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Definition df-oc 25846
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 25874 and chocvali 25893 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 25523 . 2  class  _|_
2 vx . . 3  setvar  x
3 chil 25512 . . . 4  class  ~H
43cpw 4010 . . 3  class  ~P ~H
5 vy . . . . . . . 8  setvar  y
65cv 1378 . . . . . . 7  class  y
7 vz . . . . . . . 8  setvar  z
87cv 1378 . . . . . . 7  class  z
9 csp 25515 . . . . . . 7  class  .ih
106, 8, 9co 6282 . . . . . 6  class  ( y 
.ih  z )
11 cc0 9488 . . . . . 6  class  0
1210, 11wceq 1379 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1378 . . . . 5  class  x
1412, 7, 13wral 2814 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2818 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4505 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1379 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  25874
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