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Theorem dochffval 38604
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 3487 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6652 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dochval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2875 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6652 . . . . . . . 8 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
65fveq1d 6654 . . . . . . 7 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
76fveq2d 6656 . . . . . 6 (𝑘 = 𝐾 → (Base‘((DVecH‘𝑘)‘𝑤)) = (Base‘((DVecH‘𝐾)‘𝑤)))
87pweqd 4530 . . . . 5 (𝑘 = 𝐾 → 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) = 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)))
9 fveq2 6652 . . . . . . 7 (𝑘 = 𝐾 → (DIsoH‘𝑘) = (DIsoH‘𝐾))
109fveq1d 6654 . . . . . 6 (𝑘 = 𝐾 → ((DIsoH‘𝑘)‘𝑤) = ((DIsoH‘𝐾)‘𝑤))
11 fveq2 6652 . . . . . . . 8 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
12 dochval.o . . . . . . . 8 = (oc‘𝐾)
1311, 12eqtr4di 2875 . . . . . . 7 (𝑘 = 𝐾 → (oc‘𝑘) = )
14 fveq2 6652 . . . . . . . . 9 (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
15 dochval.g . . . . . . . . 9 𝐺 = (glb‘𝐾)
1614, 15eqtr4di 2875 . . . . . . . 8 (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
17 fveq2 6652 . . . . . . . . . 10 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
18 dochval.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1917, 18eqtr4di 2875 . . . . . . . . 9 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
2010fveq1d 6654 . . . . . . . . . 10 (𝑘 = 𝐾 → (((DIsoH‘𝑘)‘𝑤)‘𝑦) = (((DIsoH‘𝐾)‘𝑤)‘𝑦))
2120sseq2d 3974 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦) ↔ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)))
2219, 21rabeqbidv 3461 . . . . . . . 8 (𝑘 = 𝐾 → {𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)} = {𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})
2316, 22fveq12d 6659 . . . . . . 7 (𝑘 = 𝐾 → ((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}) = (𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))
2413, 23fveq12d 6659 . . . . . 6 (𝑘 = 𝐾 → ((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})) = ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))
2510, 24fveq12d 6659 . . . . 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 5127 . . . 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 5127 . . 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 38603 . . 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 6973 . 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 1538  wcel 2114  {crab 3134  Vcvv 3469  wss 3908  𝒫 cpw 4511  cmpt 5122  cfv 6334  Basecbs 16474  occoc 16564  glbcglb 17544  LHypclh 37239  DVecHcdvh 38333  DIsoHcdih 38483  ocHcoch 38602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-doch 38603
This theorem is referenced by:  dochfval  38605
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