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 35916
Description: An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)
Hypotheses
Ref Expression
cvlexch.b 𝐵 = (Base‘𝐾)
cvlexch.l = (le‘𝐾)
cvlexch.j = (join‘𝐾)
cvlexch.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cvlexch2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) → 𝑄 (𝑃 𝑋)))

Proof of Theorem cvlexch2
StepHypRef Expression
1 cvlexch.b . . 3 𝐵 = (Base‘𝐾)
2 cvlexch.l . . 3 = (le‘𝐾)
3 cvlexch.j . . 3 = (join‘𝐾)
4 cvlexch.a . . 3 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4cvlexch1 35915 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))
6 cvllat 35913 . . . . 5 (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
763ad2ant1 1113 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝐾 ∈ Lat)
8 simp22 1187 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑄𝐴)
91, 4atbase 35876 . . . . 5 (𝑄𝐴𝑄𝐵)
108, 9syl 17 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑄𝐵)
11 simp23 1188 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑋𝐵)
121, 3latjcom 17527 . . . 4 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋) = (𝑋 𝑄))
137, 10, 11, 12syl3anc 1351 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑄 𝑋) = (𝑋 𝑄))
1413breq2d 4941 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) ↔ 𝑃 (𝑋 𝑄)))
15 simp21 1186 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑃𝐴)
161, 4atbase 35876 . . . . 5 (𝑃𝐴𝑃𝐵)
1715, 16syl 17 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑃𝐵)
181, 3latjcom 17527 . . . 4 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) = (𝑋 𝑃))
197, 17, 11, 18syl3anc 1351 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 𝑋) = (𝑋 𝑃))
2019breq2d 4941 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑄 (𝑃 𝑋) ↔ 𝑄 (𝑋 𝑃)))
215, 14, 203imtr4d 286 1 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) → 𝑄 (𝑃 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1068   = wceq 1507  wcel 2050   class class class wbr 4929  cfv 6188  (class class class)co 6976  Basecbs 16339  lecple 16428  joincjn 17412  Latclat 17513  Atomscatm 35850  CvLatclc 35852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-lub 17442  df-join 17444  df-lat 17514  df-ats 35854  df-atl 35885  df-cvlat 35909
This theorem is referenced by:  hlexch2  35970
  Copyright terms: Public domain W3C validator