HomeHome Metamath Proof Explorer
Theorem List (p. 225 of 325)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-22374)
  Hilbert Space Explorer  Hilbert Space Explorer
(22375-23897)
  Users' Mathboxes  Users' Mathboxes
(23898-32447)
 

Theorem List for Metamath Proof Explorer - 22401-22500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcnop 22401 Extend class notation with the operator norm function.
 class  normop
 
Syntaxccop 22402 Extend class notation with set of continuous Hilbert space operators.
 class  ConOp
 
Syntaxclo 22403 Extend class notation with set of linear Hilbert space operators.
 class  LinOp
 
Syntaxcbo 22404 Extend class notation with set of bounded linear operators.
 class  BndLinOp
 
Syntaxcuo 22405 Extend class notation with set of unitary Hilbert space operators.
 class  UniOp
 
Syntaxcho 22406 Extend class notation with set of Hermitian Hilbert space operators.
 class  HrmOp
 
Syntaxcnmf 22407 Extend class notation with the functional norm function.
 class  normfn
 
Syntaxcnl 22408 Extend class notation with the functional nullspace function.
 class  null
 
Syntaxccnfn 22409 Extend class notation with set of continuous Hilbert space functionals.
 class  ConFn
 
Syntaxclf 22410 Extend class notation with set of linear Hilbert space functionals.
 class  LinFn
 
Syntaxcado 22411 Extend class notation with Hilbert space adjoint function.
 class  adjh
 
Syntaxcbr 22412 Extend class notation with the bra of a vector in Dirac bra-ket notation.
 class  bra
 
Syntaxck 22413 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
 class  ketbra
 
Syntaxcleo 22414 Extend class notation with positive operator ordering.
 class  <_op
 
Syntaxcei 22415 Extend class notation with Hilbert space eigenvector function.
 class  eigvec
 
Syntaxcel 22416 Extend class notation with Hilbert space eigenvalue function.
 class  eigval
 
Syntaxcspc 22417 Extend class notation with the spectrum of an operator.
 class  Lambda
 
Syntaxcst 22418 Extend class notation with set of states on a Hilbert lattice.
 class  States
 
Syntaxchst 22419 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
 class  CHStates
 
Syntaxccv 22420 Extend class notation with the covers relation on a Hilbert lattice.
 class  <oH
 
Syntaxcat 22421 Extend class notation with set of atoms on a Hilbert lattice.
 class HAtoms
 
Syntaxcmd 22422 Extend class notation with the modular pair relation on a Hilbert lattice.
 class  MH
 
Syntaxcdmd 22423 Extend class notation with the dual modular pair relation on a Hilbert lattice.
 class  MH*
 
18.1.2  Preliminary ZFC lemmas
 
Definitiondf-hnorm 22424 Define the function for the norm of a vector of Hilbert space. See normval 22579 for its value and normcl 22580 for its closure. Theorems norm-i-i 22588, norm-ii-i 22592, and norm-iii-i 22594 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 22425 Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 22455). 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 22622. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
 
Definitiondf-h0v 22426 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 22623. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  0h  =  ( 0vec `  <. <.  +h  ,  .h  >. ,  normh >. )
 
Definitiondf-hvsub 22427* Define vector subtraction. See hvsubvali 22476 for its value and hvsubcli 22477 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 22428* Define the limit relation for Hilbert space. See hlimi 22643 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 22429* Define the set of Cauchy sequences on a Hilbert space. See hcau 22639 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 22430 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 22431 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  =  ( .s
 OLD `  U )
 
Theoremh2hnm 22432 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 22433 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 22434 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 22435 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 22436 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 ) )
 
18.1.3  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 22437 through axhcompl-zf 22454, 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 22424 above. See also the comment in ax-hilex 22455.

 
Theoremaxhilex-zf 22437 Derive axiom ax-hilex 22455 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |- 
 ~H  e.  _V
 
Theoremaxhfvadd-zf 22438 Derive axiom ax-hfvadd 22456 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |- 
 +h  : ( ~H 
 X.  ~H ) --> ~H
 
Theoremaxhvcom-zf 22439 Derive axiom ax-hvcom 22457 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( A  +h  B )  =  ( B  +h  A ) )
 
Theoremaxhvass-zf 22440 Derive axiom ax-hvass 22458 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) ) )
 
Theoremaxhv0cl-zf 22441 Derive axiom ax-hv0cl 22459 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |- 
 0h  e.  ~H
 
Theoremaxhvaddid-zf 22442 Derive axiom ax-hvaddid 22460 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
 
Theoremaxhfvmul-zf 22443 Derive axiom ax-hfvmul 22461 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |- 
 .h  : ( CC 
 X.  ~H ) --> ~H
 
Theoremaxhvmulid-zf 22444 Derive axiom ax-hvmulid 22462 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( A  e.  ~H  ->  ( 1  .h  A )  =  A )
 
