Theorem scaffvalg 14319

Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)

Hypotheses

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

Assertion

Ref	Expression
scaffvalg	|- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )

Distinct variable groups:   x, y, B   x, K, y   x, .x. , y   x, W, y   x, V, y

Allowed substitution hints:   .xb ( x, y)   F( x, y)

Proof of Theorem scaffvalg

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		scaffval.a	. 2 |- .xb = ( .sf ` W )
2		elex 2814	. . 3 |- ( W e. V -> W e. _V )
3		df-scaf 14303	. . . 4 |- .sf = ( w e. _V |-> ( x e. ( Base ` (Scalar ` w ) ) , y e. ( Base ` w ) |-> ( x ( .s ` w ) y ) ) )
4		fveq2 5639	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
5		scaffval.f	. . . . . . . 8 |- F = (Scalar ` W )
6	4, 5	eqtr4di 2282	. . . . . . 7 |- ( w = W -> (Scalar ` w ) = F )
7	6	fveq2d 5643	. . . . . 6 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
8		scaffval.k	. . . . . 6 |- K = ( Base ` F )
9	7, 8	eqtr4di 2282	. . . . 5 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
10		fveq2 5639	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
11		scaffval.b	. . . . . 6 |- B = ( Base ` W )
12	10, 11	eqtr4di 2282	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
13		fveq2 5639	. . . . . . 7 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
14		scaffval.s	. . . . . . 7 |- .x. = ( .s ` W )
15	13, 14	eqtr4di 2282	. . . . . 6 |- ( w = W -> ( .s ` w ) = .x. )
16	15	oveqd 6034	. . . . 5 |- ( w = W -> ( x ( .s ` w ) y ) = ( x .x. y ) )
17	9, 12, 16	mpoeq123dv 6082	. . . 4 |- ( w = W -> ( x e. ( Base ` (Scalar ` w ) ) , y e. ( Base ` w ) |-> ( x ( .s ` w ) y ) ) = ( x e. K , y e. B |-> ( x .x. y ) ) )
18		elex 2814	. . . 4 |- ( W e. _V -> W e. _V )
19		basfn 13140	. . . . . . 7 |- Base Fn _V
20		scaslid 13235	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
21	20	slotex 13108	. . . . . . . 8 |- ( W e. _V -> (Scalar ` W ) e. _V )
22	5, 21	eqeltrid 2318	. . . . . . 7 |- ( W e. _V -> F e. _V )
23		funfvex 5656	. . . . . . . 8 |- ( ( Fun Base /\ F e. dom Base ) -> ( Base ` F ) e. _V )
24	23	funfni 5432	. . . . . . 7 |- ( ( Base Fn _V /\ F e. _V ) -> ( Base ` F ) e. _V )
25	19, 22, 24	sylancr 414	. . . . . 6 |- ( W e. _V -> ( Base ` F ) e. _V )
26	8, 25	eqeltrid 2318	. . . . 5 |- ( W e. _V -> K e. _V )
27		funfvex 5656	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
28	27	funfni 5432	. . . . . . 7 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
29	19, 28	mpan 424	. . . . . 6 |- ( W e. _V -> ( Base ` W ) e. _V )
30	11, 29	eqeltrid 2318	. . . . 5 |- ( W e. _V -> B e. _V )
31		mpoexga 6376	. . . . 5 |- ( ( K e. _V /\ B e. _V ) -> ( x e. K , y e. B |-> ( x .x. y ) ) e. _V )
32	26, 30, 31	syl2anc 411	. . . 4 |- ( W e. _V -> ( x e. K , y e. B |-> ( x .x. y ) ) e. _V )
33	3, 17, 18, 32	fvmptd3 5740	. . 3 |- ( W e. _V -> ( .sf ` W ) = ( x e. K , y e. B |-> ( x .x. y ) ) )
34	2, 33	syl 14	. 2 |- ( W e. V -> ( .sf ` W ) = ( x e. K , y e. B |-> ( x .x. y ) ) )
35	1, 34	eqtrid 2276	1 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   Basecbs 13081  Scalarcsca 13162   .scvsca 13163   .sfcscaf 14301

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-ndx 13084  df-slot 13085  df-base 13087  df-sca 13175  df-scaf 14303

This theorem is referenced by:  scafvalg  14320  scafeqg  14321  scaffng  14322  lmodscaf  14323