Definition df-oc

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 and chocvali 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.)

	Ref	Expression
Assertion	df-oc	⊢ ⊥ = ( 𝑥 ∈ 𝒫 ℋ ↦ { 𝑦 ∈ ℋ ∣ ∀ 𝑧 ∈ 𝑥 ( 𝑦 ·ih 𝑧 ) = 0 } )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cort	⊢ ⊥
1		vx	⊢ 𝑥
2		chba	⊢ ℋ
3	2	cpw	⊢ 𝒫 ℋ
4		vy	⊢ 𝑦
5		vz	⊢ 𝑧
6	1	cv	⊢ 𝑥
7	4	cv	⊢ 𝑦
8		csp	⊢ ·ih
9	5	cv	⊢ 𝑧
10	7 9 8	co	⊢ ( 𝑦 ·ih 𝑧 )
11		cc0	⊢ 0
12	10 11	wceq	⊢ ( 𝑦 ·ih 𝑧 ) = 0
13	12 5 6	wral	⊢ ∀ 𝑧 ∈ 𝑥 ( 𝑦 ·ih 𝑧 ) = 0
14	13 4 2	crab	⊢ { 𝑦 ∈ ℋ ∣ ∀ 𝑧 ∈ 𝑥 ( 𝑦 ·ih 𝑧 ) = 0 }
15	1 3 14	cmpt	⊢ ( 𝑥 ∈ 𝒫 ℋ ↦ { 𝑦 ∈ ℋ ∣ ∀ 𝑧 ∈ 𝑥 ( 𝑦 ·ih 𝑧 ) = 0 } )
16	0 15	wceq	⊢ ⊥ = ( 𝑥 ∈ 𝒫 ℋ ↦ { 𝑦 ∈ ℋ ∣ ∀ 𝑧 ∈ 𝑥 ( 𝑦 ·ih 𝑧 ) = 0 } )