Theorem isgrpinv 13126

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 13115	. . . . . . . . 9 |- ( x e. B -> ( N ` x ) = ( iota_ e e. B ( e .+ x ) = .0. ) )
6	5	ad2antlr 489	. . . . . . . 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 534	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> M : B --> B )
9		simplr 528	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> x e. B )
10	8, 9	ffvelcdmd 5694	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( M ` x ) e. B )
11	1, 2, 3	grpinveu 13110	. . . . . . . . . . 11 |- ( ( G e. Grp /\ x e. B ) -> E! e e. B ( e .+ x ) = .0. )
12	11	ad4ant13 513	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> E! e e. B ( e .+ x ) = .0. )
13		oveq1 5925	. . . . . . . . . . . 12 |- ( e = ( M ` x ) -> ( e .+ x ) = ( ( M ` x ) .+ x ) )
14	13	eqeq1d 2202	. . . . . . . . . . 11 |- ( e = ( M ` x ) -> ( ( e .+ x ) = .0. <-> ( ( M ` x ) .+ x ) = .0. ) )
15	14	riota2 5896	. . . . . . . . . 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 411	. . . . . . . . 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 2226	. . . . . . 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 2561	. . . . 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 13116	. . . . 5 |- ( G e. Grp -> N Fn B )
23		ffn 5403	. . . . . 6 |- ( M : B --> B -> M Fn B )
24	23	ad2antrl 490	. . . . 5 |- ( ( G e. Grp /\ ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) ) -> M Fn B )
25		eqfnfv 5655	. . . . 5 |- ( ( N Fn B /\ M Fn B ) -> ( N = M <-> A. x e. B ( N ` x ) = ( M ` x ) ) )
26	22, 24, 25	syl2an2r 595	. . . 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 13119	. . . 4 |- ( G e. Grp -> N : B --> B )
30	1, 2, 3, 4	grplinv 13122	. . . . 5 |- ( ( G e. Grp /\ x e. B ) -> ( ( N ` x ) .+ x ) = .0. )
31	30	ralrimiva 2567	. . . 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 5386	. . . 4 |- ( N = M -> ( N : B --> B <-> M : B --> B ) )
34		fveq1 5553	. . . . . . 7 |- ( N = M -> ( N ` x ) = ( M ` x ) )
35	34	oveq1d 5933	. . . . . 6 |- ( N = M -> ( ( N ` x ) .+ x ) = ( ( M ` x ) .+ x ) )
36	35	eqeq1d 2202	. . . . 5 |- ( N = M -> ( ( ( N ` x ) .+ x ) = .0. <-> ( ( M ` x ) .+ x ) = .0. ) )
37	36	ralbidv 2494	. . . 4 |- ( N = M -> ( A. x e. B ( ( N ` x ) .+ x ) = .0. <-> A. x e. B ( ( M ` x ) .+ x ) = .0. ) )
38	33, 37	anbi12d 473	. . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   E!wreu 2474   Fn wfn 5249   -->wf 5250   ` cfv 5254   iota_crio 5872  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867   Grpcgrp 13072   invgcminusg 13073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076

This theorem is referenced by: (None)