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Theorem fn0g 12673
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 df-riota 5824 . . 3  |-  ( iota_ e  e.  ( Base `  g
) A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) )  =  ( iota e ( e  e.  ( Base `  g )  /\  A. x  e.  ( Base `  g ) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) ) )
2 basfn 12489 . . . . 5  |-  Base  Fn  _V
3 vex 2740 . . . . 5  |-  g  e. 
_V
4 funfvex 5527 . . . . . 6  |-  ( ( Fun  Base  /\  g  e.  dom  Base )  ->  ( Base `  g )  e. 
_V )
54funfni 5311 . . . . 5  |-  ( (
Base  Fn  _V  /\  g  e.  _V )  ->  ( Base `  g )  e. 
_V )
62, 3, 5mp2an 426 . . . 4  |-  ( Base `  g )  e.  _V
7 riotaexg 5828 . . . 4  |-  ( (
Base `  g )  e.  _V  ->  ( iota_ e  e.  ( Base `  g
) A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) )  e. 
_V )
86, 7ax-mp 5 . . 3  |-  ( iota_ e  e.  ( Base `  g
) A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) )  e. 
_V
91, 8eqeltrri 2251 . 2  |-  ( iota e ( e  e.  ( Base `  g
)  /\  A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) ) )  e.  _V
10 df-0g 12642 . 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 ) ) ) )
119, 10fnmpti 5339 1  |-  0g  Fn  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2737   iotacio 5171    Fn wfn 5206   ` cfv 5211   iota_crio 5823  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   0gc0g 12640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-inn 8896  df-ndx 12435  df-slot 12436  df-base 12438  df-0g 12642
This theorem is referenced by:  0mhm  12750  mulgval  12862  mulgfng  12863  issrg  12961
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