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Theorem grpinvfval 16214
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  =  ( invg `  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  =  ( invg `  G )
2 fveq2 5872 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpinvval.b . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2516 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5872 . . . . . . . . 9  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 grpinvval.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2516 . . . . . . . 8  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
87oveqd 6313 . . . . . . 7  |-  ( g  =  G  ->  (
y ( +g  `  g
) x )  =  ( y  .+  x
) )
9 fveq2 5872 . . . . . . . 8  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
10 grpinvval.o . . . . . . . 8  |-  .0.  =  ( 0g `  G )
119, 10syl6eqr 2516 . . . . . . 7  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
128, 11eqeq12d 2479 . . . . . 6  |-  ( g  =  G  ->  (
( y ( +g  `  g ) x )  =  ( 0g `  g )  <->  ( y  .+  x )  =  .0.  ) )
134, 12riotaeqbidv 6261 . . . . 5  |-  ( g  =  G  ->  ( iota_ y  e.  ( Base `  g ) ( y ( +g  `  g
) x )  =  ( 0g `  g
) )  =  (
iota_ y  e.  B  ( y  .+  x
)  =  .0.  )
)
144, 13mpteq12dv 4535 . . . 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 16184 . . . 4  |-  invg 
=  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
)  |->  ( iota_ y  e.  ( Base `  g
) ( y ( +g  `  g ) x )  =  ( 0g `  g ) ) ) )
16 fvex 5882 . . . . . 6  |-  ( Base `  G )  e.  _V
173, 16eqeltri 2541 . . . . 5  |-  B  e. 
_V
1817mptex 6144 . . . 4  |-  ( x  e.  B  |->  ( iota_ y  e.  B  ( y 
.+  x )  =  .0.  ) )  e. 
_V
1914, 15, 18fvmpt 5956 . . 3  |-  ( G  e.  _V  ->  ( invg `  G )  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x )  =  .0.  ) ) )
20 fvprc 5866 . . . . 5  |-  ( -.  G  e.  _V  ->  ( invg `  G
)  =  (/) )
21 mpt0 5714 . . . . 5  |-  ( x  e.  (/)  |->  ( iota_ y  e.  B  ( y  .+  x )  =  .0.  ) )  =  (/)
2220, 21syl6eqr 2516 . . . 4  |-  ( -.  G  e.  _V  ->  ( invg `  G
)  =  ( x  e.  (/)  |->  ( iota_ y  e.  B  ( y  .+  x )  =  .0.  ) ) )
23 fvprc 5866 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2510 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
2524mpteq1d 4538 . . . 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 2501 . . 3  |-  ( -.  G  e.  _V  ->  ( invg `  G
)  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y 
.+  x )  =  .0.  ) ) )
2719, 26pm2.61i 164 . 2  |-  ( invg `  G )  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x )  =  .0.  ) )
281, 27eqtri 2486 1  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x
)  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793    |-> cmpt 4515   ` cfv 5594   iota_crio 6257  (class class class)co 6296   Basecbs 14643   +g cplusg 14711   0gc0g 14856   invgcminusg 16180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-minusg 16184
This theorem is referenced by:  grpinvval  16215  grpinvfn  16216  grpinvf  16220  grpinvpropd  16239  opprneg  17410
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