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Theorem djhffval 38412
Description: Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)
Hypothesis
Ref Expression
djhval.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
djhffval (𝐾𝑋 → (joinH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))))))
Distinct variable groups:   𝑤,𝐻   𝑥,𝑤,𝑦,𝐾
Allowed substitution hints:   𝐻(𝑥,𝑦)   𝑋(𝑥,𝑦,𝑤)

Proof of Theorem djhffval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3510 . 2 (𝐾𝑋𝐾 ∈ V)
2 fveq2 6663 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 djhval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2871 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6663 . . . . . . . 8 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
65fveq1d 6665 . . . . . . 7 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
76fveq2d 6667 . . . . . 6 (𝑘 = 𝐾 → (Base‘((DVecH‘𝑘)‘𝑤)) = (Base‘((DVecH‘𝐾)‘𝑤)))
87pweqd 4540 . . . . 5 (𝑘 = 𝐾 → 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) = 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)))
9 fveq2 6663 . . . . . . 7 (𝑘 = 𝐾 → (ocH‘𝑘) = (ocH‘𝐾))
109fveq1d 6665 . . . . . 6 (𝑘 = 𝐾 → ((ocH‘𝑘)‘𝑤) = ((ocH‘𝐾)‘𝑤))
1110fveq1d 6665 . . . . . . 7 (𝑘 = 𝐾 → (((ocH‘𝑘)‘𝑤)‘𝑥) = (((ocH‘𝐾)‘𝑤)‘𝑥))
1210fveq1d 6665 . . . . . . 7 (𝑘 = 𝐾 → (((ocH‘𝑘)‘𝑤)‘𝑦) = (((ocH‘𝐾)‘𝑤)‘𝑦))
1311, 12ineq12d 4187 . . . . . 6 (𝑘 = 𝐾 → ((((ocH‘𝑘)‘𝑤)‘𝑥) ∩ (((ocH‘𝑘)‘𝑤)‘𝑦)) = ((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))
1410, 13fveq12d 6670 . . . . 5 (𝑘 = 𝐾 → (((ocH‘𝑘)‘𝑤)‘((((ocH‘𝑘)‘𝑤)‘𝑥) ∩ (((ocH‘𝑘)‘𝑤)‘𝑦))) = (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))))
158, 8, 14mpoeq123dv 7218 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((ocH‘𝑘)‘𝑤)‘((((ocH‘𝑘)‘𝑤)‘𝑥) ∩ (((ocH‘𝑘)‘𝑤)‘𝑦)))) = (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))))
164, 15mpteq12dv 5142 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((ocH‘𝑘)‘𝑤)‘((((ocH‘𝑘)‘𝑤)‘𝑥) ∩ (((ocH‘𝑘)‘𝑤)‘𝑦))))) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))))))
17 df-djh 38411 . . 3 joinH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((ocH‘𝑘)‘𝑤)‘((((ocH‘𝑘)‘𝑤)‘𝑥) ∩ (((ocH‘𝑘)‘𝑤)‘𝑦))))))
1816, 17, 3mptfvmpt 6981 . 2 (𝐾 ∈ V → (joinH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))))))
191, 18syl 17 1 (𝐾𝑋 → (joinH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  𝒫 cpw 4535  cmpt 5137  cfv 6348  cmpo 7147  Basecbs 16471  LHypclh 37000  DVecHcdvh 38094  ocHcoch 38363  joinHcdjh 38410
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-pw 4537  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-oprab 7149  df-mpo 7150  df-djh 38411
This theorem is referenced by:  djhfval  38413
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