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Theorem grpinvval 14772
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 6029 . . . 4  |-  ( x  =  X  ->  (
y  .+  x )  =  ( y  .+  X ) )
21eqeq1d 2396 . . 3  |-  ( x  =  X  ->  (
( y  .+  x
)  =  .0.  <->  ( y  .+  X )  =  .0.  ) )
32riotabidv 6488 . 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 14771 . 2  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
9 riotaex 6490 . 2  |-  ( iota_ y  e.  B ( y 
.+  X )  =  .0.  )  e.  _V
103, 8, 9fvmpt 5746 1  |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B ( y  .+  X )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   iota_crio 6479   Basecbs 13397   +g cplusg 13457   0gc0g 13651   inv gcminusg 14614
This theorem is referenced by:  grplinv  14779  isgrpinv  14783  xrsinvgval  24033  rnginvval  24058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-riota 6486  df-minusg 14741
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