Definition df-oc 24590

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 24618 and chocvali 24637 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 24267	. 2 class _|_
2		vx	. . 3 setvar x
3		chil 24256	. . . 4 class ~H
4	3	cpw 3857	. . 3 class ~P ~H
5		vy	. . . . . . . 8 setvar y
6	5	cv 1363	. . . . . . 7 class y
7		vz	. . . . . . . 8 setvar z
8	7	cv 1363	. . . . . . 7 class z
9		csp 24259	. . . . . . 7 class .ih
10	6, 8, 9	co 6090	. . . . . 6 class ( y .ih z )
11		cc0 9278	. . . . . 6 class 0
12	10, 11	wceq 1364	. . . . 5 wff ( y .ih z ) = 0
13	2	cv 1363	. . . . 5 class x
14	12, 7, 13	wral 2713	. . . 4 wff A. z e. x ( y .ih z ) = 0
15	14, 5, 3	crab 2717	. . 3 class { y e. ~H | A. z e. x ( y .ih z ) = 0 }
16	2, 4, 15	cmpt 4347	. 2 class ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )
17	1, 16	wceq 1364	1 wff _|_ = ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )

This definition is referenced by:  ocval  24618