Theorem grpinvcnv 13601

Description: The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Hypotheses

Ref	Expression
grpinvinv.b	|- B = ( Base ` G )
grpinvinv.n	|- N = ( invg ` G )

Assertion

Ref	Expression
grpinvcnv	|- ( G e. Grp -> `' N = N )

Proof of Theorem grpinvcnv

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2229	. . . 4 |- ( x e. B |-> ( N ` x ) ) = ( x e. B |-> ( N ` x ) )
2		grpinvinv.b	. . . . 5 |- B = ( Base ` G )
3		grpinvinv.n	. . . . 5 |- N = ( invg ` G )
4	2, 3	grpinvcl 13581	. . . 4 |- ( ( G e. Grp /\ x e. B ) -> ( N ` x ) e. B )
5	2, 3	grpinvcl 13581	. . . 4 |- ( ( G e. Grp /\ y e. B ) -> ( N ` y ) e. B )
6		eqid 2229	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
7		eqid 2229	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
8	2, 6, 7, 3	grpinvid1 13585	. . . . . . . 8 |- ( ( G e. Grp /\ y e. B /\ x e. B ) -> ( ( N ` y ) = x <-> ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
9	8	3com23 1233	. . . . . . 7 |- ( ( G e. Grp /\ x e. B /\ y e. B ) -> ( ( N ` y ) = x <-> ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
10	2, 6, 7, 3	grpinvid2 13586	. . . . . . 7 |- ( ( G e. Grp /\ x e. B /\ y e. B ) -> ( ( N ` x ) = y <-> ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
11	9, 10	bitr4d 191	. . . . . 6 |- ( ( G e. Grp /\ x e. B /\ y e. B ) -> ( ( N ` y ) = x <-> ( N ` x ) = y ) )
12	11	3expb 1228	. . . . 5 |- ( ( G e. Grp /\ ( x e. B /\ y e. B ) ) -> ( ( N ` y ) = x <-> ( N ` x ) = y ) )
13		eqcom 2231	. . . . 5 |- ( x = ( N ` y ) <-> ( N ` y ) = x )
14		eqcom 2231	. . . . 5 |- ( y = ( N ` x ) <-> ( N ` x ) = y )
15	12, 13, 14	3bitr4g 223	. . . 4 |- ( ( G e. Grp /\ ( x e. B /\ y e. B ) ) -> ( x = ( N ` y ) <-> y = ( N ` x ) ) )
16	1, 4, 5, 15	f1ocnv2d 6210	. . 3 |- ( G e. Grp -> ( ( x e. B |-> ( N ` x ) ) : B -1-1-onto-> B /\ `' ( x e. B |-> ( N ` x ) ) = ( y e. B |-> ( N ` y ) ) ) )
17	16	simprd 114	. 2 |- ( G e. Grp -> `' ( x e. B |-> ( N ` x ) ) = ( y e. B |-> ( N ` y ) ) )
18	2, 3	grpinvf 13580	. . . 4 |- ( G e. Grp -> N : B --> B )
19	18	feqmptd 5687	. . 3 |- ( G e. Grp -> N = ( x e. B |-> ( N ` x ) ) )
20	19	cnveqd 4898	. 2 |- ( G e. Grp -> `' N = `' ( x e. B |-> ( N ` x ) ) )
21	18	feqmptd 5687	. 2 |- ( G e. Grp -> N = ( y e. B |-> ( N ` y ) ) )
22	17, 20, 21	3eqtr4d 2272	1 |- ( G e. Grp -> `' N = N )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   |-> cmpt 4145   `'ccnv 4718   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   0gc0g 13289   Grpcgrp 13533   invgcminusg 13534

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537

This theorem is referenced by:  grpinvf1o  13603