Definition df-oc 26035

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 26063 and chocvali 26082 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 25712	. 2 class _|_
2		vx	. . 3 setvar x
3		chil 25701	. . . 4 class ~H
4	3	cpw 3993	. . 3 class ~P ~H
5		vy	. . . . . . . 8 setvar y
6	5	cv 1380	. . . . . . 7 class y
7		vz	. . . . . . . 8 setvar z
8	7	cv 1380	. . . . . . 7 class z
9		csp 25704	. . . . . . 7 class .ih
10	6, 8, 9	co 6277	. . . . . 6 class ( y .ih z )
11		cc0 9490	. . . . . 6 class 0
12	10, 11	wceq 1381	. . . . 5 wff ( y .ih z ) = 0
13	2	cv 1380	. . . . 5 class x
14	12, 7, 13	wral 2791	. . . 4 wff A. z e. x ( y .ih z ) = 0
15	14, 5, 3	crab 2795	. . 3 class { y e. ~H | A. z e. x ( y .ih z ) = 0 }
16	2, 4, 15	cmpt 4491	. 2 class ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )
17	1, 16	wceq 1381	1 wff _|_ = ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )

This definition is referenced by:  ocval  26063