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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 38961 | . 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 38930 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) |
7 | 1, 6 | syl3an1 1160 | 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 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 lecple 17243 joincjn 18306 Atomscatm 38865 CvLatclc 38867 HLchlt 38952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 df-cvlat 38924 df-hlat 38953 |
This theorem is referenced by: cvratlem 39024 4noncolr3 39056 3dimlem4a 39066 3dimlem4OLDN 39068 ps-2 39081 4atlem0a 39196 |
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