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Theorem cvlexch3 38202
Description: An atomic covering lattice has the exchange property. (atexch 31634 analog.) (Contributed by NM, 5-Nov-2012.)
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
cvlexch3 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))
Proof of Theorem cvlexch3
StepHypRef Expression
1 cvlatl 38195 . . . . 5 (𝐾 ∈ CvLat β†’ 𝐾 ∈ AtLat)
21adantr 482 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ 𝐾 ∈ AtLat)
3 simpr1 1195 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ 𝑃 ∈ 𝐴)
4 simpr3 1197 . . . 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 38187 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (Β¬ 𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))
112, 3, 4, 10syl3anc 1372 . . 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, 9cvlexch1 38198 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))
14133expia 1122 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ (Β¬ 𝑃 ≀ 𝑋 β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃))))
1511, 14sylbird 260 . 2 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑃 ∧ 𝑋) = 0 β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃))))
16153impia 1118 1 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  joincjn 18264  meetcmee 18265  0.cp0 18376  Atomscatm 38133  AtLatcal 38134  CvLatclc 38135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-lat 18385  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192
This theorem is referenced by:  hlexch3  38262
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