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Theorem hlsupr 38859
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 2728 . . . 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 38854 . . 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 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2937  βˆ€wral 3058  βˆƒwrex 3067   class class class wbr 5148  β€˜cfv 6548  (class class class)co 7420  Basecbs 17180  lecple 17240  joincjn 18303  Atomscatm 38735  HLchlt 38822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-cvlat 38794  df-hlat 38823
This theorem is referenced by:  hlsupr2  38860  atbtwnexOLDN  38920  atbtwnex  38921  cdlemb  39267  lhpexle2lem  39482  lhpexle3lem  39484  cdlemf1  40034  cdlemg35  40186
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