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Theorem grpinvval 12939
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  =  ( invg `  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 grpinvval.b . . . . 5  |-  B  =  ( Base `  G
)
21basmex 12534 . . . 4  |-  ( X  e.  B  ->  G  e.  _V )
3 grpinvval.p . . . . 5  |-  .+  =  ( +g  `  G )
4 grpinvval.o . . . . 5  |-  .0.  =  ( 0g `  G )
5 grpinvval.n . . . . 5  |-  N  =  ( invg `  G )
61, 3, 4, 5grpinvfvalg 12938 . . . 4  |-  ( G  e.  _V  ->  N  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x )  =  .0.  ) ) )
72, 6syl 14 . . 3  |-  ( X  e.  B  ->  N  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x )  =  .0.  ) ) )
87fveq1d 5529 . 2  |-  ( X  e.  B  ->  ( N `  X )  =  ( ( x  e.  B  |->  ( iota_ y  e.  B  ( y 
.+  x )  =  .0.  ) ) `  X ) )
9 eqid 2187 . . 3  |-  ( x  e.  B  |->  ( iota_ y  e.  B  ( y 
.+  x )  =  .0.  ) )  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x
)  =  .0.  )
)
10 oveq2 5896 . . . . 5  |-  ( x  =  X  ->  (
y  .+  x )  =  ( y  .+  X ) )
1110eqeq1d 2196 . . . 4  |-  ( x  =  X  ->  (
( y  .+  x
)  =  .0.  <->  ( y  .+  X )  =  .0.  ) )
1211riotabidv 5846 . . 3  |-  ( x  =  X  ->  ( iota_ y  e.  B  ( y  .+  x )  =  .0.  )  =  ( iota_ y  e.  B  ( y  .+  X
)  =  .0.  )
)
13 id 19 . . 3  |-  ( X  e.  B  ->  X  e.  B )
14 basfn 12533 . . . . . 6  |-  Base  Fn  _V
15 funfvex 5544 . . . . . . 7  |-  ( ( Fun  Base  /\  G  e. 
dom  Base )  ->  ( Base `  G )  e. 
_V )
1615funfni 5328 . . . . . 6  |-  ( (
Base  Fn  _V  /\  G  e.  _V )  ->  ( Base `  G )  e. 
_V )
1714, 2, 16sylancr 414 . . . . 5  |-  ( X  e.  B  ->  ( Base `  G )  e. 
_V )
181, 17eqeltrid 2274 . . . 4  |-  ( X  e.  B  ->  B  e.  _V )
19 riotaexg 5848 . . . 4  |-  ( B  e.  _V  ->  ( iota_ y  e.  B  ( y  .+  X )  =  .0.  )  e. 
_V )
2018, 19syl 14 . . 3  |-  ( X  e.  B  ->  ( iota_ y  e.  B  ( y  .+  X )  =  .0.  )  e. 
_V )
219, 12, 13, 20fvmptd3 5622 . 2  |-  ( X  e.  B  ->  (
( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x
)  =  .0.  )
) `  X )  =  ( iota_ y  e.  B  ( y  .+  X )  =  .0.  ) )
228, 21eqtrd 2220 1  |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B  ( y  .+  X )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    e. wcel 2158   _Vcvv 2749    |-> cmpt 4076    Fn wfn 5223   ` cfv 5228   iota_crio 5843  (class class class)co 5888   Basecbs 12475   +g cplusg 12550   0gc0g 12722   invgcminusg 12899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-inn 8933  df-ndx 12478  df-slot 12479  df-base 12481  df-minusg 12902
This theorem is referenced by:  grplinv  12946  isgrpinv  12950
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