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Theorem cvlexch3 30144
Description: An atomic covering lattice has the exchange property. (atexch 22977 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 30137 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
21adantr 451 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  K  e.  AtLat
)
3 simpr1 961 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  e.  A )
4 simpr3 963 . . . 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 30129 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( -.  P  .<_  X  <->  ( P  ./\ 
X )  =  .0.  ) )
112, 3, 4, 10syl3anc 1182 . . 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 30140 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) )
14133expia 1153 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( -.  P  .<_  X  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X  .\/  P ) ) ) )
1511, 14sylbird 226 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( P  ./\  X )  =  .0.  ->  ( P  .<_  ( X  .\/  Q
)  ->  Q  .<_  ( X  .\/  P ) ) ) )
16153impia 1148 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   0.cp0 14159   Atomscatm 30075   AtLatcal 30076   CvLatclc 30077
This theorem is referenced by:  hlexch3  30202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-glb 14125  df-meet 14127  df-p0 14161  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134
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