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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 37110 | . 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 37079 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) |
7 | 1, 6 | syl3an1 1165 | 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 1089 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 lecple 16809 joincjn 17818 Atomscatm 37014 CvLatclc 37016 HLchlt 37101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-iota 6338 df-fv 6388 df-ov 7216 df-cvlat 37073 df-hlat 37102 |
This theorem is referenced by: cvratlem 37172 4noncolr3 37204 3dimlem4a 37214 3dimlem4OLDN 37216 ps-2 37229 4atlem0a 37344 |
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