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Theorem hlexch1 38985
Description: A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)
Hypotheses
Ref Expression
hlsuprexch.b 𝐵 = (Base‘𝐾)
hlsuprexch.l = (le‘𝐾)
hlsuprexch.j = (join‘𝐾)
hlsuprexch.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
hlexch1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Proof of Theorem hlexch1
StepHypRef Expression
1 hlcvl 38961 . 2 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
2 hlsuprexch.b . . 3 𝐵 = (Base‘𝐾)
3 hlsuprexch.l . . 3 = (le‘𝐾)
4 hlsuprexch.j . . 3 = (join‘𝐾)
5 hlsuprexch.a . . 3 𝐴 = (Atoms‘𝐾)
62, 3, 4, 5cvlexch1 38930 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))
71, 6syl3an1 1160 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  lecple 17243  joincjn 18306  Atomscatm 38865  CvLatclc 38867  HLchlt 38952
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 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-cvlat 38924  df-hlat 38953
This theorem is referenced by:  cvratlem  39024  4noncolr3  39056  3dimlem4a  39066  3dimlem4OLDN  39068  ps-2  39081  4atlem0a  39196
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