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Theorem atllat 38772
Description: An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
Assertion
Ref Expression
atllat (๐พ โˆˆ AtLat โ†’ ๐พ โˆˆ Lat)

Proof of Theorem atllat
Dummy variables ๐‘ฅ ๐‘ 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 (glbโ€˜๐พ) = (glbโ€˜๐พ)
3 eqid 2728 . . 3 (leโ€˜๐พ) = (leโ€˜๐พ)
4 eqid 2728 . . 3 (0.โ€˜๐พ) = (0.โ€˜๐พ)
5 eqid 2728 . . 3 (Atomsโ€˜๐พ) = (Atomsโ€˜๐พ)
61, 2, 3, 4, 5isatl 38771 . 2 (๐พ โˆˆ AtLat โ†” (๐พ โˆˆ Lat โˆง (Baseโ€˜๐พ) โˆˆ dom (glbโ€˜๐พ) โˆง โˆ€๐‘ฅ โˆˆ (Baseโ€˜๐พ)(๐‘ฅ โ‰  (0.โ€˜๐พ) โ†’ โˆƒ๐‘ โˆˆ (Atomsโ€˜๐พ)๐‘(leโ€˜๐พ)๐‘ฅ)))
76simp1bi 1143 1 (๐พ โˆˆ AtLat โ†’ ๐พ โˆˆ Lat)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆˆ wcel 2099   โ‰  wne 2937  โˆ€wral 3058  โˆƒwrex 3067   class class class wbr 5148  dom cdm 5678  โ€˜cfv 6548  Basecbs 17180  lecple 17240  glbcglb 18302  0.cp0 18415  Latclat 18423  Atomscatm 38735  AtLatcal 38736
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-ne 2938  df-ral 3059  df-rex 3068  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-dm 5688  df-iota 6500  df-fv 6556  df-atl 38770
This theorem is referenced by:  atlpos  38773  atnle  38789  atlatmstc  38791  cvllat  38798  hllat  38835  snatpsubN  39223
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