proposition: Lemma rank_geP n G : reflect (exists E, E \in 'E^n(G)) (n <= 'r(G)).
proof: Proof. apply: (iffP idP) => [|[E]]. have [p _ ->] := rank_witness G; case/p_rank_geP=> E. by rewrite def_pnElem; case/setIP; exists E. case/nElemP=> p /[1!inE] /andP[EpG_E /eqP <-]. by rewrite (leq_trans (logn_le_p_rank EpG_E)) ?p_rank_le_rank. Qed.
imports: mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path choice) mathcomp(div fintype finfun bigop finset prime binomial) mathcomp(fingroup morphism perm automorphism action) mathcomp(quotient gfunctor gproduct ssralg countalg finalg zmodp) mathcomp(cyclic pgroup gseries nilpotent sylow)