Theorem fn0g 13521

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 5981	. . 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 13204	. . . . 5 |- Base Fn _V
3		vex 2806	. . . . 5 |- g e. _V
4		funfvex 5665	. . . . . 6 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
5	4	funfni 5439	. . . . 5 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
6	2, 3, 5	mp2an 426	. . . 4 |- ( Base ` g ) e. _V
7		riotaexg 5985	. . . 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 2305	. 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 13404	. 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 5468	1 |- 0g Fn _V

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   iotacio 5291   Fn wfn 5328   ` cfv 5333   iota_crio 5980  (class class class)co 6028   Basecbs 13145   +g cplusg 13223   0gc0g 13402

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151  df-0g 13404

This theorem is referenced by:  fngsum  13534  igsumvalx  13535  gsumfzval  13537  gsum0g  13542  prdsidlem  13593  pws0g  13597  0mhm  13632  prdsinvlem  13754  mulgval  13772  mulgfng  13774  issrg  14042  isdomn  14348