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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremistopxc 24901* Product of  I topology  J. (Contributed by FL, 15-Sep-2013.)
 |-  X  =  U. J   =>    |-  ( ( F : I
 --> { J }  /\  I  e.  A  /\  J  e.  Top )  ->  (  topX  `  F )  =  |^| { t  e. 
 Top  |  ( U. t  =  ( X  ^m  I )  /\  A. i  e.  I  (
 ( X  ^m  I
 )  pr  i )  e.  ( t  Cn  J ) ) }
 )
 
Theoremprtoptop 24902 The product topology of a family  F of topologies is a topology. (Contributed by FL, 5-Dec-2011.) (Proof shortened by Mario Carneiro, 26-Jan-2015.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  (  topX  `  F )  e.  Top )
 
Theoremusptop 24903* The underlying set of a product topology. (Contributed by FL, 5-Dec-2011.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  U. (  topX  `  F )  =  X_ x  e.  I  U. ( F `
  x ) )
 
Theoremprcnt 24904* The projections are continuous. (Contributed by FL, 18-Apr-2012.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  A. i  e.  I  ( X_ x  e.  I  U. ( F `  x )  pr  i )  e.  ( (  topX  `  F )  Cn  ( F `  i ) ) )
 
16.12.31  Filters
 
Theoremefilcp 24905* A filter containing a set  A exists iff  A has the finite intersection property (i.e. no finite intersection of elements of  A is empty). Bourbaki TG I.37 prop. 1. (Contributed by FL, 20-Nov-2007.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  (
 ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  ->  ( -.  (/)  e.  ( fi
 `  A )  <->  E. x  e.  ( Fil `  B ) A 
 C_  x ) )
 
Theoremfilint2 24906 A filter is closed under taking finite intersections. (Contributed by FL, 27-Apr-2008.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  (
 ( A  C_  F  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  |^| A  e.  F ) )
 
Theoremfisub 24907* If a set has the finite intersection property, its subsets have also this property. (Contributed by FL, 27-Apr-2008.)
 |-  B  =  { z  |  E. y ( y  C_  A  /\  y  e.  Fin  /\  z  =  |^| y
 ) }   &    |-  D  =  {
 z  |  E. y
 ( y  C_  C  /\  y  e.  Fin  /\  z  =  |^| y
 ) }   =>    |-  ( C  C_  A  ->  ( -.  (/)  e.  B  ->  -.  (/)  e.  D ) )
 
Theoremfgsb2 24908* Filter generated by a subbasis  A. Bourbaki TG I.37 paragraph above prop. 1. (The theorem has been slightly modified because the definitions of the empty set are different in Bourbaki and Metamath.) (Contributed by FL, 27-Apr-2008.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  (
 ( A  C_  ~P X  /\  X  e.  _V  /\  A  =/=  (/) )  ->  ( -.  (/)  e.  ( fi
 `  A )  ->  { x  e.  ~P X  |  E. y  e.  ( fi `  A ) y  C_  x }  e.  ( Fil `  X ) ) )
 
Theoremcnfilca 24909* Condition to have a filter finer than a given filter and containing a set  A. Bourbaki T.G. I.37 cor. 1 (Contributed by FL, 27-Apr-2008.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  (
 ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/= 
 (/) )  ->  ( E. g  e.  ( Fil `  X ) ( A  e.  g  /\  F  C_  g )  <->  A. x  e.  F  ( x  i^i  A )  =/=  (/) ) )
 
Theoremfil2ss 24910* A condition for a filter to be finer than another filter. Compare fgss2 17517. (Contributed by FL, 8-Aug-2011.) (Revised by Mario Carneiro, 6-Aug-2015.)
 |-  (
 ( F  e.  ( Fil `  X )  /\  G  e.  ( Fil `  X ) )  ->  ( F  C_  G  <->  A. x  e.  F  E. y  e.  G  y  C_  x ) )
 
16.12.32  Limits
 
Theoremplimfil 24911 The predicate "is a limit of a filter". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  X )  /\  L  e.  X )  ->  ( L  e.  ( J  fLim  F )  <->  ( ( nei `  J ) `  { L } )  C_  F ) )
 
Theoremlimvinlv 24912 The limit value of a convergent function whose values are in a Hausdorff space belongs to the set of the limit values. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   &    |-  A  =  U. ( ( J  fLimf  L ) `  F )   =>    |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  ( ( J 
 fLimf  L ) `  F )  =/=  (/) )  ->  A  e.  ( ( J  fLimf  L ) `  F ) )
 
