Definition df-oc 26954

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 26982 and chocvali 27001 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 syntactic 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 26632	. 2 class _|_
2		vx	. . 3 setvar x
3		chil 26621	. . . 4 class ~H
4	3	cpw 3963	. . 3 class ~P ~H
5		vy	. . . . . . . 8 setvar y
6	5	cv 1454	. . . . . . 7 class y
7		vz	. . . . . . . 8 setvar z
8	7	cv 1454	. . . . . . 7 class z
9		csp 26624	. . . . . . 7 class .ih
10	6, 8, 9	co 6315	. . . . . 6 class ( y .ih z )
11		cc0 9565	. . . . . 6 class 0
12	10, 11	wceq 1455	. . . . 5 wff ( y .ih z ) = 0
13	2	cv 1454	. . . . 5 class x
14	12, 7, 13	wral 2749	. . . 4 wff A. z e. x ( y .ih z ) = 0
15	14, 5, 3	crab 2753	. . 3 class { y e. ~H | A. z e. x ( y .ih z ) = 0 }
16	2, 4, 15	cmpt 4475	. 2 class ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )
17	1, 16	wceq 1455	1 wff _|_ = ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )

This definition is referenced by:  ocval  26982