Theorem grpidinv 8052

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 8045	. 2 |- (G e. Grp -> E.u e. X A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))
3		opreq2 3969	. . . . . . . . . . . 12 |- (z = x -> (uGz) = (uGx))
4		id 59	. . . . . . . . . . . 12 |- (z = x -> z = x)
5	3, 4	eqeq12d 1489	. . . . . . . . . . 11 |- (z = x -> ((uGz) = z <-> (uGx) = x))
6	5	rcla4cva 1876	. . . . . . . . . 10 |- ((A.z e. X (uGz) = z /\ x e. X) -> (uGx) = x)
7		pm3.26 319	. . . . . . . . . . 11 |- (((uGz) = z /\ E.w e. X (wGz) = u) -> (uGz) = z)
8	7	r19.20si 1706	. . . . . . . . . 10 |- (A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u) -> A.z e. X (uGz) = z)
9	6, 8	sylan 448	. . . . . . . . 9 |- ((A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u) /\ x e. X) -> (uGx) = x)
10	9	adantll 392	. . . . . . . 8 |- (((u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u)) /\ x e. X) -> (uGx) = x)
11	10	adantll 392	. . . . . . 7 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> (uGx) = x)
12	1	grpidinvlem4 8051	. . . . . . . . 9 |- (((G e. Grp /\ x e. X) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> (xGu) = (uGx))
13		pm3.26 319	. . . . . . . . . 10 |- ((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) -> G e. Grp)
14	13	anim1i 334	. . . . . . . . 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 /\ x e. X))
15		id 59	. . . . . . . . . . . . 13 |- ((G e. Grp /\ u e. X) -> (G e. Grp /\ u e. X))
16	15	adantrr 395	. . . . . . . . . . . 12 |- ((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) -> (G e. Grp /\ u e. X))
17	16	adantr 389	. . . . . . . . . . 11 |- (((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))
18	8	adantl 388	. . . . . . . . . . . . 13 |- ((u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u)) -> A.z e. X (uGz) = z)
19	18	ad2antlr 405	. . . . . . . . . . . 12 |- (((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)
20		pm3.27 323	. . . . . . . . . . . . . . 15 |- (((uGz) = z /\ E.w e. X (wGz) = u) -> E.w e. X (wGz) = u)
21	20	r19.20si 1706	. . . . . . . . . . . . . 14 |- (A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u) -> A.z e. X E.w e. X (wGz) = u)
22	21	adantl 388	. . . . . . . . . . . . 13 |- ((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)
23	22	ad2antlr 405	. . . . . . . . . . . 12 |- (((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)
24	19, 23	jca 288	. . . . . . . . . . 11 |- (((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 /\ A.z e. X E.w e. X (wGz) = u))
25	17, 24	jca 288	. . . . . . . . . 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) /\ (A.z e. X (uGz) = z /\ A.z e. X E.w e. X (wGz) = u)))
26		pm4.2 170	. . . . . . . . . . 11 |- (A.z e. X (uGz) = z <-> A.z e. X (uGz) = z)
27		pm4.2 170	. . . . . . . . . . 11 |- (A.z e. X E.w e. X (wGz) = u <-> A.z e. X E.w e. X (wGz) = u)
28	1, 26, 27	grpidinvlem3 8050	. . . . . . . . . 10 |- ((((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))
29	25, 28	sylancom 475	. . . . . . . . 9 |- (((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))
30	12, 14, 29	sylanc 471	. . . . . . . 8 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> (xGu) = (uGx))
31	30, 11	eqtrd 1507	. . . . . . 7 |- (((G e. Grp /\ (u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))) /\ x e. X) -> (xGu) = x)
32	11, 31	jca 288	. . . . . 6 |- (((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))
33	32, 29	jca 288	. . . . 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)))
34	33	r19.21aiva 1714	. . . 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)))
35	34	exp32 377	. . 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)))))
36	35	r19.22dv 1737	. 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))))
37	2, 36	mpd 26	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 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  ran crn 3171  (class class class)co 3963  Grpcgr 8033

This theorem is referenced by:  grpideu 8053  grpidinv2 8060

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037

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