Theorem mplvscaval

Description: The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015)

	Ref	Expression
Hypotheses	mplvsca.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
	mplvsca.n	⊢ ∙ = ( ·𝑠 ‘ 𝑃 )
	mplvsca.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	mplvsca.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
	mplvsca.m	⊢ · = ( .r ‘ 𝑅 )
	mplvsca.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
	mplvsca.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
	mplvsca.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
	mplvscaval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐷 )
Assertion	mplvscaval	⊢ ( 𝜑 → ( ( 𝑋 ∙ 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹 ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mplvsca.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
2		mplvsca.n	⊢ ∙ = ( ·𝑠 ‘ 𝑃 )
3		mplvsca.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
4		mplvsca.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
5		mplvsca.m	⊢ · = ( .r ‘ 𝑅 )
6		mplvsca.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
7		mplvsca.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
8		mplvsca.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
9		mplvscaval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐷 )
10	1 2 3 4 5 6 7 8	mplvsca	⊢ ( 𝜑 → ( 𝑋 ∙ 𝐹 ) = ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) )
11	10	fveq1d	⊢ ( 𝜑 → ( ( 𝑋 ∙ 𝐹 ) ‘ 𝑌 ) = ( ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) ‘ 𝑌 ) )
12		ovex	⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
13	6 12	rabex2	⊢ 𝐷 ∈ V
14	13	a1i	⊢ ( 𝜑 → 𝐷 ∈ V )
15		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
16	1 15 4 6 8	mplelf	⊢ ( 𝜑 → 𝐹 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
17	16	ffnd	⊢ ( 𝜑 → 𝐹 Fn 𝐷 )
18		eqidd	⊢ ( ( 𝜑 ∧ 𝑌 ∈ 𝐷 ) → ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑌 ) )
19	14 7 17 18	ofc1	⊢ ( ( 𝜑 ∧ 𝑌 ∈ 𝐷 ) → ( ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹 ‘ 𝑌 ) ) )
20	9 19	mpdan	⊢ ( 𝜑 → ( ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹 ‘ 𝑌 ) ) )
21	11 20	eqtrd	⊢ ( 𝜑 → ( ( 𝑋 ∙ 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹 ‘ 𝑌 ) ) )