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Theorem cvlexch1 39330
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 39325 . . . . 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 5145 . . . . . . . 8 (𝑝 = 𝑃 → (𝑝 𝑥𝑃 𝑥))
87notbid 318 . . . . . . 7 (𝑝 = 𝑃 → (¬ 𝑝 𝑥 ↔ ¬ 𝑃 𝑥))
9 breq1 5145 . . . . . . 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 7440 . . . . . . 7 (𝑝 = 𝑃 → (𝑥 𝑝) = (𝑥 𝑃))
1211breq2d 5154 . . . . . 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 7440 . . . . . . . 8 (𝑞 = 𝑄 → (𝑥 𝑞) = (𝑥 𝑄))
1514breq2d 5154 . . . . . . 7 (𝑞 = 𝑄 → (𝑃 (𝑥 𝑞) ↔ 𝑃 (𝑥 𝑄)))
1615anbi2d 630 . . . . . 6 (𝑞 = 𝑄 → ((¬ 𝑃 𝑥𝑃 (𝑥 𝑞)) ↔ (¬ 𝑃 𝑥𝑃 (𝑥 𝑄))))
17 breq1 5145 . . . . . 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 5146 . . . . . . . 8 (𝑥 = 𝑋 → (𝑃 𝑥𝑃 𝑋))
2019notbid 318 . . . . . . 7 (𝑥 = 𝑋 → (¬ 𝑃 𝑥 ↔ ¬ 𝑃 𝑋))
21 oveq1 7439 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 𝑄) = (𝑋 𝑄))
2221breq2d 5154 . . . . . . 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 7439 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑃) = (𝑋 𝑃))
2524breq2d 5154 . . . . . 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 3637 . . . 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 1539  wcel 2107  wral 3060   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  joincjn 18358  Atomscatm 39265  AtLatcal 39266  CvLatclc 39267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-cvlat 39324
This theorem is referenced by:  cvlexch2  39331  cvlexchb1  39332  cvlexch3  39334  cvlcvr1  39341  hlexch1  39385
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