HomeHome Metamath Proof Explorer
Theorem List (p. 258 of 328)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21514)
  Hilbert Space Explorer  Hilbert Space Explorer
(21515-23037)
  Users' Mathboxes  Users' Mathboxes
(23038-32776)
 

Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-topfld 25701* 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 25702 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 25703* 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 25704 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 25705* 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 25706 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 25707 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 25708 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 25709 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 25710 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 25711 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 25712 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 25713* Lemma for iint 25715. (Contributed by FL, 27-Dec-2007.)
 |-  (
 ( A  e.  RR  /\  y  e.  |^|_ x  e.  RR+  ( ( A  -  x ) (,) ( A  +  x )
 ) )  ->  (
 y  e.  RR  ->  y  =  A ) )
 
Theoremiintlem2 25714* Lemma for iint 25715. (Contributed by FL, 23-Dec-2007.)
 |-  (
 y  e.  |^|_ x  e.  RR+  ( ( A  -  x ) (,) ( A  +  x )
 )  ->  y  e.  RR )
 
Theoremiint 25715* 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 25716* Domain of a translation. (Contributed by FL, 17-Feb-2008.)
 |-  F  =  ( x  e.  RR  |->  ( x  +  A ) )   =>    |-  ( A  e.  RR  ->  dom  F  =  RR )
 
Theoremtrran 25717* Range of a translation. (Contributed by FL, 17-Feb-2008.)
 |-  F  =  ( x  e.  RR  |->  ( x  +  A ) )   =>    |-  ( A  e.  RR  ->  ran  F  =  RR )
 
Theoremtrnij 25718* 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 25719* 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 25720 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 25721 Transitive law for ordering on extended reals ( compare xrletr 10505). (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 25722* 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 25723* 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 25724* 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 25725 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 25726 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 25727 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 25728* 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 25729* 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 25730* 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 25731* 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 25732 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 25733* 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 25734 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 25735 Extend class notation to include the supremum of the class B.
 class  sup _  x  e.  A B
 
Syntaxclinfp 25736 Extend class notation to include the infimum of the class B.
 class  inf _  x  e.  A B
 
Definitiondf-supp 25737 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 25738 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 25739* 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 25740 Extends class notation with Frechet's filter.
 class  Frf
 
Definitiondf-frf 25741* 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 25742* 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 25743 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 25744* 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 25745 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 25746* 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 25747 Extends class notation with addition of complex vectors.
 class  + cv
 
Definitiondf-addcv 25748* 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 25749* 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 25750 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 25751 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 25752 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 25753 Extends class notation with null vector.
 class  0 cv
 
Definitiondf-nullcv 25754* 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 25755* 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 25756 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 25757 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 25758 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 25759 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 25760 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 25761 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 25762 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 25763* 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 25764* 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 25765* 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 25766* 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 25767 Extends class notation with substraction of complex vectors.
 class  - cv
 
Definitiondf-subcatv 25768* 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 25769 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 25770 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 25771* 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 25772* 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 25773* 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 25774 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 25775* 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 25776 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 25777 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 25778 Extends class notation with the negative of a complex vector.
 class  - cv
 
Definitiondf-ucv 25779* 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 25780 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 25781 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 25782 Extends class notation with scalar multiplication of complex vectors.
 class  . cv
 
Definitiondf-mulcv 25783* 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 25784* 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 25785 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 25786 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 25787 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 25788 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 25789 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 25790 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 25791 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 25792 "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 ) ) )
 
Theoremtcnvec 25793 Nuples of complex numbers has a structure of vector space. (Contributed by FL, 29-May-2014.)
 |-  + w  =  (  + cv `  N )   &    |-  . t  =  ( . cv `  N )   =>    |-  ( N  e.  NN  ->  <. + w ,  . t >.  e.  CVec OLD )
 
Syntaxcdivcv 25794 Extends class notation with scalar division of complex vectors.
 class  / cv
 
Definitiondf-divcv 25795* Division of a complex vector by a scalar in a space of dimension  n. Experimental. (Contributed by FL, 29-May-2014.)
 |-  / cv  =  ( n  e.  NN  |->  ( u  e.  ( CC  ^m  ( 1 ... n ) ) ,  s  e.  ( CC  \  { 0 } )  |->  ( ( 1  /  s ) ( .
 cv `  n ) u ) ) )
 
Theoremisdivcv2 25796 Division of complex vectors by a scalar in a space of dimension  N. (Contributed by FL, 29-May-2014.)
 |-  / t  =  ( / cv `  N )   &    |- 
 . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  S  e.  ( CC  \  { 0 } )
 )  ->  ( U / t S )  =  ( ( 1  /  S ) . t U ) )
 
Theoremdivclcvd 25797 Closure law for vector division by a scalar. (Contributed by FL, 29-May-2014.)
 |-  / t  =  ( / cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  S  e.  ( CC  \  { 0 } )
 )  ->  ( U / t S )  e.  ( CC  ^m  (
 1 ... N ) ) )
 
Theoremdivclrvd 25798 Closure law for vector division by a scalar. (Contributed by FL, 29-May-2014.)
 |-  / t  =  ( / cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ...
 N ) )  /\  S  e.  ( RR  \  { 0 } )
 )  ->  ( U / t S )  e.  ( RR  ^m  (
 1 ... N ) ) )
 
18.13.44  Calculus
 
Syntaxcintvl 25799 Extend class notation to include intervals.
 class  Intvl
 
Definitiondf-intvl 25800 The intervals of  RR. (Contributed by FL, 29-May-2014.)
 |-  Intvl  =  ( ( ran  (,)  u.  ( ran  (,]  u.  ( ran  [,)  u.  ran  [,] ) ) )  i^i 
 ~P RR )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32776
  Copyright terms: Public domain < Previous  Next >