Theorem grpidinv2 9344

Description: A group's properties using the explicit identity element.

Hypotheses

Ref	Expression
grpidval.1	|- X = ran G
grpidval.2	|- U = (Id` G)

Assertion

Ref	Expression
grpidinv2	|- ((G e. Grp /\ A e. X) -> (((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)))

Distinct variable groups:   y,A   y,G   y,U   y,X

Proof of Theorem grpidinv2

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . 6 |- (x = A -> (UGx) = (UGA))
2		id 73	. . . . . 6 |- (x = A -> x = A)
3	1, 2	eqeq12d 1899	. . . . 5 |- (x = A -> ((UGx) = x <-> (UGA) = A))
4		opreq1 4889	. . . . . 6 |- (x = A -> (xGU) = (AGU))
5	4, 2	eqeq12d 1899	. . . . 5 |- (x = A -> ((xGU) = x <-> (AGU) = A))
6	3, 5	anbi12d 690	. . . 4 |- (x = A -> (((UGx) = x /\ (xGU) = x) <-> ((UGA) = A /\ (AGU) = A)))
7		opreq2 4890	. . . . . . 7 |- (x = A -> (yGx) = (yGA))
8	7	eqeq1d 1892	. . . . . 6 |- (x = A -> ((yGx) = U <-> (yGA) = U))
9		opreq1 4889	. . . . . . 7 |- (x = A -> (xGy) = (AGy))
10	9	eqeq1d 1892	. . . . . 6 |- (x = A -> ((xGy) = U <-> (AGy) = U))
11	8, 10	anbi12d 690	. . . . 5 |- (x = A -> (((yGx) = U /\ (xGy) = U) <-> ((yGA) = U /\ (AGy) = U)))
12	11	rexbidv 2124	. . . 4 |- (x = A -> (E.y e. X ((yGx) = U /\ (xGy) = U) <-> E.y e. X ((yGA) = U /\ (AGy) = U)))
13	6, 12	anbi12d 690	. . 3 |- (x = A -> ((((UGx) = x /\ (xGU) = x) /\ E.y e. X ((yGx) = U /\ (xGy) = U)) <-> (((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U))))
14	13	rcla4cva 2379	. 2 |- ((A.x e. X (((UGx) = x /\ (xGU) = x) /\ E.y e. X ((yGx) = U /\ (xGy) = U)) /\ A e. X) -> (((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)))
15		grpidval.1	. . . . . 6 |- X = ran G
16	15	grpidinv 9332	. . . . 5 |- (G e. Grp -> E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))
17	15	grpideu 9333	. . . . 5 |- (G e. Grp -> E!u e. X A.x e. X (uGx) = x)
18		ssid 2634	. . . . . 6 |- X C_ X
19		simpll 448	. . . . . . . . 9 |- ((((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> (uGx) = x)
20	19	ralimi 2168	. . . . . . . 8 |- (A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> A.x e. X (uGx) = x)
21	20	a1i 8	. . . . . . 7 |- (u e. X -> (A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> A.x e. X (uGx) = x))
22	21	rgen 2159	. . . . . 6 |- A.u e. X (A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> A.x e. X (uGx) = x)
23		reuuniss2 3817	. . . . . 6 |- (((X C_ X /\ A.u e. X (A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> A.x e. X (uGx) = x)) /\ (E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) /\ E!u e. X A.x e. X (uGx) = x)) -> U.{u e. X | A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u))} = U.{u e. X | A.x e. X (uGx) = x})
24	18, 22, 23	mpanl12 773	. . . . 5 |- ((E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) /\ E!u e. X A.x e. X (uGx) = x) -> U.{u e. X | A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u))} = U.{u e. X | A.x e. X (uGx) = x})
25	16, 17, 24	syl11anc 524	. . . 4 |- (G e. Grp -> U.{u e. X | A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u))} = U.{u e. X | A.x e. X (uGx) = x})
26		grpidval.2	. . . . 5 |- U = (Id` G)
