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Theorem hlsupr 36654
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 2824 . . . 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 36649 . . 3 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ((𝑃𝑄 → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) ∧ ∀𝑟 ∈ (Base‘𝐾)((¬ 𝑃 𝑟𝑃 (𝑟 𝑄)) → 𝑄 (𝑟 𝑃))))
65simpld 498 . 2 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄 → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))))
76imp 410 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134   class class class wbr 5053  cfv 6345  (class class class)co 7151  Basecbs 16485  lecple 16574  joincjn 17556  Atomscatm 36531  HLchlt 36618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-iota 6304  df-fv 6353  df-ov 7154  df-cvlat 36590  df-hlat 36619
This theorem is referenced by:  hlsupr2  36655  atbtwnexOLDN  36715  atbtwnex  36716  cdlemb  37062  lhpexle2lem  37277  lhpexle3lem  37279  cdlemf1  37829  cdlemg35  37981
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