (1)) B D his5t mp3an (-u (1)) (x.) opreq2i ax1re renegcl (-u (1)) cjret ax-mp (-u (1)) (x.) opreq2i 1cn 1cn mul2neg 1cn mulid2 3eqtr (x.) (opr B (.i) D) opreq1i 1cn negcl 1cn negcl cjcl normlem8.2 normlem8.4 hicl mulass normlem8.2 normlem8.4 hicl mulid2 3eqtr3 3eqtr (opr A (.i) C) (+) opreq2i 1cn negcl normlem8.1 normlem8.4 (-u (1)) A D his5t mp3an ax1re renegcl (-u (1)) cjret ax-mp (x.) (opr A (.i) D) opreq1i normlem8.1 normlem8.4 hicl mulm1 3eqtr 1cn negcl normlem8.2 normlem8.3 (-u (1)) B C ax-his3 mp3an normlem8.2 normlem8.3 hicl mulm1 eqtr (+) opreq12i normlem8.1 normlem8.4 hicl normlem8.2 normlem8.3 hicl negdi eqtr4 (+) opreq12i normlem8.1 normlem8.3 hicl normlem8.2 normlem8.4 hicl addcl normlem8.1 normlem8.4 hicl normlem8.2 normlem8.3 hicl addcl negsub eqtr 3eqtr)) thm (normlem7t () ((normlem7t.1 (e. A (H~))) (normlem7t.2 (e. B (H~)))) (-> (/\ (e. S (CC)) (= (` (abs) S) (1))) (br (opr (opr (` (*) S) (x.) (opr A (.i) B)) (+) (opr S (x.) (opr B (.i) A))) (<_) (opr (2) (x.) (opr (` (sqr) (opr B (.i) B)) (x.) (` (sqr) (opr A (.i) A)))))) (S (if (/\ (e. S (CC)) (= (` (abs) S) (1))) S (1)) (*) fveq2 (x.) (opr A (.i) B) opreq1d S (if (/\ (e. S (CC)) (= (` (abs) S) (1))) S (1)) (x.) (opr B (.i) A) opreq1 (+) opreq12d (<_) (opr (2) (x.) (opr (` (sqr) (opr B (.i) B)) (x.) (` (sqr) (opr A (.i) A)))) breq1d S (if (/\ (e. S (CC)) (= (` (abs) S) (1))) S (1)) (CC) eleq1 S (if (/\ (e. S (CC)) (= (` (abs) S) (1))) S (1)) (abs) fveq2 (1) eqeq1d anbi12d (1) (if (/\ (e. S (CC)) (= (` (abs) S) (1))) S (1)) (CC) eleq1 (1) (if (/\ (e. S (CC)) (= (` (abs) S) (1))) S (1)) (abs) fveq2 (1) eqeq1d anbi12d 1cn 0re ax1re lt01 ltlei ax1re absid ax-mp pm3.2i elimhyp pm3.26i normlem7t.1 normlem7t.2 S (if (/\ (e. S (CC)) (= (` (abs) S) (1))) S (1)) (CC) eleq1 S (if (/\ (e. S (CC)) (= (` (abs) S) (1))) S (1)) (abs) fveq2 (1) eqeq1d anbi12d (1) (if (/\ (e. S (CC)) (= (` (abs) S) (1))) S (1)) (CC) eleq1 (1) (if (/\ (e. S (CC)) (= (` (abs) S) (1))) S (1)) (abs) fveq2 (1) eqeq1d anbi12d 1cn 0re ax1re lt01 ltlei ax1re absid ax-mp pm3.2i elimhyp pm3.27i normlem7 dedth)) thm (bcseq () ((normlem7t.1 (e. A (H~))) (normlem7t.2 (e. B (H~)))) (<-> (= (opr (opr A (.i) B) (x.) (opr B (.i) A)) (opr (opr A (.i) A) (x.) (opr B (.i) B))) (= (opr (opr B (.i) B) (.s) A) (opr (opr A (.i) B) (.s) B))) ((opr (opr A (.i) B) (x.) (opr B (.i) A)) (opr (opr A (.i) A) (x.) (opr B (.i) B)) (x.) (opr B (.i) B) opreq1 eqcomd normlem7t.2 normlem7t.2 hicl normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 (opr B (.i) B) (opr (opr B (.i) B) (.s) A) A his5t mp3an normlem7t.2 B hiidrclt ax-mp (opr B (.i) B) cjret ax-mp normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.1 (opr B (.i) B) A A ax-his3 mp3an (x.) opreq12i normlem7t.2 normlem7t.2 hicl normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.1 hicl mulcl mulcom normlem7t.1 normlem7t.1 hicl normlem7t.2 normlem7t.2 hicl mulcom (x.) (opr B (.i) B) opreq1i eqtr4 3eqtr normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.2 (opr A (.i) B) (opr (opr B (.i) B) (.s) A) B his5t mp3an normlem7t.2 normlem7t.1 his1 eqcomi normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.2 (opr B (.i) B) A B ax-his3 mp3an (x.) opreq12i normlem7t.2 normlem7t.1 hicl normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.2 hicl mulcl mulcom normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.1 hicl mulass normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.1 hicl mulcl mulcom 3eqtr 3eqtr 3eqtr4g normlem7t.2 B hiidrclt ax-mp (opr B (.i) B) cjret ax-mp normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.1 (opr A (.i) B) B A ax-his3 mp3an (x.) opreq12i normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl normlem7t.1 (opr B (.i) B) (opr (opr A (.i) B) (.s) B) A his5t mp3an normlem7t.1 normlem7t.2 hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl normlem7t.2 (opr A (.i) B) (opr (opr A (.i) B) (.s) B) B his5t mp3an normlem7t.2 normlem7t.1 his1 eqcomi normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.2 (opr A (.i) B) B B ax-his3 mp3an (x.) opreq12i normlem7t.2 normlem7t.1 hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.2 hicl mulcl mulcom normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.2 hicl normlem7t.2 normlem7t.1 hicl mul23 normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.1 hicl mulcl normlem7t.2 normlem7t.2 hicl mulcom 3eqtr 3eqtr 3eqtr4r (= (opr (opr A (.i) B) (x.) (opr B (.i) A)) (opr (opr A (.i) A) (x.) (opr B (.i) B))) a1i (+) opreq12d (-) (opr (opr (opr (opr B (.i) B) (.s) A) (.i) (opr (opr A (.i) B) (.s) B)) (+) (opr (opr (opr A (.i) B) (.s) B) (.i) (opr (opr B (.i) B) (.s) A))) opreq1d normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl hicl addcl subid syl6eq normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl normlem9 syl5eq normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl hvsubcl (opr (opr (opr B (.i) B) (.s) A) (-v) (opr (opr A (.i) B) (.s) B)) his6t ax-mp sylib normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl hvsubeq0 sylib (opr (opr B (.i) B) (.s) A) (opr (opr A (.i) B) (.s) B) (.i) A opreq1 normlem7t.1 normlem7t.1 hicl normlem7t.2 normlem7t.2 hicl mulcom normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.1 (opr B (.i) B) A A ax-his3 mp3an eqtr4 normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.1 (opr A (.i) B) B A ax-his3 mp3an eqcomi 3eqtr4g eqcomd impbi)) thm (normlem9at () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (opr A (-v) B) (.i) (opr A (-v) B)) (opr (opr (opr A (.i) A) (+) (opr B (.i) B)) (-) (opr (opr A (.i) B) (+) (opr B (.i) A))))) (A (if (e. A (H~)) A (0v)) (-v) B opreq1 A (if (e. A (H~)) A (0v)) (-v) B opreq1 (.i) opreq12d (= A (if (e. A (H~)) A (0v))) id (= A (if (e. A (H~)) A (0v))) id (.i) opreq12d (+) (opr B (.i) B) opreq1d A (if (e. A (H~)) A (0v)) (.i) B opreq1 A (if (e. A (H~)) A (0v)) B (.i) opreq2 (+) opreq12d (-) opreq12d eqeq12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (.i) opreq12d (= B (if (e. B (H~)) B (0v))) id (= B (if (e. B (H~)) B (0v))) id (.i) opreq12d (opr (if (e. A (H~)) A (0v)) (.i) (if (e. A (H~)) A (0v))) (+) opreq2d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (.i) opreq2 B (if (e. B (H~)) B (0v)) (.i) (if (e. A (H~)) A (0v)) opreq1 (+) opreq12d (-) opreq12d eqeq12d ax-hv0cl A elimel ax-hv0cl B elimel ax-hv0cl A elimel ax-hv0cl B elimel normlem9 dedth2h)) thm (normvalt ((x y) (A x) (A y)) () (-> (e. A (H~)) (= (` (norm) A) (` (sqr) (opr A (.i) A)))) ((cv x) A (cv x) A (.i) opreq12 anidms (sqr) fveq2d x y df-hnorm (sqr) (opr A (.i) A) fvex fvopab4)) thm (normclt () () (-> (e. A (H~)) (e. (` (norm) A) (RR))) (A normvalt (opr A (.i) A) sqrclt A hiidrclt A hiidge0t sylanc eqeltrd)) thm (normge0t () () (-> (e. A (H~)) (br (0) (<_) (` (norm) A))) ((opr A (.i) A) sqrge0t A hiidrclt A hiidge0t sylanc A normvalt breqtrrd)) thm (normgt0tOLD () () (-> (e. A (H~)) (<-> (-. (= A (0v))) (br (0) (<) (` (norm) A)))) ((opr A (.i) A) sqrgt0t A hiidrclt (-. (= A (0v))) adantr A ax-his4OLD sylanc ex A (0v) (.i) A opreq1 A hi01t sylan9eqr (sqr) fveq2d sqr0 syl6eq ex (opr A (.i) A) sqrclt A hiidrclt A hiidge0t sylanc 0re jctir (` (sqr) (opr A (.i) A)) (0) lttri3t syl (-. (br (` (sqr) (opr A (.i) A)) (<) (0))) (-. (br (0) (<) (` (sqr) (opr A (.i) A)))) pm3.27 syl6bi syld con2d impbid A normvalt (0) (<) breq2d bitr4d)) thm (normgt0t () () (-> (e. A (H~)) (<-> (=/= A (0v)) (br (0) (<) (` (norm) A)))) ((opr A (.i) A) sqrgt0t A hiidrclt (=/= A (0v)) adantr A ax-his4 sylanc ex A (0v) (.i) A opreq1 A hi01t sylan9eqr (sqr) fveq2d sqr0 syl6eq ex (opr A (.i) A) sqrclt A hiidrclt A hiidge0t sylanc 0re jctir (` (sqr) (opr A (.i) A)) (0) lttri3t syl (-. (br (` (sqr) (opr A (.i) A)) (<) (0))) (-. (br (0) (<) (` (sqr) (opr A (.i) A)))) pm3.27 syl6bi syld con2d A (0v) df-ne syl6ibr impbid A normvalt (0) (<) breq2d bitr4d)) thm (norm0 () () (= (` (norm) (0v)) (0)) (ax-hv0cl (0v) normvalt ax-mp ax-hv0cl (0v) hi01t (sqr) fveq2d ax-mp sqr0 3eqtr)) thm (norm-it () () (-> (e. A (H~)) (<-> (= (` (norm) A) (0)) (= A (0v)))) (A normgt0tOLD 0re (0) (` (norm) A) leltnet mp3an1 A normclt A normge0t sylanc bitrd con4bid (0) (` (norm) A) eqcom syl6rbb)) thm (normne0t () () (-> (e. A (H~)) (<-> (=/= (` (norm) A) (0)) (=/= A (0v)))) (A norm-it eqneqd)) thm (normcl () ((normcl.1 (e. A (H~)))) (e. (` (norm) A) (RR)) (normcl.1 A normclt ax-mp)) thm (normsq () ((normcl.1 (e. A (H~)))) (= (opr (` (norm) A) (^) (2)) (opr A (.i) A)) (normcl.1 A normvalt ax-mp (^) (2) opreq1i normcl.1 A hiidge0t ax-mp normcl.1 A hiidrclt ax-mp sqsqr ax-mp eqtr)) thm (norm-i () ((normcl.1 (e. A (H~)))) (<-> (= (` (norm) A) (0)) (= A (0v))) (normcl.1 A norm-it ax-mp)) thm (normsqt () () (-> (e. A (H~)) (= (opr (` (norm) A) (^) (2)) (opr A (.i) A))) (A (if (e. A (H~)) A (0v)) (norm) fveq2 (^) (2) opreq1d (= A (if (e. A (H~)) A (0v))) id (= A (if (e. A (H~)) A (0v))) id (.i) opreq12d eqeq12d ax-hv0cl A elimel normsq dedth)) thm (normsub0 () ((normsub0.1 (e. A (H~))) (normsub0.2 (e. B (H~)))) (<-> (= (` (norm) (opr A (-v) B)) (0)) (= A B)) (normsub0.1 normsub0.2 hvsubcl norm-i normsub0.1 normsub0.2 hvsubeq0 bitr)) thm (normsub0t () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (= (` (norm) (opr A (-v) B)) (0)) (= A B))) (A (if (e. A (H~)) A (0v)) (-v) B opreq1 (norm) fveq2d (0) eqeq1d A (if (e. A (H~)) A (0v)) B eqeq1 bibi12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (norm) fveq2d (0) eqeq1d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) eqeq2 bibi12d ax-hv0cl A elimel ax-hv0cl B elimel normsub0 dedth2h)) thm (norm-ii () ((norm-ii.1 (e. A (H~))) (norm-ii.2 (e. B (H~)))) (br (` (norm) (opr A (+v) B)) (<_) (opr (` (norm) A) (+) (` (norm) B))) (ax1re 1cn cjreb mpbi (x.) (opr B (.i) A) opreq1i norm-ii.2 norm-ii.1 hicl mulid2 eqtr norm-ii.1 norm-ii.2 hicl mulid2 (+) opreq12i 1cn norm-ii.2 norm-ii.1 0re ax1re lt01 ltlei ax1re absid ax-mp normlem7 eqbrtrr ax1re 1cn cjreb mpbi (x.) (opr B (.i) A) opreq1i norm-ii.2 norm-ii.1 hicl mulid2 eqtr norm-ii.1 norm-ii.2 hicl mulid2 (+) opreq12i 1cn norm-ii.2 norm-ii.1 (-u (opr (opr (` (*) (1)) (x.) (opr B (.i) A)) (+) (opr (1) (x.) (opr A (.i) B)))) eqid normlem2 1cn cjcl norm-ii.2 norm-ii.1 hicl mulcl 1cn norm-ii.1 norm-ii.2 hicl mulcl addcl negreb mpbi eqeltrr 2re norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp remulcl remulcl norm-ii.1 A hiidrclt ax-mp norm-ii.2 B hiidrclt ax-mp readdcl leadd2 mpbi norm-ii.1 norm-ii.2 norm-ii.1 norm-ii.2 normlem8 norm-ii.1 norm-ii.2 hicl norm-ii.2 norm-ii.1 hicl addcom (opr (opr A (.i) A) (+) (opr B (.i) B)) (+) opreq2i eqtr norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp recn norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp recn binom2 norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp recn sqcl 2cn norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp recn norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp recn mulcl mulcl norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp recn sqcl add23 norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqsqr ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqsqr ax-mp (+) opreq12i (+) (opr (2) (x.) (opr (` (sqr) (opr A (.i) A)) (x.) (` (sqr) (opr B (.i) B)))) opreq1i 3eqtr 3brtr4 norm-ii.1 norm-ii.2 hvaddcl (opr A (+v) B) hiidge0t ax-mp norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp readdcl sqge0 norm-ii.1 norm-ii.2 hvaddcl (opr A (+v) B) hiidrclt ax-mp norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp readdcl resqcl sqrle mp2an mpbi norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrge0 ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrge0 ax-mp norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp addge0 mp2an norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp readdcl sqrsq ax-mp breqtr norm-ii.1 norm-ii.2 hvaddcl (opr A (+v) B) normvalt ax-mp norm-ii.1 A normvalt ax-mp norm-ii.2 B normvalt ax-mp (+) opreq12i 3brtr4)) thm (norm-iit () () (-> (/\ (e. A (H~)) (e. B (H~))) (br (` (norm) (opr A (+v) B)) (<_) (opr (` (norm) A) (+) (` (norm) B)))) (A (if (e. A (H~)) A (0v)) (+v) B opreq1 (norm) fveq2d A (if (e. A (H~)) A (0v)) (norm) fveq2 (+) (` (norm) B) opreq1d (<_) breq12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (+v) opreq2 (norm) fveq2d B (if (e. B (H~)) B (0v)) (norm) fveq2 (` (norm) (if (e. A (H~)) A (0v))) (+) opreq2d (<_) breq12d ax-hv0cl A elimel ax-hv0cl B elimel norm-ii dedth2h)) thm (norm-iii () ((norm-iii.1 (e. A (CC))) (norm-iii.2 (e. B (H~)))) (= (` (norm) (opr A (.s) B)) (opr (` (abs) A) (x.) (` (norm) B))) (norm-iii.1 norm-iii.1 norm-iii.2 norm-iii.2 his35 (sqr) fveq2i norm-iii.1 cjmulrcl norm-iii.2 B hiidrclt ax-mp norm-iii.1 cjmulge0 norm-iii.2 B hiidge0t ax-mp sqrmuli eqtr norm-iii.1 norm-iii.2 hvmulcl (opr A (.s) B) normvalt ax-mp norm-iii.1 A absvalt ax-mp norm-iii.2 B normvalt ax-mp (x.) opreq12i 3eqtr4)) thm (norm-iiit () () (-> (/\ (e. A (CC)) (e. B (H~))) (= (` (norm) (opr A (.s) B)) (opr (` (abs) A) (x.) (` (norm) B)))) (A (if (e. A (CC)) A (0)) (.s) B opreq1 (norm) fveq2d A (if (e. A (CC)) A (0)) (abs) fveq2 (x.) (` (norm) B) opreq1d eqeq12d B (if (e. B (H~)) B (0v)) (if (e. A (CC)) A (0)) (.s) opreq2 (norm) fveq2d B (if (e. B (H~)) B (0v)) (norm) fveq2 (` (abs) (if (e. A (CC)) A (0))) (x.) opreq2d eqeq12d 0cn A elimel ax-hv0cl B elimel norm-iii dedth2h)) thm (normsub () ((normsub.1 (e. A (H~))) (normsub.2 (e. B (H~)))) (= (` (norm) (opr A (-v) B)) (` (norm) (opr B (-v) A))) (normsub.2 normsub.1 hvnegdi (norm) fveq2i 1cn negcl normsub.2 normsub.1 hvsubcl norm-iii eqtr3 1cn absneg 0re ax1re lt01 ltlei ax1re absid ax-mp eqtr (x.) (` (norm) (opr B (-v) A)) opreq1i normsub.2 normsub.1 hvsubcl normcl recn mulid2 3eqtr)) thm (normpyth () ((normsub.1 (e. A (H~))) (normsub.2 (e. B (H~)))) (-> (= (opr A (.i) B) (0)) (= (opr (` (norm) (opr A (+v) B)) (^) (2)) (opr (opr (` (norm) A) (^) (2)) (+) (opr (` (norm) B) (^) (2))))) ((= (opr A (.i) B) (0)) id normsub.1 normsub.2 A B orthcom mp2an biimp (+) opreq12d 0cn addid1 syl6eq (opr (opr A (.i) A) (+) (opr B (.i) B)) (+) opreq2d normsub.1 normsub.1 hicl normsub.2 normsub.2 hicl addcl addid1 syl6eq normsub.1 normsub.2 normsub.1 normsub.2 normlem8 syl5eq normsub.1 normsub.2 hvaddcl normsq normsub.1 normsq normsub.2 normsq (+) opreq12i 3eqtr4g)) thm (normsubt () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (` (norm) (opr A (-v) B)) (` (norm) (opr B (-v) A)))) (A (if (e. A (H~)) A (0v)) (-v) B opreq1 (norm) fveq2d A (if (e. A (H~)) A (0v)) B (-v) opreq2 (norm) fveq2d eqeq12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (norm) fveq2d B (if (e. B (H~)) B (0v)) (-v) (if (e. A (H~)) A (0v)) opreq1 (norm) fveq2d eqeq12d ax-hv0cl A elimel ax-hv0cl B elimel normsub dedth2h)) thm (normnegt () () (-> (e. A (H~)) (= (` (norm) (opr (-u (1)) (.s) A)) (` (norm) A))) (ax-hv0cl (0v) A normsubt mpan A hv2negt (norm) fveq2d A hvsub0t (norm) fveq2d 3eqtr3d)) thm (normpytht () () (-> (/\ (e. A (H~)) (e. B (H~))) (-> (= (opr A (.i) B) (0)) (= (opr (` (norm) (opr A (+v) B)) (^) (2)) (opr (opr (` (norm) A) (^) (2)) (+) (opr (` (norm) B) (^) (2)))))) (A (if (e. A (H~)) A (0v)) (.i) B opreq1 (0) eqeq1d A (if (e. A (H~)) A (0v)) (+v) B opreq1 (norm) fveq2d (^) (2) opreq1d A (if (e. A (H~)) A (0v)) (norm) fveq2 (^) (2) opreq1d (+) (opr (` (norm) B) (^) (2)) opreq1d eqeq12d imbi12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (.i) opreq2 (0) eqeq1d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (+v) opreq2 (norm) fveq2d (^) (2) opreq1d B (if (e. B (H~)) B (0v)) (norm) fveq2 (^) (2) opreq1d (opr (` (norm) (if (e. A (H~)) A (0v))) (^) (2)) (+) opreq2d eqeq12d imbi12d ax-hv0cl A elimel ax-hv0cl B elimel normpyth dedth2h)) thm (normpyct () () (-> (/\ (e. A (H~)) (e. B (H~))) (-> (= (opr A (.i) B) (0)) (br (` (norm) A) (<_) (` (norm) (opr A (+v) B))))) (A normclt (` (norm) A) resqclt syl recnd (opr (` (norm) A) (^) (2)) ax0id syl (e. B (H~)) adantr B normclt (` (norm) B) sqge0t syl (e. A (H~)) adantl 0re (0) (opr (` (norm) B) (^) (2)) (opr (` (norm) A) (^) (2)) leadd2t mp3an1 ancoms A normclt (` (norm) A) resqclt syl B normclt (` (norm) B) resqclt syl syl2an mpbid eqbrtrrd (= (opr A (.i) B) (0)) adantr A B normpytht imp breqtrrd ex (` (norm) A) (` (norm) (opr A (+v) B)) le2sqt A normclt (e. B (H~)) adantr A normge0t (e. B (H~)) adantr jca A B ax-hvaddcl (opr A (+v) B) normclt syl A B ax-hvaddcl (opr A (+v) B) normge0t syl jca sylanc sylibrd)) thm (norm3dif () ((norm3dif.1 (e. A (H~))) (norm3dif.2 (e. B (H~))) (norm3dif.3 (e. C (H~)))) (br (` (norm) (opr A (-v) B)) (<_) (opr (` (norm) (opr A (-v) C)) (+) (` (norm) (opr C (-v) B)))) (norm3dif.1 1cn negcl norm3dif.3 hvmulcl norm3dif.3 1cn negcl norm3dif.2 hvmulcl hvaddcl hvass 1cn negcl norm3dif.3 hvmulcl norm3dif.3 1cn negcl norm3dif.2 hvmulcl hvass 1cn negcl norm3dif.3 hvmulcl norm3dif.3 hvcom norm3dif.3 norm3dif.3 hvsubval norm3dif.3 C hvsubidt ax-mp 3eqtr2 (+v) (opr (-u (1)) (.s) B) opreq1i ax-hv0cl 1cn negcl norm3dif.2 hvmulcl hvcom 1cn negcl norm3dif.2 hvmulcl (opr (-u (1)) (.s) B) ax-hvaddid ax-mp 3eqtr eqtr3 A (+v) opreq2i eqtr2 norm3dif.1 norm3dif.2 hvsubval norm3dif.1 norm3dif.3 hvsubval norm3dif.3 norm3dif.2 hvsubval (+v) opreq12i 3eqtr4 (norm) fveq2i norm3dif.1 norm3dif.3 hvsubcl norm3dif.3 norm3dif.2 hvsubcl norm-ii eqbrtr)) thm (norm3adif () ((norm3dif.1 (e. A (H~))) (norm3dif.2 (e. B (H~))) (norm3dif.3 (e. C (H~)))) (br (` (abs) (opr (` (norm) (opr A (-v) C)) (-) (` (norm) (opr B (-v) C)))) (<_) (` (norm) (opr A (-v) B))) (norm3dif.1 norm3dif.3 hvsubcl normcl norm3dif.2 norm3dif.3 hvsubcl normcl resubcl norm3dif.1 norm3dif.2 hvsubcl normcl absleOLD norm3dif.1 norm3dif.3 norm3dif.2 norm3dif norm3dif.1 norm3dif.3 hvsubcl normcl norm3dif.2 norm3dif.3 hvsubcl normcl norm3dif.1 norm3dif.2 hvsubcl normcl lesubadd mpbir norm3dif.1 norm3dif.3 hvsubcl normcl recn norm3dif.2 norm3dif.3 hvsubcl normcl recn negsubdi2 norm3dif.2 norm3dif.3 norm3dif.1 norm3dif norm3dif.2 norm3dif.1 normsub (+) (` (norm) (opr A (-v) C)) opreq1i breqtr norm3dif.2 norm3dif.3 hvsubcl normcl norm3dif.1 norm3dif.3 hvsubcl normcl norm3dif.1 norm3dif.2 hvsubcl normcl lesubadd mpbir eqbrtr mpbir2an)) thm (norm3lem () ((norm3dif.1 (e. A (H~))) (norm3dif.2 (e. B (H~))) (norm3dif.3 (e. C (H~))) (norm3lem.4 (e. D (RR)))) (-> (/\ (br (` (norm) (opr A (-v) C)) (<) (opr D (/) (2))) (br (` (norm) (opr C (-v) B)) (<) (opr D (/) (2)))) (br (` (norm) (opr A (-v) B)) (<) D)) (norm3dif.1 norm3dif.3 hvsubcl normcl norm3dif.3 norm3dif.2 hvsubcl normcl norm3lem.4 2re 2re 2pos gt0ne0i redivcl norm3lem.4 2re 2re 2pos gt0ne0i redivcl lt2add norm3dif.1 norm3dif.2 norm3dif.3 norm3dif norm3dif.1 norm3dif.2 hvsubcl normcl norm3dif.1 norm3dif.3 hvsubcl normcl norm3dif.3 norm3dif.2 hvsubcl normcl readdcl norm3lem.4 2re 2re 2pos gt0ne0i redivcl norm3lem.4 2re 2re 2pos gt0ne0i redivcl readdcl lelttr mpan syl norm3lem.4 recn 2cn 2re 2pos gt0ne0i divcl 2times 2cn norm3lem.4 recn 2re 2pos gt0ne0i divcan2 eqtr3 syl6breq)) thm (norm3dift () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (br (` (norm) (opr A (-v) B)) (<_) (opr (` (norm) (opr A (-v) C)) (+) (` (norm) (opr C (-v) B))))) (A (if (e. A (H~)) A (0v)) (-v) B opreq1 (norm) fveq2d A (if (e. A (H~)) A (0v)) (-v) C opreq1 (norm) fveq2d (+) (` (norm) (opr C (-v) B)) opreq1d (<_) breq12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (norm) fveq2d B (if (e. B (H~)) B (0v)) C (-v) opreq2 (norm) fveq2d (` (norm) (opr (if (e. A (H~)) A (0v)) (-v) C)) (+) opreq2d (<_) breq12d C (if (e. C (H~)) C (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (norm) fveq2d C (if (e. C (H~)) C (0v)) (-v) (if (e. B (H~)) B (0v)) opreq1 (norm) fveq2d (+) opreq12d (` (norm) (opr (if (e. A (H~)) A (0v)) (-v) (if (e. B (H~)) B (0v)))) (<_) breq2d ax-hv0cl A elimel ax-hv0cl B elimel ax-hv0cl C elimel norm3dif dedth3h)) thm (norm3lemt () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (RR)))) (-> (/\ (br (` (norm) (opr A (-v) C)) (<) (opr D (/) (2))) (br (` (norm) (opr C (-v) B)) (<) (opr D (/) (2)))) (br (` (norm) (opr A (-v) B)) (<) D))) (A (if (e. A (H~)) A (0v)) (-v) C opreq1 (norm) fveq2d (<) (opr D (/) (2)) breq1d (br (` (norm) (opr C (-v) B)) (<) (opr D (/) (2))) anbi1d A (if (e. A (H~)) A (0v)) (-v) B opreq1 (norm) fveq2d (<) D breq1d imbi12d B (if (e. B (H~)) B (0v)) C (-v) opreq2 (norm) fveq2d (<) (opr D (/) (2)) breq1d (br (` (norm) (opr (if (e. A (H~)) A (0v)) (-v) C)) (<) (opr D (/) (2))) anbi2d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (norm) fveq2d (<) D breq1d imbi12d C (if (e. C (H~)) C (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (norm) fveq2d (<) (opr D (/) (2)) breq1d C (if (e. C (H~)) C (0v)) (-v) (if (e. B (H~)) B (0v)) opreq1 (norm) fveq2d (<) (opr D (/) (2)) breq1d anbi12d (br (` (norm) (opr (if (e. A (H~)) A (0v)) (-v) (if (e. B (H~)) B (0v)))) (<) D) imbi1d D (if (e. D (RR)) D (2)) (/) (2) opreq1 (` (norm) (opr (if (e. A (H~)) A (0v)) (-v) (if (e. C (H~)) C (0v)))) (<) breq2d D (if (e. D (RR)) D (2)) (/) (2) opreq1 (` (norm) (opr (if (e. C (H~)) C (0v)) (-v) (if (e. B (H~)) B (0v)))) (<) breq2d anbi12d D (if (e. D (RR)) D (2)) (` (norm) (opr (if (e. A (H~)) A (0v)) (-v) (if (e. B (H~)) B (0v)))) (<) breq2 imbi12d ax-hv0cl A elimel ax-hv0cl B elimel ax-hv0cl C elimel 2re D elimel norm3lem dedth4h)) thm (norm3adift () ((norm3adift.1 (e. C (H~)))) (-> (/\ (e. A (H~)) (e. B (H~))) (br (` (abs) (opr (` (norm) (opr A (-v) C)) (-) (` (norm) (opr B (-v) C)))) (<_) (` (norm) (opr A (-v) B)))) (A (if (e. A (H~)) A (0v)) (-v) C opreq1 (norm) fveq2d (-) (` (norm) (opr B (-v) C)) opreq1d (abs) fveq2d A (if (e. A (H~)) A (0v)) (-v) B opreq1 (norm) fveq2d (<_) breq12d B (if (e. B (H~)) B (0v)) (-v) C opreq1 (norm) fveq2d (` (norm) (opr (if (e. A (H~)) A (0v)) (-v) C)) (-) opreq2d (abs) fveq2d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (norm) fveq2d (<_) breq12d ax-hv0cl A elimel ax-hv0cl B elimel norm3adift.1 norm3adif dedth2h)) thm (normpar () ((normpar.1 (e. A (H~))) (normpar.2 (e. B (H~)))) (= (opr (opr (` (norm) (opr A (-v) B)) (^) (2)) (+) (opr (` (norm) (opr A (+v) B)) (^) (2))) (opr (opr (2) (x.) (opr (` (norm) A) (^) (2))) (+) (opr (2) (x.) (opr (` (norm) B) (^) (2))))) (normpar.1 normpar.2 normpar.1 normpar.2 normlem9 normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl addcl normpar.1 normpar.2 hicl normpar.2 normpar.1 hicl addcl negsub eqtr4 normpar.1 normpar.2 normpar.1 normpar.2 normlem8 (+) opreq12i normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl addcl normpar.1 normpar.2 hicl normpar.2 normpar.1 hicl addcl negcl normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl addcl normpar.1 normpar.2 hicl normpar.2 normpar.1 hicl addcl add42 normpar.1 normpar.2 hicl normpar.2 normpar.1 hicl addcl negid (opr (opr (opr A (.i) A) (+) (opr B (.i) B)) (+) (opr (opr A (.i) A) (+) (opr B (.i) B))) (+) opreq2i normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl addcl normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl addcl addcl addid1 normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl add4 3eqtr 3eqtr normpar.1 normpar.2 hvsubcl normsq normpar.1 normpar.2 hvaddcl normsq (+) opreq12i normpar.1 normsq (2) (x.) opreq2i normpar.1 normpar.1 hicl 2times eqtr normpar.2 normsq (2) (x.) opreq2i normpar.2 normpar.2 hicl 2times eqtr (+) opreq12i 3eqtr4)) thm (normpar2 () ((normpar2.1 (e. A (H~))) (normpar2.2 (e. B (H~))) (normpar2.3 (e. C (H~)))) (= (opr (` (norm) (opr A (-v) B)) (^) (2)) (opr (opr (opr (2) (x.) (opr (` (norm) (opr A (-v) C)) (^) (2))) (+) (opr (2) (x.) (opr (` (norm) (opr B (-v) C)) (^) (2)))) (-) (opr (` (norm) (opr (opr A (+v) B) (-v) (opr (2) (.s) C))) (^) (2)))) (4re recn normpar2.1 normpar2.3 hvsubcl normcl resqcl recn mulcl 4re recn normpar2.2 normpar2.3 hvsubcl normcl resqcl recn mulcl 2cn 2re 2pos gt0ne0i divdir 4re recn normpar2.1 normpar2.3 hvsubcl normcl resqcl recn mulcl 4re recn normpar2.2 normpar2.3 hvsubcl normcl resqcl recn mulcl addcom normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubcl normpar2.1 normpar2.2 hvsubcl hvsubval normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubval (+v) (opr (-u (1)) (.s) (opr A (-v) B)) opreq1i normpar2.1 normpar2.2 hvcom normpar2.1 normpar2.2 hvnegdi (+v) opreq12i normpar2.2 normpar2.1 hvsubcan2 eqtr (+v) (opr (-u (1)) (.s) (opr (2) (.s) C)) opreq1i normpar2.1 normpar2.2 hvaddcl 1cn negcl 2cn normpar2.3 hvmulcl hvmulcl 1cn negcl normpar2.1 normpar2.2 hvsubcl hvmulcl hvadd23 2cn normpar2.2 hvmulcl 2cn normpar2.3 hvmulcl hvsubval 3eqtr4 3eqtr 2cn normpar2.2 normpar2.3 hvsubdistr1 eqtr4 (norm) fveq2i 2cn normpar2.2 normpar2.3 hvsubcl norm-iii 0re 2re 2pos ltlei 2re absid ax-mp (x.) (` (norm) (opr B (-v) C)) opreq1i 3eqtr (^) (2) opreq1i 2cn normpar2.2 normpar2.3 hvsubcl normcl recn sqmul sq2 (x.) (opr (` (norm) (opr B (-v) C)) (^) (2)) opreq1i 3eqtr normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubval (+v) (opr A (-v) B) opreq1i normpar2.1 normpar2.2 hvaddcl 1cn negcl 2cn normpar2.3 hvmulcl hvmulcl normpar2.1 normpar2.2 hvsubcl hvadd23 normpar2.1 normpar2.2 hvsubcan2 (+v) (opr (-u (1)) (.s) (opr (2) (.s) C)) opreq1i 2cn normpar2.1 hvmulcl 2cn normpar2.3 hvmulcl hvsubval eqtr4 3eqtr 2cn normpar2.1 normpar2.3 hvsubdistr1 eqtr4 (norm) fveq2i 2cn normpar2.1 normpar2.3 hvsubcl norm-iii 0re 2re 2pos ltlei 2re absid ax-mp (x.) (` (norm) (opr A (-v) C)) opreq1i 3eqtr (^) (2) opreq1i 2cn normpar2.1 normpar2.3 hvsubcl normcl recn sqmul sq2 (x.) (opr (` (norm) (opr A (-v) C)) (^) (2)) opreq1i 3eqtr (+) opreq12i normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubcl normpar2.1 normpar2.2 hvsubcl normpar 3eqtr2 (/) (2) opreq1i 2cn normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubcl normcl resqcl recn mulcl 2cn normpar2.1 normpar2.2 hvsubcl normcl resqcl recn mulcl 2cn 2re 2pos gt0ne0i divdir 2cn normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubcl normcl resqcl recn 2re 2pos gt0ne0i divcan3 2cn normpar2.1 normpar2.2 hvsubcl normcl resqcl recn 2re 2pos gt0ne0i divcan3 (+) opreq12i 3eqtr 4re recn normpar2.1 normpar2.3 hvsubcl normcl resqcl recn 2cn 2re 2pos gt0ne0i div23 4d2e2 (x.) (opr (` (norm) (opr A (-v) C)) (^) (2)) opreq1i eqtr 4re recn normpar2.2 normpar2.3 hvsubcl normcl resqcl recn 2cn 2re 2pos gt0ne0i div23 4d2e2 (x.) (opr (` (norm) (opr B (-v) C)) (^) (2)) opreq1i eqtr (+) opreq12i 3eqtr3 2re normpar2.1 normpar2.3 hvsubcl normcl resqcl remulcl 2re normpar2.2 normpar2.3 hvsubcl normcl resqcl remulcl readdcl recn normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubcl normcl resqcl recn normpar2.1 normpar2.2 hvsubcl normcl resqcl recn subadd mpbir eqcomi)) thm (polid2 () ((polid2.1 (e. A (H~))) (polid2.2 (e. B (H~))) (polid2.3 (e. C (H~))) (polid2.4 (e. D (H~)))) (= (opr A (.i) B) (opr (opr (opr (opr (opr A (+v) C) (.i) (opr D (+v) B)) (-) (opr (opr A (-v) C) (.i) (opr D (-v) B))) (+) (opr (i) (x.) (opr (opr (opr A (+v) (opr (i) (.s) C)) (.i) (opr D (+v) (opr (i) (.s) B))) (-) (opr (opr A (-v) (opr (i) (.s) C)) (.i) (opr D (-v) (opr (i) (.s) B)))))) (/) (4))) (4re recn polid2.1 polid2.2 hicl 4re 4pos gt0ne0i divcan3 2cn polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl addcl polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl subcl adddi polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl polid2.1 polid2.2 hicl (opr A (.i) B) (opr C (.i) D) (opr A (.i) B) ppncant mp3an polid2.1 polid2.2 hicl 2times eqtr4 (2) (x.) opreq2i 2cn 2cn polid2.1 polid2.2 hicl mulass 2t2e4 (x.) (opr A (.i) B) opreq1i 3eqtr2r polid2.1 polid2.4 hicl polid2.3 polid2.2 hicl addcl polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl addcl polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl addcl pnncan polid2.1 polid2.3 polid2.4 polid2.2 normlem8 polid2.1 polid2.3 polid2.4 polid2.2 normlem9 (-) opreq12i polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl addcl 2times 3eqtr4 polid2.1 axicn polid2.3 hvmulcl polid2.4 axicn polid2.2 hvmulcl normlem8 polid2.1 axicn polid2.3 hvmulcl polid2.4 axicn polid2.2 hvmulcl normlem9 (-) opreq12i polid2.1 polid2.4 hicl axicn polid2.3 hvmulcl axicn polid2.2 hvmulcl hicl addcl polid2.1 axicn polid2.2 hvmulcl hicl axicn polid2.3 hvmulcl polid2.4 hicl addcl polid2.1 axicn polid2.2 hvmulcl hicl axicn polid2.3 hvmulcl polid2.4 hicl addcl pnncan polid2.1 axicn polid2.2 hvmulcl hicl axicn polid2.3 hvmulcl polid2.4 hicl addcl 2times axicn polid2.1 polid2.2 (i) A B his5t mp3an cji (x.) (opr A (.i) B) opreq1i eqtr axicn polid2.3 polid2.4 (i) C D ax-his3 mp3an (+) opreq12i (2) (x.) opreq2i eqtr3 3eqtr (i) (x.) opreq2i 2cn axicn axicn negcl polid2.1 polid2.2 hicl mulcl axicn polid2.3 polid2.4 hicl mulcl addcl mul12 axicn axicn negcl polid2.1 polid2.2 hicl mulcl axicn polid2.3 polid2.4 hicl mulcl adddi axicn axicn mulneg2 itimesi negeqi 1cn negneg 3eqtr (x.) (opr A (.i) B) opreq1i axicn axicn negcl polid2.1 polid2.2 hicl mulass polid2.1 polid2.2 hicl mulid2 3eqtr3 itimesi (x.) (opr C (.i) D) opreq1i axicn axicn polid2.3 polid2.4 hicl mulass polid2.3 polid2.4 hicl mulm1 3eqtr3 (+) opreq12i polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl negsub 3eqtr (2) (x.) opreq2i 3eqtr2 (+) opreq12i 3eqtr4 (/) (4) opreq1i eqtr3)) thm (polid () ((polid.1 (e. A (H~))) (polid.2 (e. B (H~)))) (= (opr A (.i) B) (opr (opr (opr (opr (` (norm) (opr A (+v) B)) (^) (2)) (-) (opr (` (norm) (opr A (-v) B)) (^) (2))) (+) (opr (i) (x.) (opr (opr (` (norm) (opr A (+v) (opr (i) (.s) B))) (^) (2)) (-) (opr (` (norm) (opr A (-v) (opr (i) (.s) B))) (^) (2))))) (/) (4))) (polid.1 polid.2 polid.2 polid.1 polid2 polid.1 polid.2 hvaddcl normsq polid.1 polid.2 hvsubcl normsq (-) opreq12i polid.1 axicn polid.2 hvmulcl hvaddcl normsq polid.1 axicn polid.2 hvmulcl hvsubcl normsq (-) opreq12i (i) (x.) opreq2i (+) opreq12i (/) (4) opreq1i eqtr4)) thm (bcs () ((bcs.1 (e. A (H~))) (bcs.2 (e. B (H~)))) (br (` (abs) (opr A (.i) B)) (<_) (opr (` (norm) A) (x.) (` (norm) B))) ((opr A (.i) B) (0) (abs) fveq2 abs0 bcs.1 A normge0t ax-mp bcs.2 B normge0t ax-mp bcs.1 normcl bcs.2 normcl mulge0 mp2an eqbrtr syl6eqbr (opr A (.i) B) (0) df-ne bcs.1 bcs.2 hicl abslem2 bcs.2 bcs.1 his1 (opr (opr A (.i) B) (/) (` (abs) (opr A (.i) B))) (x.) opreq2i (opr (` (*) (opr (opr A (.i) B) (/) (` (abs) (opr A (.i) B)))) (x.) (opr A (.i) B)) (+) opreq2i syl5req bcs.1 bcs.2 hicl abs00 eqneqi bcs.1 bcs.2 hicl bcs.1 bcs.2 hicl abscl recn divclz bcs.1 bcs.2 hicl bcs.1 bcs.2 hicl abscl recn divrecz (abs) fveq2d bcs.1 bcs.2 hicl abscl recn recclz bcs.1 bcs.2 hicl jctil (opr A (.i) B) (opr (1) (/) (` (abs) (opr A (.i) B))) absmult syl (opr (1) (/) (` (abs) (opr A (.i) B))) absidt bcs.1 bcs.2 hicl abscl rerecclz (0) (opr (1) (/) (` (abs) (opr A (.i) B))) ltlet bcs.1 bcs.2 hicl abscl rerecclz 0re jctil bcs.1 bcs.2 hicl abs00 eqneqi bcs.1 bcs.2 hicl absgt0 bitr bcs.1 bcs.2 hicl abscl recgt0 sylbi sylc sylanc (` (abs) (opr A (.i) B)) (x.) opreq2d eqtrd bcs.1 bcs.2 hicl abscl recn recidz 3eqtrd jca sylbir bcs.1 bcs.2 (opr (opr A (.i) B) (/) (` (abs) (opr A (.i) B))) normlem7t syl eqbrtrd sylbir bcs.1 normcl recn bcs.2 normcl recn mulcom bcs.2 B normvalt ax-mp bcs.1 A normvalt ax-mp (x.) opreq12i eqtr (` (abs) (opr A (.i) B)) (<_) breq2i 2pos bcs.1 bcs.2 hicl abscl bcs.2 B hiidge0t ax-mp bcs.2 B hiidrclt ax-mp sqrcl ax-mp bcs.1 A hiidge0t ax-mp bcs.1 A hiidrclt ax-mp sqrcl ax-mp remulcl 2re lemul2 ax-mp bitr sylibr pm2.61i)) thm (bcst () () (-> (/\ (e. A (H~)) (e. B (H~))) (br (` (abs) (opr A (.i) B)) (<_) (opr (` (norm) A) (x.) (` (norm) B)))) (A (if (e. A (H~)) A (0v)) (.i) B opreq1 (abs) fveq2d A (if (e. A (H~)) A (0v)) (norm) fveq2 (x.) (` (norm) B) opreq1d (<_) breq12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (.i) opreq2 (abs) fveq2d B (if (e. B (H~)) B (0v)) (norm) fveq2 (` (norm) (if (e. A (H~)) A (0v))) (x.) opreq2d (<_) breq12d ax-hv0cl A elimel ax-hv0cl B elimel bcs dedth2h)) thm (bcs2t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (br (` (norm) A) (<_) (1))) (br (` (abs) (opr A (.i) B)) (<_) (` (norm) B))) (A B ax-hicl (opr A (.i) B) absclt syl (br (` (norm) A) (<_) (1)) 3adant3 (` (norm) A) (` (norm) B) axmulrcl A normclt B normclt syl2an (br (` (norm) A) (<_) (1)) 3adant3 B normclt (e. A (H~)) (br (` (norm) A) (<_) (1)) 3ad2ant2 A B bcst (br (` (norm) A) (<_) (1)) 3adant3 ax1re (` (norm) A) (1) (` (norm) B) lemul1it mp3anl2 A normclt (e. B (H~)) (br (` (norm) A) (<_) (1)) 3ad2ant1 B normclt (e. A (H~)) (br (` (norm) A) (<_) (1)) 3ad2ant2 jca B normge0t (e. A (H~)) (br (` (norm) A) (<_) (1)) 3ad2ant2 (e. A (H~)) (e. B (H~)) (br (` (norm) A) (<_) (1)) 3simp3 jca sylanc B normclt recnd (` (norm) B) mulid2t syl (e. A (H~)) (br (` (norm) A) (<_) (1)) 3ad2ant2 breqtrd letrd)) thm (bcs3t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (br (` (norm) B) (<_) (1))) (br (` (abs) (opr A (.i) B)) (<_) (` (norm) A))) (A B abshicomt (br (` (norm) B) (<_) (1)) 3adant3 B A bcs2t 3com12 eqbrtrd)) thm (hcau ((x y) (x z) (w x) (f x) (F x) (y z) (w y) (f y) (F y) (w z) (f z) (F z) (f w) (F w) (F f)) () (<-> (e. F (Cauchy)) (/\ (:--> F (NN) (H~)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (A.e. w (NN) (-> (/\ (br (cv y) (<_) (cv z)) (br (cv y) (<_) (cv w))) (br (` (norm) (opr (` F (cv z)) (-v) (` F (cv w)))) (<) (cv x)))))))))) (F (Cauchy) elisset nnex F (NN) (H~) (V) fex mpan2 (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (A.e. w (NN) (-> (/\ (br (cv y) (<_) (cv z)) (br (cv y) (<_) (cv w))) (br (` (norm) (opr (` F (cv z)) (-v) (` F (cv w)))) (<) (cv x)))))))) adantr (cv f) F (NN) (H~) feq1 (cv f) F (cv z) fveq1 (cv f) F (cv w) fveq1 (-v) opreq12d (norm) fveq2d (<) (cv x) breq1d (/\ (br (cv y) (<_) (cv z)) (br (cv y) (<_) (cv w))) imbi2d z (NN) w (NN) 2ralbidv y (NN) rexbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv anbi12d f x y z w df-hcau (V) elab2g pm5.21nii)) thm (hcauseq ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w)) () (-> (e. F (Cauchy)) (:--> F (NN) (H~))) (F x y z w hcau pm3.26bd)) thm (hcaucvg ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w)) () (-> (e. F (Cauchy)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (A.e. w (NN) (-> (/\ (br (cv y) (<_) (cv z)) (br (cv y) (<_) (cv w))) (br (` (norm) (opr (` F (cv z)) (-v) (` F (cv w)))) (<) (cv x))))))))) (F x y z w hcau pm3.27bd)) thm (seq1hcau ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w)) () (-> (:--> F (NN) (H~)) (<-> (e. F (Cauchy)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (A.e. w (NN) (-> (/\ (br (cv y) (<_) (cv z)) (br (cv y) (<_) (cv w))) (br (` (norm) (opr (` F (cv z)) (-v) (` F (cv w)))) (<) (cv x)))))))))) (F x y z w hcau baib)) thm (hlim ((x y) (x z) (w x) (f x) (F x) (y z) (w y) (f y) (F y) (w z) (f z) (F z) (f w) (F w) (F f) (A x) (A y) (A z) (A w) (A f)) ((hlim.1 (e. F (V))) (hlim.2 (e. A (V)))) (<-> (br F (~~>v) A) (/\ (/\ (:--> F (NN) (H~)) (e. A (H~))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (cv x))))))))) (hlim.1 hlim.2 (cv f) F (NN) (H~) feq1 (e. (cv w) (H~)) anbi1d (cv f) F (cv z) fveq1 (-v) (cv w) opreq1d (norm) fveq2d (<) (cv x) breq1d (br (cv y) (<_) (cv z)) imbi2d y (NN) z (NN) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv anbi12d (cv w) A (H~) eleq1 (:--> F (NN) (H~)) anbi2d (cv w) A (` F (cv z)) (-v) opreq2 (norm) fveq2d (<) (cv x) breq1d (br (cv y) (<_) (cv z)) imbi2d y (NN) z (NN) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv anbi12d f w x y z df-hlim brab)) thm (hlimseq ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((hlim.1 (e. F (V))) (hlim.2 (e. A (V)))) (-> (br F (~~>v) A) (:--> F (NN) (H~))) (hlim.1 hlim.2 x y z hlim pm3.26bd pm3.26d)) thm (hlimvec ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((hlim.1 (e. F (V))) (hlim.2 (e. A (V)))) (-> (br F (~~>v) A) (e. A (H~))) (hlim.1 hlim.2 x y z hlim pm3.26bd pm3.27d)) thm (hlimconv ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((hlim.1 (e. F (V))) (hlim.2 (e. A (V)))) (-> (br F (~~>v) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (cv x)))))))) (hlim.1 hlim.2 x y z hlim pm3.27bd)) thm (hlim2 ((x y) (x z) (w x) (f x) (F x) (y z) (w y) (f y) (F y) (w z) (f z) (F z) (f w) (F w) (F f) (A x) (A y) (A z) (A w) (A f)) () (-> (/\ (:--> F (NN) (H~)) (e. A (H~))) (<-> (br F (~~>v) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (cv x))))))))) ((cv f) F (NN) (H~) feq1 (e. (cv w) (H~)) anbi1d (cv f) F (cv z) fveq1 (-v) (cv w) opreq1d (norm) fveq2d (<) (cv x) breq1d (br (cv y) (<_) (cv z)) imbi2d y (NN) z (NN) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv anbi12d (cv w) A (H~) eleq1 (:--> F (NN) (H~)) anbi2d (cv w) A (` F (cv z)) (-v) opreq2 (norm) fveq2d (<) (cv x) breq1d (br (cv y) (<_) (cv z)) imbi2d y (NN) z (NN) rexralbidv (br (0) (<) (cv x)) imbi2d x (RR) ralbidv anbi12d f w x y z df-hlim (V) (H~) brabg nnex F (NN) (H~) (V) fex mpan2 sylan (/\ (:--> F (NN) (H~)) (e. A (H~))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (cv x))))))) ibar bitr4d)) thm (shex ((x y) (h x) (h y)) () (e. (SH) (V)) (h x y df-sh (H~) h df-pw ax-hilex pwex eqeltrr (C_ (cv h) (H~)) (e. (0v) (cv h)) pm3.26 (/\ (A.e. x (cv h) (A.e. y (cv h) (e. (opr (cv x) (+v) (cv y)) (cv h)))) (A.e. x (CC) (A.e. y (cv h) (e. (opr (cv x) (.s) (cv y)) (cv h))))) adantr h ss2abi ssexi eqeltr)) thm (sh ((x y) (h x) (H x) (h y) (H y) (H h)) () (<-> (e. H (SH)) (/\ (/\ (C_ H (H~)) (e. (0v) H)) (/\ (A.e. x H (A.e. y H (e. (opr (cv x) (+v) (cv y)) H))) (A.e. x (CC) (A.e. y H (e. (opr (cv x) (.s) (cv y)) H)))))) (H (SH) elisset ax-hilex H ssex (e. (0v) H) (/\ (A.e. x H (A.e. y H (e. (opr (cv x) (+v) (cv y)) H))) (A.e. x (CC) (A.e. y H (e. (opr (cv x) (.s) (cv y)) H)))) ad2antrr (cv h) H (H~) sseq1 (cv h) H (0v) eleq2 anbi12d (cv h) H (opr (cv x) (+v) (cv y)) eleq2 y raleqd x raleqd (cv h) H (opr (cv x) (.s) (cv y)) eleq2 y raleqd x (CC) ralbidv anbi12d anbi12d h x y df-sh (V) elab2g pm5.21nii)) thm (shss ((x y) (H x) (H y)) () (-> (e. H (SH)) (C_ H (H~))) (H x y sh pm3.26bd pm3.26d)) thm (shelt () () (-> (/\ (e. H (SH)) (e. A H)) (e. A (H~))) (H shss A sseld imp)) thm (shssi () ((shssi.1 (e. H (SH)))) (C_ H (H~)) (shssi.1 H shss ax-mp)) thm (shel () ((shssi.1 (e. H (SH)))) (-> (e. A H) (e. A (H~))) (shssi.1 shssi A sseli)) thm (sheli () ((shssi.1 (e. H (SH))) (sheli.1 (e. A H))) (e. A (H~)) (shssi.1 shssi sheli.1 sselii)) thm (sh0 ((x y) (H x) (H y)) () (-> (e. H (SH)) (e. (0v) H)) (H x y sh pm3.26bd pm3.27d)) thm (shaddclt ((x y) (H x) (H y) (A x) (A y) (B x) (B y)) () (-> (e. H (SH)) (-> (/\ (e. A H) (e. B H)) (e. (opr A (+v) B) H))) ((cv x) A (+v) (cv y) opreq1 H eleq1d (cv y) B A (+v) opreq2 H eleq1d H H rcla42v H x y sh pm3.27bd pm3.26d syl5com)) thm (shmulclt ((x y) (H x) (H y) (A x) (A y) (B x) (B y)) () (-> (e. H (SH)) (-> (/\ (e. A (CC)) (e. B H)) (e. (opr A (.s) B) H))) ((cv x) A (.s) (cv y) opreq1 H eleq1d (cv y) B A (.s) opreq2 H eleq1d (CC) H rcla42v H x y sh pm3.27bd pm3.27d syl5com)) thm (shsubclt () () (-> (e. H (SH)) (-> (/\ (e. A H) (e. B H)) (e. (opr A (-v) B) H))) (H shss A sseld H shss B sseld anim12d imp A B hvsubvalt syl H A (opr (-u (1)) (.s) B) shaddclt 1cn negcl H (-u (1)) B shmulclt mpani sylan2d imp eqeltrd ex)) thm (sh2 ((x y) (H x) (H y)) () (-> (C_ H (H~)) (<-> (e. H (SH)) (/\ (e. (0v) H) (/\ (A.e. x H (A.e. y H (e. (opr (cv x) (+v) (cv y)) H))) (A.e. x (CC) (A.e. y H (e. (opr (cv x) (.s) (cv y)) H))))))) ((C_ H (H~)) (e. (0v) H) (/\ (A.e. x H (A.e. y H (e. (opr (cv x) (+v) (cv y)) H))) (A.e. x (CC) (A.e. y H (e. (opr (cv x) (.s) (cv y)) H)))) anass baib H x y sh syl5bb)) thm (closedsub ((f x) (h x) (H x) (f h) (H f) (H h)) () (<-> (e. H (CH)) (/\ (e. H (SH)) (A. f (A. x (-> (/\ (:--> (cv f) (NN) H) (br (cv f) (~~>v) (cv x))) (e. (cv x) H)))))) (H (CH) elisset H (SH) elisset (A. f (A. x (-> (/\ (:--> (cv f) (NN) H) (br (cv f) (~~>v) (cv x))) (e. (cv x) H)))) adantr (cv h) H (SH) eleq1 (cv h) H (cv f) (NN) feq3 (br (cv f) (~~>v) (cv x)) anbi1d (cv h) H (cv x) eleq2 imbi12d f x 2albidv anbi12d h f x df-ch (V) elab2g pm5.21nii)) thm (chsssh ((f x) (h x) (f h)) () (C_ (CH) (SH)) (h f x df-ch h (SH) (A. f (A. x (-> (/\ (:--> (cv f) (NN) (cv h)) (br (cv f) (~~>v) (cv x))) (e. (cv x) (cv h))))) ssab2 eqsstr)) thm (chex () () (e. (CH) (V)) (shex chsssh ssexi)) thm (chsh () () (-> (e. H (CH)) (e. H (SH))) (chsssh H sseli)) thm (chshi () ((chshi.1 (e. H (CH)))) (e. H (SH)) (chshi.1 H chsh ax-mp)) thm (ch0 () () (-> (e. H (CH)) (e. (0v) H)) (H chsh H sh0 syl)) thm (chss () () (-> (e. H (CH)) (C_ H (H~))) (H chsh H shss syl)) thm (chelt () () (-> (/\ (e. H (CH)) (e. A H)) (e. A (H~))) (H chss A sseld imp)) thm (chssi () ((chssi.1 (e. H (CH)))) (C_ H (H~)) (chssi.1 chshi shssi)) thm (chel () ((chssi.1 (e. H (CH)))) (-> (e. A H) (e. A (H~))) (chssi.1 chssi A sseli)) thm (cheli () ((chssi.1 (e. H (CH))) (cheli.1 (e. A H))) (e. A (H~)) (chssi.1 chssi cheli.1 sselii)) thm (chlim ((f x) (F x) (F f) (H x) (H f) (A x)) ((chlim.1 (e. A (V)))) (-> (e. H (CH)) (-> (/\ (:--> F (NN) H) (br F (~~>v) A)) (e. A H))) (H f x closedsub pm3.27bd nnex F (NN) H (V) fex mpan2 (br F (~~>v) A) adantr (cv f) F (NN) H feq1 (cv f) F (~~>v) (cv x) breq1 anbi12d (e. (cv x) H) imbi1d x albidv (V) cla4gv chlim.1 (cv x) A F (~~>v) breq2 (:--> F (NN) H) anbi2d (cv x) A H eleq1 imbi12d cla4v syl6 syl pm2.43b syl)) thm (hlim0 ((x y) (x z) (y z)) () (br (X. (NN) ({} (0v))) (~~>v) (0v)) (nnex (0v) snex xpex ax-hv0cl elisseti x y z hlim ax-hv0cl elisseti (NN) fconst ax-hv0cl (0v) (H~) snssi ax-mp (X. (NN) ({} (0v))) (NN) ({} (0v)) (H~) fss mp2an ax-hv0cl pm3.2i ax-hv0cl elisseti (cv z) (NN) fvconst2 (-v) (0v) opreq1d ax-hv0cl (0v) hvsubidt ax-mp syl6eq (norm) fveq2d norm0 syl6eq (<) (cv x) breq1d biimprd (br (1) (<_) (cv z)) a1d com3r r19.21aiv 1nn jctil (cv y) (1) (<_) (cv z) breq1 (br (` (norm) (opr (` (X. (NN) ({} (0v))) (cv z)) (-v) (0v))) (<) (cv x)) imbi1d z (NN) ralbidv (NN) rcla4ev syl (e. (cv x) (RR)) a1i rgen mpbir2an)) thm (hlimcaui ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F x) (F y) (F z) (F w) (F v)) ((hlimcau.1 (e. A (V))) (hlimcau.2 (e. F (V))) (hlimcaui.4 (br F (~~>v) A))) (e. F (Cauchy)) (F x y z w hcau hlimcaui.4 hlimcau.2 hlimcau.1 hlimseq ax-mp (cv v) (opr (cv x) (/) (2)) (0) (<) breq2 (cv v) (opr (cv x) (/) (2)) (` (norm) (opr (` F (cv z)) (-v) A)) (<) breq2 (br (cv y) (<_) (cv z)) imbi2d y (NN) z (NN) rexralbidv imbi12d hlimcaui.4 hlimcau.2 hlimcau.1 hlimseq ax-mp hlimcaui.4 hlimcau.2 hlimcau.1 hlimvec ax-mp hlimcaui.4 F A v y z hlim2 mpbii mp2an vtoclri (cv x) rehalfclt (br (0) (<) (cv x)) adantr (cv x) (if (e. (cv x) (RR)) (cv x) (0)) (0) (<) breq2 (cv x) (if (e. (cv x) (RR)) (cv x) (0)) (/) (2) opreq1 (0) (<) breq2d imbi12d 0re (cv x) elimel 2re 2pos divgt0i2 dedth imp sylc (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (opr (cv x) (/) (2))) (br (cv y) (<_) (cv w)) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr (cv x) (/) (2))) prth A (` F (cv w)) normsubt (<) (opr (cv x) (/) (2)) breq1d (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (opr (cv x) (/) (2))) anbi2d (/\ (e. (cv x) (RR)) (e. (cv z) (NN))) adantl hlimcaui.4 hlimcau.2 hlimcau.1 hlimseq ax-mp (cv z) ffvrni (e. (` F (cv w)) (H~)) anim1i ancoms (/\ (e. A (H~)) (e. (cv x) (RR))) anim1i ancoms an4s ancoms (` F (cv z)) (` F (cv w)) A (cv x) norm3lemt syl sylbird hlimcaui.4 hlimcau.2 hlimcau.1 hlimseq ax-mp (cv w) ffvrni hlimcaui.4 hlimcau.2 hlimcau.1 hlimvec ax-mp jctil sylan2 syl9r r19.20dva r19.20dva z (NN) w (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (opr (cv x) (/) (2)))) (-> (br (cv y) (<_) (cv w)) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr (cv x) (/) (2)))) raaan (cv w) (cv z) (cv y) (<_) breq2 (cv w) (cv z) F fveq2 (-v) A opreq1d (norm) fveq2d (<) (opr (cv x) (/) (2)) breq1d imbi12d (NN) cbvralv (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (opr (cv x) (/) (2))))) anbi2i (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (opr (cv x) (/) (2))))) anidm 3bitr syl5ibr y (NN) r19.22sdv (br (0) (<) (cv x)) adantr mpd ex rgen mpbir2an)) thm (hlimcau () ((hlimcau.1 (e. A (V))) (hlimcau.2 (e. F (V)))) (-> (br F (~~>v) A) (e. F (Cauchy))) (F (if (br F (~~>v) A) F (X. (NN) ({} (0v)))) (Cauchy) eleq1 hlimcau.1 ax-hv0cl elisseti (br F (~~>v) A) ifex hlimcau.2 nnex (0v) snex xpex (br F (~~>v) A) ifex F (if (br F (~~>v) A) F (X. (NN) ({} (0v)))) (~~>v) A breq1 A (if (br F (~~>v) A) A (0v)) (if (br F (~~>v) A) F (X. (NN) ({} (0v)))) (~~>v) breq2 (X. (NN) ({} (0v))) (if (br F (~~>v) A) F (X. (NN) ({} (0v)))) (~~>v) (0v) breq1 (0v) (if (br F (~~>v) A) A (0v)) (if (br F (~~>v) A) F (X. (NN) ({} (0v)))) (~~>v) breq2 hlim0 elimhyp2v hlimcaui dedth)) thm (hlimunii ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w)) ((hlimuni.1 (e. A (V))) (hlimuni.2 (e. B (V))) (hlimuni.3 (e. F (V))) (hlimunii.3 (/\ (br F (~~>v) A) (br F (~~>v) B)))) (= A B) ((cv y) (cv w) z nn2get rgen2 hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl normcl 2re 2pos divgt0i2 hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl normcl 2re 2re 2pos gt0ne0i redivcl hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 x y z hlimconv ax-mp (cv x) (opr (` (norm) (opr A (-v) B)) (/) (2)) (0) (<) breq2 (cv x) (opr (` (norm) (opr A (-v) B)) (/) (2)) (` (norm) (opr (` F (cv z)) (-v) A)) (<) breq2 (br (cv y) (<_) (cv z)) imbi2d y (NN) z (NN) rexralbidv imbi12d (RR) rcla4v mp2 hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl normcl 2re 2re 2pos gt0ne0i redivcl hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 x w z hlimconv ax-mp (cv x) (opr (` (norm) (opr A (-v) B)) (/) (2)) (0) (<) breq2 (cv x) (opr (` (norm) (opr A (-v) B)) (/) (2)) (` (norm) (opr (` F (cv z)) (-v) B)) (<) breq2 (br (cv w) (<_) (cv z)) imbi2d w (NN) z (NN) rexralbidv imbi12d (RR) rcla4v mp2 jca z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (opr (` (norm) (opr A (-v) B)) (/) (2)))) (-> (br (cv w) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) B)) (<) (opr (` (norm) (opr A (-v) B)) (/) (2)))) r19.26 (/\ (br (cv y) (<_) (cv z)) (br (cv w) (<_) (cv z))) (/\ (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (opr (` (norm) (opr A (-v) B)) (/) (2))) (br (` (norm) (opr (` F (cv z)) (-v) B)) (<) (opr (` (norm) (opr A (-v) B)) (/) (2)))) con3 hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl normcl ltnr hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimseq ax-mp (cv z) ffvrni hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp (` F (cv z)) A normsubt mpan2 syl (<) (opr (` (norm) (opr A (-v) B)) (/) (2)) breq1d (br (` (norm) (opr (` F (cv z)) (-v) B)) (<) (opr (` (norm) (opr A (-v) B)) (/) (2))) anbi1d hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimseq ax-mp (cv z) ffvrni hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl normcl hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp A B (` F (cv z)) (` (norm) (opr A (-v) B)) norm3lemt mpanl12 mpan2 syl sylbid mtoi syl5com (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (opr (` (norm) (opr A (-v) B)) (/) (2))) (br (cv w) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) B)) (<) (opr (` (norm) (opr A (-v) B)) (/) (2))) prth syl5 r19.20i sylbir w (NN) r19.22si y (NN) r19.22si y (NN) w (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (opr (` (norm) (opr A (-v) B)) (/) (2))))) (A.e. z (NN) (-> (br (cv w) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) B)) (<) (opr (` (norm) (opr A (-v) B)) (/) (2))))) reeanv z (NN) (/\ (br (cv y) (<_) (cv z)) (br (cv w) (<_) (cv z))) ralnex w (NN) rexbii w (NN) (E.e. z (NN) (/\ (br (cv y) (<_) (cv z)) (br (cv w) (<_) (cv z)))) rexnal bitr y (NN) rexbii y (NN) (A.e. w (NN) (E.e. z (NN) (/\ (br (cv y) (<_) (cv z)) (br (cv w) (<_) (cv z))))) rexnal bitr 3imtr3 3syl mt2 hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl (opr A (-v) B) normgt0tOLD ax-mp con1bii mpbi hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubeq0 mpbi)) thm (hlimuni () ((hlimuni.1 (e. A (V))) (hlimuni.2 (e. B (V))) (hlimuni.3 (e. F (V)))) (-> (/\ (br F (~~>v) A) (br F (~~>v) B)) (= A B)) (A (if (/\ (br F (~~>v) A) (br F (~~>v) B)) A (0v)) B eqeq1 B (if (/\ (br F (~~>v) A) (br F (~~>v) B)) B (0v)) (if (/\ (br F (~~>v) A) (br F (~~>v) B)) A (0v)) eqeq2 hlimuni.1 ax-hv0cl elisseti (/\ (br F (~~>v) A) (br F (~~>v) B)) ifex hlimuni.2 ax-hv0cl elisseti (/\ (br F (~~>v) A) (br F (~~>v) B)) ifex hlimuni.3 nnex (0v) snex xpex (/\ (br F (~~>v) A) (br F (~~>v) B)) ifex A (if (/\ (br F (~~>v) A) (br F (~~>v) B)) A (0v)) F (~~>v) breq2 (br F (~~>v) B) anbi1d B (if (/\ (br F (~~>v) A) (br F (~~>v) B)) B (0v)) F (~~>v) breq2 (br F (~~>v) (if (/\ (br F (~~>v) A) (br F (~~>v) B)) A (0v))) anbi2d F (if (/\ (br F (~~>v) A) (br F (~~>v) B)) F (X. (NN) ({} (0v)))) (~~>v) (if (/\ (br F (~~>v) A) (br F (~~>v) B)) A (0v)) breq1 F (if (/\ (br F (~~>v) A) (br F (~~>v) B)) F (X. (NN) ({} (0v)))) (~~>v) (if (/\ (br F (~~>v) A) (br F (~~>v) B)) B (0v)) breq1 anbi12d (0v) (if (/\ (br F (~~>v) A) (br F (~~>v) B)) A (0v)) (X. (NN) ({} (0v))) (~~>v) breq2 (br (X. (NN) ({} (0v))) (~~>v) (0v)) anbi1d (0v) (if (/\ (br F (~~>v) A) (br F (~~>v) B)) B (0v)) (X. (NN) ({} (0v))) (~~>v) breq2 (br (X. (NN) ({} (0v))) (~~>v) (if (/\ (br F (~~>v) A) (br F (~~>v) B)) A (0v))) anbi2d (X. (NN) ({} (0v))) (if (/\ (br F (~~>v) A) (br F (~~>v) B)) F (X. (NN) ({} (0v)))) (~~>v) (if (/\ (br F (~~>v) A) (br F (~~>v) B)) A (0v)) breq1 (X. (NN) ({} (0v))) (if (/\ (br F (~~>v) A) (br F (~~>v) B)) F (X. (NN) ({} (0v)))) (~~>v) (if (/\ (br F (~~>v) A) (br F (~~>v) B)) B (0v)) breq1 anbi12d hlim0 hlim0 pm3.2i elimhyp3v hlimunii dedth2v)) thm (hlimreu ((x y) (F x) (F y) (H x) (H y)) ((hlimreu.1 (e. F (V)))) (<-> (E.e. x H (br F (~~>v) (cv x))) (E!e. x H (br F (~~>v) (cv x)))) (x visset y visset hlimreu.1 hlimuni (/\ (e. (cv x) H) (e. (cv y) H)) a1i rgen2 (E.e. x H (br F (~~>v) (cv x))) jctr (cv x) (cv y) F (~~>v) breq2 H reu4 sylibr x H (br F (~~>v) (cv x)) reurex impbi)) thm (hlimeu ((x y) (F x) (F y)) ((hlimeu.1 (e. F (V)))) (<-> (E. x (br F (~~>v) (cv x))) (E! x (br F (~~>v) (cv x)))) (x visset y visset hlimeu.1 hlimuni x y gen2 (E. x (br F (~~>v) (cv x))) jctr (cv x) (cv y) F (~~>v) breq2 eu4 sylibr x (br F (~~>v) (cv x)) euex impbi)) thm (chsscm ((f x) (h x) (f h) (C h)) ((cmh.1 (= C ({|} h (/\ (e. (cv h) (SH)) (A.e. f (Cauchy) (-> (:--> (cv f) (NN) (cv h)) (E.e. x (cv h) (br (cv f) (~~>v) (cv x)))))))))) (C_ (CH) C) ((:--> (cv f) (NN) (cv h)) (br (cv f) (~~>v) (cv x)) (e. (cv x) (cv h)) impexp (br (cv f) (~~>v) (cv x)) (e. (cv x) (cv h)) ancr (e. (cv x) (H~)) adantld (:--> (cv f) (NN) (cv h)) imim2i sylbi com12 x 19.20dv impcom x (/\ (e. (cv x) (H~)) (br (cv f) (~~>v) (cv x))) (/\ (e. (cv x) (cv h)) (br (cv f) (~~>v) (cv x))) 19.22 syl x (H~) (br (cv f) (~~>v) (cv x)) df-rex x (cv h) (br (cv f) (~~>v) (cv x)) df-rex 3imtr4g (cv f) x ax-hcompl syl5 ex com23 f 19.20i f (Cauchy) (-> (:--> (cv f) (NN) (cv h)) (E.e. x (cv h) (br (cv f) (~~>v) (cv x)))) df-ral sylibr (e. (cv h) (SH)) anim2i (cv h) f x closedsub cmh.1 abeq2i 3imtr4 ssriv)) thm (chcmh ((f x) (h x) (f h) (C h)) ((cmh.1 (= C ({|} h (/\ (e. (cv h) (SH)) (A.e. f (Cauchy) (-> (:--> (cv f) (NN) (cv h)) (E.e. x (cv h) (br (cv f) (~~>v) (cv x)))))))))) (= (CH) C) (cmh.1 chsscm f (Cauchy) (-> (:--> (cv f) (NN) (cv h)) (E.e. x (cv h) (br (cv f) (~~>v) (cv x)))) df-ral (e. (cv f) (Cauchy)) x ax-17 (:--> (cv f) (NN) (cv h)) x ax-17 x (cv h) (br (cv f) (~~>v) (cv x)) hbre1 hbim hbim x visset f visset hlimcau (E.e. x (cv h) (br (cv f) (~~>v) (cv x))) imim1i (br (cv f) (~~>v) (cv x)) x 19.8a f visset x hlimeu sylib x (br (cv f) (~~>v) (cv x)) (e. (cv x) (cv h)) eupick ex x (cv h) (br (cv f) (~~>v) (cv x)) df-rex x (e. (cv x) (cv h)) (br (cv f) (~~>v) (cv x)) exancom bitr syl5ib syl pm2.43a sylcom (:--> (cv f) (NN) (cv h)) imim2i (e. (cv f) (Cauchy)) (:--> (cv f) (NN) (cv h)) (E.e. x (cv h) (br (cv f) (~~>v) (cv x))) bi2.04 (:--> (cv f) (NN) (cv h)) (br (cv f) (~~>v) (cv x)) (e. (cv x) (cv h)) impexp 3imtr4 19.21ai f 19.20i sylbi (e. (cv h) (SH)) anim2i cmh.1 abeq2i (cv h) f x closedsub 3imtr4 ssriv eqssi)) thm (ch2 ((f x) (h x) (H x) (f h) (H f) (H h) (g x) (f g) (g h)) () (<-> (e. H (CH)) (/\ (e. H (SH)) (A.e. f (Cauchy) (-> (:--> (cv f) (NN) H) (E.e. x H (br (cv f) (~~>v) (cv x))))))) (H (CH) elisset H (SH) elisset (A.e. f (Cauchy) (-> (:--> (cv f) (NN) H) (E.e. x H (br (cv f) (~~>v) (cv x))))) adantr (cv h) H (SH) eleq1 (cv h) H (cv f) (NN) feq3 (cv h) H x (br (cv f) (~~>v) (cv x)) rexeq1 imbi12d f (Cauchy) ralbidv anbi12d (cv h) (cv g) (SH) eleq1 (cv h) (cv g) (cv f) (NN) feq3 (cv h) (cv g) x (br (cv f) (~~>v) (cv x)) rexeq1 imbi12d f (Cauchy) ralbidv anbi12d cbvabv chcmh (V) elab2g pm5.21nii)) thm (chcompl ((f x) (H x) (H f) (F x) (F f)) () (-> (e. H (CH)) (-> (/\ (e. F (Cauchy)) (:--> F (NN) H)) (E.e. x H (br F (~~>v) (cv x))))) (H f x ch2 pm3.27bd (cv f) F (NN) H feq1 (cv f) F (~~>v) (cv x) breq1 x H rexbidv imbi12d (Cauchy) rcla4cv imp3a syl)) thm (helch ((x y) (f x) (f y)) () (e. (H~) (CH)) ((H~) f x closedsub (H~) x y sh (H~) ssid ax-hv0cl pm3.2i (cv x) (cv y) ax-hvaddcl rgen2 (cv x) (cv y) ax-hvmulcl rgen2a pm3.2i mpbir2an f visset x visset hlimvec (:--> (cv f) (NN) (H~)) adantl f x gen2 mpbir2an)) thm (helsh () () (e. (H~) (SH)) (helch chshi)) thm (shsspwh () () (C_ (SH) (P~ (H~))) ((SH) pwuni helsh (cv x) shss rgen (H~) (SH) x ssunieq mp2an (H~) (U. (SH)) pweq ax-mp sseqtr4)) thm (chsspwh () () (C_ (CH) (P~ (H~))) (chsssh shsspwh sstri)) thm (hsn0elch ((x y) (f x) (f y)) () (e. ({} (0v)) (CH)) (({} (0v)) f x closedsub ({} (0v)) x y sh ax-hv0cl (0v) (H~) snssi ax-mp ax-hv0cl elisseti snid pm3.2i (cv x) (0v) (cv y) (0v) (+v) opreq12 ax-hv0cl hvaddid2 syl6eq (cv x) (+v) (cv y) oprex (0v) elsnc sylibr x (0v) elsn y (0v) elsn syl2anb rgen2 (cv y) (0v) (cv x) (.s) opreq2 (cv x) hvmul0t sylan9eqr (cv x) (.s) (cv y) oprex (0v) elsnc sylibr y (0v) elsn sylan2b rgen2a pm3.2i mpbir2an ax-hv0cl elisseti snid ax-hv0cl elisseti x visset f visset hlimuni ({} (0v)) eleq1d ax-hv0cl elisseti (cv f) (NN) fconst2 hlim0 (cv f) (X. (NN) ({} (0v))) (~~>v) (0v) breq1 mpbiri sylbi sylan mpbii f x gen2 mpbir2an)) thm (norm1t () () (-> (/\ (e. A (H~)) (=/= A (0v))) (= (` (norm) (opr (opr (1) (/) (` (norm) A)) (.s) A)) (1))) ((opr (1) (/) (` (norm) A)) A norm-iiit (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc recnd (e. A (H~)) (=/= A (0v)) pm3.26 sylanc (opr (1) (/) (` (norm) A)) absidt (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc (1) (` (norm) A) divge0t A normclt (=/= A (0v)) adantr ax1re jctil A normgt0t biimpa 0re ax1re lt01 ltlei jctil sylanc sylanc (x.) (` (norm) A) opreq1d (opr (1) (/) (` (norm) A)) (` (norm) A) axmulcom (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc recnd A normclt recnd (=/= A (0v)) adantr sylanc (` (norm) A) recidt A normclt recnd (=/= A (0v)) adantr A normne0t biimpar sylanc eqtrd 3eqtrd)) thm (norm1ex ((x z) (H x) (H z) (y z) (H y)) ((norm1ex.1 (e. H (SH)))) (<-> (E.e. x H (=/= (cv x) (0v))) (E.e. y H (= (` (norm) (cv y)) (1)))) ((cv x) (cv z) (0v) neeq1 H cbvrexv (cv y) (opr (opr (1) (/) (` (norm) (cv z))) (.s) (cv z)) (norm) fveq2 (1) eqeq1d H rcla4ev norm1ex.1 H (opr (1) (/) (` (norm) (cv z))) (cv z) shmulclt ax-mp (` (norm) (cv z)) rerecclt norm1ex.1 (cv z) shel (cv z) normclt syl (=/= (cv z) (0v)) adantr norm1ex.1 (cv z) shel (cv z) normne0t syl biimpar sylanc recnd (e. (cv z) H) (=/= (cv z) (0v)) pm3.26 sylanc (cv z) norm1t norm1ex.1 (cv z) shel sylan sylanc ex r19.23aiv norm1ex.1 (cv y) shel (cv y) norm-it syl negbid (cv y) (0v) df-ne syl6bbr ax1ne0 (1) (0) df-ne mpbi (` (norm) (cv y)) (1) (0) eqeq1 mtbiri syl5bi r19.22i (cv y) (cv z) (0v) neeq1 H cbvrexv sylib impbi bitr)) thm (norm1hext () () (<-> (E.e. x (H~) (=/= (cv x) (0v))) (E.e. y (H~) (= (` (norm) (cv y)) (1)))) (helsh x y norm1ex)) thm (elch0 () () (<-> (e. A (0H)) (= A (0v))) (df-ch0 A eleq2i ax-hv0cl elisseti A elsnc2 bitr)) thm (h0elch () () (e. (0H) (CH)) (df-ch0 hsn0elch eqeltr)) thm (h0elsh () () (e. (0H) (SH)) (h0elch chshi)) thm (ocvalt ((x y) (x z) (w x) (H x) (y z) (w y) (H y) (w z) (H z) (H w)) () (-> (C_ H (H~)) (= (` (_|_) H) ({e.|} x (H~) (A.e. y H (= (opr (cv x) (.i) (cv y)) (0)))))) (ax-hilex (H~) (V) H elpw2g ax-mp (cv z) H y (= (opr (cv x) (.i) (cv y)) (0)) raleq1 x (H~) rabbisdv z w x y df-oc z visset (H~) elpw (= (cv w) ({e.|} x (H~) (A.e. y (cv z) (= (opr (cv x) (.i) (cv y)) (0))))) anbi1i z w opabbii eqtr4 ax-hilex x (A.e. y H (= (opr (cv x) (.i) (cv y)) (0))) rabex fvopab4 sylbir)) thm (ocelt ((x y) (H x) (H y) (A x) (A y)) () (-> (C_ H (H~)) (<-> (e. A (` (_|_) H)) (/\ (e. A (H~)) (A.e. x H (= (opr A (.i) (cv x)) (0)))))) (H y x ocvalt A eleq2d (cv y) A (.i) (cv x) opreq1 (0) eqeq1d x H ralbidv (H~) elrab syl6bb)) thm (shocelt ((H x) (A x)) () (-> (e. H (SH)) (<-> (e. A (` (_|_) H)) (/\ (e. A (H~)) (A.e. x H (= (opr A (.i) (cv x)) (0)))))) (H shss H A x ocelt syl)) thm (ocsh ((x y) (x z) (y z) (A x) (A y) (A z)) () (-> (C_ A (H~)) (e. (` (_|_) A) (SH))) (x (H~) (A.e. y A (= (opr (cv x) (.i) (cv y)) (0))) ssrab2 A x y ocvalt (H~) sseq1d mpbiri A (H~) (cv y) ssel (cv y) hi01t syl6 r19.21aiv ax-hv0cl jctil A (0v) y ocelt mpbird jca (cv x) (cv y) (cv z) ax-his2 3expa (opr (cv x) (.i) (cv z)) (0) (opr (cv y) (.i) (cv z)) (0) (+) opreq12 0cn addid1 syl6eq sylan9eq ex ancoms A (H~) (cv z) ssel2 sylan an1rs r19.20dva ex imdistand (cv x) (cv y) ax-hvaddcl (A.e. z A (= (opr (opr (cv x) (+v) (cv y)) (.i) (cv z)) (0))) anim1i syl6 A (cv x) z ocelt A (cv y) z ocelt anbi12d (e. (cv x) (H~)) (A.e. z A (= (opr (cv x) (.i) (cv z)) (0))) (e. (cv y) (H~)) (A.e. z A (= (opr (cv y) (.i) (cv z)) (0))) an4 z A (= (opr (cv x) (.i) (cv z)) (0)) (= (opr (cv y) (.i) (cv z)) (0)) r19.26 (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) anbi2i bitr4 syl6bb A (opr (cv x) (+v) (cv y)) z ocelt 3imtr4d exp3a r19.21adv r19.21aiv (opr (cv y) (.i) (cv z)) (0) (cv x) (x.) opreq2 (0) eqeq1d (cv x) mul01t syl5bir com12 (e. (cv z) (H~)) (e. (cv y) (H~)) ad2antrl (cv x) (cv y) (cv z) ax-his3 (0) eqeq1d 3expa ancoms sylibrd A (H~) (cv z) ssel2 sylan an1rs r19.20dva ex imdistand (cv x) (cv y) ax-hvmulcl (A.e. z A (= (opr (opr (cv x) (.s) (cv y)) (.i) (cv z)) (0))) anim1i syl6 A (cv y) z ocelt (e. (cv x) (CC)) anbi2d (e. (cv x) (CC)) (e. (cv y) (H~)) (A.e. z A (= (opr (cv y) (.i) (cv z)) (0))) anass syl6bbr A (opr (cv x) (.s) (cv y)) z ocelt 3imtr4d exp3a r19.21adv r19.21aiv jca jca (` (_|_) A) x y sh sylibr)) thm (shocsh () () (-> (e. A (SH)) (e. (` (_|_) A) (SH))) (A shss A ocsh syl)) thm (ocss () () (-> (C_ A (H~)) (C_ (` (_|_) A) (H~))) (A ocsh (` (_|_) A) shss syl)) thm (shocss () () (-> (e. A (SH)) (C_ (` (_|_) A) (H~))) (A shss A ocss syl)) thm (occont ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (C_ A (H~)) (C_ B (H~))) (-> (C_ A B) (C_ (` (_|_) B) (` (_|_) A)))) (A B (cv y) ssel (= (opr (cv x) (.i) (cv y)) (0)) imim1d y 19.20dv y B (= (opr (cv x) (.i) (cv y)) (0)) df-ral y A (= (opr (cv x) (.i) (cv y)) (0)) df-ral 3imtr4g (e. (cv x) (H~)) a1d r19.21aiv x (H~) (A.e. y B (= (opr (cv x) (.i) (cv y)) (0))) (A.e. y A (= (opr (cv x) (.i) (cv y)) (0))) ss2rab sylibr (/\ (C_ A (H~)) (C_ B (H~))) adantl B x y ocvalt (C_ A (H~)) (C_ A B) ad2antlr A x y ocvalt (C_ B (H~)) (C_ A B) ad2antrr 3sstr4d ex)) thm (occon2t () () (-> (/\ (C_ A (H~)) (C_ B (H~))) (-> (C_ A B) (C_ (` (_|_) (` (_|_) A)) (` (_|_) (` (_|_) B))))) (A B occont B ocss A ocss anim12i ancoms (` (_|_) B) (` (_|_) A) occont syl syld)) thm (occon2 () ((occon2.1 (C_ A (H~))) (occon2.2 (C_ B (H~)))) (-> (C_ A B) (C_ (` (_|_) (` (_|_) A)) (` (_|_) (` (_|_) B)))) (occon2.1 occon2.2 A B occon2t mp2an)) thm (oc0 () () (-> (e. H (SH)) (e. (0v) (` (_|_) H))) (H shocsh (` (_|_) H) sh0 syl)) thm (ocorth ((H x) (A x) (B x)) () (-> (C_ H (H~)) (-> (/\ (e. A H) (e. B (` (_|_) H))) (= (opr A (.i) B) (0)))) (H B x ocelt (e. B (H~)) (A.e. x H (= (opr B (.i) (cv x)) (0))) pm3.27 syl6bi imp (/\ (C_ H (H~)) (e. A H)) adantl (cv x) A B (.i) opreq2 (0) eqeq1d H rcla4v (C_ H (H~)) (/\ (C_ H (H~)) (e. B (` (_|_) H))) ad2antlr A B orthcom H (H~) A ssel2 H ocss B sseld imp syl2an sylibrd mpd anandis ex)) thm (shocorth () () (-> (e. H (SH)) (-> (/\ (e. A H) (e. B (` (_|_) H))) (= (opr A (.i) B) (0)))) (H shss H A B ocorth syl)) thm (ococss ((x y) (A x) (A y)) () (-> (C_ A (H~)) (C_ A (` (_|_) (` (_|_) A)))) (A (H~) (cv y) ssel A (cv y) (cv x) ocorth exp3a r19.21adv jcad A ocss (` (_|_) A) (cv y) x ocelt syl sylibrd ssrdv)) thm (shococss () () (-> (e. A (SH)) (C_ A (` (_|_) (` (_|_) A)))) (A shss A ococss syl)) thm (shorth () () (-> (e. H (SH)) (-> (C_ G (` (_|_) H)) (-> (/\ (e. A G) (e. B H)) (= (opr A (.i) B) (0))))) (H B A shocorth imp H shss B sseld H shocss A sseld anim12d imp B A orthcom syl mpbid G (` (_|_) H) A ssel (e. B H) anim1d imp (e. A (` (_|_) H)) (e. B H) ancom sylib sylan2 exp32)) thm (ocin ((x y) (A x) (A y)) () (-> (e. A (SH)) (= (i^i A (` (_|_) A)) (0H))) (A (cv x) y shocelt (cv y) (cv x) (cv x) (.i) opreq2 (0) eqeq1d A rcla4cv (cv x) his6t biimpd sylan9r syl6bi com23 imp3a (cv x) (0v) A eleq1 (cv x) (0v) (` (_|_) A) eleq1 anbi12d A sh0 A oc0 jca syl5bir com12 impbid (cv x) A (` (_|_) A) elin (cv x) elch0 3bitr4g eqrdv)) thm (ocnelt () () (-> (/\/\ (e. H (SH)) (e. A (` (_|_) H)) (=/= A (0v))) (-. (e. A H))) (H ocin A eleq2d biimpd A H (` (_|_) H) elin syl5ibr exp3a com23 imp A elch0 syl6ib con3d A (0v) df-ne syl5ib 3impia)) thm (chocval ((x y) (A x) (A y)) ((chocval.1 (e. A (CH)))) (= (` (_|_) A) ({e.|} x (H~) (A.e. y A (= (opr (cv x) (.i) (cv y)) (0))))) (chocval.1 chssi A x y ocvalt ax-mp)) thm (chocuni () ((chocuni.1 (e. H (CH)))) (-> (/\ (/\ (e. A H) (e. B (` (_|_) H))) (/\ (e. C H) (e. D (` (_|_) H)))) (-> (/\ (= R (opr A (+v) B)) (= R (opr C (+v) D))) (/\ (= A C) (= B D)))) ((e. A (H~)) (e. B (H~)) pm3.26 (e. D (H~)) anim1i A B A D hvsub4t syldan A hvsubidt (e. B (H~)) (e. D (H~)) ad2antrr (+v) (opr B (-v) D) opreq1d B D hvsubclt (opr B (-v) D) hvaddid2t syl (e. A (H~)) adantll 3eqtrd (e. C (H~)) adantrl C D A D hvsub4t (e. A (H~)) (/\ (e. C (H~)) (e. D (H~))) pm3.27 (e. C (H~)) (e. D (H~)) pm3.27 (e. A (H~)) anim2i sylanc D hvsubidt (e. A (H~)) (e. C (H~)) ad2antll (opr C (-v) A) (+v) opreq2d C A hvsubclt (opr C (-v) A) ax-hvaddid syl ancoms (e. D (H~)) adantrr 3eqtrd (e. B (H~)) adantlr eqeq12d chocuni.1 A chel chocuni.1 chshi H shocsh ax-mp B shel anim12i chocuni.1 C chel chocuni.1 chshi H shocsh ax-mp D shel anim12i syl2an chocuni.1 chshi H C A shsubclt ax-mp ancoms (= (opr B (-v) D) (opr C (-v) A)) a1d (e. B (` (_|_) H)) (e. D (` (_|_) H)) ad2ant2r chocuni.1 chshi H shocsh ax-mp (` (_|_) H) B D shsubclt ax-mp (opr B (-v) D) (opr C (-v) A) (` (_|_) H) eleq1 biimpcd syl (e. A H) (e. C H) ad2ant2l jcad C A hvsubeq0t ancoms (e. B (H~)) (e. D (H~)) ad2ant2r chocuni.1 A chel chocuni.1 chshi H shocsh ax-mp B shel anim12i chocuni.1 C chel chocuni.1 chshi H shocsh ax-mp D shel anim12i syl2an C A eqcom syl6bb chocuni.1 chshi H ocin ax-mp (opr C (-v) A) eleq2i (opr C (-v) A) H (` (_|_) H) elin (opr C (-v) A) elch0 3bitr3 syl5bb sylibd chocuni.1 chshi H C A shsubclt ax-mp ancoms (opr C (-v) A) H (opr B (-v) D) eleq1a syl (e. B (` (_|_) H)) (e. D (` (_|_) H)) ad2ant2r chocuni.1 chshi H shocsh ax-mp (` (_|_) H) B D shsubclt ax-mp (= (opr B (-v) D) (opr C (-v) A)) a1d (e. A H) (e. C H) ad2ant2l jcad B D hvsubeq0t (e. A (H~)) (e. C (H~)) ad2ant2l chocuni.1 A chel chocuni.1 chshi H shocsh ax-mp B shel anim12i chocuni.1 C chel chocuni.1 chshi H shocsh ax-mp D shel anim12i syl2an chocuni.1 chshi H ocin ax-mp (opr B (-v) D) eleq2i (opr B (-v) D) H (` (_|_) H) elin (opr B (-v) D) elch0 3bitr3 syl5bb sylibd jcad sylbid R (opr A (+v) B) (opr C (+v) D) eqtr2t (-v) (opr A (+v) D) opreq1d syl5)) thm (occllem1 () ((occllem1.1 (e. A (H~))) (occllem1.2 (e. B (H~))) (occllem1.3 (e. S (H~)))) (br (` (abs) (opr (opr B (.i) S) (-) (opr A (.i) S))) (<_) (opr (` (norm) (opr B (-v) A)) (x.) (` (norm) S))) (occllem1.2 1cn negcl occllem1.1 hvmulcl occllem1.3 B (opr (-u (1)) (.s) A) S ax-his2 mp3an 1cn negcl occllem1.1 occllem1.3 (-u (1)) A S ax-his3 mp3an (opr B (.i) S) (+) opreq2i eqtr2 occllem1.1 occllem1.3 hicl mulm1 (opr B (.i) S) (+) opreq2i occllem1.2 occllem1.3 hicl occllem1.1 occllem1.3 hicl negsub eqtr2 occllem1.2 occllem1.1 hvsubval (.i) S opreq1i 3eqtr4 (abs) fveq2i occllem1.2 occllem1.1 hvsubcl occllem1.3 bcs eqbrtr)) thm (occllem2 () ((occllem2.1 (e. S (H~)))) (-> (/\ (e. A (H~)) (e. B (H~))) (br (` (abs) (opr (opr B (.i) S) (-) (opr A (.i) S))) (<_) (opr (` (norm) (opr B (-v) A)) (x.) (` (norm) S)))) (A (if (e. A (H~)) A (0v)) (.i) S opreq1 (opr B (.i) S) (-) opreq2d (abs) fveq2d A (if (e. A (H~)) A (0v)) B (-v) opreq2 (norm) fveq2d (x.) (` (norm) S) opreq1d (<_) breq12d B (if (e. B (H~)) B (0v)) (.i) S opreq1 (-) (opr (if (e. A (H~)) A (0v)) (.i) S) opreq1d (abs) fveq2d B (if (e. B (H~)) B (0v)) (-v) (if (e. A (H~)) A (0v)) opreq1 (norm) fveq2d (x.) (` (norm) S) opreq1d (<_) breq12d ax-hv0cl A elimel ax-hv0cl B elimel occllem2.1 occllem1 dedth2h)) thm (occllem3 ((x y) (F x) (F y) (S x) (S y) (D x) (D y)) ((occllem3.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (` F (cv x)) (.i) S))))))) (-> (e. D (NN)) (= (` G D) (opr (` F D) (.i) S))) ((cv x) D F fveq2 (.i) S opreq1d occllem3.1 (` F D) (.i) S oprex fvopab4)) thm (occllem4 ((x y) (F x) (F y) (S x) (S y)) ((occllem3.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (` F (cv x)) (.i) S)))))) (occllem4.2 (e. S (H~)))) (-> (:--> F (NN) (H~)) (:--> G (NN) (CC))) (F (NN) (H~) (cv x) ffvrn occllem4.2 jctir (` F (cv x)) S ax-hicl syl r19.21aiva occllem3.1 (CC) fopab2 sylib)) thm (occllem5 ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (G z)) ((occllem3.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (` F (cv x)) (.i) S))))))) (-> (A.e. z (NN) (= (opr (` F (cv z)) (.i) S) (0))) (br G (~~>) (0))) ((cv z) (cv x) F fveq2 (.i) S opreq1d (0) eqeq1d (NN) rcla4v (opr (` F (cv x)) (.i) S) (0) (cv y) eqeq2 syl6com pm5.32d x y opabbidv occllem3.1 (NN) (0) x y fconstopab 3eqtr4g nnuz (NN) (` (ZZ>) (1)) ({} (0)) xpeq1 ax-mp 1z climuz0 eqbrtr syl6eqbr)) thm (occllem6 ((w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (x y) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (F z) (F w) (F v) (F u) (G w) (G u) (G v) (S x) (S y) (S z) (S w) (S v) (S u)) ((occllem6.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (` F (cv x)) (.i) S)))))) (occllem6.2 (e. A (H~))) (occllem6.3 (e. S (H~))) (occllem6.4 (e. F (V)))) (-> (-. (= S (0v))) (-> (br F (~~>v) A) (br G (~~>) (opr A (.i) S)))) (G (NN) (CC) (cv v) ffvrn occllem6.1 occllem6.3 occllem4 sylan r19.21aiva occllem6.2 occllem6.3 hicl jctil (e. A (H~)) adantr (-. (= S (0v))) a1i (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. v (NN) (-> (br (cv w) (<_) (cv v)) (br (` (norm) (opr (` F (cv v)) (-v) A)) (<) (cv z))))))) adantrd occllem6.3 normcl (cv u) (` (norm) S) redivclt mp3an2 occllem6.3 normcl gt0ne0 sylan2 ex (br (0) (<) (cv u)) adantr occllem6.3 normcl (cv u) (` (norm) S) divgt0t mpanl2 exp32 imp jcad com12 (:--> F (NN) (H~)) adantr (E.e. w (NN) (A.e. v (NN) (-> (br (cv w) (<_) (cv v)) (br (` (norm) (opr (` F (cv v)) (-v) A)) (<) (opr (cv u) (/) (` (norm) S)))))) imim1d occllem6.3 normcl (` (norm) (opr (` F (cv v)) (-v) A)) (` (norm) S) (cv u) ltmuldivt mp3anl2 F (NN) (H~) (cv v) ffvrn occllem6.2 jctir (` F (cv v)) A hvsubclt (opr (` F (cv v)) (-v) A) normclt 3syl (br (0) (<) (` (norm) S)) adantll (/\ (e. (cv u) (RR)) (br (0) (<) (cv u))) adantlr (e. (cv u) (RR)) (br (0) (<) (cv u)) pm3.26 (/\ (br (0) (<) (` (norm) S)) (:--> F (NN) (H~))) (e. (cv v) (NN)) ad2antlr jca (br (0) (<) (` (norm) S)) (:--> F (NN) (H~)) pm3.26 (/\ (e. (cv u) (RR)) (br (0) (<) (cv u))) (e. (cv v) (NN)) ad2antrr sylanc F (NN) (H~) (cv v) ffvrn occllem6.2 jctil occllem6.3 A (` F (cv v)) occllem2 syl occllem6.1 (cv v) occllem3 (-) (opr A (.i) S) opreq1d (abs) fveq2d (<_) (opr (` (norm) (opr (` F (cv v)) (-v) A)) (x.) (` (norm) S)) breq1d (:--> F (NN) (H~)) adantl mpbird (br (0) (<) (` (norm) S)) adantll (e. (cv u) (RR)) adantr (` (abs) (opr (` G (cv v)) (-) (opr A (.i) S))) (opr (` (norm) (opr (` F (cv v)) (-v) A)) (x.) (` (norm) S)) (cv u) lelttrt 3expa G (NN) (CC) (cv v) ffvrn occllem6.1 occllem6.3 occllem4 sylan occllem6.2 occllem6.3 hicl jctir (` G (cv v)) (opr A (.i) S) subclt (opr (` G (cv v)) (-) (opr A (.i) S)) absclt 3syl F (NN) (H~) (cv v) ffvrn occllem6.2 jctir (` F (cv v)) A hvsubclt (opr (` F (cv v)) (-v) A) normclt 3syl occllem6.3 normcl jctir (` (norm) (opr (` F (cv v)) (-v) A)) (` (norm) S) axmulrcl syl jca (br (0) (<) (` (norm) S)) adantll sylan mpand (br (0) (<) (cv u)) adantrr an1rs sylbird (br (cv w) (<_) (cv v)) imim2d r19.20dva w (NN) r19.22sdv ex a2d syld exp4a occllem6.3 S normgt0tOLD ax-mp sylanb (e. A (H~)) adantrr (cv z) (opr (cv u) (/) (` (norm) S)) (0) (<) breq2 (cv z) (opr (cv u) (/) (` (norm) S)) (` (norm) (opr (` F (cv v)) (-v) A)) (<) breq2 (br (cv w) (<_) (cv v)) imbi2d w (NN) v (NN) rexralbidv imbi12d (RR) rcla4cv imp3a syl5 r19.21adv ex imp3a jcad occllem6.4 occllem6.2 elisseti z w v hlim syl5ib occllem6.1 nnex x y (opr (` F (cv x)) (.i) S) funopabex2 eqeltr (opr A (.i) S) v u w climnn biimpar syl6)) thm (occllem7 ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z)) ((occllem7.1 (e. A (H~))) (occllem7.2 (e. S (H~))) (occllem7.3 (e. F (V)))) (-> (/\ (br F (~~>v) A) (A.e. x (NN) (= (opr (` F (cv x)) (.i) S) (0)))) (= (opr A (.i) S) (0))) (S (0v) A (.i) opreq2 occllem7.1 A hi02t ax-mp syl6eq (/\ (br F (~~>v) A) (A.e. x (NN) (= (opr (` F (cv x)) (.i) S) (0)))) a1d ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (` F (cv y)) (.i) S)))) eqid occllem7.1 occllem7.2 occllem7.3 occllem6 ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (` F (cv y)) (.i) S)))) eqid x occllem5 (-. (= S (0v))) a1i anim12d A (.i) S oprex 0re elisseti ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (` F (cv y)) (.i) S)))) climuni syl6 pm2.61i)) thm (occllem8 ((F x) (S x)) ((occllem8.1 (e. F (V)))) (-> (/\ (e. A (H~)) (e. S (H~))) (-> (/\ (br F (~~>v) A) (A.e. x (NN) (= (opr (` F (cv x)) (.i) S) (0)))) (= (opr A (.i) S) (0)))) (A (if (e. A (H~)) A (0v)) F (~~>v) breq2 (A.e. x (NN) (= (opr (` F (cv x)) (.i) S) (0))) anbi1d A (if (e. A (H~)) A (0v)) (.i) S opreq1 (0) eqeq1d imbi12d S (if (e. S (H~)) S (0v)) (` F (cv x)) (.i) opreq2 (0) eqeq1d x (NN) ralbidv (br F (~~>v) (if (e. A (H~)) A (0v))) anbi2d S (if (e. S (H~)) S (0v)) (if (e. A (H~)) A (0v)) (.i) opreq2 (0) eqeq1d imbi12d ax-hv0cl A elimel ax-hv0cl S elimel occllem8.1 x occllem7 dedth2h)) thm (occl ((x y) (x z) (f x) (A x) (y z) (f y) (A y) (f z) (A z) (A f)) ((occl.1 (C_ A (H~)))) (e. (` (_|_) A) (CH)) ((` (_|_) A) f x closedsub occl.1 A ocsh ax-mp f visset x visset hlimvec (:--> (cv f) (NN) (` (_|_) A)) adantl f visset x visset hlimvec f visset (cv x) (cv y) z occllem8 exp4b occl.1 (cv y) sseli syl5 com23 mpcom imp r19.20dva (cv f) (NN) (` (_|_) A) (cv z) ffvrn occl.1 A (` (cv f) (cv z)) y ocelt ax-mp pm3.27bd syl r19.21aiva y A z (NN) (= (opr (` (cv f) (cv z)) (.i) (cv y)) (0)) ralcom sylibr syl5 impcom jca occl.1 A (cv x) y ocelt ax-mp sylibr f x gen2 mpbir2an)) thm (occlt () () (-> (C_ A (H~)) (e. (` (_|_) A) (CH))) (A (if (C_ A (H~)) A (H~)) (_|_) fveq2 (CH) eleq1d A (if (C_ A (H~)) A (H~)) (H~) sseq1 (H~) (if (C_ A (H~)) A (H~)) (H~) sseq1 (H~) ssid elimhyp occl dedth)) thm (shocclt () () (-> (e. A (SH)) (e. (` (_|_) A) (CH))) (A shss A occlt syl)) thm (chocclt () () (-> (e. A (CH)) (e. (` (_|_) A) (CH))) (A chsh A shocclt syl)) thm (choccl () ((choccl.1 (e. A (CH)))) (e. (` (_|_) A) (CH)) (choccl.1 chssi occl)) thm (projlem1 ((R z) (D z)) ((projlem1.1 (e. R (RR))) (projlem1.2 (e. D (RR)))) (-> (br (0) (<) D) (E.e. z (NN) (br (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (cv z)) (<) (opr D (^) (2))))) (projlem1.2 gt0ne0 projlem1.2 sqgt0 projlem1.2 resqcl gt0ne0 4re 2re projlem1.1 remulcl ax1re readdcl remulcl projlem1.2 resqcl redivclz (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (opr D (^) (2))) z arch 3syl (cv z) nnret (cv z) (if (e. (cv z) (RR)) (cv z) (0)) (0) (<) breq2 (br (0) (<) (opr D (^) (2))) anbi1d (cv z) (if (e. (cv z) (RR)) (cv z) (0)) (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) opreq2 (<) (opr D (^) (2)) breq1d (cv z) (if (e. (cv z) (RR)) (cv z) (0)) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (opr D (^) (2))) (<) breq2 bibi12d imbi12d 4re 2re projlem1.1 remulcl ax1re readdcl remulcl 0re (cv z) elimel projlem1.2 resqcl ltdiv23 dedth (cv z) nngt0t sylani syl anabsi8 rexbidva mpbird 3syl)) thm (projlem2 ((R z) (D z)) ((projlem1.1 (e. R (RR))) (projlem1.2 (e. D (RR))) (projlem2.3 (br (0) (<_) R))) (-> (br (0) (<) D) (E.e. z (NN) (br (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (cv z))) (<) D))) (projlem1.1 projlem1.2 z projlem1 (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (cv z))) D lt2sqtOLD (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (cv z)) sqrclt (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (cv z) redivclt 3expa (cv z) nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil (cv z) nnne0t sylanc (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (cv z) divge0t (cv z) nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil (cv z) nngt0t 0re 4re 2re projlem1.1 remulcl ax1re readdcl remulcl 4re 2re projlem1.1 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem2.3 2re projlem1.1 mulge0 mp2an lt01 2re projlem1.1 remulcl ax1re addgegt0 mp2an mulgt0i ltlei jctil sylanc sylanc projlem1.2 jctir (br (0) (<) D) adantl (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (cv z)) sqrge0t (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (cv z) redivclt 3expa (cv z) nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil (cv z) nnne0t sylanc (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (cv z) divge0t (cv z) nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil (cv z) nngt0t 0re 4re 2re projlem1.1 remulcl ax1re readdcl remulcl 4re 2re projlem1.1 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem2.3 2re projlem1.1 mulge0 mp2an lt01 2re projlem1.1 remulcl ax1re addgegt0 mp2an mulgt0i ltlei jctil sylanc sylanc 0re projlem1.2 ltle anim12i ancoms sylc (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (cv z)) sqsqrt (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (cv z) redivclt 3expa (cv z) nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil (cv z) nnne0t sylanc (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (cv z) divge0t (cv z) nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil (cv z) nngt0t 0re 4re 2re projlem1.1 remulcl ax1re readdcl remulcl 4re 2re projlem1.1 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem2.3 2re projlem1.1 mulge0 mp2an lt01 2re projlem1.1 remulcl ax1re addgegt0 mp2an mulgt0i ltlei jctil sylanc sylanc (<) (opr D (^) (2)) breq1d (br (0) (<) D) adantl bitrd rexbidva mpbird)) thm (projlem3 () ((projlem3.1 (e. R (RR))) (projlem3.2 (e. D (NN))) (projlem3.3 (e. G (NN)))) (br (opr (opr (opr (2) (x.) (opr (opr R (+) (opr (1) (/) D)) (^) (2))) (+) (opr (2) (x.) (opr (opr R (+) (opr (1) (/) G)) (^) (2)))) (-) (opr (4) (x.) (opr R (^) (2)))) (<_) (opr (opr (opr (4) (x.) R) (+) (2)) (x.) (opr (opr (1) (/) D) (+) (opr (1) (/) G)))) (2cn projlem3.1 projlem3.2 nnre projlem3.2 nnne0 rereccl readdcl resqcl recn projlem3.1 projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl resqcl recn adddi projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn binom2 projlem3.1 recn sqcl 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl addass eqtr projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn binom2 projlem3.1 recn sqcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl addass eqtr (+) opreq12i projlem3.1 recn sqcl 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl addcl projlem3.1 recn sqcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl addcl add4 eqtr (2) (x.) opreq2i eqtr3 2cn 2cn projlem3.1 recn sqcl mulass 2t2e4 (x.) (opr R (^) (2)) opreq1i projlem3.1 recn sqcl 2times (2) (x.) opreq2i 3eqtr3 (-) opreq12i 2cn projlem3.1 recn sqcl projlem3.1 recn sqcl addcl 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl addcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl addcl addcl addcl projlem3.1 recn sqcl projlem3.1 recn sqcl addcl subdi projlem3.1 recn sqcl projlem3.1 recn sqcl addcl 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl addcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl addcl addcl projlem3.1 recn sqcl projlem3.1 recn sqcl addcl addsubass projlem3.1 recn sqcl projlem3.1 recn sqcl addcl 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl addcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl addcl addcl pncan3 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl add4 projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn adddi (2) (x.) opreq2i 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl adddi eqtr (+) (opr (opr (opr (1) (/) D) (^) (2)) (+) (opr (opr (1) (/) G) (^) (2))) opreq1i eqtr4 3eqtr (2) (x.) opreq2i 3eqtr2 projlem3.2 nncn projlem3.2 nnne0 sqreci projlem3.2 nnlesq projlem3.2 nngt0 projlem3.2 nnne0 projlem3.2 nnre sqgt0 ax-mp projlem3.2 nnre projlem3.2 nnre resqcl lerec mp2an mpbi eqbrtr projlem3.3 nncn projlem3.3 nnne0 sqreci projlem3.3 nnlesq projlem3.3 nngt0 projlem3.3 nnne0 projlem3.3 nnre sqgt0 ax-mp projlem3.3 nnre projlem3.3 nnre resqcl lerec mp2an mpbi eqbrtr projlem3.2 nnre projlem3.2 nnne0 rereccl resqcl projlem3.3 nnre projlem3.3 nnne0 rereccl resqcl projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl le2add mp2an projlem3.2 nnre projlem3.2 nnne0 rereccl resqcl projlem3.3 nnre projlem3.3 nnne0 rereccl resqcl readdcl projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl 2re projlem3.1 projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl remulcl remulcl leadd2 mpbi 2pos 2re projlem3.1 projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl remulcl remulcl projlem3.2 nnre projlem3.2 nnne0 rereccl resqcl projlem3.3 nnre projlem3.3 nnne0 rereccl resqcl readdcl readdcl 2re projlem3.1 projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl remulcl remulcl projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl readdcl 2re lemul2 ax-mp mpbi 4re recn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl recn mulass 2t2e4 (x.) (opr R (x.) (opr (opr (1) (/) D) (+) (opr (1) (/) G))) opreq1i 2cn 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl recn mulcl mulass 3eqtr2r (+) (opr (2) (x.) (opr (opr (1) (/) D) (+) (opr (1) (/) G))) opreq1i 2cn 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl recn adddi 4re projlem3.1 remulcl recn 2cn projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl recn adddir 3eqtr4 breqtr eqbrtr)) thm (projlem4 () ((projlem4.1 (e. R (RR))) (projlem4.2 (br (0) (<_) R)) (projlem4.3 (e. D (NN))) (projlem4.4 (e. G (NN))) (projlem4.5 (e. B (NN)))) (-> (/\ (br B (<_) D) (br B (<_) G)) (br (opr (opr (opr (4) (x.) R) (+) (2)) (x.) (opr (opr (1) (/) D) (+) (opr (1) (/) G))) (<_) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) B))) (projlem4.3 nnre projlem4.3 nnne0 rereccl projlem4.4 nnre projlem4.4 nnne0 rereccl projlem4.5 nnre projlem4.5 nnne0 rereccl projlem4.5 nnre projlem4.5 nnne0 rereccl le2add projlem4.5 nngt0 projlem4.3 nngt0 projlem4.5 nnre projlem4.3 nnre lerec mp2an projlem4.5 nngt0 projlem4.4 nngt0 projlem4.5 nnre projlem4.4 nnre lerec mp2an syl2anb 2cn 2re projlem4.1 remulcl recn 1cn adddi 2cn 2cn projlem4.1 recn mulass 2t2e4 (x.) R opreq1i eqtr3 2cn mulid1 (+) opreq12i eqtr2 2cn mulid1 (x.) opreq12i 2cn 2cn 2re projlem4.1 remulcl ax1re readdcl recn mul23 2t2e4 (x.) (opr (opr (2) (x.) R) (+) (1)) opreq1i 3eqtr2r (/) B opreq1i 4re projlem4.1 remulcl 2re readdcl recn 2cn 1cn mulcl projlem4.5 nncn projlem4.5 nnne0 divass 2cn 1cn projlem4.5 nncn projlem4.5 nnne0 divass projlem4.5 nnre projlem4.5 nnne0 rereccl recn 2times eqtr (opr (opr (4) (x.) R) (+) (2)) (x.) opreq2i 3eqtr (opr (opr (opr (4) (x.) R) (+) (2)) (x.) (opr (opr (1) (/) D) (+) (opr (1) (/) G))) (<_) breq2i 0re 4re 4pos ltlei projlem4.2 4re projlem4.1 mulge0 mp2an 2pos 4re projlem4.1 remulcl 2re addgegt0 mp2an projlem4.3 nnre projlem4.3 nnne0 rereccl projlem4.4 nnre projlem4.4 nnne0 rereccl readdcl projlem4.5 nnre projlem4.5 nnne0 rereccl projlem4.5 nnre projlem4.5 nnne0 rereccl readdcl 4re projlem4.1 remulcl 2re readdcl lemul2 ax-mp bitr4 sylibr)) thm (projlem5 () ((projlem5.1 (e. A (H~))) (projlem5.2 (e. B (H~))) (projlem5.3 (e. C (H~))) (projlem5.4 (e. R (RR))) (projlem5.5 (br (0) (<_) R)) (projlem5.6 (br (opr (4) (x.) (opr R (^) (2))) (<_) (opr (` (norm) (opr (opr B (+v) C) (-v) (opr (2) (.s) A))) (^) (2)))) (projlem5.7 (e. D (NN))) (projlem5.8 (e. G (NN))) (projlem5.9 (e. N (NN))) (projlem5.10 (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D)))) (projlem5.11 (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G))))) (br (opr (` (norm) (opr B (-v) C)) (^) (2)) (<) (opr (opr (opr (2) (x.) (opr (opr R (+) (opr (1) (/) D)) (^) (2))) (+) (opr (2) (x.) (opr (opr R (+) (opr (1) (/) G)) (^) (2)))) (-) (opr (4) (x.) (opr R (^) (2))))) (projlem5.2 projlem5.3 projlem5.1 normpar2 projlem5.10 projlem5.2 projlem5.1 hvsubcl (opr B (-v) A) normge0t ax-mp projlem5.5 0re projlem5.7 nnre projlem5.7 nnne0 rereccl projlem5.7 D nnrecgt0t ax-mp ltlei projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl addge0 mp2an projlem5.2 projlem5.1 hvsubcl normcl projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl lt2sq mp2an mpbi 2pos projlem5.2 projlem5.1 hvsubcl normcl resqcl projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl resqcl 2re ltmul2 ax-mp mpbi projlem5.11 projlem5.3 projlem5.1 hvsubcl (opr C (-v) A) normge0t ax-mp projlem5.5 0re projlem5.8 nnre projlem5.8 nnne0 rereccl projlem5.8 G nnrecgt0t ax-mp ltlei projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl addge0 mp2an projlem5.3 projlem5.1 hvsubcl normcl projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl lt2sq mp2an mpbi 2pos projlem5.3 projlem5.1 hvsubcl normcl resqcl projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl resqcl 2re ltmul2 ax-mp mpbi 2re projlem5.2 projlem5.1 hvsubcl normcl resqcl remulcl 2re projlem5.3 projlem5.1 hvsubcl normcl resqcl remulcl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl resqcl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl resqcl remulcl lt2add mp2an 2re projlem5.2 projlem5.1 hvsubcl normcl resqcl remulcl 2re projlem5.3 projlem5.1 hvsubcl normcl resqcl remulcl readdcl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl resqcl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl resqcl remulcl readdcl projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl resqcl renegcl ltadd1 mpbi projlem5.6 4re projlem5.4 resqcl remulcl projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl resqcl leneg mpbi projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl resqcl renegcl 4re projlem5.4 resqcl remulcl renegcl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl resqcl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl resqcl remulcl readdcl leadd2 mpbi 2re projlem5.2 projlem5.1 hvsubcl normcl resqcl remulcl 2re projlem5.3 projlem5.1 hvsubcl normcl resqcl remulcl readdcl projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl resqcl renegcl readdcl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl resqcl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl resqcl remulcl readdcl projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl resqcl renegcl readdcl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl resqcl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl resqcl remulcl readdcl 4re projlem5.4 resqcl remulcl renegcl readdcl ltletr mp2an 2re projlem5.2 projlem5.1 hvsubcl normcl resqcl remulcl 2re projlem5.3 projlem5.1 hvsubcl normcl resqcl remulcl readdcl recn projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl resqcl recn negsub 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl resqcl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl resqcl remulcl readdcl recn 4re projlem5.4 resqcl remulcl recn negsub 3brtr3 eqbrtr)) thm (projlem6 () ((projlem5.1 (e. A (H~))) (projlem5.2 (e. B (H~))) (projlem5.3 (e. C (H~))) (projlem5.4 (e. R (RR))) (projlem5.5 (br (0) (<_) R)) (projlem5.6 (br (opr (4) (x.) (opr R (^) (2))) (<_) (opr (` (norm) (opr (opr B (+v) C) (-v) (opr (2) (.s) A))) (^) (2)))) (projlem5.7 (e. D (NN))) (projlem5.8 (e. G (NN))) (projlem5.9 (e. N (NN))) (projlem5.10 (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D)))) (projlem5.11 (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G))))) (-> (/\ (br N (<_) D) (br N (<_) G)) (br (` (norm) (opr B (-v) C)) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N)))) (projlem5.4 projlem5.5 projlem5.7 projlem5.8 projlem5.9 projlem4 projlem5.1 projlem5.2 projlem5.3 projlem5.4 projlem5.5 projlem5.6 projlem5.7 projlem5.8 projlem5.7 projlem5.10 projlem5.11 projlem5 projlem5.4 projlem5.7 projlem5.8 projlem3 projlem5.2 projlem5.3 hvsubcl normcl resqcl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl resqcl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl resqcl remulcl readdcl 4re projlem5.4 resqcl remulcl resubcl 4re projlem5.4 remulcl 2re readdcl projlem5.7 nnre projlem5.7 nnne0 rereccl projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl remulcl ltletr mp2an projlem5.2 projlem5.3 hvsubcl normcl resqcl 4re projlem5.4 remulcl 2re readdcl projlem5.7 nnre projlem5.7 nnne0 rereccl projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl remulcl 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl ltletr mpan syl 0re 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre 4re 2re projlem5.4 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem5.5 2re projlem5.4 mulge0 mp2an lt01 2re projlem5.4 remulcl ax1re addgegt0 mp2an mulgt0i projlem5.9 nngt0 divgt0i ltlei 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl sqsqr ax-mp syl6breqr projlem5.2 projlem5.3 hvsubcl (opr B (-v) C) normge0t ax-mp 0re 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre 4re 2re projlem5.4 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem5.5 2re projlem5.4 mulge0 mp2an lt01 2re projlem5.4 remulcl ax1re addgegt0 mp2an mulgt0i projlem5.9 nngt0 divgt0i ltlei 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl sqrge0 ax-mp projlem5.2 projlem5.3 hvsubcl normcl 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre 4re 2re projlem5.4 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem5.5 2re projlem5.4 mulge0 mp2an lt01 2re projlem5.4 remulcl ax1re addgegt0 mp2an mulgt0i projlem5.9 nngt0 divgt0i sqrlem24 lt2sq mp2an sylibr)) thm (projlem7 () ((projlem5.1 (e. A (H~))) (projlem5.2 (e. B (H~))) (projlem5.3 (e. C (H~))) (projlem5.4 (e. R (RR))) (projlem5.5 (br (0) (<_) R)) (projlem5.6 (br (opr (4) (x.) (opr R (^) (2))) (<_) (opr (` (norm) (opr (opr B (+v) C) (-v) (opr (2) (.s) A))) (^) (2)))) (projlem5.7 (e. D (NN))) (projlem5.8 (e. G (NN))) (projlem5.9 (e. N (NN)))) (-> (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) (-> (/\ (br N (<_) D) (br N (<_) G)) (br (` (norm) (opr B (-v) C)) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N))))) (B (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) C opreq1 (norm) fveq2d (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N)) breq1d (/\ (br N (<_) D) (br N (<_) G)) imbi2d C (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) opreq2 (norm) fveq2d (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N)) breq1d (/\ (br N (<_) D) (br N (<_) G)) imbi2d R (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (2) (x.) opreq2 (+) (1) opreq1d (4) (x.) opreq2d (/) N opreq1d (sqr) fveq2d (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A))) (<) breq2d (/\ (br N (<_) D) (br N (<_) G)) imbi2d projlem5.1 projlem5.2 projlem5.1 (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) keepel projlem5.3 projlem5.1 (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) keepel projlem5.4 0re (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) keepel R (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (0) (<_) breq2 (0) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (0) (<_) breq2 projlem5.5 0re leid keephyp B (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (+v) C opreq1 (-v) (opr (2) (.s) A) opreq1d (norm) fveq2d (^) (2) opreq1d (opr (4) (x.) (opr R (^) (2))) (<_) breq2d C (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (+v) opreq2 (-v) (opr (2) (.s) A) opreq1d (norm) fveq2d (^) (2) opreq1d (opr (4) (x.) (opr R (^) (2))) (<_) breq2d R (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (^) (2) opreq1 (4) (x.) opreq2d (<_) (opr (` (norm) (opr (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (+v) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A)) (-v) (opr (2) (.s) A))) (^) (2)) breq1d A (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (+v) A opreq1 (-v) (opr (2) (.s) A) opreq1d (norm) fveq2d (^) (2) opreq1d (opr (4) (x.) (opr (0) (^) (2))) (<_) breq2d A (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (+v) opreq2 (-v) (opr (2) (.s) A) opreq1d (norm) fveq2d (^) (2) opreq1d (opr (4) (x.) (opr (0) (^) (2))) (<_) breq2d (0) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (^) (2) opreq1 (4) (x.) opreq2d (<_) (opr (` (norm) (opr (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (+v) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A)) (-v) (opr (2) (.s) A))) (^) (2)) breq1d projlem5.6 sq0 (4) (x.) opreq2i 4re recn mul01 eqtr projlem5.1 projlem5.1 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl sqge0 eqbrtr keephyp3v projlem5.7 projlem5.8 projlem5.9 B (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A opreq1 (norm) fveq2d (<) (opr R (+) (opr (1) (/) D)) breq1d (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G))) anbi1d C (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A) (-v) A opreq1 (norm) fveq2d (<) (opr R (+) (opr (1) (/) G)) breq1d (br (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A)) (<) (opr R (+) (opr (1) (/) D))) anbi2d R (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (+) (opr (1) (/) D) opreq1 (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A)) (<) breq2d R (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (+) (opr (1) (/) G) opreq1 (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A) (-v) A)) (<) breq2d anbi12d A (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A opreq1 (norm) fveq2d (<) (opr (0) (+) (opr (1) (/) D)) breq1d (br (` (norm) (opr A (-v) A)) (<) (opr (0) (+) (opr (1) (/) G))) anbi1d A (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A) (-v) A opreq1 (norm) fveq2d (<) (opr (0) (+) (opr (1) (/) G)) breq1d (br (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A)) (<) (opr (0) (+) (opr (1) (/) D))) anbi2d (0) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (+) (opr (1) (/) D) opreq1 (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A)) (<) breq2d (0) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (+) (opr (1) (/) G) opreq1 (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A) (-v) A)) (<) breq2d anbi12d projlem5.7 nnre projlem5.7 nngt0 recgt0i projlem5.1 A hvsubidt ax-mp (norm) fveq2i norm0 eqtr projlem5.7 nncn projlem5.7 nnne0 reccl addid2 3brtr4 projlem5.8 nnre projlem5.8 nngt0 recgt0i projlem5.1 A hvsubidt ax-mp (norm) fveq2i norm0 eqtr projlem5.8 nncn projlem5.8 nnne0 reccl addid2 3brtr4 pm3.2i elimhyp3v pm3.26i B (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A opreq1 (norm) fveq2d (<) (opr R (+) (opr (1) (/) D)) breq1d (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G))) anbi1d C (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A) (-v) A opreq1 (norm) fveq2d (<) (opr R (+) (opr (1) (/) G)) breq1d (br (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A)) (<) (opr R (+) (opr (1) (/) D))) anbi2d R (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (+) (opr (1) (/) D) opreq1 (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A)) (<) breq2d R (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (+) (opr (1) (/) G) opreq1 (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A) (-v) A)) (<) breq2d anbi12d A (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A opreq1 (norm) fveq2d (<) (opr (0) (+) (opr (1) (/) D)) breq1d (br (` (norm) (opr A (-v) A)) (<) (opr (0) (+) (opr (1) (/) G))) anbi1d A (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A) (-v) A opreq1 (norm) fveq2d (<) (opr (0) (+) (opr (1) (/) G)) breq1d (br (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A)) (<) (opr (0) (+) (opr (1) (/) D))) anbi2d (0) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (+) (opr (1) (/) D) opreq1 (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) B A) (-v) A)) (<) breq2d (0) (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) R (0)) (+) (opr (1) (/) G) opreq1 (` (norm) (opr (if (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) C A) (-v) A)) (<) breq2d anbi12d projlem5.7 nnre projlem5.7 nngt0 recgt0i projlem5.1 A hvsubidt ax-mp (norm) fveq2i norm0 eqtr projlem5.7 nncn projlem5.7 nnne0 reccl addid2 3brtr4 projlem5.8 nnre projlem5.8 nngt0 recgt0i projlem5.1 A hvsubidt ax-mp (norm) fveq2i norm0 eqtr projlem5.8 nncn projlem5.8 nnne0 reccl addid2 3brtr4 pm3.2i elimhyp3v pm3.27i projlem6 dedth3v)) thm (projlem8 ((u v) (u z) (u w) (A u) (v z) (v w) (A v) (w z) (A z) (A w) (H u) (H v) (H z) (H w) (S w) (S z)) ((projlem8.1 (e. A (H~))) (projlem8.2 (e. H (CH))) (projlem8.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A))))))))) (/\/\ (C_ S (RR)) (-. (= S ({/}))) (E.e. z (RR) (A.e. w S (br (cv w) (<_) (cv z))))) (projlem8.3 u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A))))) ssrab2 eqsstr (cv u) (-u (` (norm) (opr (0v) (-v) A))) (-u (` (norm) (opr (cv v) (-v) A))) eqeq1 v H rexbidv (RR) elrab ax-hv0cl projlem8.1 hvsubcl normcl renegcl projlem8.2 H ch0 ax-mp (-u (` (norm) (opr (0v) (-v) A))) eqid (cv v) (0v) (-v) A opreq1 (norm) fveq2d negeqd (-u (` (norm) (opr (0v) (-v) A))) eqeq2d H rcla4ev mp2an mpbir2an projlem8.3 eleqtrr (-u (` (norm) (opr (0v) (-v) A))) S n0i ax-mp 0re projlem8.3 (cv w) eleq2i (cv u) (cv w) (-u (` (norm) (opr (cv v) (-v) A))) eqeq1 v H rexbidv (RR) elrab bitr (cv w) (-u (` (norm) (opr (cv v) (-v) A))) (<_) (0) breq1 projlem8.2 (cv v) chel projlem8.1 jctir (cv v) A hvsubclt (opr (cv v) (-v) A) normge0t (opr (cv v) (-v) A) normclt (` (norm) (opr (cv v) (-v) A)) le0neg2t syl mpbid 3syl syl5bir com12 r19.23aiv (e. (cv w) (RR)) adantl sylbi rgen (cv z) (0) (cv w) (<_) breq2 w S ralbidv (RR) rcla4ev mp2an 3pm3.2i)) thm (projlem9 ((u v) (u x) (u z) (u w) (A u) (v x) (v z) (v w) (A v) (x z) (w x) (A x) (w z) (A z) (A w) (H u) (H v) (H x) (H z) (H w) (S w) (S z) (S x)) ((projlem8.1 (e. A (H~))) (projlem8.2 (e. H (CH))) (projlem8.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A))))))))) (e. (sup S (RR) (<)) (RR)) (ltso projlem8.1 projlem8.2 projlem8.3 z w projlem8 x sup3i supcli)) thm (projlem10 ((u v) (u x) (A u) (v x) (A v) (A x) (H u) (H v) (H x) (S x) (B x)) ((projlem8.1 (e. A (H~))) (projlem8.2 (e. H (CH))) (projlem8.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A))))))))) (-> (e. B H) (e. (-u (` (norm) (opr B (-v) A))) S)) ((cv x) B (-v) A opreq1 (norm) fveq2d negeqd S eleq1d projlem8.2 (cv x) chel projlem8.1 (cv x) A hvsubclt mpan2 syl (opr (cv x) (-v) A) normclt (` (norm) (opr (cv x) (-v) A)) renegclt 3syl (-u (` (norm) (opr (cv x) (-v) A))) eqid (cv v) (cv x) (-v) A opreq1 (norm) fveq2d negeqd (-u (` (norm) (opr (cv x) (-v) A))) eqeq2d H rcla4ev mpan2 jca projlem8.3 (-u (` (norm) (opr (cv x) (-v) A))) eleq2i (cv u) (-u (` (norm) (opr (cv x) (-v) A))) (-u (` (norm) (opr (cv v) (-v) A))) eqeq1 v H rexbidv (RR) elrab bitr sylibr vtoclga)) thm (projlem11 ((u v) (A u) (A v) (H u) (H v)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<)))))) (e. R (RR)) (projlem11.4 projlem11.1 projlem11.2 projlem11.3 projlem9 renegcl eqeltr)) thm (projlem12 ((u v) (u x) (u y) (A u) (v x) (v y) (A v) (x y) (A x) (A y) (H u) (H v) (H x) (H y) (S x) (S y) (B x)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<)))))) (-> (e. B H) (br R (<_) (` (norm) (opr B (-v) A)))) (projlem11.1 projlem11.2 projlem11.3 B projlem10 projlem11.1 projlem11.2 projlem11.3 x y projlem8 (-u (` (norm) (opr B (-v) A))) suprubi syl projlem11.4 projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 recn projlem11.1 projlem11.2 projlem11.3 projlem9 recn negcon2 mpbi syl6breq projlem11.2 B chel projlem11.1 B A hvsubclt mpan2 (opr B (-v) A) normclt 3syl projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 jctil R (` (norm) (opr B (-v) A)) lenegt syl mpbird)) thm (projlem13 ((u v) (u x) (u y) (u z) (A u) (v x) (v y) (v z) (A v) (x y) (x z) (A x) (y z) (A y) (A z) (H u) (H v) (H x) (H y) (H z) (S z) (S x) (S y) (R z)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<)))))) (br (0) (<_) R) (projlem11.4 projlem11.1 projlem11.2 projlem11.3 projlem9 recn projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 recn negcon2 mpbir 0re renegcl projlem11.3 (cv z) eleq2i (cv u) (cv z) (-u (` (norm) (opr (cv v) (-v) A))) eqeq1 v H rexbidv (RR) elrab bitr 0re renegcl (cv z) (-u (0)) lenltt mpan2 (cv z) (-u (` (norm) (opr (cv v) (-v) A))) (<_) (-u (0)) breq1 projlem11.2 (cv v) chel projlem11.1 jctir (cv v) A hvsubclt (opr (cv v) (-v) A) normge0t 3syl projlem11.2 (cv v) chel projlem11.1 jctir (cv v) A hvsubclt (opr (cv v) (-v) A) normclt 3syl 0re jctil (0) (` (norm) (opr (cv v) (-v) A)) lenegt syl mpbid syl5bir impcom syl5bi exp3a r19.23adv imp sylbi rgen ltso projlem11.1 projlem11.2 projlem11.3 x y projlem8 z sup3i (-u (0)) supnubi mp2an projlem11.1 projlem11.2 projlem11.3 projlem9 0re renegcl lenlt mpbir eqbrtrr 0re projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 leneg mpbir)) thm (projlem14 ((u v) (A u) (A v) (H u) (H v)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem14.5 (e. C (NN))) (projlem14.6 (e. B H))) (br (opr R (-) (opr (1) (/) C)) (<) (` (norm) (opr B (-v) A))) (projlem14.5 C nnrecgt0t ax-mp projlem14.5 nnre projlem14.5 nnne0 rereccl projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 ltaddpos mpbi projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem14.5 nnre projlem14.5 nnne0 rereccl projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 ltsubadd mpbir projlem14.6 projlem11.1 projlem11.2 projlem11.3 projlem11.4 B projlem12 ax-mp projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem14.5 nnre projlem14.5 nnne0 rereccl resubcl projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem11.2 projlem14.6 cheli projlem11.1 hvsubcl normcl ltletr mp2an)) thm (projlem15 ((v w) (u w) (w z) (C w) (u v) (v z) (C v) (u z) (C u) (C z) (u v) (u x) (u y) (u z) (u w) (A u) (v x) (v y) (v z) (v w) (A v) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H u) (H v) (H x) (H y) (H z) (H w) (S w) (S z) (S x) (S y) (R z) (R w)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem15.5 (e. C (NN)))) (E.e. z H (br (` (norm) (opr (cv z) (-v) A)) (<) (opr R (+) (opr (1) (/) C)))) (projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem15.5 nnre projlem15.5 nnne0 rereccl readdcl renegcl projlem11.4 projlem15.5 C nnrecgt0t ax-mp projlem15.5 nnre projlem15.5 nnne0 rereccl projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 ltaddpos mpbi eqbrtrr projlem11.1 projlem11.2 projlem11.3 projlem9 projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem15.5 nnre projlem15.5 nnne0 rereccl readdcl ltnegcon1 mpbi ltso projlem11.1 projlem11.2 projlem11.3 x y projlem8 w sup3i (-u (opr R (+) (opr (1) (/) C))) suplubi mp2an (cv w) (-u (` (norm) (opr (cv z) (-v) A))) (-u (opr R (+) (opr (1) (/) C))) (<) breq2 biimpd projlem11.2 (cv z) chel projlem11.1 jctir (cv z) A hvsubclt syl (opr (cv z) (-v) A) normclt projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem15.5 nnre projlem15.5 nnne0 rereccl readdcl (` (norm) (opr (cv z) (-v) A)) (opr R (+) (opr (1) (/) C)) ltnegt mpan2 3syl biimprd syl9 com13 imp r19.22dva projlem11.3 (cv w) eleq2i (cv u) (cv w) (-u (` (norm) (opr (cv z) (-v) A))) eqeq1 z H rexbidv (cv v) (cv z) (-v) A opreq1 (norm) fveq2d negeqd (cv u) eqeq2d H cbvrexv syl5bb (RR) elrab bitr pm3.27bd syl5com r19.23aiv ax-mp)) thm (projlem16 ((u v) (v z) (C v) (u z) (C u) (C z) (u v) (u z) (A u) (v z) (A v) (A z) (H u) (H v) (H z) (S z) (R z)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem15.5 (e. C (NN)))) (E.e. z H (/\ (br (opr R (-) (opr (1) (/) C)) (<) (` (norm) (opr (cv z) (-v) A))) (br (` (norm) (opr (cv z) (-v) A)) (<) (opr R (+) (opr (1) (/) C))))) (projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem15.5 z projlem15 (cv z) (if (e. (cv z) H) (cv z) (0v)) (-v) A opreq1 (norm) fveq2d (opr R (-) (opr (1) (/) C)) (<) breq2d projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem15.5 projlem11.2 H ch0 ax-mp (cv z) elimel projlem14 dedth (br (` (norm) (opr (cv z) (-v) A)) (<) (opr R (+) (opr (1) (/) C))) anim1i ex r19.22i ax-mp)) thm (projlem17 ((u v) (u z) (u w) (f u) (A u) (v z) (v w) (f v) (A v) (w z) (f z) (A z) (f w) (A w) (A f) (H u) (H v) (H z) (H w) (H f) (S w) (S z) (R z) (R w) (R f)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<)))))) (E. f (/\ (:--> (cv f) (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` (cv f) (cv w)) (-v) A))) (br (` (norm) (opr (` (cv f) (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))) (nnex projlem11.2 elisseti (cv z) (` (cv f) (cv w)) (-v) A opreq1 (norm) fveq2d (opr R (-) (opr (1) (/) (cv w))) (<) breq2d (cv z) (` (cv f) (cv w)) (-v) A opreq1 (norm) fveq2d (<) (opr R (+) (opr (1) (/) (cv w))) breq1d anbi12d ac6 (cv w) (if (e. (cv w) (NN)) (cv w) (1)) (1) (/) opreq2 R (-) opreq2d (<) (` (norm) (opr (cv z) (-v) A)) breq1d (cv w) (if (e. (cv w) (NN)) (cv w) (1)) (1) (/) opreq2 R (+) opreq2d (` (norm) (opr (cv z) (-v) A)) (<) breq2d anbi12d z H rexbidv projlem11.1 projlem11.2 projlem11.3 projlem11.4 1nn (cv w) elimel z projlem16 dedth mprg)) thm (projlem18 ((u v) (A u) (A v) (H u) (H v)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem18.5 (e. B H)) (projlem18.6 (e. C H))) (br (opr (4) (x.) (opr R (^) (2))) (<_) (opr (` (norm) (opr (opr B (+v) C) (-v) (opr (2) (.s) A))) (^) (2))) (2cn projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 recn sqmul sq2 (x.) (opr R (^) (2)) opreq1i eqtr2 2cn 2re 2pos gt0ne0i reccl projlem18.5 projlem18.6 projlem11.2 chshi H B C shaddclt ax-mp mp2an projlem11.2 chshi H (opr (1) (/) (2)) (opr B (+v) C) shmulclt ax-mp mp2an projlem11.1 projlem11.2 projlem11.3 projlem11.4 (opr (opr (1) (/) (2)) (.s) (opr B (+v) C)) projlem12 ax-mp 2pos projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 2cn 2re 2pos gt0ne0i reccl projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl hvmulcl projlem11.1 hvsubcl normcl 2re lemul2 ax-mp mpbi 2cn 2cn 2re 2pos gt0ne0i reccl projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl hvmulcl projlem11.1 hvsubcl norm-iii 2cn 2cn 2re 2pos gt0ne0i reccl projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl hvmulcl projlem11.1 hvsubdistr1 2cn 2re 2pos gt0ne0i recid (.s) (opr B (+v) C) opreq1i 2cn 2cn 2re 2pos gt0ne0i reccl projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl hvmulass projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl (opr B (+v) C) ax-hvmulid ax-mp 3eqtr3 (-v) (opr (2) (.s) A) opreq1i eqtr (norm) fveq2i 0re 2re 2pos ltlei 2re absid ax-mp (x.) (` (norm) (opr (opr (opr (1) (/) (2)) (.s) (opr B (+v) C)) (-v) A)) opreq1i 3eqtr3r breqtr 0re 2re 2pos ltlei projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem13 2re projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 mulge0 mp2an projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl 2cn projlem11.1 hvmulcl hvsubcl (opr (opr B (+v) C) (-v) (opr (2) (.s) A)) normge0t ax-mp 2re projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 remulcl projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl 2cn projlem11.1 hvmulcl hvsubcl normcl le2sq mp2an mpbi eqbrtr)) thm (projlem19 ((u v) (A u) (A v) (H u) (H v)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem19.5 (e. B H)) (projlem19.6 (e. C H)) (projlem19.7 (e. D (NN))) (projlem19.8 (e. G (NN))) (projlem19.9 (e. N (NN)))) (-> (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) (-> (/\ (br N (<_) D) (br N (<_) G)) (br (` (norm) (opr B (-v) C)) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N))))) (projlem11.1 projlem11.2 projlem19.5 cheli projlem11.2 projlem19.6 cheli projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem13 projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem19.5 projlem19.6 projlem18 projlem19.7 projlem19.8 projlem19.9 projlem7)) thm (projlem20 ((u v) (A u) (A v) (H u) (H v)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem20.5 (e. N (NN)))) (-> (/\ (/\ (e. B H) (e. C H)) (/\ (e. D (NN)) (e. G (NN)))) (-> (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) (-> (/\ (br N (<_) D) (br N (<_) G)) (br (` (norm) (opr B (-v) C)) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N)))))) (B (if (e. B H) B (0v)) (-v) A opreq1 (norm) fveq2d (<) (opr R (+) (opr (1) (/) D)) breq1d (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G))) anbi1d B (if (e. B H) B (0v)) (-v) C opreq1 (norm) fveq2d (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N)) breq1d (/\ (br N (<_) D) (br N (<_) G)) imbi2d imbi12d C (if (e. C H) C (0v)) (-v) A opreq1 (norm) fveq2d (<) (opr R (+) (opr (1) (/) G)) breq1d (br (` (norm) (opr (if (e. B H) B (0v)) (-v) A)) (<) (opr R (+) (opr (1) (/) D))) anbi2d C (if (e. C H) C (0v)) (if (e. B H) B (0v)) (-v) opreq2 (norm) fveq2d (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N)) breq1d (/\ (br N (<_) D) (br N (<_) G)) imbi2d imbi12d D (if (e. D (NN)) D (1)) (1) (/) opreq2 R (+) opreq2d (` (norm) (opr (if (e. B H) B (0v)) (-v) A)) (<) breq2d (br (` (norm) (opr (if (e. C H) C (0v)) (-v) A)) (<) (opr R (+) (opr (1) (/) G))) anbi1d D (if (e. D (NN)) D (1)) N (<_) breq2 (br N (<_) G) anbi1d (br (` (norm) (opr (if (e. B H) B (0v)) (-v) (if (e. C H) C (0v)))) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N))) imbi1d imbi12d G (if (e. G (NN)) G (1)) (1) (/) opreq2 R (+) opreq2d (` (norm) (opr (if (e. C H) C (0v)) (-v) A)) (<) breq2d (br (` (norm) (opr (if (e. B H) B (0v)) (-v) A)) (<) (opr R (+) (opr (1) (/) (if (e. D (NN)) D (1))))) anbi2d G (if (e. G (NN)) G (1)) N (<_) breq2 (br N (<_) (if (e. D (NN)) D (1))) anbi2d (br (` (norm) (opr (if (e. B H) B (0v)) (-v) (if (e. C H) C (0v)))) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N))) imbi1d imbi12d projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11.2 H ch0 ax-mp B elimel projlem11.2 H ch0 ax-mp C elimel 1nn D elimel 1nn G elimel projlem20.5 projlem19 dedth4h)) thm (projlem21 ((A w) (D w) (F w) (R w)) ((projlem21.1 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))))) (-> ph (-> (e. D (NN)) (e. (` F D) H))) (F (NN) H D ffvrn projlem21.1 pm3.26bd sylan ex)) thm (projlem22 ((A w) (D w) (F w) (R w)) ((projlem21.1 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))))) (-> ph (-> (e. D (NN)) (br (` (norm) (opr (` F D) (-v) A)) (<) (opr R (+) (opr (1) (/) D))))) ((cv w) D F fveq2 (-v) A opreq1d (norm) fveq2d (cv w) D (1) (/) opreq2 R (+) opreq2d (<) breq12d (NN) rcla4v projlem21.1 pm3.27bd (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))) pm3.27 w (NN) r19.20si syl syl5com)) thm (projlem23 ((x y) (D x) (D y) (A x) (A y) (F x) (F y)) ((projlem23.