Theorem fn0g 12606

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 5798	. . 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 12451	. . . . 5 |- Base Fn _V
3		vex 2729	. . . . 5 |- g e. _V
4		funfvex 5503	. . . . . 6 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
5	4	funfni 5288	. . . . 5 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
6	2, 3, 5	mp2an 423	. . . 4 |- ( Base ` g ) e. _V
7		riotaexg 5802	. . . 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 2240	. 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 12575	. 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 5316	1 |- 0g Fn _V

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   iotacio 5151   Fn wfn 5183   ` cfv 5188   iota_crio 5797  (class class class)co 5842   Basecbs 12394   +g cplusg 12457   0gc0g 12573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196  df-riota 5798  df-inn 8858  df-ndx 12397  df-slot 12398  df-base 12400  df-0g 12575

This theorem is referenced by: (None)