Theorem grpinvcnv 13140

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 2193	. . . 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 13120	. . . 4 |- ( ( G e. Grp /\ x e. B ) -> ( N ` x ) e. B )
5	2, 3	grpinvcl 13120	. . . 4 |- ( ( G e. Grp /\ y e. B ) -> ( N ` y ) e. B )
6		eqid 2193	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
7		eqid 2193	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
8	2, 6, 7, 3	grpinvid1 13124	. . . . . . . 8 |- ( ( G e. Grp /\ y e. B /\ x e. B ) -> ( ( N ` y ) = x <-> ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
9	8	3com23 1211	. . . . . . 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 13125	. . . . . . 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 1206	. . . . 5 |- ( ( G e. Grp /\ ( x e. B /\ y e. B ) ) -> ( ( N ` y ) = x <-> ( N ` x ) = y ) )
13		eqcom 2195	. . . . 5 |- ( x = ( N ` y ) <-> ( N ` y ) = x )
14		eqcom 2195	. . . . 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 6122	. . 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 13119	. . . 4 |- ( G e. Grp -> N : B --> B )
19	18	feqmptd 5610	. . 3 |- ( G e. Grp -> N = ( x e. B |-> ( N ` x ) ) )
20	19	cnveqd 4838	. 2 |- ( G e. Grp -> `' N = `' ( x e. B |-> ( N ` x ) ) )
21	18	feqmptd 5610	. 2 |- ( G e. Grp -> N = ( y e. B |-> ( N ` y ) ) )
22	17, 20, 21	3eqtr4d 2236	1 |- ( G e. Grp -> `' N = N )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   |-> cmpt 4090   `'ccnv 4658   -1-1-onto->wf1o 5253   ` cfv 5254  (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:  grpinvf1o  13142