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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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