Theorem dgradd

Description: The degree of a sum of polynomials is at most the maximum of the 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	dgradd	$$⊢\left(F\in \mathrm{Poly}\left(S\right)\wedge G\in \mathrm{Poly}\left(S\right)\right)\to \mathrm{deg}\left(F{+}_{f}G\right)\le if\left(M\le N,N,M\right)$$

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	coeaddlem	$$⊢\left(F\in \mathrm{Poly}\left(S\right)\wedge G\in \mathrm{Poly}\left(S\right)\right)\to \left(\mathrm{coeff}\left(F{+}_{f}G\right)=\mathrm{coeff}\left(F\right){+}_f\mathrm{coeff}\left(G\right)\wedge \mathrm{deg}\left(F{+}_{f}G\right)\le if\left(M\le N,N,M\right)\right)$$
6	5	simprd	$$⊢\left(F\in \mathrm{Poly}\left(S\right)\wedge G\in \mathrm{Poly}\left(S\right)\right)\to \mathrm{deg}\left(F{+}_{f}G\right)\le if\left(M\le N,N,M\right)$$