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Theorem cvlexch1 34092
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 34087 . . . . 5 (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
65simprbi 480 . . . 4 (𝐾 ∈ CvLat → ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝)))
7 breq1 4616 . . . . . . . 8 (𝑝 = 𝑃 → (𝑝 𝑥𝑃 𝑥))
87notbid 308 . . . . . . 7 (𝑝 = 𝑃 → (¬ 𝑝 𝑥 ↔ ¬ 𝑃 𝑥))
9 breq1 4616 . . . . . . 7 (𝑝 = 𝑃 → (𝑝 (𝑥 𝑞) ↔ 𝑃 (𝑥 𝑞)))
108, 9anbi12d 746 . . . . . 6 (𝑝 = 𝑃 → ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) ↔ (¬ 𝑃 𝑥𝑃 (𝑥 𝑞))))
11 oveq2 6612 . . . . . . 7 (𝑝 = 𝑃 → (𝑥 𝑝) = (𝑥 𝑃))
1211breq2d 4625 . . . . . 6 (𝑝 = 𝑃 → (𝑞 (𝑥 𝑝) ↔ 𝑞 (𝑥 𝑃)))
1310, 12imbi12d 334 . . . . 5 (𝑝 = 𝑃 → (((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝)) ↔ ((¬ 𝑃 𝑥𝑃 (𝑥 𝑞)) → 𝑞 (𝑥 𝑃))))
14 oveq2 6612 . . . . . . . 8 (𝑞 = 𝑄 → (𝑥 𝑞) = (𝑥 𝑄))
1514breq2d 4625 . . . . . . 7 (𝑞 = 𝑄 → (𝑃 (𝑥 𝑞) ↔ 𝑃 (𝑥 𝑄)))
1615anbi2d 739 . . . . . 6 (𝑞 = 𝑄 → ((¬ 𝑃 𝑥𝑃 (𝑥 𝑞)) ↔ (¬ 𝑃 𝑥𝑃 (𝑥 𝑄))))
17 breq1 4616 . . . . . 6 (𝑞 = 𝑄 → (𝑞 (𝑥 𝑃) ↔ 𝑄 (𝑥 𝑃)))
1816, 17imbi12d 334 . . . . 5 (𝑞 = 𝑄 → (((¬ 𝑃 𝑥𝑃 (𝑥 𝑞)) → 𝑞 (𝑥 𝑃)) ↔ ((¬ 𝑃 𝑥𝑃 (𝑥 𝑄)) → 𝑄 (𝑥 𝑃))))
19 breq2 4617 . . . . . . . 8 (𝑥 = 𝑋 → (𝑃 𝑥𝑃 𝑋))
2019notbid 308 . . . . . . 7 (𝑥 = 𝑋 → (¬ 𝑃 𝑥 ↔ ¬ 𝑃 𝑋))
21 oveq1 6611 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 𝑄) = (𝑋 𝑄))
2221breq2d 4625 . . . . . . 7 (𝑥 = 𝑋 → (𝑃 (𝑥 𝑄) ↔ 𝑃 (𝑋 𝑄)))
2320, 22anbi12d 746 . . . . . 6 (𝑥 = 𝑋 → ((¬ 𝑃 𝑥𝑃 (𝑥 𝑄)) ↔ (¬ 𝑃 𝑋𝑃 (𝑋 𝑄))))
24 oveq1 6611 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑃) = (𝑋 𝑃))
2524breq2d 4625 . . . . . 6 (𝑥 = 𝑋 → (𝑄 (𝑥 𝑃) ↔ 𝑄 (𝑋 𝑃)))
2623, 25imbi12d 334 . . . . 5 (𝑥 = 𝑋 → (((¬ 𝑃 𝑥𝑃 (𝑥 𝑄)) → 𝑄 (𝑥 𝑃)) ↔ ((¬ 𝑃 𝑋𝑃 (𝑋 𝑄)) → 𝑄 (𝑋 𝑃))))
2713, 18, 26rspc3v 3309 . . . 4 ((𝑃𝐴𝑄𝐴𝑋𝐵) → (∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝)) → ((¬ 𝑃 𝑋𝑃 (𝑋 𝑄)) → 𝑄 (𝑋 𝑃))))
286, 27mpan9 486 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → ((¬ 𝑃 𝑋𝑃 (𝑋 𝑄)) → 𝑄 (𝑋 𝑃)))
2928exp4b 631 . 2 (𝐾 ∈ CvLat → ((𝑃𝐴𝑄𝐴𝑋𝐵) → (¬ 𝑃 𝑋 → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))))
30293imp 1254 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  joincjn 16865  Atomscatm 34027  AtLatcal 34028  CvLatclc 34029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-cvlat 34086
This theorem is referenced by:  cvlexch2  34093  cvlexchb1  34094  cvlexch3  34096  cvlcvr1  34103  hlexch1  34145
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