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Theorem grpinvfval 14770
Description: The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
Hypotheses
Ref Expression
grpinvval.b  |-  B  =  ( Base `  G
)
grpinvval.p  |-  .+  =  ( +g  `  G )
grpinvval.o  |-  .0.  =  ( 0g `  G )
grpinvval.n  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
grpinvfval  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
Distinct variable groups:    x, y, B    x, G, y    x,  .0.    x,  .+
Allowed substitution hints:    .+ ( y)    N( x, y)    .0. ( y)

Proof of Theorem grpinvfval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 grpinvval.n . 2  |-  N  =  ( inv g `  G )
2 fveq2 5668 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpinvval.b . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2437 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5668 . . . . . . . . 9  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 grpinvval.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2437 . . . . . . . 8  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
87oveqd 6037 . . . . . . 7  |-  ( g  =  G  ->  (
y ( +g  `  g
) x )  =  ( y  .+  x
) )
9 fveq2 5668 . . . . . . . 8  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
10 grpinvval.o . . . . . . . 8  |-  .0.  =  ( 0g `  G )
119, 10syl6eqr 2437 . . . . . . 7  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
128, 11eqeq12d 2401 . . . . . 6  |-  ( g  =  G  ->  (
( y ( +g  `  g ) x )  =  ( 0g `  g )  <->  ( y  .+  x )  =  .0.  ) )
134, 12riotaeqbidv 6488 . . . . 5  |-  ( g  =  G  ->  ( iota_ y  e.  ( Base `  g ) ( y ( +g  `  g
) x )  =  ( 0g `  g
) )  =  (
iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
144, 13mpteq12dv 4228 . . . 4  |-  ( g  =  G  ->  (
x  e.  ( Base `  g )  |->  ( iota_ y  e.  ( Base `  g
) ( y ( +g  `  g ) x )  =  ( 0g `  g ) ) )  =  ( x  e.  B  |->  (
iota_ y  e.  B
( y  .+  x
)  =  .0.  )
) )
15 df-minusg 14740 . . . 4  |-  inv g  =  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
)  |->  ( iota_ y  e.  ( Base `  g
) ( y ( +g  `  g ) x )  =  ( 0g `  g ) ) ) )
16 fvex 5682 . . . . . 6  |-  ( Base `  G )  e.  _V
173, 16eqeltri 2457 . . . . 5  |-  B  e. 
_V
1817mptex 5905 . . . 4  |-  ( x  e.  B  |->  ( iota_ y  e.  B ( y 
.+  x )  =  .0.  ) )  e. 
_V
1914, 15, 18fvmpt 5745 . . 3  |-  ( G  e.  _V  ->  ( inv g `  G )  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) ) )
20 fvprc 5662 . . . . 5  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  (/) )
21 mpt0 5512 . . . . 5  |-  ( x  e.  (/)  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) )  =  (/)
2220, 21syl6eqr 2437 . . . 4  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  ( x  e.  (/)  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) ) )
23 fvprc 5662 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2431 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
2524mpteq1d 4231 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  B  |->  (
iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)  =  ( x  e.  (/)  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) ) )
2622, 25eqtr4d 2422 . . 3  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y 
.+  x )  =  .0.  ) ) )
2719, 26pm2.61i 158 . 2  |-  ( inv g `  G )  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) )
281, 27eqtri 2407 1  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1717   _Vcvv 2899   (/)c0 3571    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   iota_crio 6478   Basecbs 13396   +g cplusg 13456   0gc0g 13650   inv gcminusg 14613
This theorem is referenced by:  grpinvval  14771  grpinvfn  14772  grpinvf  14776  grpinvpropd  14793  opprneg  15667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-riota 6485  df-minusg 14740
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