Theorem isgrpinv 16728

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 16717	. . . . . . . . 9 |- ( x e. B -> ( N ` x ) = ( iota_ e e. B ( e .+ x ) = .0. ) )
6	5	ad2antlr 734	. . . . . . . 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 463	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( ( M ` x ) .+ x ) = .0. )
8		simpllr 770	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> M : B --> B )
9		simplr 763	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> x e. B )
10	8, 9	ffvelrnd 6028	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( M ` x ) e. B )
11		simplll 769	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> G e. Grp )
12	1, 2, 3	grpinveu 16712	. . . . . . . . . . 11 |- ( ( G e. Grp /\ x e. B ) -> E! e e. B ( e .+ x ) = .0. )
13	11, 9, 12	syl2anc 667	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> E! e e. B ( e .+ x ) = .0. )
14		oveq1 6302	. . . . . . . . . . . 12 |- ( e = ( M ` x ) -> ( e .+ x ) = ( ( M ` x ) .+ x ) )
15	14	eqeq1d 2455	. . . . . . . . . . 11 |- ( e = ( M ` x ) -> ( ( e .+ x ) = .0. <-> ( ( M ` x ) .+ x ) = .0. ) )
16	15	riota2 6279	. . . . . . . . . 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 667	. . . . . . . . 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 214	. . . . . . . 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 2487	. . . . . . 7 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( N ` x ) = ( M ` x ) )
20	19	ex 436	. . . . . 6 |- ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) -> ( ( ( M ` x ) .+ x ) = .0. -> ( N ` x ) = ( M ` x ) ) )
21	20	ralimdva 2798	. . . . 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 625	. . . 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 16718	. . . . 5 |- N Fn B
24		ffn 5733	. . . . . 6 |- ( M : B --> B -> M Fn B )
25	24	ad2antrl 735	. . . . 5 |- ( ( G e. Grp /\ ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) -> M Fn B )
26		eqfnfv 5981	. . . . 5 |- ( ( N Fn B /\ M Fn B ) -> ( N = M <-> A. x e. B ( N ` x ) = ( M ` x ) ) )
27	23, 25, 26	sylancr 670	. . . 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 236	. . 3 |- ( ( G e. Grp /\ ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) -> N = M )
29	28	ex 436	. 2 |- ( G e. Grp -> ( ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) -> N = M ) )
30	1, 4	grpinvf 16722	. . . 4 |- ( G e. Grp -> N : B --> B )
31	1, 2, 3, 4	grplinv 16724	. . . . 5 |- ( ( G e. Grp /\ x e. B ) -> ( ( N ` x ) .+ x ) = .0. )
32	31	ralrimiva 2804	. . . 4 |- ( G e. Grp -> A. x e. B ( ( N ` x ) .+ x ) = .0. )
33	30, 32	jca 535	. . 3 |- ( G e. Grp -> ( N : B --> B /\ A. x e. B ( ( N ` x ) .+ x ) = .0. ) )
34		feq1 5715	. . . 4 |- ( N = M -> ( N : B --> B <-> M : B --> B ) )
35		fveq1 5869	. . . . . . 7 |- ( N = M -> ( N ` x ) = ( M ` x ) )
36	35	oveq1d 6310	. . . . . 6 |- ( N = M -> ( ( N ` x ) .+ x ) = ( ( M ` x ) .+ x ) )
37	36	eqeq1d 2455	. . . . 5 |- ( N = M -> ( ( ( N ` x ) .+ x ) = .0. <-> ( ( M ` x ) .+ x ) = .0. ) )
38	37	ralbidv 2829	. . . 4 |- ( N = M -> ( A. x e. B ( ( N ` x ) .+ x ) = .0. <-> A. x e. B ( ( M ` x ) .+ x ) = .0. ) )
39	34, 38	anbi12d 718	. . 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 224	. 2 |- ( G e. Grp -> ( N = M -> ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) )
41	29, 40	impbid 194	1 |- ( G e. Grp -> ( ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) <-> N = M ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1446   e. wcel 1889   A.wral 2739   E!wreu 2741   Fn wfn 5580   -->wf 5581   ` cfv 5585   iota_crio 6256  (class class class)co 6295   Basecbs 15133   +g cplusg 15202   0gc0g 15350   Grpcgrp 16681   invgcminusg 16682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-0g 15352  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-grp 16685  df-minusg 16686

This theorem is referenced by:  oppginv  17022