Theorem atl0dm 39803

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 2739	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2739	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2739	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 39800	. 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥)))
7	6	simp2bi 1152	1 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063   class class class wbr 5073  dom cdm 5619  ‘cfv 6486  Basecbs 17171  lecple 17219  lubclub 18267  glbcglb 18268  0.cp0 18379  Latclat 18389  Atomscatm 39764  AtLatcal 39765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-dm 5629  df-iota 6442  df-fv 6494  df-atl 39799

This theorem is referenced by:  atl0cl  39804  atl0le  39805