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Theorem isatl 29998
Description: The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
isatlat.b  |-  B  =  ( Base `  K
)
isatlat.l  |-  .<_  =  ( le `  K )
isatlat.z  |-  .0.  =  ( 0. `  K )
isatlat.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
isatl  |-  ( K  e.  AtLat 
<->  ( K  e.  Lat  /\  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x ) ) )
Distinct variable groups:    y, A    x, B    x, y, K
Allowed substitution hints:    A( x)    B( y)   
.<_ ( x, y)    .0. ( x, y)

Proof of Theorem isatl
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . . . 6  |-  ( k  =  K  ->  ( 0. `  k )  =  ( 0. `  K
) )
2 isatlat.z . . . . . 6  |-  .0.  =  ( 0. `  K )
31, 2syl6eqr 2485 . . . . 5  |-  ( k  =  K  ->  ( 0. `  k )  =  .0.  )
4 fveq2 5720 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
5 isatlat.b . . . . . 6  |-  B  =  ( Base `  K
)
64, 5syl6eqr 2485 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
73, 6eleq12d 2503 . . . 4  |-  ( k  =  K  ->  (
( 0. `  k
)  e.  ( Base `  k )  <->  .0.  e.  B ) )
83neeq2d 2612 . . . . . 6  |-  ( k  =  K  ->  (
x  =/=  ( 0.
`  k )  <->  x  =/=  .0.  ) )
9 fveq2 5720 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
10 isatlat.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
119, 10syl6eqr 2485 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
12 fveq2 5720 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
13 isatlat.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
1412, 13syl6eqr 2485 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1514breqd 4215 . . . . . . 7  |-  ( k  =  K  ->  (
y ( le `  k ) x  <->  y  .<_  x ) )
1611, 15rexeqbidv 2909 . . . . . 6  |-  ( k  =  K  ->  ( E. y  e.  ( Atoms `  k ) y ( le `  k
) x  <->  E. y  e.  A  y  .<_  x ) )
178, 16imbi12d 312 . . . . 5  |-  ( k  =  K  ->  (
( x  =/=  ( 0. `  k )  ->  E. y  e.  ( Atoms `  k ) y ( le `  k
) x )  <->  ( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x ) ) )
186, 17raleqbidv 2908 . . . 4  |-  ( k  =  K  ->  ( A. x  e.  ( Base `  k ) ( x  =/=  ( 0.
`  k )  ->  E. y  e.  ( Atoms `  k ) y ( le `  k
) x )  <->  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x ) ) )
197, 18anbi12d 692 . . 3  |-  ( k  =  K  ->  (
( ( 0. `  k )  e.  (
Base `  k )  /\  A. x  e.  (
Base `  k )
( x  =/=  ( 0. `  k )  ->  E. y  e.  ( Atoms `  k ) y ( le `  k
) x ) )  <-> 
(  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x ) ) ) )
20 df-atl 29997 . . 3  |-  AtLat  =  {
k  e.  Lat  | 
( ( 0. `  k )  e.  (
Base `  k )  /\  A. x  e.  (
Base `  k )
( x  =/=  ( 0. `  k )  ->  E. y  e.  ( Atoms `  k ) y ( le `  k
) x ) ) }
2119, 20elrab2 3086 . 2  |-  ( K  e.  AtLat 
<->  ( K  e.  Lat  /\  (  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x ) ) ) )
22 3anass 940 . 2  |-  ( ( K  e.  Lat  /\  .0.  e.  B  /\  A. x  e.  B  (
x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x ) )  <->  ( K  e.  Lat  /\  (  .0. 
e.  B  /\  A. x  e.  B  (
x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x ) ) ) )
2321, 22bitr4i 244 1  |-  ( K  e.  AtLat 
<->  ( K  e.  Lat  /\  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   class class class wbr 4204   ` cfv 5446   Basecbs 13459   lecple 13526   0.cp0 14456   Latclat 14464   Atomscatm 29962   AtLatcal 29963
This theorem is referenced by:  atllat  29999  isatliN  30001  atl0cl  30002  atlex  30015
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-ne 2600  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-atl 29997
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