Theorem scaffvalg 14112

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 2784	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
3		df-scaf 14096	. . . 4 ⊢ ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)))
4		fveq2 5583	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
5		scaffval.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
6	4, 5	eqtr4di 2257	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
7	6	fveq2d 5587	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
8		scaffval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
9	7, 8	eqtr4di 2257	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
10		fveq2 5583	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
11		scaffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
12	10, 11	eqtr4di 2257	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
13		fveq2 5583	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
14		scaffval.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
15	13, 14	eqtr4di 2257	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
16	15	oveqd 5968	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
17	9, 12, 16	mpoeq123dv 6014	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
18		elex 2784	. . . 4 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
19		basfn 12934	. . . . . . 7 ⊢ Base Fn V
20		scaslid 13029	. . . . . . . . 9 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
21	20	slotex 12903	. . . . . . . 8 ⊢ (𝑊 ∈ V → (Scalar‘𝑊) ∈ V)
22	5, 21	eqeltrid 2293	. . . . . . 7 ⊢ (𝑊 ∈ V → 𝐹 ∈ V)
23		funfvex 5600	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐹 ∈ dom Base) → (Base‘𝐹) ∈ V)
24	23	funfni 5381	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐹 ∈ V) → (Base‘𝐹) ∈ V)
25	19, 22, 24	sylancr 414	. . . . . 6 ⊢ (𝑊 ∈ V → (Base‘𝐹) ∈ V)
26	8, 25	eqeltrid 2293	. . . . 5 ⊢ (𝑊 ∈ V → 𝐾 ∈ V)
27		funfvex 5600	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
28	27	funfni 5381	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
29	19, 28	mpan 424	. . . . . 6 ⊢ (𝑊 ∈ V → (Base‘𝑊) ∈ V)
30	11, 29	eqeltrid 2293	. . . . 5 ⊢ (𝑊 ∈ V → 𝐵 ∈ V)
31		mpoexga 6305	. . . . 5 ⊢ ((𝐾 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V)
32	26, 30, 31	syl2anc 411	. . . 4 ⊢ (𝑊 ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V)
33	3, 17, 18, 32	fvmptd3 5680	. . 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 2251	1 ⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2177  Vcvv 2773   Fn wfn 5271  ‘cfv 5276  (class class class)co 5951   ∈ cmpo 5953  Basecbs 12876  Scalarcsca 12956   ·_𝑠 cvsca 12957   ·_sf cscaf 14094

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-ndx 12879  df-slot 12880  df-base 12882  df-sca 12969  df-scaf 14096

This theorem is referenced by:  scafvalg  14113  scafeqg  14114  scaffng  14115  lmodscaf  14116