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Theorem grpinvfn 16207
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 6162 . 2  |-  ( iota_ y  e.  B  ( y ( +g  `  G
) x )  =  ( 0g `  G
) )  e.  _V
2 grpinvfn.b . . 3  |-  B  =  ( Base `  G
)
3 eqid 2382 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2382 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
5 grpinvfn.n . . 3  |-  N  =  ( invg `  G )
62, 3, 4, 5grpinvfval 16205 . 2  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
71, 6fnmpti 5617 1  |-  N  Fn  B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399    Fn wfn 5491   ` cfv 5496   iota_crio 6157  (class class class)co 6196   Basecbs 14634   +g cplusg 14702   0gc0g 14847   invgcminusg 16171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-minusg 16175
This theorem is referenced by:  grpinvfvi  16208  isgrpinv  16217  invrfval  17435
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