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Theorem hlsupr 38761
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 ≀ = (leβ€˜πΎ)
hlsupr.j ∨ = (joinβ€˜πΎ)
hlsupr.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
hlsupr (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
Distinct variable groups:   𝐴,π‘Ÿ   𝐾,π‘Ÿ   𝑃,π‘Ÿ   𝑄,π‘Ÿ
Allowed substitution hints:   ∨ (π‘Ÿ)   ≀ (π‘Ÿ)

Proof of Theorem hlsupr
StepHypRef Expression
1 eqid 2724 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 hlsupr.l . . . 4 ≀ = (leβ€˜πΎ)
3 hlsupr.j . . . 4 ∨ = (joinβ€˜πΎ)
4 hlsupr.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
51, 2, 3, 4hlsuprexch 38756 . . 3 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ ((𝑃 β‰  𝑄 β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) ∧ βˆ€π‘Ÿ ∈ (Baseβ€˜πΎ)((Β¬ 𝑃 ≀ π‘Ÿ ∧ 𝑃 ≀ (π‘Ÿ ∨ 𝑄)) β†’ 𝑄 ≀ (π‘Ÿ ∨ 𝑃))))
65simpld 494 . 2 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 β‰  𝑄 β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))))
76imp 406 1 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆ€wral 3053  βˆƒwrex 3062   class class class wbr 5139  β€˜cfv 6534  (class class class)co 7402  Basecbs 17149  lecple 17209  joincjn 18272  Atomscatm 38637  HLchlt 38724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-ov 7405  df-cvlat 38696  df-hlat 38725
This theorem is referenced by:  hlsupr2  38762  atbtwnexOLDN  38822  atbtwnex  38823  cdlemb  39169  lhpexle2lem  39384  lhpexle3lem  39386  cdlemf1  39936  cdlemg35  40088
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