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Theorem grpinvcnv 16305
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 2454 . . . 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 16294 . . . 4  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( N `  x
)  e.  B )
52, 3grpinvcl 16294 . . . 4  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( N `  y
)  e.  B )
6 eqid 2454 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
7 eqid 2454 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
82, 6, 7, 3grpinvid1 16297 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
983com23 1200 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( N `  y )  =  x  <-> 
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
102, 6, 7, 3grpinvid2 16298 . . . . . . 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 1195 . . . . 5  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
( N `  y
)  =  x  <->  ( N `  x )  =  y ) )
13 eqcom 2463 . . . . 5  |-  ( x  =  ( N `  y )  <->  ( N `  y )  =  x )
14 eqcom 2463 . . . . 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 6499 . . 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 461 . 2  |-  ( G  e.  Grp  ->  `' ( x  e.  B  |->  ( N `  x
) )  =  ( y  e.  B  |->  ( N `  y ) ) )
182, 3grpinvf 16293 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
1918feqmptd 5901 . . 3  |-  ( G  e.  Grp  ->  N  =  ( x  e.  B  |->  ( N `  x ) ) )
2019cnveqd 5167 . 2  |-  ( G  e.  Grp  ->  `' N  =  `' (
x  e.  B  |->  ( N `  x ) ) )
2118feqmptd 5901 . 2  |-  ( G  e.  Grp  ->  N  =  ( y  e.  B  |->  ( N `  y ) ) )
2217, 20, 213eqtr4d 2505 1  |-  ( G  e.  Grp  ->  `' N  =  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    |-> cmpt 4497   `'ccnv 4987   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   0gc0g 14929   Grpcgrp 16252   invgcminusg 16253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257
This theorem is referenced by:  grpinvf1o  16307  grpinvhmeo  20751
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