Theorem grpideu 9333

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

Hypothesis

Ref	Expression
grpfo.1	|- X = ran G

Assertion

Ref	Expression
grpideu	|- (G e. Grp -> E!u e. X A.x e. X (uGx) = x)

Distinct variable groups:   x,u,G   u,X,x

Proof of Theorem grpideu

Step	Hyp	Ref	Expression
1		grpfo.1	. . . 4 |- X = ran G
2	1	grpidinv 9332	. . 3 |- (G e. Grp -> E.u e. X A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u)))
3		simpll 448	. . . . . . . . 9 |- ((((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u)) -> (uGz) = z)
4	3	ralimi 2168	. . . . . . . 8 |- (A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u)) -> A.z e. X (uGz) = z)
5		opreq2 4890	. . . . . . . . . 10 |- (z = x -> (uGz) = (uGx))
6		id 73	. . . . . . . . . 10 |- (z = x -> z = x)
7	5, 6	eqeq12d 1899	. . . . . . . . 9 |- (z = x -> ((uGz) = z <-> (uGx) = x))
8	7	cbvralv 2280	. . . . . . . 8 |- (A.z e. X (uGz) = z <-> A.x e. X (uGx) = x)
9	4, 8	sylib 215	. . . . . . 7 |- (A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u)) -> A.x e. X (uGx) = x)
10	9	adantl 424	. . . . . 6 |- (((G e. Grp /\ u e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) -> A.x e. X (uGx) = x)
11	9	ad2antlr 441	. . . . . . . 8 |- ((((G e. Grp /\ u e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) /\ w e. X) -> A.x e. X (uGx) = x)
12		opreq2 4890	. . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (yGz) = (yGw))
13	12	eqeq1d 1892	. . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((yGz) = u <-> (yGw) = u))
14		opreq1 4889	. . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (zGy) = (wGy))
15	14	eqeq1d 1892	. . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((zGy) = u <-> (wGy) = u))
16	13, 15	anbi12d 690	. . . . . . . . . . . . . . . . . 18 |- (z = w -> (((yGz) = u /\ (zGy) = u) <-> ((yGw) = u /\ (wGy) = u)))
17	16	rexbidv 2124	. . . . . . . . . . . . . . . . 17 |- (z = w -> (E.y e. X ((yGz) = u /\ (zGy) = u) <-> E.y e. X ((yGw) = u /\ (wGy) = u)))
18	17	rcla4va 2378	. . . . . . . . . . . . . . . 16 |- ((w e. X /\ A.z e. X E.y e. X ((yGz) = u /\ (zGy) = u)) -> E.y e. X ((yGw) = u /\ (wGy) = u))
19	18	adantll 428	. . . . . . . . . . . . . . 15 |- (((G e. Grp /\ w e. X) /\ A.z e. X E.y e. X ((yGz) = u /\ (zGy) = u)) -> E.y e. X ((yGw) = u /\ (wGy) = u))
20		simpr 350	. . . . . . . . . . . . . . . 16 |- ((((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u)) -> E.y e. X ((yGz) = u /\ (zGy) = u))
21	20	ralimi 2168	. . . . . . . . . . . . . . 15 |- (A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u)) -> A.z e. X E.y e. X ((yGz) = u /\ (zGy) = u))
22	19, 21	sylan2 500	. . . . . . . . . . . . . 14 |- (((G e. Grp /\ w e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) -> E.y e. X ((yGw) = u /\ (wGy) = u))
23	1	grpidinvlem4 9331	. . . . . . . . . . . . . 14 |- (((G e. Grp /\ w e. X) /\ E.y e. X ((yGw) = u /\ (wGy) = u)) -> (wGu) = (uGw))
24	22, 23	syldan 516	. . . . . . . . . . . . 13 |- (((G e. Grp /\ w e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) -> (wGu) = (uGw))
25	24	an1rs 547	. . . . . . . . . . . 12 |- (((G e. Grp /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) /\ w e. X) -> (wGu) = (uGw))
26	25	adantllr 433	. . . . . . . . . . 11 |- ((((G e. Grp /\ u e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) /\ w e. X) -> (wGu) = (uGw))
27	26	adantr 425	. . . . . . . . . 10 |- (((((G e. Grp /\ u e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) /\ w e. X) /\ (A.x e. X (uGx) = x /\ A.x e. X (wGx) = x)) -> (wGu) = (uGw))
