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Theorem dicffval 38190
Description: The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 15-Dec-2013.)
Hypotheses
Ref Expression
dicval.l = (le‘𝐾)
dicval.a 𝐴 = (Atoms‘𝐾)
dicval.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
dicffval (𝐾𝑉 → (DIsoC‘𝐾) = (𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})))
Distinct variable groups:   𝐴,𝑟   𝑤,𝐻   𝑓,𝑔,𝑞,𝑟,𝑠,𝑤,𝐾
Allowed substitution hints:   𝐴(𝑤,𝑓,𝑔,𝑠,𝑞)   𝐻(𝑓,𝑔,𝑠,𝑟,𝑞)   (𝑤,𝑓,𝑔,𝑠,𝑟,𝑞)   𝑉(𝑤,𝑓,𝑔,𝑠,𝑟,𝑞)

Proof of Theorem dicffval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3510 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6663 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dicval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2871 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6663 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
6 dicval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
75, 6syl6eqr 2871 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
8 fveq2 6663 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9 dicval.l . . . . . . . . 9 = (le‘𝐾)
108, 9syl6eqr 2871 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1110breqd 5068 . . . . . . 7 (𝑘 = 𝐾 → (𝑟(le‘𝑘)𝑤𝑟 𝑤))
1211notbid 319 . . . . . 6 (𝑘 = 𝐾 → (¬ 𝑟(le‘𝑘)𝑤 ↔ ¬ 𝑟 𝑤))
137, 12rabeqbidv 3483 . . . . 5 (𝑘 = 𝐾 → {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} = {𝑟𝐴 ∣ ¬ 𝑟 𝑤})
14 fveq2 6663 . . . . . . . . . . 11 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
1514fveq1d 6665 . . . . . . . . . 10 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
16 fveq2 6663 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
1716fveq1d 6665 . . . . . . . . . . 11 (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑤) = ((oc‘𝐾)‘𝑤))
1817fveqeq2d 6671 . . . . . . . . . 10 (𝑘 = 𝐾 → ((𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞 ↔ (𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞))
1915, 18riotaeqbidv 7106 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞) = (𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞))
2019fveq2d 6667 . . . . . . . 8 (𝑘 = 𝐾 → (𝑠‘(𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)))
2120eqeq2d 2829 . . . . . . 7 (𝑘 = 𝐾 → (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ↔ 𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞))))
22 fveq2 6663 . . . . . . . . 9 (𝑘 = 𝐾 → (TEndo‘𝑘) = (TEndo‘𝐾))
2322fveq1d 6665 . . . . . . . 8 (𝑘 = 𝐾 → ((TEndo‘𝑘)‘𝑤) = ((TEndo‘𝐾)‘𝑤))
2423eleq2d 2895 . . . . . . 7 (𝑘 = 𝐾 → (𝑠 ∈ ((TEndo‘𝑘)‘𝑤) ↔ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤)))
2521, 24anbi12d 630 . . . . . 6 (𝑘 = 𝐾 → ((𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤)) ↔ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))))
2625opabbidv 5123 . . . . 5 (𝑘 = 𝐾 → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤))} = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})
2713, 26mpteq12dv 5142 . . . 4 (𝑘 = 𝐾 → (𝑞 ∈ {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤))}) = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))}))
284, 27mpteq12dv 5142 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑞 ∈ {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤))})) = (𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})))
29 df-dic 38189 . . 3 DIsoC = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑞 ∈ {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤))})))
3028, 29, 3mptfvmpt 6981 . 2 (𝐾 ∈ V → (DIsoC‘𝐾) = (𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})))
311, 30syl 17 1 (𝐾𝑉 → (DIsoC‘𝐾) = (𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492   class class class wbr 5057  {copab 5119  cmpt 5137  cfv 6348  crio 7102  lecple 16560  occoc 16561  Atomscatm 36279  LHypclh 37000  LTrncltrn 37117  TEndoctendo 37768  DIsoCcdic 38188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-dic 38189
This theorem is referenced by:  dicfval  38191
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