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Theorem hlhgt4 39390
Description: A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.)
Hypotheses
Ref Expression
hlhgt4.b 𝐵 = (Base‘𝐾)
hlhgt4.s < = (lt‘𝐾)
hlhgt4.z 0 = (0.‘𝐾)
hlhgt4.u 1 = (1.‘𝐾)
Assertion
Ref Expression
hlhgt4 (𝐾 ∈ HL → ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐾,𝑦,𝑧
Allowed substitution hints:   < (𝑥,𝑦,𝑧)   1 (𝑥,𝑦,𝑧)   0 (𝑥,𝑦,𝑧)

Proof of Theorem hlhgt4
StepHypRef 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)
81, 2, 3, 4, 5, 6, 7ishlat2 39354 . 2 (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ (Atoms‘𝐾)∀𝑦 ∈ (Atoms‘𝐾)((𝑥𝑦 → ∃𝑧 ∈ (Atoms‘𝐾)(𝑧𝑥𝑧𝑦𝑧(le‘𝐾)(𝑥(join‘𝐾)𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥(le‘𝐾)𝑧𝑥(le‘𝐾)(𝑧(join‘𝐾)𝑦)) → 𝑦(le‘𝐾)(𝑧(join‘𝐾)𝑥))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))
9 simprr 773 . 2 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ (Atoms‘𝐾)∀𝑦 ∈ (Atoms‘𝐾)((𝑥𝑦 → ∃𝑧 ∈ (Atoms‘𝐾)(𝑧𝑥𝑧𝑦𝑧(le‘𝐾)(𝑥(join‘𝐾)𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥(le‘𝐾)𝑧𝑥(le‘𝐾)(𝑧(join‘𝐾)𝑦)) → 𝑦(le‘𝐾)(𝑧(join‘𝐾)𝑥))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))) → ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))
108, 9sylbi 217 1 (𝐾 ∈ HL → ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 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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