Metamath Proof Explorer


Theorem cvlatexchb2

Description: A version of cvlexchb2 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 cvlatexchb2 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 ( 𝑄 𝑅 ) ↔ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) )

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 ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 ( 𝑅 𝑄 ) ↔ ( 𝑅 𝑃 ) = ( 𝑅 𝑄 ) ) )
5 cvllat ( 𝐾 ∈ CvLat → 𝐾 ∈ Lat )
6 5 3ad2ant1 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → 𝐾 ∈ Lat )
7 simp22 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → 𝑄𝐴 )
8 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
9 8 3 atbase ( 𝑄𝐴𝑄 ∈ ( Base ‘ 𝐾 ) )
10 7 9 syl ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
11 simp23 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → 𝑅𝐴 )
12 8 3 atbase ( 𝑅𝐴𝑅 ∈ ( Base ‘ 𝐾 ) )
13 11 12 syl ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
14 8 2 latjcom ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑄 𝑅 ) = ( 𝑅 𝑄 ) )
15 6 10 13 14 syl3anc ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑄 𝑅 ) = ( 𝑅 𝑄 ) )
16 15 breq2d ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 ( 𝑄 𝑅 ) ↔ 𝑃 ( 𝑅 𝑄 ) ) )
17 simp21 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → 𝑃𝐴 )
18 8 3 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
19 17 18 syl ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
20 8 2 latjcom ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 𝑅 ) = ( 𝑅 𝑃 ) )
21 6 19 13 20 syl3anc ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 𝑅 ) = ( 𝑅 𝑃 ) )
22 21 15 eqeq12d ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ↔ ( 𝑅 𝑃 ) = ( 𝑅 𝑄 ) ) )
23 4 16 22 3bitr4d ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 ( 𝑄 𝑅 ) ↔ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) )