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 H ⊆ k ∧ P pGrp G ↾ 𝑠 k ↔ H = k
2 1 simp2bi ⊢ H ∈ P pSyl G → H ∈ SubGrp ⁡ G