Theorem scaffvalg 14618

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 2833	. . 3 |- ( W e. V -> W e. _V )
3		df-scaf 14602	. . . 4 |- .sf = ( w e. _V |-> ( x e. ( Base ` (Scalar ` w ) ) , y e. ( Base ` w ) |-> ( x ( .s ` w ) y ) ) )
4		fveq2 5693	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
5		scaffval.f	. . . . . . . 8 |- F = (Scalar ` W )
6	4, 5	eqtr4di 2289	. . . . . . 7 |- ( w = W -> (Scalar ` w ) = F )
7	6	fveq2d 5697	. . . . . 6 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
8		scaffval.k	. . . . . 6 |- K = ( Base ` F )
9	7, 8	eqtr4di 2289	. . . . 5 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
10		fveq2 5693	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
11		scaffval.b	. . . . . 6 |- B = ( Base ` W )
12	10, 11	eqtr4di 2289	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
13		fveq2 5693	. . . . . . 7 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
14		scaffval.s	. . . . . . 7 |- .x. = ( .s ` W )
15	13, 14	eqtr4di 2289	. . . . . 6 |- ( w = W -> ( .s ` w ) = .x. )
16	15	oveqd 6095	. . . . 5 |- ( w = W -> ( x ( .s ` w ) y ) = ( x .x. y ) )
17	9, 12, 16	mpoeq123dv 6143	. . . 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 2833	. . . 4 |- ( W e. _V -> W e. _V )
19		basfn 13392	. . . . . . 7 |- Base Fn _V
20		scaslid 13487	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
21	20	slotex 13360	. . . . . . . 8 |- ( W e. _V -> (Scalar ` W ) e. _V )
22	5, 21	eqeltrid 2325	. . . . . . 7 |- ( W e. _V -> F e. _V )
23		funfvex 5710	. . . . . . . 8 |- ( ( Fun Base /\ F e. dom Base ) -> ( Base ` F ) e. _V )
24	23	funfni 5481	. . . . . . 7 |- ( ( Base Fn _V /\ F e. _V ) -> ( Base ` F ) e. _V )
25	19, 22, 24	sylancr 418	. . . . . 6 |- ( W e. _V -> ( Base ` F ) e. _V )
26	8, 25	eqeltrid 2325	. . . . 5 |- ( W e. _V -> K e. _V )
27		funfvex 5710	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
28	27	funfni 5481	. . . . . . 7 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
29	19, 28	mpan 428	. . . . . 6 |- ( W e. _V -> ( Base ` W ) e. _V )
30	11, 29	eqeltrid 2325	. . . . 5 |- ( W e. _V -> B e. _V )
31		mpoexga 6441	. . . . 5 |- ( ( K e. _V /\ B e. _V ) -> ( x e. K , y e. B |-> ( x .x. y ) ) e. _V )
32	26, 30, 31	syl2anc 415	. . . 4 |- ( W e. _V -> ( x e. K , y e. B |-> ( x .x. y ) ) e. _V )
33	3, 17, 18, 32	fvmptd3 5796	. . 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 2283	1 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   Fn wfn 5370   ` cfv 5375  (class class class)co 6078   e. cmpo 6080   Basecbs 13333  Scalarcsca 13414   .scvsca 13415   .sfcscaf 14600

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8263  ax-resscn 8264  ax-1re 8266  ax-addrcl 8269

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-inn 9287  df-2 9345  df-3 9346  df-4 9347  df-5 9348  df-ndx 13336  df-slot 13337  df-base 13339  df-sca 13427  df-scaf 14602

This theorem is referenced by:  scafvalg  14619  scafeqg  14620  scaffng  14621  lmodscaf  14622