Theorem hlomcmcv 38958

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.

Step	Hyp	Ref	Expression
1		eqid 2725	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2725	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2725	. . 3 ⊢ (lt‘𝐾) = (lt‘𝐾)
4		eqid 2725	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2725	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		eqid 2725	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
7		eqid 2725	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat1 38954	. 2 ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥 ∈ (Atoms‘𝐾)∀𝑦 ∈ (Atoms‘𝐾)(𝑥 ≠ 𝑦 → ∃𝑧 ∈ (Atoms‘𝐾)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝐾)(𝑥(join‘𝐾)𝑦))) ∧ ∃𝑥 ∈ (Base‘𝐾)∃𝑦 ∈ (Base‘𝐾)∃𝑧 ∈ (Base‘𝐾)(((0.‘𝐾)(lt‘𝐾)𝑥 ∧ 𝑥(lt‘𝐾)𝑦) ∧ (𝑦(lt‘𝐾)𝑧 ∧ 𝑧(lt‘𝐾)(1.‘𝐾))))))
9	8	simplbi 496	1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050  ∃wrex 3059   class class class wbr 5149  ‘cfv 6549  (class class class)co 7419  Basecbs 17183  lecple 17243  ltcplt 18303  joincjn 18306  0.cp0 18418  1.cp1 18419  CLatccla 18493  OMLcoml 38777  Atomscatm 38865  CvLatclc 38867  HLchlt 38952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-hlat 38953

This theorem is referenced by:  hloml  38959  hlclat  38960  hlcvl  38961  cvr1  39013  cvrp  39019  atcvr1  39020  atcvr2  39021