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Theorem hlsupr 34491
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 2620 . . . 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 34486 . . 3 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ((𝑃𝑄 → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) ∧ ∀𝑟 ∈ (Base‘𝐾)((¬ 𝑃 𝑟𝑃 (𝑟 𝑄)) → 𝑄 (𝑟 𝑃))))
65simpld 475 . 2 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄 → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))))
76imp 445 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910   class class class wbr 4644  cfv 5876  (class class class)co 6635  Basecbs 15838  lecple 15929  joincjn 16925  Atomscatm 34369  HLchlt 34456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-cvlat 34428  df-hlat 34457
This theorem is referenced by:  hlsupr2  34492  atbtwnexOLDN  34552  atbtwnex  34553  cdlemb  34899  lhpexle2lem  35114  lhpexle3lem  35116  cdlemf1  35668  cdlemg35  35820
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