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Theorem fn0g 14713
Description: The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
Assertion
Ref Expression
fn0g  |-  0g  Fn  _V

Proof of Theorem fn0g
Dummy variables  e 
g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iotaex 5438 . 2  |-  ( iota e ( e  e.  ( Base `  g
)  /\  A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) ) )  e.  _V
2 df-0g 13732 . 2  |-  0g  =  ( g  e.  _V  |->  ( iota e ( e  e.  ( Base `  g
)  /\  A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) ) ) )
31, 2fnmpti 5576 1  |-  0g  Fn  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958   iotacio 5419    Fn wfn 5452   ` cfv 5457  (class class class)co 6084   Basecbs 13474   +g cplusg 13534   0gc0g 13728
This theorem is referenced by:  prdsidlem  14732  pws0g  14736  prdsinvlem  14931  pws1  15727  dsmmbas2  27194  frlmbas  27214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fn 5460  df-0g 13732
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