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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlexch1 | Structured version Visualization version GIF version |
Description: A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.) |
Ref | Expression |
---|---|
hlsuprexch.b | ⊢ 𝐵 = (Base‘𝐾) |
hlsuprexch.l | ⊢ ≤ = (le‘𝐾) |
hlsuprexch.j | ⊢ ∨ = (join‘𝐾) |
hlsuprexch.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
hlexch1 | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcvl 39341 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat) | |
2 | hlsuprexch.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | hlsuprexch.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | hlsuprexch.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
5 | hlsuprexch.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 2, 3, 4, 5 | cvlexch1 39310 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) |
7 | 1, 6 | syl3an1 1162 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 joincjn 18369 Atomscatm 39245 CvLatclc 39247 HLchlt 39332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-cvlat 39304 df-hlat 39333 |
This theorem is referenced by: cvratlem 39404 4noncolr3 39436 3dimlem4a 39446 3dimlem4OLDN 39448 ps-2 39461 4atlem0a 39576 |
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