Theorem hlomcmcv 35158

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 2770	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2770	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2770	. . 3 ⊢ (lt‘𝐾) = (lt‘𝐾)
4		eqid 2770	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2770	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		eqid 2770	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
7		eqid 2770	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat1 35154	. 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 479	1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))

Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1070   ∈ wcel 2144   ≠ wne 2942  ∀wral 3060  ∃wrex 3061   class class class wbr 4784  ‘cfv 6031  (class class class)co 6792  Basecbs 16063  lecple 16155  ltcplt 17148  joincjn 17151  0.cp0 17244  1.cp1 17245  CLatccla 17314  OMLcoml 34977  Atomscatm 35065  CvLatclc 35067  HLchlt 35152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ov 6795  df-hlat 35153

This theorem is referenced by:  hloml  35159  hlclat  35160  hlcvl  35161  cvr1  35211  cvrp  35217  atcvr1  35218  atcvr2  35219