Theorem grpoidinv2 27369

Description: A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpoidval.1	|- X = ran G
grpoidval.2	|- U = (GId ` G )

Assertion

Ref	Expression
grpoidinv2	|- ( ( G e. GrpOp /\ A e. X ) -> ( ( ( U G A ) = A /\ ( A G U ) = A ) /\ E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) ) )

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

Proof of Theorem grpoidinv2

Dummy variables x u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpoidval.1	. . . . . . 7 |- X = ran G
2		grpoidval.2	. . . . . . 7 |- U = (GId ` G )
3	1, 2	grpoidval 27367	. . . . . 6 |- ( G e. GrpOp -> U = ( iota_ u e. X A. x e. X ( u G x ) = x ) )
4	1	grpoideu 27363	. . . . . . 7 |- ( G e. GrpOp -> E! u e. X A. x e. X ( u G x ) = x )
5		riotacl2 6624	. . . . . . 7 |- ( E! u e. X A. x e. X ( u G x ) = x -> ( iota_ u e. X A. x e. X ( u G x ) = x ) e. { u e. X | A. x e. X ( u G x ) = x } )
6	4, 5	syl 17	. . . . . 6 |- ( G e. GrpOp -> ( iota_ u e. X A. x e. X ( u G x ) = x ) e. { u e. X | A. x e. X ( u G x ) = x } )
7	3, 6	eqeltrd 2701	. . . . 5 |- ( G e. GrpOp -> U e. { u e. X | A. x e. X ( u G x ) = x } )
8		simpll 790	. . . . . . . . . . 11 |- ( ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) -> ( u G x ) = x )
9	8	ralimi 2952	. . . . . . . . . 10 |- ( A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) -> A. x e. X ( u G x ) = x )
10	9	rgenw 2924	. . . . . . . . 9 |- A. u e. X ( A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) -> A. x e. X ( u G x ) = x )
11	10	a1i 11	. . . . . . . 8 |- ( G e. GrpOp -> A. u e. X ( A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) -> A. x e. X ( u G x ) = x ) )
12	1	grpoidinv 27362	. . . . . . . 8 |- ( G e. GrpOp -> E. u e. X A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) )
13	11, 12, 4	3jca 1242	. . . . . . 7 |- ( G e. GrpOp -> ( A. u e. X ( A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) -> A. x e. X ( u G x ) = x ) /\ E. u e. X A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) /\ E! u e. X A. x e. X ( u G x ) = x ) )
14		reupick2 3913	. . . . . . 7 |- ( ( ( A. u e. X ( A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) -> A. x e. X ( u G x ) = x ) /\ E. u e. X A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) /\ E! u e. X A. x e. X ( u G x ) = x ) /\ u e. X ) -> ( A. x e. X ( u G x ) = x <-> A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) ) )
15	13, 14	sylan 488	. . . . . 6 |- ( ( G e. GrpOp /\ u e. X ) -> ( A. x e. X ( u G x ) = x <-> A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) ) )
16	15	rabbidva 3188	. . . . 5 |- ( G e. GrpOp -> { u e. X | A. x e. X ( u G x ) = x } = { u e. X | A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) } )
17	7, 16	eleqtrd 2703	. . . 4 |- ( G e. GrpOp -> U e. { u e. X | A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) } )
18		oveq1 6657	. . . . . . . . 9 |- ( u = U -> ( u G x ) = ( U G x ) )
19	18	eqeq1d 2624	. . . . . . . 8 |- ( u = U -> ( ( u G x ) = x <-> ( U G x ) = x ) )
20		oveq2 6658	. . . . . . . . 9 |- ( u = U -> ( x G u ) = ( x G U ) )
21	20	eqeq1d 2624	. . . . . . . 8 |- ( u = U -> ( ( x G u ) = x <-> ( x G U ) = x ) )
22	19, 21	anbi12d 747	. . . . . . 7 |- ( u = U -> ( ( ( u G x ) = x /\ ( x G u ) = x ) <-> ( ( U G x ) = x /\ ( x G U ) = x ) ) )
23		eqeq2 2633	. . . . . . . . 9 |- ( u = U -> ( ( y G x ) = u <-> ( y G x ) = U ) )
