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Theorem atl0cl 29786
Description: An atomic lattice has a zero element. We can use this in place of op0cl 29667 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 2404 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 atl0cl.z . . 3  |-  .0.  =  ( 0. `  K )
4 eqid 2404 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4isatl 29782 . 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 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   0.cp0 14421   Latclat 14429   Atomscatm 29746   AtLatcal 29747
This theorem is referenced by:  atl0le  29787  atlle0  29788  atlltn0  29789  isat3  29790  atnle0  29792  atlen0  29793  atcmp  29794  atcvreq0  29797  pmap0  30247  dia0  31535  dih0cnv  31766
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-atl 29781
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