proposition: Lemma sdprod_cfker phi : K ><| cfker phi = cfker (cfSdprod phi).
proof: Proof. have [skerH [_ _ nKH tiKH]] := (cfker_sub phi, sdprodP defG). rewrite unlock cfker_morph ?normal_norm // cfker_isom restrmEsub //=. rewrite -(sdprod_modl defG) ?sub_cosetpre //=; congr (_ ><| _). by rewrite quotientK ?(subset_trans skerH) // -group_modr //= setIC tiKH mul1g. 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)