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Theorem atlex 35123
Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 29559 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 2771 . . . . 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 35106 . . . 4 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
76simp3bi 1141 . . 3 (𝐾 ∈ AtLat → ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))
8 neeq1 3005 . . . . 5 (𝑥 = 𝑋 → (𝑥0𝑋0 ))
9 breq2 4791 . . . . . 6 (𝑥 = 𝑋 → (𝑦 𝑥𝑦 𝑋))
109rexbidv 3200 . . . . 5 (𝑥 = 𝑋 → (∃𝑦𝐴 𝑦 𝑥 ↔ ∃𝑦𝐴 𝑦 𝑋))
118, 10imbi12d 333 . . . 4 (𝑥 = 𝑋 → ((𝑥0 → ∃𝑦𝐴 𝑦 𝑥) ↔ (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
1211rspccv 3457 . . 3 (∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥) → (𝑋𝐵 → (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
137, 12syl 17 . 2 (𝐾 ∈ AtLat → (𝑋𝐵 → (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
14133imp 1101 1 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑦𝐴 𝑦 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062   class class class wbr 4787  dom cdm 5250  cfv 6030  Basecbs 16064  lecple 16156  glbcglb 17151  0.cp0 17245  Latclat 17253  Atomscatm 35070  AtLatcal 35071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-dm 5260  df-iota 5993  df-fv 6038  df-atl 35105
This theorem is referenced by:  atnle  35124  atlatmstc  35126  cvratlem  35228  cvrat4  35250  2llnmat  35331  2lnat  35591
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