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Definition df-oc 28448
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 28478 and chocvali 28497 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 28126 . 2 class
2 vx . . 3 setvar 𝑥
3 chil 28115 . . . 4 class
43cpw 4298 . . 3 class 𝒫 ℋ
5 vy . . . . . . . 8 setvar 𝑦
65cv 1630 . . . . . . 7 class 𝑦
7 vz . . . . . . . 8 setvar 𝑧
87cv 1630 . . . . . . 7 class 𝑧
9 csp 28118 . . . . . . 7 class ·ih
106, 8, 9co 6795 . . . . . 6 class (𝑦 ·ih 𝑧)
11 cc0 10141 . . . . . 6 class 0
1210, 11wceq 1631 . . . . 5 wff (𝑦 ·ih 𝑧) = 0
132cv 1630 . . . . 5 class 𝑥
1412, 7, 13wral 3061 . . . 4 wff 𝑧𝑥 (𝑦 ·ih 𝑧) = 0
1514, 5, 3crab 3065 . . 3 class {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0}
162, 4, 15cmpt 4864 . 2 class (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
171, 16wceq 1631 1 wff ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
Colors of variables: wff setvar class
This definition is referenced by:  ocval  28478
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