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Theorem cvlatl 29333
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 2316 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2316 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2316 . . 3  |-  ( join `  K )  =  (
join `  K )
4 eqid 2316 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
51, 2, 3, 4iscvlat 29331 . 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 1701   A.wral 2577   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127   Atomscatm 29271   AtLatcal 29272   CvLatclc 29273
This theorem is referenced by:  cvllat  29334  cvlexch3  29340  cvlexch4N  29341  cvlatexchb1  29342  cvlcvr1  29347  cvlcvrp  29348  cvlatcvr1  29349  cvlsupr2  29351  hlatl  29368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903  df-cvlat 29330
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