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Theorem ishlatiN 38030
Description: Properties that determine a Hilbert lattice. (Contributed by NM, 13-Nov-2011.) (New usage is discouraged.)
Hypotheses
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 𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥)))
ishlati.10 𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 ))
Assertion
Ref Expression
ishlatiN 𝐾 ∈ HL
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧
Allowed substitution hints:   < (𝑥,𝑦,𝑧)   1 (𝑥,𝑦,𝑧)   (𝑥,𝑦,𝑧)   (𝑥,𝑦,𝑧)   0 (𝑥,𝑦,𝑧)

Proof of Theorem ishlatiN
StepHypRef Expression
1 ishlati.1 . . 3 𝐾 ∈ OML
2 ishlati.2 . . 3 𝐾 ∈ CLat
3 ishlati.3 . . 3 𝐾 ∈ AtLat
41, 2, 33pm3.2i 1339 . 2 (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)
5 ishlati.9 . . 3 𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥)))
6 ishlati.10 . . 3 𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 ))
75, 6pm3.2i 471 . 2 (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 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)
158, 9, 10, 11, 12, 13, 14ishlat2 38028 . 2 (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))
164, 7, 15mpbir2an 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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