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Theorem ltrnfset 37868
Description: The set of all lattice translations for a lattice 𝐾. (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
ltrnset.l = (le‘𝐾)
ltrnset.j = (join‘𝐾)
ltrnset.m = (meet‘𝐾)
ltrnset.a 𝐴 = (Atoms‘𝐾)
ltrnset.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
ltrnfset (𝐾𝐶 → (LTrn‘𝐾) = (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))}))
Distinct variable groups:   𝑞,𝑝,𝐴   𝑤,𝐻   𝑓,𝑝,𝑞,𝑤,𝐾
Allowed substitution hints:   𝐴(𝑤,𝑓)   𝐶(𝑤,𝑓,𝑞,𝑝)   𝐻(𝑓,𝑞,𝑝)   (𝑤,𝑓,𝑞,𝑝)   (𝑤,𝑓,𝑞,𝑝)   (𝑤,𝑓,𝑞,𝑝)

Proof of Theorem ltrnfset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3426 . 2 (𝐾𝐶𝐾 ∈ V)
2 fveq2 6717 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 ltrnset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2796 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6717 . . . . . 6 (𝑘 = 𝐾 → (LDil‘𝑘) = (LDil‘𝐾))
65fveq1d 6719 . . . . 5 (𝑘 = 𝐾 → ((LDil‘𝑘)‘𝑤) = ((LDil‘𝐾)‘𝑤))
7 fveq2 6717 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
8 ltrnset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
97, 8eqtr4di 2796 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
10 fveq2 6717 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
11 ltrnset.l . . . . . . . . . . . 12 = (le‘𝐾)
1210, 11eqtr4di 2796 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
1312breqd 5064 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑤𝑝 𝑤))
1413notbid 321 . . . . . . . . 9 (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑤 ↔ ¬ 𝑝 𝑤))
1512breqd 5064 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑞(le‘𝑘)𝑤𝑞 𝑤))
1615notbid 321 . . . . . . . . 9 (𝑘 = 𝐾 → (¬ 𝑞(le‘𝑘)𝑤 ↔ ¬ 𝑞 𝑤))
1714, 16anbi12d 634 . . . . . . . 8 (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) ↔ (¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤)))
18 fveq2 6717 . . . . . . . . . . 11 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
19 ltrnset.m . . . . . . . . . . 11 = (meet‘𝐾)
2018, 19eqtr4di 2796 . . . . . . . . . 10 (𝑘 = 𝐾 → (meet‘𝑘) = )
21 fveq2 6717 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
22 ltrnset.j . . . . . . . . . . . 12 = (join‘𝐾)
2321, 22eqtr4di 2796 . . . . . . . . . . 11 (𝑘 = 𝐾 → (join‘𝑘) = )
2423oveqd 7230 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑝(join‘𝑘)(𝑓𝑝)) = (𝑝 (𝑓𝑝)))
25 eqidd 2738 . . . . . . . . . 10 (𝑘 = 𝐾𝑤 = 𝑤)
2620, 24, 25oveq123d 7234 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) = ((𝑝 (𝑓𝑝)) 𝑤))
2723oveqd 7230 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑞(join‘𝑘)(𝑓𝑞)) = (𝑞 (𝑓𝑞)))
2820, 27, 25oveq123d 7234 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑞(join‘𝑘)(𝑓𝑞))(meet‘𝑘)𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))
2926, 28eqeq12d 2753 . . . . . . . 8 (𝑘 = 𝐾 → (((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓𝑞))(meet‘𝑘)𝑤) ↔ ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤)))
3017, 29imbi12d 348 . . . . . . 7 (𝑘 = 𝐾 → (((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) → ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓𝑞))(meet‘𝑘)𝑤)) ↔ ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))))
319, 30raleqbidv 3313 . . . . . 6 (𝑘 = 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)) ↔ ∀𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))))
329, 31raleqbidv 3313 . . . . 5 (𝑘 = 𝐾 → (∀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)) ↔ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))))
336, 32rabeqbidv 3396 . . . 4 (𝑘 = 𝐾 → {𝑓 ∈ ((LDil‘𝑘)‘𝑤) ∣ ∀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))} = {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))})
344, 33mpteq12dv 5140 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ ((LDil‘𝑘)‘𝑤) ∣ ∀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))}) = (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))}))
35 df-ltrn 37856 . . 3 LTrn = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ ((LDil‘𝑘)‘𝑤) ∣ ∀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))}))
3634, 35, 3mptfvmpt 7044 . 2 (𝐾 ∈ V → (LTrn‘𝐾) = (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))}))
371, 36syl 17 1 (𝐾𝐶 → (LTrn‘𝐾) = (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wcel 2110  wral 3061  {crab 3065  Vcvv 3408   class class class wbr 5053  cmpt 5135  cfv 6380  (class class class)co 7213  lecple 16809  joincjn 17818  meetcmee 17819  Atomscatm 37014  LHypclh 37735  LDilcldil 37851  LTrncltrn 37852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-ltrn 37856
This theorem is referenced by:  ltrnset  37869
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