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Theorem ishlatiN 37863
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 𝐡 = (Baseβ€˜πΎ)
ishlati.l ≀ = (leβ€˜πΎ)
ishlati.s < = (ltβ€˜πΎ)
ishlati.j ∨ = (joinβ€˜πΎ)
ishlati.z 0 = (0.β€˜πΎ)
ishlati.u 1 = (1.β€˜πΎ)
ishlati.a 𝐴 = (Atomsβ€˜πΎ)
ishlati.9 βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯)))
ishlati.10 βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))
Assertion
Ref Expression
ishlatiN 𝐾 ∈ HL
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐴   π‘₯,𝐡,𝑦,𝑧   π‘₯,𝐾,𝑦,𝑧
Allowed substitution hints:   < (π‘₯,𝑦,𝑧)   1 (π‘₯,𝑦,𝑧)   ∨ (π‘₯,𝑦,𝑧)   ≀ (π‘₯,𝑦,𝑧)   0 (π‘₯,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 1340 . 2 (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)
5 ishlati.9 . . 3 βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯)))
6 ishlati.10 . . 3 βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))
75, 6pm3.2i 472 . 2 (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))
8 ishlati.b . . 3 𝐡 = (Baseβ€˜πΎ)
9 ishlati.l . . 3 ≀ = (leβ€˜πΎ)
10 ishlati.s . . 3 < = (ltβ€˜πΎ)
11 ishlati.j . . 3 ∨ = (joinβ€˜πΎ)
12 ishlati.z . . 3 0 = (0.β€˜πΎ)
13 ishlati.u . . 3 1 = (1.β€˜πΎ)
14 ishlati.a . . 3 𝐴 = (Atomsβ€˜πΎ)
158, 9, 10, 11, 12, 13, 14ishlat2 37861 . 2 (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
164, 7, 15mpbir2an 710 1 𝐾 ∈ HL
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  ltcplt 18202  joincjn 18205  0.cp0 18317  1.cp1 18318  CLatccla 18392  OMLcoml 37683  Atomscatm 37771  AtLatcal 37772  HLchlt 37858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-cvlat 37830  df-hlat 37859
This theorem is referenced by: (None)
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