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Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-sfld 21001* Define the class of all star fields, which are all division rings with involutions. (Contributed by NM, 29-Aug-2010.) (New usage is discouraged.)
 |-  *-Fld  =  { <. r ,  n >.  |  (
 r  e.  DivRingOps  /\  n : ran  ( 1st `  r
 ) --> ran  ( 1st `  r )  /\  A. x  e.  dom  n A. y  e.  dom  n ( ( n `  ( x ( 1st `  r
 ) y ) )  =  ( ( n `
  x ) ( 1st `  r )
 ( n `  y
 ) )  /\  ( n `  ( x ( 2nd `  r )
 y ) )  =  ( ( n `  y ) ( 2nd `  r ) ( n `
  x ) ) 
 /\  ( n `  ( n `  x ) )  =  x ) ) }
 
15.2.5  Fields and Rings
 
Syntaxccm2 21002 Extend class notation with a class that adds commutativity to various flavors of rings.
 class  Com2
 
Definitiondf-com2 21003* A device to add commutativity to various sorts of rings. I use  ran  g because I suppose  g has a neutral element and therefore is onto. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
 |- 
 Com2  =  { <. g ,  h >.  |  A. a  e.  ran  g A. b  e.  ran  g ( a h b )  =  ( b h a ) }
 
Theoremiscom2 21004* A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
 |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( <. G ,  H >.  e.  Com2  <->  A. a  e.  ran  G
 A. b  e.  ran  G ( a H b )  =  ( b H a ) ) )
 
Syntaxcfld 21005 Extend class notation with the class of all fields.
 class  Fld
 
Definitiondf-fld 21006 Definition of a field. A field is a commutative division ring. (Contributed by FL, 6-Sep-2009.) (Revised by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.)
 |- 
 Fld  =  ( DivRingOps  i^i  Com2 )
 
Theoremflddivrng 21007 A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  ( K  e.  Fld  ->  K  e.  DivRingOps )
 
Theoremrngon0 21008 The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  X  =/= 
 (/) )
 
Theoremrngmgmbs4 21009* The range of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  ( ( G :
 ( X  X.  X )
 --> X  /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  ->  ran  G  =  X )
 
Theoremrngodm1dm2 21010 In a unital ring the domain of the first variable of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
 |-  H  =  ( 2nd `  R )   &    |-  G  =  ( 1st `  R )   =>    |-  ( R  e.  RingOps  ->  dom  dom  G  =  dom  dom  H )
 
Theoremrngorn1 21011 In a unital ring the range of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
 |-  H  =  ( 2nd `  R )   &    |-  G  =  ( 1st `  R )   =>    |-  ( R  e.  RingOps  ->  ran  G  =  dom  dom  H )
 
Theoremrngorn1eq 21012 In a unital ring the range of the addition equals the range of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
 |-  H  =  ( 2nd `  R )   &    |-  G  =  ( 1st `  R )   =>    |-  ( R  e.  RingOps  ->  ran  G  =  ran  H )
 
Theoremrngomndo 21013 In a unital ring the multiplication is a monoid. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( 2nd `  R )   =>    |-  ( R  e.  RingOps  ->  H  e. MndOp )
 
Theoremrngoablo2 21014 In a unital ring the addition is an abelian group. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
 |-  ( <. G ,  H >.  e.  RingOps  ->  G  e.  AbelOp )
 
Theoremrngoidmlem 21015 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
 |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  ( 1st `  R )   &    |-  U  =  (GId `  H )   =>    |-  (
 ( R  e.  RingOps  /\  A  e.  X )  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A )
 )
 
Theoremrngolidm 21016 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
 |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  ( 1st `  R )   &    |-  U  =  (GId `  H )   =>    |-  (
 ( R  e.  RingOps  /\  A  e.  X )  ->  ( U H A )  =  A )
 
Theoremrngoridm 21017 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
 |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  ( 1st `  R )   &    |-  U  =  (GId `  H )   =>    |-  (
 ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
 
Theoremrngosn3 21018 The only unital ring with a base set consisting in one element is the zero ring. (Contributed by FL, 13-Feb-2010.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  A  e.  B ) 
 ->  ( X  =  { A }  <->  R  =  <. {
 <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >. ) )
 
