proposition: Lemma sdprod_modl A B G H : A ><| B = G -> A \subset H -> A ><| (B :&: H) = G :&: H.
proof: Proof. case/sdprodP=> {A B} [[A B -> ->]] <- nAB tiAB sAH. rewrite -group_modl ?sdprodE ?subIset ?nAB //. by rewrite setIA tiAB (setIidPl _) ?sub1G. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq div) mathcomp(choice fintype bigop finset fingroup morphism) mathcomp(quotient action)