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Metamath Proof Explorer

Theorem plymulcl

Description: The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014)

Ref Expression
Assertion plymulcl ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐹 ∘f · 𝐺 ) ∈ ( Poly ‘ ℂ ) )

Proof

Step Hyp Ref Expression
1 plyssc ⊢ ( Poly ‘ 𝑆 ) ⊆ ( Poly ‘ ℂ )
2 simpl ⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐹 ∈ ( Poly ‘ 𝑆 ) )
3 1 2 sselid ⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐹 ∈ ( Poly ‘ ℂ ) )
4 simpr ⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐺 ∈ ( Poly ‘ 𝑆 ) )
5 1 4 sselid ⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐺 ∈ ( Poly ‘ ℂ ) )
6 addcl ⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 + 𝑦 ) ∈ ℂ )
7 6 adantl ⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 + 𝑦 ) ∈ ℂ )
8 mulcl ⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 · 𝑦 ) ∈ ℂ )
9 8 adantl ⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 · 𝑦 ) ∈ ℂ )
10 3 5 7 9 plymul ⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐹 ∘f · 𝐺 ) ∈ ( Poly ‘ ℂ ) )