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Theorem iscvlat 29438
Description: The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
iscvlat.b  |-  B  =  ( Base `  K
)
iscvlat.l  |-  .<_  =  ( le `  K )
iscvlat.j  |-  .\/  =  ( join `  K )
iscvlat.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
iscvlat  |-  ( 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 ) ) ) )
Distinct variable groups:    q, p, A    x, B    x, p, K, q
Allowed substitution hints:    A( x)    B( q, p)    .\/ ( x, q, p)    .<_ ( x, q, p)

Proof of Theorem iscvlat
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fveq2 5668 . . . 4  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
2 iscvlat.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2syl6eqr 2437 . . 3  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
4 fveq2 5668 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
5 iscvlat.b . . . . . 6  |-  B  =  ( Base `  K
)
64, 5syl6eqr 2437 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
7 fveq2 5668 . . . . . . . . . 10  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
8 iscvlat.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
97, 8syl6eqr 2437 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
109breqd 4164 . . . . . . . 8  |-  ( k  =  K  ->  (
p ( le `  k ) x  <->  p  .<_  x ) )
1110notbid 286 . . . . . . 7  |-  ( k  =  K  ->  ( -.  p ( le `  k ) x  <->  -.  p  .<_  x ) )
12 eqidd 2388 . . . . . . . 8  |-  ( k  =  K  ->  p  =  p )
13 fveq2 5668 . . . . . . . . . 10  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
14 iscvlat.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
1513, 14syl6eqr 2437 . . . . . . . . 9  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1615oveqd 6037 . . . . . . . 8  |-  ( k  =  K  ->  (
x ( join `  k
) q )  =  ( x  .\/  q
) )
1712, 9, 16breq123d 4167 . . . . . . 7  |-  ( k  =  K  ->  (
p ( le `  k ) ( x ( join `  k
) q )  <->  p  .<_  ( x  .\/  q ) ) )
1811, 17anbi12d 692 . . . . . 6  |-  ( k  =  K  ->  (
( -.  p ( le `  k ) x  /\  p ( le `  k ) ( x ( join `  k ) q ) )  <->  ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) ) ) )
19 eqidd 2388 . . . . . . 7  |-  ( k  =  K  ->  q  =  q )
2015oveqd 6037 . . . . . . 7  |-  ( k  =  K  ->  (
x ( join `  k
) p )  =  ( x  .\/  p
) )
2119, 9, 20breq123d 4167 . . . . . 6  |-  ( k  =  K  ->  (
q ( le `  k ) ( x ( join `  k
) p )  <->  q  .<_  ( x  .\/  p ) ) )
2218, 21imbi12d 312 . . . . 5  |-  ( k  =  K  ->  (
( ( -.  p
( le `  k
) x  /\  p
( le `  k
) ( x (
join `  k )
q ) )  -> 
q ( le `  k ) ( x ( join `  k
) p ) )  <-> 
( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
236, 22raleqbidv 2859 . . . 4  |-  ( k  =  K  ->  ( A. x  e.  ( Base `  k ) ( ( -.  p ( le `  k ) x  /\  p ( le `  k ) ( x ( join `  k ) q ) )  ->  q ( le `  k ) ( x ( join `  k
) p ) )  <->  A. x  e.  B  ( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
243, 23raleqbidv 2859 . . 3  |-  ( k  =  K  ->  ( A. q  e.  ( Atoms `  k ) A. x  e.  ( Base `  k ) ( ( -.  p ( le
`  k ) x  /\  p ( le
`  k ) ( x ( join `  k
) q ) )  ->  q ( le
`  k ) ( x ( join `  k
) p ) )  <->  A. q  e.  A  A. x  e.  B  ( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
253, 24raleqbidv 2859 . 2  |-  ( k  =  K  ->  ( A. p  e.  ( Atoms `  k ) A. q  e.  ( Atoms `  k ) A. x  e.  ( Base `  k
) ( ( -.  p ( le `  k ) x  /\  p ( le `  k ) ( x ( join `  k
) q ) )  ->  q ( le
`  k ) ( x ( join `  k
) p ) )  <->  A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
26 df-cvlat 29437 . 2  |-  CvLat  =  {
k  e.  AtLat  |  A. p  e.  ( Atoms `  k ) A. q  e.  ( Atoms `  k ) A. x  e.  ( Base `  k ) ( ( -.  p ( le `  k ) x  /\  p ( le `  k ) ( x ( join `  k ) q ) )  ->  q ( le `  k ) ( x ( join `  k
) p ) ) }
2725, 26elrab2 3037 1  |-  ( 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 ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   Atomscatm 29378   AtLatcal 29379   CvLatclc 29380
This theorem is referenced by:  iscvlat2N  29439  cvlatl  29440  cvlexch1  29443  ishlat2  29468
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
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-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023  df-cvlat 29437
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