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Theorem List for Metamath Proof Explorer - 22201-22300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem5oai 22201 Orthoarguesian law 5OA. This 8-variable inference is called 5OA because it can be converted to a 5-variable equation (see Quantum Logic Explorer). (Contributed by NM, 5-May-2000.) (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   &    |-  S  e.  CH   &    |-  A  C_  ( _|_ `  B )   &    |-  C  C_  ( _|_ `  D )   &    |-  F  C_  ( _|_ `  G )   &    |-  R  C_  ( _|_ `  S )   =>    |-  ( ( ( A 
 vH  B )  i^i  ( C  vH  D ) )  i^i  ( ( F  vH  G )  i^i  ( R  vH  S ) ) ) 
 C_  ( B  vH  ( A  i^i  ( C 
 vH  ( ( ( ( A  vH  C )  i^i  ( B  vH  D ) )  i^i  ( ( ( A 
 vH  R )  i^i  ( B  vH  S ) )  vH  ( ( C  vH  R )  i^i  ( D  vH  S ) ) ) )  i^i  ( ( ( ( A  vH  F )  i^i  ( B 
 vH  G ) )  i^i  ( ( ( A  vH  R )  i^i  ( B  vH  S ) )  vH  ( ( F  vH  R )  i^i  ( G 
 vH  S ) ) ) )  vH  (
 ( ( C  vH  F )  i^i  ( D 
 vH  G ) )  i^i  ( ( ( C  vH  R )  i^i  ( D  vH  S ) )  vH  ( ( F  vH  R )  i^i  ( G 
 vH  S ) ) ) ) ) ) ) ) )
 
Theorem3oalem1 22202* Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  e.  CH   &    |-  S  e.  CH   =>    |-  (
 ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y
 ) )  /\  (
 ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  (
 ( ( x  e. 
 ~H  /\  y  e.  ~H )  /\  v  e. 
 ~H )  /\  (
 z  e.  ~H  /\  w  e.  ~H )
 ) )
 
Theorem3oalem2 22203* Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  e.  CH   &    |-  S  e.  CH   =>    |-  (
 ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y
 ) )  /\  (
 ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
 
Theorem3oalem3 22204 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  e.  CH   &    |-  S  e.  CH   =>    |-  (
 ( B  +H  R )  i^i  ( C  +H  S ) )  C_  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) )
 
Theorem3oalem4 22205 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  R  =  ( ( _|_ `  B )  i^i  ( B  vH  A ) )   =>    |-  R  C_  ( _|_ `  B )
 
Theorem3oalem5 22206 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  =  ( ( _|_ `  B )  i^i  ( B  vH  A ) )   &    |-  S  =  ( ( _|_ `  C )  i^i  ( C  vH  A ) )   =>    |-  ( ( B  +H  R )  i^i  ( C  +H  S ) )  =  (
 ( B  vH  R )  i^i  ( C  vH  S ) )
 
Theorem3oalem6 22207 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  =  ( ( _|_ `  B )  i^i  ( B  vH  A ) )   &    |-  S  =  ( ( _|_ `  C )  i^i  ( C  vH  A ) )   =>    |-  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) )  C_  ( B  vH  ( R  i^i  ( S  vH  ( ( B 
 vH  C )  i^i  ( R  vH  S ) ) ) ) )
 
Theorem3oai 22208 3OA (weak) orthoarguesian law. Equation IV of [GodowskiGreechie] p. 249. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  =  ( ( _|_ `  B )  i^i  ( B  vH  A ) )   &    |-  S  =  ( ( _|_ `  C )  i^i  ( C  vH  A ) )   =>    |-  ( ( B 
 vH  R )  i^i  ( C  vH  S ) )  C_  ( B 
 vH  ( R  i^i  ( S  vH  ( ( B  vH  C )  i^i  ( R  vH  S ) ) ) ) )
 
17.5.10  Projectors (cont.)
 
Theorempjorthi 22209 Projection components on orthocomplemented subspaces are orthogonal. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( H  e.  CH  ->  (
 ( ( proj  h `  H ) `  A )  .ih  ( ( proj  h `
  ( _|_ `  H ) ) `  B ) )  =  0
 )
 
Theorempjch1 22210 Property of identity projection. Remark in [Beran] p. 111. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( ( proj  h `  ~H ) `  A )  =  A )
 
