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Theorem fn0g 16213
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 5550 . 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 15056 . 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 5692 1  |-  0g  Fn  _V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   _Vcvv 3059   iotacio 5531    Fn wfn 5564   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   0gc0g 15054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fn 5572  df-0g 15056
This theorem is referenced by:  prdsidlem  16276  pws0g  16280  prdsinvlem  16502  pws1  17585  dsmmbas2  19066  frlmbas  19084  frlmbasOLD  19085
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