Definition df-oc 25846

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 25874 and chocvali 25893 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. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)

Assertion

Ref	Expression
df-oc	|- _|_ = ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )

Distinct variable group:   x, y, z

Detailed syntax breakdown of Definition df-oc

Step	Hyp	Ref	Expression
1		cort 25523	. 2 class _|_
2		vx	. . 3 setvar x
3		chil 25512	. . . 4 class ~H
4	3	cpw 4010	. . 3 class ~P ~H
5		vy	. . . . . . . 8 setvar y
6	5	cv 1378	. . . . . . 7 class y
7		vz	. . . . . . . 8 setvar z
8	7	cv 1378	. . . . . . 7 class z
9		csp 25515	. . . . . . 7 class .ih
10	6, 8, 9	co 6282	. . . . . 6 class ( y .ih z )
11		cc0 9488	. . . . . 6 class 0
12	10, 11	wceq 1379	. . . . 5 wff ( y .ih z ) = 0
13	2	cv 1378	. . . . 5 class x
14	12, 7, 13	wral 2814	. . . 4 wff A. z e. x ( y .ih z ) = 0
15	14, 5, 3	crab 2818	. . 3 class { y e. ~H | A. z e. x ( y .ih z ) = 0 }
16	2, 4, 15	cmpt 4505	. 2 class ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )
17	1, 16	wceq 1379	1 wff _|_ = ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )

This definition is referenced by:  ocval  25874