Theorem slwpgp

Description: A Sylow P -subgroup is a P -group. (Contributed by Mario Carneiro, 16-Jan-2015)

	Ref	Expression
Hypothesis	slwpgp.1	|- S = ( G |`s H )
Assertion	slwpgp	|- ( H e. ( P pSyl G ) -> P pGrp S )

Proof

Step	Hyp	Ref	Expression
1		slwpgp.1	|- S = ( G |`s H )
2		eqid	|- H = H
3		slwsubg	|- ( H e. ( P pSyl G ) -> H e. ( SubGrp ` G ) )
4	1	slwispgp	|- ( ( H e. ( P pSyl G ) /\ H e. ( SubGrp ` G ) ) -> ( ( H C_ H /\ P pGrp S ) <-> H = H ) )
5	3 4	mpdan	|- ( H e. ( P pSyl G ) -> ( ( H C_ H /\ P pGrp S ) <-> H = H ) )
6	2 5	mpbiri	|- ( H e. ( P pSyl G ) -> ( H C_ H /\ P pGrp S ) )
7	6	simprd	|- ( H e. ( P pSyl G ) -> P pGrp S )