proposition: Lemma p_rank_gt0 p H : ('r_p(H) > 0) = (p \in \pi(H)).
proof: Proof. rewrite mem_primes cardG_gt0 /=; apply/p_rank_geP/andP=> [[E] | [p_pr]]. case/pnElemP=> sEG _; rewrite lognE; case: and3P => // [[-> _ pE] _]. by rewrite (dvdn_trans _ (cardSg sEG)). case/Cauchy=> // x Hx ox; exists <[x]>%G; rewrite 2!inE [#|_|]ox cycle_subG. by rewrite Hx (pfactorK 1) ?abelemE // cycle_abelian -ox exponent_dvdn. 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)