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Theorem dicffval 36983
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 3352 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6353 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dicval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2812 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6353 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
6 dicval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
75, 6syl6eqr 2812 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
8 fveq2 6353 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9 dicval.l . . . . . . . . 9 = (le‘𝐾)
108, 9syl6eqr 2812 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1110breqd 4815 . . . . . . 7 (𝑘 = 𝐾 → (𝑟(le‘𝑘)𝑤𝑟 𝑤))
1211notbid 307 . . . . . 6 (𝑘 = 𝐾 → (¬ 𝑟(le‘𝑘)𝑤 ↔ ¬ 𝑟 𝑤))
137, 12rabeqbidv 3335 . . . . 5 (𝑘 = 𝐾 → {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} = {𝑟𝐴 ∣ ¬ 𝑟 𝑤})
14 fveq2 6353 . . . . . . . . . . 11 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
1514fveq1d 6355 . . . . . . . . . 10 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
16 fveq2 6353 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
1716fveq1d 6355 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑤) = ((oc‘𝐾)‘𝑤))
1817fveq2d 6357 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑔‘((oc‘𝑘)‘𝑤)) = (𝑔‘((oc‘𝐾)‘𝑤)))
1918eqeq1d 2762 . . . . . . . . . 10 (𝑘 = 𝐾 → ((𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞 ↔ (𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞))
2015, 19riotaeqbidv 6778 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞) = (𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞))
2120fveq2d 6357 . . . . . . . 8 (𝑘 = 𝐾 → (𝑠‘(𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)))
2221eqeq2d 2770 . . . . . . 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 6353 . . . . . . . . 9 (𝑘 = 𝐾 → (TEndo‘𝑘) = (TEndo‘𝐾))
2423fveq1d 6355 . . . . . . . 8 (𝑘 = 𝐾 → ((TEndo‘𝑘)‘𝑤) = ((TEndo‘𝐾)‘𝑤))
2524eleq2d 2825 . . . . . . 7 (𝑘 = 𝐾 → (𝑠 ∈ ((TEndo‘𝑘)‘𝑤) ↔ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤)))
2622, 25anbi12d 749 . . . . . 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 4868 . . . . 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 4885 . . . 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 4885 . . 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 36982 . . 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 6363 . . . . 5 (LHyp‘𝐾) ∈ V
323, 31eqeltri 2835 . . . 4 𝐻 ∈ V
3332mptex 6651 . . 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 6445 . 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 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340   class class class wbr 4804  {copab 4864  cmpt 4881  cfv 6049  crio 6774  lecple 16170  occoc 16171  Atomscatm 35071  LHypclh 35791  LTrncltrn 35908  TEndoctendo 36560  DIsoCcdic 36981
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-riota 6775  df-dic 36982
This theorem is referenced by:  dicfval  36984
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