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Theorem List for Intuitionistic Logic Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnconst 12101 A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  ( B  e.  Y  /\  F : X --> { B } ) )  ->  F  e.  ( J  Cn  K ) )
 
Theoremcnrest 12102 Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
 
Theoremcnrest2 12103 Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( K  e.  (TopOn `  Y )  /\  ran 
 F  C_  B  /\  B  C_  Y )  ->  ( F  e.  ( J  Cn  K )  <->  F  e.  ( J  Cn  ( Kt  B ) ) ) )
 
Theoremcnrest2r 12104 Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
 |-  ( K  e.  Top  ->  ( J  Cn  ( Kt  B ) )  C_  ( J  Cn  K ) )
 
Theoremcnptopresti 12105 One direction of cnptoprest 12106 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 31-Mar-2023.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Top )  /\  ( A  C_  X  /\  P  e.  A  /\  F  e.  ( ( J  CnP  K ) `
  P ) ) )  ->  ( F  |`  A )  e.  (
 ( ( Jt  A ) 
 CnP  K ) `  P ) )
 
Theoremcnptoprest 12106 Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 5-Apr-2023.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  A  C_  X )  /\  ( P  e.  (
 ( int `  J ) `  A )  /\  F : X --> Y ) ) 
 ->  ( F  e.  (
 ( J  CnP  K ) `  P )  <->  ( F  |`  A )  e.  ( ( ( Jt  A )  CnP  K ) `  P ) ) )
 
Theoremcnptoprest2 12107 Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : X --> B  /\  B  C_  Y ) ) 
 ->  ( F  e.  (
 ( J  CnP  K ) `  P )  <->  F  e.  (
 ( J  CnP  ( Kt  B ) ) `  P ) ) )
 
Theoremcndis 12108 Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  ->  ( ~P A  Cn  J )  =  ( X  ^m  A ) )
 
Theoremcnpdis 12109 If  A is an isolated point in  X (or equivalently, the singleton  { A } is open in  X), then every function is continuous at  A. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  { A }  e.  J )  ->  ( ( J 
 CnP  K ) `  A )  =  ( Y  ^m  X ) )
 
Theoremlmfpm 12110 If  F converges, then  F is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F ( ~~> t `  J ) P ) 
 ->  F  e.  ( X 
 ^pm  CC ) )
 
Theoremlmfss 12111 Inclusion of a function having a limit (used to ensure the limit relation is a set, under our definition). (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F ( ~~> t `  J ) P ) 
 ->  F  C_  ( CC  X.  X ) )
 
Theoremlmcl 12112 Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F ( ~~> t `  J ) P ) 
 ->  P  e.  X )
 
Theoremlmss 12113 Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
 |-  K  =  ( Jt  Y )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  P  e.  Y )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> Y )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  F ( ~~> t `  K ) P ) )
 
Theoremsslm 12114 A finer topology has fewer convergent sequences (but the sequences that do converge, converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( ~~> t `  K )  C_  ( ~~> t `  J ) )
 
Theoremlmres 12115 A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F  e.  ( X  ^pm  CC ) )   &    |-  ( ph  ->  M  e.  ZZ )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  ( F  |`  ( ZZ>= `  M ) ) ( ~~> t `  J ) P ) )
 
Theoremlmff 12116* If  F converges, there is some upper integer set on which 
F is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  dom  (
 ~~> t `  J ) )   =>    |-  ( ph  ->  E. j  e.  Z  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
 
Theoremlmtopcnp 12117 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
 |-  ( ph  ->  F (
 ~~> t `  J ) P )   &    |-  ( ph  ->  K  e.  Top )   &    |-  ( ph  ->  G  e.  (
 ( J  CnP  K ) `  P ) )   =>    |-  ( ph  ->  ( G  o.  F ) ( ~~> t `  K ) ( G `
  P ) )
 
Theoremlmcn 12118 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)
 |-  ( ph  ->  F (
 ~~> t `  J ) P )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( G  o.  F ) ( ~~> t `  K ) ( G `  P ) )
 
6.1.8  Continuous function-builders
 
Theoremcnmptid 12119* The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   =>    |-  ( ph  ->  ( x  e.  X  |->  x )  e.  ( J  Cn  J ) )
 
Theoremcnmptc 12120* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  P  e.  Y )   =>    |-  ( ph  ->  ( x  e.  X  |->  P )  e.  ( J  Cn  K ) )
 
Theoremcnmpt11 12121* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( y  e.  Y  |->  B )  e.  ( K  Cn  L ) )   &    |-  ( y  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  C )  e.  ( J  Cn  L ) )
 
