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Definition df-oc 28698
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 28728 and chocvali 28747 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 28376 . 2 class
2 vx . . 3 setvar 𝑥
3 chba 28365 . . . 4 class
43cpw 4379 . . 3 class 𝒫 ℋ
5 vy . . . . . . . 8 setvar 𝑦
65cv 1600 . . . . . . 7 class 𝑦
7 vz . . . . . . . 8 setvar 𝑧
87cv 1600 . . . . . . 7 class 𝑧
9 csp 28368 . . . . . . 7 class ·ih
106, 8, 9co 6924 . . . . . 6 class (𝑦 ·ih 𝑧)
11 cc0 10274 . . . . . 6 class 0
1210, 11wceq 1601 . . . . 5 wff (𝑦 ·ih 𝑧) = 0
132cv 1600 . . . . 5 class 𝑥
1412, 7, 13wral 3090 . . . 4 wff 𝑧𝑥 (𝑦 ·ih 𝑧) = 0
1514, 5, 3crab 3094 . . 3 class {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0}
162, 4, 15cmpt 4967 . 2 class (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
171, 16wceq 1601 1 wff ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
Colors of variables: wff setvar class
This definition is referenced by:  ocval  28728
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