Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvlexchb1 Unicode version

Theorem cvlexchb1 29447
Description: An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
Hypotheses
Ref Expression
cvlexch.b  |-  B  =  ( Base `  K
)
cvlexch.l  |-  .<_  =  ( le `  K )
cvlexch.j  |-  .\/  =  ( join `  K )
cvlexch.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlexchb1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  <-> 
( X  .\/  P
)  =  ( X 
.\/  Q ) ) )

Proof of Theorem cvlexchb1
StepHypRef Expression
1 cvllat 29443 . . . . . . . . 9  |-  ( K  e.  CvLat  ->  K  e.  Lat )
21adantr 452 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  K  e.  Lat )
3 simpr3 965 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  e.  B )
4 simpr2 964 . . . . . . . . 9  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  Q  e.  A )
5 cvlexch.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
6 cvlexch.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
75, 6atbase 29406 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  B )
84, 7syl 16 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  Q  e.  B )
9 cvlexch.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
10 cvlexch.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
115, 9, 10latlej1 14418 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  X  .<_  ( X  .\/  Q ) )
122, 3, 8, 11syl3anc 1184 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  .<_  ( X  .\/  Q ) )
13123adant3 977 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  X  .<_  ( X 
.\/  Q ) )
1413adantr 452 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  X  .<_  ( X  .\/  Q
) )
15 simpr 448 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  P  .<_  ( X  .\/  Q
) )
16 simpr1 963 . . . . . . . . 9  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  e.  A )
175, 6atbase 29406 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  B )
1816, 17syl 16 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  e.  B )
195, 10latjcl 14408 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
202, 3, 8, 19syl3anc 1184 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( X  .\/  Q )  e.  B
)
215, 9, 10latjle12 14420 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  P  e.  B  /\  ( X  .\/  Q
)  e.  B ) )  ->  ( ( X  .<_  ( X  .\/  Q )  /\  P  .<_  ( X  .\/  Q ) )  <->  ( X  .\/  P )  .<_  ( X  .\/  Q ) ) )
222, 3, 18, 20, 21syl13anc 1186 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .<_  ( X  .\/  Q )  /\  P  .<_  ( X  .\/  Q ) )  <->  ( X  .\/  P )  .<_  ( X  .\/  Q ) ) )
23223adant3 977 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( X 
.<_  ( X  .\/  Q
)  /\  P  .<_  ( X  .\/  Q ) )  <->  ( X  .\/  P )  .<_  ( X  .\/  Q ) ) )
2423adantr 452 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  (
( X  .<_  ( X 
.\/  Q )  /\  P  .<_  ( X  .\/  Q ) )  <->  ( X  .\/  P )  .<_  ( X 
.\/  Q ) ) )
2514, 15, 24mpbi2and 888 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  ( X  .\/  P )  .<_  ( X  .\/  Q ) )
265, 9, 10latlej1 14418 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  X  .<_  ( X  .\/  P ) )
272, 3, 18, 26syl3anc 1184 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  .<_  ( X  .\/  P ) )
28273adant3 977 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  X  .<_  ( X 
.\/  P ) )
2928adantr 452 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  X  .<_  ( X  .\/  P
) )
305, 9, 10, 6cvlexch1 29445 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) )
3130imp 419 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  Q  .<_  ( X  .\/  P
) )
325, 10latjcl 14408 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
332, 3, 18, 32syl3anc 1184 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( X  .\/  P )  e.  B
)
345, 9, 10latjle12 14420 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Q  e.  B  /\  ( X  .\/  P
)  e.  B ) )  ->  ( ( X  .<_  ( X  .\/  P )  /\  Q  .<_  ( X  .\/  P ) )  <->  ( X  .\/  Q )  .<_  ( X  .\/  P ) ) )
352, 3, 8, 33, 34syl13anc 1186 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .<_  ( X  .\/  P )  /\  Q  .<_  ( X  .\/  P ) )  <->  ( X  .\/  Q )  .<_  ( X  .\/  P ) ) )
36353adant3 977 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( X 
.<_  ( X  .\/  P
)  /\  Q  .<_  ( X  .\/  P ) )  <->  ( X  .\/  Q )  .<_  ( X  .\/  P ) ) )
3736adantr 452 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  (
( X  .<_  ( X 
.\/  P )  /\  Q  .<_  ( X  .\/  P ) )  <->  ( X  .\/  Q )  .<_  ( X 
.\/  P ) ) )
3829, 31, 37mpbi2and 888 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  ( X  .\/  Q )  .<_  ( X  .\/  P ) )
395, 9latasymb 14412 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( X  .\/  P )  e.  B  /\  ( X  .\/  Q )  e.  B )  ->  (
( ( X  .\/  P )  .<_  ( X  .\/  Q )  /\  ( X  .\/  Q )  .<_  ( X  .\/  P ) )  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )
402, 33, 20, 39syl3anc 1184 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( (
( X  .\/  P
)  .<_  ( X  .\/  Q )  /\  ( X 
.\/  Q )  .<_  ( X  .\/  P ) )  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )
41403adant3 977 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( ( X  .\/  P ) 
.<_  ( X  .\/  Q
)  /\  ( X  .\/  Q )  .<_  ( X 
.\/  P ) )  <-> 
( X  .\/  P
)  =  ( X 
.\/  Q ) ) )
4241adantr 452 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  (
( ( X  .\/  P )  .<_  ( X  .\/  Q )  /\  ( X  .\/  Q )  .<_  ( X  .\/  P ) )  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )
4325, 38, 42mpbi2and 888 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  ( X  .\/  P )  =  ( X  .\/  Q
) )
4443ex 424 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )
455, 9, 10latlej2 14419 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  P  .<_  ( X  .\/  P ) )
462, 3, 18, 45syl3anc 1184 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  .<_  ( X  .\/  P ) )
47 breq2 4159 . . . 4  |-  ( ( X  .\/  P )  =  ( X  .\/  Q )  ->  ( P  .<_  ( X  .\/  P
)  <->  P  .<_  ( X 
.\/  Q ) ) )
4846, 47syl5ibcom 212 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .\/  P )  =  ( X  .\/  Q
)  ->  P  .<_  ( X  .\/  Q ) ) )
49483adant3 977 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( X 
.\/  P )  =  ( X  .\/  Q
)  ->  P  .<_  ( X  .\/  Q ) ) )
5044, 49impbid 184 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( 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 1649    e. wcel 1717   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330   Latclat 14403   Atomscatm 29380   CvLatclc 29382
This theorem is referenced by:  cvlexchb2  29448  cvlexch4N  29450  cvlatexchb1  29451  cvlcvr1  29456  hlexchb1  29500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-lub 14360  df-join 14362  df-lat 14404  df-ats 29384  df-atl 29415  df-cvlat 29439
  Copyright terms: Public domain W3C validator