Definition df-oc 5156

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 ocvalt 5161 and chocval 5178 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 syntactical form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9.

Assertion

Ref	Expression
df-oc	⊢ ⊥ = {⟨x, y⟩∣(x ⊆ ℋ ∧ y = {z ∈ ℋ ∣∀w ∈ x (z ·i w) = 0})}

Distinct variable group(s):   x,y,z,w

Detailed syntax breakdown of Definition df-oc

Step	Hyp	Ref	Expression
1		cort 4969	. 2 class ⊥
2		vx	. . . . . 6 set x
3	2	cv 1089	. . . . 5 class x
4		chil 4958	. . . . 5 class ℋ
5	3, 4	wss 1487	. . . 4 wff x ⊆ ℋ
6		vy	. . . . . 6 set y
7	6	cv 1089	. . . . 5 class y
8		vz	. . . . . . . . . 10 set z
9	8	cv 1089	. . . . . . . . 9 class z
10		vw	. . . . . . . . . 10 set w
11	10	cv 1089	. . . . . . . . 9 class w
12		csp 4963	. . . . . . . . 9 class ·i
13	9, 11, 12	co 3001	. . . . . . . 8 class (z ·i w)
14		cc0 4028	. . . . . . . 8 class 0
15	13, 14	wceq 1091	. . . . . . 7 wff (z ·i w) = 0
16	15, 10, 3	wral 1201	. . . . . 6 wff ∀w ∈ x (z ·i w) = 0
17	16, 8, 4	crab 1204	. . . . 5 class {z ∈ ℋ ∣∀w ∈ x (z ·i w) = 0}
18	7, 17	wceq 1091	. . . 4 wff y = {z ∈ ℋ ∣∀w ∈ x (z ·i w) = 0}
19	5, 18	wa 196	. . 3 wff (x ⊆ ℋ ∧ y = {z ∈ ℋ ∣∀w ∈ x (z ·i w) = 0})
20	19, 2, 6	copab 2055	. 2 class {⟨x, y⟩∣(x ⊆ ℋ ∧ y = {z ∈ ℋ ∣∀w ∈ x (z ·i w) = 0})}
21	1, 20	wceq 1091	1 wff ⊥ = {⟨x, y⟩∣(x ⊆ ℋ ∧ y = {z ∈ ℋ ∣∀w ∈ x (z ·i w) = 0})}

This definition is referenced by:  ocvalt 5161

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