proposition: Lemma subid_scaleid (x: R) m n (I: 'cV[R]_m) (J: 'cV[R]_n): x != 0 -> (I <= J)%IS = (x *: I <= x *: J)%IS.
proof: Proof. move=> x0. apply/idP/idP; case/subidP => D hD; apply/subidP; exists D. by rewrite hD scalemxAr. by apply/eqP; rewrite -(inj_eq (scalemx_inj x0)) scalemxAr hD. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat div seq path) mathcomp(ssralg fintype perm choice fingroup) mathcomp(matrix bigop zmodp mxalgebra poly) ssrcomplements