Theorem cvlatexchb1

Description: A version of cvlexchb1 for atoms. (Contributed by NM, 5-Nov-2012)

	Ref	Expression
Hypotheses	cvlatexch.l	⊢ ≤ = ( le ‘ 𝐾 )
	cvlatexch.j	⊢ ∨ = ( join ‘ 𝐾 )
	cvlatexch.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	cvlatexchb1	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ↔ ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cvlatexch.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cvlatexch.j	⊢ ∨ = ( join ‘ 𝐾 )
3		cvlatexch.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		cvlatl	⊢ ( 𝐾 ∈ CvLat → 𝐾 ∈ AtLat )
5	4	adantr	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝐾 ∈ AtLat )
6		simpr1	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 )
7		simpr3	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑅 ∈ 𝐴 )
8	1 3	atncmp	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( ¬ 𝑃 ≤ 𝑅 ↔ 𝑃 ≠ 𝑅 ) )
9	5 6 7 8	syl3anc	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ¬ 𝑃 ≤ 𝑅 ↔ 𝑃 ≠ 𝑅 ) )
10		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
11	10 3	atbase	⊢ ( 𝑅 ∈ 𝐴 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
12	10 1 2 3	cvlexchb1	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ 𝑃 ≤ 𝑅 ) → ( 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ↔ ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) ) )
13	12	3expia	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) ) → ( ¬ 𝑃 ≤ 𝑅 → ( 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ↔ ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) ) ) )
14	11 13	syl3anr3	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ¬ 𝑃 ≤ 𝑅 → ( 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ↔ ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) ) ) )
15	9 14	sylbird	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( 𝑃 ≠ 𝑅 → ( 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ↔ ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) ) ) )
16	15	3impia	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ↔ ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) ) )