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Definition df-oc 29014
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 29042 and chocvali 29061 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 28692 . 2 class
2 vx . . 3 setvar 𝑥
3 chba 28681 . . . 4 class
43cpw 4512 . . 3 class 𝒫 ℋ
5 vy . . . . . . . 8 setvar 𝑦
65cv 1537 . . . . . . 7 class 𝑦
7 vz . . . . . . . 8 setvar 𝑧
87cv 1537 . . . . . . 7 class 𝑧
9 csp 28684 . . . . . . 7 class ·ih
106, 8, 9co 7130 . . . . . 6 class (𝑦 ·ih 𝑧)
11 cc0 10514 . . . . . 6 class 0
1210, 11wceq 1538 . . . . 5 wff (𝑦 ·ih 𝑧) = 0
132cv 1537 . . . . 5 class 𝑥
1412, 7, 13wral 3126 . . . 4 wff 𝑧𝑥 (𝑦 ·ih 𝑧) = 0
1514, 5, 3crab 3130 . . 3 class {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0}
162, 4, 15cmpt 5119 . 2 class (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
171, 16wceq 1538 1 wff ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
Colors of variables: wff setvar class
This definition is referenced by:  ocval  29042
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