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Theorem dochffval 37959
Description: Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)
Hypotheses
Ref Expression
dochval.b 𝐵 = (Base‘𝐾)
dochval.g 𝐺 = (glb‘𝐾)
dochval.o = (oc‘𝐾)
dochval.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
dochffval (𝐾𝑉 → (ocH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
Distinct variable groups:   𝑦,𝐵   𝑤,𝐻   𝑥,𝑤,𝑦,𝐾
Allowed substitution hints:   𝐵(𝑥,𝑤)   𝐺(𝑥,𝑦,𝑤)   𝐻(𝑥,𝑦)   (𝑥,𝑦,𝑤)   𝑉(𝑥,𝑦,𝑤)

Proof of Theorem dochffval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3427 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6496 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dochval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2826 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6496 . . . . . . . 8 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
65fveq1d 6498 . . . . . . 7 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
76fveq2d 6500 . . . . . 6 (𝑘 = 𝐾 → (Base‘((DVecH‘𝑘)‘𝑤)) = (Base‘((DVecH‘𝐾)‘𝑤)))
87pweqd 4421 . . . . 5 (𝑘 = 𝐾 → 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) = 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)))
9 fveq2 6496 . . . . . . 7 (𝑘 = 𝐾 → (DIsoH‘𝑘) = (DIsoH‘𝐾))
109fveq1d 6498 . . . . . 6 (𝑘 = 𝐾 → ((DIsoH‘𝑘)‘𝑤) = ((DIsoH‘𝐾)‘𝑤))
11 fveq2 6496 . . . . . . . 8 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
12 dochval.o . . . . . . . 8 = (oc‘𝐾)
1311, 12syl6eqr 2826 . . . . . . 7 (𝑘 = 𝐾 → (oc‘𝑘) = )
14 fveq2 6496 . . . . . . . . 9 (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
15 dochval.g . . . . . . . . 9 𝐺 = (glb‘𝐾)
1614, 15syl6eqr 2826 . . . . . . . 8 (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
17 fveq2 6496 . . . . . . . . . 10 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
18 dochval.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1917, 18syl6eqr 2826 . . . . . . . . 9 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
2010fveq1d 6498 . . . . . . . . . 10 (𝑘 = 𝐾 → (((DIsoH‘𝑘)‘𝑤)‘𝑦) = (((DIsoH‘𝐾)‘𝑤)‘𝑦))
2120sseq2d 3883 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦) ↔ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)))
2219, 21rabeqbidv 3402 . . . . . . . 8 (𝑘 = 𝐾 → {𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)} = {𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})
2316, 22fveq12d 6503 . . . . . . 7 (𝑘 = 𝐾 → ((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}) = (𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))
2413, 23fveq12d 6503 . . . . . 6 (𝑘 = 𝐾 → ((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})) = ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))
2510, 24fveq12d 6503 . . . . 5 (𝑘 = 𝐾 → (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))) = (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))
268, 25mpteq12dv 5008 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})))) = (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))
274, 26mpteq12dv 5008 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
28 df-doch 37958 . . 3 ocH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))))
2927, 28, 3mptfvmpt 6814 . 2 (𝐾 ∈ V → (ocH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
301, 29syl 17 1 (𝐾𝑉 → (ocH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  {crab 3086  Vcvv 3409  wss 3823  𝒫 cpw 4416  cmpt 5004  cfv 6185  Basecbs 16337  occoc 16427  glbcglb 17423  LHypclh 36594  DVecHcdvh 37688  DIsoHcdih 37838  ocHcoch 37957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-doch 37958
This theorem is referenced by:  dochfval  37960
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