proposition: Lemma socle_rsimP (W1 W2 : sG) : reflect (mx_rsim (socle_repr W1) (socle_repr W2)) (W1 == W2).
proof: Proof. have [simW1 simW2] := (socle_simple W1, socle_simple W2). by apply: (iffP (component_mx_isoP simW1 simW2)); move/mx_rsim_iso; apply. 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)