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Theorem trlle 38650
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 2737 . . . . 5 (ocβ€˜πΎ) = (ocβ€˜πΎ)
3 eqid 2737 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
4 trlle.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
51, 2, 3, 4lhpocnel 38484 . . . 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 482 . . 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 2737 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
8 eqid 2737 . . . 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 38629 . . 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 1463 . 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 37828 . . . 4 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1413ad2antrr 725 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐾 ∈ Lat)
15 hlop 37827 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
1615ad2antrr 725 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐾 ∈ OP)
17 eqid 2737 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1817, 4lhpbase 38464 . . . . . 6 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
1918ad2antlr 726 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ π‘Š ∈ (Baseβ€˜πΎ))
2017, 2opoccl 37659 . . . . 5 ((𝐾 ∈ OP ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ))
2116, 19, 20syl2anc 585 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ))
2217, 4, 9ltrncl 38591 . . . . 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 1463 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (πΉβ€˜((ocβ€˜πΎ)β€˜π‘Š)) ∈ (Baseβ€˜πΎ))
2417, 7latjcl 18329 . . . 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 1372 . . 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 18355 . . 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 1372 . 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 5128 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 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  lecple 17141  occoc 17142  joincjn 18201  meetcmee 18202  Latclat 18321  OPcops 37637  Atomscatm 37728  HLchlt 37815  LHypclh 38450  LTrncltrn 38567  trLctrl 38624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8768  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-p1 18316  df-lat 18322  df-oposet 37641  df-ol 37643  df-oml 37644  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-lhyp 38454  df-laut 38455  df-ldil 38570  df-ltrn 38571  df-trl 38625
This theorem is referenced by:  trlne  38651  cdlemc5  38661  cdlemg6c  39086  cdlemg10c  39105  cdlemg10  39107  cdlemg17dALTN  39130  cdlemg27a  39158  cdlemg31b0N  39160  cdlemg31b0a  39161  cdlemg27b  39162  cdlemg31c  39165  cdlemg35  39179  cdlemh2  39282  cdlemh  39283  cdlemk3  39299  cdlemk9  39305  cdlemk9bN  39306  cdlemk10  39309  cdlemk12  39316  cdlemk14  39320  cdlemk12u  39338  cdlemkfid1N  39387  cdlemk47  39415  dia1N  39519  dia1dim  39527  dia2dimlem1  39530  dia2dimlem10  39539  dib1dim  39631  cdlemn2a  39662  dih1dimb  39706  dihopelvalcpre  39714  dihwN  39755  dihglblem5apreN  39757  dih1dimatlem  39795
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