Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atllat Unicode version

Theorem atllat 29783
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 2404 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2404 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2404 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 eqid 2404 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4isatl 29782 . 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 972 1  |-  ( K  e.  AtLat  ->  K  e.  Lat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    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:  atlpos  29784  atnle  29800  atlatmstc  29802  cvllat  29809  hllat  29846  snatpsubN  30232
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
  Copyright terms: Public domain W3C validator