Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvlatl Structured version   Visualization version   GIF version

Theorem cvlatl 38797
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 2728 . . 3 (Baseโ€˜๐พ) = (Baseโ€˜๐พ)
2 eqid 2728 . . 3 (leโ€˜๐พ) = (leโ€˜๐พ)
3 eqid 2728 . . 3 (joinโ€˜๐พ) = (joinโ€˜๐พ)
4 eqid 2728 . . 3 (Atomsโ€˜๐พ) = (Atomsโ€˜๐พ)
51, 2, 3, 4iscvlat 38795 . 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 2099  โˆ€wral 3058   class class class wbr 5148  โ€˜cfv 6548  (class class class)co 7420  Basecbs 17180  lecple 17240  joincjn 18303  Atomscatm 38735  AtLatcal 38736  CvLatclc 38737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-cvlat 38794
This theorem is referenced by:  cvllat  38798  cvlexch3  38804  cvlexch4N  38805  cvlatexchb1  38806  cvlcvr1  38811  cvlcvrp  38812  cvlatcvr1  38813  cvlsupr2  38815  hlatl  38832
  Copyright terms: Public domain W3C validator