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Theorem List for Metamath Proof Explorer - 25001-25100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtopunfincomp 25001 A topology whose underlying set is finite is compact. (Contributed by FL, 22-Dec-2008.)
 |-  (
 ( J  e.  Top  /\ 
 U. J  e.  Fin )  ->  J  e.  Comp )
 
Theoremstfincomp 25002 The subspace topology induced by a finite part of the underlying set of a topology is compact. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  (
 ( J  e.  Top  /\  A  e.  Fin )  ->  ( Jt  A )  e.  Comp )
 
Theorembwt2 25003* The glorious Bolzano-Weierstrass theorem. Certainly the first general topology theorem ever proved. In his course Weierstrass called it a lemma. He certainly didn't know how famous this theorem would be. He used an euclidian space instead of a general compact space. And he was not conscious of the Heine-Borel property. Cantor was one of his students. He used the concept of neighborhood and limit point invented by his master when he studied the linear point sets and the rest of the general topology followed from that. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Comp  /\  A  C_  X  /\  -.  A  e.  Fin )  ->  E. x  e.  X  x  e.  ( ( limPt `  J ) `  A ) )
 
18.13.36  Connectedness
 
Theoremsingempcon 25004 The singleton of the empty set is a connected topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  { (/) }  e.  Con
 
Theoremusinuniopb 25005 If a topology is connected, its underlying set can't be partitioned into two non empty non-overlapping open sets. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 8-Apr-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Con  /\  A  e.  J  /\  B  e.  J )  ->  ( ( A  =/=  (/)  /\  B  =/=  (/)  /\  ( A  i^i  B )  =  (/) )  ->  X  =/=  ( A  u.  B ) ) )
 
Syntaxcopfn 25006 Extend class notation with an operator that derives an operation on functions from an operation on the elements of the common range of those functions.
 class  opfn
 
Definitiondf-opfn 25007* Multiplication or addition of two functions  x and  y derived from the operation  g on the elements of the common range of 
x and  y. The functions  x and  y must also have the same domain  i. (Contributed by FL, 15-Oct-2012.)
 |-  opfn  =  ( g  e.  _V ,  i  e.  _V  |->  ( x  e.  ( dom  dom  g  ^m  i
 ) ,  y  e.  ( dom  dom  g  ^m  i )  |->  ( a  e.  dom  x  |->  ( ( x `  a
 ) g ( y `
  a ) ) ) ) )
 
18.13.37  Topological fields
 
Syntaxctopfld 25008 Extend class notation to include TopFld.
 class  TopFld
 
Definitiondf-topfld 25009* A topological field is a field whose addition, multiplication and inverse are continuous. (Contributed by FL, 21-May-2012.)
 |-  TopFld  =  { <.
 <. g ,  h >. ,  j >.  |  ( <. g ,  h >.  e. 
 Fld  /\  <. <. g ,  h >. ,  j >.  e.  TopRing  /\  ( h  |`  ( ( ran  g  \  {
 (GId `  g ) } )  X.  ( ran  g  \  { (GId `  g ) } )
 ) )  e.  (
 ( j  tX  j
 )  Cn  j )
 ) }
 
18.13.38  Standard topology on RR
 
Theoremintrn 25010 Condition for an interval to belong to the range of  (,) (Contributed by FL, 5-Jan-2009.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e.  ran  (,) )
 
Theoremaltretop 25011* Alternate definition of the standard topology of the reals. (Morris. Def. 2.1.1 p. 34). Morris calls the standard topology of the reals the euclidean topology. (Contributed by FL, 26-Jan-2009.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. y  e.  A  E. a  e. 
 RR  E. b  e.  RR  ( y  e.  (
 a (,) b )  /\  ( a (,) b
 )  C_  A )
 ) )
 
18.13.39  Standard topology of intervals of RR
 
Theoremstoi 25012 The underlying set of the standard topology on an open interval is the open interval itself. (Contributed by FL, 31-May-2007.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  { <. (
 Base `  ndx ) ,  ( A (,) B ) >. ,  <. (TopSet `  ndx ) ,  ( ( topGen `
  ran  (,) )t  ( A (,) B ) )
 >. }  e.  TopSp
 
