Theorem isgrpinv 15894

Description: Properties showing that a function M is the inverse function of a group. (Contributed by NM, 7-Aug-2013.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
grpinv.b	|- B = ( Base ` G )
grpinv.p	|- .+ = ( +g ` G )
grpinv.u	|- .0. = ( 0g ` G )
grpinv.n	|- N = ( invg ` G )

Assertion

Ref	Expression
isgrpinv	|- ( G e. Grp -> ( ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) <-> N = M ) )

Distinct variable groups:   x, B   x, G   x, .0.   x, .+   x, M   x, N

Proof of Theorem isgrpinv

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinv.b	. . . . . . . . . 10 |- B = ( Base ` G )
2		grpinv.p	. . . . . . . . . 10 |- .+ = ( +g ` G )
3		grpinv.u	. . . . . . . . . 10 |- .0. = ( 0g ` G )
4		grpinv.n	. . . . . . . . . 10 |- N = ( invg ` G )
5	1, 2, 3, 4	grpinvval 15883	. . . . . . . . 9 |- ( x e. B -> ( N ` x ) = ( iota_ e e. B ( e .+ x ) = .0. ) )
6	5	ad2antlr 726	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( N ` x ) = ( iota_ e e. B ( e .+ x ) = .0. ) )
7		simpr 461	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( ( M ` x ) .+ x ) = .0. )
8		simpllr 758	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> M : B --> B )
9		simplr 754	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> x e. B )
10	8, 9	ffvelrnd 6013	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( M ` x ) e. B )
11		simplll 757	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> G e. Grp )
12	1, 2, 3	grpinveu 15878	. . . . . . . . . . 11 |- ( ( G e. Grp /\ x e. B ) -> E! e e. B ( e .+ x ) = .0. )
13	11, 9, 12	syl2anc 661	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> E! e e. B ( e .+ x ) = .0. )
14		oveq1 6282	. . . . . . . . . . . 12 |- ( e = ( M ` x ) -> ( e .+ x ) = ( ( M ` x ) .+ x ) )
15	14	eqeq1d 2462	. . . . . . . . . . 11 |- ( e = ( M ` x ) -> ( ( e .+ x ) = .0. <-> ( ( M ` x ) .+ x ) = .0. ) )
16	15	riota2 6259	. . . . . . . . . 10 |- ( ( ( M ` x ) e. B /\ E! e e. B ( e .+ x ) = .0. ) -> ( ( ( M ` x ) .+ x ) = .0. <-> ( iota_ e e. B ( e .+ x ) = .0. ) = ( M ` x ) ) )
17	10, 13, 16	syl2anc 661	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( ( ( M ` x ) .+ x ) = .0. <-> ( iota_ e e. B ( e .+ x ) = .0. ) = ( M ` x ) ) )
18	7, 17	mpbid 210	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( iota_ e e. B ( e .+ x ) = .0. ) = ( M ` x ) )
19	6, 18	eqtrd 2501	. . . . . . 7 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( N ` x ) = ( M ` x ) )
20	19	ex 434	. . . . . 6 |- ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) -> ( ( ( M ` x ) .+ x ) = .0. -> ( N ` x ) = ( M ` x ) ) )
21	20	ralimdva 2865	. . . . 5 |- ( ( G e. Grp /\ M : B --> B ) -> ( A. x e. B ( ( M ` x ) .+ x ) = .0. -> A. x e. B ( N ` x ) = ( M ` x ) ) )
22	21	impr 619	. . . 4 |- ( ( G e. Grp /\ ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) -> A. x e. B ( N ` x ) = ( M ` x ) )
23	1, 4	grpinvfn 15884	. . . . 5 |- N Fn B
24		ffn 5722	. . . . . 6 |- ( M : B --> B -> M Fn B )
25	24	ad2antrl 727	. . . . 5 |- ( ( G e. Grp /\ ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) -> M Fn B )
26		eqfnfv 5966	. . . . 5 |- ( ( N Fn B /\ M Fn B ) -> ( N = M <-> A. x e. B ( N ` x ) = ( M ` x ) ) )
27	23, 25, 26	sylancr 663	. . . 4 |- ( ( G e. Grp /\ ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) -> ( N = M <-> A. x e. B ( N ` x ) = ( M ` x ) ) )
28	22, 27	mpbird 232	. . 3 |- ( ( G e. Grp /\ ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) -> N = M )
29	28	ex 434	. 2 |- ( G e. Grp -> ( ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) -> N = M ) )
30	1, 4	grpinvf 15888	. . . 4 |- ( G e. Grp -> N : B --> B )
31	1, 2, 3, 4	grplinv 15890	. . . . 5 |- ( ( G e. Grp /\ x e. B ) -> ( ( N ` x ) .+ x ) = .0. )
32	31	ralrimiva 2871	. . . 4 |- ( G e. Grp -> A. x e. B ( ( N ` x ) .+ x ) = .0. )
33	30, 32	jca 532	. . 3 |- ( G e. Grp -> ( N : B --> B /\ A. x e. B ( ( N ` x ) .+ x ) = .0. ) )
34		feq1 5704	. . . 4 |- ( N = M -> ( N : B --> B <-> M : B --> B ) )
35		fveq1 5856	. . . . . . 7 |- ( N = M -> ( N ` x ) = ( M ` x ) )
36	35	oveq1d 6290	. . . . . 6 |- ( N = M -> ( ( N ` x ) .+ x ) = ( ( M ` x ) .+ x ) )
37	36	eqeq1d 2462	. . . . 5 |- ( N = M -> ( ( ( N ` x ) .+ x ) = .0. <-> ( ( M ` x ) .+ x ) = .0. ) )
38	37	ralbidv 2896	. . . 4 |- ( N = M -> ( A. x e. B ( ( N ` x ) .+ x ) = .0. <-> A. x e. B ( ( M ` x ) .+ x ) = .0. ) )
39	34, 38	anbi12d 710	. . 3 |- ( N = M -> ( ( N : B --> B /\ A. x e. B ( ( N ` x ) .+ x ) = .0. ) <-> ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) )
40	33, 39	syl5ibcom 220	. 2 |- ( G e. Grp -> ( N = M -> ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) )
41	29, 40	impbid 191	1 |- ( G e. Grp -> ( ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) <-> N = M ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   e. wcel 1762   A.wral 2807   E!wreu 2809   Fn wfn 5574   -->wf 5575   ` cfv 5579   iota_crio 6235  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   0gc0g 14684   Grpcgrp 15716   invgcminusg 15717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852

This theorem is referenced by:  oppginv  16182