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Definition df-oc 22144
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 22172 and chocvali 22191 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
StepHypRef Expression
1 cort 21823 . 2  class  _|_
2 vx . . 3  set  x
3 chil 21812 . . . 4  class  ~H
43cpw 3714 . . 3  class  ~P ~H
5 vy . . . . . . . 8  set  y
65cv 1646 . . . . . . 7  class  y
7 vz . . . . . . . 8  set  z
87cv 1646 . . . . . . 7  class  z
9 csp 21815 . . . . . . 7  class  .ih
106, 8, 9co 5981 . . . . . 6  class  ( y 
.ih  z )
11 cc0 8884 . . . . . 6  class  0
1210, 11wceq 1647 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1646 . . . . 5  class  x
1412, 7, 13wral 2628 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2632 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4179 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1647 1  wff  _|_  =  ( x  e.  ~P ~H  |->  { y  e. 
~H  |  A. z  e.  x  ( y  .ih  z )  =  0 } )
Colors of variables: wff set class
This definition is referenced by:  ocval  22172
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