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Theorem atllat 29490
Description: An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
Assertion
Ref Expression
atllat  |-  ( K  e.  AtLat  ->  K  e.  Lat )

Proof of Theorem atllat
Dummy variables  x  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2283 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 eqid 2283 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4isatl 29489 . 2  |-  ( K  e.  AtLat 
<->  ( K  e.  Lat  /\  ( 0. `  K
)  e.  ( Base `  K )  /\  A. x  e.  ( Base `  K ) ( x  =/=  ( 0. `  K )  ->  E. p  e.  ( Atoms `  K )
p ( le `  K ) x ) ) )
65simp1bi 970 1  |-  ( K  e.  AtLat  ->  K  e.  Lat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    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:  atlpos  29491  atnle  29507  atlatmstc  29509  cvllat  29516  hllat  29553  snatpsubN  29939
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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