Metamath Proof Explorer


Theorem hlatexch1

Description: Atom exchange property. (Contributed by NM, 7-Jan-2012)

Ref Expression
Hypotheses hlatexchb.l = ( le ‘ 𝐾 )
hlatexchb.j = ( join ‘ 𝐾 )
hlatexchb.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion hlatexch1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 ( 𝑅 𝑄 ) → 𝑄 ( 𝑅 𝑃 ) ) )

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 cvlatexch1 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 ( 𝑅 𝑄 ) → 𝑄 ( 𝑅 𝑃 ) ) )
6 4 5 syl3an1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 ( 𝑅 𝑄 ) → 𝑄 ( 𝑅 𝑃 ) ) )