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Theorem hlexch1 29868
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 29846 . 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 29815 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) )
71, 6syl3an1 1217 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 936    = wceq 1649    e. wcel 1721   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   Basecbs 13428   lecple 13495   joincjn 14360   Atomscatm 29750   CvLatclc 29752   HLchlt 29837
This theorem is referenced by:  cvratlem  29907  4noncolr3  29939  3dimlem4a  29949  3dimlem4OLDN  29951  ps-2  29964  4atlem0a  30079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-iota 5381  df-fv 5425  df-ov 6047  df-cvlat 29809  df-hlat 29838
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