| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhgt4 | Structured version Visualization version GIF version | ||
| Description: A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.) |
| Ref | Expression |
|---|---|
| hlhgt4.b | ⊢ 𝐵 = (Base‘𝐾) |
| hlhgt4.s | ⊢ < = (lt‘𝐾) |
| hlhgt4.z | ⊢ 0 = (0.‘𝐾) |
| hlhgt4.u | ⊢ 1 = (1.‘𝐾) |
| Ref | Expression |
|---|---|
| hlhgt4 | ⊢ (𝐾 ∈ HL → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlhgt4.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | eqid 2737 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 3 | hlhgt4.s | . . 3 ⊢ < = (lt‘𝐾) | |
| 4 | eqid 2737 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
| 5 | hlhgt4.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
| 6 | hlhgt4.u | . . 3 ⊢ 1 = (1.‘𝐾) | |
| 7 | eqid 2737 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | ishlat2 39354 | . 2 ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ (Atoms‘𝐾)∀𝑦 ∈ (Atoms‘𝐾)((𝑥 ≠ 𝑦 → ∃𝑧 ∈ (Atoms‘𝐾)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝐾)(𝑥(join‘𝐾)𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥(le‘𝐾)𝑧 ∧ 𝑥(le‘𝐾)(𝑧(join‘𝐾)𝑦)) → 𝑦(le‘𝐾)(𝑧(join‘𝐾)𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))))) |
| 9 | simprr 773 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ (Atoms‘𝐾)∀𝑦 ∈ (Atoms‘𝐾)((𝑥 ≠ 𝑦 → ∃𝑧 ∈ (Atoms‘𝐾)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝐾)(𝑥(join‘𝐾)𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥(le‘𝐾)𝑧 ∧ 𝑥(le‘𝐾)(𝑧(join‘𝐾)𝑦)) → 𝑦(le‘𝐾)(𝑧(join‘𝐾)𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))) | |
| 10 | 8, 9 | sylbi 217 | 1 ⊢ (𝐾 ∈ HL → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 ltcplt 18354 joincjn 18357 0.cp0 18468 1.cp1 18469 CLatccla 18543 OMLcoml 39176 Atomscatm 39264 AtLatcal 39265 HLchlt 39351 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-cvlat 39323 df-hlat 39352 |
| This theorem is referenced by: hlhgt2 39391 athgt 39458 |
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