Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trljat1 Structured version   Visualization version   GIF version

Theorem trljat1 40149
Description: The value of a translation of an atom 𝑃 not under the fiducial co-atom 𝑊, joined with trace. Equation above Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 39847? (Contributed by NM, 22-May-2012.)
Hypotheses
Ref Expression
trljat.l = (le‘𝐾)
trljat.j = (join‘𝐾)
trljat.a 𝐴 = (Atoms‘𝐾)
trljat.h 𝐻 = (LHyp‘𝐾)
trljat.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
trljat.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
trljat1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝑅𝐹)) = (𝑃 (𝐹𝑃)))

Proof of Theorem trljat1
StepHypRef Expression
1 trljat.l . . . 4 = (le‘𝐾)
2 trljat.j . . . 4 = (join‘𝐾)
3 eqid 2735 . . . 4 (meet‘𝐾) = (meet‘𝐾)
4 trljat.a . . . 4 𝐴 = (Atoms‘𝐾)
5 trljat.h . . . 4 𝐻 = (LHyp‘𝐾)
6 trljat.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
7 trljat.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7trlval2 40146 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊))
98oveq1d 7446 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑅𝐹) 𝑃) = (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃))
10 simp1l 1196 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ HL)
1110hllatd 39346 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ Lat)
12 simp3l 1200 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃𝐴)
13 eqid 2735 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
1413, 4atbase 39271 . . . 4 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1512, 14syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 ∈ (Base‘𝐾))
1613, 5, 6, 7trlcl 40147 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
17163adant3 1131 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) ∈ (Base‘𝐾))
1813, 2latjcom 18505 . . 3 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ (Base‘𝐾)) → (𝑃 (𝑅𝐹)) = ((𝑅𝐹) 𝑃))
1911, 15, 17, 18syl3anc 1370 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝑅𝐹)) = ((𝑅𝐹) 𝑃))
2013, 5, 6ltrncl 40108 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
2115, 20syld3an3 1408 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑃) ∈ (Base‘𝐾))
2213, 2latjcl 18497 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
2311, 15, 21, 22syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
24 simp1r 1197 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊𝐻)
2513, 5lhpbase 39981 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2624, 25syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊 ∈ (Base‘𝐾))
2713, 1, 2latlej1 18506 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → 𝑃 (𝑃 (𝐹𝑃)))
2811, 15, 21, 27syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 (𝑃 (𝐹𝑃)))
2913, 1, 2, 3, 4atmod2i1 39844 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴 ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 (𝑃 (𝐹𝑃))) → (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 𝑃)))
3010, 12, 23, 26, 28, 29syl131anc 1382 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 𝑃)))
31 eqid 2735 . . . . . 6 (1.‘𝐾) = (1.‘𝐾)
321, 2, 31, 4, 5lhpjat1 40003 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 𝑃) = (1.‘𝐾))
33323adant2 1130 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 𝑃) = (1.‘𝐾))
3433oveq2d 7447 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 𝑃)) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)))
35 hlol 39343 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
3610, 35syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ OL)
3713, 3, 31olm11 39209 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 (𝐹𝑃)))
3836, 23, 37syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 (𝐹𝑃)))
3930, 34, 383eqtrrd 2780 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) = (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃))
409, 19, 393eqtr4d 2785 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝑅𝐹)) = (𝑃 (𝐹𝑃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  1.cp1 18482  Latclat 18489  OLcol 39156  Atomscatm 39245  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  trLctrl 40141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142
This theorem is referenced by:  trljat3  40151  trlval4  40171  trlval5  40172  cdlemc5  40178  cdlemk1  40814  cdlemk8  40821  cdlemki  40824  cdlemksv2  40830  cdlemk7  40831  cdlemk12  40833  cdlemk15  40838  cdlemk7u  40853  cdlemk12u  40855  cdlemk21N  40856  cdlemk20  40857  cdlemk22  40876  cdlemm10N  41101
  Copyright terms: Public domain W3C validator