Theoremrngosn4 21019 The only unital ring with one element is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( X  ~~  1o  <->  R  =  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
 >. ) )
 
Theoremrngosn6 21020 The only unital ring with one element is the zero ring. (Contributed by FL, 15-Feb-2010.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  RingOps  ->  ( X  ~~  1o  <->  R  =  <. {
 <. <. Z ,  Z >. ,  Z >. } ,  { <. <. Z ,  Z >. ,  Z >. } >. ) )
 
Theoremrngo1cl 21021 The unit of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.)
 |-  X  =  ran  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  RingOps  ->  U  e.  X )
 
Theoremrngoueqz 21022 In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  ( X  ~~  1o  <->  U  =  Z ) )
 
Theoremisdivrngo 21023 The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
 |-  ( H  e.  A  ->  ( <. G ,  H >.  e.  DivRingOps 
 <->  ( <. G ,  H >.  e.  RingOps  /\  ( H  |`  ( ( ran  G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId `  G ) } )
 ) )  e.  GrpOp ) ) )
 
Theoremzrdivrng 21024 The zero ring is not a division ring. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  -.  <. { <. <. A ,  A >. ,  A >. } ,  { <.
 <. A ,  A >. ,  A >. } >.  e.  DivRingOps
 
Theoremdvrunz 21025 In a division ring the unit is different from the zero. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  DivRingOps  ->  U  =/=  Z )
 
15.3  Complex vector spaces
 
15.3.1  Definition and basic properties
 
Syntaxcvc 21026 Extend class notation with the class of all complex vector spaces.
 class  CVec OLD
 
Definitiondf-vc 21027* Define the class of all complex vector spaces. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
 |- 
 CVec OLD  =  { <. g ,  s >.  |  ( g  e.  AbelOp  /\  s : ( CC  X.  ran  g ) --> ran  g  /\  A. x  e.  ran  g ( ( 1 s x )  =  x  /\  A. y  e.  CC  ( A. z  e.  ran  g ( y s ( x g z ) )  =  ( ( y s x ) g ( y s z ) )  /\  A. z  e.  CC  ( ( ( y  +  z ) s x )  =  ( ( y s x ) g ( z s x ) )  /\  ( ( y  x.  z ) s x )  =  ( y s ( z s x ) ) ) ) ) ) }
 
Theoremvcrel 21028 The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CVec OLD
 
Theoremvci 21029* The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. The variable  W was chosen because  _V is already used for the universal class. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( W  e.  CVec OLD 
 ->  ( G  e.  AbelOp  /\  S : ( CC 
 X.  X ) --> X  /\  A. x  e.  X  ( ( 1 S x )  =  x  /\  A. y  e.  CC  ( A. z  e.  X  ( y S ( x G z ) )  =  ( ( y S x ) G ( y S z ) )  /\  A. z  e.  CC  (
 ( ( y  +  z ) S x )  =  ( ( y S x ) G ( z S x ) )  /\  ( ( y  x.  z ) S x )  =  ( y S ( z S x ) ) ) ) ) ) )
 
Theoremvcsm 21030 Functionality of th scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( W  e.  CVec OLD 
 ->  S : ( CC 
 X.  X ) --> X )
 
Theoremvccl 21031 Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  CC  /\  B  e.  X ) 
 ->  ( A S B )  e.  X )
 
Theoremvcid 21032 Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
 
Theoremvcdi 21033 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
 
Theoremvcdir 21034 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
 
Theoremvcass 21035 Associative law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  x.  B ) S C )  =  ( A S ( B S C ) ) )
 
Theoremvc2 21036 A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G A )  =  (
 2 S A ) )
 
Theoremvcsubdir 21037 Subtractive distributive law for the scalar product of a complex vector space. (Contributed by NM, 31-Jul-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  -  B ) S C )  =  ( ( A S C ) G ( -u 1 S ( B S C ) ) ) )
 
Theoremvcablo 21038 Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   =>    |-  ( W  e.  CVec OLD 
 ->  G  e.  AbelOp )
 
