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Theorem List for Metamath Proof Explorer - 26001-26100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxck 26001 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
 class  ketbra
 
Syntaxcleo 26002 Extend class notation with positive operator ordering.
 class  <_op
 
Syntaxcei 26003 Extend class notation with Hilbert space eigenvector function.
 class  eigvec
 
Syntaxcel 26004 Extend class notation with Hilbert space eigenvalue function.
 class  eigval
 
Syntaxcspc 26005 Extend class notation with the spectrum of an operator.
 class  Lambda
 
Syntaxcst 26006 Extend class notation with set of states on a Hilbert lattice.
 class  States
 
Syntaxchst 26007 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
 class  CHStates
 
Syntaxccv 26008 Extend class notation with the covers relation on a Hilbert lattice.
 class  <oH
 
Syntaxcat 26009 Extend class notation with set of atoms on a Hilbert lattice.
 class HAtoms
 
Syntaxcmd 26010 Extend class notation with the modular pair relation on a Hilbert lattice.
 class  MH
 
Syntaxcdmd 26011 Extend class notation with the dual modular pair relation on a Hilbert lattice.
 class  MH*
 
20.1.2  Preliminary ZFC lemmas
 
Definitiondf-hnorm 26012 Define the function for the norm of a vector of Hilbert space. See normval 26168 for its value and normcl 26169 for its closure. Theorems norm-i-i 26177, norm-ii-i 26181, and norm-iii-i 26183 show it has the expected properties of a norm. In the literature, the norm of  A is usually written "||  A ||", but we use function notation to take advantage of our existing theorems about functions. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  normh  =  ( x  e.  dom  dom  .ih  |->  ( sqr `  ( x  .ih  x ) ) )
 
Definitiondf-hba 26013 Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 26043). Note that  ~H is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. This definition can be proved independently from those axioms as theorem hhba 26211. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
 
Definitiondf-h0v 26014 Define the zero vector of Hilbert space. Note that  0vec is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. It is proved from the axioms as theorem hh0v 26212. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  0h  =  ( 0vec `  <. <.  +h  ,  .h  >. ,  normh >. )
 
Definitiondf-hvsub 26015* Define vector subtraction. See hvsubvali 26064 for its value and hvsubcli 26065 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  -h  =  ( x  e.  ~H ,  y  e.  ~H  |->  ( x  +h  ( -u 1  .h  y ) ) )
 
Definitiondf-hlim 26016* Define the limit relation for Hilbert space. See hlimi 26232 for its relational expression. Note that  f : NN --> ~H is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of converge in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  ~~>v  =  { <. f ,  w >.  |  ( ( f : NN --> ~H  /\  w  e. 
 ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
 ( normh `  ( (
 f `  z )  -h  w ) )  < 
 x ) }
 
Definitiondf-hcau 26017* Define the set of Cauchy sequences on a Hilbert space. See hcau 26228 for its membership relation. Note that  f : NN --> ~H is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of Cauchy sequence in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  Cauchy  =  {
 f  e.  ( ~H 
 ^m  NN )  |  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
 ( normh `  ( (
 f `  y )  -h  ( f `  z
 ) ) )  < 
 x }
 
Theoremh2hva 26018 The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   =>    |- 
 +h  =  ( +v
 `  U )
 
Theoremh2hsm 26019 The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   =>    |- 
 .h  =  ( .sOLD `  U )
 
Theoremh2hnm 26020 The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   =>    |- 
 normh  =  ( normCV `  U )
 
Theoremh2hvs 26021 The vector subtraction operation of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   &    |- 
 ~H  =  ( BaseSet `  U )   =>    |- 
 -h  =  ( -v
 `  U )
 
Theoremh2hmetdval 26022 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   &    |- 
 ~H  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A D B )  =  ( normh `  ( A  -h  B ) ) )
 
Theoremh2hcau 26023 The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   &    |- 
 ~H  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  Cauchy  =  ( ( Cau `  D )  i^i  ( ~H  ^m  NN ) )
 
Theoremh2hlm 26024 The limit sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   &    |- 
 ~H  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   =>    |-  ~~>v  =  ( ( ~~> t `  J )  |`  ( ~H  ^m  NN ) )
 
20.1.3  Derive the Hilbert space axioms from ZFC set theory

Before introducing the 18 axioms for Hilbert space, we first prove them as the conclusions of theorems axhilex-zf 26025 through axhcompl-zf 26042, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space  U  =  <. <.  +h  ,  .h  >. ,  normh >. that satisfies their hypotheses, then we can derive the Hilbert space axioms as theorems of ZFC set theory. In practice, in order to use these theorems to convert the Hilbert Space explorer to a ZFC-only subtheory, we would also have to provide definitions for the 3 (otherwise primitive) class constants  +h,  .h, and  .ih before df-hnorm 26012 above. See also the comment in ax-hilex 26043.

