Theorem scaffvalg 14285

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 2811	. . 3 |- ( W e. V -> W e. _V )
3		df-scaf 14269	. . . 4 |- .sf = ( w e. _V |-> ( x e. ( Base ` (Scalar ` w ) ) , y e. ( Base ` w ) |-> ( x ( .s ` w ) y ) ) )
4		fveq2 5629	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
5		scaffval.f	. . . . . . . 8 |- F = (Scalar ` W )
6	4, 5	eqtr4di 2280	. . . . . . 7 |- ( w = W -> (Scalar ` w ) = F )
7	6	fveq2d 5633	. . . . . 6 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
8		scaffval.k	. . . . . 6 |- K = ( Base ` F )
9	7, 8	eqtr4di 2280	. . . . 5 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
10		fveq2 5629	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
11		scaffval.b	. . . . . 6 |- B = ( Base ` W )
12	10, 11	eqtr4di 2280	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
13		fveq2 5629	. . . . . . 7 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
14		scaffval.s	. . . . . . 7 |- .x. = ( .s ` W )
15	13, 14	eqtr4di 2280	. . . . . 6 |- ( w = W -> ( .s ` w ) = .x. )
16	15	oveqd 6024	. . . . 5 |- ( w = W -> ( x ( .s ` w ) y ) = ( x .x. y ) )
17	9, 12, 16	mpoeq123dv 6072	. . . 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 2811	. . . 4 |- ( W e. _V -> W e. _V )
19		basfn 13106	. . . . . . 7 |- Base Fn _V
20		scaslid 13201	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
21	20	slotex 13074	. . . . . . . 8 |- ( W e. _V -> (Scalar ` W ) e. _V )
22	5, 21	eqeltrid 2316	. . . . . . 7 |- ( W e. _V -> F e. _V )
23		funfvex 5646	. . . . . . . 8 |- ( ( Fun Base /\ F e. dom Base ) -> ( Base ` F ) e. _V )
24	23	funfni 5423	. . . . . . 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 2316	. . . . 5 |- ( W e. _V -> K e. _V )
27		funfvex 5646	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
28	27	funfni 5423	. . . . . . 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 2316	. . . . 5 |- ( W e. _V -> B e. _V )
31		mpoexga 6364	. . . . 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 5730	. . 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 2274	1 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   Fn wfn 5313   ` cfv 5318  (class class class)co 6007   e. cmpo 6009   Basecbs 13047  Scalarcsca 13128   .scvsca 13129   .sfcscaf 14267

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-ndx 13050  df-slot 13051  df-base 13053  df-sca 13141  df-scaf 14269

This theorem is referenced by:  scafvalg  14286  scafeqg  14287  scaffng  14288  lmodscaf  14289