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Metamath Proof Explorer

Theorem hlexchb2

Description: A Hilbert lattice has the exchange property. (Contributed by NM, 22-Jun-2012)

Ref Expression
Hypotheses hlsuprexch.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
hlsuprexch.l ⊢ ≤ = ( le ‘ 𝐾 )
hlsuprexch.j ⊢ ∨ = ( join ‘ 𝐾 )
hlsuprexch.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion hlexchb2 ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ 𝑃 ≤ 𝑋 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑋 ) ↔ ( 𝑃 ∨ 𝑋 ) = ( 𝑄 ∨ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 hlsuprexch.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 hlsuprexch.l ⊢ ≤ = ( le ‘ 𝐾 )
3 hlsuprexch.j ⊢ ∨ = ( join ‘ 𝐾 )
4 hlsuprexch.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 hlcvl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CvLat )
6 1 2 3 4 cvlexchb2 ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ 𝑃 ≤ 𝑋 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑋 ) ↔ ( 𝑃 ∨ 𝑋 ) = ( 𝑄 ∨ 𝑋 ) ) )
7 5 6 syl3an1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ 𝑃 ≤ 𝑋 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑋 ) ↔ ( 𝑃 ∨ 𝑋 ) = ( 𝑄 ∨ 𝑋 ) ) )