Theorem cvlatexch1

Description: Atom exchange property. (Contributed by NM, 5-Nov-2012)

	Ref	Expression
Hypotheses	cvlatexch.l	⊢ ≤ = ( le ‘ 𝐾 )
	cvlatexch.j	⊢ ∨ = ( join ‘ 𝐾 )
	cvlatexch.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	cvlatexch1	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) → 𝑄 ≤ ( 𝑅 ∨ 𝑃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cvlatexch.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cvlatexch.j	⊢ ∨ = ( join ‘ 𝐾 )
3		cvlatexch.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4	1 2 3	cvlatexchb1	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ↔ ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) ) )
5		cvllat	⊢ ( 𝐾 ∈ CvLat → 𝐾 ∈ Lat )
6	5	3ad2ant1	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → 𝐾 ∈ Lat )
7		simp23	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → 𝑅 ∈ 𝐴 )
8		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
9	8 3	atbase	⊢ ( 𝑅 ∈ 𝐴 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
10	7 9	syl	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
11		simp22	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → 𝑄 ∈ 𝐴 )
12	8 3	atbase	⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
13	11 12	syl	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
14	8 1 2	latlej2	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → 𝑄 ≤ ( 𝑅 ∨ 𝑄 ) )
15	6 10 13 14	syl3anc	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → 𝑄 ≤ ( 𝑅 ∨ 𝑄 ) )
16		breq2	⊢ ( ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) → ( 𝑄 ≤ ( 𝑅 ∨ 𝑃 ) ↔ 𝑄 ≤ ( 𝑅 ∨ 𝑄 ) ) )
17	15 16	syl5ibrcom	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) → 𝑄 ≤ ( 𝑅 ∨ 𝑃 ) ) )
18	4 17	sylbid	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) → 𝑄 ≤ ( 𝑅 ∨ 𝑃 ) ) )