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Theorem grpinvval 12793
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 12490 . . . 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 12792 . . . 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 5512 . 2  |-  ( X  e.  B  ->  ( N `  X )  =  ( ( x  e.  B  |->  ( iota_ y  e.  B  ( y 
.+  x )  =  .0.  ) ) `  X ) )
9 eqid 2177 . . 3  |-  ( x  e.  B  |->  ( iota_ y  e.  B  ( y 
.+  x )  =  .0.  ) )  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x
)  =  .0.  )
)
10 oveq2 5876 . . . . 5  |-  ( x  =  X  ->  (
y  .+  x )  =  ( y  .+  X ) )
1110eqeq1d 2186 . . . 4  |-  ( x  =  X  ->  (
( y  .+  x
)  =  .0.  <->  ( y  .+  X )  =  .0.  ) )
1211riotabidv 5826 . . 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 12489 . . . . . 6  |-  Base  Fn  _V
15 funfvex 5527 . . . . . . 7  |-  ( ( Fun  Base  /\  G  e. 
dom  Base )  ->  ( Base `  G )  e. 
_V )
1615funfni 5311 . . . . . 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 2264 . . . 4  |-  ( X  e.  B  ->  B  e.  _V )
19 riotaexg 5828 . . . 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 5604 . 2  |-  ( X  e.  B  ->  (
( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x
)  =  .0.  )
) `  X )  =  ( iota_ y  e.  B  ( y  .+  X )  =  .0.  ) )
228, 21eqtrd 2210 1  |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B  ( y  .+  X )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   _Vcvv 2737    |-> cmpt 4061    Fn wfn 5206   ` cfv 5211   iota_crio 5823  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   0gc0g 12640   invgcminusg 12755
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-ndx 12435  df-slot 12436  df-base 12438  df-minusg 12758
This theorem is referenced by:  grplinv  12799  isgrpinv  12803
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