Theoremvcgrp 21039 Vector addition is a group operation. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   =>    |-  ( W  e.  CVec OLD 
 ->  G  e.  GrpOp )
 
Theoremvcgcl 21040 Closure law for the vector addition (group) operation of a complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A G B )  e.  X )
 
Theoremvccom 21041 Vector addition is commutative. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A G B )  =  ( B G A ) )
 
Theoremvcaass 21042 Vector addition is associative. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Theoremvca32 21043 Commutative/associative law that swaps the last two terms in a triple vector sum. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )
 
Theoremvca4 21044 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( ( A G B ) G ( C G D ) )  =  ( ( A G C ) G ( B G D ) ) )
 
Theoremvcrcan 21045 Right cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G C )  =  ( B G C ) 
 <->  A  =  B ) )
 
Theoremvclcan 21046 Left cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( C G A )  =  ( C G B ) 
 <->  A  =  B ) )
 
Theoremvczcl 21047 The zero vector is a vector. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( W  e.  CVec OLD 
 ->  Z  e.  X )
 
Theoremvc0rid 21048 The zero vector is a right identity element. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G Z )  =  A )
 
Theoremvc0lid 21049 The zero vector is a left identity element. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( Z G A )  =  A )
 
Theoremvc0 21050 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  =  Z )
 
Theoremvcz 21051 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
 
Theoremvcm 21052 Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 25-Nov-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   &    |-  M  =  ( inv `  G )   =>    |-  (
 ( W  e.  CVec OLD  /\  A  e.  X ) 
 ->  ( -u 1 S A )  =  ( M `  A ) )
 
Theoremvcrinv 21053 A vector minus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G ( -u 1 S A ) )  =  Z )
 
Theoremvclinv 21054 Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u 1 S A ) G A )  =  Z )
 
Theoremvcnegneg 21055 Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( -u 1 S ( -u 1 S A ) )  =  A )
 
Theoremvcnegsubdi2 21056 Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( -u 1 S ( A G ( -u 1 S B ) ) )  =  ( B G ( -u 1 S A ) ) )
 
Theoremvcsub4 21057 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( ( W  e.  CVec OLD  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( ( A G B ) G ( -u 1 S ( C G D ) ) )  =  ( ( A G ( -u 1 S C ) ) G ( B G (
 -u 1 S D ) ) ) )
 
Theoremisvclem 21058* Lemma for isvc 21062. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  _V  /\  S  e.  _V )  ->  ( <. G ,  S >.  e.  CVec OLD  <->  ( G  e.  AbelOp  /\  S : ( CC 
 X.  X ) --> X  /\  A. x  e.  X  ( ( 1 S x )  =  x  /\  A. y  e.  CC  ( A. z  e.  X  ( y S ( x G z ) )  =  ( ( y S x ) G ( y S z ) )  /\  A. z  e.  CC  (
 ( ( y  +  z ) S x )  =  ( ( y S x ) G ( z S x ) )  /\  ( ( y  x.  z ) S x )  =  ( y S ( z S x ) ) ) ) ) ) ) )
 
Theoremvcoprnelem 21059 Lemma for vcoprne 21060. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ( <. G ,  G >.  e.  CVec OLD  ->  G :
 ( CC  X.  CC )
 --> CC )
 
Theoremvcoprne 21060 The operations of a complex vector space cannot be identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ( <. G ,  S >.  e.  CVec OLD  ->  G  =/=  S )
 
Theoremvcex 21061 The components of a complex vector space are sets. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ( <. G ,  S >.  e.  CVec OLD  ->  ( G  e.  _V  /\  S  e.  _V ) )
 
Theoremisvc 21062* The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( <. G ,  S >.  e. 
 CVec OLD  <->  ( G  e.  AbelOp  /\  S : ( CC 
 X.  X ) --> X  /\  A. x  e.  X  ( ( 1 S x )  =  x  /\  A. y  e.  CC  ( A. z  e.  X  ( y S ( x G z ) )  =  ( ( y S x ) G ( y S z ) )  /\  A. z  e.  CC  (
 ( ( y  +  z ) S x )  =  ( ( y S x ) G ( z S x ) )  /\  ( ( y  x.  z ) S x )  =  ( y S ( z S x ) ) ) ) ) ) )
 