18.13.40  Cantor's set
 
Theoremcntrset 25013* Cantor's set is between  0 and  1. Viro p. 15. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Mario Carneiro, 2-Jun-2014.)
 |-  C  =  { x  |  E. f  e.  ( {
 0 ,  2 } 
 ^m  NN ) x  = 
 sum_ k  e.  NN  ( ( f `  k )  /  (
 3 ^ k ) ) }   =>    |-  C  C_  ( 0 [,] 1 )
 
18.13.41  Pre-calculus and Cartesian geometry
 
Theoremdmse1 25014 Distance between the middle of a segment and one of its extremities is a positive real. (Contributed by FL, 27-Dec-2007.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  A  =/=  B )  ->  ( ( abs `  ( A  -  B ) ) 
 /  2 )  e.  RR+ )
 
Theoremdmse2 25015 Distance between the middle of a segment and one of its extremities is a positive real. (Contributed by FL, 27-Dec-2007.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( ( abs `  ( A  -  B ) ) 
 /  2 )  e.  RR+ )
 
Theoremmsr3 25016 The midpoint of a segment AB of the real line is a real. (Contributed by FL, 27-Dec-2007.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( B  -  (
 ( abs `  ( A  -  B ) )  / 
 2 ) )  e. 
 RR )
 
Theoremmsr4 25017 The midpoint of a segment AB of the real line is a real. (To FL: The proof was shortened. Also, it is too specialized, and set.mm size will be reduced if it is placed directly in the proof using it. --NM) (Contributed by FL, 27-Dec-2007.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  (
 ( abs `  ( B  -  A ) )  / 
 2 ) )  e. 
 RR )
 
Theoremmslb1 25018 The midpoint of a segment AB of the real line is on the "left" of  B. (Contributed by FL, 2-Jan-2008.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  +  (
 ( abs `  ( B  -  A ) )  / 
 2 ) )  <  B )
 
Theorem2wsms 25019 Two ways to state the midpoint of a segment. (Contributed by FL, 3-Jan-2008.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( ( A  +  B )  /  2
 )  =  ( B  -  ( ( abs `  ( A  -  B ) )  /  2
 ) ) )
 
Theoremmsra3 25020 The midpoint of a segment AB of the real line is on the "right" of  A. (Contributed by FL, 3-Jan-2008.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  <  ( B  -  ( ( abs `  ( A  -  B ) ) 
 /  2 ) ) )
 
Theoremiintlem1 25021* Lemma for iint 25023. (Contributed by FL, 27-Dec-2007.)
 |-  (
 ( A  e.  RR  /\  y  e.  |^|_ x  e.  RR+  ( ( A  -  x ) (,) ( A  +  x )
 ) )  ->  (
 y  e.  RR  ->  y  =  A ) )
 
Theoremiintlem2 25022* Lemma for iint 25023. (Contributed by FL, 23-Dec-2007.)
 |-  (
 y  e.  |^|_ x  e.  RR+  ( ( A  -  x ) (,) ( A  +  x )
 )  ->  y  e.  RR )
 
Theoremiint 25023* Indexed intersection of a set of open intervals centered on  A. This theorem is a rough justification for taking finite intersections in the definition of a topology. If we consider we are in the standard topology of  RR this theorem means a non finite intersection of open sets can result in a closed set. (Contributed by FL, 27-Dec-2007.)
 |-  ( A  e.  RR  ->  |^|_
 x  e.  RR+  ( ( A  -  x ) (,) ( A  +  x ) )  =  { A } )
 
Theoremtrdom 25024* Domain of a translation. (Contributed by FL, 17-Feb-2008.)
 |-  F  =  ( x  e.  RR  |->  ( x  +  A ) )   =>    |-  ( A  e.  RR  ->  dom  F  =  RR )
 
Theoremtrran 25025* Range of a translation. (Contributed by FL, 17-Feb-2008.)
 |-  F  =  ( x  e.  RR  |->  ( x  +  A ) )   =>    |-  ( A  e.  RR  ->  ran  F  =  RR )
 
Theoremtrnij 25026* A translation is 1-1-onto. (Contributed by FL, 17-Feb-2008.)
 |-  F  =  ( x  e.  RR  |->  ( x  +  A ) )   =>    |-  ( A  e.  RR  ->  F : RR -1-1-onto-> RR )
 
Theoremcnvtr 25027* Converse of a translation. (Contributed by FL, 3-Aug-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  ( A  e.  RR  ->  `' ( x  e.  RR  |->  ( x  +  A ) )  =  ( x  e.  RR  |->  ( x  -  A ) ) )
 
