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Theorem cvlexch1 30140
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 30135 . . . . 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 450 . . . 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 4042 . . . . . . . 8  |-  ( p  =  P  ->  (
p  .<_  x  <->  P  .<_  x ) )
87notbid 285 . . . . . . 7  |-  ( p  =  P  ->  ( -.  p  .<_  x  <->  -.  P  .<_  x ) )
9 breq1 4042 . . . . . . 7  |-  ( p  =  P  ->  (
p  .<_  ( x  .\/  q )  <->  P  .<_  ( x  .\/  q ) ) )
108, 9anbi12d 691 . . . . . 6  |-  ( p  =  P  ->  (
( -.  p  .<_  x  /\  p  .<_  ( x 
.\/  q ) )  <-> 
( -.  P  .<_  x  /\  P  .<_  ( x 
.\/  q ) ) ) )
11 oveq2 5882 . . . . . . 7  |-  ( p  =  P  ->  (
x  .\/  p )  =  ( x  .\/  P ) )
1211breq2d 4051 . . . . . 6  |-  ( p  =  P  ->  (
q  .<_  ( x  .\/  p )  <->  q  .<_  ( x  .\/  P ) ) )
1310, 12imbi12d 311 . . . . 5  |-  ( p  =  P  ->  (
( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) )  <->  ( ( -.  P  .<_  x  /\  P  .<_  ( x  .\/  q ) )  -> 
q  .<_  ( x  .\/  P ) ) ) )
14 oveq2 5882 . . . . . . . 8  |-  ( q  =  Q  ->  (
x  .\/  q )  =  ( x  .\/  Q ) )
1514breq2d 4051 . . . . . . 7  |-  ( q  =  Q  ->  ( P  .<_  ( x  .\/  q )  <->  P  .<_  ( x  .\/  Q ) ) )
1615anbi2d 684 . . . . . 6  |-  ( q  =  Q  ->  (
( -.  P  .<_  x  /\  P  .<_  ( x 
.\/  q ) )  <-> 
( -.  P  .<_  x  /\  P  .<_  ( x 
.\/  Q ) ) ) )
17 breq1 4042 . . . . . 6  |-  ( q  =  Q  ->  (
q  .<_  ( x  .\/  P )  <->  Q  .<_  ( x 
.\/  P ) ) )
1816, 17imbi12d 311 . . . . 5  |-  ( q  =  Q  ->  (
( ( -.  P  .<_  x  /\  P  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  P ) )  <->  ( ( -.  P  .<_  x  /\  P  .<_  ( x  .\/  Q ) )  ->  Q  .<_  ( x  .\/  P
) ) ) )
19 breq2 4043 . . . . . . . 8  |-  ( x  =  X  ->  ( P  .<_  x  <->  P  .<_  X ) )
2019notbid 285 . . . . . . 7  |-  ( x  =  X  ->  ( -.  P  .<_  x  <->  -.  P  .<_  X ) )
21 oveq1 5881 . . . . . . . 8  |-  ( x  =  X  ->  (
x  .\/  Q )  =  ( X  .\/  Q ) )
2221breq2d 4051 . . . . . . 7  |-  ( x  =  X  ->  ( P  .<_  ( x  .\/  Q )  <->  P  .<_  ( X 
.\/  Q ) ) )
2320, 22anbi12d 691 . . . . . 6  |-  ( x  =  X  ->  (
( -.  P  .<_  x  /\  P  .<_  ( x 
.\/  Q ) )  <-> 
( -.  P  .<_  X  /\  P  .<_  ( X 
.\/  Q ) ) ) )
24 oveq1 5881 . . . . . . 7  |-  ( x  =  X  ->  (
x  .\/  P )  =  ( X  .\/  P ) )
2524breq2d 4051 . . . . . 6  |-  ( x  =  X  ->  ( Q  .<_  ( x  .\/  P )  <->  Q  .<_  ( X 
.\/  P ) ) )
2623, 25imbi12d 311 . . . . 5  |-  ( x  =  X  ->  (
( ( -.  P  .<_  x  /\  P  .<_  ( x  .\/  Q ) )  ->  Q  .<_  ( x  .\/  P ) )  <->  ( ( -.  P  .<_  X  /\  P  .<_  ( X  .\/  Q ) )  ->  Q  .<_  ( X  .\/  P
) ) ) )
2713, 18, 26rspc3v 2906 . . . 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 455 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( -.  P  .<_  X  /\  P  .<_  ( X  .\/  Q ) )  ->  Q  .<_  ( X  .\/  P
) ) )
2928exp4b 590 . 2  |-  ( K  e.  CvLat  ->  ( ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  ->  ( -.  P  .<_  X  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) ) ) )
30293imp 1145 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Atomscatm 30075   AtLatcal 30076   CvLatclc 30077
This theorem is referenced by:  cvlexch2  30141  cvlexchb1  30142  cvlexch3  30144  cvlcvr1  30151  hlexch1  30193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-cvlat 30134
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