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Theorem grpinvcnv 14552
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 2296 . . . 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 14543 . . . 4  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( N `  x
)  e.  B )
52, 3grpinvcl 14543 . . . 4  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( N `  y
)  e.  B )
6 eqid 2296 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
7 eqid 2296 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
82, 6, 7, 3grpinvid1 14546 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
983com23 1157 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
102, 6, 7, 3grpinvid2 14547 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  x )  =  y  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
119, 10bitr4d 247 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( N `  x
)  =  y ) )
12113expb 1152 . . . . 5  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
( N `  y
)  =  x  <->  ( N `  x )  =  y ) )
13 eqcom 2298 . . . . 5  |-  ( x  =  ( N `  y )  <->  ( N `  y )  =  x )
14 eqcom 2298 . . . . 5  |-  ( y  =  ( N `  x )  <->  ( N `  x )  =  y )
1512, 13, 143bitr4g 279 . . . 4  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  =  ( N `
 y )  <->  y  =  ( N `  x ) ) )
161, 4, 5, 15f1ocnv2d 6084 . . 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 449 . 2  |-  ( G  e.  Grp  ->  `' ( x  e.  B  |->  ( N `  x
) )  =  ( y  e.  B  |->  ( N `  y ) ) )
182, 3grpinvf 14542 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
1918feqmptd 5591 . . 3  |-  ( G  e.  Grp  ->  N  =  ( x  e.  B  |->  ( N `  x ) ) )
2019cnveqd 4873 . 2  |-  ( G  e.  Grp  ->  `' N  =  `' (
x  e.  B  |->  ( N `  x ) ) )
2118feqmptd 5591 . 2  |-  ( G  e.  Grp  ->  N  =  ( y  e.  B  |->  ( N `  y ) ) )
2217, 20, 213eqtr4d 2338 1  |-  ( G  e.  Grp  ->  `' N  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    e. cmpt 4093   `'ccnv 4704   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379
This theorem is referenced by:  grpinvf1o  14554  grpinvhmeo  17785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506
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