Theorem atllat 35449

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 2777	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2777	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		eqid 2777	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2777	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2777	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 35448	. 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 1136	1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2106   ≠ wne 2968  ∀wral 3089  ∃wrex 3090   class class class wbr 4886  dom cdm 5355  ‘cfv 6135  Basecbs 16255  lecple 16345  glbcglb 17329  0.cp0 17423  Latclat 17431  Atomscatm 35412  AtLatcal 35413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-dm 5365  df-iota 6099  df-fv 6143  df-atl 35447

This theorem is referenced by:  atlpos  35450  atnle  35466  atlatmstc  35468  cvllat  35475  hllat  35512  snatpsubN  35899