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Theorem cvlexch1 39310
Description: An atomic covering lattice has the exchange property. (Contributed by NM, 6-Nov-2011.)
Hypotheses
Ref Expression
cvlexch.b 𝐵 = (Base‘𝐾)
cvlexch.l = (le‘𝐾)
cvlexch.j = (join‘𝐾)
cvlexch.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cvlexch1 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Proof of Theorem cvlexch1
Dummy variables 𝑞 𝑝 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvlexch.b . . . . . 6 𝐵 = (Base‘𝐾)
2 cvlexch.l . . . . . 6 = (le‘𝐾)
3 cvlexch.j . . . . . 6 = (join‘𝐾)
4 cvlexch.a . . . . . 6 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4iscvlat 39305 . . . . 5 (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
65simprbi 496 . . . 4 (𝐾 ∈ CvLat → ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝)))
7 breq1 5151 . . . . . . . 8 (𝑝 = 𝑃 → (𝑝 𝑥𝑃 𝑥))
87notbid 318 . . . . . . 7 (𝑝 = 𝑃 → (¬ 𝑝 𝑥 ↔ ¬ 𝑃 𝑥))
9 breq1 5151 . . . . . . 7 (𝑝 = 𝑃 → (𝑝 (𝑥 𝑞) ↔ 𝑃 (𝑥 𝑞)))
108, 9anbi12d 632 . . . . . 6 (𝑝 = 𝑃 → ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) ↔ (¬ 𝑃 𝑥𝑃 (𝑥 𝑞))))
11 oveq2 7439 . . . . . . 7 (𝑝 = 𝑃 → (𝑥 𝑝) = (𝑥 𝑃))
1211breq2d 5160 . . . . . 6 (𝑝 = 𝑃 → (𝑞 (𝑥 𝑝) ↔ 𝑞 (𝑥 𝑃)))
1310, 12imbi12d 344 . . . . 5 (𝑝 = 𝑃 → (((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝)) ↔ ((¬ 𝑃 𝑥𝑃 (𝑥 𝑞)) → 𝑞 (𝑥 𝑃))))
14 oveq2 7439 . . . . . . . 8 (𝑞 = 𝑄 → (𝑥 𝑞) = (𝑥 𝑄))
1514breq2d 5160 . . . . . . 7 (𝑞 = 𝑄 → (𝑃 (𝑥 𝑞) ↔ 𝑃 (𝑥 𝑄)))
1615anbi2d 630 . . . . . 6 (𝑞 = 𝑄 → ((¬ 𝑃 𝑥𝑃 (𝑥 𝑞)) ↔ (¬ 𝑃 𝑥𝑃 (𝑥 𝑄))))
17 breq1 5151 . . . . . 6 (𝑞 = 𝑄 → (𝑞 (𝑥 𝑃) ↔ 𝑄 (𝑥 𝑃)))
1816, 17imbi12d 344 . . . . 5 (𝑞 = 𝑄 → (((¬ 𝑃 𝑥𝑃 (𝑥 𝑞)) → 𝑞 (𝑥 𝑃)) ↔ ((¬ 𝑃 𝑥𝑃 (𝑥 𝑄)) → 𝑄 (𝑥 𝑃))))
19 breq2 5152 . . . . . . . 8 (𝑥 = 𝑋 → (𝑃 𝑥𝑃 𝑋))
2019notbid 318 . . . . . . 7 (𝑥 = 𝑋 → (¬ 𝑃 𝑥 ↔ ¬ 𝑃 𝑋))
21 oveq1 7438 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 𝑄) = (𝑋 𝑄))
2221breq2d 5160 . . . . . . 7 (𝑥 = 𝑋 → (𝑃 (𝑥 𝑄) ↔ 𝑃 (𝑋 𝑄)))
2320, 22anbi12d 632 . . . . . 6 (𝑥 = 𝑋 → ((¬ 𝑃 𝑥𝑃 (𝑥 𝑄)) ↔ (¬ 𝑃 𝑋𝑃 (𝑋 𝑄))))
24 oveq1 7438 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑃) = (𝑋 𝑃))
2524breq2d 5160 . . . . . 6 (𝑥 = 𝑋 → (𝑄 (𝑥 𝑃) ↔ 𝑄 (𝑋 𝑃)))
2623, 25imbi12d 344 . . . . 5 (𝑥 = 𝑋 → (((¬ 𝑃 𝑥𝑃 (𝑥 𝑄)) → 𝑄 (𝑥 𝑃)) ↔ ((¬ 𝑃 𝑋𝑃 (𝑋 𝑄)) → 𝑄 (𝑋 𝑃))))
2713, 18, 26rspc3v 3638 . . . 4 ((𝑃𝐴𝑄𝐴𝑋𝐵) → (∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝)) → ((¬ 𝑃 𝑋𝑃 (𝑋 𝑄)) → 𝑄 (𝑋 𝑃))))
286, 27mpan9 506 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → ((¬ 𝑃 𝑋𝑃 (𝑋 𝑄)) → 𝑄 (𝑋 𝑃)))
2928exp4b 430 . 2 (𝐾 ∈ CvLat → ((𝑃𝐴𝑄𝐴𝑋𝐵) → (¬ 𝑃 𝑋 → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))))
30293imp 1110 1 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Atomscatm 39245  AtLatcal 39246  CvLatclc 39247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-cvlat 39304
This theorem is referenced by:  cvlexch2  39311  cvlexchb1  39312  cvlexch3  39314  cvlcvr1  39321  hlexch1  39365
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