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Theorem hloml 32375
Description: A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
Assertion
Ref Expression
hloml  |-  ( K  e.  HL  ->  K  e.  OML )

Proof of Theorem hloml
StepHypRef Expression
1 hlomcmcv 32374 . 2  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
) )
21simp1d 1009 1  |-  ( K  e.  HL  ->  K  e.  OML )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   CLatccla 16061   OMLcoml 32193   CvLatclc 32283   HLchlt 32368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-ov 6281  df-hlat 32369
This theorem is referenced by:  hlol  32379  hlomcmat  32382  poml4N  32970  doca2N  34146  djajN  34157  dihoml4c  34396
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