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Definition df-oc 31082
Description: Define orthogonal complement of a subset (usually a subspace) of Hilbert space. The orthogonal complement is the set of all vectors orthogonal to all vectors in the subset. See ocval 31110 and chocvali 31129 for its value. Textbooks usually denote this unary operation with the symbol as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation) rather than introducing a new syntactic form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
Assertion
Ref Expression
df-oc ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-oc
StepHypRef Expression
1 cort 30760 . 2 class
2 vx . . 3 setvar 𝑥
3 chba 30749 . . . 4 class
43cpw 4606 . . 3 class 𝒫 ℋ
5 vy . . . . . . . 8 setvar 𝑦
65cv 1532 . . . . . . 7 class 𝑦
7 vz . . . . . . . 8 setvar 𝑧
87cv 1532 . . . . . . 7 class 𝑧
9 csp 30752 . . . . . . 7 class ·ih
106, 8, 9co 7426 . . . . . 6 class (𝑦 ·ih 𝑧)
11 cc0 11146 . . . . . 6 class 0
1210, 11wceq 1533 . . . . 5 wff (𝑦 ·ih 𝑧) = 0
132cv 1532 . . . . 5 class 𝑥
1412, 7, 13wral 3058 . . . 4 wff 𝑧𝑥 (𝑦 ·ih 𝑧) = 0
1514, 5, 3crab 3430 . . 3 class {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0}
162, 4, 15cmpt 5235 . 2 class (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
171, 16wceq 1533 1 wff ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
Colors of variables: wff setvar class
This definition is referenced by:  ocval  31110
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