Theoremmlteqer 25028 The members of a 'less than or equal' relationship are extended reals. (Contributed by FL, 31-Jul-2009.) (Proof shortened by Mario Carneiro, 4-May-2015.)
 |-  ( A  <_  B  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
 
Theoremxrletr2 25029 Transitive law for ordering on extended reals ( compare xrletr 10484). (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  (
 ( A  <_  B  /\  B  <_  C )  ->  A  <_  C )
 
18.13.42  Extended Real numbers
 
Theoremnolimf 25030* A numerical function has at most one limit value. (Contributed by FL, 14-Nov-2010.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  (
 ( L  e.  ( Fil `  Y )  /\  F : Y --> RR )  ->  E* x  x  e.  ( ( J  fLimf  L ) `  F ) )
 
Theoremnolimf2 25031* A numerical convergent function has one limit value. (Contributed by FL, 14-Nov-2010.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  (
 ( L  e.  ( Fil `  Y )  /\  F : Y --> RR  /\  ( ( J  fLimf  L ) `  F )  =/=  (/) )  ->  E! x  x  e.  (
 ( J  fLimf  L ) `
  F ) )
 
Theoremflfnein 25032* A neighborhood of the limit value 
A of a convergent numerical function  F contains the image of a filter element. (Contributed by FL, 18-Nov-2010.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  A  =  U. ( ( J 
 fLimf  L ) `  F )   &    |-  J  =  ( topGen `  ran  (,) )   =>    |-  ( ( ( L  e.  ( Fil `  Y )  /\  F : Y --> RR )  /\  ( ( J  fLimf  L ) `  F )  =/=  (/)  /\  N  e.  ( ( nei `  J ) `  { A }
 ) )  ->  E. s  e.  L  ( F "
 s )  C_  N )
 
Theoremlimnumrr 25033 The limit of a numerical convergent function belongs to  RR. (Contributed by FL, 22-Nov-2010.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  A  =  U. ( ( J 
 fLimf  L ) `  F )   &    |-  J  =  ( topGen `  ran  (,) )   =>    |-  ( ( L  e.  ( Fil `  Y )  /\  F : Y --> RR  /\  ( ( J  fLimf  L ) `  F )  =/=  (/) )  ->  A  e.  RR )
 
Theoremcinei 25034 A centered interval is a neighborhood of its center. (Contributed by FL, 18-Nov-2010.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  -  B ) (,) ( A  +  B )
 )  e.  ( ( nei `  J ) `  { A } )
 )
 
Theoremflfneic 25035 A centered interval of the limit value  A of a convergent numerical function  F contains the image of a filter element. (Contributed by FL, 18-Nov-2010.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  X  =  U. J   &    |-  A  =  U. ( ( J  fLimf  L ) `  F )   &    |-  J  =  ( topGen `  ran  (,) )   &    |-  N  =  ( ( A  -  B ) (,) ( A  +  B ) )   =>    |-  ( ( ( L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( ( J  fLimf  L ) `  F )  =/=  (/)  /\  B  e.  RR+ )  ->  A  e.  ( ( J  fLimf  L ) `  F ) )
 
Theoremflfneicn 25036* A centered interval of the limit value  A of a convergent numerical function  F contains the image of a filter element. (Contributed by FL, 18-Nov-2010.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  A  =  U. ( ( J 
 fLimf  L ) `  F )   &    |-  J  =  ( topGen `  ran  (,) )   &    |-  N  =  ( ( A  -  B ) (,) ( A  +  B ) )   =>    |-  ( ( ( L  e.  ( Fil `  Y )  /\  F : Y --> RR )  /\  ( ( J  fLimf  L ) `  F )  =/=  (/)  /\  B  e.  RR+ )  ->  E. s  e.  L  ( F "
 s )  C_  N )
 
