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Theorem trl0 39535
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 1133 . . 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 1198 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ 𝐹 ∈ 𝑇)
3 simp2 1134 . . 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 2724 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
6 eqid 2724 . . . 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 39528 . . 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 1368 . 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 1199 . . . . 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 7418 . . . 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 1194 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ 𝐾 ∈ HL)
16 simp2l 1196 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ 𝑃 ∈ 𝐴)
175, 7hlatjidm 38733 . . . . 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 2764 . . 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 7417 . 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 39395 . . 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 2768 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 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5139  β€˜cfv 6534  (class class class)co 7402  lecple 17205  joincjn 18268  meetcmee 18269  0.cp0 18380  Atomscatm 38627  HLchlt 38714  LHypclh 39349  LTrncltrn 39466  trLctrl 39523
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-lat 18389  df-covers 38630  df-ats 38631  df-atl 38662  df-cvlat 38686  df-hlat 38715  df-lhyp 39353  df-laut 39354  df-ldil 39469  df-ltrn 39470  df-trl 39524
This theorem is referenced by:  trlator0  39536  ltrnnidn  39539  trlid0  39541  trlnidatb  39542  trlnle  39551  trlval3  39552  trlval4  39553  cdlemc6  39561  cdlemg31d  40065
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