Theorem grpidinv 9332

Description: A group has a left and right identity element, and every member has a left and right inverse.

Hypothesis

Ref	Expression
grpfo.1	|- X = ran G

Assertion

Ref	Expression
grpidinv	|- (G e. Grp -> E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))

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

Proof of Theorem grpidinv

Step	Hyp	Ref	Expression
1		grpfo.1	. . 3 |- X = ran G
2	1	grplidinv 9325	. 2 |- (G e. Grp -> E.u e. X A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))
3		opreq2 4890	. . . . . . . . . . 11 |- (z = x -> (uGz) = (uGx))
4		id 73	. . . . . . . . . . 11 |- (z = x -> z = x)
5	3, 4	eqeq12d 1899	. . . . . . . . . 10 |- (z = x -> ((uGz) = z <-> (uGx) = x))
6	5	rcla4cva 2379	. . . . . . . . 9 |- ((A.z e. X (uGz) = z /\ x e. X) -> (uGx) = x)
7		simpl 346	. . . . . . . . . 10 |- (((uGz) = z /\ E.w e. X (wGz) = u) -> (uGz) = z)
8	7	ralimi 2168	. . . . . . . . 9 |- (A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u) -> A.z e. X (uGz) = z)
9	6, 8	sylan 497	. . . . . . . 8 |- ((A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u) /\ x e. X) -> (uGx) = x)
10	9	adantll 428	. . . . . . 7 |- (((u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u)) /\ x e. X) -> (uGx) = x)
11	10	adantll 428	. . . . . 6 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> (uGx) = x)
12		simpl 346	. . . . . . . . 9 |- ((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) -> G e. Grp)
13	12	anim1i 361	. . . . . . . 8 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> (G e. Grp /\ x e. X))
14		id 73	. . . . . . . . . . . 12 |- ((G e. Grp /\ u e. X) -> (G e. Grp /\ u e. X))
15	14	adantrr 431	. . . . . . . . . . 11 |- ((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) -> (G e. Grp /\ u e. X))
16	15	adantr 425	. . . . . . . . . 10 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> (G e. Grp /\ u e. X))
17	8	adantl 424	. . . . . . . . . . 11 |- ((u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u)) -> A.z e. X (uGz) = z)
18	17	ad2antlr 441	. . . . . . . . . 10 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> A.z e. X (uGz) = z)
19		simpr 350	. . . . . . . . . . . . 13 |- (((uGz) = z /\ E.w e. X (wGz) = u) -> E.w e. X (wGz) = u)
20	19	ralimi 2168	. . . . . . . . . . . 12 |- (A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u) -> A.z e. X E.w e. X (wGz) = u)
21	20	adantl 424	. . . . . . . . . . 11 |- ((u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u)) -> A.z e. X E.w e. X (wGz) = u)
22	21	ad2antlr 441	. . . . . . . . . 10 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> A.z e. X E.w e. X (wGz) = u)
23	16, 18, 22	jca32 312	. . . . . . . . 9 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> ((G e. Grp /\ u e. X) /\ (A.z e. X (uGz) = z /\ A.z e. X E.w e. X (wGz) = u)))
24		biid 187	. . . . . . . . . 10 |- (A.z e. X (uGz) = z <-> A.z e. X (uGz) = z)
25		biid 187	. . . . . . . . . 10 |- (A.z e. X E.w e. X (wGz) = u <-> A.z e. X E.w e. X (wGz) = u)
26	1, 24, 25	grpidinvlem3 9330	. . . . . . . . 9 |- ((((G e. Grp /\ u e. X) /\ (A.z e. X (uGz) = z /\ A.z e. X E.w e. X (wGz) = u)) /\ x e. X) -> E.y e. X ((yGx) = u /\ (xGy) = u))
27	23, 26	sylancom 531	. . . . . . . 8 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> E.y e. X ((yGx) = u /\ (xGy) = u))
28	1	grpidinvlem4 9331	. . . . . . . 8 |- (((G e. Grp /\ x e. X) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> (xGu) = (uGx))
29	13, 27, 28	syl11anc 524	. . . . . . 7 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> (xGu) = (uGx))
30	29, 11	eqtrd 1925	. . . . . 6 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> (xGu) = x)
31	11, 30, 27	jca31 311	. . . . 5 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))
32	31	r19.21aiva 2176	. . . 4 |- ((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) -> A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))
33	32	exp32 408	. . 3 |- (G e. Grp -> (u e. X -> (A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u) -> A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))))
34	33	reximdvai 2201	. 2 |- (G e. Grp -> (E.u e. X A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u) -> E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u))))
35	2, 34	mpd 29	1 |- (G e. Grp -> E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))

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

This theorem is referenced by:  grpideu 9333  grprlidrid 9337  grpidinv2 9344  grpmnd 10393

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-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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