proposition: Lemma mul_mpolyC c p : c%:MP * p = c *: p.
proof: Proof. have [->|nz_c] := eqVneq c 0; first by rewrite scale0r mul0r. apply/mpoly_eqP=> /=; rewrite big_allpairs msuppC (negbTE nz_c) big_seq1. by apply: eq_bigr => i _; rewrite mpolyCK !simpm. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq path) mathcomp(choice fintype tuple finfun bigop finset binomial) mathcomp(order fingroup perm ssralg zmodp poly ssrint) mathcomp(matrix vector) mathcomp(bigenough) ssrcomplements freeg