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 35492
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 35491 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))
6 cvllat 35489 . . . . 5 (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
763ad2ant1 1124 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝐾 ∈ Lat)
8 simp22 1221 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑄𝐴)
91, 4atbase 35452 . . . . 5 (𝑄𝐴𝑄𝐵)
108, 9syl 17 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑄𝐵)
11 simp23 1222 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑋𝐵)
121, 3latjcom 17456 . . . 4 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋) = (𝑋 𝑄))
137, 10, 11, 12syl3anc 1439 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑄 𝑋) = (𝑋 𝑄))
1413breq2d 4900 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) ↔ 𝑃 (𝑋 𝑄)))
15 simp21 1220 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑃𝐴)
161, 4atbase 35452 . . . . 5 (𝑃𝐴𝑃𝐵)
1715, 16syl 17 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑃𝐵)
181, 3latjcom 17456 . . . 4 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) = (𝑋 𝑃))
197, 17, 11, 18syl3anc 1439 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 𝑋) = (𝑋 𝑃))
2019breq2d 4900 . 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 1071   = wceq 1601  wcel 2107   class class class wbr 4888  cfv 6137  (class class class)co 6924  Basecbs 16266  lecple 16356  joincjn 17341  Latclat 17442  Atomscatm 35426  CvLatclc 35428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-lub 17371  df-join 17373  df-lat 17443  df-ats 35430  df-atl 35461  df-cvlat 35485
This theorem is referenced by:  hlexch2  35546
  Copyright terms: Public domain W3C validator