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Theorem dochffval 40952
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 3480 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6896 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dochval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2783 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6896 . . . . . . . 8 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
65fveq1d 6898 . . . . . . 7 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
76fveq2d 6900 . . . . . 6 (𝑘 = 𝐾 → (Base‘((DVecH‘𝑘)‘𝑤)) = (Base‘((DVecH‘𝐾)‘𝑤)))
87pweqd 4621 . . . . 5 (𝑘 = 𝐾 → 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) = 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)))
9 fveq2 6896 . . . . . . 7 (𝑘 = 𝐾 → (DIsoH‘𝑘) = (DIsoH‘𝐾))
109fveq1d 6898 . . . . . 6 (𝑘 = 𝐾 → ((DIsoH‘𝑘)‘𝑤) = ((DIsoH‘𝐾)‘𝑤))
11 fveq2 6896 . . . . . . . 8 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
12 dochval.o . . . . . . . 8 = (oc‘𝐾)
1311, 12eqtr4di 2783 . . . . . . 7 (𝑘 = 𝐾 → (oc‘𝑘) = )
14 fveq2 6896 . . . . . . . . 9 (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
15 dochval.g . . . . . . . . 9 𝐺 = (glb‘𝐾)
1614, 15eqtr4di 2783 . . . . . . . 8 (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
17 fveq2 6896 . . . . . . . . . 10 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
18 dochval.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1917, 18eqtr4di 2783 . . . . . . . . 9 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
2010fveq1d 6898 . . . . . . . . . 10 (𝑘 = 𝐾 → (((DIsoH‘𝑘)‘𝑤)‘𝑦) = (((DIsoH‘𝐾)‘𝑤)‘𝑦))
2120sseq2d 4009 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦) ↔ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)))
2219, 21rabeqbidv 3436 . . . . . . . 8 (𝑘 = 𝐾 → {𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)} = {𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})
2316, 22fveq12d 6903 . . . . . . 7 (𝑘 = 𝐾 → ((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}) = (𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))
2413, 23fveq12d 6903 . . . . . 6 (𝑘 = 𝐾 → ((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})) = ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))
2510, 24fveq12d 6903 . . . . 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 5240 . . . 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 5240 . . 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 40951 . . 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 7240 . 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 1533  wcel 2098  {crab 3418  Vcvv 3461  wss 3944  𝒫 cpw 4604  cmpt 5232  cfv 6549  Basecbs 17183  occoc 17244  glbcglb 18305  LHypclh 39587  DVecHcdvh 40681  DIsoHcdih 40831  ocHcoch 40950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-doch 40951
This theorem is referenced by:  dochfval  40953
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