Theorem hlatexch2

Description: Atom exchange property. (Contributed by NM, 8-Jan-2012)

	Ref	Expression
Hypotheses	hlatexchb.l	⊢ ≤ = ( le ‘ 𝐾 )
	hlatexchb.j	⊢ ∨ = ( join ‘ 𝐾 )
	hlatexchb.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	hlatexch2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) → 𝑄 ≤ ( 𝑃 ∨ 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hlatexchb.l	⊢ ≤ = ( le ‘ 𝐾 )
2		hlatexchb.j	⊢ ∨ = ( join ‘ 𝐾 )
3		hlatexchb.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		hlcvl	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CvLat )
5	1 2 3	cvlatexch2	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) → 𝑄 ≤ ( 𝑃 ∨ 𝑅 ) ) )
6	4 5	syl3an1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) → 𝑄 ≤ ( 𝑃 ∨ 𝑅 ) ) )