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Theorem trljat1 37461
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 37159? (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 2801 . . . 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 37458 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊))
98oveq1d 7154 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑅𝐹) 𝑃) = (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃))
10 simp1l 1194 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ HL)
1110hllatd 36659 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ Lat)
12 simp3l 1198 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃𝐴)
13 eqid 2801 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
1413, 4atbase 36584 . . . 4 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1512, 14syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 ∈ (Base‘𝐾))
1613, 5, 6, 7trlcl 37459 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
17163adant3 1129 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) ∈ (Base‘𝐾))
1813, 2latjcom 17665 . . 3 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ (Base‘𝐾)) → (𝑃 (𝑅𝐹)) = ((𝑅𝐹) 𝑃))
1911, 15, 17, 18syl3anc 1368 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝑅𝐹)) = ((𝑅𝐹) 𝑃))
2013, 5, 6ltrncl 37420 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
2115, 20syld3an3 1406 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑃) ∈ (Base‘𝐾))
2213, 2latjcl 17657 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
2311, 15, 21, 22syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
24 simp1r 1195 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊𝐻)
2513, 5lhpbase 37293 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2624, 25syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊 ∈ (Base‘𝐾))
2713, 1, 2latlej1 17666 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → 𝑃 (𝑃 (𝐹𝑃)))
2811, 15, 21, 27syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 (𝑃 (𝐹𝑃)))
2913, 1, 2, 3, 4atmod2i1 37156 . . . 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 1380 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 𝑃)))
31 eqid 2801 . . . . . 6 (1.‘𝐾) = (1.‘𝐾)
321, 2, 31, 4, 5lhpjat1 37315 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 𝑃) = (1.‘𝐾))
33323adant2 1128 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 𝑃) = (1.‘𝐾))
3433oveq2d 7155 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 𝑃)) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)))
35 hlol 36656 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
3610, 35syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ OL)
3713, 3, 31olm11 36522 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 (𝐹𝑃)))
3836, 23, 37syl2anc 587 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 (𝐹𝑃)))
3930, 34, 383eqtrrd 2841 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) = (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃))
409, 19, 393eqtr4d 2846 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 399  w3a 1084   = wceq 1538  wcel 2112   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16479  lecple 16568  joincjn 17550  meetcmee 17551  1.cp1 17644  Latclat 17651  OLcol 36469  Atomscatm 36558  HLchlt 36645  LHypclh 37279  LTrncltrn 37396  trLctrl 37453
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-proset 17534  df-poset 17552  df-plt 17564  df-lub 17580  df-glb 17581  df-join 17582  df-meet 17583  df-p0 17645  df-p1 17646  df-lat 17652  df-clat 17714  df-oposet 36471  df-ol 36473  df-oml 36474  df-covers 36561  df-ats 36562  df-atl 36593  df-cvlat 36617  df-hlat 36646  df-psubsp 36798  df-pmap 36799  df-padd 37091  df-lhyp 37283  df-laut 37284  df-ldil 37399  df-ltrn 37400  df-trl 37454
This theorem is referenced by:  trljat3  37463  trlval4  37483  trlval5  37484  cdlemc5  37490  cdlemk1  38126  cdlemk8  38133  cdlemki  38136  cdlemksv2  38142  cdlemk7  38143  cdlemk12  38145  cdlemk15  38150  cdlemk7u  38165  cdlemk12u  38167  cdlemk21N  38168  cdlemk20  38169  cdlemk22  38188  cdlemm10N  38413
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