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Theorem iscvlat 29513
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 5525 . . . 4  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
2 iscvlat.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2syl6eqr 2333 . . 3  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
4 fveq2 5525 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
5 iscvlat.b . . . . . 6  |-  B  =  ( Base `  K
)
64, 5syl6eqr 2333 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
7 fveq2 5525 . . . . . . . . . 10  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
8 iscvlat.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
97, 8syl6eqr 2333 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
109breqd 4034 . . . . . . . 8  |-  ( k  =  K  ->  (
p ( le `  k ) x  <->  p  .<_  x ) )
1110notbid 285 . . . . . . 7  |-  ( k  =  K  ->  ( -.  p ( le `  k ) x  <->  -.  p  .<_  x ) )
12 eqidd 2284 . . . . . . . 8  |-  ( k  =  K  ->  p  =  p )
13 fveq2 5525 . . . . . . . . . 10  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
14 iscvlat.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
1513, 14syl6eqr 2333 . . . . . . . . 9  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1615oveqd 5875 . . . . . . . 8  |-  ( k  =  K  ->  (
x ( join `  k
) q )  =  ( x  .\/  q
) )
1712, 9, 16breq123d 4037 . . . . . . 7  |-  ( k  =  K  ->  (
p ( le `  k ) ( x ( join `  k
) q )  <->  p  .<_  ( x  .\/  q ) ) )
1811, 17anbi12d 691 . . . . . 6  |-  ( k  =  K  ->  (
( -.  p ( le `  k ) x  /\  p ( le `  k ) ( x ( join `  k ) q ) )  <->  ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) ) ) )
19 eqidd 2284 . . . . . . 7  |-  ( k  =  K  ->  q  =  q )
2015oveqd 5875 . . . . . . 7  |-  ( k  =  K  ->  (
x ( join `  k
) p )  =  ( x  .\/  p
) )
2119, 9, 20breq123d 4037 . . . . . 6  |-  ( k  =  K  ->  (
q ( le `  k ) ( x ( join `  k
) p )  <->  q  .<_  ( x  .\/  p ) ) )
2218, 21imbi12d 311 . . . . 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 2748 . . . 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 2748 . . 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 2748 . 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 29512 . 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 2925 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Atomscatm 29453   AtLatcal 29454   CvLatclc 29455
This theorem is referenced by:  iscvlat2N  29514  cvlatl  29515  cvlexch1  29518  ishlat2  29543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-cvlat 29512
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