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Theorem trl0 37465
 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 1133 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp3l 1198 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → 𝐹𝑇)
3 simp2 1134 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 trl0.l . . . 4 = (le‘𝐾)
5 eqid 2801 . . . 4 (join‘𝐾) = (join‘𝐾)
6 eqid 2801 . . . 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 37458 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃(join‘𝐾)(𝐹𝑃))(meet‘𝐾)𝑊))
121, 2, 3, 11syl3anc 1368 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = ((𝑃(join‘𝐾)(𝐹𝑃))(meet‘𝐾)𝑊))
13 simp3r 1199 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝐹𝑃) = 𝑃)
1413oveq2d 7155 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(join‘𝐾)(𝐹𝑃)) = (𝑃(join‘𝐾)𝑃))
15 simp1l 1194 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → 𝐾 ∈ HL)
16 simp2l 1196 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → 𝑃𝐴)
175, 7hlatjidm 36664 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
1815, 16, 17syl2anc 587 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(join‘𝐾)𝑃) = 𝑃)
1914, 18eqtrd 2836 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(join‘𝐾)(𝐹𝑃)) = 𝑃)
2019oveq1d 7154 . 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 37325 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃(meet‘𝐾)𝑊) = 0 )
231, 3, 22syl2anc 587 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑃(meet‘𝐾)𝑊) = 0 )
2412, 20, 233eqtrd 2840 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139  lecple 16568  joincjn 17550  meetcmee 17551  0.cp0 17643  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-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-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-lat 17652  df-covers 36561  df-ats 36562  df-atl 36593  df-cvlat 36617  df-hlat 36646  df-lhyp 37283  df-laut 37284  df-ldil 37399  df-ltrn 37400  df-trl 37454 This theorem is referenced by:  trlator0  37466  ltrnnidn  37469  trlid0  37471  trlnidatb  37472  trlnle  37481  trlval3  37482  trlval4  37483  cdlemc6  37491  cdlemg31d  37995
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