proposition: Lemma rfix_submod (H : {set gT}) : H \subset G -> (rfix_mx rU H :=: in_submod U (U :&: rfix_mx rG H))%MS.
proof: Proof. move=> sHG; apply/eqmxP/andP; split; last first. apply/rfix_mxP=> x Hx; rewrite -in_submodJ ?capmxSl //. by rewrite (rfix_mxP H _) ?capmxSr. rewrite -val_submodS in_submodK ?capmxSl // sub_capmx val_submodP //=. apply/rfix_mxP=> x Hx. by rewrite -(val_submodJ Umod) ?(subsetP sHG) ?rfix_mx_id. 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)