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Theorem hlexch4N 38897
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 𝐡 = (Baseβ€˜πΎ)
hlexch3.l ≀ = (leβ€˜πΎ)
hlexch3.j ∨ = (joinβ€˜πΎ)
hlexch3.m ∧ = (meetβ€˜πΎ)
hlexch3.z 0 = (0.β€˜πΎ)
hlexch3.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
hlexch4N ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))

Proof of Theorem hlexch4N
StepHypRef Expression
1 hlcvl 38863 . 2 (𝐾 ∈ HL β†’ 𝐾 ∈ CvLat)
2 hlexch3.b . . 3 𝐡 = (Baseβ€˜πΎ)
3 hlexch3.l . . 3 ≀ = (leβ€˜πΎ)
4 hlexch3.j . . 3 ∨ = (joinβ€˜πΎ)
5 hlexch3.m . . 3 ∧ = (meetβ€˜πΎ)
6 hlexch3.z . . 3 0 = (0.β€˜πΎ)
7 hlexch3.a . . 3 𝐴 = (Atomsβ€˜πΎ)
82, 3, 4, 5, 6, 7cvlexch4N 38837 . 2 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
91, 8syl3an1 1160 1 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  lecple 17247  joincjn 18310  meetcmee 18311  0.cp0 18422  Atomscatm 38767  CvLatclc 38769  HLchlt 38854
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-lat 18431  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855
This theorem is referenced by: (None)
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