Metamath Proof Explorer


Theorem cvlatexch3

Description: Atom exchange property. (Contributed by NM, 29-Nov-2012)

Ref Expression
Hypotheses cvlatexch.l = ( le ‘ 𝐾 )
cvlatexch.j = ( join ‘ 𝐾 )
cvlatexch.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion cvlatexch3 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) → ( 𝑃 ( 𝑄 𝑅 ) → ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 cvlatexch.l = ( le ‘ 𝐾 )
2 cvlatexch.j = ( join ‘ 𝐾 )
3 cvlatexch.a 𝐴 = ( Atoms ‘ 𝐾 )
4 simp1 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) → 𝐾 ∈ CvLat )
5 simp21 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) → 𝑃𝐴 )
6 simp23 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) → 𝑅𝐴 )
7 simp22 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) → 𝑄𝐴 )
8 simp3l ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) → 𝑃𝑄 )
9 1 2 3 cvlatexchb1 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑅𝐴𝑄𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑃 ( 𝑄 𝑅 ) ↔ ( 𝑄 𝑃 ) = ( 𝑄 𝑅 ) ) )
10 4 5 6 7 8 9 syl131anc ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) → ( 𝑃 ( 𝑄 𝑅 ) ↔ ( 𝑄 𝑃 ) = ( 𝑄 𝑅 ) ) )
11 10 biimpa ( ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) ∧ 𝑃 ( 𝑄 𝑅 ) ) → ( 𝑄 𝑃 ) = ( 𝑄 𝑅 ) )
12 simpl1 ( ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) ∧ 𝑃 ( 𝑄 𝑅 ) ) → 𝐾 ∈ CvLat )
13 cvllat ( 𝐾 ∈ CvLat → 𝐾 ∈ Lat )
14 12 13 syl ( ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) ∧ 𝑃 ( 𝑄 𝑅 ) ) → 𝐾 ∈ Lat )
15 simpl21 ( ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) ∧ 𝑃 ( 𝑄 𝑅 ) ) → 𝑃𝐴 )
16 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
17 16 3 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
18 15 17 syl ( ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) ∧ 𝑃 ( 𝑄 𝑅 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
19 simpl22 ( ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) ∧ 𝑃 ( 𝑄 𝑅 ) ) → 𝑄𝐴 )
20 16 3 atbase ( 𝑄𝐴𝑄 ∈ ( Base ‘ 𝐾 ) )
21 19 20 syl ( ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) ∧ 𝑃 ( 𝑄 𝑅 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
22 16 2 latjcom ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
23 14 18 21 22 syl3anc ( ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) ∧ 𝑃 ( 𝑄 𝑅 ) ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
24 1 2 3 cvlatexchb2 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 ( 𝑄 𝑅 ) ↔ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) )
25 24 3adant3l ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) → ( 𝑃 ( 𝑄 𝑅 ) ↔ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) )
26 25 biimpa ( ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) ∧ 𝑃 ( 𝑄 𝑅 ) ) → ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) )
27 11 23 26 3eqtr4d ( ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) ∧ 𝑃 ( 𝑄 𝑅 ) ) → ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) )
28 27 ex ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑃𝑄𝑃𝑅 ) ) → ( 𝑃 ( 𝑄 𝑅 ) → ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) )