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Theorem List for Metamath Proof Explorer - 24701-24800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhmeogrplem 24701* Lemma for hmeogrp 24703. (Contributed by FL, 30-May-2014.)
 |-  G  =  ( x  e.  ( J  Homeo  J ) ,  y  e.  ( J 
 Homeo  J )  |->  ( x  o.  y ) )   =>    |-  ( ( A  e.  ( J  Homeo  J ) 
 /\  B  e.  ( J  Homeo  J ) ) 
 ->  ( A G B )  =  ( A  o.  B ) )
 
Theoremhmeogrpi 24702* Lemma for hmeogrp 24703. (Contributed by FL, 31-May-2014.)
 |-  G  =  ( x  e.  ( J  Homeo  J ) ,  y  e.  ( J 
 Homeo  J )  |->  ( x  o.  y ) )   &    |-  J  e.  Top   =>    |-  G  e.  GrpOp
 
Theoremhmeogrp 24703* Homeomorphisms on a topology  J is a group for composition. This means from Felix Klein's point of view that a set equipped with a topology is a geometry, namely the so-called rubber sheet geometry. (Contributed by FL, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 31-May-2014.)
 |-  G  =  ( x  e.  ( J  Homeo  J ) ,  y  e.  ( J 
 Homeo  J )  |->  ( x  o.  y ) )   =>    |-  ( J  e.  Top  ->  G  e.  GrpOp )
 
16.11.30  Initial and final topologies
 
Theoremintopcoaconlem3b 24704* The underlying set of the initial topology is the domain of the mappings  F. (Contributed by FL, 24-Apr-2012.) (Revised by Mario Carneiro, 25-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   =>    |-  ( ( ( I  e.  A  /\  X  e.  B )  /\  I  =/= 
 (/)  /\  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K ) )  ->  U. J  =  X )
 
Theoremintopcoaconlem3 24705* The underlying set of the initial topology is the domain of the mappings  F. (Contributed by FL, 21-Apr-2012.) (Revised by Mario Carneiro, 25-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   &    |-  I  e.  A   &    |-  X  e.  B   &    |-  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K )   &    |-  I  =/=  (/)   =>    |- 
 U. J  =  X
 
Theoremintopcoaconb 24706* The initial topology is the coarsest one making the functions  F continuous . (Contributed by FL, 14-May-2012.) (Revised by Mario Carneiro, 26-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   &    |-  I  e.  A   &    |-  X  e.  B   &    |-  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K )   &    |-  I  =/=  (/)   =>    |-  J  =  |^| { t  e.  Top  |  A. i  e.  I  F  e.  ( t  Cn  K ) }
 
Theoremintopcoaconc 24707* The initial topology makes the functions  F continuous. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 26-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   &    |-  I  e.  A   &    |-  X  e.  B   &    |-  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K )   &    |-  I  =/=  (/)   =>    |- 
 A. i  e.  I  F  e.  ( J  Cn  K )
 
Theoremqusp 24708* A quotient space is a topology. (Contributed by FL, 4-Jun-2007.)
 |-  X  =  U. J   &    |-  R  Er  A   =>    |-  ( J  e.  Top  ->  { x  |  ( x  C_  ( X /. R )  /\  U. x  e.  J ) }  e.  Top )
 
Theoremintcont 24709 If  F is continous over two topologies  J and  K then it is continuous over  ( J  i^i  K
). (Contributed by FL, 27-Nov-2011.)
 |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  /\  U. J  =  U. K  /\  ( F  e.  ( J  Cn  L )  /\  F  e.  ( K  Cn  L ) ) ) 
 ->  F  e.  ( ( J  i^i  K )  Cn  L ) )
 
Syntaxctopx 24710 Extend class notation with a function whose value is a product topology.
 class  topX
 
Definitiondf-prtop 24711* The product topology of a family  f of topologies is the coarsest topology over the product of the underlying sets that makes the projections continuous. (Bourbaki TG I.14 ex. 3) Experimental. (Contributed by FL, 4-Dec-2011.)
 |-  topX  =  { <. f ,  y >.  |  ( f : dom  f --> Top  /\  y  =  |^| { t  e.  Top  |  ( U. t  =  X_ x  e. 
 dom  f U. (
 f `  x )  /\  A. i  e.  dom  f ( X_ x  e.  dom  f U. (
 f `  x )  pr  i )  e.  (
 t  Cn  ( f `  i ) ) ) } ) }
 
Theoremusptoplem 24712* Lemma for usptop 24716. (Contributed by FL, 5-Dec-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  ~P X_ x  e.  I  U. ( F `  x )  e.  { t  e.  Top  |  ( U. t  =  X_ x  e.  I  U. ( F `
  x )  /\  A. i  e.  I  (
 X_ x  e.  I  U. ( F `  x )  pr  i )  e.  ( t  Cn  ( F `  i ) ) ) } )
 
Theoremistopx 24713* Definition of the product topology of a family of topologies  F. (Contributed by FL, 4-Dec-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  (  topX  `  F )  =  |^| { t  e.  Top  |  ( U. t  =  X_ x  e.  I  U. ( F `
  x )  /\  A. i  e.  I  (
 X_ x  e.  I  U. ( F `  x )  pr  i )  e.  ( t  Cn  ( F `  i ) ) ) } )
 
