proposition: Lemma mxrank_leqif_sup m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) : (A <= B)%MS -> \rank A <= \rank B ?= iff (B <= A)%MS.
proof: Proof. move=> sAB; split; first by rewrite mxrankS. apply/idP/idP=> [| sBA]; last by rewrite eqn_leq !mxrankS. case/submxP: sAB => D ->; set r := \rank B; rewrite -(mulmx_base B) mulmxA. rewrite mxrankMfree // => /row_fullP[E kE]. by rewrite -[rB in _ *m rB]mul1mx -kE -(mulmxA E) (mulmxA _ E) submxMl. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq choice) mathcomp(fintype finfun bigop finset fingroup perm order) mathcomp(div prime binomial ssralg finalg zmodp matrix)