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Theorem cvlexch3 29449
Description: An atomic covering lattice has the exchange property. (atexch 23734 analog.) (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
cvlexch3.b  |-  B  =  ( Base `  K
)
cvlexch3.l  |-  .<_  =  ( le `  K )
cvlexch3.j  |-  .\/  =  ( join `  K )
cvlexch3.m  |-  ./\  =  ( meet `  K )
cvlexch3.z  |-  .0.  =  ( 0. `  K )
cvlexch3.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlexch3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( P  .<_  ( X 
.\/  Q )  ->  Q  .<_  ( X  .\/  P ) ) )

Proof of Theorem cvlexch3
StepHypRef Expression
1 cvlatl 29442 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
21adantr 452 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  K  e.  AtLat
)
3 simpr1 963 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  e.  A )
4 simpr3 965 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  e.  B )
5 cvlexch3.b . . . . 5  |-  B  =  ( Base `  K
)
6 cvlexch3.l . . . . 5  |-  .<_  =  ( le `  K )
7 cvlexch3.m . . . . 5  |-  ./\  =  ( meet `  K )
8 cvlexch3.z . . . . 5  |-  .0.  =  ( 0. `  K )
9 cvlexch3.a . . . . 5  |-  A  =  ( Atoms `  K )
105, 6, 7, 8, 9atnle 29434 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( -.  P  .<_  X  <->  ( P  ./\ 
X )  =  .0.  ) )
112, 3, 4, 10syl3anc 1184 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( -.  P  .<_  X  <->  ( P  ./\ 
X )  =  .0.  ) )
12 cvlexch3.j . . . . 5  |-  .\/  =  ( join `  K )
135, 6, 12, 9cvlexch1 29445 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) )
14133expia 1155 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( -.  P  .<_  X  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X  .\/  P ) ) ) )
1511, 14sylbird 227 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( P  ./\  X )  =  .0.  ->  ( P  .<_  ( X  .\/  Q
)  ->  Q  .<_  ( X  .\/  P ) ) ) )
16153impia 1150 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( P  .<_  ( X 
.\/  Q )  ->  Q  .<_  ( X  .\/  P ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330   meetcmee 14331   0.cp0 14395   Atomscatm 29380   AtLatcal 29381   CvLatclc 29382
This theorem is referenced by:  hlexch3  29507
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-glb 14361  df-meet 14363  df-p0 14397  df-lat 14404  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439
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