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Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrngodm1dm2 21101 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 21102 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 21103 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 21104 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 21105 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 21106 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 21107 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 21108 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 21109 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 21110 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 21111 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 21112 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 21113 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 21114 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 21115 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 21116 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 )
 
PART 16  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
 
16.1  Complex vector spaces
 
16.1.1  Definition and basic properties
 
Syntaxcvc 21117 Extend class notation with the class of all complex vector spaces.
 class  CVec OLD
 
Definitiondf-vc 21118* 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 21119 The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CVec OLD
 
Theoremvci 21120* 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 21121 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 21122 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 21123 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 21124 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 21125 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 21126 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 21127 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 21128 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 21129 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 21130 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 21131 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 21132 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 21133 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 21134 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 21135 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 21136 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 21137 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 21138 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 21139 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 21140 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 21141 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 21142 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 21143 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 21144 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 21145 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 21146 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 21147 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 21148 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 21149* Lemma for isvc 21153. (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 21150 Lemma for vcoprne 21151. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ( <. G ,  G >.  e.  CVec OLD  ->  G :
 ( CC  X.  CC )
 --> CC )
 
Theoremvcoprne 21151 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 21152 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 21153* 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 21154* 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
 
16.1.2  Examples of complex vector spaces
 
Theoremcncvc 21155 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
 
16.2  Normed complex vector spaces
 
16.2.1  Definition and basic properties
 
Syntaxcnv 21156 Extend class notation with the class of all normed complex vector spaces.
 class  NrmCVec
 
Syntaxcpv 21157 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 8756.
 class  +v
 
Syntaxcba 21158 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 21159 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 21160 Extend class notation with zero vector in a normed complex vector space.
 class  0vec
 
Syntaxcnsb 21161 Extend class notation with vector subtraction in a normed complex vector space.
 class  -v
 
Syntaxcnmcv 21162 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 21163 Extend class notation with the class of the induced metrics on normed complex vector spaces.
 class  IndMet
 
Definitiondf-nv 21164* 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 21165 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 21166 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 21167 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 21168 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 21169 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 21170 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 21171 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 21172 Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
 |- 
 normCV  =  2nd
 
Definitiondf-ims 21173 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 21174 The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)
 |- 
 Rel  NrmCVec
 
Theoremvafval 21175 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 21176 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 21177 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 21178 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 21179 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 21180 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 21181 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 21182* Lemma for isnv 21184. (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 21183 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 21184* 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 21185* 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 21186* 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 21187 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 21188 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 21189 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 21190 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 21191 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 )
 
Theoremnvgcl 21192 Closure law for the vector addition (group) operation of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
 
Theoremnvcom 21193 The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
 
Theoremnvass 21194 The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Theoremnvadd12 21195 Commutative/associative law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A G ( B G C ) )  =  ( B G ( A G C ) ) )
 
Theoremnvadd32 21196 Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )
 
Theoremnvrcan 21197 Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G C )  =  ( B G C ) 
 <->  A  =  B ) )
 
Theoremnvlcan 21198 Left cancellation law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( C G A )  =  ( C G B ) 
 <->  A  =  B ) )
 
Theoremnvadd4 21199 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( 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 ) ) )
 
Theoremnvscl 21200 Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
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