proposition: Lemma fful_lin_char_inj : cfaithful xi -> {in G &, injective xi}.
proof: Proof. move=> fful_phi x y Gx Gy xi_xy; apply/eqP; rewrite eq_mulgV1 -in_set1. rewrite (subsetP fful_phi) // inE groupM ?groupV //=; apply/forallP=> z. have [Gz | G'z] := boolP (z \in G); last by rewrite !cfun0 ?groupMl ?groupV. by rewrite -mulgA lin_charM ?xi_xy -?lin_charM ?groupM ?groupV // mulKVg. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype choice ssrnat seq) mathcomp(path div fintype tuple finfun bigop prime order) mathcomp(ssralg poly finset gproduct fingroup morphism) mathcomp(perm automorphism quotient finalg action zmodp) mathcomp(commutator cyclic center pgroup nilpotent sylow) mathcomp(abelian matrix mxalgebra mxpoly mxrepresentation) mathcomp(vector ssrnum algC classfun archimedean)