Theorempjo 22211 The orthogonal projection. Lemma 4.4(i) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( ( proj  h `  ( _|_ `  H )
 ) `  A )  =  ( ( ( proj  h `
  ~H ) `  A )  -h  (
 ( proj  h `  H ) `  A ) ) )
 
Theorempjcompi 22212 Component of a projection. (Contributed by NM, 31-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) ) 
 ->  ( ( proj  h `  H ) `  ( A  +h  B ) )  =  A )
 
Theorempjidmi 22213 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  (
 ( proj  h `  H ) `  ( ( proj  h `
  H ) `  A ) )  =  ( ( proj  h `  H ) `  A )
 
Theorempjadjii 22214 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( ( ( proj  h `
  H ) `  A )  .ih  B )  =  ( A  .ih  ( ( proj  h `  H ) `  B ) )
 
Theorempjaddii 22215 Projection of vector sum is sum of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( ( proj  h `  H ) `  ( A  +h  B ) )  =  ( ( (
 proj  h `  H ) `
  A )  +h  ( ( proj  h `  H ) `  B ) )
 
Theorempjinormii 22216 The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  (
 ( ( proj  h `  H ) `  A )  .ih  A )  =  ( ( normh `  (
 ( proj  h `  H ) `  A ) ) ^ 2 )
 
Theorempjmulii 22217 Projection of (scalar) product is product of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  C  e.  CC   =>    |-  ( ( proj  h `  H ) `  ( C  .h  A ) )  =  ( C  .h  ( ( proj  h `  H ) `  A ) )
 
Theorempjsubii 22218 Projection of vector difference is difference of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( ( proj  h `  H ) `  ( A  -h  B ) )  =  ( ( (
 proj  h `  H ) `
  A )  -h  ( ( proj  h `  H ) `  B ) )
 
Theorempjsslem 22219 Lemma for subset relationships of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( ( ( proj  h `
  ( _|_ `  H ) ) `  A )  -h  ( ( proj  h `
  ( _|_ `  G ) ) `  A ) )  =  (
 ( ( proj  h `  G ) `  A )  -h  ( ( proj  h `
  H ) `  A ) )
 
Theorempjss2i 22220 Subset relationship for projections. Theorem 4.5(i)->(ii) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( H  C_  G  ->  ( ( proj  h `  H ) `  (
 ( proj  h `  G ) `  A ) )  =  ( ( proj  h `
  H ) `  A ) )
 
Theorempjssmii 22221 Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( H  C_  G  ->  ( ( ( proj  h `
  G ) `  A )  -h  (
 ( proj  h `  H ) `  A ) )  =  ( ( proj  h `
  ( G  i^i  ( _|_ `  H )
 ) ) `  A ) )
 
Theorempjssge0ii 22222 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( ( ( (
 proj  h `  G ) `
  A )  -h  ( ( proj  h `  H ) `  A ) )  =  (
 ( proj  h `  ( G  i^i  ( _|_ `  H ) ) ) `  A )  ->  0  <_  ( ( ( (
 proj  h `  G ) `
  A )  -h  ( ( proj  h `  H ) `  A ) )  .ih  A ) )
 
Theorempjdifnormii 22223 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( 0  <_  (
 ( ( ( proj  h `
  G ) `  A )  -h  (
 ( proj  h `  H ) `  A ) ) 
 .ih  A )  <->  ( normh `  (
 ( proj  h `  H ) `  A ) ) 
 <_  ( normh `  ( ( proj  h `  G ) `
  A ) ) )
 
Theorempjcji 22224 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( H  C_  ( _|_ `  G )  ->  ( ( proj  h `  ( H  vH  G ) ) `  A )  =  ( ( (
 proj  h `  H ) `
  A )  +h  ( ( proj  h `  G ) `  A ) ) )
 
Theorempjadji 22225 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( ( (
 proj  h `  H ) `
  A )  .ih  B )  =  ( A 
 .ih  ( ( proj  h `
  H ) `  B ) ) )
 
Theorempjaddi 22226 Projection of vector sum is sum of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( ( proj  h `
  H ) `  ( A  +h  B ) )  =  ( ( ( proj  h `  H ) `  A )  +h  ( ( proj  h `  H ) `  B ) ) )
 
Theorempjinormi 22227 The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( ( ( proj  h `
  H ) `  A )  .ih  A )  =  ( ( normh `  ( ( proj  h `  H ) `  A ) ) ^ 2
 ) )
 
