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Theorem grpinvfval 14536
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 5541 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpinvval.b . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2346 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5541 . . . . . . . . 9  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 grpinvval.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2346 . . . . . . . 8  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
87oveqd 5891 . . . . . . 7  |-  ( g  =  G  ->  (
y ( +g  `  g
) x )  =  ( y  .+  x
) )
9 fveq2 5541 . . . . . . . 8  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
10 grpinvval.o . . . . . . . 8  |-  .0.  =  ( 0g `  G )
119, 10syl6eqr 2346 . . . . . . 7  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
128, 11eqeq12d 2310 . . . . . 6  |-  ( g  =  G  ->  (
( y ( +g  `  g ) x )  =  ( 0g `  g )  <->  ( y  .+  x )  =  .0.  ) )
134, 12riotaeqbidv 6323 . . . . 5  |-  ( g  =  G  ->  ( iota_ y  e.  ( Base `  g ) ( y ( +g  `  g
) x )  =  ( 0g `  g
) )  =  (
iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
144, 13mpteq12dv 4114 . . . 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 14506 . . . 4  |-  inv g  =  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
)  |->  ( iota_ y  e.  ( Base `  g
) ( y ( +g  `  g ) x )  =  ( 0g `  g ) ) ) )
16 fvex 5555 . . . . . 6  |-  ( Base `  G )  e.  _V
173, 16eqeltri 2366 . . . . 5  |-  B  e. 
_V
1817mptex 5762 . . . 4  |-  ( x  e.  B  |->  ( iota_ y  e.  B ( y 
.+  x )  =  .0.  ) )  e. 
_V
1914, 15, 18fvmpt 5618 . . 3  |-  ( G  e.  _V  ->  ( inv g `  G )  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) ) )
20 fvprc 5535 . . . . 5  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  (/) )
21 mpt0 5387 . . . . 5  |-  ( x  e.  (/)  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) )  =  (/)
2220, 21syl6eqr 2346 . . . 4  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  ( x  e.  (/)  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) ) )
23 fvprc 5535 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2340 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
25 mpteq1 4116 . . . . 5  |-  ( B  =  (/)  ->  ( x  e.  B  |->  ( iota_ y  e.  B ( y 
.+  x )  =  .0.  ) )  =  ( x  e.  (/)  |->  ( iota_ y  e.  B
( y  .+  x
)  =  .0.  )
) )
2624, 25syl 15 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  B  |->  (
iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)  =  ( x  e.  (/)  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) ) )
2722, 26eqtr4d 2331 . . 3  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y 
.+  x )  =  .0.  ) ) )
2819, 27pm2.61i 156 . 2  |-  ( inv g `  G )  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) )
291, 28eqtri 2316 1  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   +g cplusg 13224   0gc0g 13416   inv gcminusg 14379
This theorem is referenced by:  grpinvval  14537  grpinvfn  14538  grpinvf  14542  grpinvpropd  14559  opprneg  15433
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-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-minusg 14506
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