Definition df-oc 26287

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 26315 and chocvali 26334 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 25964	. 2 class _|_
2		vx	. . 3 setvar x
3		chil 25953	. . . 4 class ~H
4	3	cpw 3927	. . 3 class ~P ~H
5		vy	. . . . . . . 8 setvar y
6	5	cv 1398	. . . . . . 7 class y
7		vz	. . . . . . . 8 setvar z
8	7	cv 1398	. . . . . . 7 class z
9		csp 25956	. . . . . . 7 class .ih
10	6, 8, 9	co 6196	. . . . . 6 class ( y .ih z )
11		cc0 9403	. . . . . 6 class 0
12	10, 11	wceq 1399	. . . . 5 wff ( y .ih z ) = 0
13	2	cv 1398	. . . . 5 class x
14	12, 7, 13	wral 2732	. . . 4 wff A. z e. x ( y .ih z ) = 0
15	14, 5, 3	crab 2736	. . 3 class { y e. ~H | A. z e. x ( y .ih z ) = 0 }
16	2, 4, 15	cmpt 4425	. 2 class ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )
17	1, 16	wceq 1399	1 wff _|_ = ( x e. ~P ~H |-> { y e. ~H | A. z e. x ( y .ih z ) = 0 } )

This definition is referenced by:  ocval  26315