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Theorem cvlatl 34339
Description: An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
cvlatl  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)

Proof of Theorem cvlatl
Dummy variables  q  p  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2467 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2467 . . 3  |-  ( join `  K )  =  (
join `  K )
4 eqid 2467 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4iscvlat 34337 . 2  |-  ( K  e.  CvLat 
<->  ( K  e.  AtLat  /\ 
A. p  e.  (
Atoms `  K ) A. q  e.  ( Atoms `  K ) A. x  e.  ( Base `  K
) ( ( -.  p ( le `  K ) x  /\  p ( le `  K ) ( x ( join `  K
) q ) )  ->  q ( le
`  K ) ( x ( join `  K
) p ) ) ) )
65simplbi 460 1  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1767   A.wral 2814   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   Atomscatm 34277   AtLatcal 34278   CvLatclc 34279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6288  df-cvlat 34336
This theorem is referenced by:  cvllat  34340  cvlexch3  34346  cvlexch4N  34347  cvlatexchb1  34348  cvlcvr1  34353  cvlcvrp  34354  cvlatcvr1  34355  cvlsupr2  34357  hlatl  34374
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