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Theorem hlomcmat 29479
Description: A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
hlomcmat  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )

Proof of Theorem hlomcmat
StepHypRef Expression
1 hloml 29472 . 2  |-  ( K  e.  HL  ->  K  e.  OML )
2 hlclat 29473 . 2  |-  ( K  e.  HL  ->  K  e.  CLat )
3 hlatl 29475 . 2  |-  ( K  e.  HL  ->  K  e.  AtLat )
41, 2, 33jca 1134 1  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    e. wcel 1717   CLatccla 14463   OMLcoml 29290   AtLatcal 29379   HLchlt 29465
This theorem is referenced by:  hlatmstcOLDN  29511  hlatle  29512  hlrelat1  29514  pmaple  29875  pol1N  30024  polpmapN  30026  pmaplubN  30038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023  df-cvlat 29437  df-hlat 29466
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