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Theorem fn0g 14663
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 5394 . 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 13682 . 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 5532 1  |-  0g  Fn  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   iotacio 5375    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   0gc0g 13678
This theorem is referenced by:  prdsidlem  14682  pws0g  14686  prdsinvlem  14881  pws1  15677  dsmmbas2  27071  frlmbas  27091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fn 5416  df-0g 13682
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