Theorem grpidinv 8049

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 8042	. 2 |- (G e. Grp -> E.u e. X A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u))
3		opreq2 3975	. . . . . . . . . . . 12 |- (z = x -> (uGz) = (uGx))
4		id 59	. . . . . . . . . . . 12 |- (z = x -> z = x)
5	3, 4	eqeq12d 1492	. . . . . . . . . . 11 |- (z = x -> ((uGz) = z <-> (uGx) = x))
6	5	rcla4cva 1879	. . . . . . . . . 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 1709	. . . . . . . . . 10 |- (A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u) -> A.z e. X (uGz) = z)
9	6, 8	sylan 450	. . . . . . . . 9 |- ((A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u) /\ x e. X) -> (uGx) = x)
10	9	adantll 394	. . . . . . . 8 |- (((u e. X /\ A.z e. X ((uGz) = z /\ E.w e. X (wGz) = u)) /\ x e. X) -> (uGx) = x)
11	10	adantll 394	. . . . . . 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 8048	. . . . . . . . 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 397	. . . . . . . . . . . 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 391	. . . . . . . . . . 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 390	. . . . . . . . . . . . 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 407	. . . . . . . . . . . 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 1709	. . . . . . . . . . . . . 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 390	. . . . . . . . . . . . 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 407	. . . . . . . . . . . 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 8047	. . . . . . . . . 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 477	. . . . . . . . 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 473	. . . . . . . 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 1510	. . . . . . 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 1717	. . . 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 379	. . 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 1740	. 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 958   e. wcel 960  A.wral 1648  E.wrex 1649  ran crn 3177  (class class class)co 3969  Grpcgr 8030

This theorem is referenced by:  grpideu 8050  grpidinv2 8056

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-grp 8034

Copyright terms: Public domain