Theorem 4atexlempsb

Description: Lemma for 4atexlem7 . (Contributed by NM, 23-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	4atexlempsb	⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ∈ ( 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	4atexlemp	⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
6	1	4atexlems	⊢ ( 𝜑 → 𝑆 ∈ 𝐴 )
7		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8	7 2 3	hlatjcl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
9	4 5 6 8	syl3anc	⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )