Theorem hlomcmcv 35377

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 2799	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2799	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2799	. . 3 ⊢ (lt‘𝐾) = (lt‘𝐾)
4		eqid 2799	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2799	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		eqid 2799	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
7		eqid 2799	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat1 35373	. 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 492	1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))

Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108   ∈ wcel 2157   ≠ wne 2971  ∀wral 3089  ∃wrex 3090   class class class wbr 4843  ‘cfv 6101  (class class class)co 6878  Basecbs 16184  lecple 16274  ltcplt 17256  joincjn 17259  0.cp0 17352  1.cp1 17353  CLatccla 17422  OMLcoml 35196  Atomscatm 35284  CvLatclc 35286  HLchlt 35371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881  df-hlat 35372

This theorem is referenced by:  hloml  35378  hlclat  35379  hlcvl  35380  cvr1  35431  cvrp  35437  atcvr1  35438  atcvr2  35439