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Definition df-oc 22754
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 22782 and chocvali 22801 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 22433 . 2  class  _|_
2 vx . . 3  set  x
3 chil 22422 . . . 4  class  ~H
43cpw 3799 . . 3  class  ~P ~H
5 vy . . . . . . . 8  set  y
65cv 1651 . . . . . . 7  class  y
7 vz . . . . . . . 8  set  z
87cv 1651 . . . . . . 7  class  z
9 csp 22425 . . . . . . 7  class  .ih
106, 8, 9co 6081 . . . . . 6  class  ( y 
.ih  z )
11 cc0 8990 . . . . . 6  class  0
1210, 11wceq 1652 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1651 . . . . 5  class  x
1412, 7, 13wral 2705 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2709 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4266 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1652 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  22782
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