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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 = ( Poly1 ‘ 𝑅 )
ply1vscl.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
ply1vscl.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
ply1vscl.s ⊢ · = ( ·𝑠 ‘ 𝑃 )
ply1vscl.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
ply1vscl.c ⊢ ( 𝜑 → 𝐶 ∈ 𝐾 )
ply1vscl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion ply1vscl ( 𝜑 → ( 𝐶 · 𝑋 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 ply1vscl.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2 ply1vscl.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
3 ply1vscl.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
4 ply1vscl.s ⊢ · = ( ·𝑠 ‘ 𝑃 )
5 ply1vscl.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
6 ply1vscl.c ⊢ ( 𝜑 → 𝐶 ∈ 𝐾 )
7 ply1vscl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
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 ⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ ( 1o mPoly 𝑅 ) ) )
17 16 fveq2d ⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) )
18 3 17 eqtrid ⊢ ( 𝜑 → 𝐾 = ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) )
19 6 18 eleqtrd ⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) )
20 8 9 11 12 15 19 7 lmodvscld ⊢ ( 𝜑 → ( 𝐶 · 𝑋 ) ∈ 𝐵 )