Definition df-oc 31338

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 31366 and chocvali 31385 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 31016	. 2 class ⊥
2		vx	. . 3 setvar 𝑥
3		chba 31005	. . . 4 class ℋ
4	3	cpw 4542	. . 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 31008	. . . . . . 7 class ·ih
10	6, 8, 9	co 7360	. . . . . 6 class (𝑦 ·ih 𝑧)
11		cc0 11029	. . . . . 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 3390	. . 3 class {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0}
16	2, 4, 15	cmpt 5167	. 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  31366