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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremchj0 22001 Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  vH  0H )  =  A )
 
Theoremchslej 22002 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  +H  B )  C_  ( A  vH  B ) )
 
Theoremchincl 22003 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  B )  e.  CH )
 
Theoremchsscon3 22004 Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( _|_ `  B )  C_  ( _|_ `  A ) ) )
 
Theoremchsscon1 22005 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( ( _|_ `  A )  C_  B  <->  ( _|_ `  B )  C_  A ) )
 
Theoremchsscon2 22006 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  ( _|_ `  B )  <->  B  C_  ( _|_ `  A ) ) )
 
Theoremchpsscon3 22007 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C.  B  <->  ( _|_ `  B )  C.  ( _|_ `  A ) ) )
 
Theoremchpsscon1 22008 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( ( _|_ `  A )  C.  B  <->  ( _|_ `  B )  C.  A ) )
 
Theoremchpsscon2 22009 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C.  ( _|_ `  B )  <->  B  C.  ( _|_ `  A ) ) )
 
Theoremchjcom 22010 Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B )  =  ( B  vH  A ) )
 
Theoremchub1 22011 Hilbert lattice join is greater than or equal to its first argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  A  C_  ( A  vH  B ) )
 
Theoremchub2 22012 Hilbert lattice join is greater than or equal to its second argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  A  C_  ( B  vH  A ) )
 
Theoremchlub 22013 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  vH  B )  C_  C ) )
 
Theoremchlej1 22014 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  A  C_  B )  ->  ( A  vH  C )  C_  ( B  vH  C ) )
 
Theoremchlej2 22015 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  A  C_  B )  ->  ( C  vH  A )  C_  ( C  vH  B ) )
 
Theoremchlejb1 22016 Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( A  vH  B )  =  B ) )
 
Theoremchlejb2 22017 Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( B  vH  A )  =  B ) )
 
Theoremchnle 22018 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( -.  B  C_  A 
 <->  A  C.  ( A 
 vH  B ) ) )
 
Theoremchjo 22019 The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  vH  ( _|_ `  A ) )  =  ~H )
 
Theoremchabs1 22020 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  ( A  i^i  B ) )  =  A )
 
Theoremchabs2 22021 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  ( A  vH  B ) )  =  A )
 
Theoremchabs1i 22022 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  ( A  i^i  B ) )  =  A
 
Theoremchabs2i 22023 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  i^i  ( A  vH  B ) )  =  A
 
Theoremchjidm 22024 Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  vH  A )  =  A )
 
Theoremchjidmi 22025 Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  vH  A )  =  A
 
Theoremchj12i 22026 A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( A  vH  ( B  vH  C ) )  =  ( B  vH  ( A  vH  C ) )
 
Theoremchj4i 22027 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( A  vH  B )  vH  ( C  vH  D ) )  =  ( ( A  vH  C )  vH  ( B 
 vH  D ) )
 
Theoremchjjdiri 22028 Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  vH  B )  vH  C )  =  ( ( A 
 vH  C )  vH  ( B  vH  C ) )
 
Theoremchdmm1 22029 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  i^i  B ) )  =  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) )
 
Theoremchdmm2 22030 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  i^i  B ) )  =  ( A  vH  ( _|_ `  B ) ) )
 
Theoremchdmm3 22031 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  i^i  ( _|_ `  B ) ) )  =  ( ( _|_ `  A )  vH  B ) )
 
Theoremchdmm4 22032 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  i^i  ( _|_ `  B ) ) )  =  ( A  vH  B ) )
 
Theoremchdmj1 22033 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  vH  B ) )  =  ( ( _|_ `  A )  i^i  ( _|_ `  B ) ) )
 
Theoremchdmj2 22034 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  vH  B ) )  =  ( A  i^i  ( _|_ `  B ) ) )
 
Theoremchdmj3 22035 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  vH  ( _|_ `  B ) ) )  =  ( ( _|_ `  A )  i^i  B ) )
 
Theoremchdmj4 22036 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  vH  ( _|_ `  B ) ) )  =  ( A  i^i  B ) )
 
Theoremchjass 22037 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( ( A  vH  B )  vH  C )  =  ( A  vH  ( B  vH  C ) ) )
 
Theoremchj12 22038 A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  vH  ( B 
 vH  C ) )  =  ( B  vH  ( A  vH  C ) ) )
 
Theoremchj4 22039 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH )  /\  ( C  e.  CH 
 /\  D  e.  CH ) )  ->  ( ( A  vH  B ) 
 vH  ( C  vH  D ) )  =  ( ( A  vH  C )  vH  ( B 
 vH  D ) ) )
 
Theoremledii 22040 An ortholattice is distributive in one ordering direction. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  i^i  B )  vH  ( A  i^i  C ) ) 
 C_  ( A  i^i  ( B  vH  C ) )
 
