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Metamath Proof Explorer

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 sselid ⊢ ( 𝜑 → 𝐹 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) )
14 9 10 3 11 5 6 7 13 psrvsca ⊢ ( 𝜑 → ( 𝑋 ∙ 𝐹 ) = ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) )