Theorem atlex 40353

Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 32955 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 2761	. . . . 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 40336	. . . 4 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))
7	6	simp3bi 1165	. . 3 ⊢ (𝐾 ∈ AtLat → ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
8		neeq1 3018	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ 0 ↔ 𝑋 ≠ 0 ))
9		breq2 5107	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋))
10	9	rexbidv 3187	. . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋))
11	8, 10	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ↔ (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)))
12	11	rspccv 3574	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → (𝑋 ∈ 𝐵 → (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)))
13	7, 12	syl 18	. 2 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝐵 → (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)))
14	13	3imp 1128	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   class class class wbr 5103  dom cdm 5651  ‘cfv 6537  Basecbs 17380  lecple 17428  glbcglb 18477  0.cp0 18588  Latclat 18598  Atomscatm 40300  AtLatcal 40301

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-dm 5661  df-iota 6493  df-fv 6545  df-atl 40335

This theorem is used by:  atnle  40354  atlatmstc  40356  cvratlem  40458  cvrat4  40480  2llnmat  40561  2lnat  40821