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Theorem grpinvfng 12849
Description: Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypotheses
Ref Expression
grpinvfn.b  |-  B  =  ( Base `  G
)
grpinvfn.n  |-  N  =  ( invg `  G )
Assertion
Ref Expression
grpinvfng  |-  ( G  e.  V  ->  N  Fn  B )

Proof of Theorem grpinvfng
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpinvfn.b . . . . . 6  |-  B  =  ( Base `  G
)
2 basfn 12512 . . . . . . 7  |-  Base  Fn  _V
3 elex 2748 . . . . . . 7  |-  ( G  e.  V  ->  G  e.  _V )
4 funfvex 5531 . . . . . . . 8  |-  ( ( Fun  Base  /\  G  e. 
dom  Base )  ->  ( Base `  G )  e. 
_V )
54funfni 5315 . . . . . . 7  |-  ( (
Base  Fn  _V  /\  G  e.  _V )  ->  ( Base `  G )  e. 
_V )
62, 3, 5sylancr 414 . . . . . 6  |-  ( G  e.  V  ->  ( Base `  G )  e. 
_V )
71, 6eqeltrid 2264 . . . . 5  |-  ( G  e.  V  ->  B  e.  _V )
8 riotaexg 5832 . . . . 5  |-  ( B  e.  _V  ->  ( iota_ y  e.  B  ( y ( +g  `  G
) x )  =  ( 0g `  G
) )  e.  _V )
97, 8syl 14 . . . 4  |-  ( G  e.  V  ->  ( iota_ y  e.  B  ( y ( +g  `  G
) x )  =  ( 0g `  G
) )  e.  _V )
109ralrimivw 2551 . . 3  |-  ( G  e.  V  ->  A. x  e.  B  ( iota_ y  e.  B  ( y ( +g  `  G
) x )  =  ( 0g `  G
) )  e.  _V )
11 eqid 2177 . . . 4  |-  ( x  e.  B  |->  ( iota_ y  e.  B  ( y ( +g  `  G
) x )  =  ( 0g `  G
) ) )  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
1211fnmpt 5341 . . 3  |-  ( A. x  e.  B  ( iota_ y  e.  B  ( y ( +g  `  G
) x )  =  ( 0g `  G
) )  e.  _V  ->  ( x  e.  B  |->  ( iota_ y  e.  B  ( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )  Fn  B )
1310, 12syl 14 . 2  |-  ( G  e.  V  ->  (
x  e.  B  |->  (
iota_ y  e.  B  ( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )  Fn  B )
14 eqid 2177 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
15 eqid 2177 . . . 4  |-  ( 0g
`  G )  =  ( 0g `  G
)
16 grpinvfn.n . . . 4  |-  N  =  ( invg `  G )
171, 14, 15, 16grpinvfvalg 12847 . . 3  |-  ( G  e.  V  ->  N  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y ( +g  `  G ) x )  =  ( 0g `  G ) ) ) )
1817fneq1d 5305 . 2  |-  ( G  e.  V  ->  ( N  Fn  B  <->  ( x  e.  B  |->  ( iota_ y  e.  B  ( y ( +g  `  G
) x )  =  ( 0g `  G
) ) )  Fn  B ) )
1913, 18mpbird 167 1  |-  ( G  e.  V  ->  N  Fn  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2737    |-> cmpt 4063    Fn wfn 5210   ` cfv 5215   iota_crio 5827  (class class class)co 5872   Basecbs 12454   +g cplusg 12528   0gc0g 12693   invgcminusg 12810
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 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-inn 8916  df-ndx 12457  df-slot 12458  df-base 12460  df-minusg 12813
This theorem is referenced by:  isgrpinv  12858  mulgval  12918  mulgfng  12919  invrfvald  13222
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