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (norm) (opr (` F (cv x)) (-v) A)))))))) (-> (e. D (NN)) (= (` G D) (` (norm) (opr (` F D) (-v) A)))) ((cv x) D F fveq2 (-v) A opreq1d (norm) fveq2d projlem23.1 (norm) (opr (` F D) (-v) A) fvex fvopab4)) thm (projlem24 ((x y) (A x) (A y) (F x) (F y)) ((projlem23.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (norm) (opr (` F (cv x)) (-v) A))))))) (projlem24.2 (e. A (H~)))) (-> (:--> F (NN) (H~)) (:--> G (NN) (CC))) (F (NN) (H~) (cv x) ffvrn projlem24.2 jctir (` F (cv x)) A hvsubclt syl (opr (` F (cv x)) (-v) A) normclt (` (norm) (opr (` F (cv x)) (-v) A)) recnt 3syl r19.21aiva projlem23.1 (CC) fopab2 sylib)) thm (projlem25 ((x y) (u v) (u w) (u x) (u y) (A u) (v w) (v x) (v y) (A v) (w x) (w y) (A w) (A x) (A y) (u z) (F u) (v z) (F v) (w z) (F w) (x z) (F x) (y z) (F y) (F z) (G u) (G v) (G z) (G w)) ((projlem23.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (norm) (opr (` F (cv x)) (-v) A))))))) (projlem25.2 (e. A (H~))) (projlem25.3 (e. F (V)))) (-> (br F (~~>v) (cv z)) (br G (~~>) (` (norm) (opr (cv z) (-v) A)))) (projlem25.3 z visset w v u hlimconv (` (abs) (opr (` (norm) (opr (` F (cv u)) (-v) A)) (-) (` (norm) (opr (cv z) (-v) A)))) (` (norm) (opr (` F (cv u)) (-v) (cv z))) (cv w) lelttrt 3expa (` (norm) (opr (` F (cv u)) (-v) A)) (` (norm) (opr (cv z) (-v) A)) resubclt F (NN) (H~) (cv u) ffvrn projlem25.3 z visset hlimseq sylan projlem25.2 (` F (cv u)) A hvsubclt mpan2 (opr (` F (cv u)) (-v) A) normclt 3syl projlem25.3 z visset hlimvec (e. (cv u) (NN)) adantr projlem25.2 (cv z) A hvsubclt mpan2 (opr (cv z) (-v) A) normclt 3syl sylanc recnd (opr (` (norm) (opr (` F (cv u)) (-v) A)) (-) (` (norm) (opr (cv z) (-v) A))) absclt syl (` F (cv u)) (cv z) hvsubclt F (NN) (H~) (cv u) ffvrn projlem25.3 z visset hlimseq sylan projlem25.3 z visset hlimvec (e. (cv u) (NN)) adantr sylanc (opr (` F (cv u)) (-v) (cv z)) normclt syl jca sylan an1rs projlem25.2 (` F (cv u)) (cv z) norm3adift F (NN) (H~) (cv u) ffvrn projlem25.3 z visset hlimseq sylan projlem25.3 z visset hlimvec (e. (cv u) (NN)) adantr sylanc (e. (cv w) (RR)) adantlr sylani anabsi5 projlem23.1 (cv u) projlem23 (-) (` (norm) (opr (cv z) (-v) A)) opreq1d (abs) fveq2d (<) (cv w) breq1d (/\ (br F (~~>v) (cv z)) (e. (cv w) (RR))) (br (` (norm) (opr (` F (cv u)) (-v) (cv z))) (<) (cv w)) ad2antlr mpbird ex (br (cv v) (<_) (cv u)) imim2d r19.20dva v (NN) r19.22sdv (br (0) (<) (cv w)) imim2d r19.20dva mpd projlem23.1 nnex x y (` (norm) (opr (` F (cv x)) (-v) A)) funopabex2 eqeltr (` (norm) (opr (cv z) (-v) A)) u w v climnn projlem25.3 z visset hlimvec projlem25.2 jctir (cv z) A hvsubclt (opr (cv z) (-v) A) normclt 3syl recnd G (NN) (CC) (cv u) ffvrn projlem25.3 z visset hlimseq projlem23.1 projlem25.2 projlem24 syl sylan r19.21aiva sylanc mpbird)) thm (projlem26 ((u w) (v w) (w x) (w y) (A w) (u v) (u x) (u y) (A u) (v x) (v y) (A v) (x y) (A x) (A y) (F w) (F x) (F y) (f w) (g w) (w z) (R w) (f g) (f z) (R f) (g z) (R g) (R z) (G f) (G g) (G z) (f x) (f ph) (g x) (g ph) (x z) (ph x) (ph z) (H u) (H v) (g y)) ((projlem26.1 (e. A (H~))) (projlem26.2 (e. H (CH))) (projlem26.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem26.4 (= R (-u (sup S (RR) (<))))) (projlem26.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w))))))))) (projlem26.6 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (norm) (opr (` F (cv x)) (-v) A)))))))) (-> ph (br G (~~>) R)) ((cv z) gt0ne0t (cv z) rerecclt syldan (opr (1) (/) (cv z)) f arch syl (cv z) (cv f) ltrec1t (e. (cv z) (RR)) (br (0) (<) (cv z)) pm3.26 (cv f) nnret anim12i (e. (cv z) (RR)) (br (0) (<) (cv z)) pm3.27 (cv f) nngt0t anim12i sylanc rexbidva mpbid ph adantl (cv f) (cv g) lerectOLD (cv f) nnret (cv g) nnret anim12i (cv f) nngt0t (cv g) nngt0t anim12i sylanc ancoms (e. (cv z) (RR)) 3adant3 (opr (1) (/) (cv g)) (opr (1) (/) (cv f)) (cv z) lelttrt exp3a (cv f) rerecclt (cv f) nnret (cv f) nnne0t sylanc syl3an2 (cv g) rerecclt (cv g) nnret (cv g) nnne0t sylanc syl3an1 sylbid com23 imp3a 3exp com13 ph (br (0) (<) (cv z)) ad2antrl imp31 (` (abs) (opr (` (norm) (opr (` F (cv g)) (-v) A)) (-) R)) (opr (1) (/) (cv g)) (cv z) axlttrn 3expa exp3a F (NN) H (cv g) ffvrn projlem26.5 pm3.26bd sylan projlem26.2 (` F (cv g)) chel syl projlem26.1 jctir (` F (cv g)) A hvsubclt (opr (` F (cv g)) (-v) A) normclt 3syl projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 jctir (` (norm) (opr (` F (cv g)) (-v) A)) R resubclt syl recnd (opr (` (norm) (opr (` F (cv g)) (-v) A)) (-) R) absclt syl (cv g) rerecclt (cv g) nnret (cv g) nnne0t sylanc ph adantl jca (e. (cv z) (RR)) (br (0) (<) (cv z)) pm3.26 anim12i (cv w) (cv g) (1) (/) opreq2 R (-) opreq2d (cv w) (cv g) F fveq2 (-v) A opreq1d (norm) fveq2d (<) breq12d (cv w) (cv g) F fveq2 (-v) A opreq1d (norm) fveq2d (cv w) (cv g) (1) (/) opreq2 R (+) opreq2d (<) breq12d anbi12d (NN) rcla4v projlem26.5 pm3.27bd syl5com imp pm3.27d projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 (` (norm) (opr (` F (cv g)) (-v) A)) R (opr (1) (/) (cv g)) ltsubadd2t mp3an2 F (NN) H (cv g) ffvrn projlem26.5 pm3.26bd sylan projlem26.2 (` F (cv g)) chel syl projlem26.1 jctir (` F (cv g)) A hvsubclt (opr (` F (cv g)) (-v) A) normclt 3syl (cv g) rerecclt (cv g) nnret (cv g) nnne0t sylanc ph adantl sylanc mpbird F (NN) H (cv g) ffvrn projlem26.5 pm3.26bd sylan projlem26.2 (` F (cv g)) chel syl projlem26.1 jctir (` F (cv g)) A hvsubclt (opr (` F (cv g)) (-v) A) normclt 3syl recnd projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 recn jctir (` (norm) (opr (` F (cv g)) (-v) A)) R negsubdi2t syl (cv w) (cv g) (1) (/) opreq2 R (-) opreq2d (cv w) (cv g) F fveq2 (-v) A opreq1d (norm) fveq2d (<) breq12d (cv w) (cv g) F fveq2 (-v) A opreq1d (norm) fveq2d (cv w) (cv g) (1) (/) opreq2 R (+) opreq2d (<) breq12d anbi12d (NN) rcla4v projlem26.5 pm3.27bd syl5com imp pm3.26d projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 R (` (norm) (opr (` F (cv g)) (-v) A)) (opr (1) (/) (cv g)) ltsub23t mp3an1 F (NN) H (cv g) ffvrn projlem26.5 pm3.26bd sylan projlem26.2 (` F (cv g)) chel syl projlem26.1 jctir (` F (cv g)) A hvsubclt (opr (` F (cv g)) (-v) A) normclt 3syl (cv g) rerecclt (cv g) nnret (cv g) nnne0t sylanc ph adantl sylanc mpbird eqbrtrd jca (opr (` (norm) (opr (` F (cv g)) (-v) A)) (-) R) (opr (1) (/) (cv g)) abslttOLD F (NN) H (cv g) ffvrn projlem26.5 pm3.26bd sylan projlem26.2 (` F (cv g)) chel syl projlem26.1 jctir (` F (cv g)) A hvsubclt (opr (` F (cv g)) (-v) A) normclt 3syl projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 jctir (` (norm) (opr (` F (cv g)) (-v) A)) R resubclt syl (cv g) rerecclt (cv g) nnret (cv g) nnne0t sylanc ph adantl sylanc mpbird (/\ (e. (cv z) (RR)) (br (0) (<) (cv z))) adantr sylc an1rs (e. (cv f) (NN)) adantlr syld projlem26.6 (cv g) projlem23 (-) R opreq1d (abs) fveq2d (<) (cv z) breq1d (/\ (/\ ph (/\ (e. (cv z) (RR)) (br (0) (<) (cv z)))) (e. (cv f) (NN))) adantl sylibrd exp4b com23 r19.21adv r19.22dva mpd exp32 r19.21aiv G (NN) (CC) (cv g) ffvrn projlem26.5 pm3.26bd projlem26.2 chssi F (NN) H (H~) fss mpan2 projlem26.6 projlem26.1 projlem24 3syl sylan r19.21aiva projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 recn projlem26.6 nnex x y (` (norm) (opr (` F (cv x)) (-v) A)) funopabex2 eqeltr R g z f climnn mpan syl mpbird)) thm (projlem27 ((u v) (u w) (A u) (v w) (A v) (A w) (H u) (H v) (D w) (G w) (F w) (R w)) ((projlem27.1 (e. A (H~))) (projlem27.2 (e. H (CH))) (projlem27.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem27.4 (= R (-u (sup S (RR) (<))))) (projlem27.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w))))))))) (projlem27.6 (e. N (NN)))) (-> (/\ ph (/\ (e. D (NN)) (e. G (NN)))) (-> (/\ (br N (<_) D) (br N (<_) G)) (br (` (norm) (opr (` F D) (-v) (` F G))) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N))))) (projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.6 (` F D) (` F G) D G projlem20 projlem27.5 D projlem21 projlem27.5 G projlem21 anim12d impac projlem27.5 D projlem22 projlem27.5 G projlem22 anim12d imp sylc)) thm (projlem28 ((u v) (u x) (u y) (u z) (u w) (A u) (v x) (v y) (v z) (v w) (A v) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H u) (H v) (H x) (H z) (D x) (D y) (D z) (D w) (F x) (F y) (F w) (F z) (R x) (R y) (R z) (R w) (ph x) (ph y) (ph z)) ((projlem27.1 (e. A (H~))) (projlem27.2 (e. H (CH))) (projlem27.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem27.4 (= R (-u (sup S (RR) (<))))) (projlem27.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w))))))))) (projlem28.6 (e. D (RR)))) (-> ph (-> (br (0) (<) D) (E.e. z (NN) (A.e. x (NN) (A.e. y (NN) (-> (/\ (br (cv z) (<_) (cv x)) (br (cv z) (<_) (cv y))) (br (` (norm) (opr (` F (cv x)) (-v) (` F (cv y)))) (<) D))))))) ((cv z) (if (e. (cv z) (NN)) (cv z) (1)) (<_) (cv x) breq1 (cv z) (if (e. (cv z) (NN)) (cv z) (1)) (<_) (cv y) breq1 anbi12d (cv z) (if (e. (cv z) (NN)) (cv z) (1)) (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) opreq2 (sqr) fveq2d (` (norm) (opr (` F (cv x)) (-v) (` F (cv y)))) (<) breq2d imbi12d (/\ ph (/\ (e. (cv x) (NN)) (e. (cv y) (NN)))) imbi2d projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.5 1nn (cv z) elimel (cv x) (cv y) projlem27 dedth exp3a com12 imp31 (br (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (cv z))) (<) D) adantlr projlem28.6 (` (norm) (opr (` F (cv x)) (-v) (` F (cv y)))) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (cv z))) D axlttrn mp3an3 (` F (cv x)) (` F (cv y)) hvsubclt projlem27.5 (cv x) projlem21 imp projlem27.2 (` F (cv x)) chel syl projlem27.5 (cv y) projlem21 imp projlem27.2 (` F (cv y)) chel syl syl2an anandis (opr (` F (cv x)) (-v) (` F (cv y))) normclt syl (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (cv z)) sqrclt (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (cv z) redivclt 3expa (cv z) nnret 4re 2re projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 remulcl ax1re readdcl remulcl jctil (cv z) nnne0t sylanc (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (cv z) divge0t (cv z) nnret 4re 2re projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 remulcl ax1re readdcl remulcl jctil (cv z) nngt0t 0re 4re 4pos ltlei 0re 2re 2pos ltlei projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem13 2re projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 mulge0 mp2an 0re ax1re lt01 ltlei 2re projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 remulcl ax1re addge0 mp2an 4re 2re projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 remulcl ax1re readdcl mulge0 mp2an jctil sylanc sylanc syl2an exp4b com4r exp3a com4t com14 imp41 syld ex x y 19.21aivv x (NN) y (NN) (-> (/\ (br (cv z) (<_) (cv x)) (br (cv z) (<_) (cv y))) (br (` (norm) (opr (` F (cv x)) (-v) (` F (cv y)))) (<) D)) r2al sylibr ex r19.22dva projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 projlem28.6 projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem13 z projlem2 syl5)) thm (projlem29 ((u v) (u x) (u y) (u z) (u w) (A u) (v x) (v y) (v z) (v w) (A v) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (g u) (H u) (g v) (H v) (g x) (g z) (H g) (H x) (H z) (g y) (g w) (F g) (F x) (F y) (F w) (F z) (R x) (R y) (R z) (R w) (g ph) (ph x) (ph y) (ph z)) ((projlem27.1 (e. A (H~))) (projlem27.2 (e. H (CH))) (projlem27.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem27.4 (= R (-u (sup S (RR) (<))))) (projlem27.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))))) (-> ph (e. F (Cauchy))) (projlem27.5 pm3.26bd projlem27.2 chssi jctir F (NN) H (H~) fss syl (cv g) (if (e. (cv g) (RR)) (cv g) (1)) (0) (<) breq2 (cv g) (if (e. (cv g) (RR)) (cv g) (1)) (` (norm) (opr (` F (cv x)) (-v) (` F (cv y)))) (<) breq2 (/\ (br (cv z) (<_) (cv x)) (br (cv z) (<_) (cv y))) imbi2d x (NN) y (NN) 2ralbidv z (NN) rexbidv imbi12d ph imbi2d projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.5 ax1re (cv g) elimel z x y projlem28 dedth com12 r19.21aiv jca F g z x y hcau sylibr)) thm (projlem30 ((u v) (u z) (u w) (A u) (v z) (v w) (A v) (w z) (A z) (A w) (H u) (H v) (H z) (F w) (F z) (R z) (R w) (ph z)) ((projlem27.1 (e. A (H~))) (projlem27.2 (e. H (CH))) (projlem27.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem27.4 (= R (-u (sup S (RR) (<))))) (projlem27.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))))) (-> ph (E.e. z H (br F (~~>v) (cv z)))) (projlem27.2 H F z chcompl ax-mp projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.5 projlem29 projlem27.5 pm3.26bd sylanc)) thm (projlem31 ((u v) (u x) (u y) (u z) (u w) (A u) (v x) (v y) (v z) (v w) (A v) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H u) (H v) (H x) (H z) (F x) (F y) (F w) (F z) (R x) (R y) (R z) (R w) (ph x) (ph y) (ph z)) ((projlem27.1 (e. A (H~))) (projlem27.2 (e. H (CH))) (projlem27.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem27.4 (= R (-u (sup S (RR) (<))))) (projlem27.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w))))))))) (projlem31.6 (e. F (V)))) (-> ph (E.e. x H (A.e. y H (br (` (norm) (opr (cv x) (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A)))))) (projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.5 x projlem30 projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 elisseti (norm) (opr (cv x) (-v) A) fvex ({<,>|} z y (/\ (e. (cv z) (NN)) (= (cv y) (` (norm) (opr (` F (cv z)) (-v) A))))) climuni projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.5 ({<,>|} z y (/\ (e. (cv z) (NN)) (= (cv y) (` (norm) (opr (` F (cv z)) (-v) A))))) eqid projlem26 ({<,>|} z y (/\ (e. (cv z) (NN)) (= (cv y) (` (norm) (opr (` F (cv z)) (-v) A))))) eqid projlem27.1 projlem31.6 x projlem25 syl2an (<_) (` (norm) (opr (cv y) (-v) A)) breq1d projlem27.1 projlem27.2 projlem27.3 projlem27.4 (cv y) projlem12 syl5bi r19.21aiv ex x H r19.22sdv mpd)) thm (projlem ((x y) (f x) (u x) (v x) (w x) (H x) (f y) (u y) (v y) (w y) (H y) (f u) (f v) (f w) (H f) (u v) (u w) (H u) (v w) (H v) (H w) (A x) (A y) (A f) (A u) (A v) (A w)) ((projlem.1 (e. A (H~))) (projlem.2 (e. H (CH)))) (E.e. x H (A.e. y H (br (` (norm) (opr (cv x) (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A))))) (projlem.1 projlem.2 ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))) eqid (-u (sup ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))) (RR) (<))) eqid f w projlem17 projlem.1 projlem.2 ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))) eqid (-u (sup ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))) (RR) (<))) eqid (/\ (:--> (cv f) (NN) H) (A.e. w (NN) (/\ (br (opr (-u (sup ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))) (RR) (<))) (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` (cv f) (cv w)) (-v) A))) (br (` (norm) (opr (` (cv f) (cv w)) (-v) A)) (<) (opr (-u (sup ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))) (RR) (<))) (+) (opr (1) (/) (cv w))))))) pm4.2 f visset x y projlem31 f 19.23aiv ax-mp)) thm (pjthlem1 () ((pjthlem1.1 (e. A (H~))) (pjthlem1.2 (e. B (H~))) (pjthlem1.3 (e. D (H~))) (pjthlem1.4 (e. S (CC))) (pjthlem1.5 (= C (opr A (-v) B)))) (<-> (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A))) (br (opr (` (norm) C) (^) (2)) (<_) (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2)))) (pjthlem1.2 pjthlem1.1 normsub pjthlem1.5 (norm) fveq2i eqtr4 pjthlem1.2 pjthlem1.4 pjthlem1.3 hvmulcl hvaddcl pjthlem1.1 normsub pjthlem1.5 (-v) (opr S (.s) D) opreq1i pjthlem1.1 pjthlem1.2 pjthlem1.4 pjthlem1.3 hvmulcl hvsubass eqtr2 (norm) fveq2i eqtr (<_) breq12i pjthlem1.5 pjthlem1.1 pjthlem1.2 hvsubcl eqeltr C normge0t ax-mp pjthlem1.5 pjthlem1.1 pjthlem1.2 hvsubcl eqeltr pjthlem1.4 pjthlem1.3 hvmulcl hvsubcl (opr C (-v) (opr S (.s) D)) normge0t ax-mp pjthlem1.5 pjthlem1.1 pjthlem1.2 hvsubcl eqeltr normcl pjthlem1.5 pjthlem1.1 pjthlem1.2 hvsubcl eqeltr pjthlem1.4 pjthlem1.3 hvmulcl hvsubcl normcl le2sq mp2an bitr)) thm (pjthlem2 () ((pjthlem2.1 (e. D (H~))) (pjthlem2.2 (= R (opr (1) (/) (opr D (.i) D))))) (-> (-. (= D (0v))) (e. R (RR))) (pjthlem2.1 D ax-his4OLD mpan pjthlem2.1 D hiidrclt ax-mp gt0ne0 pjthlem2.1 D hiidrclt ax-mp rerecclz 3syl pjthlem2.2 syl5eqel)) thm (pjthlem3 () ((pjthlem2.1 (e. D (H~))) (pjthlem2.2 (= R (opr (1) (/) (opr D (.i) D))))) (-> (-. (= D (0v))) (br (0) (<) R)) (pjthlem2.1 D ax-his4OLD mpan pjthlem2.1 D hiidrclt ax-mp recgt0 syl pjthlem2.2 syl6breqr)) thm (pjthlem4 () ((pjthlem2.1 (e. D (H~))) (pjthlem2.2 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem4.3 (e. C (H~))) (pjthlem4.4 (= S (opr R (x.) (opr C (.i) D))))) (-> (-. (= D (0v))) (e. S (CC))) (pjthlem2.1 pjthlem2.2 pjthlem2 recnd pjthlem4.3 pjthlem2.1 hicl jctir R (opr C (.i) D) axmulcl syl pjthlem4.4 syl5eqel)) thm (pjthlem5 () ((pjthlem5.1 (e. D (H~))) (pjthlem5.2 (e. C (H~))) (pjthlem5.3 (e. S (CC)))) (= (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2)) (opr (opr (opr (` (norm) C) (^) (2)) (-) (` (*) (opr S (x.) (opr D (.i) C)))) (+) (opr (opr (opr S (x.) (` (*) S)) (x.) (opr D (.i) D)) (-) (opr S (x.) (opr D (.i) C))))) (pjthlem5.2 pjthlem5.3 pjthlem5.1 hvmulcl hvsubcl normsq pjthlem5.3 pjthlem5.2 pjthlem5.1 normlem0 pjthlem5.2 normsq pjthlem5.3 pjthlem5.1 pjthlem5.2 hicl cjmul pjthlem5.2 pjthlem5.1 his1 (` (*) S) (x.) opreq2i eqtr4 negeqi pjthlem5.3 cjcl pjthlem5.2 pjthlem5.1 hicl mulneg1 eqtr4 (+) opreq12i pjthlem5.2 normcl resqcl recn pjthlem5.3 pjthlem5.1 pjthlem5.2 hicl mulcl cjcl negsub eqtr3 pjthlem5.3 pjthlem5.1 pjthlem5.2 hicl mulneg1 (+) (opr (opr S (x.) (` (*) S)) (x.) (opr D (.i) D)) opreq1i pjthlem5.3 pjthlem5.1 pjthlem5.2 hicl mulcl negcl pjthlem5.3 cjmulrcl recn pjthlem5.1 pjthlem5.1 hicl mulcl addcom pjthlem5.3 cjmulrcl recn pjthlem5.1 pjthlem5.1 hicl mulcl pjthlem5.3 pjthlem5.1 pjthlem5.2 hicl mulcl negsub 3eqtr (+) opreq12i 3eqtr)) thm (pjthlem6 () ((pjthlem6.1 (e. D (H~))) (pjthlem6.2 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem6.3 (e. C (H~))) (pjthlem6.4 (= S (opr R (x.) (opr C (.i) D))))) (-> (-. (= D (0v))) (= (opr S (x.) (opr D (.i) C)) (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))))) (pjthlem6.1 pjthlem6.2 pjthlem2 recnd pjthlem6.3 pjthlem6.1 hicl pjthlem6.1 pjthlem6.3 hicl R (opr C (.i) D) (opr D (.i) C) axmulass mp3an23 syl pjthlem6.4 (x.) (opr D (.i) C) opreq1i pjthlem6.3 pjthlem6.1 hicl absvalsq pjthlem6.1 pjthlem6.3 his1 (opr C (.i) D) (x.) opreq2i eqtr4 R (x.) opreq2i 3eqtr4g)) thm (pjthlem7 () ((pjthlem6.1 (e. D (H~))) (pjthlem6.2 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem6.3 (e. C (H~))) (pjthlem6.4 (= S (opr R (x.) (opr C (.i) D))))) (-> (-. (= D (0v))) (= (opr (opr S (x.) (` (*) S)) (x.) (opr D (.i) D)) (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))))) (pjthlem6.1 pjthlem6.2 pjthlem2 pjthlem6.4 (e. R (RR)) a1i R recnt pjthlem6.3 pjthlem6.1 hicl R (opr C (.i) D) cjmult mpan2 syl R cjret (x.) (` (*) (opr C (.i) D)) opreq1d eqtrd pjthlem6.4 (*) fveq2i syl5eq (x.) opreq12d R (opr C (.i) D) R (` (*) (opr C (.i) D)) mul4t R recnt pjthlem6.3 pjthlem6.1 hicl jctir R recnt pjthlem6.3 pjthlem6.1 hicl cjcl jctir sylanc eqtrd (x.) (opr D (.i) D) opreq1d R R axmulcl R recnt R recnt sylanc pjthlem6.3 pjthlem6.1 hicl pjthlem6.3 pjthlem6.1 hicl cjcl mulcl pjthlem6.1 pjthlem6.1 hicl (opr R (x.) R) (opr (opr C (.i) D) (x.) (` (*) (opr C (.i) D))) (opr D (.i) D) mul23t mp3an23 syl eqtrd syl pjthlem6.1 pjthlem6.2 pjthlem2 pjthlem6.1 pjthlem6.1 hicl R R (opr D (.i) D) mul23t mp3an3 R recnt R recnt sylanc syl pjthlem6.1 D ax-his4OLD mpan pjthlem6.1 D hiidrclt ax-mp gt0ne0 syl pjthlem6.1 pjthlem6.1 hicl recclz pjthlem6.1 pjthlem6.1 hicl (opr (1) (/) (opr D (.i) D)) (opr D (.i) D) axmulcom mpan2 3syl pjthlem6.1 D ax-his4OLD mpan pjthlem6.1 D hiidrclt ax-mp gt0ne0 pjthlem6.1 pjthlem6.1 hicl recidz 3syl eqtrd pjthlem6.2 (x.) (opr D (.i) D) opreq1i syl5eq (x.) R opreq1d pjthlem6.1 pjthlem6.2 pjthlem2 R recnt R mulid2t 3syl 3eqtrd (x.) (opr (opr C (.i) D) (x.) (` (*) (opr C (.i) D))) opreq1d eqtrd pjthlem6.3 pjthlem6.1 hicl absvalsq R (x.) opreq2i syl6eqr)) thm (pjthlem8 () ((pjthlem6.1 (e. D (H~))) (pjthlem6.2 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem6.3 (e. C (H~))) (pjthlem6.4 (= S (opr R (x.) (opr C (.i) D))))) (-> (-. (= D (0v))) (= (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2)) (opr (opr (` (norm) C) (^) (2)) (-) (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2)))))) (pjthlem6.1 pjthlem6.2 pjthlem6.3 pjthlem6.4 pjthlem4 S (if (e. S (CC)) S (0)) (.s) D opreq1 C (-v) opreq2d (norm) fveq2d (^) (2) opreq1d S (if (e. S (CC)) S (0)) (x.) (opr D (.i) C) opreq1 (*) fveq2d (opr (` (norm) C) (^) (2)) (-) opreq2d (= S (if (e. S (CC)) S (0))) id S (if (e. S (CC)) S (0)) (*) fveq2 (x.) opreq12d (x.) (opr D (.i) D) opreq1d S (if (e. S (CC)) S (0)) (x.) (opr D (.i) C) opreq1 (-) opreq12d (+) opreq12d eqeq12d pjthlem6.1 pjthlem6.3 0cn S elimel pjthlem5 dedth syl pjthlem6.1 pjthlem6.2 pjthlem6.3 pjthlem6.4 pjthlem6 (*) fveq2d pjthlem6.1 pjthlem6.2 pjthlem2 pjthlem6.3 pjthlem6.1 hicl abscl resqcl R (opr (` (abs) (opr C (.i) D)) (^) (2)) axmulrcl mpan2 (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))) cjret 3syl eqtrd (opr (` (norm) C) (^) (2)) (-) opreq2d pjthlem6.1 pjthlem6.2 pjthlem6.3 pjthlem6.4 pjthlem7 pjthlem6.1 pjthlem6.2 pjthlem6.3 pjthlem6.4 pjthlem6 (-) opreq12d pjthlem6.1 pjthlem6.2 pjthlem2 pjthlem6.3 pjthlem6.1 hicl abscl resqcl R (opr (` (abs) (opr C (.i) D)) (^) (2)) axmulrcl mpan2 syl recnd (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))) subidt syl eqtrd (+) opreq12d pjthlem6.1 pjthlem6.2 pjthlem2 pjthlem6.3 pjthlem6.1 hicl abscl resqcl R (opr (` (abs) (opr C (.i) D)) (^) (2)) axmulrcl mpan2 syl recnd pjthlem6.3 normcl resqcl recn (opr (` (norm) C) (^) (2)) (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))) subclt mpan (opr (opr (` (norm) C) (^) (2)) (-) (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2)))) ax0id 3syl 3eqtrd)) thm (pjthlem9 () ((pjthlem9.1 (e. A (H~))) (pjthlem9.2 (e. B (H~))) (pjthlem9.3 (e. D (H~))) (pjthlem9.4 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem9.5 (= S (opr R (x.) (opr C (.i) D)))) (pjthlem9.6 (= C (opr A (-v) B)))) (-> (-. (= D (0v))) (<-> (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A))) (br (opr (` (norm) C) (^) (2)) (<_) (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2))))) (pjthlem9.3 pjthlem9.4 pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.5 pjthlem4 S (if (e. S (CC)) S (0)) (.s) D opreq1 B (+v) opreq2d (-v) A opreq1d (norm) fveq2d (` (norm) (opr B (-v) A)) (<_) breq2d S (if (e. S (CC)) S (0)) (.s) D opreq1 C (-v) opreq2d (norm) fveq2d (^) (2) opreq1d (opr (` (norm) C) (^) (2)) (<_) breq2d bibi12d pjthlem9.1 pjthlem9.2 pjthlem9.3 0cn S elimel pjthlem9.6 pjthlem1 dedth syl)) thm (pjthlem10 () ((pjthlem9.1 (e. A (H~))) (pjthlem9.2 (e. B (H~))) (pjthlem9.3 (e. D (H~))) (pjthlem9.4 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem9.5 (= S (opr R (x.) (opr C (.i) D)))) (pjthlem9.6 (= C (opr A (-v) B)))) (-> (/\ (-. (= D (0v))) (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A)))) (= (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))) (0))) (R (opr (` (abs) (opr C (.i) D)) (^) (2)) mulge0t pjthlem9.3 pjthlem9.4 pjthlem2 pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abscl resqcl jctir (0) R ltlet pjthlem9.3 pjthlem9.4 pjthlem2 0re jctil pjthlem9.3 pjthlem9.4 pjthlem3 sylc pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abscl sqge0 jctir sylanc (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A))) adantr pjthlem9.1 pjthlem9.2 pjthlem9.3 pjthlem9.4 pjthlem9.5 pjthlem9.6 pjthlem9 biimpa pjthlem9.3 pjthlem9.4 pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.5 pjthlem8 (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A))) adantr breqtrd jca pjthlem9.3 pjthlem9.4 pjthlem2 pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abscl resqcl jctir R (opr (` (abs) (opr C (.i) D)) (^) (2)) axmulrcl syl pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr normcl resqcl jctir (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))) (opr (` (norm) C) (^) (2)) lesub0t syl (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A))) adantr mpbid)) thm (pjthlem11 () ((pjthlem9.1 (e. A (H~))) (pjthlem9.2 (e. B (H~))) (pjthlem9.3 (e. D (H~))) (pjthlem9.4 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem9.5 (= S (opr R (x.) (opr C (.i) D)))) (pjthlem9.6 (= C (opr A (-v) B)))) (-> (/\ (-. (= D (0v))) (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A)))) (= (opr C (.i) D) (0))) (pjthlem9.3 pjthlem9.4 pjthlem3 pjthlem9.3 pjthlem9.4 pjthlem2 0re jctir R (0) lttri2t syl (br (0) (<) R) (br R (<) (0)) olc syl5bir mpd (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A))) adantr pjthlem9.1 pjthlem9.2 pjthlem9.3 pjthlem9.4 pjthlem9.5 pjthlem9.6 pjthlem10 pjthlem9.3 pjthlem9.4 pjthlem2 recnd pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abscl resqcl recn R (opr (` (abs) (opr C (.i) D)) (^) (2)) mul0ort mpan2 syl (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A))) adantr mpbid ord mpd pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abscl recn sq00 sylib pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abs00 sylib)) thm (pjthlem12 ((A y) (B y) (D y) (S y) (H y)) ((pjthlem12.1 (e. A (H~))) (pjthlem12.2 (e. H (CH))) (pjthlem12.3 (e. B H)) (pjthlem12.4 (e. D H)) (pjthlem12.5 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem12.6 (= S (opr R (x.) (opr C (.i) D)))) (pjthlem12.7 (= C (opr A (-v) B)))) (-> (A.e. y H (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A)))) (-> (-. (= D (0v))) (= (opr C (.i) D) (0)))) ((cv y) (opr B (+v) (opr S (.s) D)) (-v) A opreq1 (norm) fveq2d (` (norm) (opr B (-v) A)) (<_) breq2d H rcla4cv pjthlem12.2 pjthlem12.4 cheli pjthlem12.5 pjthlem12.7 pjthlem12.1 pjthlem12.2 pjthlem12.3 cheli hvsubcl eqeltr pjthlem12.6 pjthlem4 pjthlem12.4 jctir pjthlem12.2 chshi H S D shmulclt ax-mp syl pjthlem12.3 jctil pjthlem12.2 chshi H B (opr S (.s) D) shaddclt ax-mp syl syl5 ancld pjthlem12.1 pjthlem12.2 pjthlem12.3 cheli pjthlem12.2 pjthlem12.4 cheli pjthlem12.5 pjthlem12.6 pjthlem12.7 pjthlem11 syl6)) thm (pjthlem13 ((A y) (B y) (D y) (C y) (H y)) ((pjthlem13.1 (e. A (H~))) (pjthlem13.2 (e. H (CH))) (pjthlem13.3 (e. B H)) (pjthlem13.4 (e. D H)) (pjthlem13.5 (= C (opr A (-v) B)))) (-> (A.e. y H (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A)))) (= (opr C (.i) D) (0))) (pjthlem13.1 pjthlem13.2 pjthlem13.3 pjthlem13.4 (opr (1) (/) (opr D (.i) D)) eqid (opr (opr (1) (/) (opr D (.i) D)) (x.) (opr C (.i) D)) eqid pjthlem13.5 y pjthlem12 D (0v) C (.i) opreq2 pjthlem13.5 pjthlem13.1 pjthlem13.2 pjthlem13.3 cheli hvsubcl eqeltr ax-hv0cl his1 pjthlem13.5 pjthlem13.1 pjthlem13.2 pjthlem13.3 cheli hvsubcl eqeltr C hi01t ax-mp (*) fveq2i cj0 3eqtr syl6eq pm2.61d2)) thm (pjthlem14 ((x z) (y z) (A z) (x y) (A x) (A y) (B z) (B x) (B y) (C z) (C x) (C y) (H z) (H x) (H y)) ((pjthlem14.1 (e. A (H~))) (pjthlem14.2 (e. H (CH))) (pjthlem14.3 (e. B H)) (pjthlem14.4 (= C (opr A (-v) B)))) (-> (A.e. z H (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (cv z) (-v) A)))) (E.e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))) ((cv x) (if (e. (cv x) H) (cv x) (0v)) C (.i) opreq2 (0) eqeq1d (A.e. z H (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (cv z) (-v) A)))) imbi2d pjthlem14.1 pjthlem14.2 pjthlem14.3 pjthlem14.2 H ch0 ax-mp (cv x) elimel pjthlem14.4 z pjthlem13 dedth com12 r19.21aiv pjthlem14.2 chshi H C x shocelt ax-mp pjthlem14.4 pjthlem14.1 pjthlem14.2 pjthlem14.3 cheli hvsubcl eqeltr mpbiran sylibr pjthlem14.3 jctil pjthlem14.1 pjthlem14.2 pjthlem14.3 cheli 1cn negcl pjthlem14.2 pjthlem14.3 cheli hvmulcl hvadd12 pjthlem14.2 pjthlem14.3 cheli hvnegid A (+v) opreq2i pjthlem14.1 A ax-hvaddid ax-mp eqtr2 pjthlem14.1 pjthlem14.2 pjthlem14.3 cheli hvsubval B (+v) opreq2i 3eqtr4 pjthlem14.4 B (+v) opreq2i eqtr4 jctir (cv x) B (+v) (cv y) opreq1 A eqeq2d (cv y) C B (+v) opreq2 A eqeq2d H (` (_|_) H) rcla42ev syl)) thm (pjth ((w z) (x z) (y z) (A z) (w x) (w y) (A w) (x y) (A x) (A y) (H z) (H w) (H x) (H y)) ((pjth.1 (e. A (H~))) (pjth.2 (e. H (CH)))) (E.e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y))))) (pjth.1 pjth.2 z w projlem (cv z) (if (e. (cv z) H) (cv z) (0v)) (-v) A opreq1 (norm) fveq2d (<_) (` (norm) (opr (cv w) (-v) A)) breq1d w H ralbidv (E.e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y))))) imbi1d pjth.1 pjth.2 pjth.2 H ch0 ax-mp (cv z) elimel (opr A (-v) (if (e. (cv z) H) (cv z) (0v))) eqid w x y pjthlem14 dedth r19.23aiv ax-mp)) thm (pjtht ((x y) (A x) (A y) (H x) (H y)) () (-> (/\ (e. H (CH)) (e. A (H~))) (E.e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))) (H (if (e. H (CH)) H (H~)) (_|_) fveq2 (` (_|_) H) (` (_|_) (if (e. H (CH)) H (H~))) y (= A (opr (cv x) (+v) (cv y))) rexeq1 syl x rexeqd A (if (e. A (H~)) A (0v)) (opr (cv x) (+v) (cv y)) eqeq1 x (if (e. H (CH)) H (H~)) y (` (_|_) (if (e. H (CH)) H (H~))) 2rexbidv ax-hv0cl A elimel helch H elimel x y pjth dedth2h)) thm (pjthu ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H x) (H y) (H z) (H w)) ((pjthu.1 (e. H (CH)))) (-> (e. A (H~)) (E!e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))) (pjthu.1 H A x y pjtht mpan pjthu.1 (cv x) (cv y) (cv z) (cv w) A chocuni an4s (= (cv x) (cv z)) (= (cv y) (cv w)) pm3.26 syl6 ex imp3a (e. A (H~)) adantl y w 19.23advv (cv y) (cv w) (cv z) (+v) opreq2 A eqeq2d (` (_|_) H) cbvrexv (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))) anbi2i y (` (_|_) H) w (` (_|_) H) (= A (opr (cv x) (+v) (cv y))) (= A (opr (cv z) (+v) (cv w))) reeanv y (` (_|_) H) w (` (_|_) H) (/\ (= A (opr (cv x) (+v) (cv y))) (= A (opr (cv z) (+v) (cv w)))) r2ex 3bitr2 syl5ib exp32 r19.21adv r19.21aiv jca (cv x) (cv z) (+v) (cv y) opreq1 A eqeq2d y (` (_|_) H) rexbidv H reu4 sylibr)) thm (pjthu2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H x) (H y) (H z) (H w)) ((pjthu2.1 (e. H (CH)))) (-> (e. A (H~)) (E!e. y (` (_|_) H) (E.e. x H (= A (opr (cv x) (+v) (cv y)))))) (pjthu2.1 H A x y pjtht mpan y (` (_|_) H) x H (= A (opr (cv x) (+v) (cv y))) rexcom sylibr pjthu2.1 (cv x) (cv y) (cv z) (cv w) A chocuni an4s (= (cv x) (cv z)) (= (cv y) (cv w)) pm3.27 syl6 expcom imp3a (e. A (H~)) adantl x z 19.23advv (cv x) (cv z) (+v) (cv w) opreq1 A eqeq2d H cbvrexv (E.e. x H (= A (opr (cv x) (+v) (cv y)))) anbi2i x H z H (= A (opr (cv x) (+v) (cv y))) (= A (opr (cv z) (+v) (cv w))) reeanv x H z H (/\ (= A (opr (cv x) (+v) (cv y))) (= A (opr (cv z) (+v) (cv w)))) r2ex 3bitr2 syl5ib exp32 r19.21adv r19.21aiv jca (cv y) (cv w) (cv x) (+v) opreq2 A eqeq2d x H rexbidv (` (_|_) H) reu4 sylibr)) thm (pjthut ((x y) (A x) (A y) (H x) (H y)) () (-> (/\ (e. H (CH)) (e. A (H~))) (E!e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))) (H (if (e. H (CH)) H (H~)) (cv x) eleq2 H (if (e. H (CH)) H (H~)) (_|_) fveq2 (` (_|_) H) (` (_|_) (if (e. H (CH)) H (H~))) y (= A (opr (cv x) (+v) (cv y))) rexeq1 syl anbi12d x eubidv x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))) df-reu x (if (e. H (CH)) H (H~)) (E.e. y (` (_|_) (if (e. H (CH)) H (H~))) (= A (opr (cv x) (+v) (cv y)))) df-reu 3bitr4g (e. A (H~)) imbi2d helch H elimel A x y pjthu dedth imp)) thm (pjmvalt ((x y) (x z) (w x) (f x) (h x) (H x) (y z) (w y) (f y) (h y) (H y) (w z) (f z) (h z) (H z) (f w) (h w) (H w) (f h) (H f) (H h)) () (-> (e. H (CH)) (= (` (proj) H) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (U. ({e.|} z H (E.e. w (` (_|_) H) (= (cv x) (opr (cv z) (+v) (cv w))))))))))) ((cv h) H (cv z) eleq2 (cv h) H (_|_) fveq2 (` (_|_) (cv h)) (` (_|_) H) w (= (cv x) (opr (cv z) (+v) (cv w))) rexeq1 syl anbi12d z abbidv z (cv h) (E.e. w (` (_|_) (cv h)) (= (cv x) (opr (cv z) (+v) (cv w)))) df-rab z H (E.e. w (` (_|_) H) (= (cv x) (opr (cv z) (+v) (cv w)))) df-rab 3eqtr4g unieqd (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv h f x y z w df-pj ax-hilex x y (U. ({e.|} z H (E.e. w (` (_|_) H) (= (cv x) (opr (cv z) (+v) (cv w)))))) funopabex2 fvopab4)) thm (pjvalt ((x y) (x z) (w x) (H x) (y z) (w y) (H y) (w z) (H z) (H w) (A x) (A y) (A z) (A w)) () (-> (/\ (e. H (CH)) (e. A (H~))) (= (` (` (proj) H) A) (U. ({e.|} x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))))) (H z w x y pjmvalt A fveq1d (e. A (H~)) adantr (cv z) A (opr (cv x) (+v) (cv y)) eqeq1 y (` (_|_) H) rexbidv x H rabbisdv unieqd ({<,>|} z w (/\ (e. (cv z) (H~)) (= (cv w) (U. ({e.|} x H (E.e. y (` (_|_) H) (= (cv z) (opr (cv x) (+v) (cv y))))))))) eqid (V) fvopab4g H (CH) x (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))) rabexg ({e.|} x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y))))) (V) uniexg syl sylan2 ancoms eqtrd)) thm (axpjclt ((x y) (H x) (H y) (A x) (A y)) () (-> (/\ (e. H (CH)) (e. A (H~))) (e. (` (` (proj) H) A) H)) (H A x y pjvalt H A x y pjthut x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))) reucl syl eqeltrd)) thm (pjhclt () () (-> (/\ (e. H (CH)) (e. A (H~))) (e. (` (` (proj) H) A) (H~))) (H chss (e. A (H~)) adantr H A axpjclt sseldd)) thm (omlsilem () ((omlsilem.1 (e. G (SH))) (omlsilem.2 (e. H (SH))) (omlsilem.3 (C_ G H)) (omlsilem.4 (= (i^i H (` (_|_) G)) (0H))) (omlsilem.5 (e. A H)) (omlsilem.6 (e. B G)) (omlsilem.7 (e. C (` (_|_) G)))) (-> (= A (opr B (+v) C)) (e. A G)) (omlsilem.2 omlsilem.5 sheli omlsilem.1 omlsilem.6 sheli omlsilem.1 G shocss ax-mp omlsilem.7 sselii hvsubadd (opr B (+v) C) A eqcom bitr omlsilem.5 omlsilem.3 omlsilem.6 sselii omlsilem.2 H A B shsubclt ax-mp mp2an (opr A (-v) B) C H eleq1 mpbii sylbir omlsilem.7 omlsilem.4 C eleq2i C H (` (_|_) G) elin C elch0 3bitr3 biimp mpan2 syl B (+v) opreq2d omlsilem.1 omlsilem.6 sheli B ax-hvaddid ax-mp syl6eq omlsilem.6 syl6eqel A (opr B (+v) C) G eleq1 mpbird)) thm (omlsi ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((omlsi.1 (e. A (CH))) (omlsi.2 (e. B (SH))) (omlsi.3 (C_ A B)) (omlsi.4 (= (i^i B (` (_|_) A)) (0H)))) (= A B) (omlsi.3 omlsi.2 (cv x) shel omlsi.1 A (cv x) y z pjtht mpan syl (cv x) (if (e. (cv x) B) (cv x) (0v)) (opr (cv y) (+v) (cv z)) eqeq1 (cv x) (if (e. (cv x) B) (cv x) (0v)) A eleq1 imbi12d (cv y) (if (e. (cv y) A) (cv y) (0v)) (+v) (cv z) opreq1 (if (e. (cv x) B) (cv x) (0v)) eqeq2d (e. (if (e. (cv x) B) (cv x) (0v)) A) imbi1d (cv z) (if (e. (cv z) (` (_|_) A)) (cv z) (0v)) (if (e. (cv y) A) (cv y) (0v)) (+v) opreq2 (if (e. (cv x) B) (cv x) (0v)) eqeq2d (e. (if (e. (cv x) B) (cv x) (0v)) A) imbi1d omlsi.1 chshi omlsi.2 omlsi.3 omlsi.4 omlsi.2 B sh0 ax-mp (cv x) elimel omlsi.1 A ch0 ax-mp (cv y) elimel omlsi.1 chshi A shocsh ax-mp (` (_|_) A) sh0 ax-mp (cv z) elimel omlsilem dedth3h 3exp imp r19.23adv ex r19.23adv mpd ssriv eqssi)) thm (omls () ((omls.1 (e. A (CH))) (omls.2 (e. B (SH)))) (-> (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) (= A B)) (A (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) B eqeq1 B (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) B (0H)) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) eqeq2 omls.1 h0elch (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) keepel omls.2 h0elsh (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) keepel A (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) B sseq1 A (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) (_|_) fveq2 B ineq2d (0H) eqeq1d anbi12d B (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) B (0H)) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) sseq2 B (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) B (0H)) (` (_|_) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H))) ineq1 (0H) eqeq1d anbi12d (0H) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) (0H) sseq1 (0H) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) (_|_) fveq2 (0H) ineq2d (0H) eqeq1d anbi12d (0H) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) B (0H)) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) sseq2 (0H) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) B (0H)) (` (_|_) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H))) ineq1 (0H) eqeq1d anbi12d (0H) ssid h0elsh (0H) ocin ax-mp pm3.2i elimhyp2v pm3.26i A (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) B sseq1 A (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) (_|_) fveq2 B ineq2d (0H) eqeq1d anbi12d B (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) B (0H)) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) sseq2 B (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) B (0H)) (` (_|_) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H))) ineq1 (0H) eqeq1d anbi12d (0H) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) (0H) sseq1 (0H) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) (_|_) fveq2 (0H) ineq2d (0H) eqeq1d anbi12d (0H) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) B (0H)) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H)) sseq2 (0H) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) B (0H)) (` (_|_) (if (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) A (0H))) ineq1 (0H) eqeq1d anbi12d (0H) ssid h0elsh (0H) ocin ax-mp pm3.2i elimhyp2v pm3.27i omlsi dedth2v)) thm (ococ () ((ococ.1 (e. A (CH)))) (= (` (_|_) (` (_|_) A)) A) (ococ.1 ococ.1 chshi A shocsh ax-mp (` (_|_) A) shocsh ax-mp ococ.1 chshi A shococss ax-mp (` (_|_) (` (_|_) A)) (` (_|_) A) incom ococ.1 chshi A shocsh ax-mp (` (_|_) A) ocin ax-mp eqtr omlsi eqcomi)) thm (ococt () () (-> (e. A (CH)) (= (` (_|_) (` (_|_) A)) A)) (A (if (e. A (CH)) A (H~)) (_|_) fveq2 (_|_) fveq2d (= A (if (e. A (CH)) A (H~))) id eqeq12d helch A elimel ococ dedth)) thm (dfch2 () () (= (CH) ({|} x (/\ (C_ (cv x) (H~)) (= (` (_|_) (` (_|_) (cv x))) (cv x))))) ((cv x) chss (cv x) ococt jca (` (_|_) (` (_|_) (cv x))) (cv x) (CH) eleq1 (cv x) occlt (` (_|_) (cv x)) chss (` (_|_) (cv x)) occlt 3syl syl5bi impcom impbi abbi2i)) thm (pjcl () ((pjcl.1 (e. H (CH)))) (-> (e. A (H~)) (e. (` (` (proj) H) A) H)) (pjcl.1 H A axpjclt mpan)) thm (pjhcl () ((pjcl.1 (e. H (CH)))) (-> (e. A (H~)) (e. (` (` (proj) H) A) (H~))) (pjcl.1 H A pjhclt mpan)) thm (pjcli () ((pjcli.1 (e. H (CH))) (pjcli.2 (e. A (H~)))) (e. (` (` (proj) H) A) H) (pjcli.2 pjcli.1 A pjcl ax-mp)) thm (pjhcli () ((pjcli.1 (e. H (CH))) (pjcli.2 (e. A (H~)))) (e. (` (` (proj) H) A) (H~)) (pjcli.2 pjcli.1 A pjhcl ax-mp)) thm (pjpj0 ((x y) (A x) (A y) (H x) (H y)) ((pjcli.1 (e. H (CH))) (pjcli.2 (e. A (H~)))) (= A (opr (` (` (proj) H) A) (+v) (` (` (proj) (` (_|_) H)) A))) (x H y (` (_|_) H) (/\ (= A (opr (cv x) (+v) (` (` (proj) (` (_|_) H)) A))) (= A (opr (` (` (proj) H) A) (+v) (cv y)))) r2ex x H y (` (_|_) H) (= A (opr (cv x) (+v) (` (` (proj) (` (_|_) H)) A))) (= A (opr (` (` (proj) H) A) (+v) (cv y))) reeanv bitr3 pjcli.1 choccl pjcli.2 (` (_|_) H) A y x pjvalt mp2an eqcomi pjcli.1 choccl pjcli.2 pjcli pjcli.2 pjcli.1 choccl A y x pjthu ax-mp (cv y) (` (` (proj) (` (_|_) H)) A) (+v) (cv x) opreq1 A eqeq2d x (` (_|_) (` (_|_) H)) rexbidv (` (_|_) H) reuuni2 mp2an mpbir pjcli.1 ococ (` (_|_) (` (_|_) H)) H x (= A (opr (cv x) (+v) (` (` (proj) (` (_|_) H)) A))) rexeq1 ax-mp pjcli.1 choccl choccl (cv x) chel pjcli.1 choccl pjcli.2 pjhcli (cv x) (` (` (proj) (` (_|_) H)) A) ax-hvcom mpan2 A eqeq2d syl rexbiia bitr3 mpbir pjcli.1 pjcli.2 H A x y pjvalt mp2an eqcomi pjcli.1 pjcli.2 pjcli pjcli.2 pjcli.1 A x y pjthu ax-mp (cv x) (` (` (proj) H) A) (+v) (cv y) opreq1 A eqeq2d y (` (_|_) H) rexbidv H reuuni2 mp2an mpbir mpbir2an pjcli.1 (cv x) (` (` (proj) (` (_|_) H)) A) (` (` (proj) H) A) (cv y) A chocuni pjcli.1 choccl pjcli.2 pjcli (e. (cv x) H) jctr pjcli.1 pjcli.2 pjcli (e. (cv y) (` (_|_) H)) jctl syl2an imp pm3.26d (cv x) (` (` (proj) H) A) (+v) (` (` (proj) (` (_|_) H)) A) opreq1 A eqeq2d biimpcd (/\ (e. (cv x) H) (e. (cv y) (` (_|_) H))) (= A (opr (` (` (proj) H) A) (+v) (cv y))) ad2antrl mpd x y 19.23aivv ax-mp)) thm (axpjpjt () () (-> (/\ (e. H (CH)) (e. A (H~))) (= A (opr (` (` (proj) H) A) (+v) (` (` (proj) (` (_|_) H)) A)))) (H (if (e. H (CH)) H (H~)) (proj) fveq2 A fveq1d H (if (e. H (CH)) H (H~)) (_|_) fveq2 (proj) fveq2d A fveq1d (+v) opreq12d A eqeq2d (= A (if (e. A (H~)) A (0v))) id A (if (e. A (H~)) A (0v)) (` (proj) (if (e. H (CH)) H (H~))) fveq2 A (if (e. A (H~)) A (0v)) (` (proj) (` (_|_) (if (e. H (CH)) H (H~)))) fveq2 (+v) opreq12d eqeq12d helch H elimel ax-hv0cl A elimel pjpj0 dedth2h)) thm (pjpj () ((pjpj.1 (e. H (CH))) (pjpj.2 (e. A (H~)))) (= A (opr (` (` (proj) H) A) (+v) (` (` (proj) (` (_|_) H)) A))) (pjpj.1 pjpj.2 pjpj0)) thm (pjpjtht ((x y) (A x) (A y) (H x) (H y)) () (-> (/\ (e. H (CH)) (e. A (H~))) (E.e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))) ((cv x) (` (` (proj) H) A) (+v) (cv y) opreq1 A eqeq2d (cv y) (` (` (proj) (` (_|_) H)) A) (` (` (proj) H) A) (+v) opreq2 A eqeq2d H (` (_|_) H) rcla42ev H A axpjclt (` (_|_) H) A axpjclt H chocclt sylan jca H A axpjpjt sylanc)) thm (pjpjth ((x y) (A x) (A y) (H x) (H y)) ((pjpjth.1 (e. A (H~))) (pjpjth.2 (e. H (CH)))) (E.e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y))))) (pjpjth.2 pjpjth.1 H A x y pjpjtht mp2an)) thm (pjopt () () (-> (/\ (e. H (CH)) (e. A (H~))) (= (` (` (proj) (` (_|_) H)) A) (opr A (-v) (` (` (proj) H) A)))) (H A axpjpjt eqcomd A (` (` (proj) H) A) (` (` (proj) (` (_|_) H)) A) hvsubaddt (e. H (CH)) (e. A (H~)) pm3.27 H A pjhclt (` (_|_) H) A pjhclt H chocclt sylan syl3anc mpbird eqcomd)) thm (pjpot () () (-> (/\ (e. H (CH)) (e. A (H~))) (= (` (` (proj) H) A) (opr A (-v) (` (` (proj) (` (_|_) H)) A)))) ((` (` (proj) (` (_|_) H)) A) (` (` (proj) H) A) ax-hvcom (` (_|_) H) A pjhclt H chocclt sylan H A pjhclt sylanc H A axpjpjt eqtr4d A (` (` (proj) (` (_|_) H)) A) (` (` (proj) H) A) hvsubaddt (e. H (CH)) (e. A (H~)) pm3.27 (` (_|_) H) A pjhclt H chocclt sylan H A pjhclt syl3anc mpbird eqcomd)) thm (pjop () ((pjop.1 (e. H (CH))) (pjop.2 (e. A (H~)))) (= (` (` (proj) (` (_|_) H)) A) (opr A (-v) (` (` (proj) H) A))) (pjop.1 pjop.2 H A pjopt mp2an)) thm (pjpo () ((pjop.1 (e. H (CH))) (pjop.2 (e. A (H~)))) (= (` (` (proj) H) A) (opr A (-v) (` (` (proj) (` (_|_) H)) A))) (pjop.1 pjop.2 H A pjpot mp2an)) thm (pjoc1 () ((pjop.1 (e. H (CH))) (pjop.2 (e. A (H~)))) (<-> (e. A H) (= (` (` (proj) (` (_|_) H)) A) (0v))) (pjop.1 pjop.2 pjcli pjop.1 chshi H A (` (` (proj) H) A) shsubclt ax-mp mpan2 pjop.1 pjop.2 pjop syl5eqel pjop.1 choccl pjop.2 pjcli jctir (` (` (proj) (` (_|_) H)) A) H (` (_|_) H) elin sylibr pjop.1 chshi H ocin ax-mp syl6eleq (` (` (proj) (` (_|_) H)) A) elch0 sylib (` (` (proj) (` (_|_) H)) A) (0v) (` (` (proj) H) A) (+v) opreq2 pjop.1 pjop.2 pjpj syl5eq pjop.1 pjop.2 pjhcli (` (` (proj) H) A) ax-hvaddid ax-mp syl6eq pjop.1 pjop.2 pjcli syl6eqel impbi)) thm (pjch () ((pjop.1 (e. H (CH))) (pjop.2 (e. A (H~)))) (<-> (e. A H) (= (` (` (proj) H) A) A)) (pjop.1 pjop.2 pjoc1 biimp (` (` (proj) H) A) (+v) opreq2d pjop.1 pjop.2 pjpj syl5req pjop.1 pjop.2 pjhcli (` (` (proj) H) A) ax-hvaddid ax-mp syl5eqr pjop.1 pjop.2 pjcli (` (` (proj) H) A) A H eleq1 mpbii impbi)) thm (pjocclt () () (-> (/\ (e. H (CH)) (e. A (H~))) (e. (opr A (-v) (` (` (proj) H) A)) (` (_|_) H))) (H A pjopt (` (_|_) H) A axpjclt H chocclt sylan eqeltrrd)) thm (pjoc1t () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (e. A H) (= (` (` (proj) (` (_|_) H)) A) (0v)))) (H (if (e. H (CH)) H (H~)) A eleq2 H (if (e. H (CH)) H (H~)) (_|_) fveq2 (proj) fveq2d A fveq1d (0v) eqeq1d bibi12d A (if (e. A (H~)) A (0v)) (if (e. H (CH)) H (H~)) eleq1 A (if (e. A (H~)) A (0v)) (` (proj) (` (_|_) (if (e. H (CH)) H (H~)))) fveq2 (0v) eqeq1d bibi12d helch H elimel ax-hv0cl A elimel pjoc1 dedth2h)) thm (pjoml () ((pjoml.1 (e. A (CH))) (pjoml.2 (e. B (SH)))) (-> (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) (= A B)) (pjoml.1 pjoml.2 omls)) thm (pjomlt () () (-> (/\ (/\ (e. A (CH)) (e. B (SH))) (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H)))) (= A B)) (A (if (e. A (CH)) A (0H)) B sseq1 A (if (e. A (CH)) A (0H)) (_|_) fveq2 B ineq2d (0H) eqeq1d anbi12d A (if (e. A (CH)) A (0H)) B eqeq1 imbi12d B (if (e. B (SH)) B (0H)) (if (e. A (CH)) A (0H)) sseq2 B (if (e. B (SH)) B (0H)) (` (_|_) (if (e. A (CH)) A (0H))) ineq1 (0H) eqeq1d anbi12d B (if (e. B (SH)) B (0H)) (if (e. A (CH)) A (0H)) eqeq2 imbi12d h0elch A elimel h0elsh B elimel pjoml dedth2h imp)) thm (pjococ () ((pjococ.1 (e. H (CH)))) (= (` (_|_) (` (_|_) H)) H) (pjococ.1 ococ)) thm (pjoc2 () ((pjoc2.1 (e. H (CH))) (pjoc2.2 (e. A (H~)))) (<-> (e. A (` (_|_) H)) (= (` (` (proj) H) A) (0v))) (pjoc2.1 choccl pjoc2.2 pjoc1 pjoc2.1 pjococ (proj) fveq2i A fveq1i (0v) eqeq1i bitr)) thm (pjoc2tOLD () ((pjoc2.1 (e. H (CH)))) (-> (e. A (H~)) (<-> (e. A (` (_|_) H)) (= (` (` (proj) H) A) (0v)))) (A (if (e. A (H~)) A (0v)) (` (_|_) H) eleq1 A (if (e. A (H~)) A (0v)) (` (proj) H) fveq2 (0v) eqeq1d bibi12d pjoc2.1 ax-hv0cl A elimel pjoc2 dedth)) thm (pjoc2t () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (e. A (` (_|_) H)) (= (` (` (proj) H) A) (0v)))) (H (if (e. H (CH)) H (0H)) (_|_) fveq2 A eleq2d H (if (e. H (CH)) H (0H)) (proj) fveq2 A fveq1d (0v) eqeq1d bibi12d A (if (e. A (H~)) A (0v)) (` (_|_) (if (e. H (CH)) H (0H))) eleq1 A (if (e. A (H~)) A (0v)) (` (proj) (if (e. H (CH)) H (0H))) fveq2 (0v) eqeq1d bibi12d h0elch H elimel ax-hv0cl A elimel pjoc2 dedth2h)) thm (shsumvalt ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (SH)) (e. B (SH))) (= (opr A (+H) B) ({e.|} x (H~) (E.e. y A (E.e. z B (= (cv x) (opr (cv y) (+v) (cv z)))))))) (ax-hilex x (E.e. y A (E.e. z B (= (cv x) (opr (cv y) (+v) (cv z))))) rabex (cv w) A y (E.e. z (cv v) (= (cv x) (opr (cv y) (+v) (cv z)))) rexeq1 x (H~) rabbisdv (cv v) B z (= (cv x) (opr (cv y) (+v) (cv z))) rexeq1 y A rexbidv x (H~) rabbisdv w v u x y z df-shsum oprabval2)) thm (shselt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (/\ (e. A (SH)) (e. B (SH))) (<-> (e. C (opr A (+H) B)) (E.e. x A (E.e. y B (= C (opr (cv x) (+v) (cv y))))))) (A B z x y shsumvalt C eleq2d (cv z) C (opr (cv x) (+v) (cv y)) eqeq1 x A y B 2rexbidv (H~) elrab syl6bb A shss (cv x) sseld B shss (cv y) sseld im2anan9 (cv x) (cv y) ax-hvaddcl (opr (cv x) (+v) (cv y)) (H~) C eleq1a syl syl6 r19.23advv pm4.71rd bitr4d)) thm (shsel ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) ((shscl.1 (e. A (SH))) (shscl.2 (e. B (SH)))) (<-> (e. C (opr A (+H) B)) (E.e. x A (E.e. y B (= C (opr (cv x) (+v) (cv y)))))) (shscl.1 shscl.2 A B C x y shselt mp2an)) thm (shscl ((x y) (x z) (w x) (f x) (g x) (v x) (u x) (A x) (y z) (w y) (f y) (g y) (v y) (u y) (A y) (w z) (f z) (g z) (v z) (u z) (A z) (f w) (g w) (v w) (u w) (A w) (f g) (f v) (f u) (A f) (g v) (g u) (A g) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B f) (B g) (B v) (B u)) ((shscl.1 (e. A (SH))) (shscl.2 (e. B (SH)))) (e. (opr A (+H) B) (SH)) ((opr A (+H) B) x y sh shscl.1 shscl.2 A B z x y shsumvalt mp2an z (H~) (E.e. x A (E.e. y B (= (cv z) (opr (cv x) (+v) (cv y))))) ssrab2 eqsstr shscl.1 A sh0 ax-mp shscl.2 B sh0 ax-mp pm3.2i ax-hv0cl hvaddid2 eqcomi (cv x) (0v) (+v) (cv y) opreq1 (0v) eqeq2d (cv y) (0v) (0v) (+v) opreq2 (0v) eqeq2d A B rcla42ev mp2an shscl.1 shscl.2 (0v) x y shsel mpbir pm3.2i (cv f) (opr (cv z) (+v) (cv v)) (+v) (cv g) opreq1 (opr (cv x) (+v) (cv y)) eqeq2d (cv g) (opr (cv w) (+v) (cv u)) (opr (cv z) (+v) (cv v)) (+v) opreq2 (opr (cv x) (+v) (cv y)) eqeq2d A B rcla42ev shscl.1 A (cv z) (cv v) shaddclt ax-mp shscl.2 B (cv w) (cv u) shaddclt ax-mp anim12i (/\ (= (cv x) (opr (cv z) (+v) (cv w))) (= (cv y) (opr (cv v) (+v) (cv u)))) adantrr (cv x) (opr (cv z) (+v) (cv w)) (cv y) (opr (cv v) (+v) (cv u)) (+v) opreq12 (cv z) (cv v) (cv w) (cv u) hvadd4t shscl.1 (cv z) shel shscl.1 (cv v) shel anim12i shscl.2 (cv w) shel shscl.2 (cv u) shel anim12i syl2an eqcomd sylan9eqr anasss sylanc (e. (cv w) B) (= (cv x) (opr (cv z) (+v) (cv w))) (e. (cv u) B) (= (cv y) (opr (cv v) (+v) (cv u))) an4 sylan2b an4s exp42 imp com3r exp32 imp r19.23aivv com3l r19.23aivv imp shscl.1 shscl.2 (cv x) z w shsel shscl.1 shscl.2 (cv y) v u shsel anbi12i shscl.1 shscl.2 (opr (cv x) (+v) (cv y)) f g shsel 3imtr4 rgen2 shscl.1 A (cv x) (cv v) shmulclt ax-mp shscl.2 B (cv x) (cv u) shmulclt ax-mp anim12i anandis (= (cv y) (opr (cv v) (+v) (cv u))) adantlr (cv y) (opr (cv v) (+v) (cv u)) (cv x) (.s) opreq2 (cv x) (cv v) (cv u) ax-hvdistr1 3expb shscl.1 (cv v) shel shscl.2 (cv u) shel anim12i sylan2 sylan9eqr an1rs jca ancoms an4s an1rs (cv f) (opr (cv x) (.s) (cv v)) (+v) (cv g) opreq1 (opr (cv x) (.s) (cv y)) eqeq2d (cv g) (opr (cv x) (.s) (cv u)) (opr (cv x) (.s) (cv v)) (+v) opreq2 (opr (cv x) (.s) (cv y)) eqeq2d A B rcla42ev syl exp42 imp r19.23aivv impcom shscl.1 shscl.2 (cv y) v u shsel (e. (cv x) (CC)) anbi2i shscl.1 shscl.2 (opr (cv x) (.s) (cv y)) f g shsel 3imtr4 rgen2a pm3.2i mpbir2an)) thm (shsclt () () (-> (/\ (e. A (SH)) (e. B (SH))) (e. (opr A (+H) B) (SH))) (A (if (e. A (SH)) A (H~)) (+H) B opreq1 (SH) eleq1d B (if (e. B (SH)) B (H~)) (if (e. A (SH)) A (H~)) (+H) opreq2 (SH) eleq1d helsh A elimel helsh B elimel shscl dedth2h)) thm (shscomt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (SH)) (e. B (SH))) (= (opr A (+H) B) (opr B (+H) A))) (A (cv y) shelt B (cv z) shelt anim12i an4s (cv y) (cv z) ax-hvcom syl (cv x) eqeq2d 2rexbidva y A z B (= (cv x) (opr (cv z) (+v) (cv y))) rexcom syl6bb A B (cv x) y z shselt B A (cv x) z y shselt ancoms 3bitr4d eqrdv)) thm (shsvat ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) () (-> (/\ (e. A (SH)) (e. B (SH))) (-> (/\ (e. C A) (e. D B)) (e. (opr C (+v) D) (opr A (+H) B)))) (A B (opr C (+v) D) x y shselt (opr C (+v) D) eqid (cv x) C (+v) (cv y) opreq1 (opr C (+v) D) eqeq2d (cv y) D C (+v) opreq2 (opr C (+v) D) eqeq2d A B rcla42ev mpan2 syl5bir)) thm (shsel1t () () (-> (/\ (e. A (SH)) (e. B (SH))) (-> (e. C A) (e. C (opr A (+H) B)))) (A C shelt C ax-hvaddid syl (e. B (SH)) adantlr B sh0 (e. A (SH)) adantl A B C (0v) shsvat mpan2d imp eqeltrrd ex)) thm (shsel2t () () (-> (/\ (e. A (SH)) (e. B (SH))) (-> (e. C B) (e. C (opr A (+H) B)))) (B A C shsel1t ancoms A B shscomt C eleq2d sylibrd)) thm (shsvst () () (-> (/\ (e. A (SH)) (e. B (SH))) (-> (/\ (e. C A) (e. D B)) (e. (opr C (-v) D) (opr A (+H) B)))) (A B shsclt (opr A (+H) B) C D shsubclt syl A B C shsel1t A B D shsel2t syl2and)) thm (shsub1t ((A x) (B x)) () (-> (/\ (e. A (SH)) (e. B (SH))) (C_ A (opr A (+H) B))) (A B (cv x) shsel1t ssrdv)) thm (shsub2t () () (-> (/\ (e. A (SH)) (e. B (SH))) (C_ A (opr B (+H) A))) (A B shsub1t A B shscomt sseqtrd)) thm (choc0 ((x y)) () (= (` (_|_) (0H)) (H~)) ((e. (cv x) (H~)) (A.e. y (0H) (= (opr (cv x) (.i) (cv y)) (0))) abai h0elsh (0H) (cv x) y shocelt ax-mp (cv x) hi02t y (0H) (= (opr (cv x) (.i) (cv y)) (0)) df-ral (cv y) elch0 (= (opr (cv x) (.i) (cv y)) (0)) imbi1i y albii ax-hv0cl elisseti (cv y) (0v) (cv x) (.i) opreq2 (0) eqeq1d ceqsalv 3bitr sylibr (e. (cv x) (H~)) biantru 3bitr4 eqriv)) thm (choc1 ((x y)) () (= (` (_|_) (H~)) (0H)) (helsh (H~) (cv x) y shocelt ax-mp pm3.27bd helsh (H~) shocss ax-mp (cv x) sseli (cv x) y hial0 syl mpbid (cv x) elch0 sylibr ssriv h0elsh (0H) shococss ax-mp choc0 (_|_) fveq2i sseqtr eqssi)) thm (chocnul ((x y) (x y)) () (= (` (_|_) ({/})) (H~)) ((H~) 0ss ({/}) (cv x) y ocelt ax-mp y (= (opr (cv x) (.i) (cv y)) (0)) ral0 mpbiran2 eqriv)) thm (shintcl ((x y) (x z) (A x) (y z) (A y) (A z)) ((shintcl.1 (/\ (C_ A (SH)) (-. (= A ({/})))))) (e. (|^| A) (SH)) ((|^| A) x y sh shintcl.1 pm3.27i A z n0 (cv z) A intss1 shintcl.1 pm3.26i (cv z) sseli (cv z) shss syl sstrd z 19.23aiv sylbi ax-mp ax-hv0cl elisseti A z elint2 shintcl.1 pm3.26i (cv z) sseli (cv z) sh0 syl mprgbir pm3.2i shintcl.1 pm3.26i (cv z) sseli (cv z) (cv x) (cv y) shaddclt syl (cv x) A (cv z) elinti com12 (cv y) A (cv z) elinti com12 syl2and com12 r19.21aiv (cv x) (+v) (cv y) oprex A z elint2 sylibr rgen2 shintcl.1 pm3.26i (cv z) sseli (cv z) (cv x) (cv y) shmulclt syl (cv y) A (cv z) elinti com12 sylan2d com12 r19.21aiv (cv x) (.s) (cv y) oprex A z elint2 sylibr rgen2a pm3.2i mpbir2an)) thm (shintclt () () (-> (/\ (C_ A (SH)) (-. (= A ({/})))) (e. (|^| A) (SH))) (A (if (/\ (C_ A (SH)) (-. (= A ({/})))) A (SH)) inteq (SH) eleq1d A (if (/\ (C_ A (SH)) (-. (= A ({/})))) A (SH)) (SH) sseq1 A (if (/\ (C_ A (SH)) (-. (= A ({/})))) A (SH)) ({/}) eqeq1 negbid anbi12d (SH) (if (/\ (C_ A (SH)) (-. (= A ({/})))) A (SH)) (SH) sseq1 (SH) (if (/\ (C_ A (SH)) (-. (= A ({/})))) A (SH)) ({/}) eqeq1 negbid anbi12d (SH) ssid h0elsh (0H) (SH) n0i ax-mp pm3.2i elimhyp shintcl dedth)) thm (chintcl ((x y) (f x) (A x) (f y) (A y) (A f)) ((chintcl.1 (/\ (C_ A (CH)) (-. (= A ({/})))))) (e. (|^| A) (CH)) ((|^| A) f x closedsub chintcl.1 pm3.26i chsssh sstri chintcl.1 pm3.27i pm3.2i shintcl chintcl.1 pm3.26i (cv y) sseli x visset (cv y) (cv f) chlim syl exp3a com3r imp r19.20dva chintcl.1 pm3.27i (cv f) (NN) y fint x visset A y elint2 3imtr4g impcom f x gen2 mpbir2an)) thm (chintclt () () (-> (/\ (C_ A (CH)) (-. (= A ({/})))) (e. (|^| A) (CH))) (A (if (/\ (C_ A (CH)) (-. (= A ({/})))) A (CH)) inteq (CH) eleq1d A (if (/\ (C_ A (CH)) (-. (= A ({/})))) A (CH)) (CH) sseq1 A (if (/\ (C_ A (CH)) (-. (= A ({/})))) A (CH)) ({/}) eqeq1 negbid anbi12d (CH) (if (/\ (C_ A (CH)) (-. (= A ({/})))) A (CH)) (CH) sseq1 (CH) (if (/\ (C_ A (CH)) (-. (= A ({/})))) A (CH)) ({/}) eqeq1 negbid anbi12d (CH) ssid h0elch (0H) (CH) n0i ax-mp pm3.2i elimhyp chintcl dedth)) thm (ococint ((A x)) () (-> (C_ A (H~)) (= (` (_|_) (` (_|_) A)) (|^| ({e.|} x (CH) (C_ A (cv x)))))) ((` (_|_) (|^| ({e.|} x (CH) (C_ A (cv x))))) (` (_|_) A) occont helch (C_ A (H~)) jctl (cv x) (H~) A sseq2 (CH) elrab sylibr (H~) ({e.|} x (CH) (C_ A (cv x))) intss1 (|^| ({e.|} x (CH) (C_ A (cv x)))) ocss 3syl A ocss jca A x (CH) ssintub helch (C_ A (H~)) jctl (cv x) (H~) A sseq2 (CH) elrab sylibr (H~) ({e.|} x (CH) (C_ A (cv x))) intss1 syl A (|^| ({e.|} x (CH) (C_ A (cv x)))) occont mpdan mpi sylc helch (cv x) (H~) A sseq2 (CH) rcla4ev mpan x (CH) (C_ A (cv x)) rabn0 sylibr x (CH) (C_ A (cv x)) ssrab2 jctil ({e.|} x (CH) (C_ A (cv x))) chintclt (|^| ({e.|} x (CH) (C_ A (cv x)))) ococt 3syl sseqtrd A ocss (` (_|_) A) occlt syl A ococss jca (cv x) (` (_|_) (` (_|_) A)) A sseq2 (CH) elrab sylibr (` (_|_) (` (_|_) A)) ({e.|} x (CH) (C_ A (cv x))) intss1 syl eqssd)) thm (dfchsup2 ((x y) (x z) (y z)) () (= (\/H) ({<,>|} x y (/\ (C_ (cv x) (P~ (H~))) (= (cv y) (|^| ({e.|} z (CH) (C_ (U. (cv x)) (cv z)))))))) (x y df-chsup (cv x) (P~ (H~)) uniss (H~) unipw syl6ss (U. (cv x)) z ococint syl (cv y) eqeq2d pm5.32i x y opabbii eqtr)) thm (spanvalt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (H~)) (= (` (span) A) (|^| ({e.|} x (SH) (C_ A (cv x)))))) (ax-hilex (H~) (V) A elpw2g ax-mp ax-hilex (H~) (V) A elpw2g ax-mp helsh (cv x) (H~) A sseq2 (SH) rcla4ev mpan sylbi x (SH) (C_ A (cv x)) intexrab sylib (cv y) A (cv x) sseq1 x (SH) rabbisdv inteqd y z x df-span y visset (H~) elpw (= (cv z) (|^| ({e.|} x (SH) (C_ (cv y) (cv x))))) anbi1i y z opabbii eqtr4 (V) fvopab4g mpdan sylbir)) thm (hsupval2t ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (P~ (H~))) (= (` (\/H) A) (|^| ({e.|} x (CH) (C_ (U. A) (cv x)))))) ((cv z) A unieq (cv x) sseq1d x (CH) rabbisdv inteqd z y x dfchsup2 z visset (P~ (H~)) elpw (= (cv y) (|^| ({e.|} x (CH) (C_ (U. (cv z)) (cv x))))) anbi1i z y opabbii eqtr4 (V) fvopab4g ax-hilex pwex (P~ (H~)) (V) A elpw2g ax-mp biimpr A (P~ (H~)) uniss (H~) unipw syl6ss helch jctil (cv x) (H~) (U. A) sseq2 (CH) elrab sylibr (H~) ({e.|} x (CH) (C_ (U. A) (cv x))) n0i syl ({e.|} x (CH) (C_ (U. A) (cv x))) intex sylib sylanc)) thm (hsupvalt ((A x)) () (-> (C_ A (P~ (H~))) (= (` (\/H) A) (` (_|_) (` (_|_) (U. A))))) (A x hsupval2t A (P~ (H~)) uniss (H~) unipw syl6ss (U. A) x ococint syl eqtr4d)) thm (chsupval2t ((A x)) () (-> (C_ A (CH)) (= (` (\/H) A) (|^| ({e.|} x (CH) (C_ (U. A) (cv x)))))) (chsspwh A (CH) (P~ (H~)) sstr2 mpi A x hsupval2t syl)) thm (chsupvalt () () (-> (C_ A (CH)) (= (` (\/H) A) (` (_|_) (` (_|_) (U. A))))) (chsspwh A (CH) (P~ (H~)) sstr2 mpi A hsupvalt syl)) thm (spanclt ((A x)) () (-> (C_ A (H~)) (e. (` (span) A) (SH))) (A x spanvalt helsh (cv x) (H~) A sseq2 (SH) rcla4ev mpan x (SH) (C_ A (cv x)) rabn0 sylibr x (SH) (C_ A (cv x)) ssrab2 jctil ({e.|} x (SH) (C_ A (cv x))) shintclt syl eqeltrd)) thm (elspanclt () () (-> (/\ (C_ A (H~)) (e. B (` (span) A))) (e. B (H~))) ((` (span) A) B shelt A spanclt sylan)) thm (shsupclt () () (-> (C_ A (P~ (H~))) (e. (` (span) (U. A)) (SH))) (A (P~ (H~)) uniss (H~) unipw syl6ss (U. A) spanclt syl)) thm (hsupclt ((A x)) () (-> (C_ A (P~ (H~))) (e. (` (\/H) A) (CH))) (A x hsupval2t A x hsupval2t (\/H) A fvex syl6eqelr ({e.|} x (CH) (C_ (U. A) (cv x))) intex sylibr x (CH) (C_ (U. A) (cv x)) ssrab2 jctil ({e.|} x (CH) (C_ (U. A) (cv x))) chintclt syl eqeltrd)) thm (chsupclt () () (-> (C_ A (CH)) (e. (` (\/H) A) (CH))) (chsspwh A (CH) (P~ (H~)) sstr2 mpi A hsupclt syl)) thm (hsupss ((A x) (B x)) () (-> (/\ (C_ A (P~ (H~))) (C_ B (P~ (H~)))) (-> (C_ A B) (C_ (` (\/H) A) (` (\/H) B)))) (A x hsupval2t (C_ B (P~ (H~))) adantr B x hsupval2t (C_ A (P~ (H~))) adantl sseq12d A B uniss (U. A) (U. B) (cv x) sstr2 syl (e. (cv x) (CH)) a1d r19.21aiv x (CH) (C_ (U. B) (cv x)) (C_ (U. A) (cv x)) ss2rab sylibr ({e.|} x (CH) (C_ (U. B) (cv x))) ({e.|} x (CH) (C_ (U. A) (cv x))) intss syl syl5bir)) thm (chsupss () () (-> (/\ (C_ A (CH)) (C_ B (CH))) (-> (C_ A B) (C_ (` (\/H) A) (` (\/H) B)))) (A B hsupss chsspwh A (CH) (P~ (H~)) sstr2 mpi chsspwh B (CH) (P~ (H~)) sstr2 mpi syl2an)) thm (chsupid ((x y) (A x) (A y)) () (-> (e. A (CH)) (= (` (\/H) ({e.|} x (CH) (C_ (cv x) A))) A)) (A (CH) x unimax (cv y) sseq1d y (CH) rabbisdv inteqd A (CH) y intmin eqtrd x (CH) (C_ (cv x) A) ssrab2 ({e.|} x (CH) (C_ (cv x) A)) y chsupval2t ax-mp syl5eq)) thm (chsupsn ((A x)) () (-> (e. A (CH)) (= (` (\/H) ({} A)) A)) (A (CH) snssi ({} A) x chsupval2t syl A (CH) unisng (U. ({} A)) A eqimss syl ancli (cv x) A (U. ({} A)) sseq2 (CH) elrab sylibr A ({e.|} x (CH) (C_ (U. ({} A)) (cv x))) intss1 syl (U. ({} A)) x (CH) ssintub A (CH) unisng (|^| ({e.|} x (CH) (C_ (U. ({} A)) (cv x)))) sseq1d mpbii eqssd eqtrd)) thm (hsupunss ((A x)) () (-> (C_ A (P~ (H~))) (C_ (U. A) (` (\/H) A))) ((U. A) x (CH) ssintub A x hsupval2t (U. A) sseq2d mpbiri)) thm (spanss2 ((A x)) () (-> (C_ A (H~)) (C_ A (` (span) A))) (A x (SH) ssintub A x spanvalt A sseq2d mpbiri)) thm (shsupunss () () (-> (C_ A (SH)) (C_ (U. A) (` (span) (U. A)))) (shsspwh A (SH) (P~ (H~)) sstr mpan2 A (P~ (H~)) uniss syl (H~) unipw syl6ss (U. A) spanss2 syl)) thm (chsupunss () () (-> (C_ A (CH)) (C_ (U. A) (` (\/H) A))) (chsspwh A (CH) (P~ (H~)) sstr mpan2 A hsupunss syl)) thm (spanid ((A x)) () (-> (e. A (SH)) (= (` (span) A) A)) (A shss A x spanvalt syl A (SH) x intmin eqtrd)) thm (spanss ((A x) (B x)) () (-> (/\ (C_ B (H~)) (C_ A B)) (C_ (` (span) A) (` (span) B))) (A B (cv x) sstr2 (e. (cv x) (SH)) a1d r19.21aiv x (SH) (C_ B (cv x)) (C_ A (cv x)) ss2rab sylibr ({e.|} x (SH) (C_ B (cv x))) ({e.|} x (SH) (C_ A (cv x))) intss syl (C_ B (H~)) adantl A B (H~) sstr ancoms A x spanvalt syl B x spanvalt (C_ A B) adantr 3sstr4d)) thm (spanssoc () () (-> (C_ A (H~)) (C_ (` (span) A) (` (_|_) (` (_|_) A)))) ((` (_|_) (` (_|_) A)) A spanss A ocss (` (_|_) A) ocss syl A ococss sylanc A ocss (` (_|_) A) ocsh (` (_|_) (` (_|_) A)) spanid 3syl sseqtrd)) thm (sshjvalt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (C_ A (H~)) (C_ B (H~))) (= (opr A (vH) B) (` (_|_) (` (_|_) (u. A B))))) ((_|_) (` (_|_) (u. A B)) fvex (cv x) A (cv y) uneq1 (_|_) fveq2d (_|_) fveq2d (cv y) B A uneq2 (_|_) fveq2d (_|_) fveq2d x y z df-chj x visset (H~) elpw y visset (H~) elpw anbi12i (= (cv z) (` (_|_) (` (_|_) (u. (cv x) (cv y))))) anbi1i x y z oprabbii eqtr4 oprabval2 ax-hilex (H~) (V) A elpw2g ax-mp ax-hilex (H~) (V) B elpw2g ax-mp syl2anbr)) thm (shjvalt () () (-> (/\ (e. A (SH)) (e. B (SH))) (= (opr A (vH) B) (` (_|_) (` (_|_) (u. A B))))) (A B sshjvalt A shss B shss syl2an)) thm (chjvalt () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (opr A (vH) B) (` (_|_) (` (_|_) (u. A B))))) (A B shjvalt A chsh B chsh syl2an)) thm (chjval () ((chjval.1 (e. A (CH))) (chjval.2 (e. B (CH)))) (= (opr A (vH) B) (` (_|_) (` (_|_) (u. A B)))) (chjval.1 chjval.2 A B chjvalt mp2an)) thm (dfchj2 ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (vH) ({<<,>,>|} x y z (/\ (/\ (C_ (cv x) (H~)) (C_ (cv y) (H~))) (= (cv z) (|^| ({e.|} w (CH) (C_ (u. (cv x) (cv y)) (cv w)))))))) (x y z df-chj (cv x) (H~) (cv y) unss (u. (cv x) (cv y)) w ococint sylbi (cv z) eqeq2d pm5.32i x y z oprabbii eqtr)) thm (dfchj3 ((x y) (x z) (y z)) () (= (vH) ({<<,>,>|} x y z (/\ (/\ (C_ (cv x) (H~)) (C_ (cv y) (H~))) (= (cv z) (` (\/H) ({,} (cv x) (cv y))))))) (x y z df-chj x visset y visset (H~) prsspw ({,} (cv x) (cv y)) hsupvalt sylbir x visset y visset unipr (_|_) fveq2i (_|_) fveq2i syl6req (cv z) eqeq2d pm5.32i x y z oprabbii eqtr)) thm (sshjval3t ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (C_ A (H~)) (C_ B (H~))) (= (opr A (vH) B) (` (\/H) ({,} A B)))) ((\/H) ({,} A B) fvex (cv x) A (cv y) preq1 (\/H) fveq2d (cv y) B A preq2 (\/H) fveq2d x y z dfchj3 x visset (H~) elpw y visset (H~) elpw anbi12i (= (cv z) (` (\/H) ({,} (cv x) (cv y)))) anbi1i x y z oprabbii eqtr4 oprabval2 ax-hilex (H~) (V) A elpw2g ax-mp ax-hilex (H~) (V) B elpw2g ax-mp syl2anbr)) thm (sshjclt () () (-> (/\ (C_ A (H~)) (C_ B (H~))) (e. (opr A (vH) B) (CH))) (A B sshjvalt A (H~) B unss (u. A B) ocss (` (_|_) (u. A B)) occlt syl sylbi eqeltrd)) thm (shjclt () () (-> (/\ (e. A (SH)) (e. B (SH))) (e. (opr A (vH) B) (CH))) (A B sshjclt A shss B shss syl2an)) thm (chjclt () () (-> (/\ (e. A (CH)) (e. B (CH))) (e. (opr A (vH) B) (CH))) (A B shjclt A chsh B chsh syl2an)) thm (shjcomt () () (-> (/\ (e. A (SH)) (e. B (SH))) (= (opr A (vH) B) (opr B (vH) A))) (A B shjvalt B A shjvalt ancoms B A uncom (_|_) fveq2i (_|_) fveq2i syl6eq eqtr4d)) thm (shincl () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (e. (i^i A B) (SH)) (shincl.1 elisseti shincl.2 elisseti intpr shincl.1 shincl.2 pm3.2i shincl.1 elisseti shincl.2 elisseti (SH) prss mpbi shincl.1 elisseti B prnz pm3.2i shintcl eqeltrr)) thm (shscom () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (= (opr A (+H) B) (opr B (+H) A)) (shincl.1 shincl.2 A B shscomt mp2an)) thm (shsva () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (-> (/\ (e. C A) (e. D B)) (e. (opr C (+v) D) (opr A (+H) B))) (shincl.1 shincl.2 A B C D shsvat mp2an)) thm (shsel1 () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (-> (e. C A) (e. C (opr A (+H) B))) (shincl.1 shincl.2 A B C shsel1t mp2an)) thm (shsel2 () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (-> (e. C B) (e. C (opr A (+H) B))) (shincl.1 shincl.2 A B C shsel2t mp2an)) thm (shsvs () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (-> (/\ (e. C A) (e. D B)) (e. (opr C (-v) D) (opr A (+H) B))) (shincl.1 shincl.2 shscl (opr A (+H) B) C D shsubclt ax-mp shincl.1 shincl.2 C shsel1 shincl.1 shincl.2 D shsel2 syl2an)) thm (shunss ((x y) (x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ (u. A B) (opr A (+H) B)) ((cv y) (cv x) (+v) (cv z) opreq1 (cv x) eqeq2d (cv z) (0v) (cv x) (+v) opreq2 (cv x) eqeq2d A B rcla42ev shincl.2 B sh0 ax-mp (e. (cv x) A) jctr shincl.1 (cv x) shel (cv x) ax-hvaddid syl eqcomd sylanc (cv y) (0v) (+v) (cv z) opreq1 (cv x) eqeq2d (cv z) (cv x) (0v) (+v) opreq2 (cv x) eqeq2d A B rcla42ev shincl.1 A sh0 ax-mp (e. (cv x) B) jctl shincl.2 (cv x) shel (cv x) hvaddid2t syl eqcomd sylanc jaoi (cv x) A B elun shincl.1 shincl.2 (cv x) y z shsel 3imtr4 ssriv)) thm (shslej ((x y) (x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ (opr A (+H) B) (opr A (vH) B)) ((cv x) (opr (cv y) (+v) (cv z)) (` (_|_) (` (_|_) (u. A B))) eleq1 shincl.1 shssi shincl.2 shssi unssi (u. A B) ocss ax-mp (` (_|_) (u. A B)) ocsh ax-mp (` (_|_) (` (_|_) (u. A B))) (cv y) (cv z) shaddclt ax-mp (cv y) A B elun1 shincl.1 shssi shincl.2 shssi unssi (u. A B) ococss ax-mp (cv y) sseli syl (cv z) B A elun2 shincl.1 shssi shincl.2 shssi unssi (u. A B) ococss ax-mp (cv z) sseli syl syl2an syl5bir com12 r19.23aivv shincl.1 shincl.2 (cv x) y z shsel shincl.1 shincl.2 A B shjvalt mp2an (cv x) eleq2i 3imtr4 ssriv)) thm (shunssj () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ (u. A B) (opr A (vH) B)) (shincl.1 shincl.2 shunss shincl.1 shincl.2 shslej sstri)) thm (shjcom () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (= (opr A (vH) B) (opr B (vH) A)) (shincl.1 shincl.2 A B shjcomt mp2an)) thm (shsub1 ((A x) (B x)) ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ A (opr A (+H) B)) (shincl.1 shincl.2 (cv x) shsel1 ssriv)) thm (shsub2 ((A x) (B x)) ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ A (opr B (+H) A)) (shincl.2 shincl.1 (cv x) shsel2 ssriv)) thm (shub1 () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ A (opr A (vH) B)) (shincl.1 shincl.2 shsub1 shincl.1 shincl.2 shslej sstri)) thm (shjcl () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (e. (opr A (vH) B) (CH)) (shincl.1 shincl.2 A B shjclt mp2an)) thm (shjshcl () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (e. (opr A (vH) B) (SH)) (shincl.1 shincl.2 shjcl chshi)) thm (shlub () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH))) (shlub.1 (e. C (CH)))) (<-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (vH) B) C)) (shincl.1 shssi shincl.2 shssi unssi shlub.1 chssi occon2 A C B unss shincl.1 shincl.2 A B shjvalt mp2an shlub.1 pjococ eqcomi sseq12i 3imtr4 shincl.1 shincl.2 shub1 A (opr A (vH) B) C sstr mpan B A ssun2 shincl.1 shincl.2 shunssj sstri B (opr A (vH) B) C sstr mpan jca impbi)) thm (shless ((x y) (x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH))) (shless.1 (e. C (SH)))) (-> (C_ A B) (C_ (opr A (+H) C) (opr B (+H) C))) (A B (cv y) ssel (E.e. z C (= (cv x) (opr (cv y) (+v) (cv z)))) anim1d y 19.22dv y A (E.e. z C (= (cv x) (opr (cv y) (+v) (cv z)))) df-rex y B (E.e. z C (= (cv x) (opr (cv y) (+v) (cv z)))) df-rex 3imtr4g shincl.1 shless.1 (cv x) y z shsel shincl.2 shless.1 (cv x) y z shsel 3imtr4g ssrdv)) thm (shlej1 () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH))) (shless.1 (e. C (SH)))) (-> (C_ A B) (C_ (opr A (vH) C) (opr B (vH) C))) (shincl.2 shless.1 shub1 A B (opr B (vH) C) sstr2 mpi shless.1 shincl.2 shub1 shless.1 shincl.2 shjcom sseqtr jctir shincl.1 shless.1 shincl.2 shless.1 shjcl shlub sylib)) thm (shlej2 () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH))) (shless.1 (e. C (SH)))) (-> (C_ A B) (C_ (opr C (vH) A) (opr C (vH) B))) (shincl.1 shincl.2 shless.1 shlej1 shless.1 shincl.1 shjcom shless.1 shincl.2 shjcom 3sstr4g)) thm (shslejt () () (-> (/\ (e. A (SH)) (e. B (SH))) (C_ (opr A (+H) B) (opr A (vH) B))) (A (if (e. A (SH)) A (H~)) (+H) B opreq1 A (if (e. A (SH)) A (H~)) (vH) B opreq1 sseq12d B (if (e. B (SH)) B (H~)) (if (e. A (SH)) A (H~)) (+H) opreq2 B (if (e. B (SH)) B (H~)) (if (e. A (SH)) A (H~)) (vH) opreq2 sseq12d helsh A elimel helsh B elimel shslej dedth2h)) thm (shinclt () () (-> (/\ (e. A (SH)) (e. B (SH))) (e. (i^i A B) (SH))) (A (if (e. A (SH)) A (H~)) B ineq1 (SH) eleq1d B (if (e. B (SH)) B (H~)) (if (e. A (SH)) A (H~)) ineq2 (SH) eleq1d helsh A elimel helsh B elimel shincl dedth2h)) thm (shub1t () () (-> (/\ (e. A (SH)) (e. B (SH))) (C_ A (opr A (vH) B))) (A B shsub1t A B shslejt sstrd)) thm (shub2t () () (-> (/\ (e. A (SH)) (e. B (SH))) (C_ A (opr B (vH) A))) (A B shub1t A B shjcomt sseqtrd)) thm (shlubt () () (-> (/\/\ (e. A (SH)) (e. B (SH)) (e. C (CH))) (<-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (vH) B) C))) (A (if (e. A (SH)) A (H~)) C sseq1 (C_ B C) anbi1d A (if (e. A (SH)) A (H~)) (vH) B opreq1 C sseq1d bibi12d B (if (e. B (SH)) B (H~)) C sseq1 (C_ (if (e. A (SH)) A (H~)) C) anbi2d B (if (e. B (SH)) B (H~)) (if (e. A (SH)) A (H~)) (vH) opreq2 C sseq1d bibi12d C (if (e. C (CH)) C (H~)) (if (e. A (SH)) A (H~)) sseq2 C (if (e. C (CH)) C (H~)) (if (e. B (SH)) B (H~)) sseq2 anbi12d C (if (e. C (CH)) C (H~)) (opr (if (e. A (SH)) A (H~)) (vH) (if (e. B (SH)) B (H~))) sseq2 bibi12d helsh A elimel helsh B elimel helch C elimel shlub dedth3h)) thm (shlej1t () () (-> (/\/\ (e. A (SH)) (e. B (SH)) (e. C (SH))) (-> (C_ A B) (C_ (opr A (vH) C) (opr B (vH) C)))) (A B (opr B (vH) C) sstr2 B C shub1t syl5com C B shub2t ancoms jctird (e. A (SH)) 3adant1 A C (opr B (vH) C) shlubt (e. A (SH)) (e. B (SH)) (e. C (SH)) 3simp1 (e. A (SH)) (e. B (SH)) (e. C (SH)) 3simp3 B C shjclt (e. A (SH)) 3adant1 syl3anc sylibd)) thm (shlej2t () () (-> (/\/\ (e. A (SH)) (e. B (SH)) (e. C (SH))) (-> (C_ A B) (C_ (opr C (vH) A) (opr C (vH) B)))) (A B C shlej1t imp A C shjcomt (e. B (SH)) 3adant2 (C_ A B) adantr B C shjcomt (e. A (SH)) 3adant1 (C_ A B) adantr 3sstr3d ex)) thm (shsidm ((x y) (x z) (A x) (y z) (A y) (A z)) ((shsidm.1 (e. A (SH)))) (= (opr A (+H) A) A) (shsidm.1 shsidm.1 (cv x) y z shsel (cv x) (opr (cv y) (+v) (cv z)) A eleq1 shsidm.1 A (cv y) (cv z) shaddclt ax-mp syl5bir com12 r19.23aivv sylbi ssriv shsidm.1 shsidm.1 shsub1 eqssi)) thm (shslub () ((shslub.1 (e. A (SH))) (shslub.2 (e. B (SH))) (shslub.3 (e. C (SH)))) (<-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (+H) B) C)) (shslub.1 shslub.3 shslub.2 shless shslub.3 shslub.2 shscom syl6ss shslub.2 shslub.3 shslub.3 shless shslub.3 shsidm syl6ss sylan9ss shslub.1 shslub.2 shsub1 A (opr A (+H) B) C sstr mpan shslub.2 shslub.1 shsub2 B (opr A (+H) B) C sstr mpan jca impbi)) thm (shlesb1 () ((shlesb1.1 (e. A (SH))) (shlesb1.2 (e. B (SH)))) (<-> (C_ A B) (= (opr A (+H) B) B)) (B ssid (C_ A B) biantrur shlesb1.2 shlesb1.1 shlesb1.2 shslub (opr A (+H) B) B eqss shlesb1.2 shlesb1.1 shsub2 mpbiran2 shlesb1.1 shlesb1.2 shscom B sseq1i bitr2 3bitr)) thm (shsumval2 ((A x) (B x)) ((shlesb1.1 (e. A (SH))) (shlesb1.2 (e. B (SH)))) (= (opr A (+H) B) (|^| ({e.|} x (SH) (C_ (u. A B) (cv x))))) (A B ssun1 (u. A B) x (SH) ssintub sstri B A ssun2 (u. A B) x (SH) ssintub sstri pm3.2i shlesb1.1 shlesb1.2 x (SH) (C_ (u. A B) (cv x)) ssrab2 shlesb1.1 shlesb1.2 shscl shlesb1.1 shlesb1.2 shunss (cv x) (opr A (+H) B) (u. A B) sseq2 (SH) rcla4ev mp2an x (SH) (C_ (u. A B) (cv x)) rabn0 mpbir ({e.|} x (SH) (C_ (u. A B) (cv x))) shintclt mp2an shslub mpbi (cv x) (opr A (+H) B) (u. A B) sseq2 (SH) elrab shlesb1.1 shlesb1.2 shscl shlesb1.1 shlesb1.2 shunss mpbir2an (opr A (+H) B) ({e.|} x (SH) (C_ (u. A B) (cv x))) intss1 ax-mp eqssi)) thm (shsumval3 ((A x) (B x)) ((shlesb1.1 (e. A (SH))) (shlesb1.2 (e. B (SH)))) (= (opr A (+H) B) (` (span) (u. A B))) (shlesb1.1 shlesb1.2 x shsumval2 shlesb1.1 shssi shlesb1.2 shssi unssi (u. A B) x spanvalt ax-mp eqtr4)) thm (shmods ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((shmod.1 (e. A (SH))) (shmod.2 (e. B (SH))) (shmod.3 (e. C (SH)))) (-> (C_ A C) (C_ (i^i (opr A (+H) B) C) (opr A (+H) (i^i B C)))) ((cv z) (cv x) (cv y) hvsubaddt shmod.3 (cv z) shel shmod.1 (cv x) shel shmod.2 (cv y) shel syl3an (opr (cv x) (+v) (cv y)) (cv z) eqcom syl6bb 3expb shmod.1 shmod.3 shlesb1 biimp (opr (cv z) (-v) (cv x)) eleq2d shmod.3 shmod.1 (cv z) (cv x) shsvs shmod.3 shmod.1 shscom syl6eleq syl5bi (opr (cv z) (-v) (cv x)) (cv y) C eleq1 biimpd sylan9 (e. (cv y) B) anim2d (cv y) B C elin syl6ibr ex com13 ancoms anasss sylbird imp (cv z) (opr (cv x) (+v) (cv y)) (opr A (+H) (i^i B C)) eleq1 shmod.1 shmod.2 shmod.3 shincl (cv x) (cv y) shsva syl5bir exp3a com12 (e. (cv z) C) (e. (cv y) B) ad2antrl imp syld exp31 r19.23advv shmod.1 shmod.2 (cv z) x y shsel syl5ib com13 imp3a (cv z) (opr A (+H) B) C elin syl5ib ssrdv)) thm (shmod () ((shmod.1 (e. A (SH))) (shmod.2 (e. B (SH))) (shmod.3 (e. C (SH)))) (-> (/\ (= (opr A (+H) B) (opr A (vH) B)) (C_ A C)) (C_ (i^i (opr A (vH) B) C) (opr A (vH) (i^i B C)))) ((opr A (+H) B) (opr A (vH) B) C ineq1 (opr A (+H) (i^i B C)) sseq1d shmod.1 shmod.2 shmod.3 shmods syl5bi imp shmod.1 shmod.2 shmod.3 shincl shslej (i^i (opr A (vH) B) C) (opr A (+H) (i^i B C)) (opr A (vH) (i^i B C)) sstr mpan2 syl)) thm (sh0let () () (-> (e. A (SH)) (C_ (0H) A)) (A sh0 (0v) A snssi syl df-ch0 syl5ss)) thm (ch0let () () (-> (e. A (CH)) (C_ (0H) A)) (A chsh A sh0let syl)) thm (shle0t () () (-> (e. A (SH)) (<-> (C_ A (0H)) (= A (0H)))) (A sh0let (C_ A (0H)) biantrud A (0H) eqss syl6bbr)) thm (chle0t () () (-> (e. A (CH)) (<-> (C_ A (0H)) (= A (0H)))) (A chsh A shle0t syl)) thm (chnlen0 () () (-> (e. B (CH)) (-> (-. (C_ A B)) (-. (= A (0H))))) (A (0H) B sseq1 B ch0let syl5bir com12 con3d)) thm (ch0psst () () (-> (e. A (CH)) (<-> (C: (0H) A) (=/= A (0H)))) (A ch0let (=/= (0H) A) biantrurd (0H) A necom syl5bbr (0H) A df-pss syl6rbbr)) thm (ch0psstOLD () () (-> (e. A (CH)) (<-> (C: (0H) A) (-. (= A (0H))))) (A chle0t negbid A ch0let (-. (C_ A (0H))) biantrurd bitr3d (0H) A dfpss3 syl6rbbr)) thm (orthin () () (-> (/\ (e. A (SH)) (e. B (SH))) (-> (C_ A (` (_|_) B)) (= (i^i A B) (0H)))) (B ocin (i^i A B) sseq2d A (` (_|_) B) B ssrin (` (_|_) B) B incom syl6ss syl5bi (e. A (SH)) adantl A B shinclt (i^i A B) sh0let syl jctird (i^i A B) (0H) eqss syl6ibr)) thm (shne0OLD ((A x)) ((shne0.1 (e. A (SH)))) (<-> (-. (= A (0H))) (E.e. x A (-. (= (cv x) (0v))))) (shne0.1 A shle0t ax-mp negbii A (0H) x nss bitr3 x A (-. (e. (cv x) (0H))) df-rex (cv x) elch0 negbii x A rexbii 3bitr2)) thm (shne0 ((A x)) ((shne0.1 (e. A (SH)))) (<-> (=/= A (0H)) (E.e. x A (=/= (cv x) (0v)))) (A (0H) df-ne x A (-. (e. (cv x) (0H))) df-rex A (0H) x nss shne0.1 A shle0t ax-mp negbii 3bitr2r (cv x) elch0 negbii (cv x) (0v) df-ne bitr4 x A rexbii 3bitr)) thm (shs0 () ((shne0.1 (e. A (SH)))) (= (opr A (+H) (0H)) A) (shne0.1 h0elsh shsumval3 shne0.1 A sh0let ax-mp (0H) A ssequn2 mpbi (span) fveq2i shne0.1 A spanid ax-mp 3eqtr)) thm (shs00 () ((shne0.1 (e. A (SH))) (shs00.2 (e. B (SH)))) (<-> (/\ (= A (0H)) (= B (0H))) (= (opr A (+H) B) (0H))) (A (0H) B (0H) (+H) opreq12 h0elsh shs0 syl6eq shne0.1 shs00.2 shsub1 (opr A (+H) B) (0H) A sseq2 mpbii shne0.1 A shle0t ax-mp sylib shs00.2 shne0.1 shsub2 (opr A (+H) B) (0H) B sseq2 mpbii shs00.2 B shle0t ax-mp sylib jca impbi)) thm (ch0le () ((ch0le.1 (e. A (CH)))) (C_ (0H) A) (ch0le.1 A ch0let ax-mp)) thm (chle0 () ((ch0le.1 (e. A (CH)))) (<-> (C_ A (0H)) (= A (0H))) (ch0le.1 A chle0t ax-mp)) thm (chne0OLD ((A x)) ((ch0le.1 (e. A (CH)))) (<-> (-. (= A (0H))) (E.e. x A (-. (= (cv x) (0v))))) (ch0le.1 chshi x shne0OLD)) thm (chne0 ((A x)) ((ch0le.1 (e. A (CH)))) (<-> (=/= A (0H)) (E.e. x A (=/= (cv x) (0v)))) (ch0le.1 chshi x shne0)) thm (chocin () ((ch0le.1 (e. A (CH)))) (= (i^i A (` (_|_) A)) (0H)) (ch0le.1 chshi A ocin ax-mp)) thm (chj0 () ((ch0le.1 (e. A (CH)))) (= (opr A (vH) (0H)) A) (ch0le.1 h0elch chjval ch0le.1 ch0le (0H) A ssequn2 mpbi (_|_) fveq2i (_|_) fveq2i ch0le.1 pjococ 3eqtr)) thm (chm1 () ((ch0le.1 (e. A (CH)))) (= (i^i A (H~)) A) (A (H~) inss1 A ssid ch0le.1 chssi ssini eqssi)) thm (chjcl () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (e. (opr A (vH) B) (CH)) (ch0le.1 chshi chjcl.2 chshi shjcl)) thm (chslej () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (C_ (opr A (+H) B) (opr A (vH) B)) (ch0le.1 chshi chjcl.2 chshi shslej)) thm (chsel ((x y) (B x) (B y) (C x) (C y) (x y) (A x) (A y)) ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (e. C (opr A (+H) B)) (E.e. x A (E.e. y B (= C (opr (cv x) (+v) (cv y)))))) (ch0le.1 chshi chjcl.2 chshi C x y shsel)) thm (chincl () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (e. (i^i A B) (CH)) (ch0le.1 elisseti chjcl.2 elisseti intpr ch0le.1 chjcl.2 pm3.2i ch0le.1 elisseti chjcl.2 elisseti (CH) prss mpbi ch0le.1 elisseti B prnz pm3.2i chintcl eqeltrr)) thm (chsscon3 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (C_ A B) (C_ (` (_|_) B) (` (_|_) A))) (ch0le.1 chssi chjcl.2 chssi A B occont mp2an chjcl.2 choccl chssi ch0le.1 choccl chssi (` (_|_) B) (` (_|_) A) occont mp2an ch0le.1 pjococ chjcl.2 pjococ 3sstr3g impbi)) thm (chsscon1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (C_ (` (_|_) A) B) (C_ (` (_|_) B) A)) (ch0le.1 choccl chjcl.2 chsscon3 ch0le.1 pjococ (` (_|_) B) sseq2i bitr)) thm (chsscon2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (C_ A (` (_|_) B)) (C_ B (` (_|_) A))) (ch0le.1 chjcl.2 choccl chsscon3 chjcl.2 pjococ (` (_|_) A) sseq1i bitr)) thm (chcon2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (= A (` (_|_) B)) (= B (` (_|_) A))) (ch0le.1 chjcl.2 chsscon2 chjcl.2 ch0le.1 chsscon1 anbi12i A (` (_|_) B) eqss B (` (_|_) A) eqss 3bitr4)) thm (chcon1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (= (` (_|_) A) B) (= (` (_|_) B) A)) (chjcl.2 ch0le.1 chcon2 (` (_|_) A) B eqcom (` (_|_) B) A eqcom 3bitr4)) thm (chcon3 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (= A B) (= (` (_|_) B) (` (_|_) A))) (ch0le.1 chjcl.2 chsscon3 chjcl.2 ch0le.1 chsscon3 anbi12i A B eqss (` (_|_) B) (` (_|_) A) eqss 3bitr4)) thm (chunssj () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (C_ (u. A B) (opr A (vH) B)) (ch0le.1 chshi chjcl.2 chshi shunssj)) thm (chjcom () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (opr A (vH) B) (opr B (vH) A)) (ch0le.1 chshi chjcl.2 chshi shjcom)) thm (chub1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (C_ A (opr A (vH) B)) (ch0le.1 chshi chjcl.2 chshi shub1)) thm (chub2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (C_ A (opr B (vH) A)) (ch0le.1 chjcl.2 chub1 ch0le.1 chjcl.2 chjcom sseqtr)) thm (chlub () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chlub.1 (e. C (CH)))) (<-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (vH) B) C)) (ch0le.1 chshi chjcl.2 chshi chlub.1 shlub)) thm (chlubi () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chlub.1 (e. C (CH)))) (-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (vH) B) C)) (ch0le.1 chjcl.2 chlub.1 chlub biimp)) thm (chlej1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chlub.1 (e. C (CH)))) (-> (C_ A B) (C_ (opr A (vH) C) (opr B (vH) C))) (ch0le.1 chshi chjcl.2 chshi chlub.1 chshi shlej1)) thm (chlej2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chlub.1 (e. C (CH)))) (-> (C_ A B) (C_ (opr C (vH) A) (opr C (vH) B))) (ch0le.1 chshi chjcl.2 chshi chlub.1 chshi shlej2)) thm (chlej12 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chlub.1 (e. C (CH))) (chlej12.4 (e. D (CH)))) (-> (/\ (C_ A B) (C_ C D)) (C_ (opr A (vH) C) (opr B (vH) D))) (ch0le.1 chjcl.2 chlub.1 chlej1 chlub.1 chlej12.4 chjcl.2 chlej2 sylan9ss)) thm (chlejb1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (C_ A B) (= (opr A (vH) B) B)) (B ssid ch0le.1 chjcl.2 chjcl.2 chlubi mpan2 chjcl.2 ch0le.1 chub2 jctir (opr A (vH) B) B eqss sylibr ch0le.1 chjcl.2 chub1 A (opr A (vH) B) B sstr2 (opr A (vH) B) B eqimss syl5 ax-mp impbi)) thm (chdmm1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (i^i A B)) (opr (` (_|_) A) (vH) (` (_|_) B))) (ch0le.1 choccl chjcl.2 choccl chub1 ch0le.1 ch0le.1 choccl chjcl.2 choccl chjcl chsscon1 mpbi chjcl.2 choccl ch0le.1 choccl chub2 chjcl.2 ch0le.1 choccl chjcl.2 choccl chjcl chsscon1 mpbi ssini ch0le.1 choccl chjcl.2 choccl chjcl ch0le.1 chjcl.2 chincl chsscon1 mpbi A B inss1 ch0le.1 chjcl.2 chincl ch0le.1 chsscon3 mpbi A B inss2 ch0le.1 chjcl.2 chincl chjcl.2 chsscon3 mpbi ch0le.1 choccl chjcl.2 choccl ch0le.1 chjcl.2 chincl choccl chlubi mp2an eqssi)) thm (chdmm2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (i^i (` (_|_) A) B)) (opr A (vH) (` (_|_) B))) (ch0le.1 choccl chjcl.2 chdmm1 ch0le.1 pjococ (vH) (` (_|_) B) opreq1i eqtr)) thm (chdmm3 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (i^i A (` (_|_) B))) (opr (` (_|_) A) (vH) B)) (ch0le.1 chjcl.2 choccl chdmm1 chjcl.2 pjococ (` (_|_) A) (vH) opreq2i eqtr)) thm (chdmm4 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (i^i (` (_|_) A) (` (_|_) B))) (opr A (vH) B)) (ch0le.1 chjcl.2 choccl chdmm2 chjcl.2 pjococ A (vH) opreq2i eqtr)) thm (chdmj1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (opr A (vH) B)) (i^i (` (_|_) A) (` (_|_) B))) (ch0le.1 chjcl.2 chdmm4 (_|_) fveq2i ch0le.1 choccl chjcl.2 choccl chincl pjococ eqtr3)) thm (chdmj2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (opr (` (_|_) A) (vH) B)) (i^i A (` (_|_) B))) (ch0le.1 choccl chjcl.2 chdmj1 ch0le.1 pjococ (` (_|_) B) ineq1i eqtr)) thm (chdmj3 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (opr A (vH) (` (_|_) B))) (i^i (` (_|_) A) B)) (ch0le.1 chjcl.2 choccl chdmj1 chjcl.2 pjococ (` (_|_) A) ineq2i eqtr)) thm (chdmj4 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (opr (` (_|_) A) (vH) (` (_|_) B))) (i^i A B)) (ch0le.1 chjcl.2 choccl chdmj2 chjcl.2 pjococ A ineq2i eqtr)) thm (chnle () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (-. (C_ B A)) (C: A (opr A (vH) B))) (ch0le.1 chjcl.2 chub1 (-. (= A (opr A (vH) B))) biantrur chjcl.2 ch0le.1 chlejb1 (opr B (vH) A) A eqcom chjcl.2 ch0le.1 chjcom A eqeq2i 3bitr negbii A (opr A (vH) B) dfpss2 3bitr4)) thm (chjass () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chjass.3 (e. C (CH)))) (= (opr (opr A (vH) B) (vH) C) (opr A (vH) (opr B (vH) C))) ((` (_|_) A) (` (_|_) B) (` (_|_) C) inass ch0le.1 chjcl.2 chdmj1 (` (_|_) C) ineq1i chjcl.2 chjass.3 chdmj1 (` (_|_) A) ineq2i 3eqtr4 (_|_) fveq2i ch0le.1 chjcl.2 chjcl chjass.3 chdmm4 ch0le.1 chjcl.2 chjass.3 chjcl chdmm4 3eqtr3)) thm (chj00 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (/\ (= A (0H)) (= B (0H))) (= (opr A (vH) B) (0H))) (A (0H) B (0H) (vH) opreq12 h0elch chj0 syl6eq ch0le.1 chjcl.2 chub1 (opr A (vH) B) (0H) A sseq2 mpbii ch0le.1 chle0 sylib chjcl.2 ch0le.1 chub2 (opr A (vH) B) (0H) B sseq2 mpbii chjcl.2 chle0 sylib jca impbi)) thm (chjo () ((ch0le.1 (e. A (CH)))) (= (opr A (vH) (` (_|_) A)) (H~)) ((` (_|_) A) A incom ch0le.1 chocin eqtr (_|_) fveq2i ch0le.1 ch0le.1 chdmm2 choc0 3eqtr3)) thm (chj1 () ((ch0le.1 (e. A (CH)))) (= (opr A (vH) (H~)) (H~)) (ch0le.1 helch chjcl chssi helch ch0le.1 chub2 eqssi)) thm (chm0 () ((ch0le.1 (e. A (CH)))) (= (i^i A (0H)) (0H)) (A (0H) inss2 ch0le.1 ch0le (0H) ssid ssini eqssi)) thm (chm0t () () (-> (e. A (CH)) (= (i^i A (0H)) (0H))) (A (if (e. A (CH)) A (0H)) (0H) ineq1 (0H) eqeq1d h0elch A elimel chm0 dedth)) thm (shjshs () ((shjshs.1 (e. A (SH))) (shjshs.2 (e. B (SH)))) (= (opr A (vH) B) (` (_|_) (` (_|_) (opr A (+H) B)))) (shjshs.1 shjshs.2 A B shjvalt mp2an shjshs.1 shjshs.2 shunss shjshs.1 shssi shjshs.2 shssi unssi shjshs.1 shjshs.2 shscl shssi occon2 ax-mp eqsstr shjshs.1 shjshs.2 shslej shjshs.1 shjshs.2 shscl shssi shjshs.1 shjshs.2 shjcl chssi (opr A (+H) B) (opr A (vH) B) occont mp2an ax-mp shjshs.1 shjshs.2 shjcl shjshs.1 shjshs.2 shscl shssi occl chsscon1 mpbi eqssi)) thm (shjshsel () ((shjshs.1 (e. A (SH))) (shjshs.2 (e. B (SH)))) (<-> (e. (opr A (+H) B) (CH)) (= (opr A (+H) B) (opr A (vH) B))) ((opr A (+H) B) ococt shjshs.1 shjshs.2 shjshs syl5req shjshs.1 shjshs.2 shjcl (opr A (+H) B) (opr A (vH) B) (CH) eleq1 mpbiri impbi)) thm (chne0tOLD ((A x)) () (-> (e. A (CH)) (<-> (-. (= A (0H))) (E.e. x A (-. (= (cv x) (0v)))))) (A (if (e. A (CH)) A (0H)) (0H) eqeq1 negbid A (if (e. A (CH)) A (0H)) x (-. (= (cv x) (0v))) rexeq1 bibi12d h0elch A elimel x chne0OLD dedth)) thm (chne0t ((A x)) () (-> (e. A (CH)) (<-> (=/= A (0H)) (E.e. x A (=/= (cv x) (0v))))) (A (if (e. A (CH)) A (0H)) (0H) neeq1 A (if (e. A (CH)) A (0H)) x (=/= (cv x) (0v)) rexeq1 bibi12d h0elch A elimel x chne0 dedth)) thm (chocint () () (-> (e. A (CH)) (= (i^i A (` (_|_) A)) (0H))) ((= A (if (e. A (CH)) A (0H))) id A (if (e. A (CH)) A (0H)) (_|_) fveq2 ineq12d (0H) eqeq1d h0elch A elimel chocin dedth)) thm (chssoct () () (-> (e. A (CH)) (<-> (C_ A (` (_|_) A)) (= A (0H)))) (A chocint A sseq2d A chle0t bitrd A (` (_|_) A) A sslin A inidm syl5ssr syl5bi (e. A (CH)) (= A (0H)) pm3.27 A chocclt (` (_|_) A) ch0let syl (= A (0H)) adantr eqsstrd ex impbid)) thm (chj0t () () (-> (e. A (CH)) (= (opr A (vH) (0H)) A)) (A (if (e. A (CH)) A (0H)) (vH) (0H) opreq1 (= A (if (e. A (CH)) A (0H))) id eqeq12d h0elch A elimel chj0 dedth)) thm (chslejt () () (-> (/\ (e. A (CH)) (e. B (CH))) (C_ (opr A (+H) B) (opr A (vH) B))) (A B shslejt A chsh B chsh syl2an)) thm (chinclt () () (-> (/\ (e. A (CH)) (e. B (CH))) (e. (i^i A B) (CH))) (A (if (e. A (CH)) A (H~)) B ineq1 (CH) eleq1d B (if (e. B (CH)) B (H~)) (if (e. A (CH)) A (H~)) ineq2 (CH) eleq1d helch A elimel helch B elimel chincl dedth2h)) thm (chsscon3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ A B) (C_ (` (_|_) B) (` (_|_) A)))) (A (if (e. A (CH)) A (H~)) B sseq1 A (if (e. A (CH)) A (H~)) (_|_) fveq2 (` (_|_) B) sseq2d bibi12d B (if (e. B (CH)) B (H~)) (if (e. A (CH)) A (H~)) sseq2 B (if (e. B (CH)) B (H~)) (_|_) fveq2 (` (_|_) (if (e. A (CH)) A (H~))) sseq1d bibi12d helch A elimel helch B elimel chsscon3 dedth2h)) thm (chsscon1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ (` (_|_) A) B) (C_ (` (_|_) B) A))) ((` (_|_) A) B chsscon3t A chocclt sylan A ococt (e. B (CH)) adantr (` (_|_) B) sseq2d bitrd)) thm (chsscon2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ A (` (_|_) B)) (C_ B (` (_|_) A)))) (A (` (_|_) B) chsscon3t B chocclt sylan2 B ococt (e. A (CH)) adantl (` (_|_) A) sseq1d bitrd)) thm (chpsscon3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C: A B) (C: (` (_|_) B) (` (_|_) A)))) (A B chsscon3t B A chsscon3t ancoms negbid anbi12d A B dfpss3 (` (_|_) B) (` (_|_) A) dfpss3 3bitr4g)) thm (chpsscon1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C: (` (_|_) A) B) (C: (` (_|_) B) A))) ((` (_|_) A) B chpsscon3t A chocclt sylan A ococt (e. B (CH)) adantr (` (_|_) B) psseq2d bitrd)) thm (chpsscon2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C: A (` (_|_) B)) (C: B (` (_|_) A)))) (A (` (_|_) B) chpsscon3t B chocclt sylan2 B ococt (e. A (CH)) adantl (` (_|_) A) psseq1d bitrd)) thm (chjcomt () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (opr A (vH) B) (opr B (vH) A))) (A B shjcomt A chsh B chsh syl2an)) thm (chub1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (C_ A (opr A (vH) B))) (A B shub1t A chsh B chsh syl2an)) thm (chub2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (C_ A (opr B (vH) A))) (A B chub1t A B chjcomt sseqtrd)) thm (chlubt () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (<-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (vH) B) C))) (A B C shlubt B chsh syl3an2 A chsh syl3an1)) thm (chlej1t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (C_ A B) (C_ (opr A (vH) C) (opr B (vH) C)))) (A B C shlej1t A chsh B chsh C chsh syl3an)) thm (chlej2t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (C_ A B) (C_ (opr C (vH) A) (opr C (vH) B)))) (A B C shlej2t A chsh B chsh C chsh syl3an)) thm (chlejb1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ A B) (= (opr A (vH) B) B))) (A (if (e. A (CH)) A (0H)) B sseq1 A (if (e. A (CH)) A (0H)) (vH) B opreq1 B eqeq1d bibi12d B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) sseq2 B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) (vH) opreq2 (= B (if (e. B (CH)) B (0H))) id eqeq12d bibi12d h0elch A elimel h0elch B elimel chlejb1 dedth2h)) thm (chlejb2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ A B) (= (opr B (vH) A) B))) (A B chlejb1t A B chjcomt B eqeq1d bitrd)) thm (chnlet () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (-. (C_ B A)) (C: A (opr A (vH) B)))) (A (if (e. A (CH)) A (0H)) B sseq2 negbid (= A (if (e. A (CH)) A (0H))) id A (if (e. A (CH)) A (0H)) (vH) B opreq1 psseq12d bibi12d B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) sseq1 negbid B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) (vH) opreq2 (if (e. A (CH)) A (0H)) psseq2d bibi12d h0elch A elimel h0elch B elimel chnle dedth2h)) thm (chjot () () (-> (e. A (CH)) (= (opr A (vH) (` (_|_) A)) (H~))) ((= A (if (e. A (CH)) A (H~))) id A (if (e. A (CH)) A (H~)) (_|_) fveq2 (vH) opreq12d (H~) eqeq1d helch A elimel chjo dedth)) thm (chabs1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (opr A (vH) (i^i A B)) A)) (A ssid A B inss1 pm3.2i A (i^i A B) A chlubt (e. A (CH)) (e. B (CH)) pm3.26 A B chinclt (e. A (CH)) (e. B (CH)) pm3.26 syl3anc mpbii A B chinclt A (i^i A B) chub1t syldan eqssd)) thm (chabs2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (i^i A (opr A (vH) B)) A)) (A B chub1t A ssid jctil A A (opr A (vH) B) ssin sylib A (opr A (vH) B) inss1 jctil (i^i A (opr A (vH) B)) A eqss sylibr)) thm (chabs1 () ((chabs.1 (e. A (CH))) (chabs.2 (e. B (CH)))) (= (opr A (vH) (i^i A B)) A) (chabs.1 chabs.2 A B chabs1t mp2an)) thm (chabs2 () ((chabs.1 (e. A (CH))) (chabs.2 (e. B (CH)))) (= (i^i A (opr A (vH) B)) A) (chabs.1 chabs.2 A B chabs2t mp2an)) thm (chjidmt () () (-> (e. A (CH)) (= (opr A (vH) A) A)) (A A chabs1t anidms A inidm A (vH) opreq2i syl5eqr)) thm (chjidm () ((chjidm.1 (e. A (CH)))) (= (opr A (vH) A) A) (chjidm.1 A chjidmt ax-mp)) thm (chj12 () ((chj12.1 (e. A (CH))) (chj12.2 (e. B (CH))) (chj12.3 (e. C (CH)))) (= (opr A (vH) (opr B (vH) C)) (opr B (vH) (opr A (vH) C))) (chj12.1 chj12.2 chjcom (vH) C opreq1i chj12.1 chj12.2 chj12.3 chjass chj12.2 chj12.1 chj12.3 chjass 3eqtr3)) thm (chj4 () ((chj12.1 (e. A (CH))) (chj12.2 (e. B (CH))) (chj12.3 (e. C (CH))) (chj4.4 (e. D (CH)))) (= (opr (opr A (vH) B) (vH) (opr C (vH) D)) (opr (opr A (vH) C) (vH) (opr B (vH) D))) (chj12.2 chj12.3 chj4.4 chj12 A (vH) opreq2i chj12.1 chj12.2 chj12.3 chj4.4 chjcl chjass chj12.1 chj12.3 chj12.2 chj4.4 chjcl chjass 3eqtr4)) thm (chjjdir () ((chj12.1 (e. A (CH))) (chj12.2 (e. B (CH))) (chj12.3 (e. C (CH)))) (= (opr (opr A (vH) B) (vH) C) (opr (opr A (vH) C) (vH) (opr B (vH) C))) (chj12.3 chjidm (opr A (vH) B) (vH) opreq2i chj12.1 chj12.2 chj12.3 chj12.3 chj4 eqtr3)) thm (chdmm1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (i^i A B)) (opr (` (_|_) A) (vH) (` (_|_) B)))) (A (if (e. A (CH)) A (H~)) B ineq1 (_|_) fveq2d A (if (e. A (CH)) A (H~)) (_|_) fveq2 (vH) (` (_|_) B) opreq1d eqeq12d B (if (e. B (CH)) B (H~)) (if (e. A (CH)) A (H~)) ineq2 (_|_) fveq2d B (if (e. B (CH)) B (H~)) (_|_) fveq2 (` (_|_) (if (e. A (CH)) A (H~))) (vH) opreq2d eqeq12d helch A elimel helch B elimel chdmm1 dedth2h)) thm (chdmm2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (i^i (` (_|_) A) B)) (opr A (vH) (` (_|_) B)))) ((` (_|_) A) B chdmm1t A chocclt sylan A ococt (e. B (CH)) adantr (vH) (` (_|_) B) opreq1d eqtrd)) thm (chdmm3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (i^i A (` (_|_) B))) (opr (` (_|_) A) (vH) B))) (A (` (_|_) B) chdmm1t B chocclt sylan2 B ococt (e. A (CH)) adantl (` (_|_) A) (vH) opreq2d eqtrd)) thm (chdmm4t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (i^i (` (_|_) A) (` (_|_) B))) (opr A (vH) B))) (A (` (_|_) B) chdmm2t B chocclt sylan2 B ococt (e. A (CH)) adantl A (vH) opreq2d eqtrd)) thm (chdmj1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (opr A (vH) B)) (i^i (` (_|_) A) (` (_|_) B)))) (A B chdmm4t (_|_) fveq2d (` (_|_) A) (` (_|_) B) chinclt A chocclt B chocclt syl2an (i^i (` (_|_) A) (` (_|_) B)) ococt syl eqtr3d)) thm (chdmj2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (opr (` (_|_) A) (vH) B)) (i^i A (` (_|_) B)))) ((` (_|_) A) B chdmj1t A chocclt sylan A ococt (` (_|_) B) ineq1d (e. B (CH)) adantr eqtrd)) thm (chdmj3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (opr A (vH) (` (_|_) B))) (i^i (` (_|_) A) B))) (A (` (_|_) B) chdmj1t B chocclt sylan2 B ococt (e. A (CH)) adantl (` (_|_) A) ineq2d eqtrd)) thm (chdmj4t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (opr (` (_|_) A) (vH) (` (_|_) B))) (i^i A B))) (A (` (_|_) B) chdmj2t B chocclt sylan2 B ococt (e. A (CH)) adantl A ineq2d eqtrd)) thm (chjasst () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (= (opr (opr A (vH) B) (vH) C) (opr A (vH) (opr B (vH) C)))) (A (if (e. A (CH)) A (H~)) (vH) B opreq1 (vH) C opreq1d A (if (e. A (CH)) A (H~)) (vH) (opr B (vH) C) opreq1 eqeq12d B (if (e. B (CH)) B (H~)) (if (e. A (CH)) A (H~)) (vH) opreq2 (vH) C opreq1d B (if (e. B (CH)) B (H~)) (vH) C opreq1 (if (e. A (CH)) A (H~)) (vH) opreq2d eqeq12d C (if (e. C (CH)) C (H~)) (opr (if (e. A (CH)) A (H~)) (vH) (if (e. B (CH)) B (H~))) (vH) opreq2 C (if (e. C (CH)) C (H~)) (if (e. B (CH)) B (H~)) (vH) opreq2 (if (e. A (CH)) A (H~)) (vH) opreq2d eqeq12d helch A elimel helch B elimel helch C elimel chjass dedth3h)) thm (chj12t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (= (opr A (vH) (opr B (vH) C)) (opr B (vH) (opr A (vH) C)))) (A B chjcomt (e. C (CH)) 3adant3 (vH) C opreq1d A B C chjasst B A C chjasst 3com12 3eqtr3d)) thm (chj4t () () (-> (/\ (/\ (e. A (CH)) (e. B (CH))) (/\ (e. C (CH)) (e. D (CH)))) (= (opr (opr A (vH) B) (vH) (opr C (vH) D)) (opr (opr A (vH) C) (vH) (opr B (vH) D)))) (B C D chj12t 3expb (e. A (CH)) adantll A (vH) opreq2d A B (opr C (vH) D) chjasst 3expa C D chjclt sylan2 A C (opr B (vH) D) chjasst 3expa B D chjclt sylan2 an4s 3eqtr4d)) thm (ledi () ((ledi.1 (e. A (CH))) (ledi.2 (e. B (CH))) (ledi.3 (e. C (CH)))) (C_ (opr (i^i A B) (vH) (i^i A C)) (i^i A (opr B (vH) C))) (A B inss1 A C inss1 ledi.1 ledi.2 chincl ledi.1 ledi.3 chincl ledi.1 chlubi mp2an A B inss2 A C inss2 ledi.1 ledi.2 chincl ledi.2 ledi.1 ledi.3 chincl ledi.3 chlej12 mp2an ssini)) thm (ledir () ((ledi.1 (e. A (CH))) (ledi.2 (e. B (CH))) (ledi.3 (e. C (CH)))) (C_ (opr (i^i A C) (vH) (i^i B C)) (i^i (opr A (vH) B) C)) (ledi.3 ledi.1 ledi.2 ledi A C incom B C incom (vH) opreq12i (opr A (vH) B) C incom 3sstr4)) thm (lejdi () ((ledi.1 (e. A (CH))) (ledi.2 (e. B (CH))) (ledi.3 (e. C (CH)))) (C_ (opr A (vH) (i^i B C)) (i^i (opr A (vH) B) (opr A (vH) C))) (ledi.1 ledi.2 ledi.3 chincl ledi.1 ledi.2 chjcl ledi.1 ledi.3 chjcl chincl chlub bicomi ledi.1 ledi.2 chub1 ledi.1 ledi.3 chub1 ssini B C inss1 ledi.2 ledi.1 chub2 sstri B C inss2 ledi.3 ledi.1 chub2 sstri ssini mpbir2an)) thm (lejdir () ((ledi.1 (e. A (CH))) (ledi.2 (e. B (CH))) (ledi.3 (e. C (CH)))) (C_ (opr (i^i A B) (vH) C) (i^i (opr A (vH) C) (opr B (vH) C))) (ledi.3 ledi.1 ledi.2 lejdi ledi.1 ledi.2 chincl ledi.3 chjcom ledi.1 ledi.3 chjcom ledi.2 ledi.3 chjcom ineq12i 3sstr4)) thm (ledit () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (C_ (opr (i^i A B) (vH) (i^i A C)) (i^i A (opr B (vH) C)))) (A (if (e. A (CH)) A (0H)) B ineq1 A (if (e. A (CH)) A (0H)) C ineq1 (vH) opreq12d A (if (e. A (CH)) A (0H)) (opr B (vH) C) ineq1 sseq12d B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) ineq2 (vH) (i^i (if (e. A (CH)) A (0H)) C) opreq1d B (if (e. B (CH)) B (0H)) (vH) C opreq1 (if (e. A (CH)) A (0H)) ineq2d sseq12d C (if (e. C (CH)) C (0H)) (if (e. A (CH)) A (0H)) ineq2 (i^i (if (e. A (CH)) A (0H)) (if (e. B (CH)) B (0H))) (vH) opreq2d C (if (e. C (CH)) C (0H)) (if (e. B (CH)) B (0H)) (vH) opreq2 (if (e. A (CH)) A (0H)) ineq2d sseq12d h0elch A elimel h0elch B elimel h0elch C elimel ledi dedth3h)) thm (spansn0 () () (= (` (span) ({} (0v))) (0H)) (df-ch0 (span) fveq2i h0elsh (0H) spanid ax-mp eqtr3)) thm (span0 () () (= (` (span) ({/})) (0H)) (h0elsh shssi (0H) 0ss (0H) ({/}) spanss mp2an h0elsh (0H) spanid ax-mp sseqtr (H~) 0ss ({/}) spanclt ax-mp (` (span) ({/})) sh0let ax-mp eqssi)) thm (elspan ((A x) (B x)) ((elspan.1 (e. B (V)))) (-> (C_ A (H~)) (<-> (e. B (` (span) A)) (A.e. x (SH) (-> (C_ A (cv x)) (e. B (cv x)))))) (A x spanvalt B eleq2d elspan.1 x (SH) (C_ A (cv x)) elintrab syl6bb)) thm (spanun ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((spanun.1 (C_ A (H~))) (spanun.2 (C_ B (H~)))) (= (` (span) (u. A B)) (opr (` (span) A) (+H) (` (span) B))) (spanun.1 A spanclt ax-mp spanun.2 B spanclt ax-mp shscl shssi spanun.1 A spanss2 ax-mp spanun.2 B spanss2 ax-mp A (` (span) A) B (` (span) B) unss12 mp2an spanun.1 A spanclt ax-mp spanun.2 B spanclt ax-mp shunss sstri (opr (` (span) A) (+H) (` (span) B)) (u. A B) spanss mp2an spanun.1 A spanclt ax-mp spanun.2 B spanclt ax-mp shscl (opr (` (span) A) (+H) (` (span) B)) spanid ax-mp sseqtr spanun.1 A spanclt ax-mp spanun.2 B spanclt ax-mp (cv x) z w shsel z (` (span) A) w (` (span) B) (= (cv x) (opr (cv z) (+v) (cv w))) r2ex bitr y (SH) (/\ (-> (C_ A (cv y)) (e. (cv z) (cv y))) (-> (C_ B (cv y)) (e. (cv w) (cv y)))) (= (cv x) (opr (cv z) (+v) (cv w))) r19.27av spanun.1 z visset A y elspan ax-mp spanun.2 w visset B y elspan ax-mp anbi12i y (SH) (-> (C_ A (cv y)) (e. (cv z) (cv y))) (-> (C_ B (cv y)) (e. (cv w) (cv y))) r19.26 bitr4 sylanb (C_ A (cv y)) (e. (cv z) (cv y)) (C_ B (cv y)) (e. (cv w) (cv y)) prth A (cv y) B unss syl5ibr (cv y) (cv z) (cv w) shaddclt sylan9r (cv x) (opr (cv z) (+v) (cv w)) (cv y) eleq1 biimprd sylan9 exp42 imp4c r19.20i spanun.1 spanun.2 unssi x visset (u. A B) y elspan ax-mp sylibr syl z w 19.23aivv sylbi ssriv eqssi)) thm (spanunt () () (-> (/\ (C_ A (H~)) (C_ B (H~))) (= (` (span) (u. A B)) (opr (` (span) A) (+H) (` (span) B)))) (A (if (C_ A (H~)) A (H~)) B uneq1 (span) fveq2d A (if (C_ A (H~)) A (H~)) (span) fveq2 (+H) (` (span) B) opreq1d eqeq12d B (if (C_ B (H~)) B (H~)) (if (C_ A (H~)) A (H~)) uneq2 (span) fveq2d B (if (C_ B (H~)) B (H~)) (span) fveq2 (` (span) (if (C_ A (H~)) A (H~))) (+H) opreq2d eqeq12d A (if (C_ A (H~)) A (H~)) (H~) sseq1 (H~) (if (C_ A (H~)) A (H~)) (H~) sseq1 (H~) ssid elimhyp B (if (C_ B (H~)) B (H~)) (H~) sseq1 (H~) (if (C_ B (H~)) B (H~)) (H~) sseq1 (H~) ssid elimhyp spanun dedth2h)) thm (sshhococ () ((sshjococ.1 (C_ A (H~))) (sshjococ.2 (C_ B (H~)))) (= (opr A (vH) B) (opr (` (_|_) (` (_|_) A)) (vH) (` (_|_) (` (_|_) B)))) (sshjococ.1 A ococss ax-mp sshjococ.2 B ococss ax-mp A (` (_|_) (` (_|_) A)) B (` (_|_) (` (_|_) B)) unss12 mp2an sshjococ.1 sshjococ.2 unssi sshjococ.1 occl choccl chssi sshjococ.2 occl choccl chssi unssi occon2 ax-mp sshjococ.1 sshjococ.2 A B sshjvalt mp2an sshjococ.1 occl choccl sshjococ.2 occl choccl chjval 3sstr4 A B ssun1 sshjococ.1 sshjococ.2 unssi (u. A B) ococss ax-mp sstri sshjococ.1 sshjococ.2 A B sshjvalt mp2an sseqtr4 sshjococ.1 sshjococ.1 sshjococ.2 A B sshjclt mp2an chssi occon2 ax-mp B A ssun2 sshjococ.1 sshjococ.2 unssi (u. A B) ococss ax-mp sstri sshjococ.1 sshjococ.2 A B sshjvalt mp2an sseqtr4 sshjococ.2 sshjococ.1 sshjococ.2 A B sshjclt mp2an chssi occon2 ax-mp sshjococ.1 occl choccl sshjococ.2 occl choccl sshjococ.1 sshjococ.2 A B sshjclt mp2an choccl choccl chlubi mp2an sshjococ.1 sshjococ.2 A B sshjclt mp2an ococ sseqtr eqssi)) thm (hne0 () () (<-> (=/= (H~) (0H)) (E.e. x (H~) (=/= (cv x) (0v)))) (helch x chne0)) thm (chsup0 () () (= (` (\/H) ({/})) (0H)) (({} (0H)) 0ss (CH) 0ss h0elch (0H) (CH) snssi ax-mp ({/}) ({} (0H)) chsupss mp2an ax-mp h0elch (0H) chsupsn ax-mp sseqtr (CH) 0ss ({/}) chsupclt ax-mp chle0 mpbi)) thm (h1deot ((A x) (B x)) ((h1deot.1 (e. B (H~)))) (<-> (e. A (` (_|_) ({} B))) (/\ (e. A (H~)) (= (opr A (.i) B) (0)))) (h1deot.1 B (H~) snssi ax-mp ({} B) A x ocelt ax-mp x ({} B) (= (opr A (.i) (cv x)) (0)) df-ral x B elsn (= (opr A (.i) (cv x)) (0)) imbi1i x albii h1deot.1 elisseti (cv x) B A (.i) opreq2 (0) eqeq1d ceqsalv 3bitr (e. A (H~)) anbi2i bitr)) thm (h1det ((A x) (B x)) ((h1deot.1 (e. B (H~)))) (<-> (e. A (` (_|_) (` (_|_) ({} B)))) (/\ (e. A (H~)) (A.e. x (H~) (-> (= (opr B (.i) (cv x)) (0)) (= (opr A (.i) (cv x)) (0)))))) (h1deot.1 B (H~) snssi ax-mp occl chssi (` (_|_) ({} B)) A x ocelt ax-mp h1deot.1 (cv x) h1deot h1deot.1 (cv x) B orthcom mpan2 pm5.32i bitr (= (opr A (.i) (cv x)) (0)) imbi1i (e. (cv x) (H~)) (= (opr B (.i) (cv x)) (0)) (= (opr A (.i) (cv x)) (0)) impexp bitr ralbii2 (e. A (H~)) anbi2i bitr)) thm (h1did () () (-> (e. A (H~)) (e. A (` (_|_) (` (_|_) ({} A))))) (A (H~) snssi ({} A) ococss syl A (H~) (` (_|_) (` (_|_) ({} A))) snssg mpbird)) thm (h1dn0 () () (-> (/\ (e. A (H~)) (=/= A (0v))) (=/= (` (_|_) (` (_|_) ({} A))) (0H))) ((` (_|_) (` (_|_) ({} A))) (0H) A eleq2 A h1did syl5bi com12 A elch0 syl6ib necon3d imp)) thm (h1dn0OLD () () (-> (/\ (e. A (H~)) (=/= A (0v))) (-. (= (` (_|_) (` (_|_) ({} A))) (0H)))) ((` (_|_) (` (_|_) ({} A))) (0H) A eleq2 A h1did syl5bi com12 A elch0 syl6ib con3d imp A (0v) df-ne sylan2b)) thm (h1dn0OLDOLD () () (-> (/\ (e. A (H~)) (-. (= A (0v)))) (-. (= (` (_|_) (` (_|_) ({} A))) (0H)))) ((` (_|_) (` (_|_) ({} A))) (0H) A eleq2 A h1did syl5bi com12 A elch0 syl6ib con3d imp)) thm (h1de2 ((A x) (B x)) ((h1de2.1 (e. A (H~))) (h1de2.2 (e. B (H~)))) (-> (e. A (` (_|_) (` (_|_) ({} B)))) (= (opr (opr B (.i) B) (.s) A) (opr (opr A (.i) B) (.s) B))) (h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl h1de2.2 (opr (opr B (.i) B) (.s) A) (opr (opr A (.i) B) (.s) B) B his2subt mp3an h1de2.2 h1de2.2 hicl h1de2.1 h1de2.2 hicl mulcom h1de2.2 h1de2.2 hicl h1de2.1 h1de2.2 (opr B (.i) B) A B ax-his3 mp3an h1de2.1 h1de2.2 hicl h1de2.2 h1de2.2 (opr A (.i) B) B B ax-his3 mp3an 3eqtr4 h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.2 hicl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl h1de2.2 hicl subeq0 mpbir eqtr h1de2.2 A x h1det h1de2.1 mpbiran h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl hvsubcl (cv x) (opr (opr (opr B (.i) B) (.s) A) (-v) (opr (opr A (.i) B) (.s) B)) B (.i) opreq2 (0) eqeq1d (cv x) (opr (opr (opr B (.i) B) (.s) A) (-v) (opr (opr A (.i) B) (.s) B)) A (.i) opreq2 (0) eqeq1d imbi12d (H~) rcla4v ax-mp sylbi h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl hvsubcl h1de2.2 (opr (opr (opr B (.i) B) (.s) A) (-v) (opr (opr A (.i) B) (.s) B)) B orthcom mp2an h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl hvsubcl h1de2.1 (opr (opr (opr B (.i) B) (.s) A) (-v) (opr (opr A (.i) B) (.s) B)) A orthcom mp2an 3imtr4g mpi h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl h1de2.1 (opr (opr B (.i) B) (.s) A) (opr (opr A (.i) B) (.s) B) A his2subt mp3an h1de2.2 h1de2.2 hicl h1de2.1 h1de2.1 (opr B (.i) B) A A ax-his3 mp3an h1de2.2 h1de2.2 hicl h1de2.1 h1de2.1 hicl mulcom eqtr h1de2.1 h1de2.2 hicl h1de2.2 h1de2.1 (opr A (.i) B) B A ax-his3 mp3an (-) opreq12i eqtr2 syl5eq h1de2.1 h1de2.1 hicl h1de2.2 h1de2.2 hicl mulcl h1de2.1 h1de2.2 hicl h1de2.2 h1de2.1 hicl mulcl subeq0 sylib eqcomd h1de2.1 h1de2.2 bcseq sylib)) thm (h1de2b () ((h1de2.1 (e. A (H~))) (h1de2.2 (e. B (H~)))) (-> (=/= B (0v)) (<-> (e. A (` (_|_) (` (_|_) ({} B)))) (= A (opr (opr (opr A (.i) B) (/) (opr B (.i) B)) (.s) B)))) (h1de2.2 B his6t ax-mp eqneqi h1de2.1 h1de2.2 h1de2 (=/= (opr B (.i) B) (0)) adantl (opr (1) (/) (opr B (.i) B)) (.s) opreq2d h1de2.2 h1de2.2 hicl recclz h1de2.2 h1de2.2 hicl h1de2.1 (opr (1) (/) (opr B (.i) B)) (opr B (.i) B) A ax-hvmulass mp3an23 syl h1de2.2 h1de2.2 hicl 1cn divcan1z (.s) A opreq1d eqtr3d h1de2.1 A ax-hvmulid ax-mp syl6eq (e. A (` (_|_) (` (_|_) ({} B)))) adantr h1de2.2 h1de2.2 hicl recclz h1de2.1 h1de2.2 hicl h1de2.2 (opr (1) (/) (opr B (.i) B)) (opr A (.i) B) B ax-hvmulass mp3an23 syl h1de2.2 h1de2.2 hicl recclz h1de2.1 h1de2.2 hicl (opr (1) (/) (opr B (.i) B)) (opr A (.i) B) axmulcom mpan2 syl h1de2.1 h1de2.2 hicl h1de2.2 h1de2.2 hicl divrecz eqtr4d (.s) B opreq1d eqtr3d (e. A (` (_|_) (` (_|_) ({} B)))) adantr 3eqtr3d ex A (opr (opr (opr A (.i) B) (/) (opr B (.i) B)) (.s) B) (` (_|_) (` (_|_) ({} B))) eleq1 h1de2.1 h1de2.2 hicl h1de2.2 h1de2.2 hicl divclz h1de2.2 B h1did ax-mp h1de2.2 B (H~) snssi ax-mp occl choccl chshi (` (_|_) (` (_|_) ({} B))) (opr (opr A (.i) B) (/) (opr B (.i) B)) B shmulclt ax-mp mpan2 syl syl5bir com12 impbid sylbir)) thm (h1de2bOLD () ((h1de2.1 (e. A (H~))) (h1de2.2 (e. B (H~)))) (-> (-. (= B (0v))) (<-> (e. A (` (_|_) (` (_|_) ({} B)))) (= A (opr (opr (opr A (.i) B) (/) (opr B (.i) B)) (.s) B)))) ((opr B (.i) B) (0) df-ne h1de2.2 B his6t ax-mp negbii bitr h1de2.1 h1de2.2 h1de2 (=/= (opr B (.i) B) (0)) adantl (opr (1) (/) (opr B (.i) B)) (.s) opreq2d h1de2.2 h1de2.2 hicl recclz h1de2.2 h1de2.2 hicl h1de2.1 (opr (1) (/) (opr B (.i) B)) (opr B (.i) B) A ax-hvmulass mp3an23 syl h1de2.2 h1de2.2 hicl 1cn divcan1z (.s) A opreq1d eqtr3d h1de2.1 A ax-hvmulid ax-mp syl6eq (e. A (` (_|_) (` (_|_) ({} B)))) adantr h1de2.2 h1de2.2 hicl recclz h1de2.1 h1de2.2 hicl h1de2.2 (opr (1) (/) (opr B (.i) B)) (opr A (.i) B) B ax-hvmulass mp3an23 syl h1de2.2 h1de2.2 hicl recclz h1de2.1 h1de2.2 hicl (opr (1) (/) (opr B (.i) B)) (opr A (.i) B) axmulcom mpan2 syl h1de2.1 h1de2.2 hicl h1de2.2 h1de2.2 hicl divrecz eqtr4d (.s) B opreq1d eqtr3d (e. A (` (_|_) (` (_|_) ({} B)))) adantr 3eqtr3d ex A (opr (opr (opr A (.i) B) (/) (opr B (.i) B)) (.s) B) (` (_|_) (` (_|_) ({} B))) eleq1 h1de2.1 h1de2.2 hicl h1de2.2 h1de2.2 hicl divclz h1de2.2 B h1did ax-mp h1de2.2 B (H~) snssi ax-mp occl choccl chshi (` (_|_) (` (_|_) ({} B))) (opr (opr A (.i) B) (/) (opr B (.i) B)) B shmulclt ax-mp mpan2 syl syl5bir com12 impbid sylbir)) thm (h1de2ctlem ((A x) (B x)) ((h1de2.1 (e. A (H~))) (h1de2.2 (e. B (H~)))) (<-> (e. A (` (_|_) (` (_|_) ({} B)))) (E.e. x (CC) (= A (opr (cv x) (.s) B)))) (B (0v) sneq (_|_) fveq2d (_|_) fveq2d A eleq2d h1de2.1 elisseti (0v) elsnc hsn0elch ococ A eleq2i h1de2.2 B ax-hvmul0 ax-mp A eqeq2i 3bitr4r syl6rbbr 0cn (cv x) (0) (.s) B opreq1 A eqeq2d (CC) rcla4ev mpan syl6bir h1de2.1 h1de2.2 h1de2bOLD (cv x) (opr (opr A (.i) B) (/) (opr B (.i) B)) (.s) B opreq1 A eqeq2d (CC) rcla4ev (opr B (.i) B) (0) df-ne h1de2.2 B his6t ax-mp negbii bitr h1de2.1 h1de2.2 hicl h1de2.2 h1de2.2 hicl divclz sylbir sylan ex sylbid pm2.61i A (opr (cv x) (.s) B) (` (_|_) (` (_|_) ({} B))) eleq1 h1de2.2 B h1did ax-mp h1de2.2 B (H~) snssi ax-mp occl choccl chshi (` (_|_) (` (_|_) ({} B))) (cv x) B shmulclt ax-mp mpan2 syl5bir com12 r19.23aiv impbi)) thm (h1de2ct ((A x) (B x)) ((h1de2ct.1 (e. B (H~)))) (<-> (e. A (` (_|_) (` (_|_) ({} B)))) (E.e. x (CC) (= A (opr (cv x) (.s) B)))) (h1de2ct.1 B (H~) snssi ax-mp occl choccl A chel A (opr (cv x) (.s) B) (H~) eleq1 h1de2ct.1 (cv x) B ax-hvmulcl mpan2 syl5bir com12 r19.23aiv A (if (e. A (H~)) A (0v)) (` (_|_) (` (_|_) ({} B))) eleq1 A (if (e. A (H~)) A (0v)) (opr (cv x) (.s) B) eqeq1 x (CC) rexbidv bibi12d ax-hv0cl A elimel h1de2ct.1 x h1de2ctlem dedth pm5.21nii)) thm (spansn ((x y) (x z) (A x) (y z) (A y) (A z)) ((spansn.1 (e. A (H~)))) (= (` (span) ({} A)) (` (_|_) (` (_|_) ({} A)))) (spansn.1 A (H~) snssi ax-mp ({} A) spanssoc ax-mp spansn.1 (cv x) z h1de2ct (cv y) (cv z) A shmulclt (opr (cv z) (.s) A) (cv y) (cv x) eleq1a com12 syl9r exp4a com3r r19.23aiv sylbi imp spansn.1 elisseti (cv y) snss syl5ibr r19.21aiva spansn.1 A (H~) snssi ax-mp x visset ({} A) y elspan ax-mp sylibr ssriv eqssi)) thm (elspansn ((A x) (B x)) ((spansn.1 (e. A (H~)))) (<-> (e. B (` (span) ({} A))) (E.e. x (CC) (= B (opr (cv x) (.s) A)))) (spansn.1 spansn B eleq2i spansn.1 B x h1de2ct bitr)) thm (spansnt () () (-> (e. A (H~)) (= (` (span) ({} A)) (` (_|_) (` (_|_) ({} A))))) (A (if (e. A (H~)) A (0v)) sneq (span) fveq2d A (if (e. A (H~)) A (0v)) sneq (_|_) fveq2d (_|_) fveq2d eqeq12d ax-hv0cl A elimel spansn dedth)) thm (spansncht () () (-> (e. A (H~)) (e. (` (span) ({} A)) (CH))) (A spansnt A (H~) snssi ({} A) occlt (` (_|_) ({} A)) chocclt 3syl eqeltrd)) thm (spansnsht () () (-> (e. A (H~)) (e. (` (span) ({} A)) (SH))) (A spansncht (` (span) ({} A)) chsh syl)) thm (spansnch () ((spansnch.1 (e. A (H~)))) (e. (` (span) ({} A)) (CH)) (spansnch.1 A spansncht ax-mp)) thm (spansnid () () (-> (e. A (H~)) (e. A (` (span) ({} A)))) (A h1did A spansnt eleqtrrd)) thm (spansnmul () () (-> (/\ (e. A (H~)) (e. B (CC))) (e. (opr B (.s) A) (` (span) ({} A)))) ((` (span) ({} A)) B A shmulclt imp an1s A spansnsht A spansnid jca sylan2 ancoms)) thm (elspansnclt () () (-> (/\ (e. A (H~)) (e. B (` (span) ({} A)))) (e. B (H~))) (({} A) B elspanclt A (H~) snssi sylan)) thm (elspansnt ((A x) (B x)) () (-> (e. A (H~)) (<-> (e. B (` (span) ({} A))) (E.e. x (CC) (= B (opr (cv x) (.s) A))))) (A (if (e. A (H~)) A (0v)) sneq (span) fveq2d B eleq2d A (if (e. A (H~)) A (0v)) (cv x) (.s) opreq2 B eqeq2d x (CC) rexbidv bibi12d ax-hv0cl A elimel B x elspansn dedth)) thm (elspansn2t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (=/= B (0v))) (<-> (e. A (` (span) ({} B))) (= A (opr (opr (opr A (.i) B) (/) (opr B (.i) B)) (.s) B)))) (B spansnt A eleq2d (e. A (H~)) (=/= B (0v)) 3ad2ant2 A (if (e. A (H~)) A (0v)) (` (_|_) (` (_|_) ({} B))) eleq1 (= A (if (e. A (H~)) A (0v))) id A (if (e. A (H~)) A (0v)) (.i) B opreq1 (/) (opr B (.i) B) opreq1d (.s) B opreq1d eqeq12d bibi12d (=/= B (0v)) imbi2d B (if (e. B (H~)) B (0v)) (0v) neeq1 B (if (e. B (H~)) B (0v)) sneq (_|_) fveq2d (_|_) fveq2d (if (e. A (H~)) A (0v)) eleq2d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (.i) opreq2 B (if (e. B (H~)) B (0v)) (.i) B opreq1 B (if (e. B (H~)) B (0v)) (if (e. B (H~)) B (0v)) (.i) opreq2 eqtrd (/) opreq12d (= B (if (e. B (H~)) B (0v))) id (.s) opreq12d (if (e. A (H~)) A (0v)) eqeq2d bibi12d imbi12d ax-hv0cl A elimel ax-hv0cl B elimel h1de2b dedth2h 3impia bitrd)) thm (spansncol ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\/\ (e. A (H~)) (e. B (CC)) (=/= B (0))) (= (` (span) ({} (opr B (.s) A))) (` (span) ({} A)))) ((cv y) B A ax-hvmulass 3com13 3expa (cv x) eqeq2d biimprd (cv y) B axmulcl ancoms (e. A (H~)) adantll jctild (cv z) (opr (cv y) (x.) B) (.s) A opreq1 (cv x) eqeq2d (CC) rcla4ev syl6 ex r19.23adv (=/= B (0)) 3adant3 (opr (cv z) (/) B) B A ax-hvmulass (cv z) B divclt 3expb (e. A (H~)) adantlr (e. B (CC)) (=/= B (0)) pm3.26 (/\ (e. (cv z) (CC)) (e. A (H~))) adantl (e. (cv z) (CC)) (e. A (H~)) pm3.27 (/\ (e. B (CC)) (=/= B (0))) adantr syl3anc B (cv z) divcan1t 3exp com12 imp32 (e. A (H~)) adantlr (.s) A opreq1d eqtr3d (cv x) eqeq2d biimprd (cv z) B divclt 3expb (e. A (H~)) adantlr jctild (cv y) (opr (cv z) (/) B) (.s) (opr B (.s) A) opreq1 (cv x) eqeq2d (CC) rcla4ev syl6 exp43 com4l 3imp r19.23adv impbid B A ax-hvmulcl ancoms (opr B (.s) A) (cv x) y elspansnt syl (=/= B (0)) 3adant3 A (cv x) z elspansnt (e. B (CC)) (=/= B (0)) 3ad2ant1 3bitr4d eqrdv)) thm (spansneleqi () () (-> (e. A (H~)) (-> (= (` (span) ({} A)) (` (span) ({} B))) (e. A (` (span) ({} B))))) ((` (span) ({} A)) (` (span) ({} B)) A eleq2 A spansnid syl5bi com12)) thm (spansneleq ((A x) (B x)) () (-> (/\ (e. B (H~)) (=/= A (0v))) (-> (e. A (` (span) ({} B))) (= (` (span) ({} A)) (` (span) ({} B))))) (B A x elspansnt (=/= A (0v)) adantr A (opr (cv x) (.s) B) sneq (span) fveq2d (/\ (e. B (H~)) (=/= A (0v))) (e. (cv x) (CC)) ad2antll (cv x) (0) (.s) B opreq1 B ax-hvmul0 sylan9eqr ex A (opr (cv x) (.s) B) (0v) eqeq1 biimprd sylan9 necon3d ex com23 imp3a (e. (cv x) (CC)) adantr B (cv x) spansncol 3exp imp syld exp4b com23 imp43 eqtrd exp32 r19.23adv sylbid)) thm (spansneleqOLD ((A x) (B x)) () (-> (/\ (e. B (H~)) (-. (= A (0v)))) (-> (e. A (` (span) ({} B))) (= (` (span) ({} A)) (` (span) ({} B))))) (B A x elspansnt (-. (= A (0v))) adantr A (opr (cv x) (.s) B) sneq (span) fveq2d (/\ (e. B (H~)) (-. (= A (0v)))) (e. (cv x) (CC)) ad2antll (cv x) (0) (.s) B opreq1 B ax-hvmul0 sylan9eqr ex A (opr (cv x) (.s) B) (0v) eqeq1 biimprd sylan9 con3d ex com23 imp3a (cv x) (0) df-ne syl6ibr (e. (cv x) (CC)) adantr B (cv x) spansncol 3exp imp syld exp4b com23 imp43 eqtrd exp32 r19.23adv sylbid)) thm (spansnsst ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (SH)) (e. B A)) (C_ (` (span) ({} B)) A)) (A B shelt B (cv x) y elspansnt syl A (cv y) B shmulclt (cv x) (opr (cv y) (.s) B) A eleq1 biimprd syl9 com23 exp3a com23 imp r19.23adv sylbid ssrdv)) thm (elspansn3t () () (-> (/\/\ (e. A (SH)) (e. B A) (e. C (` (span) ({} B)))) (e. C A)) (A B spansnsst C sseld 3impia)) thm (elspansn3tOLD () () (-> (e. A (SH)) (-> (/\ (e. B A) (e. C (` (span) ({} B)))) (e. C A))) (A B spansnsst C sseld ex imp3a)) thm (elspansn4t () () (-> (/\ (/\ (e. A (SH)) (e. B (H~))) (/\ (e. C (` (span) ({} B))) (-. (= C (0v))))) (<-> (e. B A) (e. C A))) (A B C elspansn3tOLD exp3a com23 imp (e. B (H~)) (-. (= C (0v))) ad2ant2r A C B elspansn3tOLD exp3a imp B spansnid (-. (= C (0v))) (e. C (` (span) ({} B))) ad2antrr B C spansneleqOLD imp eleqtrrd syl5 exp4b com24 exp4a com23 imp43 impbid)) thm (elspansn5t () () (-> (e. A (SH)) (-> (/\ (/\ (e. B (H~)) (-. (e. B A))) (/\ (e. C (` (span) ({} B))) (e. C A))) (= C (0v)))) (A B C elspansn4t biimprd exp32 com34 imp3a ex (= C (0v)) (e. B A) con1 syl8 com34 imp4c)) thm (spansnss2t () () (-> (/\ (e. A (SH)) (e. B (H~))) (<-> (e. B A) (C_ (` (span) ({} B)) A))) (A B spansnsst ex (e. B (H~)) adantr (` (span) ({} B)) A B ssel B spansnid syl5com (e. A (SH)) adantl impbid)) thm (normcant ((A x) (B x)) () (-> (/\/\ (e. B (H~)) (=/= B (0v)) (e. A (` (span) ({} B)))) (= (opr (opr (opr A (.i) B) (/) (opr (` (norm) B) (^) (2))) (.s) B) A)) (B A x elspansnt (=/= B (0v)) adantr A (opr (cv x) (.s) B) (.i) B opreq1 (cv x) B B ax-his3 (e. B (H~)) (e. (cv x) (CC)) pm3.27 (e. B (H~)) (e. (cv x) (CC)) pm3.26 (e. B (H~)) (e. (cv x) (CC)) pm3.26 syl3anc sylan9eqr B normsqt (e. (cv x) (CC)) (= A (opr (cv x) (.s) B)) ad2antrr (/) opreq12d (=/= B (0v)) adantllr (opr B (.i) B) (cv x) divcan4t B B ax-hicl anidms (=/= B (0v)) (e. (cv x) (CC)) ad2antrr (/\ (e. B (H~)) (=/= B (0v))) (e. (cv x) (CC)) pm3.27 B his6t eqneqd biimpar (e. (cv x) (CC)) adantr syl3anc (= A (opr (cv x) (.s) B)) adantr eqtrd (.s) B opreq1d (/\ (/\ (e. B (H~)) (=/= B (0v))) (e. (cv x) (CC))) (= A (opr (cv x) (.s) B)) pm3.27 eqtr4d exp31 r19.23adv sylbid 3impia)) thm (pjspansnt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\/\ (e. A (H~)) (e. B (H~)) (=/= A (0v))) (= (` (` (proj) (` (span) ({} A))) B) (opr (opr (opr B (.i) A) (/) (opr (` (norm) A) (^) (2))) (.s) A))) ((` (span) ({} A)) B x y pjvalt A spansncht sylan (=/= A (0v)) 3adant3 B (opr (cv x) (+v) (cv y)) (.i) A opreq1 (/\ (e. A (H~)) (e. (cv x) (` (span) ({} A)))) (e. (cv y) (` (_|_) (` (span) ({} A)))) ad2antll (` (_|_) (` (span) ({} A))) (cv y) chelt A spansncht (` (span) ({} A)) chocclt syl sylan (e. (cv x) (H~)) adantlr (cv x) (cv y) A ax-his2 3comr 3expa syldan (` (span) ({} A)) A (cv y) shocorth 3impib A spansnsht (e. (cv y) (` (_|_) (` (span) ({} A)))) adantr A spansnid (e. (cv y) (` (_|_) (` (span) ({} A)))) adantr (e. A (H~)) (e. (cv y) (` (_|_) (` (span) ({} A)))) pm3.27 syl3anc (` (_|_) (` (span) ({} A))) (cv y) chelt A spansncht (` (span) ({} A)) chocclt syl sylan A (cv y) orthcom syldan mpbid (e. (cv x) (H~)) adantlr (opr (cv x) (.i) A) (+) opreq2d (cv x) A ax-hicl ancoms (opr (cv x) (.i) A) ax0id syl (e. (cv y) (` (_|_) (` (span) ({} A)))) adantr 3eqtrd (e. A (H~)) (e. (cv x) (` (span) ({} A))) pm3.26 A (cv x) elspansnclt jca sylan (= B (opr (cv x) (+v) (cv y))) adantrr eqtrd (=/= A (0v)) adantllr (/) (opr (` (norm) A) (^) (2)) opreq1d (.s) A opreq1d A (cv x) normcant 3expa (/\ (e. (cv y) (` (_|_) (` (span) ({} A)))) (= B (opr (cv x) (+v) (cv y)))) adantr eqtr2d exp32 r19.23adv (e. B (H~)) 3adantl2 ss2rabdv (` (span) ({} A)) B x y pjthut A spansncht sylan (=/= A (0v)) 3adant3 x (` (span) ({} A)) (E.e. y (` (_|_) (` (span) ({} A))) (= B (opr (cv x) (+v) (cv y)))) reuunisn syl A (opr (opr B (.i) A) (/) (opr (` (norm) A) (^) (2))) spansnmul (e. A (H~)) (e. B (H~)) (=/= A (0v)) 3simp1 (opr B (.i) A) (opr (` (norm) A) (^) (2)) divclt B A ax-hicl ancoms (=/= A (0v)) 3adant3 A normclt recnd (` (norm) A) sqclt syl (e. B (H~)) (=/= A (0v)) 3ad2ant1 A normclt recnd (` (norm) A) sqne0t syl A normne0t bitr2d biimpa (e. B (H~)) 3adant2 syl3anc sylanc (opr (opr (opr B (.i) A) (/) (opr (` (norm) A) (^) (2))) (.s) A) (` (span) ({} A)) x rabsn syl 3sstr3d (span) ({} A) fvex x (E.e. y (` (_|_) (` (span) ({} A))) (= B (opr (cv x) (+v) (cv y)))) rabex uniex (opr (opr (opr B (.i) A) (/) (opr (` (norm) A) (^) (2))) (.s) A) snsssn syl eqtrd)) thm (spansnpj () ((spansnpj.1 (C_ A (H~))) (spansnpj.2 (e. B (H~)))) (C_ A (` (_|_) (` (span) ({} (` (` (proj) (` (_|_) A)) B))))) (spansnpj.1 A ococss ax-mp spansnpj.1 occl chssi spansnpj.1 occl spansnpj.2 pjcli (` (` (proj) (` (_|_) A)) B) (` (_|_) A) snssi ax-mp (` (_|_) A) ({} (` (` (proj) (` (_|_) A)) B)) spanss mp2an spansnpj.1 occl chshi (` (_|_) A) spanid ax-mp sseqtr spansnpj.1 occl spansnpj.2 pjhcli spansnch spansnpj.1 occl chsscon3 mpbi sstri)) thm (spanunsn ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) ((spanunsn.1 (e. A (CH))) (spanunsn.2 (e. B (H~)))) (= (` (span) (u. A ({} B))) (` (span) (u. A ({} (` (` (proj) (` (_|_) A)) B))))) (spanunsn.1 chshi spanunsn.2 B (H~) snssi ax-mp ({} B) spanclt ax-mp (cv x) y z shsel (cv x) (opr (cv y) (+v) (opr (cv w) (.s) B)) (opr A (+H) (` (span) ({} (` (` (proj) (` (_|_) A)) B)))) eleq1 biimparc (cv v) (opr (cv y) (+v) (opr (cv w) (.s) (` (` (proj) A) B))) (+v) (cv u) opreq1 (opr (cv y) (+v) (opr (cv w) (.s) B)) eqeq2d (cv u) (opr (cv w) (.s) (` (` (proj) (` (_|_) A)) B)) (opr (cv y) (+v) (opr (cv w) (.s) (` (` (proj) A) B))) (+v) opreq2 (opr (cv y) (+v) (opr (cv w) (.s) B)) eqeq2d A (` (span) ({} (` (` (proj) (` (_|_) A)) B))) rcla42ev spanunsn.1 chshi A (cv y) (opr (cv w) (.s) (` (` (proj) A) B)) shaddclt spanunsn.1 spanunsn.2 pjcli spanunsn.1 chshi A (cv w) (` (` (proj) A) B) shmulclt ax-mp mpan2 sylan2i ax-mp spanunsn.1 choccl spanunsn.2 pjhcli (` (` (proj) (` (_|_) A)) B) (cv w) spansnmul mpan (e. (cv y) A) adantl jca spanunsn.1 spanunsn.2 pjhcli spanunsn.1 choccl spanunsn.2 pjhcli (cv w) (` (` (proj) A) B) (` (` (proj) (` (_|_) A)) B) ax-hvdistr1 mp3an23 spanunsn.1 spanunsn.2 pjpj (cv w) (.s) opreq2i syl5eq (e. (cv y) A) adantl (cv y) (+v) opreq2d (cv y) (opr (cv w) (.s) (` (` (proj) A) B)) (opr (cv w) (.s) (` (` (proj) (` (_|_) A)) B)) ax-hvass 3expb spanunsn.1 (cv y) chel spanunsn.1 spanunsn.2 pjhcli (cv w) (` (` (proj) A) B) ax-hvmulcl mpan2 spanunsn.1 choccl spanunsn.2 pjhcli (cv w) (` (` (proj) (` (_|_) A)) B) ax-hvmulcl mpan2 jca syl2an eqtr4d sylanc spanunsn.1 chshi spanunsn.1 choccl spanunsn.2 pjhcli (` (` (proj) (` (_|_) A)) B) (H~) snssi ax-mp ({} (` (` (proj) (` (_|_) A)) B)) spanclt ax-mp (opr (cv y) (+v) (opr (cv w) (.s) B)) v u shsel sylibr (cv z) (opr (cv w) (.s) B) (cv y) (+v) opreq2 (cv x) eqeq2d biimpa syl2an exp43 r19.23adv spanunsn.2 (cv z) w elspansn syl5ib r19.23adv r19.23aiv sylbi spanunsn.1 chshi spanunsn.1 choccl spanunsn.2 pjhcli (` (` (proj) (` (_|_) A)) B) (H~) snssi ax-mp ({} (` (` (proj) (` (_|_) A)) B)) spanclt ax-mp (cv x) y z shsel (cv x) (opr (cv y) (+v) (opr (cv w) (.s) (` (` (proj) (` (_|_) A)) B))) (opr A (+H) (` (span) ({} B))) eleq1 biimparc (cv v) (opr (opr (-u (cv w)) (.s) (` (` (proj) A) B)) (+v) (cv y)) (+v) (cv u) opreq1 (opr (cv y) (+v) (opr (cv w) (.s) (` (` (proj) (` (_|_) A)) B))) eqeq2d (cv u) (opr (cv w) (.s) B) (opr (opr (-u (cv w)) (.s) (` (` (proj) A) B)) (+v) (cv y)) (+v) opreq2 (opr (cv y) (+v) (opr (cv w) (.s) (` (` (proj) (` (_|_) A)) B))) eqeq2d A (` (span) ({} B)) rcla42ev spanunsn.1 chshi A (opr (-u (cv w)) (.s) (` (` (proj) A) B)) (cv y) shaddclt (cv w) negclt spanunsn.1 spanunsn.2 pjcli spanunsn.1 chshi A (-u (cv w)) (` (` (proj) A) B) shmulclt ax-mp mpan2 syl sylani ax-mp ancoms spanunsn.2 B (cv w) spansnmul mpan (e. (cv y) A) adantl jca spanunsn.1 spanunsn.2 pjhcli (cv w) (` (` (proj) A) B) hvm1negt mpan2 (opr (cv w) (.s) (` (` (proj) A) B)) (+v) opreq2d spanunsn.1 spanunsn.2 pjhcli (cv w) (` (` (proj) A) B) ax-hvmulcl mpan2 (opr (cv w) (.s) (` (` (proj) A) B)) hvnegidt syl (opr (cv w) (.s) (` (` (proj) A) B)) (opr (-u (cv w)) (.s) (` (` (proj) A) B)) ax-hvcom spanunsn.1 spanunsn.2 pjhcli (cv w) (` (` (proj) A) B) ax-hvmulcl mpan2 (cv w) negclt spanunsn.1 spanunsn.2 pjhcli (-u (cv w)) (` (` (proj) A) B) ax-hvmulcl mpan2 syl sylanc 3eqtr3d (e. (cv y) A) adantl (+v) (opr (cv y) (+v) (opr (cv w) (.s) (` (` (proj) (` (_|_) A)) B))) opreq1d (cv y) (opr (cv w) (.s) (` (` (proj) (` (_|_) A)) B)) ax-hvaddcl spanunsn.1 (cv y) chel spanunsn.1 choccl spanunsn.2 pjhcli (cv w) (` (` (proj) (` (_|_) A)) B) ax-hvmulcl mpan2 syl2an (opr (cv y) (+v) (opr (cv w) (.s) (` (` (proj) (` (_|_) A)) B))) hvaddid2t syl (opr (-u (cv w)) (.s) (` (` (proj) A) B)) (opr (cv w) (.s) (` (` (proj) A) B)) (cv y) (opr (cv w) (.s) (` (` (proj) (` (_|_) A)) B)) hvadd4t (cv w) negclt spanunsn.1 spanunsn.2 pjhcli (-u (cv w)) (` (` (proj) A) B) ax-hvmulcl mpan2 syl spanunsn.1 spanunsn.2 pjhcli (cv w) (` (` (proj) A) B) ax-hvmulcl mpan2 jca (e. (cv y) A) adantl spanunsn.1 (cv y) chel spanunsn.1 choccl spanunsn.2 pjhcli (cv w) (` (` (proj) (` (_|_) A)) B) ax-hvmulcl mpan2 anim12i sylanc 3eqtr3d spanunsn.1 spanunsn.2 pjhcli spanunsn.1 choccl spanunsn.2 pjhcli (cv w) (` (` (proj) A) B) (` (` (proj) (` (_|_) A)) B) ax-hvdistr1 mp3an23 spanunsn.1 spanunsn.2 pjpj (cv w) (.s) opreq2i syl5eq (e. (cv y) A) adantl (opr (opr (-u (cv w)) (.s) (` (` (proj) A) B)) (+v) (cv y)) (+v) opreq2d eqtr4d sylanc spanunsn.1 chshi spanunsn.2 B (H~) snssi ax-mp ({} B) spanclt ax-mp (opr (cv y) (+v) (opr (cv w) (.s) (` (` (proj) (` (_|_) A)) B))) v u shsel sylibr (cv z) (opr (cv w) (.s) (` (` (proj) (` (_|_) A)) B)) (cv y) (+v) opreq2 (cv x) eqeq2d biimpa syl2an exp43 r19.23adv spanunsn.1 choccl spanunsn.2 pjhcli (cv z) w elspansn syl5ib r19.23adv r19.23aiv sylbi impbi eqriv spanunsn.1 chssi spanunsn.2 B (H~) snssi ax-mp spanun spanunsn.1 chshi A spanid ax-mp (+H) (` (span) ({} B)) opreq1i eqtr spanunsn.1 chssi spanunsn.1 choccl spanunsn.2 pjhcli (` (` (proj) (` (_|_) A)) B) (H~) snssi ax-mp spanun spanunsn.1 chshi A spanid ax-mp (+H) (` (span) ({} (` (` (proj) (` (_|_) A)) B))) opreq1i eqtr 3eqtr4)) thm (spanpr ((A x) (B x)) () (-> (/\ (e. A (H~)) (e. B (H~))) (C_ (` (span) ({} (opr A (+v) B))) (` (span) ({,} A B)))) ((opr (` (span) ({} A)) (+H) (` (span) ({} B))) (opr A (+v) B) (cv x) elspansn3t (` (span) ({} A)) (` (span) ({} B)) shsclt A spansnsht B spansnsht syl2an (e. (cv x) (` (span) ({} (opr A (+v) B)))) adantr (` (span) ({} A)) (` (span) ({} B)) A B shsvat A spansnsht B spansnsht anim12i A spansnid B spansnid anim12i sylc (e. (cv x) (` (span) ({} (opr A (+v) B)))) adantr (/\ (e. A (H~)) (e. B (H~))) (e. (cv x) (` (span) ({} (opr A (+v) B)))) pm3.27 syl3anc ex ssrdv ({} A) ({} B) spanunt A (H~) snssi B (H~) snssi syl2an A B df-pr (span) fveq2i syl5req sseqtrd)) thm (h1datom ((x y) (A x) (A y) (B x) (B y)) ((h1datom.1 (e. A (CH))) (h1datom.2 (e. B (H~)))) (-> (C_ A (` (_|_) (` (_|_) ({} B)))) (\/ (= A (` (_|_) (` (_|_) ({} B)))) (= A (0H)))) (A (` (_|_) (` (_|_) ({} B))) (cv x) ssel (cv x) (opr (cv y) (.s) B) (0v) eqeq1 (cv y) (0) (.s) B opreq1 h1datom.2 B ax-hvmul0 ax-mp syl6eq syl5bir con3d (cv y) (0) df-ne syl6ibr (e. (cv y) (CC)) adantl (cv y) recclt h1datom.1 chshi A (opr (1) (/) (cv y)) (cv x) shmulclt ax-mp ex syl (= (cv x) (opr (cv y) (.s) B)) adantr (cv x) (opr (cv y) (.s) B) (opr (1) (/) (cv y)) (.s) opreq2 h1datom.2 (opr (1) (/) (cv y)) (cv y) B ax-hvmulass mp3an3 (cv y) recclt (e. (cv y) (CC)) (=/= (cv y) (0)) pm3.26 sylanc (cv y) recid2t (.s) B opreq1d eqtr3d h1datom.2 B ax-hvmulid ax-mp syl6eq sylan9eqr A eleq1d sylibd exp31 com23 imp syld com3r exp3a r19.23adv h1datom.2 (cv x) y h1de2ct syl5ib sylcom r19.23adv h1datom.1 x chne0OLD syl5ib B A snssi h1datom.2 B (H~) snssi ax-mp h1datom.1 chssi occon2 syl h1datom.1 ococ syl6ss syl6 anc2li A (` (_|_) (` (_|_) ({} B))) eqss syl6ibr con1d orrd)) thm (h1datomt () () (-> (/\ (e. A (CH)) (e. B (H~))) (-> (C_ A (` (_|_) (` (_|_) ({} B)))) (\/ (= A (` (_|_) (` (_|_) ({} B)))) (= A (0H))))) (A (if (e. A (CH)) A (0H)) (` (_|_) (` (_|_) ({} B))) sseq1 A (if (e. A (CH)) A (0H)) (` (_|_) (` (_|_) ({} B))) eqeq1 A (if (e. A (CH)) A (0H)) (0H) eqeq1 orbi12d imbi12d B (if (e. B (H~)) B (0v)) sneq (_|_) fveq2d (_|_) fveq2d (if (e. A (CH)) A (0H)) sseq2d B (if (e. B (H~)) B (0v)) sneq (_|_) fveq2d (_|_) fveq2d (if (e. A (CH)) A (0H)) eqeq2d (= (if (e. A (CH)) A (0H)) (0H)) orbi1d imbi12d h0elch A elimel ax-hv0cl B elimel h1datom dedth2h)) thm (hosmvalt ((f g) (f h) (f x) (f y) (g h) (g x) (g y) (h x) (h y) (x y) (S x) (S y) (S f) (S g) (S h) (T x) (T y) (T f) (T g) (T h)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (+op) T) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (+v) (` T (cv x)))))))) (ax-hilex x y (opr (` S (cv x)) (+v) (` T (cv x))) funopabex2 (cv f) S (cv x) fveq1 (+v) (` (cv g) (cv x)) opreq1d (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv (cv g) T (cv x) fveq1 (` S (cv x)) (+v) opreq2d (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv f g h x y df-hosum ax-hilex ax-hilex (cv f) elmap ax-hilex ax-hilex (cv g) elmap anbi12i (= (cv h) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` (cv f) (cv x)) (+v) (` (cv g) (cv x))))))) anbi1i f g h oprabbii eqtr4 oprabval2 ax-hilex ax-hilex S elmap ax-hilex ax-hilex T elmap syl2anbr)) thm (hommvalt ((f g) (f h) (f x) (f y) (A f) (g h) (g x) (g y) (A g) (h x) (h y) (A h) (x y) (A x) (A y) (T x) (T y) (T f) (T g) (T h)) () (-> (/\ (e. A (CC)) (:--> T (H~) (H~))) (= (opr A (.op) T) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr A (.s) (` T (cv x)))))))) (ax-hilex x y (opr A (.s) (` T (cv x))) funopabex2 (cv f) A (.s) (` (cv g) (cv x)) opreq1 (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv (cv g) T (cv x) fveq1 A (.s) opreq2d (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv f g h x y df-homul ax-hilex ax-hilex (cv g) elmap (e. (cv f) (CC)) anbi2i (= (cv h) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (cv f) (.s) (` (cv g) (cv x))))))) anbi1i f g h oprabbii eqtr4 oprabval2 ax-hilex ax-hilex T elmap sylan2br)) thm (hodmvalt ((f g) (f h) (f x) (f y) (g h) (g x) (g y) (h x) (h y) (x y) (S x) (S y) (S f) (S g) (S h) (T x) (T y) (T f) (T g) (T h)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (-op) T) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (-v) (` T (cv x)))))))) (ax-hilex x y (opr (` S (cv x)) (-v) (` T (cv x))) funopabex2 (cv f) S (cv x) fveq1 (-v) (` (cv g) (cv x)) opreq1d (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv (cv g) T (cv x) fveq1 (` S (cv x)) (-v) opreq2d (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv f g h x y df-hodif ax-hilex ax-hilex (cv f) elmap ax-hilex ax-hilex (cv g) elmap anbi12i (= (cv h) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` (cv f) (cv x)) (-v) (` (cv g) (cv x))))))) anbi1i f g h oprabbii eqtr4 oprabval2 ax-hilex ax-hilex S elmap ax-hilex ax-hilex T elmap syl2anbr)) thm (hfsmvalt ((f g) (f h) (f x) (f y) (g h) (g x) (g y) (h x) (h y) (x y) (S x) (S y) (S f) (S g) (S h) (T x) (T y) (T f) (T g) (T h)) () (-> (/\ (:--> S (H~) (CC)) (:--> T (H~) (CC))) (= (opr S (+fn) T) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (+) (` T (cv x)))))))) (ax-hilex x y (opr (` S (cv x)) (+) (` T (cv x))) funopabex2 (cv f) S (cv x) fveq1 (+) (` (cv g) (cv x)) opreq1d (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv (cv g) T (cv x) fveq1 (` S (cv x)) (+) opreq2d (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv f g h x y df-hfsum axcnex ax-hilex (cv f) elmap axcnex ax-hilex (cv g) elmap anbi12i (= (cv h) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` (cv f) (cv x)) (+) (` (cv g) (cv x))))))) anbi1i f g h oprabbii eqtr4 oprabval2 axcnex ax-hilex S elmap axcnex ax-hilex T elmap syl2anbr)) thm (hfmmvalt ((f g) (f h) (f x) (f y) (A f) (g h) (g x) (g y) (A g) (h x) (h y) (A h) (x y) (A x) (A y) (T x) (T y) (T f) (T g) (T h)) () (-> (/\ (e. A (CC)) (:--> T (H~) (CC))) (= (opr A (.fn) T) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr A (x.) (` T (cv x)))))))) (ax-hilex x y (opr A (x.) (` T (cv x))) funopabex2 (cv f) A (x.) (` (cv g) (cv x)) opreq1 (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv (cv g) T (cv x) fveq1 A (x.) opreq2d (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv f g h x y df-hfmul axcnex ax-hilex (cv g) elmap (e. (cv f) (CC)) anbi2i (= (cv h) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (cv f) (x.) (` (cv g) (cv x))))))) anbi1i f g h oprabbii eqtr4 oprabval2 axcnex ax-hilex T elmap sylan2br)) thm (hosvalt ((x y) (A x) (A y) (S x) (S y) (T x) (T y)) () (-> (/\/\ (:--> S (H~) (H~)) (:--> T (H~) (H~)) (e. A (H~))) (= (` (opr S (+op) T) A) (opr (` S A) (+v) (` T A)))) (S T x y hosmvalt A fveq1d (cv x) A S fveq2 (cv x) A T fveq2 (+v) opreq12d ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (+v) (` T (cv x)))))) eqid (` S A) (+v) (` T A) oprex fvopab4 sylan9eq 3impa)) thm (hosvaltOLD ((x y) (A x) (A y) (S x) (S y) (T x) (T y)) () (-> (/\ (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (e. A (H~))) (= (` (opr S (+op) T) A) (opr (` S A) (+v) (` T A)))) (S T x y hosmvalt A fveq1d (cv x) A S fveq2 (cv x) A T fveq2 (+v) opreq12d ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (+v) (` T (cv x)))))) eqid (` S A) (+v) (` T A) oprex fvopab4 sylan9eq)) thm (homvalt ((x y) (A x) (A y) (B x) (B y) (T x) (T y)) () (-> (/\/\ (e. A (CC)) (:--> T (H~) (H~)) (e. B (H~))) (= (` (opr A (.op) T) B) (opr A (.s) (` T B)))) (A T x y hommvalt B fveq1d (cv x) B T fveq2 A (.s) opreq2d ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr A (.s) (` T (cv x)))))) eqid A (.s) (` T B) oprex fvopab4 sylan9eq 3impa)) thm (hodvalt ((x y) (A x) (A y) (S x) (S y) (T x) (T y)) () (-> (/\/\ (:--> S (H~) (H~)) (:--> T (H~) (H~)) (e. A (H~))) (= (` (opr S (-op) T) A) (opr (` S A) (-v) (` T A)))) (S T x y hodmvalt A fveq1d (cv x) A S fveq2 (cv x) A T fveq2 (-v) opreq12d ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (-v) (` T (cv x)))))) eqid (` S A) (-v) (` T A) oprex fvopab4 sylan9eq 3impa)) thm (hodvaltOLD ((x y) (A x) (A y) (S x) (S y) (T x) (T y)) () (-> (/\ (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (e. A (H~))) (= (` (opr S (-op) T) A) (opr (` S A) (-v) (` T A)))) (S T x y hodmvalt A fveq1d (cv x) A S fveq2 (cv x) A T fveq2 (-v) opreq12d ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (-v) (` T (cv x)))))) eqid (` S A) (-v) (` T A) oprex fvopab4 sylan9eq)) thm (hfsvalt ((x y) (A x) (A y) (S x) (S y) (T x) (T y)) () (-> (/\ (/\ (:--> S (H~) (CC)) (:--> T (H~) (CC))) (e. A (H~))) (= (` (opr S (+fn) T) A) (opr (` S A) (+) (` T A)))) (S T x y hfsmvalt A fveq1d (cv x) A S fveq2 (cv x) A T fveq2 (+) opreq12d ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (+) (` T (cv x)))))) eqid (` S A) (+) (` T A) oprex fvopab4 sylan9eq)) thm (hfmvalt ((x y) (A x) (A y) (B x) (B y) (T x) (T y)) () (-> (/\/\ (e. A (CC)) (:--> T (H~) (CC)) (e. B (H~))) (= (` (opr A (.fn) T) B) (opr A (x.) (` T B)))) (A T x y hfmmvalt B fveq1d (cv x) B T fveq2 A (x.) opreq2d ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr A (x.) (` T (cv x)))))) eqid A (x.) (` T B) oprex fvopab4 sylan9eq 3impa)) thm (hosclt () () (-> (/\ (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (e. A (H~))) (e. (` (opr S (+op) T) A) (H~))) (S T A hosvaltOLD S (H~) (H~) A ffvrn T (H~) (H~) A ffvrn anim12i anandirs (` S A) (` T A) ax-hvaddcl syl eqeltrd)) thm (homclt () () (-> (/\/\ (e. A (CC)) (:--> T (H~) (H~)) (e. B (H~))) (e. (` (opr A (.op) T) B) (H~))) (A T B homvalt T (H~) (H~) B ffvrn (e. A (CC)) anim2i 3impb A (` T B) ax-hvmulcl syl eqeltrd)) thm (hodclt () () (-> (/\ (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (e. A (H~))) (e. (` (opr S (-op) T) A) (H~))) (S T A hodvaltOLD S (H~) (H~) A ffvrn T (H~) (H~) A ffvrn anim12i anandirs (` S A) (` T A) hvsubclt syl eqeltrd)) thm (cmbrt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (C_H) B) (= A (opr (i^i A B) (vH) (i^i A (` (_|_) B)))))) ((cv x) A (CH) eleq1 (e. (cv y) (CH)) anbi1d (= (cv x) A) id (cv x) A (cv y) ineq1 (cv x) A (` (_|_) (cv y)) ineq1 (vH) opreq12d eqeq12d anbi12d (cv y) B (CH) eleq1 (e. A (CH)) anbi2d (cv y) B A ineq2 (cv y) B (_|_) fveq2 A ineq2d (vH) opreq12d A eqeq2d anbi12d x y df-cm (CH) (CH) brabg (/\ (e. A (CH)) (e. B (CH))) (= A (opr (i^i A B) (vH) (i^i A (` (_|_) B)))) ibar bitr4d)) thm (pjoml2 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (-> (C_ A B) (= (opr A (vH) (i^i (` (_|_) A) B)) B)) ((` (_|_) A) B inss2 pjoml2.1 pjoml2.1 choccl pjoml2.2 chincl pjoml2.2 chlubi mpan2 pjoml2.1 pjoml2.1 choccl pjoml2.2 chincl chdmj1 B ineq2i B (` (_|_) A) (` (_|_) (i^i (` (_|_) A) B)) inass B (` (_|_) A) incom (` (_|_) (i^i (` (_|_) A) B)) ineq1i eqtr3 pjoml2.1 choccl pjoml2.2 chincl chocin 3eqtr pjoml2.1 pjoml2.1 choccl pjoml2.2 chincl chjcl pjoml2.2 chshi pjoml mpan2 syl)) thm (pjoml3 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (-> (C_ B A) (= (i^i A (opr (` (_|_) A) (vH) B)) B)) (pjoml2.1 choccl pjoml2.2 choccl pjoml2 pjoml2.2 pjoml2.1 chsscon3 (` (_|_) (opr (` (_|_) A) (vH) (i^i (` (_|_) (` (_|_) A)) (` (_|_) B)))) B eqcom pjoml2.1 pjoml2.1 choccl choccl pjoml2.2 choccl chincl chdmj2 pjoml2.1 choccl pjoml2.2 chdmm4 A ineq2i eqtr B eqeq1i pjoml2.2 pjoml2.1 choccl pjoml2.1 choccl choccl pjoml2.2 choccl chincl chjcl chcon2 3bitr3 3imtr4)) thm (pjoml4 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (= (opr A (vH) (i^i B (opr (` (_|_) A) (vH) (` (_|_) B)))) (opr A (vH) B)) (B (opr (` (_|_) A) (vH) (` (_|_) B)) inss1 pjoml2.2 pjoml2.1 choccl pjoml2.2 choccl chjcl chincl pjoml2.2 pjoml2.1 chlej2 ax-mp pjoml2.1 pjoml2.2 pjoml2.1 choccl pjoml2.2 choccl chjcl chincl chub1 pjoml2.1 pjoml2.2 chdmm1 B ineq1i (opr (` (_|_) A) (vH) (` (_|_) B)) B incom eqtr (i^i A B) (vH) opreq2i A B inss2 pjoml2.1 pjoml2.2 chincl pjoml2.2 pjoml2 ax-mp eqtr3 A B inss1 pjoml2.1 pjoml2.2 chincl pjoml2.1 pjoml2.2 pjoml2.1 choccl pjoml2.2 choccl chjcl chincl chlej1 ax-mp eqsstr3 pjoml2.1 pjoml2.2 pjoml2.1 pjoml2.2 pjoml2.1 choccl pjoml2.2 choccl chjcl chincl chjcl chlubi mp2an eqssi)) thm (pjoml5 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (= (opr A (vH) (i^i (` (_|_) A) (opr A (vH) B))) (opr A (vH) B)) (pjoml2.1 pjoml2.2 chub1 pjoml2.1 pjoml2.1 pjoml2.2 chjcl pjoml2 ax-mp)) thm (pjoml6 ((A x) (B x)) ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (-> (C_ A B) (E.e. x (CH) (/\ (C_ A (` (_|_) (cv x))) (= (opr A (vH) (cv x)) B)))) (pjoml2.1 pjoml2.2 pjoml2 pjoml2.1 pjoml2.2 choccl chub1 pjoml2.1 pjoml2.2 chdmm2 sseqtr4 jctil pjoml2.1 choccl pjoml2.2 chincl (cv x) (i^i (` (_|_) A) B) (_|_) fveq2 A sseq2d (cv x) (i^i (` (_|_) A) B) A (vH) opreq2 B eqeq1d anbi12d (CH) rcla4ev mpan syl)) thm (cmbr () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (= A (opr (i^i A B) (vH) (i^i A (` (_|_) B))))) (pjoml2.1 pjoml2.2 A B cmbrt mp2an)) thm (cmcmlem () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (-> (br A (C_H) B) (br B (C_H) A)) (pjoml2.1 pjoml2.2 chdmj4 pjoml2.1 pjoml2.2 chdmj2 (vH) opreq12i A eqeq2i biimpr (_|_) fveq2d pjoml2.1 choccl pjoml2.2 choccl chjcl pjoml2.1 choccl pjoml2.2 chjcl chdmj4 syl6req B ineq1d pjoml2.2 pjoml2.1 choccl chub2 B (opr (` (_|_) A) (vH) B) sseqin2 mpbi (opr (` (_|_) A) (vH) (` (_|_) B)) ineq2i (opr (` (_|_) A) (vH) (` (_|_) B)) (opr (` (_|_) A) (vH) B) B inass pjoml2.1 pjoml2.2 chdmm1 B ineq1i 3eqtr4r syl5eq (i^i A B) (vH) opreq2d A B inss2 pjoml2.1 pjoml2.2 chincl pjoml2.2 pjoml2 ax-mp A B incom (` (_|_) A) B incom (vH) opreq12i 3eqtr3g pjoml2.1 pjoml2.2 cmbr pjoml2.2 pjoml2.1 cmbr 3imtr4)) thm (cmcm () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (br B (C_H) A)) (pjoml2.1 pjoml2.2 cmcmlem pjoml2.2 pjoml2.1 cmcmlem impbi)) thm (cmcm2 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (br A (C_H) (` (_|_) B))) (pjoml2.1 pjoml2.2 chincl pjoml2.1 pjoml2.2 choccl chincl chjcom pjoml2.2 pjococ A ineq2i (i^i A (` (_|_) B)) (vH) opreq2i eqtr4 A eqeq2i pjoml2.1 pjoml2.2 cmbr pjoml2.1 pjoml2.2 choccl cmbr 3bitr4)) thm (cmcm3 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (br (` (_|_) A) (C_H) B)) (pjoml2.2 pjoml2.1 cmcm2 pjoml2.1 pjoml2.2 cmcm pjoml2.1 choccl pjoml2.2 cmcm 3bitr4)) thm (cmcm4 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (br (` (_|_) A) (C_H) (` (_|_) B))) (pjoml2.1 pjoml2.2 cmcm2 pjoml2.1 pjoml2.2 choccl cmcm3 bitr)) thm (cmbr2 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (= A (i^i (opr A (vH) B) (opr A (vH) (` (_|_) B))))) (pjoml2.1 pjoml2.2 cmcm4 pjoml2.1 choccl pjoml2.2 choccl cmbr A (i^i (opr A (vH) B) (opr A (vH) (` (_|_) B))) eqcom pjoml2.1 pjoml2.2 chjcl pjoml2.1 pjoml2.2 choccl chjcl chincl pjoml2.1 chcon3 pjoml2.1 pjoml2.2 chjcl pjoml2.1 pjoml2.2 choccl chjcl chdmm1 pjoml2.1 pjoml2.2 chdmj1 pjoml2.1 pjoml2.2 choccl chdmj1 (vH) opreq12i eqtr (` (_|_) A) eqeq2i 3bitrr 3bitr)) thm (cmcmi () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH))) (cmcmi.1 (br A (C_H) B))) (br B (C_H) A) (cmcmi.1 pjoml2.1 pjoml2.2 cmcm mpbi)) thm (cmcm2i () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH))) (cmcmi.1 (br A (C_H) B))) (br A (C_H) (` (_|_) B)) (cmcmi.1 pjoml2.1 pjoml2.2 cmcm2 mpbi)) thm (cmcm3i () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH))) (cmcmi.1 (br A (C_H) B))) (br (` (_|_) A) (C_H) B) (cmcmi.1 pjoml2.1 pjoml2.2 cmcm3 mpbi)) thm (cmbr3 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (= (i^i A (opr (` (_|_) A) (vH) B)) (i^i A B))) (pjoml2.1 pjoml2.2 cmcm pjoml2.2 pjoml2.1 cmbr2 bitr B (i^i (opr B (vH) A) (opr B (vH) (` (_|_) A))) A ineq2 A (opr B (vH) A) (opr B (vH) (` (_|_) A)) inass pjoml2.2 pjoml2.1 chjcom A ineq2i pjoml2.1 pjoml2.2 chabs2 eqtr pjoml2.2 pjoml2.1 choccl chjcom ineq12i eqtr3 syl6req sylbi pjoml2.1 pjoml2.2 chdmm3 A ineq2i A (` (_|_) (i^i A (` (_|_) B))) incom eqtr3 (i^i A B) eqeq1i (i^i (` (_|_) (i^i A (` (_|_) B))) A) (i^i A B) (vH) (i^i A (` (_|_) B)) opreq1 sylbi A (` (_|_) B) inss1 pjoml2.1 pjoml2.2 choccl chincl pjoml2.1 pjoml2 ax-mp pjoml2.1 pjoml2.2 choccl chincl pjoml2.1 pjoml2.2 choccl chincl choccl pjoml2.1 chincl chjcom eqtr3 syl5eq pjoml2.1 pjoml2.2 cmbr sylibr impbi)) thm (cmbr4 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (C_ (i^i A (opr (` (_|_) A) (vH) B)) B)) (pjoml2.1 pjoml2.2 cmbr3 A B inss2 (i^i A (opr (` (_|_) A) (vH) B)) (i^i A B) B sseq1 mpbiri A (opr (` (_|_) A) (vH) B) inss1 (C_ (i^i A (opr (` (_|_) A) (vH) B)) B) jctl (i^i A (opr (` (_|_) A) (vH) B)) A B ssin sylib pjoml2.2 pjoml2.1 choccl chub2 B (opr (` (_|_) A) (vH) B) A sslin ax-mp jctir (i^i A (opr (` (_|_) A) (vH) B)) (i^i A B) eqss sylibr impbi bitr)) thm (lecm () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (-> (C_ A B) (br A (C_H) B)) (A B (opr (` (_|_) A) (vH) B) ssinss1 pjoml2.1 pjoml2.2 cmbr4 sylibr)) thm (lecmi () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH))) (lecmi.1 (C_ A B))) (br A (C_H) B) (lecmi.1 pjoml2.1 pjoml2.2 lecm ax-mp)) thm (cmj1 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (br A (C_H) (opr A (vH) B)) (pjoml2.1 pjoml2.1 pjoml2.2 chjcl pjoml2.1 pjoml2.2 chub1 lecmi)) thm (cmj2 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (br B (C_H) (opr A (vH) B)) (pjoml2.2 pjoml2.1 pjoml2.2 chjcl pjoml2.2 pjoml2.1 chub2 lecmi)) thm (cmm1 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (br A (C_H) (i^i A B)) (pjoml2.1 pjoml2.2 chincl pjoml2.1 pjoml2.1 pjoml2.2 chincl pjoml2.1 A B inss1 lecmi cmcmi)) thm (cmm2 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (br B (C_H) (i^i A B)) (pjoml2.2 pjoml2.1 cmm1 B A incom breqtr)) thm (cmbr3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (C_H) B) (= (i^i A (opr (` (_|_) A) (vH) B)) (i^i A B)))) (A (if (e. A (CH)) A (0H)) (C_H) B breq1 (= A (if (e. A (CH)) A (0H))) id A (if (e. A (CH)) A (0H)) (_|_) fveq2 (vH) B opreq1d ineq12d A (if (e. A (CH)) A (0H)) B ineq1 eqeq12d bibi12d B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) (C_H) breq2 B (if (e. B (CH)) B (0H)) (` (_|_) (if (e. A (CH)) A (0H))) (vH) opreq2 (if (e. A (CH)) A (0H)) ineq2d B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) ineq2 eqeq12d bibi12d h0elch A elimel h0elch B elimel cmbr3 dedth2h)) thm (cm0t () () (-> (e. A (CH)) (br (0H) (C_H) A)) (h0elch choccl (` (_|_) (0H)) A chjclt mpan (opr (` (_|_) (0H)) (vH) A) chm0t syl A chm0t eqtr4d (0H) (opr (` (_|_) (0H)) (vH) A) incom (0H) A incom 3eqtr4g h0elch (0H) A cmbr3t mpan mpbird)) thm (cmid () ((cmid.1 (e. A (CH)))) (br A (C_H) A) (cmid.1 cmid.1 A ssid lecmi)) thm (pjoml2t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (C_ A B)) (= (opr A (vH) (i^i (` (_|_) A) B)) B)) (A (if (e. A (CH)) A (0H)) B sseq1 (= A (if (e. A (CH)) A (0H))) id A (if (e. A (CH)) A (0H)) (_|_) fveq2 B ineq1d (vH) opreq12d B eqeq1d imbi12d B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) sseq2 B (if (e. B (CH)) B (0H)) (` (_|_) (if (e. A (CH)) A (0H))) ineq2 (if (e. A (CH)) A (0H)) (vH) opreq2d (= B (if (e. B (CH)) B (0H))) id eqeq12d imbi12d h0elch A elimel h0elch B elimel pjoml2 dedth2h 3impia)) thm (pjoml3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ B A) (= (i^i A (opr (` (_|_) A) (vH) B)) B))) (A (if (e. A (CH)) A (H~)) B sseq2 (= A (if (e. A (CH)) A (H~))) id A (if (e. A (CH)) A (H~)) (_|_) fveq2 (vH) B opreq1d ineq12d B eqeq1d imbi12d B (if (e. B (CH)) B (H~)) (if (e. A (CH)) A (H~)) sseq1 B (if (e. B (CH)) B (H~)) (` (_|_) (if (e. A (CH)) A (H~))) (vH) opreq2 (if (e. A (CH)) A (H~)) ineq2d (= B (if (e. B (CH)) B (H~))) id eqeq12d imbi12d helch A elimel helch B elimel pjoml3 dedth2h)) thm (pjoml5t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (opr A (vH) (i^i (` (_|_) A) (opr A (vH) B))) (opr A (vH) B))) (A (opr A (vH) B) pjoml2t (e. A (CH)) (e. B (CH)) pm3.26 A B chjclt A B chub1t syl3anc)) thm (cmcmt () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (C_H) B) (br B (C_H) A))) (A (if (e. A (CH)) A (0H)) (C_H) B breq1 A (if (e. A (CH)) A (0H)) B (C_H) breq2 bibi12d B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) (C_H) breq2 B (if (e. B (CH)) B (0H)) (C_H) (if (e. A (CH)) A (0H)) breq1 bibi12d h0elch A elimel h0elch B elimel cmcm dedth2h)) thm (cmcm3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (C_H) B) (br (` (_|_) A) (C_H) B))) (A (if (e. A (CH)) A (0H)) (C_H) B breq1 A (if (e. A (CH)) A (0H)) (_|_) fveq2 (C_H) B breq1d bibi12d B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) (C_H) breq2 B (if (e. B (CH)) B (0H)) (` (_|_) (if (e. A (CH)) A (0H))) (C_H) breq2 bibi12d h0elch A elimel h0elch B elimel cmcm3 dedth2h)) thm (cmcm2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (C_H) B) (br A (C_H) (` (_|_) B)))) (B A cmcm3t ancoms A B cmcmt A (` (_|_) B) cmcmt B chocclt sylan2 3bitr4d)) thm (lecmt () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (C_ A B)) (br A (C_H) B)) (A (if (e. A (CH)) A (0H)) B sseq1 A (if (e. A (CH)) A (0H)) (C_H) B breq1 imbi12d B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) sseq2 B (if (e. B (CH)) B (0H)) (if (e. A (CH)) A (0H)) (C_H) breq2 imbi12d h0elch A elimel h0elch B elimel lecm dedth2h 3impia)) thm (fh1t () () (-> (/\ (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (/\ (br A (C_H) B) (br A (C_H) C))) (= (i^i A (opr B (vH) C)) (opr (i^i A B) (vH) (i^i A C)))) ((opr (i^i A B) (vH) (i^i A C)) (i^i A (opr B (vH) C)) pjomlt (i^i A B) (i^i A C) chjclt A B chinclt A C chinclt syl2an anandis A (opr B (vH) C) chinclt B C chjclt sylan2 (i^i A (opr B (vH) C)) chsh syl jca 3impb (/\ (br A (C_H) B) (br A (C_H) C)) adantr A B C ledit (/\ (br A (C_H) B) (br A (C_H) C)) adantr A (opr B (vH) C) incom (/\ (/\ (e. A (CH)) (e. B (CH))) (/\ (e. A (CH)) (e. C (CH)))) a1i (i^i A B) (i^i A C) chdmj1t A B chinclt A C chinclt syl2an A B chdmm1t (/\ (e. A (CH)) (e. C (CH))) adantr A C chdmm1t (/\ (e. A (CH)) (e. B (CH))) adantl ineq12d eqtrd ineq12d 3impdi (/\ (br A (C_H) B) (br A (C_H) C)) adantr A B cmcm2t A (` (_|_) B) cmbr3t B chocclt sylan2 bitrd biimpa (e. C (CH)) 3adantl3 (br A (C_H) C) adantrr A C cmcm2t A (` (_|_) C) cmbr3t C chocclt sylan2 bitrd biimpa (e. B (CH)) 3adantl2 (br A (C_H) B) adantrl ineq12d A (opr (` (_|_) A) (vH) (` (_|_) B)) (opr (` (_|_) A) (vH) (` (_|_) C)) inindi A (` (_|_) B) (` (_|_) C) inindi 3eqtr4g (opr B (vH) C) ineq2d (opr B (vH) C) A (i^i (opr (` (_|_) A) (vH) (` (_|_) B)) (opr (` (_|_) A) (vH) (` (_|_) C))) inass syl5eq (opr B (vH) C) A (i^i (` (_|_) B) (` (_|_) C)) in12 syl6eq B C chdmj1t (opr B (vH) C) ineq2d B C chjclt (opr B (vH) C) chocint syl eqtr3d A ineq2d A chm0t sylan9eqr 3impb (/\ (br A (C_H) B) (br A (C_H) C)) adantr 3eqtrd jca sylanc eqcomd)) thm (fh2t () () (-> (/\ (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (/\ (br B (C_H) A) (br B (C_H) C))) (= (i^i A (opr B (vH) C)) (opr (i^i A B) (vH) (i^i A C)))) ((opr (i^i A B) (vH) (i^i A C)) (i^i A (opr B (vH) C)) pjomlt (i^i A B) (i^i A C) chjclt A B chinclt A C chinclt syl2an anandis A (opr B (vH) C) chinclt B C chjclt sylan2 (i^i A (opr B (vH) C)) chsh syl jca 3impb (/\ (br B (C_H) A) (br B (C_H) C)) adantr A B C ledit (/\ (br B (C_H) A) (br B (C_H) C)) adantr (i^i A B) (i^i A C) chdmj1t A B chinclt A C chinclt syl2an A B chdmm1t (/\ (e. A (CH)) (e. C (CH))) adantr (` (_|_) (i^i A C)) ineq1d eqtrd 3impdi (i^i A (opr B (vH) C)) ineq2d (/\ (br B (C_H) A) (br B (C_H) C)) adantr A B cmcm2t A B cmcmt A (` (_|_) B) cmbr3t B chocclt sylan2 3bitr3d biimpa A (` (_|_) B) incom syl6eq (e. C (CH)) 3adantl3 (br B (C_H) C) adantrr (i^i (opr B (vH) C) (` (_|_) (i^i A C))) ineq1d A (opr B (vH) C) (opr (` (_|_) A) (vH) (` (_|_) B)) (` (_|_) (i^i A C)) in4 syl5eq eqtrd (` (_|_) B) A (opr B (vH) C) (` (_|_) (i^i A C)) in4 syl6eq B ococt (vH) C opreq1d (` (_|_) B) ineq2d (e. A (CH)) (e. C (CH)) 3ad2ant2 (/\ (br B (C_H) A) (br B (C_H) C)) adantr B C cmcm3t (` (_|_) B) C cmbr3t B chocclt sylan bitrd biimpa (e. A (CH)) 3adantl1 (br B (C_H) A) adantrl eqtr3d (i^i A (` (_|_) (i^i A C))) ineq1d A C chinclt (i^i A C) chocint syl C A (` (_|_) (i^i A C)) in12 A C (` (_|_) (i^i A C)) inass eqtr4 syl5eq (` (_|_) B) ineq2d (` (_|_) B) C (i^i A (` (_|_) (i^i A C))) inass syl5eq (e. B (CH)) 3adant2 B chocclt (` (_|_) B) chm0t syl (e. A (CH)) (e. C (CH)) 3ad2ant2 eqtrd (/\ (br B (C_H) A) (br B (C_H) C)) adantr 3eqtrd jca sylanc eqcomd)) thm (cm2jt () () (-> (/\ (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (/\ (br A (C_H) B) (br A (C_H) C))) (br A (C_H) (opr B (vH) C))) (A B cmcmt B A cmbrt ancoms bitrd biimpa B A incom B (` (_|_) A) incom (vH) opreq12i syl6eq (e. C (CH)) 3adantl3 (br A (C_H) C) adantrr A C cmcmt C A cmbrt ancoms bitrd biimpa C A incom C (` (_|_) A) incom (vH) opreq12i syl6eq (e. B (CH)) 3adantl2 (br A (C_H) B) adantrl (vH) opreq12d (i^i A B) (i^i (` (_|_) A) B) (i^i A C) (i^i (` (_|_) A) C) chj4t A B chinclt (` (_|_) A) B chinclt A chocclt sylan jca A C chinclt (` (_|_) A) C chinclt A chocclt sylan jca syl2an 3impdi (/\ (br A (C_H) B) (br A (C_H) C)) adantr A B C fh1t A (opr B (vH) C) incom syl5reqr (` (_|_) A) B C fh1t A chocclt (e. B (CH)) id (e. C (CH)) id 3anim123i (/\ (br A (C_H) B) (br A (C_H) C)) adantr A B cmcm3t (e. C (CH)) 3adant3 A C cmcm3t (e. B (CH)) 3adant2 anbi12d biimpa sylanc (` (_|_) A) (opr B (vH) C) incom syl5reqr (vH) opreq12d 3eqtrd ex A (opr B (vH) C) cmcmt (opr B (vH) C) A cmbrt ancoms bitrd B C chjclt sylan2 3impb sylibrd imp)) thm (fh1 () ((fh1.1 (e. A (CH))) (fh1.2 (e. B (CH))) (fh1.3 (e. C (CH))) (fh1.4 (br A (C_H) B)) (fh1.5 (br A (C_H) C))) (= (i^i A (opr B (vH) C)) (opr (i^i A B) (vH) (i^i A C))) (fh1.1 fh1.2 fh1.3 3pm3.2i fh1.4 fh1.5 pm3.2i A B C fh1t mp2an)) thm (fh2 () ((fh1.1 (e. A (CH))) (fh1.2 (e. B (CH))) (fh1.3 (e. C (CH))) (fh1.4 (br A (C_H) B)) (fh1.5 (br A (C_H) C))) (= (i^i B (opr A (vH) C)) (opr (i^i B A) (vH) (i^i B C))) (fh1.2 fh1.1 fh1.3 3pm3.2i fh1.4 fh1.5 pm3.2i B A C fh2t mp2an)) thm (fh3 () ((fh1.1 (e. A (CH))) (fh1.2 (e. B (CH))) (fh1.3 (e. C (CH))) (fh1.4 (br A (C_H) B)) (fh1.5 (br A (C_H) C))) (= (opr A (vH) (i^i B C)) (i^i (opr A (vH) B) (opr A (vH) C))) (fh1.1 choccl fh1.2 choccl fh1.3 choccl fh1.1 choccl fh1.2 fh1.1 fh1.2 fh1.4 cmcm3i cmcm2i fh1.1 choccl fh1.3 fh1.1 fh1.3 fh1.5 cmcm3i cmcm2i fh1 fh1.2 fh1.3 chdmm1 (` (_|_) A) ineq2i fh1.1 fh1.2 chdmj1 fh1.1 fh1.3 chdmj1 (vH) opreq12i 3eqtr4r fh1.1 fh1.2 chjcl fh1.1 fh1.3 chjcl chdmm1 fh1.1 fh1.2 fh1.3 chincl chdmj1 3eqtr4 fh1.1 fh1.2 fh1.3 chincl chjcl fh1.1 fh1.2 chjcl fh1.1 fh1.3 chjcl chincl chcon3 mpbir)) thm (fh4 () ((fh1.1 (e. A (CH))) (fh1.2 (e. B (CH))) (fh1.3 (e. C (CH))) (fh1.4 (br A (C_H) B)) (fh1.5 (br A (C_H) C))) (= (opr B (vH) (i^i A C)) (i^i (opr B (vH) A) (opr B (vH) C))) (fh1.1 choccl fh1.2 choccl fh1.3 choccl fh1.1 choccl fh1.2 fh1.1 fh1.2 fh1.4 cmcm3i cmcm2i fh1.1 choccl fh1.3 fh1.1 fh1.3 fh1.5 cmcm3i cmcm2i fh2 fh1.1 fh1.3 chdmm1 (` (_|_) B) ineq2i fh1.2 fh1.1 chdmj1 fh1.2 fh1.3 chdmj1 (vH) opreq12i 3eqtr4r fh1.2 fh1.1 chjcl fh1.2 fh1.3 chjcl chdmm1 fh1.2 fh1.1 fh1.3 chincl chdmj1 3eqtr4 fh1.2 fh1.1 fh1.3 chincl chjcl fh1.2 fh1.1 chjcl fh1.2 fh1.3 chjcl chincl chcon3 mpbir)) thm (qlax1 () ((qlax1.1 (e. A (CH)))) (= A (` (_|_) (` (_|_) A))) (qlax1.1 ococ eqcomi)) thm (qlax2 () ((qlax.1 (e. A (CH))) (qlax.2 (e. B (CH)))) (= (opr A (vH) B) (opr B (vH) A)) (qlax.1 qlax.2 chjcom)) thm (qlax3 () ((qlax.1 (e. A (CH))) (qlax.2 (e. B (CH))) (qlax.3 (e. C (CH)))) (= (opr (opr A (vH) B) (vH) C) (opr A (vH) (opr B (vH) C))) (qlax.1 qlax.2 qlax.3 chjass)) thm (qlax4 () ((qlax.1 (e. A (CH))) (qlax.2 (e. B (CH)))) (= (opr A (vH) (opr B (vH) (` (_|_) B))) (opr B (vH) (` (_|_) B))) (qlax.1 chj1 qlax.2 chjo A (vH) opreq2i qlax.2 chjo 3eqtr4)) thm (qlax5 () ((qlax.1 (e. A (CH))) (qlax.2 (e. B (CH)))) (= (opr A (vH) (` (_|_) (opr (` (_|_) A) (vH) B))) A) (qlax.1 qlax.2 chdmj2 A (vH) opreq2i qlax.1 qlax.2 choccl chabs1 eqtr)) thm (qlaxr1 () ((qlaxr1.1 (e. A (CH))) (qlaxr1.2 (e. B (CH))) (qlaxr1.3 (= A B))) (= B A) (qlaxr1.3 eqcomi)) thm (qlaxr2 () ((qlaxr2.1 (e. A (CH))) (qlaxr2.2 (e. B (CH))) (qlaxr2.3 (e. C (CH))) (qlaxr2.4 (= A B)) (qlaxr2.5 (= B C))) (= A C) (qlaxr2.4 qlaxr2.5 eqtr)) thm (qlaxr4 () ((qlaxr4.1 (e. A (CH))) (qlaxr4.2 (e. B (CH))) (qlaxr4.3 (= A B))) (= (` (_|_) A) (` (_|_) B)) (qlaxr4.3 (_|_) fveq2i)) thm (qlaxr5 () ((qlaxr5.1 (e. A (CH))) (qlaxr5.2 (e. B (CH))) (qlaxr5.3 (e. C (CH))) (qlaxr5.4 (= A B))) (= (opr A (vH) C) (opr B (vH) C)) (qlaxr5.4 (vH) C opreq1i)) thm (qlaxr3 () ((qlaxr3.1 (e. A (CH))) (qlaxr3.2 (e. B (CH))) (qlaxr3.3 (e. C (CH))) (qlaxr3.4 (= (opr C (vH) (` (_|_) C)) (opr (` (_|_) (opr (` (_|_) A) (vH) (` (_|_) B))) (vH) (` (_|_) (opr A (vH) B)))))) (= A B) (qlaxr3.1 qlaxr3.1 qlaxr3.2 chjcl chshi qlaxr3.1 qlaxr3.2 chub1 (opr A (vH) B) (opr (` (_|_) A) (vH) (` (_|_) B)) incom qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 choccl qlaxr3.2 choccl qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.1 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.2 cmj1 cmcmi cmcm2i qlaxr3.1 qlaxr3.2 chjcl qlaxr3.2 qlaxr3.2 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.2 cmj2 cmcmi cmcm2i fh1 qlaxr3.3 chjo qlaxr3.4 eqtr3 choc0 qlaxr3.1 choccl qlaxr3.2 choccl chjcl qlaxr3.1 qlaxr3.2 chjcl chdmm1 3eqtr4 qlaxr3.1 choccl qlaxr3.2 choccl chjcl qlaxr3.1 qlaxr3.2 chjcl chincl h0elch chcon3 mpbir 3eqtr3 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 choccl chincl qlaxr3.1 qlaxr3.2 chjcl qlaxr3.2 choccl chincl chj00 mpbir pm3.26i omlsi qlaxr3.2 qlaxr3.1 qlaxr3.2 chjcl chshi qlaxr3.2 qlaxr3.1 chub2 (opr A (vH) B) (opr (` (_|_) A) (vH) (` (_|_) B)) incom qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 choccl qlaxr3.2 choccl qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.1 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.2 cmj1 cmcmi cmcm2i qlaxr3.1 qlaxr3.2 chjcl qlaxr3.2 qlaxr3.2 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.2 cmj2 cmcmi cmcm2i fh1 qlaxr3.3 chjo qlaxr3.4 eqtr3 choc0 qlaxr3.1 choccl qlaxr3.2 choccl chjcl qlaxr3.1 qlaxr3.2 chjcl chdmm1 3eqtr4 qlaxr3.1 choccl qlaxr3.2 choccl chjcl qlaxr3.1 qlaxr3.2 chjcl chincl h0elch chcon3 mpbir 3eqtr3 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 choccl chincl qlaxr3.1 qlaxr3.2 chjcl qlaxr3.2 choccl chincl chj00 mpbir pm3.27i omlsi eqtr4)) thm (osumlem1 () ((osumlem1.1 (e. A (CH))) (osumlem1.2 (e. B (CH))) (osumlem1.3 (C_ B (` (_|_) A))) (osumlem1.4 (<-> ph (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y)))))))) (-> ph (/\ (/\ (/\ (e. C (H~)) (e. D (H~))) (e. R (H~))) (/\ (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) (e. (cv z) (H~))))) (osumlem1.4 R (opr C (+v) D) (H~) eleq1 C D ax-hvaddcl syl5bir com12 imdistani osumlem1.1 C chel osumlem1.2 D chel anim12i sylan (cv z) (opr (cv x) (+v) (cv y)) (H~) eleq1 (cv x) (cv y) ax-hvaddcl syl5bir com12 imdistani osumlem1.1 (cv x) chel osumlem1.1 choccl (cv y) chel anim12i sylan anim12i sylbi)) thm (osumlem2 () ((osumlem1.1 (e. A (CH))) (osumlem1.2 (e. B (CH))) (osumlem1.3 (C_ B (` (_|_) A))) (osumlem1.4 (<-> ph (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y)))))))) (-> ph (= (opr (opr (` (norm) (opr C (-v) (cv x))) (^) (2)) (+) (opr (` (norm) (opr D (-v) (cv y))) (^) (2))) (opr (` (norm) (opr R (-v) (cv z))) (^) (2)))) (osumlem1.4 (/\ (e. C A) (e. D B)) (= R (opr C (+v) D)) pm3.27 (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))) pm3.27 (-v) opreqan12d osumlem1.1 osumlem1.2 osumlem1.3 (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) pm4.2 osumlem1 C D (cv x) (cv y) hvsub4t (e. R (H~)) (e. (cv z) (H~)) ad2ant2r syl eqtrd (norm) fveq2d (^) (2) opreq1d (opr C (-v) (cv x)) (opr D (-v) (cv y)) normpytht osumlem1.1 osumlem1.2 osumlem1.3 (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) pm4.2 osumlem1 C (cv x) hvsubclt D (cv y) hvsubclt anim12i an4s (e. R (H~)) (e. (cv z) (H~)) ad2ant2r syl osumlem1.1 chshi A (opr C (-v) (cv x)) (opr D (-v) (cv y)) shocorth ax-mp osumlem1.1 chshi A C (cv x) shsubclt ax-mp osumlem1.1 choccl chshi (` (_|_) A) D (cv y) shsubclt ax-mp osumlem1.3 D sseli sylan syl2an an4s (= R (opr C (+v) D)) (= (cv z) (opr (cv x) (+v) (cv y))) ad2ant2r sylc eqtr2d sylbi)) thm (osumlem3 () ((osumlem1.1 (e. A (CH))) (osumlem1.2 (e. B (CH))) (osumlem1.3 (C_ B (` (_|_) A))) (osumlem1.4 (<-> ph (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y)))))))) (-> ph (br (` (norm) (opr D (-v) (cv y))) (<_) (` (norm) (opr R (-v) (cv z))))) (osumlem1.4 osumlem1.1 osumlem1.2 osumlem1.3 (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) pm4.2 osumlem1 C (cv x) hvsubclt (e. D (H~)) (e. (cv y) (H~)) ad2ant2r (e. R (H~)) (e. (cv z) (H~)) ad2ant2r (opr C (-v) (cv x)) normclt syl (` (norm) (opr C (-v) (cv x))) sqge0t 3syl osumlem1.1 osumlem1.2 osumlem1.3 (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) pm4.2 osumlem1 0re (/\ (/\ (/\ (e. C (H~)) (e. D (H~))) (e. R (H~))) (/\ (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) (e. (cv z) (H~)))) a1i C (cv x) hvsubclt (e. D (H~)) (e. (cv y) (H~)) ad2ant2r (e. R (H~)) (e. (cv z) (H~)) ad2ant2r (opr C (-v) (cv x)) normclt (` (norm) (opr C (-v) (cv x))) resqclt 3syl D (cv y) hvsubclt (e. C (H~)) (e. (cv x) (H~)) ad2ant2l (e. R (H~)) (e. (cv z) (H~)) ad2ant2r (opr D (-v) (cv y)) normclt (` (norm) (opr D (-v) (cv y))) resqclt 3syl 3jca (0) (opr (` (norm) (opr C (-v) (cv x))) (^) (2)) (opr (` (norm) (opr D (-v) (cv y))) (^) (2)) leadd1t 3syl mpbid osumlem1.1 osumlem1.2 osumlem1.3 (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) pm4.2 osumlem1 D (cv y) hvsubclt (e. C (H~)) (e. (cv x) (H~)) ad2ant2l (e. R (H~)) (e. (cv z) (H~)) ad2ant2r (opr D (-v) (cv y)) normclt (` (norm) (opr D (-v) (cv y))) resqclt 3syl recnd (opr (` (norm) (opr D (-v) (cv y))) (^) (2)) addid2t 3syl osumlem1.1 osumlem1.2 osumlem1.3 (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) pm4.2 osumlem2 3brtr3d osumlem1.1 osumlem1.2 osumlem1.3 (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) pm4.2 osumlem1 D (cv y) hvsubclt (e. C (H~)) (e. (cv x) (H~)) ad2ant2l (e. R (H~)) (e. (cv z) (H~)) ad2ant2r R (cv z) hvsubclt (/\ (e. C (H~)) (e. D (H~))) (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) ad2ant2l jca (` (norm) (opr D (-v) (cv y))) (` (norm) (opr R (-v) (cv z))) le2sqt (opr D (-v) (cv y)) normclt (opr D (-v) (cv y)) normge0t jca (opr R (-v) (cv z)) normclt (opr R (-v) (cv z)) normge0t jca syl2an 3syl mpbird sylbi)) thm (osumlem4 ((u v) (u w) (A u) (v w) (A v) (A w) (B u) (B v) (B w) (F u) (F v) (F w) (G u) (G v) (G w) (H u) (H v) (H w) (u x) (v x) (w x) (u y) (v y) (w y) (u z) (v z) (w z)) ((osumlem4.1 (e. A (CH))) (osumlem4.2 (e. B (CH))) (osumlem4.3 (C_ B (` (_|_) A))) (osumlem4.4 (e. G (V))) (osumlem4.5 (e. H (V)))) (-> (/\ (/\ (/\ (:--> F (NN) A) (:--> G (NN) B)) (A.e. w (NN) (= (` H (cv w)) (opr (` F (cv w)) (+v) (` G (cv w)))))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) (-> (br H (~~>v) (cv z)) (br G (~~>v) (cv y)))) (F (NN) A (cv w) ffvrn expcom G (NN) B (cv w) ffvrn expcom anim12d osumlem4.1 osumlem4.2 osumlem4.3 (/\ (/\ (/\ (e. (` F (cv w)) A) (e. (` G (cv w)) B)) (= (` H (cv w)) (opr (` F (cv w)) (+v) (` G (cv w))))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) pm4.2 osumlem3 (e. (cv u) (RR)) adantr osumlem4.1 osumlem4.2 osumlem4.3 (/\ (/\ (/\ (e. (` F (cv w)) A) (e. (` G (cv w)) B)) (= (` H (cv w)) (opr (` F (cv w)) (+v) (` G (cv w))))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) pm4.2 osumlem1 (` (norm) (opr (` G (cv w)) (-v) (cv y))) (` (norm) (opr (` H (cv w)) (-v) (cv z))) (cv u) lelttrt 3exp (` G (cv w)) (cv y) hvsubclt (opr (` G (cv w)) (-v) (cv y)) normclt syl (e. (` F (cv w)) (H~)) (e. (cv x) (H~)) ad2ant2l (e. (` H (cv w)) (H~)) (e. (cv z) (H~)) ad2ant2r (` H (cv w)) (cv z) hvsubclt (opr (` H (cv w)) (-v) (cv z)) normclt syl (/\ (e. (` F (cv w)) (H~)) (e. (` G (cv w)) (H~))) (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) ad2ant2l sylc syl imp mpand exp41 com24 syl6 com4l imp41 (br (` (norm) (opr (` H (cv w)) (-v) (cv z))) (<) (cv u)) (br (` (norm) (opr (` G (cv w)) (-v) (cv y))) (<) (cv u)) (br (cv v) (<_) (cv w)) imim2 syl6 r19.20dva exp31 com24 imp41 w (NN) (-> (br (cv v) (<_) (cv w)) (br (` (norm) (opr (` H (cv w)) (-v) (cv z))) (<) (cv u))) (-> (br (cv v) (<_) (cv w)) (br (` (norm) (opr (` G (cv w)) (-v) (cv y))) (<) (cv u))) r19.20 syl v (NN) r19.22sdv (br (0) (<) (cv u)) imim2d r19.20dva osumlem4.5 z visset u v w hlimconv syl5 osumlem4.2 chssi G (NN) B (H~) fss mpan2 (:--> F (NN) A) (A.e. w (NN) (= (` H (cv w)) (opr (` F (cv w)) (+v) (` G (cv w))))) ad2antlr osumlem4.1 chshi A shocss ax-mp (cv y) sseli (e. (cv x) A) (= (cv z) (opr (cv x) (+v) (cv y))) ad2antlr anim12i jctild osumlem4.4 y visset u v w hlim syl6ibr)) thm (osumlem5 ((x y) (w x) (f x) (g x) (A x) (w y) (f y) (g y) (A y) (f w) (g w) (A w) (f g) (A f) (A g) (B x) (B y) (B w) (B f) (B g) (H x) (H y) (H w) (H f) (H g)) ((osumlem5.1 (e. A (CH))) (osumlem5.2 (e. B (CH)))) (-> (:--> H (NN) (opr A (+H) B)) (E. f (E. g (/\ (/\ (:--> (cv f) (NN) A) (:--> (cv g) (NN) B)) (A.e. w (NN) (= (` H (cv w)) (opr (` (cv f) (cv w)) (+v) (` (cv g) (cv w))))))))) (H (NN) (opr A (+H) B) (cv w) ffvrn ex osumlem5.1 chshi osumlem5.2 chshi (` H (cv w)) x y shsel syl6ib r19.21aiv nnex osumlem5.1 elisseti (cv x) (` (cv f) (cv w)) (+v) (cv y) opreq1 (` H (cv w)) eqeq2d y B rexbidv ac6 nnex osumlem5.2 elisseti (cv y) (` (cv g) (cv w)) (` (cv f) (cv w)) (+v) opreq2 (` H (cv w)) eqeq2d ac6 (:--> (cv f) (NN) A) anim2i (:--> (cv f) (NN) A) (:--> (cv g) (NN) B) (A.e. w (NN) (= (` H (cv w)) (opr (` (cv f) (cv w)) (+v) (` (cv g) (cv w))))) anass g exbii g (:--> (cv f) (NN) A) (/\ (:--> (cv g) (NN) B) (A.e. w (NN) (= (` H (cv w)) (opr (` (cv f) (cv w)) (+v) (` (cv g) (cv w)))))) 19.42v bitr sylibr f 19.22i 3syl)) thm (osumlem6 ((f w) (g w) (A w) (f g) (A f) (A g) (B w) (B f) (B g) (H w) (H f) (H g) (w x) (f x) (g x) (w y) (f y) (g y) (w z) (f z) (g z)) ((osumlem6.1 (e. A (CH))) (osumlem6.2 (e. B (CH))) (osumlem6.3 (C_ B (` (_|_) A))) (osumlem6.4 (e. H (V)))) (-> (/\ (/\ (:--> H (NN) (opr A (+H) B)) (br H (~~>v) (cv z))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) (e. (cv y) B)) (osumlem6.1 osumlem6.2 H f g w osumlem5 osumlem6.2 y visset B (cv g) chlim ax-mp (:--> (cv f) (NN) A) (:--> (cv g) (NN) B) pm3.27 (A.e. w (NN) (= (` H (cv w)) (opr (` (cv f) (cv w)) (+v) (` (cv g) (cv w))))) adantr (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y)))) (br H (~~>v) (cv z)) ad2antrr osumlem6.1 osumlem6.2 osumlem6.3 g visset osumlem6.4 (cv f) w x y z osumlem4 imp sylanc exp31 com23 f g 19.23aivv syl imp31)) thm (osumlem7 ((x y) (x z) (h x) (A x) (y z) (h y) (A y) (h z) (A z) (A h) (B x) (B y) (B z) (B h)) ((osumlem7.1 (e. A (CH))) (osumlem7.2 (e. B (CH))) (osumlem7.3 (C_ B (` (_|_) A)))) (e. (opr A (+H) B) (CH)) ((opr A (+H) B) h z closedsub osumlem7.1 chshi osumlem7.2 chshi shscl h visset z visset hlimvec osumlem7.1 A (cv z) x y pjpjtht mpan syl (:--> (cv h) (NN) (opr A (+H) B)) adantl osumlem7.1 chshi osumlem7.2 chshi (cv x) (cv y) shsva (e. (cv x) A) (e. (cv y) (` (_|_) A)) pm3.26 (/\ (:--> (cv h) (NN) (opr A (+H) B)) (br (cv h) (~~>v) (cv z))) (= (cv z) (opr (cv x) (+v) (cv y))) ad2antrl osumlem7.1 osumlem7.2 osumlem7.3 h visset z x y osumlem6 sylanc (cv z) (opr (cv x) (+v) (cv y)) (opr A (+H) B) eleq1 (/\ (:--> (cv h) (NN) (opr A (+H) B)) (br (cv h) (~~>v) (cv z))) (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) ad2antll mpbird ex x y 19.23advv x A y (` (_|_) A) (= (cv z) (opr (cv x) (+v) (cv y))) r2ex syl5ib mpd h z gen2 mpbir2an)) thm (osumlem8 () ((osumlem7.1 (e. A (CH))) (osumlem7.2 (e. B (CH)))) (-> (C_ B (` (_|_) A)) (e. (opr A (+H) B) (CH))) (A (if (C_ B (` (_|_) A)) A (0H)) (+H) B opreq1 (CH) eleq1d B (if (C_ B (` (_|_) A)) B (0H)) (if (C_ B (` (_|_) A)) A (0H)) (+H) opreq2 (CH) eleq1d osumlem7.1 h0elch (C_ B (` (_|_) A)) keepel osumlem7.2 h0elch (C_ B (` (_|_) A)) keepel A (if (C_ B (` (_|_) A)) A (0H)) (_|_) fveq2 B sseq2d B (if (C_ B (` (_|_) A)) B (0H)) (` (_|_) (if (C_ B (` (_|_) A)) A (0H))) sseq1 (0H) (if (C_ B (` (_|_) A)) A (0H)) (_|_) fveq2 (0H) sseq2d (0H) (if (C_ B (` (_|_) A)) B (0H)) (` (_|_) (if (C_ B (` (_|_) A)) A (0H))) sseq1 h0elch chssi choc0 sseqtr4 elimhyp2v osumlem7 dedth2v)) thm (osum () ((osumlem7.1 (e. A (CH))) (osumlem7.2 (e. B (CH)))) (-> (C_ A (` (_|_) B)) (= (opr A (+H) B) (opr A (vH) B))) (osumlem7.2 osumlem7.1 osumlem8 osumlem7.1 chshi osumlem7.2 chshi shscom syl5eqel osumlem7.1 chshi osumlem7.2 chshi shunss osumlem7.1 chssi osumlem7.2 chssi unssi osumlem7.1 chshi osumlem7.2 chshi shscl shssi occon2 ax-mp osumlem7.1 osumlem7.2 chjval eqcomi (e. (opr A (+H) B) (CH)) a1i (opr A (+H) B) ococt sseq12d mpbii syl osumlem7.1 osumlem7.2 chslej jctil (opr A (+H) B) (opr A (vH) B) eqss sylibr)) thm (osumcor () ((osumlem7.1 (e. A (CH))) (osumlem7.2 (e. B (CH)))) (= (opr (i^i A B) (+H) (i^i A (` (_|_) B))) (opr (i^i A B) (vH) (i^i A (` (_|_) B)))) (A B inss2 osumlem7.2 osumlem7.1 choccl chub2 sstri osumlem7.1 osumlem7.2 chdmm3 sseqtr4 osumlem7.1 osumlem7.2 chincl osumlem7.1 osumlem7.2 choccl chincl osum ax-mp)) thm (osumt () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (C_ A (` (_|_) B))) (= (opr A (+H) B) (opr A (vH) B))) (A (if (e. A (CH)) A (H~)) (` (_|_) B) sseq1 A (if (e. A (CH)) A (H~)) (+H) B opreq1 A (if (e. A (CH)) A (H~)) (vH) B opreq1 eqeq12d imbi12d B (if (e. B (CH)) B (H~)) (_|_) fveq2 (if (e. A (CH)) A (H~)) sseq2d B (if (e. B (CH)) B (H~)) (if (e. A (CH)) A (H~)) (+H) opreq2 B (if (e. B (CH)) B (H~)) (if (e. A (CH)) A (H~)) (vH) opreq2 eqeq12d imbi12d helch A elimel helch B elimel osum dedth2h 3impia)) thm (chsot () () (-> (e. A (CH)) (= (opr A (+H) (` (_|_) A)) (H~))) (A (` (_|_) A) osumt (e. A (CH)) id A chocclt A chsh A shococss syl syl3anc A chjot eqtrd)) thm (osumcor2 () ((osumcor2.1 (e. A (CH))) (osumcor2.2 (e. B (CH)))) (-> (br A (C_H) B) (= (opr A (+H) B) (opr A (vH) B))) (osumcor2.1 osumcor2.2 cmcm2 osumcor2.1 osumcor2.2 choccl cmbr4 bitr osumcor2.1 osumcor2.1 choccl osumcor2.2 choccl chjcl chincl osumcor2.2 osum osumcor2.1 choccl osumcor2.2 choccl chjcom A ineq2i (vH) B opreq1i osumcor2.1 osumcor2.2 choccl osumcor2.1 choccl chjcl chincl osumcor2.2 chjcom eqtr osumcor2.2 osumcor2.1 pjoml4 osumcor2.2 osumcor2.1 chjcom 3eqtr (opr (i^i A (opr (` (_|_) A) (vH) (` (_|_) B))) (+H) B) eqeq2i A (opr (` (_|_) A) (vH) (` (_|_) B)) inss1 osumcor2.1 osumcor2.1 choccl osumcor2.2 choccl chjcl chincl chshi osumcor2.1 chshi osumcor2.2 chshi shless ax-mp (opr (i^i A (opr (` (_|_) A) (vH) (` (_|_) B))) (+H) B) (opr A (vH) B) (opr A (+H) B) sseq1 mpbii sylbi syl sylbi osumcor2.1 osumcor2.2 chslej jctil (opr A (+H) B) (opr A (vH) B) eqss sylibr)) thm (spansnj () ((spansnj.1 (e. A (CH))) (spansnj.2 (e. B (H~)))) (= (opr A (+H) (` (span) ({} B))) (opr A (vH) (` (span) ({} B)))) (spansnj.1 chshi spansnj.2 spansnch chshi shjshs spansnj.1 chssi spansnj.1 choccl spansnj.2 pjhcli (` (` (proj) (` (_|_) A)) B) (H~) snssi ax-mp spanun spansnj.1 spansnj.2 spanunsn spansnj.1 chshi A spanid ax-mp (+H) (` (span) ({} (` (` (proj) (` (_|_) A)) B))) opreq1i spansnj.1 chssi spansnj.2 spansnpj spansnj.1 spansnj.1 choccl spansnj.2 pjhcli spansnch osum ax-mp eqtr2 3eqtr4r spansnj.1 chssi spansnj.2 B (H~) snssi ax-mp spanun spansnj.1 chshi A spanid ax-mp (+H) (` (span) ({} B)) opreq1i 3eqtrr spansnj.1 spansnj.1 choccl spansnj.2 pjhcli spansnch chjcl eqeltr ococ eqtr2)) thm (spansnjt () () (-> (/\ (e. A (CH)) (e. B (H~))) (= (opr A (+H) (` (span) ({} B))) (opr A (vH) (` (span) ({} B))))) (A (if (e. A (CH)) A (H~)) (+H) (` (span) ({} B)) opreq1 A (if (e. A (CH)) A (H~)) (vH) (` (span) ({} B)) opreq1 eqeq12d B (if (e. B (H~)) B (0v)) sneq (span) fveq2d (if (e. A (CH)) A (H~)) (+H) opreq2d B (if (e. B (H~)) B (0v)) sneq (span) fveq2d (if (e. A (CH)) A (H~)) (vH) opreq2d eqeq12d helch A elimel ax-hv0cl B elimel spansnj dedth2h)) thm (spansnsclt () () (-> (/\ (e. A (CH)) (e. B (H~))) (e. (opr A (+H) (` (span) ({} B))) (CH))) (A B spansnjt A (` (span) ({} B)) chjclt B spansncht sylan2 eqeltrd)) thm (sumspansnt () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (e. (opr A (+v) B) (` (span) ({} A))) (e. B (` (span) ({} A))))) (A spansnid A spansnsht (` (span) ({} A)) (opr A (+v) B) A shsubclt syl mpan2d (e. B (H~)) adantr A B hvpncan2t (` (span) ({} A)) eleq1d sylibd A spansnid A spansnsht (` (span) ({} A)) A B shaddclt syl mpand (e. B (H~)) adantr impbid)) thm (spansnm0 ((A x) (B x)) ((spansnm0.1 (e. A (H~))) (spansnm0.2 (e. B (H~)))) (-> (-. (e. A (` (span) ({} B)))) (= (i^i (` (span) ({} A)) (` (span) ({} B))) (0H))) (spansnm0.1 spansnm0.2 spansnch chshi (` (span) ({} B)) A (cv x) elspansn5t ax-mp mpanl1 ex (cv x) (` (span) ({} A)) (` (span) ({} B)) elin (cv x) elch0 3imtr4g ssrdv spansnm0.1 spansnch spansnm0.2 spansnch chincl chle0 sylib)) thm (nonbool () ((nonbool.1 (e. A (H~))) (nonbool.2 (e. B (H~))) (nonbool.3 (= F (` (span) ({} A)))) (nonbool.4 (= G (` (span) ({} B)))) (nonbool.5 (= H (` (span) ({} (opr A (+v) B)))))) (-> (-. (\/ (e. A G) (e. B F))) (=/= (i^i H (opr F (vH) G)) (opr (i^i H F) (vH) (i^i H G)))) ((opr (i^i H F) (vH) (i^i H G)) (0H) (i^i H (opr F (vH) G)) eqeq2 negbid biimparc (opr A (+v) B) H (opr F (vH) G) elin nonbool.1 nonbool.2 hvaddcl (opr A (+v) B) spansnid ax-mp nonbool.5 eleqtrr nonbool.3 nonbool.1 spansnch chshi eqeltr nonbool.4 nonbool.2 spansnch chshi eqeltr shslej nonbool.1 A spansnid ax-mp nonbool.3 eleqtrr nonbool.2 B spansnid ax-mp nonbool.4 eleqtrr nonbool.3 nonbool.1 spansnch chshi eqeltr nonbool.4 nonbool.2 spansnch chshi eqeltr A B shsva mp2an sselii mpbir2an (i^i H (opr F (vH) G)) (0H) (opr A (+v) B) eleq2 mpbii (opr A (+v) B) elch0 sylib nonbool.1 spansnch (` (span) ({} A)) ch0 ax-mp syl6eqel nonbool.3 B eleq2i nonbool.1 nonbool.2 A B sumspansnt mp2an bitr4 sylibr con3i (-. (e. A G)) adantl nonbool.1 nonbool.2 hvaddcl nonbool.1 spansnm0 nonbool.3 B eleq2i nonbool.1 nonbool.2 A B sumspansnt mp2an bitr4 negbii nonbool.5 nonbool.3 ineq12i (0H) eqeq1i 3imtr4 nonbool.1 nonbool.2 hvaddcl nonbool.2 spansnm0 nonbool.2 nonbool.1 B A sumspansnt mp2an nonbool.1 nonbool.2 hvcom (` (span) ({} B)) eleq1i nonbool.4 A eleq2i 3bitr4r negbii nonbool.5 nonbool.4 ineq12i (0H) eqeq1i 3imtr4 (vH) opreqan12rd h0elch chj0 syl6eq sylanc (e. A G) (e. B F) ioran (i^i H (opr F (vH) G)) (opr (i^i H F) (vH) (i^i H G)) df-ne 3imtr4)) thm (spansncv ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((spansncv.1 (e. A (CH))) (spansncv.2 (e. B (CH))) (spansncv.3 (e. C (H~)))) (-> (/\ (C: A B) (C_ B (opr A (vH) (` (span) ({} C))))) (= B (opr A (vH) (` (span) ({} C))))) ((C: A B) (C_ B (opr A (vH) (` (span) ({} C)))) pm3.27 spansncv.1 spansncv.3 spansnch spansncv.2 chlubi A B pssss (C_ B (opr A (vH) (` (span) ({} C)))) adantr B (opr A (vH) (` (span) ({} C))) (cv x) ssel2 spansncv.1 spansncv.3 spansnj (cv x) eleq2i spansncv.1 spansncv.3 spansnch (cv x) y z chsel bitr3 (cv y) (cv z) hvpncan2t spansncv.1 (cv y) chel spansncv.3 spansnch (cv z) chel syl2an B eleq1d spansncv.2 chshi B (opr (cv y) (+v) (cv z)) (cv y) shsubclt ax-mp (cv x) (opr (cv y) (+v) (cv z)) B eleq1 biimpac A B (cv y) ssel2 A B pssss sylan syl2an exp43 com14 imp45 syl5bi imp anandis exp45 imp41 (-. (e. (cv x) A)) adantrr spansncv.3 C (cv z) spansneleqOLD mpan imp B sseq1d spansncv.2 chshi B (cv z) spansnsst mpan syl5bi ancoms (cv z) (0v) (cv y) (+v) opreq2 spansncv.1 (cv y) chel (cv y) ax-hvaddid syl sylan9eqr (cv x) eqeq2d (cv y) A (cv x) eleq1a (= (cv z) (0v)) adantr sylbid ex com23 imp con3d imp sylan2 exp44 com12 imp41 (/\ (C: A B) (e. (cv x) B)) adantrl mpd exp43 r19.23aivv sylbi syl imp anandirs ex imp3a x 19.23adv A B x pssnel syl5 ex pm2.43d impcom sylanc eqssd)) thm (spansncvt () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (H~))) (-> (/\ (C: A B) (C_ B (opr A (vH) (` (span) ({} C))))) (= B (opr A (vH) (` (span) ({} C)))))) (A (if (e. A (CH)) A (H~)) B psseq1 A (if (e. A (CH)) A (H~)) (vH) (` (span) ({} C)) opreq1 B sseq2d anbi12d A (if (e. A (CH)) A (H~)) (vH) (` (span) ({} C)) opreq1 B eqeq2d imbi12d B (if (e. B (CH)) B (H~)) (if (e. A (CH)) A (H~)) psseq2 B (if (e. B (CH)) B (H~)) (opr (if (e. A (CH)) A (H~)) (vH) (` (span) ({} C))) sseq1 anbi12d B (if (e. B (CH)) B (H~)) (opr (if (e. A (CH)) A (H~)) (vH) (` (span) ({} C))) eqeq1 imbi12d C (if (e. C (H~)) C (0v)) sneq (span) fveq2d (if (e. A (CH)) A (H~)) (vH) opreq2d (if (e. B (CH)) B (H~)) sseq2d (C: (if (e. A (CH)) A (H~)) (if (e. B (CH)) B (H~))) anbi2d C (if (e. C (H~)) C (0v)) sneq (span) fveq2d (if (e. A (CH)) A (H~)) (vH) opreq2d (if (e. B (CH)) B (H~)) eqeq2d imbi12d helch A elimel helch B elimel ax-hv0cl C elimel spansncv dedth3h)) thm (5oalem1 () ((5oalem1.1 (e. A (SH))) (5oalem1.2 (e. B (SH))) (5oalem1.3 (e. C (SH))) (5oalem1.4 (e. R (SH)))) (-> (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv v) (opr (cv x) (+v) (cv y)))) (/\ (e. (cv z) C) (e. (opr (cv x) (-v) (cv z)) R))) (e. (cv v) (opr B (+H) (i^i A (opr C (+H) R))))) (5oalem1.1 5oalem1.3 5oalem1.4 shscl shincl 5oalem1.2 (cv x) (cv y) shsva 5oalem1.1 5oalem1.3 5oalem1.4 shscl shincl 5oalem1.2 shscom syl6eleq (e. (cv x) A) (e. (cv y) B) pm3.26 (= (cv v) (opr (cv x) (+v) (cv y))) (/\ (e. (cv z) C) (e. (opr (cv x) (-v) (cv z)) R)) ad2antrr (cv x) (cv z) (cv z) hvaddsub12t 3expb anabsan2 (cv z) hvsubidt (cv x) (+v) opreq2d (cv x) ax-hvaddid sylan9eqr eqtr3d 5oalem1.1 (cv x) shel (e. (cv y) B) (= (cv v) (opr (cv x) (+v) (cv y))) ad2antrr 5oalem1.3 (cv z) shel (e. (opr (cv x) (-v) (cv z)) R) adantr syl2an 5oalem1.3 5oalem1.4 (cv z) (opr (cv x) (-v) (cv z)) shsva (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv v) (opr (cv x) (+v) (cv y)))) adantl eqeltrrd jca (cv x) A (opr C (+H) R) elin sylibr (e. (cv x) A) (e. (cv y) B) pm3.27 (= (cv v) (opr (cv x) (+v) (cv y))) (/\ (e. (cv z) C) (e. (opr (cv x) (-v) (cv z)) R)) ad2antrr sylanc (cv v) (opr (cv x) (+v) (cv y)) (opr B (+H) (i^i A (opr C (+H) R))) eleq1 (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (opr (cv x) (-v) (cv z)) R)) ad2antlr mpbird)) thm (5oalem2 () ((5oalem2.1 (e. A (SH))) (5oalem2.2 (e. B (SH))) (5oalem2.3 (e. C (SH))) (5oalem2.4 (e. D (SH)))) (-> (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (= (opr (cv x) (+v) (cv y)) (opr (cv z) (+v) (cv w)))) (e. (opr (cv x) (-v) (cv z)) (i^i (opr A (+H) C) (opr B (+H) D)))) (5oalem2.1 5oalem2.3 (cv x) (cv z) shsvs (e. (cv y) B) (e. (cv w) D) ad2ant2r (= (opr (cv x) (+v) (cv y)) (opr (cv z) (+v) (cv w))) adantr 5oalem2.4 5oalem2.2 (cv w) (cv y) shsvs (e. (cv y) B) (e. (cv w) D) ancom 5oalem2.2 5oalem2.4 shscom (opr (cv w) (-v) (cv y)) eleq2i 3imtr4 (e. (cv x) A) (e. (cv z) C) ad2ant2l (= (opr (cv x) (+v) (cv y)) (opr (cv z) (+v) (cv w))) adantr (opr (cv x) (+v) (cv y)) (opr (cv z) (+v) (cv w)) (-v) (opr (cv z) (+v) (cv y)) opreq1 (/\ (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) (/\ (e. (cv z) (H~)) (e. (cv w) (H~)))) adantl (e. (cv x) (H~)) (e. (cv y) (H~)) pm3.27 (e. (cv z) (H~)) anim2i ancoms (cv x) (cv y) (cv z) (cv y) hvsub4t syldan (cv y) hvsubidt (opr (cv x) (-v) (cv z)) (+v) opreq2d (e. (cv x) (H~)) (e. (cv z) (H~)) ad2antlr (cv x) (cv z) hvsubclt (opr (cv x) (-v) (cv z)) ax-hvaddid syl (e. (cv y) (H~)) adantlr 3eqtrd (e. (cv w) (H~)) adantrr (= (opr (cv x) (+v) (cv y)) (opr (cv z) (+v) (cv w))) adantr (cv z) (cv w) (cv z) (cv y) hvsub4t (e. (cv y) (H~)) (/\ (e. (cv z) (H~)) (e. (cv w) (H~))) pm3.27 (e. (cv z) (H~)) (e. (cv w) (H~)) pm3.26 (e. (cv y) (H~)) anim1i ancoms sylanc (cv z) hvsubidt (+v) (opr (cv w) (-v) (cv y)) opreq1d (e. (cv y) (H~)) (e. (cv w) (H~)) ad2antrl (cv w) (cv y) hvsubclt (opr (cv w) (-v) (cv y)) hvaddid2t syl ancoms (e. (cv z) (H~)) adantrl 3eqtrd (e. (cv x) (H~)) adantll (= (opr (cv x) (+v) (cv y)) (opr (cv z) (+v) (cv w))) adantr 3eqtr3d (opr B (+H) D) eleq1d 5oalem2.1 (cv x) shel 5oalem2.2 (cv y) shel anim12i 5oalem2.3 (cv z) shel 5oalem2.4 (cv w) shel anim12i anim12i sylan mpbird jca (opr (cv x) (-v) (cv z)) (opr A (+H) C) (opr B (+H) D) elin sylibr)) thm (5oalem3 () ((5oalem3.1 (e. A (SH))) (5oalem3.2 (e. B (SH))) (5oalem3.3 (e. C (SH))) (5oalem3.4 (e. D (SH))) (5oalem3.5 (e. F (SH))) (5oalem3.6 (e. G (SH)))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (e. (cv f) F) (e. (cv g) G))) (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv f) (+v) (cv g))) (= (opr (cv z) (+v) (cv w)) (opr (cv f) (+v) (cv g))))) (e. (opr (cv x) (-v) (cv z)) (opr (i^i (opr A (+H) F) (opr B (+H) G)) (+H) (i^i (opr C (+H) F) (opr D (+H) G))))) (5oalem3.1 5oalem3.2 5oalem3.5 5oalem3.6 x y f g 5oalem2 5oalem3.3 5oalem3.4 5oalem3.5 5oalem3.6 z w f g 5oalem2 anim12i an4s (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D)) (/\ (e. (cv f) F) (e. (cv g) G)) anandir sylanb 5oalem3.1 5oalem3.5 shscl 5oalem3.2 5oalem3.6 shscl shincl 5oalem3.3 5oalem3.5 shscl 5oalem3.4 5oalem3.6 shscl shincl (opr (cv x) (-v) (cv f)) (opr (cv z) (-v) (cv f)) shsvs syl (cv x) (cv f) (cv z) (cv f) hvsubsub4t anandirs (cv f) hvsubidt (opr (cv x) (-v) (cv z)) (-v) opreq2d (/\ (e. (cv x) (H~)) (e. (cv z) (H~))) adantl (cv x) (cv z) hvsubclt (opr (cv x) (-v) (cv z)) hvsub0t syl (e. (cv f) (H~)) adantr 3eqtrd 5oalem3.1 (cv x) shel (e. (cv y) B) adantr 5oalem3.3 (cv z) shel (e. (cv w) D) adantr anim12i 5oalem3.5 (cv f) shel (e. (cv g) G) adantr syl2an (opr (i^i (opr A (+H) F) (opr B (+H) G)) (+H) (i^i (opr C (+H) F) (opr D (+H) G))) eleq1d (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv f) (+v) (cv g))) (= (opr (cv z) (+v) (cv w)) (opr (cv f) (+v) (cv g)))) adantr mpbid)) thm (5oalem4 () ((5oalem3.1 (e. A (SH))) (5oalem3.2 (e. B (SH))) (5oalem3.3 (e. C (SH))) (5oalem3.4 (e. D (SH))) (5oalem3.5 (e. F (SH))) (5oalem3.6 (e. G (SH)))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (e. (cv f) F) (e. (cv g) G))) (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv f) (+v) (cv g))) (= (opr (cv z) (+v) (cv w)) (opr (cv f) (+v) (cv g))))) (e. (opr (cv x) (-v) (cv z)) (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) F) (opr B (+H) G)) (+H) (i^i (opr C (+H) F) (opr D (+H) G)))))) (5oalem3.1 5oalem3.2 5oalem3.3 5oalem3.4 x y z w 5oalem2 (opr (cv x) (+v) (cv y)) (opr (cv f) (+v) (cv g)) (opr (cv z) (+v) (cv w)) eqtr3t sylan2 (/\ (e. (cv f) F) (e. (cv g) G)) adantlr 5oalem3.1 5oalem3.2 5oalem3.3 5oalem3.4 5oalem3.5 5oalem3.6 x y z w f g 5oalem3 jca (opr (cv x) (-v) (cv z)) (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) F) (opr B (+H) G)) (+H) (i^i (opr C (+H) F) (opr D (+H) G))) elin sylibr)) thm (5oalem5 () ((5oalem5.1 (e. A (SH))) (5oalem5.2 (e. B (SH))) (5oalem5.3 (e. C (SH))) (5oalem5.4 (e. D (SH))) (5oalem5.5 (e. F (SH))) (5oalem5.6 (e. G (SH))) (5oalem5.7 (e. R (SH))) (5oalem5.8 (e. S (SH)))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (/\ (e. (cv v) R) (e. (cv u) S)))) (/\ (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv v) (+v) (cv u))) (= (opr (cv z) (+v) (cv w)) (opr (cv v) (+v) (cv u)))) (= (opr (cv f) (+v) (cv g)) (opr (cv v) (+v) (cv u))))) (e. (opr (cv x) (-v) (cv z)) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))))))) (5oalem5.1 5oalem5.2 5oalem5.3 5oalem5.4 5oalem5.7 5oalem5.8 x y z w v u 5oalem4 (/\ (e. (cv f) F) (e. (cv g) G)) (/\ (e. (cv v) R) (e. (cv u) S)) pm3.27 (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) anim2i (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv v) (+v) (cv u))) (= (opr (cv z) (+v) (cv w)) (opr (cv v) (+v) (cv u)))) (= (opr (cv f) (+v) (cv g)) (opr (cv v) (+v) (cv u))) pm3.26 syl2an (cv x) (cv f) (cv z) (cv f) hvsubsub4t anandirs (cv f) hvsubidt (opr (cv x) (-v) (cv z)) (-v) opreq2d (/\ (e. (cv x) (H~)) (e. (cv z) (H~))) adantl (cv x) (cv z) hvsubclt (opr (cv x) (-v) (cv z)) hvsub0t syl (e. (cv f) (H~)) adantr 3eqtrd 5oalem5.1 (cv x) shel (e. (cv y) B) adantr 5oalem5.3 (cv z) shel (e. (cv w) D) adantr anim12i 5oalem5.5 (cv f) shel (e. (cv g) G) adantr syl2an (/\ (e. (cv v) R) (e. (cv u) S)) adantrr (/\ (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv v) (+v) (cv u))) (= (opr (cv z) (+v) (cv w)) (opr (cv v) (+v) (cv u)))) (= (opr (cv f) (+v) (cv g)) (opr (cv v) (+v) (cv u)))) adantr (/\ (e. (cv f) F) (e. (cv g) G)) (/\ (e. (cv v) R) (e. (cv u) S)) pm3.26 (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) anim2i (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D)) (/\ (e. (cv f) F) (e. (cv g) G)) anandir sylib (/\ (e. (cv f) F) (e. (cv g) G)) (/\ (e. (cv v) R) (e. (cv u) S)) pm3.27 (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) adantl jca (= (opr (cv x) (+v) (cv y)) (opr (cv v) (+v) (cv u))) (= (opr (cv z) (+v) (cv w)) (opr (cv v) (+v) (cv u))) pm3.26 (= (opr (cv f) (+v) (cv g)) (opr (cv v) (+v) (cv u))) anim1i (= (opr (cv x) (+v) (cv y)) (opr (cv v) (+v) (cv u))) (= (opr (cv z) (+v) (cv w)) (opr (cv v) (+v) (cv u))) pm3.27 (= (opr (cv f) (+v) (cv g)) (opr (cv v) (+v) (cv u))) anim1i jca anim12i 5oalem5.1 5oalem5.2 5oalem5.5 5oalem5.6 5oalem5.7 5oalem5.8 x y f g v u 5oalem4 5oalem5.3 5oalem5.4 5oalem5.5 5oalem5.6 5oalem5.7 5oalem5.8 z w f g v u 5oalem4 anim12i an4s (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv f) F) (e. (cv g) G))) (/\ (/\ (e. (cv z) C) (e. (cv w) D)) (/\ (e. (cv f) F) (e. (cv g) G))) (/\ (e. (cv v) R) (e. (cv u) S)) anandir sylanb 5oalem5.1 5oalem5.5 shscl 5oalem5.2 5oalem5.6 shscl shincl 5oalem5.1 5oalem5.7 shscl 5oalem5.2 5oalem5.8 shscl shincl 5oalem5.5 5oalem5.7 shscl 5oalem5.6 5oalem5.8 shscl shincl shscl shincl 5oalem5.3 5oalem5.5 shscl 5oalem5.4 5oalem5.6 shscl shincl 5oalem5.3 5oalem5.7 shscl 5oalem5.4 5oalem5.8 shscl shincl 5oalem5.5 5oalem5.7 shscl 5oalem5.6 5oalem5.8 shscl shincl shscl shincl (opr (cv x) (-v) (cv f)) (opr (cv z) (-v) (cv f)) shsvs 3syl eqeltrrd jca (opr (cv x) (-v) (cv z)) (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S))))) elin sylibr)) thm (5oalem6 () ((5oalem5.1 (e. A (SH))) (5oalem5.2 (e. B (SH))) (5oalem5.3 (e. C (SH))) (5oalem5.4 (e. D (SH))) (5oalem5.5 (e. F (SH))) (5oalem5.6 (e. G (SH))) (5oalem5.7 (e. R (SH))) (5oalem5.8 (e. S (SH)))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv h) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) D)) (= (cv h) (opr (cv z) (+v) (cv w))))) (/\ (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (= (cv h) (opr (cv f) (+v) (cv g)))) (/\ (/\ (e. (cv v) R) (e. (cv u) S)) (= (cv h) (opr (cv v) (+v) (cv u)))))) (e. (cv h) (opr B (+H) (i^i A (opr C (+H) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S))))))))))) ((cv h) (opr (cv x) (+v) (cv y)) (opr (cv v) (+v) (cv u)) eqeq1 biimpcd (cv h) (opr (cv z) (+v) (cv w)) (opr (cv v) (+v) (cv u)) eqeq1 biimpcd anim12d (cv h) (opr (cv f) (+v) (cv g)) (opr (cv v) (+v) (cv u)) eqeq1 biimpcd anim12d exp3a com3l imp32 (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (/\ (e. (cv v) R) (e. (cv u) S)))) anim2i an4s (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv h) (opr (cv x) (+v) (cv y))) (/\ (e. (cv z) C) (e. (cv w) D)) (= (cv h) (opr (cv z) (+v) (cv w))) an4 (/\ (e. (cv f) F) (e. (cv g) G)) (= (cv h) (opr (cv f) (+v) (cv g))) (/\ (e. (cv v) R) (e. (cv u) S)) (= (cv h) (opr (cv v) (+v) (cv u))) an4 syl2anb 5oalem5.1 5oalem5.2 5oalem5.3 5oalem5.4 5oalem5.5 5oalem5.6 5oalem5.7 5oalem5.8 x y z w f g v u 5oalem5 syl 5oalem5.1 5oalem5.2 5oalem5.3 5oalem5.1 5oalem5.3 shscl 5oalem5.2 5oalem5.4 shscl shincl 5oalem5.1 5oalem5.7 shscl 5oalem5.2 5oalem5.8 shscl shincl 5oalem5.3 5oalem5.7 shscl 5oalem5.4 5oalem5.8 shscl shincl shscl shincl 5oalem5.1 5oalem5.5 shscl 5oalem5.2 5oalem5.6 shscl shincl 5oalem5.1 5oalem5.7 shscl 5oalem5.2 5oalem5.8 shscl shincl 5oalem5.5 5oalem5.7 shscl 5oalem5.6 5oalem5.8 shscl shincl shscl shincl 5oalem5.3 5oalem5.5 shscl 5oalem5.4 5oalem5.6 shscl shincl 5oalem5.3 5oalem5.7 shscl 5oalem5.4 5oalem5.8 shscl shincl 5oalem5.5 5oalem5.7 shscl 5oalem5.6 5oalem5.8 shscl shincl shscl shincl shscl shincl x y h z 5oalem1 exp32 imp (e. (cv w) D) adantrr (= (cv h) (opr (cv z) (+v) (cv w))) adantrr (/\ (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (= (cv h) (opr (cv f) (+v) (cv g)))) (/\ (/\ (e. (cv v) R) (e. (cv u) S)) (= (cv h) (opr (cv v) (+v) (cv u))))) adantr mpd)) thm (5oalem7 ((A h) (A f) (A g) (A u) (A v) (A w) (A x) (A y) (A z) (f h) (g h) (h u) (h v) (h w) (h x) (h y) (h z) (f g) (f u) (f v) (f w) (f x) (f y) (f z) (g u) (g v) (g w) (g x) (g y) (g z) (u v) (u w) (u x) (u y) (u z) (v w) (v x) (v y) (v z) (w x) (w y) (w z) (x y) (x z) (y z) (B h) (B f) (B g) (B u) (B v) (B w) (B x) (B y) (B z) (C h) (C f) (C g) (C u) (C v) (C w) (C x) (C y) (C z) (D h) (D f) (D g) (D u) (D v) (D w) (D x) (D y) (D z) (F h) (F f) (F g) (F u) (F v) (F w) (F x) (F y) (F z) (G h) (G f) (G g) (G u) (G v) (G w) (G x) (G y) (G z) (R h) (R f) (R g) (R u) (R v) (R w) (R x) (R y) (R z) (S h) (S f) (S g) (S u) (S v) (S w) (S x) (S y) (S z)) ((5oalem5.1 (e. A (SH))) (5oalem5.2 (e. B (SH))) (5oalem5.3 (e. C (SH))) (5oalem5.4 (e. D (SH))) (5oalem5.5 (e. F (SH))) (5oalem5.6 (e. G (SH))) (5oalem5.7 (e. R (SH))) (5oalem5.8 (e. S (SH)))) (C_ (i^i (i^i (opr A (+H) B) (opr C (+H) D)) (i^i (opr F (+H) G) (opr R (+H) S))) (opr B (+H) (i^i A (opr C (+H) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))))))))) (x y f g (E. z (E. w (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv h) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) D)) (= (cv h) (opr (cv z) (+v) (cv w))))))) (E. v (E. u (/\ (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (= (cv h) (opr (cv f) (+v) (cv g)))) (/\ (/\ (e. (cv v) R) (e. (cv u) S)) (= (cv h) (opr (cv v) (+v) (cv u))))))) ee4anv z w f g (E. v (E. u (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv h) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) D)) (= (cv h) (opr (cv z) (+v) (cv w))))) (/\ (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (= (cv h) (opr (cv f) (+v) (cv g)))) (/\ (/\ (e. (cv v) R) (e. (cv u) S)) (= (cv h) (opr (cv v) (+v) (cv u)))))))) exrot4 z w v u (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv h) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) D)) (= (cv h) (opr (cv z) (+v) (cv w))))) (/\ (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (= (cv h) (opr (cv f) (+v) (cv g)))) (/\ (/\ (e. (cv v) R) (e. (cv u) S)) (= (cv h) (opr (cv v) (+v) (cv u))))) ee4anv f g 2exbii bitr x y 2exbii (cv h) (i^i (opr A (+H) B) (opr C (+H) D)) (i^i (opr F (+H) G) (opr R (+H) S)) elin 5oalem5.1 5oalem5.2 (cv h) x y shsel x A y B (= (cv h) (opr (cv x) (+v) (cv y))) r2ex bitr 5oalem5.3 5oalem5.4 (cv h) z w shsel z C w D (= (cv h) (opr (cv z) (+v) (cv w))) r2ex bitr anbi12i (cv h) (opr A (+H) B) (opr C (+H) D) elin x y z w (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv h) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) D)) (= (cv h) (opr (cv z) (+v) (cv w)))) ee4anv 3bitr4r 5oalem5.5 5oalem5.6 (cv h) f g shsel f F g G (= (cv h) (opr (cv f) (+v) (cv g))) r2ex bitr 5oalem5.7 5oalem5.8 (cv h) v u shsel v R u S (= (cv h) (opr (cv v) (+v) (cv u))) r2ex bitr anbi12i (cv h) (opr F (+H) G) (opr R (+H) S) elin f g v u (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (= (cv h) (opr (cv f) (+v) (cv g)))) (/\ (/\ (e. (cv v) R) (e. (cv u) S)) (= (cv h) (opr (cv v) (+v) (cv u)))) ee4anv 3bitr4r anbi12i bitr4 3bitr4r 5oalem5.1 5oalem5.2 5oalem5.3 5oalem5.4 5oalem5.5 5oalem5.6 5oalem5.7 5oalem5.8 x y h z w f g v u 5oalem6 v u 19.23aivv f g 19.23aivv z w 19.23aivv x y 19.23aivv sylbi ssriv)) thm (5oa () ((5oa.1 (e. A (CH))) (5oa.2 (e. B (CH))) (5oa.3 (e. C (CH))) (5oa.4 (e. D (CH))) (5oa.5 (e. F (CH))) (5oa.6 (e. G (CH))) (5oa.7 (e. R (CH))) (5oa.8 (e. S (CH))) (5oa.9 (C_ A (` (_|_) B))) (5oa.10 (C_ C (` (_|_) D))) (5oa.11 (C_ F (` (_|_) G))) (5oa.12 (C_ R (` (_|_) S)))) (C_ (i^i (i^i (opr A (vH) B) (opr C (vH) D)) (i^i (opr F (vH) G) (opr R (vH) S))) (opr B (vH) (i^i A (opr C (vH) (i^i (i^i (i^i (opr A (vH) C) (opr B (vH) D)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr C (vH) R) (opr D (vH) S)))) (opr (i^i (i^i (opr A (vH) F) (opr B (vH) G)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S)))) (vH) (i^i (i^i (opr C (vH) F) (opr D (vH) G)) (opr (i^i (opr C (vH) R) (opr D (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S)))))))))) (5oa.9 5oa.1 5oa.2 osum ax-mp 5oa.10 5oa.3 5oa.4 osum ax-mp ineq12i 5oa.11 5oa.5 5oa.6 osum ax-mp 5oa.12 5oa.7 5oa.8 osum ax-mp ineq12i ineq12i 5oa.1 chshi 5oa.2 chshi 5oa.3 chshi 5oa.4 chshi 5oa.5 chshi 5oa.6 chshi 5oa.7 chshi 5oa.8 chshi 5oalem7 eqsstr3 5oa.2 chshi 5oa.1 chshi 5oa.3 chshi 5oa.1 chshi 5oa.3 chshi shscl 5oa.2 chshi 5oa.4 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shscl shincl shscl shincl shslej 5oa.3 chshi 5oa.1 chshi 5oa.3 chshi shscl 5oa.2 chshi 5oa.4 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shscl shincl shslej 5oa.1 5oa.3 chslej 5oa.2 5oa.4 chslej (opr A (+H) C) (opr A (vH) C) (opr B (+H) D) (opr B (vH) D) ss2in mp2an 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shslej 5oa.3 5oa.7 chslej 5oa.4 5oa.8 chslej (opr C (+H) R) (opr C (vH) R) (opr D (+H) S) (opr D (vH) S) ss2in mp2an 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl shlej2 ax-mp 5oa.1 5oa.7 chslej 5oa.2 5oa.8 chslej (opr A (+H) R) (opr A (vH) R) (opr B (+H) S) (opr B (vH) S) ss2in mp2an 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl shlej1 ax-mp sstri sstri (i^i (opr A (+H) C) (opr B (+H) D)) (i^i (opr A (vH) C) (opr B (vH) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S))) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr C (vH) R) (opr D (vH) S))) ss2in mp2an 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shslej 5oa.3 5oa.5 chslej 5oa.4 5oa.6 chslej (opr C (+H) F) (opr C (vH) F) (opr D (+H) G) (opr D (vH) G) ss2in mp2an 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shslej 5oa.5 5oa.7 chslej 5oa.6 5oa.8 chslej (opr F (+H) R) (opr F (vH) R) (opr G (+H) S) (opr G (vH) S) ss2in mp2an 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shlej2 ax-mp 5oa.3 5oa.7 chslej 5oa.4 5oa.8 chslej (opr C (+H) R) (opr C (vH) R) (opr D (+H) S) (opr D (vH) S) ss2in mp2an 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shlej1 ax-mp sstri sstri (i^i (opr C (+H) F) (opr D (+H) G)) (i^i (opr C (vH) F) (opr D (vH) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S))) (opr (i^i (opr C (vH) R) (opr D (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S))) ss2in mp2an 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 5oa.5 chjcl 5oa.4 5oa.6 chjcl chincl chshi 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shlej2 ax-mp 5oa.1 5oa.5 chslej 5oa.2 5oa.6 chslej (opr A (+H) F) (opr A (vH) F) (opr B (+H) G) (opr B (vH) G) ss2in mp2an 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shslej 5oa.5 5oa.7 chslej 5oa.6 5oa.8 chslej (opr F (+H) R) (opr F (vH) R) (opr G (+H) S) (opr G (vH) S) ss2in mp2an 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl shlej2 ax-mp 5oa.1 5oa.7 chslej 5oa.2 5oa.8 chslej (opr A (+H) R) (opr A (vH) R) (opr B (+H) S) (opr B (vH) S) ss2in mp2an 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shlej1 ax-mp sstri sstri (i^i (opr A (+H) F) (opr B (+H) G)) (i^i (opr A (vH) F) (opr B (vH) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S))) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S))) ss2in mp2an 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.1 5oa.5 chjcl 5oa.2 5oa.6 chjcl chincl chshi 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl 5oa.3 5oa.5 chjcl 5oa.4 5oa.6 chjcl chincl chshi 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl shlej1 ax-mp sstri sstri (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (i^i (i^i (opr A (vH) C) (opr B (vH) D)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr C (vH) R) (opr D (vH) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S))))) (opr (i^i (i^i (opr A (vH) F) (opr B (vH) G)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S)))) (vH) (i^i (i^i (opr C (vH) F) (opr D (vH) G)) (opr (i^i (opr C (vH) R) (opr D (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S))))) ss2in mp2an 5oa.1 chshi 5oa.3 chshi shscl 5oa.2 chshi 5oa.4 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shscl shincl 5oa.1 5oa.3 chjcl 5oa.2 5oa.4 chjcl chincl 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl shjcl chincl chshi 5oa.1 5oa.5 chjcl 5oa.2 5oa.6 chjcl chincl chshi 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl 5oa.3 5oa.5 chjcl 5oa.4 5oa.6 chjcl chincl chshi 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl shjshcl shincl 5oa.3 chshi shlej2 ax-mp sstri (opr C (+H) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S))))))) (opr C (vH) (i^i (i^i (i^i (opr A (vH) C) (opr B (vH) D)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr C (vH) R) (opr D (vH) S)))) (opr (i^i (i^i (opr A (vH) F) (opr B (vH) G)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S)))) (vH) (i^i (i^i (opr C (vH) F) (opr D (vH) G)) (opr (i^i (opr C (vH) R) (opr D (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S))))))) A sslin ax-mp 5oa.1 chshi 5oa.3 chshi 5oa.1 chshi 5oa.3 chshi shscl 5oa.2 chshi 5oa.4 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shscl shincl shscl shincl 5oa.1 chshi 5oa.3 chshi 5oa.1 5oa.3 chjcl 5oa.2 5oa.4 chjcl chincl 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl shjcl chincl chshi 5oa.1 5oa.5 chjcl 5oa.2 5oa.6 chjcl chincl chshi 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl 5oa.3 5oa.5 chjcl 5oa.4 5oa.6 chjcl chincl chshi 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl shjshcl shincl shjshcl shincl 5oa.2 chshi shlej2 ax-mp sstri sstri)) thm (3oalem1 ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (C x) (C y) (C z) (C w) (C v) (R x) (R y) (R z) (R w) (R v) (S x) (S y) (S z) (S w) (S v)) ((3oalem1.1 (e. B (CH))) (3oalem1.2 (e. C (CH))) (3oalem1.3 (e. R (CH))) (3oalem1.4 (e. S (CH)))) (-> (/\ (/\ (/\ (e. (cv x) B) (e. (cv y) R)) (= (cv v) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) S)) (= (cv v) (opr (cv z) (+v) (cv w))))) (/\ (/\ (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) (e. (cv v) (H~))) (/\ (e. (cv z) (H~)) (e. (cv w) (H~))))) ((cv v) (opr (cv x) (+v) (cv y)) (H~) eleq1 (cv x) (cv y) ax-hvaddcl syl5bir com12 imdistani 3oalem1.1 (cv x) chel 3oalem1.3 (cv y) chel anim12i sylan 3oalem1.2 (cv z) chel 3oalem1.4 (cv w) chel anim12i (= (cv v) (opr (cv z) (+v) (cv w))) adantr anim12i)) thm (3oalem2 ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (C x) (C y) (C z) (C w) (C v) (R x) (R y) (R z) (R w) (R v) (S x) (S y) (S z) (S w) (S v)) ((3oalem1.1 (e. B (CH))) (3oalem1.2 (e. C (CH))) (3oalem1.3 (e. R (CH))) (3oalem1.4 (e. S (CH)))) (-> (/\ (/\ (/\ (e. (cv x) B) (e. (cv y) R)) (= (cv v) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) S)) (= (cv v) (opr (cv z) (+v) (cv w))))) (e. (cv v) (opr B (+H) (i^i R (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S))))))) (3oalem1.1 chshi 3oalem1.3 chshi 3oalem1.4 chshi 3oalem1.1 chshi 3oalem1.2 chshi shscl 3oalem1.3 chshi 3oalem1.4 chshi shscl shincl shscl shincl (cv x) (cv y) shsva (e. (cv x) B) (e. (cv y) R) pm3.26 (= (cv v) (opr (cv x) (+v) (cv y))) (/\ (/\ (e. (cv z) C) (e. (cv w) S)) (= (cv v) (opr (cv z) (+v) (cv w)))) ad2antrr (e. (cv x) B) (e. (cv y) R) pm3.27 (= (cv v) (opr (cv x) (+v) (cv y))) (/\ (/\ (e. (cv z) C) (e. (cv w) S)) (= (cv v) (opr (cv z) (+v) (cv w)))) ad2antrr 3oalem1.1 3oalem1.2 3oalem1.3 3oalem1.4 x y v z w 3oalem1 (cv y) (cv w) (cv w) hvaddsub12t 3expb anabsan2 (cv w) hvsubidt (cv y) (+v) opreq2d (cv y) ax-hvaddid sylan9eqr eqtr3d (e. (cv x) (H~)) (e. (cv z) (H~)) ad2ant2l (e. (cv v) (H~)) adantlr syl 3oalem1.4 chshi 3oalem1.1 chshi 3oalem1.2 chshi shscl 3oalem1.3 chshi 3oalem1.4 chshi shscl shincl (cv w) (opr (cv y) (-v) (cv w)) shsva (e. (cv z) C) (e. (cv w) S) pm3.27 (/\ (/\ (e. (cv x) B) (e. (cv y) R)) (= (cv v) (opr (cv x) (+v) (cv y)))) (= (cv v) (opr (cv z) (+v) (cv w))) ad2antrl (cv v) (opr (cv x) (+v) (cv y)) (opr (cv z) (+v) (cv w)) eqtr2t (-v) (opr (cv x) (+v) (cv w)) opreq1d (/\ (e. (cv x) B) (e. (cv y) R)) (/\ (e. (cv z) C) (e. (cv w) S)) ad2ant2l 3oalem1.1 3oalem1.2 3oalem1.3 3oalem1.4 x y v z w 3oalem1 (e. (cv x) (H~)) (e. (cv y) (H~)) pm3.26 (e. (cv w) (H~)) anim1i (cv x) (cv y) (cv x) (cv w) hvsub4t syldan (cv x) hvsubidt (e. (cv y) (H~)) (e. (cv w) (H~)) ad2antrr (+v) (opr (cv y) (-v) (cv w)) opreq1d (cv y) (cv w) hvsubclt (opr (cv y) (-v) (cv w)) hvaddid2t syl (e. (cv x) (H~)) adantll 3eqtrd (e. (cv z) (H~)) adantrl (e. (cv v) (H~)) adantlr syl 3oalem1.1 3oalem1.2 3oalem1.3 3oalem1.4 x y v z w 3oalem1 (cv z) (cv w) (cv x) (cv w) hvsub4t (e. (cv x) (H~)) (/\ (e. (cv z) (H~)) (e. (cv w) (H~))) pm3.27 (e. (cv z) (H~)) (e. (cv w) (H~)) pm3.27 (e. (cv x) (H~)) anim2i sylanc (cv w) hvsubidt (e. (cv x) (H~)) (e. (cv z) (H~)) ad2antll (opr (cv z) (-v) (cv x)) (+v) opreq2d (cv z) (cv x) hvsubclt (opr (cv z) (-v) (cv x)) ax-hvaddid syl ancoms (e. (cv w) (H~)) adantrr 3eqtrd (e. (cv y) (H~)) adantlr (e. (cv v) (H~)) adantlr syl 3eqtr3d 3oalem1.2 chshi 3oalem1.1 chshi (cv z) (cv x) shsvs (e. (cv x) B) (e. (cv z) C) ancom 3oalem1.1 chshi 3oalem1.2 chshi shscom (opr (cv z) (-v) (cv x)) eleq2i 3imtr4 (e. (cv x) B) (e. (cv y) R) pm3.26 (= (cv v) (opr (cv x) (+v) (cv y))) adantr (e. (cv z) C) (e. (cv w) S) pm3.26 (= (cv v) (opr (cv z) (+v) (cv w))) adantr syl2an eqeltrd 3oalem1.3 chshi 3oalem1.4 chshi (cv y) (cv w) shsvs (e. (cv x) B) (e. (cv y) R) pm3.27 (= (cv v) (opr (cv x) (+v) (cv y))) adantr (e. (cv z) C) (e. (cv w) S) pm3.27 (= (cv v) (opr (cv z) (+v) (cv w))) adantr syl2an jca (opr (cv y) (-v) (cv w)) (opr B (+H) C) (opr R (+H) S) elin sylibr sylanc eqeltrrd jca (cv y) R (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S))) elin sylibr sylanc (cv v) (opr (cv x) (+v) (cv y)) (opr B (+H) (i^i R (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S))))) eleq1 (/\ (e. (cv x) B) (e. (cv y) R)) (/\ (/\ (e. (cv z) C) (e. (cv w) S)) (= (cv v) (opr (cv z) (+v) (cv w)))) ad2antlr mpbird)) thm (3oalem3 ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (C x) (C y) (C z) (C w) (C v) (R x) (R y) (R z) (R w) (R v) (S x) (S y) (S z) (S w) (S v)) ((3oalem1.1 (e. B (CH))) (3oalem1.2 (e. C (CH))) (3oalem1.3 (e. R (CH))) (3oalem1.4 (e. S (CH)))) (C_ (i^i (opr B (+H) R) (opr C (+H) S)) (opr B (+H) (i^i R (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S)))))) (3oalem1.1 3oalem1.3 (cv v) x y chsel x B y R (= (cv v) (opr (cv x) (+v) (cv y))) r2ex bitr 3oalem1.2 3oalem1.4 (cv v) z w chsel z C w S (= (cv v) (opr (cv z) (+v) (cv w))) r2ex bitr anbi12i (cv v) (opr B (+H) R) (opr C (+H) S) elin x y z w (/\ (/\ (e. (cv x) B) (e. (cv y) R)) (= (cv v) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) S)) (= (cv v) (opr (cv z) (+v) (cv w)))) ee4anv 3bitr4 3oalem1.1 3oalem1.2 3oalem1.3 3oalem1.4 x y v z w 3oalem2 z w 19.23aivv x y 19.23aivv sylbi ssriv)) thm (3oalem4 () ((3oalem4.3 (= R (i^i (` (_|_) B) (opr B (vH) A))))) (C_ R (` (_|_) B)) (3oalem4.3 (` (_|_) B) (opr B (vH) A) inss1 eqsstr)) thm (3oalem5 () ((3oa.1 (e. A (CH))) (3oa.2 (e. B (CH))) (3oa.3 (e. C (CH))) (3oa.4 (= R (i^i (` (_|_) B) (opr B (vH) A)))) (3oa.5 (= S (i^i (` (_|_) C) (opr C (vH) A))))) (= (i^i (opr B (+H) R) (opr C (+H) S)) (i^i (opr B (vH) R) (opr C (vH) S))) (3oa.4 3oalem4 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.2 osum ax-mp 3oa.2 chshi 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr chshi shscom 3oa.2 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr chjcom 3eqtr4 3oa.5 3oalem4 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr 3oa.3 osum ax-mp 3oa.3 chshi 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shscom 3oa.3 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chjcom 3eqtr4 ineq12i)) thm (3oalem6 () ((3oa.1 (e. A (CH))) (3oa.2 (e. B (CH))) (3oa.3 (e. C (CH))) (3oa.4 (= R (i^i (` (_|_) B) (opr B (vH) A)))) (3oa.5 (= S (i^i (` (_|_) C) (opr C (vH) A))))) (C_ (opr B (+H) (i^i R (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S))))) (opr B (vH) (i^i R (opr S (vH) (i^i (opr B (vH) C) (opr R (vH) S)))))) (3oa.2 chshi 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chshi 3oa.2 chshi 3oa.3 chshi shscl 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shscl shincl shscl shincl shslej 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chshi 3oa.2 chshi 3oa.3 chshi shscl 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shscl shincl shslej 3oa.2 3oa.3 chslej (opr B (+H) C) (opr B (vH) C) (opr R (+H) S) ssrin ax-mp 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chslej (opr R (+H) S) (opr R (vH) S) (opr B (vH) C) sslin ax-mp sstri 3oa.2 chshi 3oa.3 chshi shscl 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shscl shincl 3oa.2 3oa.3 chjcl 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chjcl chincl chshi 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shlej2 ax-mp sstri (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S))) (opr S (vH) (i^i (opr B (vH) C) (opr R (vH) S))) R sslin ax-mp 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chshi 3oa.2 chshi 3oa.3 chshi shscl 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shscl shincl shscl shincl 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr 3oa.2 3oa.3 chjcl 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr chjcl chincl chjcl chincl chshi 3oa.2 chshi shlej2 ax-mp sstri)) thm (3oa () ((3oa.1 (e. A (CH))) (3oa.2 (e. B (CH))) (3oa.3 (e. C (CH))) (3oa.4 (= R (i^i (` (_|_) B) (opr B (vH) A)))) (3oa.5 (= S (i^i (` (_|_) C) (opr C (vH) A))))) (C_ (i^i (opr B (vH) R) (opr C (vH) S)) (opr B (vH) (i^i R (opr S (vH) (i^i (opr B (vH) C) (opr R (vH) S)))))) (3oa.1 3oa.2 3oa.3 3oa.4 3oa.5 3oalem5 3oa.2 3oa.3 3oa.4 3oa.2 choccl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.5 3oa.3 choccl 3oa.3 3oa.1 chjcl chincl eqeltr 3oalem3 3oa.1 3oa.2 3oa.3 3oa.4 3oa.5 3oalem6 sstri eqsstr3)) thm (pjorth () ((pjorth.1 (e. A (H~))) (pjorth.2 (e. B (H~)))) (-> (e. H (CH)) (= (opr (` (` (proj) H) A) (.i) (` (` (proj) (` (_|_) H)) B)) (0))) (H (` (` (proj) H) A) (` (` (proj) (` (_|_) H)) B) shocorth H chsh pjorth.1 H A axpjclt mpan2 H chocclt pjorth.2 (` (_|_) H) B axpjclt mpan2 syl jca sylc)) thm (pjch1t () () (-> (e. A (H~)) (= (` (` (proj) (H~)) A) A)) (A (if (e. A (H~)) A (0v)) (H~) eleq1 A (if (e. A (H~)) A (0v)) (` (proj) (H~)) fveq2 (= A (if (e. A (H~)) A (0v))) id eqeq12d bibi12d helch ax-hv0cl A elimel pjch dedth ibi)) thm (pjot () () (-> (/\ (e. H (CH)) (e. A (H~))) (= (` (` (proj) (` (_|_) H)) A) (opr (` (` (proj) (H~)) A) (-v) (` (` (proj) H) A)))) (A pjch1t (e. H (CH)) adantl H A axpjpjt eqtr2d (` (` (proj) (H~)) A) (` (` (proj) H) A) (` (` (proj) (` (_|_) H)) A) hvsubaddt helch A pjcl (e. H (CH)) adantl H A pjhclt (` (_|_) H) A pjhclt H chocclt sylan syl3anc mpbird eqcomd)) thm (pjch0t () () (-> (e. A (H~)) (= (` (` (proj) (0H)) A) (0v))) (helch (H~) A pjot mpan choc1 (proj) fveq2i A fveq1i syl5eqr helch A pjhcl (` (` (proj) (H~)) A) hvsubidt syl eqtrd)) thm (pjidm () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~)))) (= (` (` (proj) H) (` (` (proj) H) A)) (` (` (proj) H) A)) (pjidm.1 pjidm.2 pjcli pjidm.1 pjidm.1 pjidm.2 pjhcli pjch mpbi)) thm (pjadj () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjadj.3 (e. B (H~)))) (= (opr (` (` (proj) H) A) (.i) B) (opr A (.i) (` (` (proj) H) B))) (pjidm.1 pjadj.3 pjidm.2 H pjorth ax-mp (*) fveq2i cj0 eqtr pjidm.1 choccl pjidm.2 pjhcli pjidm.1 pjadj.3 pjhcli his1 pjidm.1 pjidm.2 pjadj.3 H pjorth ax-mp 3eqtr4r (opr (` (` (proj) H) A) (.i) (` (` (proj) H) B)) (+) opreq2i pjidm.1 pjidm.2 pjhcli pjidm.1 pjadj.3 pjhcli pjidm.1 choccl pjadj.3 pjhcli (` (` (proj) H) A) (` (` (proj) H) B) (` (` (proj) (` (_|_) H)) B) his7t mp3an pjidm.1 pjidm.2 pjhcli pjidm.1 choccl pjidm.2 pjhcli pjidm.1 pjadj.3 pjhcli (` (` (proj) H) A) (` (` (proj) (` (_|_) H)) A) (` (` (proj) H) B) ax-his2 mp3an 3eqtr4 pjidm.1 pjadj.3 pjpj (` (` (proj) H) A) (.i) opreq2i pjidm.1 pjidm.2 pjpj (.i) (` (` (proj) H) B) opreq1i 3eqtr4)) thm (pjcomp () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjadj.3 (e. B (H~)))) (-> (/\ (e. A H) (e. B (` (_|_) H))) (= (` (` (proj) H) (opr A (+v) B)) A)) (pjidm.1 pjidm.2 pjadj.3 hvaddcl pjcli pjidm.1 choccl pjidm.2 pjadj.3 hvaddcl pjcli pjidm.1 pjidm.2 pjadj.3 hvaddcl pjpj (opr A (+v) B) eqid pjidm.1 (` (` (proj) H) (opr A (+v) B)) (` (` (proj) (` (_|_) H)) (opr A (+v) B)) A B (opr A (+v) B) chocuni mp2ani mpanl12 pm3.26d)) thm (pjadd () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjadj.3 (e. B (H~)))) (= (` (` (proj) H) (opr A (+v) B)) (opr (` (` (proj) H) A) (+v) (` (` (proj) H) B))) (pjidm.1 pjidm.2 pjpj pjidm.1 pjadj.3 pjpj (+v) opreq12i pjidm.1 pjidm.2 pjhcli pjidm.1 choccl pjidm.2 pjhcli pjidm.1 pjadj.3 pjhcli pjidm.1 choccl pjadj.3 pjhcli hvadd4 eqtr (` (proj) H) fveq2i pjidm.1 pjidm.2 pjcli pjidm.1 pjadj.3 pjcli pjidm.1 chshi H (` (` (proj) H) A) (` (` (proj) H) B) shaddclt ax-mp mp2an pjidm.1 choccl pjidm.2 pjcli pjidm.1 choccl pjadj.3 pjcli pjidm.1 choccl chshi (` (_|_) H) (` (` (proj) (` (_|_) H)) A) (` (` (proj) (` (_|_) H)) B) shaddclt ax-mp mp2an pjidm.1 pjidm.1 pjidm.2 pjhcli pjidm.1 pjadj.3 pjhcli hvaddcl pjidm.1 choccl pjidm.2 pjhcli pjidm.1 choccl pjadj.3 pjhcli hvaddcl pjcomp mp2an eqtr)) thm (pjinorm () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~)))) (= (opr (` (` (proj) H) A) (.i) A) (opr (` (norm) (` (` (proj) H) A)) (^) (2))) (pjidm.1 pjidm.2 pjhcli normsq pjidm.1 pjidm.1 pjidm.2 pjhcli pjidm.2 pjadj pjidm.1 pjidm.2 pjidm (.i) A opreq1i 3eqtr2r)) thm (pjmul () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjmul.3 (e. C (CC)))) (= (` (` (proj) H) (opr C (.s) A)) (opr C (.s) (` (` (proj) H) A))) (pjidm.1 pjidm.2 pjpj C (.s) opreq2i pjmul.3 pjidm.1 pjidm.2 pjhcli pjidm.1 choccl pjidm.2 pjhcli hvdistr1 eqtr (` (proj) H) fveq2i pjmul.3 pjidm.1 pjidm.2 pjcli pjidm.1 chshi H C (` (` (proj) H) A) shmulclt ax-mp mp2an pjmul.3 pjidm.1 choccl pjidm.2 pjcli pjidm.1 choccl chshi (` (_|_) H) C (` (` (proj) (` (_|_) H)) A) shmulclt ax-mp mp2an pjidm.1 pjmul.3 pjidm.1 pjidm.2 pjhcli hvmulcl pjmul.3 pjidm.1 choccl pjidm.2 pjhcli hvmulcl pjcomp mp2an eqtr)) thm (pjsub () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsub.3 (e. B (H~)))) (= (` (` (proj) H) (opr A (-v) B)) (opr (` (` (proj) H) A) (-v) (` (` (proj) H) B))) (pjidm.1 pjidm.2 1cn negcl pjsub.3 hvmulcl pjadd pjidm.1 pjsub.3 1cn negcl pjmul (` (` (proj) H) A) (+v) opreq2i eqtr pjidm.2 pjsub.3 hvsubval (` (proj) H) fveq2i pjidm.1 pjidm.2 pjhcli pjidm.1 pjsub.3 pjhcli hvsubval 3eqtr4)) thm (pjsslem () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (= (opr (` (` (proj) (` (_|_) H)) A) (-v) (` (` (proj) (` (_|_) G)) A)) (opr (` (` (proj) G) A) (-v) (` (` (proj) H) A))) (pjidm.1 pjidm.2 H A pjot mp2an pjsslem.1 pjidm.2 G A pjot mp2an (-v) opreq12i helch pjidm.2 pjcli pjidm.1 pjidm.2 pjhcli helch pjidm.2 pjcli pjsslem.1 pjidm.2 pjhcli hvsubsub4 helch pjidm.2 pjcli (` (` (proj) (H~)) A) hvsubidt ax-mp (-v) (opr (` (` (proj) H) A) (-v) (` (` (proj) G) A)) opreq1i 3eqtr pjidm.1 pjidm.2 pjhcli pjsslem.1 pjidm.2 pjhcli hvsubcl hv2neg pjidm.1 pjidm.2 pjhcli pjsslem.1 pjidm.2 pjhcli hvnegdi 3eqtr)) thm (pjss2 () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (-> (C_ H G) (= (` (` (proj) H) (` (` (proj) G) A)) (` (` (proj) H) A))) (pjidm.1 pjsslem.1 chsscon3 pjsslem.1 choccl pjidm.2 pjcli (` (_|_) G) (` (_|_) H) (` (` (proj) (` (_|_) G)) A) ssel mpi sylbi pjidm.1 choccl pjidm.2 pjcli jctil pjidm.1 choccl chshi (` (_|_) H) (` (` (proj) (` (_|_) H)) A) (` (` (proj) (` (_|_) G)) A) shsubclt ax-mp syl pjidm.1 pjidm.2 pjsslem.1 pjsslem (` (_|_) H) eleq1i pjidm.1 pjsslem.1 pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli hvsubcl pjoc2 bitr pjidm.1 pjsslem.1 pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli pjsub (0v) eqeq1i pjidm.1 pjsslem.1 pjidm.2 pjhcli pjhcli pjidm.1 pjidm.1 pjidm.2 pjhcli pjhcli hvsubeq0 3bitr pjidm.1 pjidm.2 pjidm (` (` (proj) H) (` (` (proj) G) A)) eqeq2i bitr2 sylibr)) thm (pjssm () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (-> (C_ H G) (= (opr (` (` (proj) G) A) (-v) (` (` (proj) H) A)) (` (` (proj) (i^i G (` (_|_) H))) A))) (pjidm.1 pjidm.2 pjcli H G (` (` (proj) H) A) ssel mpi pjsslem.1 pjidm.2 pjcli jctil pjsslem.1 chshi G (` (` (proj) G) A) (` (` (proj) H) A) shsubclt ax-mp syl pjidm.1 pjsslem.1 chsscon3 pjsslem.1 choccl pjidm.2 pjcli (` (_|_) G) (` (_|_) H) (` (` (proj) (` (_|_) G)) A) ssel mpi pjidm.1 choccl pjidm.2 pjcli jctil pjidm.1 choccl chshi (` (_|_) H) (` (` (proj) (` (_|_) H)) A) (` (` (proj) (` (_|_) G)) A) shsubclt ax-mp syl sylbi pjidm.1 pjidm.2 pjsslem.1 pjsslem syl5eqelr jca (opr (` (` (proj) G) A) (-v) (` (` (proj) H) A)) G (` (_|_) H) elin pjsslem.1 pjidm.1 choccl chincl pjsslem.1 pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli hvsubcl pjch bitr3 sylib pjsslem.1 pjidm.1 choccl chincl pjsslem.1 pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli pjsub pjsslem.1 pjidm.1 choccl chincl pjsslem.1 pjidm.2 pjhcli pjhcli pjsslem.1 pjidm.1 choccl chincl pjidm.1 pjidm.2 pjhcli pjhcli hvsubval G (` (_|_) H) inss1 pjsslem.1 pjidm.1 choccl chincl pjidm.2 pjsslem.1 pjss2 ax-mp pjidm.1 chshi H shococss ax-mp G (` (_|_) H) inss2 pjsslem.1 pjidm.1 choccl chincl pjidm.1 choccl chsscon3 mpbi sstri pjidm.1 pjidm.2 pjcli sselii pjsslem.1 pjidm.1 choccl chincl pjidm.1 pjidm.2 pjhcli pjoc2 mpbi (-u (1)) (.s) opreq2i 1cn negcl (-u (1)) hvmul0t ax-mp eqtr (+v) opreq12i pjsslem.1 pjidm.1 choccl chincl pjidm.2 pjhcli (` (` (proj) (i^i G (` (_|_) H))) A) ax-hvaddid ax-mp eqtr 3eqtr syl5reqr)) thm (pjssge0 () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (-> (= (opr (` (` (proj) G) A) (-v) (` (` (proj) H) A)) (` (` (proj) (i^i G (` (_|_) H))) A)) (br (0) (<_) (opr (opr (` (` (proj) G) A) (-v) (` (` (proj) H) A)) (.i) A))) ((opr (` (` (proj) G) A) (-v) (` (` (proj) H) A)) (` (` (proj) (i^i G (` (_|_) H))) A) (.i) A opreq1 pjsslem.1 pjidm.1 choccl chincl pjidm.2 pjinorm syl6eq pjsslem.1 pjidm.1 choccl chincl pjidm.2 pjhcli normcl sqge0 syl5breqr)) thm (pjdifnorm () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (<-> (br (0) (<_) (opr (opr (` (` (proj) G) A) (-v) (` (` (proj) H) A)) (.i) A)) (br (` (norm) (` (` (proj) H) A)) (<_) (` (norm) (` (` (proj) G) A)))) (pjsslem.1 pjidm.2 pjhcli normcl resqcl pjidm.1 pjidm.2 pjhcli normcl resqcl subge0 pjsslem.1 pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli pjidm.2 (` (` (proj) G) A) (` (` (proj) H) A) A his2subt mp3an pjsslem.1 pjidm.2 pjinorm pjidm.1 pjidm.2 pjinorm (-) opreq12i eqtr (0) (<_) breq2i pjidm.1 pjidm.2 pjhcli (` (` (proj) H) A) normge0t ax-mp pjsslem.1 pjidm.2 pjhcli (` (` (proj) G) A) normge0t ax-mp pjidm.1 pjidm.2 pjhcli normcl pjsslem.1 pjidm.2 pjhcli normcl le2sq mp2an 3bitr4)) thm (pjcj () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (-> (C_ H (` (_|_) G)) (= (` (` (proj) (opr H (vH) G)) A) (opr (` (` (proj) H) A) (+v) (` (` (proj) G) A)))) (pjidm.1 pjidm.2 pjsslem.1 choccl pjssm A (-v) opreq2d pjidm.1 pjidm.2 pjhcli pjidm.2 pjsslem.1 choccl pjidm.2 pjhcli (` (` (proj) H) A) A (` (` (proj) (` (_|_) G)) A) hvaddsub12t mp3an pjsslem.1 pjidm.2 pjpo (` (` (proj) H) A) (+v) opreq2i pjsslem.1 choccl pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli hvnegdi A (+v) opreq2i 3eqtr4 pjidm.2 pjsslem.1 choccl pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli hvsubcl hvsubval eqtr4 pjidm.1 pjsslem.1 chjcom pjsslem.1 pjidm.1 chdmm4 eqtr4 (proj) fveq2i A fveq1i pjsslem.1 choccl pjidm.1 choccl chincl pjidm.2 pjop eqtr 3eqtr4g eqcomd)) thm (pjadjt () ((pjadjt.1 (e. H (CH)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` (` (proj) H) A) (.i) B) (opr A (.i) (` (` (proj) H) B)))) (A (if (e. A (H~)) A (0v)) (` (proj) H) fveq2 (.i) B opreq1d A (if (e. A (H~)) A (0v)) (.i) (` (` (proj) H) B) opreq1 eqeq12d B (if (e. B (H~)) B (0v)) (` (` (proj) H) (if (e. A (H~)) A (0v))) (.i) opreq2 B (if (e. B (H~)) B (0v)) (` (proj) H) fveq2 (if (e. A (H~)) A (0v)) (.i) opreq2d eqeq12d pjadjt.1 ax-hv0cl A elimel ax-hv0cl B elimel pjadj dedth2h)) thm (pjaddt () ((pjadjt.1 (e. H (CH)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` (` (proj) H) (opr A (+v) B)) (opr (` (` (proj) H) A) (+v) (` (` (proj) H) B)))) (A (if (e. A (H~)) A (0v)) (+v) B opreq1 (` (proj) H) fveq2d A (if (e. A (H~)) A (0v)) (` (proj) H) fveq2 (+v) (` (` (proj) H) B) opreq1d eqeq12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (+v) opreq2 (` (proj) H) fveq2d B (if (e. B (H~)) B (0v)) (` (proj) H) fveq2 (` (` (proj) H) (if (e. A (H~)) A (0v))) (+v) opreq2d eqeq12d pjadjt.1 ax-hv0cl A elimel ax-hv0cl B elimel pjadd dedth2h)) thm (pjinormt () ((pjadjt.1 (e. H (CH)))) (-> (e. A (H~)) (= (opr (` (` (proj) H) A) (.i) A) (opr (` (norm) (` (` (proj) H) A)) (^) (2)))) (A (if (e. A (H~)) A (0v)) (` (proj) H) fveq2 (= A (if (e. A (H~)) A (0v))) id (.i) opreq12d A (if (e. A (H~)) A (0v)) (` (proj) H) fveq2 (norm) fveq2d (^) (2) opreq1d eqeq12d pjadjt.1 ax-hv0cl A elimel pjinorm dedth)) thm (pjsubt () ((pjadjt.1 (e. H (CH)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` (` (proj) H) (opr A (-v) B)) (opr (` (` (proj) H) A) (-v) (` (` (proj) H) B)))) (A (if (e. A (H~)) A (0v)) (-v) B opreq1 (` (proj) H) fveq2d A (if (e. A (H~)) A (0v)) (` (proj) H) fveq2 (-v) (` (` (proj) H) B) opreq1d eqeq12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (` (proj) H) fveq2d B (if (e. B (H~)) B (0v)) (` (proj) H) fveq2 (` (` (proj) H) (if (e. A (H~)) A (0v))) (-v) opreq2d eqeq12d pjadjt.1 ax-hv0cl A elimel ax-hv0cl B elimel pjsub dedth2h)) thm (pjmult () ((pjadjt.1 (e. H (CH)))) (-> (/\ (e. A (CC)) (e. B (H~))) (= (` (` (proj) H) (opr A (.s) B)) (opr A (.s) (` (` (proj) H) B)))) (A (if (e. A (CC)) A (0)) (.s) B opreq1 (` (proj) H) fveq2d A (if (e. A (CC)) A (0)) (.s) (` (` (proj) H) B) opreq1 eqeq12d B (if (e. B (H~)) B (0v)) (if (e. A (CC)) A (0)) (.s) opreq2 (` (proj) H) fveq2d B (if (e. B (H~)) B (0v)) (` (proj) H) fveq2 (if (e. A (CC)) A (0)) (.s) opreq2d eqeq12d pjadjt.1 ax-hv0cl B elimel 0cn A elimel pjmul dedth2h)) thm (pjige0 () ((pjadjt.1 (e. H (CH)))) (-> (e. A (H~)) (br (0) (<_) (opr (` (` (proj) H) A) (.i) A))) (pjadjt.1 A pjhcl (` (` (proj) H) A) normclt (` (norm) (` (` (proj) H) A)) sqge0t 3syl pjadjt.1 A pjinormt breqtrrd)) thm (pjige0t () () (-> (/\ (e. H (CH)) (e. A (H~))) (br (0) (<_) (opr (` (` (proj) H) A) (.i) A))) (H (if (e. H (CH)) H (0H)) (proj) fveq2 A fveq1d (.i) A opreq1d (0) (<_) breq2d (e. A (H~)) imbi2d h0elch H elimel A pjige0 dedth imp)) thm (pjcjt2 () () (-> (/\/\ (e. H (CH)) (e. G (CH)) (e. A (H~))) (-> (C_ H (` (_|_) G)) (= (` (` (proj) (opr H (vH) G)) A) (opr (` (` (proj) H) A) (+v) (` (` (proj) G) A))))) (H (if (e. H (CH)) H (H~)) (` (_|_) G) sseq1 H (if (e. H (CH)) H (H~)) (vH) G opreq1 (proj) fveq2d A fveq1d H (if (e. H (CH)) H (H~)) (proj) fveq2 A fveq1d (+v) (` (` (proj) G) A) opreq1d eqeq12d imbi12d G (if (e. G (CH)) G (H~)) (_|_) fveq2 (if (e. H (CH)) H (H~)) sseq2d G (if (e. G (CH)) G (H~)) (if (e. H (CH)) H (H~)) (vH) opreq2 (proj) fveq2d A fveq1d G (if (e. G (CH)) G (H~)) (proj) fveq2 A fveq1d (` (` (proj) (if (e. H (CH)) H (H~))) A) (+v) opreq2d eqeq12d imbi12d A (if (e. A (H~)) A (0v)) (` (proj) (opr (if (e. H (CH)) H (H~)) (vH) (if (e. G (CH)) G (H~)))) fveq2 A (if (e. A (H~)) A (0v)) (` (proj) (if (e. H (CH)) H (H~))) fveq2 A (if (e. A (H~)) A (0v)) (` (proj) (if (e. G (CH)) G (H~))) fveq2 (+v) opreq12d eqeq12d (C_ (if (e. H (CH)) H (H~)) (` (_|_) (if (e. G (CH)) G (H~)))) imbi2d helch H elimel ax-hv0cl A elimel helch G elimel pjcj dedth3h)) thm (pj0 () ((pj0.1 (e. H (CH)))) (= (` (` (proj) H) (0v)) (0v)) (pj0.1 chshi H oc0 ax-mp pj0.1 ax-hv0cl pjoc2 mpbi)) thm (pjcht () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (e. A H) (= (` (` (proj) H) A) A))) (H (if (e. H (CH)) H (H~)) A eleq2 H (if (e. H (CH)) H (H~)) (proj) fveq2 A fveq1d A eqeq1d bibi12d A (if (e. A (H~)) A (0v)) (if (e. H (CH)) H (H~)) eleq1 A (if (e. A (H~)) A (0v)) (` (proj) (if (e. H (CH)) H (H~))) fveq2 (= A (if (e. A (H~)) A (0v))) id eqeq12d bibi12d helch H elimel ax-hv0cl A elimel pjch dedth2h)) thm (pjidt () () (-> (/\ (e. H (CH)) (e. A H)) (= (` (` (proj) H) A) A)) (H A pjcht biimpa (e. H (CH)) (e. A H) pm3.26 H A chelt jca (e. H (CH)) (e. A H) pm3.27 sylanc)) thm (pjvect ((H x)) () (-> (e. H (CH)) (= H ({e.