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Theorem isatl 29489
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 5525 . . . . . 6  |-  ( k  =  K  ->  ( 0. `  k )  =  ( 0. `  K
) )
2 isatlat.z . . . . . 6  |-  .0.  =  ( 0. `  K )
31, 2syl6eqr 2333 . . . . 5  |-  ( k  =  K  ->  ( 0. `  k )  =  .0.  )
4 fveq2 5525 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
5 isatlat.b . . . . . 6  |-  B  =  ( Base `  K
)
64, 5syl6eqr 2333 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
73, 6eleq12d 2351 . . . 4  |-  ( k  =  K  ->  (
( 0. `  k
)  e.  ( Base `  k )  <->  .0.  e.  B ) )
83neeq2d 2460 . . . . . 6  |-  ( k  =  K  ->  (
x  =/=  ( 0.
`  k )  <->  x  =/=  .0.  ) )
9 fveq2 5525 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
10 isatlat.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
119, 10syl6eqr 2333 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
12 fveq2 5525 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
13 isatlat.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
1412, 13syl6eqr 2333 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1514breqd 4034 . . . . . . 7  |-  ( k  =  K  ->  (
y ( le `  k ) x  <->  y  .<_  x ) )
1611, 15rexeqbidv 2749 . . . . . 6  |-  ( k  =  K  ->  ( E. y  e.  ( Atoms `  k ) y ( le `  k
) x  <->  E. y  e.  A  y  .<_  x ) )
178, 16imbi12d 311 . . . . 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 2748 . . . 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 691 . . 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 29488 . . 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 2925 . 2  |-  ( K  e.  AtLat 
<->  ( K  e.  Lat  /\  (  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x ) ) ) )
22 3anass 938 . 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 243 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   0.cp0 14143   Latclat 14151   Atomscatm 29453   AtLatcal 29454
This theorem is referenced by:  atllat  29490  isatliN  29492  atl0cl  29493  atlex  29506
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-ne 2448  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-atl 29488
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