Metamath Proof Explorer


Theorem 4atexlemk

Description: Lemma for 4atexlem7 . (Contributed by NM, 23-Nov-2012)

Ref Expression
Hypothesis 4thatlem.ph ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) ∧ ( 𝑇𝐴 ∧ ( 𝑈 𝑇 ) = ( 𝑉 𝑇 ) ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) )
Assertion 4atexlemk ( 𝜑𝐾 ∈ HL )

Proof

Step Hyp Ref Expression
1 4thatlem.ph ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) ∧ ( 𝑇𝐴 ∧ ( 𝑈 𝑇 ) = ( 𝑉 𝑇 ) ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) )
2 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) ∧ ( 𝑇𝐴 ∧ ( 𝑈 𝑇 ) = ( 𝑉 𝑇 ) ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ HL )
3 1 2 sylbi ( 𝜑𝐾 ∈ HL )