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Theorem cvlexch1 37836
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 37831 . . . . 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 498 . . . 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 5109 . . . . . . . 8 (𝑝 = 𝑃 β†’ (𝑝 ≀ π‘₯ ↔ 𝑃 ≀ π‘₯))
87notbid 318 . . . . . . 7 (𝑝 = 𝑃 β†’ (Β¬ 𝑝 ≀ π‘₯ ↔ Β¬ 𝑃 ≀ π‘₯))
9 breq1 5109 . . . . . . 7 (𝑝 = 𝑃 β†’ (𝑝 ≀ (π‘₯ ∨ π‘ž) ↔ 𝑃 ≀ (π‘₯ ∨ π‘ž)))
108, 9anbi12d 632 . . . . . 6 (𝑝 = 𝑃 β†’ ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) ↔ (Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ π‘ž))))
11 oveq2 7366 . . . . . . 7 (𝑝 = 𝑃 β†’ (π‘₯ ∨ 𝑝) = (π‘₯ ∨ 𝑃))
1211breq2d 5118 . . . . . 6 (𝑝 = 𝑃 β†’ (π‘ž ≀ (π‘₯ ∨ 𝑝) ↔ π‘ž ≀ (π‘₯ ∨ 𝑃)))
1310, 12imbi12d 345 . . . . 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 7366 . . . . . . . 8 (π‘ž = 𝑄 β†’ (π‘₯ ∨ π‘ž) = (π‘₯ ∨ 𝑄))
1514breq2d 5118 . . . . . . 7 (π‘ž = 𝑄 β†’ (𝑃 ≀ (π‘₯ ∨ π‘ž) ↔ 𝑃 ≀ (π‘₯ ∨ 𝑄)))
1615anbi2d 630 . . . . . 6 (π‘ž = 𝑄 β†’ ((Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ π‘ž)) ↔ (Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ 𝑄))))
17 breq1 5109 . . . . . 6 (π‘ž = 𝑄 β†’ (π‘ž ≀ (π‘₯ ∨ 𝑃) ↔ 𝑄 ≀ (π‘₯ ∨ 𝑃)))
1816, 17imbi12d 345 . . . . 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 5110 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (𝑃 ≀ π‘₯ ↔ 𝑃 ≀ 𝑋))
2019notbid 318 . . . . . . 7 (π‘₯ = 𝑋 β†’ (Β¬ 𝑃 ≀ π‘₯ ↔ Β¬ 𝑃 ≀ 𝑋))
21 oveq1 7365 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (π‘₯ ∨ 𝑄) = (𝑋 ∨ 𝑄))
2221breq2d 5118 . . . . . . 7 (π‘₯ = 𝑋 β†’ (𝑃 ≀ (π‘₯ ∨ 𝑄) ↔ 𝑃 ≀ (𝑋 ∨ 𝑄)))
2320, 22anbi12d 632 . . . . . 6 (π‘₯ = 𝑋 β†’ ((Β¬ 𝑃 ≀ π‘₯ ∧ 𝑃 ≀ (π‘₯ ∨ 𝑄)) ↔ (Β¬ 𝑃 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))))
24 oveq1 7365 . . . . . . 7 (π‘₯ = 𝑋 β†’ (π‘₯ ∨ 𝑃) = (𝑋 ∨ 𝑃))
2524breq2d 5118 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝑄 ≀ (π‘₯ ∨ 𝑃) ↔ 𝑄 ≀ (𝑋 ∨ 𝑃)))
2623, 25imbi12d 345 . . . . 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 3592 . . . 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 508 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡)) β†’ ((Β¬ 𝑃 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))
2928exp4b 432 . 2 (𝐾 ∈ CvLat β†’ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (Β¬ 𝑃 ≀ 𝑋 β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))))
30293imp 1112 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  joincjn 18205  Atomscatm 37771  AtLatcal 37772  CvLatclc 37773
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-ext 2704
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-cvlat 37830
This theorem is referenced by:  cvlexch2  37837  cvlexchb1  37838  cvlexch3  37840  cvlcvr1  37847  hlexch1  37891
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