Metamath Proof Explorer


Theorem 4atexlemqtb

Description: Lemma for 4atexlem7 . (Contributed by NM, 24-Nov-2012)

Ref Expression
Hypotheses 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 ) ) ) )
4thatlempqb.j = ( join ‘ 𝐾 )
4thatlempqb.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion 4atexlemqtb ( 𝜑 → ( 𝑄 𝑇 ) ∈ ( Base ‘ 𝐾 ) )

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 4thatlempqb.j = ( join ‘ 𝐾 )
3 4thatlempqb.a 𝐴 = ( Atoms ‘ 𝐾 )
4 1 4atexlemk ( 𝜑𝐾 ∈ HL )
5 1 4atexlemq ( 𝜑𝑄𝐴 )
6 1 4atexlemt ( 𝜑𝑇𝐴 )
7 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8 7 2 3 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴 ) → ( 𝑄 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
9 4 5 6 8 syl3anc ( 𝜑 → ( 𝑄 𝑇 ) ∈ ( Base ‘ 𝐾 ) )