Theorem hlomcmcv 40111

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 2763	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2763	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2763	. . 3 ⊢ (lt‘𝐾) = (lt‘𝐾)
4		eqid 2763	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2763	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		eqid 2763	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
7		eqid 2763	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat1 40107	. 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 501	1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  lecple 17318  ltcplt 18365  joincjn 18368  0.cp0 18478  1.cp1 18479  CLatccla 18555  OMLcoml 39930  Atomscatm 40018  CvLatclc 40020  HLchlt 40105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415  df-hlat 40106

This theorem is referenced by:  hloml  40112  hlclat  40113  hlcvl  40114  cvr1  40165  cvrp  40171  atcvr1  40172  atcvr2  40173