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 ) ) |
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 ) ) |