Theorempjsubi 22228 Projection of vector difference is difference of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( ( proj  h `
  H ) `  ( A  -h  B ) )  =  ( ( ( proj  h `  H ) `  A )  -h  ( ( proj  h `  H ) `  B ) ) )
 
Theorempjmuli 22229 Projection of scalar product is scalar product of projection. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( proj  h `
  H ) `  ( A  .h  B ) )  =  ( A  .h  ( ( proj  h `
  H ) `  B ) ) )
 
Theorempjige0i 22230 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  0  <_  ( (
 ( proj  h `  H ) `  A )  .ih  A ) )
 
Theorempjige0 22231 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  0  <_  ( (
 ( proj  h `  H ) `  A )  .ih  A ) )
 
Theorempjcjt2 22232 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  G  e.  CH  /\  A  e.  ~H )  ->  ( H  C_  ( _|_ `  G )  ->  ( ( proj  h `  ( H  vH  G ) ) `  A )  =  ( ( (
 proj  h `  H ) `
  A )  +h  ( ( proj  h `  G ) `  A ) ) ) )
 
Theorempj0i 22233 The projection of the zero vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( proj  h `  H ) `  0h )  =  0h
 
Theorempjch 22234 Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( A  e.  H  <->  ( ( proj  h `  H ) `  A )  =  A ) )
 
Theorempjid 22235 The projection of a vector in the projection subspace is itself. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  H ) 
 ->  ( ( proj  h `  H ) `  A )  =  A )
 
Theorempjvec 22236* The set of vectors belonging to the subspace of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  H  =  { x  e.  ~H  |  ( ( proj  h `  H ) `  x )  =  x }
 )
 
Theorempjocvec 22237* The set of vectors belonging to the orthocomplemented subspace of a projection. Second part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  ( _|_ `  H )  =  { x  e.  ~H  |  ( ( proj  h `  H ) `  x )  =  0h } )
 
Theorempjocini 22238 Membership of projection in orthocomplement of intersection. (Contributed by NM, 21-Apr-2001.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ( _|_ `  ( G  i^i  H ) )  ->  ( (
 proj  h `  G ) `
  A )  e.  ( _|_ `  ( G  i^i  H ) ) )
 
Theorempjini 22239 Membership of projection in an intersection. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ( G  i^i  H )  ->  (
 ( proj  h `  G ) `  A )  e.  ( G  i^i  H ) )
 
Theorempjjsi 22240* A sufficient condition for subspace join to be equal to subspace sum. (Contributed by NM, 29-May-2004.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  SH   =>    |-  ( A. x  e.  ( G  vH  H ) ( ( proj  h `  ( _|_ `  G ) ) `
  x )  e.  H  ->  ( G  vH  H )  =  ( G  +H  H ) )
 
Theorempjfni 22241 Functionality of a projection. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H )  Fn  ~H
 
Theorempjrni 22242 The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |- 
 ran  ( proj  h `  H )  =  H
 
Theorempjfoi 22243 A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H ) : ~H -onto-> H
 
Theorempjfi 22244 The mapping of a projection. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H ) : ~H --> ~H
 
Theorempjvi 22245 The value of a projection in terms of components. (Contributed by NM, 28-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) ) 
 ->  ( ( proj  h `  H ) `  ( A  +h  B ) )  =  A )
 
Theorempjhfo 22246 A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  ( proj  h `  H ) : ~H -onto-> H )
 
Theorempjrn 22247 The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  ran  ( proj  h `  H )  =  H )
 
Theorempjhf 22248 The mapping of a projection. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  ( proj  h `  H ) : ~H --> ~H )
 
Theorempjfn 22249 Functionality of a projection. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  ( proj  h `  H )  Fn  ~H )
 
Theorempjsumi 22250 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( G  C_  ( _|_ `  H )  ->  (
 ( proj  h `  ( G  +H  H ) ) `
  A )  =  ( ( ( proj  h `
  G ) `  A )  +h  (
 ( proj  h `  H ) `  A ) ) ) )
 
Theorempj11i 22251 One-to-one correspondence of projection and subspace. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  (
 ( proj  h `  G )  =  ( proj  h `
  H )  <->  G  =  H )
 
