Theorem grpidinv2 8056

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 3975	. . . . . 6 |- (x = A -> (UGx) = (UGA))
2		id 59	. . . . . 6 |- (x = A -> x = A)
3	1, 2	eqeq12d 1492	. . . . 5 |- (x = A -> ((UGx) = x <-> (UGA) = A))
4		opreq1 3974	. . . . . 6 |- (x = A -> (xGU) = (AGU))
5	4, 2	eqeq12d 1492	. . . . 5 |- (x = A -> ((xGU) = x <-> (AGU) = A))
6	3, 5	anbi12d 630	. . . 4 |- (x = A -> (((UGx) = x /\ (xGU) = x) <-> ((UGA) = A /\ (AGU) = A)))
7		opreq2 3975	. . . . . . 7 |- (x = A -> (yGx) = (yGA))
8	7	eqeq1d 1486	. . . . . 6 |- (x = A -> ((yGx) = U <-> (yGA) = U))
9		opreq1 3974	. . . . . . 7 |- (x = A -> (xGy) = (AGy))
10	9	eqeq1d 1486	. . . . . 6 |- (x = A -> ((xGy) = U <-> (AGy) = U))
11	8, 10	anbi12d 630	. . . . 5 |- (x = A -> (((yGx) = U /\ (xGy) = U) <-> ((yGA) = U /\ (AGy) = U)))
12	11	rexbidv 1667	. . . 4 |- (x = A -> (E.y e. X ((yGx) = U /\ (xGy) = U) <-> E.y e. X ((yGA) = U /\ (AGy) = U)))
13	6, 12	anbi12d 630	. . 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 1879	. 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		ssid 2083	. . . . . 6 |- X (_ X
16		simpll 414	. . . . . . . . 9 |- ((((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> (uGx) = x)
17	16	r19.20si 1709	. . . . . . . 8 |- (A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> A.x e. X (uGx) = x)
18	17	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))
19	18	rgen 1701	. . . . . 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)
20		reuuniss2 2897	. . . . . 6 |- (((X (_ 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})
21	15, 19, 20	mpanl12 710	. . . . 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})
22		grpidval.1	. . . . . 6 |- X = ran G
23	22	grpidinv 8049	. . . . 5 |- (G e. Grp -> E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))
24	22	grpideu 8050	. . . . 5 |- (G e. Grp -> E!u e. X A.x e. X (uGx) = x)
25	21, 23, 24	sylanc 473	. . . 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	22, 26	grpidval 8054	. . . 4 |- (G e. Grp -> U = U.{u e. X | A.x e. X (uGx) = x})
28	25, 27	eqtr4d 1513	. . 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		opreq1 3974	. . . . . . . . 9 |- (u = U -> (uGx) = (UGx))
30	29	eqeq1d 1486	. . . . . . . 8 |- (u = U -> ((uGx) = x <-> (UGx) = x))
31		opreq2 3975	. . . . . . . . 9 |- (u = U -> (xGu) = (xGU))
32	31	eqeq1d 1486	. . . . . . . 8 |- (u = U -> ((xGu) = x <-> (xGU) = x))
33	30, 32	anbi12d 630	. . . . . . 7 |- (u = U -> (((uGx) = x /\ (xGu) = x) <-> ((UGx) = x /\ (xGU) = x)))
34		eqeq2 1487	. . . . . . . . 9 |- (u = U -> ((yGx) = u <-> (yGx) = U))
35		eqeq2 1487	. . . . . . . . 9 |- (u = U -> ((xGy) = u <-> (xGy) = U))
36	34, 35	anbi12d 630	. . . . . . . 8 |- (u = U -> (((yGx) = u /\ (xGy) = u) <-> ((yGx) = U /\ (xGy) = U)))
37	36	rexbidv 1667	. . . . . . 7 |- (u = U -> (E.y e. X ((yGx) = u /\ (xGy) = u) <-> E.y e. X ((yGx) = U /\ (xGy) = U)))
38	33, 37	anbi12d 630	. . . . . 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))))
39	38	ralbidv 1666	. . . . 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))))
40	39	reuuni2 2890	. . . 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))
41	22, 26	grpidcl 8055	. . . 4 |- (G e. Grp -> U e. X)
42		reuss2 2278	. . . . . 6 |- (((X (_ X /\ A.u e. X (A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y <