Theorem atllat 40451

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 2760	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2760	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		eqid 2760	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2760	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2760	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 40450	. 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 1163	1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   class class class wbr 5102  dom cdm 5650  ‘cfv 6539  Basecbs 17393  lecple 17441  glbcglb 18490  0.cp0 18601  Latclat 18611  Atomscatm 40414  AtLatcal 40415

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-dm 5660  df-iota 6494  df-fv 6547  df-atl 40449

This theorem is used by:  atlpos  40452  atnle  40468  atlatmstc  40470  cvllat  40477  hllat  40514  snatpsubN  40901