Definition df-oc 31339

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 31367 and chocvali 31386 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 31017	. 2 class ⊥
2		vx	. . 3 setvar 𝑥
3		chba 31006	. . . 4 class ℋ
4	3	cpw 4556	. . 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 31009	. . . . . . 7 class ·ih
10	6, 8, 9	co 7368	. . . . . 6 class (𝑦 ·ih 𝑧)
11		cc0 11038	. . . . . 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 3052	. . . 4 wff ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0
15	14, 5, 3	crab 3401	. . 3 class {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0}
16	2, 4, 15	cmpt 5181	. 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  31367