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Theorem grpideu 10255
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 10254 . . 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 721 . . . . . . . . 9 |- ((((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> (uPz) = z)
54ralimi 2212 . . . . . . . 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 4933 . . . . . . . . . 10 |- (z = x -> (uPz) = (uPx))
7 id 18 . . . . . . . . . 10 |- (z = x -> z = x)
86, 7eqeq12d 1952 . . . . . . . . 9 |- (z = x -> ((uPz) = z <-> (uPx) = x))
98cbvralv 2327 . . . . . . . 8 |- (A.z e. B (uPz) = z <-> A.x e. B (uPx) = x)
105, 9sylib 186 . . . . . . 7 |- (A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> A.x e. B (uPx) = x)
1110adantl 453 . . . . . 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 714 . . . . . . . 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 447 . . . . . . . . . . . . . . . 16 |- ((((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> E.y e. B ((yPz) = u /\ (zPy) = u))
1413ralimi 2212 . . . . . . . . . . . . . . 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 4933 . . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (yPz) = (yPw))
1615eqeq1d 1946 . . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((yPz) = u <-> (yPw) = u))
17 oveq1 4932 . . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (zPy) = (wPy))
1817eqeq1d 1946 . . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((zPy) = u <-> (wPy) = u))
1916, 18anbi12d 697 . . . . . . . . . . . . . . . . . 18 |- (z = w -> (((yPz) = u /\ (zPy) = u) <-> ((yPw) = u /\ (wPy) = u)))
2019rexbidv 2166 . . . . . . . . . . . . . . . . 17 |- (z = w -> (E.y e. B ((yPz) = u /\ (zPy) = u) <-> E.y e. B ((yPw) = u /\ (wPy) = u)))
2120rcla4va 2427 . . . . . . . . . . . . . . . 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 700 . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . 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 10253 . . . . . . . . . . . . . 14 |- (((G e. Grp /\ w e. B) /\ E.y e. B ((yPw) = u /\ (wPy) = u)) -> (wPu) = (uPw))
2523, 24syldan 457 . . . . . . . . . . . . 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 754 . . . . . . . . . . . 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 705 . . . . . . . . . . 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 452 . . . . . . . . . 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 4933 . . . . . . . . . . . . . . 15 |- (x = u -> (wPx) = (wPu))
30 id 18 . . . . . . . . . . . . . . 15 |- (x = u -> x = u)
3129, 30eqeq12d 1952 . . . . . . . . . . . . . 14 |- (x = u -> ((wPx) = x <-> (wPu) = u))
3231rcla4va 2427 . . . . . . . . . . . . 13 |- ((u e. B /\ A.x e. B (wPx) = x) -> (wPu) = u)
3332adantll 700 . . . . . . . . . . . 12 |- (((G e. Grp /\ u e. B) /\ A.x e. B (wPx) = x) -> (wPu) = u)
3433ad2ant2rl 720 . . . . . . . . . . 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 705 . . . . . . . . . 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 4933 . . . . . . . . . . . . 13 |- (x = w -> (uPx) = (uPw))
37 id 18 . . . . . . . . . . . . 13 |- (x = w -> x = w)
3836, 37eqeq12d 1952 . . . . . . . . . . . 12 |- (x = w -> ((uPx) = x <-> (uPw) = w))
3938rcla4va 2427 . . . . . . . . . . 11 |- ((w e. B /\ A.x e. B (uPx) = x) -> (uPw) = w)
4039ad2ant2lr 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)) -> (uPw) = w)
4128, 35, 403eqtr3d 1978 . . . . . . . . 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 426 . . . . . . . 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 659 . . . . . . 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 2220 . . . . . 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 521 . . . . 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 426 . . . 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 2249 . . 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 4932 . . . . 5 |- (u = w -> (uPx) = (wPx))
5049eqeq1d 1946 . . . 4 |- (u = w -> ((uPx) = x <-> (wPx) = x))
5150ralbidv 2165 . . 3 |- (u = w -> (A.x e. B (uPx) = x <-> A.x e. B (wPx) = x))
5251reu8 2506 . 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 201 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 361   = wceq 1434   e. wcel 1436  A.wral 2147  E.wrex 2148  E!wreu 2149  ` cfv 4008  (class class class)co 4927  Basecbs 10013  +g cplusg 10221  Grpcgrp 10222
This theorem is referenced by:  grpidcl 10264  grpidinv2 10265  isgrpid 10268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1351  ax-6 1352  ax-7 1353  ax-gen 1354  ax-8 1438  ax-10 1439  ax-11 1440  ax-12 1441  ax-13 1442  ax-14 1443  ax-17 1450  ax-9 1465  ax-4 1471  ax-16 1649  ax-ext 1920  ax-sep 3459  ax-nul 3468  ax-pr 3528  ax-un 3800
This theorem depends on definitions:  df-bi 175  df-or 362  df-an 363  df-3an 923  df-ex 1356  df-sb 1611  df-eu 1838  df-mo 1839  df-clab 1926  df-cleq 1931  df-clel 1934  df-ne 2058  df-ral 2151  df-rex 2152  df-reu 2153  df-v 2345  df-dif 2645  df-un 2647  df-in 2649  df-ss 2651  df-nul 2907  df-sn 3085  df-pr 3086  df-op 3088  df-uni 3219  df-br 3364  df-opab 3418  df-xp 4010  df-cnv 4012  df-dm 4014  df-rn 4015  df-res 4016  df-ima 4017  df-fv 4024  df-ov 4929  df-grp 10228
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