Theoremlediri 22041 An ortholattice is distributive in one ordering direction. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  i^i  C )  vH  ( B  i^i  C ) ) 
 C_  ( ( A 
 vH  B )  i^i 
 C )
 
Theoremlejdii 22042 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( A  vH  ( B  i^i  C ) ) 
 C_  ( ( A 
 vH  B )  i^i  ( A  vH  C ) )
 
Theoremlejdiri 22043 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  i^i  B )  vH  C ) 
 C_  ( ( A 
 vH  C )  i^i  ( B  vH  C ) )
 
Theoremledi 22044 An ortholattice is distributive in one ordering direction. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) ) 
 C_  ( A  i^i  ( B  vH  C ) ) )
 
15.9.24  Span (cont.) and one-dimensional subspaces
 
Theoremspansn0 22045 The span of the singleton of the zero vector is the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
 |-  ( span `  { 0h }
 )  =  0H
 
Theoremspan0 22046 The span of the empty set is the zero subspace. Remark 11.6.e of [Schechter] p. 276. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  ( span `  (/) )  =  0H
 
Theoremelspani 22047* Membership in the span of a subset of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  B  e.  _V   =>    |-  ( A  C_  ~H  ->  ( B  e.  ( span `  A )  <->  A. x  e.  SH  ( A  C_  x  ->  B  e.  x )
 ) )
 
Theoremspanuni 22048 The span of a union is the subspace sum of spans. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  A  C_ 
 ~H   &    |-  B  C_  ~H   =>    |-  ( span `  ( A  u.  B ) )  =  ( ( span `  A )  +H  ( span `  B ) )
 
Theoremspanun 22049 The span of a union is the subspace sum of spans. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( span `  ( A  u.  B ) )  =  ( ( span `  A )  +H  ( span `  B ) ) )
 
Theoremsshhococi 22050 The join of two Hilbert space subsets (not necessarily closed subspaces) equals the join of their closures (double orthocomplements). (Contributed by NM, 1-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  C_ 
 ~H   &    |-  B  C_  ~H   =>    |-  ( A  vH  B )  =  ( ( _|_ `  ( _|_ `  A ) )  vH  ( _|_ `  ( _|_ `  B ) ) )
 
Theoremhne0 22051 Hilbert space has a nonzero vector iff it is not trivial. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
 |-  ( ~H  =/=  0H  <->  E. x  e.  ~H  x  =/=  0h )
 
Theoremchsup0 22052 The supremum of the empty set. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  (  \/H  `  (/) )  =  0H
 
Theoremh1deoi 22053 Membership in orthocomplement of 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (New usage is discouraged.)
 |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  { B }
 ) 
 <->  ( A  e.  ~H  /\  ( A  .ih  B )  =  0 )
 )
 
Theoremh1dei 22054* Membership in 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  ( A  e.  ~H 
 /\  A. x  e.  ~H  ( ( B  .ih  x )  =  0  ->  ( A  .ih  x )  =  0 ) ) )
 
Theoremh1did 22055 A generating vector belongs to the 1-dimensional subspace it generates. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  A  e.  ( _|_ `  ( _|_ `  { A }
 ) ) )
 
Theoremh1dn0 22056 A nonzero vector generates a (nonzero) 1-dimensional subspace. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  ( _|_ `  ( _|_ `  { A }
 ) )  =/=  0H )
 
Theoremh1de2i 22057 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 17-Jul-2001.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  ->  ( ( B  .ih  B )  .h  A )  =  ( ( A 
 .ih  B )  .h  B ) )
 
Theoremh1de2bi 22058 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( B  =/=  0h  ->  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  A  =  (
 ( ( A  .ih  B )  /  ( B 
 .ih  B ) )  .h  B ) ) )
 
Theoremh1de2ctlem 22059* Lemma for h1de2ci 22060. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  E. x  e.  CC  A  =  ( x  .h  B ) )
 
Theoremh1de2ci 22060* Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 21-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  E. x  e.  CC  A  =  ( x  .h  B ) )
 
Theoremspansni 22061 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( span `  { A }
 )  =  ( _|_ `  ( _|_ `  { A } ) )
 
Theoremelspansni 22062* Membership in the span of a singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( B  e.  ( span `  { A }
 ) 
 <-> 
 E. x  e.  CC  B  =  ( x  .h  A ) )
 
Theoremspansn 22063 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 span `  { A }
 )  =  ( _|_ `  ( _|_ `  { A } ) ) )
 
Theoremspansnch 22064 The span of a Hilbert space singleton belongs to the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 span `  { A }
 )  e.  CH )
 
Theoremspansnsh 22065 The span of a Hilbert space singleton is a subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 span `  { A }
 )  e.  SH )
 
Theoremspansnchi 22066 The span of a singleton in Hilbert space is a closed subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( span `  { A }
 )  e.  CH
 
Theoremspansnid 22067 A vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  A  e.  ( span `  { A } ) )
 
