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Theorem cvlexchb1 34443
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 34439 . . . . . . . . 9 (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
21adantr 481 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝐾 ∈ Lat)
3 simpr3 1068 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑋𝐵)
4 simpr2 1067 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑄𝐴)
5 cvlexch.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
6 cvlexch.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
75, 6atbase 34402 . . . . . . . . 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 17054 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → 𝑋 (𝑋 𝑄))
122, 3, 8, 11syl3anc 1325 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑋 (𝑋 𝑄))
13123adant3 1080 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑋 (𝑋 𝑄))
1413adantr 481 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑋 (𝑋 𝑄))
15 simpr 477 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑃 (𝑋 𝑄))
16 simpr1 1066 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑃𝐴)
175, 6atbase 34402 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
1816, 17syl 17 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑃𝐵)
195, 10latjcl 17045 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → (𝑋 𝑄) ∈ 𝐵)
202, 3, 8, 19syl3anc 1325 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → (𝑋 𝑄) ∈ 𝐵)
215, 9, 10latjle12 17056 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑃𝐵 ∧ (𝑋 𝑄) ∈ 𝐵)) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
222, 3, 18, 20, 21syl13anc 1327 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
23223adant3 1080 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
2423adantr 481 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
2514, 15, 24mpbi2and 956 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 𝑃) (𝑋 𝑄))
265, 9, 10latlej1 17054 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑃𝐵) → 𝑋 (𝑋 𝑃))
272, 3, 18, 26syl3anc 1325 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑋 (𝑋 𝑃))
28273adant3 1080 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑋 (𝑋 𝑃))
2928adantr 481 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑋 (𝑋 𝑃))
305, 9, 10, 6cvlexch1 34441 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))
3130imp 445 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑄 (𝑋 𝑃))
325, 10latjcl 17045 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑃𝐵) → (𝑋 𝑃) ∈ 𝐵)
332, 3, 18, 32syl3anc 1325 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → (𝑋 𝑃) ∈ 𝐵)
345, 9, 10latjle12 17056 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑄𝐵 ∧ (𝑋 𝑃) ∈ 𝐵)) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
352, 3, 8, 33, 34syl13anc 1327 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
36353adant3 1080 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
3736adantr 481 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
3829, 31, 37mpbi2and 956 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 𝑄) (𝑋 𝑃))
395, 9latasymb 17048 . . . . . . 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 1325 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → (((𝑋 𝑃) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 𝑃)) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
41403adant3 1080 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (((𝑋 𝑃) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 𝑃)) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
4241adantr 481 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (((𝑋 𝑃) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 𝑃)) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
4325, 38, 42mpbi2and 956 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 𝑃) = (𝑋 𝑄))
4443ex 450 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → (𝑋 𝑃) = (𝑋 𝑄)))
455, 9, 10latlej2 17055 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑃𝐵) → 𝑃 (𝑋 𝑃))
462, 3, 18, 45syl3anc 1325 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑃 (𝑋 𝑃))
47 breq2 4655 . . . 4 ((𝑋 𝑃) = (𝑋 𝑄) → (𝑃 (𝑋 𝑃) ↔ 𝑃 (𝑋 𝑄)))
4846, 47syl5ibcom 235 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → ((𝑋 𝑃) = (𝑋 𝑄) → 𝑃 (𝑋 𝑄)))
49483adant3 1080 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → ((𝑋 𝑃) = (𝑋 𝑄) → 𝑃 (𝑋 𝑄)))
5044, 49impbid 202 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 196  wa 384  w3a 1037   = wceq 1482  wcel 1989   class class class wbr 4651  cfv 5886  (class class class)co 6647  Basecbs 15851  lecple 15942  joincjn 16938  Latclat 17039  Atomscatm 34376  CvLatclc 34378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-preset 16922  df-poset 16940  df-lub 16968  df-glb 16969  df-join 16970  df-meet 16971  df-lat 17040  df-ats 34380  df-atl 34411  df-cvlat 34435
This theorem is referenced by:  cvlexchb2  34444  cvlexch4N  34446  cvlatexchb1  34447  cvlcvr1  34452  hlexchb1  34496
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