Definition df-oc 31327

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 31355 and chocvali 31374 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 31005	. 2 class ⊥
2		vx	. . 3 setvar 𝑥
3		chba 30994	. . . 4 class ℋ
4	3	cpw 4554	. . 3 class 𝒫 ℋ
5		vy	. . . . . . . 8 setvar 𝑦
6	5	cv 1540	. . . . . . 7 class 𝑦
7		vz	. . . . . . . 8 setvar 𝑧
8	7	cv 1540	. . . . . . 7 class 𝑧
9		csp 30997	. . . . . . 7 class ·ih
10	6, 8, 9	co 7358	. . . . . 6 class (𝑦 ·ih 𝑧)
11		cc0 11026	. . . . . 6 class 0
12	10, 11	wceq 1541	. . . . 5 wff (𝑦 ·ih 𝑧) = 0
13	2	cv 1540	. . . . 5 class 𝑥
14	12, 7, 13	wral 3051	. . . 4 wff ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0
15	14, 5, 3	crab 3399	. . 3 class {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0}
16	2, 4, 15	cmpt 5179	. 2 class (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0})
17	1, 16	wceq 1541	1 wff ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0})

This definition is referenced by:  ocval  31355