|} x (H~) (= (` (` (proj) H) (cv x)) (cv x))))) (H chss H (H~) sseqin2 sylib H (cv x) pjcht ex rabbirdv eqtr3d)) thm (pjocvect ((H x)) () (-> (e. H (CH)) (= (` (_|_) H) ({e.|} x (H~) (= (` (` (proj) H) (cv x)) (0v))))) (H chocclt (` (_|_) H) chss syl (` (_|_) H) (H~) sseqin2 sylib H (cv x) pjoc2t ex rabbirdv eqtr3d)) thm (pjocin () ((pjocin.1 (e. G (CH))) (pjocin.2 (e. H (CH)))) (-> (e. A (` (_|_) (i^i G H))) (e. (` (` (proj) G) A) (` (_|_) (i^i G H)))) (pjocin.1 pjocin.2 chincl choccl A chel pjocin.1 G A pjpot mpan syl pjocin.1 pjocin.2 chincl choccl A chel pjocin.1 choccl A pjcl G H inss1 pjocin.1 pjocin.2 chincl pjocin.1 chsscon3 mpbi (` (` (proj) (` (_|_) G)) A) sseli 3syl pjocin.1 pjocin.2 chincl choccl chshi (` (_|_) (i^i G H)) A (` (` (proj) (` (_|_) G)) A) shsubclt ax-mp mpdan eqeltrd)) thm (pjin () ((pjocin.1 (e. G (CH))) (pjocin.2 (e. H (CH)))) (-> (e. A (i^i G H)) (e. (` (` (proj) G) A) (i^i G H))) (G H inss1 A sseli pjocin.1 G A pjidt mpan syl (i^i G H) eleq1d ibir)) thm (pjjs ((x y) (x z) (w x) (G x) (y z) (w y) (G y) (w z) (G z) (G w) (H x) (H y) (H z) (H w)) ((pjjs.1 (e. G (CH))) (pjjs.2 (e. H (SH)))) (-> (A.e. x (opr G (vH) H) (e. (` (` (proj) (` (_|_) G)) (cv x)) H)) (= (opr G (vH) H) (opr G (+H) H))) ((cv x) (cv w) (` (proj) (` (_|_) G)) fveq2 H eleq1d (opr G (vH) H) rcla4v pjjs.1 (cv w) pjcl (e. (` (` (proj) (` (_|_) G)) (cv w)) H) anim1i pjjs.1 G (cv w) axpjpjt mpan (e. (` (` (proj) (` (_|_) G)) (cv w)) H) adantr jca pjjs.1 chshi pjjs.2 shjcl (cv w) chel sylan (cv y) (` (` (proj) G) (cv w)) (+v) (cv z) opreq1 (cv w) eqeq2d (cv z) (` (` (proj) (` (_|_) G)) (cv w)) (` (` (proj) G) (cv w)) (+v) opreq2 (cv w) eqeq2d G H rcla42ev syl pjjs.1 chshi pjjs.2 (cv w) y z shsel sylibr ex syld com12 ssrdv pjjs.1 chshi pjjs.2 shslej jctir (opr G (vH) H) (opr G (+H) H) eqss sylibr)) thm (pjfn ((x y) (x z) (w x) (H x) (y z) (w y) (H y) (w z) (H z) (H w)) ((pjfn.1 (e. H (CH)))) (Fn (` (proj) H) (H~)) (pjfn.1 elisseti z (E.e. w (` (_|_) H) (= (cv x) (opr (cv z) (+v) (cv w)))) rabex uniex pjfn.1 H x y z w pjmvalt ax-mp fnopab2)) thm (pjrn ((x y) (H x) (H y)) ((pjfn.1 (e. H (CH)))) (= (ran (` (proj) H)) H) (x y (/\ (e. (cv x) (H~)) (= (cv y) (` (` (proj) H) (cv x)))) rnopab pjfn.1 pjfn (` (proj) H) (H~) x y fnopabfv mpbi rneqi y visset (cv x) (cv y) (H~) eleq1 (cv x) (cv y) (` (proj) H) fveq2 (cv y) eqeq2d anbi12d cla4ev pjfn.1 (cv y) chel pjfn.1 H (cv y) pjidt mpan eqcomd sylanc (cv y) (` (` (proj) H) (cv x)) H eleq1 pjfn.1 (cv x) pjcl syl5bir impcom x 19.23aiv impbi abbi2i 3eqtr4)) thm (pjfo () ((pjfn.1 (e. H (CH)))) (:-onto-> (` (proj) H) (H~) H) ((` (proj) H) (H~) H df-fo pjfn.1 pjfn pjfn.1 pjrn mpbir2an)) thm (pjf () ((pjfn.1 (e. H (CH)))) (:--> (` (proj) H) (H~) (H~)) ((` (proj) H) (H~) (H~) df-f pjfn.1 pjfn pjfn.1 pjrn pjfn.1 chssi eqsstr mpbir2an)) thm (pjv () ((pjfn.1 (e. H (CH)))) (-> (/\ (e. A H) (e. B (` (_|_) H))) (= (` (` (proj) H) (opr A (+v) B)) A)) (pjfn.1 A B pjaddt pjfn.1 A chel pjfn.1 choccl B chel syl2an pjfn.1 H A pjidt mpan pjfn.1 choccl B chel pjfn.1 B pjoc2tOLD syl ibi (+v) opreqan12d pjfn.1 A chel A ax-hvaddid syl (e. B (` (_|_) H)) adantr 3eqtrd)) thm (pjfot () () (-> (e. H (CH)) (:-onto-> (` (proj) H) (H~) H)) (H (if (e. H (CH)) H (0H)) (proj) fveq2 (` (proj) H) (` (proj) (if (e. H (CH)) H (0H))) (H~) H foeq1 syl H (if (e. H (CH)) H (0H)) (` (proj) (if (e. H (CH)) H (0H))) (H~) foeq3 h0elch H elimel pjfo dedth2v)) thm (pjrnt () () (-> (e. H (CH)) (= (ran (` (proj) H)) H)) (H pjfot (` (proj) H) (H~) H forn syl)) thm (pjft () () (-> (e. H (CH)) (:--> (` (proj) H) (H~) (H~))) ((` (proj) H) (H~) H (H~) fss H pjfot (` (proj) H) (H~) H fof syl H chss sylanc)) thm (pjfnt () () (-> (e. H (CH)) (Fn (` (proj) H) (H~))) (H pjft (` (proj) H) (H~) (H~) ffn syl)) thm (pjsumt () ((pjsumt.1 (e. G (CH))) (pjsumt.2 (e. H (CH)))) (-> (e. A (H~)) (-> (C_ G (` (_|_) H)) (= (` (` (proj) (opr G (+H) H)) A) (opr (` (` (proj) G) A) (+v) (` (` (proj) H) A))))) (pjsumt.1 pjsumt.2 osum (proj) fveq2d A fveq1d (e. A (H~)) adantl pjsumt.1 pjsumt.2 G H A pjcjt2 mp3an12 imp eqtrd ex)) thm (pj11 () ((pjsumt.1 (e. G (CH))) (pjsumt.2 (e. H (CH)))) (<-> (= (` (proj) G) (` (proj) H)) (= G H)) ((` (proj) G) (` (proj) H) rneq pjsumt.1 pjrn pjsumt.2 pjrn 3eqtr3g G H (proj) fveq2 impbi)) thm (pjds () ((pjsumt.1 (e. G (CH))) (pjsumt.2 (e. H (CH)))) (-> (/\ (e. A (opr G (vH) H)) (C_ G (` (_|_) H))) (= A (opr (` (` (proj) G) A) (+v) (` (` (proj) H) A)))) (pjsumt.1 pjsumt.2 osum (proj) fveq2d A fveq1d pjsumt.1 pjsumt.2 chjcl (opr G (vH) H) A pjidt mpan sylan9eqr pjsumt.1 pjsumt.2 A pjsumt imp pjsumt.1 pjsumt.2 chjcl A chel sylan eqtr3d)) thm (pjds3 () ((pjds3.1 (e. F (CH))) (pjds3.2 (e. G (CH))) (pjds3.3 (e. H (CH)))) (-> (/\ (/\ (e. A (opr (opr F (vH) G) (vH) H)) (C_ F (` (_|_) G))) (/\ (C_ F (` (_|_) H)) (C_ G (` (_|_) H)))) (= A (opr (opr (` (` (proj) F) A) (+v) (` (` (proj) G) A)) (+v) (` (` (proj) H) A)))) (pjds3.1 pjds3.2 chjcl pjds3.3 A pjds (e. A (opr (opr F (vH) G) (vH) H)) (C_ F (` (_|_) G)) pm3.26 pjds3.1 pjds3.2 pjds3.3 choccl chlubi syl2an pjds3.1 pjds3.2 osum (proj) fveq2d A fveq1d (+v) (` (` (proj) H) A) opreq1d (e. A (opr (opr F (vH) G) (vH) H)) (/\ (C_ F (` (_|_) H)) (C_ G (` (_|_) H))) ad2antlr pjds3.1 pjds3.2 A pjsumt imp pjds3.1 pjds3.2 chjcl pjds3.3 chjcl A chel sylan (+v) (` (` (proj) H) A) opreq1d (/\ (C_ F (` (_|_) H)) (C_ G (` (_|_) H))) adantr 3eqtr2d)) thm (pj11t () () (-> (/\ (e. G (CH)) (e. H (CH))) (<-> (= (` (proj) G) (` (proj) H)) (= G H))) (G (if (e. G (CH)) G (0H)) (proj) fveq2 (` (proj) H) eqeq1d G (if (e. G (CH)) G (0H)) H eqeq1 bibi12d H (if (e. H (CH)) H (0H)) (proj) fveq2 (` (proj) (if (e. G (CH)) G (0H))) eqeq2d H (if (e. H (CH)) H (0H)) (if (e. G (CH)) G (0H)) eqeq2 bibi12d h0elch G elimel h0elch H elimel pj11 dedth2h)) thm (pjmfn ((f h) (f w) (f x) (f y) (f z) (h w) (h x) (h y) (h z) (w x) (w y) (w z) (x y) (x z) (y z)) () (Fn (proj) (CH)) (ax-hilex x y (U. ({e.|} z (cv h) (E.e. w (` (_|_) (cv h)) (= (cv x) (opr (cv z) (+v) (cv w)))))) funopabex2 h f x y z w df-pj fnopab2)) thm (pjmf1 ((x y)) () (:-1-1-> (proj) (CH) (opr (H~) (^m) (H~))) ((proj) (CH) (opr (H~) (^m) (H~)) x y f1fv (proj) (CH) (opr (H~) (^m) (H~)) x ffnfv pjmfn (cv x) pjft ax-hilex ax-hilex (` (proj) (cv x)) elmap sylibr rgen mpbir2an (cv x) (cv y) pj11t biimpd rgen2 mpbir2an)) thm (pjoi0t () () (-> (/\ (/\/\ (e. G (CH)) (e. H (CH)) (e. A (H~))) (C_ G (` (_|_) H))) (= (opr (` (` (proj) G) A) (.i) (` (` (proj) H) A)) (0))) (G pjrnt (e. H (CH)) adantr H pjrnt (_|_) fveq2d (e. G (CH)) adantl sseq12d biimpar (e. A (H~)) 3adantl3 (ran (` (proj) H)) (ran (` (proj) G)) (` (` (proj) G) A) (` (` (proj) H) A) shorth 3imp H pjrnt (e. H (CH)) id eqeltrd (ran (` (proj) H)) chsh syl (e. G (CH)) (e. A (H~)) 3ad2ant2 (C_ (ran (` (proj) G)) (` (_|_) (ran (` (proj) H)))) adantr (/\/\ (e. G (CH)) (e. H (CH)) (e. A (H~))) (C_ (ran (` (proj) G)) (` (_|_) (ran (` (proj) H)))) pm3.27 (` (proj) G) (H~) A fnfvrn G pjfnt sylan (e. H (CH)) 3adant2 (` (proj) H) (H~) A fnfvrn H pjfnt sylan (e. G (CH)) 3adant1 jca (C_ (ran (` (proj) G)) (` (_|_) (ran (` (proj) H)))) adantr syl3anc syldan)) thm (pjoi0 () ((pjoi0.1 (e. G (CH))) (pjoi0.2 (e. H (CH))) (pjoi0.3 (e. A (H~)))) (-> (C_ G (` (_|_) H)) (= (opr (` (` (proj) G) A) (.i) (` (` (proj) H) A)) (0))) (pjoi0.1 pjoi0.2 pjoi0.3 3pm3.2i G H A pjoi0t mpan)) thm (pjopyth () ((pjoi0.1 (e. G (CH))) (pjoi0.2 (e. H (CH))) (pjoi0.3 (e. A (H~)))) (-> (C_ G (` (_|_) H)) (= (opr (` (norm) (opr (` (` (proj) G) A) (+v) (` (` (proj) H) A))) (^) (2)) (opr (opr (` (norm) (` (` (proj) G) A)) (^) (2)) (+) (opr (` (norm) (` (` (proj) H) A)) (^) (2))))) (pjoi0.1 pjoi0.2 pjoi0.3 pjoi0 pjoi0.1 pjoi0.3 pjhcli pjoi0.2 pjoi0.3 pjhcli normpyth syl)) thm (pjopytht () () (-> (/\/\ (e. H (CH)) (e. G (CH)) (e. A (H~))) (-> (C_ H (` (_|_) G)) (= (opr (` (norm) (opr (` (` (proj) H) A) (+v) (` (` (proj) G) A))) (^) (2)) (opr (opr (` (norm) (` (` (proj) H) A)) (^) (2)) (+) (opr (` (norm) (` (` (proj) G) A)) (^) (2)))))) (H (if (e. H (CH)) H (H~)) (` (_|_) G) sseq1 H (if (e. H (CH)) H (H~)) (proj) fveq2 A fveq1d (+v) (` (` (proj) G) A) opreq1d (norm) fveq2d (^) (2) opreq1d H (if (e. H (CH)) H (H~)) (proj) fveq2 A fveq1d (norm) fveq2d (^) (2) opreq1d (+) (opr (` (norm) (` (` (proj) G) A)) (^) (2)) opreq1d eqeq12d imbi12d G (if (e. G (CH)) G (H~)) (_|_) fveq2 (if (e. H (CH)) H (H~)) sseq2d G (if (e. G (CH)) G (H~)) (proj) fveq2 A fveq1d (` (` (proj) (if (e. H (CH)) H (H~))) A) (+v) opreq2d (norm) fveq2d (^) (2) opreq1d G (if (e. G (CH)) G (H~)) (proj) fveq2 A fveq1d (norm) fveq2d (^) (2) opreq1d (opr (` (norm) (` (` (proj) (if (e. H (CH)) H (H~))) A)) (^) (2)) (+) opreq2d eqeq12d imbi12d A (if (e. A (H~)) A (0v)) (` (proj) (if (e. H (CH)) H (H~))) fveq2 A (if (e. A (H~)) A (0v)) (` (proj) (if (e. G (CH)) G (H~))) fveq2 (+v) opreq12d (norm) fveq2d (^) (2) opreq1d A (if (e. A (H~)) A (0v)) (` (proj) (if (e. H (CH)) H (H~))) fveq2 (norm) fveq2d (^) (2) opreq1d A (if (e. A (H~)) A (0v)) (` (proj) (if (e. G (CH)) G (H~))) fveq2 (norm) fveq2d (^) (2) opreq1d (+) opreq12d eqeq12d (C_ (if (e. H (CH)) H (H~)) (` (_|_) (if (e. G (CH)) G (H~)))) imbi2d helch H elimel helch G elimel ax-hv0cl A elimel pjopyth dedth3h)) thm (pjnorm () ((pjnorm.1 (e. H (CH))) (pjnorm.2 (e. A (H~)))) (br (` (norm) (` (` (proj) H) A)) (<_) (` (norm) A)) (pjnorm.1 pjnorm.2 pjhcli pjnorm.1 choccl pjnorm.2 pjhcli pm3.2i pjnorm.1 pjnorm.2 pjnorm.2 H pjorth ax-mp (` (` (proj) H) A) (` (` (proj) (` (_|_) H)) A) normpyct mp2 pjnorm.1 pjnorm.2 pjpj (norm) fveq2i breqtrr)) thm (pjpyth () ((pjnorm.1 (e. H (CH))) (pjnorm.2 (e. A (H~)))) (= (opr (` (norm) A) (^) (2)) (opr (opr (` (norm) (` (` (proj) H) A)) (^) (2)) (+) (opr (` (norm) (` (` (proj) (` (_|_) H)) A)) (^) (2)))) (pjnorm.1 pjnorm.2 pjpj (norm) fveq2i (^) (2) opreq1i pjnorm.1 chshi H shococss ax-mp pjnorm.1 pjnorm.1 choccl pjnorm.2 pjopyth ax-mp eqtr)) thm (pjnel () ((pjnorm.1 (e. H (CH))) (pjnorm.2 (e. A (H~)))) (<-> (-. (e. A H)) (br (` (norm) (` (` (proj) H) A)) (<) (` (norm) A))) (pjnorm.1 pjnorm.2 pjnorm (-. (= (` (norm) (` (` (proj) H) A)) (` (norm) A))) biantrur pjnorm.1 pjnorm.2 pjoc1 pjnorm.1 pjnorm.2 pjhcli normcl resqcl recn 0cn sqcl pjnorm.1 choccl pjnorm.2 pjhcli normcl resqcl recn addcan sq0 (opr (` (norm) (` (` (proj) H) A)) (^) (2)) (+) opreq2i pjnorm.1 pjnorm.2 pjhcli normcl resqcl recn addid1 eqtr2 pjnorm.1 pjnorm.2 pjpyth eqeq12i 0re leid pjnorm.1 choccl pjnorm.2 pjhcli (` (` (proj) (` (_|_) H)) A) normge0t ax-mp 0re pjnorm.1 choccl pjnorm.2 pjhcli normcl sq11 mp2an (` (norm) (` (` (proj) (` (_|_) H)) A)) (0) eqcom pjnorm.1 choccl pjnorm.2 pjhcli norm-i 3bitr2r 3bitr4r pjnorm.1 pjnorm.2 pjhcli (` (` (proj) H) A) normge0t ax-mp pjnorm.2 A normge0t ax-mp pjnorm.1 pjnorm.2 pjhcli normcl pjnorm.2 normcl sq11 mp2an 3bitr negbii pjnorm.1 pjnorm.2 pjhcli normcl pjnorm.2 normcl ltlen 3bitr4)) thm (pjnormt () () (-> (/\ (e. H (CH)) (e. A (H~))) (br (` (norm) (` (` (proj) H) A)) (<_) (` (norm) A))) (H (if (e. H (CH)) H (H~)) (proj) fveq2 A fveq1d (norm) fveq2d (<_) (` (norm) A) breq1d A (if (e. A (H~)) A (0v)) (` (proj) (if (e. H (CH)) H (H~))) fveq2 (norm) fveq2d A (if (e. A (H~)) A (0v)) (norm) fveq2 (<_) breq12d helch H elimel ax-hv0cl A elimel pjnorm dedth2h)) thm (pjpytht () () (-> (/\ (e. H (CH)) (e. A (H~))) (= (opr (` (norm) A) (^) (2)) (opr (opr (` (norm) (` (` (proj) H) A)) (^) (2)) (+) (opr (` (norm) (` (` (proj) (` (_|_) H)) A)) (^) (2))))) (H (if (e. H (CH)) H (H~)) (proj) fveq2 A fveq1d (norm) fveq2d (^) (2) opreq1d H (if (e. H (CH)) H (H~)) (_|_) fveq2 (proj) fveq2d A fveq1d (norm) fveq2d (^) (2) opreq1d (+) opreq12d (opr (` (norm) A) (^) (2)) eqeq2d A (if (e. A (H~)) A (0v)) (norm) fveq2 (^) (2) opreq1d A (if (e. A (H~)) A (0v)) (` (proj) (if (e. H (CH)) H (H~))) fveq2 (norm) fveq2d (^) (2) opreq1d A (if (e. A (H~)) A (0v)) (` (proj) (` (_|_) (if (e. H (CH)) H (H~)))) fveq2 (norm) fveq2d (^) (2) opreq1d (+) opreq12d eqeq12d helch H elimel ax-hv0cl A elimel pjpyth dedth2h)) thm (pjnelt () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (-. (e. A H)) (br (` (norm) (` (` (proj) H) A)) (<) (` (norm) A)))) (H (if (e. H (CH)) H (H~)) A eleq2 negbid H (if (e. H (CH)) H (H~)) (proj) fveq2 A fveq1d (norm) fveq2d (<) (` (norm) A) breq1d bibi12d A (if (e. A (H~)) A (0v)) (if (e. H (CH)) H (H~)) eleq1 negbid A (if (e. A (H~)) A (0v)) (` (proj) (if (e. H (CH)) H (H~))) fveq2 (norm) fveq2d A (if (e. A (H~)) A (0v)) (norm) fveq2 (<) breq12d bibi12d helch H elimel ax-hv0cl A elimel pjnel dedth2h)) thm (pjnorm2t () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (e. A H) (= (` (norm) (` (` (proj) H) A)) (` (norm) A)))) ((` (norm) (` (` (proj) H) A)) (` (norm) A) eqleltt H A pjhclt (` (` (proj) H) A) normclt syl A normclt (e. H (CH)) adantl sylanc H A pjnormt (-. (br (` (norm) (` (` (proj) H) A)) (<) (` (norm) A))) biantrurd H A pjnelt con1bid 3bitr2rd)) thm (mayete3 ((D x) (F x) (G x) (R x) (S x)) ((mayete3.1 (e. A (CH))) (mayete3.2 (e. B (CH))) (mayete3.3 (e. C (CH))) (mayete3.4 (e. D (CH))) (mayete3.5 (e. F (CH))) (mayete3.6 (e. G (CH))) (mayete3.7 (C_ A (` (_|_) B))) (mayete3.8 (C_ A (` (_|_) C))) (mayete3.9 (C_ B (` (_|_) C))) (mayete3.10 (C_ A (` (_|_) D))) (mayete3.11 (C_ B (` (_|_) F))) (mayete3.12 (C_ C (` (_|_) G))) (mayete3.13 (= R (opr (opr A (vH) B) (vH) C))) (mayete3.14 (= S (i^i (i^i (opr A (vH) D) (opr B (vH) F)) (opr C (vH) G)))) (mayete3.15 (= T (opr (opr D (vH) F) (vH) G)))) (C_ (i^i R S) T) ((cv x) R S elin mayete3.13 (cv x) eleq2i mayete3.1 mayete3.2 chjcl mayete3.3 chjcl (cv x) chel sylbi (e. (cv x) S) adantr sylbi (cv x) ax-hvmulid 2cn 2re 2pos gt0ne0i reccl 2cn (opr (1) (/) (2)) (2) (cv x) ax-hvmulass mp3an12 2cn 2re 2pos gt0ne0i (2) recid2t mp2an (.s) (cv x) opreq1i syl5eqr eqtr3d syl (cv x) R S elin mayete3.13 (cv x) eleq2i mayete3.1 mayete3.2 chjcl mayete3.3 chjcl (cv x) chel sylbi (e. (cv x) S) adantr sylbi (cv x) hv2timest (+v) (cv x) opreq1d syl R S inss2 (cv x) sseli mayete3.14 (cv x) eleq2i (cv x) (i^i (opr A (vH) D) (opr B (vH) F)) (opr C (vH) G) elin bitr (cv x) (opr A (vH) D) (opr B (vH) F) elin mayete3.10 mayete3.1 mayete3.4 (cv x) pjds mpan2 mayete3.11 mayete3.2 mayete3.5 (cv x) pjds mpan2 (+v) opreqan12d sylbi (opr A (vH) D) (opr B (vH) F) inss1 (cv x) sseli mayete3.1 mayete3.4 chjcl (cv x) chel syl (` (` (proj) A) (cv x)) (` (` (proj) D) (cv x)) (` (` (proj) B) (cv x)) (` (` (proj) F) (cv x)) hvadd4t mayete3.1 (cv x) pjhcl mayete3.4 (cv x) pjhcl jca mayete3.2 (cv x) pjhcl mayete3.5 (cv x) pjhcl jca sylanc syl eqtrd mayete3.12 mayete3.3 mayete3.6 (cv x) pjds mpan2 (+v) opreqan12d sylbi syl (cv x) R S elin mayete3.13 (cv x) eleq2i mayete3.1 mayete3.2 chjcl mayete3.3 chjcl (cv x) chel sylbi (e. (cv x) S) adantr sylbi (opr (` (` (proj) A) (cv x)) (+v) (` (` (proj) B) (cv x))) (opr (` (` (proj) D) (cv x)) (+v) (` (` (proj) F) (cv x))) (` (` (proj) C) (cv x)) (` (` (proj) G) (cv x)) hvadd4t (` (` (proj) A) (cv x)) (` (` (proj) B) (cv x)) ax-hvaddcl mayete3.1 (cv x) pjhcl mayete3.2 (cv x) pjhcl sylanc (` (` (proj) D) (cv x)) (` (` (proj) F) (cv x)) ax-hvaddcl mayete3.4 (cv x) pjhcl mayete3.5 (cv x) pjhcl sylanc jca mayete3.3 (cv x) pjhcl mayete3.6 (cv x) pjhcl jca sylanc syl 3eqtrd R S inss1 (cv x) sseli mayete3.13 syl6eleq mayete3.7 mayete3.8 mayete3.9 pm3.2i mayete3.1 mayete3.2 mayete3.3 (cv x) pjds3 mpan2 mpan2 syl (-v) opreq12d (cv x) R S elin mayete3.13 (cv x) eleq2i mayete3.1 mayete3.2 chjcl mayete3.3 chjcl (cv x) chel sylbi (e. (cv x) S) adantr sylbi 2cn (2) (cv x) ax-hvmulcl mpan (opr (2) (.s) (cv x)) (cv x) hvpncant mpancom syl (cv x) R S elin mayete3.13 (cv x) eleq2i mayete3.1 mayete3.2 chjcl mayete3.3 chjcl (cv x) chel sylbi (e. (cv x) S) adantr sylbi (opr (opr (` (` (proj) A) (cv x)) (+v) (` (` (proj) B) (cv x))) (+v) (` (` (proj) C) (cv x))) (opr (opr (` (` (proj) D) (cv x)) (+v) (` (` (proj) F) (cv x))) (+v) (` (` (proj) G) (cv x))) hvpncan2t (opr (` (` (proj) A) (cv x)) (+v) (` (` (proj) B) (cv x))) (` (` (proj) C) (cv x)) ax-hvaddcl (` (` (proj) A) (cv x)) (` (` (proj) B) (cv x)) ax-hvaddcl mayete3.1 (cv x) pjhcl mayete3.2 (cv x) pjhcl sylanc mayete3.3 (cv x) pjhcl sylanc (opr (` (` (proj) D) (cv x)) (+v) (` (` (proj) F) (cv x))) (` (` (proj) G) (cv x)) ax-hvaddcl (` (` (proj) D) (cv x)) (` (` (proj) F) (cv x)) ax-hvaddcl mayete3.4 (cv x) pjhcl mayete3.5 (cv x) pjhcl sylanc mayete3.6 (cv x) pjhcl sylanc sylanc syl 3eqtr3d (cv x) R S elin mayete3.13 (cv x) eleq2i mayete3.1 mayete3.2 chjcl mayete3.3 chjcl (cv x) chel sylbi (e. (cv x) S) adantr sylbi mayete3.4 chshi mayete3.5 chshi shscl mayete3.6 chshi (opr (` (` (proj) D) (cv x)) (+v) (` (` (proj) F) (cv x))) (` (` (proj) G) (cv x)) shsva mayete3.4 chshi mayete3.5 chshi (` (` (proj) D) (cv x)) (` (` (proj) F) (cv x)) shsva mayete3.4 (cv x) pjcl mayete3.5 (cv x) pjcl sylanc mayete3.6 (cv x) pjcl sylanc syl eqeltrd 2cn 2re 2pos gt0ne0i reccl mayete3.4 chshi mayete3.5 chshi shscl mayete3.6 chshi shscl (opr (opr D (+H) F) (+H) G) (opr (1) (/) (2)) (opr (2) (.s) (cv x)) shmulclt ax-mp mpan syl eqeltrd ssriv mayete3.4 mayete3.5 chslej mayete3.4 chshi mayete3.5 chshi shscl mayete3.4 mayete3.5 chjcl chshi mayete3.6 chshi shless ax-mp sstri mayete3.4 mayete3.5 chjcl mayete3.6 chslej sstri mayete3.15 sseqtr4)) thm (df0op2 () () (= (0op) (X. (H~) (0H))) ((` (proj) (0H)) (H~) (0v) x fconstfv h0elch pjfn (cv x) pjch0t rgen mpbir2an ax-hv0cl elisseti (` (proj) (0H)) (H~) fconst2 mpbi df-0op df-ch0 (0H) ({} (0v)) (H~) xpeq2 ax-mp 3eqtr4)) thm (dfiop2 () () (= (Iop) (|` (I) (H~))) (df-iop helch pjfn (H~) fnresi (H~) eqid (cv x) pjch1t (cv x) (H~) fvresi eqtr4d rgen pm3.2i (` (proj) (H~)) (H~) (|` (I) (H~)) (H~) x eqfnfv mpbiri mp2an eqtr)) thm (ho0f () () (:--> (0op) (H~) (H~)) (h0elch pjf df-0op (0op) (` (proj) (0H)) (H~) (H~) feq1 ax-mp mpbir)) thm (ho0fOLD () () (:--> (X. (H~) (0H)) (H~) (H~)) (ax-hv0cl elisseti (H~) fconst ax-hv0cl (0v) (H~) snssi ax-mp (X. (H~) ({} (0v))) (H~) ({} (0v)) (H~) fss mp2an df-ch0 (0H) ({} (0v)) (H~) xpeq2 (X. (H~) (0H)) (X. (H~) ({} (0v))) (H~) (H~) feq1 syl ax-mp mpbir)) thm (hoif () () (:-1-1-onto-> (Iop) (H~) (H~)) ((H~) f1oi dfiop2 (Iop) (|` (I) (H~)) (H~) (H~) f1oeq1 ax-mp mpbir)) thm (ho0OLD () () (= (` (proj) (0H)) (X. (H~) (0H))) ((` (proj) (0H)) (H~) (0v) x fconstfv h0elch pjfn (cv x) pjch0t rgen mpbir2an ax-hv0cl elisseti (` (proj) (0H)) (H~) fconst2 mpbi df-ch0 (0H) ({} (0v)) (H~) xpeq2 ax-mp eqtr4)) thm (ho0valt () () (-> (e. A (H~)) (= (` (0op) A) (0v))) (A pjch0t df-0op A fveq1i syl5eq)) thm (ho0valtOLD () () (-> (e. A (H~)) (= (` (X. (H~) (0H)) A) (0v))) (ax-hv0cl elisseti A (H~) fvconst2 df-ch0 (0H) ({} (0v)) (H~) xpeq2 ax-mp A fveq1i syl5eq)) thm (hoivalt () () (-> (e. A (H~)) (= (` (Iop) A) A)) (A pjch1t df-iop A fveq1i syl5eq)) thm (hoico1t () () (-> (:--> T (H~) (H~)) (= (o. T (Iop)) T)) (T (H~) (H~) fcoi1 dfiop2 T coeq2i syl5eq)) thm (hoico2t () () (-> (:--> T (H~) (H~)) (= (o. (Iop) T) T)) (T (H~) (H~) fcoi2 dfiop2 T coeq1i syl5eq)) thm (hoaddclt ((x y) (S x) (S y) (T x) (T y)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (:--> (opr S (+op) T) (H~) (H~))) ((` S (cv x)) (` T (cv x)) ax-hvaddcl S (H~) (H~) (cv x) ffvrn T (H~) (H~) (cv x) ffvrn syl2an anandirs r19.21aiva ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (+v) (` T (cv x)))))) eqid (H~) fopab2 sylib S T x y hosmvalt (opr S (+op) T) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (+v) (` T (cv x)))))) (H~) (H~) feq1 syl mpbird)) thm (homulclt ((x y) (A x) (A y) (T x) (T y)) () (-> (/\ (e. A (CC)) (:--> T (H~) (H~))) (:--> (opr A (.op) T) (H~) (H~))) (A (` T (cv x)) ax-hvmulcl T (H~) (H~) (cv x) ffvrn sylan2 anassrs r19.21aiva ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr A (.s) (` T (cv x)))))) eqid (H~) fopab2 sylib A T x y hommvalt (opr A (.op) T) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr A (.s) (` T (cv x)))))) (H~) (H~) feq1 syl mpbird)) thm (hoeqt ((T x) (U x)) () (-> (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) (<-> (A.e. x (H~) (= (` T (cv x)) (` U (cv x)))) (= T U))) (T (H~) U (H~) x eqfnfv (H~) eqid (A.e. x (H~) (= (` T (cv x)) (` U (cv x)))) biantrur syl6rbbr T (H~) (H~) ffn U (H~) (H~) ffn syl2an)) thm (hoeq ((S x) (T x)) ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (<-> (A.e. x (H~) (= (` S (cv x)) (` T (cv x)))) (= S T)) (hoeq.1 hoeq.2 S T x hoeqt mp2an)) thm (hoscl () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (-> (e. A (H~)) (e. (` (opr S (+op) T) A) (H~))) (hoeq.1 hoeq.2 S T A hosclt mpanl12)) thm (hodcl () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (-> (e. A (H~)) (e. (` (opr S (-op) T) A) (H~))) (hoeq.1 hoeq.2 S T A hodclt mpanl12)) thm (hoco () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (-> (e. A (H~)) (= (` (o. S T) A) (` S (` T A)))) (hoeq.2 T (H~) (H~) fdm ax-mp A eleq2i hoeq.1 S (H~) (H~) ffun ax-mp hoeq.2 T (H~) (H~) ffun ax-mp S T A fvco mp3an12 sylbir)) thm (hococl () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (-> (e. A (H~)) (e. (` (o. S T) A) (H~))) (hoeq.1 hoeq.2 A hoco hoeq.2 A ffvrni hoeq.1 (` T A) ffvrni syl eqeltrd)) thm (hocof () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (:--> (o. S T) (H~) (H~)) (hoeq.1 hoeq.2 S (H~) (H~) T (H~) fco mp2an)) thm (hocofn () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (Fn (o. S T) (H~)) (hoeq.1 hoeq.2 hocof (o. S T) (H~) (H~) ffn ax-mp)) thm (hoaddcl () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (:--> (opr S (+op) T) (H~) (H~)) (hoeq.1 hoeq.2 S T hoaddclt mp2an)) thm (hosubcl ((x y) (S x) (S y) (T x) (T y)) ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (:--> (opr S (-op) T) (H~) (H~)) (hoeq.1 hoeq.2 S T x y hodmvalt mp2an (` S (cv x)) (` T (cv x)) hvsubclt hoeq.1 (cv x) ffvrni hoeq.2 (cv x) ffvrni sylanc fopab)) thm (hoaddfn () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (Fn (opr S (+op) T) (H~)) (hoeq.1 hoeq.2 hoaddcl (opr S (+op) T) (H~) (H~) ffn ax-mp)) thm (hosubfn () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (Fn (opr S (-op) T) (H~)) (hoeq.1 hoeq.2 hosubcl (opr S (-op) T) (H~) (H~) ffn ax-mp)) thm (hoaddcom ((S x) (T x)) ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (= (opr S (+op) T) (opr T (+op) S)) ((` S (cv x)) (` T (cv x)) ax-hvcom hoeq.1 (cv x) ffvrni hoeq.2 (cv x) ffvrni sylanc hoeq.1 hoeq.2 S T (cv x) hosvaltOLD mpanl12 hoeq.2 hoeq.1 T S (cv x) hosvaltOLD mpanl12 3eqtr4d rgen hoeq.1 hoeq.2 hoaddcl hoeq.2 hoeq.1 hoaddcl x hoeq mpbi)) thm (hosubclt () () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (:--> (opr S (-op) T) (H~) (H~))) (S (if (:--> S (H~) (H~)) S (0op)) (-op) T opreq1 (opr S (-op) T) (opr (if (:--> S (H~) (H~)) S (0op)) (-op) T) (H~) (H~) feq1 syl T (if (:--> T (H~) (H~)) T (0op)) (if (:--> S (H~) (H~)) S (0op)) (-op) opreq2 (opr (if (:--> S (H~) (H~)) S (0op)) (-op) T) (opr (if (:--> S (H~) (H~)) S (0op)) (-op) (if (:--> T (H~) (H~)) T (0op))) (H~) (H~) feq1 syl ho0f S elimf ho0f T elimf hosubcl dedth2h)) thm (hoaddcomt () () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (+op) T) (opr T (+op) S))) (S (if (:--> S (H~) (H~)) S (0op)) (+op) T opreq1 S (if (:--> S (H~) (H~)) S (0op)) T (+op) opreq2 eqeq12d T (if (:--> T (H~) (H~)) T (0op)) (if (:--> S (H~) (H~)) S (0op)) (+op) opreq2 T (if (:--> T (H~) (H~)) T (0op)) (+op) (if (:--> S (H~) (H~)) S (0op)) opreq1 eqeq12d ho0f S elimf ho0f T elimf hoaddcom dedth2h)) thm (hods ((R x) (S x) (T x)) ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (<-> (= (opr R (-op) S) T) (= (opr S (+op) T) R)) ((` R (cv x)) (` S (cv x)) (` T (cv x)) hvsubaddt hods.1 (cv x) ffvrni hods.2 (cv x) ffvrni hods.3 (cv x) ffvrni syl3anc hods.1 hods.2 R S (cv x) hodvaltOLD mpanl12 (` T (cv x)) eqeq1d hods.2 hods.3 S T (cv x) hosvaltOLD mpanl12 (` R (cv x)) eqeq1d 3bitr4d ralbiia hods.1 hods.2 hosubcl hods.3 x hoeq hods.2 hods.3 hoaddcl hods.1 x hoeq 3bitr3)) thm (hoaddass ((R x) (S x) (T x)) ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (= (opr (opr R (+op) S) (+op) T) (opr R (+op) (opr S (+op) T))) ((` R (cv x)) (` S (cv x)) (` T (cv x)) ax-hvass hods.1 (cv x) ffvrni hods.2 (cv x) ffvrni hods.3 (cv x) ffvrni syl3anc hods.1 hods.2 hoaddcl hods.3 (opr R (+op) S) T (cv x) hosvaltOLD mpanl12 hods.1 hods.2 R S (cv x) hosvaltOLD mpanl12 (+v) (` T (cv x)) opreq1d eqtrd hods.1 hods.2 hods.3 hoaddcl R (opr S (+op) T) (cv x) hosvaltOLD mpanl12 hods.2 hods.3 S T (cv x) hosvaltOLD mpanl12 (` R (cv x)) (+v) opreq2d eqtrd 3eqtr4d rgen hods.1 hods.2 hoaddcl hods.3 hoaddcl hods.1 hods.2 hods.3 hoaddcl hoaddcl x hoeq mpbi)) thm (hoadd12 () ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (= (opr R (+op) (opr S (+op) T)) (opr S (+op) (opr R (+op) T))) (hods.1 hods.2 hoaddcom (+op) T opreq1i hods.1 hods.2 hods.3 hoaddass hods.2 hods.1 hods.3 hoaddass 3eqtr3)) thm (hoadd23 () ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (= (opr (opr R (+op) S) (+op) T) (opr (opr R (+op) T) (+op) S)) (hods.2 hods.3 hoaddcom R (+op) opreq2i hods.1 hods.2 hods.3 hoaddass hods.1 hods.3 hods.2 hoaddass 3eqtr4)) thm (hocadddir ((R x) (S x) (T x)) ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (= (o. (opr R (+op) S) T) (opr (o. R T) (+op) (o. S T))) (hods.3 (cv x) ffvrni hods.1 hods.2 R S (` T (cv x)) hosvaltOLD mpanl12 syl hods.1 hods.3 (cv x) hoco hods.2 hods.3 (cv x) hoco (+v) opreq12d eqtr4d hods.1 hods.2 hoaddcl hods.3 (cv x) hoco hods.1 hods.3 hocof hods.2 hods.3 hocof (o. R T) (o. S T) (cv x) hosvaltOLD mpanl12 3eqtr4d rgen hods.1 hods.2 hoaddcl hods.3 hocof hods.1 hods.3 hocof hods.2 hods.3 hocof hoaddcl x hoeq mpbi)) thm (hocsubdir ((R x) (S x) (T x)) ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (= (o. (opr R (-op) S) T) (opr (o. R T) (-op) (o. S T))) (hods.3 (cv x) ffvrni hods.1 hods.2 R S (` T (cv x)) hodvaltOLD mpanl12 syl hods.1 hods.3 (cv x) hoco hods.2 hods.3 (cv x) hoco (-v) opreq12d eqtr4d hods.1 hods.2 hosubcl hods.3 (cv x) hoco hods.1 hods.3 hocof hods.2 hods.3 hocof (o. R T) (o. S T) (cv x) hodvaltOLD mpanl12 3eqtr4d rgen hods.1 hods.2 hosubcl hods.3 hocof hods.1 hods.3 hocof hods.2 hods.3 hocof hosubcl x hoeq mpbi)) thm (ho2co () ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (-> (e. A (H~)) (= (` (o. (o. R S) T) A) (` R (` S (` T A))))) (hods.1 hods.2 hocof hods.3 A hoco hods.3 A ffvrni hods.1 hods.2 (` T A) hoco syl eqtrd)) thm (hoaddasst () () (-> (/\/\ (:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr (opr R (+op) S) (+op) T) (opr R (+op) (opr S (+op) T)))) (R (if (:--> R (H~) (H~)) R (0op)) (+op) S opreq1 (+op) T opreq1d R (if (:--> R (H~) (H~)) R (0op)) (+op) (opr S (+op) T) opreq1 eqeq12d S (if (:--> S (H~) (H~)) S (0op)) (if (:--> R (H~) (H~)) R (0op)) (+op) opreq2 (+op) T opreq1d S (if (:--> S (H~) (H~)) S (0op)) (+op) T opreq1 (if (:--> R (H~) (H~)) R (0op)) (+op) opreq2d eqeq12d T (if (:--> T (H~) (H~)) T (0op)) (opr (if (:--> R (H~) (H~)) R (0op)) (+op) (if (:--> S (H~) (H~)) S (0op))) (+op) opreq2 T (if (:--> T (H~) (H~)) T (0op)) (if (:--> S (H~) (H~)) S (0op)) (+op) opreq2 (if (:--> R (H~) (H~)) R (0op)) (+op) opreq2d eqeq12d ho0f R elimf ho0f S elimf ho0f T elimf hoaddass dedth3h)) thm (hoadd23t () () (-> (/\/\ (:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr (opr R (+op) S) (+op) T) (opr (opr R (+op) T) (+op) S))) (S T hoaddcomt (:--> R (H~) (H~)) 3adant1 R (+op) opreq2d R S T hoaddasst R T S hoaddasst 3com23 3eqtr4d)) thm (hoadd4t () () (-> (/\ (/\ (:--> R (H~) (H~)) (:--> S (H~) (H~))) (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~)))) (= (opr (opr R (+op) S) (+op) (opr T (+op) U)) (opr (opr R (+op) T) (+op) (opr S (+op) U)))) (R S T hoadd23t (+op) U opreq1d 3expa (:--> U (H~) (H~)) adantrr (opr R (+op) S) T U hoaddasst 3expb R S hoaddclt sylan (opr R (+op) T) S U hoaddasst 3expb R T hoaddclt sylan an4s 3eqtr3d)) thm (hocsubdirt () () (-> (/\/\ (:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (o. (opr R (-op) S) T) (opr (o. R T) (-op) (o. S T)))) (R (if (:--> R (H~) (H~)) R (0op)) (-op) S opreq1 (opr R (-op) S) (opr (if (:--> R (H~) (H~)) R (0op)) (-op) S) T coeq1 syl R (if (:--> R (H~) (H~)) R (0op)) T coeq1 (-op) (o. S T) opreq1d eqeq12d S (if (:--> S (H~) (H~)) S (0op)) (if (:--> R (H~) (H~)) R (0op)) (-op) opreq2 (opr (if (:--> R (H~) (H~)) R (0op)) (-op) S) (opr (if (:--> R (H~) (H~)) R (0op)) (-op) (if (:--> S (H~) (H~)) S (0op))) T coeq1 syl S (if (:--> S (H~) (H~)) S (0op)) T coeq1 (o. (if (:--> R (H~) (H~)) R (0op)) T) (-op) opreq2d eqeq12d T (if (:--> T (H~) (H~)) T (0op)) (opr (if (:--> R (H~) (H~)) R (0op)) (-op) (if (:--> S (H~) (H~)) S (0op))) coeq2 T (if (:--> T (H~) (H~)) T (0op)) (if (:--> R (H~) (H~)) R (0op)) coeq2 T (if (:--> T (H~) (H~)) T (0op)) (if (:--> S (H~) (H~)) S (0op)) coeq2 (-op) opreq12d eqeq12d ho0f R elimf ho0f S elimf ho0f T elimf hocsubdir dedth3h)) thm (hoaddid1 ((T x)) ((hoaddid1.1 (:--> T (H~) (H~)))) (= (opr T (+op) (0op)) T) (df-0op T (+op) opreq2i hoaddid1.1 h0elch pjf T (` (proj) (0H)) (cv x) hosvalt mp3an12 (cv x) pjch0t (` T (cv x)) (+v) opreq2d hoaddid1.1 (cv x) ffvrni (` T (cv x)) ax-hvaddid syl 3eqtrd rgen hoaddid1.1 h0elch pjf hoaddcl hoaddid1.1 x hoeq mpbi eqtr)) thm (hoid0OLD ((T x)) ((hoaddid1.1 (:--> T (H~) (H~)))) (= (opr T (+op) (` (proj) (0H))) T) (hoaddid1.1 h0elch pjf T (` (proj) (0H)) (cv x) hosvaltOLD mpanl12 (cv x) pjch0t (` T (cv x)) (+v) opreq2d hoaddid1.1 (cv x) ffvrni (` T (cv x)) ax-hvaddid syl 3eqtrd rgen hoaddid1.1 h0elch pjf hoaddcl hoaddid1.1 x hoeq mpbi)) thm (hoid02OLD () ((hoaddid1.1 (:--> T (H~) (H~)))) (= (opr T (+op) (X. (H~) (0H))) T) (ho0OLD T (+op) opreq2i hoaddid1.1 hoid0OLD eqtr3)) thm (hodidOLD () ((hoaddid1.1 (:--> T (H~) (H~)))) (= (opr T (-op) T) (` (proj) (0H))) (hoaddid1.1 hoid0OLD hoaddid1.1 hoaddid1.1 h0elch pjf hods mpbir)) thm (hodid2OLD () ((hoaddid1.1 (:--> T (H~) (H~)))) (= (opr T (-op) T) (X. (H~) (0H))) (hoaddid1.1 hodidOLD ho0OLD eqtr)) thm (ho0co ((T x)) ((hoaddid1.1 (:--> T (H~) (H~)))) (= (o. (0op) T) (0op)) (hoaddid1.1 (cv x) ffvrni (` T (cv x)) ho0valt syl ho0f hoaddid1.1 (cv x) hoco (cv x) ho0valt 3eqtr4d rgen ho0f hoaddid1.1 hocof ho0f x hoeq mpbi)) thm (hoid1OLD ((T x)) ((hoaddid1.1 (:--> T (H~) (H~)))) (= (o. T (` (proj) (H~))) T) (hoaddid1.1 helch pjf (cv x) hoco (cv x) pjch1t T fveq2d eqtrd rgen hoaddid1.1 helch pjf hocof hoaddid1.1 x hoeq mpbi)) thm (hoid1rOLD ((T x)) ((hoaddid1.1 (:--> T (H~) (H~)))) (= (o. (` (proj) (H~)) T) T) (helch pjf hoaddid1.1 (cv x) hoco hoaddid1.1 (cv x) ffvrni (` T (cv x)) pjch1t syl eqtrd rgen helch pjf hoaddid1.1 hocof hoaddid1.1 x hoeq mpbi)) thm (hoid02tOLD () () (-> (:--> T (H~) (H~)) (= (opr T (+op) (X. (H~) (0H))) T)) (T (if (:--> T (H~) (H~)) T (X. (H~) (0H))) (+op) (X. (H~) (0H)) opreq1 (= T (if (:--> T (H~) (H~)) T (X. (H~) (0H)))) id eqeq12d T (if (:--> T (H~) (H~)) T (X. (H~) (0H))) (H~) (H~) feq1 (X. (H~) (0H)) (if (:--> T (H~) (H~)) T (X. (H~) (0H))) (H~) (H~) feq1 ho0fOLD elimhyp hoid02OLD dedth)) thm (hodid2tOLD () () (-> (:--> T (H~) (H~)) (= (opr T (-op) T) (X. (H~) (0H)))) ((= T (if (:--> T (H~) (H~)) T (X. (H~) (0H)))) id (= T (if (:--> T (H~) (H~)) T (X. (H~) (0H)))) id (-op) opreq12d (X. (H~) (0H)) eqeq1d T (if (:--> T (H~) (H~)) T (X. (H~) (0H))) (H~) (H~) feq1 (X. (H~) (0H)) (if (:--> T (H~) (H~)) T (X. (H~) (0H))) (H~) (H~) feq1 ho0fOLD elimhyp hodid2OLD dedth)) thm (hon0 () () (-> (:--> T (H~) (H~)) (-. (= T ({/})))) (ax-hv0cl (0v) (H~) n0i ax-mp T (H~) (H~) ffn T (H~) ({/}) fndmu ex syl T fn0 syl5ibr mtoi)) thm (hodseq () ((hodseq.2 (:--> S (H~) (H~))) (hodseq.3 (:--> T (H~) (H~)))) (= (opr S (+op) (opr T (-op) S)) T) ((opr T (-op) S) eqid hodseq.3 hodseq.2 hodseq.3 hodseq.2 hosubcl hods mpbi)) thm (ho0sub () ((hodseq.2 (:--> S (H~) (H~))) (hodseq.3 (:--> T (H~) (H~)))) (= (opr S (-op) T) (opr S (+op) (opr (0op) (-op) T))) (hodseq.3 hodseq.2 ho0f hodseq.3 hosubcl hoadd12 hodseq.3 ho0f hodseq S (+op) opreq2i hodseq.2 hoaddid1 3eqtr hodseq.2 hodseq.3 hodseq.2 ho0f hodseq.3 hosubcl hoaddcl hods mpbir)) thm (ho0subOLD () ((hodseq.2 (:--> S (H~) (H~))) (hodseq.3 (:--> T (H~) (H~)))) (= (opr S (-op) T) (opr S (+op) (opr (` (proj) (0H)) (-op) T))) (hodseq.3 hodseq.2 h0elch pjf hodseq.3 hosubcl hoadd12 hodseq.3 h0elch pjf hodseq S (+op) opreq2i hodseq.2 hoid0OLD 3eqtr hodseq.2 hodseq.3 hodseq.2 h0elch pjf hodseq.3 hosubcl hoaddcl hods mpbir)) thm (ho0sub2OLD () ((hodseq.2 (:--> S (H~) (H~))) (hodseq.3 (:--> T (H~) (H~)))) (= (opr S (-op) T) (opr S (+op) (opr (X. (H~) (0H)) (-op) T))) (hodseq.2 hodseq.3 ho0subOLD ho0OLD (-op) T opreq1i S (+op) opreq2i eqtr)) thm (honegsubOLD ((S x) (T x)) ((hodseq.2 (:--> S (H~) (H~))) (hodseq.3 (:--> T (H~) (H~)))) (= (opr S (-op) T) (opr S (+op) (opr (-u (1)) (.op) T))) ((` S (cv x)) (` T (cv x)) hvsubvalt hodseq.2 (cv x) ffvrni hodseq.3 (cv x) ffvrni sylanc 1cn negcl hodseq.3 (-u (1)) T (cv x) homvalt mp3an12 (` S (cv x)) (+v) opreq2d eqtr4d hodseq.2 hodseq.3 S T (cv x) hodvaltOLD mpanl12 hodseq.2 1cn negcl hodseq.3 (-u (1)) T homulclt mp2an S (opr (-u (1)) (.op) T) (cv x) hosvaltOLD mpanl12 3eqtr4d rgen hodseq.2 hodseq.3 hosubcl hodseq.2 1cn negcl hodseq.3 (-u (1)) T homulclt mp2an hoaddcl x hoeq mpbi)) thm (honegsub ((S x) (T x)) ((hodseq.2 (:--> S (H~) (H~))) (hodseq.3 (:--> T (H~) (H~)))) (= (opr S (+op) (opr (-u (1)) (.op) T)) (opr S (-op) T)) ((` S (cv x)) (` T (cv x)) hvsubvalt hodseq.2 (cv x) ffvrni hodseq.3 (cv x) ffvrni sylanc 1cn negcl hodseq.3 (-u (1)) T (cv x) homvalt mp3an12 (` S (cv x)) (+v) opreq2d eqtr4d hodseq.2 hodseq.3 S T (cv x) hodvalt mp3an12 hodseq.2 1cn negcl hodseq.3 (-u (1)) T homulclt mp2an S (opr (-u (1)) (.op) T) (cv x) hosvalt mp3an12 3eqtr4rd rgen hodseq.2 1cn negcl hodseq.3 (-u (1)) T homulclt mp2an hoaddcl hodseq.2 hodseq.3 hosubcl x hoeq mpbi)) thm (ho0subt () () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (-op) T) (opr S (+op) (opr (0op) (-op) T)))) (S (if (:--> S (H~) (H~)) S (0op)) (-op) T opreq1 S (if (:--> S (H~) (H~)) S (0op)) (+op) (opr (0op) (-op) T) opreq1 eqeq12d T (if (:--> T (H~) (H~)) T (0op)) (if (:--> S (H~) (H~)) S (0op)) (-op) opreq2 T (if (:--> T (H~) (H~)) T (0op)) (0op) (-op) opreq2 (if (:--> S (H~) (H~)) S (0op)) (+op) opreq2d eqeq12d ho0f S elimf ho0f T elimf ho0sub dedth2h)) thm (ho0sub2tOLD () () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (-op) T) (opr S (+op) (opr (X. (H~) (0H)) (-op) T)))) (S (if (:--> S (H~) (H~)) S (X. (H~) (0H))) (-op) T opreq1 S (if (:--> S (H~) (H~)) S (X. (H~) (0H))) (+op) (opr (X. (H~) (0H)) (-op) T) opreq1 eqeq12d T (if (:--> T (H~) (H~)) T (X. (H~) (0H))) (if (:--> S (H~) (H~)) S (X. (H~) (0H))) (-op) opreq2 T (if (:--> T (H~) (H~)) T (X. (H~) (0H))) (X. (H~) (0H)) (-op) opreq2 (if (:--> S (H~) (H~)) S (X. (H~) (0H))) (+op) opreq2d eqeq12d S (if (:--> S (H~) (H~)) S (X. (H~) (0H))) (H~) (H~) feq1 (X. (H~) (0H)) (if (:--> S (H~) (H~)) S (X. (H~) (0H))) (H~) (H~) feq1 ho0fOLD elimhyp T (if (:--> T (H~) (H~)) T (X. (H~) (0H))) (H~) (H~) feq1 (X. (H~) (0H)) (if (:--> T (H~) (H~)) T (X. (H~) (0H))) (H~) (H~) feq1 ho0fOLD elimhyp ho0sub2OLD dedth2h)) thm (hosubid1t () () (-> (:--> T (H~) (H~)) (= (opr T (-op) (0op)) T)) (ho0f T (0op) ho0subt mpan2 ho0f hodid2OLD T (+op) opreq2i (:--> T (H~) (H~)) a1i T hoid02tOLD 3eqtrd)) thm (honegsubtOLD () () (-> (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr T (-op) U) (opr T (+op) (opr (-u (1)) (.op) U)))) (T (if (:--> T (H~) (H~)) T (0op)) (-op) U opreq1 T (if (:--> T (H~) (H~)) T (0op)) (+op) (opr (-u (1)) (.op) U) opreq1 eqeq12d U (if (:--> U (H~) (H~)) U (0op)) (if (:--> T (H~) (H~)) T (0op)) (-op) opreq2 U (if (:--> U (H~) (H~)) U (0op)) (-u (1)) (.op) opreq2 (if (:--> T (H~) (H~)) T (0op)) (+op) opreq2d eqeq12d ho0f T elimf ho0f U elimf honegsubOLD dedth2h)) thm (honegsubt () () (-> (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr T (+op) (opr (-u (1)) (.op) U)) (opr T (-op) U))) (T (if (:--> T (H~) (H~)) T (0op)) (+op) (opr (-u (1)) (.op) U) opreq1 T (if (:--> T (H~) (H~)) T (0op)) (-op) U opreq1 eqeq12d U (if (:--> U (H~) (H~)) U (0op)) (-u (1)) (.op) opreq2 (if (:--> T (H~) (H~)) T (0op)) (+op) opreq2d U (if (:--> U (H~) (H~)) U (0op)) (if (:--> T (H~) (H~)) T (0op)) (-op) opreq2 eqeq12d ho0f T elimf ho0f U elimf honegsub dedth2h)) thm (homulid2t ((T x)) () (-> (:--> T (H~) (H~)) (= (opr (1) (.op) T) T)) (1cn (1) T (cv x) homvalt mp3an1 T (H~) (H~) (cv x) ffvrn (` T (cv x)) ax-hvmulid syl eqtrd r19.21aiva 1cn (1) T homulclt mpan (opr (1) (.op) T) T x hoeqt mpancom mpbid)) thm (homco1t ((A x) (T x) (U x)) () (-> (/\/\ (e. A (CC)) (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (o. (opr A (.op) T) U) (opr A (.op) (o. T U)))) (A T (` U (cv x)) homvalt U (H~) (H~) (cv x) ffvrn syl3an3 3expa exp43 3imp1 (opr A (.op) T) U (H~) (H~) (cv x) fvco3 A T homulclt (opr A (.op) T) (H~) (H~) ffun syl syl3an1 3expa ex 3impa imp T U (H~) (H~) (cv x) fvco3 T (H~) (H~) ffun syl3an1 3expa A (.s) opreq2d (e. A (CC)) 3adantl1 3eqtr4d A (o. T U) (cv x) homvalt T (H~) (H~) U (H~) fco syl3an2 3expa ex 3impb imp eqtr4d r19.21aiva (o. (opr A (.op) T) U) (opr A (.op) (o. T U)) x hoeqt (opr A (.op) T) (H~) (H~) U (H~) fco A T homulclt sylan 3impa A (o. T U) homulclt T (H~) (H~) U (H~) fco sylan2 3impb sylanc mpbid)) thm (homulasst ((A x) (B x) (T x)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (:--> T (H~) (H~))) (= (opr (opr A (x.) B) (.op) T) (opr A (.op) (opr B (.op) T)))) (A B (` T (cv x)) ax-hvmulass T (H~) (H~) (cv x) ffvrn syl3an3 3expa exp43 3imp1 (opr A (x.) B) T (cv x) homvalt A B axmulcl syl3an1 3expa ex 3impa imp B T (cv x) homvalt A (.s) opreq2d 3expa (e. A (CC)) 3adantl1 3eqtr4d A (opr B (.op) T) (cv x) homvalt B T homulclt syl3an2 3expa ex 3impb imp eqtr4d r19.21aiva (opr (opr A (x.) B) (.op) T) (opr A (.op) (opr B (.op) T)) x hoeqt (opr A (x.) B) T homulclt A B axmulcl sylan 3impa A (opr B (.op) T) homulclt B T homulclt sylan2 3impb sylanc mpbid)) thm (hoadddit ((A x) (T x) (U x)) () (-> (/\/\ (e. A (CC)) (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr A (.op) (opr T (+op) U)) (opr (opr A (.op) T) (+op) (opr A (.op) U)))) (A (opr T (+op) U) (cv x) homvalt 3expa T U hoaddclt (e. A (CC)) anim2i 3impb sylan T U (cv x) hosvalt A (.s) opreq2d 3expa (e. A (CC)) 3adantl1 A (` T (cv x)) (` U (cv x)) ax-hvdistr1 (e. A (CC)) (:--> T (H~) (H~)) (:--> U (H~) (H~)) 3simp1 (e. (cv x) (H~)) adantr T (H~) (H~) (cv x) ffvrn (:--> U (H~) (H~)) adantlr (e. A (CC)) 3adantl1 U (H~) (H~) (cv x) ffvrn (:--> T (H~) (H~)) adantll (e. A (CC)) 3adantl1 syl3anc A T (cv x) homvalt 3expa (:--> U (H~) (H~)) 3adantl3 A U (cv x) homvalt 3expa (:--> T (H~) (H~)) 3adantl2 (+v) opreq12d eqtr4d 3eqtrd (opr A (.op) T) (opr A (.op) U) (cv x) hosvalt 3expa A T homulclt A U homulclt anim12i 3impdi sylan eqtr4d r19.21aiva (opr A (.op) (opr T (+op) U)) (opr (opr A (.op) T) (+op) (opr A (.op) U)) x hoeqt A (opr T (+op) U) homulclt T U hoaddclt sylan2 3impb (opr A (.op) T) (opr A (.op) U) hoaddclt A T homulclt A U homulclt syl2an 3impdi sylanc mpbid)) thm (hoadddirt ((A x) (B x) (T x)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (:--> T (H~) (H~))) (= (opr (opr A (+) B) (.op) T) (opr (opr A (.op) T) (+op) (opr B (.op) T)))) (A B (` T (cv x)) ax-hvdistr2 T (H~) (H~) (cv x) ffvrn syl3an3 3exp exp4a 3imp1 (opr A (+) B) T (cv x) homvalt 3expa A B axaddcl (:--> T (H~) (H~)) anim1i 3impa sylan A T (cv x) homvalt 3expa (e. B (CC)) 3adantl2 B T (cv x) homvalt 3expa (e. A (CC)) 3adantl1 (+v) opreq12d 3eqtr4d (opr A (.op) T) (opr B (.op) T) (cv x) hosvalt 3expa A T homulclt B T homulclt anim12i 3impdir sylan eqtr4d r19.21aiva (opr (opr A (+) B) (.op) T) (opr (opr A (.op) T) (+op) (opr B (.op) T)) x hoeqt (opr A (+) B) T homulclt A B axaddcl sylan 3impa (opr A (.op) T) (opr B (.op) T) hoaddclt A T homulclt B T homulclt syl2an 3impdir sylanc mpbid)) thm (homul12t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (:--> T (H~) (H~))) (= (opr A (.op) (opr B (.op) T)) (opr B (.op) (opr A (.op) T)))) (A B axmulcom (.op) T opreq1d (:--> T (H~) (H~)) 3adant3 A B T homulasst B A T homulasst 3com12 3eqtr3d)) thm (honegnegt () () (-> (:--> T (H~) (H~)) (= (opr (-u (1)) (.op) (opr (-u (1)) (.op) T)) T)) (1cn 1cn mul2neg 1cn mulid1 eqtr (.op) T opreq1i (:--> T (H~) (H~)) a1i 1cn negcl 1cn negcl (-u (1)) (-u (1)) T homulasst mp3an12 T homulid2t 3eqtr3d)) thm (hosubnegt () () (-> (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr T (-op) (opr (-u (1)) (.op) U)) (opr T (+op) U))) (T (opr (-u (1)) (.op) U) honegsubt 1cn negcl (-u (1)) U homulclt mpan sylan2 U honegnegt T (+op) opreq2d (:--> T (H~) (H~)) adantl eqtr3d)) thm (hosubdit () () (-> (/\/\ (e. A (CC)) (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr A (.op) (opr T (-op) U)) (opr (opr A (.op) T) (-op) (opr A (.op) U)))) (A T (opr (-u (1)) (.op) U) hoadddit 1cn negcl (-u (1)) U homulclt mpan syl3an3 1cn negcl A (-u (1)) U homul12t mp3an2 (:--> T (H~) (H~)) 3adant2 (opr A (.op) T) (+op) opreq2d eqtrd T U honegsubtOLD A (.op) opreq2d (e. A (CC)) 3adant1 (opr A (.op) T) (opr A (.op) U) honegsubtOLD A T homulclt A U homulclt syl2an 3impdi 3eqtr4d)) thm (honegdit () () (-> (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr (-u (1)) (.op) (opr T (+op) U)) (opr (opr (-u (1)) (.op) T) (+op) (opr (-u (1)) (.op) U)))) (1cn negcl (-u (1)) T U hoadddit mp3an1)) thm (honegsubdit () () (-> (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr (-u (1)) (.op) (opr T (-op) U)) (opr (opr (-u (1)) (.op) T) (+op) U))) (T U honegsubtOLD (-u (1)) (.op) opreq2d T (opr (-u (1)) (.op) U) honegdit 1cn negcl (-u (1)) U homulclt mpan sylan2 U honegnegt (:--> T (H~) (H~)) adantl (opr (-u (1)) (.op) T) (+op) opreq2d 3eqtrd)) thm (honegsubdi2t () () (-> (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr (-u (1)) (.op) (opr T (-op) U)) (opr U (-op) T))) ((opr (-u (1)) (.op) T) U hoaddcomt 1cn negcl (-u (1)) T homulclt mpan sylan T U honegsubdit U T honegsubtOLD ancoms 3eqtr4d)) thm (hosubsub2t () () (-> (/\/\ (:--> S (H~) (H~)) (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr S (-op) (opr T (-op) U)) (opr S (+op) (opr U (-op) T)))) (S (opr T (-op) U) honegsubtOLD T U hosubclt sylan2 3impb T U honegsubdi2t S (+op) opreq2d (:--> S (H~) (H~)) 3adant1 eqtrd)) thm (hosub4t () () (-> (/\ (/\ (:--> R (H~) (H~)) (:--> S (H~) (H~))) (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~)))) (= (opr (opr R (+op) S) (-op) (opr T (+op) U)) (opr (opr R (-op) T) (+op) (opr S (-op) U)))) (T U honegdit (/\ (:--> R (H~) (H~)) (:--> S (H~) (H~))) adantl (opr R (+op) S) (+op) opreq2d R S (opr (-u (1)) (.op) T) (opr (-u (1)) (.op) U) hoadd4t 1cn negcl (-u (1)) T homulclt mpan 1cn negcl (-u (1)) U homulclt mpan anim12i sylan2 eqtrd (opr R (+op) S) (opr T (+op) U) honegsubt R S hoaddclt T U hoaddclt syl2an R T honegsubt (:--> S (H~) (H~)) (:--> U (H~) (H~)) ad2ant2r S U honegsubt (:--> R (H~) (H~)) (:--> T (H~) (H~)) ad2ant2l (+op) opreq12d 3eqtr3d)) thm (hosubadd4t () () (-> (/\ (/\ (:--> R (H~) (H~)) (:--> S (H~) (H~))) (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~)))) (= (opr (opr R (-op) S) (-op) (opr T (-op) U)) (opr (opr R (+op) U) (-op) (opr S (+op) T)))) ((opr R (-op) S) T U hosubsub2t 3expb R S hosubclt sylan R U S T hosub4t an42s eqtr4d)) thm (hoaddsubasst () () (-> (/\/\ (:--> S (H~) (H~)) (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr (opr S (+op) T) (-op) U) (opr S (+op) (opr T (-op) U)))) (S T (opr (0op) (-op) U) hoaddasst ho0f (0op) U hosubclt mpan syl3an3 (opr S (+op) T) U ho0subt S T hoaddclt sylan 3impa T U ho0subt (:--> S (H~) (H~)) 3adant1 S (+op) opreq2d 3eqtr4d)) thm (hoaddsubt () () (-> (/\/\ (:--> S (H~) (H~)) (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr (opr S (+op) T) (-op) U) (opr (opr S (-op) U) (+op) T))) (S T hoaddcomt (-op) U opreq1d (:--> U (H~) (H~)) 3adant3 T S U hoaddsubasst 3com12 T (opr S (-op) U) hoaddcomt ex S U hosubclt syl5 exp3a com12 3imp 3eqtrd)) thm (hosubsubt () () (-> (/\/\ (:--> S (H~) (H~)) (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr S (-op) (opr T (-op) U)) (opr (opr S (-op) T) (+op) U))) (S T U hosubsub2t S U T hoaddsubasst S U T hoaddsubt eqtr3d 3com23 eqtrd)) thm (hosubsub4t () () (-> (/\/\ (:--> S (H~) (H~)) (:--> T (H~) (H~)) (:--> U (H~) (H~))) (= (opr (opr S (-op) T) (-op) U) (opr S (-op) (opr T (+op) U)))) (S T (opr (-u (1)) (.op) U) hosubsubt 1cn negcl (-u (1)) U homulclt mpan syl3an3 T U hosubnegt (:--> S (H~) (H~)) 3adant1 S (-op) opreq2d (opr S (-op) T) U honegsubt S T hosubclt sylan 3impa 3eqtr3rd)) thm (ho2timest () () (-> (:--> T (H~) (H~)) (= (opr (2) (.op) T) (opr T (+op) T))) (1cn 1cn (1) (1) T hoadddirt mp3an12 df-2 (.op) T opreq1i syl5eq 1cn (1) T T hoadddit mp3an1 anidms T T hoaddclt anidms (opr T (+op) T) homulid2t syl 3eqtr2d)) thm (hoaddsubass () ((hoaddsubass.1 (:--> R (H~) (H~))) (hoaddsubass.2 (:--> S (H~) (H~))) (hoaddsubass.3 (:--> T (H~) (H~)))) (= (opr (opr R (+op) S) (-op) T) (opr R (+op) (opr S (-op) T))) (hoaddsubass.1 hoaddsubass.2 hoaddsubass.3 R S T hoaddsubasst mp3an)) thm (hoaddsub () ((hoaddsubass.1 (:--> R (H~) (H~))) (hoaddsubass.2 (:--> S (H~) (H~))) (hoaddsubass.3 (:--> T (H~) (H~)))) (= (opr (opr R (+op) S) (-op) T) (opr (opr R (-op) T) (+op) S)) (hoaddsubass.1 hoaddsubass.2 hoaddcom (-op) T opreq1i hoaddsubass.2 hoaddsubass.1 hoaddsubass.3 hoaddsubass hoaddsubass.2 hoaddsubass.1 hoaddsubass.3 hosubcl hoaddcom 3eqtr)) thm (hosd1 () ((hosd1.2 (:--> T (H~) (H~))) (hosd1.3 (:--> U (H~) (H~)))) (= (opr T (+op) U) (opr T (-op) (opr (` (proj) (0H)) (-op) U))) (h0elch pjf hosd1.3 hosubcl hosd1.2 hosd1.3 hoaddcl hoaddcom hosd1.2 hosd1.3 hoaddcl h0elch pjf hosd1.3 hoaddsubass hosd1.2 hosd1.3 hoaddcl hoid0OLD (-op) U opreq1i hosd1.2 hosd1.3 hosd1.3 hoaddsub hosd1.2 hosd1.3 hosubcl hosd1.3 hoaddcom hosd1.3 hosd1.2 hodseq eqtr 3eqtr 3eqtr2 hosd1.2 h0elch pjf hosd1.3 hosubcl hosd1.2 hosd1.3 hoaddcl hods mpbir eqcomi)) thm (hosd2 () ((hosd1.2 (:--> T (H~) (H~))) (hosd1.3 (:--> U (H~) (H~)))) (= (opr T (+op) U) (opr T (-op) (opr (opr U (-op) U) (-op) U))) (hosd1.2 hosd1.3 hosd1 hosd1.3 hodidOLD (-op) U opreq1i T (-op) opreq2i eqtr4)) thm (hopncan () ((hosd1.2 (:--> T (H~) (H~))) (hosd1.3 (:--> U (H~) (H~)))) (= (opr (opr T (+op) U) (-op) U) T) (hosd1.2 hosd1.3 hosd1.3 hoaddsubass hosd1.3 hodid2OLD T (+op) opreq2i hosd1.2 hoid02OLD 3eqtr)) thm (honpcan () ((hosd1.2 (:--> T (H~) (H~))) (hosd1.3 (:--> U (H~) (H~)))) (= (opr (opr T (-op) U) (+op) U) T) (hosd1.2 hosd1.3 hosd1.3 hoaddsub hosd1.2 hosd1.3 hopncan eqtr3)) thm (hosubeq0OLD () ((hosd1.2 (:--> T (H~) (H~))) (hosd1.3 (:--> U (H~) (H~)))) (<-> (= (opr T (-op) U) (X. (H~) (0H))) (= T U)) (hosd1.2 hosd1.3 honegsubOLD (X. (H~) (0H)) eqeq1i (opr T (+op) (opr (-u (1)) (.op) U)) (X. (H~) (0H)) (+op) U opreq1 sylbi hosd1.2 1cn negcl hosd1.3 (-u (1)) U homulclt mp2an hosd1.3 hoadd23 hosd1.2 hosd1.3 1cn negcl hosd1.3 (-u (1)) U homulclt mp2an hoaddass hosd1.3 hosd1.3 honegsubOLD hosd1.3 hodid2OLD eqtr3 T (+op) opreq2i hosd1.2 hoid02OLD eqtr 3eqtr ho0fOLD hosd1.3 hoaddcom hosd1.3 hoid02OLD eqtr 3eqtr3g T U (-op) U opreq1 hosd1.3 hodid2OLD syl6eq impbi)) thm (honpncan () ((honpncan.1 (:--> R (H~) (H~))) (honpncan.2 (:--> S (H~) (H~))) (honpncan.3 (:--> T (H~) (H~)))) (= (opr (opr R (-op) S) (+op) (opr S (-op) T)) (opr R (-op) T)) (honpncan.1 honpncan.2 hosubcl honpncan.2 honpncan.3 hoaddsubass honpncan.1 honpncan.2 honpcan (-op) T opreq1i eqtr3)) thm (ho01 ((x y) (T x) (T y)) ((ho0.1 (:--> T (H~) (H~)))) (<-> (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (0)))) (= T (X. (H~) (0H)))) (ho0.1 T (H~) (H~) ffn ax-mp ax-hv0cl elisseti (H~) fconst (X. (H~) ({} (0v))) (H~) ({} (0v)) ffn ax-mp T (H~) (X. (H~) ({} (0v))) (H~) x eqfnfv mp2an (H~) eqid mpbiran df-ch0 (0H) ({} (0v)) (H~) xpeq2 ax-mp T eqeq2i ho0.1 (cv x) ffvrni (` T (cv x)) y hial0 syl ax-hv0cl elisseti (cv x) (H~) fvconst2 (` T (cv x)) eqeq2d bitr4d ralbiia 3bitr4r)) thm (ho02 ((x y) (T x) (T y)) ((ho0.1 (:--> T (H~) (H~)))) (<-> (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` T (cv y))) (0)))) (= T (X. (H~) (0H)))) (x (H~) y (H~) (= (opr (cv x) (.i) (` T (cv y))) (0)) ralcom ho0.1 (cv y) ffvrni (` T (cv y)) x hial02 (` T (cv y)) x hial0 bitr4d syl ralbiia ho0.1 y x ho01 3bitr)) thm (hoeq1t ((x y) (S x) (S y) (T x) (T y)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (<-> (A.e. x (H~) (A.e. y (H~) (= (opr (` S (cv x)) (.i) (cv y)) (opr (` T (cv x)) (.i) (cv y))))) (= S T))) ((` S (cv x)) (` T (cv x)) y hial2eqt S (H~) (H~) (cv x) ffvrn T (H~) (H~) (cv x) ffvrn syl2an anandirs ralbidva S (H~) T (H~) x eqfnfv (H~) eqid (A.e. x (H~) (= (` S (cv x)) (` T (cv x)))) biantrur syl6bbr S (H~) (H~) ffn T (H~) (H~) ffn syl2an bitr4d)) thm (hoeq2t ((x y) (S x) (S y) (T x) (T y)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (<-> (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` S (cv y))) (opr (cv x) (.i) (` T (cv y)))))) (= S T))) (x (H~) y (H~) (= (opr (cv x) (.i) (` S (cv y))) (opr (cv x) (.i) (` T (cv y)))) ralcom (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) a1i (` S (cv y)) (` T (cv y)) x hial2eq2t (` S (cv y)) (` T (cv y)) x hial2eqt bitr4d S (H~) (H~) (cv y) ffvrn T (H~) (H~) (cv y) ffvrn syl2an anandirs ralbidva S T y x hoeq1t 3bitrd)) thm (adjmo ((x y) (u v) (u x) (u y) (T u) (v x) (v y) (T v) (T x) (T y)) () (E* u (/\ (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` T (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y))))))) ((cv u) (cv v) x y hoeq1t biimpa x (H~) y (H~) (= (opr (cv x) (.i) (` T (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y))) (= (opr (cv x) (.i) (` T (cv y))) (opr (` (cv v) (cv x)) (.i) (cv y))) r19.26-2 (opr (cv x) (.i) (` T (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y)) (opr (` (cv v) (cv x)) (.i) (cv y)) eqtr2t y (H~) r19.20si x (H~) r19.20si sylbir sylan2 an4s u v gen2 (cv u) (cv v) (H~) (H~) feq1 (cv u) (cv v) (cv x) fveq1 (.i) (cv y) opreq1d (opr (cv x) (.i) (` T (cv y))) eqeq2d x (H~) y (H~) 2ralbidv anbi12d mo4 mpbir)) thm (adjsymt ((x y) (x z) (S x) (y z) (S y) (S z) (T x) (T y) (T z)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (<-> (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` S (cv y))) (opr (` T (cv x)) (.i) (cv y))))) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` T (cv y))) (opr (` S (cv x)) (.i) (cv y))))))) ((` T (cv y)) (cv x) ax-his1 T (H~) (H~) (cv y) ffvrn sylan (:--> S (H~) (H~)) adantrl (cv y) (` S (cv x)) ax-his1 S (H~) (H~) (cv x) ffvrn sylan2 (:--> T (H~) (H~)) adantll eqeq12d ancoms (opr (cv x) (.i) (` T (cv y))) (opr (` S (cv x)) (.i) (cv y)) cj11t (cv x) (` T (cv y)) ax-hicl T (H~) (H~) (cv y) ffvrn sylan2 (:--> S (H~) (H~)) adantll (` S (cv x)) (cv y) ax-hicl S (H~) (H~) (cv x) ffvrn sylan (:--> T (H~) (H~)) adantrl sylanc bitr2d an4s anassrs (opr (` T (cv y)) (.i) (cv x)) (opr (cv y) (.i) (` S (cv x))) eqcom syl6bb ralbidva ralbidva x (H~) y (H~) (= (opr (cv x) (.i) (` S (cv y))) (opr (` T (cv x)) (.i) (cv y))) ralcom (cv z) (cv y) S fveq2 (cv x) (.i) opreq2d (cv z) (cv y) (` T (cv x)) (.i) opreq2 eqeq12d x (H~) ralbidv (H~) cbvralv bitr4 (cv x) (cv y) (.i) (` S (cv z)) opreq1 (cv x) (cv y) T fveq2 (.i) (cv z) opreq1d eqeq12d (H~) cbvralv z (H~) ralbii (cv z) (cv x) S fveq2 (cv y) (.i) opreq2d (cv z) (cv x) (` T (cv y)) (.i) opreq2 eqeq12d y (H~) ralbidv (H~) cbvralv 3bitr syl6rbbr)) thm (eigre () ((eigre.1 (e. A (H~))) (eigre.2 (e. B (CC)))) (-> (/\ (= (` T A) (opr B (.s) A)) (=/= A (0v))) (<-> (= (opr A (.i) (` T A)) (opr (` T A) (.i) A)) (e. B (RR)))) ((` T A) (opr B (.s) A) A (.i) opreq2 eigre.2 eigre.1 eigre.1 B A A his5t mp3an syl6eq (` T A) (opr B (.s) A) (.i) A opreq1 eigre.2 eigre.1 eigre.1 B A A ax-his3 mp3an syl6eq eqeq12d eigre.1 A ax-his4 mpan eigre.1 A hiidrclt ax-mp gt0ne0 eigre.2 cjcl eigre.2 eigre.1 eigre.1 hicl 3pm3.2i (` (*) B) B (opr A (.i) A) mulcan2t mpan 3syl sylan9bb eigre.2 cjreb syl6bbr)) thm (eigret () () (-> (/\ (/\ (e. A (H~)) (e. B (CC))) (/\ (= (` T A) (opr B (.s) A)) (=/= A (0v)))) (<-> (= (opr A (.i) (` T A)) (opr (` T A) (.i) A)) (e. B (RR)))) (A (if (e. A (H~)) A (0v)) T fveq2 A (if (e. A (H~)) A (0v)) B (.s) opreq2 eqeq12d A (if (e. A (H~)) A (0v)) (0v) neeq1 anbi12d (= A (if (e. A (H~)) A (0v))) id A (if (e. A (H~)) A (0v)) T fveq2 (.i) opreq12d A (if (e. A (H~)) A (0v)) T fveq2 (= A (if (e. A (H~)) A (0v))) id (.i) opreq12d eqeq12d (e. B (RR)) bibi1d imbi12d B (if (e. B (CC)) B (0)) (.s) (if (e. A (H~)) A (0v)) opreq1 (` T (if (e. A (H~)) A (0v))) eqeq2d (=/= (if (e. A (H~)) A (0v)) (0v)) anbi1d B (if (e. B (CC)) B (0)) (RR) eleq1 (= (opr (if (e. A (H~)) A (0v)) (.i) (` T (if (e. A (H~)) A (0v)))) (opr (` T (if (e. A (H~)) A (0v))) (.i) (if (e. A (H~)) A (0v)))) bibi2d imbi12d ax-hv0cl A elimel 0cn B elimel T eigre dedth2h imp)) thm (eigpos () ((eigpos.1 (e. A (H~))) (eigpos.2 (e. B (CC)))) (-> (/\ (/\ (e. (opr A (.i) (` T A)) (RR)) (br (0) (<_) (opr A (.i) (` T A)))) (/\ (= (` T A) (opr B (.s) A)) (=/= A (0v)))) (/\ (e. B (RR)) (br (0) (<_) B))) (eigpos.1 eigpos.2 eigpos.1 hvmulcl A (opr B (.s) A) hiret mp2an (= (` T A) (opr B (.s) A)) a1i (` T A) (opr B (.s) A) A (.i) opreq2 (RR) eleq1d (` T A) (opr B (.s) A) A (.i) opreq2 (` T A) (opr B (.s) A) (.i) A opreq1 eqeq12d 3bitr4d (=/= A (0v)) adantr eigpos.1 eigpos.2 T eigre bitrd biimpac (br (0) (<_) (opr A (.i) (` T A))) adantlr eigpos.1 A hiidrclt ax-mp B (opr A (.i) A) prodge02t mpanl2 eigpos.1 eigpos.2 eigpos.1 hvmulcl A (opr B (.s) A) hiret mp2an (= (` T A) (opr B (.s) A)) a1i (` T A) (opr B (.s) A) A (.i) opreq2 (RR) eleq1d (` T A) (opr B (.s) A) A (.i) opreq2 (` T A) (opr B (.s) A) (.i) A opreq1 eqeq12d 3bitr4d (=/= A (0v)) adantr eigpos.1 eigpos.2 T eigre bitrd biimpac (br (0) (<_) (opr A (.i) (` T A))) adantlr eigpos.1 A ax-his4 mpan (/\ (e. (opr A (.i) (` T A)) (RR)) (br (0) (<_) (opr A (.i) (` T A)))) (= (` T A) (opr B (.s) A)) ad2antll (e. (opr A (.i) (` T A)) (RR)) (br (0) (<_) (opr A (.i) (` T A))) pm3.27 (/\ (= (` T A) (opr B (.s) A)) (=/= A (0v))) adantr (` T A) (opr B (.s) A) A (.i) opreq2 (/\ (e. (opr A (.i) (` T A)) (RR)) (br (0) (<_) (opr A (.i) (` T A)))) (=/= A (0v)) ad2antrl eigpos.1 eigpos.2 eigpos.1 hvmulcl A (opr B (.s) A) hiret mp2an (= (` T A) (opr B (.s) A)) a1i (` T A) (opr B (.s) A) A (.i) opreq2 (RR) eleq1d (` T A) (opr B (.s) A) A (.i) opreq2 (` T A) (opr B (.s) A) (.i) A opreq1 eqeq12d 3bitr4d (=/= A (0v)) adantr eigpos.1 eigpos.2 T eigre bitrd biimpac (br (0) (<_) (opr A (.i) (` T A))) adantlr eigpos.2 cjreb sylib (x.) (opr A (.i) A) opreq1d eigpos.2 eigpos.1 eigpos.1 B A A his5t mp3an syl5eq eqtrd breqtrd jca sylanc jca)) thm (eigorth () ((eigorth.1 (e. A (H~))) (eigorth.2 (e. B (H~))) (eigorth.3 (e. C (CC))) (eigorth.4 (e. D (CC)))) (-> (/\ (/\ (= (` T A) (opr C (.s) A)) (= (` T B) (opr D (.s) B))) (-. (= C (` (*) D)))) (<-> (= (opr A (.i) (` T B)) (opr (` T A) (.i) B)) (= (opr A (.i) B) (0)))) ((` T B) (opr D (.s) B) A (.i) opreq2 eigorth.4 eigorth.1 eigorth.2 D A B his5t mp3an syl6eq (` T A) (opr C (.s) A) (.i) B opreq1 eigorth.3 eigorth.1 eigorth.2 C A B ax-his3 mp3an syl6eq eqeqan12rd eigorth.4 cjcl eigorth.3 eigorth.1 eigorth.2 hicl (` (*) D) C (opr A (.i) B) mulcan2t (opr A (.i) B) (0) df-ne sylan2br ex mp3an (` (*) D) C eqcom syl6bb biimpcd con1d com12 (opr A (.i) B) (0) (` (*) D) (x.) opreq2 (opr A (.i) B) (0) C (x.) opreq2 eigorth.3 mul01 eigorth.4 cjcl mul01 eqtr4 syl6eq eqtr4d (-. (= C (` (*) D))) a1i impbid sylan9bb)) thm (eigortht () () (-> (/\ (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (CC)) (e. D (CC)))) (/\ (/\ (= (` T A) (opr C (.s) A)) (= (` T B) (opr D (.s) B))) (-. (= C (` (*) D))))) (<-> (= (opr A (.i) (` T B)) (opr (` T A) (.i) B)) (= (opr A (.i) B) (0)))) (A (if (e. A (H~)) A (0v)) T fveq2 A (if (e. A (H~)) A (0v)) C (.s) opreq2 eqeq12d (= (` T B) (opr D (.s) B)) anbi1d (-. (= C (` (*) D))) anbi1d A (if (e. A (H~)) A (0v)) (.i) (` T B) opreq1 A (if (e. A (H~)) A (0v)) T fveq2 (.i) B opreq1d eqeq12d A (if (e. A (H~)) A (0v)) (.i) B opreq1 (0) eqeq1d bibi12d imbi12d B (if (e. B (H~)) B (0v)) T fveq2 B (if (e. B (H~)) B (0v)) D (.s) opreq2 eqeq12d (= (` T (if (e. A (H~)) A (0v))) (opr C (.s) (if (e. A (H~)) A (0v)))) anbi2d (-. (= C (` (*) D))) anbi1d B (if (e. B (H~)) B (0v)) T fveq2 (if (e. A (H~)) A (0v)) (.i) opreq2d B (if (e. B (H~)) B (0v)) (` T (if (e. A (H~)) A (0v))) (.i) opreq2 eqeq12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (.i) opreq2 (0) eqeq1d bibi12d imbi12d C (if (e. C (CC)) C (0)) (.s) (if (e. A (H~)) A (0v)) opreq1 (` T (if (e. A (H~)) A (0v))) eqeq2d (= (` T (if (e. B (H~)) B (0v))) (opr D (.s) (if (e. B (H~)) B (0v)))) anbi1d C (if (e. C (CC)) C (0)) (` (*) D) eqeq1 negbid anbi12d (<-> (= (opr (if (e. A (H~)) A (0v)) (.i) (` T (if (e. B (H~)) B (0v)))) (opr (` T (if (e. A (H~)) A (0v))) (.i) (if (e. B (H~)) B (0v)))) (= (opr (if (e. A (H~)) A (0v)) (.i) (if (e. B (H~)) B (0v))) (0))) imbi1d D (if (e. D (CC)) D (0)) (.s) (if (e. B (H~)) B (0v)) opreq1 (` T (if (e. B (H~)) B (0v))) eqeq2d (= (` T (if (e. A (H~)) A (0v))) (opr (if (e. C (CC)) C (0)) (.s) (if (e. A (H~)) A (0v)))) anbi2d D (if (e. D (CC)) D (0)) (*) fveq2 (if (e. C (CC)) C (0)) eqeq2d negbid anbi12d (<-> (= (opr (if (e. A (H~)) A (0v)) (.i) (` T (if (e. B (H~)) B (0v)))) (opr (` T (if (e. A (H~)) A (0v))) (.i) (if (e. B (H~)) B (0v)))) (= (opr (if (e. A (H~)) A (0v)) (.i) (if (e. B (H~)) B (0v))) (0))) imbi1d ax-hv0cl A elimel ax-hv0cl B elimel 0cn C elimel 0cn D elimel T eigorth dedth4h imp)) thm (nmopvalt ((t x) (t y) (t z) (T t) (x y) (x z) (T x) (y z) (T y) (T z)) () (-> (:--> T (H~) (H~)) (= (` (normop) T) (sup ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) (RR*) (<)))) (ax-hilex ax-hilex T elmap (cv t) T (cv y) fveq1 (norm) fveq2d (cv x) eqeq2d (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv x abbidv ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` (cv t) (cv y))))))) ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) (RR*) (<) supeq1 syl t z x y df-nmop ax-hilex ax-hilex (cv t) elmap (= (cv z) (sup ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` (cv t) (cv y))))))) (RR*) (<))) anbi1i t z opabbii eqtr4 xrltso ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) supex fvopab4 sylbir)) thm (elcnopt ((t w) (t x) (t y) (t z) (T t) (w x) (w y) (w z) (T w) (x y) (x z) (T x) (y z) (T y) (T z)) () (<-> (e. T (ConOp)) (/\ (:--> T (H~) (H~)) (A.e. x (H~) (A.e. y (RR) (-> (br (0) (<) (cv y)) (E.e. z (RR) (/\ (br (0) (<) (cv z)) (A.e. w (H~) (-> (br (` (norm) (opr (cv w) (-v) (cv x))) (<) (cv z)) (br (` (norm) (opr (` T (cv w)) (-v) (` T (cv x)))) (<) (cv y))))))))))) (T (ConOp) elisset ax-hilex T (H~) (H~) (V) fex mpan2 (A.e. x (H~) (A.e. y (RR) (-> (br (0) (<) (cv y)) (E.e. z (RR) (/\ (br (0) (<) (cv z)) (A.e. w (H~) (-> (br (` (norm) (opr (cv w) (-v) (cv x))) (<) (cv z)) (br (` (norm) (opr (` T (cv w)) (-v) (` T (cv x)))) (<) (cv y))))))))) adantr (cv t) T (H~) (H~) feq1 (cv t) T (cv w) fveq1 (cv t) T (cv x) fveq1 (-v) opreq12d (norm) fveq2d (<) (cv y) breq1d (br (` (norm) (opr (cv w) (-v) (cv x))) (<) (cv z)) imbi2d w (H~) ralbidv (br (0) (<) (cv z)) anbi2d z (RR) rexbidv (br (0) (<) (cv y)) imbi2d x (H~) y (RR) 2ralbidv anbi12d t x y z w df-cnop (V) elab2g pm5.21nii)) thm (ellnopt ((t x) (t y) (t z) (T t) (x y) (x z) (T x) (y z) (T y) (T z)) () (<-> (e. T (LinOp)) (/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))))))))) (T (LinOp) elisset ax-hilex T (H~) (H~) (V) fex mpan2 (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))))))) adantr (cv t) T (H~) (H~) feq1 (cv t) T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) fveq1 (cv t) T (cv y) fveq1 (cv x) (.s) opreq2d (cv t) T (cv z) fveq1 (+v) opreq12d eqeq12d z (H~) ralbidv x (CC) y (H~) 2ralbidv anbi12d t x y z df-lnop (V) elab2g pm5.21nii)) thm (lnopft ((x y) (x z) (T x) (y z) (T y) (T z)) () (-> (e. T (LinOp)) (:--> T (H~) (H~))) (T x y z ellnopt pm3.26bd)) thm (elbdopt ((T t)) () (<-> (e. T (BndLinOp)) (/\ (e. T (LinOp)) (br (` (normop) T) (<) (+oo)))) (t df-bdop T eleq2i T (LinOp) ({|} t (/\ (:--> (cv t) (H~) (H~)) (br (` (normop) (cv t)) (<) (+oo)))) elin (e. T (LinOp)) (:--> T (H~) (H~)) (br (` (normop) T) (<) (+oo)) anass T lnopft pm4.71i (br (` (normop) T) (<) (+oo)) anbi1i ax-hilex T (H~) (H~) (V) fex mpan2 (br (` (normop) T) (<) (+oo)) adantr (cv t) T (H~) (H~) feq1 (cv t) T (normop) fveq2 (<) (+oo) breq1d anbi12d elab3 (e. T (LinOp)) anbi2i 3bitr4r 3bitr)) thm (bdoplnt () () (-> (e. T (BndLinOp)) (e. T (LinOp))) (T elbdopt pm3.26bd)) thm (bdopft () () (-> (e. T (BndLinOp)) (:--> T (H~) (H~))) (T bdoplnt T lnopft syl)) thm (nmopsetret ((x y) (T x) (T y)) () (-> (:--> T (H~) (H~)) (C_ ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) (RR))) ((cv x) (` (norm) (` T (cv y))) (RR) eleq1 T (H~) (H~) (cv y) ffvrn (` T (cv y)) normclt syl syl5bir impcom (br (` (norm) (cv y)) (<_) (1)) adantrl exp31 r19.23adv abssdv)) thm (nmopsetn0 ((x y) (T x) (T y)) () (e. (` (norm) (` T (0v))) ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y)))))))) (ax-hv0cl norm0 0re ax1re lt01 ltlei eqbrtr (` (norm) (` T (0v))) eqid pm3.2i (cv y) (0v) (norm) fveq2 (<_) (1) breq1d (cv y) (0v) T fveq2 (norm) fveq2d (` (norm) (` T (0v))) eqeq2d anbi12d (H~) rcla4ev mp2an (norm) (` T (0v)) fvex (cv x) (` (norm) (` T (0v))) (` (norm) (` T (cv y))) eqeq1 (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv elab mpbir)) thm (nmopxrt ((x y) (T x) (T y)) () (-> (:--> T (H~) (H~)) (e. (` (normop) T) (RR*))) (T x y nmopvalt T x y nmopsetret ressxr (:--> T (H~) (H~)) a1i sstrd ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) supxrcl syl eqeltrd)) thm (nmoprepnf ((x y) (T x) (T y)) () (-> (:--> T (H~) (H~)) (<-> (e. (` (normop) T) (RR)) (-. (= (` (normop) T) (+oo))))) (T x y nmopsetret T x y nmopsetn0 (` (norm) (` T (0v))) ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) n0i ax-mp ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) supxrre2 mpan2 syl T x y nmopvalt (RR) eleq1d T x y nmopvalt (+oo) eqeq1d negbid 3bitr4d)) thm (nmopgtmnf () () (-> (:--> T (H~) (H~)) (br (-oo) (<) (` (normop) T))) (T nmoprepnf (e. (` (normop) T) (RR)) (= (` (normop) T) (+oo)) xor3 (e. (` (normop) T) (RR)) (= (` (normop) T) (+oo)) xor2 pm3.26bd sylbir (` (normop) T) mnfltxrt 3syl)) thm (nmopreltpnf () () (-> (:--> T (H~) (H~)) (<-> (e. (` (normop) T) (RR)) (br (` (normop) T) (<) (+oo)))) (T nmoprepnf T nmopxrt (` (normop) T) nltpnftt syl con2bid bitr4d)) thm (nmopret () () (-> (e. T (BndLinOp)) (e. (` (normop) T) (RR))) (T bdopft T nmopgtmnf syl T elbdopt pm3.27bd jca T bdopft T nmopxrt (` (normop) T) xrrebndt 3syl mpbird)) thm (elbdop2t () () (<-> (e. T (BndLinOp)) (/\ (e. T (LinOp)) (e. (` (normop) T) (RR)))) (T elbdopt T lnopft T nmopreltpnf syl pm5.32i bitr4)) thm (elunopt ((t x) (t y) (T t) (x y) (T x) (T y)) () (<-> (e. T (UniOp)) (/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y))))))) (T (UniOp) elisset T (H~) (H~) fof ax-hilex T (H~) (H~) (V) fex mpan2 syl (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y))))) adantr (cv t) T (H~) (H~) foeq1 (cv t) T (cv x) fveq1 (cv t) T (cv y) fveq1 (.i) opreq12d (opr (cv x) (.i) (cv y)) eqeq1d x (H~) y (H~) 2ralbidv anbi12d t x y df-unop (V) elab2g pm5.21nii)) thm (elhmopt ((t x) (t y) (T t) (x y) (T x) (T y)) () (<-> (e. T (HrmOp)) (/\ (:--> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` T (cv y))) (opr (` T (cv x)) (.i) (cv y))))))) (T (HrmOp) elisset ax-hilex T (H~) (H~) (V) fex mpan2 (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` T (cv y))) (opr (` T (cv x)) (.i) (cv y))))) adantr (cv t) T (H~) (H~) feq1 (cv t) T (cv y) fveq1 (cv x) (.i) opreq2d (cv t) T (cv x) fveq1 (.i) (cv y) opreq1d eqeq12d x (H~) y (H~) 2ralbidv anbi12d t x y df-hmop (V) elab2g pm5.21nii)) thm (hmopft ((x y) (T x) (T y)) () (-> (e. T (HrmOp)) (:--> T (H~) (H~))) (T x y elhmopt pm3.26bd)) thm (hmopex () () (e. (HrmOp) (V)) ((H~) (^m) (H~) oprex (cv t) hmopft ax-hilex ax-hilex (cv t) elmap sylibr ssriv ssexi)) thm (nmfnvalt ((t x) (t y) (t z) (T t) (x y) (x z) (T x) (y z) (T y) (T z)) () (-> (:--> T (H~) (CC)) (= (` (normfn) T) (sup ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y))))))) (RR*) (<)))) (axcnex ax-hilex T elmap (cv t) T (cv y) fveq1 (abs) fveq2d (cv x) eqeq2d (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv x abbidv ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` (cv t) (cv y))))))) ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y))))))) (RR*) (<) supeq1 syl t z x y df-nmfn axcnex ax-hilex (cv t) elmap (= (cv z) (sup ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` (cv t) (cv y))))))) (RR*) (<))) anbi1i t z opabbii eqtr4 xrltso ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y))))))) supex fvopab4 sylbir)) thm (nmfnsetret ((x y) (T x) (T y)) () (-> (:--> T (H~) (CC)) (C_ ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y))))))) (RR))) ((cv x) (` (abs) (` T (cv y))) (RR) eleq1 T (H~) (CC) (cv y) ffvrn (` T (cv y)) absclt syl syl5bir impcom (br (` (norm) (cv y)) (<_) (1)) adantrl exp31 r19.23adv abssdv)) thm (nmfnsetn0 ((x y) (T x) (T y)) () (e. (` (abs) (` T (0v))) ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y)))))))) (ax-hv0cl norm0 0re ax1re lt01 ltlei eqbrtr (` (abs) (` T (0v))) eqid pm3.2i (cv y) (0v) (norm) fveq2 (<_) (1) breq1d (cv y) (0v) T fveq2 (abs) fveq2d (` (abs) (` T (0v))) eqeq2d anbi12d (H~) rcla4ev mp2an (abs) (` T (0v)) fvex (cv x) (` (abs) (` T (0v))) (` (abs) (` T (cv y))) eqeq1 (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv elab mpbir)) thm (nmfnxrt ((x y) (T x) (T y)) () (-> (:--> T (H~) (CC)) (e. (` (normfn) T) (RR*))) (T x y nmfnvalt T x y nmfnsetret ressxr (:--> T (H~) (CC)) a1i sstrd ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y))))))) supxrcl syl eqeltrd)) thm (nmfnrepnf ((x y) (T x) (T y)) () (-> (:--> T (H~) (CC)) (<-> (e. (` (normfn) T) (RR)) (-. (= (` (normfn) T) (+oo))))) (T x y nmfnsetret T x y nmfnsetn0 (` (abs) (` T (0v))) ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y))))))) n0i ax-mp ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y))))))) supxrre2 mpan2 syl T x y nmfnvalt (RR) eleq1d T x y nmfnvalt (+oo) eqeq1d negbid 3bitr4d)) thm (nlfnvalt ((t x) (t y) (T t) (x y) (T x) (T y)) () (-> (:--> T (H~) (CC)) (= (` (null) T) ({e.|} x (H~) (= (` T (cv x)) (0))))) (axcnex ax-hilex T elmap (cv t) T (cv x) fveq1 (0) eqeq1d x (H~) rabbisdv t y x df-nlfn axcnex ax-hilex (cv t) elmap (= (cv y) ({e.|} x (H~) (= (` (cv t) (cv x)) (0)))) anbi1i t y opabbii eqtr4 ax-hilex x (= (` T (cv x)) (0)) rabex fvopab4 sylbir)) thm (elcnfnt ((t w) (t x) (t y) (t z) (T t) (w x) (w y) (w z) (T w) (x y) (x z) (T x) (y z) (T y) (T z)) () (<-> (e. T (ConFn)) (/\ (:--> T (H~) (CC)) (A.e. x (H~) (A.e. y (RR) (-> (br (0) (<) (cv y)) (E.e. z (RR) (/\ (br (0) (<) (cv z)) (A.e. w (H~) (-> (br (` (norm) (opr (cv w) (-v) (cv x))) (<) (cv z)) (br (` (abs) (opr (` T (cv w)) (-) (` T (cv x)))) (<) (cv y))))))))))) (T (ConFn) elisset ax-hilex T (H~) (CC) (V) fex mpan2 (A.e. x (H~) (A.e. y (RR) (-> (br (0) (<) (cv y)) (E.e. z (RR) (/\ (br (0) (<) (cv z)) (A.e. w (H~) (-> (br (` (norm) (opr (cv w) (-v) (cv x))) (<) (cv z)) (br (` (abs) (opr (` T (cv w)) (-) (` T (cv x)))) (<) (cv y))))))))) adantr (cv t) T (H~) (CC) feq1 (cv t) T (cv w) fveq1 (cv t) T (cv x) fveq1 (-) opreq12d (abs) fveq2d (<) (cv y) breq1d (br (` (norm) (opr (cv w) (-v) (cv x))) (<) (cv z)) imbi2d w (H~) ralbidv (br (0) (<) (cv z)) anbi2d z (RR) rexbidv (br (0) (<) (cv y)) imbi2d x (H~) y (RR) 2ralbidv anbi12d t x y z w df-cnfn (V) elab2g pm5.21nii)) thm (ellnfnt ((t x) (t y) (t z) (T t) (x y) (x z) (T x) (y z) (T y) (T z)) () (<-> (e. T (LinFn)) (/\ (:--> T (H~) (CC)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (x.) (` T (cv y))) (+) (` T (cv z))))))))) (T (LinFn) elisset ax-hilex T (H~) (CC) (V) fex mpan2 (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (x.) (` T (cv y))) (+) (` T (cv z))))))) adantr (cv t) T (H~) (CC) feq1 (cv t) T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) fveq1 (cv t) T (cv y) fveq1 (cv x) (x.) opreq2d (cv t) T (cv z) fveq1 (+) opreq12d eqeq12d z (H~) ralbidv x (CC) y (H~) 2ralbidv anbi12d t x y z df-lnfn (V) elab2g pm5.21nii)) thm (lnfnft ((x y) (x z) (T x) (y z) (T y) (T z)) () (-> (e. T (LinFn)) (:--> T (H~) (CC))) (T x y z ellnfnt pm3.26bd)) thm (dfadj2 ((t u) (t x) (t y) (u x) (u y) (x y)) () (= (adj) ({<,>|} t u (/\/\ (:--> (cv t) (H~) (H~)) (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv t) (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y)))))))) (t u x y df-adj (cv t) (cv u) x y adjsymt (opr (` (cv t) (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv u) (cv y))) eqcom x (H~) y (H~) 2ralbii syl6rbbr pm5.32i (:--> (cv t) (H~) (H~)) (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` (cv t) (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv u) (cv y)))))) df-3an (:--> (cv t) (H~) (H~)) (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv t) (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y))))) df-3an 3bitr4 t u opabbii eqtr)) thm (funadj ((t u) (t x) (t y) (u x) (u y) (x y)) () (Fun (adj)) (t u (/\/\ (:--> (cv t) (H~) (H~)) (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv t) (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y)))))) funopab u x y (cv t) adjmo (:--> (cv t) (H~) (H~)) (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv t) (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y))))) 3simpc u immoi ax-mp mpgbir t u x y dfadj2 (adj) ({<,>|} t u (/\/\ (:--> (cv t) (H~) (H~)) (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv t) (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y))))))) funeq ax-mp mpbir)) thm (adjvalt ((t u) (t x) (t y) (T t) (u x) (u y) (T u) (x y) (T x) (T y)) () (-> (:--> T (H~) (H~)) (= (` (adj) T) (U. ({|} u (/\ (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` T (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y)))))))))) (ax-hilex T (H~) (H~) (V) fex mpan2 funadj t u x y dfadj2 (cv t) T (H~) (H~) feq1 (cv t) T (cv y) fveq1 (cv x) (.i) opreq2d (opr (` (cv u) (cv x)) (.i) (cv y)) eqeq1d x (H~) y (H~) 2ralbidv (:--> (cv u) (H~) (H~)) 3anbi13d (V) fvopab5 mpan syl (:--> T (H~) (H~)) (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` T (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y))))) 3anass baib u abbidv unieqd eqtrd)) thm (adjval2t ((u x) (u y) (T u) (x y) (T x) (T y)) () (-> (:--> T (H~) (H~)) (= (` (adj) T) (U. ({|} u (/\ (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv u) (cv y))))))))))) (T u x y adjvalt T (cv u) x y adjsymt (opr (cv x) (.i) (` (cv u) (cv y))) (opr (` T (cv x)) (.i) (cv y)) eqcom x (H~) y (H~) 2ralbii syl6bb ex pm5.32d u abbidv unieqd eqtrd)) thm (cnvadj ((t u) (t x) (t y) (u x) (u y) (x y)) () (= (`' (adj)) (adj)) (u t (/\/\ (:--> (cv u) (H~) (H~)) (:--> (cv t) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv u) (cv y))) (opr (` (cv t) (cv x)) (.i) (cv y)))))) cnvopab (:--> (cv u) (H~) (H~)) (:--> (cv t) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv u) (cv y))) (opr (` (cv t) (cv x)) (.i) (cv y))))) 3ancoma (` (cv u) (cv y)) (cv x) ax-his1 (cv u) (H~) (H~) (cv y) ffvrn sylan (:--> (cv t) (H~) (H~)) adantrl (cv y) (` (cv t) (cv x)) ax-his1 (cv t) (H~) (H~) (cv x) ffvrn sylan2 (:--> (cv u) (H~) (H~)) adantll eqeq12d ancoms (opr (cv x) (.i) (` (cv u) (cv y))) (opr (` (cv t) (cv x)) (.i) (cv y)) cj11t (cv x) (` (cv u) (cv y)) ax-hicl (cv u) (H~) (H~) (cv y) ffvrn sylan2 (:--> (cv t) (H~) (H~)) adantll (` (cv t) (cv x)) (cv y) ax-hicl (cv t) (H~) (H~) (cv x) ffvrn sylan (:--> (cv u) (H~) (H~)) adantrl sylanc bitr2d an4s anassrs (opr (` (cv u) (cv y)) (.i) (cv x)) (opr (cv y) (.i) (` (cv t) (cv x))) eqcom syl6bb ralbidva ralbidva x (H~) y (H~) (= (opr (cv y) (.i) (` (cv t) (cv x))) (opr (` (cv u) (cv y)) (.i) (cv x))) ralcom syl6bb pm5.32i (:--> (cv t) (H~) (H~)) (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv u) (cv y))) (opr (` (cv t) (cv x)) (.i) (cv y))))) df-3an (:--> (cv t) (H~) (H~)) (:--> (cv u) (H~) (H~)) (A.e. y (H~) (A.e. x (H~) (= (opr (cv y) (.i) (` (cv t) (cv x))) (opr (` (cv u) (cv y)) (.i) (cv x))))) df-3an 3bitr4 bitr t u opabbii eqtr u t x y dfadj2 (adj) ({<,>|} u t (/\/\ (:--> (cv u) (H~) (H~)) (:--> (cv t) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv u) (cv y))) (opr (` (cv t) (cv x)) (.i) (cv y))))))) cnveq ax-mp t u y x dfadj2 3eqtr4)) thm (funcnvadj () () (Fun (`' (adj))) (funadj cnvadj (`' (adj)) (adj) funeq ax-mp mpbir)) thm (adj1o () () (:-1-1-onto-> (adj) (dom (adj)) (dom (adj))) (funadj (adj) funfn mpbi funcnvadj (adj) df-rn cnvadj dmeqi eqtr 3pm3.2i (adj) (dom (adj)) (dom (adj)) f1o2 mpbir)) thm (dmadjss ((t u) (t x) (t y) (u x) (u y) (x y)) () (C_ (dom (adj)) (opr (H~) (^m) (H~))) (t u x y dfadj2 (:--> (cv t) (H~) (H~)) (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv t) (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y))))) 3anass ax-hilex ax-hilex (cv t) elmap (/\ (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv t) (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y)))))) anbi1i bitr4 t u opabbii eqtr dmeqi t u (opr (H~) (^m) (H~)) (/\ (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` (cv t) (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y)))))) dmopabss eqsstr)) thm (dmadjopt () () (-> (e. T (dom (adj))) (:--> T (H~) (H~))) (dmadjss T sseli ax-hilex ax-hilex T elmap sylib)) thm (dmadjrnt () () (-> (e. T (dom (adj))) (e. (` (adj) T) (dom (adj)))) (adj1o (adj) (dom (adj)) (dom (adj)) f1of ax-mp T ffvrni)) thm (eigvecvalt ((t x) (t y) (t z) (T t) (x y) (x z) (T x) (y z) (T y) (T z)) () (-> (:--> T (H~) (H~)) (= (` (eigvec) T) ({e.|} x (H~) (/\ (=/= (cv x) (0v)) (E.e. y (CC) (= (` T (cv x)) (opr (cv y) (.s) (cv x)))))))) (ax-hilex ax-hilex T elmap (cv t) T (cv x) fveq1 (opr (cv y) (.s) (cv x)) eqeq1d y (CC) rexbidv (=/= (cv x) (0v)) anbi2d x (H~) rabbisdv t z x y df-eigvec ax-hilex ax-hilex (cv t) elmap (= (cv z) ({e.|} x (H~) (/\ (=/= (cv x) (0v)) (E.e. y (CC) (= (` (cv t) (cv x)) (opr (cv y) (.s) (cv x))))))) anbi1i t z opabbii eqtr4 ax-hilex x (/\ (=/= (cv x) (0v)) (E.e. y (CC) (= (` T (cv x)) (opr (cv y) (.s) (cv x))))) rabex fvopab4 sylbir)) thm (eigvalvalt ((t x) (t y) (t z) (T t) (x y) (x z) (T x) (y z) (T y) (T z)) () (-> (:--> T (H~) (H~)) (= (` (eigval) T) ({<,>|} x y (/\ (e. (cv x) (` (eigvec) T)) (= (cv y) (opr (opr (` T (cv x)) (.i) (cv x)) (/) (opr (` (norm) (cv x)) (^) (2)))))))) (ax-hilex ax-hilex T elmap (cv t) T (eigvec) fveq2 (cv x) eleq2d (cv t) T (cv x) fveq1 (.i) (cv x) opreq1d (/) (opr (` (norm) (cv x)) (^) (2)) opreq1d (cv y) eqeq2d anbi12d x y opabbidv t z x y df-eigval ax-hilex ax-hilex (cv t) elmap (= (cv z) ({<,>|} x y (/\ (e. (cv x) (` (eigvec) (cv t))) (= (cv y) (opr (opr (` (cv t) (cv x)) (.i) (cv x)) (/) (opr (` (norm) (cv x)) (^) (2))))))) anbi1i t z opabbii eqtr4 (eigvec) T fvex x y (opr (opr (` T (cv x)) (.i) (cv x)) (/) (opr (` (norm) (cv x)) (^) (2))) funopabex2 fvopab4 sylbir)) thm (specvalt ((t x) (t y) (T t) (x y) (T x) (T y)) () (-> (:--> T (H~) (H~)) (= (` (Lambda) T) ({e.|} x (CC) (-. (:-1-1-> (opr T (-op) (opr (cv x) (.op) (|` (I) (H~)))) (H~) (H~)))))) (ax-hilex ax-hilex T elmap (cv t) T (-op) (opr (cv x) (.op) (|` (I) (H~))) opreq1 (opr (cv t) (-op) (opr (cv x) (.op) (|` (I) (H~)))) (opr T (-op) (opr (cv x) (.op) (|` (I) (H~)))) (H~) (H~) f1eq1 syl negbid x (CC) rabbisdv t y x df-spec ax-hilex ax-hilex (cv t) elmap (= (cv y) ({e.|} x (CC) (-. (:-1-1-> (opr (cv t) (-op) (opr (cv x) (.op) (|` (I) (H~)))) (H~) (H~))))) anbi1i t y opabbii eqtr4 axcnex x (-. (:-1-1-> (opr T (-op) (opr (cv x) (.op) (|` (I) (H~)))) (H~) (H~))) rabex fvopab4 sylbir)) thm (dmadjrnb () () (<-> (e. T (dom (adj))) (e. (` (adj) T) (dom (adj)))) (T dmadjrnt ax-hv0cl (0v) (H~) n0i ax-mp ({/}) (H~) eqcom mtbir dm0 (H~) eqeq1i mtbir ({/}) (H~) (H~) fdm mto ({/}) dmadjopt mto T (adj) ndmfv (dom (adj)) eleq1d mtbiri a3i impbi)) thm (nmoplbt ((x y) (A x) (A y) (T x) (T y)) () (-> (/\/\ (:--> T (H~) (H~)) (e. A (H~)) (br (` (norm) A) (<_) (1))) (br (` (norm) (` T A)) (<_) (` (normop) T))) (({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) (` (norm) (` T A)) supxrub T x y nmopsetret ressxr (:--> T (H~) (H~)) a1i sstrd (e. A (H~)) (br (` (norm) A) (<_) (1)) 3ad2ant1 (cv y) A (norm) fveq2 (<_) (1) breq1d (cv y) A T fveq2 (norm) fveq2d (` (norm) (` T A)) eqeq2d anbi12d (` (norm) (` T A)) eqid (br (` (norm) A) (<_) (1)) biantru syl6bbr (H~) rcla4ev (norm) (` T A) fvex (cv x) (` (norm) (` T A)) (` (norm) (` T (cv y))) eqeq1 (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv elab sylibr (:--> T (H~) (H~)) 3adant1 sylanc T x y nmopvalt (e. A (H~)) (br (` (norm) A) (<_) (1)) 3ad2ant1 breqtrrd)) thm (nmopleubt ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (T x) (T y) (T z) (T w)) () (-> (/\/\ (:--> T (H~) (H~)) (e. A (RR*)) (A.e. x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (br (` (norm) (` T (cv x))) (<_) A)))) (br (` (normop) T) (<_) A)) (T y z nmopvalt (e. A (RR*)) (A.e. x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (br (` (norm) (` T (cv x))) (<_) A))) 3ad2ant1 ({|} y (E.e. z (H~) (/\ (br (` (norm) (cv z)) (<_) (1)) (= (cv y) (` (norm) (` T (cv z))))))) A w supxrleub T y z nmopsetret ressxr (:--> T (H~) (H~)) a1i sstrd (e. A (RR*)) (A.e. x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (br (` (norm) (` T (cv x))) (<_) A))) 3ad2ant1 (:--> T (H~) (H~)) (e. A (RR*)) (A.e. x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (br (` (norm) (` T (cv x))) (<_) A))) 3simp2 (cv x) (cv z) (norm) fveq2 (<_) (1) breq1d (cv x) (cv z) T fveq2 (norm) fveq2d (<_) A breq1d imbi12d (H~) rcla4cv (cv w) (` (norm) (` T (cv z))) (<_) A breq1 biimprcd syl8 imp4a r19.23adv w visset (cv y) (cv w) (` (norm) (` T (cv z))) eqeq1 (br (` (norm) (cv z)) (<_) (1)) anbi2d z (H~) rexbidv elab syl5ib r19.21aiv (:--> T (H~) (H~)) (e. A (RR*)) 3ad2ant3 syl3anc eqbrtrd)) thm (nmopge0t () () (-> (:--> T (H~) (H~)) (br (0) (<_) (` (normop) T))) (ax-hv0cl T (H~) (H~) (0v) ffvrn mpan2 (` T (0v)) normge0t syl ax-hv0cl norm0 0re ax1re lt01 ltlei eqbrtr T (0v) nmoplbt mp3an23 0re (0) rexrt ax-mp (0) (` (norm) (` T (0v))) (` (normop) T) xrletrt mp3an1 ax-hv0cl T (H~) (H~) (0v) ffvrn mpan2 (` T (0v)) normclt (` (norm) (` T (0v))) rexrt 3syl T nmopxrt sylanc mp2and)) thm (nmopgt0t () () (-> (:--> T (H~) (H~)) (<-> (=/= (` (normop) T) (0)) (br (0) (<) (` (normop) T)))) (0re (0) rexrt ax-mp (0) (` (normop) T) xrleltnet mp3an1 (0) (` (normop) T) df-ne syl6bbr T nmopxrt T nmopge0t sylanc (` (normop) T) (0) necom syl6rbbr)) thm (nmopgt0tOLD () () (-> (:--> T (H~) (H~)) (<-> (-. (= (` (normop) T) (0))) (br (0) (<) (` (normop) T)))) (0re (0) rexrt ax-mp (0) (` (normop) T) xrleltnet mp3an1 T nmopxrt T nmopge0t sylanc (` (normop) T) (0) eqcom negbii syl6rbbr)) thm (cnopct ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (B w) (B x) (B y) (B z) (T x) (T y) (T z) (T w)) () (-> (/\ (/\ (e. T (ConOp)) (e. A (H~))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (RR) (/\ (br (0) (<) (cv x)) (A.e. y (H~) (-> (br (` (norm) (opr (cv y) (-v) A)) (<) (cv x)) (br (` (norm) (opr (` T (cv y)) (-v) (` T A))) (<) B)))))) (T z w x y elcnopt pm3.27bd (cv z) A (cv y) (-v) opreq2 (norm) fveq2d (<) (cv x) breq1d (cv z) A T fveq2 (` T (cv y)) (-v) opreq2d (norm) fveq2d (<) (cv w) breq1d imbi12d y (H~) ralbidv (br (0) (<) (cv x)) anbi2d x (RR) rexbidv (br (0) (<) (cv w)) imbi2d w (RR) ralbidv (H~) rcla4cv (cv w) B (0) (<) breq2 (cv w) B (` (norm) (opr (` T (cv y)) (-v) (` T A))) (<) breq2 (br (` (norm) (opr (cv y) (-v) A)) (<) (cv x)) imbi2d y (H~) ralbidv (br (0) (<) (cv x)) anbi2d x (RR) rexbidv imbi12d (RR) rcla4cv syl6 syl imp43)) thm (lnoplt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (T x) (T y) (T z)) () (-> (/\ (/\ (e. T (LinOp)) (e. A (CC))) (/\ (e. B (H~)) (e. C (H~)))) (= (` T (opr (opr A (.s) B) (+v) C)) (opr (opr A (.s) (` T B)) (+v) (` T C)))) ((cv x) A (.s) (cv y) opreq1 (+v) (cv z) opreq1d T fveq2d (cv x) A (.s) (` T (cv y)) opreq1 (+v) (` T (cv z)) opreq1d eqeq12d (cv y) B A (.s) opreq2 (+v) (cv z) opreq1d T fveq2d (cv y) B T fveq2 A (.s) opreq2d (+v) (` T (cv z)) opreq1d eqeq12d (cv z) C (opr A (.s) B) (+v) opreq2 T fveq2d (cv z) C T fveq2 (opr A (.s) (` T B)) (+v) opreq2d eqeq12d (CC) (H~) (H~) rcla43v T x y z ellnopt pm3.27bd syl5 3expb impcom anassrs)) thm (unopt ((x y) (A x) (A y) (B x) (B y) (T x) (T y)) () (-> (/\/\ (e. T (UniOp)) (e. A (H~)) (e. B (H~))) (= (opr (` T A) (.i) (` T B)) (opr A (.i) B))) (T x y elunopt pm3.27bd (e. A (H~)) (e. B (H~)) 3ad2ant1 (cv x) A T fveq2 (.i) (` T (cv y)) opreq1d (cv x) A (.i) (cv y) opreq1 eqeq12d (cv y) B T fveq2 (` T A) (.i) opreq2d (cv y) B A (.i) opreq2 eqeq12d (H~) (H~) rcla42v (e. T (UniOp)) 3adant1 mpd)) thm (unopf1ot ((x y) (T x) (T y)) () (-> (e. T (UniOp)) (:-1-1-onto-> T (H~) (H~))) (T x y elunopt pm3.26bd T (H~) (H~) fof syl T (cv x) (cv x) unopt (e. T (UniOp)) (e. (cv x) (H~)) pm3.26 (e. T (UniOp)) (e. (cv x) (H~)) pm3.27 (e. T (UniOp)) (e. (cv x) (H~)) pm3.27 syl3anc (e. (cv y) (H~)) 3adant3 T (cv y) (cv y) unopt (e. T (UniOp)) (e. (cv y) (H~)) pm3.26 (e. T (UniOp)) (e. (cv y) (H~)) pm3.27 (e. T (UniOp)) (e. (cv y) (H~)) pm3.27 syl3anc (e. (cv x) (H~)) 3adant2 (+) opreq12d T (cv x) (cv y) unopt T (cv y) (cv x) unopt 3com23 (+) opreq12d (-) opreq12d 3expb T (H~) (H~) (cv x) ffvrn T (H~) (H~) (cv y) ffvrn anim12i anandis T x y elunopt pm3.26bd T (H~) (H~) fof syl sylan (` T (cv x)) (` T (cv y)) normlem9at syl (cv x) (cv y) normlem9at (e. T (UniOp)) adantl 3eqtr4rd (0) eqeq1d (cv x) (cv y) hvsubclt (opr (cv x) (-v) (cv y)) his6t syl (cv x) (cv y) hvsubeq0t bitrd (e. T (UniOp)) adantl T (H~) (H~) (cv x) ffvrn T (H~) (H~) (cv y) ffvrn anim12i anandis T x y elunopt pm3.26bd T (H~) (H~) fof syl sylan (` T (cv x)) (` T (cv y)) hvsubclt (opr (` T (cv x)) (-v) (` T (cv y))) his6t syl (` T (cv x)) (` T (cv y)) hvsubeq0t bitrd syl 3bitr3rd biimpd ex r19.21aivv jca T (H~) (H~) x y f1fv sylibr T x y elunopt pm3.26bd jca T (H~) (H~) df-f1o sylibr)) thm (unopnormt () () (-> (/\ (e. T (UniOp)) (e. A (H~))) (= (` (norm) (` T A)) (` (norm) A))) (T A A unopt (e. T (UniOp)) (e. A (H~)) pm3.26 (e. T (UniOp)) (e. A (H~)) pm3.27 (e. T (UniOp)) (e. A (H~)) pm3.27 syl3anc T (H~) (H~) A ffvrn T unopf1ot T (H~) (H~) f1of syl sylan (` T A) normsqt syl A normsqt (e. T (UniOp)) adantl 3eqtr4d (` (norm) (` T A)) (` (norm) A) sq11t T (H~) (H~) A ffvrn T unopf1ot T (H~) (H~) f1of syl sylan (` T A) normclt syl T (H~) (H~) A ffvrn T unopf1ot T (H~) (H~) f1of syl sylan (` T A) normge0t syl jca A normclt (e. T (UniOp)) adantl A normge0t (e. T (UniOp)) adantl jca sylanc mpbid)) thm (cnvunopt ((x y) (T x) (T y)) () (-> (e. T (UniOp)) (e. (`' T) (UniOp))) (T unopf1ot T (H~) (H~) f1ocnv (`' T) (H~) (H~) f1ofo syl syl T (` (`' T) (cv x)) (` (`' T) (cv y)) unopt (e. T (UniOp)) (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) pm3.26 (`' T) (H~) (H~) (cv x) ffvrn T unopf1ot T (H~) (H~) f1ocnv (`' T) (H~) (H~) f1ofo syl syl (`' T) (H~) (H~) fof syl sylan (e. (cv y) (H~)) adantrr (`' T) (H~) (H~) (cv y) ffvrn T unopf1ot T (H~) (H~) f1ocnv (`' T) (H~) (H~) f1ofo syl syl (`' T) (H~) (H~) fof syl sylan (e. (cv x) (H~)) adantrl syl3anc T (H~) (H~) (cv x) f1ocnvfv2 (e. (cv y) (H~)) adantrr T (H~) (H~) (cv y) f1ocnvfv2 (e. (cv x) (H~)) adantrl (.i) opreq12d T unopf1ot sylan eqtr3d ex r19.21aivv jca (`' T) x y elunopt sylibr)) thm (unopadjt () () (-> (/\/\ (e. T (UniOp)) (e. A (H~)) (e. B (H~))) (= (opr (` T A) (.i) B) (opr A (.i) (` (`' T) B)))) (T (H~) (H~) B f1ocnvfv2 T unopf1ot sylan (e. A (H~)) 3adant2 (` T A) (.i) opreq2d T A (` (`' T) B) unopt (e. T (UniOp)) (e. A (H~)) (e. B (H~)) 3simp1 (e. T (UniOp)) (e. A (H~)) (e. B (H~)) 3simp2 (`' T) (H~) (H~) B ffvrn T unopf1ot T (H~) (H~) f1ocnv (`' T) (H~) (H~) f1of 3syl sylan (e. A (H~)) 3adant2 syl3anc eqtr3d)) thm (unoplint ((w x) (w y) (w z) (x y) (x z) (y z) (T x) (T y) (T z) (T w)) () (-> (e. T (UniOp)) (e. T (LinOp))) (T unopf1ot T (H~) (H~) f1of syl T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) (cv w) unopadjt (e. T (UniOp)) (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) pm3.26 (e. (cv z) (H~)) (e. (cv w) (H~)) ad2antrr (opr (cv x) (.s) (cv y)) (cv z) ax-hvaddcl (cv x) (cv y) ax-hvmulcl sylan (e. T (UniOp)) adantll (e. (cv w) (H~)) adantr (/\ (/\ (e. T (UniOp)) (/\ (e. (cv x) (CC)) (e. (cv y) (H~)))) (e. (cv z) (H~))) (e. (cv w) (H~)) pm3.27 syl3anc (cv x) (cv y) (cv z) (` (`' T) (cv w)) hiassdit (e. (cv x) (CC)) (e. (cv y) (H~)) pm3.26 (e. T (UniOp)) adantl (e. (cv z) (H~)) (e. (cv w) (H~)) ad2antrr (e. (cv x) (CC)) (e. (cv y) (H~)) pm3.27 (e. T (UniOp)) adantl (e. (cv z) (H~)) (e. (cv w) (H~)) ad2antrr jca (/\ (e. T (UniOp)) (/\ (e. (cv x) (CC)) (e. (cv y) (H~)))) (e. (cv z) (H~)) pm3.27 (e. (cv w) (H~)) adantr (`' T) (H~) (H~) (cv w) ffvrn T cnvunopt (`' T) unopf1ot (`' T) (H~) (H~) f1of 3syl sylan (e. (cv z) (H~)) adantlr (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) adantllr jca sylanc (cv x) (` T (cv y)) (` T (cv z)) (cv w) hiassdit (e. (cv x) (CC)) (e. (cv y) (H~)) pm3.26 (e. T (UniOp)) adantl (e. (cv z) (H~)) (e. (cv w) (H~)) ad2antrr T (H~) (H~) (cv y) ffvrn T unopf1ot T (H~) (H~) f1of syl sylan (e. (cv x) (CC)) adantrl (e. (cv z) (H~)) (e. (cv w) (H~)) ad2antrr jca T (H~) (H~) (cv z) ffvrn T unopf1ot T (H~) (H~) f1of syl sylan (e. (cv w) (H~)) adantr (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) adantllr (/\ (/\ (e. T (UniOp)) (/\ (e. (cv x) (CC)) (e. (cv y) (H~)))) (e. (cv z) (H~))) (e. (cv w) (H~)) pm3.27 jca sylanc T (cv y) (cv w) unopadjt 3expa (cv x) (x.) opreq2d (e. (cv x) (CC)) adantlrl (e. (cv z) (H~)) adantlr T (cv z) (cv w) unopadjt 3expa (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) adantllr (+) opreq12d eqtr2d 3eqtrd r19.21aiva (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))) w hial2eqt T (H~) (H~) (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) ffvrn (opr (cv x) (.s) (cv y)) (cv z) ax-hvaddcl (cv x) (cv y) ax-hvmulcl sylan sylan2 anassrs (opr (cv x) (.s) (` T (cv y))) (` T (cv z)) ax-hvaddcl (cv x) (` T (cv y)) ax-hvmulcl T (H~) (H~) (cv y) ffvrn sylan2 an1s (e. (cv z) (H~)) adantr T (H~) (H~) (cv z) ffvrn (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) adantlr sylanc sylanc T unopf1ot T (H~) (H~) f1of syl sylanl1 mpbid r19.21aiva ex r19.21aivv jca T x y z ellnopt sylibr)) thm (hmopt ((x y) (A x) (A y) (B x) (B y) (T x) (T y)) () (-> (/\/\ (e. T (HrmOp)) (e. A (H~)) (e. B (H~))) (= (opr A (.i) (` T B)) (opr (` T A) (.i) B))) (T x y elhmopt pm3.27bd (e. A (H~)) (e. B (H~)) 3ad2ant1 (cv x) A (.i) (` T (cv y)) opreq1 (cv x) A T fveq2 (.i) (cv y) opreq1d eqeq12d (cv y) B T fveq2 A (.i) opreq2d (cv y) B (` T A) (.i) opreq2 eqeq12d (H~) (H~) rcla42v (e. T (HrmOp)) 3adant1 mpd)) thm (hmopret () () (-> (/\ (e. T (HrmOp)) (e. A (H~))) (e. (opr (` T A) (.i) A) (RR))) (T A A hmopt (e. T (HrmOp)) (e. A (H~)) pm3.26 (e. T (HrmOp)) (e. A (H~)) pm3.27 (e. T (HrmOp)) (e. A (H~)) pm3.27 syl3anc eqcomd (` T A) A hiret T (H~) (H~) A ffvrn T hmopft sylan (e. T (HrmOp)) (e. A (H~)) pm3.27 sylanc mpbird)) thm (nmfnlbt ((x y) (A x) (A y) (T x) (T y)) () (-> (/\/\ (:--> T (H~) (CC)) (e. A (H~)) (br (` (norm) A) (<_) (1))) (br (` (abs) (` T A)) (<_) (` (normfn) T))) (({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y))))))) (` (abs) (` T A)) supxrub T x y nmfnsetret ressxr (:--> T (H~) (CC)) a1i sstrd (e. A (H~)) (br (` (norm) A) (<_) (1)) 3ad2ant1 (cv y) A (norm) fveq2 (<_) (1) breq1d (cv y) A T fveq2 (abs) fveq2d (` (abs) (` T A)) eqeq2d anbi12d (` (abs) (` T A)) eqid (br (` (norm) A) (<_) (1)) biantru syl6bbr (H~) rcla4ev (abs) (` T A) fvex (cv x) (` (abs) (` T A)) (` (abs) (` T (cv y))) eqeq1 (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv elab sylibr (:--> T (H~) (CC)) 3adant1 sylanc T x y nmfnvalt (e. A (H~)) (br (` (norm) A) (<_) (1)) 3ad2ant1 breqtrrd)) thm (nmfnleubt ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (T x) (T y) (T z) (T w)) () (-> (/\/\ (:--> T (H~) (CC)) (e. A (RR*)) (A.e. x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (br (` (abs) (` T (cv x))) (<_) A)))) (br (` (normfn) T) (<_) A)) (T y z nmfnvalt (e. A (RR*)) (A.e. x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (br (` (abs) (` T (cv x))) (<_) A))) 3ad2ant1 ({|} y (E.e. z (H~) (/\ (br (` (norm) (cv z)) (<_) (1)) (= (cv y) (` (abs) (` T (cv z))))))) A w supxrleub T y z nmfnsetret ressxr (:--> T (H~) (CC)) a1i sstrd (e. A (RR*)) (A.e. x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (br (` (abs) (` T (cv x))) (<_) A))) 3ad2ant1 (:--> T (H~) (CC)) (e. A (RR*)) (A.e. x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (br (` (abs) (` T (cv x))) (<_) A))) 3simp2 (cv x) (cv z) (norm) fveq2 (<_) (1) breq1d (cv x) (cv z) T fveq2 (abs) fveq2d (<_) A breq1d imbi12d (H~) rcla4cv (cv w) (` (abs) (` T (cv z))) (<_) A breq1 biimprcd syl8 imp4a r19.23adv w visset (cv y) (cv w) (` (abs) (` T (cv z))) eqeq1 (br (` (norm) (cv z)) (<_) (1)) anbi2d z (H~) rexbidv elab syl5ib r19.21aiv (:--> T (H~) (CC)) (e. A (RR*)) 3ad2ant3 syl3anc eqbrtrd)) thm (elnlfnt ((A x) (T x)) () (-> (/\ (:--> T (H~) (CC)) (e. A (H~))) (<-> (e. A (` (null) T)) (= (` T A) (0)))) (T x nlfnvalt A eleq2d (cv x) A T fveq2 (0) eqeq1d (H~) elrab syl6bb (e. A (H~)) (= (` T A) (0)) ibar bicomd sylan9bb)) thm (elnlfn2t ((A x) (T x)) () (-> (/\ (:--> T (H~) (CC)) (e. A (` (null) T))) (= (` T A) (0))) ((:--> T (H~) (CC)) (e. A (` (null) T)) pm3.27 T x nlfnvalt x (H~) (= (` T (cv x)) (0)) ssrab2 (:--> T (H~) (CC)) a1i eqsstrd A sseld imp T A elnlfnt syldan mpbid)) thm (cnfnct ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (B w) (B x) (B y) (B z) (T x) (T y) (T z) (T w)) () (-> (/\ (/\ (e. T (ConFn)) (e. A (H~))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (RR) (/\ (br (0) (<) (cv x)) (A.e. y (H~) (-> (br (` (norm) (opr (cv y) (-v) A)) (<) (cv x)) (br (` (abs) (opr (` T (cv y)) (-) (` T A))) (<) B)))))) (T z w x y elcnfnt pm3.27bd (cv z) A (cv y) (-v) opreq2 (norm) fveq2d (<) (cv x) breq1d (cv z) A T fveq2 (` T (cv y)) (-) opreq2d (abs) fveq2d (<) (cv w) breq1d imbi12d y (H~) ralbidv (br (0) (<) (cv x)) anbi2d x (RR) rexbidv (br (0) (<) (cv w)) imbi2d w (RR) ralbidv (H~) rcla4cv (cv w) B (0) (<) breq2 (cv w) B (` (abs) (opr (` T (cv y)) (-) (` T A))) (<) breq2 (br (` (norm) (opr (cv y) (-v) A)) (<) (cv x)) imbi2d y (H~) ralbidv (br (0) (<) (cv x)) anbi2d x (RR) rexbidv imbi12d (RR) rcla4cv syl6 syl imp43)) thm (lnfnlt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (T x) (T y) (T z)) () (-> (/\ (/\ (e. T (LinFn)) (e. A (CC))) (/\ (e. B (H~)) (e. C (H~)))) (= (` T (opr (opr A (.s) B) (+v) C)) (opr (opr A (x.) (` T B)) (+) (` T C)))) ((cv x) A (.s) (cv y) opreq1 (+v) (cv z) opreq1d T fveq2d (cv x) A (x.) (` T (cv y)) opreq1 (+) (` T (cv z)) opreq1d eqeq12d (cv y) B A (.s) opreq2 (+v) (cv z) opreq1d T fveq2d (cv y) B T fveq2 A (x.) opreq2d (+) (` T (cv z)) opreq1d eqeq12d (cv z) C (opr A (.s) B) (+v) opreq2 T fveq2d (cv z) C T fveq2 (opr A (x.) (` T B)) (+) opreq2d eqeq12d (CC) (H~) (H~) rcla43v T x y z ellnfnt pm3.27bd syl5 3expb impcom anassrs)) thm (adjclt () () (-> (/\ (e. T (dom (adj))) (e. A (H~))) (e. (` (` (adj) T) A) (H~))) ((` (adj) T) (H~) (H~) A ffvrn T dmadjrnt (` (adj) T) dmadjopt syl sylan)) thm (adjt ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (B w) (B x) (B y) (B z) (T x) (T y) (T z) (T w)) () (-> (/\/\ (e. T (dom (adj))) (e. A (H~)) (e. B (H~))) (= (opr A (.i) (` T B)) (opr (` (` (adj) T) A) (.i) B))) ((cv x) A (.i) (` T (cv y)) opreq1 (cv x) A (` (adj) T) fveq2 (.i) (cv y) opreq1d eqeq12d (cv y) B T fveq2 A (.i) opreq2d (cv y) B (` (` (adj) T) A) (.i) opreq2 eqeq12d (H~) (H~) rcla42v funadj (adj) T funfvop mpan z w x y dfadj2 syl6eleq (adj) T fvex (cv z) T (H~) (H~) feq1 (cv z) T (cv y) fveq1 (cv x) (.i) opreq2d (opr (` (cv w) (cv x)) (.i) (cv y)) eqeq1d x (H~) y (H~) 2ralbidv (:--> (cv w) (H~) (H~)) 3anbi13d (cv w) (` (adj) T) (H~) (H~) feq1 (cv w) (` (adj) T) (cv x) fveq1 (.i) (cv y) opreq1d (opr (cv x) (.i) (` T (cv y))) eqeq2d x (H~) y (H~) 2ralbidv (:--> T (H~) (H~)) 3anbi23d (dom (adj)) (V) opelopabg mpan2 mpbid 3simp3d syl5com 3impib)) thm (adj2tOLD () () (-> (/\/\ (e. T (dom (adj))) (e. A (H~)) (e. B (H~))) (= (opr A (.i) (` (` (adj) T) B)) (opr (` T A) (.i) B))) (T B A adjt B (` T A) ax-his1 (e. T (dom (adj))) (e. B (H~)) (e. A (H~)) 3simp2 T (H~) (H~) A ffvrn T dmadjopt sylan (e. B (H~)) 3adant2 sylanc (` (` (adj) T) B) A ax-his1 (` (adj) T) (H~) (H~) B ffvrn T dmadjrnt (` (adj) T) dmadjopt syl sylan (e. A (H~)) 3adant3 (e. T (dom (adj))) (e. B (H~)) (e. A (H~)) 3simp3 sylanc 3eqtr3rd (opr A (.i) (` (` (adj) T) B)) (opr (` T A) (.i) B) cj11t A (` (` (adj) T) B) ax-hicl (e. T (dom (adj))) (e. B (H~)) (e. A (H~)) 3simp3 (` (adj) T) (H~) (H~) B ffvrn T dmadjrnt (` (adj) T) dmadjopt syl sylan (e. A (H~)) 3adant3 sylanc (` T A) B ax-hicl T (H~) (H~) A ffvrn T dmadjopt sylan (e. B (H~)) 3adant2 (e. T (dom (adj))) (e. B (H~)) (e. A (H~)) 3simp2 sylanc sylanc mpbid 3com23)) thm (adj2t () () (-> (/\/\ (e. T (dom (adj))) (e. A (H~)) (e. B (H~))) (= (opr (` T A) (.i) B) (opr A (.i) (` (` (adj) T) B)))) (T B A adjt B (` T A) ax-his1 (e. T (dom (adj))) (e. B (H~)) (e. A (H~)) 3simp2 T (H~) (H~) A ffvrn T dmadjopt sylan (e. B (H~)) 3adant2 sylanc (` (` (adj) T) B) A ax-his1 T B adjclt (e. A (H~)) 3adant3 (e. T (dom (adj))) (e. B (H~)) (e. A (H~)) 3simp3 sylanc 3eqtr3d (opr (` T A) (.i) B) (opr A (.i) (` (` (adj) T) B)) cj11t (` T A) B ax-hicl T (H~) (H~) A ffvrn T dmadjopt sylan (e. B (H~)) 3adant2 (e. T (dom (adj))) (e. B (H~)) (e. A (H~)) 3simp2 sylanc A (` (` (adj) T) B) ax-hicl (e. T (dom (adj))) (e. B (H~)) (e. A (H~)) 3simp3 T B adjclt (e. A (H~)) 3adant3 sylanc sylanc mpbid 3com23)) thm (adjeqt ((w x) (w y) (w z) (x y) (x z) (y z) (S w) (S x) (S y) (S z) (T x) (T y) (T z) (T w)) () (-> (/\/\ (:--> T (H~) (H~)) (:--> S (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` S (cv y))))))) (= (` (adj) T) S)) (funadj S (V) (adj) T funopfvg mpan2 ax-hilex S (H~) (H~) (V) fex mpan2 (:--> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` S (cv y)))))) 3ad2ant2 (cv z) T (H~) (H~) feq1 (cv z) T (cv x) fveq1 (.i) (cv y) opreq1d (opr (cv x) (.i) (` (cv w) (cv y))) eqeq1d x (H~) y (H~) 2ralbidv (:--> (cv w) (H~) (H~)) 3anbi13d (cv w) S (H~) (H~) feq1 (cv w) S (cv y) fveq1 (cv x) (.i) opreq2d (opr (` T (cv x)) (.i) (cv y)) eqeq2d x (H~) y (H~) 2ralbidv (:--> T (H~) (H~)) 3anbi23d (V) (V) opelopabg ax-hilex T (H~) (H~) (V) fex mpan2 ax-hilex S (H~) (H~) (V) fex mpan2 syl2an z w x y df-adj (<,> T S) eleq2i syl5bb (:--> T (H~) (H~)) (:--> S (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` S (cv y)))))) df-3an baibr bitr4d biimpar 3impa sylc)) thm (adjadjt ((x y) (T x) (T y)) () (-> (e. T (dom (adj))) (= (` (adj) (` (adj) T)) T)) ((` (adj) T) (cv x) (cv y) adjt T dmadjrnt syl3an1 T (cv x) (cv y) adj2tOLD eqtr3d 3expb ex r19.21aivv (` (adj) (` (adj) T)) T x y hoeq1t T dmadjrnt (` (adj) T) dmadjrnt (` (adj) (` (adj) T)) dmadjopt 3syl T dmadjopt sylanc mpbid)) thm (adjvalvalt ((w x) (w y) (A w) (x y) (A x) (A y) (T x) (T y) (T w)) () (-> (/\ (e. T (dom (adj))) (e. A (H~))) (= (` (` (adj) T) A) (U. ({e.