proposition: Lemma cfQuo_char G H (chi : 'CF(G)) : chi \is a character -> (chi / H)%CF \is a character.
proof: Proof. move=> Nchi; without loss kerH: / H \subset cfker chi. move/contraNF=> IHchi; apply/wlog_neg=> N'chiH. suffices ->: (chi / H)%CF = (chi 1%g)%:A. by rewrite rpredZ_nat ?Cnat_char1 ?rpred1. by apply/cfunP=> x; rewrite cfunE cfun1E mulr_natr cfunElock IHchi. without loss nsHG: G chi Nchi kerH / H <| G. move=> IHchi; have nsHN := normalSG (subset_trans kerH (cfker_sub chi)). by rewrite cfQuoInorm ?(cfRes_char, IHchi) ?sub_cfker_Res // ?normal_sub. have [rG Dchi] := char_reprP Nchi; rewrite Dchi cfker_repr in kerH. apply/char_reprP; exists (Representation (quo_repr kerH (normal_norm nsHG))). apply/cfun_inP=> _ /morphimP[x nHx Gx ->]; rewrite Dchi cfQuoE ?cfker_repr //=. by rewrite !cfunE Gx quo_repr_coset ?mem_quotient. 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)