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Theorem atl0cl 29420
Description: An atomic lattice has a zero element. We can use this in place of op0cl 29301 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
atl0cl.b  |-  B  =  ( Base `  K
)
atl0cl.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
atl0cl  |-  ( K  e.  AtLat  ->  .0.  e.  B )

Proof of Theorem atl0cl
Dummy variables  x  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 atl0cl.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2389 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 atl0cl.z . . 3  |-  .0.  =  ( 0. `  K )
4 eqid 2389 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4isatl 29416 . 2  |-  ( K  e.  AtLat 
<->  ( K  e.  Lat  /\  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. p  e.  ( Atoms `  K ) p ( le `  K ) x ) ) )
65simp2bi 973 1  |-  ( K  e.  AtLat  ->  .0.  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   E.wrex 2652   class class class wbr 4155   ` cfv 5396   Basecbs 13398   lecple 13465   0.cp0 14395   Latclat 14403   Atomscatm 29380   AtLatcal 29381
This theorem is referenced by:  atl0le  29421  atlle0  29422  atlltn0  29423  isat3  29424  atnle0  29426  atlen0  29427  atcmp  29428  atcvreq0  29431  pmap0  29881  dia0  31169  dih0cnv  31400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-iota 5360  df-fv 5404  df-atl 29415
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