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Theorem hlexch4N 29654
Description: A Hilbert lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 15-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
hlexch3.b  |-  B  =  ( Base `  K
)
hlexch3.l  |-  .<_  =  ( le `  K )
hlexch3.j  |-  .\/  =  ( join `  K )
hlexch3.m  |-  ./\  =  ( meet `  K )
hlexch3.z  |-  .0.  =  ( 0. `  K )
hlexch3.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlexch4N  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  ./\ 
X )  =  .0.  )  ->  ( P  .<_  ( X  .\/  Q
)  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )

Proof of Theorem hlexch4N
StepHypRef Expression
1 hlcvl 29622 . 2  |-  ( K  e.  HL  ->  K  e.  CvLat )
2 hlexch3.b . . 3  |-  B  =  ( Base `  K
)
3 hlexch3.l . . 3  |-  .<_  =  ( le `  K )
4 hlexch3.j . . 3  |-  .\/  =  ( join `  K )
5 hlexch3.m . . 3  |-  ./\  =  ( meet `  K )
6 hlexch3.z . . 3  |-  .0.  =  ( 0. `  K )
7 hlexch3.a . . 3  |-  A  =  ( Atoms `  K )
82, 3, 4, 5, 6, 7cvlexch4N 29596 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( P  .<_  ( X 
.\/  Q )  <->  ( X  .\/  P )  =  ( X  .\/  Q ) ) )
91, 8syl3an1 1215 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  ./\ 
X )  =  .0.  )  ->  ( P  .<_  ( X  .\/  Q
)  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1625    e. wcel 1686   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   Basecbs 13150   lecple 13217   joincjn 14080   meetcmee 14081   0.cp0 14145   Atomscatm 29526   CvLatclc 29528   HLchlt 29613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-lat 14154  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614
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