 
Theoremaxhilex-zf 26025 Derive axiom ax-hilex 26043 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  ~H  e.  _V
 
Theoremaxhfvadd-zf 26026 Derive axiom ax-hfvadd 26044 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  +h  : ( ~H  X.  ~H ) --> ~H
 
Theoremaxhvcom-zf 26027 Derive axiom ax-hvcom 26045 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B )  =  ( B  +h  A ) )
 
Theoremaxhvass-zf 26028 Derive axiom ax-hvass 26046 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) ) )
 
Theoremaxhv0cl-zf 26029 Derive axiom ax-hv0cl 26047 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  0h  e.  ~H
 
Theoremaxhvaddid-zf 26030 Derive axiom ax-hvaddid 26048 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  ( A  e.  ~H 
 ->  ( A  +h  0h )  =  A )
 
Theoremaxhfvmul-zf 26031 Derive axiom ax-hfvmul 26049 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  .h  : ( CC  X.  ~H ) --> ~H
 
Theoremaxhvmulid-zf 26032 Derive axiom ax-hvmulid 26050 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  ( A  e.  ~H 
 ->  ( 1  .h  A )  =  A )
 
Theoremaxhvmulass-zf 26033 Derive axiom ax-hvmulass 26051 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( ( A  x.  B )  .h  C )  =  ( A  .h  ( B  .h  C ) ) )
 
Theoremaxhvdistr1-zf 26034 Derive axiom ax-hvdistr1 26052 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  +h  C ) )  =  (
 ( A  .h  B )  +h  ( A  .h  C ) ) )
 
Theoremaxhvdistr2-zf 26035 Derive axiom ax-hvdistr2 26053 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( ( A  +  B )  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C ) ) )
 
Theoremaxhvmul0-zf 26036 Derive axiom ax-hvmul0 26054 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  ( A  e.  ~H 
 ->  ( 0  .h  A )  =  0h )
 
Theoremaxhfi-zf 26037 Derive axiom ax-hfi 26123 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   &    |-  .ih  =  ( .iOLD `  U )   =>    |- 
 .ih  : ( ~H  X.  ~H ) --> CC
 
Theoremaxhis1-zf 26038 Derive axiom ax-his1 26126 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   &    |-  .ih  =  ( .iOLD `  U )   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( A  .ih  B )  =  ( * `
  ( B  .ih  A ) ) )
 
Theoremaxhis2-zf 26039 Derive axiom ax-his2 26127 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   &    |-  .ih  =  ( .iOLD `  U )   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  .ih  C )  =  ( ( A 
 .ih  C )  +  ( B  .ih  C ) ) )
 
Theoremaxhis3-zf 26040 Derive axiom ax-his3 26128 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   &    |-  .ih  =  ( .iOLD `  U )   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  .h  B )  .ih  C )  =  ( A  x.  ( B  .ih  C ) ) )
 
Theoremaxhis4-zf 26041 Derive axiom ax-his4 26129 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   &    |-  .ih  =  ( .iOLD `  U )   =>    |-  ( ( A  e.  ~H 
 /\  A  =/=  0h )  ->  0  <  ( A  .ih  A ) )
 
Theoremaxhcompl-zf 26042* Derive axiom ax-hcompl 26246 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHilOLD   =>    |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
 
20.1.4  Introduce the vector space axioms for a Hilbert space

Here we introduce the axioms a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. The 18 axioms for a complex Hilbert space consist of ax-hilex 26043, ax-hfvadd 26044, ax-hvcom 26045, ax-hvass 26046, ax-hv0cl 26047, ax-hvaddid 26048, ax-hfvmul 26049, ax-hvmulid 26050, ax-hvmulass 26051, ax-hvdistr1 26052, ax-hvdistr2 26053, ax-hvmul0 26054, ax-hfi 26123, ax-his1 26126, ax-his2 26127, ax-his3 26128, ax-his4 26129, and ax-hcompl 26246.