Theoremlvsovso 25037* If the limit values of two convergent numerical functions are strictly ordered, the values of the functions are strictly ordered for some element of the filter. Bourbaki TG IV.18 prop. 2. (Contributed by FL, 6-Dec-2010.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  L1  =  U. ( ( J  fLimf  F ) `  F1 )   &    |-  L 2  =  U. ( ( J  fLimf  F ) `  F 2 )   &    |-  J  =  (
 topGen `  ran  (,) )   =>    |-  (
 ( ( F  e.  ( Fil `  Y )  /\  F1 : Y --> RR  /\  F 2 : Y --> RR )  /\  ( ( ( J 
 fLimf  F ) `  F1 )  =/= 
 (/)  /\  ( ( J 
 fLimf  F ) `  F 2 )  =/=  (/)  /\  L1  <  L 2 ) )  ->  E. a  e.  F  A. x  e.  a  (
 F1 `  x )  <  ( F 2 `  x ) )
 
Theoremlvsovso2 25038* Condition on the elements of the filter so that the limits are weakly ordered. Bourbaki TG IV.18 prop. 1. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  L1  =  U. ( ( J  fLimf  F ) `  F1 )   &    |-  L 2  =  U. ( ( J  fLimf  F ) `  F 2 )   &    |-  J  =  (
 topGen `  ran  (,) )   =>    |-  (
 ( ( F  e.  ( Fil `  Y )  /\  F1 : Y --> RR  /\  F 2 : Y --> RR )  /\  ( ( ( J 
 fLimf  F ) `  F1 )  =/= 
 (/)  /\  ( ( J 
 fLimf  F ) `  F 2 )  =/=  (/)  /\  A. a  e.  F  E. x  e.  a  ( F1 `  x )  <_  ( F 2 `  x ) ) )  ->  L1  <_  L 2 )
 
Theoremlvsovso3 25039* Condition on the values of two numerical functions so that their limits are weakly ordered. Bourbaki TG IV.18 th. 1. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  L1  =  U. ( ( J  fLimf  F ) `  F1 )   &    |-  L 2  =  U. ( ( J  fLimf  F ) `  F 2 )   &    |-  J  =  (
 topGen `  ran  (,) )   =>    |-  (
 ( ( F  e.  ( Fil `  Y )  /\  F1 : Y --> RR  /\  F 2 : Y --> RR )  /\  ( ( ( J 
 fLimf  F ) `  F1 )  =/= 
 (/)  /\  ( ( J 
 fLimf  F ) `  F 2 )  =/=  (/)  /\  A. x  e.  Y  ( F1
 `  x )  <_  ( F 2 `  x ) ) )  ->  L1 
 <_  L 2 )
 
Theoremsupnuf 25040 The supremum of a numerical function  F is greater or equal to every element of  ( F `  A ). Bourbaki TG IV.20. (Contributed by FL, 25-Dec-2011.)
 |-  (
 ( F : A --> RR*  /\  A  e.  _V  /\  C  e.  A )  ->  ( F `  C )  <_  (  <_  sup w  ran  F ) )
 
Theoremsupnufb 25041* The supremum of a numerical function  F is greater or equal to every element of  ( F `  A ). Bourbaki TG IV.20. (Contributed by FL, 25-Dec-2011.)
 |-  F  =  ( x  e.  A  |->  U )   &    |-  ( x  =  C  ->  U  =  V )   =>    |-  ( ( A. x  e.  A  U  e.  RR*  /\  A  e.  M  /\  ( C  e.  A  /\  V  e.  N ) )  ->  V  <_  ( 
 <_  sup w  ran  F ) )
 
Theoremsupexr 25042 Two ways to express the supremum of a set of extended reals. (Contributed by FL, 25-Dec-2011.) (Revised by Mario Carneiro, 20-Nov-2013.)
 |-  ( A  C_  RR*  ->  (  <_  sup w  A )  = 
 sup ( A ,  RR*
 ,  <  ) )
 
Syntaxclsupp 25043 Extend class notation to include the supremum of the class B.
 class  sup _  x  e.  A B
 
Syntaxclinfp 25044 Extend class notation to include the infimum of the class B.
 class  inf _  x  e.  A B
 
Definitiondf-supp 25045 Definition of the supremum of an indexed class of extended reals. (Contributed by FL, 16-Apr-2012.)
 |-  sup _  x  e.  A B  =  (  <_  sup w  ( ( x  e.  A  |->  B ) " A ) )
 
Definitiondf-infp 25046 Definition of the infimum of an indexed class of extended reals. (Contributed by FL, 21-May-2012.)
 |-  inf _  x  e.  A B  =  (  <_  inf w  ( ( x  e.  A  |->  B ) " A ) )
 
