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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 ∧ 𝑋𝐵𝑋0 ) → ∃𝑦𝐴 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.
StepHypRef 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)
61, 2, 3, 4, 5isatl 39257 . . . 4 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
76simp3bi 1147 . . 3 (𝐾 ∈ AtLat → ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))
8 neeq1 3009 . . . . 5 (𝑥 = 𝑋 → (𝑥0𝑋0 ))
9 breq2 5170 . . . . . 6 (𝑥 = 𝑋 → (𝑦 𝑥𝑦 𝑋))
109rexbidv 3185 . . . . 5 (𝑥 = 𝑋 → (∃𝑦𝐴 𝑦 𝑥 ↔ ∃𝑦𝐴 𝑦 𝑋))
118, 10imbi12d 344 . . . 4 (𝑥 = 𝑋 → ((𝑥0 → ∃𝑦𝐴 𝑦 𝑥) ↔ (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
1211rspccv 3632 . . 3 (∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥) → (𝑋𝐵 → (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
137, 12syl 17 . 2 (𝐾 ∈ AtLat → (𝑋𝐵 → (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
14133imp 1111 1 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑦𝐴 𝑦 𝑋)
Colors of variables: wff setvar class
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
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