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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlexchb1 | Structured version Visualization version GIF version |
Description: A Hilbert lattice has the exchange property. (Contributed by NM, 16-Nov-2011.) |
Ref | Expression |
---|---|
hlsuprexch.b | ⊢ 𝐵 = (Base‘𝐾) |
hlsuprexch.l | ⊢ ≤ = (le‘𝐾) |
hlsuprexch.j | ⊢ ∨ = (join‘𝐾) |
hlsuprexch.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
hlexchb1 | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcvl 36375 | . 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 | cvlexchb1 36346 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))) |
7 | 1, 6 | syl3an1 1155 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lecple 16560 joincjn 17542 Atomscatm 36279 CvLatclc 36281 HLchlt 36366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-proset 17526 df-poset 17544 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-lat 17644 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 |
This theorem is referenced by: 3dimlem3a 36476 3dimlem3OLDN 36478 3atlem2 36500 |
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