Theorem atllat 40107

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 2765	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2765	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		eqid 2765	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2765	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2765	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 40106	. 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 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   class class class wbr 5111  dom cdm 5663  ‘cfv 6540  Basecbs 17287  lecple 17335  glbcglb 18384  0.cp0 18495  Latclat 18505  Atomscatm 40070  AtLatcal 40071

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-dm 5673  df-iota 6496  df-fv 6548  df-atl 40105

This theorem is used by:  atlpos  40108  atnle  40124  atlatmstc  40126  cvllat  40133  hllat  40170  snatpsubN  40557