Theorem atlex 37330

Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 30722 analog.) (Contributed by NM, 21-Oct-2011.)

Hypotheses

Ref	Expression
atlex.b	⊢ 𝐵 = (Base‘𝐾)
atlex.l	⊢ ≤ = (le‘𝐾)
atlex.z	⊢ 0 = (0.‘𝐾)
atlex.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
atlex	⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐾   𝑦,𝑋

Allowed substitution hints:   𝐵(𝑦)   ≤ (𝑦)   0 (𝑦)

Proof of Theorem atlex

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		atlex.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2738	. . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		atlex.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		atlex.z	. . . . 5 ⊢ 0 = (0.‘𝐾)
5		atlex.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	isatl 37313	. . . 4 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))
7	6	simp3bi 1146	. . 3 ⊢ (𝐾 ∈ AtLat → ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
8		neeq1 3006	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ 0 ↔ 𝑋 ≠ 0 ))
9		breq2 5078	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋))
10	9	rexbidv 3226	. . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋))
11	8, 10	imbi12d 345	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ↔ (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)))
12	11	rspccv 3558	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → (𝑋 ∈ 𝐵 → (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)))
13	7, 12	syl 17	. 2 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝐵 → (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)))
14	13	3imp 1110	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   class class class wbr 5074  dom cdm 5589  ‘cfv 6433  Basecbs 16912  lecple 16969  glbcglb 18028  0.cp0 18141  Latclat 18149  Atomscatm 37277  AtLatcal 37278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-dm 5599  df-iota 6391  df-fv 6441  df-atl 37312

This theorem is referenced by:  atnle  37331  atlatmstc  37333  cvratlem  37435  cvrat4  37457  2llnmat  37538  2lnat  37798