Metamath Proof Explorer


Theorem ply1vscl

Description: Closure of scalar multiplication for univariate polynomials. (Contributed by SN, 20-May-2025)

Ref Expression
Hypotheses ply1vscl.p 𝑃 = ( Poly1𝑅 )
ply1vscl.b 𝐵 = ( Base ‘ 𝑃 )
ply1vscl.k 𝐾 = ( Base ‘ 𝑅 )
ply1vscl.s · = ( ·𝑠𝑃 )
ply1vscl.r ( 𝜑𝑅 ∈ Ring )
ply1vscl.c ( 𝜑𝐶𝐾 )
ply1vscl.x ( 𝜑𝑋𝐵 )
Assertion ply1vscl ( 𝜑 → ( 𝐶 · 𝑋 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 ply1vscl.p 𝑃 = ( Poly1𝑅 )
2 ply1vscl.b 𝐵 = ( Base ‘ 𝑃 )
3 ply1vscl.k 𝐾 = ( Base ‘ 𝑅 )
4 ply1vscl.s · = ( ·𝑠𝑃 )
5 ply1vscl.r ( 𝜑𝑅 ∈ Ring )
6 ply1vscl.c ( 𝜑𝐶𝐾 )
7 ply1vscl.x ( 𝜑𝑋𝐵 )
8 1 2 ply1bas 𝐵 = ( Base ‘ ( 1o mPoly 𝑅 ) )
9 eqid ( Scalar ‘ ( 1o mPoly 𝑅 ) ) = ( Scalar ‘ ( 1o mPoly 𝑅 ) )
10 eqid ( 1o mPoly 𝑅 ) = ( 1o mPoly 𝑅 )
11 1 10 4 ply1vsca · = ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) )
12 eqid ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) = ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) )
13 1oex 1o ∈ V
14 13 a1i ( 𝜑 → 1o ∈ V )
15 10 14 5 mpllmodd ( 𝜑 → ( 1o mPoly 𝑅 ) ∈ LMod )
16 10 14 5 mplsca ( 𝜑𝑅 = ( Scalar ‘ ( 1o mPoly 𝑅 ) ) )
17 16 fveq2d ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) )
18 3 17 eqtrid ( 𝜑𝐾 = ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) )
19 6 18 eleqtrd ( 𝜑𝐶 ∈ ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) )
20 8 9 11 12 15 19 7 lmodvscld ( 𝜑 → ( 𝐶 · 𝑋 ) ∈ 𝐵 )