Theorem fn0g 13423

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 5960	. . 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 13106	. . . . 5 |- Base Fn _V
3		vex 2802	. . . . 5 |- g e. _V
4		funfvex 5646	. . . . . 6 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
5	4	funfni 5423	. . . . 5 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
6	2, 3, 5	mp2an 426	. . . 4 |- ( Base ` g ) e. _V
7		riotaexg 5964	. . . 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 2303	. 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 13306	. 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 5452	1 |- 0g Fn _V

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   iotacio 5276   Fn wfn 5313   ` cfv 5318   iota_crio 5959  (class class class)co 6007   Basecbs 13047   +g cplusg 13125   0gc0g 13304

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5960  df-inn 9122  df-ndx 13050  df-slot 13051  df-base 13053  df-0g 13306

This theorem is referenced by:  fngsum  13436  igsumvalx  13437  gsumfzval  13439  gsum0g  13444  prdsidlem  13495  pws0g  13499  0mhm  13534  prdsinvlem  13656  mulgval  13674  mulgfng  13676  issrg  13943  isdomn  14248