proposition: Lemma cfIndEsdprod (phi : 'CF(K)) x : K ><| H = G -> 'Ind[G] phi x = \sum_(w in H) phi (x ^ w)%g.
proof: Proof. move=> defG; have [/andP[sKG _] _ mulKH nKH _] := sdprod_context defG. rewrite cfIndE //; apply: canLR (mulKf (neq0CG _)) _; rewrite -mulKH mulr_sumr. rewrite (set_partition_big _ (rcosets_partition_mul H K)) ?big_imset /=. apply: eq_bigr => y Hy; rewrite rcosetE norm_rlcoset ?(subsetP nKH) //. rewrite -lcosetE mulr_natl big_imset /=; last exact: in2W (mulgI _). by rewrite -sumr_const; apply: eq_bigr => z Kz; rewrite conjgM cfunJ. have [{}nKH /isomP[injf _]] := sdprod_isom defG. apply: can_in_inj (fun Ky => invm injf (coset K (repr Ky))) _ => y Hy. by rewrite rcosetE -val_coset ?(subsetP nKH) // coset_reprK invmE. 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)