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Theorem diaffval 38046
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 3510 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6663 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 diaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2871 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6663 . . . . . . 7 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
6 diaval.b . . . . . . 7 𝐵 = (Base‘𝐾)
75, 6syl6eqr 2871 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
8 fveq2 6663 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9 diaval.l . . . . . . . 8 = (le‘𝐾)
108, 9syl6eqr 2871 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = )
1110breqd 5068 . . . . . 6 (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑤𝑦 𝑤))
127, 11rabeqbidv 3483 . . . . 5 (𝑘 = 𝐾 → {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} = {𝑦𝐵𝑦 𝑤})
13 fveq2 6663 . . . . . . 7 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
1413fveq1d 6665 . . . . . 6 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
15 fveq2 6663 . . . . . . . . 9 (𝑘 = 𝐾 → (trL‘𝑘) = (trL‘𝐾))
1615fveq1d 6665 . . . . . . . 8 (𝑘 = 𝐾 → ((trL‘𝑘)‘𝑤) = ((trL‘𝐾)‘𝑤))
1716fveq1d 6665 . . . . . . 7 (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘𝑓) = (((trL‘𝐾)‘𝑤)‘𝑓))
18 eqidd 2819 . . . . . . 7 (𝑘 = 𝐾𝑥 = 𝑥)
1917, 10, 18breq123d 5071 . . . . . 6 (𝑘 = 𝐾 → ((((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥 ↔ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥))
2014, 19rabeqbidv 3483 . . . . 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 5142 . . . 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 5142 . . 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 38045 . . 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 6981 . 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 1528  wcel 2105  {crab 3139  Vcvv 3492   class class class wbr 5057  cmpt 5137  cfv 6348  Basecbs 16471  lecple 16560  LHypclh 37000  LTrncltrn 37117  trLctrl 37174  DIsoAcdia 38044
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-disoa 38045
This theorem is referenced by:  diafval  38047
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