Definition df-oc 30483

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 30511 and chocvali 30530 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 ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0})

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-oc

Step	Hyp	Ref	Expression
1		cort 30161	. 2 class ⊥
2		vx	. . 3 setvar 𝑥
3		chba 30150	. . . 4 class ℋ
4	3	cpw 4601	. . 3 class 𝒫 ℋ
5		vy	. . . . . . . 8 setvar 𝑦
6	5	cv 1541	. . . . . . 7 class 𝑦
7		vz	. . . . . . . 8 setvar 𝑧
8	7	cv 1541	. . . . . . 7 class 𝑧
9		csp 30153	. . . . . . 7 class ·ih
10	6, 8, 9	co 7404	. . . . . 6 class (𝑦 ·ih 𝑧)
11		cc0 11106	. . . . . 6 class 0
12	10, 11	wceq 1542	. . . . 5 wff (𝑦 ·ih 𝑧) = 0
13	2	cv 1541	. . . . 5 class 𝑥
14	12, 7, 13	wral 3062	. . . 4 wff ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0
15	14, 5, 3	crab 3433	. . 3 class {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0}
16	2, 4, 15	cmpt 5230	. 2 class (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0})
17	1, 16	wceq 1542	1 wff ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0})

This definition is referenced by:  ocval  30511