Theoremisvci 21063* Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
 |-  G  e.  AbelOp   &    |- 
 dom  G  =  ( X  X.  X )   &    |-  S : ( CC  X.  X ) --> X   &    |-  ( x  e.  X  ->  ( 1 S x )  =  x )   &    |-  (
 ( y  e.  CC  /\  x  e.  X  /\  z  e.  X )  ->  ( y S ( x G z ) )  =  ( ( y S x ) G ( y S z ) ) )   &    |-  ( ( y  e. 
 CC  /\  z  e.  CC  /\  x  e.  X )  ->  ( ( y  +  z ) S x )  =  ( ( y S x ) G ( z S x ) ) )   &    |-  ( ( y  e.  CC  /\  z  e.  CC  /\  x  e.  X )  ->  (
 ( y  x.  z
 ) S x )  =  ( y S ( z S x ) ) )   &    |-  W  =  <. G ,  S >.   =>    |-  W  e.  CVec OLD
 
15.3.2  Examples of complex vector spaces
 
Theoremcncvc 21064 The set of complex numbers is a complex vector space. The vector operation is  +, and the scalar product is  x.. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
 |- 
 <.  +  ,  x.  >.  e. 
 CVec OLD
 
15.4  Normed complex vector spaces
 
15.4.1  Definition and basic properties
 
Syntaxcnv 21065 Extend class notation with the class of all normed complex vector spaces.
 class  NrmCVec
 
Syntaxcpv 21066 Extend class notation with vector addition in a normed complex vector space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition  + caddc 8673.
 class  +v
 
Syntaxcba 21067 Extend class notation with the base set of a normed complex vector space. (Note that  BaseSet is capitalized because, once it is fixed for a particular vector space  U, it is not a function, unlike e.g.  normCV. This is our typical convention.) (New usage is discouraged.)
 class  BaseSet
 
Syntaxcns 21068 Extend class notation with scalar multiplication in a normed complex vector space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.
 class  .s OLD
 
Syntaxcn0v 21069 Extend class notation with zero vector in a normed complex vector space.
 class  0vec
 
Syntaxcnsb 21070 Extend class notation with vector subtraction in a normed complex vector space.
 class  -v
 
Syntaxcnmcv 21071 Extend class notation with the norm function in a normed complex vector space. 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.
 class  normCV
 
Syntaxcims 21072 Extend class notation with the class of the induced metrics on normed complex vector spaces.
 class  IndMet
 
Definitiondf-nv 21073* Define the class of all normed complex vector spaces. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
 |-  NrmCVec  =  { <. <. g ,  s >. ,  n >.  |  ( <. g ,  s >.  e.  CVec OLD  /\  n : ran  g --> RR  /\  A. x  e.  ran  g ( ( ( n `  x )  =  0  ->  x  =  (GId `  g
 ) )  /\  A. y  e.  CC  ( n `  ( y s x ) )  =  ( ( abs `  y
 )  x.  ( n `
  x ) ) 
 /\  A. y  e.  ran  g ( n `  ( x g y ) )  <_  ( ( n `  x )  +  ( n `  y ) ) ) ) }
 
Theoremnvss 21074 Structure of the class of all normed complex vectors spaces. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
 |-  NrmCVec  C_  ( CVec OLD  X.  _V )
 
Theoremnvvcop 21075 A normed complex vector space is a vector space. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
 |-  ( <. W ,  N >.  e.  NrmCVec  ->  W  e.  CVec OLD )
 
Definitiondf-va 21076 Define vector addition on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
 |- 
 +v  =  ( 1st 
 o.  1st )
 
Definitiondf-ba 21077 Define the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
 |-  BaseSet  =  ( x  e. 
 _V  |->  ran  ( +v `  x ) )
 
Definitiondf-sm 21078 Define scalar multiplication on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
 |- 
 .s OLD  =  ( 2nd  o.  1st )
 
Definitiondf-0v 21079 Define the zero vector in a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
 |- 
 0vec  =  (GId  o.  +v )
 
