Theorem atlex 36612

Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 30143 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 2798	. . . . 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 36595	. . . 4 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))
7	6	simp3bi 1144	. . 3 ⊢ (𝐾 ∈ AtLat → ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
8		neeq1 3049	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ 0 ↔ 𝑋 ≠ 0 ))
9		breq2 5034	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋))
10	9	rexbidv 3256	. . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋))
11	8, 10	imbi12d 348	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ↔ (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)))
12	11	rspccv 3568	. . 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 1108	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   class class class wbr 5030  dom cdm 5519  ‘cfv 6324  Basecbs 16475  lecple 16564  glbcglb 17545  0.cp0 17639  Latclat 17647  Atomscatm 36559  AtLatcal 36560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-dm 5529  df-iota 6283  df-fv 6332  df-atl 36594

This theorem is referenced by:  atnle  36613  atlatmstc  36615  cvratlem  36717  cvrat4  36739  2llnmat  36820  2lnat  37080