Definition df-oc 24660

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 24688 and chocvali 24707 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 24337	. 2 class _|_
2		vx	. . 3 setvar x
3		chil 24326	. . . 4 class ~H
4	3	cpw 3865	. . 3 class ~P ~H
5		vy	. . . . . . . 8 setvar y
6	5	cv 1368	. . . . . . 7 class y
7		vz	. . . . . . . 8 setvar z
8	7	cv 1368	. . . . . . 7 class z
9		csp 24329	. . . . . . 7 class .ih
10	6, 8, 9	co 6096	. . . . . 6 class ( y .ih z )
11		cc0 9287	. . . . . 6 class 0
12	10, 11	wceq 1369	. . . . 5 wff ( y .ih z ) = 0
13	2	cv 1368	. . . . 5 class x
14	12, 7, 13	wral 2720	. . . 4 wff A. z e. x ( y .ih z ) = 0
15	14, 5, 3	crab 2724	. . 3 class { y e. ~H | A. z e. x ( y .ih z ) = 0 }
16	2, 4, 15	cmpt 4355	. 2 class ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )
17	1, 16	wceq 1369	1 wff _|_ = ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )

This definition is referenced by:  ocval  24688