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Theorem List for Metamath Proof Explorer - 22301-22400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempjpythi 22301 Pythagorean theorem for projections. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  (
 ( normh `  A ) ^ 2 )  =  ( ( ( normh `  ( ( proj  h `  H ) `  A ) ) ^ 2
 )  +  ( (
 normh `  ( ( proj  h `
  ( _|_ `  H ) ) `  A ) ) ^ 2
 ) )
 
Theorempjneli 22302 If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  ( -.  A  e.  H  <->  ( normh `  (
 ( proj  h `  H ) `  A ) )  <  ( normh `  A ) )
 
Theorempjnorm 22303 The norm of the projection is less than or equal to the norm. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( normh `  ( ( proj  h `  H ) `
  A ) ) 
 <_  ( normh `  A )
 )
 
Theorempjpyth 22304 Pythagorean theorem for projectors. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( ( normh `  A ) ^ 2 )  =  ( ( ( normh `  ( ( proj  h `  H ) `  A ) ) ^ 2
 )  +  ( (
 normh `  ( ( proj  h `
  ( _|_ `  H ) ) `  A ) ) ^ 2
 ) ) )
 
Theorempjnel 22305 If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( -.  A  e.  H 
 <->  ( normh `  ( ( proj  h `  H ) `
  A ) )  <  ( normh `  A ) ) )
 
Theorempjnorm2 22306 A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch 22273 yield Theorem 26.3 of [Halmos] p. 44. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( A  e.  H  <->  (
 normh `  ( ( proj  h `
  H ) `  A ) )  =  ( normh `  A )
 ) )
 
17.5.11  Mayet's equation E_3
 
Theoremmayete3i 22307 Mayet's equation E3. Part of Theorem 4.1 of [Mayet3] p. 1223. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   &    |-  F  e.  CH   &    |-  G  e.  CH   &    |-  A  C_  ( _|_ `  C )   &    |-  A  C_  ( _|_ `  F )   &    |-  C  C_  ( _|_ `  F )   &    |-  A  C_  ( _|_ `  B )   &    |-  C  C_  ( _|_ `  D )   &    |-  F  C_  ( _|_ `  G )   &    |-  X  =  ( ( A  vH  C )  vH  F )   &    |-  Y  =  ( (
 ( A  vH  B )  i^i  ( C  vH  D ) )  i^i  ( F  vH  G ) )   &    |-  Z  =  ( ( B  vH  D )  vH  G )   =>    |-  ( X  i^i  Y )  C_  Z
 
Theoremmayete3iOLD 22308 Mayet's equation E3. Part of Theorem 4.1 of [Mayet3] p. 7. (Contributed by NM, 22-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   &    |-  F  e.  CH   &    |-  G  e.  CH   &    |-  A  C_  ( _|_ `  B )   &    |-  A  C_  ( _|_ `  C )   &    |-  B  C_  ( _|_ `  C )   &    |-  A  C_  ( _|_ `  D )   &    |-  B  C_  ( _|_ `  F )   &    |-  C  C_  ( _|_ `  G )   &    |-  R  =  ( ( A  vH  B )  vH  C )   &    |-  S  =  ( (
 ( A  vH  D )  i^i  ( B  vH  F ) )  i^i  ( C  vH  G ) )   &    |-  T  =  ( ( D  vH  F )  vH  G )   =>    |-  ( R  i^i  S )  C_  T
 
Theoremmayetes3i 22309 Mayet's equation E^*3, derived from E3. Solution, for n = 3, to open problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. (Contributed by NM, 10-May-2009.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   &    |-  F  e.  CH   &    |-  G  e.  CH   &    |-  R  e.  CH   &    |-  A  C_  ( _|_ `  C )   &    |-  A  C_  ( _|_ `  F )   &    |-  C  C_  ( _|_ `  F )   &    |-  A  C_  ( _|_ `  B )   &    |-  C  C_  ( _|_ `  D )   &    |-  F  C_  ( _|_ `  G )   &    |-  R  C_  ( _|_ `  X )   &    |-  X  =  ( ( A  vH  C )  vH  F )   &    |-  Y  =  ( (
 ( A  vH  B )  i^i  ( C  vH  D ) )  i^i  ( F  vH  G ) )   &    |-  Z  =  ( ( B  vH  D )  vH  G )   =>    |-  ( ( X 
 vH  R )  i^i 
 Y )  C_  ( Z  vH  R )
 
17.6  Operators on Hilbert spaces
 
17.6.1  Operator sum, difference, and scalar multiplication

Note on operators. Unlike some authors, we use the term "operator" to mean any function from  ~H to  ~H. This is the definition of operator in [Hughes] p. 14, the definition of operator in [AkhiezerGlazman] p. 30, and the definition of operator in [Goldberg] p. 10. For Reed and Simon, an operator is linear (definition of operator in [ReedSimon] p. 2). For Halmos, an operator is bounded and linear (definition of operator in [Halmos] p. 35). For Kalmbach and Beran, an operator is continuous and linear (definition of operator in [Kalmbach] p. 353; definition of operator in [Beran] p. 99). Note that "bounded and linear" and "continuous and linear" are equivalent by lncnbd 22618.

