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Theorem dicffval 35940
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 3198 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6148 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dicval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2673 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6148 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
6 dicval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
75, 6syl6eqr 2673 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
8 fveq2 6148 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9 dicval.l . . . . . . . . 9 = (le‘𝐾)
108, 9syl6eqr 2673 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1110breqd 4624 . . . . . . 7 (𝑘 = 𝐾 → (𝑟(le‘𝑘)𝑤𝑟 𝑤))
1211notbid 308 . . . . . 6 (𝑘 = 𝐾 → (¬ 𝑟(le‘𝑘)𝑤 ↔ ¬ 𝑟 𝑤))
137, 12rabeqbidv 3181 . . . . 5 (𝑘 = 𝐾 → {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} = {𝑟𝐴 ∣ ¬ 𝑟 𝑤})
14 fveq2 6148 . . . . . . . . . . 11 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
1514fveq1d 6150 . . . . . . . . . 10 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
16 fveq2 6148 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
1716fveq1d 6150 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑤) = ((oc‘𝐾)‘𝑤))
1817fveq2d 6152 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑔‘((oc‘𝑘)‘𝑤)) = (𝑔‘((oc‘𝐾)‘𝑤)))
1918eqeq1d 2623 . . . . . . . . . 10 (𝑘 = 𝐾 → ((𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞 ↔ (𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞))
2015, 19riotaeqbidv 6568 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞) = (𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞))
2120fveq2d 6152 . . . . . . . 8 (𝑘 = 𝐾 → (𝑠‘(𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)))
2221eqeq2d 2631 . . . . . . 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))))
23 fveq2 6148 . . . . . . . . 9 (𝑘 = 𝐾 → (TEndo‘𝑘) = (TEndo‘𝐾))
2423fveq1d 6150 . . . . . . . 8 (𝑘 = 𝐾 → ((TEndo‘𝑘)‘𝑤) = ((TEndo‘𝐾)‘𝑤))
2524eleq2d 2684 . . . . . . 7 (𝑘 = 𝐾 → (𝑠 ∈ ((TEndo‘𝑘)‘𝑤) ↔ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤)))
2622, 25anbi12d 746 . . . . . 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))))
2726opabbidv 4678 . . . . 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))})
2813, 27mpteq12dv 4693 . . . 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))}))
294, 28mpteq12dv 4693 . . 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))})))
30 df-dic 35939 . . 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))})))
31 fvex 6158 . . . . 5 (LHyp‘𝐾) ∈ V
323, 31eqeltri 2694 . . . 4 𝐻 ∈ V
3332mptex 6440 . . 3 (𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})) ∈ V
3429, 30, 33fvmpt 6239 . 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))})))
351, 34syl 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 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186   class class class wbr 4613  {copab 4672  cmpt 4673  cfv 5847  crio 6564  lecple 15869  occoc 15870  Atomscatm 34027  LHypclh 34747  LTrncltrn 34864  TEndoctendo 35517  DIsoCcdic 35938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-dic 35939
This theorem is referenced by:  dicfval  35941
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