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Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremvcdir 21101 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 21102 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 21103 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 21104 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 21105 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 21106 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 21107 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 21108 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 21109 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 21110 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 21111 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 21112 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 21113 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 21114 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 21115 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 21116 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 21117 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 21118 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 21119 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 21120 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 21121 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 21122 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 21123 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 21124 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 21125* Lemma for isvc 21129. (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 21126 Lemma for vcoprne 21127. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ( <. G ,  G >.  e.  CVec OLD  ->  G :
 ( CC  X.  CC )
 --> CC )
 
Theoremvcoprne 21127 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 21128 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 21129* 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 21130* 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 21131 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 21132 Extend class notation with the class of all normed complex vector spaces.
 class  NrmCVec
 
Syntaxcpv 21133 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 8735.
 class  +v
 
Syntaxcba 21134 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 21135 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 21136 Extend class notation with zero vector in a normed complex vector space.
 class  0vec
 
Syntaxcnsb 21137 Extend class notation with vector subtraction in a normed complex vector space.
 class  -v
 
Syntaxcnmcv 21138 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 21139 Extend class notation with the class of the induced metrics on normed complex vector spaces.
 class  IndMet
 
Definitiondf-nv 21140* 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 21141 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 21142 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 21143 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 21144 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 21145 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 21146 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 21147 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 21148 Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
 |- 
 normCV  =  2nd
 
Definitiondf-ims 21149 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 21150 The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)
 |- 
 Rel  NrmCVec
 
Theoremvafval 21151 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 21152 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 21153 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 21154 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 21155 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 21156 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 21157 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 21158* Lemma for isnv 21160. (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 21159 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 21160* 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 21161* 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 21162* 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 21163 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 21164 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 21165 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 21166 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 21167 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 21168 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 21169 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 21170 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 21171 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 21172 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 21173 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 21174 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 21175 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 21176 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 )
 
Theoremnvsid 21177 Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( 1 S A )  =  A )
 
Theoremnvsass 21178 Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
 )  ->  ( ( A  x.  B ) S C )  =  ( A S ( B S C ) ) )
 
Theoremnvscom 21179 Commutative law for the scalar product of a normed complex vector space. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
 )  ->  ( A S ( B S C ) )  =  ( B S ( A S C ) ) )
 
Theoremnvdi 21180 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
 
Theoremnvdir 21181 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
 
Theoremnv2 21182 A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )
 
Theoremvsfval 21183 Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  M  =  ( -v `  U )   =>    |-  M  =  (  /g  `  G )
 
Theoremnvzcl 21184 Closure law for the zero vector of a normed complex vector space. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   =>    |-  ( U  e.  NrmCVec  ->  Z  e.  X )
 
Theoremnv0rid 21185 The zero vector is a right identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( A G Z )  =  A )
 
Theoremnv0lid 21186 The zero vector is a left identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( Z G A )  =  A )
 
Theoremnv0 21187 Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( 0 S A )  =  Z )
 
Theoremnvsz 21188 Anything times the zero vector is the zero vector. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  S  =  ( .s
 OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
 
Theoremnvinv 21189 Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  M  =  ( inv `  G )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( -u 1 S A )  =  ( M `  A ) )
 
Theoremnvinvfval 21190 Function for the negative of a vector on a normed complex vector space, in terms of the underlying addition group inverse. (We currently do not have a separate notation for the negative of a vector.) (Contributed by NM, 27-Mar-2008.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( S  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )   =>    |-  ( U  e.  NrmCVec  ->  N  =  ( inv `  G ) )
 
Theoremnvm 21191 Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( 
 /g  `  G )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A N B ) )
 
Theoremnvmval 21192 Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G ( -u 1 S B ) ) )
 
Theoremnvmval2 21193 Value of vector subtraction on a normed complex vector space. (Contributed by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( ( -u 1 S B ) G A ) )
 
Theoremnvmfval 21194* Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( U  e.  NrmCVec  ->  M  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( -u 1 S y ) ) ) )
 
Theoremnvzs 21195 Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( Z M A )  =  ( -u 1 S A ) )
 
Theoremnvmf 21196 Mapping for the vector subtraction operation. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( U  e.  NrmCVec  ->  M : ( X  X.  X ) --> X )
 
Theoremnvmcl 21197 Closure law for the vector subtraction operation of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
 
Theoremnvnnncan1 21198 Cancellation law for vector subtraction. (nnncan1 9078 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A M B ) M ( A M C ) )  =  ( C M B ) )
 
Theoremnvnnncan2 21199 Cancellation law for vector subtraction. (nnncan2 9079 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A M C ) M ( B M C ) )  =  ( A M B ) )
 
Theoremnvmdi 21200 Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A S ( B M C ) )  =  ( ( A S B ) M ( A S C ) ) )
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