proposition: Lemma regular_op_inj : {in [pred A | (A \in 'R_iG)%MS] &, injective (gring_op rG)}.
proof: Proof. move=> A B RnA RnB /= eqAB; apply/eqP; rewrite -subr_eq0 -mxvec_eq0 -submx0. rewrite -ker_irr_comp_op sub_capmx (sameP sub_kermxP eqP) mul_vec_lin. by rewrite 2!linearB /= eqAB subrr linear0 addmx_sub ?eqmx_opp /=. 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)