Theoremflfneih 24913* A neighborhood of the limit value of a convergent function whose values are in a Hausdorff space contains the image of a filter element. (Contributed by FL, 18-Nov-2010.) (Revised by Mario Carneiro, 6-Aug-2015.)
 |-  X  =  U. J   &    |-  A  =  U. ( ( J  fLimf  L ) `  F )   =>    |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  ( ( J 
 fLimf  L ) `  F )  =/=  (/)  /\  N  e.  ( ( nei `  J ) `  { A }
 ) )  ->  E. s  e.  L  ( F "
 s )  C_  N )
 
Theoremlimfilnei 24914  A is a limit of the filter of the neighborhoods of  A. (Contributed by FL, 27-May-2011.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   &    |-  F  =  ( ( nei `  J ) `  { A }
 )   =>    |-  ( ( J  e.  Top  /\  A  e.  X ) 
 ->  A  e.  ( J 
 fLim  F ) )
 
Theoremconttnf2 24915  F is continous at point  A iff  ( F `  A ) is a limit of the image filter of the neighborhoods of  A. (Contributed by FL, 7-Aug-2011.) (Revised by Mario Carneiro, 6-Aug-2015.)
 |-  L  =  ( ( nei `  J ) `  { A }
 )   &    |-  X  =  U. K   &    |-  Y  =  U. J   =>    |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
 ( F `  A )  e.  ( K  fLim  ( ( X  FilMap  F ) `  L ) ) ) )
 
Theoremiscnp4 24916* The predicate " F is a continuous function from topology  J to topology  K at point  P." in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  (
 ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X ) 
 ->  ( F  e.  (
 ( J  CnP  K ) `  P )  <->  ( F : X
 --> Y  /\  A. y  e.  ( ( nei `  K ) `  { ( F `
  P ) }
 ) E. x  e.  ( ( nei `  J ) `  { P }
 ) ( F " x )  C_  y ) ) )
 
Theoremcnpflf4 24917 If  G is continuous at point  A, and the filter base  F converges to  A then  G ( F ) converges to  G ( A ). Bourbaki TG I.50 cor 1. (Contributed by FL, 19-Sep-2011.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( K  e.  Top  /\  F  e.  ( fBas `  X )
 )  /\  ( G  e.  ( ( J  CnP  K ) `  A ) 
 /\  A  e.  ( J  fLim  ( X filGen F ) ) ) ) 
 ->  ( G `  A )  e.  ( K  fLim  ( ( Y  FilMap  G ) `  F ) ) )
 
Theoremlimfn 24918 The limits of a function are elements of its range. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( ( J  fLimf  L ) `  F ) 
 C_  X )
 
Theoremlimfn2 24919 If  A is a limit of a function  F,  A is an element of the range of  F. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  ->  A  e.  X )
 )
 
Theoremlimfn3 24920 If  A is the limit of a convergent function in a Hausdorff space,  A is an element of the range of the function. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 7-Aug-2015.)
 |-  X  =  U. J   &    |-  A  =  U. ( ( J  fLimf  L ) `  F )   =>    |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( ( ( J 
 fLimf  L ) `  F )  =/=  (/)  ->  A  e.  X ) )
 
Theoremcmptdst 24921  G  o.  F tends to  G ( A ) if  G is continuous at point 
A and  F tends to A . Bourbaki TG I.50 cor. 2. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  Y  =  U. J   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  ( Fil `  X ) )  /\  ( F : X --> Y  /\  A  e.  ( ( J  fLimf  L ) `  F )  /\  G  e.  ( ( J  CnP  K ) `  A ) ) )  ->  ( G `  A )  e.  ( ( K  fLimf  L ) `  ( G  o.  F ) ) )
 
Theoremunexun 24922* If  A is an element of  B and  B has a unique element,  U. B  =  A. (Contributed by FL, 15-Oct-2012.)
 |-  (
 ( A  e.  B  /\  E! x  x  e.  B )  ->  U. B  =  A )
 
Theoremlimhun 24923 In a Hausdorff space if  A is a limit of a convergent function  F, then  A is the unique limit of  F. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  X  =  U. J   &    |-  L1  =  U. ( ( J  fLimf  L ) `  F )   =>    |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  A  e.  (
 ( J  fLimf  L ) `
  F ) ) 
 ->  L1  =  A )
 
