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Theorem trljat1 34471
 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 34168? (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 2610 . . . 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 34468 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊))
98oveq1d 6564 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑅𝐹) 𝑃) = (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃))
10 simp1l 1078 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ HL)
11 hllat 33668 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1210, 11syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ Lat)
13 simp3l 1082 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃𝐴)
14 eqid 2610 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
1514, 4atbase 33594 . . . 4 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1613, 15syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 ∈ (Base‘𝐾))
1714, 5, 6, 7trlcl 34469 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
18173adant3 1074 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) ∈ (Base‘𝐾))
1914, 2latjcom 16882 . . 3 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ (Base‘𝐾)) → (𝑃 (𝑅𝐹)) = ((𝑅𝐹) 𝑃))
2012, 16, 18, 19syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝑅𝐹)) = ((𝑅𝐹) 𝑃))
2114, 5, 6ltrncl 34429 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
2216, 21syld3an3 1363 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑃) ∈ (Base‘𝐾))
2314, 2latjcl 16874 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
2412, 16, 22, 23syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
25 simp1r 1079 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊𝐻)
2614, 5lhpbase 34302 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2725, 26syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊 ∈ (Base‘𝐾))
2814, 1, 2latlej1 16883 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → 𝑃 (𝑃 (𝐹𝑃)))
2912, 16, 22, 28syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 (𝑃 (𝐹𝑃)))
3014, 1, 2, 3, 4atmod2i1 34165 . . . 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)))
3110, 13, 24, 27, 29, 30syl131anc 1331 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 𝑃)))
32 eqid 2610 . . . . . 6 (1.‘𝐾) = (1.‘𝐾)
331, 2, 32, 4, 5lhpjat1 34324 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 𝑃) = (1.‘𝐾))
34333adant2 1073 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 𝑃) = (1.‘𝐾))
3534oveq2d 6565 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 𝑃)) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)))
36 hlol 33666 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
3710, 36syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ OL)
3814, 3, 32olm11 33532 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 (𝐹𝑃)))
3937, 24, 38syl2anc 691 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 (𝐹𝑃)))
4031, 35, 393eqtrrd 2649 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) = (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) 𝑃))
419, 20, 403eqtr4d 2654 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  1.cp1 16861  Latclat 16868  OLcol 33479  Atomscatm 33568  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  trLctrl 34463 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464 This theorem is referenced by:  trljat3  34473  trlval4  34493  trlval5  34494  cdlemc5  34500  cdlemk1  35137  cdlemk8  35144  cdlemki  35147  cdlemksv2  35153  cdlemk7  35154  cdlemk12  35156  cdlemk15  35161  cdlemk7u  35176  cdlemk12u  35178  cdlemk21N  35179  cdlemk20  35180  cdlemk22  35199  cdlemm10N  35425
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