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Theorem hlsupr 37878
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 2737 . . . 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 37873 . . 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 496 . 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 408 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  lecple 17147  joincjn 18207  Atomscatm 37754  HLchlt 37841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-ov 7365  df-cvlat 37813  df-hlat 37842
This theorem is referenced by:  hlsupr2  37879  atbtwnexOLDN  37939  atbtwnex  37940  cdlemb  38286  lhpexle2lem  38501  lhpexle3lem  38503  cdlemf1  39053  cdlemg35  39205
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