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Theorem cvlatl 38706
Description: An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
cvlatl (๐พ โˆˆ CvLat โ†’ ๐พ โˆˆ AtLat)

Proof of Theorem cvlatl
Dummy variables ๐‘ž ๐‘ ๐‘ฅ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . . 3 (Baseโ€˜๐พ) = (Baseโ€˜๐พ)
2 eqid 2726 . . 3 (leโ€˜๐พ) = (leโ€˜๐พ)
3 eqid 2726 . . 3 (joinโ€˜๐พ) = (joinโ€˜๐พ)
4 eqid 2726 . . 3 (Atomsโ€˜๐พ) = (Atomsโ€˜๐พ)
51, 2, 3, 4iscvlat 38704 . 2 (๐พ โˆˆ CvLat โ†” (๐พ โˆˆ AtLat โˆง โˆ€๐‘ โˆˆ (Atomsโ€˜๐พ)โˆ€๐‘ž โˆˆ (Atomsโ€˜๐พ)โˆ€๐‘ฅ โˆˆ (Baseโ€˜๐พ)((ยฌ ๐‘(leโ€˜๐พ)๐‘ฅ โˆง ๐‘(leโ€˜๐พ)(๐‘ฅ(joinโ€˜๐พ)๐‘ž)) โ†’ ๐‘ž(leโ€˜๐พ)(๐‘ฅ(joinโ€˜๐พ)๐‘))))
65simplbi 497 1 (๐พ โˆˆ CvLat โ†’ ๐พ โˆˆ AtLat)
Colors of variables: wff setvar class
Syntax hints:  ยฌ wn 3   โ†’ wi 4   โˆง wa 395   โˆˆ wcel 2098  โˆ€wral 3055   class class class wbr 5141  โ€˜cfv 6536  (class class class)co 7404  Basecbs 17151  lecple 17211  joincjn 18274  Atomscatm 38644  AtLatcal 38645  CvLatclc 38646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-cvlat 38703
This theorem is referenced by:  cvllat  38707  cvlexch3  38713  cvlexch4N  38714  cvlatexchb1  38715  cvlcvr1  38720  cvlcvrp  38721  cvlatcvr1  38722  cvlsupr2  38724  hlatl  38741
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