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Theorem iscvlat 30058
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 5720 . . . 4  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
2 iscvlat.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2syl6eqr 2485 . . 3  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
4 fveq2 5720 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
5 iscvlat.b . . . . . 6  |-  B  =  ( Base `  K
)
64, 5syl6eqr 2485 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
7 fveq2 5720 . . . . . . . . . 10  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
8 iscvlat.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
97, 8syl6eqr 2485 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
109breqd 4215 . . . . . . . 8  |-  ( k  =  K  ->  (
p ( le `  k ) x  <->  p  .<_  x ) )
1110notbid 286 . . . . . . 7  |-  ( k  =  K  ->  ( -.  p ( le `  k ) x  <->  -.  p  .<_  x ) )
12 eqidd 2436 . . . . . . . 8  |-  ( k  =  K  ->  p  =  p )
13 fveq2 5720 . . . . . . . . . 10  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
14 iscvlat.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
1513, 14syl6eqr 2485 . . . . . . . . 9  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1615oveqd 6090 . . . . . . . 8  |-  ( k  =  K  ->  (
x ( join `  k
) q )  =  ( x  .\/  q
) )
1712, 9, 16breq123d 4218 . . . . . . 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 2436 . . . . . . 7  |-  ( k  =  K  ->  q  =  q )
2015oveqd 6090 . . . . . . 7  |-  ( k  =  K  ->  (
x ( join `  k
) p )  =  ( x  .\/  p
) )
2119, 9, 20breq123d 4218 . . . . . 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 2908 . . . 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 2908 . . 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 2908 . 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 30057 . 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 3086 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 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   Atomscatm 29998   AtLatcal 29999   CvLatclc 30000
This theorem is referenced by:  iscvlat2N  30059  cvlatl  30060  cvlexch1  30063  ishlat2  30088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-cvlat 30057
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