Theorem slwsubg

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

	Ref	Expression
Assertion	slwsubg	⊢ ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) → 𝐻 ∈ ( SubGrp ‘ 𝐺 ) )

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	simp2bi	⊢ ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) → 𝐻 ∈ ( SubGrp ‘ 𝐺 ) )