Metamath Proof Explorer
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 ) ) ) |