Theorempjdsi 22252 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 21-Jun-2006.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  (
 ( A  e.  ( G  vH  H )  /\  G  C_  ( _|_ `  H ) )  ->  A  =  ( ( ( proj  h `
  G ) `  A )  +h  (
 ( proj  h `  H ) `  A ) ) )
 
Theorempjds3i 22253 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
 |-  F  e.  CH   &    |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( ( ( A  e.  ( ( F 
 vH  G )  vH  H )  /\  F  C_  ( _|_ `  G )
 )  /\  ( F  C_  ( _|_ `  H )  /\  G  C_  ( _|_ `  H ) ) )  ->  A  =  ( ( ( (
 proj  h `  F ) `
  A )  +h  ( ( proj  h `  G ) `  A ) )  +h  (
 ( proj  h `  H ) `  A ) ) )
 
Theorempj11 22254 One-to-one correspondence of projection and subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( G  e.  CH  /\  H  e.  CH )  ->  ( ( proj  h `  G )  =  ( proj  h `  H )  <->  G  =  H )
 )
 
Theorempjmfn 22255 Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  proj  h  Fn  CH
 
Theorempjmf1 22256 The projector function maps one-to-one into the set of Hilbert space operators. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  proj  h : CH -1-1-> ( ~H 
 ^m  ~H )
 
Theorempjoi0 22257 The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( G  e.  CH 
 /\  H  e.  CH  /\  A  e.  ~H )  /\  G  C_  ( _|_ `  H ) )  ->  ( ( ( proj  h `
  G ) `  A )  .ih  ( (
 proj  h `  H ) `
  A ) )  =  0 )
 
Theorempjoi0i 22258 The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  ( G  C_  ( _|_ `  H )  ->  ( ( ( proj  h `
  G ) `  A )  .ih  ( (
 proj  h `  H ) `
  A ) )  =  0 )
 
Theorempjopythi 22259 Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  ( G  C_  ( _|_ `  H )  ->  ( ( normh `  (
 ( ( proj  h `  G ) `  A )  +h  ( ( proj  h `
  H ) `  A ) ) ) ^ 2 )  =  ( ( ( normh `  ( ( proj  h `  G ) `  A ) ) ^ 2
 )  +  ( (
 normh `  ( ( proj  h `
  H ) `  A ) ) ^
 2 ) ) )
 
Theorempjopyth 22260 Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  G  e.  CH  /\  A  e.  ~H )  ->  ( H  C_  ( _|_ `  G )  ->  ( ( normh `  (
 ( ( proj  h `  H ) `  A )  +h  ( ( proj  h `
  G ) `  A ) ) ) ^ 2 )  =  ( ( ( normh `  ( ( proj  h `  H ) `  A ) ) ^ 2
 )  +  ( (
 normh `  ( ( proj  h `
  G ) `  A ) ) ^
 2 ) ) ) )
 
Theorempjnormi 22261 The norm of the projection is less than or equal to the norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  ( normh `  ( ( proj  h `
  H ) `  A ) )  <_  ( normh `  A )
 
Theorempjpythi 22262 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 22263 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 22264 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 22265 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 22266 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 22267 A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch 22234 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 22268 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 22269 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 22270 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 22579.

 
Definitiondf-hosum 22271* 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 22272* 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 22273* 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 22274* 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 22275* 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 22276* 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 22277* 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 22278* 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 22279* 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 22280* 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 22281 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 22282 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 22283 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 22284 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 22285 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 22286 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 22287 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 22288 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 22289 Define the Hilbert space zero operator. See df0op2 22293 for alternate definition. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  0hop  =  ( proj  h `  0H )
 
Definitiondf-iop 22290 Define the Hilbert space identity operator. See dfiop2 22294 for alternate definition. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
 |-  Iop  =  ( proj  h `  ~H )
 
Theoremho0val 22291 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 22292 Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  0hop : ~H --> ~H
 
Theoremdf0op2 22293 Alternate definition of Hilbert space zero operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
 |-  0hop  =  ( ~H  X.  0H )
 
Theoremdfiop2 22294 Alternate definition of Hilbert space identity operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
 |-  Iop  =  (  _I  |`  ~H )
 
Theoremhoif 22295 Functionality of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
 |-  Iop  : ~H
 -1-1-onto-> ~H
 
Theoremhoival 22296 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 22297 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 22298 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 22299 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 22300 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 )
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