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Theorem hlomcmat 30100
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 30093 . 2  |-  ( K  e.  HL  ->  K  e.  OML )
2 hlclat 30094 . 2  |-  ( K  e.  HL  ->  K  e.  CLat )
3 hlatl 30096 . 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 1725   CLatccla 14529   OMLcoml 29911   AtLatcal 30000   HLchlt 30086
This theorem is referenced by:  hlatmstcOLDN  30132  hlatle  30133  hlrelat1  30135  pmaple  30496  pol1N  30645  polpmapN  30647  pmaplubN  30659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-iota 5411  df-fv 5455  df-ov 6077  df-cvlat 30058  df-hlat 30087
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