Definition df-oc 31585

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 31613 and chocvali 31632 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 31263	. 2 class ⊥
2		vx	. . 3 setvar 𝑥
3		chba 31252	. . . 4 class ℋ
4	3	cpw 4563	. . 3 class 𝒫 ℋ
5		vy	. . . . . . . 8 setvar 𝑦
6	5	cv 1569	. . . . . . 7 class 𝑦
7		vz	. . . . . . . 8 setvar 𝑧
8	7	cv 1569	. . . . . . 7 class 𝑧
9		csp 31255	. . . . . . 7 class ·ih
10	6, 8, 9	co 7412	. . . . . 6 class (𝑦 ·ih 𝑧)
11		cc0 11101	. . . . . 6 class 0
12	10, 11	wceq 1570	. . . . 5 wff (𝑦 ·ih 𝑧) = 0
13	2	cv 1569	. . . . 5 class 𝑥
14	12, 7, 13	wral 3079	. . . 4 wff ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0
15	14, 5, 3	crab 3416	. . 3 class {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0}
16	2, 4, 15	cmpt 5193	. 2 class (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0})
17	1, 16	wceq 1570	1 wff ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0})

This definition is referenced by:  ocval  31613