Theoremsupbrr 25047* The supremum of a set of extended reals always exists. (Contributed by FL, 16-Apr-2012.)
 |-  B  e.  C   =>    |-  ( A. x  e.  A  B  e.  RR*  ->  sup _  x  e.  A B  e.  RR* )
 
Syntaxcfrf 25048 Extends class notation with Frechet's filter.
 class  Frf
 
Definitiondf-frf 25049* Frechet's filter. Used to define the limit of a sequence. (Contributed by FL, 21-May-2012.)
 |-  Frf  =  { x  |  E. b ( b  C_  NN  /\  b  e.  Fin  /\  x  =  ( NN  \  b ) ) }
 
Theorembsi2 25050* Membership to the set of closed intervals. (Contributed by FL, 29-May-2014.)
 |-  ( A  e.  ran  [,]  <->  E. x  e.  RR*  E. y  e.  RR*  A  =  ( x [,] y ) )
 
Theoremicof 25051 The set of closed-below, open-above intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 29-May-2014.)
 |-  [,) : ( RR*  X.  RR* ) --> ~P RR*
 
Theorembsi3 25052* Membership to the set of closed-above, open-below intervals. (Contributed by FL, 29-May-2014.)
 |-  ( A  e.  ran  [,)  <->  E. x  e.  RR*  E. y  e.  RR*  A  =  ( x [,) y ) )
 
Theoremiocf 25053 The set of closed-below, open-above intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 29-May-2014.)
 |-  (,] : ( RR*  X.  RR* ) --> ~P RR*
 
Theorembsi4 25054* Membership to the set of open-below, closed-above intervals. (Contributed by FL, 29-May-2014.)
 |-  ( A  e.  ran  (,]  <->  E. x  e.  RR*  E. y  e.  RR*  A  =  ( x (,] y ) )
 
18.13.43  ( RR ^ N ) and ( CC ^ N )
 
Syntaxcplcv 25055 Extends class notation with addition of complex vectors.
 class  + cv
 
Definitiondf-addcv 25056* Addition of complex vectors in a space of dimension  n. Experimental. (Contributed by FL, 14-Sep-2013.)
 |-  + cv  =  ( n  e.  NN  |->  ( u  e.  ( CC  ^m  (
 1 ... n ) ) ,  v  e.  ( CC  ^m  ( 1 ... n ) )  |->  ( x  e.  ( 1
 ... n )  |->  ( ( u `  x )  +  ( v `  x ) ) ) ) )
 
Theoremisaddrv 25057* Addition of complex vectors. Experimental. (Contributed by FL, 14-Sep-2013.)
 |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  (
 1 ... N ) ) 
 /\  V  e.  ( CC  ^m  ( 1 ...
 N ) ) ) 
 ->  ( U + w V )  =  ( x  e.  ( 1 ... N )  |->  ( ( U `  x )  +  ( V `  x ) ) ) )
 
Theoremcladdrv 25058 Closure of addition of complex vectors. (Contributed by FL, 14-Sep-2013.)
 |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  (
 1 ... N ) ) 
 /\  V  e.  ( CC  ^m  ( 1 ...
 N ) ) ) 
 ->  ( U + w V )  e.  ( CC  ^m  ( 1 ...
 N ) ) )
 
Theoremcladdrvr 25059 Closure of addition of real vectors. (Contributed by FL, 29-May-2014.)
 |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  (
 1 ... N ) ) 
 /\  V  e.  ( RR  ^m  ( 1 ...
 N ) ) ) 
 ->  ( U + w V )  e.  ( RR  ^m  ( 1 ...
 N ) ) )
 
Theoremsigadd 25060 Functionality of vector addition. (Contributed by FL, 29-May-2014.)
 |-  + w  =  (  + cv `  N )   =>    |-  ( N  e.  NN  ->  + w :
 ( ( CC  ^m  ( 1 ... N ) )  X.  ( CC  ^m  ( 1 ...
 N ) ) ) --> ( CC  ^m  (
 1 ... N ) ) )
 
Syntaxc0cv 25061 Extends class notation with null vector.
 class  0 cv
 
Definitiondf-nullcv 25062* The null vector in a space of dimension  n. Experimental. (Contributed by FL, 15-Sep-2013.)
 |-  0 cv  =  ( n  e.  NN  |->  ( x  e.  ( 1 ... n )  |->  0 ) )
 