27	15, 26	grpidval 9342	. . . 4 |- (G e. Grp -> U = U.{u e. X | A.x e. X (uGx) = x})
28	25, 27	eqtr4d 1928	. . 3 |- (G e. Grp -> U.{u e. X | A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u))} = U)
29	15, 26	grpidcl 9343	. . . 4 |- (G e. Grp -> U e. X)
30		reuss2 2870	. . . . . 6 |- (((X C_ X /\ A.u e. X (A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> A.x e. X (uGx) = x)) /\ (E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) /\ E!u e. X A.x e. X (uGx) = x)) -> E!u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))
31	18, 22, 30	mpanl12 773	. . . . 5 |- ((E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) /\ E!u e. X A.x e. X (uGx) = x) -> E!u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))
32	16, 17, 31	syl11anc 524	. . . 4 |- (G e. Grp -> E!u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))
33		opreq1 4889	. . . . . . . . 9 |- (u = U -> (uGx) = (UGx))
34	33	eqeq1d 1892	. . . . . . . 8 |- (u = U -> ((uGx) = x <-> (UGx) = x))
35		opreq2 4890	. . . . . . . . 9 |- (u = U -> (xGu) = (xGU))
36	35	eqeq1d 1892	. . . . . . . 8 |- (u = U -> ((xGu) = x <-> (xGU) = x))
37	34, 36	anbi12d 690	. . . . . . 7 |- (u = U -> (((uGx) = x /\ (xGu) = x) <-> ((UGx) = x /\ (xGU) = x)))
38		eqeq2 1893	. . . . . . . . 9 |- (u = U -> ((yGx) = u <-> (yGx) = U))
39		eqeq2 1893	. . . . . . . . 9 |- (u = U -> ((xGy) = u <-> (xGy) = U))
40	38, 39	anbi12d 690	. . . . . . . 8 |- (u = U -> (((yGx) = u /\ (xGy) = u) <-> ((yGx) = U /\ (xGy) = U)))
41	40	rexbidv 2124	. . . . . . 7 |- (u = U -> (E.y e. X ((yGx) = u /\ (xGy) = u) <-> E.y e. X ((yGx) = U /\ (xGy) = U)))
42	37, 41	anbi12d 690	. . . . . 6 |- (u = U -> ((((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) <-> (((UGx) = x /\ (xGU) = x) /\ E.y e. X ((yGx) = U /\ (xGy) = U))))
43	42	ralbidv 2123	. . . . 5 |- (u = U -> (A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) <-> A.x e. X (((UGx) = x /\ (xGU) = x) /\ E.y e. X ((yGx) = U /\ (xGy) = U))))
44	43	reuuni2 3811	. . . 4 |- ((U e. X /\ E!u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u))) -> (A.x e. X (((UGx) = x /\ (xGU) = x) /\ E.y e. X ((yGx) = U /\ (xGy) = U)) <-> U.{u e. X | A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u))} = U))
45	29, 32, 44	syl11anc 524	. . 3 |- (G e. Grp -> (A.x e. X (((UGx) = x /\ (xGU) = x) /\ E.y e. X ((yGx) = U /\ (xGy) = U)) <-> U.{u e. X | A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u))} = U))
46	28, 45	mpbird 213	. 2 |- (G e. Grp -> A.x e. X (((UGx) = x /\ (xGU) = x) /\ E.y e. X ((yGx) = U /\ (xGy) = U)))
47	14, 46	sylan 497	1 |- ((G e. Grp /\ A e. X) -> (((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107  {crab 2108   C_ wss 2593  U.cuni 3177  ran crn 3987  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312

This theorem is referenced by:  grplid 9345  grprid 9346  grprcan 9347  grpinveu 9348  grpinv 9353

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-rep 3428  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-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-grp 9316  df-gid 9317

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