Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvlexch4N Unicode version

Theorem cvlexch4N 28790
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 28782 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
21adantr 453 . . . 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 28774 . . . 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 28787 . . . 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 228 . 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   0.cp0 14137   Atomscatm 28720   AtLatcal 28721   CvLatclc 28722
This theorem is referenced by:  hlexch4N  28848
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-lat 14146  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779
  Copyright terms: Public domain W3C validator