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Definition df-oc 21825
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 21853 and chocvali 21872 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 21504 . 2  class  _|_
2 vx . . 3  set  x
3 chil 21493 . . . 4  class  ~H
43cpw 3628 . . 3  class  ~P ~H
5 vy . . . . . . . 8  set  y
65cv 1624 . . . . . . 7  class  y
7 vz . . . . . . . 8  set  z
87cv 1624 . . . . . . 7  class  z
9 csp 21496 . . . . . . 7  class  .ih
106, 8, 9co 5821 . . . . . 6  class  ( y 
.ih  z )
11 cc0 8734 . . . . . 6  class  0
1210, 11wceq 1625 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1624 . . . . 5  class  x
1412, 7, 13wral 2546 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2550 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4080 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1625 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  21853
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