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Theorem hlsupr 33690
Description: A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)
Hypotheses
Ref Expression
hlsupr.l = (le‘𝐾)
hlsupr.j = (join‘𝐾)
hlsupr.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
hlsupr (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄)))
Distinct variable groups:   𝐴,𝑟   𝐾,𝑟   𝑃,𝑟   𝑄,𝑟
Allowed substitution hints:   (𝑟)   (𝑟)

Proof of Theorem hlsupr
StepHypRef Expression
1 eqid 2610 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 hlsupr.l . . . 4 = (le‘𝐾)
3 hlsupr.j . . . 4 = (join‘𝐾)
4 hlsupr.a . . . 4 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4hlsuprexch 33685 . . 3 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ((𝑃𝑄 → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) ∧ ∀𝑟 ∈ (Base‘𝐾)((¬ 𝑃 𝑟𝑃 (𝑟 𝑄)) → 𝑄 (𝑟 𝑃))))
65simpld 474 . 2 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄 → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))))
76imp 444 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897   class class class wbr 4583  cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Atomscatm 33568  HLchlt 33655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-cvlat 33627  df-hlat 33656
This theorem is referenced by:  hlsupr2  33691  atbtwnexOLDN  33751  atbtwnex  33752  cdlemb  34098  lhpexle2lem  34313  lhpexle3lem  34315  cdlemf1  34867  cdlemg35  35019
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