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Theorem cvlexch4N 29970
Description: An atomic covering lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
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
cvlexch4N  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( P  .<_  ( X 
.\/  Q )  <->  ( X  .\/  P )  =  ( X  .\/  Q ) ) )

Proof of Theorem cvlexch4N
StepHypRef Expression
1 cvlatl 29962 . . . . 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 29954 . . . 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, 9cvlexchb1 29967 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  <-> 
( X  .\/  P
)  =  ( X 
.\/  Q ) ) )
14133expia 1155 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( -.  P  .<_  X  ->  ( P  .<_  ( X  .\/  Q )  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) ) )
1511, 14sylbird 227 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( P  ./\  X )  =  .0.  ->  ( P  .<_  ( X  .\/  Q
)  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) ) )
16153impia 1150 1  |-  ( ( K  e.  CvLat  /\  ( 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:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   0.cp0 14454   Atomscatm 29900   AtLatcal 29901   CvLatclc 29902
This theorem is referenced by:  hlexch4N  30028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-lat 14463  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959
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