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Theorem djaffvalN 39147
Description: Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
djaval.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
djaffvalN (𝐾𝑉 → (vA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))
Distinct variable groups:   𝑤,𝐻   𝑥,𝑤,𝑦,𝐾
Allowed substitution hints:   𝐻(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑤)

Proof of Theorem djaffvalN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3450 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6774 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 djaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2796 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6774 . . . . . . 7 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
65fveq1d 6776 . . . . . 6 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
76pweqd 4552 . . . . 5 (𝑘 = 𝐾 → 𝒫 ((LTrn‘𝑘)‘𝑤) = 𝒫 ((LTrn‘𝐾)‘𝑤))
8 fveq2 6774 . . . . . . 7 (𝑘 = 𝐾 → (ocA‘𝑘) = (ocA‘𝐾))
98fveq1d 6776 . . . . . 6 (𝑘 = 𝐾 → ((ocA‘𝑘)‘𝑤) = ((ocA‘𝐾)‘𝑤))
109fveq1d 6776 . . . . . . 7 (𝑘 = 𝐾 → (((ocA‘𝑘)‘𝑤)‘𝑥) = (((ocA‘𝐾)‘𝑤)‘𝑥))
119fveq1d 6776 . . . . . . 7 (𝑘 = 𝐾 → (((ocA‘𝑘)‘𝑤)‘𝑦) = (((ocA‘𝐾)‘𝑤)‘𝑦))
1210, 11ineq12d 4147 . . . . . 6 (𝑘 = 𝐾 → ((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦)) = ((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))
139, 12fveq12d 6781 . . . . 5 (𝑘 = 𝐾 → (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦))) = (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))
147, 7, 13mpoeq123dv 7350 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦)))) = (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))))
154, 14mpteq12dv 5165 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦))))) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))
16 df-djaN 39146 . . 3 vA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦))))))
1715, 16, 3mptfvmpt 7104 . 2 (𝐾 ∈ V → (vA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))
181, 17syl 17 1 (𝐾𝑉 → (vA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  Vcvv 3432  cin 3886  𝒫 cpw 4533  cmpt 5157  cfv 6433  cmpo 7277  LHypclh 37998  LTrncltrn 38115  ocAcocaN 39133  vAcdjaN 39145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-oprab 7279  df-mpo 7280  df-djaN 39146
This theorem is referenced by:  djafvalN  39148
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