HomeHome Metamath Proof Explorer
Theorem List (p. 216 of 314)
< 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
(1-21444)
  Hilbert Space Explorer  Hilbert Space Explorer
(21445-22967)
  Users' Mathboxes  Users' Mathboxes
(22968-31305)
 

Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremh2hsm 21501 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 21502 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 21503 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 21504 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 21505 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 21506 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 ) )
 
15.9.3  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 21507 through axhcompl-zf 21524, 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 21494 above. See also the comment in ax-hilex 21525.

 
Theoremaxhilex-zf 21507 Derive axiom ax-hilex 21525 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 21508 Derive axiom ax-hfvadd 21526 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 21509 Derive axiom ax-hvcom 21527 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 21510 Derive axiom ax-hvass 21528 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 21511 Derive axiom ax-hv0cl 21529 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 21512 Derive axiom ax-hvaddid 21530 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 21513 Derive axiom ax-hfvmul 21531 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 21514 Derive axiom ax-hvmulid 21532 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 21515 Derive axiom ax-hvmulass 21533 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 21516 Derive axiom ax-hvdistr1 21534 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 21517 Derive axiom ax-hvdistr2 21535 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 21518 Derive axiom ax-hvmul0 21536 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 21519 Derive axiom ax-hfi 21604 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 21520 Derive axiom ax-his1 21607 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 21521 Derive axiom ax-his2 21608 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 21522 Derive axiom ax-his3 21609 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 21523 Derive axiom ax-his4 21610 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 21524* Derive axiom ax-hcompl 21727 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 )
 
15.9.4  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 21525, ax-hfvadd 21526, ax-hvcom 21527, ax-hvass 21528, ax-hv0cl 21529, ax-hvaddid 21530, ax-hfvmul 21531, ax-hvmulid 21532, ax-hvmulass 21533, ax-hvdistr1 21534, ax-hvdistr2 21535, ax-hvmul0 21536, ax-hfi 21604, ax-his1 21607, ax-his2 21608, ax-his3 21609, ax-his4 21610, and ax-hcompl 21727.

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 21507, axhfvadd-zf 21508, axhvcom-zf 21509, axhvass-zf 21510, axhv0cl-zf 21511, axhvaddid-zf 21512, axhfvmul-zf 21513, axhvmulid-zf 21514, axhvmulass-zf 21515, axhvdistr1-zf 21516, axhvdistr2-zf 21517, axhvmul0-zf 21518, axhfi-zf 21519, axhis1-zf 21520, axhis2-zf 21521, axhis3-zf 21522, axhis4-zf 21523, and axhcompl-zf 21524 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 21507.

 
Axiomax-hilex 21525 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 21526 Vector addition is an operation on 
~H. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  +h  : ( ~H  X.  ~H )
 --> ~H
 
Axiomax-hvcom 21527 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 21528 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 21529 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  0h  e.  ~H
 
Axiomax-hvaddid 21530 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
 
Axiomax-hfvmul 21531 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 21532 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 1  .h  A )  =  A )
 
Axiomax-hvmulass 21533 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 21534 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 21535 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 21536 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 21551 and hvsubval 21542). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 0  .h  A )  =  0h )
 
15.9.5  Vector operations
 
Theoremhvmulex 21537 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
 |-  .h  e.  _V
 
Theoremhvaddcl 21538 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 21539 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 21540 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 21541 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
 |-  -h  : ( ~H  X.  ~H )
 --> ~H
 
Theoremhvsubval 21542 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 21543 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 21544 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 21545 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 21546 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 21547 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 21548 Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 0h  +h  A )  =  A )
 
Theoremhvmul0 21549 Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  CC  ->  ( A  .h  0h )  =  0h )
 
Theoremhvmul0or 21550 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 21551 Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  -h  A )  =  0h )
 
Theoremhvnegid 21552 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 21553 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 21554 Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( 0h  +h  A )  =  A
 
Theoremhvnegidi 21555 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 21556 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 21557 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 21558 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 21559 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 21560 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 21561 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 21562 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 21563 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 21564 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 21565 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 21566 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 21567 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 21568 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 21569 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 21570 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 21571 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 21572 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 21573 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 21574 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 21575 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 21576 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 21577 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 21578 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 21579 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 21580 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 21581 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 )
 
Theoremhvadd12i 21582 Hilbert vector space commutative/associative law. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( A  +h  ( B  +h  C ) )  =  ( B  +h  ( A  +h  C ) )
 
Theoremhvadd4i 21583 Hilbert vector space addition law. (Contributed by NM, 3-Sep-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 ) )
 
Theoremhvsubsub4i 21584 Hilbert vector space addition law. (Contributed by NM, 31-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 ) )
 
Theoremhvsubsub4 21585 Hilbert vector space addition/subtraction law. (Contributed by NM, 2-Apr-2000.) (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 ) ) )
 
Theoremhv2times 21586 Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 2  .h  A )  =  ( A  +h  A ) )
 
Theoremhvnegdii 21587 Distribution of negative over subtraction. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( -u 1  .h  ( A  -h  B ) )  =  ( B  -h  A )
 
Theoremhvsubeq0i 21588 If the difference between two vectors is zero, they are equal. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( A  -h  B )  =  0h  <->  A  =  B )
 
Theoremhvsubcan2i 21589 Vector cancellation law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( A  +h  B )  +h  ( A  -h  B ) )  =  ( 2  .h  A )
 
Theoremhvaddcani 21590 Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( ( A  +h  B )  =  ( A  +h  C )  <->  B  =  C )
 
Theoremhvsubaddi 21591 Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( ( A  -h  B )  =  C  <->  ( B  +h  C )  =  A )
 
Theoremhvnegdi 21592 Distribution of negative over subtraction. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( -u 1  .h  ( A  -h  B ) )  =  ( B  -h  A ) )
 
Theoremhvsubeq0 21593 If the difference between two vectors is zero, they are equal. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  -h  B )  =  0h  <->  A  =  B ) )
 
Theoremhvaddeq0 21594 If the sum of two vectors is zero, one is the negative of the other. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  =  0h  <->  A  =  ( -u 1  .h  B ) ) )
 
Theoremhvaddcan 21595 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  =  ( A  +h  C )  <->  B  =  C ) )
 
Theoremhvaddcan2 21596 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  C )  =  ( B  +h  C )  <->  A  =  B ) )
 
Theoremhvmulcan 21597 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CC  /\  A  =/=  0
 )  /\  B  e.  ~H 
 /\  C  e.  ~H )  ->  ( ( A  .h  B )  =  ( A  .h  C ) 
 <->  B  =  C ) )
 
Theoremhvmulcan2 21598 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  ~H  /\  C  =/=  0h )
 )  ->  ( ( A  .h  C )  =  ( B  .h  C ) 
 <->  A  =  B ) )
 
Theoremhvsubcan 21599 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  B )  =  ( A  -h  C )  <->  B  =  C ) )
 
Theoremhvsubcan2 21600 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  C )  =  ( B  -h  C )  <->  A  =  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 31301-31305
  Copyright terms: Public domain < Previous  Next >