 
Definitiondf-hosum 22310* Define the sum of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)
 |-  +op  =  ( f  e.  ( ~H  ^m  ~H ) ,  g  e.  ( ~H 
 ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x )  +h  ( g `  x ) ) ) )
 
Definitiondf-homul 22311* Define the scalar product with a Hilbert space operator. Definition of [Beran] p. 111. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  .op  =  ( f  e.  CC ,  g  e.  ( ~H  ^m  ~H )  |->  ( x  e.  ~H  |->  ( f  .h  ( g `
  x ) ) ) )
 
Definitiondf-hodif 22312* Define the difference of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)
 |-  -op  =  ( f  e.  ( ~H  ^m  ~H ) ,  g  e.  ( ~H 
 ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x )  -h  ( g `  x ) ) ) )
 
Definitiondf-hfsum 22313* Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from  ~H to  CC, not just linear (or bounded linear) ones. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
 |-  +fn  =  ( f  e.  ( CC  ^m  ~H ) ,  g  e.  ( CC 
 ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x )  +  ( g `  x ) ) ) )
 
Definitiondf-hfmul 22314* Define the scalar product with a Hilbert space functional. Definition of [Beran] p. 111. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
 |-  .fn  =  ( f  e.  CC ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( f  x.  ( g `
  x ) ) ) )
 
Theoremhosmval 22315* Value of the sum of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 +op  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
 
Theoremhommval 22316* Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T )  =  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) )
 
Theoremhodmval 22317* Value of the difference of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 -op  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) )
 
Theoremhfsmval 22318* Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> CC  /\  T : ~H --> CC )  ->  ( S 
 +fn  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  +  ( T `  x ) ) ) )
 
Theoremhfmmval 22319* Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> CC )  ->  ( A  .fn  T )  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
 
Theoremhosval 22320 Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S 
 +op  T ) `  A )  =  ( ( S `  A )  +h  ( T `  A ) ) )
 
Theoremhomval 22321 Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  =  ( A  .h  ( T `  B ) ) )
 
Theoremhodval 22322 Value of the difference of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S 
 -op  T ) `  A )  =  ( ( S `  A )  -h  ( T `  A ) ) )
 
Theoremhfsval 22323 Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> CC  /\  T : ~H --> CC  /\  A  e.  ~H )  ->  ( ( S 
 +fn  T ) `  A )  =  ( ( S `  A )  +  ( T `  A ) ) )
 
Theoremhfmval 22324 Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> CC  /\  B  e.  ~H )  ->  ( ( A  .fn  T ) `  B )  =  ( A  x.  ( T `  B ) ) )
 
Theoremhoscl 22325 Closure of the sum of two Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  (
 ( ( S : ~H
 --> ~H  /\  T : ~H
 --> ~H )  /\  A  e.  ~H )  ->  (
 ( S  +op  T ) `  A )  e. 
 ~H )
 
Theoremhomcl 22326 Closure of the scalar product of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  e.  ~H )
 
Theoremhodcl 22327 Closure of the difference of two Hilbert space operators. (Contributed by NM, 15-Nov-2002.) (New usage is discouraged.)
 |-  (
 ( ( S : ~H
 --> ~H  /\  T : ~H
 --> ~H )  /\  A  e.  ~H )  ->  (
 ( S  -op  T ) `  A )  e. 
 ~H )
 
17.6.2  Zero and identity operators
 
Definitiondf-h0op 22328 Define the Hilbert space zero operator. See df0op2 22332 for alternate definition. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  0hop  =  ( proj  h `  0H )
 
Definitiondf-iop 22329 Define the Hilbert space identity operator. See dfiop2 22333 for alternate definition. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
 |-  Iop  =  ( proj  h `  ~H )
 
Theoremho0val 22330 Value of the zero Hilbert space operator (null projector). Remark in [Beran] p. 111. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 0hop `  A )  =  0h )
 
Theoremho0f 22331 Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  0hop : ~H --> ~H
 
Theoremdf0op2 22332 Alternate definition of Hilbert space zero operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
 |-  0hop  =  ( ~H  X.  0H )
 
Theoremdfiop2 22333 Alternate definition of Hilbert space identity operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
 |-  Iop  =  (  _I  |`  ~H )
 
