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Theorem hlomcmat 34161
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 34154 . 2  |-  ( K  e.  HL  ->  K  e.  OML )
2 hlclat 34155 . 2  |-  ( K  e.  HL  ->  K  e.  CLat )
3 hlatl 34157 . 2  |-  ( K  e.  HL  ->  K  e.  AtLat )
41, 2, 33jca 1176 1  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    e. wcel 1767   CLatccla 15590   OMLcoml 33972   AtLatcal 34061   HLchlt 34147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-cvlat 34119  df-hlat 34148
This theorem is referenced by:  hlatmstcOLDN  34193  hlatle  34194  hlrelat1  34196  pmaple  34557  pol1N  34706  polpmapN  34708  pmaplubN  34720
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