Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvlexchb2 Structured version   Unicode version

Theorem cvlexchb2 30066
Description: An atomic covering lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
Hypotheses
Ref Expression
cvlexch.b  |-  B  =  ( Base `  K
)
cvlexch.l  |-  .<_  =  ( le `  K )
cvlexch.j  |-  .\/  =  ( join `  K )
cvlexch.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlexchb2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X )  <-> 
( P  .\/  X
)  =  ( Q 
.\/  X ) ) )

Proof of Theorem cvlexchb2
StepHypRef Expression
1 cvlexch.b . . 3  |-  B  =  ( Base `  K
)
2 cvlexch.l . . 3  |-  .<_  =  ( le `  K )
3 cvlexch.j . . 3  |-  .\/  =  ( join `  K )
4 cvlexch.a . . 3  |-  A  =  ( Atoms `  K )
51, 2, 3, 4cvlexchb1 30065 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  <-> 
( X  .\/  P
)  =  ( X 
.\/  Q ) ) )
6 cvllat 30061 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  Lat )
763ad2ant1 978 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  K  e.  Lat )
8 simp22 991 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  Q  e.  A
)
91, 4atbase 30024 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
108, 9syl 16 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  Q  e.  B
)
11 simp23 992 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  X  e.  B
)
121, 3latjcom 14480 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  ( Q  .\/  X
)  =  ( X 
.\/  Q ) )
137, 10, 11, 12syl3anc 1184 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( Q  .\/  X )  =  ( X 
.\/  Q ) )
1413breq2d 4216 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X )  <-> 
P  .<_  ( X  .\/  Q ) ) )
15 simp21 990 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  P  e.  A
)
161, 4atbase 30024 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
1715, 16syl 16 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  P  e.  B
)
181, 3latjcom 14480 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .\/  X
)  =  ( X 
.\/  P ) )
197, 17, 11, 18syl3anc 1184 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .\/  X )  =  ( X 
.\/  P ) )
2019, 13eqeq12d 2449 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( P 
.\/  X )  =  ( Q  .\/  X
)  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )
215, 14, 203bitr4d 277 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X )  <-> 
( P  .\/  X
)  =  ( Q 
.\/  X ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   Latclat 14466   Atomscatm 29998   CvLatclc 30000
This theorem is referenced by:  hlexchb2  30119
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 4693
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 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-lub 14423  df-join 14425  df-lat 14467  df-ats 30002  df-atl 30033  df-cvlat 30057
  Copyright terms: Public domain W3C validator