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Theorem diaffval 41013
Description: The partial isomorphism A for a lattice 𝐾. (Contributed by NM, 15-Oct-2013.)
Hypotheses
Ref Expression
diaval.b 𝐵 = (Base‘𝐾)
diaval.l = (le‘𝐾)
diaval.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
diaffval (𝐾𝑉 → (DIsoA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥})))
Distinct variable groups:   𝑥,𝑤,𝑦,   𝑤,𝐵,𝑥,𝑦   𝑤,𝐻   𝑤,𝑓,𝑥,𝑦,𝐾
Allowed substitution hints:   𝐵(𝑓)   𝐻(𝑥,𝑦,𝑓)   (𝑓)   𝑉(𝑥,𝑦,𝑤,𝑓)

Proof of Theorem diaffval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3499 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6907 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 diaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2793 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6907 . . . . . . 7 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
6 diaval.b . . . . . . 7 𝐵 = (Base‘𝐾)
75, 6eqtr4di 2793 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
8 fveq2 6907 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9 diaval.l . . . . . . . 8 = (le‘𝐾)
108, 9eqtr4di 2793 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = )
1110breqd 5159 . . . . . 6 (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑤𝑦 𝑤))
127, 11rabeqbidv 3452 . . . . 5 (𝑘 = 𝐾 → {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} = {𝑦𝐵𝑦 𝑤})
13 fveq2 6907 . . . . . . 7 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
1413fveq1d 6909 . . . . . 6 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
15 fveq2 6907 . . . . . . . . 9 (𝑘 = 𝐾 → (trL‘𝑘) = (trL‘𝐾))
1615fveq1d 6909 . . . . . . . 8 (𝑘 = 𝐾 → ((trL‘𝑘)‘𝑤) = ((trL‘𝐾)‘𝑤))
1716fveq1d 6909 . . . . . . 7 (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘𝑓) = (((trL‘𝐾)‘𝑤)‘𝑓))
18 eqidd 2736 . . . . . . 7 (𝑘 = 𝐾𝑥 = 𝑥)
1917, 10, 18breq123d 5162 . . . . . 6 (𝑘 = 𝐾 → ((((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥 ↔ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥))
2014, 19rabeqbidv 3452 . . . . 5 (𝑘 = 𝐾 → {𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ∣ (((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥} = {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥})
2112, 20mpteq12dv 5239 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} ↦ {𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ∣ (((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥}) = (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}))
224, 21mpteq12dv 5239 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} ↦ {𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ∣ (((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥})) = (𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥})))
23 df-disoa 41012 . . 3 DIsoA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} ↦ {𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ∣ (((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥})))
2422, 23, 3mptfvmpt 7248 . 2 (𝐾 ∈ V → (DIsoA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥})))
251, 24syl 17 1 (𝐾𝑉 → (DIsoA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478   class class class wbr 5148  cmpt 5231  cfv 6563  Basecbs 17245  lecple 17305  LHypclh 39967  LTrncltrn 40084  trLctrl 40141  DIsoAcdia 41011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-disoa 41012
This theorem is referenced by:  diafval  41014
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