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Theorem hlomcmat 32395
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 32388 . 2  |-  ( K  e.  HL  ->  K  e.  OML )
2 hlclat 32389 . 2  |-  ( K  e.  HL  ->  K  e.  CLat )
3 hlatl 32391 . 2  |-  ( K  e.  HL  ->  K  e.  AtLat )
41, 2, 33jca 1179 1  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 976    e. wcel 1844   CLatccla 16063   OMLcoml 32206   AtLatcal 32295   HLchlt 32381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-iota 5535  df-fv 5579  df-ov 6283  df-cvlat 32353  df-hlat 32382
This theorem is referenced by:  hlatmstcOLDN  32427  hlatle  32428  hlrelat1  32430  pmaple  32791  pol1N  32940  polpmapN  32942  pmaplubN  32954
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