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Definition df-oc 21826
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 21854 and chocvali 21873 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 21505 . 2  class  _|_
2 vx . . 3  set  x
3 chil 21494 . . . 4  class  ~H
43cpw 3625 . . 3  class  ~P ~H
5 vy . . . . . . . 8  set  y
65cv 1622 . . . . . . 7  class  y
7 vz . . . . . . . 8  set  z
87cv 1622 . . . . . . 7  class  z
9 csp 21497 . . . . . . 7  class  .ih
106, 8, 9co 5819 . . . . . 6  class  ( y 
.ih  z )
11 cc0 8732 . . . . . 6  class  0
1210, 11wceq 1623 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1622 . . . . 5  class  x
1412, 7, 13wral 2543 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2547 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4077 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1623 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  21854
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