Theorem atlex 39274

Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 32394 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 2740	. . . . 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 39257	. . . 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 3009	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ 0 ↔ 𝑋 ≠ 0 ))
9		breq2 5170	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋))
10	9	rexbidv 3185	. . . . 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 3632	. . 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 1111	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   class class class wbr 5166  dom cdm 5700  ‘cfv 6575  Basecbs 17260  lecple 17320  glbcglb 18382  0.cp0 18495  Latclat 18503  Atomscatm 39221  AtLatcal 39222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-dm 5710  df-iota 6527  df-fv 6583  df-atl 39256

This theorem is referenced by:  atnle  39275  atlatmstc  39277  cvratlem  39380  cvrat4  39402  2llnmat  39483  2lnat  39743