24		eqeq2 2633	. . . . . . . . 9 |- ( u = U -> ( ( x G y ) = u <-> ( x G y ) = U ) )
25	23, 24	anbi12d 747	. . . . . . . 8 |- ( u = U -> ( ( ( y G x ) = u /\ ( x G y ) = u ) <-> ( ( y G x ) = U /\ ( x G y ) = U ) ) )
26	25	rexbidv 3052	. . . . . . 7 |- ( u = U -> ( E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) <-> E. y e. X ( ( y G x ) = U /\ ( x G y ) = U ) ) )
27	22, 26	anbi12d 747	. . . . . 6 |- ( u = U -> ( ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) <-> ( ( ( U G x ) = x /\ ( x G U ) = x ) /\ E. y e. X ( ( y G x ) = U /\ ( x G y ) = U ) ) ) )
28	27	ralbidv 2986	. . . . 5 |- ( u = U -> ( A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) <-> A. x e. X ( ( ( U G x ) = x /\ ( x G U ) = x ) /\ E. y e. X ( ( y G x ) = U /\ ( x G y ) = U ) ) ) )
29	28	elrab 3363	. . . 4 |- ( U e. { u e. X | A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) } <-> ( U e. X /\ A. x e. X ( ( ( U G x ) = x /\ ( x G U ) = x ) /\ E. y e. X ( ( y G x ) = U /\ ( x G y ) = U ) ) ) )
30	17, 29	sylib 208	. . 3 |- ( G e. GrpOp -> ( U e. X /\ A. x e. X ( ( ( U G x ) = x /\ ( x G U ) = x ) /\ E. y e. X ( ( y G x ) = U /\ ( x G y ) = U ) ) ) )
31	30	simprd 479	. 2 |- ( G e. GrpOp -> A. x e. X ( ( ( U G x ) = x /\ ( x G U ) = x ) /\ E. y e. X ( ( y G x ) = U /\ ( x G y ) = U ) ) )
32		oveq2 6658	. . . . . 6 |- ( x = A -> ( U G x ) = ( U G A ) )
33		id 22	. . . . . 6 |- ( x = A -> x = A )
34	32, 33	eqeq12d 2637	. . . . 5 |- ( x = A -> ( ( U G x ) = x <-> ( U G A ) = A ) )
35		oveq1 6657	. . . . . 6 |- ( x = A -> ( x G U ) = ( A G U ) )
36	35, 33	eqeq12d 2637	. . . . 5 |- ( x = A -> ( ( x G U ) = x <-> ( A G U ) = A ) )
37	34, 36	anbi12d 747	. . . 4 |- ( x = A -> ( ( ( U G x ) = x /\ ( x G U ) = x ) <-> ( ( U G A ) = A /\ ( A G U ) = A ) ) )
38		oveq2 6658	. . . . . . 7 |- ( x = A -> ( y G x ) = ( y G A ) )
39	38	eqeq1d 2624	. . . . . 6 |- ( x = A -> ( ( y G x ) = U <-> ( y G A ) = U ) )
40		oveq1 6657	. . . . . . 7 |- ( x = A -> ( x G y ) = ( A G y ) )
41	40	eqeq1d 2624	. . . . . 6 |- ( x = A -> ( ( x G y ) = U <-> ( A G y ) = U ) )
42	39, 41	anbi12d 747	. . . . 5 |- ( x = A -> ( ( ( y G x ) = U /\ ( x G y ) = U ) <-> ( ( y G A ) = U /\ ( A G y ) = U ) ) )
43	42	rexbidv 3052	. . . 4 |- ( x = A -> ( E. y e. X ( ( y G x ) = U /\ ( x G y ) = U ) <-> E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) ) )
44	37, 43	anbi12d 747	. . 3 |- ( x = A -> ( ( ( ( U G x ) = x /\ ( x G U ) = x ) /\ E. y e. X ( ( y G x ) = U /\ ( x G y ) = U ) ) <-> ( ( ( U G A ) = A /\ ( A G U ) = A ) /\ E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) ) ) )
45	44	rspccva 3308	. 2 |- ( ( A. x e. X ( ( ( U G x ) = x /\ ( x G U ) = x ) /\ E. y e. X ( ( y G x ) = U /\ ( x G y ) = U ) ) /\ A e. X ) -> ( ( ( U G A ) = A /\ ( A G U ) = A ) /\ E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) ) )
46	31, 45	sylan 488	1 |- ( ( G e. GrpOp /\ A e. X ) -> ( ( ( U G A ) = A /\ ( A G U ) = A ) /\ E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   {crab 2916   ran crn 5115   ` cfv 5888   iota_crio 6610  (class class class)co 6650   GrpOpcgr 27343  GIdcgi 27344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-grpo 27347  df-gid 27348

This theorem is referenced by:  grpolid  27370  grporid  27371  grporcan  27372  grpoinveu  27373  grpoinv  27379