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Theorem ltrnldil 34222
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 2609 . . 3 (le‘𝐾) = (le‘𝐾)
2 eqid 2609 . . 3 (join‘𝐾) = (join‘𝐾)
3 eqid 2609 . . 3 (meet‘𝐾) = (meet‘𝐾)
4 eqid 2609 . . 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 34219 . 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 650 1 (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) → 𝐹𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895   class class class wbr 4577  cfv 5790  (class class class)co 6527  lecple 15721  joincjn 16713  meetcmee 16714  Atomscatm 33364  LHypclh 34084  LDilcldil 34200  LTrncltrn 34201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-ltrn 34205
This theorem is referenced by:  ltrnlaut  34223  ltrnval1  34234  ltrncnv  34246  ltrnco  34821
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