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Theorem cvlatl 33309
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 2454 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2454 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2454 . . 3  |-  ( join `  K )  =  (
join `  K )
4 eqid 2454 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4iscvlat 33307 . 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 1758   A.wral 2799   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   Atomscatm 33247   AtLatcal 33248   CvLatclc 33249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-iota 5490  df-fv 5535  df-ov 6204  df-cvlat 33306
This theorem is referenced by:  cvllat  33310  cvlexch3  33316  cvlexch4N  33317  cvlatexchb1  33318  cvlcvr1  33323  cvlcvrp  33324  cvlatcvr1  33325  cvlsupr2  33327  hlatl  33344
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