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Theorem ltrnldil 40124
Description: A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ltrnldil.h 𝐻 = (LHyp‘𝐾)
ltrnldil.d 𝐷 = ((LDil‘𝐾)‘𝑊)
ltrnldil.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
ltrnldil (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) → 𝐹𝐷)

Proof of Theorem ltrnldil
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 (le‘𝐾) = (le‘𝐾)
2 eqid 2737 . . 3 (join‘𝐾) = (join‘𝐾)
3 eqid 2737 . . 3 (meet‘𝐾) = (meet‘𝐾)
4 eqid 2737 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
5 ltrnldil.h . . 3 𝐻 = (LHyp‘𝐾)
6 ltrnldil.d . . 3 𝐷 = ((LDil‘𝐾)‘𝑊)
7 ltrnldil.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7isltrn 40121 . 2 ((𝐾𝑉𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(𝐹𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(𝐹𝑞))(meet‘𝐾)𝑊)))))
98simprbda 498 1 (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) → 𝐹𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061   class class class wbr 5143  cfv 6561  (class class class)co 7431  lecple 17304  joincjn 18357  meetcmee 18358  Atomscatm 39264  LHypclh 39986  LDilcldil 40102  LTrncltrn 40103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-ltrn 40107
This theorem is referenced by:  ltrnlaut  40125  ltrnval1  40136  ltrncnv  40148  ltrnco  40721
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