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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ishlatiN | Structured version Visualization version GIF version |
Description: Properties that determine a Hilbert lattice. (Contributed by NM, 13-Nov-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ishlati.1 | ⊢ 𝐾 ∈ OML |
ishlati.2 | ⊢ 𝐾 ∈ CLat |
ishlati.3 | ⊢ 𝐾 ∈ AtLat |
ishlati.b | ⊢ 𝐵 = (Base‘𝐾) |
ishlati.l | ⊢ ≤ = (le‘𝐾) |
ishlati.s | ⊢ < = (lt‘𝐾) |
ishlati.j | ⊢ ∨ = (join‘𝐾) |
ishlati.z | ⊢ 0 = (0.‘𝐾) |
ishlati.u | ⊢ 1 = (1.‘𝐾) |
ishlati.a | ⊢ 𝐴 = (Atoms‘𝐾) |
ishlati.9 | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) |
ishlati.10 | ⊢ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )) |
Ref | Expression |
---|---|
ishlatiN | ⊢ 𝐾 ∈ HL |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishlati.1 | . . 3 ⊢ 𝐾 ∈ OML | |
2 | ishlati.2 | . . 3 ⊢ 𝐾 ∈ CLat | |
3 | ishlati.3 | . . 3 ⊢ 𝐾 ∈ AtLat | |
4 | 1, 2, 3 | 3pm3.2i 1339 | . 2 ⊢ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) |
5 | ishlati.9 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) | |
6 | ishlati.10 | . . 3 ⊢ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )) | |
7 | 5, 6 | pm3.2i 471 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))) |
8 | ishlati.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
9 | ishlati.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
10 | ishlati.s | . . 3 ⊢ < = (lt‘𝐾) | |
11 | ishlati.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
12 | ishlati.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
13 | ishlati.u | . . 3 ⊢ 1 = (1.‘𝐾) | |
14 | ishlati.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
15 | 8, 9, 10, 11, 12, 13, 14 | ishlat2 38028 | . 2 ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))))) |
16 | 4, 7, 15 | mpbir2an 709 | 1 ⊢ 𝐾 ∈ HL |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 lecple 17186 ltcplt 18243 joincjn 18246 0.cp0 18358 1.cp1 18359 CLatccla 18433 OMLcoml 37850 Atomscatm 37938 AtLatcal 37939 HLchlt 38025 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-iota 6484 df-fv 6540 df-ov 7396 df-cvlat 37997 df-hlat 38026 |
This theorem is referenced by: (None) |
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