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Theorem ltrnfset 35924
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 3352 . 2 (𝐾𝐶𝐾 ∈ V)
2 fveq2 6353 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 ltrnset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2812 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6353 . . . . . 6 (𝑘 = 𝐾 → (LDil‘𝑘) = (LDil‘𝐾))
65fveq1d 6355 . . . . 5 (𝑘 = 𝐾 → ((LDil‘𝑘)‘𝑤) = ((LDil‘𝐾)‘𝑤))
7 fveq2 6353 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
8 ltrnset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
97, 8syl6eqr 2812 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
10 fveq2 6353 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
11 ltrnset.l . . . . . . . . . . . 12 = (le‘𝐾)
1210, 11syl6eqr 2812 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
1312breqd 4815 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑤𝑝 𝑤))
1413notbid 307 . . . . . . . . 9 (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑤 ↔ ¬ 𝑝 𝑤))
1512breqd 4815 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑞(le‘𝑘)𝑤𝑞 𝑤))
1615notbid 307 . . . . . . . . 9 (𝑘 = 𝐾 → (¬ 𝑞(le‘𝑘)𝑤 ↔ ¬ 𝑞 𝑤))
1714, 16anbi12d 749 . . . . . . . 8 (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) ↔ (¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤)))
18 fveq2 6353 . . . . . . . . . . 11 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
19 ltrnset.m . . . . . . . . . . 11 = (meet‘𝐾)
2018, 19syl6eqr 2812 . . . . . . . . . 10 (𝑘 = 𝐾 → (meet‘𝑘) = )
21 fveq2 6353 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
22 ltrnset.j . . . . . . . . . . . 12 = (join‘𝐾)
2321, 22syl6eqr 2812 . . . . . . . . . . 11 (𝑘 = 𝐾 → (join‘𝑘) = )
2423oveqd 6831 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑝(join‘𝑘)(𝑓𝑝)) = (𝑝 (𝑓𝑝)))
25 eqidd 2761 . . . . . . . . . 10 (𝑘 = 𝐾𝑤 = 𝑤)
2620, 24, 25oveq123d 6835 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) = ((𝑝 (𝑓𝑝)) 𝑤))
2723oveqd 6831 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑞(join‘𝑘)(𝑓𝑞)) = (𝑞 (𝑓𝑞)))
2820, 27, 25oveq123d 6835 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑞(join‘𝑘)(𝑓𝑞))(meet‘𝑘)𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))
2926, 28eqeq12d 2775 . . . . . . . 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 333 . . . . . . 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 3291 . . . . . 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 3291 . . . . 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 3335 . . . 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 4885 . . 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 35912 . . 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))}))
36 fvex 6363 . . . . 5 (LHyp‘𝐾) ∈ V
373, 36eqeltri 2835 . . . 4 𝐻 ∈ V
3837mptex 6651 . . 3 (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))}) ∈ V
3934, 35, 38fvmpt 6445 . 2 (𝐾 ∈ V → (LTrn‘𝐾) = (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))}))
401, 39syl 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 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340   class class class wbr 4804  cmpt 4881  cfv 6049  (class class class)co 6814  lecple 16170  joincjn 17165  meetcmee 17166  Atomscatm 35071  LHypclh 35791  LDilcldil 35907  LTrncltrn 35908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-ltrn 35912
This theorem is referenced by:  ltrnset  35925
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