|} w (H~) (A.e. x (H~) (= (opr (` T (cv x)) (.i) A) (opr (cv x) (.i) (cv w)))))))) (T (cv x) A adj2t 3com23 3expa r19.21aiva (cv w) (` (` (adj) T) A) (cv x) (.i) opreq2 (opr (` T (cv x)) (.i) A) eqeq2d x (H~) ralbidv (H~) reuuni2 T A adjclt (cv w) (` (` (adj) T) A) (cv x) (.i) opreq2 (opr (` T (cv x)) (.i) A) eqeq2d x (H~) ralbidv (H~) rcla4ev T A adjclt T (cv x) A adj2t 3com23 3expa r19.21aiva sylanc (cv w) (cv y) x hial2eq2t x (H~) (= (opr (` T (cv x)) (.i) A) (opr (cv x) (.i) (cv w))) (= (opr (` T (cv x)) (.i) A) (opr (cv x) (.i) (cv y))) r19.26 (opr (` T (cv x)) (.i) A) (opr (cv x) (.i) (cv w)) (opr (cv x) (.i) (cv y)) eqtr2t x (H~) r19.20si sylbir syl5bi rgen2 jctir (cv w) (cv y) (cv x) (.i) opreq2 (opr (` T (cv x)) (.i) A) eqeq2d x (H~) ralbidv (H~) reu4 sylibr sylanc mpbid eqcomd)) thm (unopadj2t ((x y) (T x) (T y)) () (-> (e. T (UniOp)) (= (` (adj) T) (`' T))) (T (`' T) x y adjeqt T unoplint T lnopft syl T cnvunopt (`' T) unoplint (`' T) lnopft 3syl T (cv x) (cv y) unopadjt 3expb ex r19.21aivv syl3anc)) thm (hmopadjt ((x y) (T x) (T y)) () (-> (e. T (HrmOp)) (= (` (adj) T) T)) (T T x y adjeqt T hmopft T hmopft T (cv x) (cv y) hmopt eqcomd 3expb ex r19.21aivv syl3anc)) thm (hmdmadjt () () (-> (e. T (HrmOp)) (e. T (dom (adj)))) (T hmopft T hon0 syl T hmopadjt ({/}) eqeq1d mtbird T (adj) ndmfv nsyl2)) thm (hmoplint ((w x) (w y) (w z) (x y) (x z) (y z) (T x) (T y) (T z) (T w)) () (-> (e. T (HrmOp)) (e. T (LinOp))) (T hmopft T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) (cv w) hmopt eqcomd (e. T (HrmOp)) (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) pm3.26 (e. (cv z) (H~)) (e. (cv w) (H~)) ad2antrr (opr (cv x) (.s) (cv y)) (cv z) ax-hvaddcl (cv x) (cv y) ax-hvmulcl sylan (e. T (HrmOp)) adantll (e. (cv w) (H~)) adantr (/\ (/\ (e. T (HrmOp)) (/\ (e. (cv x) (CC)) (e. (cv y) (H~)))) (e. (cv z) (H~))) (e. (cv w) (H~)) pm3.27 syl3anc (cv x) (cv y) (cv z) (` T (cv w)) hiassdit (e. (cv x) (CC)) (e. (cv y) (H~)) pm3.26 (e. T (HrmOp)) adantl (e. (cv z) (H~)) (e. (cv w) (H~)) ad2antrr (e. (cv x) (CC)) (e. (cv y) (H~)) pm3.27 (e. T (HrmOp)) adantl (e. (cv z) (H~)) (e. (cv w) (H~)) ad2antrr jca (/\ (e. T (HrmOp)) (/\ (e. (cv x) (CC)) (e. (cv y) (H~)))) (e. (cv z) (H~)) pm3.27 (e. (cv w) (H~)) adantr T (H~) (H~) (cv w) ffvrn T hmopft sylan (e. (cv z) (H~)) adantlr (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) adantllr jca sylanc (cv x) (` T (cv y)) (` T (cv z)) (cv w) hiassdit (e. (cv x) (CC)) (e. (cv y) (H~)) pm3.26 (e. T (HrmOp)) adantl (e. (cv z) (H~)) (e. (cv w) (H~)) ad2antrr T (H~) (H~) (cv y) ffvrn T hmopft sylan (e. (cv x) (CC)) adantrl (e. (cv z) (H~)) (e. (cv w) (H~)) ad2antrr jca T (H~) (H~) (cv z) ffvrn T hmopft sylan (e. (cv w) (H~)) adantr (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) adantllr (/\ (/\ (e. T (HrmOp)) (/\ (e. (cv x) (CC)) (e. (cv y) (H~)))) (e. (cv z) (H~))) (e. (cv w) (H~)) pm3.27 jca sylanc T (cv y) (cv w) hmopt eqcomd 3expa (cv x) (x.) opreq2d (e. (cv x) (CC)) adantlrl (e. (cv z) (H~)) adantlr T (cv z) (cv w) hmopt eqcomd 3expa (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) adantllr (+) opreq12d eqtr2d 3eqtrd r19.21aiva (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))) w hial2eqt T (H~) (H~) (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) ffvrn (opr (cv x) (.s) (cv y)) (cv z) ax-hvaddcl (cv x) (cv y) ax-hvmulcl sylan sylan2 anassrs (opr (cv x) (.s) (` T (cv y))) (` T (cv z)) ax-hvaddcl (cv x) (` T (cv y)) ax-hvmulcl T (H~) (H~) (cv y) ffvrn sylan2 an1s (e. (cv z) (H~)) adantr T (H~) (H~) (cv z) ffvrn (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) adantlr sylanc sylanc T hmopft sylanl1 mpbid r19.21aiva ex r19.21aivv jca T x y z ellnopt sylibr)) thm (bravalt ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (H~)) (= (` (bra) A) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (cv x) (.i) A)))))) ((cv z) A (cv x) (.i) opreq2 (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv z w x y df-bra ax-hilex x y (opr (cv x) (.i) A) funopabex2 fvopab4)) thm (bravalvalt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (H~)) (e. B (H~))) (= (` (` (bra) A) B) (opr B (.i) A))) (A x y bravalt B fveq1d (cv x) B (.i) A opreq1 ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (cv x) (.i) A)))) eqid B (.i) A oprex fvopab4 sylan9eq)) thm (braaddt () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (` (` (bra) A) (opr B (+v) C)) (opr (` (` (bra) A) B) (+) (` (` (bra) A) C)))) (B C A ax-his2 3comr A (opr B (+v) C) bravalvalt B C ax-hvaddcl sylan2 3impb A B bravalvalt (e. C (H~)) 3adant3 A C bravalvalt (e. B (H~)) 3adant2 (+) opreq12d 3eqtr4d)) thm (bramult () () (-> (/\/\ (e. A (H~)) (e. B (CC)) (e. C (H~))) (= (` (` (bra) A) (opr B (.s) C)) (opr B (x.) (` (` (bra) A) C)))) (B C A ax-his3 3comr A (opr B (.s) C) bravalvalt B C ax-hvmulcl sylan2 3impb A C bravalvalt (e. B (CC)) 3adant2 B (x.) opreq2d 3eqtr4d)) thm (brafnt ((x y) (A x) (A y)) () (-> (e. A (H~)) (:--> (` (bra) A) (H~) (CC))) ((cv x) A ax-hicl ancoms r19.21aiva ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (cv x) (.i) A)))) eqid (CC) fopab2 sylib A x y bravalt (` (bra) A) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (cv x) (.i) A)))) (H~) (CC) feq1 syl mpbird)) thm (bralnfnt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (H~)) (e. (` (bra) A) (LinFn))) (A brafnt A (opr (cv x) (.s) (cv y)) (cv z) braaddt (e. A (H~)) (e. (cv x) (CC)) pm3.26 (/\ (e. (cv y) (H~)) (e. (cv z) (H~))) adantr (cv x) (cv y) ax-hvmulcl (e. (cv z) (H~)) adantrr (e. A (H~)) adantll (e. (cv y) (H~)) (e. (cv z) (H~)) pm3.27 (/\ (e. A (H~)) (e. (cv x) (CC))) adantl syl3anc A (cv x) (cv y) bramult 3expa (e. (cv z) (H~)) adantrr (+) (` (` (bra) A) (cv z)) opreq1d eqtrd ex r19.21aivv r19.21aiva jca (` (bra) A) x y z ellnfnt sylibr)) thm (braclt () () (-> (/\ (e. A (H~)) (e. B (H~))) (e. (` (` (bra) A) B) (CC))) ((` (bra) A) (H~) (CC) B ffvrn A brafnt sylan)) thm (bra0 ((x y)) () (= (` (bra) (0v)) (X. (H~) ({} (0)))) ((cv x) hi02t (cv y) eqeq2d pm5.32i x y opabbii ax-hv0cl (0v) x y bravalt ax-mp (H~) (0) x y fconstopab 3eqtr4)) thm (brafnmult ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CC)) (e. B (H~))) (= (` (bra) (opr A (.s) B)) (opr (` (*) A) (.fn) (` (bra) B)))) (A (cv x) B his5t 3com23 3expa B (cv x) bravalvalt (e. A (CC)) adantll (` (*) A) (x.) opreq2d eqtr4d (cv y) eqeq2d ex pm5.32d x y opabbidv A B ax-hvmulcl (opr A (.s) B) x y bravalt syl (` (*) A) (` (bra) B) x y hfmmvalt A cjclt B brafnt syl2an 3eqtr4d)) thm (kbvalt ((t w) (t x) (t y) (t z) (A t) (w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (B t) (B w) (B x) (B y) (B z)) () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr A (ketbra) B) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (opr (cv x) (.i) B) (.s) A)))))) (ax-hilex x y (opr (opr (cv x) (.i) B) (.s) A) funopabex2 (cv z) A (opr (cv x) (.i) (cv w)) (.s) opreq2 (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv (cv w) B (cv x) (.i) opreq2 (.s) A opreq1d (cv y) eqeq2d (e. (cv x) (H~)) anbi2d x y opabbidv z w t x y df-kb oprabval2)) thm (kbopt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (H~)) (e. B (H~))) (:--> (opr A (ketbra) B) (H~) (H~))) ((opr (cv x) (.i) B) A ax-hvmulcl (cv x) B ax-hicl sylan ancom31s r19.21aiva ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (opr (cv x) (.i) B) (.s) A)))) eqid (H~) fopab2 sylib A B x y kbvalt (opr A (ketbra) B) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (opr (cv x) (.i) B) (.s) A)))) (H~) (H~) feq1 syl mpbird)) thm (kbvalvalt ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (` (opr A (ketbra) B) C) (opr (opr C (.i) B) (.s) A))) (A B x y kbvalt C fveq1d (cv x) C (.i) B opreq1 (.s) A opreq1d ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (opr (cv x) (.i) B) (.s) A)))) eqid (opr C (.i) B) (.s) A oprex fvopab4 sylan9eq 3impa)) thm (kbmult ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (.s) B) (ketbra) C) (opr B (ketbra) (opr (` (*) A) (.s) C)))) ((opr (cv x) (.i) C) A B ax-hvmulass (cv x) C ax-hicl ancoms (e. B (H~)) adantll (e. A (CC)) 3adantl1 (e. A (CC)) (e. B (H~)) (e. C (H~)) 3simp1 (e. (cv x) (H~)) adantr (e. A (CC)) (e. B (H~)) (e. C (H~)) 3simp2 (e. (cv x) (H~)) adantr syl3anc (opr (cv x) (.i) C) A axmulcom (cv x) C ax-hicl sylan A (cv x) C his52t 3expb ancoms eqtr4d ancom31s (e. B (H~)) 3adantl2 (.s) B opreq1d eqtr3d (cv y) eqeq2d ex pm5.32d x y opabbidv (opr A (.s) B) C x y kbvalt A B ax-hvmulcl sylan 3impa B (opr (` (*) A) (.s) C) x y kbvalt (` (*) A) C ax-hvmulcl A cjclt sylan sylan2 3impb 3com12 3eqtr4d)) thm (kbpjt ((A x)) () (-> (/\ (e. A (H~)) (= (` (norm) A) (1))) (= (opr A (ketbra) A) (` (proj) (` (span) ({} A))))) ((` (norm) A) (1) (^) (2) opreq1 sq1 syl6eq (opr (cv x) (.i) A) (/) opreq2d (cv x) A ax-hicl ancoms (opr (cv x) (.i) A) div1t syl sylan9eqr an1rs (.s) A opreq1d A (cv x) pjspansnt (e. A (H~)) (= (` (norm) A) (1)) pm3.26 (e. (cv x) (H~)) adantr (/\ (e. A (H~)) (= (` (norm) A) (1))) (e. (cv x) (H~)) pm3.27 A normne0t ax1ne0 (` (norm) A) (1) (0) neeq1 mpbiri syl5bi imp (e. (cv x) (H~)) adantr syl3anc A A (cv x) kbvalvalt (e. A (H~)) (e. (cv x) (H~)) pm3.26 (e. A (H~)) (e. (cv x) (H~)) pm3.26 (e. A (H~)) (e. (cv x) (H~)) pm3.27 syl3anc (= (` (norm) A) (1)) adantlr 3eqtr4rd r19.21aiva (H~) eqid jctil (opr A (ketbra) A) (H~) (` (proj) (` (span) ({} A))) (H~) x eqfnfv A A kbopt anidms (opr A (ketbra) A) (H~) (H~) ffn syl A spansncht (` (span) ({} A)) pjfnt syl sylanc (= (` (norm) A) (1)) adantr mpbird)) thm (kbass1t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (` (opr A (ketbra) B) C) (opr (` (` (bra) B) C) (.s) A))) (A B C kbvalvalt B C bravalvalt (e. A (H~)) 3adant1 (.s) A opreq1d eqtr4d)) thm (kbass2t ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (` (` (bra) A) B) (.fn) (` (bra) C)) (o. (` (bra) A) (opr B (ketbra) C)))) ((` (` (bra) A) B) (` (bra) C) x y hfmmvalt A B braclt C brafnt syl2an 3impa (opr B (.i) A) (opr (cv x) (.i) C) axmulcom B A ax-hicl ancoms (e. C (H~)) 3adant3 (e. (cv x) (H~)) adantr (cv x) C ax-hicl ancoms (e. B (H~)) adantll (e. A (H~)) 3adantl1 sylanc (opr (cv x) (.i) C) B A ax-his3 (cv x) C ax-hicl ancoms (e. B (H~)) adantll (e. A (H~)) 3adantl1 (e. A (H~)) (e. B (H~)) (e. C (H~)) 3simp2 (e. (cv x) (H~)) adantr (e. A (H~)) (e. B (H~)) (e. C (H~)) 3simp1 (e. (cv x) (H~)) adantr syl3anc eqtr4d A B bravalvalt (e. C (H~)) 3adant3 (e. (cv x) (H~)) adantr C (cv x) bravalvalt (e. B (H~)) adantll (e. A (H~)) 3adantl1 (x.) opreq12d (opr (cv x) (.i) C) (.s) B oprex (opr (opr (cv x) (.i) C) (.s) B) (V) x (cv x) (.i) A csbopr1g ax-mp (opr (cv x) (.i) C) (.s) B oprex (opr (opr (cv x) (.i) C) (.s) B) (V) x csbvarg ax-mp (.i) A opreq1i eqtr (/\ (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (e. (cv x) (H~))) a1i 3eqtr4d (cv y) eqeq2d ex pm5.32d x y opabbidv A x y bravalt (opr B (ketbra) C) coeq1d B C x y kbvalt ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (cv x) (.i) A)))) coeq2d sylan9eq 3impb (opr (cv x) (.i) C) B ax-hvmulcl (cv x) C ax-hicl sylan ancom31s r19.21aiva ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (opr (cv x) (.i) C) (.s) B)))) eqid (opr (cv x) (.i) C) (.s) B oprex (H~) rnssopab sylib (cv x) (.i) A oprex (opr (cv x) (.i) C) (.s) B oprex ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (cv x) (.i) A)))) eqid ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (opr (cv x) (.i) C) (.s) B)))) eqid fopabco syl (e. A (H~)) 3adant1 eqtr2d 3eqtrd)) thm (eleigvect ((x y) (A x) (A y) (T x) (T y)) () (-> (:--> T (H~) (H~)) (<-> (e. A (` (eigvec) T)) (/\/\ (e. A (H~)) (=/= A (0v)) (E.e. x (CC) (= (` T A) (opr (cv x) (.s) A)))))) (T y x eigvecvalt A eleq2d (cv y) A (0v) neeq1 (cv y) A T fveq2 (cv y) A (cv x) (.s) opreq2 eqeq12d x (CC) rexbidv anbi12d (H~) elrab (e. A (H~)) (=/= A (0v)) (E.e. x (CC) (= (` T A) (opr (cv x) (.s) A))) 3anass bitr4 syl6bb)) thm (eleigvec2t ((A x) (T x)) () (-> (:--> T (H~) (H~)) (<-> (e. A (` (eigvec) T)) (/\/\ (e. A (H~)) (=/= A (0v)) (e. (` T A) (` (span) ({} A)))))) (T A x eleigvect A (` T A) x elspansnt (=/= A (0v)) adantr pm5.32i (e. A (H~)) (=/= A (0v)) (e. (` T A) (` (span) ({} A))) df-3an (e. A (H~)) (=/= A (0v)) (E.e. x (CC) (= (` T A) (opr (cv x) (.s) A))) df-3an 3bitr4 syl6bbr)) thm (eleigvecclt () () (-> (/\ (:--> T (H~) (H~)) (e. A (` (eigvec) T))) (e. A (H~))) (T A eleigvec2t biimpa 3simp1d)) thm (eigvalt ((x y) (A x) (A y) (T x) (T y)) () (-> (/\ (:--> T (H~) (H~)) (e. A (` (eigvec) T))) (= (` (` (eigval) T) A) (opr (opr (` T A) (.i) A) (/) (opr (` (norm) A) (^) (2))))) (T x y eigvalvalt A fveq1d (cv x) A T fveq2 (= (cv x) A) id (.i) opreq12d (cv x) A (norm) fveq2 (^) (2) opreq1d (/) opreq12d ({<,>|} x y (/\ (e. (cv x) (` (eigvec) T)) (= (cv y) (opr (opr (` T (cv x)) (.i) (cv x)) (/) (opr (` (norm) (cv x)) (^) (2)))))) eqid (opr (` T A) (.i) A) (/) (opr (` (norm) A) (^) (2)) oprex fvopab4 sylan9eq)) thm (eigvalclt ((A x) (T x)) () (-> (/\ (:--> T (H~) (H~)) (e. A (` (eigvec) T))) (e. (` (` (eigval) T) A) (CC))) (T A eigvalt (opr (` T A) (.i) A) (opr (` (norm) A) (^) (2)) divclt T A eleigvecclt (` T A) A ax-hicl T (H~) (H~) A ffvrn (:--> T (H~) (H~)) (e. A (H~)) pm3.27 sylanc syldan T A eleigvecclt A normclt recnd (` (norm) A) sqclt 3syl T A x eleigvect biimpa A normclt recnd (` (norm) A) sqne0t syl A normne0t bitr2d biimpa (E.e. x (CC) (= (` T A) (opr (cv x) (.s) A))) 3adant3 syl syl3anc eqeltrd)) thm (eigvect () () (-> (/\ (:--> T (H~) (H~)) (e. A (` (eigvec) T))) (/\ (= (` T A) (opr (` (` (eigval) T) A) (.s) A)) (=/= A (0v)))) (T A eigvalt (.s) A opreq1d T A eleigvec2t biimpa A (` T A) normcant syl eqtr2d T A eleigvec2t biimpa 3simp2d jca)) thm (eighmret () () (-> (/\ (e. T (HrmOp)) (e. A (` (eigvec) T))) (e. (` (` (eigval) T) A) (RR))) (A (` (` (eigval) T) A) T eigret biimpa T A eleigvecclt T A eigvalclt jca T A eigvect jca T hmopft sylan T A eleigvecclt T A eleigvecclt jca T hmopft sylan T A A hmopt 3expb syldan sylanc)) thm (eighmortht () () (-> (/\ (/\ (e. T (HrmOp)) (e. A (` (eigvec) T))) (/\ (e. B (` (eigvec) T)) (-. (= (` (` (eigval) T) A) (` (` (eigval) T) B))))) (= (opr A (.i) B) (0))) (A B (` (` (eigval) T) A) (` (` (eigval) T) B) T eigortht biimpa T A eleigvecclt T hmopft sylan (e. B (` (eigvec) T)) adantr T B eleigvecclt T hmopft sylan (e. A (` (eigvec) T)) adantlr jca T A eighmret recnd (e. B (` (eigvec) T)) adantr T B eighmret recnd (e. A (` (eigvec) T)) adantlr jca jca (-. (= (` (` (eigval) T) A) (` (` (eigval) T) B))) adantrr T A eigvect pm3.26d T hmopft sylan (e. B (` (eigvec) T)) adantr T B eigvect pm3.26d T hmopft sylan (e. A (` (eigvec) T)) adantlr jca (-. (= (` (` (eigval) T) A) (` (` (eigval) T) B))) adantrr T B eighmret (` (` (eigval) T) B) cjret syl (` (` (eigval) T) A) eqeq2d negbid biimpar anasss (e. A (` (eigvec) T)) adantlr jca jca T A B hmopt (e. T (HrmOp)) (e. A (` (eigvec) T)) pm3.26 (e. B (` (eigvec) T)) adantr T A eleigvecclt T hmopft sylan (e. B (` (eigvec) T)) adantr T B eleigvecclt T hmopft sylan (e. A (` (eigvec) T)) adantlr syl3anc (-. (= (` (` (eigval) T) A) (` (` (eigval) T) B))) adantrr sylanc)) thm (nmopneg ((x y) (T x) (T y)) ((nmopneg.1 (:--> T (H~) (H~)))) (= (` (normop) (opr (-u (1)) (.op) T)) (` (normop) T)) (1cn negcl nmopneg.1 (-u (1)) T (cv y) homvalt mp3an12 (norm) fveq2d nmopneg.1 (cv y) ffvrni (` T (cv y)) normnegt syl eqtrd (cv x) eqeq2d (br (` (norm) (cv y)) (<_) (1)) anbi2d rexbiia x abbii ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` (opr (-u (1)) (.op) T) (cv y))))))) ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) (RR*) (<) supeq1 ax-mp 1cn negcl nmopneg.1 (-u (1)) T homulclt mp2an (opr (-u (1)) (.op) T) x y nmopvalt ax-mp nmopneg.1 T x y nmopvalt ax-mp 3eqtr4)) thm (lnop0t () () (-> (e. T (LinOp)) (= (` T (0v)) (0v))) (1cn ax-hv0cl ax-hv0cl pm3.2i T (1) (0v) (0v) lnoplt mpan2 mpan2 1cn ax-hv0cl hvmulcl (opr (1) (.s) (0v)) ax-hvaddid ax-mp ax-hv0cl (0v) ax-hvmulid ax-mp eqtr T fveq2i syl5eqr T lnopft ax-hv0cl T (H~) (H~) (0v) ffvrn mpan2 syl (` T (0v)) ax-hvmulid syl (+v) (` T (0v)) opreq1d eqtrd (-v) (` T (0v)) opreq1d T lnopft ax-hv0cl T (H~) (H~) (0v) ffvrn mpan2 syl (` T (0v)) hvsubidt syl T lnopft ax-hv0cl T (H~) (H~) (0v) ffvrn mpan2 syl (` T (0v)) (` T (0v)) hvpncant anidms syl 3eqtr3rd)) thm (lnopmult () () (-> (/\/\ (e. T (LinOp)) (e. A (CC)) (e. B (H~))) (= (` T (opr A (.s) B)) (opr A (.s) (` T B)))) (ax-hv0cl T A B (0v) lnoplt mpanr2 3impa A B ax-hvmulcl (opr A (.s) B) ax-hvaddid syl (e. T (LinOp)) 3adant1 T fveq2d T lnop0t (opr A (.s) (` T B)) (+v) opreq2d (e. A (CC)) (e. B (H~)) 3ad2ant1 A (` T B) ax-hvmulcl T (H~) (H~) B ffvrn T lnopft sylan sylan2 3impb 3com12 (opr A (.s) (` T B)) ax-hvaddid syl eqtrd 3eqtr3d)) thm (lnopl () ((lnopl.1 (e. T (LinOp)))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr (opr A (.s) B) (+v) C)) (opr (opr A (.s) (` T B)) (+v) (` T C)))) (lnopl.1 T A B C lnoplt mpanl1 3impb)) thm (lnopf () ((lnopl.1 (e. T (LinOp)))) (:--> T (H~) (H~)) (lnopl.1 T lnopft ax-mp)) thm (lnop0 () ((lnopl.1 (e. T (LinOp)))) (= (` T (0v)) (0v)) (lnopl.1 T lnop0t ax-mp)) thm (lnopadd () ((lnopl.1 (e. T (LinOp)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` T (opr A (+v) B)) (opr (` T A) (+v) (` T B)))) (1cn lnopl.1 (1) A B lnopl mp3an1 A ax-hvmulid (+v) B opreq1d T fveq2d (e. B (H~)) adantr lnopl.1 lnopf A ffvrni (` T A) ax-hvmulid syl (e. B (H~)) adantr (+v) (` T B) opreq1d 3eqtr3d)) thm (lnopmul () ((lnopl.1 (e. T (LinOp)))) (-> (/\ (e. A (CC)) (e. B (H~))) (= (` T (opr A (.s) B)) (opr A (.s) (` T B)))) (lnopl.1 T A B lnopmult mp3an1)) thm (lnopaddmul () ((lnopl.1 (e. T (LinOp)))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr B (+v) (opr A (.s) C))) (opr (` T B) (+v) (opr A (.s) (` T C))))) (lnopl.1 B (opr A (.s) C) lnopadd A C ax-hvmulcl sylan2 3impb 3com12 lnopl.1 A C lnopmul (e. B (H~)) 3adant2 (` T B) (+v) opreq2d eqtrd)) thm (lnopsub () ((lnopl.1 (e. T (LinOp)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` T (opr A (-v) B)) (opr (` T A) (-v) (` T B)))) (1cn negcl lnopl.1 (-u (1)) A B lnopaddmul mp3an1 A B hvsubvalt T fveq2d (` T A) (` T B) hvsubvalt lnopl.1 lnopf A ffvrni lnopl.1 lnopf B ffvrni syl2an 3eqtr4d)) thm (lnopsubmul () ((lnopl.1 (e. T (LinOp)))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr B (-v) (opr A (.s) C))) (opr (` T B) (-v) (opr A (.s) (` T C))))) (lnopl.1 B (opr A (.s) C) lnopsub A C ax-hvmulcl sylan2 3impb 3com12 lnopl.1 A C lnopmul (` T B) (-v) opreq2d (e. B (H~)) 3adant2 eqtrd)) thm (lnopmulsub () ((lnopl.1 (e. T (LinOp)))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr (opr A (.s) B) (-v) C)) (opr (opr A (.s) (` T B)) (-v) (` T C)))) (lnopl.1 (opr A (.s) B) C lnopsub A B ax-hvmulcl sylan 3impa lnopl.1 A B lnopmul (e. C (H~)) 3adant3 (-v) (` T C) opreq1d eqtrd)) thm (homco2t ((A x) (T x) (U x)) () (-> (/\/\ (e. A (CC)) (e. T (LinOp)) (:--> U (H~) (H~))) (= (o. T (opr A (.op) U)) (opr A (.op) (o. T U)))) (T A (` U (cv x)) lnopmult U (H~) (H~) (cv x) ffvrn syl3an3 3exp exp4a com12 3imp1 T (opr A (.op) U) (H~) (H~) (cv x) fvco3 T (H~) (H~) ffun syl3an1 A U homulclt syl3an2 3exp exp3a com12 3imp1 A U (cv x) homvalt 3expa (:--> T (H~) (H~)) 3adantl2 T fveq2d eqtrd T lnopft syl3anl2 A (o. T U) (cv x) homvalt T (H~) (H~) U (H~) fco syl3an2 3expa ex 3impb imp T U (H~) (H~) (cv x) fvco3 T (H~) (H~) ffun syl3an1 3expa A (.s) opreq2d (e. A (CC)) 3adantl1 eqtrd T lnopft syl3anl2 3eqtr4d r19.21aiva (o. T (opr A (.op) U)) (opr A (.op) (o. T U)) x hoeqt T (H~) (H~) (opr A (.op) U) (H~) fco A U homulclt sylan2 3impb 3com12 A (o. T U) homulclt T (H~) (H~) U (H~) fco sylan2 3impb sylanc T lnopft syl3an2 mpbid)) thm (idunop ((x y)) () (e. (|` (I) (H~)) (UniOp)) ((|` (I) (H~)) x y elunopt (H~) f1oi (|` (I) (H~)) (H~) (H~) f1ofo ax-mp (cv x) (H~) fvresi (cv y) (H~) fvresi (.i) opreqan12d rgen2 mpbir2an)) thm (0cnop ((x y) (x z) (w x) (y z) (w y) (w z)) () (e. (X. (H~) (0H)) (ConOp)) ((X. (H~) (0H)) x y z w elcnopt ho0fOLD ax1re (br (0) (<) (cv y)) a1i (/\ (e. (cv x) (H~)) (e. (cv y) (RR))) a1i lt01 (br (0) (<) (cv y)) a1i (e. (cv x) (H~)) a1i (cv w) ho0valtOLD (cv x) ho0valtOLD (-v) opreqan12rd ax-hv0cl (0v) hvsubidt ax-mp syl6eq (norm) fveq2d norm0 syl6eq (<) (cv y) breq1d biimprd (br (` (norm) (opr (cv w) (-v) (cv x))) (<) (1)) a1dd ex com23 r19.21adv jcad (e. (cv y) (RR)) adantr jcad (cv z) (1) (0) (<) breq2 (cv z) (1) (` (norm) (opr (cv w) (-v) (cv x))) (<) breq2 (br (` (norm) (opr (` (X. (H~) (0H)) (cv w)) (-v) (` (X. (H~) (0H)) (cv x)))) (<) (cv y)) imbi1d w (H~) ralbidv anbi12d (RR) rcla4ev syl6 rgen2a mpbir2an)) thm (0cnfn ((x y) (x z) (w x) (y z) (w y) (w z)) () (e. (X. (H~) ({} (0))) (ConFn)) ((X. (H~) ({} (0))) x y z w elcnfnt 0cn elisseti (H~) fconst 0cn (0) (CC) snssi ax-mp (X. (H~) ({} (0))) (H~) ({} (0)) (CC) fss mp2an ax1re (br (0) (<) (cv y)) a1i (/\ (e. (cv x) (H~)) (e. (cv y) (RR))) a1i lt01 (br (0) (<) (cv y)) a1i (e. (cv x) (H~)) a1i 0cn elisseti (cv w) (H~) fvconst2 0cn elisseti (cv x) (H~) fvconst2 (-) opreqan12rd 0cn subid syl6eq (abs) fveq2d abs0 syl6eq (<) (cv y) breq1d biimprd (br (` (norm) (opr (cv w) (-v) (cv x))) (<) (1)) a1dd ex com23 r19.21adv jcad (e. (cv y) (RR)) adantr jcad (cv z) (1) (0) (<) breq2 (cv z) (1) (` (norm) (opr (cv w) (-v) (cv x))) (<) breq2 (br (` (abs) (opr (` (X. (H~) ({} (0))) (cv w)) (-) (` (X. (H~) ({} (0))) (cv x)))) (<) (cv y)) imbi1d w (H~) ralbidv anbi12d (RR) rcla4ev syl6 rgen2a mpbir2an)) thm (idcnop ((x y) (x z) (w x) (y z) (w y) (w z)) () (e. (|` (I) (H~)) (ConOp)) ((|` (I) (H~)) x y z w elcnopt (H~) f1oi (|` (I) (H~)) (H~) (H~) f1of ax-mp (cv w) (H~) fvresi (cv x) (H~) fvresi (-v) opreqan12rd (norm) fveq2d (<) (cv y) breq1d biimprd r19.21aiva (br (0) (<) (cv y)) a1d ancld (e. (cv y) (RR)) adantr (e. (cv x) (H~)) (e. (cv y) (RR)) pm3.27 jctild (cv z) (cv y) (0) (<) breq2 (cv z) (cv y) (` (norm) (opr (cv w) (-v) (cv x))) (<) breq2 (br (` (norm) (opr (` (|` (I) (H~)) (cv w)) (-v) (` (|` (I) (H~)) (cv x)))) (<) (cv y)) imbi1d w (H~) ralbidv anbi12d (RR) rcla4ev syl6 rgen2a mpbir2an)) thm (idhmop ((x y)) () (e. (Iop) (HrmOp)) ((Iop) x y elhmopt hoif (Iop) (H~) (H~) f1of ax-mp (cv x) hoivalt eqcomd (cv y) hoivalt (.i) opreqan12d rgen2 mpbir2an)) thm (idhmopOLD ((x y)) () (e. (|` (I) (H~)) (HrmOp)) ((|` (I) (H~)) x y elhmopt (H~) f1oi (|` (I) (H~)) (H~) (H~) f1of ax-mp (cv x) (H~) fvresi eqcomd (cv y) (H~) fvresi (.i) opreqan12d rgen2 mpbir2an)) thm (0hmop ((x y)) () (e. (0op) (HrmOp)) ((0op) x y elhmopt ho0f (cv y) ho0valt (cv x) (.i) opreq2d (cv x) hi02t sylan9eqr (cv x) ho0valt (.i) (cv y) opreq1d (cv y) hi01t sylan9eq eqtr4d rgen2 mpbir2an)) thm (0hmopOLD ((x y)) () (e. (X. (H~) (0H)) (HrmOp)) ((X. (H~) (0H)) x y elhmopt ho0fOLD (cv y) ho0valtOLD (cv x) (.i) opreq2d (cv x) hi02t sylan9eqr (cv x) ho0valtOLD (.i) (cv y) opreq1d (cv y) hi01t sylan9eq eqtr4d rgen2 mpbir2an)) thm (0lnop () () (e. (0op) (LinOp)) (0hmop (0op) hmoplint ax-mp)) thm (0lnopOLD () () (e. (X. (H~) (0H)) (LinOp)) (0hmopOLD (X. (H~) (0H)) hmoplint ax-mp)) thm (0lnfn ((x y) (x z) (y z)) () (e. (X. (H~) ({} (0))) (LinFn)) ((X. (H~) ({} (0))) x y z ellnfnt 0cn elisseti (H~) fconst 0cn (0) (CC) snssi ax-mp (X. (H~) ({} (0))) (H~) ({} (0)) (CC) fss mp2an (opr (cv x) (.s) (cv y)) (cv z) ax-hvaddcl (cv x) (cv y) ax-hvmulcl sylan 0cn elisseti (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) (H~) fvconst2 syl 0cn elisseti (cv y) (H~) fvconst2 (cv x) (x.) opreq2d (cv x) mul01t sylan9eqr 0cn elisseti (cv z) (H~) fvconst2 (+) opreqan12d 0cn addid1 syl6eq eqtr4d r19.21aiva rgen2a mpbir2an)) thm (nmop0 ((x y)) () (= (` (normop) (0op)) (0)) (ho0f (0op) x y nmopvalt ax-mp y (H~) (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (0)) r19.41v (cv y) ho0valt (norm) fveq2d norm0 syl6eq (cv x) eqeq2d (br (` (norm) (cv y)) (<_) (1)) anbi2d rexbiia ax-hv0cl 0re ax1re lt01 ltlei (cv y) (0v) (norm) fveq2 norm0 syl6eq (<_) (1) breq1d (H~) rcla4ev mp2an (= (cv x) (0)) biantrur 3bitr4 x abbii (0) x df-sn eqtr4 ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` (0op) (cv y))))))) ({} (0)) (RR*) (<) supeq1 ax-mp 0re (0) rexrt ax-mp xrltso (0) supsn ax-mp 3eqtr)) thm (nmop0OLD ((x y)) () (= (` (normop) (X. (H~) (0H))) (0)) (0lnopOLD lnopf (X. (H~) (0H)) x y nmopvalt ax-mp y (H~) (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (0)) r19.41v (cv y) ho0valtOLD (norm) fveq2d norm0 syl6eq (cv x) eqeq2d (br (` (norm) (cv y)) (<_) (1)) anbi2d rexbiia ax-hv0cl 0re ax1re lt01 ltlei (cv y) (0v) (norm) fveq2 norm0 syl6eq (<_) (1) breq1d (H~) rcla4ev mp2an (= (cv x) (0)) biantrur 3bitr4 x abbii (0) x df-sn eqtr4 ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` (X. (H~) (0H)) (cv y))))))) ({} (0)) (RR*) (<) supeq1 ax-mp 0re (0) rexrt ax-mp xrltso (0) supsn ax-mp 3eqtr)) thm (nmfn0 ((x y)) () (= (` (normfn) (X. (H~) ({} (0)))) (0)) (0lnfn (X. (H~) ({} (0))) lnfnft ax-mp (X. (H~) ({} (0))) x y nmfnvalt ax-mp y (H~) (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (0)) r19.41v 0re elisseti (cv y) (H~) fvconst2 (abs) fveq2d abs0 syl6eq (cv x) eqeq2d (br (` (norm) (cv y)) (<_) (1)) anbi2d rexbiia ax-hv0cl 0re ax1re lt01 ltlei (cv y) (0v) (norm) fveq2 norm0 syl6eq (<_) (1) breq1d (H~) rcla4ev mp2an (= (cv x) (0)) biantrur 3bitr4 x abbii (0) x df-sn eqtr4 ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` (X. (H~) ({} (0))) (cv y))))))) ({} (0)) (RR*) (<) supeq1 ax-mp 0re (0) rexrt ax-mp xrltso (0) supsn ax-mp 3eqtr)) thm (hoddi ((R x) (S x) (T x)) ((hoddi.1 (e. R (LinOp))) (hoddi.2 (:--> S (H~) (H~))) (hoddi.3 (:--> T (H~) (H~)))) (= (o. R (opr S (-op) T)) (opr (o. R S) (-op) (o. R T))) (hoddi.1 (` S (cv x)) (` T (cv x)) lnopsub hoddi.2 (cv x) ffvrni hoddi.3 (cv x) ffvrni sylanc hoddi.2 hoddi.3 S T (cv x) hodvaltOLD mpanl12 R fveq2d hoddi.1 lnopf hoddi.2 (cv x) hoco hoddi.1 lnopf hoddi.3 (cv x) hoco (-v) opreq12d 3eqtr4d hoddi.1 lnopf hoddi.2 hoddi.3 hosubcl (cv x) hoco hoddi.1 lnopf hoddi.2 hocof hoddi.1 lnopf hoddi.3 hocof (o. R S) (o. R T) (cv x) hodvaltOLD mpanl12 3eqtr4d rgen hoddi.1 lnopf hoddi.2 hoddi.3 hosubcl hocof hoddi.1 lnopf hoddi.2 hocof hoddi.1 lnopf hoddi.3 hocof hosubcl x hoeq mpbi)) thm (hoddit () () (-> (/\/\ (e. R (LinOp)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (o. R (opr S (-op) T)) (opr (o. R S) (-op) (o. R T)))) (R (if (e. R (LinOp)) R (0op)) (opr S (-op) T) coeq1 R (if (e. R (LinOp)) R (0op)) S coeq1 R (if (e. R (LinOp)) R (0op)) T coeq1 (-op) opreq12d eqeq12d S (if (:--> S (H~) (H~)) S (0op)) (-op) T opreq1 (opr S (-op) T) (opr (if (:--> S (H~) (H~)) S (0op)) (-op) T) (if (e. R (LinOp)) R (0op)) coeq2 syl S (if (:--> S (H~) (H~)) S (0op)) (if (e. R (LinOp)) R (0op)) coeq2 (-op) (o. (if (e. R (LinOp)) R (0op)) T) opreq1d eqeq12d T (if (:--> T (H~) (H~)) T (0op)) (if (:--> S (H~) (H~)) S (0op)) (-op) opreq2 (opr (if (:--> S (H~) (H~)) S (0op)) (-op) T) (opr (if (:--> S (H~) (H~)) S (0op)) (-op) (if (:--> T (H~) (H~)) T (0op))) (if (e. R (LinOp)) R (0op)) coeq2 syl T (if (:--> T (H~) (H~)) T (0op)) (if (e. R (LinOp)) R (0op)) coeq2 (o. (if (e. R (LinOp)) R (0op)) (if (:--> S (H~) (H~)) S (0op))) (-op) opreq2d eqeq12d 0lnop R elimel ho0f S elimf ho0f T elimf hoddi dedth3h)) thm (nmop0h () () (-> (/\ (= (H~) (0H)) (:--> T (H~) (H~))) (= (` (normop) T) (0))) (df-ch0 (H~) eqeq2i (H~) ({} (0v)) T (H~) feq3 sylbi ax-hv0cl elisseti T (H~) fconst2 df-ch0 (0H) ({} (0v)) (H~) xpeq2 ax-mp T eqeq2i bitr4 syl6bb biimpa (normop) fveq2d nmop0OLD syl6eq)) thm (idlnop () () (e. (|` (I) (H~)) (LinOp)) (idunop (|` (I) (H~)) unoplint ax-mp)) thm (0bdop () () (e. (X. (H~) (0H)) (BndLinOp)) ((X. (H~) (0H)) elbdopt 0lnopOLD nmop0OLD 0re (0) ltpnft ax-mp eqbrtr mpbir2an)) thm (adj0 ((x y)) () (= (` (adj) (X. (H~) (0H))) (X. (H~) (0H))) (0lnopOLD lnopf 0lnopOLD lnopf (cv x) ho0valtOLD (.i) (cv y) opreq1d (cv y) hi01t sylan9eq (cv y) ho0valtOLD (cv x) (.i) opreq2d (cv x) hi02t sylan9eqr eqtr4d rgen2 (X. (H~) (0H)) (X. (H~) (0H)) x y adjeqt mp3an)) thm (nmlnop0 ((x y) (x z) (T x) (y z) (T y) (T z)) ((nmlnop0.1 (e. T (LinOp)))) (<-> (= (` (normop) T) (0)) (= T (0op))) ((` (norm) (cv x)) recne0t (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc (opr (1) (/) (` (norm) (cv x))) (0) df-ne sylib (e. (cv x) (H~)) (-. (= (` T (cv x)) (0v))) pm3.27 jca (opr (1) (/) (` (norm) (cv x))) (` T (cv x)) hvmul0ort (` (norm) (cv x)) recclt (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc nmlnop0.1 lnopf (cv x) ffvrni (-. (= (` T (cv x)) (0v))) adantr sylanc negbid (= (opr (1) (/) (` (norm) (cv x))) (0)) (= (` T (cv x)) (0v)) ioran syl6bb mpbird (opr (1) (/) (` (norm) (cv x))) (` T (cv x)) ax-hvmulcl (` (norm) (cv x)) recclt (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc nmlnop0.1 lnopf (cv x) ffvrni (-. (= (` T (cv x)) (0v))) adantr sylanc (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x))) normgt0tOLD syl mpbid (= (` (normop) T) (0)) adantll (cv z) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (cv x)) (norm) fveq2 (<_) (1) breq1d (cv z) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (cv x)) T fveq2 (norm) fveq2d (` (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x)))) eqeq2d anbi12d (H~) rcla4ev (opr (1) (/) (` (norm) (cv x))) (cv x) ax-hvmulcl (` (norm) (cv x)) recclt (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc (e. (cv x) (H~)) (-. (= (` T (cv x)) (0v))) pm3.26 sylanc (` (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (cv x))) (1) eqlet (cv x) norm1t (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq con3i (cv x) (0v) df-ne sylibr sylan2 ax1re syl6eqel (cv x) norm1t (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq con3i (cv x) (0v) df-ne sylibr sylan2 sylanc nmlnop0.1 (opr (1) (/) (` (norm) (cv x))) (cv x) lnopmul (` (norm) (cv x)) recclt (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc (e. (cv x) (H~)) (-. (= (` T (cv x)) (0v))) pm3.26 sylanc eqcomd (norm) fveq2d jca sylanc (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x))) fvex (cv y) (` (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x)))) (` (norm) (` T (cv z))) eqeq1 (br (` (norm) (cv z)) (<_) (1)) anbi2d z (H~) rexbidv elab sylibr nmlnop0.1 lnopf T y z nmopsetret ax-mp ressxr sstri ({|} y (E.e. z (H~) (/\ (br (` (norm) (cv z)) (<_) (1)) (= (cv y) (` (norm) (` T (cv z))))))) (` (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x)))) supxrub mpan syl (= (` (normop) T) (0)) adantll nmlnop0.1 lnopf T y z nmopvalt ax-mp (0) eqeq1i biimp (e. (cv x) (H~)) (-. (= (` T (cv x)) (0v))) ad2antrr breqtrd (opr (1) (/) (` (norm) (cv x))) (` T (cv x)) ax-hvmulcl (` (norm) (cv x)) recclt (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc nmlnop0.1 lnopf (cv x) ffvrni (-. (= (` T (cv x)) (0v))) adantr sylanc (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x))) normclt syl 0re (` (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x)))) (0) lenltt mpan2 syl (= (` (normop) T) (0)) adantll mpbid condan (cv x) ho0valt (= (` (normop) T) (0)) adantl eqtr4d r19.21aiva (H~) eqid jctil nmlnop0.1 lnopf T (H~) (H~) ffn ax-mp ho0f (0op) (H~) (H~) ffn ax-mp T (H~) (0op) (H~) x eqfnfv mp2an sylibr T (0op) (normop) fveq2 nmop0 syl6eq impbi)) thm (nmlnop0OLD ((x y) (x z) (T x) (y z) (T y) (T z)) ((nmlnop0.1 (e. T (LinOp)))) (<-> (= (` (normop) T) (0)) (= T (X. (H~) (0H)))) ((` (norm) (cv x)) recne0t (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc (opr (1) (/) (` (norm) (cv x))) (0) df-ne sylib (e. (cv x) (H~)) (-. (= (` T (cv x)) (0v))) pm3.27 jca (opr (1) (/) (` (norm) (cv x))) (` T (cv x)) hvmul0ort (` (norm) (cv x)) recclt (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc nmlnop0.1 lnopf (cv x) ffvrni (-. (= (` T (cv x)) (0v))) adantr sylanc negbid (= (opr (1) (/) (` (norm) (cv x))) (0)) (= (` T (cv x)) (0v)) ioran syl6bb mpbird (opr (1) (/) (` (norm) (cv x))) (` T (cv x)) ax-hvmulcl (` (norm) (cv x)) recclt (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc nmlnop0.1 lnopf (cv x) ffvrni (-. (= (` T (cv x)) (0v))) adantr sylanc (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x))) normgt0tOLD syl mpbid (= (` (normop) T) (0)) adantll (cv z) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (cv x)) (norm) fveq2 (<_) (1) breq1d (cv z) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (cv x)) T fveq2 (norm) fveq2d (` (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x)))) eqeq2d anbi12d (H~) rcla4ev (opr (1) (/) (` (norm) (cv x))) (cv x) ax-hvmulcl (` (norm) (cv x)) recclt (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc (e. (cv x) (H~)) (-. (= (` T (cv x)) (0v))) pm3.26 sylanc (` (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (cv x))) (1) eqlet (cv x) norm1t (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq con3i (cv x) (0v) df-ne sylibr sylan2 ax1re syl6eqel (cv x) norm1t (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq con3i (cv x) (0v) df-ne sylibr sylan2 sylanc nmlnop0.1 (opr (1) (/) (` (norm) (cv x))) (cv x) lnopmul (` (norm) (cv x)) recclt (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc (e. (cv x) (H~)) (-. (= (` T (cv x)) (0v))) pm3.26 sylanc eqcomd (norm) fveq2d jca sylanc (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x))) fvex (cv y) (` (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x)))) (` (norm) (` T (cv z))) eqeq1 (br (` (norm) (cv z)) (<_) (1)) anbi2d z (H~) rexbidv elab sylibr nmlnop0.1 lnopf T y z nmopsetret ax-mp ressxr sstri ({|} y (E.e. z (H~) (/\ (br (` (norm) (cv z)) (<_) (1)) (= (cv y) (` (norm) (` T (cv z))))))) (` (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x)))) supxrub mpan syl (= (` (normop) T) (0)) adantll nmlnop0.1 lnopf T y z nmopvalt ax-mp (0) eqeq1i biimp (e. (cv x) (H~)) (-. (= (` T (cv x)) (0v))) ad2antrr breqtrd (opr (1) (/) (` (norm) (cv x))) (` T (cv x)) ax-hvmulcl (` (norm) (cv x)) recclt (cv x) normclt recnd (-. (= (` T (cv x)) (0v))) adantr (cv x) norm-it (cv x) (0v) T fveq2 nmlnop0.1 lnop0 syl6eq syl6bi con3d imp (` (norm) (cv x)) (0) df-ne sylibr sylanc nmlnop0.1 lnopf (cv x) ffvrni (-. (= (` T (cv x)) (0v))) adantr sylanc (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x))) normclt syl 0re (` (norm) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (` T (cv x)))) (0) lenltt mpan2 syl (= (` (normop) T) (0)) adantll mpbid condan (cv x) ho0valtOLD (= (` (normop) T) (0)) adantl eqtr4d r19.21aiva (H~) eqid jctil nmlnop0.1 lnopf T (H~) (H~) ffn ax-mp 0lnopOLD lnopf (X. (H~) (0H)) (H~) (H~) ffn ax-mp T (H~) (X. (H~) (0H)) (H~) x eqfnfv mp2an sylibr T (X. (H~) (0H)) (normop) fveq2 nmop0OLD syl6eq impbi)) thm (nmlnopgt0 () ((nmlnop0.1 (e. T (LinOp)))) (<-> (=/= T (0op)) (br (0) (<) (` (normop) T))) (nmlnop0.1 nmlnop0 eqneqi nmlnop0.1 lnopf T nmopgt0t ax-mp bitr3)) thm (nmlnop0t () () (-> (e. T (LinOp)) (<-> (= (` (normop) T) (0)) (= T (0op)))) (T (if (e. T (LinOp)) T (0op)) (normop) fveq2 (0) eqeq1d T (if (e. T (LinOp)) T (0op)) (0op) eqeq1 bibi12d 0lnop T elimel nmlnop0 dedth)) thm (nmlnopne0t () () (-> (e. T (LinOp)) (<-> (=/= (` (normop) T) (0)) (=/= T (0op)))) (T nmlnop0t eqneqd)) thm (lnopm ((x y) (x z) (A x) (y z) (A y) (A z) (T x) (T y) (T z)) ((lnopm.1 (e. T (LinOp)))) (-> (e. A (CC)) (e. (opr A (.op) T) (LinOp))) (lnopm.1 lnopf A T homulclt mpan2 lnopm.1 (cv x) (cv y) (cv z) lnopl 3expa A (.s) opreq2d (e. A (CC)) adantl A (opr (cv x) (.s) (` T (cv y))) (` T (cv z)) ax-hvdistr1 (e. A (CC)) id (cv x) (` T (cv y)) ax-hvmulcl lnopm.1 lnopf (cv y) ffvrni sylan2 lnopm.1 lnopf (cv z) ffvrni syl3an 3expb eqtrd lnopm.1 lnopf A T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) homvalt mp3an2 (opr (cv x) (.s) (cv y)) (cv z) ax-hvaddcl (cv x) (cv y) ax-hvmulcl sylan sylan2 lnopm.1 lnopf A T (cv y) homvalt mp3an2 (e. (cv x) (CC)) adantrl (cv x) (.s) opreq2d A (cv x) (` T (cv y)) hvmulcomt lnopm.1 lnopf (cv y) ffvrni syl3an3 3expb eqtr4d lnopm.1 lnopf A T (cv z) homvalt mp3an2 (+v) opreqan12d anandis 3eqtr4d exp32 r19.21adv r19.21aivv jca (opr A (.op) T) x y z ellnopt sylibr)) thm (lnophs ((x y) (x z) (S x) (y z) (S y) (S z) (T x) (T y) (T z)) ((lnopco.1 (e. S (LinOp))) (lnopco.2 (e. T (LinOp)))) (e. (opr S (+op) T) (LinOp)) ((opr S (+op) T) x y z ellnopt lnopco.1 lnopf lnopco.2 lnopf hoaddcl lnopco.1 (opr (cv x) (.s) (cv y)) (cv z) lnopadd lnopco.2 (opr (cv x) (.s) (cv y)) (cv z) lnopadd (+v) opreq12d (cv x) (cv y) ax-hvmulcl sylan (` S (opr (cv x) (.s) (cv y))) (` S (cv z)) (` T (opr (cv x) (.s) (cv y))) (` T (cv z)) hvadd4t (cv x) (cv y) ax-hvmulcl lnopco.1 lnopf (opr (cv x) (.s) (cv y)) ffvrni syl lnopco.1 lnopf (cv z) ffvrni anim12i (cv x) (cv y) ax-hvmulcl lnopco.2 lnopf (opr (cv x) (.s) (cv y)) ffvrni syl lnopco.2 lnopf (cv z) ffvrni anim12i sylanc eqtrd (opr (cv x) (.s) (cv y)) (cv z) ax-hvaddcl (cv x) (cv y) ax-hvmulcl sylan lnopco.1 lnopf lnopco.2 lnopf S T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) hosvaltOLD mpanl12 syl (cv x) (` S (cv y)) (` T (cv y)) ax-hvdistr1 3expb lnopco.1 lnopf (cv y) ffvrni lnopco.2 lnopf (cv y) ffvrni jca sylan2 lnopco.1 lnopf lnopco.2 lnopf S T (cv y) hosvaltOLD mpanl12 (cv x) (.s) opreq2d (e. (cv x) (CC)) adantl lnopco.1 (cv x) (cv y) lnopmul lnopco.2 (cv x) (cv y) lnopmul (+v) opreq12d 3eqtr4d lnopco.1 lnopf lnopco.2 lnopf S T (cv z) hosvaltOLD mpanl12 (+v) opreqan12d 3eqtr4d r19.21aiva rgen2a mpbir2an)) thm (lnophd () ((lnopco.1 (e. S (LinOp))) (lnopco.2 (e. T (LinOp)))) (e. (opr S (-op) T) (LinOp)) (lnopco.1 lnopf lnopco.2 lnopf honegsubOLD lnopco.1 1cn negcl lnopco.2 (-u (1)) lnopm ax-mp lnophs eqeltr)) thm (lnopco ((x y) (x z) (S x) (y z) (S y) (S z) (T x) (T y) (T z)) ((lnopco.1 (e. S (LinOp))) (lnopco.2 (e. T (LinOp)))) (e. (o. S T) (LinOp)) ((o. S T) x y z ellnopt lnopco.1 lnopf lnopco.2 lnopf hocof lnopco.2 (cv x) (cv y) (cv z) lnopl S fveq2d lnopco.1 (cv x) (` T (cv y)) (` T (cv z)) lnopl (e. (cv x) (CC)) id lnopco.2 lnopf (cv y) ffvrni lnopco.2 lnopf (cv z) ffvrni syl3an eqtrd 3expa (opr (cv x) (.s) (cv y)) (cv z) ax-hvaddcl (cv x) (cv y) ax-hvmulcl sylan lnopco.2 lnopf T (H~) (H~) fdm ax-mp syl6eleqr lnopco.1 lnopf S (H~) (H~) ffun ax-mp lnopco.2 lnopf T (H~) (H~) ffun ax-mp S T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) fvco mp3an12 syl lnopco.1 lnopf lnopco.2 lnopf (cv y) hoco (cv x) (.s) opreq2d (e. (cv x) (CC)) adantl lnopco.1 lnopf lnopco.2 lnopf (cv z) hoco (+v) opreqan12d 3eqtr4d r19.21aiva rgen2a mpbir2an)) thm (lnopco0 ((S x) (T x)) ((lnopco.1 (e. S (LinOp))) (lnopco.2 (e. T (LinOp)))) (-> (= (` (normop) T) (0)) (= (` (normop) (o. S T)) (0))) (T (X. (H~) (0H)) S coeq2 lnopco.1 0lnopOLD lnopco lnopf (o. S (X. (H~) (0H))) (H~) (H~) ffn ax-mp 0lnopOLD lnopf (X. (H~) (0H)) (H~) (H~) ffn ax-mp (o. S (X. (H~) (0H))) (H~) (X. (H~) (0H)) (H~) x eqfnfv mp2an (H~) eqid (cv x) ho0valtOLD S fveq2d lnopco.1 lnop0 syl6eq lnopco.1 lnopf 0lnopOLD lnopf (cv x) hoco (cv x) ho0valtOLD 3eqtr4d rgen mpbir2an syl6eq lnopco.2 nmlnop0OLD lnopco.1 lnopco.2 lnopco nmlnop0OLD 3imtr4)) thm (lnopeq0lem1 () ((lnopeq0.1 (e. T (LinOp))) (lnopeq0lem1.2 (e. A (H~))) (lnopeq0lem1.3 (e. B (H~)))) (= (opr (` T A) (.i) B) (opr (opr (opr (opr (` T (opr A (+v) B)) (.i) (opr A (+v) B)) (-) (opr (` T (opr A (-v) B)) (.i) (opr A (-v) B))) (+) (opr (i) (x.) (opr (opr (` T (opr A (+v) (opr (i) (.s) B))) (.i) (opr A (+v) (opr (i) (.s) B))) (-) (opr (` T (opr A (-v) (opr (i) (.s) B))) (.i) (opr A (-v) (opr (i) (.s) B)))))) (/) (4))) (lnopeq0lem1.2 lnopeq0.1 lnopf A ffvrni ax-mp lnopeq0lem1.3 lnopeq0lem1.3 lnopeq0.1 lnopf B ffvrni ax-mp lnopeq0lem1.2 polid2 lnopeq0lem1.2 lnopeq0lem1.3 lnopeq0.1 A B lnopadd mp2an (.i) (opr A (+v) B) opreq1i lnopeq0lem1.2 lnopeq0lem1.3 lnopeq0.1 A B lnopsub mp2an (.i) (opr A (-v) B) opreq1i (-) opreq12i axicn lnopeq0lem1.2 lnopeq0lem1.3 lnopeq0.1 (i) A B lnopaddmul mp3an (.i) (opr A (+v) (opr (i) (.s) B)) opreq1i axicn lnopeq0lem1.2 lnopeq0lem1.3 lnopeq0.1 (i) A B lnopsubmul mp3an (.i) (opr A (-v) (opr (i) (.s) B)) opreq1i (-) opreq12i (i) (x.) opreq2i (+) opreq12i (/) (4) opreq1i eqtr4)) thm (lnopeq0lem2 () ((lnopeq0.1 (e. T (LinOp)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` T A) (.i) B) (opr (opr (opr (opr (` T (opr A (+v) B)) (.i) (opr A (+v) B)) (-) (opr (` T (opr A (-v) B)) (.i) (opr A (-v) B))) (+) (opr (i) (x.) (opr (opr (` T (opr A (+v) (opr (i) (.s) B))) (.i) (opr A (+v) (opr (i) (.s) B))) (-) (opr (` T (opr A (-v) (opr (i) (.s) B))) (.i) (opr A (-v) (opr (i) (.s) B)))))) (/) (4)))) (A (if (e. A (H~)) A (0v)) T fveq2 (.i) B opreq1d A (if (e. A (H~)) A (0v)) (+v) B opreq1 T fveq2d A (if (e. A (H~)) A (0v)) (+v) B opreq1 (.i) opreq12d A (if (e. A (H~)) A (0v)) (-v) B opreq1 T fveq2d A (if (e. A (H~)) A (0v)) (-v) B opreq1 (.i) opreq12d (-) opreq12d A (if (e. A (H~)) A (0v)) (+v) (opr (i) (.s) B) opreq1 T fveq2d A (if (e. A (H~)) A (0v)) (+v) (opr (i) (.s) B) opreq1 (.i) opreq12d A (if (e. A (H~)) A (0v)) (-v) (opr (i) (.s) B) opreq1 T fveq2d A (if (e. A (H~)) A (0v)) (-v) (opr (i) (.s) B) opreq1 (.i) opreq12d (-) opreq12d (i) (x.) opreq2d (+) opreq12d (/) (4) opreq1d eqeq12d B (if (e. B (H~)) B (0v)) (` T (if (e. A (H~)) A (0v))) (.i) opreq2 B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (+v) opreq2 T fveq2d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (+v) opreq2 (.i) opreq12d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 T fveq2d B (if (e. B (H~)) B (0v)) (if (e. A (H~)) A (0v)) (-v) opreq2 (.i) opreq12d (-) opreq12d B (if (e. B (H~)) B (0v)) (i) (.s) opreq2 (if (e. A (H~)) A (0v)) (+v) opreq2d T fveq2d B (if (e. B (H~)) B (0v)) (i) (.s) opreq2 (if (e. A (H~)) A (0v)) (+v) opreq2d (.i) opreq12d B (if (e. B (H~)) B (0v)) (i) (.s) opreq2 (if (e. A (H~)) A (0v)) (-v) opreq2d T fveq2d B (if (e. B (H~)) B (0v)) (i) (.s) opreq2 (if (e. A (H~)) A (0v)) (-v) opreq2d (.i) opreq12d (-) opreq12d (i) (x.) opreq2d (+) opreq12d (/) (4) opreq1d eqeq12d lnopeq0.1 ax-hv0cl A elimel ax-hv0cl B elimel lnopeq0lem1 dedth2h)) thm (lnopeq0 ((x y) (x z) (T x) (y z) (T y) (T z)) ((lnopeq0.1 (e. T (LinOp)))) (<-> (A.e. x (H~) (= (opr (` T (cv x)) (.i) (cv x)) (0))) (= T (X. (H~) (0H)))) (lnopeq0.1 (cv y) (cv z) lnopeq0lem2 (A.e. x (H~) (= (opr (` T (cv x)) (.i) (cv x)) (0))) adantl (cv x) (opr (cv y) (+v) (cv z)) T fveq2 (= (cv x) (opr (cv y) (+v) (cv z))) id (.i) opreq12d (0) eqeq1d (H~) rcla4cva (cv y) (cv z) ax-hvaddcl sylan2 (cv x) (opr (cv y) (-v) (cv z)) T fveq2 (= (cv x) (opr (cv y) (-v) (cv z))) id (.i) opreq12d (0) eqeq1d (H~) rcla4cva (cv y) (cv z) hvsubclt sylan2 (-) opreq12d 0cn subid syl6eq (cv x) (opr (cv y) (+v) (opr (i) (.s) (cv z))) T fveq2 (= (cv x) (opr (cv y) (+v) (opr (i) (.s) (cv z)))) id (.i) opreq12d (0) eqeq1d (H~) rcla4cva (cv y) (opr (i) (.s) (cv z)) ax-hvaddcl axicn (i) (cv z) ax-hvmulcl mpan sylan2 sylan2 (cv x) (opr (cv y) (-v) (opr (i) (.s) (cv z))) T fveq2 (= (cv x) (opr (cv y) (-v) (opr (i) (.s) (cv z)))) id (.i) opreq12d (0) eqeq1d (H~) rcla4cva (cv y) (opr (i) (.s) (cv z)) hvsubclt axicn (i) (cv z) ax-hvmulcl mpan sylan2 sylan2 (-) opreq12d 0cn subid syl6eq (i) (x.) opreq2d axicn mul01 syl6eq (+) opreq12d 0cn addid1 syl6eq (/) (4) opreq1d 4re recn 4re 4pos gt0ne0i div0 syl6eq eqtrd ex r19.21aivv lnopeq0.1 lnopf y z ho01 sylib T (X. (H~) (0H)) (cv x) fveq1 (cv x) ho0valtOLD sylan9eq (.i) (cv x) opreq1d (cv x) hi01t (= T (X. (H~) (0H))) adantl eqtrd r19.21aiva impbi)) thm (lnopeq ((T x) (U x)) ((lnopeq.1 (e. T (LinOp))) (lnopeq.2 (e. U (LinOp)))) (<-> (A.e. x (H~) (= (opr (` T (cv x)) (.i) (cv x)) (opr (` U (cv x)) (.i) (cv x)))) (= T U)) ((opr (` T (cv x)) (.i) (cv x)) (opr (` U (cv x)) (.i) (cv x)) subeq0t lnopeq.1 lnopf (cv x) ffvrni (` T (cv x)) (cv x) ax-hicl mpancom lnopeq.2 lnopf (cv x) ffvrni (` U (cv x)) (cv x) ax-hicl mpancom sylanc lnopeq.1 lnopf lnopeq.2 lnopf T U (cv x) hodvalt mp3an12 (.i) (cv x) opreq1d (` T (cv x)) (` U (cv x)) (cv x) his2subt lnopeq.1 lnopf (cv x) ffvrni lnopeq.2 lnopf (cv x) ffvrni (e. (cv x) (H~)) id syl3anc eqtr2d (0) eqeq1d bitr3d ralbiia lnopeq.1 lnopeq.2 lnophd x lnopeq0 lnopeq.1 lnopf lnopeq.2 lnopf hosubeq0OLD 3bitr)) thm (lnopeqt ((T x) (U x)) () (-> (/\ (e. T (LinOp)) (e. U (LinOp))) (<-> (A.e. x (H~) (= (opr (` T (cv x)) (.i) (cv x)) (opr (` U (cv x)) (.i) (cv x)))) (= T U))) (T (if (e. T (LinOp)) T (0op)) (cv x) fveq1 (.i) (cv x) opreq1d (opr (` U (cv x)) (.i) (cv x)) eqeq1d x (H~) ralbidv T (if (e. T (LinOp)) T (0op)) U eqeq1 bibi12d U (if (e. U (LinOp)) U (0op)) (cv x) fveq1 (.i) (cv x) opreq1d (opr (` (if (e. T (LinOp)) T (0op)) (cv x)) (.i) (cv x)) eqeq2d x (H~) ralbidv U (if (e. U (LinOp)) U (0op)) (if (e. T (LinOp)) T (0op)) eqeq2 bibi12d 0lnop T elimel 0lnop U elimel x lnopeq dedth2h)) thm (lnopunilem1 ((A y) (B y) (C y) (x y) (T x) (T y)) ((lnopunilem.1 (e. T (LinOp))) (lnopunilem.2 (A.e. x (H~) (= (` (norm) (` T (cv x))) (` (norm) (cv x))))) (lnopunilem.3 (e. A (H~))) (lnopunilem.4 (e. B (H~))) (lnopunilem1.5 (e. C (CC)))) (= (` (Re) (opr C (x.) (opr (` T A) (.i) (` T B)))) (` (Re) (opr C (x.) (opr A (.i) B)))) (lnopunilem.3 lnopunilem.2 (cv x) (cv y) T fveq2 (norm) fveq2d (cv x) (cv y) (norm) fveq2 eqeq12d (H~) cbvralv mpbi lnopunilem.1 lnopf (cv y) ffvrni (` T (cv y)) normsqt syl (cv y) normsqt eqeq12d (` (norm) (` T (cv y))) (` (norm) (cv y)) (^) (2) opreq1 syl5bi r19.20i ax-mp (cv y) A T fveq2 (cv y) A T fveq2 (.i) opreq12d (= (cv y) A) id (= (cv y) A) id (.i) opreq12d eqeq12d (H~) rcla4v mp2 (` (*) C) (x.) opreq2i C (x.) opreq2i lnopunilem.4 lnopunilem.2 (cv x) (cv y) T fveq2 (norm) fveq2d (cv x) (cv y) (norm) fveq2 eqeq12d (H~) cbvralv mpbi lnopunilem.1 lnopf (cv y) ffvrni (` T (cv y)) normsqt syl (cv y) normsqt eqeq12d (` (norm) (` T (cv y))) (` (norm) (cv y)) (^) (2) opreq1 syl5bi r19.20i ax-mp (cv y) B T fveq2 (cv y) B T fveq2 (.i) opreq12d (= (cv y) B) id (= (cv y) B) id (.i) opreq12d eqeq12d (H~) rcla4v mp2 (+) opreq12i (+) (opr (opr C (x.) (opr (` T A) (.i) (` T B))) (+) (` (*) (opr C (x.) (opr (` T A) (.i) (` T B))))) opreq1i lnopunilem1.5 lnopunilem.3 hvmulcl lnopunilem.4 hvaddcl lnopunilem.2 (cv x) (cv y) T fveq2 (norm) fveq2d (cv x) (cv y) (norm) fveq2 eqeq12d (H~) cbvralv mpbi lnopunilem.1 lnopf (cv y) ffvrni (` T (cv y)) normsqt syl (cv y) normsqt eqeq12d (` (norm) (` T (cv y))) (` (norm) (cv y)) (^) (2) opreq1 syl5bi r19.20i ax-mp (cv y) (opr (opr C (.s) A) (+v) B) T fveq2 (cv y) (opr (opr C (.s) A) (+v) B) T fveq2 (.i) opreq12d (= (cv y) (opr (opr C (.s) A) (+v) B)) id (= (cv y) (opr (opr C (.s) A) (+v) B)) id (.i) opreq12d eqeq12d (H~) rcla4v mp2 lnopunilem1.5 lnopunilem.3 lnopunilem.4 lnopunilem.1 C A B lnopl mp3an lnopunilem1.5 lnopunilem.3 lnopunilem.4 lnopunilem.1 C A B lnopl mp3an (.i) opreq12i lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp hvmulcl lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp hvmulcl lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp hvaddcl (opr C (.s) (` T A)) (` T B) (opr (opr C (.s) (` T A)) (+v) (` T B)) ax-his2 mp3an lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp hvmulcl lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp hvaddcl C (` T A) (opr (opr C (.s) (` T A)) (+v) (` T B)) ax-his3 mp3an lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp hvmulcl lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp (` T A) (opr C (.s) (` T A)) (` T B) his7t mp3an lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp C (` T A) (` T A) his5t mp3an (+) (opr (` T A) (.i) (` T B)) opreq1i eqtr C (x.) opreq2i lnopunilem1.5 lnopunilem1.5 cjcl lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp hicl mulcl lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp hicl adddi 3eqtr lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp hvmulcl lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp (` T B) (opr C (.s) (` T A)) (` T B) his7t mp3an lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp his1 (` (*) C) (x.) opreq2i lnopunilem1.5 lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp C (` T B) (` T A) his5t mp3an lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp hicl cjmul 3eqtr4 (+) (opr (` T B) (.i) (` T B)) opreq1i eqtr (+) opreq12i 3eqtrr lnopunilem1.5 lnopunilem.3 hvmulcl lnopunilem.4 lnopunilem1.5 lnopunilem.3 hvmulcl lnopunilem.4 hvaddcl (opr C (.s) A) B (opr (opr C (.s) A) (+v) B) ax-his2 mp3an lnopunilem1.5 lnopunilem.3 lnopunilem1.5 lnopunilem.3 hvmulcl lnopunilem.4 hvaddcl C A (opr (opr C (.s) A) (+v) B) ax-his3 mp3an lnopunilem.3 lnopunilem1.5 lnopunilem.3 hvmulcl lnopunilem.4 A (opr C (.s) A) B his7t mp3an lnopunilem1.5 lnopunilem.3 lnopunilem.3 C A A his5t mp3an (+) (opr A (.i) B) opreq1i eqtr C (x.) opreq2i lnopunilem1.5 lnopunilem1.5 cjcl lnopunilem.3 lnopunilem.3 hicl mulcl lnopunilem.3 lnopunilem.4 hicl adddi 3eqtr lnopunilem.4 lnopunilem1.5 lnopunilem.3 hvmulcl lnopunilem.4 B (opr C (.s) A) B his7t mp3an lnopunilem.4 lnopunilem.3 his1 (` (*) C) (x.) opreq2i lnopunilem1.5 lnopunilem.4 lnopunilem.3 C B A his5t mp3an lnopunilem1.5 lnopunilem.3 lnopunilem.4 hicl cjmul 3eqtr4 (+) (opr B (.i) B) opreq1i eqtr (+) opreq12i eqtr2 3eqtr4 lnopunilem1.5 lnopunilem1.5 cjcl lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp hicl mulcl mulcl lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp hicl lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp hicl mulcl lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp hicl mulcl cjcl add42 lnopunilem1.5 lnopunilem1.5 cjcl lnopunilem.3 lnopunilem.3 hicl mulcl mulcl lnopunilem.4 lnopunilem.4 hicl lnopunilem1.5 lnopunilem.3 lnopunilem.4 hicl mulcl lnopunilem1.5 lnopunilem.3 lnopunilem.4 hicl mulcl cjcl add42 3eqtr4 eqtr3 lnopunilem1.5 lnopunilem1.5 cjcl lnopunilem.3 lnopunilem.3 hicl mulcl mulcl lnopunilem.4 lnopunilem.4 hicl addcl lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp hicl mulcl lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp hicl mulcl cjcl addcl lnopunilem1.5 lnopunilem.3 lnopunilem.4 hicl mulcl lnopunilem1.5 lnopunilem.3 lnopunilem.4 hicl mulcl cjcl addcl addcan mpbi (/) (2) opreq1i lnopunilem1.5 lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp hicl mulcl (opr C (x.) (opr (` T A) (.i) (` T B))) recjt ax-mp lnopunilem1.5 lnopunilem.3 lnopunilem.4 hicl mulcl (opr C (x.) (opr A (.i) B)) recjt ax-mp 3eqtr4)) thm (lnopunilem2 ((A y) (B y) (x y) (T x) (T y)) ((lnopunilem.1 (e. T (LinOp))) (lnopunilem.2 (A.e. x (H~) (= (` (norm) (` T (cv x))) (` (norm) (cv x))))) (lnopunilem.3 (e. A (H~))) (lnopunilem.4 (e. B (H~)))) (= (opr (` T A) (.i) (` T B)) (opr A (.i) B)) ((cv y) (if (e. (cv y) (CC)) (cv y) (0)) (x.) (opr (` T A) (.i) (` T B)) opreq1 (Re) fveq2d (cv y) (if (e. (cv y) (CC)) (cv y) (0)) (x.) (opr A (.i) B) opreq1 (Re) fveq2d eqeq12d lnopunilem.1 lnopunilem.2 lnopunilem.3 lnopunilem.4 0cn (cv y) elimel lnopunilem1 dedth rgen lnopunilem.3 lnopunilem.1 lnopf A ffvrni ax-mp lnopunilem.4 lnopunilem.1 lnopf B ffvrni ax-mp hicl lnopunilem.3 lnopunilem.4 hicl (opr (` T A) (.i) (` T B)) (opr A (.i) B) y recant mp2an mpbi)) thm (lnopuni ((x y) (T x) (T y)) ((lnopuni.1 (e. T (LinOp))) (lnopuni.2 (:-onto-> T (H~) (H~))) (lnopuni.3 (A.e. x (H~) (= (` (norm) (` T (cv x))) (` (norm) (cv x)))))) (e. T (UniOp)) (T x y elunopt lnopuni.2 (cv x) (if (e. (cv x) (H~)) (cv x) (0v)) T fveq2 (.i) (` T (cv y)) opreq1d (cv x) (if (e. (cv x) (H~)) (cv x) (0v)) (.i) (cv y) opreq1 eqeq12d (cv y) (if (e. (cv y) (H~)) (cv y) (0v)) T fveq2 (` T (if (e. (cv x) (H~)) (cv x) (0v))) (.i) opreq2d (cv y) (if (e. (cv y) (H~)) (cv y) (0v)) (if (e. (cv x) (H~)) (cv x) (0v)) (.i) opreq2 eqeq12d lnopuni.1 lnopuni.3 ax-hv0cl (cv x) elimel ax-hv0cl (cv y) elimel lnopunilem2 dedth2h rgen2 mpbir2an)) thm (nmopunt ((w x) (w y) (w z) (T w) (x y) (x z) (T x) (y z) (T y) (T z)) () (-> (/\ (=/= (H~) (0H)) (e. T (UniOp))) (= (` (normop) T) (1))) (T unoplint T lnopft syl T x y nmopvalt syl (=/= (H~) (0H)) adantl ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) (1) z w supxr2OLD T unoplint T lnopft syl T x y nmopsetret ressxr (:--> T (H~) (H~)) a1i sstrd syl (=/= (H~) (0H)) adantl ax1re (1) rexrt ax-mp jctir T (cv y) unopnormt (cv z) eqeq2d (br (` (norm) (cv y)) (<_) (1)) anbi2d (cv z) (` (norm) (cv y)) (<_) (1) breq1 biimparc syl6bi ex r19.23adv imp z visset (cv x) (cv z) (` (norm) (` T (cv y))) eqeq1 (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv elab sylan2b r19.21aiva (=/= (H~) (0H)) adantl (cv w) (1) (cv z) (<) breq2 ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) rcla4ev y hne0 y y norm1hext bitr biimp (e. T (UniOp)) adantr ax1re leid (` (norm) (cv y)) (1) (<_) (1) breq1 mpbiri (/\ (e. T (UniOp)) (e. (cv y) (H~))) a1i T (cv y) unopnormt (= (` (norm) (cv y)) (1)) adantr (` (norm) (cv y)) (1) (` (norm) (` T (cv y))) eqeq2 (/\ (e. T (UniOp)) (e. (cv y) (H~))) adantl mpbid eqcomd ex jcad (=/= (H~) (0H)) adantll r19.22dva mpd ax1re elisseti (cv x) (1) (` (norm) (` T (cv y))) eqeq1 (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv elab sylibr (e. (cv z) (RR*)) adantr sylan ex r19.21aiva jca sylanc eqtrd)) thm (unopbdt () () (-> (e. T (UniOp)) (e. T (BndLinOp))) (T unoplint T nmop0h 0re syl6eqel T unopf1ot T (H~) (H~) f1of syl sylan2 T nmopunt ax1re syl6eqel (H~) (0H) df-ne sylanbr pm2.61ian jca T elbdop2t sylibr)) thm (lnophmlem1 ((A x) (B x) (T x)) ((lnophmlem.1 (e. A (H~))) (lnophmlem.2 (e. B (H~))) (lnophmlem.3 (e. T (LinOp))) (lnophmlem.4 (A.e. x (H~) (e. (opr (cv x) (.i) (` T (cv x))) (RR))))) (e. (opr A (.i) (` T A)) (RR)) (lnophmlem.1 lnophmlem.4 (= (cv x) A) id (cv x) A T fveq2 (.i) opreq12d (RR) eleq1d (H~) rcla4v mp2)) thm (lnophmlem2 ((A x) (B x) (T x)) ((lnophmlem.1 (e. A (H~))) (lnophmlem.2 (e. B (H~))) (lnophmlem.3 (e. T (LinOp))) (lnophmlem.4 (A.e. x (H~) (e. (opr (cv x) (.i) (` T (cv x))) (RR))))) (= (opr A (.i) (` T B)) (opr (` T A) (.i) B)) (lnophmlem.2 lnophmlem.1 lnophmlem.3 lnopf A ffvrni ax-mp lnophmlem.1 lnophmlem.2 lnophmlem.3 lnopf B ffvrni ax-mp polid2 lnophmlem.2 lnophmlem.1 hvcom lnophmlem.2 lnophmlem.3 lnopf B ffvrni ax-mp lnophmlem.1 lnophmlem.3 lnopf A ffvrni ax-mp hvcom lnophmlem.1 lnophmlem.2 lnophmlem.3 A B lnopadd mp2an eqtr4 (.i) opreq12i lnophmlem.2 lnophmlem.1 lnophmlem.2 lnophmlem.3 lnopf B ffvrni ax-mp lnophmlem.1 lnophmlem.3 lnopf A ffvrni ax-mp hisubcom lnophmlem.1 lnophmlem.2 lnophmlem.3 A B lnopsub mp2an (opr A (-v) B) (.i) opreq2i eqtr4 (-) opreq12i axicn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvsubdistr1 axicn lnophmlem.1 hvmulcl axicn axicn lnophmlem.2 hvmulcl hvmulcl hvsubval axicn axicn lnophmlem.2 hvmulass (-u (1)) (.s) opreq2i itimesi (-u (1)) (x.) opreq2i 1cn 1cn mul2neg 1cn mulid1 3eqtr (.s) B opreq1i 1cn negcl axicn axicn mulcl lnophmlem.2 hvmulass lnophmlem.2 B ax-hvmulid ax-mp 3eqtr3 eqtr3 (opr (i) (.s) A) (+v) opreq2i eqtr axicn lnophmlem.1 hvmulcl lnophmlem.2 hvcom 3eqtr axicn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvsubdistr1 axicn lnophmlem.1 hvmulcl axicn axicn lnophmlem.2 hvmulcl hvmulcl hvsubval axicn axicn lnophmlem.2 hvmulass (-u (1)) (.s) opreq2i itimesi (-u (1)) (x.) opreq2i 1cn 1cn mul2neg 1cn mulid1 3eqtr (.s) B opreq1i 1cn negcl axicn axicn mulcl lnophmlem.2 hvmulass lnophmlem.2 B ax-hvmulid ax-mp 3eqtr3 eqtr3 (opr (i) (.s) A) (+v) opreq2i eqtr axicn lnophmlem.1 hvmulcl lnophmlem.2 hvcom 3eqtr T fveq2i axicn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvsubcl lnophmlem.3 (i) (opr A (-v) (opr (i) (.s) B)) lnopmul mp2an axicn lnophmlem.2 lnophmlem.1 lnophmlem.3 (i) B A lnopaddmul mp3an 3eqtr3 (.i) opreq12i axicn axicn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvsubcl lnophmlem.1 axicn lnophmlem.2 hvmulcl hvsubcl lnophmlem.3 lnopf (opr A (-v) (opr (i) (.s) B)) ffvrni ax-mp his35 cji (i) (x.) opreq2i axicn axicn mulneg2 itimesi negeqi 1cn negneg eqtr 3eqtr (x.) (opr (opr A (-v) (opr (i) (.s) B)) (.i) (` T (opr A (-v) (opr (i) (.s) B)))) opreq1i lnophmlem.1 axicn lnophmlem.2 hvmulcl hvsubcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 recn mulid2 3eqtr eqtr3 lnophmlem.2 axicn lnophmlem.1 hvmulcl lnophmlem.2 lnophmlem.3 lnopf B ffvrni ax-mp axicn lnophmlem.1 lnophmlem.3 lnopf A ffvrni ax-mp hvmulcl hisubcom axicn axicn lnophmlem.2 hvmulass itimesi (.s) B opreq1i eqtr3 (opr (i) (.s) A) (+v) opreq2i axicn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvdistr1 axicn lnophmlem.1 hvmulcl lnophmlem.2 hvsubval 3eqtr4 axicn axicn lnophmlem.2 hvmulass itimesi (.s) B opreq1i eqtr3 (opr (i) (.s) A) (+v) opreq2i axicn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvdistr1 axicn lnophmlem.1 hvmulcl lnophmlem.2 hvsubval 3eqtr4 T fveq2i axicn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvaddcl lnophmlem.3 (i) (opr A (+v) (opr (i) (.s) B)) lnopmul mp2an axicn lnophmlem.1 lnophmlem.2 lnophmlem.3 (i) A B lnopmulsub mp3an 3eqtr3 (.i) opreq12i axicn axicn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvaddcl lnophmlem.1 axicn lnophmlem.2 hvmulcl hvaddcl lnophmlem.3 lnopf (opr A (+v) (opr (i) (.s) B)) ffvrni ax-mp his35 cji (i) (x.) opreq2i axicn axicn mulneg2 itimesi negeqi 1cn negneg eqtr 3eqtr (x.) (opr (opr A (+v) (opr (i) (.s) B)) (.i) (` T (opr A (+v) (opr (i) (.s) B)))) opreq1i lnophmlem.1 axicn lnophmlem.2 hvmulcl hvaddcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 recn mulid2 3eqtr 3eqtr2 (-) opreq12i (i) (x.) opreq2i (+) opreq12i (/) (4) opreq1i eqtr (*) fveq2i 4re 4pos gt0ne0i lnophmlem.1 lnophmlem.2 hvaddcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 lnophmlem.1 lnophmlem.2 hvsubcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 resubcl recn axicn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvsubcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 lnophmlem.1 axicn lnophmlem.2 hvmulcl hvaddcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 resubcl recn mulcl addcl 4re recn cjdiv ax-mp lnophmlem.1 lnophmlem.2 hvaddcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 lnophmlem.1 lnophmlem.2 hvsubcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 resubcl recn axicn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvsubcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 lnophmlem.1 axicn lnophmlem.2 hvmulcl hvaddcl lnophmlem.2 lnophmlem.3 lnophmlem.4 lnophmlem1 resubcl recn mulcl negsub axicn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvsubcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 lnophmlem.1 axicn lnophmlem.2 hvmulcl hvaddcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 resubcl recn mulneg2 lnophmlem.1 axicn lnophmlem.2 hvmulcl hvsubcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 recn lnophmlem.1 axicn lnophmlem.2 hvmulcl hvaddcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 recn negsubdi2 (i) (x.) opreq2i eqtr3 (opr (opr (opr A (+v) B) (.i) (` T (opr A (+v) B))) (-) (opr (opr A (-v) B) (.i) (` T (opr A (-v) B)))) (+) opreq2i lnophmlem.1 lnophmlem.2 lnophmlem.3 A B lnopadd mp2an (opr A (+v) B) (.i) opreq2i lnophmlem.1 lnophmlem.2 lnophmlem.3 A B lnopsub mp2an (opr A (-v) B) (.i) opreq2i (-) opreq12i axicn lnophmlem.1 lnophmlem.2 lnophmlem.3 (i) A B lnopaddmul mp3an (opr A (+v) (opr (i) (.s) B)) (.i) opreq2i axicn lnophmlem.1 lnophmlem.2 lnophmlem.3 (i) A B lnopsubmul mp3an (opr A (-v) (opr (i) (.s) B)) (.i) opreq2i (-) opreq12i (i) (x.) opreq2i (+) opreq12i eqtr2 lnophmlem.1 lnophmlem.2 hvaddcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 lnophmlem.1 lnophmlem.2 hvsubcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 resubcl lnophmlem.1 axicn lnophmlem.2 hvmulcl hvsubcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 lnophmlem.1 axicn lnophmlem.2 hvmulcl hvaddcl lnophmlem.1 lnophmlem.3 lnophmlem.4 lnophmlem1 resubcl (opr (opr (opr A (+v) B) (.i) (` T (opr A (+v) B))) (-) (opr (opr A (-v) B) (.i) (` T (opr A (-v) B)))) (opr (opr (opr A (-v) (opr (i) (.s) B)) (.i) (` T (opr A (-v) (opr (i) (.s) B)))) (-) (opr (opr A (+v) (opr (i) (.s) B)) (.i) (` T (opr A (+v) (opr (i) (.s) B))))) cjreimt mp2an 3eqtr4r 4re (4) cjret ax-mp (/) opreq12i 3eqtrr lnophmlem.1 lnophmlem.2 lnophmlem.3 lnopf B ffvrni ax-mp lnophmlem.2 lnophmlem.1 lnophmlem.3 lnopf A ffvrni ax-mp polid2 lnophmlem.1 lnophmlem.3 lnopf A ffvrni ax-mp lnophmlem.2 his1 3eqtr4)) thm (lnophm ((x y) (x z) (T x) (y z) (T y) (T z)) ((lnophm.1 (e. T (LinOp))) (lnophm.2 (A.e. x (H~) (e. (opr (cv x) (.i) (` T (cv x))) (RR))))) (e. T (HrmOp)) (T y z elhmopt lnophm.1 lnopf (cv y) (if (e. (cv y) (H~)) (cv y) (0v)) (.i) (` T (cv z)) opreq1 (cv y) (if (e. (cv y) (H~)) (cv y) (0v)) T fveq2 (.i) (cv z) opreq1d eqeq12d (cv z) (if (e. (cv z) (H~)) (cv z) (0v)) T fveq2 (if (e. (cv y) (H~)) (cv y) (0v)) (.i) opreq2d (cv z) (if (e. (cv z) (H~)) (cv z) (0v)) (` T (if (e. (cv y) (H~)) (cv y) (0v))) (.i) opreq2 eqeq12d ax-hv0cl (cv y) elimel ax-hv0cl (cv z) elimel lnophm.1 lnophm.2 lnophmlem2 dedth2h rgen2 mpbir2an)) thm (hmopst ((x y) (T x) (T y) (U x) (U y)) () (-> (/\ (e. T (HrmOp)) (e. U (HrmOp))) (e. (opr T (+op) U) (HrmOp))) (T U hoaddclt T hmopft U hmopft syl2an T (cv x) (cv y) hmopt 3expb U (cv x) (cv y) hmopt 3expb (+) opreqan12d anandirs T U (cv y) hosvaltOLD (cv x) (.i) opreq2d (e. (cv x) (H~)) adantrl (cv x) (` T (cv y)) (` U (cv y)) his7t (e. (cv x) (H~)) (e. (cv y) (H~)) pm3.26 (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) adantl T (H~) (H~) (cv y) ffvrn (e. (cv x) (H~)) adantrl (:--> U (H~) (H~)) adantlr U (H~) (H~) (cv y) ffvrn (:--> T (H~) (H~)) (e. (cv x) (H~)) ad2ant2l syl3anc eqtrd T hmopft U hmopft anim12i sylan T U (cv x) hosvaltOLD (.i) (cv y) opreq1d (e. (cv y) (H~)) adantrr (` T (cv x)) (` U (cv x)) (cv y) ax-his2 T (H~) (H~) (cv x) ffvrn (:--> U (H~) (H~)) (e. (cv y) (H~)) ad2ant2r U (H~) (H~) (cv x) ffvrn (e. (cv y) (H~)) adantrr (:--> T (H~) (H~)) adantll (e. (cv x) (H~)) (e. (cv y) (H~)) pm3.27 (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) adantl syl3anc eqtrd T hmopft U hmopft anim12i sylan 3eqtr4d ex r19.21aivv jca (opr T (+op) U) x y elhmopt sylibr)) thm (hmopmt ((x y) (A x) (A y) (T x) (T y)) () (-> (/\ (e. A (RR)) (e. T (HrmOp))) (e. (opr A (.op) T) (HrmOp))) (A T homulclt A recnt T hmopft syl2an A cjret T (cv x) (cv y) hmopt 3expb (x.) opreqan12d anassrs A T (cv y) homvalt 3expa (e. (cv x) (H~)) adantrl (cv x) (.i) opreq2d A (cv x) (` T (cv y)) his5t (e. A (CC)) (:--> T (H~) (H~)) pm3.26 (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) adantr (e. (cv x) (H~)) (e. (cv y) (H~)) pm3.26 (/\ (e. A (CC)) (:--> T (H~) (H~))) adantl T (H~) (H~) (cv y) ffvrn (e. A (CC)) (e. (cv x) (H~)) ad2ant2l syl3anc eqtrd A recnt T hmopft anim12i sylan A T (cv x) homvalt 3expa (e. (cv y) (H~)) adantrr (.i) (cv y) opreq1d A (` T (cv x)) (cv y) ax-his3 (e. A (CC)) (:--> T (H~) (H~)) pm3.26 (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) adantr T (H~) (H~) (cv x) ffvrn (e. (cv y) (H~)) adantrr (e. A (CC)) adantll (e. (cv x) (H~)) (e. (cv y) (H~)) pm3.27 (/\ (e. A (CC)) (:--> T (H~) (H~))) adantl syl3anc eqtrd A recnt T hmopft anim12i sylan 3eqtr4d ex r19.21aivv jca (opr A (.op) T) x y elhmopt sylibr)) thm (hmopdt () () (-> (/\ (e. T (HrmOp)) (e. U (HrmOp))) (e. (opr T (-op) U) (HrmOp))) (T U honegsubtOLD T hmopft U hmopft syl2an T (opr (-u (1)) (.op) U) hmopst ax1re renegcl (-u (1)) U hmopmt mpan sylan2 eqeltrd)) thm (hmopcot ((x y) (T x) (T y) (U x) (U y)) () (-> (/\/\ (e. T (HrmOp)) (e. U (HrmOp)) (= (o. T U) (o. U T))) (e. (o. T U) (HrmOp))) (T (H~) (H~) U (H~) fco T hmopft U hmopft syl2an (= (o. T U) (o. U T)) 3adant3 T U (H~) (H~) (cv y) fvco3 T hmopft T (H~) (H~) ffun syl U hmopft (e. (cv y) (H~)) id syl3an 3expa (cv x) (.i) opreq2d (e. (cv x) (H~)) adantrl T (cv x) (` U (cv y)) hmopt (e. T (HrmOp)) (e. U (HrmOp)) pm3.26 (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) adantr (e. (cv x) (H~)) (e. (cv y) (H~)) pm3.26 (/\ (e. T (HrmOp)) (e. U (HrmOp))) adantl U (H~) (H~) (cv y) ffvrn U hmopft sylan (e. T (HrmOp)) (e. (cv x) (H~)) ad2ant2l syl3anc U (` T (cv x)) (cv y) hmopt (e. T (HrmOp)) (e. U (HrmOp)) pm3.27 (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) adantr T (H~) (H~) (cv x) ffvrn T hmopft sylan (e. U (HrmOp)) (e. (cv y) (H~)) ad2ant2r (e. (cv x) (H~)) (e. (cv y) (H~)) pm3.27 (/\ (e. T (HrmOp)) (e. U (HrmOp))) adantl syl3anc 3eqtrd U T (H~) (H~) (cv x) fvco3 U hmopft U (H~) (H~) ffun syl T hmopft (e. (cv x) (H~)) id syl3an 3expa ancom1s (.i) (cv y) opreq1d (e. (cv y) (H~)) adantrr eqtr4d (= (o. T U) (o. U T)) 3adantl3 (o. T U) (o. U T) (cv x) fveq1 (.i) (cv y) opreq1d (e. T (HrmOp)) (e. U (HrmOp)) 3ad2ant3 (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) adantr eqtr4d ex r19.21aivv jca (o. T U) x y elhmopt sylibr)) thm (nmbdoplb () ((nmbdoplb.1 (e. T (BndLinOp)))) (-> (e. A (H~)) (br (` (norm) (` T A)) (<_) (opr (` (normop) T) (x.) (` (norm) A)))) (A (0v) T fveq2 nmbdoplb.1 T bdoplnt ax-mp lnop0 syl6eq (norm) fveq2d norm0 syl6eq A (0v) (norm) fveq2 norm0 syl6eq (` (normop) T) (x.) opreq2d nmbdoplb.1 T nmopret ax-mp recn mul01 syl6req 0re leid syl5breq eqbrtrd (e. A (H~)) adantl (` (norm) (` T A)) (` (norm) A) divrec2t nmbdoplb.1 T bdoplnt ax-mp lnopf A ffvrni (` T A) normclt syl (-. (= A (0v))) adantr recnd A normclt (-. (= A (0v))) adantr recnd A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr syl3anc nmbdoplb.1 T bdoplnt ax-mp (opr (1) (/) (` (norm) A)) A lnopmul (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd (e. A (H~)) (-. (= A (0v))) pm3.26 sylanc (norm) fveq2d (opr (1) (/) (` (norm) A)) (` T A) norm-iiit (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd nmbdoplb.1 T bdoplnt ax-mp lnopf A ffvrni (-. (= A (0v))) adantr sylanc (opr (1) (/) (` (norm) A)) absidt (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc 0re (0) (opr (1) (/) (` (norm) A)) ltlet mpan (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc (` (norm) A) recgt0t A normclt (-. (= A (0v))) adantr A normgt0tOLD biimpa sylanc sylc sylanc (x.) (` (norm) (` T A)) opreq1d 3eqtrrd eqtrd nmbdoplb.1 T bdoplnt ax-mp lnopf T (opr (opr (1) (/) (` (norm) A)) (.s) A) nmoplbt mp3an1 (opr (1) (/) (` (norm) A)) A ax-hvmulcl (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd (e. A (H~)) (-. (= A (0v))) pm3.26 sylanc (` (norm) (opr (opr (1) (/) (` (norm) A)) (.s) A)) (1) eqlet (opr (1) (/) (` (norm) A)) A ax-hvmulcl (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd (e. A (H~)) (-. (= A (0v))) pm3.26 sylanc (opr (opr (1) (/) (` (norm) A)) (.s) A) normclt syl A norm1t A (0v) df-ne sylan2br sylanc sylanc eqbrtrd (` (norm) (` T A)) (` (norm) A) (` (normop) T) ledivmul2t nmbdoplb.1 T bdoplnt ax-mp lnopf A ffvrni (` T A) normclt syl (-. (= A (0v))) adantr A normclt (-. (= A (0v))) adantr nmbdoplb.1 T nmopret ax-mp (/\ (e. A (H~)) (-. (= A (0v)))) a1i 3jca A normgt0tOLD biimpa sylanc mpbid pm2.61dan)) thm (nmbdoplbt () () (-> (/\ (e. T (BndLinOp)) (e. A (H~))) (br (` (norm) (` T A)) (<_) (opr (` (normop) T) (x.) (` (norm) A)))) (T (if (e. T (BndLinOp)) T (X. (H~) (0H))) A fveq1 (norm) fveq2d T (if (e. T (BndLinOp)) T (X. (H~) (0H))) (normop) fveq2 (x.) (` (norm) A) opreq1d (<_) breq12d (e. A (H~)) imbi2d 0bdop T elimel A nmbdoplb dedth imp)) thm (nmcopexlem1 ((x y) (x z) (T x) (y z) (T y) (T z) (n x) (n y) (n z) (T n)) ((nmcopex.1 (e. T (LinOp))) (nmcopex.2 (e. T (ConOp)))) (-> (/\ (e. (cv n) (NN)) (A.e. x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (-. (br (cv n) (<) (` (norm) (` T (cv x)))))))) (e. (` (normop) T) (RR))) (n (RR) (E.e. y ({|} z (E.e. x (H~) (/\ (br (` (norm) (cv x)) (<_) (1)) (= (cv z) (` (norm) (` T (cv x))))))) (br (cv n) (<) (cv y))) ra4 impcom (cv n) nnret nmcopex.1 lnopf T z x nmopsetret ax-mp ressxr sstri ({|} z (E.e. x (H~) (/\ (br (` (norm) (cv x)) (<_) (1)) (= (cv z) (` (norm) (` T (cv x))))))) n y supxrunb2 nmcopex.1 lnopf T z x nmopvalt ax-mp (+oo) eqeq1i syl6rbbr ax-mp biimp syl2an x (H~) y (/\ (/\ (br (` (norm) (cv x)) (<_) (1)) (= (cv y) (` (norm) (` T (cv x))))) (br (cv n) (<) (cv y))) rexcom4 y (br (` (norm) (cv x)) (<_) (1)) (/\ (= (cv y) (` (norm) (` T (cv x)))) (br (cv n) (<) (cv y))) 19.42v (br (` (norm) (cv x)) (<_) (1)) (= (cv y) (` (norm) (` T (cv x)))) (br (cv n) (<) (cv y)) anass bicomi y exbii (norm) (` T (cv x)) fvex (cv y) (` (norm) (` T (cv x))) (cv n) (<) breq2 ceqsexv (br (` (norm) (cv x)) (<_) (1)) anbi2i 3bitr3r x (H~) rexbii y visset (cv z) (cv y) (` (norm) (` T (cv x))) eqeq1 (br (` (norm) (cv x)) (<_) (1)) anbi2d x (H~) rexbidv elab (br (cv n) (<) (cv y)) anbi1i x (H~) (/\ (br (` (norm) (cv x)) (<_) (1)) (= (cv y) (` (norm) (` T (cv x))))) (br (cv n) (<) (cv y)) r19.41v bitr4 y exbii 3bitr4 y ({|} z (E.e. x (H~) (/\ (br (` (norm) (cv x)) (<_) (1)) (= (cv z) (` (norm) (` T (cv x))))))) (br (cv n) (<) (cv y)) df-rex bitr4 sylibr ex (br (` (norm) (cv x)) (<_) (1)) (br (cv n) (<) (` (norm) (` T (cv x)))) df-an x (H~) rexbii x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (-. (br (cv n) (<) (` (norm) (` T (cv x)))))) rexnal bitr syl6ib con2d imp nmcopex.1 lnopf T nmoprepnf ax-mp sylibr)) thm (nmcopexlem2 ((y z) (T y) (T z)) ((nmcopex.1 (e. T (LinOp))) (nmcopex.2 (e. T (ConOp)))) (E.e. y (RR) (/\ (br (0) (<) (cv y)) (A.e. z (H~) (-> (br (` (norm) (cv z)) (<) (cv y)) (br (` (norm) (` T (cv z))) (<) (1)))))) (nmcopex.2 ax-hv0cl pm3.2i ax1re lt01 pm3.2i T (0v) (1) y z cnopct mp2an (cv z) hvsub0t (norm) fveq2d (<) (cv y) breq1d nmcopex.1 lnopf (cv z) ffvrni (` T (cv z)) hvsub0t syl nmcopex.1 lnop0 (` T (cv z)) (-v) opreq2i syl5eq (norm) fveq2d (<) (1) breq1d imbi12d ralbiia (br (0) (<) (cv y)) anbi2i y (RR) rexbii mpbi)) thm (nmcopexlem3 ((T x) (n x) (T n)) ((nmcopex.1 (e. T (LinOp))) (nmcopex.2 (e. T (ConOp)))) (-> (/\ (e. (cv n) (NN)) (e. (cv x) (H~))) (<-> (br (` (norm) (` T (cv x))) (<) (cv n)) (br (` (norm) (` T (opr (opr (1) (/) (cv n)) (.s) (cv x)))) (<) (1)))) ((` (norm) (` T (cv x))) (cv n) (opr (1) (/) (cv n)) ltmul2t nmcopex.1 lnopf (cv x) ffvrni (` T (cv x)) normclt syl (e. (cv n) (NN)) adantl (cv n) nnret (e. (cv x) (H~)) adantr (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc (e. (cv x) (H~)) adantr 3jca (cv n) nnrecgt0t (e. (cv x) (H~)) adantr sylanc nmcopex.1 (opr (1) (/) (cv n)) (cv x) lnopmul (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd sylan (norm) fveq2d (opr (1) (/) (cv n)) (` T (cv x)) norm-iiit (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd nmcopex.1 lnopf (cv x) ffvrni syl2an (opr (1) (/) (cv n)) absidt (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc 0re (0) (opr (1) (/) (cv n)) ltlet mpan (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc (cv n) nnrecgt0t sylc sylanc (e. (cv x) (H~)) adantr (x.) (` (norm) (` T (cv x))) opreq1d 3eqtrrd (cv n) recid2t (cv n) nncnt (cv n) nnne0t sylanc (e. (cv x) (H~)) adantr (<) breq12d bitrd)) thm (nmcopexlem4 ((M k) (k y) (T k) (T y)) ((nmcopex.1 (e. T (LinOp))) (nmcopex.2 (e. T (ConOp))) (nmcopexlem4.3 (= A ({e.|} k (NN) (br (opr (1) (/) (cv k)) (<) (cv y))))) (nmcopexlem4.4 (= M (sup A (RR) (`' (<)))))) (-> (/\ (e. (cv y) (RR)) (br (0) (<) (cv y))) (/\ (e. M (NN)) (br (opr (1) (/) M) (<) (cv y)))) ((cv y) k nnreclt k (NN) (br (opr (1) (/) (cv k)) (<) (cv y)) rabn0 sylibr nmcopexlem4.3 ({/}) eqeq1i negbii sylibr nmcopexlem4.3 k (NN) (br (opr (1) (/) (cv k)) (<) (cv y)) ssrab2 eqsstr nnuz sseqtr A (1) infmssuzcl mpan syl nmcopexlem4.3 syl6eleq nmcopexlem4.4 syl5eqel (cv k) M (1) (/) opreq2 (<) (cv y) breq1d (NN) elrab sylib)) thm (nmcopexlem5 ((k x) (M k) (M x) (k y) (T k) (x y) (T x) (T y) (n x) (n y) (T n)) ((nmcopex.1 (e. T (LinOp))) (nmcopex.2 (e. T (ConOp))) (nmcopexlem4.3 (= A ({e.|} k (NN) (br (opr (1) (/) (cv k)) (<) (cv y))))) (nmcopexlem4.4 (= M (sup A (RR) (`' (<)))))) (-> (/\/\ (/\ (e. (cv y) (RR)) (br (0) (<) (cv y))) (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1)))) (br (` (norm) (opr (opr (1) (/) (cv n)) (.s) (cv x))) (<) (cv y))) ((opr (1) (/) (cv n)) (cv x) ax-hvmulcl (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd sylan (opr (opr (1) (/) (cv n)) (.s) (cv x)) normclt syl (br M (<_) (cv n)) (br (` (norm) (cv x)) (<_) (1)) ad2ant2r (/\ (e. (cv y) (RR)) (br (0) (<) (cv y))) 3adant1 nmcopex.1 nmcopex.2 nmcopexlem4.3 nmcopexlem4.4 nmcopexlem4 pm3.26d M rerecclt M nnret M nnne0t sylanc syl (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3ad2ant1 (e. (cv y) (RR)) (br (0) (<) (cv y)) pm3.26 (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3ad2ant1 (opr (1) (/) (cv n)) (cv x) ax-hvmulcl (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd sylan (opr (opr (1) (/) (cv n)) (.s) (cv x)) normclt syl (br M (<_) (cv n)) (br (` (norm) (cv x)) (<_) (1)) ad2ant2r (e. M (NN)) 3adant1 (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc (br M (<_) (cv n)) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) ad2antrr (e. M (NN)) 3adant1 M rerecclt M nnret M nnne0t sylanc (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3ad2ant1 (` (norm) (cv x)) (1) (opr (1) (/) (cv n)) lemul2it (cv x) normclt (e. M (NN)) (br (` (norm) (cv x)) (<_) (1)) ad2antrl (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) 3adant2 ax1re (/\/\ (e. M (NN)) (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1)))) a1i (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc (br M (<_) (cv n)) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) ad2antrr (e. M (NN)) 3adant1 3jca ax1re 0re ax1re lt01 ltlei (1) (cv n) divge0t an4s mpanl12 (cv n) nnret (cv n) nngt0t sylanc (e. M (NN)) (br M (<_) (cv n)) ad2antrl (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3adant3 (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1)) pm3.27 (e. M (NN)) (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) 3ad2ant3 jca sylanc (opr (1) (/) (cv n)) (cv x) norm-iiit (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd sylan (opr (1) (/) (cv n)) absidt (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc ax1re 0re ax1re lt01 ltlei (1) (cv n) divge0t an4s mpanl12 (cv n) nnret (cv n) nngt0t sylanc sylanc (e. (cv x) (H~)) adantr (x.) (` (norm) (cv x)) opreq1d eqtr2d (br M (<_) (cv n)) (br (` (norm) (cv x)) (<_) (1)) ad2ant2r (e. M (NN)) 3adant1 (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd (opr (1) (/) (cv n)) ax1id syl (e. M (NN)) (br M (<_) (cv n)) ad2antrl (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3adant3 3brtr3d M (cv n) lerect M nnret M nngt0t jca (cv n) nnret (cv n) nngt0t jca syl2an biimpa anasss (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3adant3 letrd nmcopex.1 nmcopex.2 nmcopexlem4.3 nmcopexlem4.4 nmcopexlem4 pm3.26d syl3an1 nmcopex.1 nmcopex.2 nmcopexlem4.3 nmcopexlem4.4 nmcopexlem4 pm3.27d (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3ad2ant1 lelttrd)) thm (nmcopexlem6 ((k x) (M k) (M x) (k y) (k z) (T k) (x y) (x z) (T x) (y z) (T y) (T z)) ((nmcopex.1 (e. T (LinOp))) (nmcopex.2 (e. T (ConOp))) (nmcopexlem4.3 (= A ({e.|} k (NN) (br (opr (1) (/) (cv k)) (<) (cv y))))) (nmcopexlem4.4 (= M (sup A (RR) (`' (<)))))) (e. (` (normop) T) (RR)) (nmcopex.1 nmcopex.2 y z nmcopexlem2 nmcopex.1 nmcopex.2 nmcopexlem4.3 nmcopexlem4.4 nmcopexlem4 pm3.26d ex (A.e. z (H~) (-> (br (` (norm) (cv z)) (<) (cv y)) (br (` (norm) (` T (cv z))) (<) (1)))) adantrd nmcopex.1 nmcopex.2 nmcopexlem4.3 nmcopexlem4.4 k x nmcopexlem5 3expb (opr (1) (/) (cv k)) (cv x) ax-hvmulcl (cv k) rerecclt (cv k) nnret (cv k) nnne0t sylanc recnd sylan (cv z) (opr (opr (1) (/) (cv k)) (.s) (cv x)) (norm) fveq2 (<) (cv y) breq1d (cv z) (opr (opr (1) (/) (cv k)) (.s) (cv x)) T fveq2 (norm) fveq2d (<) (1) breq1d imbi12d (H~) rcla4v syl nmcopex.1 nmcopex.2 k x nmcopexlem3 (` (norm) (` T (cv x))) (cv k) ltnsymt nmcopex.1 lnopf (cv x) ffvrni (` T (cv x)) normclt syl (cv k) nnret syl2an ancoms sylbird syl6d (br M (<_) (cv k)) (br (` (norm) (cv x)) (<_) (1)) ad2ant2r (/\ (e. (cv y) (RR)) (br (0) (<) (cv y))) adantl mpid ex com23 imp exp4d anasss imp r19.21aiv nmcopex.1 nmcopex.2 k x nmcopexlem1 (br M (<_) (cv k)) adantlr (/\ (e. (cv y) (RR)) (/\ (br (0) (<) (cv y)) (A.e. z (H~) (-> (br (` (norm) (cv z)) (<) (cv y)) (br (` (norm) (` T (cv z))) (<) (1)))))) adantll mpdan exp32 r19.23adv M nnret M leidt syl (cv k) M M (<_) breq2 (NN) rcla4ev mpdan syl5 ex mpdd r19.23aiv ax-mp)) thm (nmcopex ((k y) (k z) (T k) (y z) (T y) (T z)) ((nmcopex.1 (e. T (LinOp))) (nmcopex.2 (e. T (ConOp)))) (e. (` (normop) T) (RR)) (nmcopex.1 nmcopex.2 ({e.|} k (NN) (br (opr (1) (/) (cv k)) (<) (cv y))) eqid (cv z) (cv k) (1) (/) opreq2 (<) (cv y) breq1d (NN) cbvrabv ({e.|} z (NN) (br (opr (1) (/) (cv z)) (<) (cv y))) ({e.|} k (NN) (br (opr (1) (/) (cv k)) (<) (cv y))) (RR) (`' (<)) supeq1 ax-mp nmcopexlem6)) thm (nmcoplb () ((nmcopex.1 (e. T (LinOp))) (nmcopex.2 (e. T (ConOp)))) (-> (e. A (H~)) (br (` (norm) (` T A)) (<_) (opr (` (normop) T) (x.) (` (norm) A)))) (A (0v) T fveq2 nmcopex.1 lnop0 syl6eq (norm) fveq2d norm0 syl6eq A (0v) (norm) fveq2 norm0 syl6eq (` (normop) T) (x.) opreq2d nmcopex.1 nmcopex.2 nmcopex recn mul01 syl6req 0re leid syl5breq eqbrtrd (e. A (H~)) adantl (` (norm) (` T A)) (` (norm) A) divrec2t nmcopex.1 lnopf A ffvrni (` T A) normclt syl (-. (= A (0v))) adantr recnd A normclt (-. (= A (0v))) adantr recnd A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr syl3anc nmcopex.1 (opr (1) (/) (` (norm) A)) A lnopmul (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd (e. A (H~)) (-. (= A (0v))) pm3.26 sylanc (norm) fveq2d (opr (1) (/) (` (norm) A)) (` T A) norm-iiit (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd nmcopex.1 lnopf A ffvrni (-. (= A (0v))) adantr sylanc (opr (1) (/) (` (norm) A)) absidt (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc 0re (0) (opr (1) (/) (` (norm) A)) ltlet mpan (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc (` (norm) A) recgt0t A normclt (-. (= A (0v))) adantr A normgt0tOLD biimpa sylanc sylc sylanc (x.) (` (norm) (` T A)) opreq1d 3eqtrrd eqtrd nmcopex.1 lnopf T (opr (opr (1) (/) (` (norm) A)) (.s) A) nmoplbt mp3an1 (opr (1) (/) (` (norm) A)) A ax-hvmulcl (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd (e. A (H~)) (-. (= A (0v))) pm3.26 sylanc (` (norm) (opr (opr (1) (/) (` (norm) A)) (.s) A)) (1) eqlet (opr (1) (/) (` (norm) A)) A ax-hvmulcl (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd (e. A (H~)) (-. (= A (0v))) pm3.26 sylanc (opr (opr (1) (/) (` (norm) A)) (.s) A) normclt syl A norm1t A (0v) df-ne sylan2br sylanc sylanc eqbrtrd (` (norm) (` T A)) (` (norm) A) (` (normop) T) ledivmul2t nmcopex.1 lnopf A ffvrni (` T A) normclt syl (-. (= A (0v))) adantr A normclt (-. (= A (0v))) adantr nmcopex.1 nmcopex.2 nmcopex (/\ (e. A (H~)) (-. (= A (0v)))) a1i 3jca A normgt0tOLD biimpa sylanc mpbid pm2.61dan)) thm (nmcopext () () (-> (/\ (e. T (LinOp)) (e. T (ConOp))) (e. (` (normop) T) (RR))) (T (LinOp) (ConOp) elin T (if (e. T (i^i (LinOp) (ConOp))) T (|` (I) (H~))) (normop) fveq2 (RR) eleq1d (|` (I) (H~)) (LinOp) (ConOp) elin idlnop idcnop mpbir2an T elimel (if (e. T (i^i (LinOp) (ConOp))) T (|` (I) (H~))) (LinOp) (ConOp) elin mpbi pm3.26i (|` (I) (H~)) (LinOp) (ConOp) elin idlnop idcnop mpbir2an T elimel (if (e. T (i^i (LinOp) (ConOp))) T (|` (I) (H~))) (LinOp) (ConOp) elin mpbi pm3.27i nmcopex dedth sylbir)) thm (nmcoplbt () () (-> (/\/\ (e. T (LinOp)) (e. T (ConOp)) (e. A (H~))) (br (` (norm) (` T A)) (<_) (opr (` (normop) T) (x.) (` (norm) A)))) (T (if (e. T (i^i (LinOp) (ConOp))) T (|` (I) (H~))) A fveq1 (norm) fveq2d T (if (e. T (i^i (LinOp) (ConOp))) T (|` (I) (H~))) (normop) fveq2 (x.) (` (norm) A) opreq1d (<_) breq12d (e. A (H~)) imbi2d (|` (I) (H~)) (LinOp) (ConOp) elin idlnop idcnop mpbir2an T elimel (if (e. T (i^i (LinOp) (ConOp))) T (|` (I) (H~))) (LinOp) (ConOp) elin mpbi pm3.26i (|` (I) (H~)) (LinOp) (ConOp) elin idlnop idcnop mpbir2an T elimel (if (e. T (i^i (LinOp) (ConOp))) T (|` (I) (H~))) (LinOp) (ConOp) elin mpbi pm3.27i A nmcoplb dedth imp T (LinOp) (ConOp) elin sylanbr 3impa)) thm (nmophm ((A x) (T x)) ((nmophm.1 (e. T (BndLinOp)))) (-> (e. A (CC)) (= (` (normop) (opr A (.op) T)) (opr (` (abs) A) (x.) (` (normop) T)))) ((opr A (.op) T) (opr (` (abs) A) (x.) (` (normop) T)) x nmopleubt nmophm.1 T bdopft ax-mp A T homulclt mpan2 A absclt nmophm.1 T nmopret ax-mp (` (abs) A) (` (normop) T) axmulrcl mpan2 syl (opr (` (abs) A) (x.) (` (normop) T)) rexrt syl nmophm.1 T bdopft ax-mp A T (cv x) homvalt mp3an2 (norm) fveq2d A (` T (cv x)) norm-iiit nmophm.1 T bdopft ax-mp (cv x) ffvrni sylan2 eqtrd (br (` (norm) (cv x)) (<_) (1)) adantr (` (norm) (` T (cv x))) (` (normop) T) (` (abs) A) lemul2it nmophm.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl (e. A (CC)) adantl nmophm.1 T nmopret ax-mp (/\ (e. A (CC)) (e. (cv x) (H~))) a1i A absclt (e. (cv x) (H~)) adantr 3jca (br (` (norm) (cv x)) (<_) (1)) adantr A absge0t nmophm.1 T bdopft ax-mp T (cv x) nmoplbt mp3an1 anim12i anassrs sylanc eqbrtrd ex r19.21aiva syl3anc A (0) (abs) fveq2 abs0 syl6eq (x.) (` (normop) T) opreq1d nmophm.1 T nmopret ax-mp recn mul02 syl6eq (e. A (CC)) adantl nmophm.1 T bdopft ax-mp A T homulclt mpan2 (opr A (.op) T) nmopge0t syl (= A (0)) adantr eqbrtrd nmophm.1 T bdopft ax-mp T (opr (` (normop) (opr A (.op) T)) (/) (` (abs) A)) x nmopleubt mp3an1 (` (normop) (opr A (.op) T)) (` (abs) A) redivclt (` (normop) (opr A (.op) T)) (opr (` (abs) A) (x.) (` (normop) T)) xrret nmophm.1 T bdopft ax-mp A T homulclt mpan2 (opr A (.op) T) nmopxrt syl A absclt nmophm.1 T nmopret ax-mp (` (abs) A) (` (normop) T) axmulrcl mpan2 syl jca nmophm.1 T bdopft ax-mp A T homulclt mpan2 (opr A (.op) T) nmopgtmnf syl (opr A (.op) T) (opr (` (abs) A) (x.) (` (normop) T)) x nmopleubt nmophm.1 T bdopft ax-mp A T homulclt mpan2 A absclt nmophm.1 T nmopret ax-mp (` (abs) A) (` (normop) T) axmulrcl mpan2 syl (opr (` (abs) A) (x.) (` (normop) T)) rexrt syl nmophm.1 T bdopft ax-mp A T (cv x) homvalt mp3an2 (norm) fveq2d A (` T (cv x)) norm-iiit nmophm.1 T bdopft ax-mp (cv x) ffvrni sylan2 eqtrd (br (` (norm) (cv x)) (<_) (1)) adantr (` (norm) (` T (cv x))) (` (normop) T) (` (abs) A) lemul2it nmophm.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl (e. A (CC)) adantl nmophm.1 T nmopret ax-mp (/\ (e. A (CC)) (e. (cv x) (H~))) a1i A absclt (e. (cv x) (H~)) adantr 3jca (br (` (norm) (cv x)) (<_) (1)) adantr A absge0t nmophm.1 T bdopft ax-mp T (cv x) nmoplbt mp3an1 anim12i anassrs sylanc eqbrtrd ex r19.21aiva syl3anc jca sylanc (=/= A (0)) adantr A absclt (=/= A (0)) adantr A abs00t eqneqd biimpar syl3anc (opr (` (normop) (opr A (.op) T)) (/) (` (abs) A)) rexrt syl nmophm.1 T bdopft ax-mp A T (cv x) homvalt mp3an2 (norm) fveq2d A (` T (cv x)) norm-iiit nmophm.1 T bdopft ax-mp (cv x) ffvrni sylan2 eqtrd (br (` (norm) (cv x)) (<_) (1)) adantr (opr A (.op) T) (cv x) nmoplbt nmophm.1 T bdopft ax-mp A T homulclt mpan2 syl3an1 3expa eqbrtrrd (=/= A (0)) adantllr (` (abs) A) (` (norm) (` T (cv x))) (` (normop) (opr A (.op) T)) lemuldiv2t A absclt (=/= A (0)) (e. (cv x) (H~)) ad2antrr nmophm.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl (/\ (e. A (CC)) (=/= A (0))) adantl (` (normop) (opr A (.op) T)) (opr (` (abs) A) (x.) (` (normop) T)) xrret nmophm.1 T bdopft ax-mp A T homulclt mpan2 (opr A (.op) T) nmopxrt syl A absclt nmophm.1 T nmopret ax-mp (` (abs) A) (` (normop) T) axmulrcl mpan2 syl jca nmophm.1 T bdopft ax-mp A T homulclt mpan2 (opr A (.op) T) nmopgtmnf syl (opr A (.op) T) (opr (` (abs) A) (x.) (` (normop) T)) x nmopleubt nmophm.1 T bdopft ax-mp A T homulclt mpan2 A absclt nmophm.1 T nmopret ax-mp (` (abs) A) (` (normop) T) axmulrcl mpan2 syl (opr (` (abs) A) (x.) (` (normop) T)) rexrt syl nmophm.1 T bdopft ax-mp A T (cv x) homvalt mp3an2 (norm) fveq2d A (` T (cv x)) norm-iiit nmophm.1 T bdopft ax-mp (cv x) ffvrni sylan2 eqtrd (br (` (norm) (cv x)) (<_) (1)) adantr (` (norm) (` T (cv x))) (` (normop) T) (` (abs) A) lemul2it nmophm.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl (e. A (CC)) adantl nmophm.1 T nmopret ax-mp (/\ (e. A (CC)) (e. (cv x) (H~))) a1i A absclt (e. (cv x) (H~)) adantr 3jca (br (` (norm) (cv x)) (<_) (1)) adantr A absge0t nmophm.1 T bdopft ax-mp T (cv x) nmoplbt mp3an1 anim12i anassrs sylanc eqbrtrd ex r19.21aiva syl3anc jca sylanc (=/= A (0)) (e. (cv x) (H~)) ad2antrr 3jca A absgt0t biimpa (e. (cv x) (H~)) adantr sylanc (br (` (norm) (cv x)) (<_) (1)) adantr mpbid ex r19.21aiva sylanc (` (abs) A) (` (normop) T) (` (normop) (opr A (.op) T)) lemuldiv2t A absclt (=/= A (0)) adantr nmophm.1 T nmopret ax-mp (/\ (e. A (CC)) (=/= A (0))) a1i (` (normop) (opr A (.op) T)) (opr (` (abs) A) (x.) (` (normop) T)) xrret nmophm.1 T bdopft ax-mp A T homulclt mpan2 (opr A (.op) T) nmopxrt syl A absclt nmophm.1 T nmopret ax-mp (` (abs) A) (` (normop) T) axmulrcl mpan2 syl jca nmophm.1 T bdopft ax-mp A T homulclt mpan2 (opr A (.op) T) nmopgtmnf syl (opr A (.op) T) (opr (` (abs) A) (x.) (` (normop) T)) x nmopleubt nmophm.1 T bdopft ax-mp A T homulclt mpan2 A absclt nmophm.1 T nmopret ax-mp (` (abs) A) (` (normop) T) axmulrcl mpan2 syl (opr (` (abs) A) (x.) (` (normop) T)) rexrt syl nmophm.1 T bdopft ax-mp A T (cv x) homvalt mp3an2 (norm) fveq2d A (` T (cv x)) norm-iiit nmophm.1 T bdopft ax-mp (cv x) ffvrni sylan2 eqtrd (br (` (norm) (cv x)) (<_) (1)) adantr (` (norm) (` T (cv x))) (` (normop) T) (` (abs) A) lemul2it nmophm.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl (e. A (CC)) adantl nmophm.1 T nmopret ax-mp (/\ (e. A (CC)) (e. (cv x) (H~))) a1i A absclt (e. (cv x) (H~)) adantr 3jca (br (` (norm) (cv x)) (<_) (1)) adantr A absge0t nmophm.1 T bdopft ax-mp T (cv x) nmoplbt mp3an1 anim12i anassrs sylanc eqbrtrd ex r19.21aiva syl3anc jca sylanc (=/= A (0)) adantr 3jca A absgt0t biimpa sylanc mpbird A (0) df-ne sylan2br pm2.61dan jca (` (normop) (opr A (.op) T)) (opr (` (abs) A) (x.) (` (normop) T)) letri3t (` (normop) (opr A (.op) T)) (opr (` (abs) A) (x.) (` (normop) T)) xrret nmophm.1 T bdopft ax-mp A T homulclt mpan2 (opr A (.op) T) nmopxrt syl A absclt nmophm.1 T nmopret ax-mp (` (abs) A) (` (normop) T) axmulrcl mpan2 syl jca nmophm.1 T bdopft ax-mp A T homulclt mpan2 (opr A (.op) T) nmopgtmnf syl (opr A (.op) T) (opr (` (abs) A) (x.) (` (normop) T)) x nmopleubt nmophm.1 T bdopft ax-mp A T homulclt mpan2 A absclt nmophm.1 T nmopret ax-mp (` (abs) A) (` (normop) T) axmulrcl mpan2 syl (opr (` (abs) A) (x.) (` (normop) T)) rexrt syl nmophm.1 T bdopft ax-mp A T (cv x) homvalt mp3an2 (norm) fveq2d A (` T (cv x)) norm-iiit nmophm.1 T bdopft ax-mp (cv x) ffvrni sylan2 eqtrd (br (` (norm) (cv x)) (<_) (1)) adantr (` (norm) (` T (cv x))) (` (normop) T) (` (abs) A) lemul2it nmophm.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl (e. A (CC)) adantl nmophm.1 T nmopret ax-mp (/\ (e. A (CC)) (e. (cv x) (H~))) a1i A absclt (e. (cv x) (H~)) adantr 3jca (br (` (norm) (cv x)) (<_) (1)) adantr A absge0t nmophm.1 T bdopft ax-mp T (cv x) nmoplbt mp3an1 anim12i anassrs sylanc eqbrtrd ex r19.21aiva syl3anc jca sylanc A absclt nmophm.1 T nmopret ax-mp (` (abs) A) (` (normop) T) axmulrcl mpan2 syl sylanc mpbird)) thm (bdophm () ((nmophm.1 (e. T (BndLinOp)))) (-> (e. A (CC)) (e. (opr A (.op) T) (BndLinOp))) (nmophm.1 T bdoplnt ax-mp A lnopm nmophm.1 A nmophm A absclt nmophm.1 T nmopret ax-mp (` (abs) A) (` (normop) T) axmulrcl mpan2 syl eqeltrd jca (opr A (.op) T) elbdop2t sylibr)) thm (lnopcon ((x y) (x z) (w x) (v x) (u x) (T x) (y z) (w y) (v y) (u y) (T y) (w z) (v z) (u z) (T z) (v w) (u w) (T w) (u v) (T v) (T u)) ((lnopcon.1 (e. T (LinOp)))) (<-> (e. T (ConOp)) (E.e. x (RR) (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))))) ((cv x) (` (normop) T) (x.) (` (norm) (cv y)) opreq1 (` (norm) (` T (cv y))) (<_) breq2d y (H~) ralbidv (RR) rcla4ev lnopcon.1 T nmcopext mpan lnopcon.1 T (cv y) nmcoplbt mp3an1 r19.21aiva sylanc (cv x) (` (norm) (cv z)) prodge02t (cv z) normclt (-. (= (cv z) (0v))) adantr (e. (cv x) (RR)) anim2i ancoms (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantr (cv z) normgt0tOLD biimpa (e. (cv x) (RR)) (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) ad2antrr 0re (/\ (/\ (/\ (e. (cv z) (H~)) (-. (= (cv z) (0v)))) (e. (cv x) (RR))) (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))))) a1i lnopcon.1 lnopf (cv z) ffvrni (` T (cv z)) normclt syl (-. (= (cv z) (0v))) adantr (e. (cv x) (RR)) (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) ad2antrr (cv x) (` (norm) (cv z)) axmulrcl (cv z) normclt sylan2 ancoms (-. (= (cv z) (0v))) adantlr (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantr lnopcon.1 lnopf (cv z) ffvrni (` T (cv z)) normge0t syl (-. (= (cv z) (0v))) adantr (e. (cv x) (RR)) (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) ad2antrr (cv y) (cv z) T fveq2 (norm) fveq2d (cv y) (cv z) (norm) fveq2 (cv x) (x.) opreq2d (<_) breq12d (H~) rcla4va (-. (= (cv z) (0v))) adantlr (e. (cv x) (RR)) adantlr letrd jca sylanc 0re (0) (cv x) leloet mpan (/\ (e. (cv z) (H~)) (-. (= (cv z) (0v)))) (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) ad2antlr mpbid (cv v) (opr (cv w) (/) (cv x)) (0) (<) breq2 (cv v) (opr (cv w) (/) (cv x)) (` (norm) (opr (cv u) (-v) (cv z))) (<) breq2 (br (` (norm) (opr (` T (cv u)) (-v) (` T (cv z)))) (<) (cv w)) imbi1d u (H~) ralbidv anbi12d (RR) rcla4ev (cv w) (cv x) redivclt (e. (cv w) (RR)) (br (0) (<) (cv w)) pm3.26 (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) adantl (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (/\ (e. (cv w) (RR)) (br (0) (<) (cv w))) adantr (cv x) gt0ne0t (/\ (e. (cv w) (RR)) (br (0) (<) (cv w))) adantr syl3anc (e. (cv z) (H~)) adantrll (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantllr (cv w) (cv x) divgt0t (e. (cv w) (RR)) (br (0) (<) (cv w)) pm3.26 (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) adantl (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (/\ (e. (cv w) (RR)) (br (0) (<) (cv w))) adantr jca (e. (cv w) (RR)) (br (0) (<) (cv w)) pm3.27 (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) adantl (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.27 (/\ (e. (cv w) (RR)) (br (0) (<) (cv w))) adantr jca sylanc (e. (cv z) (H~)) adantrll (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantllr (` T (cv u)) (` T (cv z)) hvsubclt lnopcon.1 lnopf (cv u) ffvrni lnopcon.1 lnopf (cv z) ffvrni syl2an ancoms (opr (` T (cv u)) (-v) (` T (cv z))) normclt syl (e. (cv w) (RR)) adantlr (br (0) (<) (cv w)) adantlr (/\ (/\ (e. (cv x) (RR)) (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))))) (br (0) (<) (cv x))) (br (` (norm) (opr (cv u) (-v) (cv z))) (<) (opr (cv w) (/) (cv x))) ad2antrl (cv x) (` (norm) (opr (cv u) (-v) (cv z))) axmulrcl (cv u) (cv z) hvsubclt ancoms (opr (cv u) (-v) (cv z)) normclt syl sylan2 (e. (cv w) (RR)) adantrlr (br (0) (<) (cv w)) adantrlr (br (0) (<) (cv x)) (br (` (norm) (opr (cv u) (-v) (cv z))) (<) (opr (cv w) (/) (cv x))) ad2ant2r (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantllr (e. (cv z) (H~)) (e. (cv w) (RR)) pm3.27 (br (0) (<) (cv w)) adantr (e. (cv u) (H~)) (br (` (norm) (opr (cv u) (-v) (cv z))) (<) (opr (cv w) (/) (cv x))) ad2antrr (/\ (/\ (e. (cv x) (RR)) (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))))) (br (0) (<) (cv x))) adantl lnopcon.1 (cv u) (cv z) lnopsub ancoms (norm) fveq2d (/\ (e. (cv x) (RR)) (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))))) adantl (cv y) (opr (cv u) (-v) (cv z)) T fveq2 (norm) fveq2d (cv y) (opr (cv u) (-v) (cv z)) (norm) fveq2 (cv x) (x.) opreq2d (<_) breq12d (H~) rcla4va (cv u) (cv z) hvsubclt ancoms sylan ancoms (e. (cv x) (RR)) adantll eqbrtrrd (br (0) (<) (cv x)) adantlr (e. (cv w) (RR)) adantrlr (br (0) (<) (cv w)) adantrlr (br (` (norm) (opr (cv u) (-v) (cv z))) (<) (opr (cv w) (/) (cv x))) adantrr (cv x) (` (norm) (opr (cv u) (-v) (cv z))) (cv w) ltmuldiv2t (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (/\ (/\ (e. (cv z) (H~)) (e. (cv w) (RR))) (e. (cv u) (H~))) adantr (cv u) (cv z) hvsubclt ancoms (opr (cv u) (-v) (cv z)) normclt syl (e. (cv w) (RR)) adantlr (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) adantl (e. (cv z) (H~)) (e. (cv w) (RR)) pm3.27 (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) (e. (cv u) (H~)) ad2antrl 3jca (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.27 (/\ (/\ (e. (cv z) (H~)) (e. (cv w) (RR))) (e. (cv u) (H~))) adantr sylanc biimprd (br (0) (<) (cv w)) adantrlr ex imp32 (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantllr lelttrd anassrs ex anassrs r19.21aiva jca sylanc exp32 r19.21aivv T z w v u elcnopt lnopcon.1 lnopf mpbiran sylibr ex (0) (cv x) (x.) (` (norm) (cv y)) opreq1 (` (norm) (` T (cv y))) (<_) breq2d y (H~) ralbidv (cv y) normclt recnd (` (norm) (cv y)) mul02t syl (` (norm) (` T (cv y))) (<_) breq2d ralbiia lnopcon.1 lnopf (cv y) ffvrni (` T (cv y)) normge0t (br (` (norm) (` T (cv y))) (<_) (0)) biantrud (` T (cv y)) normclt 0re (` (norm) (` T (cv y))) (0) letri3t mpan2 syl (` T (cv y)) norm-it 3bitr2d syl (cv y) ho0valtOLD (` T (cv y)) eqeq2d bitr4d ralbiia lnopcon.1 lnopf T (H~) (H~) ffn ax-mp 0lnopOLD lnopf (X. (H~) (0H)) (H~) (H~) ffn ax-mp T (H~) (X. (H~) (0H)) (H~) y eqfnfv mp2an (H~) eqid mpbiran biimpr 0cnop syl6eqel sylbi sylbi syl6bir com12 (e. (cv x) (RR)) adantl jaod (/\ (e. (cv z) (H~)) (-. (= (cv z) (0v)))) adantll mpd exp31 r19.23adv ex r19.23aiv (cv y) (cv z) (0v) eqeq1 (H~) cbvralv z (H~) (= (cv z) (0v)) dfral2 bitr2 (cv x) recnt (cv x) mul01t syl (` (norm) (` T (cv y))) (<_) breq2d y (H~) ralbidv lnopcon.1 lnopf (cv y) ffvrni (` T (cv y)) normge0t (br (` (norm) (` T (cv y))) (<_) (0)) biantrud (` T (cv y)) normclt 0re (` (norm) (` T (cv y))) (0) letri3t mpan2 syl (` T (cv y)) norm-it 3bitr2d syl (cv y) ho0valtOLD (` T (cv y)) eqeq2d bitr4d ralbiia lnopcon.1 lnopf T (H~) (H~) ffn ax-mp 0lnopOLD lnopf (X. (H~) (0H)) (H~) (H~) ffn ax-mp T (H~) (X. (H~) (0H)) (H~) y eqfnfv mp2an (H~) eqid mpbiran biimpr 0cnop syl6eqel sylbi syl6bi y (H~) (= (cv y) (0v)) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))) r19.26 (cv y) (0v) (norm) fveq2 norm0 syl6eq (cv x) (x.) opreq2d (` (norm) (` T (cv y))) (<_) breq2d biimpa y (H~) r19.20si sylbir syl5 exp3a com12 r19.23adv sylbi pm2.61i impbi)) thm (lnopcont ((x y) (T x) (T y)) () (-> (e. T (LinOp)) (<-> (e. T (ConOp)) (E.e. x (RR) (A.e. y (H~) (br (` (norm) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))))))) (T (if (e. T (LinOp)) T (|` (I) (H~))) (ConOp) eleq1 T (if (e. T (LinOp)) T (|` (I) (H~))) (cv y) fveq1 (norm) fveq2d (<_) (opr (cv x) (x.) (` (norm) (cv y))) breq1d y (H~) ralbidv x (RR) rexbidv bibi12d idlnop T elimel x y lnopcon dedth)) thm (lnopcnbdt ((x y) (T x) (T y)) () (-> (e. T (LinOp)) (<-> (e. T (ConOp)) (e. T (BndLinOp)))) (T nmcopext ex T elbdop2t baibr sylibd T x y lnopcont (cv x) (` (normop) T) (x.) (` (norm) (cv y)) opreq1 (` (norm) (` T (cv y))) (<_) breq2d y (H~) ralbidv (RR) rcla4ev T nmopret T (cv y) nmbdoplbt r19.21aiva sylanc syl5bir impbid)) thm (lncnopbd () () (<-> (e. T (i^i (LinOp) (ConOp))) (e. T (BndLinOp))) (T (LinOp) (ConOp) elin T lnopcnbdt biimpa T bdoplnt T bdoplnt T lnopcnbdt biimparc mpdan jca impbi bitr)) thm (lncnbd () () (= (i^i (LinOp) (ConOp)) (BndLinOp)) ((cv t) lncnopbd eqriv)) thm (lnopcnret () () (-> (e. T (LinOp)) (<-> (e. T (ConOp)) (e. (` (normop) T) (RR)))) (T lnopcnbdt T elbdop2t baib bitrd)) thm (lnfnl () ((lnfnl.1 (e. T (LinFn)))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr (opr A (.s) B) (+v) C)) (opr (opr A (x.) (` T B)) (+) (` T C)))) (lnfnl.1 T A B C lnfnlt mpanl1 3impb)) thm (lnfnf () ((lnfnl.1 (e. T (LinFn)))) (:--> T (H~) (CC)) (lnfnl.1 T lnfnft ax-mp)) thm (lnfn0 () ((lnfnl.1 (e. T (LinFn)))) (= (` T (0v)) (0)) (1cn ax-hv0cl ax-hv0cl lnfnl.1 (1) (0v) (0v) lnfnl mp3an 1cn ax-hv0cl hvmulcl (opr (1) (.s) (0v)) ax-hvaddid ax-mp ax-hv0cl (0v) ax-hvmulid ax-mp eqtr T fveq2i ax-hv0cl lnfnl.1 lnfnf (0v) ffvrni ax-mp mulid2 (+) (` T (0v)) opreq1i 3eqtr3 (-) (` T (0v)) opreq1i ax-hv0cl lnfnl.1 lnfnf (0v) ffvrni ax-mp subid ax-hv0cl lnfnl.1 lnfnf (0v) ffvrni ax-mp ax-hv0cl lnfnl.1 lnfnf (0v) ffvrni ax-mp (` T (0v)) (` T (0v)) pncant mp2an 3eqtr3r)) thm (lnfnadd () ((lnfnl.1 (e. T (LinFn)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` T (opr A (+v) B)) (opr (` T A) (+) (` T B)))) (1cn lnfnl.1 (1) A B lnfnl mp3an1 A ax-hvmulid (+v) B opreq1d T fveq2d (e. B (H~)) adantr lnfnl.1 lnfnf A ffvrni (` T A) mulid2t syl (e. B (H~)) adantr (+) (` T B) opreq1d 3eqtr3d)) thm (lnfnmul () ((lnfnl.1 (e. T (LinFn)))) (-> (/\ (e. A (CC)) (e. B (H~))) (= (` T (opr A (.s) B)) (opr A (x.) (` T B)))) (ax-hv0cl lnfnl.1 A B (0v) lnfnl mp3an3 A B ax-hvmulcl (opr A (.s) B) ax-hvaddid syl T fveq2d A (` T B) axmulcl lnfnl.1 lnfnf B ffvrni sylan2 (opr A (x.) (` T B)) ax0id syl lnfnl.1 lnfn0 (opr A (x.) (` T B)) (+) opreq2i syl5eq 3eqtr3d)) thm (lnfnaddmul () ((lnfnl.1 (e. T (LinFn)))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr B (+v) (opr A (.s) C))) (opr (` T B) (+) (opr A (x.) (` T C))))) (lnfnl.1 B (opr A (.s) C) lnfnadd A C ax-hvmulcl sylan2 3impb 3com12 lnfnl.1 A C lnfnmul (e. B (H~)) 3adant2 (` T B) (+) opreq2d eqtrd)) thm (lnfnsub () ((lnfnl.1 (e. T (LinFn)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` T (opr A (-v) B)) (opr (` T A) (-) (` T B)))) (1cn negcl lnfnl.1 (-u (1)) A B lnfnaddmul mp3an1 A B hvsubvalt T fveq2d (` T B) mulm1t (` T A) (+) opreq2d (e. (` T A) (CC)) adantl (` T A) (` T B) negsubt eqtr2d lnfnl.1 lnfnf A ffvrni lnfnl.1 lnfnf B ffvrni syl2an 3eqtr4d)) thm (lnfnmult () () (-> (/\/\ (e. T (LinFn)) (e. A (CC)) (e. B (H~))) (= (` T (opr A (.s) B)) (opr A (x.) (` T B)))) (T (if (e. T (LinFn)) T (X. (H~) ({} (0)))) (opr A (.s) B) fveq1 T (if (e. T (LinFn)) T (X. (H~) ({} (0)))) B fveq1 A (x.) opreq2d eqeq12d (/\ (e. A (CC)) (e. B (H~))) imbi2d 0lnfn T elimel A B lnfnmul dedth 3impib)) thm (nmbdfnlb () ((nmbdfnlb.1 (/\ (e. T (LinFn)) (e. (` (normfn) T) (RR))))) (-> (e. A (H~)) (br (` (abs) (` T A)) (<_) (opr (` (normfn) T) (x.) (` (norm) A)))) (A (0v) T fveq2 nmbdfnlb.1 pm3.26i lnfn0 syl6eq (abs) fveq2d abs0 syl6eq A (0v) (norm) fveq2 norm0 syl6eq (` (normfn) T) (x.) opreq2d nmbdfnlb.1 pm3.27i recn mul01 syl6req 0re leid syl5breq eqbrtrd (e. A (H~)) adantl (` (abs) (` T A)) (` (norm) A) divrec2t nmbdfnlb.1 pm3.26i lnfnf A ffvrni (` T A) absclt syl (=/= A (0v)) adantr recnd A normclt (=/= A (0v)) adantr recnd A normne0t biimpar syl3anc nmbdfnlb.1 pm3.26i (opr (1) (/) (` (norm) A)) A lnfnmul (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc recnd (e. A (H~)) (=/= A (0v)) pm3.26 sylanc (abs) fveq2d (opr (1) (/) (` (norm) A)) (` T A) absmult (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc recnd nmbdfnlb.1 pm3.26i lnfnf A ffvrni (=/= A (0v)) adantr sylanc (opr (1) (/) (` (norm) A)) absidt (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc 0re (0) (opr (1) (/) (` (norm) A)) ltlet mpan (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc (` (norm) A) recgt0t A normclt (=/= A (0v)) adantr A normgt0t biimpa sylanc sylc sylanc (x.) (` (abs) (` T A)) opreq1d 3eqtrrd eqtrd nmbdfnlb.1 pm3.26i lnfnf T (opr (opr (1) (/) (` (norm) A)) (.s) A) nmfnlbt mp3an1 (opr (1) (/) (` (norm) A)) A ax-hvmulcl (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc recnd (e. A (H~)) (=/= A (0v)) pm3.26 sylanc (` (norm) (opr (opr (1) (/) (` (norm) A)) (.s) A)) (1) eqlet (opr (1) (/) (` (norm) A)) A ax-hvmulcl (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc recnd (e. A (H~)) (=/= A (0v)) pm3.26 sylanc (opr (opr (1) (/) (` (norm) A)) (.s) A) normclt syl A norm1t sylanc sylanc eqbrtrd (` (abs) (` T A)) (` (norm) A) (` (normfn) T) ledivmul2t nmbdfnlb.1 pm3.26i lnfnf A ffvrni (` T A) absclt syl (=/= A (0v)) adantr A normclt (=/= A (0v)) adantr nmbdfnlb.1 pm3.27i (/\ (e. A (H~)) (=/= A (0v))) a1i 3jca A normgt0t biimpa sylanc mpbid A (0v) df-ne sylan2br pm2.61dan)) thm (nmbdfnlbt () () (-> (/\/\ (e. T (LinFn)) (e. (` (normfn) T) (RR)) (e. A (H~))) (br (` (abs) (` T A)) (<_) (opr (` (normfn) T) (x.) (` (norm) A)))) (T (if (/\ (e. T (LinFn)) (e. (` (normfn) T) (RR))) T (X. (H~) ({} (0)))) A fveq1 (abs) fveq2d T (if (/\ (e. T (LinFn)) (e. (` (normfn) T) (RR))) T (X. (H~) ({} (0)))) (normfn) fveq2 (x.) (` (norm) A) opreq1d (<_) breq12d (e. A (H~)) imbi2d T (if (/\ (e. T (LinFn)) (e. (` (normfn) T) (RR))) T (X. (H~) ({} (0)))) (LinFn) eleq1 T (if (/\ (e. T (LinFn)) (e. (` (normfn) T) (RR))) T (X. (H~) ({} (0)))) (normfn) fveq2 (RR) eleq1d anbi12d (X. (H~) ({} (0))) (if (/\ (e. T (LinFn)) (e. (` (normfn) T) (RR))) T (X. (H~) ({} (0)))) (LinFn) eleq1 (X. (H~) ({} (0))) (if (/\ (e. T (LinFn)) (e. (` (normfn) T) (RR))) T (X. (H~) ({} (0)))) (normfn) fveq2 (RR) eleq1d anbi12d 0lnfn nmfn0 0re eqeltr pm3.2i elimhyp A nmbdfnlb dedth 3impia)) thm (nmcfnexlem1 ((x y) (x z) (T x) (y z) (T y) (T z) (n x) (n y) (n z) (T n)) ((nmcfnex.1 (e. T (LinFn))) (nmcfnex.2 (e. T (ConFn)))) (-> (/\ (e. (cv n) (NN)) (A.e. x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (-. (br (cv n) (<) (` (abs) (` T (cv x)))))))) (e. (` (normfn) T) (RR))) (n (RR) (E.e. y ({|} z (E.e. x (H~) (/\ (br (` (norm) (cv x)) (<_) (1)) (= (cv z) (` (abs) (` T (cv x))))))) (br (cv n) (<) (cv y))) ra4 impcom (cv n) nnret nmcfnex.1 lnfnf T z x nmfnsetret ax-mp ressxr sstri ({|} z (E.e. x (H~) (/\ (br (` (norm) (cv x)) (<_) (1)) (= (cv z) (` (abs) (` T (cv x))))))) n y supxrunb2 nmcfnex.1 lnfnf T z x nmfnvalt ax-mp (+oo) eqeq1i syl6rbbr ax-mp biimp syl2an x (H~) y (/\ (/\ (br (` (norm) (cv x)) (<_) (1)) (= (cv y) (` (abs) (` T (cv x))))) (br (cv n) (<) (cv y))) rexcom4 y (br (` (norm) (cv x)) (<_) (1)) (/\ (= (cv y) (` (abs) (` T (cv x)))) (br (cv n) (<) (cv y))) 19.42v (br (` (norm) (cv x)) (<_) (1)) (= (cv y) (` (abs) (` T (cv x)))) (br (cv n) (<) (cv y)) anass bicomi y exbii (abs) (` T (cv x)) fvex (cv y) (` (abs) (` T (cv x))) (cv n) (<) breq2 ceqsexv (br (` (norm) (cv x)) (<_) (1)) anbi2i 3bitr3r x (H~) rexbii y visset (cv z) (cv y) (` (abs) (` T (cv x))) eqeq1 (br (` (norm) (cv x)) (<_) (1)) anbi2d x (H~) rexbidv elab (br (cv n) (<) (cv y)) anbi1i x (H~) (/\ (br (` (norm) (cv x)) (<_) (1)) (= (cv y) (` (abs) (` T (cv x))))) (br (cv n) (<) (cv y)) r19.41v bitr4 y exbii 3bitr4 y ({|} z (E.e. x (H~) (/\ (br (` (norm) (cv x)) (<_) (1)) (= (cv z) (` (abs) (` T (cv x))))))) (br (cv n) (<) (cv y)) df-rex bitr4 sylibr ex (br (` (norm) (cv x)) (<_) (1)) (br (cv n) (<) (` (abs) (` T (cv x)))) df-an x (H~) rexbii x (H~) (-> (br (` (norm) (cv x)) (<_) (1)) (-. (br (cv n) (<) (` (abs) (` T (cv x)))))) rexnal bitr syl6ib con2d imp nmcfnex.1 lnfnf T nmfnrepnf ax-mp sylibr)) thm (nmcfnexlem2 ((y z) (T y) (T z)) ((nmcfnex.1 (e. T (LinFn))) (nmcfnex.2 (e. T (ConFn)))) (E.e. y (RR) (/\ (br (0) (<) (cv y)) (A.e. z (H~) (-> (br (` (norm) (cv z)) (<) (cv y)) (br (` (abs) (` T (cv z))) (<) (1)))))) (nmcfnex.2 ax-hv0cl pm3.2i ax1re lt01 pm3.2i T (0v) (1) y z cnfnct mp2an (cv z) hvsub0t (norm) fveq2d (<) (cv y) breq1d nmcfnex.1 lnfnf (cv z) ffvrni (` T (cv z)) subid1t syl nmcfnex.1 lnfn0 (` T (cv z)) (-) opreq2i syl5eq (abs) fveq2d (<) (1) breq1d imbi12d ralbiia (br (0) (<) (cv y)) anbi2i y (RR) rexbii mpbi)) thm (nmcfnexlem3 ((T x) (n x) (T n)) ((nmcfnex.1 (e. T (LinFn))) (nmcfnex.2 (e. T (ConFn)))) (-> (/\ (e. (cv n) (NN)) (e. (cv x) (H~))) (<-> (br (` (abs) (` T (cv x))) (<) (cv n)) (br (` (abs) (` T (opr (opr (1) (/) (cv n)) (.s) (cv x)))) (<) (1)))) ((` (abs) (` T (cv x))) (cv n) (opr (1) (/) (cv n)) ltmul2t nmcfnex.1 lnfnf (cv x) ffvrni (` T (cv x)) absclt syl (e. (cv n) (NN)) adantl (cv n) nnret (e. (cv x) (H~)) adantr (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc (e. (cv x) (H~)) adantr 3jca (cv n) nnrecgt0t (e. (cv x) (H~)) adantr sylanc nmcfnex.1 (opr (1) (/) (cv n)) (cv x) lnfnmul (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd sylan (abs) fveq2d (opr (1) (/) (cv n)) (` T (cv x)) absmult (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd nmcfnex.1 lnfnf (cv x) ffvrni syl2an (opr (1) (/) (cv n)) absidt (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc 0re (0) (opr (1) (/) (cv n)) ltlet mpan (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc (cv n) nnrecgt0t sylc sylanc (e. (cv x) (H~)) adantr (x.) (` (abs) (` T (cv x))) opreq1d 3eqtrrd (cv n) recid2t (cv n) nncnt (cv n) nnne0t sylanc (e. (cv x) (H~)) adantr (<) breq12d bitrd)) thm (nmcfnexlem4 ((M k) (k y) (T k) (T y)) ((nmcfnex.1 (e. T (LinFn))) (nmcfnex.2 (e. T (ConFn))) (nmcfnexlem4.3 (= A ({e.|} k (NN) (br (opr (1) (/) (cv k)) (<) (cv y))))) (nmcfnexlem4.4 (= M (sup A (RR) (`' (<)))))) (-> (/\ (e. (cv y) (RR)) (br (0) (<) (cv y))) (/\ (e. M (NN)) (br (opr (1) (/) M) (<) (cv y)))) ((cv y) k nnreclt k (NN) (br (opr (1) (/) (cv k)) (<) (cv y)) rabn0 sylibr nmcfnexlem4.3 ({/}) eqeq1i negbii sylibr nmcfnexlem4.3 k (NN) (br (opr (1) (/) (cv k)) (<) (cv y)) ssrab2 eqsstr nnuz sseqtr A (1) infmssuzcl mpan syl nmcfnexlem4.3 syl6eleq nmcfnexlem4.4 syl5eqel (cv k) M (1) (/) opreq2 (<) (cv y) breq1d (NN) elrab sylib)) thm (nmcfnexlem5 ((k x) (M k) (M x) (k y) (T k) (x y) (T x) (T y) (n x) (n y) (T n)) ((nmcfnex.1 (e. T (LinFn))) (nmcfnex.2 (e. T (ConFn))) (nmcfnexlem4.3 (= A ({e.|} k (NN) (br (opr (1) (/) (cv k)) (<) (cv y))))) (nmcfnexlem4.4 (= M (sup A (RR) (`' (<)))))) (-> (/\/\ (/\ (e. (cv y) (RR)) (br (0) (<) (cv y))) (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1)))) (br (` (norm) (opr (opr (1) (/) (cv n)) (.s) (cv x))) (<) (cv y))) ((opr (1) (/) (cv n)) (cv x) ax-hvmulcl (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd sylan (opr (opr (1) (/) (cv n)) (.s) (cv x)) normclt syl (br M (<_) (cv n)) (br (` (norm) (cv x)) (<_) (1)) ad2ant2r (/\ (e. (cv y) (RR)) (br (0) (<) (cv y))) 3adant1 nmcfnex.1 nmcfnex.2 nmcfnexlem4.3 nmcfnexlem4.4 nmcfnexlem4 pm3.26d M rerecclt M nnret M nnne0t sylanc syl (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3ad2ant1 (e. (cv y) (RR)) (br (0) (<) (cv y)) pm3.26 (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3ad2ant1 (opr (1) (/) (cv n)) (cv x) ax-hvmulcl (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd sylan (opr (opr (1) (/) (cv n)) (.s) (cv x)) normclt syl (br M (<_) (cv n)) (br (` (norm) (cv x)) (<_) (1)) ad2ant2r (e. M (NN)) 3adant1 (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc (br M (<_) (cv n)) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) ad2antrr (e. M (NN)) 3adant1 M rerecclt M nnret M nnne0t sylanc (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3ad2ant1 (` (norm) (cv x)) (1) (opr (1) (/) (cv n)) lemul2it (cv x) normclt (e. M (NN)) (br (` (norm) (cv x)) (<_) (1)) ad2antrl (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) 3adant2 ax1re (/\/\ (e. M (NN)) (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1)))) a1i (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc (br M (<_) (cv n)) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) ad2antrr (e. M (NN)) 3adant1 3jca ax1re 0re ax1re lt01 ltlei (1) (cv n) divge0t an4s mpanl12 (cv n) nnret (cv n) nngt0t sylanc (e. M (NN)) (br M (<_) (cv n)) ad2antrl (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3adant3 (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1)) pm3.27 (e. M (NN)) (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) 3ad2ant3 jca sylanc (opr (1) (/) (cv n)) (cv x) norm-iiit (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd sylan (opr (1) (/) (cv n)) absidt (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc ax1re 0re ax1re lt01 ltlei (1) (cv n) divge0t an4s mpanl12 (cv n) nnret (cv n) nngt0t sylanc sylanc (e. (cv x) (H~)) adantr (x.) (` (norm) (cv x)) opreq1d eqtr2d (br M (<_) (cv n)) (br (` (norm) (cv x)) (<_) (1)) ad2ant2r (e. M (NN)) 3adant1 (cv n) rerecclt (cv n) nnret (cv n) nnne0t sylanc recnd (opr (1) (/) (cv n)) ax1id syl (e. M (NN)) (br M (<_) (cv n)) ad2antrl (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3adant3 3brtr3d M (cv n) lerect M nnret M nngt0t jca (cv n) nnret (cv n) nngt0t jca syl2an biimpa anasss (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3adant3 letrd nmcfnex.1 nmcfnex.2 nmcfnexlem4.3 nmcfnexlem4.4 nmcfnexlem4 pm3.26d syl3an1 nmcfnex.1 nmcfnex.2 nmcfnexlem4.3 nmcfnexlem4.4 nmcfnexlem4 pm3.27d (/\ (e. (cv n) (NN)) (br M (<_) (cv n))) (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) 3ad2ant1 lelttrd)) thm (nmcfnexlem6 ((k x) (M k) (M x) (k y) (k z) (T k) (x y) (x z) (T x) (y z) (T y) (T z)) ((nmcfnex.1 (e. T (LinFn))) (nmcfnex.2 (e. T (ConFn))) (nmcfnexlem4.3 (= A ({e.