Theoremhoif 22334 Functionality of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
 |-  Iop  : ~H
 -1-1-onto-> ~H
 
Theoremhoival 22335 The value of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 
 Iop  `  A )  =  A )
 
Theoremhoico1 22336 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( T  o.  Iop  )  =  T )
 
Theoremhoico2 22337 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( 
 Iop  o.  T )  =  T )
 
17.6.3  Operations on Hilbert space operators
 
Theoremhoaddcl 22338 The sum of Hilbert space operators is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 +op  T ) : ~H --> ~H )
 
Theoremhomulcl 22339 The scalar product of a Hilbert space operator is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T ) : ~H --> ~H )
 
Theoremhoeq 22340* Equality of Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A. x  e.  ~H  ( T `  x )  =  ( U `  x ) 
 <->  T  =  U ) )
 
Theoremhoeqi 22341* Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A. x  e. 
 ~H  ( S `  x )  =  ( T `  x )  <->  S  =  T )
 
Theoremhoscli 22342 Closure of Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A  e.  ~H  ->  ( ( S  +op  T ) `  A )  e.  ~H )
 
Theoremhodcli 22343 Closure of Hilbert space operator difference. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A  e.  ~H  ->  ( ( S  -op  T ) `  A )  e.  ~H )
 
Theoremhocoi 22344 Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A  e.  ~H  ->  ( ( S  o.  T ) `  A )  =  ( S `  ( T `  A ) ) )
 
Theoremhococli 22345 Closure of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A  e.  ~H  ->  ( ( S  o.  T ) `  A )  e.  ~H )
 
Theoremhocofi 22346 Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  o.  T ) : ~H --> ~H
 
Theoremhocofni 22347 Functionality of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  o.  T )  Fn  ~H
 
Theoremhoaddcli 22348 Mapping of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  +op  T ) : ~H --> ~H
 
Theoremhosubcli 22349 Mapping of difference of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  -op  T ) : ~H --> ~H
 
Theoremhoaddfni 22350 Functionality of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  +op  T )  Fn  ~H
 
Theoremhosubfni 22351 Functionality of difference of Hilbert space operators. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  -op  T )  Fn  ~H
 
Theoremhoaddcomi 22352 Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  +op  T )  =  ( T  +op  S )
 
Theoremhosubcl 22353 Mapping of difference of Hilbert space operators. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 -op  T ) : ~H --> ~H )
 
Theoremhoaddcom 22354 Commutativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 +op  T )  =  ( T  +op  S )
 )
 
Theoremhodsi 22355 Relationship between Hilbert space operator difference and sum. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 -op  S )  =  T  <->  ( S  +op  T )  =  R )
 
Theoremhoaddassi 22356 Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 +op  S )  +op  T )  =  ( R  +op  ( S  +op  T ) )
 
Theoremhoadd12i 22357 Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( R  +op  ( S  +op  T ) )  =  ( S 
 +op  ( R  +op  T ) )
 
Theoremhoadd32i 22358 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 +op  S )  +op  T )  =  ( ( R  +op  T )  +op  S )
 
Theoremhocadddiri 22359 Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 +op  S )  o.  T )  =  ( ( R  o.  T )  +op  ( S  o.  T ) )
 
Theoremhocsubdiri 22360 Distributive law for Hilbert space operator difference. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 -op  S )  o.  T )  =  ( ( R  o.  T )  -op  ( S  o.  T ) )
 
Theoremho2coi 22361 Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A  e.  ~H 
 ->  ( ( ( R  o.  S )  o.  T ) `  A )  =  ( R `  ( S `  ( T `  A ) ) ) )
 
Theoremhoaddass 22362 Associativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( R : ~H --> ~H  /\  S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( R  +op  S )  +op  T )  =  ( R  +op  ( S  +op  T ) ) )
 
Theoremhoadd32 22363 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( R : ~H --> ~H  /\  S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( R  +op  S )  +op  T )  =  ( ( R  +op  T )  +op  S ) )
 
Theoremhoadd4 22364 Rearrangement of 4 terms in a sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( ( R : ~H
 --> ~H  /\  S : ~H
 --> ~H )  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
 )  ->  ( ( R  +op  S )  +op  ( T  +op  U ) )  =  ( ( R  +op  T )  +op  ( S  +op  U ) ) )
 
Theoremhocsubdir 22365 Distributive law for Hilbert space operator difference. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( R : ~H --> ~H  /\  S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( R  -op  S )  o.  T )  =  ( ( R  o.  T )  -op  ( S  o.  T ) ) )
 
Theoremhoaddid1i 22366 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  ( T  +op  0hop )  =  T
 
