Theorem grpinvcnv 12932

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 2177	. . . 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 12915	. . . 4 |- ( ( G e. Grp /\ x e. B ) -> ( N ` x ) e. B )
5	2, 3	grpinvcl 12915	. . . 4 |- ( ( G e. Grp /\ y e. B ) -> ( N ` y ) e. B )
6		eqid 2177	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
7		eqid 2177	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
8	2, 6, 7, 3	grpinvid1 12918	. . . . . . . 8 |- ( ( G e. Grp /\ y e. B /\ x e. B ) -> ( ( N ` y ) = x <-> ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
9	8	3com23 1209	. . . . . . 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 12919	. . . . . . 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 1204	. . . . 5 |- ( ( G e. Grp /\ ( x e. B /\ y e. B ) ) -> ( ( N ` y ) = x <-> ( N ` x ) = y ) )
13		eqcom 2179	. . . . 5 |- ( x = ( N ` y ) <-> ( N ` y ) = x )
14		eqcom 2179	. . . . 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 6074	. . 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 12914	. . . 4 |- ( G e. Grp -> N : B --> B )
19	18	feqmptd 5569	. . 3 |- ( G e. Grp -> N = ( x e. B |-> ( N ` x ) ) )
20	19	cnveqd 4803	. 2 |- ( G e. Grp -> `' N = `' ( x e. B |-> ( N ` x ) ) )
21	18	feqmptd 5569	. 2 |- ( G e. Grp -> N = ( y e. B |-> ( N ` y ) ) )
22	17, 20, 21	3eqtr4d 2220	1 |- ( G e. Grp -> `' N = N )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   |-> cmpt 4064   `'ccnv 4625   -1-1-onto->wf1o 5215   ` cfv 5216  (class class class)co 5874   Basecbs 12456   +g cplusg 12530   0gc0g 12699   Grpcgrp 12871   invgcminusg 12872

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-inn 8918  df-2 8976  df-ndx 12459  df-slot 12460  df-base 12462  df-plusg 12543  df-0g 12701  df-mgm 12769  df-sgrp 12802  df-mnd 12812  df-grp 12874  df-minusg 12875

This theorem is referenced by:  grpinvf1o  12934