Theorem atl0dm 40104

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 2762	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2762	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2762	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 40101	. 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥)))
7	6	simp2bi 1163	1 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088   class class class wbr 5108  dom cdm 5660  ‘cfv 6536  Basecbs 17275  lecple 17323  lubclub 18371  glbcglb 18372  0.cp0 18483  Latclat 18493  Atomscatm 40065  AtLatcal 40066

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-dm 5670  df-iota 6492  df-fv 6544  df-atl 40100

This theorem is used by:  atl0cl  40105  atl0le  40106