Theoremspansnmul 22068 A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  CC )  ->  ( B  .h  A )  e.  ( span ` 
 { A } )
 )
 
Theoremelspansncl 22069 A member of a span of a singleton is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ( span ` 
 { A } )
 )  ->  B  e.  ~H )
 
Theoremelspansn 22070* Membership in the span of a singleton. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( B  e.  ( span ` 
 { A } )  <->  E. x  e.  CC  B  =  ( x  .h  A ) ) )
 
Theoremelspansn2 22071 Membership in the span of a singleton. All members are collinear with the generating vector. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  B  =/=  0h )  ->  ( A  e.  ( span `  { B }
 ) 
 <->  A  =  ( ( ( A  .ih  B )  /  ( B  .ih  B ) )  .h  B ) ) )
 
Theoremspansncol 22072 The singletons of collinear vectors have the same span. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( span `  { ( B  .h  A ) }
 )  =  ( span ` 
 { A } )
 )
 
Theoremspansneleqi 22073 Membership relation implied by equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( ( span `  { A }
 )  =  ( span ` 
 { B } )  ->  A  e.  ( span ` 
 { B } )
 ) )
 
Theoremspansneleq 22074 Membership relation that implies equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( B  e.  ~H  /\  A  =/=  0h )  ->  ( A  e.  ( span `  { B }
 )  ->  ( span ` 
 { A } )  =  ( span `  { B }
 ) ) )
 
Theoremspansnss 22075 The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  A ) 
 ->  ( span `  { B }
 )  C_  A )
 
Theoremelspansn3 22076 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  A  /\  C  e.  ( span ` 
 { B } )
 )  ->  C  e.  A )
 
Theoremelspansn4 22077 A span membership condition implying two vectors belong to the same subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  SH  /\  B  e.  ~H )  /\  ( C  e.  ( span `  { B }
 )  /\  C  =/=  0h ) )  ->  ( B  e.  A  <->  C  e.  A ) )
 
Theoremelspansn5 22078 A vector belonging to both a subspace and the span of the singleton of a vector not in it must be zero. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( ( ( B  e.  ~H 
 /\  -.  B  e.  A )  /\  ( C  e.  ( span `  { B } )  /\  C  e.  A ) )  ->  C  =  0h )
 )
 
Theoremspansnss2 22079 The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  ~H )  ->  ( B  e.  A  <->  (
 span `  { B }
 )  C_  A )
 )
 
Theoremnormcan 22080 Cancellation-type law that "extracts" a vector  A from its inner product with a proportional vector  B. (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( B  e.  ~H  /\  B  =/=  0h  /\  A  e.  ( span ` 
 { B } )
 )  ->  ( (
 ( A  .ih  B )  /  ( ( normh `  B ) ^ 2
 ) )  .h  B )  =  A )
 
Theorempjspansn 22081 A projection on the span of a singleton. (The proof ws shortened by Mario Carneiro, 15-Dec-2013.) (Contributed by NM, 28-May-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  ->  ( ( proj  h `  ( span `  { A }
 ) ) `  B )  =  ( (
 ( B  .ih  A )  /  ( ( normh `  A ) ^ 2
 ) )  .h  A ) )
 
Theoremspansnpji 22082 A subset of Hilbert space is orthogonal to the span of the singleton of a projection onto its orthocomplement. (Contributed by NM, 4-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  C_ 
 ~H   &    |-  B  e.  ~H   =>    |-  A  C_  ( _|_ `  ( span `  { (
 ( proj  h `  ( _|_ `  A ) ) `
  B ) }
 ) )
 
Theoremspanunsni 22083 The span of the union of a closed subspace with a singleton equals the span of its union with an orthogonal singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  ~H   =>    |-  ( span `  ( A  u.  { B } ) )  =  ( span `  ( A  u.  { ( (
 proj  h `  ( _|_ `  A ) ) `  B ) } )
 )
 
Theoremspanpr 22084 The span of a pair of vectors. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( span `  { ( A  +h  B ) }
 )  C_  ( span ` 
 { A ,  B } ) )
 
Theoremh1datomi 22085 A 1-dimensional subspace is an atom. (Contributed by NM, 20-Jul-2001.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  ~H   =>    |-  ( A  C_  ( _|_ `  ( _|_ `  { B }
 ) )  ->  ( A  =  ( _|_ `  ( _|_ `  { B } ) )  \/  A  =  0H )
 )
 
Theoremh1datom 22086 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  ~H )  ->  ( A  C_  ( _|_ `  ( _|_ `  { B } ) )  ->  ( A  =  ( _|_ `  ( _|_ `  { B } ) )  \/  A  =  0H )
 ) )
 
15.9.25  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 22543.

 
Definitiondf-hosum 22087* 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 22088* 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 22089* 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 22090* 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 22091* 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 22092* 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 22093* 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 22094* 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 22095* 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 22096* 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 22097 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 22098 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 22099 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 22100 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 ) ) )
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