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Definition df-oc 28237
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 28267 and chocvali 28286 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 27915 . 2 class
2 vx . . 3 setvar 𝑥
3 chil 27904 . . . 4 class
43cpw 4191 . . 3 class 𝒫 ℋ
5 vy . . . . . . . 8 setvar 𝑦
65cv 1522 . . . . . . 7 class 𝑦
7 vz . . . . . . . 8 setvar 𝑧
87cv 1522 . . . . . . 7 class 𝑧
9 csp 27907 . . . . . . 7 class ·ih
106, 8, 9co 6690 . . . . . 6 class (𝑦 ·ih 𝑧)
11 cc0 9974 . . . . . 6 class 0
1210, 11wceq 1523 . . . . 5 wff (𝑦 ·ih 𝑧) = 0
132cv 1522 . . . . 5 class 𝑥
1412, 7, 13wral 2941 . . . 4 wff 𝑧𝑥 (𝑦 ·ih 𝑧) = 0
1514, 5, 3crab 2945 . . 3 class {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0}
162, 4, 15cmpt 4762 . 2 class (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
171, 16wceq 1523 1 wff ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
Colors of variables: wff setvar class
This definition is referenced by:  ocval  28267
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