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Theorem trl0 39637
Description: If an atom not under the fiducial co-atom π‘Š equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012.)
Hypotheses
Ref Expression
trl0.l ≀ = (leβ€˜πΎ)
trl0.z 0 = (0.β€˜πΎ)
trl0.a 𝐴 = (Atomsβ€˜πΎ)
trl0.h 𝐻 = (LHypβ€˜πΎ)
trl0.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
trl0.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
trl0 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (π‘…β€˜πΉ) = 0 )

Proof of Theorem trl0
StepHypRef Expression
1 simp1 1134 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 simp3l 1199 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ 𝐹 ∈ 𝑇)
3 simp2 1135 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
4 trl0.l . . . 4 ≀ = (leβ€˜πΎ)
5 eqid 2728 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
6 eqid 2728 . . . 4 (meetβ€˜πΎ) = (meetβ€˜πΎ)
7 trl0.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
8 trl0.h . . . 4 𝐻 = (LHypβ€˜πΎ)
9 trl0.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
10 trl0.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
114, 5, 6, 7, 8, 9, 10trlval2 39630 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (π‘…β€˜πΉ) = ((𝑃(joinβ€˜πΎ)(πΉβ€˜π‘ƒ))(meetβ€˜πΎ)π‘Š))
121, 2, 3, 11syl3anc 1369 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (π‘…β€˜πΉ) = ((𝑃(joinβ€˜πΎ)(πΉβ€˜π‘ƒ))(meetβ€˜πΎ)π‘Š))
13 simp3r 1200 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (πΉβ€˜π‘ƒ) = 𝑃)
1413oveq2d 7430 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (𝑃(joinβ€˜πΎ)(πΉβ€˜π‘ƒ)) = (𝑃(joinβ€˜πΎ)𝑃))
15 simp1l 1195 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ 𝐾 ∈ HL)
16 simp2l 1197 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ 𝑃 ∈ 𝐴)
175, 7hlatjidm 38835 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) β†’ (𝑃(joinβ€˜πΎ)𝑃) = 𝑃)
1815, 16, 17syl2anc 583 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (𝑃(joinβ€˜πΎ)𝑃) = 𝑃)
1914, 18eqtrd 2768 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (𝑃(joinβ€˜πΎ)(πΉβ€˜π‘ƒ)) = 𝑃)
2019oveq1d 7429 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ ((𝑃(joinβ€˜πΎ)(πΉβ€˜π‘ƒ))(meetβ€˜πΎ)π‘Š) = (𝑃(meetβ€˜πΎ)π‘Š))
21 trl0.z . . . 4 0 = (0.β€˜πΎ)
224, 6, 21, 7, 8lhpmat 39497 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (𝑃(meetβ€˜πΎ)π‘Š) = 0 )
231, 3, 22syl2anc 583 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (𝑃(meetβ€˜πΎ)π‘Š) = 0 )
2412, 20, 233eqtrd 2772 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (π‘…β€˜πΉ) = 0 )
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  lecple 17233  joincjn 18296  meetcmee 18297  0.cp0 18408  Atomscatm 38729  HLchlt 38816  LHypclh 39451  LTrncltrn 39568  trLctrl 39625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8840  df-proset 18280  df-poset 18298  df-plt 18315  df-lub 18331  df-glb 18332  df-join 18333  df-meet 18334  df-p0 18410  df-lat 18417  df-covers 38732  df-ats 38733  df-atl 38764  df-cvlat 38788  df-hlat 38817  df-lhyp 39455  df-laut 39456  df-ldil 39571  df-ltrn 39572  df-trl 39626
This theorem is referenced by:  trlator0  39638  ltrnnidn  39641  trlid0  39643  trlnidatb  39644  trlnle  39653  trlval3  39654  trlval4  39655  cdlemc6  39663  cdlemg31d  40167
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