The axioms specify the properties of 5 primitive symbols,  ~H,  +h,  .h,  0h, and  .ih.

If we can prove in ZFC set theory that a class  U  =  <. <.  +h  ,  .h  >. ,  normh >. is a complex Hilbert space, i.e. that  U  e.  CHilOLD, then these axioms can be proved as theorems axhilex-zf 26025, axhfvadd-zf 26026, axhvcom-zf 26027, axhvass-zf 26028, axhv0cl-zf 26029, axhvaddid-zf 26030, axhfvmul-zf 26031, axhvmulid-zf 26032, axhvmulass-zf 26033, axhvdistr1-zf 26034, axhvdistr2-zf 26035, axhvmul0-zf 26036, axhfi-zf 26037, axhis1-zf 26038, axhis2-zf 26039, axhis3-zf 26040, axhis4-zf 26041, and axhcompl-zf 26042 respectively. In that case, the theorems of the Hilbert Space Explorer will become theorems of ZFC set theory. See also the comments in axhilex-zf 26025.

 
Axiomax-hilex 26043 This is our first axiom for a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. We assume that there exists a primitive class,  ~H, which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  ~H  e.  _V
 
Axiomax-hfvadd 26044 Vector addition is an operation on 
~H. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  +h  : ( ~H  X.  ~H )
 --> ~H
 
Axiomax-hvcom 26045 Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B )  =  ( B  +h  A ) )
 
Axiomax-hvass 26046 Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) ) )
 
Axiomax-hv0cl 26047 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  0h  e.  ~H
 
Axiomax-hvaddid 26048 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
 
Axiomax-hfvmul 26049 Scalar multiplication is an operation on  CC and  ~H. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  .h  : ( CC  X.  ~H ) --> ~H
 
Axiomax-hvmulid 26050 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 1  .h  A )  =  A )
 
Axiomax-hvmulass 26051 Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( ( A  x.  B )  .h  C )  =  ( A  .h  ( B  .h  C ) ) )
 
Axiomax-hvdistr1 26052 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  +h  C ) )  =  ( ( A  .h  B )  +h  ( A  .h  C ) ) )
 
Axiomax-hvdistr2 26053 Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( ( A  +  B )  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C ) ) )
 
Axiomax-hvmul0 26054 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 26070 and hvsubval 26060). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 0  .h  A )  =  0h )
 
20.1.5  Vector operations
 
Theoremhvmulex 26055 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
 |-  .h  e.  _V
 
Theoremhvaddcl 26056 Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B )  e.  ~H )
 
Theoremhvmulcl 26057 Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B )  e.  ~H )
 
Theoremhvmulcli 26058 Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  CC   &    |-  B  e.  ~H   =>    |-  ( A  .h  B )  e. 
 ~H
 
Theoremhvsubf 26059 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
 |-  -h  : ( ~H  X.  ~H )
 --> ~H
 
Theoremhvsubval 26060 Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) ) )
 
Theoremhvsubcl 26061 Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B )  e.  ~H )
 
Theoremhvaddcli 26062 Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  +h  B )  e. 
 ~H
 
Theoremhvcomi 26063 Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  +h  B )  =  ( B  +h  A )
 
Theoremhvsubvali 26064 Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
 
Theoremhvsubcli 26065 Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  -h  B )  e. 
 ~H
 
Theoremifhvhv0 26066 Prove  if ( A  e.  ~H ,  A ,  0h )  e.  ~H (common case). (Contributed by David A. Wheeler, 7-Dec-2018.) (New usage is discouraged.)
 |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
 
Theoremhvaddid2 26067 Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 0h  +h  A )  =  A )
 
Theoremhvmul0 26068 Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  CC  ->  ( A  .h  0h )  =  0h )
 
Theoremhvmul0or 26069 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  .h  B )  =  0h  <->  ( A  =  0  \/  B  =  0h )
 ) )
 
Theoremhvsubid 26070 Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  -h  A )  =  0h )
 
Theoremhvnegid 26071 Addition of negative of a vector to itself. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  +h  ( -u 1  .h  A ) )  =  0h )
 
Theoremhv2neg 26072 Two ways to express the negative of a vector. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 0h  -h  A )  =  ( -u 1  .h  A ) )
 
