Theorem scaffvalg 13872

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 2774	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
3		df-scaf 13856	. . . 4 ⊢ ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)))
4		fveq2 5559	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
5		scaffval.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
6	4, 5	eqtr4di 2247	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
7	6	fveq2d 5563	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
8		scaffval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
9	7, 8	eqtr4di 2247	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
10		fveq2 5559	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
11		scaffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
12	10, 11	eqtr4di 2247	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
13		fveq2 5559	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
14		scaffval.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
15	13, 14	eqtr4di 2247	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
16	15	oveqd 5940	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
17	9, 12, 16	mpoeq123dv 5985	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
18		elex 2774	. . . 4 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
19		basfn 12746	. . . . . . 7 ⊢ Base Fn V
20		scaslid 12840	. . . . . . . . 9 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
21	20	slotex 12715	. . . . . . . 8 ⊢ (𝑊 ∈ V → (Scalar‘𝑊) ∈ V)
22	5, 21	eqeltrid 2283	. . . . . . 7 ⊢ (𝑊 ∈ V → 𝐹 ∈ V)
23		funfvex 5576	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐹 ∈ dom Base) → (Base‘𝐹) ∈ V)
24	23	funfni 5359	. . . . . . 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 2283	. . . . 5 ⊢ (𝑊 ∈ V → 𝐾 ∈ V)
27		funfvex 5576	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
28	27	funfni 5359	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
29	19, 28	mpan 424	. . . . . 6 ⊢ (𝑊 ∈ V → (Base‘𝑊) ∈ V)
30	11, 29	eqeltrid 2283	. . . . 5 ⊢ (𝑊 ∈ V → 𝐵 ∈ V)
31		mpoexga 6271	. . . . 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 5656	. . 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 2241	1 ⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  Vcvv 2763   Fn wfn 5254  ‘cfv 5259  (class class class)co 5923   ∈ cmpo 5925  Basecbs 12688  Scalarcsca 12768   ·_𝑠 cvsca 12769   ·_sf cscaf 13854

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7972  ax-resscn 7973  ax-1re 7975  ax-addrcl 7978

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-inn 8993  df-2 9051  df-3 9052  df-4 9053  df-5 9054  df-ndx 12691  df-slot 12692  df-base 12694  df-sca 12781  df-scaf 13856

This theorem is referenced by:  scafvalg  13873  scafeqg  13874  scaffng  13875  lmodscaf  13876