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Theorem grpinvval 14521
Description: The inverse of a group element. (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
grpinvval  |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B ( y  .+  X )  =  .0.  ) )
Distinct variable groups:    y, B    y, G    y, X
Allowed substitution hints:    .+ ( y)    N( y)    .0. ( y)

Proof of Theorem grpinvval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq2 5866 . . . 4  |-  ( x  =  X  ->  (
y  .+  x )  =  ( y  .+  X ) )
21eqeq1d 2291 . . 3  |-  ( x  =  X  ->  (
( y  .+  x
)  =  .0.  <->  ( y  .+  X )  =  .0.  ) )
32riotabidv 6306 . 2  |-  ( x  =  X  ->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  )  =  ( iota_ y  e.  B
( y  .+  X
)  =  .0.  )
)
4 grpinvval.b . . 3  |-  B  =  ( Base `  G
)
5 grpinvval.p . . 3  |-  .+  =  ( +g  `  G )
6 grpinvval.o . . 3  |-  .0.  =  ( 0g `  G )
7 grpinvval.n . . 3  |-  N  =  ( inv g `  G )
84, 5, 6, 7grpinvfval 14520 . 2  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
9 riotaex 6308 . 2  |-  ( iota_ y  e.  B ( y 
.+  X )  =  .0.  )  e.  _V
103, 8, 9fvmpt 5602 1  |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B ( y  .+  X )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` 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:  grplinv  14528  isgrpinv  14532  xrsinvgval  23306
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
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