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Theorem ocval 31309
Description: Value of orthogonal complement of a subset of Hilbert space. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ocval (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
Distinct variable group:   𝑥,𝑦,𝐻

Proof of Theorem ocval
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 31028 . . 3 ℋ ∈ V
21elpw2 5340 . 2 (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3 raleq 3321 . . . 4 (𝑧 = 𝐻 → (∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0))
43rabbidv 3441 . . 3 (𝑧 = 𝐻 → {𝑥 ∈ ℋ ∣ ∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0} = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
5 df-oc 31281 . . 3 ⊥ = (𝑧 ∈ 𝒫 ℋ ↦ {𝑥 ∈ ℋ ∣ ∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0})
61rabex 5345 . . 3 {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0} ∈ V
74, 5, 6fvmpt 7016 . 2 (𝐻 ∈ 𝒫 ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
82, 7sylbir 235 1 (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  {crab 3433  wss 3963  𝒫 cpw 4605  cfv 6563  (class class class)co 7431  0cc0 11153  chba 30948   ·ih csp 30951  cort 30959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-oc 31281
This theorem is referenced by:  ocel  31310  ocsh  31312  occon  31316  chocvali  31328
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