Theorem isgrpinv 13129

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 13118	. . . . . . . . 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 5695	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ M : B --> B ) /\ x e. B ) /\ ( ( M ` x ) .+ x ) = .0. ) -> ( M ` x ) e. B )
11	1, 2, 3	grpinveu 13113	. . . . . . . . . . 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 5926	. . . . . . . . . . . 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 5897	. . . . . . . . . 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 13119	. . . . 5 |- ( G e. Grp -> N Fn B )
23		ffn 5404	. . . . . 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 5656	. . . . 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 13122	. . . 4 |- ( G e. Grp -> N : B --> B )
30	1, 2, 3, 4	grplinv 13125	. . . . 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 5387	. . . 4 |- ( N = M -> ( N : B --> B <-> M : B --> B ) )
34		fveq1 5554	. . . . . . 7 |- ( N = M -> ( N ` x ) = ( M ` x ) )
35	34	oveq1d 5934	. . . . . 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 5250   -->wf 5251   ` cfv 5255   iota_crio 5873  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   0gc0g 12870   Grpcgrp 13075   invgcminusg 13076

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 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

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 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079

This theorem is referenced by: (None)