Theorem scaffvalg 14626

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	⊢ ∙ = ( ·sf ‘𝑊)
scaffval.s	⊢ · = ( ·𝑠 ‘𝑊)

Assertion

Ref	Expression
scaffvalg	⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))

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

Allowed substitution hints:   ∙ (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 ⊢ ∙ = ( ·sf ‘𝑊)
2		elex 2833	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
3		df-scaf 14609	. . . 4 ⊢ ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (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 ⊢ · = ( ·𝑠 ‘𝑊)
15	13, 14	eqtr4di 2289	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
16	15	oveqd 6096	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
17	9, 12, 16	mpoeq123dv 6144	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
18		elex 2833	. . . 4 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
19		basfn 13394	. . . . . . 7 ⊢ Base Fn V
20		scaslid 13490	. . . . . . . . 9 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
21	20	slotex 13362	. . . . . . . 8 ⊢ (𝑊 ∈ V → (Scalar‘𝑊) ∈ V)
22	5, 21	eqeltrid 2325	. . . . . . 7 ⊢ (𝑊 ∈ V → 𝐹 ∈ V)
23		funfvex 5710	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐹 ∈ dom Base) → (Base‘𝐹) ∈ V)
24	23	funfni 5481	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐹 ∈ V) → (Base‘𝐹) ∈ V)
25	19, 22, 24	sylancr 418	. . . . . 6 ⊢ (𝑊 ∈ V → (Base‘𝐹) ∈ V)
26	8, 25	eqeltrid 2325	. . . . 5 ⊢ (𝑊 ∈ V → 𝐾 ∈ V)
27		funfvex 5710	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
28	27	funfni 5481	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
29	19, 28	mpan 428	. . . . . 6 ⊢ (𝑊 ∈ V → (Base‘𝑊) ∈ V)
30	11, 29	eqeltrid 2325	. . . . 5 ⊢ (𝑊 ∈ V → 𝐵 ∈ V)
31		mpoexga 6442	. . . . 5 ⊢ ((𝐾 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V)
32	26, 30, 31	syl2anc 415	. . . 4 ⊢ (𝑊 ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V)
33	3, 17, 18, 32	fvmptd3 5796	. . 3 ⊢ (𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
34	2, 33	syl 14	. 2 ⊢ (𝑊 ∈ 𝑉 → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
35	1, 34	eqtrid 2283	1 ⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079   ∈ cmpo 6081  Basecbs 13335  Scalarcsca 13417   ·_𝑠 cvsca 13418   ·_sf cscaf 14607

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 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

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 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-ndx 13338  df-slot 13339  df-base 13341  df-sca 13430  df-scaf 14609

This theorem is referenced by:  scafvalg  14627  scafeqg  14628  scaffng  14629  lmodscaf  14630