Theoremisnullcv 25063* The null vector in a space of dimension  n. (Contributed by FL, 15-Sep-2013.)
 |-  0 w  =  ( 0 cv `  N )   =>    |-  ( N  e.  NN  ->  0 w  =  ( x  e.  (
 1 ... N )  |->  0 ) )
 
Theoremzernpl 25064 The null vector is a complex vector. (Contributed by FL, 15-Sep-2013.)
 |-  0 w  =  ( 0 cv `  N )   =>    |-  ( N  e.  NN  ->  0 w  e.  ( CC  ^m  ( 1
 ... N ) ) )
 
Theoremvalvze 25065 Value of the complex vector at a specific coordinate. (Contributed by FL, 15-Sep-2013.)
 |-  0 w  =  ( 0 cv `  N )   =>    |-  ( ( N  e.  NN  /\  A  e.  ( 1 ... N ) )  ->  ( 0 w `  A )  =  0 )
 
Theoremaddcomv 25066 Vector addition is commutative. (Contributed by FL, 15-Sep-2013.)
 |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  (
 1 ... N ) ) 
 /\  B  e.  ( CC  ^m  ( 1 ...
 N ) ) ) 
 ->  ( A + w B )  =  ( B + w A ) )
 
Theoremaddassv 25067 Vector addition is associative. (Contributed by FL, 15-Sep-2013.)
 |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  ( A  e.  ( CC  ^m  ( 1 ... N ) )  /\  B  e.  ( CC  ^m  ( 1
 ... N ) ) 
 /\  C  e.  ( CC  ^m  ( 1 ...
 N ) ) ) )  ->  ( ( A + w B ) + w C )  =  ( A + w ( B + w C ) ) )
 
Theoremaddidv2 25068 The null vector is a left identity for vector addition. (Contributed by FL, 15-Sep-2013.)
 |-  0 w  =  ( 0 cv `  N )   &    |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  (
 1 ... N ) ) )  ->  ( 0 w + w A )  =  A )
 
Theoremaddidrv2 25069 The null vector is a right identity for vector addition. (Contributed by FL, 15-Sep-2013.)
 |-  0 w  =  ( 0 cv `  N )   &    |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  (
 1 ... N ) ) )  ->  ( A + w 0 w )  =  A )
 
Theoremvecaddonto 25070 Vector addition is onto. (Contributed by FL, 29-May-2014.)
 |-  + w  =  (  + cv `  N )   =>    |-  ( N  e.  NN  ->  + w :
 ( ( CC  ^m  ( 1 ... N ) )  X.  ( CC  ^m  ( 1 ...
 N ) ) )
 -onto-> ( CC  ^m  (
 1 ... N ) ) )
 
Theoremcnegvex2 25071* Existence of a left inverse for vector addition. (Contributed by FL, 15-Sep-2013.)
 |-  + w  =  (  + cv `  N )   &    |-  0 w  =  ( 0 cv `  N )   &    |-  N  e.  NN   =>    |-  ( A  e.  ( CC  ^m  ( 1 ...
 N ) )  ->  E. x  e.  ( CC  ^m  ( 1 ...
 N ) ) ( x + w A )  =  0 w
 )
 
Theoremrnegvex2 25072* Existence of a left inverse for vector addition. (Contributed by FL, 29-May-2014.)
 |-  + w  =  (  + cv `  N )   &    |-  0 w  =  ( 0 cv `  N )   &    |-  N  e.  NN   =>    |-  ( A  e.  ( RR  ^m  ( 1 ...
 N ) )  ->  E. x  e.  ( RR  ^m  ( 1 ...
 N ) ) ( x + w A )  =  0 w
 )
 
Theoremcnegvex2b 25073* Existence of a left inverse for vector addition. (Contributed by FL, 15-Sep-2013.)
 |-  + w  =  (  + cv `  N )   &    |-  0 w  =  ( 0 cv `  N )   =>    |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  (
 1 ... N ) ) )  ->  E. x  e.  ( CC  ^m  (
 1 ... N ) ) ( x + w A )  =  0 w )
 
