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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 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 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 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) )
6 4 5 syl3an1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) )