proposition: Lemma component_mx_isoP U V : mxsimple U -> mxsimple V -> reflect (mx_iso U V) (component_mx U == component_mx V).
proof: Proof. move=> simU simV; apply: (iffP eqP) => isoUV. by apply: component_mx_iso; rewrite ?isoUV ?component_mx_id. rewrite -(genmx_component U) -(genmx_component V); apply/genmxP. wlog suffices: U V simU simV isoUV / (component_mx U <= component_mx V)%MS. by move=> IH; rewrite !IH //; apply: mx_iso_sym. have [I [W isoWU ->]] := component_mx_def simU. apply/sumsmx_subP => i _; apply: mx_iso_component => //. exact: mx_iso_trans (mx_iso_sym isoUV) (isoWU 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)