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Theorem hlomcmat 33318
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 33311 . 2  |-  ( K  e.  HL  ->  K  e.  OML )
2 hlclat 33312 . 2  |-  ( K  e.  HL  ->  K  e.  CLat )
3 hlatl 33314 . 2  |-  ( K  e.  HL  ->  K  e.  AtLat )
41, 2, 33jca 1168 1  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    e. wcel 1758   CLatccla 15388   OMLcoml 33129   AtLatcal 33218   HLchlt 33304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-ov 6196  df-cvlat 33276  df-hlat 33305
This theorem is referenced by:  hlatmstcOLDN  33350  hlatle  33351  hlrelat1  33353  pmaple  33714  pol1N  33863  polpmapN  33865  pmaplubN  33877
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