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Theorem cvlatl 32600
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 2429 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2429 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2429 . . 3  |-  ( join `  K )  =  (
join `  K )
4 eqid 2429 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4iscvlat 32598 . 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 461 1  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    e. wcel 1870   A.wral 2782   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   Atomscatm 32538   AtLatcal 32539   CvLatclc 32540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-iota 5565  df-fv 5609  df-ov 6308  df-cvlat 32597
This theorem is referenced by:  cvllat  32601  cvlexch3  32607  cvlexch4N  32608  cvlatexchb1  32609  cvlcvr1  32614  cvlcvrp  32615  cvlatcvr1  32616  cvlsupr2  32618  hlatl  32635
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