This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 4atexlemqtb

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

Ref Expression
Hypotheses 4thatlem.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
2 4thatlempqb.j ⊢ ∨ = ( join ‘ 𝐾 )
3 4thatlempqb.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 1 4atexlemk ⊢ ( 𝜑 → 𝐾 ∈ HL )
5 1 4atexlemq ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
6 1 4atexlemt ⊢ ( 𝜑 → 𝑇 ∈ 𝐴 )
7 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8 7 2 3 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
9 4 5 6 8 syl3anc ⊢ ( 𝜑 → ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )