Theorem mplvsca

Description: The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015) (Revised by Mario Carneiro, 2-Oct-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 )
Assertion	mplvsca	⊢ ( 𝜑 → ( 𝑋 ∙ 𝐹 ) = ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) )

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		eqid	⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 )
10	1 9 2	mplvsca2	⊢ ∙ = ( ·𝑠 ‘ ( 𝐼 mPwSer 𝑅 ) )
11		eqid	⊢ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑅 ) )
12	1 9 4 11	mplbasss	⊢ 𝐵 ⊆ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) )
13	12 8	sseldi	⊢ ( 𝜑 → 𝐹 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) )
14	9 10 3 11 5 6 7 13	psrvsca	⊢ ( 𝜑 → ( 𝑋 ∙ 𝐹 ) = ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) )