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Theorem hlexch1 39365
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 39341 . 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 39310 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))
71, 6syl3an1 1162 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 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Atomscatm 39245  CvLatclc 39247  HLchlt 39332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-cvlat 39304  df-hlat 39333
This theorem is referenced by:  cvratlem  39404  4noncolr3  39436  3dimlem4a  39446  3dimlem4OLDN  39448  ps-2  39461  4atlem0a  39576
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