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Theorem grpinvfn 15886
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
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 6242 . 2  |-  ( iota_ y  e.  B  ( y ( +g  `  G
) x )  =  ( 0g `  G
) )  e.  _V
2 grpinvfn.b . . 3  |-  B  =  ( Base `  G
)
3 eqid 2462 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2462 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
5 grpinvfn.n . . 3  |-  N  =  ( invg `  G )
62, 3, 4, 5grpinvfval 15884 . 2  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
71, 6fnmpti 5702 1  |-  N  Fn  B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    Fn wfn 5576   ` cfv 5581   iota_crio 6237  (class class class)co 6277   Basecbs 14481   +g cplusg 14546   0gc0g 14686   invgcminusg 15719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-minusg 15854
This theorem is referenced by:  grpinvfvi  15887  isgrpinv  15896  invrfval  17101
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