proposition: Lemma cap_cfker_normal G H : H <| G -> \bigcap_(i | H \subset cfker 'chi[G]_i) (cfker 'chi_i) = H.
proof: Proof. move=> nsHG; have [sHG nHG] := andP nsHG; set lhs := \bigcap_(i | _) _. have nHlhs: lhs \subset 'N(H) by rewrite (bigcap_min 0) ?cfker_irr0. apply/esym/eqP; rewrite eqEsubset (introT bigcapsP) //= -quotient_sub1 //. rewrite -(TI_cfker_irr (G / H)); apply/bigcapsP=> i _. rewrite sub_quotient_pre // (bigcap_min (mod_Iirr i)) ?mod_IirrE ?cfker_mod //. by rewrite cfker_morph ?subsetIr. 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)