Theorem atllat 40623

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 2763	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2763	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		eqid 2763	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2763	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2763	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 40622	. 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 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   class class class wbr 5108  dom cdm 5658  ‘cfv 6566  Basecbs 17431  lecple 17479  glbcglb 18534  0.cp0 18645  Latclat 18655  Atomscatm 40586  AtLatcal 40587

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-dm 5668  df-iota 6520  df-fv 6574  df-atl 40621

This theorem is used by:  atlpos  40624  atnle  40640  atlatmstc  40642  cvllat  40649  hllat  40686  snatpsubN  41073