Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2llnma1b Structured version   Visualization version   GIF version

Theorem 2llnma1b 38105
Description: Generalization of 2llnma1 38106. (Contributed by NM, 26-Apr-2013.)
Hypotheses
Ref Expression
2llnma1b.b 𝐵 = (Base‘𝐾)
2llnma1b.l = (le‘𝐾)
2llnma1b.j = (join‘𝐾)
2llnma1b.m = (meet‘𝐾)
2llnma1b.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
2llnma1b ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) = 𝑃)

Proof of Theorem 2llnma1b
StepHypRef Expression
1 hllat 37681 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1133 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ Lat)
3 simp22 1207 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝐴)
4 2llnma1b.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 2llnma1b.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
64, 5atbase 37607 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝐵)
8 simp21 1206 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑋𝐵)
9 2llnma1b.l . . . . . 6 = (le‘𝐾)
10 2llnma1b.j . . . . . 6 = (join‘𝐾)
114, 9, 10latlej1 18263 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → 𝑃 (𝑃 𝑋))
122, 7, 8, 11syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 (𝑃 𝑋))
13 simp23 1208 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑄𝐴)
144, 5atbase 37607 . . . . . 6 (𝑄𝐴𝑄𝐵)
1513, 14syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑄𝐵)
164, 9, 10latlej1 18263 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → 𝑃 (𝑃 𝑄))
172, 7, 15, 16syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 (𝑃 𝑄))
184, 10latjcl 18254 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
192, 7, 8, 18syl3anc 1371 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑋) ∈ 𝐵)
20 simp1 1136 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ HL)
214, 10, 5hlatjcl 37685 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
2220, 3, 13, 21syl3anc 1371 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑄) ∈ 𝐵)
23 2llnma1b.m . . . . . 6 = (meet‘𝐾)
244, 9, 23latlem12 18281 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃𝐵 ∧ (𝑃 𝑋) ∈ 𝐵 ∧ (𝑃 𝑄) ∈ 𝐵)) → ((𝑃 (𝑃 𝑋) ∧ 𝑃 (𝑃 𝑄)) ↔ 𝑃 ((𝑃 𝑋) (𝑃 𝑄))))
252, 7, 19, 22, 24syl13anc 1372 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 (𝑃 𝑋) ∧ 𝑃 (𝑃 𝑄)) ↔ 𝑃 ((𝑃 𝑋) (𝑃 𝑄))))
2612, 17, 25mpbi2and 710 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 ((𝑃 𝑋) (𝑃 𝑄)))
27 hlatl 37678 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
28273ad2ant1 1133 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ AtLat)
29 simp3 1138 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ¬ 𝑄 (𝑃 𝑋))
30 nbrne2 5116 . . . . . 6 ((𝑃 (𝑃 𝑋) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝑄)
3112, 29, 30syl2anc 585 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝑄)
324, 10latjcl 18254 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑋) ∈ 𝐵𝑄𝐵) → ((𝑃 𝑋) 𝑄) ∈ 𝐵)
332, 19, 15, 32syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) 𝑄) ∈ 𝐵)
344, 9, 10latlej1 18263 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑋) ∈ 𝐵𝑄𝐵) → (𝑃 𝑋) ((𝑃 𝑋) 𝑄))
352, 19, 15, 34syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑋) ((𝑃 𝑋) 𝑄))
364, 9, 2, 7, 19, 33, 12, 35lattrd 18261 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 ((𝑃 𝑋) 𝑄))
374, 9, 10, 23, 5cvrat3 37761 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑃 𝑋) ∈ 𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑋) ∧ 𝑃 ((𝑃 𝑋) 𝑄)) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴))
38373impia 1117 . . . . 5 ((𝐾 ∈ HL ∧ ((𝑃 𝑋) ∈ 𝐵𝑃𝐴𝑄𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑋) ∧ 𝑃 ((𝑃 𝑋) 𝑄))) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴)
3920, 19, 3, 13, 31, 29, 36, 38syl133anc 1393 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴)
409, 5atcmp 37629 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴 ∧ ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴) → (𝑃 ((𝑃 𝑋) (𝑃 𝑄)) ↔ 𝑃 = ((𝑃 𝑋) (𝑃 𝑄))))
4128, 3, 39, 40syl3anc 1371 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 ((𝑃 𝑋) (𝑃 𝑄)) ↔ 𝑃 = ((𝑃 𝑋) (𝑃 𝑄))))
4226, 41mpbid 231 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 = ((𝑃 𝑋) (𝑃 𝑄)))
4342eqcomd 2743 1 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1087   = wceq 1541  wcel 2106  wne 2941   class class class wbr 5096  cfv 6483  (class class class)co 7341  Basecbs 17009  lecple 17066  joincjn 18126  meetcmee 18127  Latclat 18246  Atomscatm 37581  AtLatcal 37582  HLchlt 37668
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-id 5522  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-riota 7297  df-ov 7344  df-oprab 7345  df-proset 18110  df-poset 18128  df-plt 18145  df-lub 18161  df-glb 18162  df-join 18163  df-meet 18164  df-p0 18240  df-lat 18247  df-clat 18314  df-oposet 37494  df-ol 37496  df-oml 37497  df-covers 37584  df-ats 37585  df-atl 37616  df-cvlat 37640  df-hlat 37669
This theorem is referenced by:  2llnma1  38106  cdlemg4  38936  cdlemkfid1N  39240
  Copyright terms: Public domain W3C validator