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	$$⊢M=\mathrm{deg}\left(F\right)$$
	dgradd.2	$$⊢N=\mathrm{deg}\left(G\right)$$
Assertion	dgrmul2	$$⊢\left(F\in \mathrm{Poly}\left(S\right)\wedge G\in \mathrm{Poly}\left(S\right)\right)\to \mathrm{deg}\left(F{\times }_{f}G\right)\le M+N$$

Proof

Step	Hyp	Ref	Expression
1		dgradd.1	$$⊢M=\mathrm{deg}\left(F\right)$$
2		dgradd.2	$$⊢N=\mathrm{deg}\left(G\right)$$
3		eqid	$$⊢\mathrm{coeff}\left(F\right)=\mathrm{coeff}\left(F\right)$$
4		eqid	$$⊢\mathrm{coeff}\left(G\right)=\mathrm{coeff}\left(G\right)$$
5	3 4 1 2	coemullem	$$⊢\left(F\in \mathrm{Poly}\left(S\right)\wedge G\in \mathrm{Poly}\left(S\right)\right)\to \left(\mathrm{coeff}\left(F{\times }_{f}G\right)=\left(n\in {\mathbb{N}}_0⟼\sum _{k=0}^n\mathrm{coeff}\left(F\right)\left(k\right)\mathrm{coeff}\left(G\right)\left(n-k\right)\right)\wedge \mathrm{deg}\left(F{\times }_{f}G\right)\le M+N\right)$$
6	5	simprd	$$⊢\left(F\in \mathrm{Poly}\left(S\right)\wedge G\in \mathrm{Poly}\left(S\right)\right)\to \mathrm{deg}\left(F{\times }_{f}G\right)\le M+N$$