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Definition df-eigvec 22379
Description: Define the eigenvector function. Theorem eleigveccl 22485 shows that  eigvec `  T, the set of eigenvectors of Hilbert space operator  T, are Hilbert space vectors. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-eigvec  |-  eigvec  =  ( t  e.  ( ~H 
^m  ~H )  |->  { x  e.  ( ~H  \  0H )  |  E. z  e.  CC  ( t `  x )  =  ( z  .h  x ) } )
Distinct variable group:    x, t, z

Detailed syntax breakdown of Definition df-eigvec
StepHypRef Expression
1 cei 21485 . 2  class  eigvec
2 vt . . 3  set  t
3 chil 21445 . . . 4  class  ~H
4 cmap 6726 . . . 4  class  ^m
53, 3, 4co 5778 . . 3  class  ( ~H 
^m  ~H )
6 vx . . . . . . . 8  set  x
76cv 1618 . . . . . . 7  class  x
82cv 1618 . . . . . . 7  class  t
97, 8cfv 4659 . . . . . 6  class  ( t `
 x )
10 vz . . . . . . . 8  set  z
1110cv 1618 . . . . . . 7  class  z
12 csm 21447 . . . . . . 7  class  .h
1311, 7, 12co 5778 . . . . . 6  class  ( z  .h  x )
149, 13wceq 1619 . . . . 5  wff  ( t `
 x )  =  ( z  .h  x
)
15 cc 8689 . . . . 5  class  CC
1614, 10, 15wrex 2517 . . . 4  wff  E. z  e.  CC  ( t `  x )  =  ( z  .h  x )
17 c0h 21461 . . . . 5  class  0H
183, 17cdif 3110 . . . 4  class  ( ~H 
\  0H )
1916, 6, 18crab 2520 . . 3  class  { x  e.  ( ~H  \  0H )  |  E. z  e.  CC  ( t `  x )  =  ( z  .h  x ) }
202, 5, 19cmpt 4037 . 2  class  ( t  e.  ( ~H  ^m  ~H )  |->  { x  e.  ( ~H  \  0H )  |  E. z  e.  CC  ( t `  x )  =  ( z  .h  x ) } )
211, 20wceq 1619 1  wff  eigvec  =  ( t  e.  ( ~H 
^m  ~H )  |->  { x  e.  ( ~H  \  0H )  |  E. z  e.  CC  ( t `  x )  =  ( z  .h  x ) } )
Colors of variables: wff set class
This definition is referenced by:  eigvecval  22422
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