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Theorem hlexch1 30253
Description: A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)
Hypotheses
Ref Expression
hlsuprexch.b  |-  B  =  ( Base `  K
)
hlsuprexch.l  |-  .<_  =  ( le `  K )
hlsuprexch.j  |-  .\/  =  ( join `  K )
hlsuprexch.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlexch1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X  .\/  P ) ) )

Proof of Theorem hlexch1
StepHypRef Expression
1 hlcvl 30231 . 2  |-  ( K  e.  HL  ->  K  e.  CvLat )
2 hlsuprexch.b . . 3  |-  B  =  ( Base `  K
)
3 hlsuprexch.l . . 3  |-  .<_  =  ( le `  K )
4 hlsuprexch.j . . 3  |-  .\/  =  ( join `  K )
5 hlsuprexch.a . . 3  |-  A  =  ( Atoms `  K )
62, 3, 4, 5cvlexch1 30200 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) )
71, 6syl3an1 1218 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X  .\/  P ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   Atomscatm 30135   CvLatclc 30137   HLchlt 30222
This theorem is referenced by:  cvratlem  30292  4noncolr3  30324  3dimlem4a  30334  3dimlem4OLDN  30336  ps-2  30349  4atlem0a  30464
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-cvlat 30194  df-hlat 30223
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