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 ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ¬ 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 ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ¬ 𝑃 𝑋 ) → ( 𝑃 ( 𝑄 𝑋 ) ↔ ( 𝑃 𝑋 ) = ( 𝑄 𝑋 ) ) )
7 5 6 syl3an1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ¬ 𝑃 𝑋 ) → ( 𝑃 ( 𝑄 𝑋 ) ↔ ( 𝑃 𝑋 ) = ( 𝑄 𝑋 ) ) )