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Theorem trl0 35979
Description: If an atom not under the fiducial co-atom 𝑊 equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012.)
Hypotheses
Ref Expression
trl0.l = (le‘𝐾)
trl0.z 0 = (0.‘𝐾)
trl0.a 𝐴 = (Atoms‘𝐾)
trl0.h 𝐻 = (LHyp‘𝐾)
trl0.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
trl0.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
trl0 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = 0 )

Proof of Theorem trl0
StepHypRef Expression
1 simp1 1130 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp3l 1243 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → 𝐹𝑇)
3 simp2 1131 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 trl0.l . . . 4 = (le‘𝐾)
5 eqid 2771 . . . 4 (join‘𝐾) = (join‘𝐾)
6 eqid 2771 . . . 4 (meet‘𝐾) = (meet‘𝐾)
7 trl0.a . . . 4 𝐴 = (Atoms‘𝐾)
8 trl0.h . . . 4 𝐻 = (LHyp‘𝐾)
9 trl0.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
10 trl0.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
114, 5, 6, 7, 8, 9, 10trlval2 35972 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃(join‘𝐾)(𝐹𝑃))(meet‘𝐾)𝑊))
121, 2, 3, 11syl3anc 1476 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = ((𝑃(join‘𝐾)(𝐹𝑃))(meet‘𝐾)𝑊))
13 simp3r 1244 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝐹𝑃) = 𝑃)
1413oveq2d 6809 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(join‘𝐾)(𝐹𝑃)) = (𝑃(join‘𝐾)𝑃))
15 simp1l 1239 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → 𝐾 ∈ HL)
16 simp2l 1241 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → 𝑃𝐴)
175, 7hlatjidm 35177 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
1815, 16, 17syl2anc 573 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(join‘𝐾)𝑃) = 𝑃)
1914, 18eqtrd 2805 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(join‘𝐾)(𝐹𝑃)) = 𝑃)
2019oveq1d 6808 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → ((𝑃(join‘𝐾)(𝐹𝑃))(meet‘𝐾)𝑊) = (𝑃(meet‘𝐾)𝑊))
21 trl0.z . . . 4 0 = (0.‘𝐾)
224, 6, 21, 7, 8lhpmat 35838 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃(meet‘𝐾)𝑊) = 0 )
231, 3, 22syl2anc 573 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(meet‘𝐾)𝑊) = 0 )
2412, 20, 233eqtrd 2809 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = 0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786  cfv 6031  (class class class)co 6793  lecple 16156  joincjn 17152  meetcmee 17153  0.cp0 17245  Atomscatm 35072  HLchlt 35159  LHypclh 35792  LTrncltrn 35909  trLctrl 35967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-lhyp 35796  df-laut 35797  df-ldil 35912  df-ltrn 35913  df-trl 35968
This theorem is referenced by:  trlator0  35980  ltrnnidn  35983  trlid0  35985  trlnidatb  35986  trlnle  35995  trlval3  35996  trlval4  35997  cdlemc6  36005  cdlemg31d  36509
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