Theorem scaffng 14294

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 2802	. . . . . 6 |- x e. _V
2		vscaslid 13217	. . . . . . 7 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
3	2	slotex 13080	. . . . . 6 |- ( W e. V -> ( .s ` W ) e. _V )
4		vex 2802	. . . . . . 7 |- y e. _V
5	4	a1i 9	. . . . . 6 |- ( W e. V -> y e. _V )
6		ovexg 6044	. . . . . 6 |- ( ( x e. _V /\ ( .s ` W ) e. _V /\ y e. _V ) -> ( x ( .s ` W ) y ) e. _V )
7	1, 3, 5, 6	mp3an2i 1376	. . . . 5 |- ( W e. V -> ( x ( .s ` W ) y ) e. _V )
8	7	ralrimivw 2604	. . . 4 |- ( W e. V -> A. y e. B ( x ( .s ` W ) y ) e. _V )
9	8	ralrimivw 2604	. . 3 |- ( W e. V -> A. x e. K A. y e. B ( x ( .s ` W ) y ) e. _V )
10		eqid 2229	. . . 4 |- ( x e. K , y e. B |-> ( x ( .s ` W ) y ) ) = ( x e. K , y e. B |-> ( x ( .s ` W ) y ) )
11	10	fnmpo 6359	. . 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 2229	. . . 4 |- ( .s ` W ) = ( .s ` W )
18	13, 14, 15, 16, 17	scaffvalg 14291	. . 3 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x ( .s ` W ) y ) ) )
19	18	fneq1d 5414	. 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 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   X. cxp 4718   Fn wfn 5316   ` cfv 5321  (class class class)co 6010   e. cmpo 6012   Basecbs 13053  Scalarcsca 13134   .scvsca 13135   .sfcscaf 14273

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-coll 4199  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-cnex 8106  ax-resscn 8107  ax-1re 8109  ax-addrcl 8112

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-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  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-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-ndx 13056  df-slot 13057  df-base 13059  df-sca 13147  df-vsca 13148  df-scaf 14275

This theorem is referenced by:  lmodfopnelem1  14309