|} k (NN) (br (opr (1) (/) (cv k)) (<) (cv y))))) (nmcfnexlem4.4 (= M (sup A (RR) (`' (<)))))) (e. (` (normfn) T) (RR)) (nmcfnex.1 nmcfnex.2 y z nmcfnexlem2 nmcfnex.1 nmcfnex.2 nmcfnexlem4.3 nmcfnexlem4.4 nmcfnexlem4 pm3.26d ex (A.e. z (H~) (-> (br (` (norm) (cv z)) (<) (cv y)) (br (` (abs) (` T (cv z))) (<) (1)))) adantrd nmcfnex.1 nmcfnex.2 nmcfnexlem4.3 nmcfnexlem4.4 k x nmcfnexlem5 3expb (opr (1) (/) (cv k)) (cv x) ax-hvmulcl (cv k) rerecclt (cv k) nnret (cv k) nnne0t sylanc recnd sylan (cv z) (opr (opr (1) (/) (cv k)) (.s) (cv x)) (norm) fveq2 (<) (cv y) breq1d (cv z) (opr (opr (1) (/) (cv k)) (.s) (cv x)) T fveq2 (abs) fveq2d (<) (1) breq1d imbi12d (H~) rcla4v syl nmcfnex.1 nmcfnex.2 k x nmcfnexlem3 (` (abs) (` T (cv x))) (cv k) ltnsymt nmcfnex.1 lnfnf (cv x) ffvrni (` T (cv x)) absclt syl (cv k) nnret syl2an ancoms sylbird syl6d (br M (<_) (cv k)) (br (` (norm) (cv x)) (<_) (1)) ad2ant2r (/\ (e. (cv y) (RR)) (br (0) (<) (cv y))) adantl mpid ex com23 imp exp4d anasss imp r19.21aiv nmcfnex.1 nmcfnex.2 k x nmcfnexlem1 (br M (<_) (cv k)) adantlr (/\ (e. (cv y) (RR)) (/\ (br (0) (<) (cv y)) (A.e. z (H~) (-> (br (` (norm) (cv z)) (<) (cv y)) (br (` (abs) (` T (cv z))) (<) (1)))))) adantll mpdan exp32 r19.23adv M nnret M leidt syl (cv k) M M (<_) breq2 (NN) rcla4ev mpdan syl5 ex mpdd r19.23aiv ax-mp)) thm (nmcfnex ((k y) (k z) (T k) (y z) (T y) (T z)) ((nmcfnex.1 (e. T (LinFn))) (nmcfnex.2 (e. T (ConFn)))) (e. (` (normfn) T) (RR)) (nmcfnex.1 nmcfnex.2 ({e.|} k (NN) (br (opr (1) (/) (cv k)) (<) (cv y))) eqid (cv z) (cv k) (1) (/) opreq2 (<) (cv y) breq1d (NN) cbvrabv ({e.|} z (NN) (br (opr (1) (/) (cv z)) (<) (cv y))) ({e.|} k (NN) (br (opr (1) (/) (cv k)) (<) (cv y))) (RR) (`' (<)) supeq1 ax-mp nmcfnexlem6)) thm (nmcfnlb () ((nmcfnex.1 (e. T (LinFn))) (nmcfnex.2 (e. T (ConFn)))) (-> (e. A (H~)) (br (` (abs) (` T A)) (<_) (opr (` (normfn) T) (x.) (` (norm) A)))) (A (0v) T fveq2 nmcfnex.1 lnfn0 syl6eq (abs) fveq2d abs0 syl6eq A (0v) (norm) fveq2 norm0 syl6eq (` (normfn) T) (x.) opreq2d nmcfnex.1 nmcfnex.2 nmcfnex recn mul01 syl6req 0re leid syl5breq eqbrtrd (e. A (H~)) adantl (` (abs) (` T A)) (` (norm) A) divrec2t nmcfnex.1 lnfnf A ffvrni (` T A) absclt syl (-. (= A (0v))) adantr recnd A normclt (-. (= A (0v))) adantr recnd A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr syl3anc nmcfnex.1 (opr (1) (/) (` (norm) A)) A lnfnmul (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd (e. A (H~)) (-. (= A (0v))) pm3.26 sylanc (abs) fveq2d (opr (1) (/) (` (norm) A)) (` T A) absmult (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd nmcfnex.1 lnfnf A ffvrni (-. (= A (0v))) adantr sylanc (opr (1) (/) (` (norm) A)) absidt (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc 0re (0) (opr (1) (/) (` (norm) A)) ltlet mpan (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc (` (norm) A) recgt0t A normclt (-. (= A (0v))) adantr A normgt0tOLD biimpa sylanc sylc sylanc (x.) (` (abs) (` T A)) opreq1d 3eqtrrd eqtrd nmcfnex.1 lnfnf T (opr (opr (1) (/) (` (norm) A)) (.s) A) nmfnlbt mp3an1 (opr (1) (/) (` (norm) A)) A ax-hvmulcl (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd (e. A (H~)) (-. (= A (0v))) pm3.26 sylanc (` (norm) (opr (opr (1) (/) (` (norm) A)) (.s) A)) (1) eqlet (opr (1) (/) (` (norm) A)) A ax-hvmulcl (` (norm) A) rerecclt A normclt (-. (= A (0v))) adantr A norm-it negbid biimpar (` (norm) A) (0) df-ne sylibr sylanc recnd (e. A (H~)) (-. (= A (0v))) pm3.26 sylanc (opr (opr (1) (/) (` (norm) A)) (.s) A) normclt syl A norm1t A (0v) df-ne sylan2br sylanc sylanc eqbrtrd (` (abs) (` T A)) (` (norm) A) (` (normfn) T) ledivmul2t nmcfnex.1 lnfnf A ffvrni (` T A) absclt syl (-. (= A (0v))) adantr A normclt (-. (= A (0v))) adantr nmcfnex.1 nmcfnex.2 nmcfnex (/\ (e. A (H~)) (-. (= A (0v)))) a1i 3jca A normgt0tOLD biimpa sylanc mpbid pm2.61dan)) thm (nmcfnext () () (-> (/\ (e. T (LinFn)) (e. T (ConFn))) (e. (` (normfn) T) (RR))) (T (LinFn) (ConFn) elin T (if (e. T (i^i (LinFn) (ConFn))) T (X. (H~) ({} (0)))) (normfn) fveq2 (RR) eleq1d (X. (H~) ({} (0))) (LinFn) (ConFn) elin 0lnfn 0cnfn mpbir2an T elimel (if (e. T (i^i (LinFn) (ConFn))) T (X. (H~) ({} (0)))) (LinFn) (ConFn) elin mpbi pm3.26i (X. (H~) ({} (0))) (LinFn) (ConFn) elin 0lnfn 0cnfn mpbir2an T elimel (if (e. T (i^i (LinFn) (ConFn))) T (X. (H~) ({} (0)))) (LinFn) (ConFn) elin mpbi pm3.27i nmcfnex dedth sylbir)) thm (nmcfnlbt () () (-> (/\/\ (e. T (LinFn)) (e. T (ConFn)) (e. A (H~))) (br (` (abs) (` T A)) (<_) (opr (` (normfn) T) (x.) (` (norm) A)))) (T (if (e. T (i^i (LinFn) (ConFn))) T (X. (H~) ({} (0)))) A fveq1 (abs) fveq2d T (if (e. T (i^i (LinFn) (ConFn))) T (X. (H~) ({} (0)))) (normfn) fveq2 (x.) (` (norm) A) opreq1d (<_) breq12d (e. A (H~)) imbi2d (X. (H~) ({} (0))) (LinFn) (ConFn) elin 0lnfn 0cnfn mpbir2an T elimel (if (e. T (i^i (LinFn) (ConFn))) T (X. (H~) ({} (0)))) (LinFn) (ConFn) elin mpbi pm3.26i (X. (H~) ({} (0))) (LinFn) (ConFn) elin 0lnfn 0cnfn mpbir2an T elimel (if (e. T (i^i (LinFn) (ConFn))) T (X. (H~) ({} (0)))) (LinFn) (ConFn) elin mpbi pm3.27i A nmcfnlb dedth imp T (LinFn) (ConFn) elin sylanbr 3impa)) thm (lnfncon ((x y) (x z) (w x) (v x) (u x) (T x) (y z) (w y) (v y) (u y) (T y) (w z) (v z) (u z) (T z) (v w) (u w) (T w) (u v) (T v) (T u)) ((lnfncon.1 (e. T (LinFn)))) (<-> (e. T (ConFn)) (E.e. x (RR) (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))))) ((cv x) (` (normfn) T) (x.) (` (norm) (cv y)) opreq1 (` (abs) (` T (cv y))) (<_) breq2d y (H~) ralbidv (RR) rcla4ev lnfncon.1 T nmcfnext mpan lnfncon.1 T (cv y) nmcfnlbt mp3an1 r19.21aiva sylanc (cv x) (` (norm) (cv z)) prodge02t (cv z) normclt (-. (= (cv z) (0v))) adantr (e. (cv x) (RR)) anim2i ancoms (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantr (cv z) normgt0tOLD biimpa (e. (cv x) (RR)) (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) ad2antrr 0re (/\ (/\ (/\ (e. (cv z) (H~)) (-. (= (cv z) (0v)))) (e. (cv x) (RR))) (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))))) a1i lnfncon.1 lnfnf (cv z) ffvrni (` T (cv z)) absclt syl (-. (= (cv z) (0v))) adantr (e. (cv x) (RR)) (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) ad2antrr (cv x) (` (norm) (cv z)) axmulrcl (cv z) normclt sylan2 ancoms (-. (= (cv z) (0v))) adantlr (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantr lnfncon.1 lnfnf (cv z) ffvrni (` T (cv z)) absge0t syl (-. (= (cv z) (0v))) adantr (e. (cv x) (RR)) (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) ad2antrr (cv y) (cv z) T fveq2 (abs) fveq2d (cv y) (cv z) (norm) fveq2 (cv x) (x.) opreq2d (<_) breq12d (H~) rcla4va (-. (= (cv z) (0v))) adantlr (e. (cv x) (RR)) adantlr letrd jca sylanc 0re (0) (cv x) leloet mpan (/\ (e. (cv z) (H~)) (-. (= (cv z) (0v)))) (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) ad2antlr mpbid (cv v) (opr (cv w) (/) (cv x)) (0) (<) breq2 (cv v) (opr (cv w) (/) (cv x)) (` (norm) (opr (cv u) (-v) (cv z))) (<) breq2 (br (` (abs) (opr (` T (cv u)) (-) (` T (cv z)))) (<) (cv w)) imbi1d u (H~) ralbidv anbi12d (RR) rcla4ev (cv w) (cv x) redivclt (e. (cv w) (RR)) (br (0) (<) (cv w)) pm3.26 (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) adantl (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (/\ (e. (cv w) (RR)) (br (0) (<) (cv w))) adantr (cv x) gt0ne0t (/\ (e. (cv w) (RR)) (br (0) (<) (cv w))) adantr syl3anc (e. (cv z) (H~)) adantrll (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantllr (cv w) (cv x) divgt0t (e. (cv w) (RR)) (br (0) (<) (cv w)) pm3.26 (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) adantl (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (/\ (e. (cv w) (RR)) (br (0) (<) (cv w))) adantr jca (e. (cv w) (RR)) (br (0) (<) (cv w)) pm3.27 (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) adantl (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.27 (/\ (e. (cv w) (RR)) (br (0) (<) (cv w))) adantr jca sylanc (e. (cv z) (H~)) adantrll (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantllr (` T (cv u)) (` T (cv z)) subclt lnfncon.1 lnfnf (cv u) ffvrni lnfncon.1 lnfnf (cv z) ffvrni syl2an ancoms (opr (` T (cv u)) (-) (` T (cv z))) absclt syl (e. (cv w) (RR)) adantlr (br (0) (<) (cv w)) adantlr (/\ (/\ (e. (cv x) (RR)) (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))))) (br (0) (<) (cv x))) (br (` (norm) (opr (cv u) (-v) (cv z))) (<) (opr (cv w) (/) (cv x))) ad2antrl (cv x) (` (norm) (opr (cv u) (-v) (cv z))) axmulrcl (cv u) (cv z) hvsubclt ancoms (opr (cv u) (-v) (cv z)) normclt syl sylan2 (e. (cv w) (RR)) adantrlr (br (0) (<) (cv w)) adantrlr (br (0) (<) (cv x)) (br (` (norm) (opr (cv u) (-v) (cv z))) (<) (opr (cv w) (/) (cv x))) ad2ant2r (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantllr (e. (cv z) (H~)) (e. (cv w) (RR)) pm3.27 (br (0) (<) (cv w)) adantr (e. (cv u) (H~)) (br (` (norm) (opr (cv u) (-v) (cv z))) (<) (opr (cv w) (/) (cv x))) ad2antrr (/\ (/\ (e. (cv x) (RR)) (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))))) (br (0) (<) (cv x))) adantl lnfncon.1 (cv u) (cv z) lnfnsub ancoms (abs) fveq2d (/\ (e. (cv x) (RR)) (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))))) adantl (cv y) (opr (cv u) (-v) (cv z)) T fveq2 (abs) fveq2d (cv y) (opr (cv u) (-v) (cv z)) (norm) fveq2 (cv x) (x.) opreq2d (<_) breq12d (H~) rcla4va (cv u) (cv z) hvsubclt ancoms sylan ancoms (e. (cv x) (RR)) adantll eqbrtrrd (br (0) (<) (cv x)) adantlr (e. (cv w) (RR)) adantrlr (br (0) (<) (cv w)) adantrlr (br (` (norm) (opr (cv u) (-v) (cv z))) (<) (opr (cv w) (/) (cv x))) adantrr (cv x) (` (norm) (opr (cv u) (-v) (cv z))) (cv w) ltmuldiv2t (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.26 (/\ (/\ (e. (cv z) (H~)) (e. (cv w) (RR))) (e. (cv u) (H~))) adantr (cv u) (cv z) hvsubclt ancoms (opr (cv u) (-v) (cv z)) normclt syl (e. (cv w) (RR)) adantlr (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) adantl (e. (cv z) (H~)) (e. (cv w) (RR)) pm3.27 (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) (e. (cv u) (H~)) ad2antrl 3jca (e. (cv x) (RR)) (br (0) (<) (cv x)) pm3.27 (/\ (/\ (e. (cv z) (H~)) (e. (cv w) (RR))) (e. (cv u) (H~))) adantr sylanc biimprd (br (0) (<) (cv w)) adantrlr ex imp32 (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))) adantllr lelttrd anassrs ex anassrs r19.21aiva jca sylanc exp32 r19.21aivv T z w v u elcnfnt lnfncon.1 lnfnf mpbiran sylibr ex (0) (cv x) (x.) (` (norm) (cv y)) opreq1 (` (abs) (` T (cv y))) (<_) breq2d y (H~) ralbidv (cv y) normclt recnd (` (norm) (cv y)) mul02t syl (` (abs) (` T (cv y))) (<_) breq2d ralbiia lnfncon.1 lnfnf (cv y) ffvrni (` T (cv y)) absge0t (br (` (abs) (` T (cv y))) (<_) (0)) biantrud (` T (cv y)) absclt 0re (` (abs) (` T (cv y))) (0) letri3t mpan2 syl (` T (cv y)) abs00t 3bitr2d syl 0cn elisseti (cv y) (H~) fvconst2 (` T (cv y)) eqeq2d bitr4d ralbiia lnfncon.1 lnfnf T (H~) (CC) ffn ax-mp 0lnfn lnfnf (X. (H~) ({} (0))) (H~) (CC) ffn ax-mp T (H~) (X. (H~) ({} (0))) (H~) y eqfnfv mp2an (H~) eqid mpbiran biimpr 0cnfn syl6eqel sylbi sylbi syl6bir com12 (e. (cv x) (RR)) adantl jaod (/\ (e. (cv z) (H~)) (-. (= (cv z) (0v)))) adantll mpd exp31 r19.23adv ex r19.23aiv (cv y) (cv z) (0v) eqeq1 (H~) cbvralv z (H~) (= (cv z) (0v)) dfral2 bitr2 (cv x) recnt (cv x) mul01t syl (` (abs) (` T (cv y))) (<_) breq2d y (H~) ralbidv lnfncon.1 lnfnf (cv y) ffvrni (` T (cv y)) absge0t (br (` (abs) (` T (cv y))) (<_) (0)) biantrud (` T (cv y)) absclt 0re (` (abs) (` T (cv y))) (0) letri3t mpan2 syl (` T (cv y)) abs00t 3bitr2d syl 0cn elisseti (cv y) (H~) fvconst2 (` T (cv y)) eqeq2d bitr4d ralbiia lnfncon.1 lnfnf T (H~) (CC) ffn ax-mp 0lnfn lnfnf (X. (H~) ({} (0))) (H~) (CC) ffn ax-mp T (H~) (X. (H~) ({} (0))) (H~) y eqfnfv mp2an (H~) eqid mpbiran biimpr 0cnfn syl6eqel sylbi syl6bi y (H~) (= (cv y) (0v)) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))) r19.26 (cv y) (0v) (norm) fveq2 norm0 syl6eq (cv x) (x.) opreq2d (` (abs) (` T (cv y))) (<_) breq2d biimpa y (H~) r19.20si sylbir syl5 exp3a com12 r19.23adv sylbi pm2.61i impbi)) thm (lnfncont ((x y) (T x) (T y)) () (-> (e. T (LinFn)) (<-> (e. T (ConFn)) (E.e. x (RR) (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y)))))))) (T (if (e. T (LinFn)) T (X. (H~) ({} (0)))) (ConFn) eleq1 T (if (e. T (LinFn)) T (X. (H~) ({} (0)))) (cv y) fveq1 (abs) fveq2d (<_) (opr (cv x) (x.) (` (norm) (cv y))) breq1d y (H~) ralbidv x (RR) rexbidv bibi12d 0lnfn T elimel x y lnfncon dedth)) thm (lnfncnbdt ((x y) (T x) (T y)) () (-> (e. T (LinFn)) (<-> (e. T (ConFn)) (e. (` (normfn) T) (RR)))) (T nmcfnext ex (cv x) (` (normfn) T) (x.) (` (norm) (cv y)) opreq1 (` (abs) (` T (cv y))) (<_) breq2d y (H~) ralbidv (RR) rcla4ev (e. T (LinFn)) (e. (` (normfn) T) (RR)) pm3.27 T (cv y) nmbdfnlbt 3expa r19.21aiva sylanc ex T x y lnfncont sylibrd impbid)) thm (nlelsh ((x y) (T x) (T y)) ((nlelsh.1 (e. T (LinFn)))) (e. (` (null) T) (SH)) (nlelsh.1 lnfnf T x nlfnvalt ax-mp x (H~) (= (` T (cv x)) (0)) ssrab2 eqsstr (` (null) T) x y sh2 ax-mp nlelsh.1 lnfn0 nlelsh.1 lnfnf ax-hv0cl T (0v) elnlfnt mp2an mpbir nlelsh.1 (cv x) (cv y) lnfnadd nlelsh.1 lnfnf T x nlfnvalt ax-mp x (H~) (= (` T (cv x)) (0)) ssrab2 eqsstr (cv x) sseli nlelsh.1 lnfnf T x nlfnvalt ax-mp x (H~) (= (` T (cv x)) (0)) ssrab2 eqsstr (cv y) sseli syl2an nlelsh.1 lnfnf T (cv x) elnlfn2t mpan nlelsh.1 lnfnf T (cv y) elnlfn2t mpan (+) opreqan12d 0cn addid1 syl6eq eqtrd (cv x) (cv y) ax-hvaddcl nlelsh.1 lnfnf T x nlfnvalt ax-mp x (H~) (= (` T (cv x)) (0)) ssrab2 eqsstr (cv x) sseli nlelsh.1 lnfnf T x nlfnvalt ax-mp x (H~) (= (` T (cv x)) (0)) ssrab2 eqsstr (cv y) sseli syl2an nlelsh.1 lnfnf T (opr (cv x) (+v) (cv y)) elnlfnt mpan syl mpbird rgen2a nlelsh.1 (cv x) (cv y) lnfnmul nlelsh.1 lnfnf T x nlfnvalt ax-mp x (H~) (= (` T (cv x)) (0)) ssrab2 eqsstr (cv y) sseli sylan2 nlelsh.1 lnfnf T (cv y) elnlfn2t mpan (cv x) (x.) opreq2d (e. (cv x) (CC)) adantl (cv x) mul01t (e. (cv y) (` (null) T)) adantr 3eqtrd (cv x) (cv y) ax-hvmulcl nlelsh.1 lnfnf T x nlfnvalt ax-mp x (H~) (= (` T (cv x)) (0)) ssrab2 eqsstr (cv y) sseli sylan2 nlelsh.1 lnfnf T (opr (cv x) (.s) (cv y)) elnlfnt mpan syl mpbird rgen2a pm3.2i mpbir2an)) thm (nlelch ((g h) (g m) (g n) (g v) (g x) (g y) (T g) (h m) (h n) (h v) (h x) (h y) (T h) (m n) (m v) (m x) (m y) (T m) (n v) (n x) (n y) (T n) (v x) (v y) (T v) (x y) (T x) (T y)) ((nlelch.1 (e. T (LinFn))) (nlelch.2 (e. T (ConFn)))) (e. (` (null) T) (CH)) ((` (null) T) g h closedsub nlelch.1 nlelsh (` (abs) (` T (cv h))) y squeeze0 g visset h visset hlimvec nlelch.1 lnfnf (cv h) ffvrni syl (` T (cv h)) absclt syl (:--> (cv g) (NN) (` (null) T)) adantl g visset h visset hlimvec nlelch.1 lnfnf (cv h) ffvrni syl (` T (cv h)) absge0t syl (:--> (cv g) (NN) (` (null) T)) adantl nlelch.2 T (cv h) (cv y) x v cnfnct mpanl1 g visset h visset hlimvec sylan (:--> (cv g) (NN) (` (null) T)) adantll g visset h visset x m n hlimconv x (RR) (-> (br (0) (<) (cv x)) (E.e. m (NN) (A.e. n (NN) (-> (br (cv m) (<_) (cv n)) (br (` (norm) (opr (` (cv g) (cv n)) (-v) (cv h))) (<) (cv x)))))) ra4 syl imp32 (/\ (e. (cv y) (RR)) (br (0) (<) (cv y))) adantlr (:--> (cv g) (NN) (` (null) T)) adantlll (A.e. v (H~) (-> (br (` (norm) (opr (cv v) (-v) (cv h))) (<) (cv x)) (br (` (abs) (opr (` T (cv v)) (-) (` T (cv h)))) (<) (cv y)))) adantrrr (cv n) (cv m) (cv m) (<_) breq2 (br (` (abs) (` T (cv h))) (<) (cv y)) imbi1d (NN) rcla4v g visset h visset hlimvec nlelch.1 lnfnf (cv h) ffvrni syl (` T (cv h)) absnegt syl (:--> (cv g) (NN) (` (null) T)) (e. (cv n) (NN)) ad2antlr (cv g) (NN) (` (null) T) (cv n) ffvrn nlelch.1 lnfnf T (` (cv g) (cv n)) elnlfn2t mpan syl (-) (` T (cv h)) opreq1d (` T (cv h)) df-neg syl6reqr (abs) fveq2d (br (cv g) (~~>v) (cv h)) adantlr eqtr3d (A.e. v (H~) (-> (br (` (norm) (opr (cv v) (-v) (cv h))) (<) (cv x)) (br (` (abs) (opr (` T (cv v)) (-) (` T (cv h)))) (<) (cv y)))) (br (` (norm) (opr (` (cv g) (cv n)) (-v) (cv h))) (<) (cv x)) ad2antrr (cv v) (` (cv g) (cv n)) (-v) (cv h) opreq1 (norm) fveq2d (<) (cv x) breq1d (cv v) (` (cv g) (cv n)) T fveq2 (-) (` T (cv h)) opreq1d (abs) fveq2d (<) (cv y) breq1d imbi12d (H~) rcla4va (cv g) (NN) (` (null) T) (cv n) ffvrn nlelch.1 nlelsh (` (cv g) (cv n)) shel syl sylan (br (cv g) (~~>v) (cv h)) adantllr imp eqbrtrd ex an1rs (br (0) (<) (cv x)) adantlrl (e. (cv x) (RR)) adantlrl (/\ (e. (cv y) (RR)) (br (0) (<) (cv y))) adantllr (br (cv m) (<_) (cv n)) imim2d r19.20dva imp syl5com imp3a ex com23 (cv m) nnret (cv m) leidt syl ancli syl5 r19.23adv mpd exp32 r19.23adv mpd anassrs ex r19.21aiva syl3anc g visset h visset hlimvec nlelch.1 lnfnf (cv h) ffvrni syl (` T (cv h)) abs00t syl (:--> (cv g) (NN) (` (null) T)) adantl mpbid g visset h visset hlimvec nlelch.1 lnfnf T (cv h) elnlfnt mpan syl (:--> (cv g) (NN) (` (null) T)) adantl mpbird g h gen2 mpbir2an)) thm (riesz3 ((u v) (u w) (T u) (v w) (T v) (T w)) ((nlelch.1 (e. T (LinFn))) (nlelch.2 (e. T (ConFn)))) (E.e. w (H~) (A.e. v (H~) (= (` T (cv v)) (opr (cv v) (.i) (cv w))))) ((` (_|_) (` (null) T)) (0H) (_|_) fveq2 nlelch.1 nlelch.2 nlelch ococ choc0 3eqtr3g (cv v) eleq2d biimpar nlelch.1 lnfnf T (cv v) elnlfn2t mpan syl (cv v) hi02t (= (` (_|_) (` (null) T)) (0H)) adantl eqtr4d r19.21aiva ax-hv0cl (cv w) (0v) (cv v) (.i) opreq2 (` T (cv v)) eqeq2d v (H~) ralbidv (H~) rcla4ev mpan syl nlelch.1 nlelch.2 nlelch choccl u chne0OLD nlelch.1 nlelch.2 nlelch choccl (cv u) chel (cv w) (opr (` (*) (opr (` T (cv u)) (/) (opr (cv u) (.i) (cv u)))) (.s) (cv u)) (cv v) (.i) opreq2 (` T (cv v)) eqeq2d v (H~) ralbidv (H~) rcla4ev (` (*) (opr (` T (cv u)) (/) (opr (cv u) (.i) (cv u)))) (cv u) ax-hvmulcl (` T (cv u)) (opr (cv u) (.i) (cv u)) divclt nlelch.1 lnfnf (cv u) ffvrni (-. (= (cv u) (0v))) adantr (cv u) (cv u) ax-hicl anidms (-. (= (cv u) (0v))) adantr (cv u) his6t negbid (opr (cv u) (.i) (cv u)) (0) df-ne syl5bb biimpar syl3anc (opr (` T (cv u)) (/) (opr (cv u) (.i) (cv u))) cjclt syl (e. (cv u) (H~)) (-. (= (cv u) (0v))) pm3.26 sylanc (e. (cv u) (` (_|_) (` (null) T))) adantll (opr (` T (cv u)) (.s) (cv v)) (opr (` T (cv v)) (.s) (cv u)) (cv u) his2subt (` T (cv u)) (cv v) ax-hvmulcl nlelch.1 lnfnf (cv u) ffvrni sylan (` T (cv v)) (cv u) ax-hvmulcl nlelch.1 lnfnf (cv v) ffvrni sylan ancoms (e. (cv u) (H~)) (e. (cv v) (H~)) pm3.26 syl3anc (` T (cv u)) (cv v) (cv u) ax-his3 nlelch.1 lnfnf (cv u) ffvrni (e. (cv v) (H~)) adantr (e. (cv u) (H~)) (e. (cv v) (H~)) pm3.27 (e. (cv u) (H~)) (e. (cv v) (H~)) pm3.26 syl3anc (` T (cv v)) (cv u) (cv u) ax-his3 nlelch.1 lnfnf (cv v) ffvrni (e. (cv u) (H~)) adantl (e. (cv u) (H~)) (e. (cv v) (H~)) pm3.26 (e. (cv u) (H~)) (e. (cv v) (H~)) pm3.26 syl3anc (-) opreq12d eqtr2d (e. (cv u) (` (_|_) (` (null) T))) adantll nlelch.1 nlelch.2 nlelch chssi (` (null) T) (opr (opr (` T (cv u)) (.s) (cv v)) (-v) (opr (` T (cv v)) (.s) (cv u))) (cv u) ocorth ax-mp nlelch.1 (opr (` T (cv u)) (.s) (cv v)) (opr (` T (cv v)) (.s) (cv u)) lnfnsub (` T (cv u)) (cv v) ax-hvmulcl nlelch.1 lnfnf (cv u) ffvrni sylan (` T (cv v)) (cv u) ax-hvmulcl nlelch.1 lnfnf (cv v) ffvrni sylan ancoms sylanc nlelch.1 (` T (cv u)) (cv v) lnfnmul nlelch.1 lnfnf (cv u) ffvrni sylan nlelch.1 (` T (cv v)) (cv u) lnfnmul (` T (cv v)) (` T (cv u)) axmulcom nlelch.1 lnfnf (cv u) ffvrni sylan2 eqtrd nlelch.1 lnfnf (cv v) ffvrni sylan ancoms (-) opreq12d (` T (cv u)) (` T (cv v)) axmulcl nlelch.1 lnfnf (cv u) ffvrni nlelch.1 lnfnf (cv v) ffvrni syl2an (opr (` T (cv u)) (x.) (` T (cv v))) subidt syl 3eqtrd (opr (` T (cv u)) (.s) (cv v)) (opr (` T (cv v)) (.s) (cv u)) hvsubclt (` T (cv u)) (cv v) ax-hvmulcl nlelch.1 lnfnf (cv u) ffvrni sylan (` T (cv v)) (cv u) ax-hvmulcl nlelch.1 lnfnf (cv v) ffvrni sylan ancoms sylanc nlelch.1 lnfnf T (opr (opr (` T (cv u)) (.s) (cv v)) (-v) (opr (` T (cv v)) (.s) (cv u))) elnlfnt mpan syl mpbird sylan ancoms anassrs eqtrd (opr (` T (cv u)) (x.) (opr (cv v) (.i) (cv u))) (opr (` T (cv v)) (x.) (opr (cv u) (.i) (cv u))) subeq0t (` T (cv u)) (opr (cv v) (.i) (cv u)) axmulcl nlelch.1 lnfnf (cv u) ffvrni (e. (cv v) (H~)) adantr (cv v) (cv u) ax-hicl ancoms sylanc (` T (cv v)) (opr (cv u) (.i) (cv u)) axmulcl nlelch.1 lnfnf (cv v) ffvrni (cv u) (cv u) ax-hicl anidms syl2an ancoms sylanc (e. (cv u) (` (_|_) (` (null) T))) adantll mpbid (-. (= (cv u) (0v))) adantlr (opr (` T (cv u)) (x.) (opr (cv v) (.i) (cv u))) (opr (cv u) (.i) (cv u)) (` T (cv v)) divmul3t (` T (cv u)) (opr (cv v) (.i) (cv u)) axmulcl nlelch.1 lnfnf (cv u) ffvrni (e. (cv v) (H~)) adantr (cv v) (cv u) ax-hicl ancoms sylanc (cv u) (cv u) ax-hicl anidms (e. (cv v) (H~)) adantr nlelch.1 lnfnf (cv v) ffvrni (e. (cv u) (H~)) adantl 3jca (-. (= (cv u) (0v))) adantlr (cv u) his6t negbid (opr (cv u) (.i) (cv u)) (0) df-ne syl5bb biimpar (e. (cv v) (H~)) adantr sylanc (e. (cv u) (` (_|_) (` (null) T))) adantlll mpbird (` T (cv u)) (opr (cv v) (.i) (cv u)) (opr (cv u) (.i) (cv u)) div23t nlelch.1 lnfnf (cv u) ffvrni (e. (cv v) (H~)) adantr (cv v) (cv u) ax-hicl ancoms (cv u) (cv u) ax-hicl anidms (e. (cv v) (H~)) adantr 3jca (-. (= (cv u) (0v))) adantlr (cv u) his6t negbid (opr (cv u) (.i) (cv u)) (0) df-ne syl5bb biimpar (e. (cv v) (H~)) adantr sylanc (opr (` T (cv u)) (/) (opr (cv u) (.i) (cv u))) (cv v) (cv u) his52t (` T (cv u)) (opr (cv u) (.i) (cv u)) divclt nlelch.1 lnfnf (cv u) ffvrni (-. (= (cv u) (0v))) adantr (cv u) (cv u) ax-hicl anidms (-. (= (cv u) (0v))) adantr (cv u) his6t negbid (opr (cv u) (.i) (cv u)) (0) df-ne syl5bb biimpar syl3anc (e. (cv v) (H~)) adantr (/\ (e. (cv u) (H~)) (-. (= (cv u) (0v)))) (e. (cv v) (H~)) pm3.27 (e. (cv u) (H~)) (-. (= (cv u) (0v))) pm3.26 (e. (cv v) (H~)) adantr syl3anc eqtr4d (e. (cv u) (` (_|_) (` (null) T))) adantlll eqtr3d r19.21aiva sylanc ex mpdan r19.23aiv sylbi pm2.61i)) thm (riesz4 ((u v) (u w) (T u) (v w) (T v) (T w)) ((nlelch.1 (e. T (LinFn))) (nlelch.2 (e. T (ConFn)))) (E!e. w (H~) (A.e. v (H~) (= (` T (cv v)) (opr (cv v) (.i) (cv w))))) ((cv w) (cv u) (cv v) (.i) opreq2 (` T (cv v)) eqeq2d v (H~) ralbidv (H~) reu4 nlelch.1 nlelch.2 w v riesz3 (cv w) (cv u) hvsubclt (cv v) (opr (cv w) (-v) (cv u)) (.i) (cv w) opreq1 (cv v) (opr (cv w) (-v) (cv u)) (.i) (cv u) opreq1 (-) opreq12d (0) eqeq1d (H~) rcla4v syl (cv w) (cv u) hvsubclt (opr (cv w) (-v) (cv u)) normclt recnd (` (norm) (opr (cv w) (-v) (cv u))) sq0t syl (opr (cv w) (-v) (cv u)) norm-it bitrd syl (cv w) (cv u) hvsubclt (opr (cv w) (-v) (cv u)) normsqt syl (opr (cv w) (-v) (cv u)) (cv w) (cv u) his2sub2t (cv w) (cv u) hvsubclt (e. (cv w) (H~)) (e. (cv u) (H~)) pm3.26 (e. (cv w) (H~)) (e. (cv u) (H~)) pm3.27 syl3anc eqtrd (0) eqeq1d (cv w) (cv u) hvsubeq0t 3bitr3d sylibd v (H~) (= (` T (cv v)) (opr (cv v) (.i) (cv w))) (= (` T (cv v)) (opr (cv v) (.i) (cv u))) r19.26 (` T (cv v)) (opr (cv v) (.i) (cv w)) (` T (cv v)) (opr (cv v) (.i) (cv u)) (-) opreq12 (e. (cv v) (H~)) adantl nlelch.1 lnfnf (cv v) ffvrni (` T (cv v)) subidt syl (/\ (= (` T (cv v)) (opr (cv v) (.i) (cv w))) (= (` T (cv v)) (opr (cv v) (.i) (cv u)))) adantr eqtr3d ex r19.20i sylbir syl5 rgen2 mpbir2an)) thm (riesz4t ((v w) (T w) (T v)) () (-> (e. T (i^i (LinFn) (ConFn))) (E!e. w (H~) (A.e. v (H~) (= (` T (cv v)) (opr (cv v) (.i) (cv w)))))) (T (if (e. T (i^i (LinFn) (ConFn))) T (X. (H~) ({} (0)))) (cv v) fveq1 (opr (cv v) (.i) (cv w)) eqeq1d v (H~) ralbidv w (H~) reubidv (LinFn) (ConFn) inss1 (X. (H~) ({} (0))) (LinFn) (ConFn) elin 0lnfn 0cnfn mpbir2an T elimel sselii (LinFn) (ConFn) inss2 (X. (H~) ({} (0))) (LinFn) (ConFn) elin 0lnfn 0cnfn mpbir2an T elimel sselii w v riesz4 dedth)) thm (riesz1t ((x y) (x z) (T x) (y z) (T y) (T z)) () (-> (e. T (LinFn)) (<-> (e. (` (normfn) T) (RR)) (E.e. y (H~) (A.e. x (H~) (= (` T (cv x)) (opr (cv x) (.i) (cv y))))))) (T lnfncnbdt T (LinFn) (ConFn) elin T (if (e. T (i^i (LinFn) (ConFn))) T (X. (H~) ({} (0)))) (cv x) fveq1 (opr (cv x) (.i) (cv y)) eqeq1d y (H~) x (H~) rexralbidv (LinFn) (ConFn) inss1 (X. (H~) ({} (0))) (LinFn) (ConFn) elin 0lnfn 0cnfn mpbir2an T elimel sselii (LinFn) (ConFn) inss2 (X. (H~) ({} (0))) (LinFn) (ConFn) elin 0lnfn 0cnfn mpbir2an T elimel sselii y x riesz3 dedth sylbir ex (` T (cv x)) (opr (cv x) (.i) (cv y)) (abs) fveq2 (/\ (/\ (e. T (LinFn)) (e. (cv x) (H~))) (e. (cv y) (H~))) adantl (cv x) (cv y) bcst (` (norm) (cv x)) (` (norm) (cv y)) axmulcom (` (norm) (cv x)) recnt (` (norm) (cv y)) recnt syl2an (cv x) normclt (cv y) normclt syl2an breqtrd (e. T (LinFn)) adantll (= (` T (cv x)) (opr (cv x) (.i) (cv y))) adantr eqbrtrd ex an1rs r19.20dva (cv y) normclt (e. T (LinFn)) adantl jctild (cv z) (` (norm) (cv y)) (x.) (` (norm) (cv x)) opreq1 (` (abs) (` T (cv x))) (<_) breq2d x (H~) ralbidv (RR) rcla4ev syl6 ex r19.23adv T z x lnfncont sylibrd impbid bitr3d)) thm (riesz2t ((x y) (T x) (T y)) () (-> (/\ (e. T (LinFn)) (e. (` (normfn) T) (RR))) (E!e. y (H~) (A.e. x (H~) (= (` T (cv x)) (opr (cv x) (.i) (cv y)))))) (T (LinFn) (ConFn) elin T lnfncnbdt pm5.32i bitr T y x riesz4t sylbir)) thm (cnlnadjlem1 ((g h) (g y) (A g) (h y) (A h) (A y) (T g) (T h) (T y)) ((cnlnadjlem.1 (e. T (LinOp))) (cnlnadjlem.2 (e. T (ConOp))) (cnlnadjlem.3 (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y)))))))) (-> (e. A (H~)) (= (` G A) (opr (` T A) (.i) (cv y)))) ((cv g) A T fveq2 (.i) (cv y) opreq1d cnlnadjlem.3 (` T A) (.i) (cv y) oprex fvopab4)) thm (cnlnadjlem2 ((g h) (g w) (g y) (g z) (h w) (h y) (h z) (w y) (w z) (y z) (w x) (x z) (G w) (G x) (G z) (g x) (T g) (h x) (T h) (T w) (x y) (T x) (T y) (T z)) ((cnlnadjlem.1 (e. T (LinOp))) (cnlnadjlem.2 (e. T (ConOp))) (cnlnadjlem.3 (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y)))))))) (-> (e. (cv y) (H~)) (/\ (e. G (LinFn)) (e. G (ConFn)))) ((` T (cv g)) (cv y) ax-hicl cnlnadjlem.1 lnopf (cv g) ffvrni sylan ancoms r19.21aiva cnlnadjlem.3 (CC) fopab2 sylib cnlnadjlem.1 (opr (cv x) (.s) (cv w)) (cv z) lnopadd (e. (cv y) (H~)) 3adant3 (.i) (cv y) opreq1d (` T (opr (cv x) (.s) (cv w))) (` T (cv z)) (cv y) ax-his2 cnlnadjlem.1 lnopf (opr (cv x) (.s) (cv w)) ffvrni cnlnadjlem.1 lnopf (cv z) ffvrni (e. (cv y) (H~)) id syl3an eqtrd 3comr 3expa (cv x) (cv w) ax-hvmulcl sylanl2 (opr (cv x) (.s) (cv w)) (cv z) ax-hvaddcl (cv x) (cv w) ax-hvmulcl sylan cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 (opr (opr (cv x) (.s) (cv w)) (+v) (cv z)) cnlnadjlem1 syl (e. (cv y) (H~)) adantll (cv x) (` T (cv w)) (cv y) ax-his3 cnlnadjlem.1 lnopf (cv w) ffvrni syl3an2 3comr 3expb cnlnadjlem.1 (cv x) (cv w) lnopmul (.i) (cv y) opreq1d (e. (cv y) (H~)) adantl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 (cv w) cnlnadjlem1 (cv x) (x.) opreq2d (e. (cv y) (H~)) (e. (cv x) (CC)) ad2antll 3eqtr4rd cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 (cv z) cnlnadjlem1 (+) opreqan12d 3eqtr4d r19.21aiva ex r19.21aivv jca G x w z ellnfnt sylibr (cv x) (opr (` (normop) T) (x.) (` (norm) (cv y))) (x.) (` (norm) (cv z)) opreq1 (` (abs) (` G (cv z))) (<_) breq2d z (H~) ralbidv (RR) rcla4ev (cv y) normclt cnlnadjlem.1 cnlnadjlem.2 nmcopex (` (normop) T) (` (norm) (cv y)) axmulrcl mpan syl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 (cv z) cnlnadjlem1 (e. (cv y) (H~)) adantr (` T (cv z)) (cv y) ax-hicl cnlnadjlem.1 lnopf (cv z) ffvrni sylan eqeltrd (` G (cv z)) absclt syl (` (norm) (` T (cv z))) (` (norm) (cv y)) axmulrcl cnlnadjlem.1 lnopf (cv z) ffvrni (` T (cv z)) normclt syl (cv y) normclt syl2an (opr (` (normop) T) (x.) (` (norm) (cv z))) (` (norm) (cv y)) axmulrcl (cv z) normclt cnlnadjlem.1 cnlnadjlem.2 nmcopex (` (normop) T) (` (norm) (cv z)) axmulrcl mpan syl (cv y) normclt syl2an cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 (cv z) cnlnadjlem1 (e. (cv y) (H~)) adantr (abs) fveq2d (` T (cv z)) (cv y) bcst cnlnadjlem.1 lnopf (cv z) ffvrni sylan eqbrtrd (` (norm) (` T (cv z))) (opr (` (normop) T) (x.) (` (norm) (cv z))) (` (norm) (cv y)) lemul1it cnlnadjlem.1 lnopf (cv z) ffvrni (` T (cv z)) normclt syl (e. (cv y) (H~)) adantr (cv z) normclt cnlnadjlem.1 cnlnadjlem.2 nmcopex (` (normop) T) (` (norm) (cv z)) axmulrcl mpan syl (e. (cv y) (H~)) adantr (cv y) normclt (e. (cv z) (H~)) adantl 3jca (cv y) normge0t (e. (cv z) (H~)) adantl cnlnadjlem.1 cnlnadjlem.2 (cv z) nmcoplb (e. (cv y) (H~)) adantr jca sylanc letrd cnlnadjlem.1 cnlnadjlem.2 nmcopex recn (` (normop) T) (` (norm) (cv z)) (` (norm) (cv y)) mul23t mp3an1 (cv z) normclt recnd (cv y) normclt recnd syl2an breqtrd ancoms r19.21aiva sylanc (` T (cv g)) (cv y) ax-hicl cnlnadjlem.1 lnopf (cv g) ffvrni sylan ancoms r19.21aiva cnlnadjlem.3 (CC) fopab2 sylib cnlnadjlem.1 (opr (cv x) (.s) (cv w)) (cv z) lnopadd (e. (cv y) (H~)) 3adant3 (.i) (cv y) opreq1d (` T (opr (cv x) (.s) (cv w))) (` T (cv z)) (cv y) ax-his2 cnlnadjlem.1 lnopf (opr (cv x) (.s) (cv w)) ffvrni cnlnadjlem.1 lnopf (cv z) ffvrni (e. (cv y) (H~)) id syl3an eqtrd 3comr 3expa (cv x) (cv w) ax-hvmulcl sylanl2 (opr (cv x) (.s) (cv w)) (cv z) ax-hvaddcl (cv x) (cv w) ax-hvmulcl sylan cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 (opr (opr (cv x) (.s) (cv w)) (+v) (cv z)) cnlnadjlem1 syl (e. (cv y) (H~)) adantll (cv x) (` T (cv w)) (cv y) ax-his3 cnlnadjlem.1 lnopf (cv w) ffvrni syl3an2 3comr 3expb cnlnadjlem.1 (cv x) (cv w) lnopmul (.i) (cv y) opreq1d (e. (cv y) (H~)) adantl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 (cv w) cnlnadjlem1 (cv x) (x.) opreq2d (e. (cv y) (H~)) (e. (cv x) (CC)) ad2antll 3eqtr4rd cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 (cv z) cnlnadjlem1 (+) opreqan12d 3eqtr4d r19.21aiva ex r19.21aivv jca G x w z ellnfnt sylibr G x z lnfncont syl mpbird jca)) thm (cnlnadjlem3 ((g h) (g u) (g v) (g w) (g y) (h u) (h v) (h w) (h y) (u v) (u w) (u y) (v w) (v y) (w y) (B u) (F w) (G v) (G w) (T g) (T h) (T u) (T v) (T w) (T y)) ((cnlnadjlem.1 (e. T (LinOp))) (cnlnadjlem.2 (e. T (ConOp))) (cnlnadjlem.3 (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y))))))) (cnlnadjlem.4 (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w)))))))) (cnlnadjlem.5 (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B)))))) (-> (e. (cv y) (H~)) (e. B (H~))) (cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem2 G (LinFn) (ConFn) elin sylibr G w v riesz4t syl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 (cv v) cnlnadjlem1 (opr (cv v) (.i) (cv w)) eqeq1d ralbiia w (H~) reubii sylib w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w)))) reucl syl cnlnadjlem.4 syl5eqel)) thm (cnlnadjlem4 ((g h) (g u) (g v) (g w) (g y) (A g) (h u) (h v) (h w) (h y) (A h) (u v) (u w) (u y) (A u) (v w) (v y) (A v) (w y) (A w) (A y) (B u) (F w) (G v) (G w) (T g) (T h) (T u) (T v) (T w) (T y)) ((cnlnadjlem.1 (e. T (LinOp))) (cnlnadjlem.2 (e. T (ConOp))) (cnlnadjlem.3 (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y))))))) (cnlnadjlem.4 (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w)))))))) (cnlnadjlem.5 (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B)))))) (-> (e. A (H~)) (e. (` F A) (H~))) ((cv y) A (` T (cv v)) (.i) opreq2 (opr (cv v) (.i) (cv w)) eqeq1d v (H~) ralbidv w (H~) rabbisdv unieqd cnlnadjlem.4 syl5eq cnlnadjlem.5 ax-hilex w (A.e. v (H~) (= (opr (` T (cv v)) (.i) A) (opr (cv v) (.i) (cv w)))) rabex uniex fvopab4 (cv y) A (` T (cv v)) (.i) opreq2 (opr (cv v) (.i) (cv w)) eqeq1d v (H~) ralbidv w (H~) reubidv cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem2 G (LinFn) (ConFn) elin sylibr G w v riesz4t syl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 (cv v) cnlnadjlem1 (opr (cv v) (.i) (cv w)) eqeq1d ralbiia w (H~) reubii sylib vtoclga w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) A) (opr (cv v) (.i) (cv w)))) reucl syl eqeltrd)) thm (cnlnadjlem5 ((f g) (f h) (f u) (f v) (f w) (f y) (f z) (A f) (g h) (g u) (g v) (g w) (g y) (g z) (A g) (h u) (h v) (h w) (h y) (h z) (A h) (u v) (u w) (u y) (u z) (A u) (v w) (v y) (v z) (A v) (w y) (w z) (A w) (y z) (A y) (A z) (B u) (C f) (F f) (F w) (F z) (G f) (G v) (G w) (G z) (T f) (T g) (T h) (T u) (T v) (T w) (T y) (T z)) ((cnlnadjlem.1 (e. T (LinOp))) (cnlnadjlem.2 (e. T (ConOp))) (cnlnadjlem.3 (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y))))))) (cnlnadjlem.4 (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w)))))))) (cnlnadjlem.5 (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B)))))) (-> (/\ (e. A (H~)) (e. C (H~))) (= (opr (` T C) (.i) A) (opr C (.i) (` F A)))) ((cv f) C T fveq2 (.i) A opreq1d (cv f) C (.i) (` F A) opreq1 eqeq12d (e. A (H~)) imbi2d (e. (cv z) A) y ax-17 (e. (cv f) (H~)) y ax-17 (e. (cv z) (opr (` T (cv f)) (.i) A)) y ax-17 (e. (cv z) (cv f)) y ax-17 (e. (cv z) (.i)) y ax-17 z y u (/\ (e. (cv y) (H~)) (= (cv u) B)) hbopab1 cnlnadjlem.5 (cv z) eleq2i cnlnadjlem.5 (cv z) eleq2i y albii 3imtr4 (e. (cv z) A) y ax-17 hbfv hbopr hbeq hbral (cv y) A (` T (cv f)) (.i) opreq2 (cv y) A F fveq2 (cv f) (.i) opreq2d eqeq12d f (H~) ralbidv cnlnadjlem.4 ax-hilex w (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w)))) rabex uniex eqeltr y (H~) B (V) u fvopab2 mpan2 cnlnadjlem.5 (cv y) fveq1i syl5eq cnlnadjlem.4 (cv v) (cv f) T fveq2 (.i) (cv y) opreq1d (cv v) (cv f) (.i) (cv w) opreq1 eqeq12d (H~) cbvralv (e. (cv w) (H~)) a1i rabbii unieqi eqtr syl6req (cv w) (` F (cv y)) (cv f) (.i) opreq2 (opr (` T (cv f)) (.i) (cv y)) eqeq2d f (H~) ralbidv (H~) reuuni2 cnlnadjlem.4 ax-hilex w (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w)))) rabex uniex eqeltr y (H~) B (V) u fvopab2 mpan2 cnlnadjlem.5 (cv y) fveq1i syl5eq cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 cnlnadjlem3 eqeltrd cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem2 G (LinFn) (ConFn) elin sylibr G w f riesz4t syl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 (cv f) cnlnadjlem1 (opr (cv f) (.i) (cv w)) eqeq1d ralbiia w (H~) reubii sylib sylanc mpbird vtoclgaf f (H~) (= (opr (` T (cv f)) (.i) A) (opr (cv f) (.i) (` F A))) ra4 syl com12 vtoclga impcom)) thm (cnlnadjlem6 ((f g) (f h) (f u) (f v) (f w) (f y) (f z) (g h) (g u) (g v) (g w) (g y) (g z) (h u) (h v) (h w) (h y) (h z) (u v) (u w) (u y) (u z) (v w) (v y) (v z) (w y) (w z) (y z) (B u) (f t) (f x) (F f) (t w) (t x) (t z) (F t) (w x) (F w) (x z) (F x) (F z) (G f) (v x) (G v) (G w) (G x) (G z) (T f) (g t) (g x) (T g) (h t) (h x) (T h) (t u) (t v) (t y) (T t) (u x) (T u) (T v) (T w) (x y) (T x) (T y) (T z)) ((cnlnadjlem.1 (e. T (LinOp))) (cnlnadjlem.2 (e. T (ConOp))) (cnlnadjlem.3 (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y))))))) (cnlnadjlem.4 (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w)))))))) (cnlnadjlem.5 (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B)))))) (e. F (LinOp)) (F x f z ellnopt cnlnadjlem.5 cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 cnlnadjlem3 fopab (` T (cv t)) (opr (cv x) (.s) (cv f)) (cv z) his7t cnlnadjlem.1 lnopf (cv t) ffvrni (/\ (/\ (e. (cv x) (CC)) (e. (cv f) (H~))) (e. (cv z) (H~))) adantl (cv x) (cv f) ax-hvmulcl (e. (cv z) (H~)) (e. (cv t) (H~)) ad2antrr (/\ (e. (cv x) (CC)) (e. (cv f) (H~))) (e. (cv z) (H~)) pm3.27 (e. (cv t) (H~)) adantr syl3anc cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 (opr (opr (cv x) (.s) (cv f)) (+v) (cv z)) (cv t) cnlnadjlem5 (opr (cv x) (.s) (cv f)) (cv z) ax-hvaddcl (cv x) (cv f) ax-hvmulcl sylan sylan cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 (cv f) (cv t) cnlnadjlem5 (e. (cv x) (CC)) adantll (` (*) (cv x)) (x.) opreq2d (cv x) (` T (cv t)) (cv f) his5t (e. (cv x) (CC)) (e. (cv f) (H~)) pm3.26 (e. (cv t) (H~)) adantr cnlnadjlem.1 lnopf (cv t) ffvrni (/\ (e. (cv x) (CC)) (e. (cv f) (H~))) adantl (e. (cv x) (CC)) (e. (cv f) (H~)) pm3.27 (e. (cv t) (H~)) adantr syl3anc (cv x) (cv t) (` F (cv f)) his5t (e. (cv x) (CC)) (e. (cv f) (H~)) pm3.26 (e. (cv t) (H~)) adantr (/\ (e. (cv x) (CC)) (e. (cv f) (H~))) (e. (cv t) (H~)) pm3.27 cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 (cv f) cnlnadjlem4 (e. (cv x) (CC)) (e. (cv t) (H~)) ad2antlr syl3anc 3eqtr4d (e. (cv z) (H~)) adantlr cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 (cv z) (cv t) cnlnadjlem5 (/\ (e. (cv x) (CC)) (e. (cv f) (H~))) adantll (+) opreq12d (cv t) (opr (cv x) (.s) (` F (cv f))) (` F (cv z)) his7t (/\ (/\ (e. (cv x) (CC)) (e. (cv f) (H~))) (e. (cv z) (H~))) (e. (cv t) (H~)) pm3.27 (cv x) (` F (cv f)) ax-hvmulcl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 (cv f) cnlnadjlem4 sylan2 (e. (cv z) (H~)) (e. (cv t) (H~)) ad2antrr cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 (cv z) cnlnadjlem4 (/\ (e. (cv x) (CC)) (e. (cv f) (H~))) (e. (cv t) (H~)) ad2antlr syl3anc eqtr4d 3eqtr3d r19.21aiva (` F (opr (opr (cv x) (.s) (cv f)) (+v) (cv z))) (opr (opr (cv x) (.s) (` F (cv f))) (+v) (` F (cv z))) t hial2eq2t (opr (cv x) (.s) (cv f)) (cv z) ax-hvaddcl (cv x) (cv f) ax-hvmulcl sylan cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 (opr (opr (cv x) (.s) (cv f)) (+v) (cv z)) cnlnadjlem4 syl (opr (cv x) (.s) (` F (cv f))) (` F (cv z)) ax-hvaddcl (cv x) (` F (cv f)) ax-hvmulcl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 (cv f) cnlnadjlem4 sylan2 cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 (cv z) cnlnadjlem4 syl2an sylanc mpbid r19.21aiva rgen2a mpbir2an)) thm (cnlnadjlem7 ((g h) (g u) (g v) (g w) (g y) (A g) (h u) (h v) (h w) (h y) (A h) (u v) (u w) (u y) (A u) (v w) (v y) (A v) (w y) (A w) (A y) (B u) (F w) (G v) (G w) (T g) (T h) (T u) (T v) (T w) (T y)) ((cnlnadjlem.1 (e. T (LinOp))) (cnlnadjlem.2 (e. T (ConOp))) (cnlnadjlem.3 (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y))))))) (cnlnadjlem.4 (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w)))))))) (cnlnadjlem.5 (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B)))))) (-> (e. A (H~)) (br (` (norm) (` F A)) (<_) (opr (` (normop) T) (x.) (` (norm) A)))) ((e. A (H~)) (= (0) (` (norm) (` F A))) pm3.27 (` (normop) T) (` (norm) A) mulge0t A normclt cnlnadjlem.1 cnlnadjlem.2 nmcopex jctil A normge0t cnlnadjlem.1 lnopf T nmopge0t ax-mp jctil sylanc (= (0) (` (norm) (` F A))) adantr eqbrtrrd cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 cnlnadjlem.1 lnopf (` F A) ffvrni syl (` T (` F A)) A ax-hicl mpancom (opr (` T (` F A)) (.i) A) absclt syl (` (norm) (` T (` F A))) (` (norm) A) axmulrcl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 cnlnadjlem.1 lnopf (` F A) ffvrni syl (` T (` F A)) normclt syl A normclt sylanc (opr (` (normop) T) (x.) (` (norm) (` F A))) (` (norm) A) axmulrcl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 (` F A) normclt syl cnlnadjlem.1 cnlnadjlem.2 nmcopex (` (normop) T) (` (norm) (` F A)) axmulrcl mpan syl A normclt sylanc cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 cnlnadjlem.1 lnopf (` F A) ffvrni syl (` T (` F A)) A bcst mpancom (` (norm) (` T (` F A))) (opr (` (normop) T) (x.) (` (norm) (` F A))) (` (norm) A) lemul1it cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 cnlnadjlem.1 lnopf (` F A) ffvrni syl (` T (` F A)) normclt syl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 (` F A) normclt syl cnlnadjlem.1 cnlnadjlem.2 nmcopex (` (normop) T) (` (norm) (` F A)) axmulrcl mpan syl A normclt 3jca A normge0t cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 cnlnadjlem.1 cnlnadjlem.2 (` F A) nmcoplb syl jca sylanc letrd cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 (` F A) normsqt syl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 (` F A) normclt syl recnd (` (norm) (` F A)) sqvalt syl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A (` F A) cnlnadjlem5 mpdan (abs) fveq2d (opr (` F A) (.i) (` F A)) absidt cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 (` F A) hiidrclt syl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 (` F A) hiidge0t syl sylanc eqtr2d 3eqtr3rd cnlnadjlem.1 cnlnadjlem.2 nmcopex recn (` (normop) T) (` (norm) (` F A)) (` (norm) A) mul23t mp3an1 cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 (` F A) normclt syl recnd A normclt recnd sylanc 3brtr3d (-. (= (0) (` (norm) (` F A)))) adantr (` (norm) (` F A)) (opr (` (normop) T) (x.) (` (norm) A)) (` (norm) (` F A)) lemul1t cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 (` F A) normclt syl A normclt cnlnadjlem.1 cnlnadjlem.2 nmcopex (` (normop) T) (` (norm) A) axmulrcl mpan syl cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 (` F A) normclt syl 3jca (-. (= (0) (` (norm) (` F A)))) adantr 0re (0) (` (norm) (` F A)) leltnet mp3an1 (` F A) normclt (` F A) normge0t sylanc biimpar cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 A cnlnadjlem4 sylan sylanc mpbird pm2.61dan)) thm (cnlnadjlem8 ((g h) (g u) (g v) (g w) (g y) (g z) (h u) (h v) (h w) (h y) (h z) (u v) (u w) (u y) (u z) (v w) (v y) (v z) (w y) (w z) (y z) (B u) (w x) (F w) (x z) (F x) (F z) (v x) (G v) (G w) (G x) (G z) (g x) (T g) (h x) (T h) (u x) (T u) (T v) (T w) (x y) (T x) (T y) (T z)) ((cnlnadjlem.1 (e. T (LinOp))) (cnlnadjlem.2 (e. T (ConOp))) (cnlnadjlem.3 (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y))))))) (cnlnadjlem.4 (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w)))))))) (cnlnadjlem.5 (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B)))))) (e. F (ConOp)) (cnlnadjlem.1 cnlnadjlem.2 nmcopex cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 (cv z) cnlnadjlem7 rgen (cv x) (` (normop) T) (x.) (` (norm) (cv z)) opreq1 (` (norm) (` F (cv z))) (<_) breq2d z (H~) ralbidv (RR) rcla4ev mp2an cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 cnlnadjlem6 x z lnopcon mpbir)) thm (cnlnadjlem9 ((g h) (g u) (g v) (g w) (g y) (g z) (h u) (h v) (h w) (h y) (h z) (u v) (u w) (u y) (u z) (v w) (v y) (v z) (w y) (w z) (y z) (B u) (t w) (t x) (t z) (F t) (w x) (F w) (x z) (F x) (F z) (v x) (G v) (G w) (G x) (G z) (g t) (g x) (T g) (h t) (h x) (T h) (t u) (t v) (t y) (T t) (u x) (T u) (T v) (T w) (x y) (T x) (T y) (T z)) ((cnlnadjlem.1 (e. T (LinOp))) (cnlnadjlem.2 (e. T (ConOp))) (cnlnadjlem.3 (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y))))))) (cnlnadjlem.4 (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w)))))))) (cnlnadjlem.5 (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B)))))) (E.e. t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. z (H~) (= (opr (` T (cv x)) (.i) (cv z)) (opr (cv x) (.i) (` (cv t) (cv z))))))) (F (LinOp) (ConOp) elin cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 cnlnadjlem6 cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 cnlnadjlem8 mpbir2an cnlnadjlem.1 cnlnadjlem.2 cnlnadjlem.3 cnlnadjlem.4 cnlnadjlem.5 (cv z) (cv x) cnlnadjlem5 ancoms rgen2 (cv t) F (cv z) fveq1 (cv x) (.i) opreq2d (opr (` T (cv x)) (.i) (cv z)) eqeq2d x (H~) z (H~) 2ralbidv (i^i (LinOp) (ConOp)) rcla4ev mp2an)) thm (cnlnadj ((f g) (f h) (f t) (f u) (f v) (f w) (f x) (f y) (f z) (T f) (g h) (g t) (g u) (g v) (g w) (g x) (g y) (g z) (T g) (h t) (h u) (h v) (h w) (h x) (h y) (h z) (T h) (t u) (t v) (t w) (t x) (t y) (t z) (T t) (u v) (u w) (u x) (u y) (u z) (T u) (v w) (v x) (v y) (v z) (T v) (w x) (w y) (w z) (T w) (x y) (x z) (T x) (y z) (T y) (T z)) ((cnlnadj.1 (e. T (LinOp))) (cnlnadj.2 (e. T (ConOp)))) (E.e. t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y))))))) (cnlnadj.1 cnlnadj.2 ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv z))))) eqid (cv f) (cv w) (cv v) (.i) opreq2 (opr (` T (cv v)) (.i) (cv z)) eqeq2d v (H~) ralbidv (H~) cbvrabv unieqi ({<,>|} z u (/\ (e. (cv z) (H~)) (= (cv u) (U. ({e.|} f (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv z)) (opr (cv v) (.i) (cv f))))))))) eqid t x y cnlnadjlem9)) thm (cnlnadjeu ((t x) (t y) (T t) (x y) (T x) (T y)) ((cnlnadj.1 (e. T (LinOp))) (cnlnadj.2 (e. T (ConOp)))) (E!e. t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y))))))) (t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))) reu5 cnlnadj.1 cnlnadj.2 t x y cnlnadj t x y T adjmo (:--> (cv t) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))) pm3.26 cnlnadj.1 lnopf (cv t) T x y adjsymt mpan2 (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y))) eqcom x (H~) y (H~) 2ralbii syl5bb biimpa jca (LinOp) (ConOp) inss1 (cv t) sseli (cv t) lnopft syl sylan t immoi ax-mp mpbir2an)) thm (cnlnadjeut ((t x) (t y) (T t) (x y) (T x) (T y)) () (-> (e. T (i^i (LinOp) (ConOp))) (E!e. t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))))) (T (if (e. T (i^i (LinOp) (ConOp))) T (X. (H~) (0H))) (cv x) fveq1 (.i) (cv y) opreq1d (opr (cv x) (.i) (` (cv t) (cv y))) eqeq1d x (H~) y (H~) 2ralbidv t (i^i (LinOp) (ConOp)) reubidv (LinOp) (ConOp) inss1 (X. (H~) (0H)) (LinOp) (ConOp) elin 0lnopOLD 0cnop mpbir2an T elimel sselii (LinOp) (ConOp) inss2 (X. (H~) (0H)) (LinOp) (ConOp) elin 0lnopOLD 0cnop mpbir2an T elimel sselii t x y cnlnadjeu dedth)) thm (cnlnadjt ((t x) (t y) (T t) (x y) (T x) (T y)) () (-> (e. T (i^i (LinOp) (ConOp))) (E.e. t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))))) (T t x y cnlnadjeut t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))) reurex syl)) thm (cnlnssadj ((t u) (t v) (t x) (t y) (t z) (u v) (u x) (u y) (u z) (v x) (v y) (v z) (x y) (x z) (y z)) () (C_ (i^i (LinOp) (ConOp)) (dom (adj))) ((cv y) t x z cnlnadjt t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. z (H~) (= (opr (` (cv y) (cv x)) (.i) (cv z)) (opr (cv x) (.i) (` (cv t) (cv z)))))) df-rex sylib (LinOp) (ConOp) inss1 (cv y) sseli (cv y) lnopft syl (/\ (e. (cv t) (i^i (LinOp) (ConOp))) (A.e. x (H~) (A.e. z (H~) (= (opr (` (cv y) (cv x)) (.i) (cv z)) (opr (cv x) (.i) (` (cv t) (cv z))))))) a1d (LinOp) (ConOp) inss1 (cv t) sseli (cv t) lnopft syl (e. (cv y) (i^i (LinOp) (ConOp))) a1i (A.e. x (H~) (A.e. z (H~) (= (opr (` (cv y) (cv x)) (.i) (cv z)) (opr (cv x) (.i) (` (cv t) (cv z)))))) adantrd (cv t) (cv y) x z adjsymt (LinOp) (ConOp) inss1 (cv t) sseli (cv t) lnopft syl (LinOp) (ConOp) inss1 (cv y) sseli (cv y) lnopft syl syl2an ancoms (opr (` (cv y) (cv x)) (.i) (cv z)) (opr (cv x) (.i) (` (cv t) (cv z))) eqcom biimp z (H~) r19.20si x (H~) r19.20si syl5bi ex imp3a 3jcad u v x z dfadj2 (<,> (cv y) (cv t)) eleq2i y visset t visset (cv u) (cv y) (H~) (H~) feq1 (cv u) (cv y) (cv z) fveq1 (cv x) (.i) opreq2d (opr (` (cv v) (cv x)) (.i) (cv z)) eqeq1d x (H~) z (H~) 2ralbidv (:--> (cv v) (H~) (H~)) 3anbi13d (cv v) (cv t) (H~) (H~) feq1 (cv v) (cv t) (cv x) fveq1 (.i) (cv z) opreq1d (opr (cv x) (.i) (` (cv y) (cv z))) eqeq2d x (H~) z (H~) 2ralbidv (:--> (cv y) (H~) (H~)) 3anbi23d opelopab bitr2 syl6ib t 19.22dv mpd y visset (adj) t eldm2 sylibr ssriv)) thm (bdopssadj () () (C_ (BndLinOp) (dom (adj))) (lncnbd cnlnssadj eqsstr3)) thm (bdopadjt () () (-> (e. T (BndLinOp)) (e. T (dom (adj)))) (bdopssadj T sseli)) thm (adjbdlnt ((t x) (t y) (T t) (x y) (T x) (T y)) () (-> (e. T (BndLinOp)) (e. (` (adj) T) (BndLinOp))) (t (opr (H~) (^m) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))) df-rab ax-hilex ax-hilex (cv t) elmap (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))) anbi1i t abbii eqtr unieqi (e. T (BndLinOp)) a1i (cv t) bdopft ax-hilex ax-hilex (cv t) elmap sylibr ssriv (BndLinOp) (opr (H~) (^m) (H~)) t (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))) mouniss mp3an1 T t x y cnlnadjt T lncnopbd lncnbd (i^i (LinOp) (ConOp)) (BndLinOp) t (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))) rexeq1 ax-mp 3imtr3 t x y T adjmo (e. T (BndLinOp)) t ax-17 T (cv t) x y adjsymt T bdopft sylan ex pm5.32d ax-hilex ax-hilex (cv t) elmap (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y))) eqcom x (H~) y (H~) 2ralbii anbi12i syl6rbbr mobid mpbiri sylanc T bdopft T t x y adjval2t syl 3eqtr4rd T t x y cnlnadjeut T lncnopbd lncnbd (i^i (LinOp) (ConOp)) (BndLinOp) t (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))) reueq1 ax-mp 3imtr3 t (BndLinOp) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))) reucl syl eqeltrd)) thm (adjbdlnb () () (<-> (e. T (BndLinOp)) (e. (` (adj) T) (BndLinOp))) (T adjbdlnt (` (adj) T) bdopadjt T dmadjrnb sylibr cnvadj (adj) coeq1i adj1o (adj) (dom (adj)) (dom (adj)) f1ococnv1 ax-mp eqtr3 T fveq1i (/\ (e. T (dom (adj))) (e. (` (adj) T) (BndLinOp))) a1i funadj funadj (adj) (adj) T fvco mp3an12 (e. (` (adj) T) (BndLinOp)) adantr T (dom (adj)) fvresi (e. (` (adj) T) (BndLinOp)) adantr 3eqtr3rd (` (adj) T) adjbdlnt (e. T (dom (adj))) adantl eqeltrd mpancom impbi)) thm (adjlnopt ((w x) (w y) (w z) (T w) (x y) (x z) (T x) (y z) (T y) (T z)) () (-> (e. T (dom (adj))) (e. (` (adj) T) (LinOp))) (T dmadjrnt (` (adj) T) dmadjopt syl (` T (cv w)) (opr (cv x) (.s) (cv y)) (cv z) his7t T (H~) (H~) (cv w) ffvrn T dmadjopt sylan (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) (e. (cv z) (H~))) 3adant3 (cv x) (cv y) ax-hvmulcl (e. (cv z) (H~)) adantr (e. T (dom (adj))) (e. (cv w) (H~)) 3ad2ant3 (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) (e. (cv z) (H~)) pm3.27 (e. T (dom (adj))) (e. (cv w) (H~)) 3ad2ant3 syl3anc T (cv w) (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) adj2t (opr (cv x) (.s) (cv y)) (cv z) ax-hvaddcl (cv x) (cv y) ax-hvmulcl sylan syl3an3 T (cv w) (cv y) adj2t (e. (cv x) (CC)) 3adant3l (` (*) (cv x)) (x.) opreq2d (cv x) (` T (cv w)) (cv y) his5t (e. (cv x) (CC)) (e. (cv y) (H~)) pm3.26 (e. T (dom (adj))) (e. (cv w) (H~)) 3ad2ant3 T (H~) (H~) (cv w) ffvrn T dmadjopt sylan (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) 3adant3 (e. (cv x) (CC)) (e. (cv y) (H~)) pm3.27 (e. T (dom (adj))) (e. (cv w) (H~)) 3ad2ant3 syl3anc (cv x) (cv w) (` (` (adj) T) (cv y)) his5t (e. (cv x) (CC)) (e. (cv y) (H~)) pm3.26 (e. T (dom (adj))) (e. (cv w) (H~)) 3ad2ant3 (e. T (dom (adj))) (e. (cv w) (H~)) (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) 3simp2 T (cv y) adjclt (e. (cv x) (CC)) adantrl (e. (cv w) (H~)) 3adant2 syl3anc 3eqtr4d (e. (cv z) (H~)) 3adant3r T (cv w) (cv z) adj2t (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) 3adant3l (+) opreq12d (cv w) (opr (cv x) (.s) (` (` (adj) T) (cv y))) (` (` (adj) T) (cv z)) his7t (e. T (dom (adj))) (e. (cv w) (H~)) (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) (e. (cv z) (H~))) 3simp2 (cv x) (` (` (adj) T) (cv y)) ax-hvmulcl T (cv y) adjclt sylan2 an1s (e. (cv z) (H~)) adantrr (e. (cv w) (H~)) 3adant2 T (cv z) adjclt (/\ (e. (cv x) (CC)) (e. (cv y) (H~))) adantrl (e. (cv w) (H~)) 3adant2 syl3anc eqtr4d 3eqtr3d 3com23 3expa r19.21aiva (` (` (adj) T) (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` (` (adj) T) (cv y))) (+v) (` (` (adj) T) (cv z))) w hial2eq2t T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z)) adjclt (opr (cv x) (.s) (cv y)) (cv z) ax-hvaddcl (cv x) (cv y) ax-hvmulcl sylan sylan2 (opr (cv x) (.s) (` (` (adj) T) (cv y))) (` (` (adj) T) (cv z)) ax-hvaddcl (cv x) (` (` (adj) T) (cv y)) ax-hvmulcl T (cv y) adjclt sylan2 an1s T (cv z) adjclt syl2an anandis sylanc mpbid exp32 r19.21adv r19.21aivv jca (` (adj) T) x y z ellnopt sylibr)) thm (nmopadjle ((f g) (f h) (f u) (f v) (f w) (f z) (A f) (g h) (g u) (g v) (g w) (g z) (A g) (h u) (h v) (h w) (h z) (A h) (u v) (u w) (u z) (A u) (v w) (v z) (A v) (w z) (A w) (A z) (T f) (T g) (T h) (T u) (T v) (T w) (T z)) ((nmopadjle.1 (e. T (BndLinOp)))) (-> (e. A (H~)) (br (` (norm) (` (` (adj) T) A)) (<_) (opr (` (normop) T) (x.) (` (norm) A)))) (bdopssadj nmopadjle.1 sselii T A f v adjvalvalt mpan (cv z) A (` T (cv v)) (.i) opreq2 (opr (cv v) (.i) (cv f)) eqeq1d v (H~) ralbidv f (H~) rabbisdv unieqd ({<,>|} z u (/\ (e. (cv z) (H~)) (= (cv u) (U. ({e.|} f (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv z)) (opr (cv v) (.i) (cv f))))))))) eqid ax-hilex f (A.e. v (H~) (= (opr (` T (cv v)) (.i) A) (opr (cv v) (.i) (cv f)))) rabex uniex fvopab4 eqtr4d (norm) fveq2d (LinOp) (ConOp) inss1 nmopadjle.1 lncnbd eleqtrr sselii (LinOp) (ConOp) inss2 nmopadjle.1 lncnbd eleqtrr sselii ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv z))))) eqid (cv f) (cv w) (cv v) (.i) opreq2 (opr (` T (cv v)) (.i) (cv z)) eqeq2d v (H~) ralbidv (H~) cbvrabv unieqi ({<,>|} z u (/\ (e. (cv z) (H~)) (= (cv u) (U. ({e.|} f (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv z)) (opr (cv v) (.i) (cv f))))))))) eqid A cnlnadjlem7 eqbrtrd)) thm (nmopadjlem ((T y)) ((nmopadjle.1 (e. T (BndLinOp)))) (br (` (normop) (` (adj) T)) (<_) (` (normop) T)) (nmopadjle.1 T adjbdlnt ax-mp (` (adj) T) bdopft ax-mp nmopadjle.1 T bdopft ax-mp T nmopxrt ax-mp nmopadjle.1 T adjbdlnt ax-mp (` (adj) T) bdopft ax-mp (cv y) ffvrni (` (` (adj) T) (cv y)) normclt syl (br (` (norm) (cv y)) (<_) (1)) adantr (cv y) normclt nmopadjle.1 T nmopret ax-mp (` (normop) T) (` (norm) (cv y)) axmulrcl mpan syl (br (` (norm) (cv y)) (<_) (1)) adantr nmopadjle.1 T nmopret ax-mp ax1re remulcl (/\ (e. (cv y) (H~)) (br (` (norm) (cv y)) (<_) (1))) a1i nmopadjle.1 (cv y) nmopadjle (br (` (norm) (cv y)) (<_) (1)) adantr (` (norm) (cv y)) (1) (` (normop) T) lemul2it (cv y) normclt (br (` (norm) (cv y)) (<_) (1)) adantr ax1re (/\ (e. (cv y) (H~)) (br (` (norm) (cv y)) (<_) (1))) a1i nmopadjle.1 T nmopret ax-mp (/\ (e. (cv y) (H~)) (br (` (norm) (cv y)) (<_) (1))) a1i 3jca (e. (cv y) (H~)) (br (` (norm) (cv y)) (<_) (1)) pm3.27 nmopadjle.1 T bdopft ax-mp T nmopge0t ax-mp jctil sylanc letrd nmopadjle.1 T nmopret ax-mp recn mulid1 syl6breq ex rgen (` (adj) T) (` (normop) T) y nmopleubt mp3an)) thm (nmopadj () ((nmopadjle.1 (e. T (BndLinOp)))) (= (` (normop) (` (adj) T)) (` (normop) T)) (nmopadjle.1 T adjbdlnt ax-mp (` (adj) T) nmopret ax-mp nmopadjle.1 T nmopret ax-mp letri3 nmopadjle.1 nmopadjlem nmopadjle.1 T bdopadjt ax-mp T adjadjt ax-mp (normop) fveq2i nmopadjle.1 T adjbdlnt ax-mp nmopadjlem eqbrtrr mpbir2an)) thm (nmoptri ((S x) (T x)) ((nmoptri.1 (e. S (BndLinOp))) (nmoptri.2 (e. T (BndLinOp)))) (br (` (normop) (opr S (+op) T)) (<_) (opr (` (normop) S) (+) (` (normop) T))) (nmoptri.1 S bdopft ax-mp nmoptri.2 T bdopft ax-mp hoaddcl nmoptri.1 S nmopret ax-mp nmoptri.2 T nmopret ax-mp readdcl (opr (` (normop) S) (+) (` (normop) T)) rexrt ax-mp nmoptri.1 S bdopft ax-mp nmoptri.2 T bdopft ax-mp (cv x) hoscl (` (opr S (+op) T) (cv x)) normclt syl (br (` (norm) (cv x)) (<_) (1)) adantr (` (norm) (` S (cv x))) (` (norm) (` T (cv x))) axaddrcl nmoptri.1 S bdopft ax-mp (cv x) ffvrni (` S (cv x)) normclt syl nmoptri.2 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl sylanc (br (` (norm) (cv x)) (<_) (1)) adantr nmoptri.1 S nmopret ax-mp nmoptri.2 T nmopret ax-mp readdcl (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) a1i nmoptri.1 S bdopft ax-mp nmoptri.2 T bdopft ax-mp S T (cv x) hosvaltOLD mpanl12 (norm) fveq2d (` S (cv x)) (` T (cv x)) norm-iit nmoptri.1 S bdopft ax-mp (cv x) ffvrni nmoptri.2 T bdopft ax-mp (cv x) ffvrni sylanc eqbrtrd (br (` (norm) (cv x)) (<_) (1)) adantr nmoptri.1 S bdopft ax-mp S (cv x) nmoplbt mp3an1 nmoptri.2 T bdopft ax-mp T (cv x) nmoplbt mp3an1 nmoptri.1 S nmopret ax-mp nmoptri.2 T nmopret ax-mp pm3.2i (` (norm) (` S (cv x))) (` (norm) (` T (cv x))) (` (normop) S) (` (normop) T) le2addt mpan2 nmoptri.1 S bdopft ax-mp (cv x) ffvrni (` S (cv x)) normclt syl nmoptri.2 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl sylanc (br (` (norm) (cv x)) (<_) (1)) adantr mp2and letrd ex rgen (opr S (+op) T) (opr (` (normop) S) (+) (` (normop) T)) x nmopleubt mp3an)) thm (nmopco ((S x) (T x)) ((nmoptri.1 (e. S (BndLinOp))) (nmoptri.2 (e. T (BndLinOp)))) (br (` (normop) (o. S T)) (<_) (opr (` (normop) S) (x.) (` (normop) T))) (nmoptri.1 S bdoplnt ax-mp nmoptri.2 T bdoplnt ax-mp lnopco lnopf nmoptri.1 S nmopret ax-mp nmoptri.2 T nmopret ax-mp remulcl (opr (` (normop) S) (x.) (` (normop) T)) rexrt ax-mp 0re leid (/\ (= (` (normop) T) (0)) (e. (cv x) (H~))) a1i (o. S T) (X. (H~) (0H)) (cv x) fveq1 (norm) fveq2d (cv x) ho0valtOLD (norm) fveq2d norm0 syl6eq sylan9eq nmoptri.1 S bdoplnt ax-mp nmoptri.2 T bdoplnt ax-mp lnopco0 nmoptri.1 S bdoplnt ax-mp nmoptri.2 T bdoplnt ax-mp lnopco nmlnop0OLD sylib sylan (` (normop) T) (0) (` (normop) S) (x.) opreq2 nmoptri.1 S nmopret ax-mp recn mul01 syl6eq (e. (cv x) (H~)) adantr 3brtr4d (br (` (norm) (cv x)) (<_) (1)) adantrr (opr (1) (/) (` (normop) T)) (` (o. S T) (cv x)) norm-iiit nmoptri.2 T nmopret ax-mp recn recclz nmoptri.1 S bdoplnt ax-mp nmoptri.2 T bdoplnt ax-mp lnopco lnopf (cv x) ffvrni syl2an nmoptri.1 S bdoplnt ax-mp (opr (1) (/) (` (normop) T)) (` T (cv x)) lnopmul nmoptri.2 T nmopret ax-mp recn recclz nmoptri.2 T bdopft ax-mp (cv x) ffvrni syl2an nmoptri.1 S bdopft ax-mp nmoptri.2 T bdopft ax-mp (cv x) hoco (opr (1) (/) (` (normop) T)) (.s) opreq2d (=/= (` (normop) T) (0)) adantl eqtr4d (norm) fveq2d nmoptri.2 T nmopret ax-mp recn (` (norm) (` (o. S T) (cv x))) (` (normop) T) divrec2t mp3an2 nmoptri.1 S bdoplnt ax-mp nmoptri.2 T bdoplnt ax-mp lnopco lnopf (cv x) ffvrni (` (o. S T) (cv x)) normclt syl recnd sylan ancoms (opr (1) (/) (` (normop) T)) absidt nmoptri.2 T nmopret ax-mp rerecclz 0re (0) (opr (1) (/) (` (normop) T)) ltlet mpan nmoptri.2 T nmopret ax-mp rerecclz nmoptri.2 T bdopft ax-mp T nmopgt0t ax-mp nmoptri.2 T nmopret ax-mp recgt0 sylbi sylc sylanc (e. (cv x) (H~)) adantr (x.) (` (norm) (` (o. S T) (cv x))) opreq1d eqtr4d 3eqtr4rd (br (` (norm) (cv x)) (<_) (1)) adantrr nmoptri.1 S bdopft ax-mp S (opr (opr (1) (/) (` (normop) T)) (.s) (` T (cv x))) nmoplbt mp3an1 (opr (1) (/) (` (normop) T)) (` T (cv x)) ax-hvmulcl nmoptri.2 T nmopret ax-mp recn recclz nmoptri.2 T bdopft ax-mp (cv x) ffvrni syl2an (br (` (norm) (cv x)) (<_) (1)) adantrr (opr (1) (/) (` (normop) T)) absidt nmoptri.2 T nmopret ax-mp rerecclz 0re (0) (opr (1) (/) (` (normop) T)) ltlet mpan nmoptri.2 T nmopret ax-mp rerecclz nmoptri.2 T bdopft ax-mp T nmopgt0t ax-mp nmoptri.2 T nmopret ax-mp recgt0 sylbi sylc sylanc (e. (cv x) (H~)) adantr (x.) (` (norm) (` T (cv x))) opreq1d (opr (1) (/) (` (normop) T)) (` T (cv x)) norm-iiit nmoptri.2 T nmopret ax-mp recn recclz nmoptri.2 T bdopft ax-mp (cv x) ffvrni syl2an nmoptri.2 T nmopret ax-mp recn (` (norm) (` T (cv x))) (` (normop) T) divrec2t mp3an2 nmoptri.2 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl recnd sylan ancoms 3eqtr4d (br (` (norm) (cv x)) (<_) (1)) adantrr nmoptri.2 T bdopft ax-mp T (cv x) nmoplbt mp3an1 nmoptri.2 T nmopret ax-mp recn mulid2 syl6breqr (=/= (` (normop) T) (0)) adantl ax1re nmoptri.2 T nmopret ax-mp (` (norm) (` T (cv x))) (` (normop) T) (1) ledivmul2t mp3anl2 mpanl2 nmoptri.2 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl nmoptri.2 T bdopft ax-mp T nmopgt0t ax-mp biimp syl2an ancoms (br (` (norm) (cv x)) (<_) (1)) adantrr mpbird eqbrtrd sylanc eqbrtrd (` (norm) (` (o. S T) (cv x))) (` (normop) T) (` (normop) S) ledivmul2t nmoptri.1 S bdoplnt ax-mp nmoptri.2 T bdoplnt ax-mp lnopco lnopf (cv x) ffvrni (` (o. S T) (cv x)) normclt syl nmoptri.2 T nmopret ax-mp (e. (cv x) (H~)) a1i nmoptri.1 S nmopret ax-mp (e. (cv x) (H~)) a1i 3jca (=/= (` (normop) T) (0)) (br (` (norm) (cv x)) (<_) (1)) ad2antrl nmoptri.2 T bdopft ax-mp T nmopgt0t ax-mp biimp (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) adantr sylanc mpbid (` (normop) T) (0) df-ne sylanbr pm2.61ian ex rgen (o. S T) (opr (` (normop) S) (x.) (` (normop) T)) x nmopleubt mp3an)) thm (bdophs () ((nmoptri.1 (e. S (BndLinOp))) (nmoptri.2 (e. T (BndLinOp)))) (e. (opr S (+op) T) (BndLinOp)) ((opr S (+op) T) elbdop2t nmoptri.1 S bdoplnt ax-mp nmoptri.2 T bdoplnt ax-mp lnophs nmoptri.1 S bdopft ax-mp nmoptri.2 T bdopft ax-mp hoaddcl (opr S (+op) T) nmopxrt ax-mp nmoptri.1 S nmopret ax-mp nmoptri.2 T nmopret ax-mp readdcl pm3.2i nmoptri.1 S bdopft ax-mp nmoptri.2 T bdopft ax-mp hoaddcl (opr S (+op) T) nmopgtmnf ax-mp nmoptri.1 nmoptri.2 nmoptri pm3.2i (` (normop) (opr S (+op) T)) (opr (` (normop) S) (+) (` (normop) T)) xrret mp2an mpbir2an)) thm (bdophd () ((nmoptri.1 (e. S (BndLinOp))) (nmoptri.2 (e. T (BndLinOp)))) (e. (opr S (-op) T) (BndLinOp)) (nmoptri.1 S bdopft ax-mp nmoptri.2 T bdopft ax-mp honegsubOLD nmoptri.1 1cn negcl nmoptri.2 (-u (1)) bdophm ax-mp bdophs eqeltr)) thm (bdopco () ((nmoptri.1 (e. S (BndLinOp))) (nmoptri.2 (e. T (BndLinOp)))) (e. (o. S T) (BndLinOp)) ((o. S T) elbdop2t nmoptri.1 S bdoplnt ax-mp nmoptri.2 T bdoplnt ax-mp lnopco nmoptri.1 S bdoplnt ax-mp lnopf nmoptri.2 T bdoplnt ax-mp lnopf hocof (o. S T) nmopxrt ax-mp nmoptri.1 S nmopret ax-mp nmoptri.2 T nmopret ax-mp remulcl pm3.2i nmoptri.1 S bdoplnt ax-mp lnopf nmoptri.2 T bdoplnt ax-mp lnopf hocof (o. S T) nmopgtmnf ax-mp nmoptri.1 nmoptri.2 nmopco pm3.2i (` (normop) (o. S T)) (opr (` (normop) S) (x.) (` (normop) T)) xrret mp2an mpbir2an)) thm (adjco ((x y) (S x) (S y) (T x) (T y)) ((nmoptri.1 (e. S (BndLinOp))) (nmoptri.2 (e. T (BndLinOp)))) (= (` (adj) (o. S T)) (o. (` (adj) T) (` (adj) S))) (nmoptri.2 T adjbdlnt ax-mp (` (adj) T) bdopft ax-mp nmoptri.1 S adjbdlnt ax-mp (` (adj) S) bdopft ax-mp (cv y) hoco (cv x) (.i) opreq2d (e. (cv x) (H~)) adantl nmoptri.1 S bdopft ax-mp nmoptri.2 T bdopft ax-mp (cv x) hoco (.i) (cv y) opreq1d (e. (cv y) (H~)) adantr nmoptri.1 S bdopadjt ax-mp S (` T (cv x)) (cv y) adj2t mp3an1 nmoptri.2 T bdopft ax-mp (cv x) ffvrni sylan nmoptri.2 T bdopadjt ax-mp T (cv x) (` (` (adj) S) (cv y)) adj2t mp3an1 nmoptri.1 S adjbdlnt ax-mp (` (adj) S) bdopft ax-mp (cv y) ffvrni sylan2 3eqtrd nmoptri.1 nmoptri.2 bdopco (o. S T) bdopadjt ax-mp (o. S T) (cv x) (cv y) adj2t mp3an1 3eqtr2rd rgen2 nmoptri.1 nmoptri.2 bdopco (o. S T) adjbdlnt ax-mp (` (adj) (o. S T)) bdopft ax-mp nmoptri.2 T adjbdlnt ax-mp (` (adj) T) bdopft ax-mp nmoptri.1 S adjbdlnt ax-mp (` (adj) S) bdopft ax-mp hocof (` (adj) (o. S T)) (o. (` (adj) T) (` (adj) S)) x y hoeq2t mp2an mpbi)) thm (nmopcoadj ((T x)) ((nmopcoadj.1 (e. T (BndLinOp)))) (= (` (normop) (o. (` (adj) T) T)) (opr (` (normop) T) (^) (2))) (nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp nmopcoadj.1 T nmopret ax-mp resqcl letri3 nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp hocof nmopcoadj.1 T nmopret ax-mp resqcl (opr (` (normop) T) (^) (2)) rexrt ax-mp nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp (cv x) hoco (norm) fveq2d (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp (` T (cv x)) ffvrni syl (` (` (adj) T) (` T (cv x))) normclt syl (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) nmopret ax-mp (` (normop) (` (adj) T)) (` (norm) (` T (cv x))) axmulrcl mpan syl (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) nmopret ax-mp nmopcoadj.1 T nmopret ax-mp remulcl (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) a1i nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni nmopcoadj.1 T adjbdlnb mpbi (` T (cv x)) nmbdoplb syl (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T nmopret ax-mp nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) nmopret ax-mp nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp (` (adj) T) nmopge0t ax-mp (` (norm) (` T (cv x))) (` (normop) T) (` (normop) (` (adj) T)) lemul2it mpanr1 mp3anl3 mpanl2 nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl (br (` (norm) (cv x)) (<_) (1)) adantr (cv x) normclt nmopcoadj.1 T nmopret ax-mp (` (normop) T) (` (norm) (cv x)) axmulrcl mpan syl (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T nmopret ax-mp (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) a1i nmopcoadj.1 (cv x) nmbdoplb (br (` (norm) (cv x)) (<_) (1)) adantr ax1re nmopcoadj.1 T nmopret ax-mp nmopcoadj.1 T bdopft ax-mp T nmopge0t ax-mp (` (norm) (cv x)) (1) (` (normop) T) lemul2it mpanr1 mp3anl3 mpanl2 (cv x) normclt sylan nmopcoadj.1 T nmopret ax-mp recn mulid1 syl6breq letrd sylanc letrd eqbrtrd nmopcoadj.1 nmopadj (x.) (` (normop) T) opreq1i nmopcoadj.1 T nmopret ax-mp recn sqval eqtr4 syl6breq ex rgen (o. (` (adj) T) T) (opr (` (normop) T) (^) (2)) x nmopleubt mp3an nmopcoadj.1 T bdopft ax-mp nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp hocof (o. (` (adj) T) T) nmopge0t ax-mp nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp sqrcl ax-mp (` (sqr) (` (normop) (o. (` (adj) T) T))) rexrt ax-mp nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp (` T (cv x)) ffvrni syl (` (` (adj) T) (` T (cv x))) (cv x) ax-hicl mpancom (opr (` (` (adj) T) (` T (cv x))) (.i) (cv x)) absclt syl (br (` (norm) (cv x)) (<_) (1)) adantr (` (norm) (` (` (adj) T) (` T (cv x)))) (` (norm) (cv x)) axmulrcl nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp (` T (cv x)) ffvrni syl (` (` (adj) T) (` T (cv x))) normclt syl (cv x) normclt sylanc (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) a1i nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp (` T (cv x)) ffvrni syl (` (` (adj) T) (` T (cv x))) (cv x) bcst mpancom (br (` (norm) (cv x)) (<_) (1)) adantr (` (norm) (` (` (adj) T) (` T (cv x)))) (` (norm) (cv x)) axmulrcl nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp (` T (cv x)) ffvrni syl (` (` (adj) T) (` T (cv x))) normclt syl (cv x) normclt sylanc (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp (cv x) hococl (` (o. (` (adj) T) T) (cv x)) normclt syl (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) a1i ax1re (` (norm) (cv x)) (1) (` (norm) (` (` (adj) T) (` T (cv x)))) lemul2it mp3anl2 (cv x) normclt nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp (` T (cv x)) ffvrni syl (` (` (adj) T) (` T (cv x))) normclt syl jca (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp (` T (cv x)) ffvrni syl (` (` (adj) T) (` T (cv x))) normge0t syl (br (` (norm) (cv x)) (<_) (1)) anim1i sylanc nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp (` T (cv x)) ffvrni syl (` (` (adj) T) (` T (cv x))) normclt syl recnd (` (norm) (` (` (adj) T) (` T (cv x)))) ax1id syl nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp (cv x) hoco (norm) fveq2d eqtr4d (br (` (norm) (cv x)) (<_) (1)) adantr breqtrd nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp (cv x) hococl (` (o. (` (adj) T) T) (cv x)) normclt syl (br (` (norm) (cv x)) (<_) (1)) adantr (cv x) normclt nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp (` (normop) (o. (` (adj) T) T)) (` (norm) (cv x)) axmulrcl mpan syl (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) a1i nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (cv x) nmbdoplb (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp hocof (o. (` (adj) T) T) nmopge0t ax-mp ax1re nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp (` (norm) (cv x)) (1) (` (normop) (o. (` (adj) T) T)) lemul2it mp3anl3 mpanl2 mpanr1 (cv x) normclt sylan nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp recn mulid1 syl6breq letrd letrd letrd nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl (opr (` (norm) (` T (cv x))) (^) (2)) absidt (` (norm) (` T (cv x))) resqclt (` (norm) (` T (cv x))) sqge0t sylanc syl nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normsqt syl nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopadjt ax-mp (` (adj) T) (` T (cv x)) (cv x) adj2tOLD mp3an1 mpancom nmopcoadj.1 T bdopadjt ax-mp T adjadjt ax-mp (cv x) fveq1i (` T (cv x)) (.i) opreq2i syl5eqr eqtrd (abs) fveq2d eqtr3d (br (` (norm) (cv x)) (<_) (1)) adantr nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp hocof (o. (` (adj) T) T) nmopge0t ax-mp nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp sqsqr ax-mp (/\ (e. (cv x) (H~)) (br (` (norm) (cv x)) (<_) (1))) a1i 3brtr4d nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp hocof (o. (` (adj) T) T) nmopge0t ax-mp nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp sqrcl ax-mp nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp hocof (o. (` (adj) T) T) nmopge0t ax-mp nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp sqrge0 ax-mp pm3.2i (` (norm) (` T (cv x))) (` (sqr) (` (normop) (o. (` (adj) T) T))) le2sqt mpan2 nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normclt syl nmopcoadj.1 T bdopft ax-mp (cv x) ffvrni (` T (cv x)) normge0t syl sylanc (br (` (norm) (cv x)) (<_) (1)) adantr mpbird ex rgen T (` (sqr) (` (normop) (o. (` (adj) T) T))) x nmopleubt mp3an nmopcoadj.1 T bdopft ax-mp T nmopge0t ax-mp nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp hocof (o. (` (adj) T) T) nmopge0t ax-mp nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp sqrge0 ax-mp nmopcoadj.1 T nmopret ax-mp nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp hocof (o. (` (adj) T) T) nmopge0t ax-mp nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp sqrcl ax-mp le2sq mp2an mpbi nmopcoadj.1 T adjbdlnb mpbi (` (adj) T) bdopft ax-mp nmopcoadj.1 T bdopft ax-mp hocof (o. (` (adj) T) T) nmopge0t ax-mp nmopcoadj.1 T adjbdlnb mpbi nmopcoadj.1 bdopco (o. (` (adj) T) T) nmopret ax-mp sqsqr ax-mp breqtr mpbir2an)) thm (nmopcoadj2 () ((nmopcoadj.1 (e. T (BndLinOp)))) (= (` (normop) (o. T (` (adj) T))) (opr (` (normop) T) (^) (2))) (nmopcoadj.1 T adjbdlnt ax-mp nmopcoadj nmopcoadj.1 T bdopadjt ax-mp T adjadjt ax-mp (` (adj) T) coeq1i (normop) fveq2i nmopcoadj.1 nmopadj (^) (2) opreq1i 3eqtr3)) thm (unierr () ((unierr.1 (e. F (UniOp))) (unierr.2 (e. G (UniOp))) (unierr.3 (e. S (UniOp))) (unierr.4 (e. T (UniOp)))) (br (` (normop) (opr (o. F G) (-op) (o. S T))) (<_) (opr (` (normop) (opr F (-op) S)) (+) (` (normop) (opr G (-op) T)))) (unierr.1 F unopbdt ax-mp F bdopft ax-mp unierr.2 G unopbdt ax-mp G bdopft ax-mp hocof unierr.3 S unopbdt ax-mp S bdopft ax-mp unierr.4 T unopbdt ax-mp T bdopft ax-mp hocof hosubcl (opr (o. F G) (-op) (o. S T)) nmop0h mpan2 unierr.1 F unopbdt ax-mp F bdopft ax-mp unierr.3 S unopbdt ax-mp S bdopft ax-mp hosubcl (opr F (-op) S) nmop0h mpan2 unierr.2 G unopbdt ax-mp G bdopft ax-mp unierr.4 T unopbdt ax-mp T bdopft ax-mp hosubcl (opr G (-op) T) nmop0h mpan2 (+) opreq12d 0re leid 0cn addid1 breqtrr syl5breqr eqbrtrd (H~) (0H) df-ne unierr.2 G nmopunt mpan2 (` (normop) (opr F (-op) S)) (x.) opreq2d unierr.1 F unopbdt ax-mp unierr.3 S unopbdt ax-mp bdophd (opr F (-op) S) nmopret ax-mp recn mulid1 syl6eq unierr.3 S nmopunt mpan2 (x.) (` (normop) (opr G (-op) T)) opreq1d unierr.2 G unopbdt ax-mp unierr.4 T unopbdt ax-mp bdophd (opr G (-op) T) nmopret ax-mp recn mulid2 syl6eq (+) opreq12d unierr.1 F unopbdt ax-mp F bdopft ax-mp unierr.2 G unopbdt ax-mp G bdopft ax-mp hocof unierr.3 S unopbdt ax-mp S bdopft ax-mp unierr.2 G unopbdt ax-mp G bdopft ax-mp hocof unierr.3 S unopbdt ax-mp S bdopft ax-mp unierr.4 T unopbdt ax-mp T bdopft ax-mp hocof honpncan (normop) fveq2i unierr.1 F unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco unierr.3 S unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco bdophd unierr.3 S unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco unierr.3 S unopbdt ax-mp unierr.4 T unopbdt ax-mp bdopco bdophd nmoptri unierr.1 F unopbdt ax-mp F bdopft ax-mp unierr.3 S unopbdt ax-mp S bdopft ax-mp unierr.2 G unopbdt ax-mp G bdopft ax-mp hocsubdir (normop) fveq2i unierr.1 F unopbdt ax-mp unierr.3 S unopbdt ax-mp bdophd unierr.2 G unopbdt ax-mp nmopco eqbrtrr unierr.3 S unopbdt ax-mp S bdoplnt ax-mp unierr.2 G unopbdt ax-mp G bdopft ax-mp unierr.4 T unopbdt ax-mp T bdopft ax-mp hoddi (normop) fveq2i unierr.3 S unopbdt ax-mp unierr.2 G unopbdt ax-mp unierr.4 T unopbdt ax-mp bdophd nmopco eqbrtrr unierr.1 F unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco unierr.3 S unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco bdophd (opr (o. F G) (-op) (o. S G)) nmopret ax-mp unierr.3 S unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco unierr.3 S unopbdt ax-mp unierr.4 T unopbdt ax-mp bdopco bdophd (opr (o. S G) (-op) (o. S T)) nmopret ax-mp unierr.1 F unopbdt ax-mp unierr.3 S unopbdt ax-mp bdophd (opr F (-op) S) nmopret ax-mp unierr.2 G unopbdt ax-mp G nmopret ax-mp remulcl unierr.3 S unopbdt ax-mp S nmopret ax-mp unierr.2 G unopbdt ax-mp unierr.4 T unopbdt ax-mp bdophd (opr G (-op) T) nmopret ax-mp remulcl le2add mp2an unierr.1 F unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco unierr.3 S unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco bdophd unierr.3 S unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco unierr.3 S unopbdt ax-mp unierr.4 T unopbdt ax-mp bdopco bdophd bdophs (opr (opr (o. F G) (-op) (o. S G)) (+op) (opr (o. S G) (-op) (o. S T))) nmopret ax-mp unierr.1 F unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco unierr.3 S unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco bdophd (opr (o. F G) (-op) (o. S G)) nmopret ax-mp unierr.3 S unopbdt ax-mp unierr.2 G unopbdt ax-mp bdopco unierr.3 S unopbdt ax-mp unierr.4 T unopbdt ax-mp bdopco bdophd (opr (o. S G) (-op) (o. S T)) nmopret ax-mp readdcl unierr.1 F unopbdt ax-mp unierr.3 S unopbdt ax-mp bdophd (opr F (-op) S) nmopret ax-mp unierr.2 G unopbdt ax-mp G nmopret ax-mp remulcl unierr.3 S unopbdt ax-mp S nmopret ax-mp unierr.2 G unopbdt ax-mp unierr.4 T unopbdt ax-mp bdophd (opr G (-op) T) nmopret ax-mp remulcl readdcl letr mp2an eqbrtrr syl5breq sylbir pm2.61i)) thm (branmfnt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (H~)) (= (` (normfn) (` (bra) A)) (` (norm) A))) (A (0v) (bra) fveq2 (normfn) fveq2d A (0v) (norm) fveq2 eqeq12d A brafnt (` (bra) A) x y nmfnvalt syl (=/= A (0v)) adantr ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` (` (bra) A) (cv y))))))) (` (norm) A) z w supxr2 A brafnt (` (bra) A) x y nmfnsetret syl ressxr (e. A (H~)) a1i sstrd A normclt (` (norm) A) rexrt syl jca (=/= A (0v)) adantr (= (cv z) (` (abs) (` (` (bra) A) (cv y)))) id A (cv y) bravalvalt (abs) fveq2d (br (` (norm) (cv y)) (<_) (1)) adantr sylan9eqr (cv y) A bcs2t 3expa ancom1s (= (cv z) (` (abs) (` (` (bra) A) (cv y)))) adantr eqbrtrd exp41 imp4a r19.23adv imp z visset (cv x) (cv z) (` (abs) (` (` (bra) A) (cv y))) eqeq1 (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv elab sylan2b r19.21aiva (=/= A (0v)) adantr (cv w) (` (norm) A) (cv z) (<) breq2 ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` (` (bra) A) (cv y))))))) rcla4ev (cv y) (opr (opr (1) (/) (` (norm) A)) (.s) A) (norm) fveq2 (<_) (1) breq1d (cv y) (opr (opr (1) (/) (` (norm) A)) (.s) A) (.i) A opreq1 (abs) fveq2d (` (norm) A) eqeq2d anbi12d (H~) rcla4ev (opr (1) (/) (` (norm) A)) A ax-hvmulcl (` (norm) A) recclt A normclt recnd (=/= A (0v)) adantr A normne0t biimpar sylanc (e. A (H~)) (=/= A (0v)) pm3.26 sylanc A norm1t ax1re leid syl6eqbr (opr (1) (/) (` (norm) A)) A A ax-his3 (` (norm) A) recclt A normclt recnd (=/= A (0v)) adantr A normne0t biimpar sylanc (e. A (H~)) (=/= A (0v)) pm3.26 (e. A (H~)) (=/= A (0v)) pm3.26 syl3anc (opr (opr (opr (1) (/) (` (norm) A)) (.s) A) (.i) A) absidt (opr (1) (/) (` (norm) A)) A A ax-his3 (` (norm) A) recclt A normclt recnd (=/= A (0v)) adantr A normne0t biimpar sylanc (e. A (H~)) (=/= A (0v)) pm3.26 (e. A (H~)) (=/= A (0v)) pm3.26 syl3anc (opr (1) (/) (` (norm) A)) (opr A (.i) A) axmulrcl (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc A hiidrclt (=/= A (0v)) adantr sylanc eqeltrd (opr (1) (/) (` (norm) A)) (opr A (.i) A) mulge0t (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc A hiidrclt (=/= A (0v)) adantr jca 0re (0) (opr (1) (/) (` (norm) A)) ltlet mpan (` (norm) A) rerecclt A normclt (=/= A (0v)) adantr A normne0t biimpar sylanc (` (norm) A) recgt0t A normclt (=/= A (0v)) adantr A normgt0t biimpa sylanc sylc A hiidge0t (=/= A (0v)) adantr jca sylanc (opr (1) (/) (` (norm) A)) A A ax-his3 (` (norm) A) recclt A normclt recnd (=/= A (0v)) adantr A normne0t biimpar sylanc (e. A (H~)) (=/= A (0v)) pm3.26 (e. A (H~)) (=/= A (0v)) pm3.26 syl3anc breqtrrd sylanc (opr (1) (/) (` (norm) A)) (` (norm) A) axmulcom (` (norm) A) recclt A normclt recnd (=/= A (0v)) adantr A normne0t biimpar sylanc A normclt recnd (=/= A (0v)) adantr sylanc (` (norm) A) recidt A normclt recnd (=/= A (0v)) adantr A normne0t biimpar sylanc eqtrd (` (norm) A) (x.) opreq2d A normclt recnd (` (norm) A) ax1id syl (=/= A (0v)) adantr eqtr2d (` (norm) A) (opr (1) (/) (` (norm) A)) (` (norm) A) mul12t A normclt recnd (=/= A (0v)) adantr (` (norm) A) recclt A normclt recnd (=/= A (0v)) adantr A normne0t biimpar sylanc A normclt recnd (=/= A (0v)) adantr syl3anc A normclt recnd (` (norm) A) sqvalt syl A normsqt eqtr3d (=/= A (0v)) adantr (opr (1) (/) (` (norm) A)) (x.) opreq2d 3eqtrd 3eqtr4rd jca sylanc A (cv y) bravalvalt (abs) fveq2d (` (norm) A) eqeq2d (br (` (norm) (cv y)) (<_) (1)) anbi2d rexbidva (=/= A (0v)) adantr mpbird (norm) A fvex (cv x) (` (norm) A) (` (abs) (` (` (bra) A) (cv y))) eqeq1 (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv elab sylibr sylan (e. (cv z) (RR)) adantlr ex r19.21aiva jca sylanc eqtrd nmfn0 bra0 (normfn) fveq2i norm0 3eqtr4 (e. A (H~)) a1i pm2.61ne)) thm (brabnt () () (-> (e. A (H~)) (e. (` (normfn) (` (bra) A)) (RR))) (A branmfnt A normclt eqeltrd)) thm (rnbra ((x y) (x z) (y z) (t x) (t y) (t z)) () (= (ran (bra)) (i^i (LinFn) (ConFn))) ((cv z) x y bravalt (cv t) eqeq2d (e. (cv z) (H~)) (= (cv t) (` (bra) (cv z))) pm3.27 (cv z) bralnfnt (= (cv t) (` (bra) (cv z))) adantr eqeltrd (cv t) (` (bra) (cv z)) (normfn) fveq2 (e. (cv z) (H~)) adantl (cv z) brabnt (= (cv t) (` (bra) (cv z))) adantr eqeltrd jca ex sylbird r19.23aiv (cv t) z x riesz1t biimpa (cv t) lnfnft (e. (` (normfn) (cv t)) (RR)) (e. (cv z) (H~)) ad2antrr (cv t) (H~) (CC) x y fopabfv pm3.26bd syl (= ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (` (cv t) (cv x))))) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (cv x) (.i) (cv z)))))) id sylan9eq ex x (H~) (= (` (cv t) (cv x)) (opr (cv x) (.i) (cv z))) hbra1 (A.e. x (H~) (= (` (cv t) (cv x)) (opr (cv x) (.i) (cv z)))) y ax-17 x (H~) (= (` (cv t) (cv x)) (opr (cv x) (.i) (cv z))) ra4 (` (cv t) (cv x)) (opr (cv x) (.i) (cv z)) (cv y) eqeq2 syl6 pm5.32d opabbid syl5 r19.22dva mpd impbi (cv t) lnfncnbdt pm5.32i bitr4 ax-hilex x y (opr (cv x) (.i) (cv z)) funopabex2 z t x y df-bra (cv t) elrnopab (cv t) (LinFn) (ConFn) elin 3bitr4 eqriv)) thm (bra11 ((x y) (x z) (y z) (t x) (t y) (t z)) () (:-1-1-onto-> (bra) (H~) (i^i (LinFn) (ConFn))) (ax-hilex x y (opr (cv x) (.i) (cv z)) funopabex2 z t x y df-bra fnopab2 (bra) t z funcnv ax-hilex x y (opr (cv x) (.i) (cv z)) funopabex2 z t x y df-bra fnopab2 t visset (bra) (H~) (cv z) fnbrfvb mpan ax-hilex x y (opr (cv x) (.i) (cv z)) funopabex2 z t x y df-bra fnopab2 t visset (bra) (H~) (cv y) fnbrfvb mpan bi2anan9 (` (bra) (cv z)) (` (bra) (cv y)) (cv x) fveq1 (/\ (/\ (e. (cv z) (H~)) (e. (cv y) (H~))) (e. (cv x) (H~))) adantl (cv z) (cv x) bravalvalt (e. (cv y) (H~)) adantlr (= (` (bra) (cv z)) (` (bra) (cv y))) adantr (cv y) (cv x) bravalvalt (e. (cv z) (H~)) adantll (= (` (bra) (cv z)) (` (bra) (cv y))) adantr 3eqtr3d exp31 com23 r19.21adv (cv z) (cv y) x hial2eq2t sylibd (` (bra) (cv z)) (cv t) (` (bra) (cv y)) eqtr3t syl5 sylbird imp an4s z y gen2 (E.e. z (H~) (= (` (bra) (cv z)) (cv t))) a1i ax-hilex x y (opr (cv x) (.i) (cv z)) funopabex2 z t x y df-bra fnopab2 (bra) (H~) (cv t) z fvelrn ax-mp z visset (bra) (cv t) breldm ax-hilex x y (opr (cv x) (.i) (cv z)) funopabex2 z t x y df-bra dmopab2 syl6eleq pm4.71ri z mobii (cv z) (cv y) (H~) eleq1 (cv z) (cv y) (bra) (cv t) breq1 anbi12d mo4 bitr 3imtr4 mprgbir rnbra 3pm3.2i (bra) (H~) (i^i (LinFn) (ConFn)) f1o2 mpbir)) thm (bracnlnt () () (-> (e. A (H~)) (e. (` (bra) A) (i^i (LinFn) (ConFn)))) (bra11 (bra) (H~) (i^i (LinFn) (ConFn)) f1of ax-mp A ffvrni)) thm (cnvbravalt ((x y) (T x) (T y)) () (-> (e. T (i^i (LinFn) (ConFn))) (= (` (`' (bra)) T) (U. ({e.|} y (H~) (A.e. x (H~) (= (` T (cv x)) (opr (cv x) (.i) (cv y)))))))) (rnbra T eleq2i biimpr bra11 (bra) (H~) (i^i (LinFn) (ConFn)) f1of ax-mp (bra) (H~) (i^i (LinFn) (ConFn)) ffn ax-mp (bra) (H~) T y fvelrn ax-mp sylib (e. (cv x) (H~)) adantr bra11 (bra) (H~) (i^i (LinFn) (ConFn)) (cv y) T f1ocnvfv mpan imp (cv x) (.i) opreq2d (/\ (e. T (i^i (LinFn) (ConFn))) (e. (cv x) (H~))) adantll (cv y) (cv x) bravalvalt ancoms (e. T (i^i (LinFn) (ConFn))) adantll (= (` (bra) (cv y)) T) adantr (` (bra) (cv y)) T (cv x) fveq1 (/\ (/\ (e. T (i^i (LinFn) (ConFn))) (e. (cv x) (H~))) (e. (cv y) (H~))) adantl 3eqtr2rd exp31 r19.23adv mpd r19.21aiva (cv y) (` (`' (bra)) T) (cv x) (.i) opreq2 (` T (cv x)) eqeq2d x (H~) ralbidv (H~) reuuni2 bra11 (bra) (H~) (i^i (LinFn) (ConFn)) T f1ocnvdm mpan T y x riesz4t sylanc mpbid eqcomd)) thm (cnvbraclt () () (-> (e. T (i^i (LinFn) (ConFn))) (e. (` (`' (bra)) T) (H~))) (bra11 (bra) (H~) (i^i (LinFn) (ConFn)) T f1ocnvdm mpan)) thm (cnvbrabrat () () (-> (e. A (H~)) (= (` (`' (bra)) (` (bra) A)) A)) (bra11 (bra) (H~) (i^i (LinFn) (ConFn)) A f1ocnvfv1 mpan)) thm (bracnvbrat () () (-> (e. T (i^i (LinFn) (ConFn))) (= (` (bra) (` (`' (bra)) T)) T)) (bra11 (bra) (H~) (i^i (LinFn) (ConFn)) T f1ocnvfv2 mpan)) thm (cnvbramult () () (-> (/\ (e. A (CC)) (e. T (i^i (LinFn) (ConFn)))) (= (` (`' (bra)) (opr A (.fn) T)) (opr (` (*) A) (.s) (` (`' (bra)) T)))) ((` (*) A) (` (`' (bra)) T) brafnmult A cjclt sylan A cjcjt (e. (` (`' (bra)) T) (H~)) adantr (.fn) (` (bra) (` (`' (bra)) T)) opreq1d eqtrd T cnvbraclt sylan2 T bracnvbrat A (.fn) opreq2d (e. A (CC)) adantl eqtrd (`' (bra)) fveq2d (` (*) A) (` (`' (bra)) T) ax-hvmulcl A cjclt T cnvbraclt syl2an (opr (` (*) A) (.s) (` (`' (bra)) T)) cnvbrabrat syl eqtr3d)) thm (kbass5t ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (o. (opr A (ketbra) B) (opr C (ketbra) D)) (opr (` (opr A (ketbra) B) C) (ketbra) D))) (C D x y kbvalt rneqd C D kbopt (opr C (ketbra) D) (H~) (H~) frn syl eqsstr3d (/\ (e. A (H~)) (e. B (H~))) adantl (opr (cv x) (.i) B) (.s) A oprex (opr (cv x) (.i) D) (.s) C oprex ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (opr (cv x) (.i) B) (.s) A)))) eqid ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (opr (cv x) (.i) D) (.s) C)))) eqid fopabco syl (opr (cv x) (.i) D) C B ax-his3 (cv x) D ax-hicl ancoms (e. C (H~)) adantll (e. B (H~)) adantll (e. C (H~)) (e. D (H~)) pm3.26 (e. B (H~)) (e. (cv x) (H~)) ad2antlr (e. B (H~)) (/\ (e. C (H~)) (e. D (H~))) pm3.26 (e. (cv x) (H~)) adantr syl3anc (e. A (H~)) adantlll (.s) A opreq1d (opr (cv x) (.i) D) (opr C (.i) B) A ax-hvmulass (cv x) D ax-hicl ancoms (e. C (H~)) adantll (/\ (e. A (H~)) (e. B (H~))) adantll C B ax-hicl ancoms (e. D (H~)) adantrr (e. A (H~)) adantll (e. (cv x) (H~)) adantr (e. A (H~)) (e. B (H~)) pm3.26 (/\ (e. C (H~)) (e. D (H~))) (e. (cv x) (H~)) ad2antrr syl3anc eqtrd (opr (cv x) (.i) D) (.s) C oprex (opr (opr (cv x) (.i) D) (.s) C) (V) x (opr (cv x) (.i) B) (.s) A csbopr1g ax-mp (opr (cv x) (.i) D) (.s) C oprex (opr (opr (cv x) (.i) D) (.s) C) (V) x (cv x) (.i) B csbopr1g ax-mp (opr (cv x) (.i) D) (.s) C oprex (opr (opr (cv x) (.i) D) (.s) C) (V) x csbvarg ax-mp (.i) B opreq1i eqtr (.s) A opreq1i eqtr syl5eq (cv y) eqeq2d ex pm5.32d x y opabbidv eqtrd A B x y kbvalt (/\ (e. C (H~)) (e. D (H~))) adantr (opr C (ketbra) D) coeq1d C D x y kbvalt (/\ (e. A (H~)) (e. B (H~))) adantl ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (opr (cv x) (.i) B) (.s) A)))) coeq2d eqtrd A B C kbvalvalt 3expa (e. D (H~)) adantr (ketbra) D opreq1d (opr (opr C (.i) B) (.s) A) D x y kbvalt (opr C (.i) B) A ax-hvmulcl C B ax-hicl sylan ancom31s sylan eqtrd anasss 3eqtr4d)) thm (leopg ((A x) (B x) (t u) (t x) (T t) (u x) (T u) (T x) (U t) (U u) (U x)) () (-> (/\ (e. T A) (e. U B)) (<-> (br T (<_op) U) (/\ (e. (opr U (-op) T) (HrmOp)) (A.e. x (H~) (br (0) (<_) (opr (` (opr U (-op) T) (cv x)) (.i) (cv x))))))) ((cv t) T (cv u) (-op) opreq2 (HrmOp) eleq1d (cv t) T (cv u) (-op) opreq2 (cv x) fveq1d (.i) (cv x) opreq1d (0) (<_) breq2d x (H~) ralbidv anbi12d (cv u) U (-op) T opreq1 (HrmOp) eleq1d (cv u) U (-op) T opreq1 (cv x) fveq1d (.i) (cv x) opreq1d (0) (<_) breq2d x (H~) ralbidv anbi12d t u x df-leop A B brabg)) thm (leopt ((T x) (U x)) () (-> (/\ (e. T (HrmOp)) (e. U (HrmOp))) (<-> (br T (<_op) U) (A.e. x (H~) (br (0) (<_) (opr (` (opr U (-op) T) (cv x)) (.i) (cv x)))))) (T (HrmOp) U (HrmOp) x leopg U T hmopdt ancoms (A.e. x (H~) (br (0) (<_) (opr (` (opr U (-op) T) (cv x)) (.i) (cv x)))) biantrurd bitr4d)) thm (leop2t ((T x) (U x)) () (-> (/\ (e. T (HrmOp)) (e. U (HrmOp))) (<-> (br T (<_op) U) (A.e. x (H~) (br (opr (` T (cv x)) (.i) (cv x)) (<_) (opr (` U (cv x)) (.i) (cv x)))))) (T U x leopt U T (cv x) hodvalt 3com12 3expa (.i) (cv x) opreq1d (` U (cv x)) (` T (cv x)) (cv x) his2subt U (H~) (H~) (cv x) ffvrn (:--> T (H~) (H~)) adantll T (H~) (H~) (cv x) ffvrn (:--> U (H~) (H~)) adantlr (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) (e. (cv x) (H~)) pm3.27 syl3anc eqtrd T hmopft U hmopft anim12i sylan (0) (<_) breq2d (opr (` U (cv x)) (.i) (cv x)) (opr (` T (cv x)) (.i) (cv x)) subge0t U (cv x) hmopret (e. T (HrmOp)) adantll T (cv x) hmopret (e. U (HrmOp)) adantlr sylanc bitrd ralbidva bitrd)) thm (leop3t ((T x) (U x)) () (-> (/\ (e. T (HrmOp)) (e. U (HrmOp))) (<-> (br T (<_op) U) (br (0op) (<_op) (opr U (-op) T)))) (T U x leopt (0op) (opr U (-op) T) x leop2t 0hmop (/\ (e. T (HrmOp)) (e. U (HrmOp))) a1i U T hmopdt ancoms sylanc (cv x) ho0valt (.i) (cv x) opreq1d (cv x) hi01t eqtrd (<_) (opr (` (opr U (-op) T) (cv x)) (.i) (cv x)) breq1d ralbiia syl6rbb bitrd)) thm (leoppost ((T x)) () (-> (e. T (HrmOp)) (<-> (br (0op) (<_op) T) (A.e. x (H~) (br (0) (<_) (opr (` T (cv x)) (.i) (cv x)))))) (0hmop (0op) T x leopt mpan T hmopft T hosubid1t syl (cv x) fveq1d (.i) (cv x) opreq1d (0) (<_) breq2d x (H~) ralbidv bitrd)) thm (leoprft ((T x)) () (-> (e. T (HrmOp)) (br T (<_op) T)) (T hmopft T hodid2tOLD syl (e. (cv x) (H~)) adantr (cv x) fveq1d (cv x) ho0valtOLD (e. T (HrmOp)) adantl eqtrd (.i) (cv x) opreq1d (cv x) hi01t (e. T (HrmOp)) adantl eqtr2d 0re leid syl5breq r19.21aiva T T x leopt anidms mpbird)) thm (leopsqt ((T x)) () (-> (e. T (HrmOp)) (br (0op) (<_op) (o. T T))) (T (H~) (H~) (cv x) ffvrn T hmopft sylan (` T (cv x)) hiidge0t syl T (` T (cv x)) (cv x) hmopt (e. T (HrmOp)) (e. (cv x) (H~)) pm3.26 T (H~) (H~) (cv x) ffvrn T hmopft sylan (e. T (HrmOp)) (e. (cv x) (H~)) pm3.27 syl3anc T T (H~) (H~) (cv x) fvco3 T hmopft T (H~) (H~) ffun syl (e. (cv x) (H~)) adantr T hmopft (e. (cv x) (H~)) adantr (e. T (HrmOp)) (e. (cv x) (H~)) pm3.27 syl3anc (.i) (cv x) opreq1d eqtr4d breqtrd r19.21aiva (o. T T) eqid T T hmopcot mp3an3 anidms (o. T T) x leoppost syl mpbird)) thm (0leop () () (br (0op) (<_op) (0op)) (0hmop (0op) leoprft ax-mp)) thm (idleop () () (br (0op) (<_op) (Iop)) (0hmop idhmop (0op) (Iop) x leop2t mp2an (cv x) hiidge0t (cv x) ho0valt (.i) (cv x) opreq1d (cv x) hi01t eqtrd (cv x) hoivalt (.i) (cv x) opreq1d 3brtr4d mprgbir)) thm (leopaddt ((T x) (U x)) () (-> (/\ (/\ (e. T (HrmOp)) (e. U (HrmOp))) (/\ (br (0op) (<_op) T) (br (0op) (<_op) U))) (br (0op) (<_op) (opr T (+op) U))) ((opr (` T (cv x)) (.i) (cv x)) (opr (` U (cv x)) (.i) (cv x)) addge0t ex T (cv x) hmopret U (cv x) hmopret syl2an anandirs T U (cv x) hosvaltOLD (.i) (cv x) opreq1d (` T (cv x)) (` U (cv x)) (cv x) ax-his2 T (H~) (H~) (cv x) ffvrn (:--> U (H~) (H~)) adantlr U (H~) (H~) (cv x) ffvrn (:--> T (H~) (H~)) adantll (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) (e. (cv x) (H~)) pm3.27 syl3anc eqtrd T hmopft U hmopft anim12i sylan (0) (<_) breq2d sylibrd r19.20dva x (H~) (br (0) (<_) (opr (` T (cv x)) (.i) (cv x))) (br (0) (<_) (opr (` U (cv x)) (.i) (cv x))) r19.26 syl5ibr T x leoppost U x leoppost bi2anan9 T U hmopst (opr T (+op) U) x leoppost syl 3imtr4d imp)) thm (leopmulit ((A x) (T x)) () (-> (/\ (/\ (e. A (RR)) (e. T (HrmOp))) (/\ (br (0) (<_) A) (br (0op) (<_op) T))) (br (0op) (<_op) (opr A (.op) T))) (A (opr (` T (cv x)) (.i) (cv x)) mulge0t (e. A (RR)) (e. T (HrmOp)) pm3.26 (e. (cv x) (H~)) adantr T (cv x) hmopret (e. A (RR)) adantll jca sylan exp32 imp an1rs A T (cv x) homvalt 3expa (.i) (cv x) opreq1d A (` T (cv x)) (cv x) ax-his3 (e. A (CC)) (:--> T (H~) (H~)) pm3.26 (e. (cv x) (H~)) adantr T (H~) (H~) (cv x) ffvrn (e. A (CC)) adantll (/\ (e. A (CC)) (:--> T (H~) (H~))) (e. (cv x) (H~)) pm3.27 syl3anc eqtrd A recnt T hmopft anim12i sylan (0) (<_) breq2d (br (0) (<_) A) adantlr sylibrd r19.20dva ex imp3a T x leoppost (e. A (RR)) adantl (br (0) (<_) A) anbi2d A T hmopmt (opr A (.op) T) x leoppost syl 3imtr4d imp)) thm (leopmult () () (-> (/\/\ (e. A (RR)) (e. T (HrmOp)) (br (0) (<) A)) (<-> (br (0op) (<_op) T) (br (0op) (<_op) (opr A (.op) T)))) (A T leopmulit (e. A (RR)) (e. T (HrmOp)) (br (0) (<) A) 3simpa (br (0op) (<_op) T) adantr 0re (0) A ltlet 3impia mp3an1 (e. T (HrmOp)) 3adant2 (br (0op) (<_op) T) anim1i sylanc ex (opr (1) (/) A) (opr A (.op) T) leopmulit anassrs A gt0ne0t A rerecclt syldan (e. T (HrmOp)) 3adant2 A T hmopmt (br (0) (<) A) 3adant3 jca A recgt0t (0) (opr (1) (/) A) ltlet 0re (/\ (e. A (RR)) (br (0) (<) A)) a1i A gt0ne0t A rerecclt syldan sylanc mpd (e. T (HrmOp)) 3adant2 jca sylan A recid2t A recnt (br (0) (<) A) adantr A gt0ne0t sylanc (.op) T opreq1d (e. T (HrmOp)) 3adant2 (opr (1) (/) A) A T homulasst A recclt A recnt (br (0) (<) A) adantr A gt0ne0t sylanc (e. T (HrmOp)) 3adant2 A recnt (e. T (HrmOp)) (br (0) (<) A) 3ad2ant1 T hmopft (e. A (RR)) (br (0) (<) A) 3ad2ant2 syl3anc T hmopft T homulid2t syl (e. A (RR)) (br (0) (<) A) 3ad2ant2 3eqtr3d (br (0op) (<_op) (opr A (.op) T)) adantr breqtrd ex impbid)) thm (leopmul2it () () (-> (/\ (/\/\ (e. A (RR)) (e. T (HrmOp)) (e. U (HrmOp))) (/\ (br (0) (<_) A) (br T (<_op) U))) (br (opr A (.op) T) (<_op) (opr A (.op) U))) (A (opr U (-op) T) leopmulit exp32 (e. A (RR)) (e. T (HrmOp)) (e. U (HrmOp)) 3simp1 U T hmopdt ancoms (e. A (RR)) 3adant1 sylanc imp T U leop3t (e. A (RR)) 3adant1 (br (0) (<_) A) adantr (opr A (.op) T) (opr A (.op) U) leop3t A T hmopmt A U hmopmt syl2an 3impdi A U T hosubdit A recnt U hmopft T hmopft syl3an 3com23 (0op) (<_op) breq2d bitr4d (br (0) (<_) A) adantr 3imtr4d ex imp32)) thm (leopnmidt ((T x)) () (-> (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) (br T (<_op) (opr (` (normop) T) (.op) (Iop)))) (T (cv x) hmopret (e. (` (normop) T) (RR)) adantlr T (cv x) hmopret recnd (opr (` T (cv x)) (.i) (cv x)) absclt syl (e. (` (normop) T) (RR)) adantlr (opr (` (normop) T) (.op) (Iop)) (cv x) hmopret idhmop (` (normop) T) (Iop) hmopmt mpan2 sylan (e. T (HrmOp)) adantll T (cv x) hmopret (opr (` T (cv x)) (.i) (cv x)) leabst syl (e. (` (normop) T) (RR)) adantlr T (cv x) hmopret recnd (opr (` T (cv x)) (.i) (cv x)) absclt syl (e. (` (normop) T) (RR)) adantlr (` (norm) (` T (cv x))) (` (norm) (cv x)) axmulrcl T (H~) (H~) (cv x) ffvrn (` T (cv x)) normclt syl T hmopft sylan (e. (` (normop) T) (RR)) adantlr (cv x) normclt (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) adantl sylanc (opr (` (normop) T) (.op) (Iop)) (cv x) hmopret idhmop (` (normop) T) (Iop) hmopmt mpan2 sylan (e. T (HrmOp)) adantll (` T (cv x)) (cv x) bcst T (H~) (H~) (cv x) ffvrn T hmopft sylan (e. T (HrmOp)) (e. (cv x) (H~)) pm3.27 sylanc (e. (` (normop) T) (RR)) adantlr (` (norm) (` T (cv x))) (opr (` (normop) T) (x.) (` (norm) (cv x))) (` (norm) (cv x)) lemul1it T (H~) (H~) (cv x) ffvrn (` T (cv x)) normclt syl T hmopft sylan (e. (` (normop) T) (RR)) adantlr (` (normop) T) (` (norm) (cv x)) axmulrcl (e. T (HrmOp)) (e. (` (normop) T) (RR)) pm3.27 (cv x) normclt syl2an (cv x) normclt (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) adantl 3jca (cv x) normge0t (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) adantl T (cv x) nmbdoplbt T elbdop2t biimpr T hmoplint sylan sylan jca sylanc (cv x) normclt recnd (` (norm) (cv x)) sqvalt syl (cv x) normsqt eqtr3d (` (normop) T) (x.) opreq2d (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) adantl (` (normop) T) (` (norm) (cv x)) (` (norm) (cv x)) axmulass (` (normop) T) recnt (e. T (HrmOp)) (e. (cv x) (H~)) ad2antlr (cv x) normclt (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) adantl recnd (cv x) normclt (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) adantl recnd syl3anc (` (normop) T) (cv x) (cv x) ax-his3 (` (normop) T) recnt (e. T (HrmOp)) (e. (cv x) (H~)) ad2antlr (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) (e. (cv x) (H~)) pm3.27 (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) (e. (cv x) (H~)) pm3.27 syl3anc 3eqtr4d (` (normop) T) (Iop) (cv x) homvalt (` (normop) T) recnt (e. T (HrmOp)) (e. (cv x) (H~)) ad2antlr hoif (Iop) (H~) (H~) f1of ax-mp (/\ (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) (e. (cv x) (H~))) a1i (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) (e. (cv x) (H~)) pm3.27 syl3anc (cv x) hoivalt (` (normop) T) (.s) opreq2d (/\ (e. T (HrmOp)) (e. (` (normop) T) (RR))) adantl eqtrd (.i) (cv x) opreq1d eqtr4d breqtrd letrd letrd r19.21aiva T (opr (` (normop) T) (.op) (Iop)) x leop2t idhmop (` (normop) T) (Iop) hmopmt mpan2 sylan2 mpbird)) thm (nmopleidt () () (-> (/\/\ (e. T (HrmOp)) (e. (` (normop) T) (RR)) (=/= T (0op))) (br (opr (opr (1) (/) (` (normop) T)) (.op) T) (<_op) (Iop))) (T nmlnopne0t biimpar T hmoplint sylan (e. (` (normop) T) (RR)) adantlr (opr (1) (/) (` (normop) T)) T (opr (` (normop) T) (.op) (Iop)) leopmul2it (` (normop) T) rerecclt (e. T (HrmOp)) adantll (e. T (HrmOp)) (e. (` (normop) T) (RR)) pm3.26 (=/= (` (normop) T) (0)) adantr idhmop (` (normop) T) (Iop) hmopmt mpan2 (e. T (HrmOp)) (=/= (` (normop) T) (0)) ad2antlr 3jca (` (normop) T) recgt0t (e. T (HrmOp)) (e. (` (normop) T) (RR)) pm3.27 (=/= (` (normop) T) (0)) adantr T nmopgt0t biimpa T hmopft sylan (e. (` (normop) T) (RR)) adantlr sylanc (` (normop) T) rerecclt 0re (0) (opr (1) (/) (` (normop) T)) ltlet mpan syl (e. T (HrmOp)) adantll mpd T leopnmidt (=/= (` (normop) T) (0)) adantr jca sylanc hoif (Iop) (H~) (H~) f1of ax-mp (opr (1) (/) (` (normop) T)) (` (normop) T) (Iop) homulasst mp3an3 (` (normop) T) recclt (e. (` (normop) T) (CC)) (=/= (` (normop) T) (0)) pm3.26 sylanc (` (normop) T) recid2t (.op) (Iop) opreq1d eqtr3d hoif (Iop) (H~) (H~) f1of ax-mp (Iop) homulid2t ax-mp syl6eq (` (normop) T) recnt sylan (e. T (HrmOp)) adantll breqtrd syldan 3impa)) thm (pjhmop ((x y) (H x) (H y)) ((pjhmop.1 (e. H (CH)))) (e. (` (proj) H) (HrmOp)) ((` (proj) H) x y elhmopt pjhmop.1 pjf pjhmop.1 (cv x) (cv y) pjadjt eqcomd rgen2 mpbir2an)) thm (pjlnop () ((pjhmop.1 (e. H (CH)))) (e. (` (proj) H) (LinOp)) (pjhmop.1 pjhmop (` (proj) H) hmoplint ax-mp)) thm (pjnmop ((w x) (w y) (w z) (H w) (x y) (x z) (H x) (y z) (H y) (H z)) ((pjhmop.1 (e. H (CH)))) (-> (=/= H (0H)) (= (` (normop) (` (proj) H)) (1))) ((cv w) (1) (cv z) (<) breq2 ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` (` (proj) H) (cv y))))))) rcla4ev pjhmop.1 (cv y) chel (= (` (norm) (cv y)) (1)) adantr (` (norm) (cv y)) (1) eqlet pjhmop.1 (cv y) chel (cv y) normclt syl sylan pjhmop.1 H (cv y) pjidt mpan (norm) fveq2d (= (` (norm) (cv y)) (1)) adantr (e. (cv y) H) (= (` (norm) (cv y)) (1)) pm3.27 eqtr2d jca jca r19.22i2 pjhmop.1 y chne0 pjhmop.1 chshi y y norm1ex bitr ax1re elisseti (cv x) (1) (` (norm) (` (` (proj) H) (cv y))) eqeq1 (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv elab 3imtr4 sylan ex (e. (cv z) (RR)) a1d r19.21aiv z visset (cv x) (cv z) (` (norm) (` (` (proj) H) (cv y))) eqeq1 (br (` (norm) (cv y)) (<_) (1)) anbi2d y (H~) rexbidv elab (cv z) (` (norm) (` (` (proj) H) (cv y))) (<_) (1) breq1 biimparc pjhmop.1 H (cv y) pjnormt mpan ax1re (` (norm) (` (` (proj) H) (cv y))) (` (norm) (cv y)) (1) letrt mp3an3 pjhmop.1 (cv y) pjhcl (` (` (proj) H) (cv y)) normclt syl (cv y) normclt sylanc mpand imp sylan exp31 imp3a r19.23aiv sylbi rgen pjhmop.1 pjf (` (proj) H) x y nmopsetret ax-mp ressxr sstri ax1re (1) rexrt ax-mp ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` (` (proj) H) (cv y))))))) (1) z w supxr2 mpanl12 mpan syl pjhmop.1 pjf (` (proj) H) x y nmopvalt ax-mp syl5eq)) thm (pjbdln () ((pjhmop.1 (e. H (CH)))) (e. (` (proj) H) (BndLinOp)) ((` (proj) H) elbdop2t pjhmop.1 pjlnop pjhmop.1 H (0H) (proj) fveq2 (normop) fveq2d (RR) eleq1d pjhmop.1 pjnmop ax1re syl6eqel (e. H (CH)) adantl ho0OLD (normop) fveq2i nmop0OLD eqtr 0re eqeltr (e. H (CH)) a1i pm2.61ne ax-mp mpbir2an)) thm (pjhmopt () () (-> (e. H (CH)) (e. (` (proj) H) (HrmOp))) (H (if (e. H (CH)) H (0H)) (proj) fveq2 (HrmOp) eleq1d h0elch H elimel pjhmop dedth)) thm (hmopidmchlem ((T x)) ((hmopidmch.1 (= H ({e.|} x (H~) (= (` T (cv x)) (cv x))))) (hmopidmch.2 (/\ (e. T (HrmOp)) (= (o. T T) T)))) (-> (e. A (H~)) (br (` (norm) (` T A)) (<_) (` (norm) A))) (hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf A ffvrni (` T A) normge0t syl hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf A ffvrni (` T A) normclt 0re (0) (` (norm) (` T A)) leloet mpan 3syl hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf A ffvrni (` T A) normsqt syl hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf A ffvrni (` T A) normclt syl recnd (` (norm) (` T A)) sqvalt syl hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf A ffvrni hmopidmch.2 pm3.26i T (` T A) A hmopt mp3an1 mpancom hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf A hoco hmopidmch.2 pm3.27i A fveq1i syl5reqr (.i) A opreq1d eqtr2d (abs) fveq2d (opr (` T A) (.i) (` T A)) absidt hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf A ffvrni (` T A) hiidrclt syl hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf A ffvrni (` T A) hiidge0t syl sylanc eqtr2d 3eqtr3d hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf A ffvrni (` T A) A bcst mpancom eqbrtrd (br (0) (<) (` (norm) (` T A))) adantr (` (norm) (` T A)) (` (norm) A) (` (norm) (` T A)) lemul2t hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf A ffvrni (` T A) normclt syl A normclt hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf A ffvrni (` T A) normclt syl 3jca sylan mpbird (e. A (H~)) (= (0) (` (norm) (` T A))) pm3.27 A normge0t (= (0) (` (norm) (` T A))) adantr eqbrtrrd jaodan ex sylbid mpd)) thm (hmopidmch ((f m) (f y) (f z) (H f) (m y) (m z) (H m) (y z) (H y) (H z) (m x) (T m) (x z) (x y) (T x) (T z) (T y) (f n) (f x) (m n) (n x) (n y) (n z)) ((hmopidmch.1 (= H ({e.|} x (H~) (= (` T (cv x)) (cv x))))) (hmopidmch.2 (/\ (e. T (HrmOp)) (= (o. T T) T)))) (e. H (CH)) (H f y closedsub hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr H y z sh2 ax-mp (cv x) (0v) T fveq2 (= (cv x) (0v)) id eqeq12d (H~) elrab ax-hv0cl hmopidmch.2 pm3.26i T hmoplint ax-mp lnop0 mpbir2an hmopidmch.1 eleqtrr (cv y) (cv z) ax-hvaddcl (= (` T (cv y)) (cv y)) (= (` T (cv z)) (cv z)) ad2ant2r hmopidmch.2 pm3.26i T hmoplint ax-mp (cv y) (cv z) lnopadd (` T (cv y)) (cv y) (` T (cv z)) (cv z) (+v) opreq12 sylan9eq an4s jca hmopidmch.1 (cv y) eleq2i (cv x) (cv y) T fveq2 (= (cv x) (cv y)) id eqeq12d (H~) elrab bitr hmopidmch.1 (cv z) eleq2i (cv x) (cv z) T fveq2 (= (cv x) (cv z)) id eqeq12d (H~) elrab bitr syl2anb (cv x) (opr (cv y) (+v) (cv z)) T fveq2 (= (cv x) (opr (cv y) (+v) (cv z))) id eqeq12d (H~) elrab sylibr hmopidmch.1 syl6eleqr rgen2 (cv y) (cv z) ax-hvmulcl (= (` T (cv z)) (cv z)) adantr hmopidmch.2 pm3.26i T hmoplint ax-mp (cv y) (cv z) lnopmul (` T (cv z)) (cv z) (cv y) (.s) opreq2 sylan9eq jca anasss hmopidmch.1 (cv z) eleq2i (cv x) (cv z) T fveq2 (= (cv x) (cv z)) id eqeq12d (H~) elrab bitr sylan2b (cv x) (opr (cv y) (.s) (cv z)) T fveq2 (= (cv x) (opr (cv y) (.s) (cv z))) id eqeq12d (H~) elrab sylibr hmopidmch.1 syl6eleqr rgen2a pm3.2i mpbir2an f visset y visset hlimvec (:--> (cv f) (NN) H) adantl (opr (` (norm) (opr (` T (cv y)) (-v) (cv y))) (/) (2)) z squeeze0 (` T (cv y)) (cv y) hvsubclt f visset y visset hlimvec hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni syl f visset y visset hlimvec sylanc (opr (` T (cv y)) (-v) (cv y)) normclt syl (` (norm) (opr (` T (cv y)) (-v) (cv y))) rehalfclt syl (:--> (cv f) (NN) H) adantl (` T (cv y)) (cv y) hvsubclt f visset y visset hlimvec hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni syl f visset y visset hlimvec sylanc (opr (` T (cv y)) (-v) (cv y)) normge0t syl (` T (cv y)) (cv y) hvsubclt f visset y visset hlimvec hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni syl f visset y visset hlimvec sylanc (opr (` T (cv y)) (-v) (cv y)) normclt syl (` (norm) (opr (` T (cv y)) (-v) (cv y))) halfnneg2t syl mpbid (:--> (cv f) (NN) H) adantl f visset y visset z m n hlimconv (:--> (cv f) (NN) H) adantl (cv m) nnret (cv m) leidt syl (cv n) (cv m) (cv m) (<_) breq2 (cv n) (cv m) (cv f) fveq2 (-v) (cv y) opreq1d (norm) fveq2d (<) (cv z) breq1d imbi12d (NN) rcla4v mpid (/\ (/\ (:--> (cv f) (NN) H) (br (cv f) (~~>v) (cv y))) (e. (cv z) (RR))) adantl hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni (` T (cv y)) (cv y) hvsubclt mpancom (opr (` T (cv y)) (-v) (cv y)) normclt syl (e. (` (cv f) (cv m)) H) adantl (` (norm) (opr (` T (cv y)) (-v) (` (cv f) (cv m)))) (` (norm) (opr (` (cv f) (cv m)) (-v) (cv y))) axaddrcl (` T (cv y)) (` (cv f) (cv m)) hvsubclt hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli syl2an ancoms (opr (` T (cv y)) (-v) (` (cv f) (cv m))) normclt syl (` (cv f) (cv m)) (cv y) hvsubclt hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli sylan (opr (` (cv f) (cv m)) (-v) (cv y)) normclt syl sylanc (` (norm) (opr (` (cv f) (cv m)) (-v) (cv y))) (` (norm) (opr (` (cv f) (cv m)) (-v) (cv y))) axaddrcl (` (cv f) (cv m)) (cv y) hvsubclt hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli sylan (opr (` (cv f) (cv m)) (-v) (cv y)) normclt syl (` (cv f) (cv m)) (cv y) hvsubclt hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli sylan (opr (` (cv f) (cv m)) (-v) (cv y)) normclt syl sylanc (` T (cv y)) (cv y) (` (cv f) (cv m)) norm3dift hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni (e. (` (cv f) (cv m)) H) adantl (e. (` (cv f) (cv m)) H) (e. (cv y) (H~)) pm3.27 hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli (e. (cv y) (H~)) adantr syl3anc (cv y) (` (cv f) (cv m)) hvsubclt ancoms hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli sylan hmopidmch.1 hmopidmch.2 (opr (cv y) (-v) (` (cv f) (cv m))) hmopidmchlem syl hmopidmch.2 pm3.26i T hmoplint ax-mp (cv y) (` (cv f) (cv m)) lnopsub ancoms hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli sylan hmopidmch.1 (` (cv f) (cv m)) eleq2i (cv x) (` (cv f) (cv m)) T fveq2 (= (cv x) (` (cv f) (cv m))) id eqeq12d (H~) elrab bitr pm3.27bd (e. (cv y) (H~)) adantr (` T (cv y)) (-v) opreq2d eqtr2d (norm) fveq2d (` (cv f) (cv m)) (cv y) normsubt hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli sylan 3brtr4d (` (norm) (opr (` T (cv y)) (-v) (` (cv f) (cv m)))) (` (norm) (opr (` (cv f) (cv m)) (-v) (cv y))) (` (norm) (opr (` (cv f) (cv m)) (-v) (cv y))) leadd1t (` T (cv y)) (` (cv f) (cv m)) hvsubclt hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli syl2an ancoms (opr (` T (cv y)) (-v) (` (cv f) (cv m))) normclt syl (` (cv f) (cv m)) (cv y) hvsubclt hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli sylan (opr (` (cv f) (cv m)) (-v) (cv y)) normclt syl (` (cv f) (cv m)) (cv y) hvsubclt hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli sylan (opr (` (cv f) (cv m)) (-v) (cv y)) normclt syl syl3anc mpbid letrd (` (cv f) (cv m)) (cv y) hvsubclt hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli sylan (opr (` (cv f) (cv m)) (-v) (cv y)) normclt syl recnd (` (norm) (opr (` (cv f) (cv m)) (-v) (cv y))) 2timest syl breqtrrd 2re 2pos (` (norm) (opr (` T (cv y)) (-v) (cv y))) (2) (` (norm) (opr (` (cv f) (cv m)) (-v) (cv y))) ledivmult mpan2 mp3an2 hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni (` T (cv y)) (cv y) hvsubclt mpancom (opr (` T (cv y)) (-v) (cv y)) normclt syl (e. (` (cv f) (cv m)) H) adantl (` (cv f) (cv m)) (cv y) hvsubclt hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli sylan (opr (` (cv f) (cv m)) (-v) (cv y)) normclt syl sylanc mpbird (e. (cv z) (RR)) adantr (opr (` (norm) (opr (` T (cv y)) (-v) (cv y))) (/) (2)) (` (norm) (opr (` (cv f) (cv m)) (-v) (cv y))) (cv z) lelttrt hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni (` T (cv y)) (cv y) hvsubclt mpancom (opr (` T (cv y)) (-v) (cv y)) normclt syl (` (norm) (opr (` T (cv y)) (-v) (cv y))) rehalfclt syl (e. (` (cv f) (cv m)) H) (e. (cv z) (RR)) ad2antlr (` (cv f) (cv m)) (cv y) hvsubclt hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (` (cv f) (cv m)) sseli sylan (opr (` (cv f) (cv m)) (-v) (cv y)) normclt syl (e. (cv z) (RR)) adantr (/\ (e. (` (cv f) (cv m)) H) (e. (cv y) (H~))) (e. (cv z) (RR)) pm3.27 syl3anc mpand (cv f) (NN) H (cv m) ffvrn f visset y visset hlimvec anim12i an1rs sylan an1rs syld ex r19.23adv (br (0) (<) (cv z)) imim2d r19.20dva mpd syl3anc (` T (cv y)) (cv y) hvsubclt f visset y visset hlimvec hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni syl f visset y visset hlimvec sylanc (opr (` T (cv y)) (-v) (cv y)) normclt syl recnd (` (norm) (opr (` T (cv y)) (-v) (cv y))) half0t syl (:--> (cv f) (NN) H) adantl mpbid (` T (cv y)) (cv y) hvsubclt f visset y visset hlimvec hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni syl f visset y visset hlimvec sylanc (opr (` T (cv y)) (-v) (cv y)) norm-it syl (:--> (cv f) (NN) H) adantl mpbid (` T (cv y)) (cv y) hvsubeq0t f visset y visset hlimvec hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni syl f visset y visset hlimvec sylanc (:--> (cv f) (NN) H) adantl mpbid jca hmopidmch.1 (cv y) eleq2i (cv x) (cv y) T fveq2 (= (cv x) (cv y)) id eqeq12d (H~) elrab bitr sylibr f y gen2 mpbir2an)) thm (hmopidmpj ((y z) (H y) (H z) (x z) (x y) (T x) (T z) (T y)) ((hmopidmch.1 (= H ({e.