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Theorem trlfset 37164
Description: The set of all traces of lattice translations for a lattice 𝐾. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
trlset.b 𝐵 = (Base‘𝐾)
trlset.l = (le‘𝐾)
trlset.j = (join‘𝐾)
trlset.m = (meet‘𝐾)
trlset.a 𝐴 = (Atoms‘𝐾)
trlset.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
trlfset (𝐾𝐶 → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
Distinct variable groups:   𝐴,𝑝   𝑥,𝐵   𝑤,𝐻   𝑓,𝑝,𝑤,𝑥,𝐾
Allowed substitution hints:   𝐴(𝑥,𝑤,𝑓)   𝐵(𝑤,𝑓,𝑝)   𝐶(𝑥,𝑤,𝑓,𝑝)   𝐻(𝑥,𝑓,𝑝)   (𝑥,𝑤,𝑓,𝑝)   (𝑥,𝑤,𝑓,𝑝)   (𝑥,𝑤,𝑓,𝑝)

Proof of Theorem trlfset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3517 . 2 (𝐾𝐶𝐾 ∈ V)
2 fveq2 6666 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 trlset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2878 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6666 . . . . . 6 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
65fveq1d 6668 . . . . 5 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7 fveq2 6666 . . . . . . 7 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
8 trlset.b . . . . . . 7 𝐵 = (Base‘𝐾)
97, 8syl6eqr 2878 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
10 fveq2 6666 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
11 trlset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
1210, 11syl6eqr 2878 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
13 fveq2 6666 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
14 trlset.l . . . . . . . . . . 11 = (le‘𝐾)
1513, 14syl6eqr 2878 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = )
1615breqd 5073 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑤𝑝 𝑤))
1716notbid 319 . . . . . . . 8 (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑤 ↔ ¬ 𝑝 𝑤))
18 fveq2 6666 . . . . . . . . . . 11 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
19 trlset.m . . . . . . . . . . 11 = (meet‘𝐾)
2018, 19syl6eqr 2878 . . . . . . . . . 10 (𝑘 = 𝐾 → (meet‘𝑘) = )
21 fveq2 6666 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
22 trlset.j . . . . . . . . . . . 12 = (join‘𝐾)
2321, 22syl6eqr 2878 . . . . . . . . . . 11 (𝑘 = 𝐾 → (join‘𝑘) = )
2423oveqd 7168 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑝(join‘𝑘)(𝑓𝑝)) = (𝑝 (𝑓𝑝)))
25 eqidd 2825 . . . . . . . . . 10 (𝑘 = 𝐾𝑤 = 𝑤)
2620, 24, 25oveq123d 7172 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) = ((𝑝 (𝑓𝑝)) 𝑤))
2726eqeq2d 2835 . . . . . . . 8 (𝑘 = 𝐾 → (𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) ↔ 𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))
2817, 27imbi12d 346 . . . . . . 7 (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤)) ↔ (¬ 𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))
2912, 28raleqbidv 3406 . . . . . 6 (𝑘 = 𝐾 → (∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤)) ↔ ∀𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))
309, 29riotaeqbidv 7112 . . . . 5 (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤))) = (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))
316, 30mpteq12dv 5147 . . . 4 (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))
324, 31mpteq12dv 5147 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤))))) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
33 df-trl 37163 . . 3 trL = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤))))))
3432, 33, 3mptfvmpt 6988 . 2 (𝐾 ∈ V → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
351, 34syl 17 1 (𝐾𝐶 → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1530  wcel 2106  wral 3142  Vcvv 3499   class class class wbr 5062  cmpt 5142  cfv 6351  crio 7108  (class class class)co 7151  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Atomscatm 36267  LHypclh 36988  LTrncltrn 37105  trLctrl 37162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-trl 37163
This theorem is referenced by:  trlset  37165
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