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Theorem cvlexchb1 36571
Description: An atomic covering lattice has the exchange property. (Contributed by NM, 16-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
cvlexchb1 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Proof of Theorem cvlexchb1
StepHypRef Expression
1 cvllat 36567 . . . . . . . . 9 (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
21adantr 484 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝐾 ∈ Lat)
3 simpr3 1193 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑋𝐵)
4 simpr2 1192 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑄𝐴)
5 cvlexch.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
6 cvlexch.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
75, 6atbase 36530 . . . . . . . . 9 (𝑄𝐴𝑄𝐵)
84, 7syl 17 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑄𝐵)
9 cvlexch.l . . . . . . . . 9 = (le‘𝐾)
10 cvlexch.j . . . . . . . . 9 = (join‘𝐾)
115, 9, 10latlej1 17670 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → 𝑋 (𝑋 𝑄))
122, 3, 8, 11syl3anc 1368 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑋 (𝑋 𝑄))
13123adant3 1129 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑋 (𝑋 𝑄))
1413adantr 484 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑋 (𝑋 𝑄))
15 simpr 488 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑃 (𝑋 𝑄))
16 simpr1 1191 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑃𝐴)
175, 6atbase 36530 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
1816, 17syl 17 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑃𝐵)
195, 10latjcl 17661 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → (𝑋 𝑄) ∈ 𝐵)
202, 3, 8, 19syl3anc 1368 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → (𝑋 𝑄) ∈ 𝐵)
215, 9, 10latjle12 17672 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑃𝐵 ∧ (𝑋 𝑄) ∈ 𝐵)) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
222, 3, 18, 20, 21syl13anc 1369 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
23223adant3 1129 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
2423adantr 484 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
2514, 15, 24mpbi2and 711 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 𝑃) (𝑋 𝑄))
265, 9, 10latlej1 17670 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑃𝐵) → 𝑋 (𝑋 𝑃))
272, 3, 18, 26syl3anc 1368 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑋 (𝑋 𝑃))
28273adant3 1129 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑋 (𝑋 𝑃))
2928adantr 484 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑋 (𝑋 𝑃))
305, 9, 10, 6cvlexch1 36569 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))
3130imp 410 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑄 (𝑋 𝑃))
325, 10latjcl 17661 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑃𝐵) → (𝑋 𝑃) ∈ 𝐵)
332, 3, 18, 32syl3anc 1368 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → (𝑋 𝑃) ∈ 𝐵)
345, 9, 10latjle12 17672 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑄𝐵 ∧ (𝑋 𝑃) ∈ 𝐵)) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
352, 3, 8, 33, 34syl13anc 1369 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
36353adant3 1129 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
3736adantr 484 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
3829, 31, 37mpbi2and 711 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 𝑄) (𝑋 𝑃))
395, 9latasymb 17664 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑋 𝑃) ∈ 𝐵 ∧ (𝑋 𝑄) ∈ 𝐵) → (((𝑋 𝑃) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 𝑃)) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
402, 33, 20, 39syl3anc 1368 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → (((𝑋 𝑃) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 𝑃)) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
41403adant3 1129 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (((𝑋 𝑃) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 𝑃)) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
4241adantr 484 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (((𝑋 𝑃) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 𝑃)) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
4325, 38, 42mpbi2and 711 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 𝑃) = (𝑋 𝑄))
4443ex 416 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → (𝑋 𝑃) = (𝑋 𝑄)))
455, 9, 10latlej2 17671 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑃𝐵) → 𝑃 (𝑋 𝑃))
462, 3, 18, 45syl3anc 1368 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑃 (𝑋 𝑃))
47 breq2 5056 . . . 4 ((𝑋 𝑃) = (𝑋 𝑄) → (𝑃 (𝑋 𝑃) ↔ 𝑃 (𝑋 𝑄)))
4846, 47syl5ibcom 248 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → ((𝑋 𝑃) = (𝑋 𝑄) → 𝑃 (𝑋 𝑄)))
49483adant3 1129 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → ((𝑋 𝑃) = (𝑋 𝑄) → 𝑃 (𝑋 𝑄)))
5044, 49impbid 215 1 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  Latclat 17655  Atomscatm 36504  CvLatclc 36506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-proset 17538  df-poset 17556  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-lat 17656  df-ats 36508  df-atl 36539  df-cvlat 36563
This theorem is referenced by:  cvlexchb2  36572  cvlexch4N  36574  cvlatexchb1  36575  cvlcvr1  36580  hlexchb1  36625
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