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Definition df-oc 26287
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 26315 and chocvali 26334 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 25964 . 2  class  _|_
2 vx . . 3  setvar  x
3 chil 25953 . . . 4  class  ~H
43cpw 3927 . . 3  class  ~P ~H
5 vy . . . . . . . 8  setvar  y
65cv 1398 . . . . . . 7  class  y
7 vz . . . . . . . 8  setvar  z
87cv 1398 . . . . . . 7  class  z
9 csp 25956 . . . . . . 7  class  .ih
106, 8, 9co 6196 . . . . . 6  class  ( y 
.ih  z )
11 cc0 9403 . . . . . 6  class  0
1210, 11wceq 1399 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1398 . . . . 5  class  x
1412, 7, 13wral 2732 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2736 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4425 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1399 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  26315
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