Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvlexch2 Structured version   Visualization version   GIF version
Theorem cvlexch2 38712
Description: An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)
Hypotheses
Ref Expression
cvlexch.b 𝐡 = (Baseβ€˜πΎ)
cvlexch.l ≀ = (leβ€˜πΎ)
cvlexch.j ∨ = (joinβ€˜πΎ)
cvlexch.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
cvlexch2 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑄 ∨ 𝑋) β†’ 𝑄 ≀ (𝑃 ∨ 𝑋)))
Proof of Theorem cvlexch2
StepHypRef Expression
1 cvlexch.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 cvlexch.l . . 3 ≀ = (leβ€˜πΎ)
3 cvlexch.j . . 3 ∨ = (joinβ€˜πΎ)
4 cvlexch.a . . 3 𝐴 = (Atomsβ€˜πΎ)
51, 2, 3, 4cvlexch1 38711 . 2 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))
6 cvllat 38709 . . . . 5 (𝐾 ∈ CvLat β†’ 𝐾 ∈ Lat)
763ad2ant1 1130 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ 𝐾 ∈ Lat)
8 simp22 1204 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ 𝑄 ∈ 𝐴)
91, 4atbase 38672 . . . . 5 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
108, 9syl 17 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ 𝑄 ∈ 𝐡)
11 simp23 1205 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ 𝑋 ∈ 𝐡)
121, 3latjcom 18412 . . . 4 ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄))
137, 10, 11, 12syl3anc 1368 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄))
1413breq2d 5153 . 2 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑄 ∨ 𝑋) ↔ 𝑃 ≀ (𝑋 ∨ 𝑄)))
15 simp21 1203 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ 𝑃 ∈ 𝐴)
161, 4atbase 38672 . . . . 5 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
1715, 16syl 17 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ 𝑃 ∈ 𝐡)
181, 3latjcom 18412 . . . 4 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ∨ 𝑋) = (𝑋 ∨ 𝑃))
197, 17, 11, 18syl3anc 1368 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ∨ 𝑋) = (𝑋 ∨ 𝑃))
2019breq2d 5153 . 2 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑄 ≀ (𝑃 ∨ 𝑋) ↔ 𝑄 ≀ (𝑋 ∨ 𝑃)))
215, 14, 203imtr4d 294 1 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑄 ∨ 𝑋) β†’ 𝑄 ≀ (𝑃 ∨ 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  Latclat 18396  Atomscatm 38646  CvLatclc 38648
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-lub 18311  df-join 18313  df-lat 18397  df-ats 38650  df-atl 38681  df-cvlat 38705
This theorem is referenced by:  hlexch2  38767
  Copyright terms: Public domain W3C validator