|} x (H~) (= (` T (cv x)) (cv x))))) (hmopidmch.2 (/\ (e. T (HrmOp)) (= (o. T T) T)))) (= T (` (proj) H)) (hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf T (H~) (H~) ffn ax-mp hmopidmch.1 hmopidmch.2 hmopidmch pjfn T (H~) (` (proj) H) (H~) y eqfnfv mp2an (H~) eqid hmopidmch.1 hmopidmch.2 hmopidmch (` T (cv y)) (opr (cv y) (-v) (` T (cv y))) pjv hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) hoco hmopidmch.2 pm3.27i (cv y) fveq1i syl5reqr jca hmopidmch.1 (` T (cv y)) eleq2i (cv x) (` T (cv y)) T fveq2 (= (cv x) (` T (cv y))) id eqeq12d (H~) elrab bitr sylibr hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni (cv y) (` T (cv y)) hvsubclt mpdan (cv y) (` T (cv y)) (cv z) his2subt (e. (cv y) (H~)) (e. (cv z) H) pm3.26 hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni (e. (cv z) H) adantr hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (cv z) sseli (e. (cv y) (H~)) adantl syl3anc hmopidmch.2 pm3.26i T (cv y) (cv z) hmopt mp3an1 hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (cv z) sseli sylan2 hmopidmch.1 (cv z) eleq2i (cv x) (cv z) T fveq2 (= (cv x) (cv z)) id eqeq12d (H~) elrab bitr pm3.27bd (cv y) (.i) opreq2d (e. (cv y) (H~)) adantl eqtr3d (opr (cv y) (.i) (cv z)) (-) opreq2d (cv y) (cv z) ax-hicl hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr (cv z) sseli sylan2 (opr (cv y) (.i) (cv z)) subidt syl 3eqtrd r19.21aiva jca hmopidmch.1 x (H~) (= (` T (cv x)) (cv x)) ssrab2 eqsstr H (opr (cv y) (-v) (` T (cv y))) z ocelt ax-mp sylibr sylanc hmopidmch.2 pm3.26i T hmoplint ax-mp lnopf (cv y) ffvrni (` T (cv y)) (cv y) hvpncan3t mpancom (` (proj) H) fveq2d eqtr3d rgen mpbir2an)) thm (hmopidmcht ((T x)) () (-> (/\ (e. T (HrmOp)) (= (o. T T) T)) (e. ({e.|} x (H~) (= (` T (cv x)) (cv x))) (CH))) (T (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (cv x) fveq1 (cv x) eqeq1d x (H~) rabbisdv (CH) eleq1d ({e.|} x (H~) (= (` (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (cv x)) (cv x))) eqid T (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (HrmOp) eleq1 T (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) T coeq1 T (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) coeq2 eqtrd (= T (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~)))) id eqeq12d anbi12d (|` (I) (H~)) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (HrmOp) eleq1 (|` (I) (H~)) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (|` (I) (H~)) coeq1 (|` (I) (H~)) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) coeq2 eqtrd (= (|` (I) (H~)) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~)))) id eqeq12d anbi12d idhmopOLD (H~) f1oi (|` (I) (H~)) (H~) (H~) f1of ax-mp (|` (I) (H~)) (H~) (H~) fcoi1 ax-mp pm3.2i elimhyp hmopidmch dedth)) thm (hmopidmpjt ((T x)) () (-> (/\ (e. T (HrmOp)) (= (o. T T) T)) (= T (` (proj) ({e.|} x (H~) (= (` T (cv x)) (cv x)))))) ((= T (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~)))) id T (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (cv x) fveq1 (cv x) eqeq1d x (H~) rabbisdv (proj) fveq2d eqeq12d ({e.|} x (H~) (= (` (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (cv x)) (cv x))) eqid T (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (HrmOp) eleq1 T (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) T coeq1 T (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) coeq2 eqtrd (= T (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~)))) id eqeq12d anbi12d (|` (I) (H~)) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (HrmOp) eleq1 (|` (I) (H~)) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (|` (I) (H~)) coeq1 (|` (I) (H~)) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~))) coeq2 eqtrd (= (|` (I) (H~)) (if (/\ (e. T (HrmOp)) (= (o. T T) T)) T (|` (I) (H~)))) id eqeq12d anbi12d idhmopOLD (H~) f1oi (|` (I) (H~)) (H~) (H~) f1of ax-mp (|` (I) (H~)) (H~) (H~) fcoi1 ax-mp pm3.2i elimhyp hmopidmpj dedth)) thm (pjsdi ((H x) (S x) (T x)) ((pjsdi.1 (e. H (CH))) (pjsdi.2 (:--> S (H~) (H~))) (pjsdi.3 (:--> T (H~) (H~)))) (= (o. (` (proj) H) (opr S (+op) T)) (opr (o. (` (proj) H) S) (+op) (o. (` (proj) H) T))) (pjsdi.1 (` S (cv x)) (` T (cv x)) pjaddt pjsdi.2 (cv x) ffvrni pjsdi.3 (cv x) ffvrni sylanc pjsdi.2 pjsdi.3 S T (cv x) hosvaltOLD mpanl12 (` (proj) H) fveq2d pjsdi.1 pjf pjsdi.2 (cv x) hoco pjsdi.1 pjf pjsdi.3 (cv x) hoco (+v) opreq12d 3eqtr4d pjsdi.1 pjf pjsdi.2 pjsdi.3 hoaddcl (cv x) hoco pjsdi.1 pjf pjsdi.2 hocof pjsdi.1 pjf pjsdi.3 hocof (o. (` (proj) H) S) (o. (` (proj) H) T) (cv x) hosvaltOLD mpanl12 3eqtr4d rgen pjsdi.1 pjf pjsdi.2 pjsdi.3 hoaddcl hocof pjsdi.1 pjf pjsdi.2 hocof pjsdi.1 pjf pjsdi.3 hocof hoaddcl x hoeq mpbi)) thm (pjddi ((H x) (S x) (T x)) ((pjsdi.1 (e. H (CH))) (pjsdi.2 (:--> S (H~) (H~))) (pjsdi.3 (:--> T (H~) (H~)))) (= (o. (` (proj) H) (opr S (-op) T)) (opr (o. (` (proj) H) S) (-op) (o. (` (proj) H) T))) (pjsdi.1 (` S (cv x)) (` T (cv x)) pjsubt pjsdi.2 (cv x) ffvrni pjsdi.3 (cv x) ffvrni sylanc pjsdi.2 pjsdi.3 S T (cv x) hodvaltOLD mpanl12 (` (proj) H) fveq2d pjsdi.1 pjf pjsdi.2 (cv x) hoco pjsdi.1 pjf pjsdi.3 (cv x) hoco (-v) opreq12d 3eqtr4d pjsdi.1 pjf pjsdi.2 pjsdi.3 hosubcl (cv x) hoco pjsdi.1 pjf pjsdi.2 hocof pjsdi.1 pjf pjsdi.3 hocof (o. (` (proj) H) S) (o. (` (proj) H) T) (cv x) hodvaltOLD mpanl12 3eqtr4d rgen pjsdi.1 pjf pjsdi.2 pjsdi.3 hosubcl hocof pjsdi.1 pjf pjsdi.2 hocof pjsdi.1 pjf pjsdi.3 hocof hosubcl x hoeq mpbi)) thm (pjsdi2 () ((pjsdi2.1 (e. H (CH))) (pjsdi2.2 (:--> R (H~) (H~))) (pjsdi2.3 (:--> S (H~) (H~))) (pjsdi2.4 (:--> T (H~) (H~)))) (-> (= (o. R (opr S (+op) T)) (opr (o. R S) (+op) (o. R T))) (= (o. (o. (` (proj) H) R) (opr S (+op) T)) (opr (o. (o. (` (proj) H) R) S) (+op) (o. (o. (` (proj) H) R) T)))) ((o. R (opr S (+op) T)) (opr (o. R S) (+op) (o. R T)) (` (proj) H) coeq2 pjsdi2.1 pjsdi2.2 pjsdi2.3 hocof pjsdi2.2 pjsdi2.4 hocof pjsdi syl6eq (` (proj) H) R (opr S (+op) T) coass (` (proj) H) R S coass (` (proj) H) R T coass (+op) opreq12i 3eqtr4g)) thm (pjco () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (= (` (o. (` (proj) G) (` (proj) H)) A) (` (` (proj) G) (` (` (proj) H) A)))) (pjco.1 pjf pjco.2 pjf A hoco)) thm (pjcocl () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (e. (` (o. (` (proj) G) (` (proj) H)) A) G)) (pjco.1 pjco.2 A pjco pjco.2 A pjhcl pjco.1 (` (` (proj) H) A) pjcl syl eqeltrd)) thm (pjcohcl () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (e. (` (o. (` (proj) G) (` (proj) H)) A) (H~))) (pjco.1 pjf pjco.2 pjf A hococl)) thm (pjadjco () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` (o. (` (proj) G) (` (proj) H)) A) (.i) B) (opr A (.i) (` (o. (` (proj) H) (` (proj) G)) B)))) (pjco.1 (` (` (proj) H) A) B pjadjt pjco.2 A pjhcl sylan pjco.2 A (` (` (proj) G) B) pjadjt pjco.1 B pjhcl sylan2 eqtrd pjco.1 pjco.2 A pjco (.i) B opreq1d (e. B (H~)) adantr pjco.2 pjco.1 B pjco A (.i) opreq2d (e. A (H~)) adantl 3eqtr4d)) thm (pjcofn () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (Fn (o. (` (proj) G) (` (proj) H)) (H~)) (pjco.1 pjf pjco.2 pjf hocofn)) thm (pjss1co ((H x) (G x)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (<-> (C_ G H) (= (o. (` (proj) H) (` (proj) G)) (` (proj) G))) (pjco.2 pjco.1 (cv x) pjco (C_ G H) adantl G H (` (` (proj) G) (cv x)) ssel2 pjco.1 (cv x) pjcl sylan2 pjco.2 H (` (` (proj) G) (cv x)) pjidt mpan syl eqtrd r19.21aiva pjco.2 pjf pjco.1 pjf hocof pjco.1 pjf x hoeq sylib (o. (` (proj) H) (` (proj) G)) (` (proj) G) rneq (` (proj) H) (` (proj) G) rnco (ran (o. (` (proj) H) (` (proj) G))) (ran (` (proj) G)) (ran (` (proj) H)) sseq1 mpbii syl pjco.1 pjrn pjco.2 pjrn 3sstr3g impbi)) thm (pjss2co ((x y) (H x) (H y) (G x) (G y)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (<-> (C_ G H) (= (o. (` (proj) G) (` (proj) H)) (` (proj) G))) (pjco.1 pjco.2 (cv x) pjco (C_ G H) adantl (cv x) (if (e. (cv x) (H~)) (cv x) (0v)) (` (proj) H) fveq2 (` (proj) G) fveq2d (cv x) (if (e. (cv x) (H~)) (cv x) (0v)) (` (proj) G) fveq2 eqeq12d (C_ G H) imbi2d pjco.1 ax-hv0cl (cv x) elimel pjco.2 pjss2 dedth impcom eqtrd r19.21aiva pjco.1 pjf pjco.2 pjf hocof pjco.1 pjf x hoeq sylib (o. (` (proj) G) (` (proj) H)) (` (proj) G) (cv y) fveq1 (cv x) (.i) opreq2d (e. (cv x) (H~)) (e. (cv y) (H~)) ad2antlr pjco.2 pjco.1 (cv x) (cv y) pjadjco (= (o. (` (proj) G) (` (proj) H)) (` (proj) G)) adantlr pjco.1 (cv x) (cv y) pjadjt (= (o. (` (proj) G) (` (proj) H)) (` (proj) G)) adantlr 3eqtr4d exp31 r19.21adv (` (o. (` (proj) H) (` (proj) G)) (cv x)) (` (` (proj) G) (cv x)) y hial2eqt pjco.2 pjco.1 (cv x) pjcohcl pjco.1 (cv x) pjhcl sylanc sylibd com12 r19.21aiv pjco.2 pjf pjco.1 pjf hocof pjco.1 pjf x hoeq sylib pjco.1 pjco.2 pjss1co sylibr impbi)) thm (pjssmt () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (-> (C_ H G) (= (opr (` (` (proj) G) A) (-v) (` (` (proj) H) A)) (` (` (proj) (i^i G (` (_|_) H))) A)))) (A (if (e. A (H~)) A (0v)) (` (proj) G) fveq2 A (if (e. A (H~)) A (0v)) (` (proj) H) fveq2 (-v) opreq12d A (if (e. A (H~)) A (0v)) (` (proj) (i^i G (` (_|_) H))) fveq2 eqeq12d (C_ H G) imbi2d pjco.2 ax-hv0cl A elimel pjco.1 pjssm dedth)) thm (pjssge0t () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (-> (= (opr (` (` (proj) G) A) (-v) (` (` (proj) H) A)) (` (` (proj) (i^i G (` (_|_) H))) A)) (br (0) (<_) (opr (opr (` (` (proj) G) A) (-v) (` (` (proj) H) A)) (.i) A)))) (A (if (e. A (H~)) A (0v)) (` (proj) G) fveq2 A (if (e. A (H~)) A (0v)) (` (proj) H) fveq2 (-v) opreq12d A (if (e. A (H~)) A (0v)) (` (proj) (i^i G (` (_|_) H))) fveq2 eqeq12d A (if (e. A (H~)) A (0v)) (` (proj) G) fveq2 A (if (e. A (H~)) A (0v)) (` (proj) H) fveq2 (-v) opreq12d (= A (if (e. A (H~)) A (0v))) id (.i) opreq12d (0) (<_) breq2d imbi12d pjco.2 ax-hv0cl A elimel pjco.1 pjssge0 dedth)) thm (pjdifnormt () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (<-> (br (0) (<_) (opr (opr (` (` (proj) G) A) (-v) (` (` (proj) H) A)) (.i) A)) (br (` (norm) (` (` (proj) H) A)) (<_) (` (norm) (` (` (proj) G) A))))) (A (if (e. A (H~)) A (0v)) (` (proj) G) fveq2 A (if (e. A (H~)) A (0v)) (` (proj) H) fveq2 (-v) opreq12d (= A (if (e. A (H~)) A (0v))) id (.i) opreq12d (0) (<_) breq2d A (if (e. A (H~)) A (0v)) (` (proj) H) fveq2 (norm) fveq2d A (if (e. A (H~)) A (0v)) (` (proj) G) fveq2 (norm) fveq2d (<_) breq12d bibi12d pjco.2 ax-hv0cl A elimel pjco.1 pjdifnorm dedth)) thm (pjnormss ((H x) (G x)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (<-> (C_ G H) (A.e. x (H~) (br (` (norm) (` (` (proj) G) (cv x))) (<_) (` (norm) (` (` (proj) H) (cv x)))))) (pjco.2 pjco.1 (cv x) pjssmt pjco.2 pjco.1 (cv x) pjssge0t syld pjco.2 pjco.1 (cv x) pjdifnormt sylibd com12 r19.21aiv pjco.1 (cv x) pjhcl (` (` (proj) G) (cv x)) normclt syl 0re jctir (` (norm) (` (` (proj) G) (cv x))) (0) letri3t biimprd syl pjco.1 (cv x) pjhcl (` (` (proj) G) (cv x)) normge0t syl sylan2i anabsi6 (` (norm) (` (` (proj) H) (cv x))) (0) (` (norm) (` (` (proj) G) (cv x))) (<_) breq2 biimpac sylan2 exp32 imp pjco.2 (cv x) pjhcl (` (` (proj) H) (cv x)) norm-it syl pjco.2 (cv x) pjoc2tOLD bitr4d (br (` (norm) (` (` (proj) G) (cv x))) (<_) (` (norm) (` (` (proj) H) (cv x)))) adantr pjco.1 (cv x) pjhcl (` (` (proj) G) (cv x)) norm-it syl pjco.1 (cv x) pjoc2tOLD bitr4d (br (` (norm) (` (` (proj) G) (cv x))) (<_) (` (norm) (` (` (proj) H) (cv x)))) adantr 3imtr3d ex a2i pjco.2 choccl (cv x) chel syl5 pm2.43d x 19.20i x (H~) (br (` (norm) (` (` (proj) G) (cv x))) (<_) (` (norm) (` (` (proj) H) (cv x)))) df-ral (` (_|_) H) (` (_|_) G) x dfss2 3imtr4 pjco.1 pjco.2 chsscon3 sylibr impbi)) thm (pjorthco ((H x) (G x)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (C_ G (` (_|_) H)) (= (o. (` (proj) G) (` (proj) H)) (` (proj) (0H)))) (pjco.1 pjco.2 chsscon2 H (` (_|_) G) (` (` (proj) H) (cv x)) ssel sylbi pjco.2 (cv x) pjcl syl5com pjco.2 (cv x) pjhcl pjco.1 (` (` (proj) H) (cv x)) pjoc2tOLD syl sylibd impcom pjco.1 pjco.2 (cv x) pjco (C_ G (` (_|_) H)) adantl (cv x) pjch0t (C_ G (` (_|_) H)) adantl 3eqtr4d r19.21aiva pjco.1 pjf pjco.2 pjf hocof h0elch pjf x hoeq sylib)) thm (pjscj ((H x) (G x)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (C_ G (` (_|_) H)) (= (` (proj) (opr G (vH) H)) (opr (` (proj) G) (+op) (` (proj) H)))) (pjco.1 pjco.2 G H (cv x) pjcjt2 mp3an12 impcom pjco.1 pjf pjco.2 pjf (` (proj) G) (` (proj) H) (cv x) hosvaltOLD mpanl12 (C_ G (` (_|_) H)) adantl eqtr4d r19.21aiva pjco.1 pjco.2 chjcl pjf pjco.1 pjf pjco.2 pjf hoaddcl x hoeq sylib)) thm (pjssum () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (C_ G (` (_|_) H)) (= (` (proj) (opr G (+H) H)) (opr (` (proj) G) (+op) (` (proj) H)))) (pjco.1 pjco.2 osum (proj) fveq2d pjco.1 pjco.2 pjscj eqtrd)) thm (pjsspos ((H x) (G x)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (<-> (A.e. x (H~) (br (0) (<_) (opr (` (opr (` (proj) H) (-op) (` (proj) G)) (cv x)) (.i) (cv x)))) (C_ G H)) ((opr (` (norm) (` (` (proj) H) (cv x))) (^) (2)) (opr (` (norm) (` (` (proj) G) (cv x))) (^) (2)) subge0t pjco.2 (cv x) pjhcl (` (` (proj) H) (cv x)) normclt (` (norm) (` (` (proj) H) (cv x))) resqclt 3syl pjco.1 (cv x) pjhcl (` (` (proj) G) (cv x)) normclt (` (norm) (` (` (proj) G) (cv x))) resqclt 3syl sylanc pjco.2 pjf pjco.1 pjf (` (proj) H) (` (proj) G) (cv x) hodvalt mp3an12 (.i) (cv x) opreq1d (` (` (proj) H) (cv x)) (` (` (proj) G) (cv x)) (cv x) his2subt pjco.2 (cv x) pjhcl pjco.1 (cv x) pjhcl (e. (cv x) (H~)) id syl3anc pjco.2 (cv x) pjinormt pjco.1 (cv x) pjinormt (-) opreq12d 3eqtrd (0) (<_) breq2d (` (norm) (` (` (proj) G) (cv x))) (` (norm) (` (` (proj) H) (cv x))) le2sqt pjco.1 (cv x) pjhcl (` (` (proj) G) (cv x)) normclt syl pjco.1 (cv x) pjhcl (` (` (proj) G) (cv x)) normge0t syl jca pjco.2 (cv x) pjhcl (` (` (proj) H) (cv x)) normclt syl pjco.2 (cv x) pjhcl (` (` (proj) H) (cv x)) normge0t syl jca sylanc 3bitr4d ralbiia pjco.1 pjco.2 x pjnormss bitr4)) thm (pjssdif2 ((H x) (G x)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (<-> (C_ G H) (= (opr (` (proj) H) (-op) (` (proj) G)) (` (proj) (i^i H (` (_|_) G))))) (pjco.2 pjf pjco.1 pjf (` (proj) H) (` (proj) G) (cv x) hodvalt mp3an12 (C_ G H) adantl pjco.2 pjco.1 (cv x) pjssmt impcom eqtrd r19.21aiva (H~) eqid jctil pjco.2 pjf pjco.1 pjf hosubfn pjco.2 pjco.1 choccl chincl pjfn (opr (` (proj) H) (-op) (` (proj) G)) (H~) (` (proj) (i^i H (` (_|_) G))) (H~) x eqfnfv mp2an sylibr pjco.2 pjco.1 choccl chincl (cv x) pjige0 (= (opr (` (proj) H) (-op) (` (proj) G)) (` (proj) (i^i H (` (_|_) G)))) adantl (opr (` (proj) H) (-op) (` (proj) G)) (` (proj) (i^i H (` (_|_) G))) (cv x) fveq1 (.i) (cv x) opreq1d (e. (cv x) (H~)) adantr breqtrrd r19.21aiva pjco.1 pjco.2 x pjsspos sylib impbi)) thm (pjidmco () ((pjidmco.1 (e. H (CH)))) (= (o. (` (proj) H) (` (proj) H)) (` (proj) H)) (H ssid pjidmco.1 pjidmco.1 pjss2co mpbi)) thm (pjocco () ((pjidmco.1 (e. H (CH)))) (= (o. (` (proj) H) (` (proj) (` (_|_) H))) (` (proj) (0H))) (pjidmco.1 chssi H ococss ax-mp pjidmco.1 pjidmco.1 choccl pjorthco ax-mp)) thm (pjtot ((H x)) ((pjidmco.1 (e. H (CH)))) (= (opr (` (proj) H) (+op) (` (proj) (` (_|_) H))) (` (proj) (H~))) (pjidmco.1 H (cv x) axpjpjt mpan (cv x) pjch1t pjidmco.1 pjf pjidmco.1 choccl pjf (` (proj) H) (` (proj) (` (_|_) H)) (cv x) hosvaltOLD mpanl12 3eqtr4rd rgen pjidmco.1 pjf pjidmco.1 choccl pjf hoaddcl helch pjf x hoeq mpbi)) thm (pjoc () ((pjidmco.1 (e. H (CH)))) (= (opr (` (proj) (H~)) (-op) (` (proj) H)) (` (proj) (` (_|_) H))) (pjidmco.1 pjtot helch pjf pjidmco.1 pjf pjidmco.1 choccl pjf hods mpbir)) thm (pjidmcot () () (-> (e. H (CH)) (= (o. (` (proj) H) (` (proj) H)) (` (proj) H))) (H (if (e. H (CH)) H (0H)) (proj) fveq2 (` (proj) H) (` (proj) (if (e. H (CH)) H (0H))) (` (proj) H) coeq1 (` (proj) H) (` (proj) (if (e. H (CH)) H (0H))) (` (proj) (if (e. H (CH)) H (0H))) coeq2 eqtrd syl H (if (e. H (CH)) H (0H)) (proj) fveq2 eqeq12d h0elch H elimel pjidmco dedth)) thm (pjhmopidm ((h t) (h u) (h x) (t u) (t x) (u x)) () (= (ran (proj)) (i^i (HrmOp) ({|} t (= (o. (cv t) (cv t)) (cv t))))) ((cv h) ({e.|} x (H~) (= (` (cv u) (cv x)) (cv x))) (proj) fveq2 (cv u) eqeq1d (CH) rcla4ev (cv u) x hmopidmcht (cv u) x hmopidmpjt eqcomd sylanc (e. (cv h) (CH)) (= (` (proj) (cv h)) (cv u)) pm3.27 (cv h) pjhmopt (= (` (proj) (cv h)) (cv u)) adantr eqeltrrd (cv h) pjidmcot (= (` (proj) (cv h)) (cv u)) adantr (` (proj) (cv h)) (cv u) (` (proj) (cv h)) coeq1 (` (proj) (cv h)) (cv u) (cv u) coeq2 eqtrd (e. (cv h) (CH)) adantl (e. (cv h) (CH)) (= (` (proj) (cv h)) (cv u)) pm3.27 3eqtr3d jca ex r19.23aiv impbi u visset (cv t) (cv u) (cv t) coeq1 (cv t) (cv u) (cv u) coeq2 eqtrd (= (cv t) (cv u)) id eqeq12d elab (e. (cv u) (HrmOp)) anbi2i pjmfn (proj) (CH) (cv u) h fvelrn ax-mp 3bitr4 ineqri eqcomi)) thm (dfpjopt ((T t)) () (<-> (e. T (ran (proj))) (/\ (e. T (HrmOp)) (= (o. T T) T))) (t pjhmopidm T eleq2i T (HrmOp) ({|} t (= (o. (cv t) (cv t)) (cv t))) elin (cv t) T (cv t) coeq1 (cv t) T T coeq2 eqtrd (= (cv t) T) id eqeq12d (HrmOp) elabg pm5.32i 3bitr)) thm (elpjhmopt () () (-> (e. T (ran (proj))) (e. T (HrmOp))) (T dfpjopt pm3.26bd)) thm (pjadj2t () () (-> (e. T (ran (proj))) (= (` (adj) T) T)) (T elpjhmopt T hmopadjt syl)) thm (pjadj3t () () (-> (e. H (CH)) (= (` (adj) (` (proj) H)) (` (proj) H))) (pjmfn (proj) (CH) H fnfvrn mpan (` (proj) H) pjadj2t syl)) thm (elpjcht ((T x)) () (-> (e. T (ran (proj))) (/\ (e. ({e.|} x (H~) (= (` T (cv x)) (cv x))) (CH)) (= T (` (proj) ({e.|} x (H~) (= (` T (cv x)) (cv x))))))) (T dfpjopt T x hmopidmcht T x hmopidmpjt jca sylbi)) thm (elpjrnt ((T x)) () (-> (e. T (ran (proj))) (= (ran T) ({e.|} x (H~) (= (` T (cv x)) (cv x))))) (T x elpjcht pm3.27d rneqd T x elpjcht pm3.26d ({e.|} x (H~) (= (` T (cv x)) (cv x))) pjrnt syl eqtrd)) thm (pjinvar ((H x) (S x) (T x)) ((pjinvar.1 (:--> S (H~) (H~))) (pjinvar.2 (e. H (CH))) (pjinvar.3 (= T (` (proj) H)))) (<-> (:--> (o. S T) (H~) H) (= (o. S T) (o. T (o. S T)))) ((o. S T) (H~) H (cv x) ffvrn pjinvar.2 H (` (o. S T) (cv x)) pjidt mpan syl pjinvar.3 (` (o. S T) (cv x)) fveq1i syl5req pjinvar.2 pjfo (` (proj) H) (H~) H fof ax-mp pjinvar.3 T (` (proj) H) (H~) H feq1 ax-mp mpbir pjinvar.2 chssi T (H~) H (H~) fss mp2an pjinvar.1 pjinvar.2 pjfo (` (proj) H) (H~) H fof ax-mp pjinvar.3 T (` (proj) H) (H~) H feq1 ax-mp mpbir pjinvar.2 chssi T (H~) H (H~) fss mp2an hocof (cv x) hoco (:--> (o. S T) (H~) H) adantl eqtr4d r19.21aiva pjinvar.1 pjinvar.2 pjfo (` (proj) H) (H~) H fof ax-mp pjinvar.3 T (` (proj) H) (H~) H feq1 ax-mp mpbir pjinvar.2 chssi T (H~) H (H~) fss mp2an hocofn pjinvar.2 pjfo (` (proj) H) (H~) H fof ax-mp pjinvar.3 T (` (proj) H) (H~) H feq1 ax-mp mpbir pjinvar.2 chssi T (H~) H (H~) fss mp2an pjinvar.1 pjinvar.2 pjfo (` (proj) H) (H~) H fof ax-mp pjinvar.3 T (` (proj) H) (H~) H feq1 ax-mp mpbir pjinvar.2 chssi T (H~) H (H~) fss mp2an hocof hocofn (o. S T) (H~) (o. T (o. S T)) (H~) x eqfnfv mp2an (H~) eqid mpbiran sylibr pjinvar.2 pjfo (` (proj) H) (H~) H fof ax-mp pjinvar.3 T (` (proj) H) (H~) H feq1 ax-mp mpbir pjinvar.1 pjinvar.2 pjfo (` (proj) H) (H~) H fof ax-mp pjinvar.3 T (` (proj) H) (H~) H feq1 ax-mp mpbir pjinvar.2 chssi T (H~) H (H~) fss mp2an hocof T (H~) H (o. S T) (H~) fco mp2an (o. S T) (o. T (o. S T)) (H~) H feq1 mpbiri impbi)) thm (pjin1 () ((pjin1.1 (e. G (CH))) (pjin1.2 (e. H (CH)))) (= (` (proj) (i^i G H)) (o. (` (proj) G) (` (proj) (i^i G H)))) (G H inss1 pjin1.1 pjin1.2 chincl pjin1.1 pjss1co mpbi eqcomi)) thm (pjin2 () ((pjin1.1 (e. G (CH))) (pjin1.2 (e. H (CH)))) (<-> (/\ (= (` (proj) G) (o. (` (proj) G) (` (proj) H))) (= (` (proj) H) (o. (` (proj) H) (` (proj) G)))) (= (` (proj) G) (` (proj) H))) (G H eqss pjin1.1 pjin1.2 pjss2co (o. (` (proj) G) (` (proj) H)) (` (proj) G) eqcom bitr pjin1.2 pjin1.1 pjss2co (o. (` (proj) H) (` (proj) G)) (` (proj) H) eqcom bitr anbi12i bitr2 G H (proj) fveq2 sylbi (` (proj) G) (` (proj) H) (` (proj) G) coeq2 pjin1.1 pjidmco syl5eqr (` (proj) G) (` (proj) H) (` (proj) H) coeq2 pjin1.2 pjidmco syl6req jca impbi)) thm (pjin3 () ((pjin3.1 (e. F (CH))) (pjin3.2 (e. G (CH))) (pjin3.3 (e. H (CH)))) (<-> (/\ (= (` (proj) F) (o. (` (proj) F) (` (proj) G))) (= (` (proj) F) (o. (` (proj) F) (` (proj) H)))) (= (` (proj) F) (o. (` (proj) F) (` (proj) (i^i G H))))) (F G H ssin pjin3.1 pjin3.2 pjss2co (o. (` (proj) F) (` (proj) G)) (` (proj) F) eqcom bitr pjin3.1 pjin3.3 pjss2co (o. (` (proj) F) (` (proj) H)) (` (proj) F) eqcom bitr anbi12i pjin3.1 pjin3.2 pjin3.3 chincl pjss2co (o. (` (proj) F) (` (proj) (i^i G H))) (` (proj) F) eqcom bitr 3bitr3)) thm (pjclem1 () ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (-> (br G (C_H) H) (= (o. (` (proj) G) (` (proj) H)) (` (proj) (i^i G H)))) (pjclem1.1 pjclem1.2 cmbr G (opr (i^i G H) (vH) (i^i G (` (_|_) H))) (proj) fveq2 sylbi G H inss2 pjclem1.2 pjclem1.1 choccl chub2 pjclem1.1 pjclem1.2 chdmm3 sseqtr4 sstri pjclem1.1 pjclem1.2 chincl pjclem1.1 pjclem1.2 choccl chincl pjscj ax-mp (` (proj) G) eqeq2i (` (proj) G) (opr (` (proj) (i^i G H)) (+op) (` (proj) (i^i G (` (_|_) H)))) (` (proj) H) coeq2 pjclem1.2 pjclem1.1 pjclem1.2 chincl pjf pjclem1.1 pjclem1.2 choccl chincl pjf pjsdi G H inss2 pjclem1.1 pjclem1.2 chincl pjclem1.2 pjss1co mpbi pjclem1.2 pjclem1.1 choccl chub2 pjclem1.1 pjclem1.2 chdmm3 sseqtr4 pjclem1.2 pjclem1.1 pjclem1.2 choccl chincl pjorthco ax-mp (+op) opreq12i pjclem1.1 pjclem1.2 chincl pjf hoid0OLD 3eqtr (o. (` (proj) H) (` (proj) G)) eqeq2i (o. (` (proj) H) (` (proj) G)) (` (proj) (i^i G H)) (` (proj) G) coeq2 G H inss1 pjclem1.1 pjclem1.2 chincl pjclem1.1 pjss1co mpbi syl6eq sylbi syl sylbi syl pjclem1.1 pjclem1.2 cmcm3 pjclem1.1 choccl pjclem1.2 cmbr bitr (` (_|_) G) (opr (i^i (` (_|_) G) H) (vH) (i^i (` (_|_) G) (` (_|_) H))) (proj) fveq2 (` (_|_) G) H inss2 pjclem1.2 pjclem1.1 chub2 pjclem1.1 pjclem1.2 chdmm4 sseqtr4 sstri pjclem1.1 choccl pjclem1.2 chincl pjclem1.1 choccl pjclem1.2 choccl chincl pjscj ax-mp (` (proj) (` (_|_) G)) eqeq2i (` (proj) (` (_|_) G)) (opr (` (proj) (i^i (` (_|_) G) H)) (+op) (` (proj) (i^i (` (_|_) G) (` (_|_) H)))) (` (proj) H) coeq2 pjclem1.2 pjclem1.1 choccl pjclem1.2 chincl pjf pjclem1.1 choccl pjclem1.2 choccl chincl pjf pjsdi (` (_|_) G) H inss2 pjclem1.1 choccl pjclem1.2 chincl pjclem1.2 pjss1co mpbi pjclem1.2 pjclem1.1 chub2 pjclem1.1 pjclem1.2 chdmm4 sseqtr4 pjclem1.2 pjclem1.1 choccl pjclem1.2 choccl chincl pjorthco ax-mp (+op) opreq12i pjclem1.1 choccl pjclem1.2 chincl pjf hoid0OLD 3eqtr (o. (` (proj) H) (` (proj) (` (_|_) G))) eqeq2i (o. (` (proj) H) (` (proj) (` (_|_) G))) (` (proj) (i^i (` (_|_) G) H)) (` (proj) G) coeq2 pjclem1.1 pjclem1.2 choccl chub1 pjclem1.1 pjclem1.2 chdmm2 sseqtr4 pjclem1.1 pjclem1.1 choccl pjclem1.2 chincl pjorthco ax-mp syl6eq sylbi syl sylbi syl sylbi (+op) opreq12d pjclem1.1 pjtot (` (proj) H) coeq2i pjclem1.2 pjclem1.1 pjf pjclem1.1 choccl pjf pjsdi pjclem1.2 pjf hoid1OLD 3eqtr3r (` (proj) G) coeq2i pjclem1.1 pjclem1.2 pjf pjclem1.1 pjf hocof pjclem1.2 pjf pjclem1.1 choccl pjf hocof pjsdi eqtr2 pjclem1.1 pjclem1.2 chincl pjf hoid0OLD 3eqtr3g)) thm (pjclem2 () ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (-> (br G (C_H) H) (= (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G)))) (pjclem1.1 pjclem1.2 pjclem1 pjclem1.1 pjclem1.2 cmcm pjclem1.2 pjclem1.1 pjclem1 sylbi G H incom (proj) fveq2i syl6reqr eqtrd)) thm (pjclem3 () ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (-> (= (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G))) (= (o. (` (proj) G) (` (proj) (` (_|_) H))) (o. (` (proj) (` (_|_) H)) (` (proj) G)))) ((o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G)) (o. (` (proj) (H~)) (` (proj) G)) (-op) opreq2 pjclem1.1 pjf hoid1OLD pjclem1.1 pjf hoid1rOLD eqtr4 (-op) (o. (` (proj) G) (` (proj) H)) opreq1i syl5eq pjclem1.1 helch pjf pjclem1.2 pjf pjddi helch pjf pjclem1.2 pjf pjclem1.1 pjf hocsubdir 3eqtr4g pjclem1.2 pjoc (` (proj) G) coeq2i pjclem1.2 pjoc (` (proj) G) coeq1i 3eqtr3g)) thm (pjclem4a () ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (-> (e. A (i^i G H)) (= (` (o. (` (proj) G) (` (proj) H)) A) A)) (A G H elin pjclem1.2 A chel (e. A G) adantl pjclem1.1 pjclem1.2 A pjco syl pjclem1.2 H A pjidt mpan (` (` (proj) H) A) A G eleq1 pjclem1.1 G (` (` (proj) H) A) pjidt mpan syl6bir (` (` (proj) H) A) A (` (` (proj) G) (` (` (proj) H) A)) eqeq2 sylibd syl impcom eqtrd sylbi)) thm (pjclem4 ((x y) (G x) (G y) (H x) (H y)) ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (-> (= (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G))) (= (o. (` (proj) G) (` (proj) H)) (` (proj) (i^i G H)))) (pjclem1.1 pjclem1.2 chincl (` (o. (` (proj) G) (` (proj) H)) (cv x)) (opr (cv x) (-v) (` (o. (` (proj) G) (` (proj) H)) (cv x))) pjv pjclem1.1 pjclem1.2 (cv x) pjcocl (= (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G))) adantl (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G)) (cv x) fveq1 H eleq1d pjclem1.2 pjclem1.1 (cv x) pjcocl syl5bir imp jca (` (o. (` (proj) G) (` (proj) H)) (cv x)) G H elin sylibr pjclem1.1 pjclem1.2 (cv x) pjcohcl (cv x) (` (o. (` (proj) G) (` (proj) H)) (cv x)) hvsubclt mpdan (= (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G))) adantl (e. (cv x) (H~)) (e. (cv y) (i^i G H)) pm3.26 pjclem1.1 pjclem1.2 (cv x) pjcohcl (e. (cv y) (i^i G H)) adantr pjclem1.1 pjclem1.2 chincl (cv y) chel (e. (cv x) (H~)) adantl 3jca (= (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G))) adantl (cv x) (` (o. (` (proj) G) (` (proj) H)) (cv x)) (cv y) his2subt syl (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G)) (cv x) fveq1 (.i) (cv y) opreq1d pjclem1.2 pjclem1.1 (cv x) (cv y) pjadjco pjclem1.1 pjclem1.2 chincl (cv y) chel sylan2 pjclem1.1 pjclem1.2 (cv y) pjclem4a (cv x) (.i) opreq2d (e. (cv x) (H~)) adantl eqtrd sylan9eq (-) (opr (` (o. (` (proj) G) (` (proj) H)) (cv x)) (.i) (cv y)) opreq1d pjclem1.1 pjclem1.2 (cv x) pjcohcl pjclem1.1 pjclem1.2 chincl (cv y) chel anim12i (= (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G))) adantl (` (o. (` (proj) G) (` (proj) H)) (cv x)) (cv y) ax-hicl (opr (` (o. (` (proj) G) (` (proj) H)) (cv x)) (.i) (cv y)) subidt 3syl 3eqtr2d exp32 imp r19.21aiv jca pjclem1.1 pjclem1.2 chincl chshi (i^i G H) (opr (cv x) (-v) (` (o. (` (proj) G) (` (proj) H)) (cv x))) y shocelt ax-mp sylibr sylanc (` (o. (` (proj) G) (` (proj) H)) (cv x)) (cv x) (` (o. (` (proj) G) (` (proj) H)) (cv x)) hvaddsub12t pjclem1.1 pjclem1.2 (cv x) pjcohcl (e. (cv x) (H~)) id pjclem1.1 pjclem1.2 (cv x) pjcohcl syl3anc pjclem1.1 pjclem1.2 (cv x) pjcohcl (` (o. (` (proj) G) (` (proj) H)) (cv x)) hvsubidt syl (cv x) (+v) opreq2d (cv x) ax-hvaddid 3eqtrd (` (proj) (i^i G H)) fveq2d (= (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G))) adantl eqtr3d r19.21aiva pjclem1.1 pjf pjclem1.2 pjf hocof pjclem1.1 pjclem1.2 chincl pjf x hoeq sylib)) thm (pjc () ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (<-> (br G (C_H) H) (= (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G)))) (pjclem1.1 pjclem1.2 pjclem2 pjclem1.1 pjclem1.2 pjclem4 pjclem1.1 pjclem1.2 pjclem3 pjclem1.1 pjclem1.2 choccl pjclem4 syl (+op) opreq12d pjclem1.2 pjtot (` (proj) G) coeq2i pjclem1.1 pjclem1.2 pjf pjclem1.2 choccl pjf pjsdi pjclem1.1 pjf hoid1OLD 3eqtr3r G H inss2 pjclem1.2 pjclem1.1 choccl chub2 sstri pjclem1.1 pjclem1.2 chdmm3 sseqtr4 pjclem1.1 pjclem1.2 chincl pjclem1.1 pjclem1.2 choccl chincl pjscj ax-mp 3eqtr4g pjclem1.1 pjclem1.1 pjclem1.2 chincl pjclem1.1 pjclem1.2 choccl chincl chjcl pj11 sylib pjclem1.1 pjclem1.2 cmbr sylibr impbi)) thm (pjcmmul1 () ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (<-> (= (o. (` (proj) G) (` (proj) H)) (o. (` (proj) H) (` (proj) G))) (e. (o. (` (proj) G) (` (proj) H)) (ran (proj)))) (pjclem1.1 pjclem1.2 pjclem4 pjmfn pjclem1.1 pjclem1.2 chincl (proj) (CH) (i^i G H) fnfvrn mp2an syl6eqel (o. (` (proj) G) (` (proj) H)) pjadj2t pjclem1.1 pjbdln pjclem1.2 pjbdln adjco pjclem1.2 H pjadj3t ax-mp (` (adj) (` (proj) G)) coeq1i pjclem1.1 G pjadj3t ax-mp (` (proj) H) coeq2i 3eqtr syl5reqr impbi)) thm (pjcohocl () ((pjcohocl.1 (e. H (CH))) (pjcohocl.2 (:--> T (H~) (H~)))) (-> (e. A (H~)) (e. (` (o. (` (proj) H) T) A) H)) (pjcohocl.1 pjf pjcohocl.2 A hoco pjcohocl.2 A ffvrni pjcohocl.1 (` T A) pjcl syl eqeltrd)) thm (pjadj2co () ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) A) (.i) B) (opr A (.i) (` (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F)) B)))) (pjadj2co.1 pjadj2co.2 (` (` (proj) H) A) B pjadjco pjadj2co.3 A pjhcl sylan pjadj2co.3 A (` (o. (` (proj) G) (` (proj) F)) B) pjadjt pjadj2co.2 pjadj2co.1 B pjcohcl sylan2 eqtrd pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf A hoco (.i) B opreq1d (e. B (H~)) adantr pjadj2co.3 pjf pjadj2co.2 pjf pjadj2co.1 pjf hocof B hoco (` (proj) H) (` (proj) G) (` (proj) F) coass B fveq1i syl5eq A (.i) opreq2d (e. A (H~)) adantl 3eqtr4d)) thm (pj2cocl () ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (e. A (H~)) (e. (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) A) F)) (pjadj2co.1 pjf pjadj2co.2 pjf pjadj2co.3 pjf A ho2co pjadj2co.3 A pjhcl pjadj2co.2 (` (` (proj) H) A) pjhcl pjadj2co.1 (` (` (proj) G) (` (` (proj) H) A)) pjcl 3syl eqeltrd)) thm (pj3lem1 () ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (e. A (i^i (i^i F G) H)) (= (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) A) A)) (A F (i^i G H) elin pjadj2co.1 A chel (e. A (i^i G H)) adantr pjadj2co.1 pjf pjadj2co.2 pjf pjadj2co.3 pjf hocof A hoco syl pjadj2co.2 pjadj2co.3 A pjclem4a (` (o. (` (proj) G) (` (proj) H)) A) A F eleq1 pjadj2co.1 F (` (o. (` (proj) G) (` (proj) H)) A) pjidt mpan syl6bir (` (o. (` (proj) G) (` (proj) H)) A) A (` (` (proj) F) (` (o. (` (proj) G) (` (proj) H)) A)) eqeq2 sylibd syl impcom eqtrd sylbi F G H inass A eleq2i (` (proj) F) (` (proj) G) (` (proj) H) coass A fveq1i A eqeq1i 3imtr4)) thm (pj3s ((x y) (F x) (F y) (G x) (G y) (H x) (H y)) ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (/\ (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F))) (C_ (ran (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H))) G)) (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (` (proj) (i^i (i^i F G) H)))) (pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) (opr (cv x) (-v) (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x))) pjv pjadj2co.1 pjadj2co.2 pjadj2co.3 (cv x) pj2cocl (C_ (ran (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H))) G) adantl (ran (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H))) G (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) ssel pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf hocofn (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (H~) (cv x) fnfvrn mpan syl5 imp jca (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) F G elin sylibr (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F))) adantll (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F)) (cv x) fveq1 H eleq1d pjadj2co.3 pjadj2co.2 pjadj2co.1 (cv x) pj2cocl syl5bir imp (C_ (ran (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H))) G) adantlr jca (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) (i^i F G) H elin sylibr pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf (cv x) hococl (cv x) (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) hvsubclt mpdan (/\ (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F))) (C_ (ran (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H))) G)) adantl (e. (cv x) (H~)) (e. (cv y) (i^i (i^i F G) H)) pm3.26 pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf (cv x) hococl (e. (cv y) (i^i (i^i F G) H)) adantr pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl (cv y) chel (e. (cv x) (H~)) adantl 3jca (/\ (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F))) (C_ (ran (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H))) G)) adantl (cv x) (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) (cv y) his2subt syl (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F)) (cv x) fveq1 (C_ (ran (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H))) G) adantr (.i) (cv y) opreq1d pjadj2co.3 pjadj2co.2 pjadj2co.1 (cv x) (cv y) pjadj2co pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl (cv y) chel sylan2 pjadj2co.1 pjadj2co.2 pjadj2co.3 (cv y) pj3lem1 (cv x) (.i) opreq2d (e. (cv x) (H~)) adantl eqtrd sylan9eq (-) (opr (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) (.i) (cv y)) opreq1d pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf (cv x) hococl pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl (cv y) chel anim12i (/\ (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F))) (C_ (ran (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H))) G)) adantl (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) (cv y) ax-hicl (opr (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) (.i) (cv y)) subidt 3syl 3eqtr2d exp32 imp r19.21aiv jca pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl chshi (i^i (i^i F G) H) (opr (cv x) (-v) (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x))) y shocelt ax-mp sylibr sylanc (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) (cv x) (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) hvaddsub12t pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf (cv x) hococl (e. (cv x) (H~)) id pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf (cv x) hococl syl3anc pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf (cv x) hococl (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) hvsubidt syl (cv x) (+v) opreq2d (cv x) ax-hvaddid 3eqtrd (` (proj) (i^i (i^i F G) H)) fveq2d (/\ (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F))) (C_ (ran (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H))) G)) adantl eqtr3d r19.21aiva pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf hocof pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl pjf x hoeq sylib)) thm (pj3 () ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (/\ (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F))) (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) G) (` (proj) F)) (` (proj) H)))) (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (` (proj) (i^i (i^i F G) H)))) (pjadj2co.1 pjadj2co.2 pjadj2co.3 pj3s (` (proj) G) (o. (` (proj) F) (` (proj) H)) rnco pjadj2co.2 pjrn sseqtr (` (proj) G) (` (proj) F) (` (proj) H) coass (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) G) (` (proj) F)) (` (proj) H)) (o. (` (proj) G) (o. (` (proj) F) (` (proj) H))) eqeq1 mpbiri rneqd G sseq1d mpbiri sylan2)) thm (pj3cor1 ((x y) (F x) (F y) (G x) (G y) (H x) (H y)) ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (/\ (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F))) (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) G) (` (proj) F)) (` (proj) H)))) (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) F)) (` (proj) G)))) ((o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) G) (` (proj) F)) (` (proj) H)) (cv y) fveq1 (cv x) (.i) opreq2d (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F))) adantl (e. (cv x) (H~)) (e. (cv y) (H~)) ad2antlr pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl (cv x) (cv y) pjadjt (/\ (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F))) (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) G) (` (proj) F)) (` (proj) H)))) adantlr pjadj2co.1 pjadj2co.2 pjadj2co.3 pj3 (cv x) fveq1d (.i) (cv y) opreq1d (e. (cv x) (H~)) (e. (cv y) (H~)) ad2antlr pjadj2co.1 pjadj2co.2 pjadj2co.3 pj3 (cv y) fveq1d (cv x) (.i) opreq2d (e. (cv x) (H~)) (e. (cv y) (H~)) ad2antlr 3eqtr4d pjadj2co.3 pjadj2co.1 pjadj2co.2 (cv x) (cv y) pjadj2co (/\ (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) H) (` (proj) G)) (` (proj) F))) (= (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (o. (o. (` (proj) G) (` (proj) F)) (` (proj) H)))) adantlr 3eqtr4d exp31 r19.21adv (` (o. (o. (` (proj) F) (` (proj) G)) (` (proj) H)) (cv x)) (` (o. (o. (` (proj) H) (` (proj) F)) (` (proj) G)) (cv x)) y hial2eqt pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf (cv x) hococl pjadj2co.3 pjf pjadj2co.1 pjf hocof pjadj2co.2 pjf (cv x) hococl sylanc sylibd com12 r19.21aiv pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf hocof pjadj2co.3 pjf pjadj2co.1 pjf hocof pjadj2co.2 pjf hocof x hoeq sylib)) thm (pjs14 () ((pjs14.1 (e. G (CH))) (pjs14.2 (e. H (CH)))) (-> (e. A (H~)) (br (` (norm) (` (o. (` (proj) H) (` (proj) G)) A)) (<_) (` (norm) (` (` (proj) G) A)))) (pjs14.2 pjs14.1 A pjco (norm) fveq2d pjs14.1 A pjhcl pjs14.2 H (` (` (proj) G) A) pjnormt mpan syl eqbrtrd)) thm (stelt ((x y) (f x) (f y) (S x) (S y) (S f)) () (<-> (e. S (States)) (/\ (/\ (:--> S (CH) (RR)) (A.e. x (CH) (/\ (br (0) (<_) (` S (cv x))) (br (` S (cv x)) (<_) (1))))) (/\ (= (` S (H~)) (1)) (A.e. x (CH) (A.e. y (CH) (-> (C_ (cv x) (` (_|_) (cv y))) (= (` S (opr (cv x) (vH) (cv y))) (opr (` S (cv x)) (+) (` S (cv y)))))))))) (S (States) elisset chex S (CH) (RR) (V) fex mpan2 (A.e. x (CH) (/\ (br (0) (<_) (` S (cv x))) (br (` S (cv x)) (<_) (1)))) (/\ (= (` S (H~)) (1)) (A.e. x (CH) (A.e. y (CH) (-> (C_ (cv x) (` (_|_) (cv y))) (= (` S (opr (cv x) (vH) (cv y))) (opr (` S (cv x)) (+) (` S (cv y)))))))) ad2antrr (cv f) S (CH) (RR) feq1 (cv f) S (cv x) fveq1 (0) (<_) breq2d (cv f) S (cv x) fveq1 (<_) (1) breq1d anbi12d x (CH) ralbidv anbi12d (cv f) S (H~) fveq1 (1) eqeq1d (cv f) S (opr (cv x) (vH) (cv y)) fveq1 (cv f) S (cv x) fveq1 (cv f) S (cv y) fveq1 (+) opreq12d eqeq12d (C_ (cv x) (` (_|_) (cv y))) imbi2d x (CH) y (CH) 2ralbidv anbi12d anbi12d f x y df-st (V) elab2g pm5.21nii)) thm (hstelt ((x y) (f x) (f y) (S x) (S y) (S f)) () (<-> (e. S (CHStates)) (/\/\ (:--> S (CH) (H~)) (= (` (norm) (` S (H~))) (1)) (A.e. x (CH) (A.e. y (CH) (-> (C_ (cv x) (` (_|_) (cv y))) (/\ (= (opr (` S (cv x)) (.i) (` S (cv y))) (0)) (= (` S (opr (cv x) (vH) (cv y))) (opr (` S (cv x)) (+v) (` S (cv y)))))))))) (S (CHStates) elisset chex S (CH) (H~) (V) fex mpan2 (= (` (norm) (` S (H~))) (1)) (A.e. x (CH) (A.e. y (CH) (-> (C_ (cv x) (` (_|_) (cv y))) (/\ (= (opr (` S (cv x)) (.i) (` S (cv y))) (0)) (= (` S (opr (cv x) (vH) (cv y))) (opr (` S (cv x)) (+v) (` S (cv y)))))))) 3ad2ant1 (cv f) S (CH) (H~) feq1 (cv f) S (H~) fveq1 (norm) fveq2d (1) eqeq1d (cv f) S (cv x) fveq1 (cv f) S (cv y) fveq1 (.i) opreq12d (0) eqeq1d (cv f) S (opr (cv x) (vH) (cv y)) fveq1 (cv f) S (cv x) fveq1 (cv f) S (cv y) fveq1 (+v) opreq12d eqeq12d anbi12d (C_ (cv x) (` (_|_) (cv y))) imbi2d x (CH) y (CH) 2ralbidv 3anbi123d f x y df-hst (V) elab2g pm5.21nii)) thm (stclt ((x y) (A x) (A y) (S x) (S y)) () (-> (e. S (States)) (-> (e. A (CH)) (e. (` S A) (RR)))) (S (CH) (RR) A ffvrn S x y stelt pm3.26bd pm3.26d sylan ex)) thm (hstclt ((x y) (A x) (A y) (S x) (S y)) () (-> (/\ (e. S (CHStates)) (e. A (CH))) (e. (` S A) (H~))) (S (CH) (H~) A ffvrn S x y hstelt (:--> S (CH) (H~)) (= (` (norm) (` S (H~))) (1)) (A.e. x (CH) (A.e. y (CH) (-> (C_ (cv x) (` (_|_) (cv y))) (/\ (= (opr (` S (cv x)) (.i) (` S (cv y))) (0)) (= (` S (opr (cv x) (vH) (cv y))) (opr (` S (cv x)) (+v) (` S (cv y)))))))) 3simp1 sylbi sylan)) thm (hst1t ((x y) (S x) (S y)) () (-> (e. S (CHStates)) (= (` (norm) (` S (H~))) (1))) (S x y hstelt (:--> S (CH) (H~)) (= (` (norm) (` S (H~))) (1)) (A.e. x (CH) (A.e. y (CH) (-> (C_ (cv x) (` (_|_) (cv y))) (/\ (= (opr (` S (cv x)) (.i) (` S (cv y))) (0)) (= (` S (opr (cv x) (vH) (cv y))) (opr (` S (cv x)) (+v) (` S (cv y)))))))) 3simp2 sylbi)) thm (hstel2t ((x y) (A x) (A y) (S x) (S y) (B y)) () (-> (/\ (/\ (e. S (CHStates)) (e. A (CH))) (/\ (e. B (CH)) (C_ A (` (_|_) B)))) (/\ (= (opr (` S A) (.i) (` S B)) (0)) (= (` S (opr A (vH) B)) (opr (` S A) (+v) (` S B))))) (S x y hstelt (:--> S (CH) (H~)) (= (` (norm) (` S (H~))) (1)) (A.e. x (CH) (A.e. y (CH) (-> (C_ (cv x) (` (_|_) (cv y))) (/\ (= (opr (` S (cv x)) (.i) (` S (cv y))) (0)) (= (` S (opr (cv x) (vH) (cv y))) (opr (` S (cv x)) (+v) (` S (cv y)))))))) 3simp3 sylbi (e. A (CH)) (/\ (e. B (CH)) (C_ A (` (_|_) B))) ad2antrr (cv x) A (` (_|_) (cv y)) sseq1 (cv x) A S fveq2 (.i) (` S (cv y)) opreq1d (0) eqeq1d (cv x) A (vH) (cv y) opreq1 S fveq2d (cv x) A S fveq2 (+v) (` S (cv y)) opreq1d eqeq12d anbi12d imbi12d (cv y) B (_|_) fveq2 A sseq2d (cv y) B S fveq2 (` S A) (.i) opreq2d (0) eqeq1d (cv y) B A (vH) opreq2 S fveq2d (cv y) B S fveq2 (` S A) (+v) opreq2d eqeq12d anbi12d imbi12d (CH) (CH) rcla42v com23 imp anasss (e. S (CHStates)) adantll mpd)) thm (hstortht () () (-> (/\ (/\ (e. S (CHStates)) (e. A (CH))) (/\ (e. B (CH)) (C_ A (` (_|_) B)))) (= (opr (` S A) (.i) (` S B)) (0))) (S A B hstel2t pm3.26d)) thm (hstosumt () () (-> (/\ (/\ (e. S (CHStates)) (e. A (CH))) (/\ (e. B (CH)) (C_ A (` (_|_) B)))) (= (` S (opr A (vH) B)) (opr (` S A) (+v) (` S B)))) (S A B hstel2t pm3.27d)) thm (hstoct () () (-> (/\ (e. S (CHStates)) (e. A (CH))) (= (opr (` S A) (+v) (` S (` (_|_) A))) (` S (H~)))) (A chocclt (e. S (CHStates)) adantl A chsh A shococss syl (e. S (CHStates)) adantl jca S A (` (_|_) A) hstosumt mpdan A chjot S fveq2d (e. S (CHStates)) adantl eqtr3d)) thm (hstnmoct () () (-> (/\ (e. S (CHStates)) (e. A (CH))) (= (opr (opr (` (norm) (` S A)) (^) (2)) (+) (opr (` (norm) (` S (` (_|_) A))) (^) (2))) (1))) (S A hstoct (norm) fveq2d (^) (2) opreq1d (` S A) (` S (` (_|_) A)) normpytht S A hstclt S (` (_|_) A) hstclt A chocclt sylan2 jca A chocclt (e. S (CHStates)) adantl A chsh A shococss syl (e. S (CHStates)) adantl jca S A (` (_|_) A) hstortht mpdan sylc S hst1t (^) (2) opreq1d sq1 syl6eq (e. A (CH)) adantr 3eqtr3d)) thm (stge0t ((x y) (A x) (A y) (S x) (S y)) () (-> (e. S (States)) (-> (e. A (CH)) (br (0) (<_) (` S A)))) ((cv x) A S fveq2 (0) (<_) breq2d (CH) rcla4v S x y stelt pm3.26bd pm3.27d (br (0) (<_) (` S (cv x))) (br (` S (cv x)) (<_) (1)) pm3.26 x (CH) r19.20si syl syl5com)) thm (stle1t ((x y) (A x) (A y) (S x) (S y)) () (-> (e. S (States)) (-> (e. A (CH)) (br (` S A) (<_) (1)))) ((cv x) A S fveq2 (<_) (1) breq1d (CH) rcla4v S x y stelt pm3.26bd pm3.27d (br (0) (<_) (` S (cv x))) (br (` S (cv x)) (<_) (1)) pm3.27 x (CH) r19.20si syl syl5com)) thm (hstle1t () () (-> (/\ (e. S (CHStates)) (e. A (CH))) (br (` (norm) (` S A)) (<_) (1))) (S (` (_|_) A) hstclt A chocclt sylan2 (` S (` (_|_) A)) normclt syl (` (norm) (` S (` (_|_) A))) sqge0t syl (opr (` (norm) (` S A)) (^) (2)) (opr (` (norm) (` S (` (_|_) A))) (^) (2)) addge01t S A hstclt (` S A) normclt syl (` (norm) (` S A)) resqclt syl S (` (_|_) A) hstclt A chocclt sylan2 (` S (` (_|_) A)) normclt syl (` (norm) (` S (` (_|_) A))) resqclt syl sylanc mpbid S A hstnmoct sq1 syl6eqr breqtrd ax1re 0re ax1re lt01 ltlei pm3.2i (` (norm) (` S A)) (1) le2sqt mpan2 S A hstclt (` S A) normclt syl S A hstclt (` S A) normge0t syl sylanc mpbird)) thm (hst0ht () () (-> (/\ (e. S (CHStates)) (e. A (CH))) (<-> (= (` (norm) (` S A)) (0)) (= (` S A) (0v)))) (S A hstclt (` S A) norm-it syl)) thm (hstpytht () () (-> (/\ (/\ (e. S (CHStates)) (e. A (CH))) (/\ (e. B (CH)) (C_ A (` (_|_) B)))) (= (opr (` (norm) (` S (opr A (vH) B))) (^) (2)) (opr (opr (` (norm) (` S A)) (^) (2)) (+) (opr (` (norm) (` S B)) (^) (2))))) (S A B hstosumt (norm) fveq2d (^) (2) opreq1d (` S A) (` S B) normpytht 3impia S A hstclt (/\ (e. B (CH)) (C_ A (` (_|_) B))) adantr S B hstclt (e. A (CH)) (C_ A (` (_|_) B)) ad2ant2r S A B hstortht syl3anc eqtrd)) thm (hstlet () () (-> (/\ (/\ (e. S (CHStates)) (e. A (CH))) (/\ (e. B (CH)) (C_ A B))) (br (` (norm) (` S A)) (<_) (` (norm) (` S B)))) (S B hstnmoct (e. A (CH)) adantlr (opr (` (norm) (` S A)) (^) (2)) (+) opreq2d (opr (` (norm) (` S A)) (^) (2)) (opr (` (norm) (` S B)) (^) (2)) (opr (` (norm) (` S (` (_|_) B))) (^) (2)) add12t S A hstclt (` S A) normclt syl (` (norm) (` S A)) resqclt syl (e. B (CH)) adantr recnd S B hstclt (` S B) normclt syl (` (norm) (` S B)) resqclt syl (e. A (CH)) adantlr recnd S (` (_|_) B) hstclt B chocclt sylan2 (` S (` (_|_) B)) normclt syl (` (norm) (` S (` (_|_) B))) resqclt syl (e. A (CH)) adantlr recnd syl3anc eqtr3d (C_ A B) adantrr S A (` (_|_) B) hstpytht B chocclt (C_ A B) adantr B ococt A sseq2d biimpar jca sylan2 S (opr A (vH) (` (_|_) B)) hstle1t A (` (_|_) B) chjclt B chocclt sylan2 sylan2 anassrs ax1re 0re ax1re lt01 ltlei pm3.2i (` (norm) (` S (opr A (vH) (` (_|_) B)))) (1) le2sqt mpan2 S (opr A (vH) (` (_|_) B)) hstclt A (` (_|_) B) chjclt B chocclt sylan2 sylan2 anassrs (` S (opr A (vH) (` (_|_) B))) normclt syl S (opr A (vH) (` (_|_) B)) hstclt A (` (_|_) B) chjclt B chocclt sylan2 sylan2 anassrs (` S (opr A (vH) (` (_|_) B))) normge0t syl sylanc mpbid sq1 syl6breq (C_ A B) adantrr eqbrtrrd ax1re (opr (opr (` (norm) (` S A)) (^) (2)) (+) (opr (` (norm) (` S (` (_|_) B))) (^) (2))) (1) (opr (` (norm) (` S B)) (^) (2)) leadd2t mp3an2 (opr (` (norm) (` S A)) (^) (2)) (opr (` (norm) (` S (` (_|_) B))) (^) (2)) axaddrcl S A hstclt (` S A) normclt syl (` (norm) (` S A)) resqclt syl (e. B (CH)) adantr S (` (_|_) B) hstclt B chocclt sylan2 (` S (` (_|_) B)) normclt syl (` (norm) (` S (` (_|_) B))) resqclt syl (e. A (CH)) adantlr sylanc S B hstclt (` S B) normclt syl (` (norm) (` S B)) resqclt syl (e. A (CH)) adantlr sylanc (C_ A B) adantrr mpbid eqbrtrd ax1re (opr (` (norm) (` S A)) (^) (2)) (opr (` (norm) (` S B)) (^) (2)) (1) leadd1t mp3an3 S A hstclt (` S A) normclt syl (` (norm) (` S A)) resqclt syl (e. B (CH)) adantr S B hstclt (` S B) normclt syl (` (norm) (` S B)) resqclt syl (e. A (CH)) adantlr sylanc (C_ A B) adantrr mpbird (` (norm) (` S A)) (` (norm) (` S B)) le2sqt S A hstclt (` S A) normclt syl S A hstclt (` S A) normge0t syl jca (e. B (CH)) adantr S B hstclt (` S B) normclt syl S B hstclt (` S B) normge0t syl jca (e. A (CH)) adantlr sylanc (C_ A B) adantrr mpbird)) thm (hstlest () () (-> (/\ (/\ (e. S (CHStates)) (e. A (CH))) (/\ (e. B (CH)) (C_ A B))) (-> (= (` (norm) (` S A)) (1)) (= (` (norm) (` S B)) (1)))) ((/\ (/\ (e. S (CHStates)) (e. A (CH))) (/\ (e. B (CH)) (C_ A B))) (= (` (norm) (` S A)) (1)) pm3.27 S A B hstlet (= (` (norm) (` S A)) (1)) adantr eqbrtrrd ex S B hstle1t (e. A (CH)) (C_ A B) ad2ant2r jctild S B hstclt (` S B) normclt ax1re (` (norm) (` S B)) (1) letri3t mpan2 3syl (e. A (CH)) (C_ A B) ad2ant2r sylibrd)) thm (hstoht () () (-> (/\/\ (e. S (CHStates)) (e. A (CH)) (= (opr (` S A) (.i) (` S (H~))) (0))) (= (` S A) (0v))) ((` S A) (` S A) (` S (` (_|_) A)) his7t S A hstclt S A hstclt S (` (_|_) A) hstclt A chocclt sylan2 syl3anc S A hstclt (` S A) normsqt syl eqcomd A chocclt A ococt (` (_|_) (` (_|_) A)) A eqimss2 syl jca (e. S (CHStates)) adantl S A (` (_|_) A) hstortht mpdan (+) opreq12d S A hstclt (` S A) normclt syl (` (norm) (` S A)) resqclt syl recnd (opr (` (norm) (` S A)) (^) (2)) ax0id syl 3eqtrrd S A hstoct (` S A) (.i) opreq2d eqtrd (= (opr (` S A) (.i) (` S (H~))) (0)) id sylan9eq 3impa S A hstclt (` S A) normclt syl recnd (` (norm) (` S A)) sq0t syl (= (opr (` S A) (.i) (` S (H~))) (0)) 3adant3 mpbid S A hst0ht (= (opr (` S A) (.i) (` S (H~))) (0)) 3adant3 mpbid)) thm (sthil ((x y) (S x) (S y)) () (-> (e. S (States)) (= (` S (H~)) (1))) (S x y stelt pm3.27bd pm3.26d)) thm (stjt ((x y) (A x) (A y) (S x) (S y) (B y)) () (-> (e. S (States)) (-> (/\ (/\ (e. A (CH)) (e. B (CH))) (C_ A (` (_|_) B))) (= (` S (opr A (vH) B)) (opr (` S A) (+) (` S B))))) ((cv x) A (` (_|_) (cv y)) sseq1 (cv x) A (vH) (cv y) opreq1 S fveq2d (cv x) A S fveq2 (+) (` S (cv y)) opreq1d eqeq12d imbi12d (cv y) B (_|_) fveq2 A sseq2d (cv y) B A (vH) opreq2 S fveq2d (cv y) B S fveq2 (` S A) (+) opreq2d eqeq12d imbi12d (CH) (CH) rcla42v S x y stelt pm3.27bd pm3.27d syl5com imp3a)) thm (sto1 () ((sto1.1 (e. A (CH)))) (-> (e. S (States)) (= (opr (` S A) (+) (` S (` (_|_) A))) (1))) (sto1.1 sto1.1 choccl pm3.2i sto1.1 chshi A shococss ax-mp S A (` (_|_) A) stjt mp2ani S sthil sto1.1 chjo S fveq2i syl5eq eqtr3d)) thm (sto2 () ((sto1.1 (e. A (CH)))) (-> (e. S (States)) (= (` S (` (_|_) A)) (opr (1) (-) (` S A)))) (sto1.1 S sto1 1cn (1) (` S A) (` S (` (_|_) A)) subaddt mp3an1 sto1.1 S A stclt mpi recnd sto1.1 choccl S (` (_|_) A) stclt mpi recnd sylanc mpbird eqcomd)) thm (stge1 () ((sto1.1 (e. A (CH)))) (-> (e. S (States)) (<-> (br (1) (<_) (` S A)) (= (` S A) (1)))) (sto1.1 S A stle1t mpi (br (1) (<_) (` S A)) anim1i ex sto1.1 S A stclt mpi ax1re jctir (` S A) (1) letri3t syl sylibrd ax1re leid (` S A) (1) (1) (<_) breq2 mpbiri (e. S (States)) a1i impbid)) thm (stle0 () ((sto1.1 (e. A (CH)))) (-> (e. S (States)) (<-> (br (` S A) (<_) (0)) (= (` S A) (0)))) (sto1.1 S A stge0t mpi (br (` S A) (<_) (0)) anim2i expcom sto1.1 S A stclt mpi 0re jctir (` S A) (0) letri3t syl sylibrd 0re leid (` S A) (0) (<_) (0) breq1 mpbiri (e. S (States)) a1i impbid)) thm (stle () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (C_ A B) (br (` S A) (<_) (` S B)))) (S A (` (_|_) B) stjt stle.2 chshi B shococss ax-mp A B (` (_|_) (` (_|_) B)) sstr2 mpi stle.1 stle.2 choccl pm3.2i jctil syl5 imp stle.1 stle.2 choccl chjcl S (opr A (vH) (` (_|_) B)) stle1t mpi stle.2 S sto1 breqtrrd (C_ A B) adantr eqbrtrrd stle.1 S A stclt mpi stle.2 S B stclt mpi stle.2 choccl S (` (_|_) B) stclt mpi 3jca (C_ A B) adantr (` S A) (` S B) (` S (` (_|_) B)) leadd1t syl mpbird ex)) thm (stles () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (C_ A B) (-> (= (` S A) (1)) (= (` S B) (1))))) (stle.2 S B stle1t mpi (/\ (C_ A B) (= (` S A) (1))) adantr stle.1 stle.2 S stle imp (= (` S A) (1)) adantrr (` S A) (1) (<_) (` S B) breq1 (e. S (States)) (C_ A B) ad2antll mpbid jca stle.2 S B stclt mpi ax1re jctir (/\ (C_ A B) (= (` S A) (1))) adantr (` S B) (1) letri3t syl mpbird exp32)) thm (stji1 () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (= (` S (opr (` (_|_) A) (vH) (i^i A B))) (opr (` S (` (_|_) A)) (+) (` S (i^i A B))))) (stle.1 choccl stle.1 stle.2 chincl pm3.2i A B inss1 stle.1 stle.2 chincl stle.1 chsscon3 mpbi S (` (_|_) A) (i^i A B) stjt mp2ani)) thm (stm1 () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (= (` S (i^i A B)) (1)) (= (` S A) (1)))) ((` S (i^i A B)) (1) (<_) (` S A) breq1 A B inss1 stle.1 stle.2 chincl stle.1 S stle mpi syl5bi com12 stle.1 S stge1 sylibd)) thm (stm1r () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (= (` S (i^i A B)) (1)) (= (` S B) (1)))) (stle.2 stle.1 S stm1 A B incom S fveq2i (1) eqeq1i syl5ib)) thm (stm1add () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (= (` S (i^i A B)) (1)) (= (opr (` S A) (+) (` S B)) (2)))) (stle.1 stle.2 S stm1 stle.1 stle.2 S stm1r jcad (` S A) (1) (` S B) (1) (+) opreq12 df-2 syl6eqr syl6)) thm (stadd () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (= (opr (` S A) (+) (` S B)) (2)) (= (` S A) (1)))) ((` S A) (` S B) axaddrcl stle.1 S A stclt mpi stle.2 S B stclt mpi sylanc 2re jctir (opr (` S A) (+) (` S B)) (2) ltnet syl con2d stle.2 S B stle1t mpi (` S B) (1) (` S A) leadd2t stle.2 S B stclt mpi ax1re (e. S (States)) a1i stle.1 S A stclt mpi syl3anc mpbid (br (` S A) (<) (1)) adantr (` S A) (1) (1) ltadd1t biimpd stle.1 S A stclt mpi ax1re (e. S (States)) a1i ax1re (e. S (States)) a1i syl3anc imp (opr (` S A) (+) (` S B)) (opr (` S A) (+) (1)) (opr (1) (+) (1)) lelttrt (` S A) (` S B) axaddrcl stle.1 S A stclt mpi stle.2 S B stclt mpi sylanc stle.1 S A stclt mpi ax1re jctir (` S A) (1) axaddrcl syl ax1re ax1re readdcl (e. S (States)) a1i syl3anc (br (` S A) (<) (1)) adantr mp2and df-2 syl6breqr ex con3d stle.1 S A stle1t mpi stle.1 S A stclt mpi ax1re jctir (` S A) (1) leloet syl mpbid ord syld syld)) thm (stm1add3 () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH))) (stm1add3.3 (e. C (CH)))) (-> (e. S (States)) (-> (= (` S (i^i (i^i A B) C)) (1)) (= (opr (opr (` S A) (+) (` S B)) (+) (` S C)) (3)))) (stle.1 stle.2 chincl stm1add3.3 S stm1 stle.1 stle.2 S stm1add syld stle.1 stle.2 chincl stm1add3.3 S stm1r jcad (opr (` S A) (+) (` S B)) (2) (` S C) (1) (+) opreq12 df-3 syl6eqr syl6)) thm (stadd3 () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH))) (stm1add3.3 (e. C (CH)))) (-> (e. S (States)) (-> (= (opr (opr (` S A) (+) (` S B)) (+) (` S C)) (3)) (= (` S A) (1)))) ((` S A) (` S B) (` S C) axaddass stle.1 S A stclt mpi recnd stle.2 S B stclt mpi recnd stm1add3.3 S C stclt mpi recnd syl3anc (3) eqeq1d (` S A) (opr (` S B) (+) (` S C)) axaddrcl stle.1 S A stclt mpi (` S B) (` S C) axaddrcl stle.2 S B stclt mpi stm1add3.3 S C stclt mpi sylanc sylanc 3re jctir (opr (` S A) (+) (opr (` S B) (+) (` S C))) (3) ltnet syl con2d (` S B) (` S C) axaddrcl stle.2 S B stclt mpi stm1add3.3 S C stclt mpi sylanc stle.2 S B stclt mpi ax1re jctir (` S B) (1) axaddrcl syl ax1re ax1re readdcl (e. S (States)) a1i stm1add3.3 S C stle1t mpi (` S C) (1) (` S B) leadd2t stm1add3.3 S C stclt mpi ax1re (e. S (States)) a1i stle.2 S B stclt mpi syl3anc mpbid stle.2 S B stle1t mpi (` S B) (1) (1) leadd1t stle.2 S B stclt mpi ax1re (e. S (States)) a1i ax1re (e. S (States)) a1i syl3anc mpbid letrd (opr (` S B) (+) (` S C)) (opr (1) (+) (1)) (` S A) leadd2t (` S B) (` S C) axaddrcl stle.2 S B stclt mpi stm1add3.3 S C stclt mpi sylanc ax1re ax1re readdcl (e. S (States)) a1i stle.1 S A stclt mpi syl3anc mpbid (br (` S A) (<) (1)) adantr (` S A) (1) (opr (1) (+) (1)) ltadd1t biimpd stle.1 S A stclt mpi ax1re (e. S (States)) a1i ax1re ax1re readdcl (e. S (States)) a1i syl3anc imp (opr (` S A) (+) (opr (` S B) (+) (` S C))) (opr (` S A) (+) (opr (1) (+) (1))) (opr (1) (+) (opr (1) (+) (1))) lelttrt (` S A) (opr (` S B) (+) (` S C)) axaddrcl stle.1 S A stclt mpi (` S B) (` S C) axaddrcl stle.2 S B stclt mpi stm1add3.3 S C stclt mpi sylanc sylanc stle.1 S A stclt mpi ax1re ax1re readdcl jctir (` S A) (opr (1) (+) (1)) axaddrcl syl ax1re ax1re ax1re readdcl readdcl (e. S (States)) a1i syl3anc (br (` S A) (<) (1)) adantr mp2and df-3 df-2 (+) (1) opreq1i 1cn 1cn 1cn addass 3eqtrr syl6breq ex con3d stle.1 S A stle1t mpi stle.1 S A stclt mpi ax1re jctir (` S A) (1) leloet syl mpbid ord syld syld sylbid)) thm (st0 () () (-> (e. S (States)) (= (` S (0H)) (0))) (helch S sto2 S sthil (1) (-) opreq2d eqtrd choc1 S fveq2i 1cn subid 3eqtr3g)) thm (strlem1 ((u x) (A u) (A x) (B u) (B x)) ((strlem1.1 (e. A (CH))) (strlem1.2 (e. B (CH)))) (-> (-. (C_ A B)) (E.e. u (\ A B) (= (` (norm) (cv u)) (1)))) (A B ssdif0 negbii (\ A B) x n0 bitr (cv u) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (cv x)) (norm) fveq2 (1) eqeq1d (\ A B) rcla4ev (` (norm) (cv x)) rerecclt (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normclt 3syl strlem1.2 B ch0 ax-mp (0v) A B eldifn mt2 (cv x) (0v) (\ A B) eleq1 mtbiri con2i (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) norm-it 3syl mtbird (` (norm) (cv x)) (0) df-ne sylibr sylanc recnd strlem1.1 chshi A (opr (1) (/) (` (norm) (cv x))) (cv x) shmulclt ax-mp ex syl (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normclt 3syl recnd strlem1.2 chshi B (` (norm) (cv x)) (opr (opr (1) (/) (` (norm) (cv x))) (.s) (cv x)) shmulclt ax-mp ex syl (` (norm) (cv x)) recidt (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normclt 3syl recnd strlem1.2 B ch0 ax-mp (0v) A B eldifn mt2 (cv x) (0v) (\ A B) eleq1 mtbiri con2i (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) norm-it 3syl mtbird (` (norm) (cv x)) (0) df-ne sylibr sylanc (.s) (cv x) opreq1d (` (norm) (cv x)) (opr (1) (/) (` (norm) (cv x))) (cv x) ax-hvmulass (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normclt 3syl recnd (` (norm) (cv x)) rerecclt (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normclt 3syl strlem1.2 B ch0 ax-mp (0v) A B eldifn mt2 (cv x) (0v) (\ A B) eleq1 mtbiri con2i (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) norm-it 3syl mtbird (` (norm) (cv x)) (0) df-ne sylibr sylanc recnd (cv x) A B eldifi strlem1.1 (cv x) chel syl syl3anc (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) ax-hvmulid 3syl 3eqtr3d B eleq1d sylibd con3d anim12d (cv x) A B eldif (opr (opr (1) (/) (` (norm) (cv x))) (.s) (cv x)) A B eldif 3imtr4g pm2.43i (opr (1) (/) (` (norm) (cv x))) (cv x) norm-iiit (` (norm) (cv x)) rerecclt (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normclt 3syl strlem1.2 B ch0 ax-mp (0v) A B eldifn mt2 (cv x) (0v) (\ A B) eleq1 mtbiri con2i (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) norm-it 3syl mtbird (` (norm) (cv x)) (0) df-ne sylibr sylanc recnd (cv x) A B eldifi strlem1.1 (cv x) chel syl sylanc (opr (1) (/) (` (norm) (cv x))) absidt (` (norm) (cv x)) rerecclt (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normclt 3syl strlem1.2 B ch0 ax-mp (0v) A B eldifn mt2 (cv x) (0v) (\ A B) eleq1 mtbiri con2i (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) norm-it 3syl mtbird (` (norm) (cv x)) (0) df-ne sylibr sylanc (1) (` (norm) (cv x)) divge0t (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normclt 3syl ax1re jctil strlem1.2 B ch0 ax-mp (0v) A B eldifn mt2 (cv x) (0v) (\ A B) eleq1 mtbiri con2i (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normgt0tOLD 3syl mpbid 0re ax1re lt01 ltlei jctil sylanc sylanc (x.) (` (norm) (cv x)) opreq1d (opr (1) (/) (` (norm) (cv x))) (` (norm) (cv x)) axmulcom (` (norm) (cv x)) rerecclt (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normclt 3syl strlem1.2 B ch0 ax-mp (0v) A B eldifn mt2 (cv x) (0v) (\ A B) eleq1 mtbiri con2i (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) norm-it 3syl mtbird (` (norm) (cv x)) (0) df-ne sylibr sylanc recnd (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normclt 3syl recnd sylanc (` (norm) (cv x)) recidt (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) normclt 3syl recnd strlem1.2 B ch0 ax-mp (0v) A B eldifn mt2 (cv x) (0v) (\ A B) eleq1 mtbiri con2i (cv x) A B eldifi strlem1.1 (cv x) chel (cv x) norm-it 3syl mtbird (` (norm) (cv x)) (0) df-ne sylibr sylanc eqtrd 3eqtrd sylanc x 19.23aiv sylbi)) thm (strlem2 ((x y) (C x) (C y) (u x) (u y)) ((strlem2.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2)))))))) (-> (e. C (CH)) (= (` S C) (opr (` (norm) (` (` (proj) C) (cv u))) (^) (2)))) ((cv x) C (proj) fveq2 (cv u) fveq1d (norm) fveq2d (^) (2) opreq1d strlem2.1 (` (norm) (` (` (proj) C) (cv u))) (^) (2) oprex fvopab4)) thm (strlem3a ((x z) (w x) (u x) (w z) (u z) (u w) (S z) (S w) (x y) (u y) (y z) (w y)) ((strlem3a.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2)))))))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (e. S (States))) ((cv x) (cv u) pjhclt (= (` (norm) (cv u)) (1)) adantrr (` (` (proj) (cv x)) (cv u)) normclt (` (norm) (` (` (proj) (cv x)) (cv u))) resqclt 3syl expcom r19.21aiv strlem3a.1 (RR) fopab2 sylib (cv z) (cv u) pjhclt (= (` (norm) (cv u)) (1)) adantrr (` (` (proj) (cv z)) (cv u)) normclt (` (norm) (` (` (proj) (cv z)) (cv u))) sqge0t 3syl strlem3a.1 (cv z) strlem2 (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) adantr breqtrrd strlem3a.1 (cv z) strlem2 (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) adantr (cv z) (cv u) pjnormt (= (` (norm) (cv u)) (1)) adantrr (e. (cv u) (H~)) (= (` (norm) (cv u)) (1)) pm3.27 (e. (cv z) (CH)) adantl breqtrd ax1re 0re ax1re lt01 ltlei pm3.2i (` (norm) (` (` (proj) (cv z)) (cv u))) (1) le2sqt mpan2 (cv z) (cv u) pjhclt (= (` (norm) (cv u)) (1)) adantrr (` (` (proj) (cv z)) (cv u)) normclt syl (cv z) (cv u) pjhclt (= (` (norm) (cv u)) (1)) adantrr (` (` (proj) (cv z)) (cv u)) normge0t syl sylanc mpbid sq1 syl6breq eqbrtrd jca expcom r19.21aiv jca (cv u) pjch1t (norm) fveq2d (^) (2) opreq1d (` (norm) (cv u)) (1) (^) (2) opreq1 sq1 syl6eq sylan9eq helch strlem3a.1 (H~) strlem2 ax-mp syl5eq (cv z) (cv w) (cv u) pjcjt2 imp (norm) fveq2d (^) (2) opreq1d (cv z) (cv w) (cv u) pjopytht imp eqtrd (cv z) (cv w) chjclt (e. (cv u) (H~)) 3adant3 (C_ (cv z) (` (_|_) (cv w))) adantr strlem3a.1 (opr (cv z) (vH) (cv w)) strlem2 syl (e. (cv z) (CH)) (e. (cv w) (CH)) (e. (cv u) (H~)) 3simpa (C_ (cv z) (` (_|_) (cv w))) adantr strlem3a.1 (cv z) strlem2 strlem3a.1 (cv w) strlem2 (+) opreqan12d syl 3eqtr4d 3exp1 com3r (= (` (norm) (cv u)) (1)) adantr r19.21adv r19.21aiv jca jca S z w stelt sylibr)) thm (strlem3 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((strlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2))))))) (strlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (strlem3.3 (e. A (CH))) (strlem3.4 (e. B (CH)))) (-> ph (e. S (States))) (strlem3.2 strlem3.1 strlem3a (cv u) A B eldifi strlem3.3 (cv u) chel syl sylan sylbi)) thm (strlem4 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((strlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2))))))) (strlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (strlem3.3 (e. A (CH))) (strlem3.4 (e. B (CH)))) (-> ph (= (` S A) (1))) (strlem3.2 (` (norm) (cv u)) (1) (` (norm) (` (` (proj) A) (cv u))) eqeq2 strlem3.3 A (cv u) pjidt mpan (norm) fveq2d syl5bi impcom (cv u) A B eldifi sylan sylbi (^) (2) opreq1d sq1 syl6eq strlem3.3 strlem3.1 A strlem2 ax-mp syl5eq)) thm (strlem5 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((strlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2))))))) (strlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (strlem3.3 (e. A (CH))) (strlem3.4 (e. B (CH)))) (-> ph (br (` S B) (<) (1))) (strlem3.2 (` (norm) (cv u)) (1) (` (norm) (` (` (proj) B) (cv u))) (<) breq2 (cv u) A B eldif strlem3.4 B (cv u) pjnelt mpan biimpa strlem3.3 (cv u) chel sylan sylbi syl5bi impcom (cv u) A B eldifi strlem3.3 (cv u) chel (` (norm) (` (` (proj) B) (cv u))) (1) lt2sqtOLD strlem3.4 (cv u) pjhcl (` (` (proj) B) (cv u)) normclt syl ax1re jctir strlem3.4 (cv u) pjhcl (` (` (proj) B) (cv u)) normge0t syl 0re ax1re lt01 ltlei jctir sylc 3syl (= (` (norm) (cv u)) (1)) adantr mpbid strlem3.4 strlem3.1 B strlem2 ax-mp syl5eqbr sq1 syl6breq sylbi)) thm (strlem6 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((strlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2))))))) (strlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (strlem3.3 (e. A (CH))) (strlem3.4 (e. B (CH)))) (-> ph (-. (-> (= (` S A) (1)) (= (` S B) (1))))) (strlem3.1 strlem3.2 strlem3.3 strlem3.4 strlem4 strlem3.1 strlem3.2 strlem3.3 strlem3.4 strlem5 strlem3.1 strlem3.2 strlem3.3 strlem3.4 strlem3 strlem3.4 S B stclt mpi ax1re (` S B) (1) ltnet mpan2 3syl mpd jca (= (` S A) (1)) (= (` S B) (1)) annim sylib)) thm (str ((x y) (u x) (f x) (A x) (u y) (f y) (A y) (f u) (A u) (A f) (B x) (B y) (B u) (B f)) ((str.1 (e. A (CH))) (str.2 (e. B (CH)))) (-> (A.e. f (States) (-> (= (` (cv f) A) (1)) (= (` (cv f) B) (1)))) (C_ A B)) (f (States) (-> (= (` (cv f) A) (1)) (= (` (cv f) B) (1))) dfral2 str.1 str.2 u strlem1 (cv f) ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2))))) A fveq1 (1) eqeq1d (cv f) ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2))))) B fveq1 (1) eqeq1d imbi12d negbid (States) rcla4ev ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2))))) eqid (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))) pm4.2 str.1 str.2 strlem3 ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2))))) eqid (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))) pm4.2 str.1 str.2 strlem6 sylanc ex r19.23aiv syl con1i sylbi)) thm (strb ((A f) (B f)) ((str.1 (e. A (CH))) (str.2 (e. B (CH)))) (<-> (A.e. f (States) (-> (= (` (cv f) A) (1)) (= (` (cv f) B) (1)))) (C_ A B)) (str.1 str.2 f str str.1 str.2 (cv f) stles com12 r19.21aiv impbi)) thm (hstrlem2 ((x y) (C x) (C y) (u x) (u y)) ((hstrlem2.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u)))))))) (-> (e. C (CH)) (= (` S C) (` (` (proj) C) (cv u)))) ((cv x) C (proj) fveq2 (cv u) fveq1d hstrlem2.1 (` (proj) C) (cv u) fvex fvopab4)) thm (hstrlem3a ((x z) (w x) (u x) (w z) (u z) (u w) (S z) (S w) (x y) (u y) (y z) (w y)) ((hstrlem3a.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u)))))))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (e. S (CHStates))) ((cv x) (cv u) pjhclt (= (` (norm) (cv u)) (1)) adantrr expcom r19.21aiv hstrlem3a.1 (H~) fopab2 sylib (cv u) pjch1t (norm) fveq2d (= (` (norm) (cv u)) (1)) id sylan9eq helch hstrlem3a.1 (H~) hstrlem2 ax-mp (norm) fveq2i syl5eq hstrlem3a.1 (cv z) hstrlem2 hstrlem3a.1 (cv w) hstrlem2 (.i) opreqan12d (e. (cv u) (H~)) 3adant3 (C_ (cv z) (` (_|_) (cv w))) adantr (cv z) (cv w) (cv u) pjoi0t eqtrd (cv z) (cv w) (cv u) pjcjt2 imp (cv z) (cv w) chjclt hstrlem3a.1 (opr (cv z) (vH) (cv w)) hstrlem2 syl (e. (cv u) (H~)) 3adant3 (C_ (cv z) (` (_|_) (cv w))) adantr hstrlem3a.1 (cv z) hstrlem2 hstrlem3a.1 (cv w) hstrlem2 (+v) opreqan12d (e. (cv u) (H~)) 3adant3 (C_ (cv z) (` (_|_) (cv w))) adantr 3eqtr4d jca 3exp1 com3r (= (` (norm) (cv u)) (1)) adantr r19.21adv r19.21aiv 3jca S z w hstelt sylibr)) thm (hstrlem3 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((hstrlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u))))))) (hstrlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (hstrlem3.3 (e. A (CH))) (hstrlem3.4 (e. B (CH)))) (-> ph (e. S (CHStates))) (hstrlem3.2 hstrlem3.1 hstrlem3a (cv u) A B eldifi hstrlem3.3 (cv u) chel syl sylan sylbi)) thm (hstrlem4 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((hstrlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u))))))) (hstrlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (hstrlem3.3 (e. A (CH))) (hstrlem3.4 (e. B (CH)))) (-> ph (= (` (norm) (` S A)) (1))) (hstrlem3.2 (` (norm) (cv u)) (1) (` (norm) (` (` (proj) A) (cv u))) eqeq2 hstrlem3.3 A (cv u) pjidt mpan (norm) fveq2d syl5bi impcom (cv u) A B eldifi sylan sylbi hstrlem3.3 hstrlem3.1 A hstrlem2 ax-mp (norm) fveq2i syl5eq)) thm (hstrlem5 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((hstrlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u))))))) (hstrlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (hstrlem3.3 (e. A (CH))) (hstrlem3.4 (e. B (CH)))) (-> ph (br (` (norm) (` S B)) (<) (1))) (hstrlem3.2 (` (norm) (cv u)) (1) (` (norm) (` (` (proj) B) (cv u))) (<) breq2 (cv u) A B eldif hstrlem3.4 B (cv u) pjnelt mpan biimpa hstrlem3.3 (cv u) chel sylan sylbi syl5bi impcom hstrlem3.4 hstrlem3.1 B hstrlem2 (norm) fveq2d ax-mp syl5eqbr sylbi)) thm (hstrlem6 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((hstrlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u))))))) (hstrlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (hstrlem3.3 (e. A (CH))) (hstrlem3.4 (e. B (CH)))) (-> ph (-. (-> (= (` (norm) (` S A)) (1)) (= (` (norm) (` S B)) (1))))) (hstrlem3.1 hstrlem3.2 hstrlem3.3 hstrlem3.4 hstrlem4 hstrlem3.1 hstrlem3.2 hstrlem3.3 hstrlem3.4 hstrlem5 hstrlem3.1 hstrlem3.2 hstrlem3.3 hstrlem3.4 hstrlem3 hstrlem3.4 S B hstclt (` S B) normclt syl mpan2 ax1re (` (norm) (` S B)) (1) ltnet mpan2 3syl mpd jca (= (` (norm) (` S A)) (1)) (= (` (norm) (` S B)) (1)) annim sylib)) thm (hstr ((x y) (u x) (f x) (A x) (u y) (f y) (A y) (f u) (A u) (A f) (B x) (B y) (B u) (B f)) ((hstr.1 (e. A (CH))) (hstr.2 (e. B (CH)))) (-> (A.e. f (CHStates) (-> (= (` (norm) (` (cv f) A)) (1)) (= (` (norm) (` (cv f) B)) (1)))) (C_ A B)) (f (CHStates) (-> (= (` (norm) (` (cv f) A)) (1)) (= (` (norm) (` (cv f) B)) (1))) dfral2 hstr.1 hstr.2 u strlem1 (cv f) ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u))))) A fveq1 (norm) fveq2d (1) eqeq1d (cv f) ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u))))) B fveq1 (norm) fveq2d (1) eqeq1d imbi12d negbid (CHStates) rcla4ev ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u))))) eqid (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))) pm4.2 hstr.1 hstr.2 hstrlem3 ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u))))) eqid (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))) pm4.2 hstr.1 hstr.2 hstrlem6 sylanc ex r19.23aiv syl con1i sylbi)) thm (hstrb ((A f) (B f)) ((hstr.1 (e. A (CH))) (hstr.2 (e. B (CH)))) (<-> (A.e. f (CHStates) (-> (= (` (norm) (` (cv f) A)) (1)) (= (` (norm) (` (cv f) B)) (1)))) (C_ A B)) (hstr.1 hstr.2 f hstr hstr.1 hstr.2 (cv f) A B hstlest mpanr1 mpanl2 expcom r19.21aiv impbi)) thm (large ((A f)) ((large.1 (e. A (CH)))) (<-> (-. (= A (0H))) (E.e. f (States) (= (` (cv f) A) (1)))) (f (States) (= (` (cv f) A) (1)) ralnex ax1ne0 (1) (0) df-ne mpbi (cv f) st0 (1) eqeq1d (0) (1) eqcom syl6bb mtbiri (= (` (cv f) (0H)) (1)) (= (` (cv f) A) (1)) mtt syl ralbiia large.1 h0elch f strb large.1 chle0 3bitr bitr3 con1bii)) thm (jplem1 () ((jplem1.1 (e. A (CH)))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (<-> (e. (cv u) A) (= (opr (` (norm) (` (` (proj) A) (cv u))) (^) (2)) (1)))) (jplem1.1 A (cv u) pjnorm2t mpan (` (norm) (cv u)) (1) (` (norm) (` (` (proj) A) (cv u))) eqeq2 sylan9bb ax1re 0re ax1re lt01 ltlei pm3.2i (` (norm) (` (` (proj) A) (cv u))) (1) sq11t mpan2 jplem1.1 (cv u) pjhcl (` (` (proj) A) (cv u)) normclt syl jplem1.1 (cv u) pjhcl (` (` (proj) A) (cv u)) normge0t syl sylanc sq1 (opr (` (norm) (` (` (proj) A) (cv u))) (^) (2)) eqeq2i syl5bbr (= (` (norm) (cv u)) (1)) adantr bitr4d)) thm (jplem2 ((u x) (x y) (u y) (A x) (A y)) ((jp.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2))))))) (jp.2 (e. A (CH)))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (<-> (e. (cv u) A) (= (` S A) (1)))) (jp.2 u jplem1 jp.2 jp.1 A strlem2 ax-mp (1) eqeq1i syl6bbr)) thm (jp ((u x) (x y) (u y) (A x) (A y) (B x) (B y)) ((jp.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2))))))) (jp.2 (e. A (CH))) (jp.3 (e. B (CH)))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (<-> (/\ (= (` S A) (1)) (= (` S B) (1))) (= (` S (i^i A B)) (1)))) (jp.1 jp.2 jplem2 jp.1 jp.3 jplem2 anbi12d (cv u) A B elin syl5bb jp.1 jp.2 jp.3 chincl jplem2 bitr3d)) thm (golem1 () ((golem1.1 (e. A (CH))) (golem1.2 (e. B (CH))) (golem1.3 (e. C (CH))) (golem1.4 (= F (opr (` (_|_) A) (vH) (i^i A B)))) (golem1.5 (= G (opr (` (_|_) B) (vH) (i^i B C)))) (golem1.6 (= H (opr (` (_|_) C) (vH) (i^i C A)))) (golem1.7 (= D (opr (` (_|_) B) (vH) (i^i B A)))) (golem1.8 (= R (opr (` (_|_) C) (vH) (i^i C B)))) (golem1.9 (= S (opr (` (_|_) A) (vH) (i^i A C))))) (-> (e. (cv f) (States)) (= (opr (opr (` (cv f) F) (+) (` (cv f) G)) (+) (` (cv f) H)) (opr (opr (` (cv f) D) (+) (` (cv f) R)) (+) (` (cv f) S)))) ((` (cv f) (` (_|_) A)) (` (cv f) (` (_|_) B)) (` (cv f) (` (_|_) C)) axaddass golem1.1 choccl (cv f) (` (_|_) A) stclt mpi recnd golem1.2 choccl (cv f) (` (_|_) B) stclt mpi recnd golem1.3 choccl (cv f) (` (_|_) C) stclt mpi recnd syl3anc (` (cv f) (` (_|_) A)) (opr (` (cv f) (` (_|_) B)) (+) (` (cv f) (` (_|_) C))) axaddcom golem1.1 choccl (cv f) (` (_|_) A) stclt mpi recnd (` (cv f) (` (_|_) B)) (` (cv f) (` (_|_) C)) axaddcl golem1.2 choccl (cv f) (` (_|_) B) stclt mpi recnd golem1.3 choccl (cv f) (` (_|_) C) stclt mpi recnd sylanc sylanc eqtrd (+) (opr (opr (` (cv f) (i^i A B)) (+) (` (cv f) (i^i B C))) (+) (` (cv f) (i^i C A))) opreq1d (opr (` (cv f) (` (_|_) A)) (+) (` (cv f) (` (_|_) B))) (opr (` (cv f) (i^i A B)) (+) (` (cv f) (i^i B C))) (` (cv f) (` (_|_) C)) (` (cv f) (i^i C A)) add4t (` (cv f) (` (_|_) A)) (` (cv f) (` (_|_) B)) axaddcl golem1.1 choccl (cv f) (` (_|_) A) stclt mpi recnd golem1.2 choccl (cv f) (` (_|_) B) stclt mpi recnd sylanc (` (cv f) (i^i A B)) (` (cv f) (i^i B C)) axaddcl golem1.1 golem1.2 chincl (cv f) (i^i A B) stclt mpi recnd golem1.2 golem1.3 chincl (cv f) (i^i B C) stclt mpi recnd sylanc jca golem1.3 choccl (cv f) (` (_|_) C) stclt mpi recnd golem1.3 golem1.1 chincl (cv f) (i^i C A) stclt mpi recnd jca sylanc (opr (` (cv f) (` (_|_) B)) (+) (` (cv f) (` (_|_) C))) (opr (` (cv f) (i^i A B)) (+) (` (cv f) (i^i B C))) (` (cv f) (` (_|_) A)) (` (cv f) (i^i C A)) add4t (` (cv f) (` (_|_) B)) (` (cv f) (` (_|_) C)) axaddcl golem1.2 choccl (cv f) (` (_|_) B) stclt mpi recnd golem1.3 choccl (cv f) (` (_|_) C) stclt mpi recnd sylanc (` (cv f) (i^i A B)) (` (cv f) (i^i B C)) axaddcl golem1.1 golem1.2 chincl (cv f) (i^i A B) stclt mpi recnd golem1.2 golem1.3 chincl (cv f) (i^i B C) stclt mpi recnd sylanc jca golem1.1 choccl (cv f) (` (_|_) A) stclt mpi recnd golem1.3 golem1.1 chincl (cv f) (i^i C A) stclt mpi recnd jca sylanc 3eqtr4d (` (cv f) (` (_|_) A)) (` (cv f) (i^i A B)) (` (cv f) (` (_|_) B)) (` (cv f) (i^i B C)) add4t golem1.1 choccl (cv f) (` (_|_) A) stclt mpi recnd golem1.1 golem1.2 chincl (cv f) (i^i A B) stclt mpi recnd jca golem1.2 choccl (cv f) (` (_|_) B) stclt mpi recnd golem1.2 golem1.3 chincl (cv f) (i^i B C) stclt mpi recnd jca sylanc (+) (opr (` (cv f) (` (_|_) C)) (+) (` (cv f) (i^i C A))) opreq1d (` (cv f) (` (_|_) B)) (` (cv f) (i^i A B)) (` (cv f) (` (_|_) C)) (` (cv f) (i^i B C)) add4t golem1.2 choccl (cv f) (` (_|_) B) stclt mpi recnd golem1.1 golem1.2 chincl (cv f) (i^i A B) stclt mpi recnd jca golem1.3 choccl (cv f) (` (_|_) C) stclt mpi recnd golem1.2 golem1.3 chincl (cv f) (i^i B C) stclt mpi recnd jca sylanc (+) (opr (` (cv f) (` (_|_) A)) (+) (` (cv f) (i^i C A))) opreq1d 3eqtr4d golem1.1 golem1.2 (cv f) stji1 golem1.2 golem1.3 (cv f) stji1 (+) opreq12d golem1.3 golem1.1 (cv f) stji1 (+) opreq12d golem1.2 golem1.1 (cv f) stji1 B A incom (cv f) fveq2i (` (cv f) (` (_|_) B)) (+) opreq2i syl6eq golem1.3 golem1.2 (cv f) stji1 C B incom (cv f) fveq2i (` (cv f) (` (_|_) C)) (+) opreq2i syl6eq (+) opreq12d golem1.1 golem1.3 (cv f) stji1 A C incom (cv f) fveq2i (` (cv f) (` (_|_) A)) (+) opreq2i syl6eq (+) opreq12d 3eqtr4d golem1.4 (cv f) fveq2i golem1.5 (cv f) fveq2i (+) opreq12i golem1.6 (cv f) fveq2i (+) opreq12i golem1.7 (cv f) fveq2i golem1.8 (cv f) fveq2i (+) opreq12i golem1.9 (cv f) fveq2i (+) opreq12i 3eqtr4g)) thm (golem2 () ((golem1.1 (e. A (CH))) (golem1.2 (e. B (CH))) (golem1.3 (e. C (CH))) (golem1.4 (= F (opr (` (_|_) A) (vH) (i^i A B)))) (golem1.5 (= G (opr (` (_|_) B) (vH) (i^i B C)))) (golem1.6 (= H (opr (` (_|_) C) (vH) (i^i C A)))) (golem1.7 (= D (opr (` (_|_) B) (vH) (i^i B A)))) (golem1.8 (= R (opr (` (_|_) C) (vH) (i^i C B)))) (golem1.9 (= S (opr (` (_|_) A) (vH) (i^i A C))))) (-> (e. (cv f) (States)) (-> (= (` (cv f) (i^i (i^i F G) H)) (1)) (= (` (cv f) D) (1)))) (golem1.4 golem1.1 choccl golem1.1 golem1.2 chincl chjcl eqeltr golem1.5 golem1.2 choccl golem1.2 golem1.3 chincl chjcl eqeltr golem1.6 golem1.3 choccl golem1.3 golem1.1 chincl chjcl eqeltr (cv f) stm1add3 golem1.1 golem1.2 golem1.3 golem1.4 golem1.5 golem1.6 golem1.7 golem1.8 golem1.9 f golem1 (3) eqeq1d sylibd golem1.7 golem1.2 choccl golem1.2 golem1.1 chincl chjcl eqeltr golem1.8 golem1.3 choccl golem1.3 golem1.2 chincl chjcl eqeltr golem1.9 golem1.1 choccl golem1.1 golem1.3 chincl chjcl eqeltr (cv f) stadd3 syld)) thm (goeq ((F f) (G f) (H f) (D f)) ((goeq.1 (e. A (CH))) (goeq.2 (e. B (CH))) (goeq.3 (e. C (CH))) (goeq.4 (= F (opr (` (_|_) A) (vH) (i^i A B)))) (goeq.5 (= G (opr (` (_|_) B) (vH) (i^i B C)))) (goeq.6 (= H (opr (` (_|_) C) (vH) (i^i C A)))) (goeq.7 (= D (opr (` (_|_) B) (vH) (i^i B A))))) (C_ (i^i (i^i F G) H) D) (goeq.4 goeq.1 choccl goeq.1 goeq.2 chincl chjcl eqeltr goeq.5 goeq.2 choccl goeq.2 goeq.3 chincl chjcl eqeltr chincl goeq.6 goeq.3 choccl goeq.3 goeq.1 chincl chjcl eqeltr chincl goeq.7 goeq.2 choccl goeq.2 goeq.1 chincl chjcl eqeltr f str goeq.1 goeq.2 goeq.3 goeq.4 goeq.5 goeq.6 goeq.7 (opr (` (_|_) C) (vH) (i^i C B)) eqid (opr (` (_|_) A) (vH) (i^i A C)) eqid f golem2 mprg)) thm (stcltr1 ((x y) (A x) (A y) (B x) (B y) (S x) (S y)) ((stcltr1.1 (<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y)))))))) (stcltr1.2 (e. A (CH))) (stcltr1.3 (e. B (CH)))) (-> ph (-> (-> (= (` S A) (1)) (= (` S B) (1))) (C_ A B))) (stcltr1.1 stcltr1.2 stcltr1.3 (cv x) A S fveq2 (1) eqeq1d (= (` S (cv y)) (1)) imbi1d (cv x) A (cv y) sseq1 imbi12d (cv y) B S fveq2 (1) eqeq1d (= (` S A) (1)) imbi2d (cv y) B A sseq2 imbi12d (CH) (CH) rcla42v mp2an (e. S (States)) adantl sylbi)) thm (stcltr2 ((x y) (A x) (A y) (S x) (S y)) ((stcltr1.1 (<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y)))))))) (stcltr1.2 (e. A (CH)))) (-> ph (-> (= (` S A) (1)) (= A (H~)))) (stcltr1.1 helch stcltr1.2 stcltr1 (= (` S A) (1)) (= (` S (H~)) (1)) ax-1 syl5 A (H~) eqss stcltr1.2 chssi mpbiran syl6ibr)) thm (stcltrlem1 ((x y) (A x) (A y) (B x) (B y) (S x) (S y)) ((stcltr1.1 (<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y)))))))) (stcltr1.2 (e. A (CH))) (stcltrlem1.3 (e. B (CH)))) (-> ph (-> (= (` S B) (1)) (= (` S (opr (` (_|_) A) (vH) (i^i A B))) (1)))) (stcltr1.1 pm3.26bd stcltr1.2 stcltrlem1.3 S stji1 syl (= (` S B) (1)) adantr stcltr1.1 stcltrlem1.3 stcltr2 imp B (H~) A ineq2 stcltr1.2 chm1 syl6eq syl S fveq2d (` S (` (_|_) A)) (+) opreq2d stcltr1.1 pm3.26bd (= (` S B) (1)) adantr (` S (` (_|_) A)) (` S A) axaddcom stcltr1.2 choccl S (` (_|_) A) stclt mpi recnd stcltr1.2 S A stclt mpi recnd sylanc stcltr1.2 S sto1 eqtrd syl 3eqtrd ex)) thm (stcltrlem2 ((x y) (A x) (A y) (B x) (B y) (S x) (S y)) ((stcltr1.1 (<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y)))))))) (stcltr1.2 (e. A (CH))) (stcltrlem1.3 (e. B (CH)))) (-> ph (C_ B (opr (` (_|_) A) (vH) (i^i A B)))) (stcltr1.1 stcltr1.2 stcltrlem1.3 stcltrlem1 stcltr1.1 stcltrlem1.3 stcltr1.2 choccl stcltr1.2 stcltrlem1.3 chincl chjcl stcltr1 mpd)) thm (stcltrth ((x y) (s x) (A x) (s y) (A y) (A s) (B x) (B y) (B s)) ((stcltrth.1 (e. A (CH))) (stcltrth.2 (e. B (CH))) (stcltrth.3 (E.e. s (States) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` (cv s) (cv x)) (1)) (= (` (cv s) (cv y)) (1))) (C_ (cv x) (cv y)))))))) (C_ B (opr (` (_|_) A) (vH) (i^i A B))) (stcltrth.3 (/\ (e. (cv s) (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` (cv s) (cv x)) (1)) (= (` (cv s) (cv y)) (1))) (C_ (cv x) (cv y)))))) pm4.2 stcltrth.1 stcltrth.2 stcltrlem2 ex r19.23aiv ax-mp)) thm (cvbrt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A ( (/\ (e. A (CH)) (e. B (CH))) (<-> (br A ( (/\ (C: A (cv x)) (C_ (cv x) B)) (= (cv x) B)))))) (A B x cvbrt (/\ (C: A (cv x)) (C_ (cv x) B)) (= (cv x) B) iman (C: A (cv x)) (C_ (cv x) B) (-. (= (cv x) B)) anass (cv x) B dfpss2 (C: A (cv x)) anbi2i bitr4 negbii bitr x (CH) ralbii x (CH) (/\ (C: A (cv x)) (C: (cv x) B)) ralnex bitr (C: A B) anbi2i syl6bbr)) thm (cvcon3t ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A ( (/\ (e. A (CH)) (e. B (CH))) (-> (br A ( (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A ( (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A ( (/\ (C: A C) (C_ C B)) (= C B)))) (A B C cvnbtwnt (/\ (C: A C) (C_ C B)) (= C B) iman (C: A C) (C_ C B) (-. (= C B)) anass C B dfpss2 (C: A C) anbi2i bitr4 negbii bitr2 syl6ib)) thm (cvnbtwn3t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A ( (/\ (C_ A C) (C: C B)) (= C A)))) (A B C cvnbtwnt (/\ (C_ A C) (C: C B)) (= A C) iman C A eqcom (/\ (C_ A C) (C: C B)) imbi2i A C dfpss2 (C: C B) anbi1i (C_ A C) (-. (= A C)) (C: C B) an23 bitr negbii 3bitr4r syl6ib)) thm (cvnbtwn4t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A ( (/\ (C_ A C) (C_ C B)) (\/ (= C A) (= C B))))) (A B C cvnbtwnt (/\ (C_ A C) (C_ C B)) (\/ (= C A) (= C B)) iman (C_ A C) (C_ C B) (-. (= A C)) (-. (= C B)) an4 (= C A) (= C B) ioran C A eqcom negbii (-. (= C B)) anbi1i bitr (/\ (C_ A C) (C_ C B)) anbi2i A C dfpss2 C B dfpss2 anbi12i 3bitr4 negbii bitr2 syl6ib)) thm (cvnsymt () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (br A ( (e. A (CH)) (-. (br A ( (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (/\ (br A ( (/\ (e. A (CH)) (e. B (H~))) (-> (-. (C_ (` (span) ({} B)) A)) (br A ( (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (-> (C_ (cv x) B) (= (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B))))))) ((cv y) A (CH) eleq1 (e. (cv z) (CH)) anbi1d (cv y) A (cv x) (vH) opreq2 (cv z) ineq1d (cv y) A (cv z) ineq1 (cv x) (vH) opreq2d eqeq12d (C_ (cv x) (cv z)) imbi2d x (CH) ralbidv anbi12d (cv z) B (CH) eleq1 (e. A (CH)) anbi2d (cv z) B (cv x) sseq2 (cv z) B (opr (cv x) (vH) A) ineq2 (cv z) B A ineq2 (cv x) (vH) opreq2d eqeq12d imbi12d x (CH) ralbidv anbi12d y z x df-md (CH) (CH) brabg (/\ (e. A (CH)) (e. B (CH))) (A.e. x (CH) (-> (C_ (cv x) B) (= (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B))))) ibar bitr4d)) thm (mdit ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A (MH) B) (-> (C_ C B) (= (i^i (opr C (vH) A) B) (opr C (vH) (i^i A B)))))) (A B x mdbrt biimpd (cv x) C B sseq1 (cv x) C (vH) A opreq1 B ineq1d (cv x) C (vH) (i^i A B) opreq1 eqeq12d imbi12d (CH) rcla4v sylan9 3impa)) thm (mdbr2 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (-> (C_ (cv x) B) (C_ (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B))))))) (A B x mdbrt (C_ (cv x) B) (C_ (cv x) (opr (cv x) (vH) A)) iba (cv x) (opr (cv x) (vH) A) B ssin syl6bb (cv x) A chub1t ancoms syl5bi com12 A (cv x) chub2t A (opr (cv x) (vH) A) B ssrin syl jctird (e. B (CH)) adantlr (cv x) (i^i A B) (i^i (opr (cv x) (vH) A) B) chlubt (/\ (e. A (CH)) (e. B (CH))) (e. (cv x) (CH)) pm3.27 A B chinclt (e. (cv x) (CH)) adantr (opr (cv x) (vH) A) B chinclt (cv x) A chjclt ancoms sylan an1rs syl3anc sylibd (C_ (opr (cv x) (vH) (i^i A B)) (i^i (opr (cv x) (vH) A) B)) (C_ (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B))) iba (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B)) eqss syl6rbbr syl6 pm5.74d ralbidva bitrd)) thm (mdbr3 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (= (i^i (opr (i^i (cv x) B) (vH) A) B) (opr (i^i (cv x) B) (vH) (i^i A B)))))) (A B y mdbrt (cv x) B chinclt (cv x) B inss2 (cv y) (i^i (cv x) B) B sseq1 (cv y) (i^i (cv x) B) (vH) A opreq1 B ineq1d (cv y) (i^i (cv x) B) (vH) (i^i A B) opreq1 eqeq12d imbi12d (CH) rcla4v mpii syl ex com3l r19.21adv (cv x) B dfss biimp (vH) A opreq1d B ineq1d (cv x) B dfss biimp (vH) (i^i A B) opreq1d eqeq12d biimprcd x (CH) r19.20si (cv x) (cv y) B sseq1 (cv x) (cv y) (vH) A opreq1 B ineq1d (cv x) (cv y) (vH) (i^i A B) opreq1 eqeq12d imbi12d (CH) cbvralv sylib (e. B (CH)) a1i impbid (e. A (CH)) adantl bitrd)) thm (mdbr4 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (C_ (i^i (opr (i^i (cv x) B) (vH) A) B) (opr (i^i (cv x) B) (vH) (i^i A B)))))) (A B y mdbr2 (cv x) B chinclt (cv x) B inss2 (cv y) (i^i (cv x) B) B sseq1 (cv y) (i^i (cv x) B) (vH) A opreq1 B ineq1d (cv y) (i^i (cv x) B) (vH) (i^i A B) opreq1 sseq12d imbi12d (CH) rcla4v mpii syl ex com3l r19.21adv (cv x) B dfss biimp (vH) A opreq1d B ineq1d (cv x) B dfss biimp (vH) (i^i A B) opreq1d sseq12d biimprcd x (CH) r19.20si (cv x) (cv y) B sseq1 (cv x) (cv y) (vH) A opreq1 B ineq1d (cv x) (cv y) (vH) (i^i A B) opreq1 sseq12d imbi12d (CH) cbvralv sylib (e. B (CH)) a1i impbid (e. A (CH)) adantl bitrd)) thm (dmdbrt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH*) B) (A.e. x (CH) (-> (C_ B (cv x)) (= (opr (i^i (cv x) A) (vH) B) (i^i (cv x) (opr A (vH) B))))))) ((cv y) A (CH) eleq1 (e. (cv z) (CH)) anbi1d (cv y) A (cv x) ineq2 (vH) (cv z) opreq1d (cv y) A (vH) (cv z) opreq1 (cv x) ineq2d eqeq12d (C_ (cv z) (cv x)) imbi2d x (CH) ralbidv anbi12d (cv z) B (CH) eleq1 (e. A (CH)) anbi2d (cv z) B (cv x) sseq1 (cv z) B (i^i (cv x) A) (vH) opreq2 (cv z) B A (vH) opreq2 (cv x) ineq2d eqeq12d imbi12d x (CH) ralbidv anbi12d y z x df-dmd (CH) (CH) brabg (/\ (e. A (CH)) (e. B (CH))) (A.e. x (CH) (-> (C_ B (cv x)) (= (opr (i^i (cv x) A) (vH) B) (i^i (cv x) (opr A (vH) B))))) ibar bitr4d)) thm (dmdmdt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH*) B) (br (` (_|_) A) (MH) (` (_|_) B)))) ((cv x) chocclt (-> (C_ (` (_|_) (cv x)) (` (_|_) B)) (= (i^i (opr (` (_|_) (cv x)) (vH) (` (_|_) A)) (` (_|_) B)) (opr (` (_|_) (cv x)) (vH) (i^i (` (_|_) A) (` (_|_) B))))) imim1i com12 (/\ (e. A (CH)) (e. B (CH))) adantl B (cv x) chsscon3t biimpd (e. A (CH)) adantll (opr (` (_|_) (cv x)) (vH) (` (_|_) A)) B chdmm3t (` (_|_) (cv x)) (` (_|_) A) chjclt (cv x) chocclt A chocclt syl2an sylan (cv x) A chdmj4t (e. B (CH)) adantr (vH) B opreq1d eqtrd anasss (cv x) (i^i (` (_|_) A) (` (_|_) B)) chdmj2t (` (_|_) A) (` (_|_) B) chinclt A chocclt B chocclt syl2an sylan2 A B chdmm4t (e. (cv x) (CH)) adantl (cv x) ineq2d eqtrd eqeq12d ancoms (i^i (opr (` (_|_) (cv x)) (vH) (` (_|_) A)) (` (_|_) B)) (opr (` (_|_) (cv x)) (vH) (i^i (` (_|_) A) (` (_|_) B))) (_|_) fveq2 syl5bi imim12d syld ex com23 (cv y) (` (_|_) (cv x)) (` (_|_) B) sseq1 (cv y) (` (_|_) (cv x)) (vH) (` (_|_) A) opreq1 (` (_|_) B) ineq1d (cv y) (` (_|_) (cv x)) (vH) (i^i (` (_|_) A) (` (_|_) B)) opreq1 eqeq12d imbi12d (CH) rcla4cv syl5 r19.21adv (cv y) chocclt (-> (C_ B (` (_|_) (cv y))) (= (opr (i^i (` (_|_) (cv y)) A) (vH) B) (i^i (` (_|_) (cv y)) (opr A (vH) B)))) imim1i com12 (/\ (e. A (CH)) (e. B (CH))) adantl B (cv y) chsscon2t biimprd (e. A (CH)) adantll (i^i (` (_|_) (cv y)) A) B chdmj1t (` (_|_) (cv y)) A chinclt (cv y) chocclt sylan sylan (cv y) A chdmm2t (e. B (CH)) adantr (` (_|_) B) ineq1d eqtrd anasss (cv y) (opr A (vH) B) chdmm2t A B chjclt sylan2 A B chdmj1t (e. (cv y) (CH)) adantl (cv y) (vH) opreq2d eqtrd eqeq12d ancoms (opr (i^i (` (_|_) (cv y)) A) (vH) B) (i^i (` (_|_) (cv y)) (opr A (vH) B)) (_|_) fveq2 syl5bi imim12d syld ex com23 (cv x) (` (_|_) (cv y)) B sseq2 (cv x) (` (_|_) (cv y)) A ineq1 (vH) B opreq1d (cv x) (` (_|_) (cv y)) (opr A (vH) B) ineq1 eqeq12d imbi12d (CH) rcla4cv syl5 r19.21adv impbid (` (_|_) A) (` (_|_) B) y mdbrt A chocclt B chocclt syl2an A B x dmdbrt 3bitr4rd)) thm (dmdbrtOLD ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (-> (C_ B (cv x)) (= (opr (i^i (cv x) A) (vH) B) (i^i (cv x) (opr A (vH) B))))))) ((` (_|_) A) (` (_|_) B) y mdbrt A chocclt B chocclt syl2an (cv x) chocclt (-> (C_ (` (_|_) (cv x)) (` (_|_) B)) (= (i^i (opr (` (_|_) (cv x)) (vH) (` (_|_) A)) (` (_|_) B)) (opr (` (_|_) (cv x)) (vH) (i^i (` (_|_) A) (` (_|_) B))))) imim1i com12 (/\ (e. A (CH)) (e. B (CH))) adantl B (cv x) chsscon3t biimpd (e. A (CH)) adantll (opr (` (_|_) (cv x)) (vH) (` (_|_) A)) B chdmm3t (` (_|_) (cv x)) (` (_|_) A) chjclt (cv x) chocclt A chocclt syl2an sylan (cv x) A chdmj4t (e. B (CH)) adantr (vH) B opreq1d eqtrd anasss (cv x) (i^i (` (_|_) A) (` (_|_) B)) chdmj2t (` (_|_) A) (` (_|_) B) chinclt A chocclt B chocclt syl2an sylan2 A B chdmm4t (e. (cv x) (CH)) adantl (cv x) ineq2d eqtrd eqeq12d ancoms (i^i (opr (` (_|_) (cv x)) (vH) (` (_|_) A)) (` (_|_) B)) (opr (` (_|_) (cv x)) (vH) (i^i (` (_|_) A) (` (_|_) B))) (_|_) fveq2 syl5bi imim12d syld ex com23 (cv y) (` (_|_) (cv x)) (` (_|_) B) sseq1 (cv y) (` (_|_) (cv x)) (vH) (` (_|_) A) opreq1 (` (_|_) B) ineq1d (cv y) (` (_|_) (cv x)) (vH) (i^i (` (_|_) A) (` (_|_) B)) opreq1 eqeq12d imbi12d (CH) rcla4cv syl5 r19.21adv (cv y) chocclt (-> (C_ B (` (_|_) (cv y))) (= (opr (i^i (` (_|_) (cv y)) A) (vH) B) (i^i (` (_|_) (cv y)) (opr A (vH) B)))) imim1i com12 (/\ (e. A (CH)) (e. B (CH))) adantl B (cv y) chsscon2t biimprd (e. A (CH)) adantll (i^i (` (_|_) (cv y)) A) B chdmj1t (` (_|_) (cv y)) A chinclt (cv y) chocclt sylan sylan (cv y) A chdmm2t (e. B (CH)) adantr (` (_|_) B) ineq1d eqtrd anasss (cv y) (opr A (vH) B) chdmm2t A B chjclt sylan2 A B chdmj1t (e. (cv y) (CH)) adantl (cv y) (vH) opreq2d eqtrd eqeq12d ancoms (opr (i^i (` (_|_) (cv y)) A) (vH) B) (i^i (` (_|_) (cv y)) (opr A (vH) B)) (_|_) fveq2 syl5bi imim12d syld ex com23 (cv x) (` (_|_) (cv y)) B sseq2 (cv x) (` (_|_) (cv y)) A ineq1 (vH) B opreq1d (cv x) (` (_|_) (cv y)) (opr A (vH) B) ineq1 eqeq12d imbi12d (CH) rcla4cv syl5 r19.21adv impbid bitrd)) thm (mddmdt () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (br (` (_|_) A) (MH*) (` (_|_) B)))) ((` (_|_) A) (` (_|_) B) dmdmdt A chocclt B chocclt syl2an A ococt B ococt (MH) breqan12d bitr2d)) thm (dmdit ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A (MH*) B) (-> (C_ B C) (= (opr (i^i C A) (vH) B) (i^i C (opr A (vH) B)))))) (A B x dmdbrt biimpd (cv x) C B sseq2 (cv x) C A ineq1 (vH) B opreq1d (cv x) C (opr A (vH) B) ineq1 eqeq12d imbi12d (CH) rcla4v sylan9 3impa)) thm (dmditOLD ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br (` (_|_) A) (MH) (` (_|_) B)) (-> (C_ B C) (= (opr (i^i C A) (vH) B) (i^i C (opr A (vH) B)))))) (A B x dmdbrtOLD biimpd (cv x) C B sseq2 (cv x) C A ineq1 (vH) B opreq1d (cv x) C (opr A (vH) B) ineq1 eqeq12d imbi12d (CH) rcla4v sylan9 3impa)) thm (dmdbr2 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (-> (C_ B (cv x)) (C_ (i^i (cv x) (opr A (vH) B)) (opr (i^i (cv x) A) (vH) B)))))) (A B x dmdbrtOLD (cv x) A inss1 (i^i (cv x) A) B (cv x) chlubt biimpd mpani (cv x) A chinclt ancoms (e. B (CH)) adantlr (e. A (CH)) (e. B (CH)) pm3.27 (e. (cv x) (CH)) adantr (/\ (e. A (CH)) (e. B (CH))) (e. (cv x) (CH)) pm3.27 syl3anc (cv x) A inss2 (i^i (cv x) A) A B chlej1t mpi (cv x) A chinclt ancoms (e. B (CH)) adantlr (e. A (CH)) (e. B (CH)) pm3.26 (e. (cv x) (CH)) adantr (e. A (CH)) (e. B (CH)) pm3.27 (e. (cv x) (CH)) adantr syl3anc jctird (opr (i^i (cv x) A) (vH) B) (cv x) (opr A (vH) B) ssin syl6ib (opr (i^i (cv x) A) (vH) B) (i^i (cv x) (opr A (vH) B)) eqss baib syl6 pm5.74d ralbidva bitrd)) thm (dmdi2 () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br (` (_|_) A) (MH) (` (_|_) B)) (-> (C_ B C) (C_ (i^i C (opr A (vH) B)) (opr (i^i C A) (vH) B))))) (A B C dmditOLD (opr (i^i C A) (vH) B) (i^i C (opr A (vH) B)) eqimss2 syl8)) thm (dmdbr3 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (= (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (i^i (opr (cv x) (vH) B) (opr A (vH) B)))))) (A B y dmdbrtOLD B (cv x) chub2t ancoms (cv x) B chjclt (cv y) (opr (cv x) (vH) B) B sseq2 (cv y) (opr (cv x) (vH) B) A ineq1 (vH) B opreq1d (cv y) (opr (cv x) (vH) B) (opr A (vH) B) ineq1 eqeq12d imbi12d (CH) rcla4v syl mpid ex com3l r19.21adv B (cv x) chlejb2t biimpa A ineq1d (vH) B opreq1d B (cv x) chlejb2t biimpa (opr A (vH) B) ineq1d eqeq12d biimpd ex com23 r19.20dva (cv x) (cv y) B sseq2 (cv x) (cv y) A ineq1 (vH) B opreq1d (cv x) (cv y) (opr A (vH) B) ineq1 eqeq12d imbi12d (CH) cbvralv syl6ib impbid (e. A (CH)) adantl bitrd)) thm (dmdbr4 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))))) (A B y dmdbr2 B (cv x) chub2t ancoms (cv x) B chjclt (cv y) (opr (cv x) (vH) B) B sseq2 (cv y) (opr (cv x) (vH) B) (opr A (vH) B) ineq1 (cv y) (opr (cv x) (vH) B) A ineq1 (vH) B opreq1d sseq12d imbi12d (CH) rcla4v syl mpid ex com3l r19.21adv B (cv x) chlejb2t biimpa (opr A (vH) B) ineq1d B (cv x) chlejb2t biimpa A ineq1d (vH) B opreq1d sseq12d biimpd ex com23 r19.20dva (cv x) (cv y) B sseq2 (cv x) (cv y) (opr A (vH) B) ineq1 (cv x) (cv y) A ineq1 (vH) B opreq1d sseq12d imbi12d (CH) cbvralv syl6ib impbid (e. A (CH)) adantl bitrd)) thm (dmdi4 ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br (` (_|_) A) (MH) (` (_|_) B)) (C_ (i^i (opr C (vH) B) (opr A (vH) B)) (opr (i^i (opr C (vH) B) A) (vH) B)))) (A B x dmdbr4 biimpd (cv x) C (vH) B opreq1 (opr A (vH) B) ineq1d (cv x) C (vH) B opreq1 A ineq1d (vH) B opreq1d sseq12d (CH) rcla4v sylan9 3impa)) thm (dmdbr5 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (-> (C_ (cv x) (opr A (vH) B)) (C_ (cv x) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)))))) (A B x dmdbr4 (cv x) (opr (cv x) (vH) B) (opr A (vH) B) ssin (cv x) (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B) sstr2 sylbi (cv x) B chub1t ancoms sylan ex com23 r19.20dva (e. A (CH)) adantl sylbid (cv y) B chjclt ancoms (e. A (CH)) adantll A B chjclt (e. (cv y) (CH)) adantr jca (opr (cv y) (vH) B) (opr A (vH) B) chinclt (opr (cv y) (vH) B) (opr A (vH) B) inss2 (e. (i^i (opr (cv y) (vH) B) (opr A (vH) B)) (CH)) (-> (C_ (i^i (opr (cv y) (vH) B) (opr A (vH) B)) (opr A (vH) B)) (C_ (i^i (opr (cv y) (vH) B) (opr A (vH) B)) (opr (i^i (opr (i^i (opr (cv y) (vH) B) (opr A (vH) B)) (vH) B) A) (vH) B))) pm2.27 mpii 3syl B (cv y) chub2t (e. A (CH)) adantll B A chub2t ancoms (e. (cv y) (CH)) adantr jca B (opr (cv y) (vH) B) (opr A (vH) B) ssin sylib B (i^i (opr (cv y) (vH) B) (opr A (vH) B)) chlejb2t (e. A (CH)) (e. B (CH)) pm3.27 (e. (cv y) (CH)) adantr (opr (cv y) (vH) B) (opr A (vH) B) chinclt (cv y) B chjclt ancoms (e. A (CH)) adantll A B chjclt (e. (cv y) (CH)) adantr sylanc sylanc mpbid A ineq1d A B chabs2t (opr A (vH) B) A incom syl5eq (opr (cv y) (vH) B) ineq2d (opr (cv y) (vH) B) (opr A (vH) B) A inass syl5eq (e. (cv y) (CH)) adantr eqtrd (vH) B opreq1d (i^i (opr (cv y) (vH) B) (opr A (vH) B)) sseq2d sylibd ex com23 (cv x) (i^i (opr (cv y) (vH) B) (opr A (vH) B)) (opr A (vH) B) sseq1 (= (cv x) (i^i (opr (cv y) (vH) B) (opr A (vH) B))) id (cv x) (i^i (opr (cv y) (vH) B) (opr A (vH) B)) (vH) B opreq1 A ineq1d (vH) B opreq1d sseq12d imbi12d (CH) rcla4cv syl5 r19.21adv A B y dmdbr4 sylibrd impbid)) thm (mddmd ((x y) (A x) (A y)) () (-> (e. A (CH)) (<-> (A.e. x (CH) (br A (MH) (cv x))) (A.e. x (CH) (br (` (_|_) A) (MH) (` (_|_) (cv x)))))) (A (cv y) x mdbrt A (cv x) chjcomt (cv y) ineq1d (opr A (vH) (cv x)) (cv y) incom syl5reqr (e. (cv y) (CH)) adantlr (i^i A (cv y)) (cv x) chjcomt A (cv y) chinclt sylan A (cv y) incom (vH) (cv x) opreq1i syl5reqr eqeq12d (i^i (cv y) (opr A (vH) (cv x))) (opr (i^i (cv y) A) (vH) (cv x)) eqcom syl6bb (C_ (cv x) (cv y)) imbi2d ralbidva bitrd ralbidva (cv x) (cv y) A (MH) breq2 (CH) cbvralv syl5bb y (CH) x (CH) (-> (C_ (cv x) (cv y)) (= (opr (i^i (cv y) A) (vH) (cv x)) (i^i (cv y) (opr A (vH) (cv x))))) ralcom syl6bb A (cv x) y dmdbrtOLD ralbidva bitr4d)) thm (mdsl0 ((A x) (B x) (C x) (D x)) () (-> (/\ (/\ (e. A (CH)) (e. B (CH))) (/\ (e. C (CH)) (e. D (CH)))) (-> (/\ (/\ (/\ (C_ C A) (C_ D B)) (= (i^i A B) (0H))) (br A (MH) B)) (br C (MH) D))) ((cv x) D B sstr2 com12 (C_ C A) (= (i^i A B) (0H)) ad2antlr (/\ (/\ (e. A (CH)) (e. B (CH))) (/\ (e. C (CH)) (e. D (CH)))) (e. (cv x) (CH)) ad2antlr (i^i (opr (cv x) (vH) C) D) (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B)) sstr2 (i^i (opr (cv x) (vH) C) D) (opr (cv x) (vH) (i^i A B)) (opr (cv x) (vH) (i^i C D)) sstr2 syl6 com23 C A (cv x) chlej2t 3expa (opr (cv x) (vH) C) (opr (cv x) (vH) A) D B ss2in ex syl6 ex ancoms (e. B (CH)) (e. D (CH)) ad2ant2r imp43 (= (i^i A B) (0H)) adantrr (i^i A B) (0H) (cv x) (vH) opreq2 (cv x) chj0t sylan9eqr (/\ (e. C (CH)) (e. D (CH))) adantl (cv x) (i^i C D) chub1t ancoms C D chinclt sylan (= (i^i A B) (0H)) adantrr eqsstrd (/\ (e. A (CH)) (e. B (CH))) adantll anassrs (/\ (C_ C A) (C_ D B)) adantrl sylc an1rs imim12d r19.20dva A B x mdbr2 (/\ (e. C (CH)) (e. D (CH))) (/\ (/\ (C_ C A) (C_ D B)) (= (i^i A B) (0H))) ad2antrr C D x mdbr2 (/\ (e. A (CH)) (e. B (CH))) (/\ (/\ (C_ C A) (C_ D B)) (= (i^i A B) (0H))) ad2antlr 3imtr4d ex imp3a)) thm (ssmd1 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br A (MH) B))) (A B x mdbr2 (opr (cv x) (vH) A) B inss1 (C_ A B) a1i A B dfss biimp (cv x) (vH) opreq2d sseqtrd (C_ (cv x) B) a1d (e. (cv x) (CH)) a1d r19.21aiv syl5bir)) thm (ssmd2 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br B (MH) A))) ((opr (cv x) (vH) B) A inss2 (/\ (e. A (CH)) (e. (cv x) (CH))) a1i A (cv x) chub2t sstrd (C_ A B) adantrl (C_ A B) (e. (cv x) (CH)) pm3.26 A B sseqin2 sylib (e. A (CH)) adantl (cv x) (vH) opreq2d sseqtr4d (C_ (cv x) A) a1d exp32 r19.21adv (e. B (CH)) adantr B A x mdbr2 ancoms sylibrd)) thm (ssdmd1 () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br (` (_|_) A) (MH) (` (_|_) B)))) (A B chsscon3t (` (_|_) B) (` (_|_) A) ssmd2 B chocclt A chocclt syl2an ancoms sylbid)) thm (ssdmd2 () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br (` (_|_) B) (MH) (` (_|_) A)))) (A B chsscon3t (` (_|_) B) (` (_|_) A) ssmd1 B chocclt A chocclt syl2an ancoms sylbid)) thm (dmdsl3t () () (-> (/\ (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (/\/\ (br B (MH*) A) (C_ A C) (C_ C (opr A (vH) B)))) (= (opr (i^i C B) (vH) A) C)) (B A C dmdit 3com12 imp32 (C_ C (opr A (vH) B)) 3adantr3 A B chjcomt C ineq2d (e. C (CH)) 3adant3 (C_ C (opr A (vH) B)) adantr C (opr A (vH) B) df-ss biimp (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) adantl eqtr3d (C_ A C) adantrl (br B (MH*) A) 3adantr1 eqtrd)) thm (mdsl3t () () (-> (/\ (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (/\/\ (br A (MH) B) (C_ (i^i A B) C) (C_ C B))) (= (i^i (opr C (vH) A) B) C)) (A B C mdit imp32 (C_ (i^i A B) C) 3adantr2 (i^i A B) C chlejb2t A B chinclt sylan 3impa biimpa (C_ C B) adantrr (br A (MH) B) 3adantr1 eqtrd)) thm (mdslle1 () ((mdslle1.1 (e. A (CH))) (mdslle1.2 (e. B (CH))) (mdslle1.3 (e. C (CH))) (mdslle1.4 (e. D (CH)))) (-> (/\/\ (br B (MH*) A) (C_ A (i^i C D)) (C_ (opr C (vH) D) (opr A (vH) B))) (<-> (C_ C D) (C_ (i^i C B) (i^i D B)))) (C D B ssrin (/\/\ (br B (MH*) A) (C_ A (i^i C D)) (C_ (opr C (vH) D) (opr A (vH) B))) a1i mdslle1.1 mdslle1.2 mdslle1.3 3pm3.2i A B C dmdsl3t mpan (br B (MH*) A) id A C D ssin bicomi pm3.26bd mdslle1.3 mdslle1.4 mdslle1.1 mdslle1.2 chjcl chlub bicomi pm3.26bd syl3an mdslle1.1 mdslle1.2 mdslle1.4 3pm3.2i A B D dmdsl3t mpan (br B (MH*) A) id A C D ssin bicomi pm3.27bd mdslle1.3 mdslle1.4 mdslle1.1 mdslle1.2 chjcl chlub bicomi pm3.27bd syl3an sseq12d mdslle1.3 mdslle1.2 chincl mdslle1.4 mdslle1.2 chincl mdslle1.1 chlej1 syl5bi impbid)) thm (mdslj1 () ((mdslle1.1 (e. A (CH))) (mdslle1.2 (e. B (CH))) (mdslle1.3 (e. C (CH))) (mdslle1.4 (e. D (CH)))) (-> (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ A (i^i C D)) (C_ (opr C (vH) D) (opr A (vH) B)))) (= (i^i (opr C (vH) D) B) (opr (i^i C B) (vH) (i^i D B)))) (mdslle1.1 mdslle1.2 mdslle1.3 3pm3.2i A B C dmdsl3t mpan (br A (MH) B) (br B (MH*) A) pm3.27 (C_ A C) (C_ A D) pm3.26 (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)) pm3.26 syl3an mdslle1.3 mdslle1.2 chincl mdslle1.4 mdslle1.2 chincl chub1 mdslle1.3 mdslle1.2 chincl mdslle1.3 mdslle1.2 chincl mdslle1.4 mdslle1.2 chincl chjcl mdslle1.1 chlej1 ax-mp (/\/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)))) a1i eqsstr3d mdslle1.1 mdslle1.2 mdslle1.4 3pm3.2i A B D dmdsl3t mpan (br A (MH) B) (br B (MH*) A) pm3.27 (C_ A C) (C_ A D) pm3.27 (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)) pm3.27 syl3an mdslle1.4 mdslle1.2 chincl mdslle1.3 mdslle1.2 chincl chub2 mdslle1.4 mdslle1.2 chincl mdslle1.3 mdslle1.2 chincl mdslle1.4 mdslle1.2 chincl chjcl mdslle1.1 chlej1 ax-mp (/\/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)))) a1i eqsstr3d jca mdslle1.3 mdslle1.4 mdslle1.3 mdslle1.2 chincl mdslle1.4 mdslle1.2 chincl chjcl mdslle1.1 chjcl chlub sylib (opr C (vH) D) (opr (opr (i^i C B) (vH) (i^i D B)) (vH) A) B ssrin syl mdslle1.1 mdslle1.2 mdslle1.3 mdslle1.2 chincl mdslle1.4 mdslle1.2 chincl chjcl 3pm3.2i A B (opr (i^i C B) (vH) (i^i D B)) mdsl3t mpan (br A (MH) B) (br B (MH*) A) pm3.26 A C B ssrin mdslle1.3 mdslle1.2 chincl mdslle1.4 mdslle1.2 chincl chub1 (i^i A B) (i^i C B) (opr (i^i C B) (vH) (i^i D B)) sstr mpan2 syl (C_ A D) adantr mdslle1.3 mdslle1.2 chincl mdslle1.4 mdslle1.2 chincl mdslle1.2 chlub bicomi C B inss2 D B inss2 mpbir2an (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) a1i syl3an sseqtrd 3expb A C D ssin bicomi mdslle1.3 mdslle1.4 mdslle1.1 mdslle1.2 chjcl chlub bicomi anbi12i sylan2b mdslle1.3 mdslle1.4 mdslle1.2 ledir (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ A (i^i C D)) (C_ (opr C (vH) D) (opr A (vH) B)))) a1i eqssd)) thm (mdslj2 () ((mdslle1.1 (e. A (CH))) (mdslle1.2 (e. B (CH))) (mdslle1.3 (e. C (CH))) (mdslle1.4 (e. D (CH)))) (-> (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i A B) (i^i C D)) (C_ (opr C (vH) D) B))) (= (opr (i^i C D) (vH) A) (i^i (opr C (vH) A) (opr D (vH) A)))) (mdslle1.3 mdslle1.4 mdslle1.1 lejdir (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i A B) (i^i C D)) (C_ (opr C (vH) D) B))) a1i mdslle1.1 mdslle1.2 mdslle1.3 mdslle1.1 chjcl mdslle1.4 mdslle1.1 chjcl chincl 3pm3.2i A B (i^i (opr C (vH) A) (opr D (vH) A)) dmdsl3t mpan (br A (MH) B) (br B (MH*) A) pm3.27 mdslle1.1 mdslle1.3 chub2 mdslle1.1 mdslle1.4 chub2 ssini (/\ (C_ (i^i A B) C) (C_ (i^i A B) D)) a1i mdslle1.3 mdslle1.2 mdslle1.1 chlej1 mdslle1.2 mdslle1.1 chjcom syl6ss (opr C (vH) A) (opr A (vH) B) (opr D (vH) A) ssinss1 syl (C_ D B) adantr syl3an (opr C (vH) A) (opr D (vH) A) inss1 (i^i (opr C (vH) A) (opr D (vH) A)) (opr C (vH) A) B ssrin ax-mp (/\/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i A B) C) (C_ (i^i A B) D)) (/\ (C_ C B) (C_ D B))) a1i mdslle1.1 mdslle1.2 mdslle1.3 3pm3.2i A B C mdsl3t mpan (br A (MH) B) (br B (MH*) A) pm3.26 (C_ (i^i A B) C) (C_ (i^i A B) D) pm3.26 (C_ C B) (C_ D B) pm3.26 syl3an sseqtrd (opr C (vH) A) (opr D (vH) A) inss2 (i^i (opr C (vH) A) (opr D (vH) A)) (opr D (vH) A) B ssrin ax-mp (/\/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i A B) C) (C_ (i^i A B) D)) (/\ (C_ C B) (C_ D B))) a1i mdslle1.1 mdslle1.2 mdslle1.4 3pm3.2i A B D mdsl3t mpan (br A (MH) B) (br B (MH*) A) pm3.26 (C_ (i^i A B) C) (C_ (i^i A B) D) pm3.27 (C_ C B) (C_ D B) pm3.27 syl3an sseqtrd jca (i^i (i^i (opr C (vH) A) (opr D (vH) A)) B) C D ssin sylib mdslle1.3 mdslle1.1 chjcl mdslle1.4 mdslle1.1 chjcl chincl mdslle1.2 chincl mdslle1.3 mdslle1.4 chincl mdslle1.1 chlej1 syl eqsstr3d 3expb (i^i A B) C D ssin bicomi mdslle1.3 mdslle1.4 mdslle1.2 chlub bicomi anbi12i sylan2b eqssd)) thm (mdsl1 ((x y) (A x) (A y) (B x) (B y)) ((mdsl.1 (e. A (CH))) (mdsl.2 (e. B (CH)))) (<-> (A.e. x (CH) (-> (/\ (C_ (i^i A B) (cv x)) (C_ (cv x) (opr A (vH) B))) (-> (C_ (cv x) B) (= (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B)))))) (br A (MH) B)) (A B inss2 mdsl.1 mdsl.2 chincl mdsl.2 (cv y) (i^i A B) B chlubt mp3an23 biimpd mpan2i mdsl.2 mdsl.1 chub2 (opr (cv y) (vH) (i^i A B)) B (opr A (vH) B) sstr mpan2 syl6 mdsl.1 mdsl.2 chincl (i^i A B) (cv y) chub2t mpan jctild mdsl.1 mdsl.2 chincl (cv y) (i^i A B) chjclt mpan2 jctild A B inss2 mdsl.1 mdsl.2 chincl mdsl.2 (cv y) (i^i A B) B chlubt mp3an23 biimpd mpan2i jcad mdsl.1 mdsl.2 chincl mdsl.1 (cv y) (i^i A B) A chjasst mp3an23 mdsl.1 mdsl.2 chincl mdsl.1 chjcom mdsl.1 mdsl.2 chabs1 eqtr (cv y) (vH) opreq2i syl6eq B ineq1d mdsl.1 mdsl.2 chincl mdsl.1 mdsl.2 chincl (cv y) (i^i A B) (i^i A B) chjasst mp3an23 mdsl.1 mdsl.2 chincl chjidm (cv y) (vH) opreq2i syl6eq eqeq12d biimpd imim12d (/\ (e. (opr (cv y) (vH) (i^i A B)) (CH)) (/\ (C_ (i^i A B) (opr (cv y) (vH) (i^i A B))) (C_ (opr (cv y) (vH) (i^i A B)) (opr A (vH) B)))) (C_ (opr (cv y) (vH) (i^i A B)) B) (= (i^i (opr (opr (cv y) (vH) (i^i A B)) (vH) A) B) (opr (opr (cv y) (vH) (i^i A B)) (vH) (i^i A B))) impexp (e. (opr (cv y) (vH) (i^i A B)) (CH)) (/\ (C_ (i^i A B) (opr (cv y) (vH) (i^i A B))) (C_ (opr (cv y) (vH) (i^i A B)) (opr A (vH) B))) (-> (C_ (opr (cv y) (vH) (i^i A B)) B) (= (i^i (opr (opr (cv y) (vH) (i^i A B)) (vH) A) B) (opr (opr (cv y) (vH) (i^i A B)) (vH) (i^i A B)))) impexp bitr2 syl5ib (cv x) (opr (cv y) (vH) (i^i A B)) (i^i A B) sseq2 (cv x) (opr (cv y) (vH) (i^i A B)) (opr A (vH) B) sseq1 anbi12d (cv x) (opr (cv y) (vH) (i^i A B)) B sseq1 (cv x) (opr (cv y) (vH) (i^i A B)) (vH) A opreq1 B ineq1d (cv x) (opr (cv y) (vH) (i^i A B)) (vH) (i^i A B) opreq1 eqeq12d imbi12d imbi12d (CH) rcla4cv syl5com r19.21aiv mdsl.1 mdsl.2 A B y mdbrt mp2an sylibr mdsl.1 mdsl.2 A B x mdbrt mp2an (-> (C_ (cv x) B) (= (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B)))) (/\ (C_ (i^i A B) (cv x)) (C_ (cv x) (opr A (vH) B))) ax-1 x (CH) r19.20si sylbi impbi)) thm (mdsl2 ((A x) (B x)) ((mdsl.1 (e. A (CH))) (mdsl.2 (e. B (CH)))) (<-> (br A (MH) B) (A.e. x (CH) (-> (/\ (C_ (i^i A B) (cv x)) (C_ (cv x) B)) (C_ (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B)))))) ((C_ (cv x) B) (C_ (cv x) (opr (cv x) (vH) A)) iba (cv x) (opr (cv x) (vH) A) B ssin syl6bb mdsl.1 (cv x) A chub1t mpan2 syl5bi com12 mdsl.1 A (cv x) chub2t mpan A (opr (cv x) (vH) A) B ssrin syl jctird mdsl.1 (cv x) A chjclt mpan2 mdsl.2 (opr (cv x) (vH) A) B chinclt mpan2 syl mdsl.1 mdsl.2 chincl (cv x) (i^i A B) (i^i (opr (cv x) (vH) A) B) chlubt mp3an2 mpdan sylibd (C_ (opr (cv x) (vH) (i^i A B)) (i^i (opr (cv x) (vH) A) B)) (C_ (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B))) iba (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B)) eqss syl6rbbr syl6 (C_ (i^i A B) (cv x)) adantld pm5.74d mdsl.2 mdsl.1 chub2 (cv x) B (opr A (vH) B) sstr mpan2 pm4.71ri (C_ (i^i A B) (cv x)) anbi2i (C_ (i^i A B) (cv x)) (C_ (cv x) (opr A (vH) B)) (C_ (cv x) B) anass bitr4 (= (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B))) imbi1i syl5rbbr (/\ (C_ (i^i A B) (cv x)) (C_ (cv x) (opr A (vH) B))) (C_ (cv x) B) (= (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B))) impexp syl6bb ralbiia mdsl.1 mdsl.2 x mdsl1 bitr2)) thm (mdsl1OLD ((x y) (A x) (A y) (B x) (B y)) ((mdsl.1 (e. A (CH))) (mdsl.2 (e. B (CH)))) (-> (A.e. x (CH) (-> (/\ (C_ (i^i A B) (cv x)) (C_ (cv x) (opr A (vH) B))) (-> (C_ (cv x) B) (= (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B)))))) (br A (MH) B)) (A B inss2 mdsl.1 mdsl.2 chincl mdsl.2 (cv y) (i^i A B) B chlubt mp3an23 biimpd mpan2i mdsl.2 mdsl.1 chub2 (opr (cv y) (vH) (i^i A