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Metamath Proof Explorer

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