Theorem atlex 39282

Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 32262 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 2729	. . . . 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 39265	. . . 4 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))
7	6	simp3bi 1147	. . 3 ⊢ (𝐾 ∈ AtLat → ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
8		neeq1 2987	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ 0 ↔ 𝑋 ≠ 0 ))
9		breq2 5106	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋))
10	9	rexbidv 3157	. . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋))
11	8, 10	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ↔ (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)))
12	11	rspccv 3582	. . 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 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   class class class wbr 5102  dom cdm 5631  ‘cfv 6499  Basecbs 17155  lecple 17203  glbcglb 18247  0.cp0 18358  Latclat 18366  Atomscatm 39229  AtLatcal 39230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-dm 5641  df-iota 6452  df-fv 6507  df-atl 39264

This theorem is referenced by:  atnle  39283  atlatmstc  39285  cvratlem  39388  cvrat4  39410  2llnmat  39491  2lnat  39751