Theoremcmptdst2 24924  G  o.  F tends to  G ( L1 ) if  G is continuous at point  L1 and  F tends to  L1. (cmptdst 24921 in the Hausdorff case.) Bourbaki TG I.50 cor. 2. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  Y  =  U. J   &    |-  L1  =  U. ( ( J  fLimf  L ) `  F )   &    |-  L 2  =  U. (
 ( K  fLimf  L ) `
  ( G  o.  F ) )   =>    |-  ( ( ( J  e.  Haus  /\  K  e.  Haus  /\  L  e.  ( Fil `  X )
 )  /\  ( F : X --> Y  /\  (
 ( J  fLimf  L ) `
  F )  =/=  (/)  /\  G  e.  (
 ( J  CnP  K ) `  L1 ) ) ) 
 ->  L 2  =  ( G `  L1 )
 )
 
Theoremexopcopn 24925* For every neighborhood  N of  <. A ,  B >. in a product topology, there exist two open sets  u and  v of the component topologies so that  ( u  X.  v ) is an open neighborhood of  <. A ,  B >. and a part of  N. (Use opelxp 4693 to have  A  e.  u and  B  e.  v.) (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  T  =  ( R  tX  S )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( A  e.  C  /\  B  e.  D )  /\  N  e.  ( ( nei `  T ) `  { <. A ,  B >. } ) ) 
 ->  E. u  e.  R  E. v  e.  S  ( <. A ,  B >.  e.  ( u  X.  v )  /\  ( u  X.  v )  C_  N ) )
 
Theoremprdnei 24926 The product of two neighborhoods is a neighborhood. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 10-Jun-2014.)
 |-  T  =  ( R  tX  S )   &    |-  X  =  U. R   &    |-  Y  =  U. S   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( A  e.  X  /\  B  e.  Y )  /\  ( U  e.  ( ( nei `  R ) `  { A } )  /\  V  e.  ( ( nei `  S ) `  { B } ) ) )  ->  ( U  X.  V )  e.  (
 ( nei `  T ) `  { <. A ,  B >. } ) )
 
Theoremlimptlimpr2lem1 24927 Lemma for limptlimpr 24929. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 9-Aug-2015.)
 |-  T  =  ( R  tX  S )   &    |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  Z  =  ( X  X.  Y )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top  /\  L  e.  ( Fil `  W ) )  /\  ( F : W --> Z  /\  L 2  e.  A ) )  ->  ( <. L1 ,  L 2 >.  e.  ( ( T  fLimf  L ) `  F ) 
 ->  ( L1  e.  (
 ( R  fLimf  L ) `
  ( 1st  o.  F ) )  /\  L 2  e.  ( ( S  fLimf  L ) `  ( 2nd  o.  F ) ) ) ) )
 
Theoremlimptlimpr2lem2 24928 Lemma for limptlimpr 24929. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 9-Aug-2015.)
 |-  T  =  ( R  tX  S )   &    |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  Z  =  ( X  X.  Y )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top  /\  L  e.  ( Fil `  W ) )  /\  F : W --> Z ) 
 ->  ( ( L1  e.  ( ( R  fLimf  L ) `  ( 1st 
 o.  F ) ) 
 /\  L 2  e.  ( ( S  fLimf  L ) `  ( 2nd 
 o.  F ) ) )  ->  <. L1 ,  L 2 >.  e.  ( ( T  fLimf  L ) `  F ) ) )
 
Theoremlimptlimpr 24929 A limit in a product topology exists iff the limits of the projections exist. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 9-Aug-2015.)
 |-  T  =  ( R  tX  S )   &    |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  Z  =  ( X  X.  Y )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top  /\  L  e.  ( Fil `  W ) )  /\  ( F : W --> Z  /\  L 2  e.  A ) )  ->  ( <. L1 ,  L 2 >.  e.  ( ( T  fLimf  L ) `  F )  <-> 
 ( L1  e.  (
 ( R  fLimf  L ) `
  ( 1st  o.  F ) )  /\  L 2  e.  ( ( S  fLimf  L ) `  ( 2nd  o.  F ) ) ) ) )
 
Theoremflfnei2 24930* The property of being a limit point of a function in terms of filter and of preimage of a neighborhood. (Contributed by FL, 13-Dec-2013.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J ) `  { A }
 ) ( `' F " n )  e.  L ) ) )
 
Syntaxcflimfrs 24931 Extend the definition of a class to include the limit of a function relatively to a subspace.
 class  fLimfrs
 
