Definition df-oc 10549

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 10578 and chocvali 10596 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 C_ ~H /\ y = {z e. ~H | A.w e. x (z .ih w) = 0})}

Distinct variable group:   x,y,z,w

Detailed syntax breakdown of Definition df-oc

Step	Hyp	Ref	Expression
1		cort 10223	. 2 class _|_
2		vx	. . . . . 6 set x
3	2	cv 1135	. . . . 5 class x
4		chil 10212	. . . . 5 class ~H
5	3, 4	wss 2426	. . . 4 wff x C_ ~H
6		vy	. . . . . 6 set y
7	6	cv 1135	. . . . 5 class y
8		vz	. . . . . . . . . 10 set z
9	8	cv 1135	. . . . . . . . 9 class z
10		vw	. . . . . . . . . 10 set w
11	10	cv 1135	. . . . . . . . 9 class w
12		csp 10217	. . . . . . . . 9 class .ih
13	9, 11, 12	co 4695	. . . . . . . 8 class (z .ih w)
14		cc0 6182	. . . . . . . 8 class 0
15	13, 14	wceq 1136	. . . . . . 7 wff (z .ih w) = 0
16	15, 10, 3	wral 1939	. . . . . 6 wff A.w e. x (z .ih w) = 0
17	16, 8, 4	crab 1942	. . . . 5 class {z e. ~H | A.w e. x (z .ih w) = 0}
18	7, 17	wceq 1136	. . . 4 wff y = {z e. ~H | A.w e. x (z .ih w) = 0}
19	5, 18	wa 239	. . 3 wff (x C_ ~H /\ y = {z e. ~H | A.w e. x (z .ih w) = 0})
20	19, 2, 6	copab 3213	. 2 class {<.x, y>. | (x C_ ~H /\ y = {z e. ~H | A.w e. x (z .ih w) = 0})}
21	1, 20	wceq 1136	1 wff _|_ = {<.x, y>. | (x C_ ~H /\ y = {z e. ~H | A.w e. x (z .ih w) = 0})}

This definition is referenced by:  ocval 10578

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