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Theorem hlexch4N 36728
 Description: A Hilbert lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 15-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
hlexch3.b 𝐵 = (Base‘𝐾)
hlexch3.l = (le‘𝐾)
hlexch3.j = (join‘𝐾)
hlexch3.m = (meet‘𝐾)
hlexch3.z 0 = (0.‘𝐾)
hlexch3.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
hlexch4N ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Proof of Theorem hlexch4N
StepHypRef Expression
1 hlcvl 36695 . 2 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
2 hlexch3.b . . 3 𝐵 = (Base‘𝐾)
3 hlexch3.l . . 3 = (le‘𝐾)
4 hlexch3.j . . 3 = (join‘𝐾)
5 hlexch3.m . . 3 = (meet‘𝐾)
6 hlexch3.z . . 3 0 = (0.‘𝐾)
7 hlexch3.a . . 3 𝐴 = (Atoms‘𝐾)
82, 3, 4, 5, 6, 7cvlexch4N 36669 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
91, 8syl3an1 1160 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5031  ‘cfv 6327  (class class class)co 7140  Basecbs 16482  lecple 16571  joincjn 17553  meetcmee 17554  0.cp0 17646  Atomscatm 36599  CvLatclc 36601  HLchlt 36686 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7098  df-ov 7143  df-oprab 7144  df-proset 17537  df-poset 17555  df-plt 17567  df-lub 17583  df-glb 17584  df-join 17585  df-meet 17586  df-p0 17648  df-lat 17655  df-covers 36602  df-ats 36603  df-atl 36634  df-cvlat 36658  df-hlat 36687 This theorem is referenced by: (None)
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