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Theorem grpinvfval 15574
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 5689 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpinvval.b . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2491 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5689 . . . . . . . . 9  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 grpinvval.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2491 . . . . . . . 8  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
87oveqd 6106 . . . . . . 7  |-  ( g  =  G  ->  (
y ( +g  `  g
) x )  =  ( y  .+  x
) )
9 fveq2 5689 . . . . . . . 8  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
10 grpinvval.o . . . . . . . 8  |-  .0.  =  ( 0g `  G )
119, 10syl6eqr 2491 . . . . . . 7  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
128, 11eqeq12d 2455 . . . . . 6  |-  ( g  =  G  ->  (
( y ( +g  `  g ) x )  =  ( 0g `  g )  <->  ( y  .+  x )  =  .0.  ) )
134, 12riotaeqbidv 6053 . . . . 5  |-  ( g  =  G  ->  ( iota_ y  e.  ( Base `  g ) ( y ( +g  `  g
) x )  =  ( 0g `  g
) )  =  (
iota_ y  e.  B  ( y  .+  x
)  =  .0.  )
)
144, 13mpteq12dv 4368 . . . 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 15544 . . . 4  |-  invg 
=  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
)  |->  ( iota_ y  e.  ( Base `  g
) ( y ( +g  `  g ) x )  =  ( 0g `  g ) ) ) )
16 fvex 5699 . . . . . 6  |-  ( Base `  G )  e.  _V
173, 16eqeltri 2511 . . . . 5  |-  B  e. 
_V
1817mptex 5946 . . . 4  |-  ( x  e.  B  |->  ( iota_ y  e.  B  ( y 
.+  x )  =  .0.  ) )  e. 
_V
1914, 15, 18fvmpt 5772 . . 3  |-  ( G  e.  _V  ->  ( invg `  G )  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x )  =  .0.  ) ) )
20 fvprc 5683 . . . . 5  |-  ( -.  G  e.  _V  ->  ( invg `  G
)  =  (/) )
21 mpt0 5536 . . . . 5  |-  ( x  e.  (/)  |->  ( iota_ y  e.  B  ( y  .+  x )  =  .0.  ) )  =  (/)
2220, 21syl6eqr 2491 . . . 4  |-  ( -.  G  e.  _V  ->  ( invg `  G
)  =  ( x  e.  (/)  |->  ( iota_ y  e.  B  ( y  .+  x )  =  .0.  ) ) )
23 fvprc 5683 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2485 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
2524mpteq1d 4371 . . . 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 2476 . . 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 2461 1  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x
)  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2970   (/)c0 3635    e. cmpt 4348   ` cfv 5416   iota_crio 6049  (class class class)co 6089   Basecbs 14172   +g cplusg 14236   0gc0g 14376   invgcminusg 15409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-minusg 15544
This theorem is referenced by:  grpinvval  15575  grpinvfn  15576  grpinvf  15580  grpinvpropd  15599  opprneg  16725
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