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Theorem trljat2 37173
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. (Contributed by NM, 25-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
trljat2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) (𝑅𝐹)) = (𝑃 (𝐹𝑃)))

Proof of Theorem trljat2
StepHypRef Expression
1 simp1l 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ HL)
2 trljat.l . . . . . 6 = (le‘𝐾)
3 trljat.a . . . . . 6 𝐴 = (Atoms‘𝐾)
4 trljat.h . . . . . 6 𝐻 = (LHyp‘𝐾)
5 trljat.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
62, 3, 4, 5ltrnat 37146 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)
763adant3r 1175 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑃) ∈ 𝐴)
81hllatd 36370 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ Lat)
9 simp3l 1195 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃𝐴)
10 eqid 2825 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1110, 3atbase 36295 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
129, 11syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 ∈ (Base‘𝐾))
13 simp1 1130 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
14 simp2 1131 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐹𝑇)
1510, 4, 5ltrncl 37131 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
1613, 14, 12, 15syl3anc 1365 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑃) ∈ (Base‘𝐾))
17 trljat.j . . . . . 6 = (join‘𝐾)
1810, 17latjcl 17654 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
198, 12, 16, 18syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
20 simp1r 1192 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊𝐻)
2110, 4lhpbase 37004 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2220, 21syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊 ∈ (Base‘𝐾))
2310, 2, 17latlej2 17664 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → (𝐹𝑃) (𝑃 (𝐹𝑃)))
248, 12, 16, 23syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑃) (𝑃 (𝐹𝑃)))
25 eqid 2825 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
2610, 2, 17, 25, 3atmod2i1 36867 . . . 4 ((𝐾 ∈ HL ∧ ((𝐹𝑃) ∈ 𝐴 ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ (𝐹𝑃) (𝑃 (𝐹𝑃))) → (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) (𝐹𝑃)) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 (𝐹𝑃))))
271, 7, 19, 22, 24, 26syl131anc 1377 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) (𝐹𝑃)) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 (𝐹𝑃))))
282, 3, 4, 5ltrnel 37145 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
29 eqid 2825 . . . . . 6 (1.‘𝐾) = (1.‘𝐾)
302, 17, 29, 3, 4lhpjat1 37026 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊)) → (𝑊 (𝐹𝑃)) = (1.‘𝐾))
311, 20, 28, 30syl21anc 835 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 (𝐹𝑃)) = (1.‘𝐾))
3231oveq2d 7167 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(𝑊 (𝐹𝑃))) = ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)))
33 hlol 36367 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
341, 33syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ OL)
3510, 25, 29olm11 36233 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 (𝐹𝑃)))
3634, 19, 35syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 (𝐹𝑃)))
3727, 32, 363eqtrrd 2865 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) = (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) (𝐹𝑃)))
38 trljat.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
392, 17, 25, 3, 4, 5, 38trlval2 37169 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊))
4039oveq1d 7166 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑅𝐹) (𝐹𝑃)) = (((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) (𝐹𝑃)))
4110, 4, 5, 38trlcl 37170 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
4213, 14, 41syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) ∈ (Base‘𝐾))
4310, 17latjcom 17662 . . 3 ((𝐾 ∈ Lat ∧ (𝑅𝐹) ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → ((𝑅𝐹) (𝐹𝑃)) = ((𝐹𝑃) (𝑅𝐹)))
448, 42, 16, 43syl3anc 1365 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑅𝐹) (𝐹𝑃)) = ((𝐹𝑃) (𝑅𝐹)))
4537, 40, 443eqtr2rd 2867 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) (𝑅𝐹)) = (𝑃 (𝐹𝑃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107   class class class wbr 5062  cfv 6351  (class class class)co 7151  Basecbs 16476  lecple 16565  joincjn 17547  meetcmee 17548  1.cp1 17641  Latclat 17648  OLcol 36180  Atomscatm 36269  HLchlt 36356  LHypclh 36990  LTrncltrn 37107  trLctrl 37164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-map 8401  df-proset 17531  df-poset 17549  df-plt 17561  df-lub 17577  df-glb 17578  df-join 17579  df-meet 17580  df-p0 17642  df-p1 17643  df-lat 17649  df-clat 17711  df-oposet 36182  df-ol 36184  df-oml 36185  df-covers 36272  df-ats 36273  df-atl 36304  df-cvlat 36328  df-hlat 36357  df-psubsp 36509  df-pmap 36510  df-padd 36802  df-lhyp 36994  df-laut 36995  df-ldil 37110  df-ltrn 37111  df-trl 37165
This theorem is referenced by:  trljat3  37174  cdlemc3  37199
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