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Definition df-oc 22715
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 22743 and chocvali 22762 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 22394 . 2  class  _|_
2 vx . . 3  set  x
3 chil 22383 . . . 4  class  ~H
43cpw 3767 . . 3  class  ~P ~H
5 vy . . . . . . . 8  set  y
65cv 1648 . . . . . . 7  class  y
7 vz . . . . . . . 8  set  z
87cv 1648 . . . . . . 7  class  z
9 csp 22386 . . . . . . 7  class  .ih
106, 8, 9co 6048 . . . . . 6  class  ( y 
.ih  z )
11 cc0 8954 . . . . . 6  class  0
1210, 11wceq 1649 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1648 . . . . 5  class  x
1412, 7, 13wral 2674 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2678 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4234 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1649 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  22743
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