Theorem slwprm

Description: Reverse closure for the first argument of a Sylow P -subgroup. (Contributed by Mario Carneiro, 16-Jan-2015) (Revised by Mario Carneiro, 2-May-2015)

	Ref	Expression
Assertion	slwprm	⊢ ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) → 𝑃 ∈ ℙ )

Proof

Step	Hyp	Ref	Expression
1		isslw	⊢ ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) ↔ ( 𝑃 ∈ ℙ ∧ 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) )
2	1	simp1bi	⊢ ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) → 𝑃 ∈ ℙ )