Theoremistopxc 24714* 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 24715 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 24716* 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 24717* 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 ) ) )
 
16.11.31  Filters
 
Theoremefilcp 24718* 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 24719 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 24720* 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 24721* 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 24722* 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 24723* A condition for a filter to be finer than another filter. Compare fgss2 17401. (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 ) )
 
16.11.32  Limits
 
Theoremplimfil 24724 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 24725 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 24726* 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 24727  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 24728  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 24729* 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 24730 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 24731 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 24732 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 24733 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 24734  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 24735* 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 24736 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 24737  G  o.  F tends to  G ( L1 ) if  G is continuous at point  L1 and  F tends to  L1. (cmptdst 24734 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 24738* 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 4626 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 24739 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 24740 Lemma for limptlimpr 24742. (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 24741 Lemma for limptlimpr 24742. (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 24742 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 24743* 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 24744 Extend the definition of a class to include the limit of a function relatively to a subspace.
 class  fLimfrs
 
Definitiondf-flimfrs 24745* 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 17419). The set  a can't be empty since its closure is not empty ( see cldifemp 24690). 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 24746 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 24745) . (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 24747 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 24748 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 24749 Extend class notation to include isolated points.
 class  IsolatedPt
 
Definitiondf-islpt 24750* 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 } } ) )
 
16.11.33  Uniform spaces
 
Syntaxcunifsp 24751 Extend class notation with the class of all uniform spaces.
 class  UnifSp
 
Definitiondf-unifsp 24752* 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 } ) } )
 ) ) }
 
16.11.34  Separated spaces: T0, T1, T2 (Hausdorff) ...
 
Theoremhst1 24753 A Hausdorff space is a T1 space. (Contributed by FL, 18-Jun-2007.)
 |-  Haus  C_ 
 Fre
 
Theoremdtt1 24754 A discrete topology is T1. Morris, Topology without tears. (Contributed by FL, 8-Jun-2007.)
 |-  A  e.  _V   =>    |- 
 ~P A  e.  Fre
 
16.11.35  Compactness
 
Theoremindcomp 24755 The indiscrete topology is compact. (Contributed by FL, 2-Aug-2009.)
 |-  { (/) ,  A }  e.  Comp
 
Theoremtopunfincomp 24756 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 24757 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 24758* 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 ) )
 
16.11.36  Connectedness
 
Theoremsingempcon 24759 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 24760 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 24761 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 24762* 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 ) ) ) ) )
 
16.11.37  Topological fields
 
Syntaxctopfld 24763 Extend class notation to include TopFld.
 class  TopFld
 
Definitiondf-topfld 24764* 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 )
 ) }
 
16.11.38  Standard topology on RR
 
Theoremintrn 24765 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 24766* 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 )
 ) )
 
16.11.39  Standard topology of intervals of RR
 
Theoremstoi 24767 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
 
16.11.40  Cantor's set
 
Theoremcntrset 24768* 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 )
 
16.11.41  Pre-calculus and Cartesian geometry
 
Theoremdmse1 24769 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 24770 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 24771 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 24772 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 24773 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 24774 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 24775 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 24776* Lemma for iint 24778. (Contributed by FL, 27-Dec-2007.)
 |-  (
 ( A  e.  RR  /\  y  e.  |^|_ x  e.  RR+  ( ( A  -  x ) (,) ( A  +  x )
 ) )  ->  (
 y  e.  RR  ->  y  =  A ) )
 
Theoremiintlem2 24777* Lemma for iint 24778. (Contributed by FL, 23-Dec-2007.)
 |-  (
 y  e.  |^|_ x  e.  RR+  ( ( A  -  x ) (,) ( A  +  x )
 )  ->  y  e.  RR )
 
Theoremiint 24778* 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 24779* Domain of a translation. (Contributed by FL, 17-Feb-2008.)
 |-  F  =  ( x  e.  RR  |->  ( x  +  A ) )   =>    |-  ( A  e.  RR  ->  dom  F  =  RR )
 
Theoremtrran 24780* Range of a translation. (Contributed by FL, 17-Feb-2008.)
 |-  F  =  ( x  e.  RR  |->  ( x  +  A ) )   =>    |-  ( A  e.  RR  ->  ran  F  =  RR )
 
Theoremtrnij 24781* 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 24782* 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 24783 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 24784 Transitive law for ordering on extended reals ( compare xrletr 10368). (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  (
 ( A  <_  B  /\  B  <_  C )  ->  A  <_  C )
 
16.11.42  Extended Real numbers
 
Theoremnolimf 24785* 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 24786* 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 24787* 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 24788 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 24789 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 24790 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 24791* 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 24792* 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 24793* 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 24794* 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 24795 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 24796* 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 24797 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 24798 Extend class notation to include the supremum of the class B.
 class  sup _  x  e.  A B
 
Syntaxclinfp 24799 Extend class notation to include the infimum of the class B.
 class  inf _  x  e.  A B
 
Definitiondf-supp 24800 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 ) )
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