Theorem atl0dm 36437

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 2821	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2821	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		eqid 2821	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 36434	. 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥)))
7	6	simp2bi 1142	1 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   class class class wbr 5065  dom cdm 5554  ‘cfv 6354  Basecbs 16482  lecple 16571  lubclub 17551  glbcglb 17552  0.cp0 17646  Latclat 17654  Atomscatm 36398  AtLatcal 36399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-dm 5564  df-iota 6313  df-fv 6362  df-atl 36433

This theorem is referenced by:  atl0cl  36438  atl0le  36439