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Theorem diaffval 37050
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 3401 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6412 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 diaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2852 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6412 . . . . . . 7 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
6 diaval.b . . . . . . 7 𝐵 = (Base‘𝐾)
75, 6syl6eqr 2852 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
8 fveq2 6412 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9 diaval.l . . . . . . . 8 = (le‘𝐾)
108, 9syl6eqr 2852 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = )
1110breqd 4855 . . . . . 6 (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑤𝑦 𝑤))
127, 11rabeqbidv 3380 . . . . 5 (𝑘 = 𝐾 → {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} = {𝑦𝐵𝑦 𝑤})
13 fveq2 6412 . . . . . . 7 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
1413fveq1d 6414 . . . . . 6 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
15 fveq2 6412 . . . . . . . . 9 (𝑘 = 𝐾 → (trL‘𝑘) = (trL‘𝐾))
1615fveq1d 6414 . . . . . . . 8 (𝑘 = 𝐾 → ((trL‘𝑘)‘𝑤) = ((trL‘𝐾)‘𝑤))
1716fveq1d 6414 . . . . . . 7 (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘𝑓) = (((trL‘𝐾)‘𝑤)‘𝑓))
18 eqidd 2801 . . . . . . 7 (𝑘 = 𝐾𝑥 = 𝑥)
1917, 10, 18breq123d 4858 . . . . . 6 (𝑘 = 𝐾 → ((((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥 ↔ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥))
2014, 19rabeqbidv 3380 . . . . 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 4927 . . . 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 4927 . . 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 37049 . . 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 6720 . 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 1653  wcel 2157  {crab 3094  Vcvv 3386   class class class wbr 4844  cmpt 4923  cfv 6102  Basecbs 16183  lecple 16273  LHypclh 36004  LTrncltrn 36121  trLctrl 36178  DIsoAcdia 37048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-disoa 37049
This theorem is referenced by:  diafval  37051
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