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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 2798 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | hlhgt4.s | . . 3 ⊢ < = (lt‘𝐾) | |
4 | eqid 2798 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | hlhgt4.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
6 | hlhgt4.u | . . 3 ⊢ 1 = (1.‘𝐾) | |
7 | eqid 2798 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ishlat2 36649 | . 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 772 | . 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 220 | 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 ltcplt 17543 joincjn 17546 0.cp0 17639 1.cp1 17640 CLatccla 17709 OMLcoml 36471 Atomscatm 36559 AtLatcal 36560 HLchlt 36646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-cvlat 36618 df-hlat 36647 |
This theorem is referenced by: hlhgt2 36685 athgt 36752 |
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