Definitiondf-flimfrs 24932* Gives the limits of a function  f : a --> U. j at a point  p  e.  ( ( cls `  k
) `  a ) relatively to a subspace  a of a topology  k. ( The condition  p  e.  ( ( cls `  k
) `  a ) ensures the traces of the neighborhoods of  p over  a is a filter ( see trnei 17535). The set  a can't be empty since its closure is not empty ( see cldifemp 24877). Experimental. (Contributed by FL, 15-Sep-2013.)
 |-  fLimfrs  =  ( j  e.  Top ,  k  e.  Top  |->  ( a  e.  ~P U. k  |->  ( p  e.  (
 ( cls `  k ) `  a ) ,  f  e.  ( U. j  ^m  a )  |->  ( ( j  fLimf  ( ( ( nei `  k ) `  { p } )t  a
 ) ) `  f
 ) ) ) )
 
Theoremislimrs 24933 The limits of  F at point  P when one only considers the traces of the neighborhoods of  P over  A.  F is a function whose domain is  A. The point  P must belong to  ( ( cls `  K ) `  A
) (see also the comments under df-flimfrs 24932) . (Contributed by FL, 15-Sep-2013.)
 |-  L  =  ( ( ( nei `  K ) `  { P } )t  A )   &    |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  (
 ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  Y  /\  P  e.  ( ( cls `  K ) `  A ) )  /\  F  e.  ( X  ^m  A ) )  ->  ( ( ( J 
 fLimfrs  K ) `  A ) `  <. P ,  F >. )  =  ( ( J  fLimf  L ) `  F ) )
 
Theoremislimrs3 24934 The limits of  F at point  P relatively to  A is a limit of  F at point  P relatively to  B  C_  A. The opposite direction doesn't hold. (Contributed by FL, 13-Dec-2013.)
 |-  L  =  ( ( ( nei `  K ) `  { P } )t  A )   &    |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  B  C_  A )  /\  ( A  C_  Y  /\  P  e.  ( ( cls `  K ) `  B ) )  /\  F  e.  ( X  ^m  A ) )  ->  ( ( ( J 
 fLimfrs  K ) `  A ) `  <. P ,  F >. )  C_  ( (
 ( J  fLimfrs  K ) `
  B ) `  <. P ,  ( F  |`  B ) >. ) )
 
Theoremislimrs4 24935 The limits of  F at point  P relatively to  A is a limit of  F at point  P relatively to  B. (Contributed by FL, 13-Dec-2013.)
 |-  L  =  ( ( ( nei `  K ) `  { P } )t  A )   &    |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  N  e.  ( ( nei `  K ) `  { P } ) ) 
 /\  ( A  C_  Y  /\  P  e.  (
 ( cls `  K ) `  A ) )  /\  F  e.  ( X  ^m  A ) )  ->  ( ( ( J 
 fLimfrs  K ) `  A ) `  <. P ,  F >. )  =  ( ( ( J  fLimfrs  K ) `
  ( A  i^i  N ) ) `  <. P ,  ( F  |`  ( A  i^i  N ) )
 >. ) )
 
Syntaxcisopt 24936 Extend class notation to include isolated points.
 class  IsolatedPt
 
Definitiondf-islpt 24937* Definition of an isolated point. Experimental. (Contributed by FL, 16-Sep-2013.)
 |-  IsolatedPt  =  (
 j  e.  Top  |->  ( a  e.  ~P U. j  |->  { x  e.  a  |  E. v  e.  (
 ( nei `  j ) `  { x } )
 ( v  i^i  a
 )  =  { x } } ) )
 
16.12.33  Uniform spaces
 
Syntaxcunifsp 24938 Extend class notation with the class of all uniform spaces.
 class  UnifSp
 
Definitiondf-unifsp 24939* Definition of a uniform space. Bourbaki TG II.1 def. 1. A uniform structure is used to give a generalization of the idea of Cauchy's sequence. We consider the space is equipped with the topology induced by the uniform structure. (Contributed by FL, 29-May-2014.)
 |-  UnifSp  =  {
 f  |  E. y E. x E. t ( ( y  =  (
 Base `  f )  /\  x  =  ( Unif `  f )  /\  t  =  ( TopOpen `  f )
 )  /\  ( x  C_ 
 ~P ( y  X.  y )  /\  ( y  X.  y )  e.  x  /\  A. v  e.  x  ( A. u  e.  ~P  (
 y  X.  y )
 ( v  C_  u  ->  u  e.  x ) 
 /\  A. w  e.  x  ( v  i^i  w )  e.  x  /\  (
 (  _I  |`  y ) 
 C_  v  /\  `' v  e.  x  /\  E. w  e.  x  ( w  o.  w ) 
 C_  v ) ) )  /\  t  =  ( iota_ s  e.  Top ( U. s  =  y 
 /\  A. a  e.  y  ( ( nei `  s
 ) `  { a } )  =  {
 z  |  E. u  e.  x  z  =  ( u " { a } ) } )
 ) ) }
 
