Theorem plysubcl

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

	Ref	Expression
Assertion	plysubcl	⊢ ( ( 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		neg1cn	⊢ - 1 ∈ ℂ
11	10	a1i	⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → - 1 ∈ ℂ )
12	3 5 7 9 11	plysub	⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐹 ∘f − 𝐺 ) ∈ ( Poly ‘ ℂ ) )