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Theorem atlex 34083
Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 29068 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 2621 . . . . 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 34066 . . . 4 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
76simp3bi 1076 . . 3 (𝐾 ∈ AtLat → ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))
8 neeq1 2852 . . . . 5 (𝑥 = 𝑋 → (𝑥0𝑋0 ))
9 breq2 4617 . . . . . 6 (𝑥 = 𝑋 → (𝑦 𝑥𝑦 𝑋))
109rexbidv 3045 . . . . 5 (𝑥 = 𝑋 → (∃𝑦𝐴 𝑦 𝑥 ↔ ∃𝑦𝐴 𝑦 𝑋))
118, 10imbi12d 334 . . . 4 (𝑥 = 𝑋 → ((𝑥0 → ∃𝑦𝐴 𝑦 𝑥) ↔ (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
1211rspccv 3292 . . 3 (∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥) → (𝑋𝐵 → (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
137, 12syl 17 . 2 (𝐾 ∈ AtLat → (𝑋𝐵 → (𝑋0 → ∃𝑦𝐴 𝑦 𝑋)))
14133imp 1254 1 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑦𝐴 𝑦 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908   class class class wbr 4613  dom cdm 5074  cfv 5847  Basecbs 15781  lecple 15869  glbcglb 16864  0.cp0 16958  Latclat 16966  Atomscatm 34030  AtLatcal 34031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-dm 5084  df-iota 5810  df-fv 5855  df-atl 34065
This theorem is referenced by:  atnle  34084  atlatmstc  34086  cvratlem  34187  cvrat4  34209  2llnmat  34290  2lnat  34550
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