Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvlexchb1 Structured version   Visualization version   GIF version

Theorem cvlexchb1 35139
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 35135 . . . . . . . . 9 (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
21adantr 466 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝐾 ∈ Lat)
3 simpr3 1237 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑋𝐵)
4 simpr2 1235 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑄𝐴)
5 cvlexch.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
6 cvlexch.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
75, 6atbase 35098 . . . . . . . . 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 17268 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → 𝑋 (𝑋 𝑄))
122, 3, 8, 11syl3anc 1476 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑋 (𝑋 𝑄))
13123adant3 1126 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑋 (𝑋 𝑄))
1413adantr 466 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑋 (𝑋 𝑄))
15 simpr 471 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑃 (𝑋 𝑄))
16 simpr1 1233 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑃𝐴)
175, 6atbase 35098 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
1816, 17syl 17 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑃𝐵)
195, 10latjcl 17259 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → (𝑋 𝑄) ∈ 𝐵)
202, 3, 8, 19syl3anc 1476 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → (𝑋 𝑄) ∈ 𝐵)
215, 9, 10latjle12 17270 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑃𝐵 ∧ (𝑋 𝑄) ∈ 𝐵)) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
222, 3, 18, 20, 21syl13anc 1478 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
23223adant3 1126 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
2423adantr 466 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 (𝑋 𝑄) ∧ 𝑃 (𝑋 𝑄)) ↔ (𝑋 𝑃) (𝑋 𝑄)))
2514, 15, 24mpbi2and 691 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 𝑃) (𝑋 𝑄))
265, 9, 10latlej1 17268 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑃𝐵) → 𝑋 (𝑋 𝑃))
272, 3, 18, 26syl3anc 1476 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑋 (𝑋 𝑃))
28273adant3 1126 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → 𝑋 (𝑋 𝑃))
2928adantr 466 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑋 (𝑋 𝑃))
305, 9, 10, 6cvlexch1 35137 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))
3130imp 393 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑄 (𝑋 𝑃))
325, 10latjcl 17259 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑃𝐵) → (𝑋 𝑃) ∈ 𝐵)
332, 3, 18, 32syl3anc 1476 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → (𝑋 𝑃) ∈ 𝐵)
345, 9, 10latjle12 17270 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑄𝐵 ∧ (𝑋 𝑃) ∈ 𝐵)) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
352, 3, 8, 33, 34syl13anc 1478 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
36353adant3 1126 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
3736adantr 466 . . . . 5 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 (𝑋 𝑃) ∧ 𝑄 (𝑋 𝑃)) ↔ (𝑋 𝑄) (𝑋 𝑃)))
3829, 31, 37mpbi2and 691 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 𝑄) (𝑋 𝑃))
395, 9latasymb 17262 . . . . . . 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 1476 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → (((𝑋 𝑃) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 𝑃)) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
41403adant3 1126 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (((𝑋 𝑃) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 𝑃)) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
4241adantr 466 . . . 4 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (((𝑋 𝑃) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 𝑃)) ↔ (𝑋 𝑃) = (𝑋 𝑄)))
4325, 38, 42mpbi2and 691 . . 3 (((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 𝑃) = (𝑋 𝑄))
4443ex 397 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → (𝑋 𝑃) = (𝑋 𝑄)))
455, 9, 10latlej2 17269 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑃𝐵) → 𝑃 (𝑋 𝑃))
462, 3, 18, 45syl3anc 1476 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵)) → 𝑃 (𝑋 𝑃))
47 breq2 4791 . . . 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 1126 . 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 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4787  cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  Latclat 17253  Atomscatm 35072  CvLatclc 35074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-preset 17136  df-poset 17154  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-lat 17254  df-ats 35076  df-atl 35107  df-cvlat 35131
This theorem is referenced by:  cvlexchb2  35140  cvlexch4N  35142  cvlatexchb1  35143  cvlcvr1  35148  hlexchb1  35193
  Copyright terms: Public domain W3C validator