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Definition df-oc 26740
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 26768 and chocvali 26787 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 26418 . 2  class  _|_
2 vx . . 3  setvar  x
3 chil 26407 . . . 4  class  ~H
43cpw 3985 . . 3  class  ~P ~H
5 vy . . . . . . . 8  setvar  y
65cv 1436 . . . . . . 7  class  y
7 vz . . . . . . . 8  setvar  z
87cv 1436 . . . . . . 7  class  z
9 csp 26410 . . . . . . 7  class  .ih
106, 8, 9co 6305 . . . . . 6  class  ( y 
.ih  z )
11 cc0 9538 . . . . . 6  class  0
1210, 11wceq 1437 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1436 . . . . 5  class  x
1412, 7, 13wral 2782 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2786 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4484 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1437 1  wff  _|_  =  ( x  e.  ~P ~H  |->  { y  e. 
~H  |  A. z  e.  x  ( y  .ih  z )  =  0 } )
Colors of variables: wff setvar class
This definition is referenced by:  ocval  26768
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