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Theorem grpinvcnv 15980
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.
StepHypRef Expression
1 eqid 2443 . . . 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 )
42, 3grpinvcl 15969 . . . 4  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( N `  x
)  e.  B )
52, 3grpinvcl 15969 . . . 4  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( N `  y
)  e.  B )
6 eqid 2443 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
7 eqid 2443 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
82, 6, 7, 3grpinvid1 15972 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
983com23 1203 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
102, 6, 7, 3grpinvid2 15973 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  x )  =  y  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
119, 10bitr4d 256 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( N `  x
)  =  y ) )
12113expb 1198 . . . . 5  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
( N `  y
)  =  x  <->  ( N `  x )  =  y ) )
13 eqcom 2452 . . . . 5  |-  ( x  =  ( N `  y )  <->  ( N `  y )  =  x )
14 eqcom 2452 . . . . 5  |-  ( y  =  ( N `  x )  <->  ( N `  x )  =  y )
1512, 13, 143bitr4g 288 . . . 4  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  =  ( N `
 y )  <->  y  =  ( N `  x ) ) )
161, 4, 5, 15f1ocnv2d 6511 . . 3  |-  ( G  e.  Grp  ->  (
( x  e.  B  |->  ( N `  x
) ) : B -1-1-onto-> B  /\  `' ( x  e.  B  |->  ( N `  x ) )  =  ( y  e.  B  |->  ( N `  y
) ) ) )
1716simprd 463 . 2  |-  ( G  e.  Grp  ->  `' ( x  e.  B  |->  ( N `  x
) )  =  ( y  e.  B  |->  ( N `  y ) ) )
182, 3grpinvf 15968 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
1918feqmptd 5911 . . 3  |-  ( G  e.  Grp  ->  N  =  ( x  e.  B  |->  ( N `  x ) ) )
2019cnveqd 5168 . 2  |-  ( G  e.  Grp  ->  `' N  =  `' (
x  e.  B  |->  ( N `  x ) ) )
2118feqmptd 5911 . 2  |-  ( G  e.  Grp  ->  N  =  ( y  e.  B  |->  ( N `  y ) ) )
2217, 20, 213eqtr4d 2494 1  |-  ( G  e.  Grp  ->  `' N  =  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    |-> cmpt 4495   `'ccnv 4988   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   Basecbs 14509   +g cplusg 14574   0gc0g 14714   Grpcgrp 15927   invgcminusg 15928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932
This theorem is referenced by:  grpinvf1o  15982  grpinvhmeo  20458
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