28		opreq2 4890	. . . . . . . . . . . . . . 15 |- (x = u -> (wGx) = (wGu))
29		id 73	. . . . . . . . . . . . . . 15 |- (x = u -> x = u)
30	28, 29	eqeq12d 1899	. . . . . . . . . . . . . 14 |- (x = u -> ((wGx) = x <-> (wGu) = u))
31	30	rcla4va 2378	. . . . . . . . . . . . 13 |- ((u e. X /\ A.x e. X (wGx) = x) -> (wGu) = u)
32	31	adantll 428	. . . . . . . . . . . 12 |- (((G e. Grp /\ u e. X) /\ A.x e. X (wGx) = x) -> (wGu) = u)
33	32	ad2ant2rl 447	. . . . . . . . . . 11 |- ((((G e. Grp /\ u e. X) /\ w e. X) /\ (A.x e. X (uGx) = x /\ A.x e. X (wGx) = x)) -> (wGu) = u)
34	33	adantllr 433	. . . . . . . . . 10 |- (((((G e. Grp /\ u e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) /\ w e. X) /\ (A.x e. X (uGx) = x /\ A.x e. X (wGx) = x)) -> (wGu) = u)
35		opreq2 4890	. . . . . . . . . . . . 13 |- (x = w -> (uGx) = (uGw))
36		id 73	. . . . . . . . . . . . 13 |- (x = w -> x = w)
37	35, 36	eqeq12d 1899	. . . . . . . . . . . 12 |- (x = w -> ((uGx) = x <-> (uGw) = w))
38	37	rcla4va 2378	. . . . . . . . . . 11 |- ((w e. X /\ A.x e. X (uGx) = x) -> (uGw) = w)
39	38	ad2ant2lr 446	. . . . . . . . . 10 |- (((((G e. Grp /\ u e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) /\ w e. X) /\ (A.x e. X (uGx) = x /\ A.x e. X (wGx) = x)) -> (uGw) = w)
40	27, 34, 39	3eqtr3d 1934	. . . . . . . . 9 |- (((((G e. Grp /\ u e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) /\ w e. X) /\ (A.x e. X (uGx) = x /\ A.x e. X (wGx) = x)) -> u = w)
41	40	ex 402	. . . . . . . 8 |- ((((G e. Grp /\ u e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) /\ w e. X) -> ((A.x e. X (uGx) = x /\ A.x e. X (wGx) = x) -> u = w))
42	11, 41	mpand 765	. . . . . . 7 |- ((((G e. Grp /\ u e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) /\ w e. X) -> (A.x e. X (wGx) = x -> u = w))
43	42	r19.21aiva 2176	. . . . . 6 |- (((G e. Grp /\ u e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) -> A.w e. X (A.x e. X (wGx) = x -> u = w))
44	10, 43	jca 310	. . . . 5 |- (((G e. Grp /\ u e. X) /\ A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u))) -> (A.x e. X (uGx) = x /\ A.w e. X (A.x e. X (wGx) = x -> u = w)))
45	44	ex 402	. . . 4 |- ((G e. Grp /\ u e. X) -> (A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u)) -> (A.x e. X (uGx) = x /\ A.w e. X (A.x e. X (wGx) = x -> u = w))))
46	45	reximdva 2203	. . 3 |- (G e. Grp -> (E.u e. X A.z e. X (((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u)) -> E.u e. X (A.x e. X (uGx) = x /\ A.w e. X (A.x e. X (wGx) = x -> u = w))))
47	2, 46	mpd 29	. 2 |- (G e. Grp -> E.u e. X (A.x e. X (uGx) = x /\ A.w e. X (A.x e. X (wGx) = x -> u = w)))
48		opreq1 4889	. . . . 5 |- (u = w -> (uGx) = (wGx))
49	48	eqeq1d 1892	. . . 4 |- (u = w -> ((uGx) = x <-> (wGx) = x))
50	49	ralbidv 2123	. . 3 |- (u = w -> (A.x e. X (uGx) = x <-> A.x e. X (wGx) = x))
51	50	reu8 2448	. 2 |- (E!u e. X A.x e. X (uGx) = x <-> E.u e. X (A.x e. X (uGx) = x /\ A.w e. X (A.x e. X (wGx) = x -> u = w)))
52	47, 51	sylibr 217	1 |- (G e. Grp -> E!u e. X A.x e. X (uGx) = x)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107  ran crn 3987  (class class class)co 4884  Grpcgr 9311

This theorem is referenced by:  grprlidrid 9337  grpidcl 9343  grpidinv2 9344  cnid 9435  mulid 9440  hilid 10661

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-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-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-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-grp 9316

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