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Theorem grpinvcnv 14859
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  =  ( inv g `  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 2436 . . . 4  |-  ( x  e.  B  |->  ( N `
 x ) )  =  ( x  e.  B  |->  ( N `  x ) )
2 grpinvinv.b . . . . 5  |-  B  =  ( Base `  G
)
3 grpinvinv.n . . . . 5  |-  N  =  ( inv g `  G )
42, 3grpinvcl 14850 . . . 4  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( N `  x
)  e.  B )
52, 3grpinvcl 14850 . . . 4  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( N `  y
)  e.  B )
6 eqid 2436 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
7 eqid 2436 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
82, 6, 7, 3grpinvid1 14853 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
983com23 1159 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
102, 6, 7, 3grpinvid2 14854 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  x )  =  y  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
119, 10bitr4d 248 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( N `  x
)  =  y ) )
12113expb 1154 . . . . 5  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
( N `  y
)  =  x  <->  ( N `  x )  =  y ) )
13 eqcom 2438 . . . . 5  |-  ( x  =  ( N `  y )  <->  ( N `  y )  =  x )
14 eqcom 2438 . . . . 5  |-  ( y  =  ( N `  x )  <->  ( N `  x )  =  y )
1512, 13, 143bitr4g 280 . . . 4  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  =  ( N `
 y )  <->  y  =  ( N `  x ) ) )
161, 4, 5, 15f1ocnv2d 6295 . . 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 450 . 2  |-  ( G  e.  Grp  ->  `' ( x  e.  B  |->  ( N `  x
) )  =  ( y  e.  B  |->  ( N `  y ) ) )
182, 3grpinvf 14849 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
1918feqmptd 5779 . . 3  |-  ( G  e.  Grp  ->  N  =  ( x  e.  B  |->  ( N `  x ) ) )
2019cnveqd 5048 . 2  |-  ( G  e.  Grp  ->  `' N  =  `' (
x  e.  B  |->  ( N `  x ) ) )
2118feqmptd 5779 . 2  |-  ( G  e.  Grp  ->  N  =  ( y  e.  B  |->  ( N `  y ) ) )
2217, 20, 213eqtr4d 2478 1  |-  ( G  e.  Grp  ->  `' N  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    e. cmpt 4266   `'ccnv 4877   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685   inv gcminusg 14686
This theorem is referenced by:  grpinvf1o  14861  grpinvhmeo  18116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813
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