Theorem hlomcmcv 39861

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 2741	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2741	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2741	. . 3 ⊢ (lt‘𝐾) = (lt‘𝐾)
4		eqid 2741	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2741	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		eqid 2741	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
7		eqid 2741	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat1 39857	. 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 498	1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  ∃wrex 3065   class class class wbr 5074  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  lecple 17222  ltcplt 18269  joincjn 18272  0.cp0 18382  1.cp1 18383  CLatccla 18459  OMLcoml 39680  Atomscatm 39768  CvLatclc 39770  HLchlt 39855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6444  df-fv 6496  df-ov 7362  df-hlat 39856

This theorem is referenced by:  hloml  39862  hlclat  39863  hlcvl  39864  cvr1  39915  cvrp  39921  atcvr1  39922  atcvr2  39923