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Theorem isatl 30111
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 5541 . . . . . 6  |-  ( k  =  K  ->  ( 0. `  k )  =  ( 0. `  K
) )
2 isatlat.z . . . . . 6  |-  .0.  =  ( 0. `  K )
31, 2syl6eqr 2346 . . . . 5  |-  ( k  =  K  ->  ( 0. `  k )  =  .0.  )
4 fveq2 5541 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
5 isatlat.b . . . . . 6  |-  B  =  ( Base `  K
)
64, 5syl6eqr 2346 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
73, 6eleq12d 2364 . . . 4  |-  ( k  =  K  ->  (
( 0. `  k
)  e.  ( Base `  k )  <->  .0.  e.  B ) )
83neeq2d 2473 . . . . . 6  |-  ( k  =  K  ->  (
x  =/=  ( 0.
`  k )  <->  x  =/=  .0.  ) )
9 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
10 isatlat.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
119, 10syl6eqr 2346 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
12 fveq2 5541 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
13 isatlat.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
1412, 13syl6eqr 2346 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1514breqd 4050 . . . . . . 7  |-  ( k  =  K  ->  (
y ( le `  k ) x  <->  y  .<_  x ) )
1611, 15rexeqbidv 2762 . . . . . 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 2761 . . . 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 30110 . . 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 2938 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   0.cp0 14159   Latclat 14167   Atomscatm 30075   AtLatcal 30076
This theorem is referenced by:  atllat  30112  isatliN  30114  atl0cl  30115  atlex  30128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-atl 30110
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