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Theorem atl0cl 29493
Description: An atomic lattice has a zero element. We can use this in place of op0cl 29374 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 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 atl0cl.z . . 3  |-  .0.  =  ( 0. `  K )
4 eqid 2283 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4isatl 29489 . 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 971 1  |-  ( K  e.  AtLat  ->  .0.  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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:  atl0le  29494  atlle0  29495  atlltn0  29496  isat3  29497  atnle0  29499  atlen0  29500  atcmp  29501  atcvreq0  29504  pmap0  29954  dia0  31242  dih0cnv  31473
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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