Theorem atllat 39293

Description: An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)

Assertion

Ref	Expression
atllat	⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)

Proof of Theorem atllat

Dummy variables 𝑥 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2729	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		eqid 2729	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2729	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2729	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 39292	. 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ (Base‘𝐾) ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥 ≠ (0.‘𝐾) → ∃𝑝 ∈ (Atoms‘𝐾)𝑝(le‘𝐾)𝑥)))
7	6	simp1bi 1145	1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   class class class wbr 5107  dom cdm 5638  ‘cfv 6511  Basecbs 17179  lecple 17227  glbcglb 18271  0.cp0 18382  Latclat 18390  Atomscatm 39256  AtLatcal 39257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-dm 5648  df-iota 6464  df-fv 6519  df-atl 39291

This theorem is referenced by:  atlpos  39294  atnle  39310  atlatmstc  39312  cvllat  39319  hllat  39356  snatpsubN  39744