MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpinvfval Unicode version

Theorem grpinvfval 14520
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 5525 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpinvval.b . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2333 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5525 . . . . . . . . 9  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 grpinvval.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2333 . . . . . . . 8  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
87oveqd 5875 . . . . . . 7  |-  ( g  =  G  ->  (
y ( +g  `  g
) x )  =  ( y  .+  x
) )
9 fveq2 5525 . . . . . . . 8  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
10 grpinvval.o . . . . . . . 8  |-  .0.  =  ( 0g `  G )
119, 10syl6eqr 2333 . . . . . . 7  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
128, 11eqeq12d 2297 . . . . . 6  |-  ( g  =  G  ->  (
( y ( +g  `  g ) x )  =  ( 0g `  g )  <->  ( y  .+  x )  =  .0.  ) )
134, 12riotaeqbidv 6307 . . . . 5  |-  ( g  =  G  ->  ( iota_ y  e.  ( Base `  g ) ( y ( +g  `  g
) x )  =  ( 0g `  g
) )  =  (
iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
144, 13mpteq12dv 4098 . . . 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 14490 . . . 4  |-  inv g  =  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
)  |->  ( iota_ y  e.  ( Base `  g
) ( y ( +g  `  g ) x )  =  ( 0g `  g ) ) ) )
16 fvex 5539 . . . . . 6  |-  ( Base `  G )  e.  _V
173, 16eqeltri 2353 . . . . 5  |-  B  e. 
_V
1817mptex 5746 . . . 4  |-  ( x  e.  B  |->  ( iota_ y  e.  B ( y 
.+  x )  =  .0.  ) )  e. 
_V
1914, 15, 18fvmpt 5602 . . 3  |-  ( G  e.  _V  ->  ( inv g `  G )  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) ) )
20 fvprc 5519 . . . . 5  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  (/) )
21 mpt0 5371 . . . . 5  |-  ( x  e.  (/)  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) )  =  (/)
2220, 21syl6eqr 2333 . . . 4  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  ( x  e.  (/)  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) ) )
23 fvprc 5519 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2327 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
25 mpteq1 4100 . . . . 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 2318 . . 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 2303 1  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   +g cplusg 13208   0gc0g 13400   inv gcminusg 14363
This theorem is referenced by:  grpinvval  14521  grpinvfn  14522  grpinvf  14526  grpinvpropd  14543  opprneg  15417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-minusg 14490
  Copyright terms: Public domain W3C validator