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Theorem cvlexch4N 29523
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 29515 . . . . 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 29507 . . . 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, 9cvlexchb1 29520 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  <-> 
( X  .\/  P
)  =  ( X 
.\/  Q ) ) )
14133expia 1153 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( -.  P  .<_  X  ->  ( P  .<_  ( X  .\/  Q )  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) ) )
1511, 14sylbird 226 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( P  ./\  X )  =  .0.  ->  ( P  .<_  ( X  .\/  Q
)  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) ) )
16153impia 1148 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   0.cp0 14143   Atomscatm 29453   AtLatcal 29454   CvLatclc 29455
This theorem is referenced by:  hlexch4N  29581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512
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