Theoremcnmpt11f 12122* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  F  e.  ( K  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( F `  A ) )  e.  ( J  Cn  L ) )
 
Theoremcnmpt1res 12123* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
 |-  K  =  ( Jt  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y 
 C_  X )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  Y  |->  A )  e.  ( K  Cn  L ) )
 
6.2  Metric spaces
 
6.2.1  Pseudometric spaces
 
Theorempsmetrel 12124 The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
 |- 
 Rel PsMet
 
Theoremispsmet 12125* Express the predicate " D is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( X  e.  V  ->  ( D  e.  (PsMet `  X )  <->  ( D :
 ( X  X.  X )
 --> RR*  /\  A. x  e.  X  ( ( x D x )  =  0  /\  A. y  e.  X  A. z  e.  X  ( x D y )  <_  (
 ( z D x ) +e ( z D y ) ) ) ) ) )
 
Theorempsmetdmdm 12126 Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  X  =  dom  dom  D )
 
Theorempsmetf 12127 The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  D : ( X  X.  X ) --> RR* )
 
Theorempsmetcl 12128 Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  RR* )
 
Theorempsmet0 12129 The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X )  ->  ( A D A )  =  0 )
 
Theorempsmettri2 12130 Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X )
 )  ->  ( A D B )  <_  (
 ( C D A ) +e ( C D B ) ) )
 
Theorempsmetsym 12131 The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( B D A ) )
 
Theorempsmettri 12132 Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C ) +e ( C D B ) ) )
 
Theorempsmetge0 12133 The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.)
 |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  0  <_  ( A D B ) )
 
Theorempsmetxrge0 12134 The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  D : ( X  X.  X ) --> ( 0 [,] +oo ) )
 
Theorempsmetres2 12135 Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  ( D  |`  ( R  X.  R ) )  e.  (PsMet `  R ) )
 
Theorempsmetlecl 12136 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  RR  /\  ( A D B )  <_  C ) )  ->  ( A D B )  e.  RR )
 
Theoremdistspace 12137 A set  X together with a (distance) function  D which is a pseudometric is a distance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set  X equipped with a distance  D, which is a mapping of two elements of the base set to the (extended) reals and which is nonnegative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021.) (Revised by AV, 5-Jul-2022.)
 |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( ( D :
 ( X  X.  X )
 --> RR*  /\  ( A D A )  =  0 )  /\  ( 0 
 <_  ( A D B )  /\  ( A D B )  =  ( B D A ) ) ) )
 
6.2.2  Basic metric space properties
 
Syntaxcxms 12138 Extend class notation with the class of extended metric spaces.
 class  *MetSp
 
Syntaxcms 12139 Extend class notation with the class of metric spaces.
 class  MetSp
 
Syntaxctms 12140 Extend class notation with the function mapping a metric to the metric space it defines.
 class toMetSp
 
Definitiondf-xms 12141 Define the (proper) class of extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- 
 *MetSp  =  { f  e.  TopSp  |  ( TopOpen `  f )  =  ( MetOpen `  ( ( dist `  f
 )  |`  ( ( Base `  f )  X.  ( Base `  f ) ) ) ) }
 
Definitiondf-ms 12142 Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.)
 |- 
 MetSp  =  { f  e.  *MetSp  |  (
 ( dist `  f )  |`  ( ( Base `  f
 )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f ) ) }
 
Definitiondf-tms 12143 Define the function mapping a metric to the metric space which it defines. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- toMetSp  =  ( d  e.  U. ran  *Met  |->  ( { <. ( Base `  ndx ) , 
 dom  dom  d >. ,  <. (
 dist `  ndx ) ,  d >. } sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >. ) )
 
Theoremmetrel 12144 The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
 |- 
 Rel  Met
 
Theoremxmetrel 12145 The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
 |- 
 Rel  *Met
 
Theoremismet 12146* Express the predicate " D is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( X  e.  A  ->  ( D  e.  ( Met `  X )  <->  ( D :
 ( X  X.  X )
 --> RR  /\  A. x  e.  X  A. y  e.  X  ( ( ( x D y )  =  0  <->  x  =  y
 )  /\  A. z  e.  X  ( x D y )  <_  (
 ( z D x )  +  ( z D y ) ) ) ) ) )
 
