Theorem scaffvalg 13401

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 2750	. . 3 |- ( W e. V -> W e. _V )
3		df-scaf 13385	. . . 4 |- .sf = ( w e. _V |-> ( x e. ( Base ` (Scalar ` w ) ) , y e. ( Base ` w ) |-> ( x ( .s ` w ) y ) ) )
4		fveq2 5517	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
5		scaffval.f	. . . . . . . 8 |- F = (Scalar ` W )
6	4, 5	eqtr4di 2228	. . . . . . 7 |- ( w = W -> (Scalar ` w ) = F )
7	6	fveq2d 5521	. . . . . 6 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
8		scaffval.k	. . . . . 6 |- K = ( Base ` F )
9	7, 8	eqtr4di 2228	. . . . 5 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
10		fveq2 5517	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
11		scaffval.b	. . . . . 6 |- B = ( Base ` W )
12	10, 11	eqtr4di 2228	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
13		fveq2 5517	. . . . . . 7 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
14		scaffval.s	. . . . . . 7 |- .x. = ( .s ` W )
15	13, 14	eqtr4di 2228	. . . . . 6 |- ( w = W -> ( .s ` w ) = .x. )
16	15	oveqd 5894	. . . . 5 |- ( w = W -> ( x ( .s ` w ) y ) = ( x .x. y ) )
17	9, 12, 16	mpoeq123dv 5939	. . . 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 2750	. . . 4 |- ( W e. _V -> W e. _V )
19		basfn 12522	. . . . . . 7 |- Base Fn _V
20		scaslid 12613	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
21	20	slotex 12491	. . . . . . . 8 |- ( W e. _V -> (Scalar ` W ) e. _V )
22	5, 21	eqeltrid 2264	. . . . . . 7 |- ( W e. _V -> F e. _V )
23		funfvex 5534	. . . . . . . 8 |- ( ( Fun Base /\ F e. dom Base ) -> ( Base ` F ) e. _V )
24	23	funfni 5318	. . . . . . 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 2264	. . . . 5 |- ( W e. _V -> K e. _V )
27		funfvex 5534	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
28	27	funfni 5318	. . . . . . 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 2264	. . . . 5 |- ( W e. _V -> B e. _V )
31		mpoexga 6215	. . . . 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 5611	. . 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 2222	1 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2739   Fn wfn 5213   ` cfv 5218  (class class class)co 5877   e. cmpo 5879   Basecbs 12464  Scalarcsca 12541   .scvsca 12542   .sfcscaf 13383

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-ndx 12467  df-slot 12468  df-base 12470  df-sca 12554  df-scaf 13385

This theorem is referenced by:  scafvalg  13402  scafeqg  13403  scaffng  13404  lmodscaf  13405