Definitiondf-vs 21080 Define vector subtraction on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |- 
 -v  =  (  /g  o.  +v )
 
Definitiondf-nmcv 21081 Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
 |- 
 normCV  =  2nd
 
Definitiondf-ims 21082 Define the induced metric on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
 |-  IndMet  =  ( u  e.  NrmCVec 
 |->  ( ( normCV `  u )  o.  ( -v `  u ) ) )
 
Theoremnvrel 21083 The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)
 |- 
 Rel  NrmCVec
 
Theoremvafval 21084 Value of the function for the vector addition (group) operation on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   =>    |-  G  =  ( 1st `  ( 1st `  U ) )
 
Theorembafval 21085 Value of the function for the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  X  =  ran  G
 
Theoremsmfval 21086 Value of the function for the scalar multiplication operation on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
 |-  S  =  ( .s
 OLD `  U )   =>    |-  S  =  ( 2nd `  ( 1st `  U ) )
 
Theorem0vfval 21087 Value of the function for the zero vector on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( U  e.  V  ->  Z  =  (GId `  G ) )
 
Theoremnmcvfval 21088 Value of the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
 |-  N  =  ( normCV `  U )   =>    |-  N  =  ( 2nd `  U )
 
Theoremnvop2 21089 A normed complex vector space is an ordered pair of a vector space and a norm operation. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
 |-  W  =  ( 1st `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( U  e.  NrmCVec  ->  U  =  <. W ,  N >. )
 
Theoremnvvop 21090 The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
 |-  W  =  ( 1st `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( U  e.  NrmCVec  ->  W  =  <. G ,  S >. )
 
Theoremisnvlem 21091* Lemma for isnv 21093. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  (
 ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )  ->  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  <->  ( <. G ,  S >.  e.  CVec OLD  /\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  Z ) 
 /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) ) 
 /\  A. y  e.  X  ( N `  ( x G y ) ) 
 <_  ( ( N `  x )  +  ( N `  y ) ) ) ) ) )
 
Theoremnvex 21092 The components of a normed complex vector space are sets. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
 |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )
 )
 
Theoremisnv 21093* The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( <.
 <. G ,  S >. ,  N >.  e.  NrmCVec  <->  ( <. G ,  S >.  e.  CVec OLD  /\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  Z ) 
 /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) ) 
 /\  A. y  e.  X  ( N `  ( x G y ) ) 
 <_  ( ( N `  x )  +  ( N `  y ) ) ) ) )
 
Theoremisnvi 21094* Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  <. G ,  S >.  e.  CVec OLD   &    |-  N : X --> RR   &    |-  (
 ( x  e.  X  /\  ( N `  x )  =  0 )  ->  x  =  Z )   &    |-  ( ( y  e. 
 CC  /\  x  e.  X )  ->  ( N `
  ( y S x ) )  =  ( ( abs `  y
 )  x.  ( N `
  x ) ) )   &    |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( N `  ( x G y ) )  <_  ( ( N `  x )  +  ( N `  y ) ) )   &    |-  U  =  <. <. G ,  S >. ,  N >.   =>    |-  U  e.  NrmCVec
 
Theoremnvi 21095* The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( U  e.  NrmCVec  ->  ( <. G ,  S >.  e.  CVec OLD  /\  N : X
 --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
 )  x.  ( N `
  x ) ) 
 /\  A. y  e.  X  ( N `  ( x G y ) ) 
 <_  ( ( N `  x )  +  ( N `  y ) ) ) ) )
 
Theoremnvvc 21096 The vector space component of a normed complex vector space. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  W  =  ( 1st `  U )   =>    |-  ( U  e.  NrmCVec  ->  W  e.  CVec OLD )
 
Theoremnvablo 21097 The vector addition operation of a normed complex vector space is an Abelian group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   =>    |-  ( U  e.  NrmCVec  ->  G  e.  AbelOp )
 
Theoremnvgrp 21098 The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   =>    |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
 
Theoremnvgf 21099 Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( U  e.  NrmCVec  ->  G : ( X  X.  X ) --> X )
 
Theoremnvsf 21100 Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  X ) --> X )
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