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Theorem hlexch2 29631
Description: A Hilbert lattice has the exchange property. (Contributed by NM, 6-May-2012.)
Hypotheses
Ref Expression
hlsuprexch.b  |-  B  =  ( Base `  K
)
hlsuprexch.l  |-  .<_  =  ( le `  K )
hlsuprexch.j  |-  .\/  =  ( join `  K )
hlsuprexch.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlexch2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X )  ->  Q  .<_  ( P  .\/  X ) ) )

Proof of Theorem hlexch2
StepHypRef Expression
1 hlcvl 29608 . 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, 5cvlexch2 29578 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X )  ->  Q  .<_  ( P 
.\/  X ) ) )
71, 6syl3an1 1216 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X )  ->  Q  .<_  ( P  .\/  X ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 935    = wceq 1647    e. wcel 1715   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   joincjn 14288   Atomscatm 29512   CvLatclc 29514   HLchlt 29599
This theorem is referenced by:  cdlemc3  30441  cdlemg4  30865  cdlemg6c  30868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-join 14320  df-lat 14362  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600
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