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Theorem dochfval 36956
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)
dochval.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dochval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochval.v 𝑉 = (Base‘𝑈)
dochval.n 𝑁 = ((ocH‘𝐾)‘𝑊)
Assertion
Ref Expression
dochfval ((𝐾𝑋𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
Distinct variable groups:   𝑦,𝐵   𝑥,𝑦,𝐾   𝑥,𝑉   𝑥,𝑊,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝑈(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝐼(𝑥,𝑦)   𝑁(𝑥,𝑦)   (𝑥,𝑦)   𝑉(𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem dochfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dochval.n . . 3 𝑁 = ((ocH‘𝐾)‘𝑊)
2 dochval.b . . . . 5 𝐵 = (Base‘𝐾)
3 dochval.g . . . . 5 𝐺 = (glb‘𝐾)
4 dochval.o . . . . 5 = (oc‘𝐾)
5 dochval.h . . . . 5 𝐻 = (LHyp‘𝐾)
62, 3, 4, 5dochffval 36955 . . . 4 (𝐾𝑋 → (ocH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
76fveq1d 6231 . . 3 (𝐾𝑋 → ((ocH‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))‘𝑊))
81, 7syl5eq 2697 . 2 (𝐾𝑋𝑁 = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))‘𝑊))
9 fveq2 6229 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
10 dochval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
119, 10syl6eqr 2703 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
1211fveq2d 6233 . . . . . 6 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘𝑈))
13 dochval.v . . . . . 6 𝑉 = (Base‘𝑈)
1412, 13syl6eqr 2703 . . . . 5 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
1514pweqd 4196 . . . 4 (𝑤 = 𝑊 → 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) = 𝒫 𝑉)
16 fveq2 6229 . . . . . 6 (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = ((DIsoH‘𝐾)‘𝑊))
17 dochval.i . . . . . 6 𝐼 = ((DIsoH‘𝐾)‘𝑊)
1816, 17syl6eqr 2703 . . . . 5 (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = 𝐼)
1918fveq1d 6231 . . . . . . . . 9 (𝑤 = 𝑊 → (((DIsoH‘𝐾)‘𝑤)‘𝑦) = (𝐼𝑦))
2019sseq2d 3666 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦) ↔ 𝑥 ⊆ (𝐼𝑦)))
2120rabbidv 3220 . . . . . . 7 (𝑤 = 𝑊 → {𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)} = {𝑦𝐵𝑥 ⊆ (𝐼𝑦)})
2221fveq2d 6233 . . . . . 6 (𝑤 = 𝑊 → (𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}) = (𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))
2322fveq2d 6233 . . . . 5 (𝑤 = 𝑊 → ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})) = ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))
2418, 23fveq12d 6235 . . . 4 (𝑤 = 𝑊 → (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))
2515, 24mpteq12dv 4766 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
26 eqid 2651 . . 3 (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))
27 fvex 6239 . . . . . 6 (Base‘𝑈) ∈ V
2813, 27eqeltri 2726 . . . . 5 𝑉 ∈ V
2928pwex 4878 . . . 4 𝒫 𝑉 ∈ V
3029mptex 6527 . . 3 (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))) ∈ V
3125, 26, 30fvmpt 6321 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))‘𝑊) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
328, 31sylan9eq 2705 1 ((𝐾𝑋𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191  cmpt 4762  cfv 5926  Basecbs 15904  occoc 15996  glbcglb 16990  LHypclh 35588  DVecHcdvh 36684  DIsoHcdih 36834  ocHcoch 36953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-doch 36954
This theorem is referenced by:  dochval  36957  dochfN  36962
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