Theorem atllat 39760

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 2737	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2737	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		eqid 2737	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2737	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2737	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 39759	. 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 1146	1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   class class class wbr 5086  dom cdm 5624  ‘cfv 6492  Basecbs 17170  lecple 17218  glbcglb 18267  0.cp0 18378  Latclat 18388  Atomscatm 39723  AtLatcal 39724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-dm 5634  df-iota 6448  df-fv 6500  df-atl 39758

This theorem is referenced by:  atlpos  39761  atnle  39777  atlatmstc  39779  cvllat  39786  hllat  39823  snatpsubN  40210