Theorem atllat 36430

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 2821	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2821	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		eqid 2821	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2821	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2821	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 36429	. 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 1141	1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   class class class wbr 5059  dom cdm 5550  ‘cfv 6350  Basecbs 16477  lecple 16566  glbcglb 17547  0.cp0 17641  Latclat 17649  Atomscatm 36393  AtLatcal 36394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-dm 5560  df-iota 6309  df-fv 6358  df-atl 36428

This theorem is referenced by:  atlpos  36431  atnle  36447  atlatmstc  36449  cvllat  36456  hllat  36493  snatpsubN  36880