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Theorem grpinvfn 14846
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  =  ( inv g `  G )
Assertion
Ref Expression
grpinvfn  |-  N  Fn  B

Proof of Theorem grpinvfn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6554 . 2  |-  ( iota_ y  e.  B ( y ( +g  `  G
) x )  =  ( 0g `  G
) )  e.  _V
2 grpinvfn.b . . 3  |-  B  =  ( Base `  G
)
3 eqid 2437 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2437 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
5 grpinvfn.n . . 3  |-  N  =  ( inv g `  G )
62, 3, 4, 5grpinvfval 14844 . 2  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
71, 6fnmpti 5574 1  |-  N  Fn  B
Colors of variables: wff set class
Syntax hints:    = wceq 1653    Fn wfn 5450   ` cfv 5455  (class class class)co 6082   iota_crio 6543   Basecbs 13470   +g cplusg 13530   0gc0g 13724   inv gcminusg 14687
This theorem is referenced by:  grpinvfvi  14847  isgrpinv  14856  invrfval  15779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-riota 6550  df-minusg 14814
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