Theoremrnegvex2b 25074* Existence of a left inverse for vector addition. (Contributed by FL, 29-May-2014.)
 |-  + w  =  (  + cv `  N )   &    |-  0 w  =  ( 0 cv `  N )   =>    |-  ( ( N  e.  NN  /\  A  e.  ( RR  ^m  (
 1 ... N ) ) )  ->  E. x  e.  ( RR  ^m  (
 1 ... N ) ) ( x + w A )  =  0 w )
 
Syntaxcmcv 25075 Extends class notation with substraction of complex vectors.
 class  - cv
 
Definitiondf-subcatv 25076* Substraction of complex vectors in a space of dimension  n. Experimental. (Contributed by FL, 15-Sep-2013.)
 |-  - cv  =  ( n  e.  NN  |->  ( u  e.  ( CC  ^m  (
 1 ... n ) ) ,  v  e.  ( CC  ^m  ( 1 ... n ) )  |->  (
 iota_ w  e.  ( CC  ^m  ( 1 ... n ) ) ( v (  + cv `  n ) w )  =  u ) ) )
 
Theoremaddcanri 25077 Cancellation law for vector addition. (Contributed by FL, 15-Sep-2013.)
 |-  + w  =  (  + cv `  N )   &    |-  N  e.  NN   &    |-  A  e.  ( CC  ^m  ( 1 ...
 N ) )   &    |-  B  e.  ( CC  ^m  (
 1 ... N ) )   &    |-  C  e.  ( CC  ^m  ( 1 ... N ) )   =>    |-  ( ( A + w B )  =  ( A + w C ) 
 <->  B  =  C )
 
Theoremaddcanrg 25078 Cancellation law for vector addition. (Contributed by FL, 15-Sep-2013.)
 |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  ( A  e.  ( CC  ^m  ( 1 ... N ) )  /\  B  e.  ( CC  ^m  ( 1
 ... N ) ) 
 /\  C  e.  ( CC  ^m  ( 1 ...
 N ) ) ) )  ->  ( ( A + w B )  =  ( A + w C )  <->  B  =  C ) )
 
Theoremnegveud 25079* Existential uniqueness of vector negatives. (Contributed by FL, 15-Sep-2013.)
 |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  (
 1 ... N ) ) 
 /\  B  e.  ( CC  ^m  ( 1 ...
 N ) ) ) 
 ->  E! x  e.  ( CC  ^m  ( 1 ...
 N ) ) ( A + w x )  =  B )
 
Theoremnegveudr 25080* Existential uniqueness of vector negatives. (Contributed by FL, 29-May-2014.)
 |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  A  e.  ( RR  ^m  (
 1 ... N ) ) 
 /\  B  e.  ( RR  ^m  ( 1 ...
 N ) ) ) 
 ->  E! x  e.  ( RR  ^m  ( 1 ...
 N ) ) ( A + w x )  =  B )
 
Theoremissubcv 25081* Substraction of complex vectors in a space of dimension  n. (Contributed by FL, 15-Sep-2013.)
 |-  + w  =  (  + cv `  N )   &    |-  - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  V  e.  ( CC  ^m  ( 1 ... N ) ) )  ->  ( U - w V )  =  ( iota_ w  e.  ( CC  ^m  (
 1 ... N ) ) ( V + w w )  =  U ) )
 
Theoremsubaddv 25082 Relationship between subtraction and addition. (Contributed by FL, 30-May-2014.)
 |-  + w  =  (  + cv `  N )   &    |-  - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  ( A  e.  ( CC  ^m  ( 1
 ... N ) ) 
 /\  B  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  C  e.  ( CC  ^m  ( 1 ... N ) ) ) ) 
 ->  ( ( A - w B )  =  C  <->  ( B + w C )  =  A )
 )
 
Theoremissubrv 25083* Addition of complex vectors. (Contributed by FL, 29-May-2014.)
 |-  - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  B  e.  ( CC  ^m  ( 1 ... N ) ) )  ->  ( A - w B )  =  ( x  e.  ( 1 ... N )  |->  ( ( A `
  x )  -  ( B `  x ) ) ) )
 
Theoremsubclcvd 25084 Closure law for vector substraction. (Contributed by FL, 15-Sep-2013.)
 |-  - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  V  e.  ( CC  ^m  ( 1 ... N ) ) )  ->  ( U - w V )  e.  ( CC  ^m  ( 1 ... N ) ) )
 
