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Theorem dochfval 35657
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 35656 . . . 4 (𝐾𝑋 → (ocH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
76fveq1d 6105 . . 3 (𝐾𝑋 → ((ocH‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))‘𝑊))
81, 7syl5eq 2656 . 2 (𝐾𝑋𝑁 = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))‘𝑊))
9 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
10 dochval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
119, 10syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
1211fveq2d 6107 . . . . . 6 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘𝑈))
13 dochval.v . . . . . 6 𝑉 = (Base‘𝑈)
1412, 13syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
1514pweqd 4113 . . . 4 (𝑤 = 𝑊 → 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) = 𝒫 𝑉)
16 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = ((DIsoH‘𝐾)‘𝑊))
17 dochval.i . . . . . 6 𝐼 = ((DIsoH‘𝐾)‘𝑊)
1816, 17syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = 𝐼)
1918fveq1d 6105 . . . . . . . . 9 (𝑤 = 𝑊 → (((DIsoH‘𝐾)‘𝑤)‘𝑦) = (𝐼𝑦))
2019sseq2d 3596 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦) ↔ 𝑥 ⊆ (𝐼𝑦)))
2120rabbidv 3164 . . . . . . 7 (𝑤 = 𝑊 → {𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)} = {𝑦𝐵𝑥 ⊆ (𝐼𝑦)})
2221fveq2d 6107 . . . . . 6 (𝑤 = 𝑊 → (𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}) = (𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))
2322fveq2d 6107 . . . . 5 (𝑤 = 𝑊 → ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})) = ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))
2418, 23fveq12d 6109 . . . 4 (𝑤 = 𝑊 → (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))
2515, 24mpteq12dv 4663 . . 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 2610 . . 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 6113 . . . . . 6 (Base‘𝑈) ∈ V
2813, 27eqeltri 2684 . . . . 5 𝑉 ∈ V
2928pwex 4774 . . . 4 𝒫 𝑉 ∈ V
3029mptex 6390 . . 3 (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))) ∈ V
3125, 26, 30fvmpt 6191 . 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 2664 1 ((𝐾𝑋𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {crab 2900  Vcvv 3173  wss 3540  𝒫 cpw 4108  cmpt 4643  cfv 5804  Basecbs 15695  occoc 15776  glbcglb 16766  LHypclh 34288  DVecHcdvh 35385  DIsoHcdih 35535  ocHcoch 35654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-doch 35655
This theorem is referenced by:  dochval  35658  dochfN  35663
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