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Theorem trlle 39559
Description: The trace of a lattice translation is less than the fiducial co-atom π‘Š. (Contributed by NM, 25-May-2012.)
Hypotheses
Ref Expression
trlle.l ≀ = (leβ€˜πΎ)
trlle.h 𝐻 = (LHypβ€˜πΎ)
trlle.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
trlle.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
trlle (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜πΉ) ≀ π‘Š)

Proof of Theorem trlle
StepHypRef Expression
1 trlle.l . . . . 5 ≀ = (leβ€˜πΎ)
2 eqid 2724 . . . . 5 (ocβ€˜πΎ) = (ocβ€˜πΎ)
3 eqid 2724 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
4 trlle.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
51, 2, 3, 4lhpocnel 39393 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (((ocβ€˜πΎ)β€˜π‘Š) ∈ (Atomsβ€˜πΎ) ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š))
65adantr 480 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (((ocβ€˜πΎ)β€˜π‘Š) ∈ (Atomsβ€˜πΎ) ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š))
7 eqid 2724 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
8 eqid 2724 . . . 4 (meetβ€˜πΎ) = (meetβ€˜πΎ)
9 trlle.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
10 trlle.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
111, 7, 8, 3, 4, 9, 10trlval2 39538 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (((ocβ€˜πΎ)β€˜π‘Š) ∈ (Atomsβ€˜πΎ) ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š)) β†’ (π‘…β€˜πΉ) = ((((ocβ€˜πΎ)β€˜π‘Š)(joinβ€˜πΎ)(πΉβ€˜((ocβ€˜πΎ)β€˜π‘Š)))(meetβ€˜πΎ)π‘Š))
126, 11mpd3an3 1458 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜πΉ) = ((((ocβ€˜πΎ)β€˜π‘Š)(joinβ€˜πΎ)(πΉβ€˜((ocβ€˜πΎ)β€˜π‘Š)))(meetβ€˜πΎ)π‘Š))
13 hllat 38737 . . . 4 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1413ad2antrr 723 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐾 ∈ Lat)
15 hlop 38736 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
1615ad2antrr 723 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐾 ∈ OP)
17 eqid 2724 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1817, 4lhpbase 39373 . . . . . 6 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
1918ad2antlr 724 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ π‘Š ∈ (Baseβ€˜πΎ))
2017, 2opoccl 38568 . . . . 5 ((𝐾 ∈ OP ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ))
2116, 19, 20syl2anc 583 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ))
2217, 4, 9ltrncl 39500 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ)) β†’ (πΉβ€˜((ocβ€˜πΎ)β€˜π‘Š)) ∈ (Baseβ€˜πΎ))
2321, 22mpd3an3 1458 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (πΉβ€˜((ocβ€˜πΎ)β€˜π‘Š)) ∈ (Baseβ€˜πΎ))
2417, 7latjcl 18400 . . . 4 ((𝐾 ∈ Lat ∧ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ) ∧ (πΉβ€˜((ocβ€˜πΎ)β€˜π‘Š)) ∈ (Baseβ€˜πΎ)) β†’ (((ocβ€˜πΎ)β€˜π‘Š)(joinβ€˜πΎ)(πΉβ€˜((ocβ€˜πΎ)β€˜π‘Š))) ∈ (Baseβ€˜πΎ))
2514, 21, 23, 24syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (((ocβ€˜πΎ)β€˜π‘Š)(joinβ€˜πΎ)(πΉβ€˜((ocβ€˜πΎ)β€˜π‘Š))) ∈ (Baseβ€˜πΎ))
2617, 1, 8latmle2 18426 . . 3 ((𝐾 ∈ Lat ∧ (((ocβ€˜πΎ)β€˜π‘Š)(joinβ€˜πΎ)(πΉβ€˜((ocβ€˜πΎ)β€˜π‘Š))) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((((ocβ€˜πΎ)β€˜π‘Š)(joinβ€˜πΎ)(πΉβ€˜((ocβ€˜πΎ)β€˜π‘Š)))(meetβ€˜πΎ)π‘Š) ≀ π‘Š)
2714, 25, 19, 26syl3anc 1368 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ ((((ocβ€˜πΎ)β€˜π‘Š)(joinβ€˜πΎ)(πΉβ€˜((ocβ€˜πΎ)β€˜π‘Š)))(meetβ€˜πΎ)π‘Š) ≀ π‘Š)
2812, 27eqbrtrd 5161 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜πΉ) ≀ π‘Š)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   class class class wbr 5139  β€˜cfv 6534  (class class class)co 7402  Basecbs 17149  lecple 17209  occoc 17210  joincjn 18272  meetcmee 18273  Latclat 18392  OPcops 38546  Atomscatm 38637  HLchlt 38724  LHypclh 39359  LTrncltrn 39476  trLctrl 39533
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 18256  df-poset 18274  df-plt 18291  df-lub 18307  df-glb 18308  df-join 18309  df-meet 18310  df-p0 18386  df-p1 18387  df-lat 18393  df-oposet 38550  df-ol 38552  df-oml 38553  df-covers 38640  df-ats 38641  df-atl 38672  df-cvlat 38696  df-hlat 38725  df-lhyp 39363  df-laut 39364  df-ldil 39479  df-ltrn 39480  df-trl 39534
This theorem is referenced by:  trlne  39560  cdlemc5  39570  cdlemg6c  39995  cdlemg10c  40014  cdlemg10  40016  cdlemg17dALTN  40039  cdlemg27a  40067  cdlemg31b0N  40069  cdlemg31b0a  40070  cdlemg27b  40071  cdlemg31c  40074  cdlemg35  40088  cdlemh2  40191  cdlemh  40192  cdlemk3  40208  cdlemk9  40214  cdlemk9bN  40215  cdlemk10  40218  cdlemk12  40225  cdlemk14  40229  cdlemk12u  40247  cdlemkfid1N  40296  cdlemk47  40324  dia1N  40428  dia1dim  40436  dia2dimlem1  40439  dia2dimlem10  40448  dib1dim  40540  cdlemn2a  40571  dih1dimb  40615  dihopelvalcpre  40623  dihwN  40664  dihglblem5apreN  40666  dih1dimatlem  40704
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