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Theorem hlomcmat 28927
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 28920 . 2  |-  ( K  e.  HL  ->  K  e.  OML )
2 hlclat 28921 . 2  |-  ( K  e.  HL  ->  K  e.  CLat )
3 hlatl 28923 . 2  |-  ( K  e.  HL  ->  K  e.  AtLat )
41, 2, 33jca 1132 1  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1684   CLatccla 14213   OMLcoml 28738   AtLatcal 28827   HLchlt 28913
This theorem is referenced by:  hlatmstcOLDN  28959  hlatle  28960  hlrelat1  28962  pmaple  29323  pol1N  29472  polpmapN  29474  pmaplubN  29486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-cvlat 28885  df-hlat 28914
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