Theorem fn0g 12958

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 5873	. . 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 12676	. . . . 5 |- Base Fn _V
3		vex 2763	. . . . 5 |- g e. _V
4		funfvex 5571	. . . . . 6 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
5	4	funfni 5354	. . . . 5 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
6	2, 3, 5	mp2an 426	. . . 4 |- ( Base ` g ) e. _V
7		riotaexg 5877	. . . 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 2267	. 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 12869	. 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 5382	1 |- 0g Fn _V

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   _Vcvv 2760   iotacio 5213   Fn wfn 5249   ` cfv 5254   iota_crio 5872  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-0g 12869

This theorem is referenced by:  fngsum  12971  igsumvalx  12972  gsumfzval  12974  gsum0g  12979  0mhm  13058  mulgval  13192  mulgfng  13194  issrg  13461  isdomn  13765