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Theorem trlfset 35969
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 3364 . 2 (𝐾𝐶𝐾 ∈ V)
2 fveq2 6332 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 trlset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2823 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6332 . . . . . 6 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
65fveq1d 6334 . . . . 5 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7 fveq2 6332 . . . . . . 7 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
8 trlset.b . . . . . . 7 𝐵 = (Base‘𝐾)
97, 8syl6eqr 2823 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
10 fveq2 6332 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
11 trlset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
1210, 11syl6eqr 2823 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
13 fveq2 6332 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
14 trlset.l . . . . . . . . . . 11 = (le‘𝐾)
1513, 14syl6eqr 2823 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = )
1615breqd 4797 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑤𝑝 𝑤))
1716notbid 307 . . . . . . . 8 (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑤 ↔ ¬ 𝑝 𝑤))
18 fveq2 6332 . . . . . . . . . . 11 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
19 trlset.m . . . . . . . . . . 11 = (meet‘𝐾)
2018, 19syl6eqr 2823 . . . . . . . . . 10 (𝑘 = 𝐾 → (meet‘𝑘) = )
21 fveq2 6332 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
22 trlset.j . . . . . . . . . . . 12 = (join‘𝐾)
2321, 22syl6eqr 2823 . . . . . . . . . . 11 (𝑘 = 𝐾 → (join‘𝑘) = )
2423oveqd 6810 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑝(join‘𝑘)(𝑓𝑝)) = (𝑝 (𝑓𝑝)))
25 eqidd 2772 . . . . . . . . . 10 (𝑘 = 𝐾𝑤 = 𝑤)
2620, 24, 25oveq123d 6814 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) = ((𝑝 (𝑓𝑝)) 𝑤))
2726eqeq2d 2781 . . . . . . . 8 (𝑘 = 𝐾 → (𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) ↔ 𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))
2817, 27imbi12d 333 . . . . . . 7 (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤)) ↔ (¬ 𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))
2912, 28raleqbidv 3301 . . . . . 6 (𝑘 = 𝐾 → (∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤)) ↔ ∀𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))
309, 29riotaeqbidv 6757 . . . . 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 4867 . . . 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 4867 . . 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 35968 . . 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))))))
34 fvex 6342 . . . . 5 (LHyp‘𝐾) ∈ V
353, 34eqeltri 2846 . . . 4 𝐻 ∈ V
3635mptex 6630 . . 3 (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))) ∈ V
3732, 33, 36fvmpt 6424 . 2 (𝐾 ∈ V → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
381, 37syl 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 1631  wcel 2145  wral 3061  Vcvv 3351   class class class wbr 4786  cmpt 4863  cfv 6031  crio 6753  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Atomscatm 35072  LHypclh 35792  LTrncltrn 35909  trLctrl 35967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-trl 35968
This theorem is referenced by:  trlset  35970
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