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Theorem trljat1 36054
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 35752? (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 2764 . . . 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 36051 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊))
98oveq1d 6856 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑅𝐹) 𝑃) = (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃))
10 simp1l 1254 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ HL)
1110hllatd 35252 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ Lat)
12 simp3l 1258 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃𝐴)
13 eqid 2764 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
1413, 4atbase 35177 . . . 4 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1512, 14syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 ∈ (Base‘𝐾))
1613, 5, 6, 7trlcl 36052 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
17163adant3 1162 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) ∈ (Base‘𝐾))
1813, 2latjcom 17326 . . 3 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ (Base‘𝐾)) → (𝑃 (𝑅𝐹)) = ((𝑅𝐹) 𝑃))
1911, 15, 17, 18syl3anc 1490 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝑅𝐹)) = ((𝑅𝐹) 𝑃))
2013, 5, 6ltrncl 36013 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
2115, 20syld3an3 1528 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑃) ∈ (Base‘𝐾))
2213, 2latjcl 17318 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
2311, 15, 21, 22syl3anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
24 simp1r 1255 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊𝐻)
2513, 5lhpbase 35886 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2624, 25syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊 ∈ (Base‘𝐾))
2713, 1, 2latlej1 17327 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → 𝑃 (𝑃 (𝐹𝑃)))
2811, 15, 21, 27syl3anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 (𝑃 (𝐹𝑃)))
2913, 1, 2, 3, 4atmod2i1 35749 . . . 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 1502 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 𝑃)))
31 eqid 2764 . . . . . 6 (1.‘𝐾) = (1.‘𝐾)
321, 2, 31, 4, 5lhpjat1 35908 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 𝑃) = (1.‘𝐾))
33323adant2 1161 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 𝑃) = (1.‘𝐾))
3433oveq2d 6857 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 𝑃)) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)))
35 hlol 35249 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
3610, 35syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ OL)
3713, 3, 31olm11 35115 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 (𝐹𝑃)))
3836, 23, 37syl2anc 579 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 (𝐹𝑃)))
3930, 34, 383eqtrrd 2803 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) = (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃))
409, 19, 393eqtr4d 2808 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 384  w3a 1107   = wceq 1652  wcel 2155   class class class wbr 4808  cfv 6067  (class class class)co 6841  Basecbs 16131  lecple 16222  joincjn 17211  meetcmee 17212  1.cp1 17305  Latclat 17312  OLcol 35062  Atomscatm 35151  HLchlt 35238  LHypclh 35872  LTrncltrn 35989  trLctrl 36046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-iin 4678  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-1st 7365  df-2nd 7366  df-map 8061  df-proset 17195  df-poset 17213  df-plt 17225  df-lub 17241  df-glb 17242  df-join 17243  df-meet 17244  df-p0 17306  df-p1 17307  df-lat 17313  df-clat 17375  df-oposet 35064  df-ol 35066  df-oml 35067  df-covers 35154  df-ats 35155  df-atl 35186  df-cvlat 35210  df-hlat 35239  df-psubsp 35391  df-pmap 35392  df-padd 35684  df-lhyp 35876  df-laut 35877  df-ldil 35992  df-ltrn 35993  df-trl 36047
This theorem is referenced by:  trljat3  36056  trlval4  36076  trlval5  36077  cdlemc5  36083  cdlemk1  36719  cdlemk8  36726  cdlemki  36729  cdlemksv2  36735  cdlemk7  36736  cdlemk12  36738  cdlemk15  36743  cdlemk7u  36758  cdlemk12u  36760  cdlemk21N  36761  cdlemk20  36762  cdlemk22  36781  cdlemm10N  37006
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