proposition: Lemma addsmx_semisimple U V : mxsemisimple U -> mxsemisimple V -> mxsemisimple (U + V)%MS.
proof: Proof. case=> [I W /= simW defU _] [J T /= simT defV _]. have defUV: (\sum_ij sum_rect (fun _ => 'M_n) W T ij :=: U + V)%MS. by rewrite big_sumType /=; apply: adds_eqmx. by apply: intro_mxsemisimple defUV _; case=> /=. 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)