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 = ( ·𝑠𝑃 )
mplvsca.k 𝐾 = ( Base ‘ 𝑅 )
mplvsca.b 𝐵 = ( Base ‘ 𝑃 )
mplvsca.m · = ( .r𝑅 )
mplvsca.d 𝐷 = { ∈ ( ℕ0m 𝐼 ) ∣ ( “ ℕ ) ∈ Fin }
mplvsca.x ( 𝜑𝑋𝐾 )
mplvsca.f ( 𝜑𝐹𝐵 )
mplvscaval.y ( 𝜑𝑌𝐷 )
Assertion mplvscaval ( 𝜑 → ( ( 𝑋 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 mplvsca.p 𝑃 = ( 𝐼 mPoly 𝑅 )
2 mplvsca.n = ( ·𝑠𝑃 )
3 mplvsca.k 𝐾 = ( Base ‘ 𝑅 )
4 mplvsca.b 𝐵 = ( Base ‘ 𝑃 )
5 mplvsca.m · = ( .r𝑅 )
6 mplvsca.d 𝐷 = { ∈ ( ℕ0m 𝐼 ) ∣ ( “ ℕ ) ∈ Fin }
7 mplvsca.x ( 𝜑𝑋𝐾 )
8 mplvsca.f ( 𝜑𝐹𝐵 )
9 mplvscaval.y ( 𝜑𝑌𝐷 )
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 ( ℕ0m 𝐼 ) ∈ V
13 6 12 rabex2 𝐷 ∈ V
14 13 a1i ( 𝜑𝐷 ∈ V )
15 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
16 1 15 4 6 8 mplelf ( 𝜑𝐹 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
17 16 ffnd ( 𝜑𝐹 Fn 𝐷 )
18 eqidd ( ( 𝜑𝑌𝐷 ) → ( 𝐹𝑌 ) = ( 𝐹𝑌 ) )
19 14 7 17 18 ofc1 ( ( 𝜑𝑌𝐷 ) → ( ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹𝑌 ) ) )
20 9 19 mpdan ( 𝜑 → ( ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹𝑌 ) ) )
21 11 20 eqtrd ( 𝜑 → ( ( 𝑋 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹𝑌 ) ) )