Theorem dgrmul2

Description: The degree of a product of polynomials is at most the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014)

	Ref	Expression
Hypotheses	dgradd.1	⊢ 𝑀 = ( deg ‘ 𝐹 )
	dgradd.2	⊢ 𝑁 = ( deg ‘ 𝐺 )
Assertion	dgrmul2	⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( deg ‘ ( 𝐹 ∘f · 𝐺 ) ) ≤ ( 𝑀 + 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		dgradd.1	⊢ 𝑀 = ( deg ‘ 𝐹 )
2		dgradd.2	⊢ 𝑁 = ( deg ‘ 𝐺 )
3		eqid	⊢ ( coeff ‘ 𝐹 ) = ( coeff ‘ 𝐹 )
4		eqid	⊢ ( coeff ‘ 𝐺 ) = ( coeff ‘ 𝐺 )
5	3 4 1 2	coemullem	⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( ( coeff ‘ ( 𝐹 ∘f · 𝐺 ) ) = ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) · ( ( coeff ‘ 𝐺 ) ‘ ( 𝑛 − 𝑘 ) ) ) ) ∧ ( deg ‘ ( 𝐹 ∘f · 𝐺 ) ) ≤ ( 𝑀 + 𝑁 ) ) )
6	5	simprd	⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( deg ‘ ( 𝐹 ∘f · 𝐺 ) ) ≤ ( 𝑀 + 𝑁 ) )