proposition: Lemma cfConjgIsom phi y : y \in G -> y \in 'N(H) -> (cfIsom isoH phi ^ g y)%CF = cfIsom isoH (phi ^ y).
proof: Proof. move=> Gy nHy; have [_ defS] := isomP isoH. rewrite morphimEdom (eq_in_imset eq_hg) -morphimEsub // in defS. apply/cfun_inP=> gx; rewrite -{1}defS => /morphimP[x Gx Hx ->] {gx}. rewrite cfConjgE; last by rewrite -defS inE -morphimJ ?(normP nHy). by rewrite -morphV -?morphJ -?eq_hg ?cfIsomE ?cfConjgE ?memJ_norm ?groupV. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq path) mathcomp(choice fintype div tuple finfun bigop prime order) mathcomp(ssralg ssrnum finset fingroup morphism perm) mathcomp(automorphism quotient action zmodp cyclic center) mathcomp(gproduct commutator gseries nilpotent pgroup) mathcomp(sylow maximal frobenius matrix mxalgebra) mathcomp(mxrepresentation vector algC classfun character) mathcomp(archimedean)