Theorem atllat 39959

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 2769	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2769	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		eqid 2769	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2769	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2769	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 39958	. 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 1161	1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   class class class wbr 5110  dom cdm 5659  ‘cfv 6533  Basecbs 17265  lecple 17313  glbcglb 18362  0.cp0 18473  Latclat 18483  Atomscatm 39922  AtLatcal 39923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-dm 5669  df-iota 6489  df-fv 6541  df-atl 39957

This theorem is referenced by:  atlpos  39960  atnle  39976  atlatmstc  39978  cvllat  39985  hllat  40022  snatpsubN  40409