|   | Mathbox for Norm Megill | < Previous  
      Next > Nearby theorems | |
| 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 39361 | . 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 39330 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) | 
| 7 | 1, 6 | syl3an1 1163 | 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 1539 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 lecple 17305 joincjn 18358 Atomscatm 39265 CvLatclc 39267 HLchlt 39352 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-cvlat 39324 df-hlat 39353 | 
| This theorem is referenced by: cvratlem 39424 4noncolr3 39456 3dimlem4a 39466 3dimlem4OLDN 39468 ps-2 39481 4atlem0a 39596 | 
| Copyright terms: Public domain | W3C validator |