Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvlexch1 Structured version   Visualization version   GIF version

Theorem cvlexch1 38501
Description: An atomic covering lattice has the exchange property. (Contributed by NM, 6-Nov-2011.)
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
cvlexch1 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))

Proof of Theorem cvlexch1
Dummy variables π‘ž 𝑝 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvlexch.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
2 cvlexch.l . . . . . 6 ≀ = (leβ€˜πΎ)
3 cvlexch.j . . . . . 6 ∨ = (joinβ€˜πΎ)
4 cvlexch.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
51, 2, 3, 4iscvlat 38496 . . . . 5 (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ βˆ€π‘ ∈ 𝐴 βˆ€π‘ž ∈ 𝐴 βˆ€π‘₯ ∈ 𝐡 ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝))))
65simprbi 495 . . . 4 (𝐾 ∈ CvLat β†’ βˆ€π‘ ∈ 𝐴 βˆ€π‘ž ∈ 𝐴 βˆ€π‘₯ ∈ 𝐡 ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝)))
7 breq1 5150 . . . . . . . 8 (𝑝 = 𝑃 β†’ (𝑝 ≀ π‘₯ ↔ 𝑃 ≀ π‘₯))
87notbid 317 . . . . . . 7 (𝑝 = 𝑃 β†’ (Β¬ 𝑝 ≀ π‘₯ ↔ Β¬ 𝑃 ≀ π‘₯))
9 breq1 5150 . . . . . . 7 (𝑝 = 𝑃 β†’ (𝑝 ≀ (π‘₯ ∨ π‘ž) ↔ 𝑃 ≀ (π‘₯ ∨ π‘ž)))
108, 9anbi12d 629 . . . . . 6 (𝑝 = 𝑃 β†’ ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) ↔ (Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ π‘ž))))
11 oveq2 7419 . . . . . . 7 (𝑝 = 𝑃 β†’ (π‘₯ ∨ 𝑝) = (π‘₯ ∨ 𝑃))
1211breq2d 5159 . . . . . 6 (𝑝 = 𝑃 β†’ (π‘ž ≀ (π‘₯ ∨ 𝑝) ↔ π‘ž ≀ (π‘₯ ∨ 𝑃)))
1310, 12imbi12d 343 . . . . 5 (𝑝 = 𝑃 β†’ (((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝)) ↔ ((Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑃))))
14 oveq2 7419 . . . . . . . 8 (π‘ž = 𝑄 β†’ (π‘₯ ∨ π‘ž) = (π‘₯ ∨ 𝑄))
1514breq2d 5159 . . . . . . 7 (π‘ž = 𝑄 β†’ (𝑃 ≀ (π‘₯ ∨ π‘ž) ↔ 𝑃 ≀ (π‘₯ ∨ 𝑄)))
1615anbi2d 627 . . . . . 6 (π‘ž = 𝑄 β†’ ((Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ π‘ž)) ↔ (Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ 𝑄))))
17 breq1 5150 . . . . . 6 (π‘ž = 𝑄 β†’ (π‘ž ≀ (π‘₯ ∨ 𝑃) ↔ 𝑄 ≀ (π‘₯ ∨ 𝑃)))
1816, 17imbi12d 343 . . . . 5 (π‘ž = 𝑄 β†’ (((Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑃)) ↔ ((Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ 𝑄)) β†’ 𝑄 ≀ (π‘₯ ∨ 𝑃))))
19 breq2 5151 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (𝑃 ≀ π‘₯ ↔ 𝑃 ≀ 𝑋))
2019notbid 317 . . . . . . 7 (π‘₯ = 𝑋 β†’ (Β¬ 𝑃 ≀ π‘₯ ↔ Β¬ 𝑃 ≀ 𝑋))
21 oveq1 7418 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (π‘₯ ∨ 𝑄) = (𝑋 ∨ 𝑄))
2221breq2d 5159 . . . . . . 7 (π‘₯ = 𝑋 β†’ (𝑃 ≀ (π‘₯ ∨ 𝑄) ↔ 𝑃 ≀ (𝑋 ∨ 𝑄)))
2320, 22anbi12d 629 . . . . . 6 (π‘₯ = 𝑋 β†’ ((Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ 𝑄)) ↔ (Β¬ 𝑃 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))))
24 oveq1 7418 . . . . . . 7 (π‘₯ = 𝑋 β†’ (π‘₯ ∨ 𝑃) = (𝑋 ∨ 𝑃))
2524breq2d 5159 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝑄 ≀ (π‘₯ ∨ 𝑃) ↔ 𝑄 ≀ (𝑋 ∨ 𝑃)))
2623, 25imbi12d 343 . . . . 5 (π‘₯ = 𝑋 β†’ (((Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ 𝑄)) β†’ 𝑄 ≀ (π‘₯ ∨ 𝑃)) ↔ ((Β¬ 𝑃 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃))))
2713, 18, 26rspc3v 3626 . . . 4 ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘ ∈ 𝐴 βˆ€π‘ž ∈ 𝐴 βˆ€π‘₯ ∈ 𝐡 ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝)) β†’ ((Β¬ 𝑃 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃))))
286, 27mpan9 505 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ ((Β¬ 𝑃 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))
2928exp4b 429 . 2 (𝐾 ∈ CvLat β†’ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (Β¬ 𝑃 ≀ 𝑋 β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))))
30293imp 1109 1 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  lecple 17208  joincjn 18268  Atomscatm 38436  AtLatcal 38437  CvLatclc 38438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-cvlat 38495
This theorem is referenced by:  cvlexch2  38502  cvlexchb1  38503  cvlexch3  38505  cvlcvr1  38512  hlexch1  38556
  Copyright terms: Public domain W3C validator