Theorem hlomcmcv 40507

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 2760	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2760	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2760	. . 3 ⊢ (lt‘𝐾) = (lt‘𝐾)
4		eqid 2760	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2760	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		eqid 2760	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
7		eqid 2760	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat1 40503	. 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 502	1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   class class class wbr 5102  ‘cfv 6539  (class class class)co 7421  Basecbs 17393  lecple 17441  ltcplt 18488  joincjn 18491  0.cp0 18601  1.cp1 18602  CLatccla 18678  OMLcoml 40326  Atomscatm 40414  CvLatclc 40416  HLchlt 40501

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6547  df-ov 7424  df-hlat 40502

This theorem is used by:  hloml  40508  hlclat  40509  hlcvl  40510  cvr1  40561  cvrp  40567  atcvr1  40568  atcvr2  40569