Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hlomcmat Structured version   Visualization version   GIF version

Theorem hlomcmat 35440
Description: A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
hlomcmat (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))

Proof of Theorem hlomcmat
StepHypRef Expression
1 hloml 35432 . 2 (𝐾 ∈ HL → 𝐾 ∈ OML)
2 hlclat 35433 . 2 (𝐾 ∈ HL → 𝐾 ∈ CLat)
3 hlatl 35435 . 2 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
41, 2, 33jca 1164 1 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1113  wcel 2166  CLatccla 17460  OMLcoml 35250  AtLatcal 35339  HLchlt 35425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-cvlat 35397  df-hlat 35426
This theorem is referenced by:  hlatmstcOLDN  35472  hlatle  35473  hlrelat1  35475  pmaple  35836  pol1N  35985  polpmapN  35987  pmaplubN  35999
  Copyright terms: Public domain W3C validator