Theorem atl0dm 39291

Description: Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.)

Hypotheses

Ref	Expression
atl01dm.b	⊢ 𝐵 = (Base‘𝐾)
atl01dm.u	⊢ 𝑈 = (lub‘𝐾)
atl01dm.g	⊢ 𝐺 = (glb‘𝐾)

Assertion

Ref	Expression
atl0dm	⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)

Proof of Theorem atl0dm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		atl01dm.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		atl01dm.g	. . 3 ⊢ 𝐺 = (glb‘𝐾)
3		eqid 2729	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2729	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2729	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 39288	. 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥)))
7	6	simp2bi 1146	1 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   class class class wbr 5092  dom cdm 5619  ‘cfv 6482  Basecbs 17120  lecple 17168  lubclub 18215  glbcglb 18216  0.cp0 18327  Latclat 18337  Atomscatm 39252  AtLatcal 39253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-dm 5629  df-iota 6438  df-fv 6490  df-atl 39287

This theorem is referenced by:  atl0cl  39292  atl0le  39293