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Metamath Proof Explorer

Theorem slwispgp

Description: Defining property of a Sylow P -subgroup. (Contributed by Mario Carneiro, 16-Jan-2015)

Ref Expression
Hypothesis slwispgp.1 ⊢ 𝑆 = ( 𝐺 ↾s 𝐾 )
Assertion slwispgp ( ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆 ) ↔ 𝐻 = 𝐾 ) )

Proof

Step Hyp Ref Expression
1 slwispgp.1 ⊢ 𝑆 = ( 𝐺 ↾s 𝐾 )
2 isslw ⊢ ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) ↔ ( 𝑃 ∈ ℙ ∧ 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) )
3 2 simp3bi ⊢ ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) → ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) )
4 sseq2 ⊢ ( 𝑘 = 𝐾 → ( 𝐻 ⊆ 𝑘 ↔ 𝐻 ⊆ 𝐾 ) )
5 oveq2 ⊢ ( 𝑘 = 𝐾 → ( 𝐺 ↾s 𝑘 ) = ( 𝐺 ↾s 𝐾 ) )
6 5 1 eqtr4di ⊢ ( 𝑘 = 𝐾 → ( 𝐺 ↾s 𝑘 ) = 𝑆 )
7 6 breq2d ⊢ ( 𝑘 = 𝐾 → ( 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ↔ 𝑃 pGrp 𝑆 ) )
8 4 7 anbi12d ⊢ ( 𝑘 = 𝐾 → ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ ( 𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆 ) ) )
9 eqeq2 ⊢ ( 𝑘 = 𝐾 → ( 𝐻 = 𝑘 ↔ 𝐻 = 𝐾 ) )
10 8 9 bibi12d ⊢ ( 𝑘 = 𝐾 → ( ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ↔ ( ( 𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆 ) ↔ 𝐻 = 𝐾 ) ) )
11 10 rspccva ⊢ ( ( ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ∧ 𝐾 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆 ) ↔ 𝐻 = 𝐾 ) )
12 3 11 sylan ⊢ ( ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆 ) ↔ 𝐻 = 𝐾 ) )