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Theorem cvlatl 29515
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 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2283 . . 3  |-  ( join `  K )  =  (
join `  K )
4 eqid 2283 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4iscvlat 29513 . 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 446 1  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Atomscatm 29453   AtLatcal 29454   CvLatclc 29455
This theorem is referenced by:  cvllat  29516  cvlexch3  29522  cvlexch4N  29523  cvlatexchb1  29524  cvlcvr1  29529  cvlcvrp  29530  cvlatcvr1  29531  cvlsupr2  29533  hlatl  29550
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-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-ov 5861  df-cvlat 29512
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