Theoremhodidi 22367 Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  ( T  -op  T )  =  0hop
 
Theoremho0coi 22368 Composition of the zero operator and a Hilbert space operator. (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  ( 0hop  o.  T )  =  0hop
 
Theoremhoid1i 22369 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  ( T  o.  Iop  )  =  T
 
Theoremhoid1ri 22370 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  (  Iop  o.  T )  =  T
 
Theoremhoaddid1 22371 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( T  +op  0hop )  =  T )
 
Theoremhodid 22372 Difference of a Hilbert space operator from itself. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( T  -op  T )  =  0hop )
 
Theoremhon0 22373 A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  -.  T  =  (/) )
 
Theoremhodseqi 22374 Subtraction and addition of equal Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  +op  ( T  -op  S ) )  =  T
 
Theoremho0subi 22375 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  -op  T )  =  ( S  +op  ( 0hop  -op  T ) )
 
Theoremhonegsubi 22376 Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  +op  ( -u 1  .op  T )
 )  =  ( S 
 -op  T )
 
Theoremho0sub 22377 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 -op  T )  =  ( S  +op  ( 0hop  -op 
 T ) ) )
 
Theoremhosubid1 22378 The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( T  -op  0hop )  =  T )
 
Theoremhonegsub 22379 Relationship between Hilbert space operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T 
 +op  ( -u 1  .op  U ) )  =  ( T  -op  U ) )
 
Theoremhomulid2 22380 An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( 1  .op  T )  =  T )
 
Theoremhomco1 22381 Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U )  =  ( A  .op  ( T  o.  U ) ) )
 
Theoremhomulass 22382 Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  x.  B )  .op  T )  =  ( A  .op  ( B  .op  T ) ) )
 
Theoremhoadddi 22383 Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) )  =  ( ( A 
 .op  T )  +op  ( A  .op  U ) ) )
 
Theoremhoadddir 22384 Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  +  B )  .op  T )  =  ( ( A 
 .op  T )  +op  ( B  .op  T ) ) )
 
Theoremhomul12 22385 Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  ( B  .op  T ) )  =  ( B  .op  ( A  .op  T ) ) )
 
Theoremhonegneg 22386 Double negative of a Hilbert space operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  (
 -u 1  .op  ( -u 1  .op  T )
 )  =  T )
 
Theoremhosubneg 22387 Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T 
 -op  ( -u 1  .op  U ) )  =  ( T  +op  U ) )
 
Theoremhosubdi 22388 Scalar product distributive law for operator difference. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  -op  U ) )  =  ( ( A 
 .op  T )  -op  ( A  .op  U ) ) )
 
Theoremhonegdi 22389 Distribution of negative over addition. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( -u 1  .op  ( T  +op  U ) )  =  ( ( -u 1  .op  T )  +op  ( -u 1  .op  U ) ) )
 
Theoremhonegsubdi 22390 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( -u 1  .op  ( T  -op  U ) )  =  ( ( -u 1  .op  T )  +op  U ) )
 
Theoremhonegsubdi2 22391 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( -u 1  .op  ( T  -op  U ) )  =  ( U  -op  T ) )
 
Theoremhosubsub2 22392 Law for double subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( S 
 -op  ( T  -op  U ) )  =  ( S  +op  ( U  -op  T ) ) )
 
Theoremhosub4 22393 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( ( R : ~H
 --> ~H  /\  S : ~H
 --> ~H )  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
 )  ->  ( ( R  +op  S )  -op  ( T  +op  U ) )  =  ( ( R  -op  T ) 
 +op  ( S  -op  U ) ) )
 
Theoremhosubadd4 22394 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( ( R : ~H
 --> ~H  /\  S : ~H
 --> ~H )  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
 )  ->  ( ( R  -op  S )  -op  ( T  -op  U ) )  =  ( ( R  +op  U )  -op  ( S  +op  T ) ) )
 
Theoremhoaddsubass 22395 Associative-type law for addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( S  +op  T )  -op  U )  =  ( S  +op  ( T  -op  U ) ) )
 
Theoremhoaddsub 22396 Law for operator addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( S  +op  T )  -op  U )  =  ( ( S  -op  U )  +op  T ) )
 
Theoremhosubsub 22397 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( S 
 -op  ( T  -op  U ) )  =  ( ( S  -op  T )  +op  U ) )
 
Theoremhosubsub4 22398 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( S  -op  T ) 
 -op  U )  =  ( S  -op  ( T 
 +op  U ) ) )
 
Theoremho2times 22399 Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( 2  .op  T )  =  ( T  +op  T ) )
 
Theoremhoaddsubassi 22400 Associativity of sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 +op  S )  -op  T )  =  ( R  +op  ( S  -op  T ) )
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