Theorem scaffng 14238

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 2782	. . . . . 6 |- x e. _V
2		vscaslid 13162	. . . . . . 7 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
3	2	slotex 13025	. . . . . 6 |- ( W e. V -> ( .s ` W ) e. _V )
4		vex 2782	. . . . . . 7 |- y e. _V
5	4	a1i 9	. . . . . 6 |- ( W e. V -> y e. _V )
6		ovexg 6008	. . . . . 6 |- ( ( x e. _V /\ ( .s ` W ) e. _V /\ y e. _V ) -> ( x ( .s ` W ) y ) e. _V )
7	1, 3, 5, 6	mp3an2i 1357	. . . . 5 |- ( W e. V -> ( x ( .s ` W ) y ) e. _V )
8	7	ralrimivw 2584	. . . 4 |- ( W e. V -> A. y e. B ( x ( .s ` W ) y ) e. _V )
9	8	ralrimivw 2584	. . 3 |- ( W e. V -> A. x e. K A. y e. B ( x ( .s ` W ) y ) e. _V )
10		eqid 2209	. . . 4 |- ( x e. K , y e. B |-> ( x ( .s ` W ) y ) ) = ( x e. K , y e. B |-> ( x ( .s ` W ) y ) )
11	10	fnmpo 6318	. . 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 2209	. . . 4 |- ( .s ` W ) = ( .s ` W )
18	13, 14, 15, 16, 17	scaffvalg 14235	. . 3 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x ( .s ` W ) y ) ) )
19	18	fneq1d 5387	. 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 1375   e. wcel 2180   A.wral 2488   _Vcvv 2779   X. cxp 4694   Fn wfn 5289   ` cfv 5294  (class class class)co 5974   e. cmpo 5976   Basecbs 12998  Scalarcsca 13079   .scvsca 13080   .sfcscaf 14217

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-cnex 8058  ax-resscn 8059  ax-1re 8061  ax-addrcl 8064

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-ndx 13001  df-slot 13002  df-base 13004  df-sca 13092  df-vsca 13093  df-scaf 14219

This theorem is referenced by:  lmodfopnelem1  14253