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Theorem grpideu 10157
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 10156 . . 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 832 . . . . . . . . 9 |- ((((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> (uPz) = z)
54ralimi 2373 . . . . . . . 8 |- (A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> A.z e. B (uPz) = z)
6 opreq2 5073 . . . . . . . . . 10 |- (z = x -> (uPz) = (uPx))
7 id 18 . . . . . . . . . 10 |- (z = x -> z = x)
86, 7eqeq12d 2106 . . . . . . . . 9 |- (z = x -> ((uPz) = z <-> (uPx) = x))
98cbvralv 2484 . . . . . . . 8 |- (A.z e. B (uPz) = z <-> A.x e. B (uPx) = x)
105, 9sylib 221 . . . . . . 7 |- (A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> A.x e. B (uPx) = x)
1110adantl 517 . . . . . 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 825 . . . . . . . 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 511 . . . . . . . . . . . . . . . 16 |- ((((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> E.y e. B ((yPz) = u /\ (zPy) = u))
1413ralimi 2373 . . . . . . . . . . . . . . 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 opreq2 5073 . . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (yPz) = (yPw))
1615eqeq1d 2100 . . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((yPz) = u <-> (yPw) = u))
17 opreq1 5072 . . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (zPy) = (wPy))
1817eqeq1d 2100 . . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((zPy) = u <-> (wPy) = u))
1916, 18anbi12d 805 . . . . . . . . . . . . . . . . . 18 |- (z = w -> (((yPz) = u /\ (zPy) = u) <-> ((yPw) = u /\ (wPy) = u)))
2019rexbidv 2329 . . . . . . . . . . . . . . . . 17 |- (z = w -> (E.y e. B ((yPz) = u /\ (zPy) = u) <-> E.y e. B ((yPw) = u /\ (wPy) = u)))
2120rcla4va 2589 . . . . . . . . . . . . . . . 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 811 . . . . . . . . . . . . . . 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 526 . . . . . . . . . . . . . 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 10155 . . . . . . . . . . . . . 14 |- (((G e. Grp /\ w e. B) /\ E.y e. B ((yPw) = u /\ (wPy) = u)) -> (wPu) = (uPw))
2523, 24syldan 521 . . . . . . . . . . . . 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 866 . . . . . . . . . . . 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 816 . . . . . . . . . . 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 516 . . . . . . . . . 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 opreq2 5073 . . . . . . . . . . . . . . 15 |- (x = u -> (wPx) = (wPu))
30 id 18 . . . . . . . . . . . . . . 15 |- (x = u -> x = u)
3129, 30eqeq12d 2106 . . . . . . . . . . . . . 14 |- (x = u -> ((wPx) = x <-> (wPu) = u))
3231rcla4va 2589 . . . . . . . . . . . . 13 |- ((u e. B /\ A.x e. B (wPx) = x) -> (wPu) = u)
3332adantll 811 . . . . . . . . . . . 12 |- (((G e. Grp /\ u e. B) /\ A.x e. B (wPx) = x) -> (wPu) = u)
3433ad2ant2rl 831 . . . . . . . . . . 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 816 . . . . . . . . . 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 opreq2 5073 . . . . . . . . . . . . 13 |- (x = w -> (uPx) = (uPw))
37 id 18 . . . . . . . . . . . . 13 |- (x = w -> x = w)
3836, 37eqeq12d 2106 . . . . . . . . . . . 12 |- (x = w -> ((uPx) = x <-> (uPw) = w))
3938rcla4va 2589 . . . . . . . . . . 11 |- ((w e. B /\ A.x e. B (uPx) = x) -> (uPw) = w)
4039ad2ant2lr 830 . . . . . . . . . 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 2132 . . . . . . . . 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 485 . . . . . . . 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 759 . . . . . . 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 2381 . . . . . 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 601 . . . . 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 485 . . . 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 2408 . . 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 12 . 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 opreq1 5072 . . . . 5 |- (u = w -> (uPx) = (wPx))
5049eqeq1d 2100 . . . 4 |- (u = w -> ((uPx) = x <-> (wPx) = x))
5150ralbidv 2328 . . 3 |- (u = w -> (A.x e. B (uPx) = x <-> A.x e. B (wPx) = x))
5251reu8 2667 . 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 241 1 |- (G e. Grp -> E!u e. B A.x e. B (uPx) = x)
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 418   = wceq 1592   e. wcel 1594  A.wral 2310  E.wrex 2311  E!wreu 2312  ` cfv 4149  (class class class)co 5067  Basecbs 9967  +gcplusg 10125  Grpcgrp 10126
This theorem is referenced by:  grpidcl 10166  grpidinv2 10167  isgrpid 10170
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1516  ax-6 1517  ax-7 1518  ax-gen 1519  ax-8 1596  ax-10 1597  ax-11 1598  ax-12 1599  ax-13 1600  ax-14 1601  ax-17 1608  ax-9 1620  ax-4 1626  ax-16 1803  ax-ext 2074  ax-sep 3609  ax-nul 3619  ax-pr 3679  ax-un 3947
This theorem depends on definitions:  df-bi 210  df-or 419  df-an 420  df-3an 1039  df-ex 1521  df-sb 1765  df-eu 1992  df-mo 1993  df-clab 2080  df-cleq 2085  df-clel 2088  df-ne 2220  df-ral 2314  df-rex 2315  df-reu 2316  df-v 2501  df-dif 2804  df-un 2806  df-in 2808  df-ss 2810  df-nul 3066  df-sn 3237  df-pr 3238  df-op 3241  df-uni 3365  df-br 3510  df-opab 3568  df-xp 4151  df-cnv 4153  df-dm 4155  df-rn 4156  df-res 4157  df-ima 4158  df-fv 4165  df-opr 5069  df-grp 10131
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