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Theorem atl0cl 30002
Description: An atomic lattice has a zero element. We can use this in place of op0cl 29883 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 2435 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 atl0cl.z . . 3  |-  .0.  =  ( 0. `  K )
4 eqid 2435 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4isatl 29998 . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   class class class wbr 4204   ` cfv 5446   Basecbs 13459   lecple 13526   0.cp0 14456   Latclat 14464   Atomscatm 29962   AtLatcal 29963
This theorem is referenced by:  atl0le  30003  atlle0  30004  atlltn0  30005  isat3  30006  atnle0  30008  atlen0  30009  atcmp  30010  atcvreq0  30013  pmap0  30463  dia0  31751  dih0cnv  31982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-atl 29997
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