Theorem grpinvcnv 13875

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 2238	. . . 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 13855	. . . 4 |- ( ( G e. Grp /\ x e. B ) -> ( N ` x ) e. B )
5	2, 3	grpinvcl 13855	. . . 4 |- ( ( G e. Grp /\ y e. B ) -> ( N ` y ) e. B )
6		eqid 2238	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
7		eqid 2238	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
8	2, 6, 7, 3	grpinvid1 13859	. . . . . . . 8 |- ( ( G e. Grp /\ y e. B /\ x e. B ) -> ( ( N ` y ) = x <-> ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
9	8	3com23 1240	. . . . . . 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 13860	. . . . . . 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 1235	. . . . 5 |- ( ( G e. Grp /\ ( x e. B /\ y e. B ) ) -> ( ( N ` y ) = x <-> ( N ` x ) = y ) )
13		eqcom 2240	. . . . 5 |- ( x = ( N ` y ) <-> ( N ` y ) = x )
14		eqcom 2240	. . . . 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 6294	. . 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 13854	. . . 4 |- ( G e. Grp -> N : B --> B )
19	18	feqmptd 5756	. . 3 |- ( G e. Grp -> N = ( x e. B |-> ( N ` x ) ) )
20	19	cnveqd 4956	. 2 |- ( G e. Grp -> `' N = `' ( x e. B |-> ( N ` x ) ) )
21	18	feqmptd 5756	. 2 |- ( G e. Grp -> N = ( y e. B |-> ( N ` y ) ) )
22	17, 20, 21	3eqtr4d 2281	1 |- ( G e. Grp -> `' N = N )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   |-> cmpt 4192   `'ccnv 4773   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   Basecbs 13354   +g cplusg 13433   0gc0g 13612   Grpcgrp 13807   invgcminusg 13808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9306  df-2 9364  df-ndx 13357  df-slot 13358  df-base 13360  df-plusg 13446  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-minusg 13811

This theorem is used by:  grpinvf1o  13877