Theoremhvaddid2i 26073 Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( 0h  +h  A )  =  A
 
Theoremhvnegidi 26074 Addition of negative of a vector to itself. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( A  +h  ( -u 1  .h  A ) )  =  0h
 
Theoremhv2negi 26075 Two ways to express the negative of a vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( 0h  -h  A )  =  ( -u 1  .h  A )
 
Theoremhvm1neg 26076 Convert minus one times a scalar product to the negative of the scalar. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H )  ->  ( -u 1  .h  ( A  .h  B ) )  =  ( -u A  .h  B ) )
 
Theoremhvaddsubval 26077 Value of vector addition in terms of vector subtraction. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B )  =  ( A  -h  ( -u 1  .h  B ) ) )
 
Theoremhvadd32 26078 Commutative/associative law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  +h  C )  =  ( ( A  +h  C )  +h  B ) )
 
Theoremhvadd12 26079 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  +h  C ) )  =  ( B  +h  ( A  +h  C ) ) )
 
Theoremhvadd4 26080 Hilbert vector space addition law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  ~H 
 /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( A  +h  B )  +h  ( C  +h  D ) )  =  ( ( A  +h  C )  +h  ( B  +h  D ) ) )
 
Theoremhvsub4 26081 Hilbert vector space addition/subtraction law. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  ~H 
 /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( A  +h  B )  -h  ( C  +h  D ) )  =  ( ( A  -h  C )  +h  ( B  -h  D ) ) )
 
Theoremhvaddsub12 26082 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  -h  C ) )  =  ( B  +h  ( A  -h  C ) ) )
 
Theoremhvpncan 26083 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  -h  B )  =  A )
 
Theoremhvpncan2 26084 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  -h  A )  =  B )
 
Theoremhvaddsubass 26085 Associativity of sum and difference of Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  -h  C )  =  ( A  +h  ( B  -h  C ) ) )
 
Theoremhvpncan3 26086 Subtraction and addition of equal Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  ( B  -h  A ) )  =  B )
 
Theoremhvmulcom 26087 Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  ( B  .h  C ) )  =  ( B  .h  ( A  .h  C ) ) )
 
Theoremhvsubass 26088 Hilbert vector space associative law for subtraction. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  B )  -h  C )  =  ( A  -h  ( B  +h  C ) ) )
 
Theoremhvsub32 26089 Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  B )  -h  C )  =  ( ( A  -h  C )  -h  B ) )
 
Theoremhvmulassi 26090 Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  ~H   =>    |-  ( ( A  x.  B )  .h  C )  =  ( A  .h  ( B  .h  C ) )
 
Theoremhvmulcomi 26091 Scalar multiplication commutative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  ~H   =>    |-  ( A  .h  ( B  .h  C ) )  =  ( B  .h  ( A  .h  C ) )
 
Theoremhvmul2negi 26092 Double negative in scalar multiplication. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  ~H   =>    |-  ( -u A  .h  ( -u B  .h  C ) )  =  ( A  .h  ( B  .h  C ) )
 
Theoremhvsubdistr1 26093 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  -h  C ) )  =  ( ( A  .h  B )  -h  ( A  .h  C ) ) )
 
Theoremhvsubdistr2 26094 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( ( A  -  B )  .h  C )  =  ( ( A  .h  C )  -h  ( B  .h  C ) ) )
 
Theoremhvdistr1i 26095 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  CC   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( A  .h  ( B  +h  C ) )  =  ( ( A  .h  B )  +h  ( A  .h  C ) )
 
Theoremhvsubdistr1i 26096 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  CC   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( A  .h  ( B  -h  C ) )  =  ( ( A  .h  B )  -h  ( A  .h  C ) )
 
Theoremhvassi 26097 Hilbert vector space associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) )
 
Theoremhvadd32i 26098 Hilbert vector space commutative/associative law. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( ( A  +h  B )  +h  C )  =  ( ( A  +h  C )  +h  B )
 
Theoremhvsubassi 26099 Hilbert vector space associative law for subtraction. (Contributed by NM, 7-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( ( A  -h  B )  -h  C )  =  ( A  -h  ( B  +h  C ) )
 
Theoremhvsub32i 26100 Hilbert vector space commutative/associative law. (Contributed by NM, 7-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( ( A  -h  B )  -h  C )  =  ( ( A  -h  C )  -h  B )
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