Theoremaxhvmulass-zf 22445 Derive axiom ax-hvmulass 22463 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( ( A  x.  B )  .h  C )  =  ( A  .h  ( B  .h  C ) ) )
 
Theoremaxhvdistr1-zf 22446 Derive axiom ax-hvdistr1 22464 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  +h  C ) )  =  ( ( A  .h  B )  +h  ( A  .h  C ) ) )
 
Theoremaxhvdistr2-zf 22447 Derive axiom ax-hvdistr2 22465 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( ( A  +  B )  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C ) ) )
 
Theoremaxhvmul0-zf 22448 Derive axiom ax-hvmul0 22466 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( A  e.  ~H  ->  ( 0  .h  A )  =  0h )
 
Theoremaxhfi-zf 22449 Derive axiom ax-hfi 22534 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   &    |- 
 .ih  =  ( .i OLD `  U )   =>    |-  .ih  : ( ~H  X.  ~H ) --> CC
 
Theoremaxhis1-zf 22450 Derive axiom ax-his1 22537 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   &    |- 
 .ih  =  ( .i OLD `  U )   =>    |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B )  =  ( * `  ( B  .ih  A ) ) )
 
Theoremaxhis2-zf 22451 Derive axiom ax-his2 22538 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   &    |- 
 .ih  =  ( .i OLD `  U )   =>    |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  .ih  C )  =  ( ( A  .ih  C )  +  ( B  .ih  C ) ) )
 
Theoremaxhis3-zf 22452 Derive axiom ax-his3 22539 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   &    |- 
 .ih  =  ( .i OLD `  U )   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  .h  B )  .ih  C )  =  ( A  x.  ( B  .ih  C ) ) )
 
Theoremaxhis4-zf 22453 Derive axiom ax-his4 22540 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   &    |- 
 .ih  =  ( .i OLD `  U )   =>    |-  ( ( A  e.  ~H  /\  A  =/=  0h )  ->  0  <  ( A  .ih  A ) )
 
Theoremaxhcompl-zf 22454* Derive axiom ax-hcompl 22657 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.  CHil OLD   =>    |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
 
18.1.4  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 22455, ax-hfvadd 22456, ax-hvcom 22457, ax-hvass 22458, ax-hv0cl 22459, ax-hvaddid 22460, ax-hfvmul 22461, ax-hvmulid 22462, ax-hvmulass 22463, ax-hvdistr1 22464, ax-hvdistr2 22465, ax-hvmul0 22466, ax-hfi 22534, ax-his1 22537, ax-his2 22538, ax-his3 22539, ax-his4 22540, and ax-hcompl 22657.

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.  CHil
OLD, then these axioms can be proved as theorems axhilex-zf 22437, axhfvadd-zf 22438, axhvcom-zf 22439, axhvass-zf 22440, axhv0cl-zf 22441, axhvaddid-zf 22442, axhfvmul-zf 22443, axhvmulid-zf 22444, axhvmulass-zf 22445, axhvdistr1-zf 22446, axhvdistr2-zf 22447, axhvmul0-zf 22448, axhfi-zf 22449, axhis1-zf 22450, axhis2-zf 22451, axhis3-zf 22452, axhis4-zf 22453, and axhcompl-zf 22454 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 22437.

 
Axiomax-hilex 22455 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 22456 Vector addition is an operation on 
~H. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  +h  : ( ~H  X.  ~H )
 --> ~H
 
Axiomax-hvcom 22457 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 22458 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 22459 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  0h  e.  ~H
 
Axiomax-hvaddid 22460 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
 
Axiomax-hfvmul 22461 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 22462 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 1  .h  A )  =  A )
 
Axiomax-hvmulass 22463 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 22464 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 22465 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 22466 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 22481 and hvsubval 22472). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 0  .h  A )  =  0h )
 
18.1.5  Vector operations
 
Theoremhvmulex 22467 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
 |-  .h  e.  _V
 
Theoremhvaddcl 22468 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 22469 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 22470 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 22471 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
 |-  -h  : ( ~H  X.  ~H )
 --> ~H
 
Theoremhvsubval 22472 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 22473 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 22474 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 22475 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 22476 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 22477 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
 
Theoremhvaddid2 22478 Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 0h  +h  A )  =  A )
 
Theoremhvmul0 22479 Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  CC  ->  ( A  .h  0h )  =  0h )
 
Theoremhvmul0or 22480 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 22481 Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  -h  A )  =  0h )
 
Theoremhvnegid 22482 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 22483 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 22484 Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( 0h  +h  A )  =  A
 
Theoremhvnegidi 22485 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 22486 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 22487 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 22488 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 22489 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 22490 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 22491 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 22492 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 22493 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 22494 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 22495 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 22496 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 22497 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 22498 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 22499 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 22500 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 ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32447
  Copyright terms: Public domain < Previous  Next >