Section LowerCentral.
Variable gT : finGroupType.
Implicit Types (A B : {set gT}) (G H : {group gT}).

Lemma bigdprod_nil I r (P : pred I) (A_ : I -> {set gT}) G :
\big[dprod/1]_(i <- r | P i) A_ i = G
-> (forall i, P i -> nilpotent (A_ i)) -> nilpotent G.