Theorem grpideu 8050

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 8049	. . 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 414	. . . . . . . . 9 |- ((((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u)) -> (uGz) = z)
4	3	r19.20si 1709	. . . . . . . 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 3975	. . . . . . . . . 10 |- (z = x -> (uGz) = (uGx))
6		id 59	. . . . . . . . . 10 |- (z = x -> z = x)
7	5, 6	eqeq12d 1492	. . . . . . . . 9 |- (z = x -> ((uGz) = z <-> (uGx) = x))
8	7	cbvralv 1803	. . . . . . . 8 |- (A.z e. X (uGz) = z <-> A.x e. X (uGx) = x)
9	4, 8	sylib 198	. . . . . . 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 390	. . . . . 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 407	. . . . . . . 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 3975	. . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (yGz) = (yGw))
13	12	eqeq1d 1486	. . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((yGz) = u <-> (yGw) = u))
14		opreq1 3974	. . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (zGy) = (wGy))
15	14	eqeq1d 1486	. . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((zGy) = u <-> (wGy) = u))
16	13, 15	anbi12d 630	. . . . . . . . . . . . . . . . . 18 |- (z = w -> (((yGz) = u /\ (zGy) = u) <-> ((yGw) = u /\ (wGy) = u)))
17	16	rexbidv 1667	. . . . . . . . . . . . . . . . 17 |- (z = w -> (E.y e. X ((yGz) = u /\ (zGy) = u) <-> E.y e. X ((yGw) = u /\ (wGy) = u)))
18	17	rcla4va 1878	. . . . . . . . . . . . . . . 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 394	. . . . . . . . . . . . . . 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		pm3.27 323	. . . . . . . . . . . . . . . 16 |- ((((uGz) = z /\ (zGu) = z) /\ E.y e. X ((yGz) = u /\ (zGy) = u)) -> E.y e. X ((yGz) = u /\ (zGy) = u))
21	20	r19.20si 1709	. . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . 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 8048	. . . . . . . . . . . . . 14 |- (((G e. Grp /\ w e. X) /\ E.y e. X ((yGw) = u /\ (wGy) = u)) -> (wGu) = (uGw))
24	22, 23	syldan 469	. . . . . . . . . . . . 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 491	. . . . . . . . . . . 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 399	. . . . . . . . . . 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 391	. . . . . . . . . 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 3975	. . . . . . . . . . . . . . 15 |- (x = u -> (wGx) = (wGu))
29		id 59	. . . . . . . . . . . . . . 15 |- (x = u -> x = u)
30	28, 29	eqeq12d 1492	. . . . . . . . . . . . . 14 |- (x = u -> ((wGx) = x <-> (wGu) = u))
31	30	rcla4va 1878	. . . . . . . . . . . . 13 |- ((u e. X /\ A.x e. X (wGx) = x) -> (wGu) = u)
32	31	adantll 394	. . . . . . . . . . . 12 |- (((G e. Grp /\ u e. X) /\ A.x e. X (wGx) = x) -> (wGu) = u)
33	32	ad2ant2rl 413	. . . . . . . . . . 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 399	. . . . . . . . . 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 3975	. . . . . . . . . . . . 13 |- (x = w -> (uGx) = (uGw))
36		id 59	. . . . . . . . . . . . 13 |- (x = w -> x = w)
37	35, 36	eqeq12d 1492	. . . . . . . . . . . 12 |- (x = w -> ((uGx) = x <-> (uGw) = w))
38	37	rcla4va 1878	. . . . . . . . . . 11 |- ((w e. X /\ A.x e. X (uGx) = x) -> (uGw) = w)
39	38	ad2ant2lr 412	. . . . . . . . . 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 1518	. . . . . . . . 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 373	. . . . . . . 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 703	. . . . . . 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