proposition: Lemma cfIndIsom phi : 'Ind[R] (cfIsom isoH phi) = cfIsom isoG ('Ind[G] phi).
proof: Proof. have [[injg defR] [_ defS]] := (isomP isoG, isomP isoH). rewrite morphimEdom (eq_in_imset eq_hg) -morphimEsub // in defS. apply/cfun_inP=> s; rewrite -{1}defR => /morphimP[x _ Gx ->]{s}. rewrite cfIsomE ?cfIndE // -defR -{1}defS ?morphimS ?card_injm // morphimEdom. congr (_ * _); rewrite big_imset //=; last exact/injmP. apply: eq_bigr => y Gy; rewrite -morphJ //. have [Hxy | H'xy] := boolP (x ^ y \in H); first by rewrite -eq_hg ?cfIsomE. by rewrite !cfun0 -?defS // -sub1set -morphim_set1 ?injmSK ?sub1set // groupJ. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path) mathcomp(div choice fintype tuple finfun bigop prime order) mathcomp(ssralg poly finset fingroup morphism perm) mathcomp(automorphism quotient finalg action gproduct) mathcomp(zmodp commutator cyclic center pgroup sylow) mathcomp(matrix vector falgebra ssrnum algC algnum) mathcomp(archimedean)