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Theorem grpideu 10152
Description: The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55.
Hypotheses
Ref Expression
grplem1.b |- B = (Base` G)
grplem1.p |- P = ( +g ` G)
Assertion
Ref Expression
grpideu |- (G e. Grp -> E!u e. B A.x e. B (uPx) = x)
Distinct variable groups:   x,u,B   u,P,x   u,G,x

Proof of Theorem grpideu
StepHypRef Expression
1 grplem1.b . . . 4 |- B = (Base` G)
2 grplem1.p . . . 4 |- P = ( +g ` G)
31, 2grpidinv 10151 . . 3 |- (G e. Grp -> E.u e. B A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)))
4 simpll 735 . . . . . . . . 9 |- ((((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> (uPz) = z)
54ralimi 2227 . . . . . . . 8 |- (A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> A.z e. B (uPz) = z)
6 oveq2 4937 . . . . . . . . . 10 |- (z = x -> (uPz) = (uPx))
7 id 18 . . . . . . . . . 10 |- (z = x -> z = x)
86, 7eqeq12d 1967 . . . . . . . . 9 |- (z = x -> ((uPz) = z <-> (uPx) = x))
98cbvralv 2342 . . . . . . . 8 |- (A.z e. B (uPz) = z <-> A.x e. B (uPx) = x)
105, 9sylib 196 . . . . . . 7 |- (A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> A.x e. B (uPx) = x)
1110adantl 469 . . . . . 6 |- (((G e. Grp /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) -> A.x e. B (uPx) = x)
1210ad2antlr 728 . . . . . . . 8 |- ((((G e. Grp /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) /\ w e. B) -> A.x e. B (uPx) = x)
13 simpr 463 . . . . . . . . . . . . . . . 16 |- ((((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> E.y e. B ((yPz) = u /\ (zPy) = u))
1413ralimi 2227 . . . . . . . . . . . . . . 15 |- (A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> A.z e. B E.y e. B ((yPz) = u /\ (zPy) = u))
15 oveq2 4937 . . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (yPz) = (yPw))
1615eqeq1d 1961 . . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((yPz) = u <-> (yPw) = u))
17 oveq1 4936 . . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (zPy) = (wPy))
1817eqeq1d 1961 . . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((zPy) = u <-> (wPy) = u))
1916, 18anbi12d 711 . . . . . . . . . . . . . . . . . 18 |- (z = w -> (((yPz) = u /\ (zPy) = u) <-> ((yPw) = u /\ (wPy) = u)))
2019rexbidv 2181 . . . . . . . . . . . . . . . . 17 |- (z = w -> (E.y e. B ((yPz) = u /\ (zPy) = u) <-> E.y e. B ((yPw) = u /\ (wPy) = u)))
2120rcla4va 2442 . . . . . . . . . . . . . . . 16 |- ((w e. B /\ A.z e. B E.y e. B ((yPz) = u /\ (zPy) = u)) -> E.y e. B ((yPw) = u /\ (wPy) = u))
2221adantll 714 . . . . . . . . . . . . . . 15 |- (((G e. Grp /\ w e. B) /\ A.z e. B E.y e. B ((yPz) = u /\ (zPy) = u)) -> E.y e. B ((yPw) = u /\ (wPy) = u))
2314, 22sylan2 477 . . . . . . . . . . . . . 14 |- (((G e. Grp /\ w e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) -> E.y e. B ((yPw) = u /\ (wPy) = u))
241, 2grpidinvlem4 10150 . . . . . . . . . . . . . 14 |- (((G e. Grp /\ w e. B) /\ E.y e. B ((yPw) = u /\ (wPy) = u)) -> (wPu) = (uPw))
2523, 24syldan 473 . . . . . . . . . . . . 13 |- (((G e. Grp /\ w e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) -> (wPu) = (uPw))
2625an32s 769 . . . . . . . . . . . 12 |- (((G e. Grp /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) /\ w e. B) -> (wPu) = (uPw))
2726adantllr 719 . . . . . . . . . . 11 |- ((((G e. Grp /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) /\ w e. B) -> (wPu) = (uPw))
2827adantr 468 . . . . . . . . . 10 |- (((((G e. Grp /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) /\ w e. B) /\ (A.x e. B (uPx) = x /\ A.x e. B (wPx) = x)) -> (wPu) = (uPw))
29 oveq2 4937 . . . . . . . . . . . . . . 15 |- (x = u -> (wPx) = (wPu))
30 id 18 . . . . . . . . . . . . . . 15 |- (x = u -> x = u)
3129, 30eqeq12d 1967 . . . . . . . . . . . . . 14 |- (x = u -> ((wPx) = x <-> (wPu) = u))
