Theorem grpideuNEW 17114

Description: The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55.

Hypotheses

Ref	Expression
grplem1.1NEW	|- B = (base` G)
grplem1.2NEW	|- P = (+g` G)

Assertion

Ref	Expression
grpideuNEW	|- (G e. GrpNEW -> 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 grpideuNEW

Step	Hyp	Ref	Expression
1		grplem1.1NEW	. . . 4 |- B = (base` G)
2		grplem1.2NEW	. . . 4 |- P = (+g` G)
3	1, 2	grpidinvNEW 17113	. . 3 |- (G e. GrpNEW -> E.u e. B A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)))
4		simpll 448	. . . . . . . . 9 |- ((((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> (uPz) = z)
5	4	ralimi 2168	. . . . . . . 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 4890	. . . . . . . . . 10 |- (z = x -> (uPz) = (uPx))
7		id 73	. . . . . . . . . 10 |- (z = x -> z = x)
8	6, 7	eqeq12d 1899	. . . . . . . . 9 |- (z = x -> ((uPz) = z <-> (uPx) = x))
9	8	cbvralv 2280	. . . . . . . 8 |- (A.z e. B (uPz) = z <-> A.x e. B (uPx) = x)
10	5, 9	sylib 215	. . . . . . 7 |- (A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> A.x e. B (uPx) = x)
11	10	adantl 424	. . . . . 6 |- (((G e. GrpNEW /\ 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)
12	10	ad2antlr 441	. . . . . . . 8 |- ((((G e. GrpNEW /\ 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		opreq2 4890	. . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (yPz) = (yPw))
14	13	eqeq1d 1892	. . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((yPz) = u <-> (yPw) = u))
15		opreq1 4889	. . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (zPy) = (wPy))
16	15	eqeq1d 1892	. . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((zPy) = u <-> (wPy) = u))
17	14, 16	anbi12d 690	. . . . . . . . . . . . . . . . . 18 |- (z = w -> (((yPz) = u /\ (zPy) = u) <-> ((yPw) = u /\ (wPy) = u)))
18	17	rexbidv 2124	. . . . . . . . . . . . . . . . 17 |- (z = w -> (E.y e. B ((yPz) = u /\ (zPy) = u) <-> E.y e. B ((yPw) = u /\ (wPy) = u)))
19	18	rcla4va 2378	. . . . . . . . . . . . . . . 16 |- ((w e. B /\ A.z e. B E.y e. B ((yPz) = u /\ (zPy) = u)) -> E.y e. B ((yPw) = u /\ (wPy) = u))
20	19	adantll 428	. . . . . . . . . . . . . . 15 |- (((G e. GrpNEW /\ w e. B) /\ A.z e. B E.y e. B ((yPz) = u /\ (zPy) = u)) -> E.y e. B ((yPw) = u /\ (wPy) = u))
21		simpr 350	. . . . . . . . . . . . . . . 16 |- ((((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u)) -> E.y e. B ((yPz) = u /\ (zPy) = u))
22	21	ralimi 2168	. . . . . . . . . . . . . . 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))
23	20, 22	sylan2 500	. . . . . . . . . . . . . 14 |- (((G e. GrpNEW /\ 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))
24	1, 2	grpidinvlem4NEW 17112	. . . . . . . . . . . . . 14 |- (((G e. GrpNEW /\ w e. B) /\ E.y e. B ((yPw) = u /\ (wPy) = u)) -> (wPu) = (uPw))
25	23, 24	syldan 516	. . . . . . . . . . . . 13 |- (((G e. GrpNEW /\ w e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) -> (wPu) = (uPw))
26	25	an1rs 547	. . . . . . . . . . . 12 |- (((G e. GrpNEW /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) /\ w e. B) -> (wPu) = (uPw))
27	26	adantllr 433	. . . . . . . . . . 11 |- ((((G e. GrpNEW /\ u e. B) /\ A.z e. B (((uPz) = z /\ (zPu) = z) /\ E.y e. B ((yPz) = u /\ (zPy) = u))) /\ w e. B) -> (wPu) = (uPw))
28	27	adantr 425	. . . . . . . . . 10 |- (((((G e. GrpNEW /\ 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 4890	. . . . . . . . . . . . . . 15 |- (x = u -> (wPx) = (wPu))
30		id 73	. . . . . . . . . . . . . . 15 |- (x = u -> x = u)
31	29, 30	eqeq12d 1899	. . . . . . . . . . . . . 14 |- (x = u -> ((wPx) = x <-> (wPu) = u))
32	31	rcla4va 2378	. . . . . . . . . . . . 13 |- ((u e. B /\ A.x e. B (wPx) = x) -> (wPu) = u)
33	32	adantll 428	. . . . . . . . . . . 12 |- (((G e. GrpNEW /\ u e. B) /\ A.x e. B (wPx) = x) -> (wPu) = u)
34	33	ad2ant2rl 447	. . . . . . . . . . 11 |- ((((G e. GrpNEW /\ u e. B) /\ w e. B) /\ (A.x e. B (uPx) = x /\ A.x e. B (wPx) = x)) -> (wPu) = u)
35	34	adantllr 433	. . . . . . . . . 10 |- (((((G e. GrpNEW /\ 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 4890	. . . . . . . . . . . . 13 |- (x = w -> (uPx) = (uPw))
37		id 73	. . . . . . . . . . . . 13 |- (x = w -> x = w)
38	36, 37	eqeq12d 1899	. . . . . . . . . . . 12 |- (x = w -> ((uPx) = x <-> (uPw) = w))
39	38	rcla4va 2378	. . . . . . . . . . 11 |- ((w e. B /\ A.x e. B (uPx) = x) -> (uPw) = w)
40	39	ad2ant2lr 446	. . . . . . . . . 10 |- (((((G e. GrpNEW /\ 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)
41	28, 35, 40	3eqtr3d 1934	. . . . . . . . 9 |- (((((G e. GrpNEW /\ 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)
42	41	ex 402	. . . . . . . 8 |- ((((G e. GrpNEW /\ 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))
43	12, 42	mpand 765	. . . . . . 7 |- ((((G e. GrpNEW /\ 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))
44	43	r19.21aiva 2176	. . . . . 6 |- (((G e. GrpNEW /\ 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))
45	11, 44	jca 310	. . . . 5 |- (((G e. GrpNEW /\ 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)))
46	45	ex 402	. . . 4 |- ((G e. GrpNEW /\ 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))))
47	46	reximdva 2203	. . 3 |- (G e. GrpNEW -> (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))))
48	3, 47	mpd 29	. 2 |- (G e. GrpNEW -> E.u e. B (A.x e. B (uPx) = x /\ A.w e. B (A.x e. B (wPx) = x -> u = w)))
49		opreq1 4889	. . . . 5 |- (u = w -> (uPx) = (wPx))
50	49	eqeq1d 1892	. . . 4 |- (u = w -> ((uPx) = x <-> (wPx) = x))
51	50	ralbidv 2123	. . 3 |- (u = w -> (A.x e. B (uPx) = x <-> A.x e. B (wPx) = x))
52	51	reu8 2448	. 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)))
53	48, 52	sylibr 217	1 |- (G e. GrpNEW -> E!u e. B A.x e. B (uPx) = x)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107  ` cfv 3998  (class class class)co 4884  basecbs 16758  +_gcplusg 17080  GrpNEWcgrp 17081

This theorem is referenced by:  grpidclNEW 17118  grpidinv2NEW 17119

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-tru 1262  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-struct 16708  df-grpNEW 17089

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