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Theorem ltrnfset 36094
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 3365 . 2 (𝐾𝐶𝐾 ∈ V)
2 fveq2 6379 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 ltrnset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2817 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6379 . . . . . 6 (𝑘 = 𝐾 → (LDil‘𝑘) = (LDil‘𝐾))
65fveq1d 6381 . . . . 5 (𝑘 = 𝐾 → ((LDil‘𝑘)‘𝑤) = ((LDil‘𝐾)‘𝑤))
7 fveq2 6379 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
8 ltrnset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
97, 8syl6eqr 2817 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
10 fveq2 6379 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
11 ltrnset.l . . . . . . . . . . . 12 = (le‘𝐾)
1210, 11syl6eqr 2817 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
1312breqd 4822 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑤𝑝 𝑤))
1413notbid 309 . . . . . . . . 9 (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑤 ↔ ¬ 𝑝 𝑤))
1512breqd 4822 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑞(le‘𝑘)𝑤𝑞 𝑤))
1615notbid 309 . . . . . . . . 9 (𝑘 = 𝐾 → (¬ 𝑞(le‘𝑘)𝑤 ↔ ¬ 𝑞 𝑤))
1714, 16anbi12d 624 . . . . . . . 8 (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) ↔ (¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤)))
18 fveq2 6379 . . . . . . . . . . 11 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
19 ltrnset.m . . . . . . . . . . 11 = (meet‘𝐾)
2018, 19syl6eqr 2817 . . . . . . . . . 10 (𝑘 = 𝐾 → (meet‘𝑘) = )
21 fveq2 6379 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
22 ltrnset.j . . . . . . . . . . . 12 = (join‘𝐾)
2321, 22syl6eqr 2817 . . . . . . . . . . 11 (𝑘 = 𝐾 → (join‘𝑘) = )
2423oveqd 6863 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑝(join‘𝑘)(𝑓𝑝)) = (𝑝 (𝑓𝑝)))
25 eqidd 2766 . . . . . . . . . 10 (𝑘 = 𝐾𝑤 = 𝑤)
2620, 24, 25oveq123d 6867 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) = ((𝑝 (𝑓𝑝)) 𝑤))
2723oveqd 6863 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑞(join‘𝑘)(𝑓𝑞)) = (𝑞 (𝑓𝑞)))
2820, 27, 25oveq123d 6867 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑞(join‘𝑘)(𝑓𝑞))(meet‘𝑘)𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))
2926, 28eqeq12d 2780 . . . . . . . 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 335 . . . . . . 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 3300 . . . . . 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 3300 . . . . 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 3344 . . . 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 4894 . . 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 36082 . . 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 6687 . 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 384   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350   class class class wbr 4811  cmpt 4890  cfv 6070  (class class class)co 6846  lecple 16235  joincjn 17224  meetcmee 17225  Atomscatm 35240  LHypclh 35961  LDilcldil 36077  LTrncltrn 36078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-ltrn 36082
This theorem is referenced by:  ltrnset  36095
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