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Theorem atl0cl 30115
Description: An atomic lattice has a zero element. We can use this in place of op0cl 29996 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 2296 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 atl0cl.z . . 3  |-  .0.  =  ( 0. `  K )
4 eqid 2296 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4isatl 30111 . 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 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:  atl0le  30116  atlle0  30117  atlltn0  30118  isat3  30119  atnle0  30121  atlen0  30122  atcmp  30123  atcvreq0  30126  pmap0  30576  dia0  31864  dih0cnv  32095
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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