Definition df-oc 26740

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 26768 and chocvali 26787 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 26418	. 2 class _|_
2		vx	. . 3 setvar x
3		chil 26407	. . . 4 class ~H
4	3	cpw 3985	. . 3 class ~P ~H
5		vy	. . . . . . . 8 setvar y
6	5	cv 1436	. . . . . . 7 class y
7		vz	. . . . . . . 8 setvar z
8	7	cv 1436	. . . . . . 7 class z
9		csp 26410	. . . . . . 7 class .ih
10	6, 8, 9	co 6305	. . . . . 6 class ( y .ih z )
11		cc0 9538	. . . . . 6 class 0
12	10, 11	wceq 1437	. . . . 5 wff ( y .ih z ) = 0
13	2	cv 1436	. . . . 5 class x
14	12, 7, 13	wral 2782	. . . 4 wff A. z e. x ( y .ih z ) = 0
15	14, 5, 3	crab 2786	. . 3 class { y e. ~H | A. z e. x ( y .ih z ) = 0 }
16	2, 4, 15	cmpt 4484	. 2 class ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )
17	1, 16	wceq 1437	1 wff _|_ = ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )

This definition is referenced by:  ocval  26768