Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalawlem2 Structured version   Visualization version   GIF version

Theorem dalawlem2 35476
Description: Lemma for dalaw 35490. Utility lemma that breaks ((𝑃 𝑄) (𝑆 𝑇)) into a join of two pieces. (Contributed by NM, 6-Oct-2012.)
Hypotheses
Ref Expression
dalawlem.l = (le‘𝐾)
dalawlem.j = (join‘𝐾)
dalawlem.m = (meet‘𝐾)
dalawlem.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
dalawlem2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)))

Proof of Theorem dalawlem2
StepHypRef Expression
1 simp1 1081 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ HL)
2 hllat 34968 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
31, 2syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ Lat)
4 simp2l 1107 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑃𝐴)
5 simp2r 1108 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑄𝐴)
6 eqid 2651 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
7 dalawlem.j . . . . . . 7 = (join‘𝐾)
8 dalawlem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
96, 7, 8hlatjcl 34971 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
101, 4, 5, 9syl3anc 1366 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
11 simp3r 1110 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑇𝐴)
126, 8atbase 34894 . . . . . 6 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
1311, 12syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑇 ∈ (Base‘𝐾))
14 dalawlem.l . . . . . 6 = (le‘𝐾)
156, 14, 7latlej1 17107 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑇))
163, 10, 13, 15syl3anc 1366 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑇))
17 simp3l 1109 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆𝐴)
186, 8atbase 34894 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆 ∈ (Base‘𝐾))
206, 14, 7latlej1 17107 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
213, 10, 19, 20syl3anc 1366 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
226, 7latjcl 17098 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾))
233, 10, 13, 22syl3anc 1366 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾))
246, 7latjcl 17098 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
253, 10, 19, 24syl3anc 1366 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
26 dalawlem.m . . . . . 6 = (meet‘𝐾)
276, 14, 26latlem12 17125 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))) → (((𝑃 𝑄) ((𝑃 𝑄) 𝑇) ∧ (𝑃 𝑄) ((𝑃 𝑄) 𝑆)) ↔ (𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆))))
283, 10, 23, 25, 27syl13anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) ((𝑃 𝑄) 𝑇) ∧ (𝑃 𝑄) ((𝑃 𝑄) 𝑆)) ↔ (𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆))))
2916, 21, 28mpbi2and 976 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)))
306, 26latmcl 17099 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) ∈ (Base‘𝐾))
313, 23, 25, 30syl3anc 1366 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) ∈ (Base‘𝐾))
326, 7, 8hlatjcl 34971 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
331, 17, 11, 32syl3anc 1366 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑆 𝑇) ∈ (Base‘𝐾))
346, 14, 26latmlem1 17128 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾))) → ((𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇))))
353, 10, 31, 33, 34syl13anc 1368 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇))))
3629, 35mpd 15 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)))
376, 14, 7latlej2 17108 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑆 ((𝑃 𝑄) 𝑆))
383, 10, 19, 37syl3anc 1366 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆 ((𝑃 𝑄) 𝑆))
396, 14, 7, 26, 8atmod3i1 35468 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆𝐴 ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) ∧ 𝑆 ((𝑃 𝑄) 𝑆)) → (𝑆 (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑆) (𝑆 𝑇)))
401, 17, 25, 13, 38, 39syl131anc 1379 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑆 (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑆) (𝑆 𝑇)))
4140oveq2d 6706 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑇) (𝑆 (((𝑃 𝑄) 𝑆) 𝑇))) = (((𝑃 𝑄) 𝑇) (((𝑃 𝑄) 𝑆) (𝑆 𝑇))))
426, 26latmcl 17099 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
433, 25, 13, 42syl3anc 1366 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
446, 14, 7, 26latmlej22 17140 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑇 ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑃 𝑄) 𝑇))
453, 13, 25, 10, 44syl13anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑃 𝑄) 𝑇))
466, 14, 7, 26, 8atmod2i2 35466 . . . 4 ((𝐾 ∈ HL ∧ (𝑆𝐴 ∧ ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾) ∧ (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾)) ∧ (((𝑃 𝑄) 𝑆) 𝑇) ((𝑃 𝑄) 𝑇)) → ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑇) (𝑆 (((𝑃 𝑄) 𝑆) 𝑇))))
471, 17, 23, 43, 45, 46syl131anc 1379 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑇) (𝑆 (((𝑃 𝑄) 𝑆) 𝑇))))
48 hlol 34966 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
491, 48syl 17 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ OL)
506, 26latmassOLD 34834 . . . 4 ((𝐾 ∈ OL ∧ (((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾))) → ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)) = (((𝑃 𝑄) 𝑇) (((𝑃 𝑄) 𝑆) (𝑆 𝑇))))
5149, 23, 25, 33, 50syl13anc 1368 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)) = (((𝑃 𝑄) 𝑇) (((𝑃 𝑄) 𝑆) (𝑆 𝑇))))
5241, 47, 513eqtr4rd 2696 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)) = ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)))
5336, 52breqtrd 4711 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Latclat 17092  OLcol 34779  Atomscatm 34868  HLchlt 34955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-psubsp 35107  df-pmap 35108  df-padd 35400
This theorem is referenced by:  dalawlem5  35479  dalawlem8  35482
  Copyright terms: Public domain W3C validator