Theorem fn0g 13695

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.

Step	Hyp	Ref	Expression
1		df-riota 6038	. . 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 13411	. . . . 5 |- Base Fn _V
3		vex 2824	. . . . 5 |- g e. _V
4		funfvex 5712	. . . . . 6 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
5	4	funfni 5483	. . . . 5 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
6	2, 3, 5	mp2an 430	. . . 4 |- ( Base ` g ) e. _V
7		riotaexg 6042	. . . 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 )
8	6, 7	ax-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
9	1, 8	eqeltrri 2312	. 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 13612	. 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 ) ) ) )
11	9, 10	fnmpti 5512	1 |- 0g Fn _V

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   iotacio 5335   Fn wfn 5372   ` cfv 5377   iota_crio 6037  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-inn 9305  df-ndx 13355  df-slot 13356  df-base 13358  df-0g 13612

This theorem is used by:  fngzsum  13708  gzsumvalx  13709  gzsumfzval  13711  gzsum0  13713  0mhm  13793  mulgval  13925  mulgfng  13927  prdsidlem  14193  prdsinvlem  14196  pws0g  14213  ringidval  14265  issrg  14269  isdomn  14578