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Theorem trl0 35775
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 1081 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp3l 1109 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → 𝐹𝑇)
3 simp2 1082 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 trl0.l . . . 4 = (le‘𝐾)
5 eqid 2651 . . . 4 (join‘𝐾) = (join‘𝐾)
6 eqid 2651 . . . 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 35768 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃(join‘𝐾)(𝐹𝑃))(meet‘𝐾)𝑊))
121, 2, 3, 11syl3anc 1366 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = ((𝑃(join‘𝐾)(𝐹𝑃))(meet‘𝐾)𝑊))
13 simp3r 1110 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝐹𝑃) = 𝑃)
1413oveq2d 6706 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(join‘𝐾)(𝐹𝑃)) = (𝑃(join‘𝐾)𝑃))
15 simp1l 1105 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → 𝐾 ∈ HL)
16 simp2l 1107 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → 𝑃𝐴)
175, 7hlatjidm 34973 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
1815, 16, 17syl2anc 694 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(join‘𝐾)𝑃) = 𝑃)
1914, 18eqtrd 2685 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(join‘𝐾)(𝐹𝑃)) = 𝑃)
2019oveq1d 6705 . 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 35634 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃(meet‘𝐾)𝑊) = 0 )
231, 3, 22syl2anc 694 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(meet‘𝐾)𝑊) = 0 )
2412, 20, 233eqtrd 2689 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 383  w3a 1054   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  (class class class)co 6690  lecple 15995  joincjn 16991  meetcmee 16992  0.cp0 17084  Atomscatm 34868  HLchlt 34955  LHypclh 35588  LTrncltrn 35705  trLctrl 35763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764
This theorem is referenced by:  trlator0  35776  ltrnnidn  35779  trlid0  35781  trlnidatb  35782  trlnle  35791  trlval3  35792  trlval4  35793  cdlemc6  35801  cdlemg31d  36305
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