proposition: Lemma linear_irr_comp i : 'n_i = 1 -> (i :=: socle_base i)%MS.
proof: Proof. move=> ni1; apply/eqmxP; rewrite andbC -mxrank_leqif_eq -/'n_i. by rewrite -(mxrankMfree _ gring_free) -genmxE rank_Wedderburn_subring ni1. exact: component_mx_id (socle_simple i). 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)