16.12.34  Separated spaces: T0, T1, T2 (Hausdorff) ...
 
Theoremhst1 24940 A Hausdorff space is a T1 space. (Contributed by FL, 18-Jun-2007.)
 |-  Haus  C_ 
 Fre
 
Theoremdtt1 24941 A discrete topology is T1. Morris, Topology without tears. (Contributed by FL, 8-Jun-2007.)
 |-  A  e.  _V   =>    |- 
 ~P A  e.  Fre
 
16.12.35  Compactness
 
Theoremindcomp 24942 The indiscrete topology is compact. (Contributed by FL, 2-Aug-2009.)
 |-  { (/) ,  A }  e.  Comp
 
Theoremtopunfincomp 24943 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 24944 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 24945* 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 ) )
 
16.12.36  Connectedness
 
Theoremsingempcon 24946 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 24947 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 24948 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 24949* 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 ) ) ) ) )
 
16.12.37  Topological fields
 
Syntaxctopfld 24950 Extend class notation to include TopFld.
 class  TopFld
 
Definitiondf-topfld 24951* 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 )
 ) }
 
16.12.38  Standard topology on RR
 
Theoremintrn 24952 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 24953* 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 )
 ) )
 
16.12.39  Standard topology of intervals of RR
 
Theoremstoi 24954 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
 
16.12.40  Cantor's set
 
Theoremcntrset 24955* 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 )
 
16.12.41  Pre-calculus and Cartesian geometry
 
Theoremdmse1 24956 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 24957 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 24958 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 24959 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 24960 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 24961 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 24962 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 24963* Lemma for iint 24965. (Contributed by FL, 27-Dec-2007.)
 |-  (
 ( A  e.  RR  /\  y  e.  |^|_ x  e.  RR+  ( ( A  -  x ) (,) ( A  +  x )
 ) )  ->  (
 y  e.  RR  ->  y  =  A ) )
 
Theoremiintlem2 24964* Lemma for iint 24965. (Contributed by FL, 23-Dec-2007.)
 |-  (
 y  e.  |^|_ x  e.  RR+  ( ( A  -  x ) (,) ( A  +  x )
 )  ->  y  e.  RR )
 
Theoremiint 24965* 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 24966* Domain of a translation. (Contributed by FL, 17-Feb-2008.)
 |-  F  =  ( x  e.  RR  |->  ( x  +  A ) )   =>    |-  ( A  e.  RR  ->  dom  F  =  RR )
 
Theoremtrran 24967* Range of a translation. (Contributed by FL, 17-Feb-2008.)
 |-  F  =  ( x  e.  RR  |->  ( x  +  A ) )   =>    |-  ( A  e.  RR  ->  ran  F  =  RR )
 
Theoremtrnij 24968* 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 24969* 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 24970 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 24971 Transitive law for ordering on extended reals ( compare xrletr 10442). (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  (
 ( A  <_  B  /\  B  <_  C )  ->  A  <_  C )
 
16.12.42  Extended Real numbers
 
Theoremnolimf 24972* 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 24973* 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 24974* 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 24975 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 24976 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 24977 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 24978* 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 24979* 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 24980* 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 24981* 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 24982 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 24983* 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 24984 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 24985 Extend class notation to include the supremum of the class B.
 class  sup _  x  e.  A B
 
Syntaxclinfp 24986 Extend class notation to include the infimum of the class B.
 class  inf _  x  e.  A B
 
Definitiondf-supp 24987 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 24988 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 24989* 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 24990 Extends class notation with Frechet's filter.
 class  Frf
 
Definitiondf-frf 24991* 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 24992* 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 24993 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 24994* 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 24995 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 24996* 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 ) )
 
16.12.43  ( RR ^ N ) and ( CC ^ N )
 
Syntaxcplcv 24997 Extends class notation with addition of complex vectors.
 class  + cv
 
Definitiondf-addcv 24998* 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 24999* 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 25000 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 ) ) )
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