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Theorem cvlexch1 34418
Description: An atomic covering lattice has the exchange property. (Contributed by NM, 6-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
cvlexch1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) )

Proof of Theorem cvlexch1
Dummy variables  q  p  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvlexch.b . . . . . 6  |-  B  =  ( Base `  K
)
2 cvlexch.l . . . . . 6  |-  .<_  =  ( le `  K )
3 cvlexch.j . . . . . 6  |-  .\/  =  ( join `  K )
4 cvlexch.a . . . . . 6  |-  A  =  ( Atoms `  K )
51, 2, 3, 4iscvlat 34413 . . . . 5  |-  ( K  e.  CvLat 
<->  ( K  e.  AtLat  /\ 
A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
65simprbi 464 . . . 4  |-  ( K  e.  CvLat  ->  A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) )  -> 
q  .<_  ( x  .\/  p ) ) )
7 breq1 4455 . . . . . . . 8  |-  ( p  =  P  ->  (
p  .<_  x  <->  P  .<_  x ) )
87notbid 294 . . . . . . 7  |-  ( p  =  P  ->  ( -.  p  .<_  x  <->  -.  P  .<_  x ) )
9 breq1 4455 . . . . . . 7  |-  ( p  =  P  ->  (
p  .<_  ( x  .\/  q )  <->  P  .<_  ( x  .\/  q ) ) )
108, 9anbi12d 710 . . . . . 6  |-  ( p  =  P  ->  (
( -.  p  .<_  x  /\  p  .<_  ( x 
.\/  q ) )  <-> 
( -.  P  .<_  x  /\  P  .<_  ( x 
.\/  q ) ) ) )
11 oveq2 6302 . . . . . . 7  |-  ( p  =  P  ->  (
x  .\/  p )  =  ( x  .\/  P ) )
1211breq2d 4464 . . . . . 6  |-  ( p  =  P  ->  (
q  .<_  ( x  .\/  p )  <->  q  .<_  ( x  .\/  P ) ) )
1310, 12imbi12d 320 . . . . 5  |-  ( p  =  P  ->  (
( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) )  <->  ( ( -.  P  .<_  x  /\  P  .<_  ( x  .\/  q ) )  -> 
q  .<_  ( x  .\/  P ) ) ) )
14 oveq2 6302 . . . . . . . 8  |-  ( q  =  Q  ->  (
x  .\/  q )  =  ( x  .\/  Q ) )
1514breq2d 4464 . . . . . . 7  |-  ( q  =  Q  ->  ( P  .<_  ( x  .\/  q )  <->  P  .<_  ( x  .\/  Q ) ) )
1615anbi2d 703 . . . . . 6  |-  ( q  =  Q  ->  (
( -.  P  .<_  x  /\  P  .<_  ( x 
.\/  q ) )  <-> 
( -.  P  .<_  x  /\  P  .<_  ( x 
.\/  Q ) ) ) )
17 breq1 4455 . . . . . 6  |-  ( q  =  Q  ->  (
q  .<_  ( x  .\/  P )  <->  Q  .<_  ( x 
.\/  P ) ) )
1816, 17imbi12d 320 . . . . 5  |-  ( q  =  Q  ->  (
( ( -.  P  .<_  x  /\  P  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  P ) )  <->  ( ( -.  P  .<_  x  /\  P  .<_  ( x  .\/  Q ) )  ->  Q  .<_  ( x  .\/  P
) ) ) )
19 breq2 4456 . . . . . . . 8  |-  ( x  =  X  ->  ( P  .<_  x  <->  P  .<_  X ) )
2019notbid 294 . . . . . . 7  |-  ( x  =  X  ->  ( -.  P  .<_  x  <->  -.  P  .<_  X ) )
21 oveq1 6301 . . . . . . . 8  |-  ( x  =  X  ->  (
x  .\/  Q )  =  ( X  .\/  Q ) )
2221breq2d 4464 . . . . . . 7  |-  ( x  =  X  ->  ( P  .<_  ( x  .\/  Q )  <->  P  .<_  ( X 
.\/  Q ) ) )
2320, 22anbi12d 710 . . . . . 6  |-  ( x  =  X  ->  (
( -.  P  .<_  x  /\  P  .<_  ( x 
.\/  Q ) )  <-> 
( -.  P  .<_  X  /\  P  .<_  ( X 
.\/  Q ) ) ) )
24 oveq1 6301 . . . . . . 7  |-  ( x  =  X  ->  (
x  .\/  P )  =  ( X  .\/  P ) )
2524breq2d 4464 . . . . . 6  |-  ( x  =  X  ->  ( Q  .<_  ( x  .\/  P )  <->  Q  .<_  ( X 
.\/  P ) ) )
2623, 25imbi12d 320 . . . . 5  |-  ( x  =  X  ->  (
( ( -.  P  .<_  x  /\  P  .<_  ( x  .\/  Q ) )  ->  Q  .<_  ( x  .\/  P ) )  <->  ( ( -.  P  .<_  X  /\  P  .<_  ( X  .\/  Q ) )  ->  Q  .<_  ( X  .\/  P
) ) ) )
2713, 18, 26rspc3v 3231 . . . 4  |-  ( ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  ->  ( A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) )  -> 
q  .<_  ( x  .\/  p ) )  -> 
( ( -.  P  .<_  X  /\  P  .<_  ( X  .\/  Q ) )  ->  Q  .<_  ( X  .\/  P ) ) ) )
286, 27mpan9 469 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( -.  P  .<_  X  /\  P  .<_  ( X  .\/  Q ) )  ->  Q  .<_  ( X  .\/  P
) ) )
2928exp4b 607 . 2  |-  ( K  e.  CvLat  ->  ( ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  ->  ( -.  P  .<_  X  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) ) ) )
30293imp 1190 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   Basecbs 14502   lecple 14574   joincjn 15443   Atomscatm 34353   AtLatcal 34354   CvLatclc 34355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-iota 5556  df-fv 5601  df-ov 6297  df-cvlat 34412
This theorem is referenced by:  cvlexch2  34419  cvlexchb1  34420  cvlexch3  34422  cvlcvr1  34429  hlexch1  34471
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