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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.
StepHypRef 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 )
42, 3grpinvcl 12915 . . . 4  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( N `  x
)  e.  B )
52, 3grpinvcl 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
)
82, 6, 7, 3grpinvid1 12918 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
983com23 1209 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
102, 6, 7, 3grpinvid2 12919 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  x )  =  y  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
119, 10bitr4d 191 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( N `  x
)  =  y ) )
12113expb 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 )
1512, 13, 143bitr4g 223 . . . 4  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  =  ( N `
 y )  <->  y  =  ( N `  x ) ) )
161, 4, 5, 15f1ocnv2d 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
) ) ) )
1716simprd 114 . 2  |-  ( G  e.  Grp  ->  `' ( x  e.  B  |->  ( N `  x
) )  =  ( y  e.  B  |->  ( N `  y ) ) )
182, 3grpinvf 12914 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
1918feqmptd 5569 . . 3  |-  ( G  e.  Grp  ->  N  =  ( x  e.  B  |->  ( N `  x ) ) )
2019cnveqd 4803 . 2  |-  ( G  e.  Grp  ->  `' N  =  `' (
x  e.  B  |->  ( N `  x ) ) )
2118feqmptd 5569 . 2  |-  ( G  e.  Grp  ->  N  =  ( y  e.  B  |->  ( N `  y ) ) )
2217, 20, 213eqtr4d 2220 1  |-  ( G  e.  Grp  ->  `' N  =  N )
Colors of variables: wff set class
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
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