Theorem grpoinv 27379

Description: The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpinv.1	|- X = ran G
grpinv.2	|- U = (GId ` G )
grpinv.3	|- N = ( inv ` G )

Assertion

Ref	Expression
grpoinv	|- ( ( G e. GrpOp /\ A e. X ) -> ( ( ( N ` A ) G A ) = U /\ ( A G ( N ` A ) ) = U ) )

Proof of Theorem grpoinv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinv.1	. . . . . 6 |- X = ran G
2		grpinv.2	. . . . . 6 |- U = (GId ` G )
3		grpinv.3	. . . . . 6 |- N = ( inv ` G )
4	1, 2, 3	grpoinvval 27377	. . . . 5 |- ( ( G e. GrpOp /\ A e. X ) -> ( N ` A ) = ( iota_ y e. X ( y G A ) = U ) )
5	1, 2	grpoinveu 27373	. . . . . 6 |- ( ( G e. GrpOp /\ A e. X ) -> E! y e. X ( y G A ) = U )
6		riotacl2 6624	. . . . . 6 |- ( E! y e. X ( y G A ) = U -> ( iota_ y e. X ( y G A ) = U ) e. { y e. X | ( y G A ) = U } )
7	5, 6	syl 17	. . . . 5 |- ( ( G e. GrpOp /\ A e. X ) -> ( iota_ y e. X ( y G A ) = U ) e. { y e. X | ( y G A ) = U } )
8	4, 7	eqeltrd 2701	. . . 4 |- ( ( G e. GrpOp /\ A e. X ) -> ( N ` A ) e. { y e. X | ( y G A ) = U } )
9		simpl 473	. . . . . . . . 9 |- ( ( ( y G A ) = U /\ ( A G y ) = U ) -> ( y G A ) = U )
10	9	rgenw 2924	. . . . . . . 8 |- A. y e. X ( ( ( y G A ) = U /\ ( A G y ) = U ) -> ( y G A ) = U )
11	10	a1i 11	. . . . . . 7 |- ( ( G e. GrpOp /\ A e. X ) -> A. y e. X ( ( ( y G A ) = U /\ ( A G y ) = U ) -> ( y G A ) = U ) )
12	1, 2	grpoidinv2 27369	. . . . . . . 8 |- ( ( 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 ) ) )
13	12	simprd 479	. . . . . . 7 |- ( ( G e. GrpOp /\ A e. X ) -> E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) )
14	11, 13, 5	3jca 1242	. . . . . 6 |- ( ( G e. GrpOp /\ A e. X ) -> ( A. y e. X ( ( ( y G A ) = U /\ ( A G y ) = U ) -> ( y G A ) = U ) /\ E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) /\ E! y e. X ( y G A ) = U ) )
15		reupick2 3913	. . . . . 6 |- ( ( ( A. y e. X ( ( ( y G A ) = U /\ ( A G y ) = U ) -> ( y G A ) = U ) /\ E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) /\ E! y e. X ( y G A ) = U ) /\ y e. X ) -> ( ( y G A ) = U <-> ( ( y G A ) = U /\ ( A G y ) = U ) ) )
16	14, 15	sylan 488	. . . . 5 |- ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) -> ( ( y G A ) = U <-> ( ( y G A ) = U /\ ( A G y ) = U ) ) )
17	16	rabbidva 3188	. . . 4 |- ( ( G e. GrpOp /\ A e. X ) -> { y e. X | ( y G A ) = U } = { y e. X | ( ( y G A ) = U /\ ( A G y ) = U ) } )
18	8, 17	eleqtrd 2703	. . 3 |- ( ( G e. GrpOp /\ A e. X ) -> ( N ` A ) e. { y e. X | ( ( y G A ) = U /\ ( A G y ) = U ) } )
19		oveq1 6657	. . . . . 6 |- ( y = ( N ` A ) -> ( y G A ) = ( ( N ` A ) G A ) )
20	19	eqeq1d 2624	. . . . 5 |- ( y = ( N ` A ) -> ( ( y G A ) = U <-> ( ( N ` A ) G A ) = U ) )
21		oveq2 6658	. . . . . 6 |- ( y = ( N ` A ) -> ( A G y ) = ( A G ( N ` A ) ) )
22	21	eqeq1d 2624	. . . . 5 |- ( y = ( N ` A ) -> ( ( A G y ) = U <-> ( A G ( N ` A ) ) = U ) )
23	20, 22	anbi12d 747	. . . 4 |- ( y = ( N ` A ) -> ( ( ( y G A ) = U /\ ( A G y ) = U ) <-> ( ( ( N ` A ) G A ) = U /\ ( A G ( N ` A ) ) = U ) ) )
24	23	elrab 3363	. . 3 |- ( ( N ` A ) e. { y e. X | ( ( y G A ) = U /\ ( A G y ) = U ) } <-> ( ( N ` A ) e. X /\ ( ( ( N ` A ) G A ) = U /\ ( A G ( N ` A ) ) = U ) ) )
25	18, 24	sylib 208	. 2 |- ( ( G e. GrpOp /\ A e. X ) -> ( ( N ` A ) e. X /\ ( ( ( N ` A ) G A ) = U /\ ( A G ( N ` A ) ) = U ) ) )
26	25	simprd 479	1 |- ( ( G e. GrpOp /\ A e. X ) -> ( ( ( N ` A ) G A ) = U /\ ( A G ( N ` A ) ) = 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   invcgn 27345

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-rep 4771  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-ne 2795  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-grpo 27347  df-gid 27348  df-ginv 27349

This theorem is referenced by:  grpolinv  27380  grporinv  27381