Metamath Proof Explorer


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 ) ( ( 𝐻𝑘𝑃 pGrp ( 𝐺s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) )
2 1 simp2bi ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) → 𝐻 ∈ ( SubGrp ‘ 𝐺 ) )