proposition: Lemma sumsmx_subP P m n (A_ : I -> 'M_n) (B : 'M_(m, n)) : reflect (forall i, P i -> A_ i <= B)%MS (\sum_(i | P i) A_ i <= B)%MS.
proof: Proof. apply: (iffP idP) => [sAB i Pi | sAB]. by apply: submx_trans sAB; apply: sumsmx_sup Pi _. by elim/big_rec: _ => [|i Ai Pi sAiB]; rewrite ?sub0mx // addsmx_sub sAB. 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)