Theorem scaffvalg 13938

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 2774	. . 3 |- ( W e. V -> W e. _V )
3		df-scaf 13922	. . . 4 |- .sf = ( w e. _V |-> ( x e. ( Base ` (Scalar ` w ) ) , y e. ( Base ` w ) |-> ( x ( .s ` w ) y ) ) )
4		fveq2 5561	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
5		scaffval.f	. . . . . . . 8 |- F = (Scalar ` W )
6	4, 5	eqtr4di 2247	. . . . . . 7 |- ( w = W -> (Scalar ` w ) = F )
7	6	fveq2d 5565	. . . . . 6 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
8		scaffval.k	. . . . . 6 |- K = ( Base ` F )
9	7, 8	eqtr4di 2247	. . . . 5 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
10		fveq2 5561	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
11		scaffval.b	. . . . . 6 |- B = ( Base ` W )
12	10, 11	eqtr4di 2247	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
13		fveq2 5561	. . . . . . 7 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
14		scaffval.s	. . . . . . 7 |- .x. = ( .s ` W )
15	13, 14	eqtr4di 2247	. . . . . 6 |- ( w = W -> ( .s ` w ) = .x. )
16	15	oveqd 5942	. . . . 5 |- ( w = W -> ( x ( .s ` w ) y ) = ( x .x. y ) )
17	9, 12, 16	mpoeq123dv 5988	. . . 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 2774	. . . 4 |- ( W e. _V -> W e. _V )
19		basfn 12761	. . . . . . 7 |- Base Fn _V
20		scaslid 12855	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
21	20	slotex 12730	. . . . . . . 8 |- ( W e. _V -> (Scalar ` W ) e. _V )
22	5, 21	eqeltrid 2283	. . . . . . 7 |- ( W e. _V -> F e. _V )
23		funfvex 5578	. . . . . . . 8 |- ( ( Fun Base /\ F e. dom Base ) -> ( Base ` F ) e. _V )
24	23	funfni 5361	. . . . . . 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 2283	. . . . 5 |- ( W e. _V -> K e. _V )
27		funfvex 5578	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
28	27	funfni 5361	. . . . . . 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 2283	. . . . 5 |- ( W e. _V -> B e. _V )
31		mpoexga 6279	. . . . 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 5658	. . 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 2241	1 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   Basecbs 12703  Scalarcsca 12783   .scvsca 12784   .sfcscaf 13920

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-ndx 12706  df-slot 12707  df-base 12709  df-sca 12796  df-scaf 13922

This theorem is referenced by:  scafvalg  13939  scafeqg  13940  scaffng  13941  lmodscaf  13942