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Theorem ocval 31299
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 31018 . . 3 ℋ ∈ V
21elpw2 5334 . 2 (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3 raleq 3323 . . . 4 (𝑧 = 𝐻 → (∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0))
43rabbidv 3444 . . 3 (𝑧 = 𝐻 → {𝑥 ∈ ℋ ∣ ∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0} = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
5 df-oc 31271 . . 3 ⊥ = (𝑧 ∈ 𝒫 ℋ ↦ {𝑥 ∈ ℋ ∣ ∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0})
61rabex 5339 . . 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 1540  wcel 2108  wral 3061  {crab 3436  wss 3951  𝒫 cpw 4600  cfv 6561  (class class class)co 7431  0cc0 11155  chba 30938   ·ih csp 30941  cort 30949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-oc 31271
This theorem is referenced by:  ocel  31300  ocsh  31302  occon  31306  chocvali  31318
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