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Definition df-oc 21777
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 21805 and chocvali 21824 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 21456 . 2  class  _|_
2 vx . . 3  set  x
3 chil 21445 . . . 4  class  ~H
43cpw 3585 . . 3  class  ~P ~H
5 vy . . . . . . . 8  set  y
65cv 1618 . . . . . . 7  class  y
7 vz . . . . . . . 8  set  z
87cv 1618 . . . . . . 7  class  z
9 csp 21448 . . . . . . 7  class  .ih
106, 8, 9co 5778 . . . . . 6  class  ( y 
.ih  z )
11 cc0 8691 . . . . . 6  class  0
1210, 11wceq 1619 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1618 . . . . 5  class  x
1412, 7, 13wral 2516 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2520 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4037 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1619 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  21805
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