proposition: Lemma Wedderburn_annihilate i j : i != j -> ('R_i * 'R_j)%MS = 0.
proof: Proof. move=> ne_ij; apply/eqP; rewrite -submx0 -(Wedderburn_disjoint ne_ij). rewrite sub_capmx; apply/andP; split. case/andP: (Wedderburn_ideal i) => _; apply: submx_trans. by rewrite mulsmxS // genmxE submxMl. case/andP: (Wedderburn_ideal j) => idlRj _; apply: submx_trans idlRj. by rewrite mulsmxS // genmxE submxMl. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path) mathcomp(div choice fintype tuple finfun bigop prime) mathcomp(ssralg poly polydiv finset fingroup morphism) mathcomp(perm automorphism quotient finalg action zmodp) mathcomp(commutator cyclic center pgroup matrix mxalgebra) mathcomp(mxpoly)