Theoremsubclrvd 25085 Closure law for vector substraction. (Contributed by FL, 29-May-2014.)
 |-  - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ...
 N ) )  /\  V  e.  ( RR  ^m  ( 1 ... N ) ) )  ->  ( U - w V )  e.  ( RR  ^m  ( 1 ... N ) ) )
 
Syntaxcnegcv 25086 Extends class notation with the negative of a complex vector.
 class  - cv
 
Definitiondf-ucv 25087* Negative of a complex vector. Experimental. (Contributed by FL, 15-Sep-2013.)
 |-  - cv  =  ( n  e.  NN  |->  ( u  e.  ( CC  ^m  ( 1 ... n ) )  |->  ( ( 0 cv `  n ) (  - cv  `  n ) u ) ) )
 
Theoremisucv 25088 Negative of a complex vector. (Contributed by FL, 15-Sep-2013.)
 |-  ~ w  =  ( - cv `  N )   &    |-  0 w  =  ( 0 cv `  N )   &    |- 
 - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) ) ) 
 ->  ( ~ w `  U )  =  (
 0 w - w U ) )
 
Theoremisucvr 25089 Negative of a complex vector. (Contributed by FL, 29-May-2014.)
 |-  ~ w  =  ( - cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) ) ) 
 ->  ( ~ w `  U )  e.  ( CC  ^m  ( 1 ...
 N ) ) )
 
Syntaxcsmcv 25090 Extends class notation with scalar multiplication of complex vectors.
 class  . cv
 
Definitiondf-mulcv 25091* Multiplication of complex vectors by a scalar in a space of dimension  n. Experimental. (Contributed by FL, 15-Sep-2013.)
 |-  . cv  =  ( n  e.  NN  |->  ( s  e. 
 CC ,  u  e.  ( CC  ^m  (
 1 ... n ) ) 
 |->  ( x  e.  (
 1 ... n )  |->  ( s  x.  ( u `
  x ) ) ) ) )
 
Theoremismulcv 25092* Multiplication of complex vectors by a scalar in a space of dimension  n. (Contributed by FL, 15-Sep-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  ( 1
 ... N ) ) )  ->  ( S . t U )  =  ( x  e.  (
 1 ... N )  |->  ( S  x.  ( U `
  x ) ) ) )
 
Theoremclsmulcv 25093 Closure of scalar multiplication. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  ( 1
 ... N ) ) )  ->  ( S . t U )  e.  ( CC  ^m  (
 1 ... N ) ) )
 
Theoremclsmulrv 25094 Closure of scalar multiplication. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  RR  /\  U  e.  ( RR  ^m  ( 1
 ... N ) ) )  ->  ( S . t U )  e.  ( RR  ^m  (
 1 ... N ) ) )
 
Theoremfnmulcv 25095 Functionality of scalar multiplication. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( N  e.  NN  ->  . t :
 ( CC  X.  ( CC  ^m  ( 1 ...
 N ) ) ) --> ( CC  ^m  (
 1 ... N ) ) )
 
Theoremmulone 25096 Multiplication of a vector by 1. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  (
 1 ... N ) ) )  ->  ( 1 . t U )  =  U )
 
Theoremvecscmonto 25097 Vector addition is onto. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( N  e.  NN  ->  . t :
 ( CC  X.  ( CC  ^m  ( 1 ...
 N ) ) )
 -onto-> ( CC  ^m  (
 1 ... N ) ) )
 
Theoremmulmulvec 25098 Connection between multiplication of complex numbers and scalar multiplication. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  CC  /\  ( T  e.  CC  /\  U  e.  ( CC  ^m  (
 1 ... N ) ) ) )  ->  (
 ( S  x.  T ) . t U )  =  ( S . t ( T . t U ) ) )
 
Theoremdistmlva 25099 Distribution of scalar multiplication over vector addition. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   &    |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  CC  /\  ( U  e.  ( CC  ^m  ( 1 ... N ) )  /\  V  e.  ( CC  ^m  ( 1
 ... N ) ) ) )  ->  ( S . t ( U + w V ) )  =  ( ( S . t U ) + w ( S . t V ) ) )
 
Theoremdistsava 25100 "Distribution" of scalar addition. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   &    |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  CC  /\  ( T  e.  CC  /\  U  e.  ( CC  ^m  (
 1 ... N ) ) ) )  ->  (
 ( S  +  T ) . t U )  =  ( ( S . t U ) + w ( T . t U ) ) )
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