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Metamath Proof Explorer

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 ) ) )