3231rcla4va 2442 . . . . . . . . . . . . 13 |- ((u e. B /\ A.x e. B (wPx) = x) -> (wPu) = u)
3332adantll 714 . . . . . . . . . . . 12 |- (((G e. Grp /\ u e. B) /\ A.x e. B (wPx) = x) -> (wPu) = u)
3433ad2ant2rl 734 . . . . . . . . . . 11 |- ((((G e. Grp /\ u e. B) /\ w e. B) /\ (A.x e. B (uPx) = x /\ A.x e. B (wPx) = x)) -> (wPu) = u)
3534adantllr 719 . . . . . . . . . 10 |- (((((G e. Grp /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) /\ w e. B) /\ (A.x e. B (uPx) = x /\ A.x e. B (wPx) = x)) -> (wPu) = u)
36 oveq2 4937 . . . . . . . . . . . . 13 |- (x = w -> (uPx) = (uPw))
37 id 18 . . . . . . . . . . . . 13 |- (x = w -> x = w)
3836, 37eqeq12d 1967 . . . . . . . . . . . 12 |- (x = w -> ((uPx) = x <-> (uPw) = w))
3938rcla4va 2442 . . . . . . . . . . 11 |- ((w e. B /\ A.x e. B (uPx) = x) -> (uPw) = w)
4039ad2ant2lr 733 . . . . . . . . . 10 |- (((((G e. Grp /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) /\ w e. B) /\ (A.x e. B (uPx) = x /\ A.x e. B (wPx) = x)) -> (uPw) = w)
4128, 35, 403eqtr3d 1993 . . . . . . . . 9 |- (((((G e. Grp /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) /\ w e. B) /\ (A.x e. B (uPx) = x /\ A.x e. B (wPx) = x)) -> u = w)
4241ex 442 . . . . . . . 8 |- ((((G e. Grp /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) /\ w e. B) -> ((A.x e. B (uPx) = x /\ A.x e. B (wPx) = x) -> u = w))
4312, 42mpand 673 . . . . . . 7 |- ((((G e. Grp /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) /\ w e. B) -> (A.x e. B (wPx) = x -> u = w))
4443ralrimiva 2235 . . . . . 6 |- (((G e. Grp /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) -> A.w e. B (A.x e. B (wPx) = x -> u = w))
4511, 44jca 535 . . . . 5 |- (((G e. Grp /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) -> (A.x e. B (uPx) = x /\ A.w e. B (A.x e. B (wPx) = x -> u = w)))
4645ex 442 . . . 4 |- ((G e. Grp /\ u e. B) -> (A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> (A.x e. B (uPx) = x /\ A.w e. B (A.x e. B (wPx) = x -> u = w))))
4746reximdva 2264 . . 3 |- (G e. Grp -> (E.u e. B A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> E.u e. B (A.x e. B (uPx) = x /\ A.w e. B (A.x e. B (wPx) = x -> u = w))))
483, 47mpd 13 . 2 |- (G e. Grp -> E.u e. B (A.x e. B (uPx) = x /\ A.w e. B (A.x e. B (wPx) = x -> u = w)))
49 oveq1 4936 . . . . 5 |- (u = w -> (uPx) = (wPx))
5049eqeq1d 1961 . . . 4 |- (u = w -> ((uPx) = x <-> (wPx) = x))
5150ralbidv 2180 . . 3 |- (u = w -> (A.x e. B (uPx) = x <-> A.x e. B (wPx) = x))
5251reu8 2521 . 2 |- (E!u e. B A.x e. B (uPx) = x <-> E.u e. B (A.x e. B (uPx) = x /\ A.w e. B (A.x e. B (wPx) = x -> u = w)))
5348, 52sylibr 216 1 |- (G e. Grp -> E!u e. B A.x e. B (uPx) = x)
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 377   = wceq 1449   e. wcel 1451  A.wral 2162  E.wrex 2163  E!wreu 2164  ` cfv 4016  (class class class)co 4931  Basecbs 9910   +g cplusg 10118  Grpcgrp 10119
This theorem is referenced by:  grpidcl 10161  grpidinv2 10162  isgrpid 10165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1367  ax-6 1368  ax-7 1369  ax-gen 1370  ax-8 1453  ax-10 1454  ax-11 1455  ax-12 1456  ax-13 1457  ax-14 1458  ax-17 1465  ax-9 1480  ax-4 1486  ax-16 1664  ax-ext 1935  ax-sep 3469  ax-nul 3478  ax-pr 3538  ax-un 3808
This theorem depends on definitions:  df-bi 185  df-or 378  df-an 379  df-3an 939  df-ex 1372  df-sb 1626  df-eu 1853  df-mo 1854  df-clab 1941  df-cleq 1946  df-clel 1949  df-ne 2073  df-ral 2166  df-rex 2167  df-reu 2168  df-v 2360  df-dif 2660  df-un 2662  df-in 2664  df-ss 2666  df-nul 2922  df-sn 3096  df-pr 3097  df-op 3100  df-uni 3229  df-br 3374  df-opab 3428  df-xp 4018  df-cnv 4020  df-dm 4022  df-rn 4023  df-res 4024  df-ima 4025  df-fv 4032  df-ov 4933  df-grp 10125
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