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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrngomndo 22001 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 22002 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 22003 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 22004 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 22005 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 22006 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 22007 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 22008 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 22009 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 22010 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 22011 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 22012 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 22013 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 )
 
Theoremzerdivemp1 22014* In a unitary ring a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  ( X 
 \  Z )  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) )
 
Theoremrngoridfz 22015* In a unitary ring a left invertible element is different from zero iff  1  =/=  0. (Contributed by FL, 18-Apr-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( A  =/=  Z  <->  U  =/=  Z ) )
 
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
 
17.1  Complex vector spaces
 
17.1.1  Definition and basic properties
 
Syntaxcvc 22016 Extend class notation with the class of all complex vector spaces.
 class  CVec OLD
 
Definitiondf-vc 22017* 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 22018 The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CVec OLD
 
Theoremvci 22019* 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 22020 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 22021 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 22022 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 22023 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 22024 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 22025 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 22026 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 22027 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 22028 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 22029 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 22030 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 22031 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 22032 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 22033 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 22034 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 22035 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 22036 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 22037 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 22038 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 22039 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 22040 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 22041 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 22042 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 22043 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 22044 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 22045 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 22046 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 22047 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 22048* Lemma for isvc 22052. (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 22049 Lemma for vcoprne 22050. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ( <. G ,  G >.  e.  CVec OLD  ->  G :
 ( CC  X.  CC )
 --> CC )
 
Theoremvcoprne 22050 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 22051 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 22052* 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 22053* 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
 
17.1.2  Examples of complex vector spaces
 
Theoremcncvc 22054 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
 
17.2  Normed complex vector spaces
 
17.2.1  Definition and basic properties
 
Syntaxcnv 22055 Extend class notation with the class of all normed complex vector spaces.
 class  NrmCVec
 
Syntaxcpv 22056 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 8985.
 class  +v
 
Syntaxcba 22057 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 22058 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 22059 Extend class notation with zero vector in a normed complex vector space.
 class  0vec
 
Syntaxcnsb 22060 Extend class notation with vector subtraction in a normed complex vector space.
 class  -v
 
Syntaxcnmcv 22061 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 22062 Extend class notation with the class of the induced metrics on normed complex vector spaces.
 class  IndMet
 
Definitiondf-nv 22063* 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 22064 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 22065 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 22066 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 22067 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 22068 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 22069 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 22070 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 22071 Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
 |- 
 normCV  =  2nd
 
Definitiondf-ims 22072 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 22073 The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)
 |- 
 Rel  NrmCVec
 
Theoremvafval 22074 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 22075 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 22076 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 22077 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 22078 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 22079 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 22080 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 22081* Lemma for isnv 22083. (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 22082 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 22083* 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 22084* 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 22085* 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 22086 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 22087 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 22088 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 22089 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 22090 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 22091 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 22092 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 22093 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 22094 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 22095 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 22096 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 22097 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 22098 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 22099 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 22100 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 )
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