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Theorem grpinvval 12959
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 12545 . . . 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 12958 . . . 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 5532 . 2  |-  ( X  e.  B  ->  ( N `  X )  =  ( ( x  e.  B  |->  ( iota_ y  e.  B  ( y 
.+  x )  =  .0.  ) ) `  X ) )
9 eqid 2189 . . 3  |-  ( x  e.  B  |->  ( iota_ y  e.  B  ( y 
.+  x )  =  .0.  ) )  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x
)  =  .0.  )
)
10 oveq2 5899 . . . . 5  |-  ( x  =  X  ->  (
y  .+  x )  =  ( y  .+  X ) )
1110eqeq1d 2198 . . . 4  |-  ( x  =  X  ->  (
( y  .+  x
)  =  .0.  <->  ( y  .+  X )  =  .0.  ) )
1211riotabidv 5849 . . 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 12544 . . . . . 6  |-  Base  Fn  _V
15 funfvex 5547 . . . . . . 7  |-  ( ( Fun  Base  /\  G  e. 
dom  Base )  ->  ( Base `  G )  e. 
_V )
1615funfni 5331 . . . . . 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 2276 . . . 4  |-  ( X  e.  B  ->  B  e.  _V )
19 riotaexg 5851 . . . 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 5625 . 2  |-  ( X  e.  B  ->  (
( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x
)  =  .0.  )
) `  X )  =  ( iota_ y  e.  B  ( y  .+  X )  =  .0.  ) )
228, 21eqtrd 2222 1  |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B  ( y  .+  X )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160   _Vcvv 2752    |-> cmpt 4079    Fn wfn 5226   ` cfv 5231   iota_crio 5846  (class class class)co 5891   Basecbs 12486   +g cplusg 12561   0gc0g 12733   invgcminusg 12918
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-inn 8939  df-ndx 12489  df-slot 12490  df-base 12492  df-minusg 12921
This theorem is referenced by:  grplinv  12966  isgrpinv  12970
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