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Theorem hlomcmcv 36486
Description: A Hilbert lattice is orthomodular, complete, and has the covering (exchange) property. (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
hlomcmcv (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))

Proof of Theorem hlomcmcv
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2821 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2821 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2821 . . 3 (lt‘𝐾) = (lt‘𝐾)
4 eqid 2821 . . 3 (join‘𝐾) = (join‘𝐾)
5 eqid 2821 . . 3 (0.‘𝐾) = (0.‘𝐾)
6 eqid 2821 . . 3 (1.‘𝐾) = (1.‘𝐾)
7 eqid 2821 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
81, 2, 3, 4, 5, 6, 7ishlat1 36482 . 2 (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥 ∈ (Atoms‘𝐾)∀𝑦 ∈ (Atoms‘𝐾)(𝑥𝑦 → ∃𝑧 ∈ (Atoms‘𝐾)(𝑧𝑥𝑧𝑦𝑧(le‘𝐾)(𝑥(join‘𝐾)𝑦))) ∧ ∃𝑥 ∈ (Base‘𝐾)∃𝑦 ∈ (Base‘𝐾)∃𝑧 ∈ (Base‘𝐾)(((0.‘𝐾)(lt‘𝐾)𝑥𝑥(lt‘𝐾)𝑦) ∧ (𝑦(lt‘𝐾)𝑧𝑧(lt‘𝐾)(1.‘𝐾))))))
98simplbi 500 1 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083  wcel 2110  wne 3016  wral 3138  wrex 3139   class class class wbr 5059  cfv 6350  (class class class)co 7150  Basecbs 16477  lecple 16566  ltcplt 17545  joincjn 17548  0.cp0 17641  1.cp1 17642  CLatccla 17711  OMLcoml 36305  Atomscatm 36393  CvLatclc 36395  HLchlt 36480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-iota 6309  df-fv 6358  df-ov 7153  df-hlat 36481
This theorem is referenced by:  hloml  36487  hlclat  36488  hlcvl  36489  cvr1  36540  cvrp  36546  atcvr1  36547  atcvr2  36548
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