Theorem atl0dm 38904

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 2725	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2725	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2725	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 38901	. 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥)))
7	6	simp2bi 1143	1 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050  ∃wrex 3059   class class class wbr 5149  dom cdm 5678  ‘cfv 6549  Basecbs 17183  lecple 17243  lubclub 18304  glbcglb 18305  0.cp0 18418  Latclat 18426  Atomscatm 38865  AtLatcal 38866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-dm 5688  df-iota 6501  df-fv 6557  df-atl 38900

This theorem is referenced by:  atl0cl  38905  atl0le  38906