Definition df-oc 22707

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 22735 and chocvali 22754 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 22386	. 2 class _|_
2		vx	. . 3 set x
3		chil 22375	. . . 4 class ~H
4	3	cpw 3759	. . 3 class ~P ~H
5		vy	. . . . . . . 8 set y
6	5	cv 1648	. . . . . . 7 class y
7		vz	. . . . . . . 8 set z
8	7	cv 1648	. . . . . . 7 class z
9		csp 22378	. . . . . . 7 class .ih
10	6, 8, 9	co 6040	. . . . . 6 class ( y .ih z )
11		cc0 8946	. . . . . 6 class 0
12	10, 11	wceq 1649	. . . . 5 wff ( y .ih z ) = 0
13	2	cv 1648	. . . . 5 class x
14	12, 7, 13	wral 2666	. . . 4 wff A. z e. x ( y .ih z ) = 0
15	14, 5, 3	crab 2670	. . 3 class { y e. ~H | A. z e. x ( y .ih z ) = 0 }
16	2, 4, 15	cmpt 4226	. 2 class ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )
17	1, 16	wceq 1649	1 wff _|_ = ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )

This definition is referenced by:  ocval  22735