Theorem scaffng 13404

Description: The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
scaffval.b	|- B = ( Base ` W )
scaffval.f	|- F = (Scalar ` W )
scaffval.k	|- K = ( Base ` F )
scaffval.a	|- .xb = ( .sf ` W )

Assertion

Ref	Expression
scaffng	|- ( W e. V -> .xb Fn ( K X. B ) )

Proof of Theorem scaffng

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2742	. . . . . 6 |- x e. _V
2		vscaslid 12623	. . . . . . 7 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
3	2	slotex 12491	. . . . . 6 |- ( W e. V -> ( .s ` W ) e. _V )
4		vex 2742	. . . . . . 7 |- y e. _V
5	4	a1i 9	. . . . . 6 |- ( W e. V -> y e. _V )
6		ovexg 5911	. . . . . 6 |- ( ( x e. _V /\ ( .s ` W ) e. _V /\ y e. _V ) -> ( x ( .s ` W ) y ) e. _V )
7	1, 3, 5, 6	mp3an2i 1342	. . . . 5 |- ( W e. V -> ( x ( .s ` W ) y ) e. _V )
8	7	ralrimivw 2551	. . . 4 |- ( W e. V -> A. y e. B ( x ( .s ` W ) y ) e. _V )
9	8	ralrimivw 2551	. . 3 |- ( W e. V -> A. x e. K A. y e. B ( x ( .s ` W ) y ) e. _V )
10		eqid 2177	. . . 4 |- ( x e. K , y e. B |-> ( x ( .s ` W ) y ) ) = ( x e. K , y e. B |-> ( x ( .s ` W ) y ) )
11	10	fnmpo 6205	. . 3 |- ( A. x e. K A. y e. B ( x ( .s ` W ) y ) e. _V -> ( x e. K , y e. B |-> ( x ( .s ` W ) y ) ) Fn ( K X. B ) )
12	9, 11	syl 14	. 2 |- ( W e. V -> ( x e. K , y e. B |-> ( x ( .s ` W ) y ) ) Fn ( K X. B ) )
13		scaffval.b	. . . 4 |- B = ( Base ` W )
14		scaffval.f	. . . 4 |- F = (Scalar ` W )
15		scaffval.k	. . . 4 |- K = ( Base ` F )
16		scaffval.a	. . . 4 |- .xb = ( .sf ` W )
17		eqid 2177	. . . 4 |- ( .s ` W ) = ( .s ` W )
18	13, 14, 15, 16, 17	scaffvalg 13401	. . 3 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x ( .s ` W ) y ) ) )
19	18	fneq1d 5308	. 2 |- ( W e. V -> ( .xb Fn ( K X. B ) <-> ( x e. K , y e. B |-> ( x ( .s ` W ) y ) ) Fn ( K X. B ) ) )
20	12, 19	mpbird 167	1 |- ( W e. V -> .xb Fn ( K X. B ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2739   X. cxp 4626   Fn wfn 5213   ` cfv 5218  (class class class)co 5877   e. cmpo 5879   Basecbs 12464  Scalarcsca 12541   .scvsca 12542   .sfcscaf 13383

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-ndx 12467  df-slot 12468  df-base 12470  df-sca 12554  df-vsca 12555  df-scaf 13385

This theorem is referenced by:  lmodfopnelem1  13419