Theorem atllat 38674

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 2724	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2724	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		eqid 2724	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2724	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2724	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 38673	. 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 1142	1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2098   ≠ wne 2932  ∀wral 3053  ∃wrex 3062   class class class wbr 5139  dom cdm 5667  ‘cfv 6534  Basecbs 17149  lecple 17209  glbcglb 18271  0.cp0 18384  Latclat 18392  Atomscatm 38637  AtLatcal 38638

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 2695

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 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-dm 5677  df-iota 6486  df-fv 6542  df-atl 38672

This theorem is referenced by:  atlpos  38675  atnle  38691  atlatmstc  38693  cvllat  38700  hllat  38737  snatpsubN  39125