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Theorem List for Metamath Proof Explorer - 21401-21500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxhvdistr2-zf 21401 Derive axiom ax-hvdistr2 21419 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 21402 Derive axiom ax-hvmul0 21420 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 21403 Derive axiom ax-hfi 21488 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 21404 Derive axiom ax-his1 21491 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 21405 Derive axiom ax-his2 21492 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 21406 Derive axiom ax-his3 21493 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 21407 Derive axiom ax-his4 21494 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 21408* Derive axiom ax-hcompl 21611 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 21409, ax-hfvadd 21410, ax-hvcom 21411, ax-hvass 21412, ax-hv0cl 21413, ax-hvaddid 21414, ax-hfvmul 21415, ax-hvmulid 21416, ax-hvmulass 21417, ax-hvdistr1 21418, ax-hvdistr2 21419, ax-hvmul0 21420, ax-hfi 21488, ax-his1 21491, ax-his2 21492, ax-his3 21493, ax-his4 21494, and ax-hcompl 21611.

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 21391, axhfvadd-zf 21392, axhvcom-zf 21393, axhvass-zf 21394, axhv0cl-zf 21395, axhvaddid-zf 21396, axhfvmul-zf 21397, axhvmulid-zf 21398, axhvmulass-zf 21399, axhvdistr1-zf 21400, axhvdistr2-zf 21401, axhvmul0-zf 21402, axhfi-zf 21403, axhis1-zf 21404, axhis2-zf 21405, axhis3-zf 21406, axhis4-zf 21407, and axhcompl-zf 21408 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 21391.

 
Axiomax-hilex 21409 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 21410 Vector addition is an operation on 
~H. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  +h  : ( ~H  X.  ~H )
 --> ~H
 
Axiomax-hvcom 21411 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 21412 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 21413 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  0h  e.  ~H
 
Axiomax-hvaddid 21414 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
 
Axiomax-hfvmul 21415 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 21416 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 1  .h  A )  =  A )
 
Axiomax-hvmulass 21417 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 21418 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 21419 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 21420 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 21435 and hvsubval 21426). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 0  .h  A )  =  0h )
 
15.9.5  Vector operations
 
Theoremhvmulex 21421 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
 |-  .h  e.  _V
 
Theoremhvaddcl 21422 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 21423 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 21424 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 21425 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)
 |-  -h  : ( ~H  X.  ~H )
 --> ~H
 
Theoremhvsubval 21426 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 21427 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 21428 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 21429 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 21430 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 21431 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 21432 Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 0h  +h  A )  =  A )
 
Theoremhvmul0 21433 Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  CC  ->  ( A  .h  0h )  =  0h )
 
Theoremhvmul0or 21434 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 21435 Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  -h  A )  =  0h )
 
Theoremhvnegid 21436 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 21437 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 21438 Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( 0h  +h  A )  =  A
 
Theoremhvnegidi 21439 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 21440 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 21441 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 21442 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 21443 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 21444 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 21445 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 21446 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 21447 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 21448 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 21449 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 21450 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 21451 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 21452 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 21453 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 21454 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 21455 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 21456 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 21457 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 21458 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 21459 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 21460 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 21461 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 21462 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 21463 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 21464 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 21465 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 21466 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 21467 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 21468 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 21469 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 21470 Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 2  .h  A )  =  ( A  +h  A ) )
 
Theoremhvnegdii 21471 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 21472 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 21473 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 21474 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 21475 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 21476 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 21477 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 21478 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 21479 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 21480 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 21481 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 21482 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 21483 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 21484 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 ) )
 
Theoremhvsub0 21485 Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  -h  0h )  =  A )
 
Theoremhvsubadd 21486 Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  B )  =  C  <->  ( B  +h  C )  =  A ) )
 
Theoremhvaddsub4 21487 Hilbert vector space addition/subtraction law. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  ~H 
 /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( A  +h  B )  =  ( C  +h  D )  <->  ( A  -h  C )  =  ( D  -h  B ) ) )
 
15.9.6  Inner product postulates for a Hilbert space
 
Axiomax-hfi 21488 Inner product maps pairs from  ~H to  CC. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  .ih  : ( ~H  X.  ~H )
 --> CC
 
Theoremhicl 21489 Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B )  e.  CC )
 
Theoremhicli 21490 Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  .ih  B )  e. 
 CC
 
Axiomax-his1 21491 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that  * `  x is the complex conjugate cjval 11464 of  x. In the literature, the inner product of  A and  B is usually written  <. A ,  B >., but our operation notation co 5710 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 3553. Physicists use  <. B  |  A >., called Dirac bra-ket notation, to represent this operation; see comments in df-bra 22260. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B )  =  ( * `  ( B  .ih  A ) ) )
 
Axiomax-his2 21492 Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  .ih  C )  =  ( ( A 
 .ih  C )  +  ( B  .ih  C ) ) )
 
Axiomax-his3 21493 Associative law for inner product. Postulate (S3) of [Beran] p. 95. Warning: Mathematics textbooks usually use our version of the axiom. Physics textbooks, on the other hand, usually replace the left-hand side with  ( B  .ih  ( A  .h  C
) ) (e.g. Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 22260 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  .h  B )  .ih  C )  =  ( A  x.  ( B  .ih  C ) ) )
 
Axiomax-his4 21494 Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  0  <  ( A 
 .ih  A ) )
 
15.9.7  Inner product
 
Theoremhis5 21495 Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( ( * `
  A )  x.  ( B  .ih  C ) ) )
 
Theoremhis52 21496 Associative law for inner product. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  (
 ( * `  A )  .h  C ) )  =  ( A  x.  ( B  .ih  C ) ) )
 
Theoremhis35 21497 Move scalar multiplication to outside of inner product. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( A  .h  C ) 
 .ih  ( B  .h  D ) )  =  ( ( A  x.  ( * `  B ) )  x.  ( C 
 .ih  D ) ) )
 
Theoremhis35i 21498 Move scalar multiplication to outside of inner product. (Contributed by NM, 1-Jul-2005.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  ~H   &    |-  D  e.  ~H   =>    |-  (
 ( A  .h  C )  .ih  ( B  .h  D ) )  =  ( ( A  x.  ( * `  B ) )  x.  ( C 
 .ih  D ) )
 
Theoremhis7 21499 Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  ( B  +h  C ) )  =  ( ( A 
 .ih  B )  +  ( A  .ih  C ) ) )
 
Theoremhiassdi 21500 Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( ( A  .h  B )  +h  C )  .ih  D )  =  ( ( A  x.  ( B 
 .ih  D ) )  +  ( C  .ih  D ) ) )
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