B)) B (opr A (vH) B) sstr mpan2 syl6 mdsl.1 mdsl.2 chincl (i^i A B) (cv y) chub2t mpan jctild mdsl.1 mdsl.2 chincl (cv y) (i^i A B) chjclt mpan2 jctild A B inss2 mdsl.1 mdsl.2 chincl mdsl.2 (cv y) (i^i A B) B chlubt mp3an23 biimpd mpan2i jcad mdsl.1 mdsl.2 chincl mdsl.1 (cv y) (i^i A B) A chjasst mp3an23 mdsl.1 mdsl.2 chincl mdsl.1 chjcom mdsl.1 mdsl.2 chabs1 eqtr (cv y) (vH) opreq2i syl6eq B ineq1d mdsl.1 mdsl.2 chincl mdsl.1 mdsl.2 chincl (cv y) (i^i A B) (i^i A B) chjasst mp3an23 mdsl.1 mdsl.2 chincl chjidm (cv y) (vH) opreq2i syl6eq eqeq12d biimpd imim12d (/\ (e. (opr (cv y) (vH) (i^i A B)) (CH)) (/\ (C_ (i^i A B) (opr (cv y) (vH) (i^i A B))) (C_ (opr (cv y) (vH) (i^i A B)) (opr A (vH) B)))) (C_ (opr (cv y) (vH) (i^i A B)) B) (= (i^i (opr (opr (cv y) (vH) (i^i A B)) (vH) A) B) (opr (opr (cv y) (vH) (i^i A B)) (vH) (i^i A B))) impexp (e. (opr (cv y) (vH) (i^i A B)) (CH)) (/\ (C_ (i^i A B) (opr (cv y) (vH) (i^i A B))) (C_ (opr (cv y) (vH) (i^i A B)) (opr A (vH) B))) (-> (C_ (opr (cv y) (vH) (i^i A B)) B) (= (i^i (opr (opr (cv y) (vH) (i^i A B)) (vH) A) B) (opr (opr (cv y) (vH) (i^i A B)) (vH) (i^i A B)))) impexp bitr2 syl5ib (cv x) (opr (cv y) (vH) (i^i A B)) (i^i A B) sseq2 (cv x) (opr (cv y) (vH) (i^i A B)) (opr A (vH) B) sseq1 anbi12d (cv x) (opr (cv y) (vH) (i^i A B)) B sseq1 (cv x) (opr (cv y) (vH) (i^i A B)) (vH) A opreq1 B ineq1d (cv x) (opr (cv y) (vH) (i^i A B)) (vH) (i^i A B) opreq1 eqeq12d imbi12d imbi12d (CH) rcla4cv syl5com r19.21aiv mdsl.1 mdsl.2 A B y mdbrt mp2an sylibr)) thm (cvmd ((A x) (B x)) ((mdsl.1 (e. A (CH))) (mdsl.2 (e. B (CH)))) (-> (br (i^i A B) ( (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) (-> (-> (C_ (opr R (vH) A) D) (C_ (i^i (opr (opr R (vH) A) (vH) C) D) (opr (opr R (vH) A) (vH) (i^i C D)))) (-> (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) (C_ (i^i (opr R (vH) (i^i C B)) (i^i D B)) (opr R (vH) (i^i (i^i C B) (i^i D B))))))) (mdslmd.1 mdslmd.2 mdslmd.4 3pm3.2i A B D dmdsl3t mpan (br A (MH) B) (br B (MH*) A) pm3.27 (C_ A C) (C_ A D) pm3.27 (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)) pm3.27 syl3an 3expb (opr R (vH) A) sseq2d mdslmd1lem.5 mdslmd.4 mdslmd.2 chincl mdslmd.1 chlej1 syl5bi (C_ (i^i (i^i C B) (i^i D B)) R) adantld (C_ (i^i (opr (opr R (vH) A) (vH) C) D) (opr (opr R (vH) A) (vH) (i^i C D))) imim1d mdslmd.1 mdslmd.2 mdslmd1lem.5 mdslmd.1 chjcl mdslmd.3 mdslj1 (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)))) pm3.26 (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) adantr (C_ A C) (C_ A D) pm3.26 (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) adantr (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antlr mdslmd.1 mdslmd1lem.5 chub2 jctil A (opr R (vH) A) C ssin sylib R D (opr A (vH) B) sstr D B inss1 R (i^i D B) D sstr mpan2 sylan ancoms (C_ C (opr A (vH) B)) adantll (/\ (C_ A C) (C_ A D)) adantll (/\ (br A (MH) B) (br B (MH*) A)) (C_ (i^i (i^i C B) (i^i D B)) R) ad2ant2l mdslmd.1 mdslmd.2 chub1 jctir mdslmd1lem.5 mdslmd.1 mdslmd.1 mdslmd.2 chjcl chlub sylib (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)) pm3.26 (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ A C) (C_ A D)) ad2antll (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) adantr jca mdslmd1lem.5 mdslmd.1 chjcl mdslmd.3 mdslmd.1 mdslmd.2 chjcl chlub sylib jca sylanc mdslmd.1 mdslmd.2 mdslmd1lem.5 3pm3.2i A B R mdsl3t mpan (br A (MH) B) (br B (MH*) A) pm3.26 (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)))) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antrr (C_ A C) (C_ A D) pm3.26 (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) adantr (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antlr (C_ A C) (C_ A D) pm3.27 (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) adantr (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antlr jca A C D ssin sylib A (i^i C D) B ssrin syl C D B inindir syl6ss (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B)) pm3.26 (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) adantl sstrd D B inss2 R (i^i D B) B sstr mpan2 (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) (C_ (i^i (i^i C B) (i^i D B)) R) ad2antll syl3anc (vH) (i^i C B) opreq1d eqtr2d (i^i D B) ineq1d (opr (opr R (vH) A) (vH) C) D B inindir syl6eqr mdslmd.1 mdslmd.2 mdslmd1lem.5 mdslmd.1 chjcl mdslmd.3 mdslmd.4 chincl mdslj1 (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)))) pm3.26 (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) adantr (C_ A C) (C_ A D) pm3.26 (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) adantr (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antlr (C_ A C) (C_ A D) pm3.27 (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) adantr (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antlr jca A C D ssin sylib mdslmd.1 mdslmd1lem.5 chub2 jctil A (opr R (vH) A) (i^i C D) ssin sylib R D (opr A (vH) B) sstr D B inss1 R (i^i D B) D sstr mpan2 sylan ancoms (C_ C (opr A (vH) B)) adantll (/\ (C_ A C) (C_ A D)) adantll (/\ (br A (MH) B) (br B (MH*) A)) (C_ (i^i (i^i C B) (i^i D B)) R) ad2ant2l mdslmd.1 mdslmd.2 chub1 jctir mdslmd1lem.5 mdslmd.1 mdslmd.1 mdslmd.2 chjcl chlub sylib C (opr A (vH) B) D ssinss1 (/\ (C_ A C) (C_ A D)) (C_ D (opr A (vH) B)) ad2antrl (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antlr jca mdslmd1lem.5 mdslmd.1 chjcl mdslmd.3 mdslmd.4 chincl mdslmd.1 mdslmd.2 chjcl chlub sylib jca sylanc mdslmd.1 mdslmd.2 mdslmd1lem.5 3pm3.2i A B R mdsl3t mpan (br A (MH) B) (br B (MH*) A) pm3.26 (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)))) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antrr (C_ A C) (C_ A D) pm3.26 (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) adantr (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antlr (C_ A C) (C_ A D) pm3.27 (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) adantr (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antlr jca A C D ssin sylib A (i^i C D) B ssrin syl C D B inindir syl6ss (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B)) pm3.26 (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) adantl sstrd D B inss2 R (i^i D B) B sstr mpan2 (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) (C_ (i^i (i^i C B) (i^i D B)) R) ad2antll syl3anc C D B inindir (/\ (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B)))) a1i (vH) opreq12d eqtr2d sseq12d mdslmd.1 mdslmd.2 mdslmd1lem.5 mdslmd.1 chjcl mdslmd.3 chjcl mdslmd.4 chincl mdslmd1lem.5 mdslmd.1 chjcl mdslmd.3 mdslmd.4 chincl chjcl mdslle1 (br A (MH) B) (br B (MH*) A) pm3.27 (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)))) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antrr (C_ A C) (C_ A D) pm3.27 (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) adantr (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antlr mdslmd.1 mdslmd1lem.5 chub2 mdslmd1lem.5 mdslmd.1 chjcl mdslmd.3 chub1 sstri jctil A (opr (opr R (vH) A) (vH) C) D ssin sylib mdslmd.1 mdslmd1lem.5 chub2 mdslmd1lem.5 mdslmd.1 chjcl mdslmd.3 mdslmd.4 chincl chub1 sstri jctir A (i^i (opr (opr R (vH) A) (vH) C) D) (opr (opr R (vH) A) (vH) (i^i C D)) ssin sylib (opr (opr R (vH) A) (vH) C) D inss2 (i^i (opr (opr R (vH) A) (vH) C) D) D (opr A (vH) B) sstr mpan (/\ (C_ A C) (C_ A D)) (C_ C (opr A (vH) B)) ad2antll (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antlr R D (opr A (vH) B) sstr D B inss1 R (i^i D B) D sstr mpan2 sylan ancoms (C_ C (opr A (vH) B)) adantll (/\ (C_ A C) (C_ A D)) adantll (/\ (br A (MH) B) (br B (MH*) A)) (C_ (i^i (i^i C B) (i^i D B)) R) ad2ant2l mdslmd.1 mdslmd.2 chub1 jctir mdslmd1lem.5 mdslmd.1 mdslmd.1 mdslmd.2 chjcl chlub sylib C (opr A (vH) B) D ssinss1 (/\ (C_ A C) (C_ A D)) (C_ D (opr A (vH) B)) ad2antrl (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i (i^i C B) (i^i D B)) R) (C_ R (i^i D B))) ad2antlr jca mdslmd1lem.5 mdslmd.1 chjcl mdslmd.3 mdslmd.4 chincl mdslmd.1 mdslmd.2 chjcl chlub sylib jca mdslmd1lem.5 mdslmd.1 chjcl mdslmd.3 chjcl mdslmd.4 chincl mdslmd1lem.5 mdslmd.1 chjcl mdslmd.3 mdslmd.4 chincl chjcl mdslmd.1 mdslmd.2 chjcl chlub sylib syl3anc bitr4d biimprd ex a2d syld)) thm (mdslmd1lem2 () ((mdslmd.1 (e. A (CH))) (mdslmd.2 (e. B (CH))) (mdslmd.3 (e. C (CH))) (mdslmd.4 (e. D (CH))) (mdslmd1lem.5 (e. R (CH)))) (-> (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) (-> (-> (C_ (i^i R B) (i^i D B)) (C_ (i^i (opr (i^i R B) (vH) (i^i C B)) (i^i D B)) (opr (i^i R B) (vH) (i^i (i^i C B) (i^i D B))))) (-> (/\ (C_ (i^i C D) R) (C_ R D)) (C_ (i^i (opr R (vH) C) D) (opr R (vH) (i^i C D)))))) (mdslmd.1 mdslmd.2 mdslmd1lem.5 mdslmd.3 chjcl mdslmd.4 chincl mdslmd1lem.5 mdslmd.3 mdslmd.4 chincl chjcl mdslle1 (br A (MH) B) (br B (MH*) A) pm3.27 (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)))) (/\ (C_ (i^i C D) R) (C_ R D)) ad2antrr mdslmd.3 mdslmd1lem.5 chub2 A C (opr R (vH) C) sstr mpan2 (C_ A D) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) ad2antrr (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i C D) R) (C_ R D)) ad2antlr (C_ A C) (C_ A D) pm3.27 (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) adantr (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i C D) R) (C_ R D)) ad2antlr jca A (opr R (vH) C) D ssin sylib A C D ssin mdslmd.3 mdslmd.4 chincl mdslmd1lem.5 chub2 A (i^i C D) (opr R (vH) (i^i C D)) sstr mpan2 sylbi (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) adantr (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i C D) R) (C_ R D)) ad2antlr jca A (i^i (opr R (vH) C) D) (opr R (vH) (i^i C D)) ssin sylib (opr R (vH) C) D inss2 (i^i (opr R (vH) C) D) D (opr A (vH) B) sstr mpan (/\ (C_ A C) (C_ A D)) (C_ C (opr A (vH) B)) ad2antll (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i C D) R) (C_ R D)) ad2antlr R D (opr A (vH) B) sstr ancoms (C_ C (opr A (vH) B)) (C_ (i^i C D) R) ad2ant2l (/\ (C_ A C) (C_ A D)) adantll (/\ (br A (MH) B) (br B (MH*) A)) adantll C (opr A (vH) B) D ssinss1 (/\ (C_ A C) (C_ A D)) (C_ D (opr A (vH) B)) ad2antrl (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i C D) R) (C_ R D)) ad2antlr jca mdslmd1lem.5 mdslmd.3 mdslmd.4 chincl mdslmd.1 mdslmd.2 chjcl chlub sylib jca mdslmd1lem.5 mdslmd.3 chjcl mdslmd.4 chincl mdslmd1lem.5 mdslmd.3 mdslmd.4 chincl chjcl mdslmd.1 mdslmd.2 chjcl chlub sylib syl3anc mdslmd.1 mdslmd.2 mdslmd1lem.5 mdslmd.3 mdslj1 A (i^i C D) R sstr A C D ssin sylanb (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) (C_ R D) ad2ant2r (C_ A C) (C_ A D) pm3.26 (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))) (/\ (C_ (i^i C D) R) (C_ R D)) ad2antrr jca A R C ssin sylib R D (opr A (vH) B) sstr ancoms (C_ C (opr A (vH) B)) (C_ (i^i C D) R) ad2ant2l (/\ (C_ A C) (C_ A D)) adantll (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)) pm3.26 (/\ (C_ A C) (C_ A D)) (/\ (C_ (i^i C D) R) (C_ R D)) ad2antlr jca mdslmd1lem.5 mdslmd.3 mdslmd.1 mdslmd.2 chjcl chlub sylib jca sylan2 anassrs (i^i D B) ineq1d (opr R (vH) C) D B inindir syl5req mdslmd.1 mdslmd.2 mdslmd1lem.5 mdslmd.3 mdslmd.4 chincl mdslj1 A (i^i C D) R sstr A C D ssin sylanb A C D ssin biimp (C_ (i^i C D) R) adantr jca A R (i^i C D) ssin sylib R D (opr A (vH) B) sstr ancoms (C_ C (opr A (vH) B)) adantll C (opr A (vH) B) D ssinss1 (C_ D (opr A (vH) B)) (C_ R D) ad2antrr jca mdslmd1lem.5 mdslmd.3 mdslmd.4 chincl mdslmd.1 mdslmd.2 chjcl chlub sylib anim12i an4s sylan2 anassrs C D B inindir (/\ (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) (/\ (C_ (i^i C D) R) (C_ R D))) a1i (i^i R B) (vH) opreq2d eqtr2d sseq12d bitr4d biimprd ex a2d R D B ssrin (C_ (i^i C D) R) adantl (C_ (i^i (opr (i^i R B) (vH) (i^i C B)) (i^i D B)) (opr (i^i R B) (vH) (i^i (i^i C B) (i^i D B)))) imim1i syl5)) thm (mdslmd1lem3 ((A x) (B x) (C x) (D x)) ((mdslmd.1 (e. A (CH))) (mdslmd.2 (e. B (CH))) (mdslmd.3 (e. C (CH))) (mdslmd.4 (e. D (CH)))) (-> (/\ (e. (cv x) (CH)) (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)))))) (-> (-> (C_ (opr (cv x) (vH) A) D) (C_ (i^i (opr (opr (cv x) (vH) A) (vH) C) D) (opr (opr (cv x) (vH) A) (vH) (i^i C D)))) (-> (/\ (C_ (i^i (i^i C B) (i^i D B)) (cv x)) (C_ (cv x) (i^i D B))) (C_ (i^i (opr (cv x) (vH) (i^i C B)) (i^i D B)) (opr (cv x) (vH) (i^i (i^i C B) (i^i D B))))))) ((cv x) (if (e. (cv x) (CH)) (cv x) (0H)) (vH) A opreq1 D sseq1d (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) (vH) A opreq1 (vH) C opreq1d D ineq1d (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) (vH) A opreq1 (vH) (i^i C D) opreq1d sseq12d imbi12d (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) (i^i (i^i C B) (i^i D B)) sseq2 (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) (i^i D B) sseq1 anbi12d (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) (vH) (i^i C B) opreq1 (i^i D B) ineq1d (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) (vH) (i^i (i^i C B) (i^i D B)) opreq1 sseq12d imbi12d imbi12d (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) imbi2d mdslmd.1 mdslmd.2 mdslmd.3 mdslmd.4 h0elch (cv x) elimel mdslmd1lem1 dedth imp)) thm (mdslmd1lem4 ((A x) (B x) (C x) (D x)) ((mdslmd.1 (e. A (CH))) (mdslmd.2 (e. B (CH))) (mdslmd.3 (e. C (CH))) (mdslmd.4 (e. D (CH)))) (-> (/\ (e. (cv x) (CH)) (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B)))))) (-> (-> (C_ (i^i (cv x) B) (i^i D B)) (C_ (i^i (opr (i^i (cv x) B) (vH) (i^i C B)) (i^i D B)) (opr (i^i (cv x) B) (vH) (i^i (i^i C B) (i^i D B))))) (-> (/\ (C_ (i^i C D) (cv x)) (C_ (cv x) D)) (C_ (i^i (opr (cv x) (vH) C) D) (opr (cv x) (vH) (i^i C D)))))) ((cv x) (if (e. (cv x) (CH)) (cv x) (0H)) B ineq1 (i^i D B) sseq1d (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) B ineq1 (vH) (i^i C B) opreq1d (i^i D B) ineq1d (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) B ineq1 (vH) (i^i (i^i C B) (i^i D B)) opreq1d sseq12d imbi12d (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) (i^i C D) sseq2 (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) D sseq1 anbi12d (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) (vH) C opreq1 D ineq1d (cv x) (if (e. (cv x) (CH)) (cv x) (0H)) (vH) (i^i C D) opreq1 sseq12d imbi12d imbi12d (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) imbi2d mdslmd.1 mdslmd.2 mdslmd.3 mdslmd.4 h0elch (cv x) elimel mdslmd1lem2 dedth imp)) thm (mdslmd1 ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) ((mdslmd.1 (e. A (CH))) (mdslmd.2 (e. B (CH))) (mdslmd.3 (e. C (CH))) (mdslmd.4 (e. D (CH)))) (-> (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ A (i^i C D)) (C_ (opr C (vH) D) (opr A (vH) B)))) (<-> (br C (MH) D) (br (i^i C B) (MH) (i^i D B)))) (mdslmd.1 (cv x) A chjclt mpan2 (cv y) (opr (cv x) (vH) A) D sseq1 (cv y) (opr (cv x) (vH) A) (vH) C opreq1 D ineq1d (cv y) (opr (cv x) (vH) A) (vH) (i^i C D) opreq1 sseq12d imbi12d (CH) rcla4v syl (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) adantr mdslmd.1 mdslmd.2 mdslmd.3 mdslmd.4 x mdslmd1lem3 syld ex com3l r19.21adv mdslmd.3 mdslmd.4 C D y mdbr2 mp2an mdslmd.3 mdslmd.2 chincl mdslmd.4 mdslmd.2 chincl x mdsl2 3imtr4g mdslmd.2 (cv x) B chinclt mpan2 (cv y) (i^i (cv x) B) (i^i D B) sseq1 (cv y) (i^i (cv x) B) (vH) (i^i C B) opreq1 (i^i D B) ineq1d (cv y) (i^i (cv x) B) (vH) (i^i (i^i C B) (i^i D B)) opreq1 sseq12d imbi12d (CH) rcla4v syl (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (/\ (C_ A C) (C_ A D)) (/\ (C_ C (opr A (vH) B)) (C_ D (opr A (vH) B))))) adantr mdslmd.1 mdslmd.2 mdslmd.3 mdslmd.4 x mdslmd1lem4 syld ex com3l r19.21adv mdslmd.3 mdslmd.2 chincl mdslmd.4 mdslmd.2 chincl (i^i C B) (i^i D B) y mdbr2 mp2an mdslmd.3 mdslmd.4 x mdsl2 3imtr4g impbid A C D ssin mdslmd.3 mdslmd.4 mdslmd.1 mdslmd.2 chjcl chlub anbi12i sylan2br)) thm (mdslmd2 () ((mdslmd.1 (e. A (CH))) (mdslmd.2 (e. B (CH))) (mdslmd.3 (e. C (CH))) (mdslmd.4 (e. D (CH)))) (-> (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i A B) (i^i C D)) (C_ (opr C (vH) D) B))) (<-> (br C (MH) D) (br (opr C (vH) A) (MH) (opr D (vH) A)))) (mdslmd.1 mdslmd.2 mdslmd.3 mdslmd.1 chjcl mdslmd.4 mdslmd.1 chjcl mdslmd1 mdslmd.3 mdslmd.4 chjcl mdslmd.2 mdslmd.1 chlej1 mdslmd.3 mdslmd.4 mdslmd.1 chjjdir mdslmd.2 mdslmd.1 chjcom 3sstr3g (C_ (i^i A B) (i^i C D)) adantl mdslmd.1 mdslmd.3 chub2 mdslmd.1 mdslmd.4 chub2 ssini jctil sylan2 mdslmd.1 mdslmd.2 mdslmd.3 3pm3.2i A B C mdsl3t mpan (br A (MH) B) id C D inss1 (i^i A B) (i^i C D) C sstr mpan2 mdslmd.3 mdslmd.4 chub1 C (opr C (vH) D) B sstr mpan syl3an mdslmd.1 mdslmd.2 mdslmd.4 3pm3.2i A B D mdsl3t mpan (br A (MH) B) id C D inss2 (i^i A B) (i^i C D) D sstr mpan2 mdslmd.4 mdslmd.3 chub2 D (opr C (vH) D) B sstr mpan syl3an (MH) breq12d 3expb (br B (MH*) A) adantlr bitr2d)) thm (mdsldmd1 () ((mdslmd.1 (e. A (CH))) (mdslmd.2 (e. B (CH))) (mdslmd.3 (e. C (CH))) (mdslmd.4 (e. D (CH)))) (-> (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ A (i^i C D)) (C_ (opr C (vH) D) (opr A (vH) B)))) (<-> (br C (MH*) D) (br (i^i C B) (MH*) (i^i D B)))) (mdslmd.2 choccl mdslmd.1 choccl mdslmd.3 choccl mdslmd.4 choccl mdslmd2 (br A (MH) B) (br B (MH*) A) ancom mdslmd.2 mdslmd.1 B A dmdmdt mp2an mdslmd.1 mdslmd.2 A B mddmdt mp2an anbi12i bitr (C_ A (i^i C D)) (C_ (opr C (vH) D) (opr A (vH) B)) ancom mdslmd.3 mdslmd.4 chjcl mdslmd.1 mdslmd.2 chjcl chsscon3 mdslmd.1 mdslmd.2 chdmj1 (` (_|_) A) (` (_|_) B) incom eqtr mdslmd.3 mdslmd.4 chdmj1 sseq12i bitr mdslmd.1 mdslmd.3 mdslmd.4 chincl chsscon3 mdslmd.3 mdslmd.4 chdmm1 (` (_|_) A) sseq1i bitr anbi12i bitr anbi12i mdslmd.3 mdslmd.4 C D dmdmdt mp2an mdslmd.3 mdslmd.2 chincl mdslmd.4 mdslmd.2 chincl (i^i C B) (i^i D B) dmdmdt mp2an mdslmd.3 mdslmd.2 chdmm1 mdslmd.4 mdslmd.2 chdmm1 (MH) breq12i bitr bibi12i 3imtr4)) thm (mdexch ((A x) (B x) (C x)) ((mdexch.1 (e. A (CH))) (mdexch.2 (e. B (CH))) (mdexch.3 (e. C (CH)))) (-> (/\/\ (br A (MH) B) (br C (MH) (opr A (vH) B)) (C_ (i^i C (opr A (vH) B)) A)) (/\ (br (opr C (vH) A) (MH) B) (= (i^i (opr C (vH) A) B) (i^i A B)))) (mdexch.3 mdexch.1 C A (cv x) chjasst mp3an12 mdexch.3 mdexch.1 chjcl (cv x) (opr C (vH) A) chjcomt mpan2 mdexch.1 A (cv x) chjclt mpan mdexch.3 (opr A (vH) (cv x)) C chjcomt mpan2 syl 3eqtr4d B ineq1d (opr (opr A (vH) (cv x)) (vH) C) (opr A (vH) B) B inass (opr A (vH) B) B incom mdexch.1 mdexch.2 chjcom B ineq2i mdexch.2 mdexch.1 chabs2 3eqtr (opr (opr A (vH) (cv x)) (vH) C) ineq2i eqtr syl6eqr (C_ (cv x) B) (/\ (br C (MH) (opr A (vH) B)) (C_ (i^i C (opr A (vH) B)) A)) ad2antrr mdexch.2 mdexch.1 (cv x) B A chlej2t mp3an23 mdexch.1 A (cv x) chjclt mpan mdexch.3 mdexch.1 mdexch.2 chjcl C (opr A (vH) B) (opr A (vH) (cv x)) mdit mp3an12 syl com23 syld imp31 (C_ (i^i C (opr A (vH) B)) A) adantrr mdexch.1 A (cv x) chjclt mpan mdexch.3 mdexch.1 mdexch.2 chjcl chincl mdexch.1 (i^i C (opr A (vH) B)) A (opr A (vH) (cv x)) chlej2t mp3an12 syl imp mdexch.1 A (cv x) chjclt mpan mdexch.1 (opr A (vH) (cv x)) A chjcomt mpan2 syl mdexch.1 mdexch.1 A A (cv x) chjasst mp3an12 mdexch.1 chjidm (vH) (cv x) opreq1i syl5reqr mdexch.1 A (cv x) chjcomt mpan 3eqtrd (C_ (i^i C (opr A (vH) B)) A) adantr sseqtrd (C_ (cv x) B) adantlr (br C (MH) (opr A (vH) B)) adantrl eqsstrd (i^i (opr (opr A (vH) (cv x)) (vH) C) (opr A (vH) B)) (opr (cv x) (vH) A) B ssrin syl eqsstrd (br A (MH) B) adantrl mdexch.1 mdexch.2 A B (cv x) mdit mp3an12 com23 imp31 mdexch.1 mdexch.3 chub2 A (opr C (vH) A) B ssrin ax-mp mdexch.1 mdexch.2 chincl mdexch.3 mdexch.1 chjcl mdexch.2 chincl (i^i A B) (i^i (opr C (vH) A) B) (cv x) chlej2t mp3an12 mpi (C_ (cv x) B) (br A (MH) B) ad2antrr eqsstrd (/\ (br C (MH) (opr A (vH) B)) (C_ (i^i C (opr A (vH) B)) A)) adantrr sstrd exp31 com3r 3impb r19.21aiv mdexch.3 mdexch.1 chjcl mdexch.2 (opr C (vH) A) B x mdbr2 mp2an sylibr mdexch.1 mdexch.2 chub1 mdexch.3 mdexch.1 mdexch.2 chjcl mdexch.1 C (opr A (vH) B) A mdit mp3an mpi mdexch.3 mdexch.1 mdexch.2 chjcl chincl mdexch.1 chlejb1 biimp mdexch.1 mdexch.3 mdexch.1 mdexch.2 chjcl chincl chjcom syl5eq sylan9eq B ineq1d (opr A (vH) C) (opr A (vH) B) B inass syl5eqr mdexch.3 mdexch.1 chjcom mdexch.1 mdexch.2 chjcom B ineq2i B (opr A (vH) B) incom mdexch.2 mdexch.1 chabs2 3eqtr3r ineq12i syl5eq (br A (MH) B) 3adant1 jca)) thm (cvmdt () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (br (i^i A B) ( (/\ (e. A (CH)) (e. B (CH))) (-> (br B ( (e. A (Atoms)) (/\ (e. A (CH)) (br (0H) ( (e. A (Atoms)) (/\ (e. A (CH)) (/\ (=/= A (0H)) (A.e. x (CH) (-> (C_ (cv x) A) (\/ (= (cv x) A) (= (cv x) (0H)))))))) (A elat h0elch (0H) A x cvbr2t mpan A ch0psst (cv x) ch0psst (= (cv x) A) imbi1d (C_ (cv x) A) imbi2d (C: (0H) (cv x)) (C_ (cv x) A) (= (cv x) A) impexp (C: (0H) (cv x)) (C_ (cv x) A) (= (cv x) A) bi2.04 bitr (= (cv x) (0H)) (= (cv x) A) df-or (= (cv x) A) (= (cv x) (0H)) orcom (cv x) (0H) df-ne (= (cv x) A) imbi1i 3bitr4 (C_ (cv x) A) imbi2i 3bitr4g ralbiia (e. A (CH)) a1i anbi12d bitr2d pm5.32i bitr4)) thm (elat2OLD ((A x)) () (<-> (e. A (Atoms)) (/\ (e. A (CH)) (/\ (-. (= A (0H))) (A.e. x (CH) (-> (C_ (cv x) A) (\/ (= (cv x) A) (= (cv x) (0H)))))))) (A elat h0elch (0H) A x cvbr2t mpan A ch0psstOLD (cv x) ch0psstOLD (= (cv x) A) imbi1d (C_ (cv x) A) imbi2d (C: (0H) (cv x)) (C_ (cv x) A) (= (cv x) A) impexp (C: (0H) (cv x)) (C_ (cv x) A) (= (cv x) A) bi2.04 bitr (= (cv x) A) (= (cv x) (0H)) orcom (= (cv x) (0H)) (= (cv x) A) df-or bitr (C_ (cv x) A) imbi2i 3bitr4g ralbiia (e. A (CH)) a1i anbi12d bitr2d pm5.32i bitr4)) thm (elatcv0 () () (-> (e. A (CH)) (<-> (e. A (Atoms)) (br (0H) ( (e. A (Atoms)) (br (0H) ( (e. A (Atoms)) (e. A (CH))) (atssch A sseli)) thm (atn0 ((A x)) () (-> (e. A (Atoms)) (=/= A (0H))) (A x elat2 (=/= A (0H)) (A.e. x (CH) (-> (C_ (cv x) A) (\/ (= (cv x) A) (= (cv x) (0H))))) pm3.26 (e. A (CH)) adantl sylbi)) thm (atn0OLD ((A x)) () (-> (e. A (Atoms)) (-. (= A (0H)))) (A x elat2OLD (-. (= A (0H))) (A.e. x (CH) (-> (C_ (cv x) A) (\/ (= (cv x) A) (= (cv x) (0H))))) pm3.26 (e. A (CH)) adantl sylbi)) thm (atss ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (Atoms))) (-> (C_ A B) (\/ (= A B) (= A (0H))))) ((cv x) A B sseq1 (cv x) A B eqeq1 (cv x) A (0H) eqeq1 orbi12d imbi12d (CH) rcla4v (-. (= B (0H))) adantld (e. B (CH)) adantld imp B x elat2OLD sylan2b)) thm (atsseq () () (-> (/\ (e. A (Atoms)) (e. B (Atoms))) (<-> (C_ A B) (= A B))) (A atn0OLD (e. B (Atoms)) (C_ A B) ad2antrr A B atss A atelch sylan imp ord mt3d ex A B eqimss (/\ (e. A (Atoms)) (e. B (Atoms))) a1i impbid)) thm (atcveq0 () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (<-> (br A ( (/\ (e. A (H~)) (=/= A (0v))) (e. (` (_|_) (` (_|_) ({} A))) (Atoms))) (A (H~) snssi ({} A) occlt (` (_|_) ({} A)) chocclt 3syl (=/= A (0v)) adantr A h1dn0OLD (cv x) A h1datomt expcom r19.21aiv (=/= A (0v)) adantr jca jca (` (_|_) (` (_|_) ({} A))) x elat2OLD sylibr)) thm (h1datOLD ((A x)) () (-> (/\ (e. A (H~)) (-. (= A (0v)))) (e. (` (_|_) (` (_|_) ({} A))) (Atoms))) (A (H~) snssi ({} A) occlt (` (_|_) ({} A)) chocclt 3syl (-. (= A (0v))) adantr A h1dn0OLDOLD (cv x) A h1datomt expcom r19.21aiv (-. (= A (0v))) adantr jca jca (` (_|_) (` (_|_) ({} A))) x elat2OLD sylibr)) thm (spansnat () () (-> (/\ (e. A (H~)) (=/= A (0v))) (e. (` (span) ({} A)) (Atoms))) (A spansnt (=/= A (0v)) adantr A h1dat eqeltrd)) thm (spansnatOLD () () (-> (/\ (e. A (H~)) (-. (= A (0v)))) (e. (` (span) ({} A)) (Atoms))) (A spansnt (-. (= A (0v))) adantr A h1datOLD eqeltrd)) thm (sh1dle () () (-> (/\ (e. A (SH)) (e. B A)) (C_ (` (_|_) (` (_|_) ({} B))) A)) (A B shelt B spansnt syl A B spansnsst eqsstr3d)) thm (ch1dle () () (-> (/\ (e. A (CH)) (e. B A)) (C_ (` (_|_) (` (_|_) ({} B))) A)) (A B sh1dle A chsh sylan)) thm (atom1d ((x y) (A x) (A y)) () (<-> (e. A (Atoms)) (E.e. x (H~) (/\ (=/= (cv x) (0v)) (= A (` (span) ({} (cv x))))))) (A y elat2 A x chne0t (e. A (CH)) x ax-17 (A.e. y (CH) (-> (C_ (cv y) A) (\/ (= (cv y) A) (= (cv y) (0H))))) x ax-17 x (H~) (/\ (=/= (cv x) (0v)) (= A (` (_|_) (` (_|_) ({} (cv x)))))) hbre1 hbim x (H~) (/\ (=/= (cv x) (0v)) (= A (` (_|_) (` (_|_) ({} (cv x)))))) ra4e A (cv x) chelt (=/= (cv x) (0v)) adantrr (A.e. y (CH) (-> (C_ (cv y) A) (\/ (= (cv y) A) (= (cv y) (0H))))) adantrr (e. (cv x) A) (=/= (cv x) (0v)) pm3.27 (e. A (CH)) (A.e. y (CH) (-> (C_ (cv y) A) (\/ (= (cv y) A) (= (cv y) (0H))))) ad2antrl (cv x) h1dn0 A (cv x) chelt sylan anasss (A.e. y (CH) (-> (C_ (cv y) A) (\/ (= (cv y) A) (= (cv y) (0H))))) adantrr A (cv x) ch1dle A (cv x) chelt (cv x) (H~) snssi ({} (cv x)) occlt 3syl (` (_|_) ({} (cv x))) chocclt (cv y) (` (_|_) (` (_|_) ({} (cv x)))) A sseq1 (cv y) (` (_|_) (` (_|_) ({} (cv x)))) A eqeq1 (cv y) (` (_|_) (` (_|_) ({} (cv x)))) (0H) eqeq1 orbi12d imbi12d (CH) rcla4v 3syl mpid ex imp32 (=/= (cv x) (0v)) adantrlr ord (` (_|_) (` (_|_) ({} (cv x)))) (0H) nne syl6ibr mt4d eqcomd jca sylanc exp44 r19.23ad sylbid imp32 sylbi A (` (_|_) (` (_|_) ({} (cv x)))) (Atoms) eleq1 (cv x) h1dat syl5bir exp3a com3l imp3a r19.23aiv impbi (cv x) spansnt A eqeq2d (=/= (cv x) (0v)) anbi2d rexbiia bitr4)) thm (atom1dOLD ((x y) (A x) (A y)) () (<-> (e. A (Atoms)) (E.e. x (H~) (/\ (-. (= (cv x) (0v))) (= A (` (span) ({} (cv x))))))) (A y elat2OLD A x chne0tOLD (e. A (CH)) x ax-17 (A.e. y (CH) (-> (C_ (cv y) A) (\/ (= (cv y) A) (= (cv y) (0H))))) x ax-17 x (H~) (/\ (-. (= (cv x) (0v))) (= A (` (_|_) (` (_|_) ({} (cv x)))))) hbre1 hbim x (H~) (/\ (-. (= (cv x) (0v))) (= A (` (_|_) (` (_|_) ({} (cv x)))))) ra4e A (cv x) chelt (-. (= (cv x) (0v))) adantrr (A.e. y (CH) (-> (C_ (cv y) A) (\/ (= (cv y) A) (= (cv y) (0H))))) adantrr (e. (cv x) A) (-. (= (cv x) (0v))) pm3.27 (e. A (CH)) (A.e. y (CH) (-> (C_ (cv y) A) (\/ (= (cv y) A) (= (cv y) (0H))))) ad2antrl (cv x) h1dn0OLDOLD A (cv x) chelt sylan anasss (A.e. y (CH) (-> (C_ (cv y) A) (\/ (= (cv y) A) (= (cv y) (0H))))) adantrr A (cv x) ch1dle A (cv x) chelt (cv x) (H~) snssi ({} (cv x)) occlt syl (` (_|_) ({} (cv x))) chocclt 3syl (cv y) (` (_|_) (` (_|_) ({} (cv x)))) A sseq1 (cv y) (` (_|_) (` (_|_) ({} (cv x)))) A eqeq1 (cv y) (` (_|_) (` (_|_) ({} (cv x)))) (0H) eqeq1 orbi12d imbi12d (CH) rcla4v syl mpid ex imp32 (-. (= (cv x) (0v))) adantrlr ord mt3d eqcomd jca sylanc exp44 r19.23ad sylbid imp32 sylbi A (` (_|_) (` (_|_) ({} (cv x)))) (Atoms) eleq1 (cv x) h1datOLD syl5bir exp3a com3l imp3a r19.23aiv impbi (cv x) spansnt A eqeq2d (-. (= (cv x) (0v))) anbi2d rexbiia bitr4)) thm (superpos ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (v w) (v x) (v y) (v z) (w x) (w y) (w z)) () (-> (/\/\ (e. A (Atoms)) (e. B (Atoms)) (=/= A B)) (E.e. x (Atoms) (/\/\ (=/= (cv x) A) (=/= (cv x) B) (C_ (cv x) (opr A (vH) B))))) (y (H~) z (H~) (/\ (=/= (cv y) (0v)) (= A (` (span) ({} (cv y))))) (/\ (=/= (cv z) (0v)) (= B (` (span) ({} (cv z))))) reeanv A (` (span) ({} (cv y))) B neeq1 B (` (span) ({} (cv z))) (` (span) ({} (cv y))) neeq2 sylan9bb (/\ (/\ (e. (cv y) (H~)) (e. (cv z) (H~))) (/\ (=/= (cv y) (0v)) (=/= (cv z) (0v)))) adantl (cv x) (` (span) ({} (opr (cv y) (+v) (cv z)))) A neeq1 (cv x) (` (span) ({} (opr (cv y) (+v) (cv z)))) B neeq1 (cv x) (` (span) ({} (opr (cv y) (+v) (cv z)))) (opr A (vH) B) sseq1 3anbi123d (Atoms) rcla4ev (opr (cv y) (+v) (cv z)) spansnat (cv y) (cv z) ax-hvaddcl (=/= (` (span) ({} (cv y))) (` (span) ({} (cv z)))) adantr (cv y) (cv z) hvaddeq0t (cv y) (opr (-u (1)) (.s) (cv z)) sneq (span) fveq2d 1cn negcl 1cn ax1ne0 negn0 (cv z) (-u (1)) spansncol mp3an23 sylan9eqr ex (e. (cv y) (H~)) adantl sylbid necon3d imp sylanc (/\ (=/= (cv y) (0v)) (=/= (cv z) (0v))) adantlr (/\ (= A (` (span) ({} (cv y)))) (= B (` (span) ({} (cv z))))) adantlr A (` (span) ({} (cv y))) (` (span) ({} (opr (cv y) (+v) (cv z)))) eqeq2 biimpd (cv y) (cv z) ax-hvaddcl (opr (cv y) (+v) (cv z)) (cv y) spansneleqi syl (cv y) (opr (cv y) (+v) (cv z)) v elspansnt (e. (cv z) (H~)) adantr (cv w) (opr (cv v) (+) (-u (1))) (.s) (cv y) opreq1 (cv z) eqeq2d (CC) rcla4ev 1cn negcl (cv v) (-u (1)) axaddcl mpan2 (/\ (e. (cv y) (H~)) (e. (cv z) (H~))) (= (opr (cv y) (+v) (cv z)) (opr (cv v) (.s) (cv y))) ad2antlr (opr (cv v) (.s) (cv y)) (cv y) (cv z) hvsubaddt (cv v) (cv y) ax-hvmulcl ancoms (e. (cv z) (H~)) adantlr (e. (cv y) (H~)) (e. (cv z) (H~)) pm3.26 (e. (cv v) (CC)) adantr (e. (cv y) (H~)) (e. (cv z) (H~)) pm3.27 (e. (cv v) (CC)) adantr syl3anc biimpar (opr (cv v) (.s) (cv y)) (cv y) hvsubvalt (cv v) (cv y) ax-hvmulcl (e. (cv v) (CC)) (e. (cv y) (H~)) pm3.27 sylanc 1cn negcl (cv v) (-u (1)) (cv y) ax-hvdistr2 mp3an2 eqtr4d ancoms (e. (cv z) (H~)) adantlr (= (opr (cv y) (+v) (cv z)) (opr (cv v) (.s) (cv y))) adantr eqtr3d sylanc exp31 r19.23adv sylbid syld (cv y) (cv z) w elspansnt (e. (cv z) (H~)) adantr sylibrd (=/= (cv z) (0v)) adantr (cv y) (cv z) spansneleq (` (span) ({} (cv z))) (` (span) ({} (cv y))) eqcom syl6ib (e. (cv z) (H~)) adantlr syld sylan9r necon3d (=/= (cv y) (0v)) adantlrl (= B (` (span) ({} (cv z)))) adantrr imp B (` (span) ({} (cv z))) (` (span) ({} (opr (cv y) (+v) (cv z)))) eqeq2 biimpd (cv y) (cv z) ax-hvaddcl (opr (cv y) (+v) (cv z)) (cv z) spansneleqi syl (cv z) (opr (cv y) (+v) (cv z)) v elspansnt (e. (cv y) (H~)) adantl (cv w) (opr (cv v) (+) (-u (1))) (.s) (cv z) opreq1 (cv y) eqeq2d (CC) rcla4ev 1cn negcl (cv v) (-u (1)) axaddcl mpan2 (/\ (e. (cv y) (H~)) (e. (cv z) (H~))) (= (opr (cv y) (+v) (cv z)) (opr (cv v) (.s) (cv z))) ad2antlr (opr (cv v) (.s) (cv z)) (cv z) (cv y) hvsubaddt (cv v) (cv z) ax-hvmulcl ancoms (e. (cv y) (H~)) adantll (e. (cv y) (H~)) (e. (cv z) (H~)) pm3.27 (e. (cv v) (CC)) adantr (e. (cv y) (H~)) (e. (cv z) (H~)) pm3.26 (e. (cv v) (CC)) adantr syl3anc (cv y) (cv z) ax-hvcom (e. (cv v) (CC)) adantr (opr (cv v) (.s) (cv z)) eqeq1d bitr4d biimpar (opr (cv v) (.s) (cv z)) (cv z) hvsubvalt (cv v) (cv z) ax-hvmulcl (e. (cv v) (CC)) (e. (cv z) (H~)) pm3.27 sylanc 1cn negcl (cv v) (-u (1)) (cv z) ax-hvdistr2 mp3an2 eqtr4d ancoms (e. (cv y) (H~)) adantll (= (opr (cv y) (+v) (cv z)) (opr (cv v) (.s) (cv z))) adantr eqtr3d sylanc exp31 r19.23adv sylbid syld (cv z) (cv y) w elspansnt (e. (cv y) (H~)) adantl sylibrd (=/= (cv y) (0v)) adantr (cv z) (cv y) spansneleq (e. (cv y) (H~)) adantll syld sylan9r necon3d (=/= (cv z) (0v)) adantlrr (= A (` (span) ({} (cv y)))) adantrl imp (cv y) (cv z) spanpr (/\ (= A (` (span) ({} (cv y)))) (= B (` (span) ({} (cv z))))) adantr A (` (span) ({} (cv y))) B (` (span) ({} (cv z))) (vH) opreq12 ({} (cv y)) ({} (cv z)) spanunt (cv y) (H~) snssi (cv z) (H~) snssi syl2an (cv y) (cv z) df-pr (span) fveq2i syl5eq (` (span) ({} (cv y))) (cv z) spansnjt (cv y) spansncht sylan eqtr2d sylan9eqr sseqtr4d (/\ (=/= (cv y) (0v)) (=/= (cv z) (0v))) adantlr (=/= (` (span) ({} (cv y))) (` (span) ({} (cv z)))) adantr 3jca sylanc ex sylbid anasss ex (=/= (cv y) (0v)) (= A (` (span) ({} (cv y)))) (=/= (cv z) (0v)) (= B (` (span) ({} (cv z)))) an4 syl5ib r19.23aivv sylbir A y atom1d B z atom1d syl2anb 3impia)) thm (chcv1t ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (Atoms))) (<-> (-. (C_ B A)) (br A ( (/\ (e. A (CH)) (e. B (Atoms))) (<-> (C: A (opr A (vH) B)) (br A ( (/\ (e. A (CH)) (e. B (Atoms))) (= (opr A (+H) B) (opr A (vH) B))) (B (` (span) ({} (cv x))) A (+H) opreq2 B (` (span) ({} (cv x))) A (vH) opreq2 eqeq12d A (cv x) spansnjt syl5bir exp3a (-. (= (cv x) (0v))) adantl com3l r19.23adv B x atom1dOLD syl5ib imp)) thm (shatomic ((x y) (A x) (A y)) ((shatomic.1 (e. A (SH)))) (-> (=/= A (0H)) (E.e. x (Atoms) (C_ (cv x) A))) (shatomic.1 y shne0 (cv x) (` (_|_) (` (_|_) ({} (cv y)))) A sseq1 (Atoms) rcla4ev (cv y) h1dat shatomic.1 (cv y) shel sylan shatomic.1 A (cv y) sh1dle mpan (=/= (cv y) (0v)) adantr sylanc ex r19.23aiv sylbi)) thm (shatomicOLD ((x y) (A x) (A y)) ((shatomic.1 (e. A (SH)))) (-> (-. (= A (0H))) (E.e. x (Atoms) (C_ (cv x) A))) (shatomic.1 y shne0OLD (cv x) (` (_|_) (` (_|_) ({} (cv y)))) A sseq1 (Atoms) rcla4ev (cv y) h1datOLD shatomic.1 (cv y) shel sylan shatomic.1 A (cv y) sh1dle mpan (-. (= (cv y) (0v))) adantr sylanc ex r19.23aiv sylbi)) thm (hatomic ((A x)) ((hatomic.1 (e. A (CH)))) (-> (=/= A (0H)) (E.e. x (Atoms) (C_ (cv x) A))) (hatomic.1 chshi x shatomic)) thm (hatomicOLD ((A x)) ((hatomic.1 (e. A (CH)))) (-> (-. (= A (0H))) (E.e. x (Atoms) (C_ (cv x) A))) (hatomic.1 chshi x shatomicOLD)) thm (hatomict ((A x)) () (-> (/\ (e. A (CH)) (=/= A (0H))) (E.e. x (Atoms) (C_ (cv x) A))) (A (if (e. A (CH)) A (0H)) (0H) neeq1 A (if (e. A (CH)) A (0H)) (cv x) sseq2 x (Atoms) rexbidv imbi12d h0elch A elimel x hatomic dedth imp)) thm (hatomictOLD ((A x)) () (-> (e. A (CH)) (-> (-. (= A (0H))) (E.e. x (Atoms) (C_ (cv x) A)))) (A (if (e. A (CH)) A (0H)) (0H) eqeq1 negbid A (if (e. A (CH)) A (0H)) (cv x) sseq2 x (Atoms) rexbidv imbi12d h0elch A elimel x hatomicOLD dedth)) thm (shatomistic ((x y) (A x) (A y)) ((shatomistic.1 (e. A (SH)))) (= A (` (span) (U. ({e.|} x (Atoms) (C_ (cv x) A))))) (shatomistic.1 (cv y) shel (cv y) spansnsht (` (span) ({} (cv y))) spanid 3syl (-. (= (cv y) (0v))) adantr (cv y) spansnatOLD shatomistic.1 (cv y) shel sylan shatomistic.1 A (cv y) spansnsst mpan (-. (= (cv y) (0v))) adantr jca (cv x) (` (span) ({} (cv y))) A sseq1 (Atoms) elrab sylibr (` (span) ({} (cv y))) ({e.|} x (Atoms) (C_ (cv x) A)) elssuni atssch chsssh sstri (Atoms) (SH) x (C_ (cv x) A) rabss2 ax-mp ({e.|} x (Atoms) (C_ (cv x) A)) ({e.|} x (SH) (C_ (cv x) A)) uniss ax-mp shatomistic.1 A (SH) x unimax ax-mp shatomistic.1 shssi eqsstr sstri (U. ({e.|} x (Atoms) (C_ (cv x) A))) (` (span) ({} (cv y))) spanss mpan 3syl eqsstr3d shatomistic.1 (cv y) shel (cv y) spansnid syl (-. (= (cv y) (0v))) adantr sseldd ex atssch chsssh sstri (Atoms) (SH) x (C_ (cv x) A) rabss2 ax-mp ({e.|} x (Atoms) (C_ (cv x) A)) ({e.|} x (SH) (C_ (cv x) A)) uniss ax-mp shatomistic.1 A (SH) x unimax ax-mp shatomistic.1 shssi eqsstr sstri (U. ({e.|} x (Atoms) (C_ (cv x) A))) spanclt ax-mp (` (span) (U. ({e.|} x (Atoms) (C_ (cv x) A)))) sh0 ax-mp (cv y) (0v) (` (span) (U. ({e.|} x (Atoms) (C_ (cv x) A)))) eleq1 mpbiri pm2.61d2 ssriv shatomistic.1 A (SH) x unimax ax-mp shatomistic.1 shssi eqsstr atssch chsssh sstri (Atoms) (SH) x (C_ (cv x) A) rabss2 ax-mp ({e.|} x (Atoms) (C_ (cv x) A)) ({e.|} x (SH) (C_ (cv x) A)) uniss ax-mp (U. ({e.|} x (SH) (C_ (cv x) A))) (U. ({e.|} x (Atoms) (C_ (cv x) A))) spanss mp2an shatomistic.1 A (SH) x unimax ax-mp (span) fveq2i shatomistic.1 A spanid ax-mp eqtr sseqtr eqssi)) thm (hatomistic ((x y) (A x) (A y)) ((hatomistic.1 (e. A (CH)))) (= A (` (\/H) ({e.|} x (Atoms) (C_ (cv x) A)))) (x (Atoms) (C_ (cv x) A) ssrab2 atssch sstri ({e.|} x (Atoms) (C_ (cv x) A)) chsupclt ax-mp hatomistic.1 chshi (cv y) atelch (C_ (cv y) A) anim1i (cv x) (cv y) A sseq1 (Atoms) elrab (cv x) (cv y) A sseq1 (CH) elrab 3imtr4 ssriv x (Atoms) (C_ (cv x) A) ssrab2 atssch sstri x (CH) (C_ (cv x) A) ssrab2 ({e.|} x (Atoms) (C_ (cv x) A)) ({e.|} x (CH) (C_ (cv x) A)) chsupss mp2an ax-mp hatomistic.1 A x chsupid ax-mp sseqtr (cv x) (cv y) A sseq1 (Atoms) elrab (cv y) ({e.|} x (Atoms) (C_ (cv x) A)) elssuni sylbir x (Atoms) (C_ (cv x) A) ssrab2 atssch sstri ({e.|} x (Atoms) (C_ (cv x) A)) chsupunss ax-mp (cv y) (U. ({e.|} x (Atoms) (C_ (cv x) A))) (` (\/H) ({e.|} x (Atoms) (C_ (cv x) A))) sstr2 mpi syl ex (cv y) atn0OLD (C_ (cv y) (` (\/H) ({e.|} x (Atoms) (C_ (cv x) A)))) adantr (cv y) atelch (cv y) chle0t syl (cv y) (` (\/H) ({e.|} x (Atoms) (C_ (cv x) A))) (` (_|_) (` (\/H) ({e.|} x (Atoms) (C_ (cv x) A)))) ssin x (Atoms) (C_ (cv x) A) ssrab2 atssch sstri ({e.|} x (Atoms) (C_ (cv x) A)) chsupclt ax-mp chocin (cv y) sseq2i bitr2 syl5bbr biimpa exp32 imp mtod ex syld (C_ (cv y) A) (C_ (cv y) (` (_|_) (` (\/H) ({e.|} x (Atoms) (C_ (cv x) A))))) imnan sylib (cv y) A (` (_|_) (` (\/H) ({e.|} x (Atoms) (C_ (cv x) A)))) ssin negbii sylib nrex hatomistic.1 x (Atoms) (C_ (cv x) A) ssrab2 atssch sstri ({e.|} x (Atoms) (C_ (cv x) A)) chsupclt ax-mp choccl chincl y hatomicOLD mt3 omlsi eqcomi)) thm (chpssat ((A x) (B x)) ((chpssat.1 (e. A (CH))) (chpssat.2 (e. B (CH)))) (-> (C: A B) (E.e. x (Atoms) (/\ (C_ (cv x) B) (-. (C_ (cv x) A))))) (A B dfpss3 pm3.27bd (C_ (cv x) B) (C_ (cv x) A) iman x (Atoms) ralbii x (Atoms) (C_ (cv x) B) (C_ (cv x) A) ss2rab x (Atoms) (C_ (cv x) B) ssrab2 atssch sstri x (Atoms) (C_ (cv x) A) ssrab2 atssch sstri ({e.|} x (Atoms) (C_ (cv x) B)) ({e.|} x (Atoms) (C_ (cv x) A)) chsupss mp2an chpssat.2 x hatomistic chpssat.1 x hatomistic 3sstr4g sylbir sylbir con3i x (Atoms) (/\ (C_ (cv x) B) (-. (C_ (cv x) A))) dfrex2 sylibr syl)) thm (chrelat ((A x) (B x)) ((chpssat.1 (e. A (CH))) (chpssat.2 (e. B (CH)))) (-> (C: A B) (E.e. x (Atoms) (/\ (C: A (opr A (vH) (cv x))) (C_ (opr A (vH) (cv x)) B)))) (chpssat.1 chpssat.2 x chpssat chpssat.1 A (cv x) chnlet mpan (C_ A B) adantl (C_ A B) (C_ (cv x) B) ibar chpssat.1 chpssat.2 A (cv x) B chlubt mp3an13 sylan9bb anbi12d A B pssss (cv x) atelch syl2an (C_ (cv x) B) (-. (C_ (cv x) A)) ancom syl5bb rexbidva mpbid)) thm (chrelat2 ((A x) (B x)) ((chpssat.1 (e. A (CH))) (chpssat.2 (e. B (CH)))) (<-> (-. (C_ A B)) (E.e. x (Atoms) (/\ (C_ (cv x) A) (-. (C_ (cv x) B))))) (A B nssinpss chpssat.1 chpssat.2 chincl chpssat.1 x chrelat (cv x) atelch chpssat.1 chpssat.2 chincl chpssat.1 (i^i A B) (cv x) A chlubt mp3an13 (C_ (i^i A B) A) (C_ (cv x) A) pm3.27 syl6bir (C: (i^i A B) (opr (i^i A B) (vH) (cv x))) adantld chpssat.1 chpssat.2 chincl (i^i A B) (cv x) chnlet mpan (cv x) A B ssin negbii syl5bb chpssat.1 chpssat.2 chincl chpssat.1 (i^i A B) (cv x) A chlubt mp3an13 anbi12d (C_ (cv x) B) (C_ (cv x) A) pm3.21 (-. (/\ (C_ (cv x) A) (C_ (cv x) B))) (C_ (cv x) A) ianor (/\ (C_ (cv x) A) (C_ (cv x) B)) pm4.13 (-. (C_ (cv x) A)) orbi1i (/\ (C_ (cv x) A) (C_ (cv x) B)) (-. (C_ (cv x) A)) orcom (C_ (cv x) A) (/\ (C_ (cv x) A) (C_ (cv x) B)) imor bitr4 3bitr2r sylib con2i (C_ (i^i A B) A) adantrl syl6bir jcad syl r19.22i syl sylbi (cv x) A B sstr2 com12 (e. (cv x) (Atoms)) a1d r19.21aiv (C_ (cv x) A) (C_ (cv x) B) iman x (Atoms) ralbii x (Atoms) (/\ (C_ (cv x) A) (-. (C_ (cv x) B))) ralnex bitr sylib con2i impbi)) thm (cvat ((A x) (B x)) ((chpssat.1 (e. A (CH))) (chpssat.2 (e. B (CH)))) (-> (br A ( (br A ( (br (i^i A B) ( (br (i^i A B) ( (/\ (e. A (CH)) (e. B (CH))) (<-> (-. (C_ A B)) (E.e. x (Atoms) (/\ (C_ (cv x) A) (-. (C_ (cv x) B)))))) (A (if (e. A (CH)) A (H~)) B sseq1 negbid A (if (e. A (CH)) A (H~)) (cv x) sseq2 (-. (C_ (cv x) B)) anbi1d x (Atoms) rexbidv bibi12d B (if (e. B (CH)) B (H~)) (if (e. A (CH)) A (H~)) sseq2 negbid B (if (e. B (CH)) B (H~)) (cv x) sseq2 negbid (C_ (cv x) (if (e. A (CH)) A (H~))) anbi2d x (Atoms) rexbidv bibi12d helch A elimel helch B elimel x chrelat2 dedth2h)) thm (chrelat3t ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ A B) (A.e. x (Atoms) (-> (C_ (cv x) A) (C_ (cv x) B))))) (A B x chrelat2t x (Atoms) (/\ (C_ (cv x) A) (-. (C_ (cv x) B))) dfrex2 syl6bb con4bid (C_ (cv x) A) (C_ (cv x) B) iman x (Atoms) ralbii syl6bbr)) thm (chrelat3 ((A x) (B x)) ((chrelat3.1 (e. A (CH))) (chrelat3.2 (e. B (CH)))) (<-> (C_ A B) (A.e. x (Atoms) (-> (C_ (cv x) A) (C_ (cv x) B)))) (chrelat3.1 chrelat3.2 A B x chrelat3t mp2an)) thm (chrelat4 ((A x) (B x)) ((chrelat3.1 (e. A (CH))) (chrelat3.2 (e. B (CH)))) (<-> (= A B) (A.e. x (Atoms) (<-> (C_ (cv x) A) (C_ (cv x) B)))) (chrelat3.1 chrelat3.2 x chrelat3 chrelat3.2 chrelat3.1 x chrelat3 anbi12i A B eqss (C_ (cv x) A) (C_ (cv x) B) bi x (Atoms) ralbii x (Atoms) (-> (C_ (cv x) A) (C_ (cv x) B)) (-> (C_ (cv x) B) (C_ (cv x) A)) r19.26 bitr 3bitr4)) thm (cvexcht () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (i^i A B) ( (/\ (e. A (CH)) (e. B (Atoms))) (<-> (= (i^i A B) (0H)) (br A ( (/\ (e. A (CH)) (e. B (Atoms))) (<-> (-. (C_ B A)) (= (i^i A B) (0H)))) (A B chcv1t A B cvp bitr4d)) thm (atnem0 () () (-> (/\ (e. A (Atoms)) (e. B (Atoms))) (<-> (-. (= A B)) (= (i^i A B) (0H)))) (B A atsseq B A eqcom syl6bb ancoms negbid A B atnssm0 A atelch sylan bitr3d)) thm (atssmat () () (-> (/\ (e. A (Atoms)) (e. B (CH))) (<-> (C_ A B) (e. (i^i A B) (Atoms)))) (A B df-ss biimp (Atoms) eleq1d biimprcd (e. B (CH)) adantr B A atnssm0 ancoms biimpd con1d A B incom (Atoms) eleq1i (i^i B A) atn0 (i^i B A) (0H) df-ne sylib sylbi syl5 impbid)) thm (atcv0eq () () (-> (/\ (e. A (Atoms)) (e. B (Atoms))) (<-> (br (0H) ( (/\ (/\/\ (e. A (CH)) (e. B (Atoms)) (e. C (Atoms))) (br A ( (= A (0H)) (= B C))) (A (0H) ( (/\/\ (e. A (CH)) (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (C_ B (opr A (vH) C)) (= (i^i A B) (0H))) (C_ C (opr A (vH) B)))) (C A chub2t ancoms C atelch sylan2 (e. B (Atoms)) 3adant2 (/\ (C_ B (opr A (vH) C)) (= (i^i A B) (0H))) adantr A B cvp A B chjclt B atelch sylan2 A (opr A (vH) B) cvpsst syldan sylbid (e. C (Atoms)) 3adant3 (C_ B (opr A (vH) C)) adantld A C chub1t (e. B (CH)) 3adant2 (C_ B (opr A (vH) C)) a1d ancrd A B (opr A (vH) C) chlubt (e. A (CH)) (e. B (CH)) (e. C (CH)) 3simp1 (e. A (CH)) (e. B (CH)) (e. C (CH)) 3simp2 A C chjclt (e. B (CH)) 3adant2 syl3anc sylibd (e. A (CH)) id B atelch C atelch syl3an (= (i^i A B) (0H)) adantrd jcad imp A B cvp A B chjclt B atelch sylan2 A (opr A (vH) B) cvpsst syldan sylbid (e. C (Atoms)) 3adant3 A C chub1t (e. B (CH)) 3adant2 (C_ B (opr A (vH) C)) a1d ancrd A B (opr A (vH) C) chlubt (e. A (CH)) (e. B (CH)) (e. C (CH)) 3simp1 (e. A (CH)) (e. B (CH)) (e. C (CH)) 3simp2 A C chjclt (e. B (CH)) 3adant2 syl3anc sylibd (e. A (CH)) id B atelch C atelch syl3an anim12d ancomsd A (opr A (vH) B) (opr A (vH) C) psssstr syl6 A C chcv2t (e. B (Atoms)) 3adant2 sylibd (e. A (CH)) (e. B (CH)) (e. C (CH)) 3simp1 A C chjclt (e. B (CH)) 3adant2 A B chjclt (e. C (CH)) 3adant3 3jca (e. A (CH)) id B atelch C atelch syl3an A (opr A (vH) C) (opr A (vH) B) cvnbtwn2t syl syld imp mpd sseqtr4d ex)) thm (atoml ((A x) (B x)) ((atoml.1 (e. A (CH)))) (-> (e. B (Atoms)) (e. (i^i (opr A (vH) B) (` (_|_) A)) (u. (Atoms) ({} (0H))))) (B atelch atoml.1 A B chjclt mpan syl atoml.1 choccl (opr A (vH) B) (` (_|_) A) chinclt mpan2 (i^i (opr A (vH) B) (` (_|_) A)) x hatomictOLD 3syl imp atoml.1 choccl (` (_|_) A) (cv x) pjoml3t mpan imp atoml.1 pjococ (vH) (cv x) opreq1i (` (_|_) A) ineq1i (opr (` (_|_) (` (_|_) A)) (vH) (cv x)) (` (_|_) A) incom eqtr3 syl5eq (cv x) atelch (opr A (vH) B) (` (_|_) A) inss2 (cv x) (i^i (opr A (vH) B) (` (_|_) A)) (` (_|_) A) sstr mpan2 syl2an (e. B (Atoms)) adantll (-. (= (i^i (opr A (vH) B) (` (_|_) A)) (0H))) adantrr atoml.1 A B chub1t mpan (e. (cv x) (CH)) adantr atoml.1 A (cv x) (opr A (vH) B) chlubt mp3an1 atoml.1 A B chjclt mpan sylan2 biimpd ancoms mpand B atelch (cv x) atelch syl2an imp (opr A (vH) B) (` (_|_) A) inss1 (cv x) (i^i (opr A (vH) B) (` (_|_) A)) (opr A (vH) B) sstr mpan2 sylan2 (-. (= (i^i (opr A (vH) B) (` (_|_) A)) (0H))) adantrr atoml.1 A B (opr A (vH) (cv x)) chlubt mp3an1 biimpd exp3a com23 imp B atelch (cv x) atelch atoml.1 A (cv x) chjclt mpan syl anim12i (/\ (C_ (cv x) (i^i (opr A (vH) B) (` (_|_) A))) (-. (= (i^i (opr A (vH) B) (` (_|_) A)) (0H)))) adantr atoml.1 A (cv x) B atexcht mp3an1 (e. B (Atoms)) (e. (cv x) (Atoms)) pm3.22 (/\ (C_ (cv x) (i^i (opr A (vH) B) (` (_|_) A))) (-. (= (i^i (opr A (vH) B) (` (_|_) A)) (0H)))) adantr (opr A (vH) B) (` (_|_) A) inss1 (cv x) (i^i (opr A (vH) B) (` (_|_) A)) (opr A (vH) B) sstr mpan2 (e. (cv x) (Atoms)) adantl (cv x) chsh atoml.1 chshi (cv x) A orthin mpan2 syl imp A (cv x) incom syl5eq (cv x) atelch (opr A (vH) B) (` (_|_) A) inss2 (cv x) (i^i (opr A (vH) B) (` (_|_) A)) (` (_|_) A) sstr mpan2 syl2an jca (e. B (Atoms)) adantll (-. (= (i^i (opr A (vH) B) (` (_|_) A)) (0H))) adantrr sylc jca (cv x) atelch atoml.1 A (cv x) chub1t mpan syl (e. B (Atoms)) (/\ (C_ (cv x) (i^i (opr A (vH) B) (` (_|_) A))) (-. (= (i^i (opr A (vH) B) (` (_|_) A)) (0H)))) ad2antlr sylc eqssd (` (_|_) A) ineq1d eqtr3d (Atoms) eleq1d exp43 com24 imp31 ibd ex com23 r19.23adv mpd ex con1d orrd (i^i (opr A (vH) B) (` (_|_) A)) (Atoms) ({} (0H)) elun (_|_) A fvex (opr A (vH) B) inex2 (0H) elsnc (e. (i^i (opr A (vH) B) (` (_|_) A)) (Atoms)) orbi2i bitr sylibr)) thm (atoml2 () ((atoml.1 (e. A (CH)))) (-> (/\ (e. B (Atoms)) (-. (C_ B A))) (e. (i^i (opr A (vH) B) (` (_|_) A)) (Atoms))) (B atelch atoml.1 A B pjoml5t mpan syl (= (i^i (opr A (vH) B) (` (_|_) A)) (0H)) adantr (opr A (vH) B) (` (_|_) A) incom (0H) eqeq1i biimp A (vH) opreq2d atoml.1 chj0 syl6eq (e. B (Atoms)) adantl eqtr3d ex B atelch atoml.1 B A chlejb2t mpan2 syl sylibrd con3d atoml.1 B atoml (i^i (opr A (vH) B) (` (_|_) A)) (Atoms) ({} (0H)) elun h0elch elisseti (i^i (opr A (vH) B) (` (_|_) A)) elsnc2 (e. (i^i (opr A (vH) B) (` (_|_) A)) (Atoms)) orbi2i (e. (i^i (opr A (vH) B) (` (_|_) A)) (Atoms)) (= (i^i (opr A (vH) B) (` (_|_) A)) (0H)) orcom 3bitr sylib ord syld imp)) thm (atord () ((atoml.1 (e. A (CH)))) (-> (/\ (e. B (Atoms)) (br A (C_H) B)) (\/ (C_ B A) (C_ B (` (_|_) A)))) (atoml.1 choccl (` (_|_) A) B chinclt mpan (i^i (` (_|_) A) B) chj0t syl (` (_|_) A) B incom syl6eq atoml.1 choccl (` (_|_) A) B chinclt mpan h0elch (i^i (` (_|_) A) B) (0H) chjcomt mpan2 syl eqtr3d (` (_|_) A) A incom atoml.1 chocin eqtr (vH) (i^i (` (_|_) A) B) opreq1i syl6eqr (br A (C_H) B) adantr atoml.1 choccl atoml.1 atoml.1 atoml.1 atoml.1 cmid cmcm2i (` (_|_) A) A B fh2t mpanr1 mp3anl2 mpanl1 eqtr4d B atelch sylan (` (_|_) A) (opr A (vH) B) incom syl6eq (-. (C_ B A)) adantr atoml.1 B atoml2 (br A (C_H) B) adantlr eqeltrd atoml.1 choccl B (` (_|_) A) atssmat mpan2 (br A (C_H) B) (-. (C_ B A)) ad2antrr mpbird ex orrd)) thm (atcvatlem ((A x) (B x) (C x)) ((atoml.1 (e. A (CH)))) (-> (/\ (/\ (e. B (Atoms)) (e. C (Atoms))) (/\ (-. (= A (0H))) (C: A (opr B (vH) C)))) (-> (-. (C_ B A)) (e. A (Atoms)))) ((cv x) B atnem0 (cv x) B A sseq1 biimpcd con3d imp (C: A (opr B (vH) C)) adantrl syl5bi (cv x) B cvp (cv x) B chjcomt B atelch sylan2 (cv x) ( (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= A (0H))) (C: A (opr B (vH) C))) (e. A (Atoms)))) (atoml.1 B C A chlubt 3comr (opr B (vH) C) A ssnpss syl6bi con2d (C_ B A) (C_ C A) ianor syl6ib mp3an1 B atelch C atelch syl2an imp (-. (= A (0H))) adantrl atoml.1 B C atcvatlem C B chjcomt C atelch B atelch syl2an A psseq2d (-. (= A (0H))) anbi2d atoml.1 C B atcvatlem ex sylbird ancoms imp jaod mpd ex)) thm (atcvat2 () ((atoml.1 (e. A (CH)))) (-> (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= B C)) (br A ( (/\/\ (e. A (CH)) (e. B (Atoms)) (br A (C_H) B)) (\/ (C_ B A) (C_ B (` (_|_) A)))) (A (if (e. A (CH)) A (0H)) (C_H) B breq1 (e. B (Atoms)) anbi2d A (if (e. A (CH)) A (0H)) B sseq2 A (if (e. A (CH)) A (0H)) (_|_) fveq2 B sseq2d orbi12d imbi12d h0elch A elimel B atord dedth 3impib)) thm (atcvat2t () () (-> (/\/\ (e. A (CH)) (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= B C)) (br A ( (/\ (/\ (e. (cv p) (Atoms)) (/\ (e. (cv q) (CH)) (C_ (cv q) (` (_|_) A)))) (/\ (/\ (e. (cv r) (Atoms)) (C_ (cv r) A)) (C_ (cv r) (opr (cv p) (vH) (cv q))))) (= (i^i (cv p) (` (_|_) (cv r))) (0H))) ((cv q) (` (_|_) A) (` (_|_) (cv r)) sstr2 irred.1 (cv r) A chsscon3t mpan2 biimpa (cv r) atelch sylan syl5 (cv r) atn0 (cv r) (0H) df-ne sylib (C_ (cv q) (` (_|_) (cv r))) adantr (/\ (e. (cv p) (CH)) (e. (cv q) (CH))) (C_ (cv r) (opr (cv p) (vH) (cv q))) ad2antrr (/\ (/\ (e. (cv r) (Atoms)) (C_ (cv q) (` (_|_) (cv r)))) (/\ (e. (cv p) (CH)) (e. (cv q) (CH)))) (C_ (cv r) (opr (cv p) (vH) (cv q))) pm3.27 (C_ (cv p) (` (_|_) (cv r))) adantr (cv p) (` (_|_) (cv r)) (cv q) chlej1t (cv r) atelch (cv r) chocclt syl syl3an2 3com12 3expb imp (C_ (cv q) (` (_|_) (cv r))) adantllr (C_ (cv r) (opr (cv p) (vH) (cv q))) adantlr sstrd (cv q) (` (_|_) (cv r)) chlejb2t ancoms biimpa (cv r) atelch (cv r) chocclt syl sylanl1 an1rs (e. (cv p) (CH)) adantrl (C_ (cv r) (opr (cv p) (vH) (cv q))) (C_ (cv p) (` (_|_) (cv r))) ad2antrr sseqtrd ex (cv r) atelch (cv r) chssoct biimpd syl (C_ (cv q) (` (_|_) (cv r))) adantr (/\ (e. (cv p) (CH)) (e. (cv q) (CH))) (C_ (cv r) (opr (cv p) (vH) (cv q))) ad2antrr syld mtod ex (cv p) atelch sylanr1 (` (_|_) (cv r)) (cv p) atnssm0 (` (_|_) (cv r)) (cv p) incom (0H) eqeq1i syl6bb (cv r) atelch (cv r) chocclt syl sylan (C_ (cv q) (` (_|_) (cv r))) (e. (cv q) (CH)) ad2ant2r sylibd exp43 (C_ (cv r) A) adantr sylcom com4t imp3a imp43)) thm (irredlem2 ((p q) (p r) (A p) (q r) (A q) (A r)) ((irred.1 (e. A (CH)))) (-> (/\ (/\ (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) (/\ (e. (cv q) (CH)) (C_ (cv q) (` (_|_) A)))) (/\ (/\ (e. (cv r) (Atoms)) (C_ (cv r) A)) (C_ (cv r) (opr (cv p) (vH) (cv q))))) (= (i^i (` (_|_) (cv r)) (opr (cv p) (vH) (cv q))) (cv q))) ((cv p) (cv q) chjcomt (cv p) atelch sylan (C_ (cv p) A) (C_ (cv q) (` (_|_) A)) ad2ant2r (/\ (/\ (e. (cv r) (Atoms)) (C_ (cv r) A)) (C_ (cv r) (opr (cv p) (vH) (cv q)))) adantr (` (_|_) (cv r)) ineq2d (` (_|_) (cv r)) (cv q) (cv p) fh2t (cv r) atelch (cv r) chocclt syl (e. (cv q) (CH)) id (cv p) atelch 3anim123i 3com13 3expa (C_ (cv p) A) adantllr (C_ (cv q) (` (_|_) A)) adantlrr (C_ (cv r) A) adantrr (C_ (cv r) (opr (cv p) (vH) (cv q))) adantrr (cv q) (` (_|_) (cv r)) lecmt (e. (cv q) (CH)) (C_ (cv q) (` (_|_) A)) pm3.26 (/\ (e. (cv r) (Atoms)) (C_ (cv r) A)) adantr (cv r) atelch (cv r) chocclt syl (/\ (e. (cv q) (CH)) (C_ (cv q) (` (_|_) A))) (C_ (cv r) A) ad2antrl (cv q) (` (_|_) A) (` (_|_) (cv r)) sstr (cv r) atelch irred.1 (cv r) A chsscon3t mpan2 syl biimpa sylan2 (e. (cv q) (CH)) adantll syl3anc (C_ (cv r) (opr (cv p) (vH) (cv q))) adantrr (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) adantll (cv q) (` (_|_) A) (` (_|_) (cv p)) sstr irred.1 (cv p) A chsscon3t mpan2 biimpa sylan2 an1s ancom2s (e. (cv q) (CH)) adantll (cv q) (` (_|_) (cv p)) lecmt (cv p) chocclt syl3an2 3expa ex (cv q) (cv p) cmcm2t sylibrd (/\ (C_ (cv p) A) (C_ (cv q) (` (_|_) A))) adantr mpd (cv p) atelch sylanl2 ancom1s an4s (/\ (/\ (e. (cv r) (Atoms)) (C_ (cv r) A)) (C_ (cv r) (opr (cv p) (vH) (cv q)))) adantr jca sylanc (cv q) (` (_|_) A) (` (_|_) (cv r)) sstr (cv r) atelch irred.1 (cv r) A chsscon3t mpan2 syl biimpa sylan2 (e. (cv q) (CH)) adantll (cv q) (` (_|_) (cv r)) sseqin2 sylib (C_ (cv r) (opr (cv p) (vH) (cv q))) adantrr (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) adantll irred.1 p q r irredlem1 (C_ (cv p) A) adantllr (` (_|_) (cv r)) (cv p) incom syl5eq (vH) opreq12d (cv q) chj0t (C_ (cv q) (` (_|_) A)) adantr (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) (/\ (/\ (e. (cv r) (Atoms)) (C_ (cv r) A)) (C_ (cv r) (opr (cv p) (vH) (cv q)))) ad2antlr eqtrd 3eqtrd)) thm (irredlem3 ((p q) (p r) (p x) (A p) (q r) (q x) (A q) (r x) (A r) (A x)) ((irred.1 (e. A (CH))) (irred.2 (-> (e. (cv x) (CH)) (br A (C_H) (cv x))))) (-> (/\ (/\ (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) (/\ (e. (cv q) (Atoms)) (C_ (cv q) (` (_|_) A)))) (/\ (e. (cv r) (Atoms)) (C_ (cv r) (opr (cv p) (vH) (cv q))))) (-> (C_ (cv r) A) (= (cv r) (cv p)))) (irred.1 p q r irredlem2 (cv r) (vH) opreq2d (cv r) (opr (cv p) (vH) (cv q)) pjoml2t (cv r) atelch (C_ (cv r) A) adantr (cv p) (cv q) chjclt (cv p) atelch sylan (C_ (cv p) A) (C_ (cv q) (` (_|_) A)) ad2ant2r (C_ (cv r) (opr (cv p) (vH) (cv q))) id syl3an 3com12 3expb eqtr3d A ineq2d (cv x) (cv r) A (C_H) breq2 irred.2 vtoclga (cv x) (cv q) A (C_H) breq2 irred.2 vtoclga anim12i irred.1 A (cv r) (cv q) fh1t mp3anl1 mpdan (cv r) atelch sylan ancoms (C_ (cv r) A) adantrr (C_ (cv q) (` (_|_) A)) (C_ (cv r) (opr (cv p) (vH) (cv q))) ad2ant2r (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) adantll (cv x) (cv p) A (C_H) breq2 irred.2 vtoclga (cv x) (cv q) A (C_H) breq2 irred.2 vtoclga anim12i irred.1 A (cv p) (cv q) fh1t mp3anl1 mpdan (cv p) atelch sylan (C_ (cv p) A) (C_ (cv q) (` (_|_) A)) ad2ant2r (/\ (/\ (e. (cv r) (Atoms)) (C_ (cv r) A)) (C_ (cv r) (opr (cv p) (vH) (cv q)))) adantr 3eqtr3d (cv r) A sseqin2 biimp (e. (cv r) (Atoms)) (C_ (cv r) (opr (cv p) (vH) (cv q))) ad2antlr (/\ (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) (/\ (e. (cv q) (CH)) (C_ (cv q) (` (_|_) A)))) adantl (cv q) chsh irred.1 chshi (cv q) A orthin mpan2 syl imp A (cv q) incom syl5eq (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) (/\ (/\ (e. (cv r) (Atoms)) (C_ (cv r) A)) (C_ (cv r) (opr (cv p) (vH) (cv q)))) ad2antlr (vH) opreq12d (cv p) A sseqin2 biimp (e. (cv p) (Atoms)) adantl (/\ (e. (cv q) (CH)) (C_ (cv q) (` (_|_) A))) (/\ (/\ (e. (cv r) (Atoms)) (C_ (cv r) A)) (C_ (cv r) (opr (cv p) (vH) (cv q)))) ad2antrr (cv q) chsh irred.1 chshi (cv q) A orthin mpan2 syl imp A (cv q) incom syl5eq (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) (/\ (/\ (e. (cv r) (Atoms)) (C_ (cv r) A)) (C_ (cv r) (opr (cv p) (vH) (cv q)))) ad2antlr (vH) opreq12d 3eqtr3d (cv r) atelch (cv r) chj0t syl (C_ (cv r) A) (C_ (cv r) (opr (cv p) (vH) (cv q))) ad2antrr (/\ (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) (/\ (e. (cv q) (CH)) (C_ (cv q) (` (_|_) A)))) adantl (cv p) atelch (cv p) chj0t syl (C_ (cv p) A) adantr (/\ (e. (cv q) (CH)) (C_ (cv q) (` (_|_) A))) (/\ (/\ (e. (cv r) (Atoms)) (C_ (cv r) A)) (C_ (cv r) (opr (cv p) (vH) (cv q)))) ad2antrr 3eqtr3d exp44 com34 (cv q) atelch sylanr1 imp32)) thm (irredlem4 ((p q) (p r) (p x) (A p) (q r) (q x) (A q) (r x) (A r) (A x)) ((irred.1 (e. A (CH))) (irred.2 (-> (e. (cv x) (CH)) (br A (C_H) (cv x))))) (-> (/\ (/\ (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) (/\ (e. (cv q) (Atoms)) (C_ (cv q) (` (_|_) A)))) (/\ (e. (cv r) (Atoms)) (C_ (cv r) (opr (cv p) (vH) (cv q))))) (\/ (= (cv r) (cv p)) (= (cv r) (cv q)))) ((cv r) atelch (cv x) (cv r) A (C_H) breq2 irred.2 vtoclga syl irred.1 (cv r) atord mpdan (/\ (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) (/\ (e. (cv q) (Atoms)) (C_ (cv q) (` (_|_) A)))) (C_ (cv r) (opr (cv p) (vH) (cv q))) ad2antrl irred.1 irred.2 p q r irredlem3 (cv q) (cv p) chjcomt (cv q) atelch (cv p) atelch syl2an (cv r) sseq2d (e. (cv r) (Atoms)) anbi2d (C_ (cv q) (` (_|_) A)) (C_ (cv p) (` (_|_) (` (_|_) A))) ad2ant2r irred.1 choccl irred.2 irred.1 A (cv x) cmcm3t mpan mpbid q p r irredlem3 ex sylbird irred.1 ococ (cv p) sseq2i biimpr sylanr2 imp ancom1s orim12d mpd)) thm (irred ((p q) (p r) (p x) (A p) (q r) (q x) (A q) (r x) (A r) (A x)) ((irred.1 (e. A (CH))) (irred.2 (-> (e. (cv x) (CH)) (br A (C_H) (cv x))))) (\/ (= A (0H)) (= A (H~))) ((0H) eqid (= A (0H)) (= (` (_|_) A) (0H)) ioran A (0H) df-ne (` (_|_) A) (0H) df-ne anbi12i bitr4 irred.1 p hatomic irred.1 choccl q hatomic anim12i p (Atoms) q (Atoms) (C_ (cv p) A) (C_ (cv q) (` (_|_) A)) reeanv sylibr (cv p) (cv q) r superpos (e. (cv p) (Atoms)) (C_ (cv p) A) pm3.26 (/\ (e. (cv q) (Atoms)) (C_ (cv q) (` (_|_) A))) adantr (e. (cv q) (Atoms)) (C_ (cv q) (` (_|_) A)) pm3.26 (/\ (e. (cv p) (Atoms)) (C_ (cv p) A)) adantl (cv q) (` (_|_) A) (` (_|_) (cv p)) sstr (cv p) atelch irred.1 (cv p) A chsscon3t mpan2 syl biimpa sylan2 ancoms (cv p) atn0 (C_ (cv q) (` (_|_) (cv p))) adantr (cv p) (cv q) (` (_|_) (cv p)) sseq1 bicomd (cv p) atelch (cv p) chssoct syl sylan9bbr biimpa an1rs ex necon3d mpd (C_ (cv p) A) adantlr syldan (e. (cv q) (Atoms)) adantrl syl3anc irred.1 irred.2 p q r irredlem4 anassrs (-. (= (0H) (0H))) pm2.21nd ex com23 imp3a (=/= (cv r) (cv p)) (=/= (cv r) (cv q)) (C_ (cv r) (opr (cv p) (vH) (cv q))) df-3an (cv r) (cv p) df-ne (cv r) (cv q) df-ne anbi12i (= (cv r) (cv p)) (= (cv r) (cv q)) ioran bitr4 (C_ (cv r) (opr (cv p) (vH) (cv q))) anbi1i bitr syl5ib ex r19.23adv mpd an4s ex r19.23aivv syl sylbi a3i ax-mp (` (_|_) A) (0H) (_|_) fveq2 irred.1 ococ choc0 3eqtr3g (= A (0H)) orim2i ax-mp)) thm (irredt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) (\/ (= A (0H)) (= A (H~)))) (A (if (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) A (0H)) (0H) eqeq1 A (if (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) A (0H)) (H~) eqeq1 orbi12d A (if (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) A (0H)) (CH) eleq1 (e. (cv y) A) x ax-17 (e. A (CH)) x ax-17 x (CH) (br A (C_H) (cv x)) hbra1 hban (e. (cv y) A) x ax-17 (e. (cv y) (0H)) x ax-17 hbif hbeq A (if (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) A (0H)) (C_H) (cv x) breq1 (CH) ralbid anbi12d (0H) (if (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) A (0H)) (CH) eleq1 (e. (cv y) (0H)) x ax-17 (e. A (CH)) x ax-17 x (CH) (br A (C_H) (cv x)) hbra1 hban (e. (cv y) A) x ax-17 (e. (cv y) (0H)) x ax-17 hbif hbeq (0H) (if (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) A (0H)) (C_H) (cv x) breq1 (CH) ralbid anbi12d h0elch (cv x) cm0t rgen pm3.2i elimhyp pm3.26i A (if (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) A (0H)) (CH) eleq1 (e. (cv y) A) x ax-17 (e. A (CH)) x ax-17 x (CH) (br A (C_H) (cv x)) hbra1 hban (e. (cv y) A) x ax-17 (e. (cv y) (0H)) x ax-17 hbif hbeq A (if (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) A (0H)) (C_H) (cv x) breq1 (CH) ralbid anbi12d (0H) (if (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) A (0H)) (CH) eleq1 (e. (cv y) (0H)) x ax-17 (e. A (CH)) x ax-17 x (CH) (br A (C_H) (cv x)) hbra1 hban (e. (cv y) A) x ax-17 (e. (cv y) (0H)) x ax-17 hbif hbeq (0H) (if (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) A (0H)) (C_H) (cv x) breq1 (CH) ralbid anbi12d h0elch (cv x) cm0t rgen pm3.2i elimhyp pm3.27i (e. A (CH)) x ax-17 x (CH) (br A (C_H) (cv x)) hbra1 hban (e. (cv z) A) x ax-17 (e. (cv z) (0H)) x ax-17 hbif (e. (cv z) (C_H)) x ax-17 (e. (cv z) (cv y)) x ax-17 hbbr (cv x) (cv y) (if (/\ (e. A (CH)) (A.e. x (CH) (br A (C_H) (cv x)))) A (0H)) (C_H) breq2 (CH) rcla4 mpi irred dedth)) thm (atcvat3 () ((atcvat3.1 (e. A (CH)))) (-> (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (/\ (-. (= B C)) (-. (C_ C A))) (C_ B (opr A (vH) C))) (e. (i^i A (opr B (vH) C)) (Atoms)))) (atcvat3.1 A C chcv1t mpan biimpa (e. B (Atoms)) adantll (C_ B (opr A (vH) C)) adantrr B C chjcomt A (vH) opreq2d atcvat3.1 A C B chjasst mp3an1 ancoms eqtr4d (C_ B (opr A (vH) C)) adantr B (opr A (vH) C) (opr A (vH) C) chlej2t (e. B (CH)) (e. C (CH)) pm3.26 atcvat3.1 A C chjclt mpan (e. B (CH)) adantl atcvat3.1 A C chjclt mpan (e. B (CH)) adantl syl3anc imp eqsstrd atcvat3.1 A C chjclt mpan (opr A (vH) C) chjidmt syl (e. B (CH)) (C_ B (opr A (vH) C)) ad2antlr sseqtrd atcvat3.1 C (opr B (vH) C) A chlej2t mp3an3 (e. B (CH)) (e. C (CH)) pm3.27 B C chjclt jca C B chub2t ancoms sylc (C_ B (opr A (vH) C)) adantr eqssd B atelch C atelch anim12i sylan A ( (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= A (0H))) (C_ B (opr A (vH) C))) (E.e. x (Atoms) (/\ (C_ (cv x) A) (C_ B (opr C (vH) (cv x))))))) (B C (opr C (vH) (cv x)) sseq1 C (cv x) chub1t C atelch (cv x) atelch syl2an syl5bir exp3a impcom (C_ (cv x) A) anim2d exp3a com23 r19.22dv atcvat3.1 x hatomicOLD syl5 ex (C_ B (opr A (vH) C)) a1i com4l imp4a (e. B (Atoms)) adantl atcvat3.1 C A chlejb2t mpan2 biimpa B sseq2d biimpa anasss ex (e. B (CH)) adantl B C chub2t jctird B atelch C atelch syl2an (e. B (Atoms)) (e. C (Atoms)) pm3.26 jctild imp anassrs (cv x) B A sseq1 (cv x) B C (vH) opreq2 B sseq2d anbi12d (Atoms) rcla4ev syl (-. (= A (0H))) adantrl exp31 atcvat3.1 B C atcvat3 C (i^i A (opr B (vH) C)) B atexcht C atelch (e. B (Atoms)) (/\ (/\ (-. (= B C)) (-. (C_ C A))) (C_ B (opr A (vH) C))) ad2antlr atcvat3.1 B C atcvat3 imp (e. B (Atoms)) (e. C (Atoms)) pm3.26 (/\ (/\ (-. (= B C)) (-. (C_ C A))) (C_ B (opr A (vH) C))) adantr 3jca A (opr B (vH) C) inss2 (/\ (e. B (Atoms)) (e. C (Atoms))) a1i B C chjcomt B atelch C atelch syl2an sseqtrd (/\ (/\ (-. (= B C)) (-. (C_ C A))) (C_ B (opr A (vH) C))) adantr atcvat3.1 A C atnssm0 mpan (e. B (Atoms)) adantl C (i^i A (opr B (vH) C)) chinclt (e. B (CH)) (e. C (CH)) pm3.27 B C chjclt atcvat3.1 A (opr B (vH) C) chinclt mpan syl sylanc B atelch C atelch syl2an (i^i C (i^i A (opr B (vH) C))) chle0t syl A (opr B (vH) C) inss1 (i^i A (opr B (vH) C)) A C sslin ax-mp C A incom sseqtr (i^i A C) (0H) (i^i C (i^i A (opr B (vH) C))) sseq2 mpbii syl5bi sylbid imp (-. (= B C)) adantrl (C_ B (opr A (vH) C)) adantrr jca sylc A (opr B (vH) C) inss1 jctil ex jcad (cv x) (i^i A (opr B (vH) C)) A sseq1 (cv x) (i^i A (opr B (vH) C)) C (vH) opreq2 B sseq2d anbi12d (Atoms) rcla4ev syl6 exp3a (= B C) (C_ C A) ioran syl5ib (-. (= A (0H))) (C_ B (opr A (vH) C)) pm3.27 syl7 ecase3d)) thm (atdmd () () (-> (/\ (e. A (Atoms)) (e. B (CH))) (br (` (_|_) A) (MH) (` (_|_) B))) (B A cvp B A chjcomt A atelch sylan2 B ( (/\ (e. A (Atoms)) (e. B (CH))) (br A (MH) B)) (A (cv x) atdmd r19.21aiva A atelch A x mddmd syl mpbird (e. B (CH)) adantr (cv x) B A (MH) breq2 (CH) rcla4v (e. A (Atoms)) adantl mpd)) thm (atmd2 () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (br A (MH) B)) (A B cvp A B cvexcht A B cvmdt sylbird B atelch sylan2 sylbid A B atnssm0 con1bid B A ssmd2 ancoms B atelch sylan2 sylbid pm2.61d)) thm (atabs () ((atabs.1 (e. A (CH))) (atabs.2 (e. B (CH)))) (-> (e. C (Atoms)) (-> (-. (C_ C (opr A (vH) B))) (= (i^i (opr A (vH) C) B) (i^i A B)))) (atabs.1 atabs.2 chub1 atabs.1 atabs.2 chjcl atabs.1 C (opr A (vH) B) A mdit mp3an23 C atelch atabs.1 atabs.2 chjcl C (opr A (vH) B) atmd mpan2 sylc mpi (-. (C_ C (opr A (vH) B))) adantr atabs.1 atabs.2 chjcl (opr A (vH) B) C atnssm0 mpan biimpa C (opr A (vH) B) incom syl5eq A (vH) opreq2d atabs.1 chj0 syl6eq eqtrd B ineq1d (opr A (vH) C) (opr A (vH) B) B inass atabs.1 atabs.2 chjcom B ineq1i (opr B (vH) A) B incom atabs.2 atabs.1 chabs2 3eqtr (opr A (vH) C) ineq2i eqtr2 syl5eq ex)) thm (atabs2 () ((atabs.1 (e. A (CH))) (atabs.2 (e. B (CH)))) (-> (e. C (Atoms)) (-> (-. (C_ C (opr A (vH) B))) (= (i^i (opr A (vH) C) (opr A (vH) B)) A))) (atabs.1 atabs.1 atabs.2 chjcl C atabs atabs.1 atabs.1 atabs.2 chjass atabs.1 chjidm (vH) B opreq1i eqtr3 C sseq2i negbii atabs.1 atabs.2 chabs2 (i^i (opr A (vH) C) (opr A (vH) B)) eqeq2i 3imtr3g)) thm (mdsymlem1 ((A p) (B p)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (/\ (e. (cv p) (CH)) (C_ (i^i B C) A)) (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (C_ (cv p) (opr A (vH) B)))) (C_ (cv p) A)) (mdsymlem1.1 (cv p) A chub2t mpan2 mdsymlem1.3 syl6ssr mdsymlem1.1 mdsymlem1.2 chjcom (cv p) sseq2i biimp anim12i (cv p) C (opr B (vH) A) ssin sylib (br (` (_|_) B) (MH) (` (_|_) A)) adantrl (C_ (i^i B C) A) adantlr mdsymlem1.1 A (cv p) chub1t mpan mdsymlem1.3 syl6ssr (br (` (_|_) B) (MH) (` (_|_) A)) adantr mdsymlem1.1 A (cv p) chjclt mpan mdsymlem1.3 syl5eqel mdsymlem1.2 mdsymlem1.1 B A C dmditOLD mp3an12 syl imp mpd (C_ (i^i B C) A) adantlr mdsymlem1.1 A (cv p) chjclt mpan mdsymlem1.3 syl5eqel mdsymlem1.2 B C chinclt mpan mdsymlem1.1 (i^i B C) A chlejb1t mpan2 3syl biimpa C B incom (vH) A opreq1i syl5eq (br (` (_|_) B) (MH) (` (_|_) A)) adantr eqtr3d (C_ (cv p) (opr A (vH) B)) adantrr sseqtrd)) thm (mdsymlem2 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (/\ (e. (cv p) (Atoms)) (C_ (i^i B C) A)) (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (C_ (cv p) (opr A (vH) B)))) (-> (-. (= B (0H))) (E.e. r (Atoms) (E.e. q (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))) ((cv q) (cv p) (vH) (cv r) opreq1 (cv p) sseq2d (cv q) (cv p) A sseq1 (C_ (cv r) B) anbi1d anbi12d (Atoms) rcla4ev (e. (cv p) (Atoms)) (C_ (i^i B C) A) pm3.26 (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (C_ (cv p) (opr A (vH) B))) (/\ (e. (cv r) (Atoms)) (C_ (cv r) B)) ad2antrr (cv p) (cv r) chub1t (cv p) atelch (cv r) atelch syl2an (C_ (i^i B C) A) adantlr (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (C_ (cv p) (opr A (vH) B))) adantlr mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 mdsymlem1 (cv p) atelch sylanl1 (e. (cv r) (Atoms)) adantr jca (C_ (cv r) B) anim1i (C_ (cv p) (opr (cv p) (vH) (cv r))) (C_ (cv p) A) (C_ (cv r) B) anass sylib anasss sylanc exp32 r19.22dv mdsymlem1.2 r hatomicOLD syl5)) thm (mdsymlem3 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (/\ (/\ (e. (cv p) (Atoms)) (-. (C_ (i^i B C) A))) (C_ (cv p) (opr A (vH) B))) (-. (= A (0H)))) (E.e. r (Atoms) (E.e. q (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B)))))) ((cv p) atelch mdsymlem1.1 A (cv p) chjclt mpan mdsymlem1.3 syl5eqel mdsymlem1.2 B C chinclt mpan syl mdsymlem1.1 (i^i B C) A r chrelat2t mpan2 3syl biimpa (C_ (cv p) (opr A (vH) B)) (-. (= A (0H))) ad2antrr mdsymlem1.1 (cv r) (cv p) q atcvat4 exp4b com34 com23 imp4b (cv r) B C ssin mdsymlem1.3 (cv r) sseq2i biimp (C_ (cv r) B) adantl sylbir sylan2 (-. (C_ (cv r) A)) adantrr com12 (-. (C_ (i^i B C) A)) adantlr (C_ (cv p) (opr A (vH) B)) adantlr imp (cv q) (cv r) atnem0 ancoms (cv q) (cv r) A sseq1 biimpcd con3d imp (C_ (cv r) (i^i B C)) adantrl syl5bi (e. (cv p) (Atoms)) adantll (C_ (cv r) (opr (cv p) (vH) (cv q))) adantr (cv p) (cv q) chjcomt (cv p) atelch (cv q) atelch syl2an (e. (cv r) (Atoms)) adantlr (cv r) sseq2d (cv q) (cv r) (cv p) atexcht (cv q) atelch syl3an1 3com13 3expa exp3a sylbid imp syld exp3a exp31 com24 imp3a com24 imp4b anasss (C_ (cv q) A) (C_ (cv r) (opr (cv p) (vH) (cv q))) pm3.26 (e. (cv q) (Atoms)) adantl (/\ (e. (cv p) (Atoms)) (/\ (e. (cv r) (Atoms)) (/\ (C_ (cv r) (i^i B C)) (-. (C_ (cv r) A))))) a1i (cv r) B C ssin (C_ (cv r) B) (C_ (cv r) C) pm3.26 sylbir (e. (cv r) (Atoms)) (-. (C_ (cv r) A)) ad2antrl (e. (cv p) (Atoms)) adantl jctird jcad exp3a (-. (C_ (i^i B C) A)) adantlr (C_ (cv p) (opr A (vH) B)) adantlr (-. (= A (0H))) adantlr r19.22dv mpd exp32 r19.22dv mpd)) thm (mdsymlem4 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (e. (cv p) (Atoms)) (-> (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (/\ (/\ (-. (= A (0H))) (-. (= B (0H)))) (C_ (cv p) (opr A (vH) B)))) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))) (mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 r q mdsymlem2 exp31 com4t ex com23 (-. (= A (0H))) a1d imp44 com3l mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 r q mdsymlem3 anasss exp31 com3r ancoms (-. (= B (0H))) adantlr (br (` (_|_) B) (MH) (` (_|_) A)) adantl com3l pm2.61d r (Atoms) q (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))) rexcom syl6ib)) thm (mdsymlem5 ((q r) (C q) (C r) (c p) (c q) (c r) (A c) (p q) (p r) (A p) (A q) (A r) (B c) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (e. (cv q) (Atoms)) (e. (cv r) (Atoms))) (-> (-. (= (cv q) (cv p))) (-> (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))) (-> (/\ (/\ (e. (cv c) (CH)) (C_ A (cv c))) (e. (cv p) (Atoms))) (-> (C_ (cv p) (cv c)) (C_ (cv p) (opr (i^i (cv c) B) (vH) A))))))) ((cv q) (cv p) atnem0 (C_ (cv p) (opr (cv q) (vH) (cv r))) anbi2d (e. (cv r) (Atoms)) 3adant3 (cv q) (cv p) (cv r) atexcht (cv q) atelch syl3an1 sylbid exp3a 3com23 3expa (e. (cv c) (CH)) adantrl (/\ (C_ (cv q) A) (C_ (cv r) B)) adantrd imp32 (C_ A (cv c)) adantrl (C_ (cv p) (cv c)) adantrr (cv q) A (opr (cv c) (vH) A) sstr mdsymlem1.1 A (cv c) chub2t mpan sylan2 (cv p) (cv c) (opr (cv c) (vH) A) sstr mdsymlem1.1 (cv c) A chub1t mpan2 sylan2 anim12i anandirs ancoms (/\ (e. (cv q) (CH)) (e. (cv p) (CH))) adantll (cv q) (cv p) (opr (cv c) (vH) A) chlubt mdsymlem1.1 (cv c) A chjclt mpan2 syl3an3 3expa (/\ (C_ (cv q) A) (C_ (cv p) (cv c))) adantr mpbid (C_ A (cv c)) adantrl mdsymlem1.1 A (cv c) chlejb2t mpan biimpa (/\ (e. (cv q) (CH)) (e. (cv p) (CH))) adantll (/\ (C_ (cv q) A) (C_ (cv p) (cv c))) adantrr sseqtrd exp45 anasss (cv q) atelch (cv p) atelch (e. (cv c) (CH)) anim1i ancoms syl2an (e. (cv r) (Atoms)) adantlr (C_ (cv q) A) (C_ (cv r) B) pm3.26 (C_ (cv p) (opr (cv q) (vH) (cv r))) (-. (= (cv q) (cv p))) ad2antlr syl7 imp44 sstrd (C_ (cv q) A) (C_ (cv r) B) pm3.27 (C_ (cv p) (opr (cv q) (vH) (cv r))) (-. (= (cv q) (cv p))) ad2antlr (C_ A (cv c)) (C_ (cv p) (cv c)) ad2antlr (/\ (/\ (e. (cv q) (Atoms)) (e. (cv r) (Atoms))) (/\ (e. (cv c) (CH)) (e. (cv p) (Atoms)))) adantl jca (cv r) (cv c) B ssin sylib (cv r) (i^i (cv c) B) (cv q) chlej1t mdsymlem1.2 (cv c) B chinclt mpan2 syl3an2 3comr 3expa (C_ (cv q) A) adantr (opr (cv r) (vH) (cv q)) (opr (i^i (cv c) B) (vH) (cv q)) (opr (i^i (cv c) B) (vH) A) sstr2 mdsymlem1.1 (cv q) A (i^i (cv c) B) chlej2t mp3an2 mdsymlem1.2 (cv c) B chinclt mpan2 sylan2 (e. (cv r) (CH)) adantlr imp syl5com (cv q) (cv r) chjcomt (e. (cv c) (CH)) (C_ (cv q) A) ad2antrr (opr (i^i (cv c) B) (vH) A) sseq1d sylibrd syld (C_ (cv p) (opr (cv q) (vH) (cv r))) adantrl (cv p) (opr (cv q) (vH) (cv r)) (opr (i^i (cv c) B) (vH) A) sstr2 (/\ (/\ (e. (cv q) (CH)) (e. (cv r) (CH))) (e. (cv c) (CH))) (C_ (cv q) A) ad2antrl syld exp32 (cv q) atelch (cv r) atelch anim12i sylan (e. (cv p) (Atoms)) adantrr imp31 (C_ (cv r) B) adantrr anasss (-. (= (cv q) (cv p))) adantrr (C_ A (cv c)) adantrl (C_ (cv p) (cv c)) adantrr mpd exp32 exp4d exp32 com34 imp4c com24)) thm (mdsymlem6 ((q r) (C q) (C r) (c p) (c q) (c r) (A c) (p q) (p r) (A p) (A q) (A r) (B c) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (A.e. p (Atoms) (-> (C_ (cv p) (opr A (vH) B)) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))) (br (` (_|_) B) (MH) (` (_|_) A))) (mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 q r c mdsymlem5 (cv q) (cv p) A sseq1 (cv p) A (opr (i^i (cv c) B) (vH) A) sstr2 mdsymlem1.2 (cv c) B chinclt mpan2 mdsymlem1.1 A (i^i (cv c) B) chub2t mpan syl syl5 syl6bi imp3a (C_ (cv p) (cv c)) a1i com13 (C_ A (cv c)) adantrr (C_ (cv r) B) (e. (cv p) (Atoms)) ad2ant2r (C_ (cv p) (opr (cv q) (vH) (cv r))) adantll com12 exp3a pm2.61d2 r19.23aivv com12 (C_ (cv p) (opr A (vH) B)) imim2d com34 imp4b mdsymlem1.1 mdsymlem1.2 chjcom (cv p) sseq2i (C_ (cv p) (cv c)) anbi2i (cv p) (cv c) (opr B (vH) A) ssin bitr syl5ibr ex r19.20dva (i^i (cv c) (opr B (vH) A)) (opr (i^i (cv c) B) (vH) A) p chrelat3t mdsymlem1.2 mdsymlem1.1 chjcl (cv c) (opr B (vH) A) chinclt mpan2 mdsymlem1.2 (cv c) B chinclt mpan2 mdsymlem1.1 (i^i (cv c) B) A chjclt mpan2 syl sylanc (C_ A (cv c)) adantr sylibrd ex com3r r19.21aiv mdsymlem1.2 mdsymlem1.1 B A c dmdbr2 mp2an sylibr)) thm (mdsymlem7 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (-. (= A (0H))) (-. (= B (0H)))) (<-> (br (` (_|_) B) (MH) (` (_|_) A)) (A.e. p (Atoms) (-> (C_ (cv p) (opr A (vH) B)) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))))) (mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 q r mdsymlem4 exp4d com13 r19.21adv mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 q r mdsymlem6 (/\ (-. (= A (0H))) (-. (= B (0H)))) a1i impbid)) thm (mdsymlem8 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (-. (= A (0H))) (-. (= B (0H)))) (<-> (br (` (_|_) B) (MH) (` (_|_) A)) (br (` (_|_) A) (MH) (` (_|_) B)))) (mdsymlem1.1 mdsymlem1.2 chjcom (cv p) sseq2i (cv q) (cv r) chjcomt (cv q) atelch (cv r) atelch syl2an (cv p) sseq2d (C_ (cv q) A) (C_ (cv r) B) ancom (/\ (e. (cv q) (Atoms)) (e. (cv r) (Atoms))) a1i anbi12d 2rexbiia q (Atoms) r (Atoms) (/\ (C_ (cv p) (opr (cv r) (vH) (cv q))) (/\ (C_ (cv r) B) (C_ (cv q) A))) rexcom bitr imbi12i p (Atoms) ralbii (/\ (-. (= A (0H))) (-. (= B (0H)))) a1i mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 q r mdsymlem7 mdsymlem1.2 mdsymlem1.1 (opr B (vH) (cv p)) eqid r q mdsymlem7 ancoms 3bitr4d)) thm (mdsym ((A x) (B x)) ((mdsym.1 (e. A (CH))) (mdsym.2 (e. B (CH)))) (<-> (br A (MH) B) (br B (MH) A)) (mdsym.2 choccl mdsym.1 choccl (opr (` (_|_) B) (vH) (cv x)) eqid mdsymlem8 mdsym.1 pjococ mdsym.2 pjococ (MH) breq12i mdsym.2 pjococ mdsym.1 pjococ (MH) breq12i 3bitr3g (= (` (_|_) B) (0H)) (= (` (_|_) A) (0H)) oran mdsym.1 chssi (` (_|_) B) (0H) (_|_) fveq2 mdsym.2 pjococ choc0 3eqtr3g A sseq2d mpbiri mdsym.1 mdsym.2 A B ssmd1 mp2an mdsym.1 mdsym.2 A B ssmd2 mp2an jca (br A (MH) B) (br B (MH) A) pm5.1 3syl mdsym.2 chssi (` (_|_) A) (0H) (_|_) fveq2 mdsym.1 pjococ choc0 3eqtr3g B sseq2d mpbiri mdsym.2 mdsym.1 B A ssmd2 mp2an mdsym.2 mdsym.1 B A ssmd1 mp2an jca (br A (MH) B) (br B (MH) A) pm5.1 3syl jaoi sylbir pm2.61i)) thm (mdsymt () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (br B (MH) A))) (A (if (e. A (CH)) A (H~)) (MH) B breq1 A (if (e. A (CH)) A (H~)) B (MH) breq2 bibi12d B (if (e. B (CH)) B (H~)) (if (e. A (CH)) A (H~)) (MH) breq2 B (if (e. B (CH)) B (H~)) (MH) (if (e. A (CH)) A (H~)) breq1 bibi12d helch A elimel helch B elimel mdsym dedth2h)) thm (atdmd2 () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (br (` (_|_) A) (MH) (` (_|_) B))) (B A atdmd ancoms (` (_|_) A) (` (_|_) B) mdsymt A chocclt B atelch B chocclt syl syl2an mpbird)) thm (sumdmdi ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (-> (= (opr A (+H) B) (opr A (vH) B)) (br (` (_|_) A) (MH) (` (_|_) B))) ((opr A (+H) B) (opr A (vH) B) (cv x) ineq2 (/\ (e. (cv x) (CH)) (C_ B (cv x))) adantr (cv x) chsh (cv x) (cv y) (cv w) shsubclt exp3a syl B (cv x) (cv w) ssel2 syl7 exp4a com23 imp41 (e. (cv z) A) adantlr (= (cv y) (opr (cv z) (+v) (cv w))) adantr (cv y) (cv w) (cv z) hvsubaddt (cv w) (cv z) ax-hvcom (cv y) eqeq1d (opr (cv z) (+v) (cv w)) (cv y) eqcom syl6bb (e. (cv y) (H~)) 3adant1 bitrd 3com23 (cv x) (cv y) chelt (C_ B (cv x)) adantlr sumdmdi.1 (cv z) chel sumdmdi.2 (cv w) chel syl3an 3expa (opr (cv y) (-v) (cv w)) (cv z) (cv x) eleq1 syl6bir imp mpbid (/\ (/\ (/\ (/\ (e. (cv x) (CH)) (C_ B (cv x))) (e. (cv y) (cv x))) (e. (cv z) A)) (e. (cv w) B)) (= (cv y) (opr (cv z) (+v) (cv w))) pm3.27 jca exp31 r19.22dv w B (e. (cv z) (cv x)) (= (cv y) (opr (cv z) (+v) (cv w))) r19.42v syl6ib r19.22dva (cv z) (cv x) A elin (e. (cv z) (cv x)) (e. (cv z) A) ancom bitr (E.e. w B (= (cv y) (opr (cv z) (+v) (cv w)))) anbi1i (e. (cv z) A) (e. (cv z) (cv x)) (E.e. w B (= (cv y) (opr (cv z) (+v) (cv w)))) anass bitr rexbii2 syl6ibr (cv x) chsh sumdmdi.1 chshi (cv x) A shinclt mpan2 syl (C_ B (cv x)) (e. (cv y) (cv x)) ad2antrr sumdmdi.2 chshi (i^i (cv x) A) B (cv y) z w shselt mpan2 syl sylibrd sumdmdi.1 sumdmdi.2 (cv y) z w chsel syl5ib ex imp3a (cv y) (cv x) (opr A (+H) B) elin syl5ib ssrdv (= (opr A (+H) B) (opr A (vH) B)) adantl eqsstr3d sumdmdi.1 (cv x) A chinclt mpan2 sumdmdi.2 (i^i (cv x) A) B chslejt mpan2 syl (= (opr A (+H) B) (opr A (vH) B)) (C_ B (cv x)) ad2antrl sstrd exp32 r19.21aiv sumdmdi.1 sumdmdi.2 A B x dmdbr2 mp2an sylibr)) thm (cmmd () ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (-> (br A (C_H) B) (br A (MH) B)) (sumdmdi.1 sumdmdi.2 cmcm4 sumdmdi.1 choccl sumdmdi.2 choccl osumcor2 sylbi sumdmdi.1 choccl sumdmdi.2 choccl sumdmdi syl sumdmdi.1 ococ sumdmdi.2 ococ 3brtr3g)) thm (cmdmd () ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (-> (br A (C_H) B) (br A (MH*) B)) (sumdmdi.1 choccl sumdmdi.2 choccl cmmd sumdmdi.1 sumdmdi.2 cmcm4 sumdmdi.1 sumdmdi.2 A B dmdmdt mp2an 3imtr4)) thm (sumdmdlem ((y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C y) (C z) (C w)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (-> (/\ (e. C (H~)) (-. (e. C (opr A (+H) B)))) (= (i^i (opr B (+H) (` (span) ({} C))) A) (i^i B A))) (C spansnsht sumdmdi.2 chshi B (` (span) ({} C)) (cv y) z w shselt mpan syl (cv y) (cv z) (cv w) hvsubaddt (opr (cv z) (+v) (cv w)) (cv y) eqcom syl6bb sumdmdi.1 (cv y) chel sumdmdi.2 (cv z) chel C (cv w) elspansnclt syl3an 3expa (opr (cv y) (-v) (cv z)) (cv w) (opr A (+H) B) eleq1 sumdmdi.1 chshi sumdmdi.2 chshi (cv y) (cv z) shsvs syl5bi com12 (/\ (e. C (H~)) (e. (cv w) (` (span) ({} C)))) adantr sylbird exp32 com4r imp31 (-. (e. C (opr A (+H) B))) adantrr sumdmdi.1 chshi sumdmdi.2 chshi shscl (opr A (+H) B) C (cv w) elspansn5t ax-mp exp32 (/\ (= (cv y) (opr (cv z) (+v) (cv w))) (/\ (e. (cv y) A) (e. (cv z) B))) adantl mpdd (cv w) (0v) (cv z) (+v) opreq2 (cv z) ax-hvaddid sylan9eqr sumdmdi.2 (cv z) chel sylan (cv y) eqeq2d (e. (cv y) A) adantll biimpac (cv y) B A elin biimpr ancoms (cv y) (cv z) B eleq1 biimparc sylan2 anassrs ex (= (cv y) (opr (cv z) (+v) (cv w))) (= (cv w) (0v)) ad2antrl mpd exp32 imp (e. (cv w) (` (span) ({} C))) a1d (/\ (e. C (H~)) (-. (e. C (opr A (+H) B)))) adantr mpdd ex com23 exp32 com4l imp4c exp4a com23 com4l exp3a r19.23advv sylbid com23 imp4b (cv y) (opr B (+H) (` (span) ({} C))) A elin syl5ib ssrdv C spansnsht sumdmdi.2 chshi B (` (span) ({} C)) shsub1t mpan B (opr B (+H) (` (span) ({} C))) A ssrin 3syl (-. (e. C (opr A (+H) B))) adantr eqssd)) thm (sumdmdlem2 ((x y) (A x) (A y) (B x) (B y)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (-> (A.e. x (Atoms) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))) (= (opr A (+H) B) (opr A (vH) B))) ((cv y) spansnsht sumdmdi.2 chshi (` (span) ({} (cv y))) B shsub2t mpan2 syl (cv y) spansnid sseldd (A.e. x (Atoms) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))) (-. (e. (cv y) (opr A (+H) B))) ad2antrl (cv y) spansnatOLD (cv x) (` (span) ({} (cv y))) (vH) B opreq1 (opr A (vH) B) ineq1d (cv x) (` (span) ({} (cv y))) (vH) B opreq1 A ineq1d (vH) B opreq1d sseq12d (Atoms) rcla4v syl sumdmdi.2 B (cv y) spansnjt B (` (span) ({} (cv y))) chjcomt (cv y) spansncht sylan2 eqtrd mpan (opr A (vH) B) ineq1d sumdmdi.2 B (cv y) spansnjt B (` (span) ({} (cv y))) chjcomt (cv y) spansncht sylan2 eqtrd mpan A ineq1d (vH) B opreq1d sseq12d (-. (= (cv y) (0v))) adantr sylibrd com12 exp3a imp B ssid (cv y) (0v) sneq (span) fveq2d spansn0 syl6eq B (+H) opreq2d sumdmdi.2 chshi shs0 syl6eq (opr A (vH) B) ineq1d B (opr A (vH) B) inss1 B ssid sumdmdi.2 sumdmdi.1 chub2 ssini eqssi syl6eq (cv y) (0v) sneq (span) fveq2d spansn0 syl6eq B (+H) opreq2d sumdmdi.2 chshi shs0 syl6eq A ineq1d (vH) B opreq1d sumdmdi.2 sumdmdi.1 chincl sumdmdi.2 chjcom sumdmdi.2 sumdmdi.1 chabs1 eqtr syl6eq sseq12d mpbiri pm2.61d2 (-. (e. (cv y) (opr A (+H) B))) adantrr sumdmdi.2 chshi sumdmdi.1 chshi shsub2 sumdmdi.1 sumdmdi.2 (cv y) sumdmdlem (vH) B opreq1d sumdmdi.2 sumdmdi.1 chincl sumdmdi.2 chjcom sumdmdi.2 sumdmdi.1 chabs1 eqtr syl6eq (opr A (+H) B) sseq1d mpbiri (A.e. x (Atoms) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))) adantl sstrd (cv y) sseld (cv y) (opr B (+H) (` (span) ({} (cv y)))) (opr A (vH) B) elin syl5ibr mpand exp32 com34 (e. (cv y) (opr A (+H) B)) pm2.18 syl8 sumdmdi.1 sumdmdi.2 chjcl (cv y) chel syl5 pm2.43d ssrdv sumdmdi.1 sumdmdi.2 chslej jctil (opr A (+H) B) (opr A (vH) B) eqss sylibr)) thm (sumdmd ((A x) (B x)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (<-> (= (opr A (+H) B) (opr A (vH) B)) (br (` (_|_) A) (MH) (` (_|_) B))) (sumdmdi.1 sumdmdi.2 sumdmdi sumdmdi.1 sumdmdi.2 A B x dmdbr4 mp2an (cv x) atelch (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)) imim1i r19.20i2 sylbi sumdmdi.1 sumdmdi.2 x sumdmdlem2 syl impbi)) thm (dmdbr4at ((A x) (B x)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (Atoms) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)))) (sumdmdi.1 sumdmdi.2 A B x dmdbr4 mp2an (cv x) atelch (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)) imim1i r19.20i2 sylbi sumdmdi.1 sumdmdi.2 x sumdmdlem2 sumdmdi.1 sumdmdi.2 sumdmd sylib impbi)) thm (dmdbr5at ((A x) (B x)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (Atoms) (-> (C_ (cv x) (opr A (vH) B)) (C_ (cv x) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))))) (sumdmdi.1 sumdmdi.2 A B (cv x) dmdi4 mp3an12 com12 (cv x) atelch syl5 (C_ (cv x) (opr A (vH) B)) a1dd r19.21aiv (cv x) atelch sumdmdi.2 B (cv x) chjcomt mpan syl (opr B (vH) A) ineq1d sumdmdi.1 sumdmdi.2 chjcom (opr (cv x) (vH) B) ineq2i syl6eqr (-. (C_ (cv x) (opr A (vH) B))) adantr sumdmdi.2 sumdmdi.1 (cv x) atabs2 imp sumdmdi.1 sumdmdi.2 chjcom (cv x) sseq2i negbii sylan2b eqtr3d (cv x) atelch sumdmdi.2 (cv x) B chjclt mpan2 syl sumdmdi.1 (opr (cv x) (vH) B) A chinclt mpan2 sumdmdi.2 B (i^i (opr (cv x) (vH) B) A) chub2t mpan 3syl (-. (C_ (cv x) (opr A (vH) B))) adantr eqsstrd ex (-> (C_ (cv x) (opr A (vH) B)) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))) biantrud (C_ (cv x) (opr A (vH) B)) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)) pm4.83 syl6bb ralbiia sumdmdi.1 sumdmdi.2 x sumdmdlem2 sylbi sumdmdi.1 sumdmdi.2 sumdmd sylib impbi (cv x) atelch sumdmdi.2 sumdmdi.1 sumdmdi.2 chjcl (cv x) B (opr A (vH) B) chlubt mp3an23 sumdmdi.2 sumdmdi.1 chub2 (C_ (cv x) (opr A (vH) B)) biantru syl5bb (opr (cv x) (vH) B) ssid (C_ (opr (cv x) (vH) B) (opr A (vH) B)) biantrur (opr (cv x) (vH) B) (opr (cv x) (vH) B) (opr A (vH) B) ssin bitr syl6bb biimpa (opr (cv x) (vH) B) (opr A (vH) B) inss1 jctil (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (cv x) (vH) B) eqss sylibr (opr (i^i (opr (cv x) (vH) B) A) (vH) B) sseq1d sumdmdi.2 (cv x) B chjclt mpan2 sumdmdi.1 (opr (cv x) (vH) B) A chinclt mpan2 sumdmdi.2 B (i^i (opr (cv x) (vH) B) A) chub2t mpan 3syl (C_ (cv x) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)) biantrud sumdmdi.2 (cv x) B chjclt mpan2 sumdmdi.1 (opr (cv x) (vH) B) A chinclt mpan2 sumdmdi.2 (i^i (opr (cv x) (vH) B) A) B chjclt mpan2 3syl sumdmdi.2 (cv x) B (opr (i^i (opr (cv x) (vH) B) A) (vH) B) chlubt mp3an2 mpdan bitrd (C_ (cv x) (opr A (vH) B)) adantr bitr4d ex pm5.74d syl ralbiia bitr)) thm (dmdbr6at ((A x) (B x)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (Atoms) (= (i^i (opr A (vH) B) (cv x)) (i^i (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (cv x))))) (sumdmdi.1 sumdmdi.2 A B x dmdbr3 mp2an sumdmdi.2 (cv x) B chabs2t mpan2 (opr A (vH) B) ineq2d (opr A (vH) B) (i^i (cv x) (opr (cv x) (vH) B)) incom (cv x) (opr (cv x) (vH) B) (opr A (vH) B) inass (cv x) (i^i (opr (cv x) (vH) B) (opr A (vH) B)) incom 3eqtr syl5reqr (= (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (i^i (opr (cv x) (vH) B) (opr A (vH) B))) adantr (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (cv x) ineq1 (e. (cv x) (CH)) adantl eqtr4d ex r19.20i sylbi (cv x) atelch (= (i^i (opr A (vH) B) (cv x)) (i^i (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (cv x))) imim1i r19.20i2 syl (cv x) (opr A (vH) B) df-ss biimp (opr A (vH) B) (cv x) incom syl5eq (opr (i^i (opr (cv x) (vH) B) A) (vH) B) sseq1d (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (cv x) inss1 (i^i (opr A (vH) B) (cv x)) (i^i (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (cv x)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B) sseq1 mpbiri syl5bi com12 x (Atoms) r19.20si sumdmdi.1 sumdmdi.2 x dmdbr5at sylibr impbi)) thm (dmdbr7at ((A x) (B x)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (Atoms) (C_ (i^i (opr A (vH) B) (cv x)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)))) (sumdmdi.1 sumdmdi.2 x dmdbr6at (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (cv x) inss1 (i^i (opr A (vH) B) (cv x)) (i^i (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (cv x)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B) sseq1 mpbiri x (Atoms) r19.20si sylbi (cv x) (opr A (vH) B) sseqin2 biimp (opr (i^i (opr (cv x) (vH) B) A) (vH) B) sseq1d biimpcd x (Atoms) r19.20si sumdmdi.1 sumdmdi.2 x dmdbr5at sylibr impbi)) thm (mdoc1 () ((mdoc1.1 (e. A (CH)))) (br A (MH) (` (_|_) A)) (mdoc1.1 mdoc1.1 mdoc1.1 cmid cmcm2i mdoc1.1 mdoc1.1 choccl cmmd ax-mp)) thm (mdoc2 () ((mdoc1.1 (e. A (CH)))) (br (` (_|_) A) (MH) A) (mdoc1.1 choccl mdoc1 mdoc1.1 ococ breqtr)) thm (dmdoc1 () ((mdoc1.1 (e. A (CH)))) (br A (MH*) (` (_|_) A)) (mdoc1.1 mdoc1.1 mdoc1.1 cmid cmcm2i mdoc1.1 mdoc1.1 choccl cmdmd ax-mp)) thm (dmdoc2 () ((mdoc1.1 (e. A (CH)))) (br (` (_|_) A) (MH*) A) (mdoc1.1 choccl dmdoc1 mdoc1.1 ococ breqtr)) thm (mdcompl () ((mdcompl.1 (e. A (CH))) (mdcompl.2 (e. B (CH)))) (<-> (br A (MH) B) (br (i^i A (` (_|_) (i^i A B))) (MH) (i^i B (` (_|_) (i^i A B))))) (mdcompl.1 mdcompl.2 chincl mdoc1 mdcompl.1 mdcompl.2 chincl dmdoc2 pm3.2i (i^i A B) ssid mdcompl.1 mdcompl.2 chjcl chssi mdcompl.1 mdcompl.2 chincl chjo sseqtr4 pm3.2i mdcompl.1 mdcompl.2 chincl mdcompl.1 mdcompl.2 chincl choccl mdcompl.1 mdcompl.2 mdslmd1 mp2an)) thm (dmdcompl () ((mdcompl.1 (e. A (CH))) (mdcompl.2 (e. B (CH)))) (<-> (br A (MH*) B) (br (i^i A (` (_|_) (i^i A B))) (MH*) (i^i B (` (_|_) (i^i A B))))) (mdcompl.1 mdcompl.2 chincl mdoc1 mdcompl.1 mdcompl.2 chincl dmdoc2 pm3.2i (i^i A B) ssid mdcompl.1 mdcompl.2 chjcl chssi mdcompl.1 mdcompl.2 chincl chjo sseqtr4 pm3.2i mdcompl.1 mdcompl.2 chincl mdcompl.1 mdcompl.2 chincl choccl mdcompl.1 mdcompl.2 mdsldmd1 mp2an)) thm (mddmdin0 ((x y) (A x) (A y) (B x) (B y)) ((mddmdin0.1 (e. A (CH))) (mddmdin0.2 (e. B (CH))) (mddmdin0.3 (A.e. x (CH) (A.e. y (CH) (-> (/\ (br (cv x) (MH*) (cv y)) (= (i^i (cv x) (cv y)) (0H))) (br (cv x) (MH) (cv y))))))) (-> (br A (MH*) B) (br A (MH) B)) (A B (` (_|_) (i^i A B)) inindir mddmdin0.1 mddmdin0.2 chincl chocin eqtr3 mddmdin0.3 mddmdin0.1 mddmdin0.1 mddmdin0.2 chincl choccl chincl mddmdin0.2 mddmdin0.1 mddmdin0.2 chincl choccl chincl (cv x) (i^i A (` (_|_) (i^i A B))) (MH*) (cv y) breq1 (cv x) (i^i A (` (_|_) (i^i A B))) (cv y) ineq1 (0H) eqeq1d anbi12d (cv x) (i^i A (` (_|_) (i^i A B))) (MH) (cv y) breq1 imbi12d (cv y) (i^i B (` (_|_) (i^i A B))) (i^i A (` (_|_) (i^i A B))) (MH*) breq2 (cv y) (i^i B (` (_|_) (i^i A B))) (i^i A (` (_|_) (i^i A B))) ineq2 (0H) eqeq1d anbi12d (cv y) (i^i B (` (_|_) (i^i A B))) (i^i A (` (_|_) (i^i A B))) (MH) breq2 imbi12d (CH) (CH) rcla42v mp2an ax-mp mpan2 mddmdin0.1 mddmdin0.2 dmdcompl mddmdin0.1 mddmdin0.2 mdcompl 3imtr4)) thm (cdjreu ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((cdjreu.1 (e. A (SH))) (cdjreu.2 (e. B (SH)))) (-> (/\ (e. C (opr A (+H) B)) (= (i^i A B) (0H))) (E!e. x A (E.e. y B (= C (opr (cv x) (+v) (cv y)))))) (cdjreu.1 cdjreu.2 C x y shsel biimp (cv x) (cv y) (cv z) (cv w) hvaddsub4t cdjreu.1 (cv x) shel cdjreu.2 (cv y) shel anim12i cdjreu.1 (cv z) shel cdjreu.2 (cv w) shel anim12i syl2an an4s (= (i^i A B) (0H)) adantll (opr (cv x) (-v) (cv z)) (opr (cv w) (-v) (cv y)) B eleq1 cdjreu.2 B (cv w) (cv y) shsubclt ax-mp ancoms syl5bir com12 (/\ (e. (cv x) A) (e. (cv z) A)) adantl cdjreu.1 A (cv x) (cv z) shsubclt ax-mp (/\ (e. (cv y) B) (e. (cv w) B)) adantr jctild (= (i^i A B) (0H)) adantll (i^i A B) (0H) (opr (cv x) (-v) (cv z)) eleq2 (opr (cv x) (-v) (cv z)) A B elin syl5bbr (/\ (e. (cv x) A) (e. (cv z) A)) (/\ (e. (cv y) B) (e. (cv w) B)) ad2antrr sylibd (cv x) (cv z) hvsubeq0t (opr (cv x) (-v) (cv z)) elch0 syl5bb cdjreu.1 (cv x) shel cdjreu.1 (cv z) shel syl2an (= (i^i A B) (0H)) (/\ (e. (cv y) B) (e. (cv w) B)) ad2antlr sylibd sylbid C (opr (cv x) (+v) (cv y)) (opr (cv z) (+v) (cv w)) eqtr2t syl5 ex r19.23advv y B w B (= C (opr (cv x) (+v) (cv y))) (= C (opr (cv z) (+v) (cv w))) reeanv syl5ibr ex r19.21aivv anim12i (cv x) (cv z) (+v) (cv y) opreq1 C eqeq2d y B rexbidv (cv y) (cv w) (cv z) (+v) opreq2 C eqeq2d B cbvrexv syl6bb A reu4 sylibr)) thm (cdj1 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B x) (B y) (B z) (B w) (B v)) ((cdj1.1 (e. A (SH))) (cdj1.2 (e. B (SH)))) (-> (E.e. w (RR) (/\ (br (0) (<) (cv w)) (A.e. y A (A.e. v B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv v))) (<_) (opr (cv w) (x.) (` (norm) (opr (cv y) (+v) (cv v))))))))) (E.e. x (RR) (/\ (br (0) (<) (cv x)) (A.e. y A (A.e. z B (-> (= (` (norm) (cv y)) (1)) (br (cv x) (<_) (` (norm) (opr (cv y) (-v) (cv z)))))))))) ((cv w) gt0ne0t (cv w) rerecclt syldan (A.e. y A (A.e. v B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv v))) (<_) (opr (cv w) (x.) (` (norm) (opr (cv y) (+v) (cv v))))))) adantrr (cv w) recgt0t (A.e. y A (A.e. v B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv v))) (<_) (opr (cv w) (x.) (` (norm) (opr (cv y) (+v) (cv v))))))) adantrr ax1re (/\ (/\ (/\ (e. (cv w) (RR)) (e. (cv y) A)) (e. (cv z) B)) (/\ (A.e. v B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv v))) (<_) (opr (cv w) (x.) (` (norm) (opr (cv y) (+v) (cv v)))))) (= (` (norm) (cv y)) (1)))) a1i cdj1.2 (cv z) shel 1cn negcl (-u (1)) (cv z) ax-hvmulcl mpan syl (opr (-u (1)) (.s) (cv z)) normclt syl (/\ (e. (cv w) (RR)) (e. (cv y) A)) adantl ax1re (1) (` (norm) (opr (-u (1)) (.s) (cv z))) axaddrcl mpan syl (/\ (A.e. v B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv v))) (<_) (opr (cv w) (x.) (` (norm) (opr (cv y) (+v) (cv v)))))) (= (` (norm) (cv y)) (1))) adantr (cv w) (` (norm) (opr (cv y) (-v) (cv z))) axmulrcl (cv y) (cv z) hvsubclt cdj1.1 (cv y) shel cdj1.2 (cv z) shel syl2an (opr (cv y) (-v) (cv z)) normclt syl sylan2 anassrs (/\ (A.e. v B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv v))) (<_) (opr (cv w) (x.) (` (norm) (opr (cv y) (+v) (cv v)))))) (= (` (norm) (cv y)) (1))) adantr ax1re (1) (` (norm) (opr (-u (1)) (.s) (cv z))) addge01tOLD mp3an1 cdj1.2 (cv z) shel 1cn negcl (-u (1)) (cv z) ax-hvmulcl mpan syl (opr (-u (1)) (.s) (cv z)) normclt syl cdj1.2 (cv z) shel 1cn negcl (-u (1)) (cv z) ax-hvmulcl mpan syl (opr (-u (1)) (.s) (cv z)) normge0t syl sylanc (/\ (e. (cv w) (RR)) (e. (cv y) A)) (/\ (A.e. v B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv v))) (<_) (opr (cv w) (x.) (` (norm) (opr (cv y) (+v) (cv v)))))) (= (` (norm) (cv y)) (1))) ad2antlr 1cn negcl cdj1.2 B (-u (1)) (cv z) shmulclt ax-mp mpan (cv v) (opr (-u (1)) (.s) (cv z)) (norm) fveq2 (` (norm) (cv y)) (+) opreq2d (cv v) (opr (-u (1)) (.s) (cv z)) (cv y) (+v) opreq2 (norm) fveq2d (cv w) (x.) opreq2d (<_) breq12d B rcla4v syl imp (/\ (e. (cv w) (RR)) (e. (cv y) A)) adantll (= (` (norm) (cv y)) (1)) adantrr (1) (` (norm) (cv y)) (+) (` (norm) (opr (-u (1)) (.s) (cv z))) opreq1 eqcoms (/\ (/\ (e. (cv w) (RR)) (e. (cv y) A)) (e. (cv z) B)) (A.e. v B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv v))) (<_) (opr (cv w) (x.) (` (norm) (opr (cv y) (+v) (cv v)))))) ad2antll (cv y) (cv z) hvsubvalt cdj1.1 (cv y) shel cdj1.2 (cv z) shel syl2an (norm) fveq2d (cv w) (x.) opreq2d (e. (cv w) (RR)) adantll (/\ (A.e. v B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv v))) (<_) (opr (cv w) (x.) (` (norm) (opr (cv y) (+v) (cv v)))))) (= (` (norm) (cv y)) (1))) adantr 3brtr4d letrd ex (br (0) (<) (cv w)) adantllr ax1re (1) (opr (cv w) (x.) (` (norm) (opr (cv y) (-v) (cv z)))) (cv w) lediv1t mp3anl1 (cv w) (` (norm) (opr (cv y) (-v) (cv z))) axmulrcl (cv y) (cv z) hvsubclt cdj1.1 (cv y) shel cdj1.2 (cv z) shel syl2an (opr (cv y) (-v) (cv z)) normclt syl sylan2 anassrs (br (0) (<) (cv w)) adantllr (e. (cv w) (RR)) (br (0) (<) (cv w)) pm3.26 (e. (cv y) A) (e. (cv z) B) ad2antrr jca (e. (cv w) (RR)) (br (0) (<) (cv w)) pm3.27 (e. (cv y) A) (e. (cv z) B) ad2antrr sylanc sylibd imp (cv w) (` (norm) (opr (cv y) (-v) (cv z))) divcan3t (cv w) recnt (br (0) (<) (cv w)) adantr (e. (cv y) A) (e. (cv z) B) ad2antrr (cv y) (cv z) hvsubclt cdj1.1 (cv y) shel cdj1.2 (cv z) shel syl2an (opr (cv y) (-v) (cv z)) normclt syl recnd (/\ (e. (cv w) (RR)) (br (0) (<) (cv w))) adantll (cv w) gt0ne0t (e. (cv y) A) (e. (cv z) B) ad2antrr syl3anc (/\ (A.e. v B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv v))) (<_) (opr (cv w) (x.) (` (norm) (opr (cv y) (+v) (cv v)))))) (= (` (norm) (cv y)) (1))) adantr breqtrd exp43 com23 r19.21adv r19.20dva ex imp32 jca jca ex (cv x) (opr (1) (/) (cv w)) (0) (<) breq2 (cv x) (opr (1) (/) (cv w)) (<_) (` (norm) (opr (cv y) (-v) (cv z))) breq1 (= (` (norm) (cv y)) (1)) imbi2d y A z B 2ralbidv anbi12d (RR) rcla4ev syl6 r19.23aiv)) thm (cdj3lem1 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((cdj1.1 (e. A (SH))) (cdj1.2 (e. B (SH)))) (-> (E.e. x (RR) (/\ (br (0) (<) (cv x)) (A.e. y A (A.e. z B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv z))) (<_) (opr (cv x) (x.) (` (norm) (opr (cv y) (+v) (cv z))))))))) (= (i^i A B) (0H))) ((cv w) A B elin 1cn negcl cdj1.2 B (-u (1)) (cv w) shmulclt ax-mp mpan (e. (cv w) A) anim2i sylbi (cv y) (cv w) (norm) fveq2 (+) (` (norm) (cv z)) opreq1d (cv y) (cv w) (+v) (cv z) opreq1 (norm) fveq2d (cv x) (x.) opreq2d (<_) breq12d (cv z) (opr (-u (1)) (.s) (cv w)) (norm) fveq2 (` (norm) (cv w)) (+) opreq2d (cv z) (opr (-u (1)) (.s) (cv w)) (cv w) (+v) opreq2 (norm) fveq2d (cv x) (x.) opreq2d (<_) breq12d A B rcla42v syl (e. (cv x) (RR)) adantl (cv w) normnegt (` (norm) (cv w)) (+) opreq2d (cv w) normclt recnd (` (norm) (cv w)) 2timest syl eqtr4d (e. (cv x) (RR)) adantl (cv w) hvnegidt (norm) fveq2d norm0 syl6eq (cv x) (x.) opreq2d (cv x) recnt (cv x) mul01t syl sylan9eqr 2cn mul01 syl6eqr (<_) breq12d (cv w) normclt 0re 2re 2pos (` (norm) (cv w)) (0) (2) lemul2t mpan2 mp3an23 syl (cv w) normge0t (br (` (norm) (cv w)) (<_) (0)) biantrud bitr3d (cv w) normclt 0re (` (norm) (cv w)) (0) letri3t mpan2 syl (cv w) norm-it 3bitr2d (e. (cv x) (RR)) adantl bitrd cdj1.1 cdj1.2 shincl (cv w) shel sylan2 sylibd ex com23 imp (cv w) elch0 syl6ibr ssrdv ex cdj1.1 cdj1.2 shincl (i^i A B) shle0t ax-mp syl6ib (br (0) (<) (cv x)) adantld r19.23aiv)) thm (cdj3lem2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w)) ((cdj3lem2.1 (e. A (SH))) (cdj3lem2.2 (e. B (SH))) (cdj3lem2.3 (= S ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} z A (E.e. w B (= (cv x) (opr (cv z) (+v) (cv w)))))))))))) (-> (/\/\ (e. C A) (e. D B) (= (i^i A B) (0H))) (= (` S (opr C (+v) D)) C)) (cdj3lem2.1 cdj3lem2.2 C D shsva (cv x) (opr C (+v) D) (opr (cv z) (+v) (cv w)) eqeq1 w B rexbidv z A rabbisdv unieqd cdj3lem2.3 cdj3lem2.1 elisseti z (E.e. w B (= (opr C (+v) D) (opr (cv z) (+v) (cv w)))) rabex uniex fvopab4 syl (= (i^i A B) (0H)) 3adant3 (opr C (+v) D) eqid (cv w) D C (+v) opreq2 (opr C (+v) D) eqeq2d B rcla4ev mpan2 (e. C A) (= (i^i A B) (0H)) 3ad2ant2 (cv z) C (+v) (cv w) opreq1 (opr C (+v) D) eqeq2d w B rexbidv A reuuni2 (e. C A) (e. D B) (= (i^i A B) (0H)) 3simp1 cdj3lem2.1 cdj3lem2.2 (opr C (+v) D) z w cdjreu cdj3lem2.1 cdj3lem2.2 C D shsva sylan 3impa sylanc mpbid eqtrd)) thm (cdj3lem2a ((u v) (S v) (S u) (x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((cdj3lem2.1 (e. A (SH))) (cdj3lem2.2 (e. B (SH))) (cdj3lem2.3 (= S ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} z A (E.e. w B (= (cv x) (opr (cv z) (+v) (cv w)))))))))))) (-> (/\ (e. C (opr A (+H) B)) (= (i^i A B) (0H))) (e. (` S C) A)) (C (opr (cv v) (+v) (cv u)) S fveq2 A eleq1d cdj3lem2.1 cdj3lem2.2 cdj3lem2.3 (cv v) (cv u) cdj3lem2 (e. (cv v) A) (e. (cv u) B) (= (i^i A B) (0H)) 3simp1 eqeltrd 3expa syl5bir exp3a com13 r19.23advv cdj3lem2.1 cdj3lem2.2 C v u shsel syl5ib impcom)) thm (cdj3lem2b ((u v) (t v) (h v) (S v) (t u) (h u) (S u) (h t) (S t) (S h) (x y) (x z) (w x) (v x) (u x) (t x) (h x) (A x) (y z) (w y) (v y) (u y) (t y) (h y) (A y) (w z) (v z) (u z) (t z) (h z) (A z) (v w) (u w) (t w) (h w) (A w) (u v) (t v) (h v) (A v) (t u) (h u) (A u) (h t) (A t) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B h)) ((cdj3lem2.1 (e. A (SH))) (cdj3lem2.2 (e. B (SH))) (cdj3lem2.3 (= S ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} z A (E.e. w B (= (cv x) (opr (cv z) (+v) (cv w)))))))))))) (-> (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. x A (A.e. y B (br (opr (` (norm) (cv x)) (+) (` (norm) (cv y))) (<_) (opr (cv v) (x.) (` (norm) (opr (cv x) (+v) (cv y))))))))) (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. u (opr A (+H) B) (br (` (norm) (` S (cv u))) (<_) (opr (cv v) (x.) (` (norm) (cv u)))))))) (cdj3lem2.1 cdj3lem2.2 v x y cdj3lem1 cdj3lem2.1 cdj3lem2.2 (cv u) t h cdjreu t A (E.e. h B (= (cv u) (opr (cv t) (+v) (cv h)))) reurex syl (e. (cv v) (RR)) adantrr (cv x) (cv t) (norm) fveq2 (+) (` (norm) (cv y)) opreq1d (cv x) (cv t) (+v) (cv y) opreq1 (norm) fveq2d (cv v) (x.) opreq2d (<_) breq12d (cv y) (cv h) (norm) fveq2 (` (norm) (cv t)) (+) opreq2d (cv y) (cv h) (cv t) (+v) opreq2 (norm) fveq2d (cv v) (x.) opreq2d (<_) breq12d A B rcla42v cdj3lem2.1 cdj3lem2.2 cdj3lem2.3 (cv t) (cv h) cdj3lem2 3expa (norm) fveq2d (br (opr (` (norm) (cv t)) (+) (` (norm) (cv h))) (<_) (opr (cv v) (x.) (` (norm) (opr (cv t) (+v) (cv h))))) (e. (cv v) (RR)) ad2ant2r (` (norm) (cv t)) (` (norm) (cv h)) addge01tOLD 3expa cdj3lem2.1 (cv t) shel (cv t) normclt syl cdj3lem2.2 (cv h) shel (cv h) normclt syl anim12i cdj3lem2.2 (cv h) shel (cv h) normge0t syl (e. (cv t) A) adantl sylanc (e. (cv v) (RR)) adantr (` (norm) (cv t)) (opr (` (norm) (cv t)) (+) (` (norm) (cv h))) (opr (cv v) (x.) (` (norm) (opr (cv t) (+v) (cv h)))) letrt cdj3lem2.1 (cv t) shel (cv t) normclt syl (e. (cv h) B) (e. (cv v) (RR)) ad2antrr (` (norm) (cv t)) (` (norm) (cv h)) axaddrcl cdj3lem2.1 (cv t) shel (cv t) normclt syl cdj3lem2.2 (cv h) shel (cv h) normclt syl syl2an (e. (cv v) (RR)) adantr (cv v) (` (norm) (opr (cv t) (+v) (cv h))) axmulrcl (cv t) (cv h) ax-hvaddcl cdj3lem2.1 (cv t) shel cdj3lem2.2 (cv h) shel syl2an (opr (cv t) (+v) (cv h)) normclt syl sylan2 ancoms syl3anc mpand imp an1rs (= (i^i A B) (0H)) adantrl eqbrtrd (cv u) (opr (cv t) (+v) (cv h)) S fveq2 (norm) fveq2d (cv u) (opr (cv t) (+v) (cv h)) (norm) fveq2 (cv v) (x.) opreq2d (<_) breq12d biimprcd syl exp31 syld com14 com4t r19.23advv (e. (cv u) (opr A (+H) B)) adantl mpd ex com3l r19.21adv (br (0) (<) (cv v)) anim2d r19.22dva mpcom)) thm (cdj3lem3 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w)) ((cdj3lem2.1 (e. A (SH))) (cdj3lem2.2 (e. B (SH))) (cdj3lem3.3 (= T ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} w B (E.e. z A (= (cv x) (opr (cv z) (+v) (cv w)))))))))))) (-> (/\/\ (e. C A) (e. D B) (= (i^i A B) (0H))) (= (` T (opr C (+v) D)) D)) (D C ax-hvcom cdj3lem2.2 D shel cdj3lem2.1 C shel syl2an T fveq2d (= (i^i B A) (0H)) 3adant3 cdj3lem2.2 cdj3lem2.1 cdj3lem3.3 cdj3lem2.2 cdj3lem2.1 shscom (cv x) eleq2i (cv w) (cv z) ax-hvcom cdj3lem2.2 (cv w) shel cdj3lem2.1 (cv z) shel syl2an (cv x) eqeq2d rexbidva rabbii unieqi (cv y) eqeq2i anbi12i x y opabbii eqtr4 D C cdj3lem2 eqtr3d A B incom (0H) eqeq1i syl3an3b 3com12)) thm (cdj3lem3a ((u v) (T v) (T u) (x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((cdj3lem2.1 (e. A (SH))) (cdj3lem2.2 (e. B (SH))) (cdj3lem3.3 (= T ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} w B (E.e. z A (= (cv x) (opr (cv z) (+v) (cv w)))))))))))) (-> (/\ (e. C (opr A (+H) B)) (= (i^i A B) (0H))) (e. (` T C) B)) (C (opr (cv v) (+v) (cv u)) T fveq2 B eleq1d cdj3lem2.1 cdj3lem2.2 cdj3lem3.3 (cv v) (cv u) cdj3lem3 (e. (cv v) A) (e. (cv u) B) (= (i^i A B) (0H)) 3simp2 eqeltrd 3expa syl5bir exp3a com13 r19.23advv cdj3lem2.1 cdj3lem2.2 C v u shsel syl5ib impcom)) thm (cdj3lem3b ((u v) (t v) (h v) (T v) (t u) (h u) (T u) (h t) (T t) (T h) (x y) (x z) (w x) (v x) (u x) (t x) (h x) (A x) (y z) (w y) (v y) (u y) (t y) (h y) (A y) (w z) (v z) (u z) (t z) (h z) (A z) (v w) (u w) (t w) (h w) (A w) (u v) (t v) (h v) (A v) (t u) (h u) (A u) (h t) (A t) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B h)) ((cdj3lem2.1 (e. A (SH))) (cdj3lem2.2 (e. B (SH))) (cdj3lem3.3 (= T ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} w B (E.e. z A (= (cv x) (opr (cv z) (+v) (cv w)))))))))))) (-> (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. x A (A.e. y B (br (opr (` (norm) (cv x)) (+) (` (norm) (cv y))) (<_) (opr (cv v) (x.) (` (norm) (opr (cv x) (+v) (cv y))))))))) (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. u (opr A (+H) B) (br (` (norm) (` T (cv u))) (<_) (opr (cv v) (x.) (` (norm) (cv u)))))))) (cdj3lem2.2 cdj3lem2.1 cdj3lem3.3 cdj3lem2.2 cdj3lem2.1 shscom (cv x) eleq2i (cv w) (cv z) ax-hvcom cdj3lem2.2 (cv w) shel cdj3lem2.1 (cv z) shel syl2an (cv x) eqeq2d rexbidva rabbii unieqi (cv y) eqeq2i anbi12i x y opabbii eqtr4 v u cdj3lem2b (cv x) (cv t) (norm) fveq2 (+) (` (norm) (cv y)) opreq1d (cv x) (cv t) (+v) (cv y) opreq1 (norm) fveq2d (cv v) (x.) opreq2d (<_) breq12d (cv y) (cv h) (norm) fveq2 (` (norm) (cv t)) (+) opreq2d (cv y) (cv h) (cv t) (+v) opreq2 (norm) fveq2d (cv v) (x.) opreq2d (<_) breq12d A B cbvral2v t A h B (br (opr (` (norm) (cv t)) (+) (` (norm) (cv h))) (<_) (opr (cv v) (x.) (` (norm) (opr (cv t) (+v) (cv h))))) ralcom (` (norm) (cv x)) (` (norm) (cv y)) axaddcom cdj3lem2.2 (cv x) shel (cv x) normclt syl recnd cdj3lem2.1 (cv y) shel (cv y) normclt syl recnd syl2an (cv x) (cv y) ax-hvcom cdj3lem2.2 (cv x) shel cdj3lem2.1 (cv y) shel syl2an (norm) fveq2d (cv v) (x.) opreq2d (<_) breq12d ralbidva ralbiia (cv x) (cv h) (norm) fveq2 (` (norm) (cv y)) (+) opreq2d (cv x) (cv h) (cv y) (+v) opreq2 (norm) fveq2d (cv v) (x.) opreq2d (<_) breq12d (cv y) (cv t) (norm) fveq2 (+) (` (norm) (cv h)) opreq1d (cv y) (cv t) (+v) (cv h) opreq1 (norm) fveq2d (cv v) (x.) opreq2d (<_) breq12d B A cbvral2v bitr2 3bitr (br (0) (<) (cv v)) anbi2i v (RR) rexbii cdj3lem2.1 cdj3lem2.2 shscom (opr A (+H) B) (opr B (+H) A) u (br (` (norm) (` T (cv u))) (<_) (opr (cv v) (x.) (` (norm) (cv u)))) raleq1 ax-mp (br (0) (<) (cv v)) anbi2i v (RR) rexbii 3imtr4)) thm (cdj3 ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (f x) (g x) (A x) (y z) (w y) (v y) (u y) (t y) (h y) (f y) (g y) (A y) (w z) (v z) (u z) (t z) (h z) (f z) (g z) (A z) (v w) (u w) (t w) (h w) (f w) (g w) (A w) (u v) (t v) (h v) (f v) (g v) (A v) (t u) (h u) (f u) (g u) (A u) (h t) (f t) (g t) (A t) (f h) (g h) (A h) (f g) (A f) (A g) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B h) (B f) (B g) (S v) (S u) (S t) (S h) (S f) (S g) (T v) (T u) (T t) (T h) (T f) (T g)) ((cdj3.1 (e. A (SH))) (cdj3.2 (e. B (SH))) (cdj3.3 (= S ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} z A (E.e. w B (= (cv x) (opr (cv z) (+v) (cv w))))))))))) (cdj3.4 (= T ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} w B (E.e. z A (= (cv x) (opr (cv z) (+v) (cv w))))))))))) (cdj3.5 (<-> ph (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. u (opr A (+H) B) (br (` (norm) (` S (cv u))) (<_) (opr (cv v) (x.) (` (norm) (cv u))))))))) (cdj3.6 (<-> ps (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. u (opr A (+H) B) (br (` (norm) (` T (cv u))) (<_) (opr (cv v) (x.) (` (norm) (cv u)))))))))) (<-> (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. x A (A.e. y B (br (opr (` (norm) (cv x)) (+) (` (norm) (cv y))) (<_) (opr (cv v) (x.) (` (norm) (opr (cv x) (+v) (cv y))))))))) (/\/\ (= (i^i A B) (0H)) ph ps)) (cdj3.1 cdj3.2 v x y cdj3lem1 cdj3.1 cdj3.2 cdj3.3 v u cdj3lem2b cdj3.5 sylibr cdj3.1 cdj3.2 cdj3.4 v u cdj3lem3b cdj3.6 sylibr 3jca (cv f) (cv g) axaddrcl (cv v) (opr (cv f) (+) (cv g)) (0) (<) breq2 (cv v) (opr (cv f) (+) (cv g)) (x.) (` (norm) (opr (cv t) (+v) (cv h))) opreq1 (opr (` (norm) (cv t)) (+) (` (norm) (cv h))) (<_) breq2d t A h B 2ralbidv (cv x) (cv t) (norm) fveq2 (+) (` (norm) (cv y)) opreq1d (cv x) (cv t) (+v) (cv y) opreq1 (norm) fveq2d (cv v) (x.) opreq2d (<_) breq12d (cv y) (cv h) (norm) fveq2 (` (norm) (cv t)) (+) opreq2d (cv y) (cv h) (cv t) (+v) opreq2 (norm) fveq2d (cv v) (x.) opreq2d (<_) breq12d A B cbvral2v syl5bb anbi12d (RR) rcla4ev ex syl (= (i^i A B) (0H)) adantl (cv f) (cv g) addgt0t ex (= (i^i A B) (0H)) adantl cdj3.1 cdj3.2 (cv t) (cv h) shsva (cv u) (opr (cv t) (+v) (cv h)) S fveq2 (norm) fveq2d (cv u) (opr (cv t) (+v) (cv h)) (norm) fveq2 (cv f) (x.) opreq2d (<_) breq12d (opr A (+H) B) rcla4v (cv u) (opr (cv t) (+v) (cv h)) T fveq2 (norm) fveq2d (cv u) (opr (cv t) (+v) (cv h)) (norm) fveq2 (cv g) (x.) opreq2d (<_) breq12d (opr A (+H) B) rcla4v anim12d syl (/\ (= (i^i A B) (0H)) (/\ (e. (cv f) (RR)) (e. (cv g) (RR)))) adantl (` (norm) (cv t)) (` (norm) (cv h)) (opr (cv f) (x.) (` (norm) (opr (cv t) (+v) (cv h)))) (opr (cv g) (x.) (` (norm) (opr (cv t) (+v) (cv h)))) le2addt cdj3.1 (cv t) shel (cv t) normclt syl cdj3.2 (cv h) shel (cv h) normclt syl anim12i (/\ (e. (cv f) (RR)) (e. (cv g) (RR))) adantl (cv f) (` (norm) (opr (cv t) (+v) (cv h))) axmulrcl (cv t) (cv h) ax-hvaddcl cdj3.1 (cv t) shel cdj3.2 (cv h) shel syl2an (opr (cv t) (+v) (cv h)) normclt syl sylan2 (e. (cv g) (RR)) adantlr (cv g) (` (norm) (opr (cv t) (+v) (cv h))) axmulrcl (cv t) (cv h) ax-hvaddcl cdj3.1 (cv t) shel cdj3.2 (cv h) shel syl2an (opr (cv t) (+v) (cv h)) normclt syl sylan2 (e. (cv f) (RR)) adantll jca sylanc (= (i^i A B) (0H)) adantll cdj3.1 cdj3.2 cdj3.3 (cv t) (cv h) cdj3lem2 (norm) fveq2d (<_) (opr (cv f) (x.) (` (norm) (opr (cv t) (+v) (cv h)))) breq1d cdj3.1 cdj3.2 cdj3.4 (cv t) (cv h) cdj3lem3 (norm) fveq2d (<_) (opr (cv g) (x.) (` (norm) (opr (cv t) (+v) (cv h)))) breq1d anbi12d 3expa ancoms (/\ (e. (cv f) (RR)) (e. (cv g) (RR))) adantlr (cv f) (cv g) (` (norm) (opr (cv t) (+v) (cv h))) adddirt (cv f) recnt (cv g) recnt (cv t) (cv h) ax-hvaddcl cdj3.1 (cv t) shel cdj3.2 (cv h) shel syl2an (opr (cv t) (+v) (cv h)) normclt syl recnd syl3an 3expa (opr (` (norm) (cv t)) (+) (` (norm) (cv h))) (<_) breq2d (= (i^i A B) (0H)) adantll 3imtr4d syld ex com23 r19.21advv syl2and (br (0) (<) (cv f)) (A.e. u (opr A (+H) B) (br (` (norm) (` S (cv u))) (<_) (opr (cv f) (x.) (` (norm) (cv u))))) (br (0) (<) (cv g)) (A.e. u (opr A (+H) B) (br (` (norm) (` T (cv u))) (<_) (opr (cv g) (x.) (` (norm) (cv u))))) an4 syl5ib ex r19.23advv cdj3.5 (cv v) (cv f) (0) (<) breq2 (cv v) (cv f) (x.) (` (norm) (cv u)) opreq1 (` (norm) (` S (cv u))) (<_) breq2d u (opr A (+H) B) ralbidv anbi12d (RR) cbvrexv bitr cdj3.6 (cv v) (cv g) (0) (<) breq2 (cv v) (cv g) (x.) (` (norm) (cv u)) opreq1 (` (norm) (` T (cv u))) (<_) breq2d u (opr A (+H) B) ralbidv anbi12d (RR) cbvrexv bitr anbi12i f (RR) g (RR) (/\ (br (0) (<) (cv f)) (A.e. u (opr A (+H) B) (br (` (norm) (` S (cv u))) (<_) (opr (cv f) (x.) (` (norm) (cv u)))))) (/\ (br (0) (<) (cv g)) (A.e. u (opr A (+H) B) (br (` (norm) (` T (cv u))) (<_) (opr (cv g) (x.) (` (norm) (cv u)))))) reeanv bitr4 syl5ib 3impib impbi)) # 2038331 Metamath steps converted to 323986 Ghilbert steps export (SET_MM zfc/set_mm (PROP SET_MM_AX) "")