Metamath Proof Explorer


Theorem hlatexchb2

Description: A version of hlexchb2 for atoms. (Contributed by NM, 7-Feb-2012)

Ref Expression
Hypotheses hlatexchb.l = ( le ‘ 𝐾 )
hlatexchb.j = ( join ‘ 𝐾 )
hlatexchb.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion hlatexchb2 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 ( 𝑄 𝑅 ) ↔ ( 𝑃 𝑅 ) = ( 𝑄 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 cvlatexchb2 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 ( 𝑄 𝑅 ) ↔ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) )
6 4 5 syl3an1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ 𝑃𝑅 ) → ( 𝑃 ( 𝑄 𝑅 ) ↔ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) )