Theorem atllat 39323

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 2736	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2736	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		eqid 2736	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2736	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2736	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 39322	. 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 2933  ∀wral 3052  ∃wrex 3061   class class class wbr 5124  dom cdm 5659  ‘cfv 6536  Basecbs 17233  lecple 17283  glbcglb 18327  0.cp0 18438  Latclat 18446  Atomscatm 39286  AtLatcal 39287

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 2708

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 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-dm 5669  df-iota 6489  df-fv 6544  df-atl 39321

This theorem is referenced by:  atlpos  39324  atnle  39340  atlatmstc  39342  cvllat  39349  hllat  39386  snatpsubN  39774