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Theorem hloml 30229
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 30228 . 2  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
) )
21simp1d 970 1  |-  ( K  e.  HL  ->  K  e.  OML )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   CLatccla 14541   OMLcoml 30047   CvLatclc 30137   HLchlt 30222
This theorem is referenced by:  hlol  30233  hlomcmat  30236  poml4N  30824  doca2N  31998  djajN  32009  dihoml4c  32248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-hlat 30223
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