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Theorem cvlexch4N 38845
Description: An atomic covering lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cvlexch3.b 𝐡 = (Baseβ€˜πΎ)
cvlexch3.l ≀ = (leβ€˜πΎ)
cvlexch3.j ∨ = (joinβ€˜πΎ)
cvlexch3.m ∧ = (meetβ€˜πΎ)
cvlexch3.z 0 = (0.β€˜πΎ)
cvlexch3.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
cvlexch4N ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
Proof of Theorem cvlexch4N
StepHypRef Expression
1 cvlatl 38837 . . . . 5 (𝐾 ∈ CvLat β†’ 𝐾 ∈ AtLat)
21adantr 479 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ 𝐾 ∈ AtLat)
3 simpr1 1191 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ 𝑃 ∈ 𝐴)
4 simpr3 1193 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ 𝑋 ∈ 𝐡)
5 cvlexch3.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
6 cvlexch3.l . . . . 5 ≀ = (leβ€˜πΎ)
7 cvlexch3.m . . . . 5 ∧ = (meetβ€˜πΎ)
8 cvlexch3.z . . . . 5 0 = (0.β€˜πΎ)
9 cvlexch3.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
105, 6, 7, 8, 9atnle 38829 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (Β¬ 𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))
112, 3, 4, 10syl3anc 1368 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ (Β¬ 𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))
12 cvlexch3.j . . . . 5 ∨ = (joinβ€˜πΎ)
135, 6, 12, 9cvlexchb1 38842 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
14133expia 1118 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ (Β¬ 𝑃 ≀ 𝑋 β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))))
1511, 14sylbird 259 . 2 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑃 ∧ 𝑋) = 0 β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))))
16153impia 1114 1 ((𝐾 ∈ CvLat ∧ (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¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17189  lecple 17249  joincjn 18312  meetcmee 18313  0.cp0 18424  Atomscatm 38775  AtLatcal 38776  CvLatclc 38777
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 7748
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 18296  df-poset 18314  df-plt 18331  df-lub 18347  df-glb 18348  df-join 18349  df-meet 18350  df-p0 18426  df-lat 18433  df-covers 38778  df-ats 38779  df-atl 38810  df-cvlat 38834
This theorem is referenced by:  hlexch4N  38905
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