Theorem isgrpinv 13859

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 13848	. . . . . . . . 9 |- ( x e. B -> ( N ` x ) = ( iota_ e e. B ( e .+ x ) = .0. ) )
6	5	ad2antlr 493	. . . . . . . 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 110	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( ( M ` x ) .+ x ) = .0. )
8		simpllr 540	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> M : B --> B )
9		simplr 533	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> x e. B )
10	8, 9	ffvelcdmd 5844	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( M ` x ) e. B )
11	1, 2, 3	grpinveu 13843	. . . . . . . . . . 11 |- ( ( G e. Grp /\ x e. B ) -> E! e e. B ( e .+ x ) = .0. )
12	11	ad4ant13 517	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> E! e e. B ( e .+ x ) = .0. )
13		oveq1 6092	. . . . . . . . . . . 12 |- ( e = ( M ` x ) -> ( e .+ x ) = ( ( M ` x ) .+ x ) )
14	13	eqeq1d 2247	. . . . . . . . . . 11 |- ( e = ( M ` x ) -> ( ( e .+ x ) = .0. <-> ( ( M ` x ) .+ x ) = .0. ) )
15	14	riota2 6062	. . . . . . . . . 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 ) ) )
16	10, 12, 15	syl2anc 415	. . . . . . . . 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 ) ) )
17	7, 16	mpbid 147	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( iota_ e e. B ( e .+ x ) = .0. ) = ( M ` x ) )
18	6, 17	eqtrd 2271	. . . . . . 7 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( N ` x ) = ( M ` x ) )
19	18	ex 115	. . . . . 6 |- ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) -> ( ( ( M ` x ) .+ x ) = .0. -> ( N ` x ) = ( M ` x ) ) )
20	19	ralimdva 2617	. . . . 5 |- ( ( G e. Grp /\ M : B --> B ) -> ( A. x e. B ( ( M ` x ) .+ x ) = .0. -> A. x e. B ( N ` x ) = ( M ` x ) ) )
21	20	impr 379	. . . 4 |- ( ( G e. Grp /\ ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) -> A. x e. B ( N ` x ) = ( M ` x ) )
22	1, 4	grpinvfng 13849	. . . . 5 |- ( G e. Grp -> N Fn B )
23		ffn 5533	. . . . . 6 |- ( M : B --> B -> M Fn B )
24	23	ad2antrl 494	. . . . 5 |- ( ( G e. Grp /\ ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) -> M Fn B )
25		eqfnfv 5806	. . . . 5 |- ( ( N Fn B /\ M Fn B ) -> ( N = M <-> A. x e. B ( N ` x ) = ( M ` x ) ) )
26	22, 24, 25	syl2an2r 603	. . . 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 ) ) )
27	21, 26	mpbird 167	. . 3 |- ( ( G e. Grp /\ ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) -> N = M )
28	27	ex 115	. 2 |- ( G e. Grp -> ( ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) -> N = M ) )
29	1, 4	grpinvf 13852	. . . 4 |- ( G e. Grp -> N : B --> B )
30	1, 2, 3, 4	grplinv 13855	. . . . 5 |- ( ( G e. Grp /\ x e. B ) -> ( ( N ` x ) .+ x ) = .0. )
31	30	ralrimiva 2623	. . . 4 |- ( G e. Grp -> A. x e. B ( ( N ` x ) .+ x ) = .0. )
32	29, 31	jca 306	. . 3 |- ( G e. Grp -> ( N : B --> B /\ A. x e. B ( ( N ` x ) .+ x ) = .0. ) )
33		feq1 5516	. . . 4 |- ( N = M -> ( N : B --> B <-> M : B --> B ) )
34		fveq1 5694	. . . . . . 7 |- ( N = M -> ( N ` x ) = ( M ` x ) )
35	34	oveq1d 6100	. . . . . 6 |- ( N = M -> ( ( N ` x ) .+ x ) = ( ( M ` x ) .+ x ) )
36	35	eqeq1d 2247	. . . . 5 |- ( N = M -> ( ( ( N ` x ) .+ x ) = .0. <-> ( ( M ` x ) .+ x ) = .0. ) )
37	36	ralbidv 2550	. . . 4 |- ( N = M -> ( A. x e. B ( ( N ` x ) .+ x ) = .0. <-> A. x e. B ( ( M ` x ) .+ x ) = .0. ) )
38	33, 37	anbi12d 477	. . 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. ) ) )
39	32, 38	syl5ibcom 155	. 2 |- ( G e. Grp -> ( N = M -> ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) )
40	28, 39	impbid 129	1 |- ( G e. Grp -> ( ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) <-> N = M ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E!wreu 2530   Fn wfn 5372   -->wf 5373   ` cfv 5377   iota_crio 6037  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610   Grpcgrp 13805   invgcminusg 13806

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809

This theorem is used by: (None)