Theoremisxmet 12147* Express the predicate " D is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( X  e.  A  ->  ( D  e.  ( *Met `  X )  <->  ( D : ( X  X.  X ) --> RR*  /\  A. x  e.  X  A. y  e.  X  ( ( ( x D y )  =  0  <->  x  =  y
 )  /\  A. z  e.  X  ( x D y )  <_  (
 ( z D x ) +e ( z D y ) ) ) ) ) )
 
Theoremismeti 12148* Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  X  e.  _V   &    |-  D : ( X  X.  X ) --> RR   &    |-  (
 ( x  e.  X  /\  y  e.  X )  ->  ( ( x D y )  =  0  <->  x  =  y
 ) )   &    |-  ( ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( x D y )  <_  ( ( z D x )  +  (
 z D y ) ) )   =>    |-  D  e.  ( Met `  X )
 
Theoremisxmetd 12149* Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  D : ( X  X.  X ) --> RR* )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( ( x D y )  =  0  <-> 
 x  =  y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x D y )  <_  ( ( z D x ) +e
 ( z D y ) ) )   =>    |-  ( ph  ->  D  e.  ( *Met `  X ) )
 
Theoremisxmet2d 12150* It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample:  D ( x ,  y )  =  if ( x  =  y ,  0 , -oo ) satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  D : ( X  X.  X ) --> RR* )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
 0  <_  ( x D y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( ( x D y )  <_ 
 0 
 <->  x  =  y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  /\  ( ( z D x )  e. 
 RR  /\  ( z D y )  e. 
 RR ) )  ->  ( x D y ) 
 <_  ( ( z D x )  +  (
 z D y ) ) )   =>    |-  ( ph  ->  D  e.  ( *Met `  X ) )
 
Theoremmetflem 12151* Lemma for metf 12153 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  ( D : ( X  X.  X ) --> RR  /\  A. x  e.  X  A. y  e.  X  (
 ( ( x D y )  =  0  <-> 
 x  =  y ) 
 /\  A. z  e.  X  ( x D y ) 
 <_  ( ( z D x )  +  (
 z D y ) ) ) ) )
 
Theoremxmetf 12152 Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
 
Theoremmetf 12153 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)
 |-  ( D  e.  ( Met `  X )  ->  D : ( X  X.  X ) --> RR )
 
Theoremxmetcl 12154 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A D B )  e.  RR* )
 
Theoremmetcl 12155 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  RR )
 
Theoremismet2 12156 An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  <->  ( D  e.  ( *Met `  X )  /\  D : ( X  X.  X ) --> RR ) )
 
Theoremmetxmet 12157 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  D  e.  ( *Met `  X ) )
 
Theoremxmetdmdm 12158 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( *Met `  X )  ->  X  =  dom  dom  D )
 
Theoremmetdmdm 12159 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  X  =  dom  dom  D )
 
Theoremxmetunirn 12160 Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( D  e.  U. ran  *Met  <->  D  e.  ( *Met `  dom  dom  D ) )
 
Theoremxmeteq0 12161 The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( ( A D B )  =  0  <->  A  =  B ) )
 
Theoremmeteq0 12162 The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A D B )  =  0  <->  A  =  B ) )
 
Theoremxmettri2 12163 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X ) )  ->  ( A D B )  <_  (
 ( C D A ) +e ( C D B ) ) )
 
Theoremmettri2 12164 Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X )
 )  ->  ( A D B )  <_  (
 ( C D A )  +  ( C D B ) ) )
 
Theoremxmet0 12165 The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X )  ->  ( A D A )  =  0
 )
 
Theoremmet0 12166 The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X ) 
 ->  ( A D A )  =  0 )
 
Theoremxmetge0 12167 The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  0  <_  ( A D B ) )
 
Theoremmetge0 12168 The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  0  <_  ( A D B ) )
 
Theoremxmetlecl 12169 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  RR  /\  ( A D B )  <_  C ) )  ->  ( A D B )  e.  RR )
 
Theoremxmetsym 12170 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A D B )  =  ( B D A ) )
 
Theoremxmetpsmet 12171 An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( D  e.  ( *Met `  X )  ->  D  e.  (PsMet `  X ) )
 
Theoremxmettpos 12172 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( *Met `  X )  -> tpos 
 D  =  D )
 
Theoremmetsym 12173 The distance function of a metric space is symmetric. Definition 14-1.1(c) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( B D A ) )
 
Theoremxmettri 12174 Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A D B )  <_  (
 ( A D C ) +e ( C D B ) ) )
 
Theoremmettri 12175 Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C )  +  ( C D B ) ) )
 
Theoremxmettri3 12176 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A D B )  <_  (
 ( A D C ) +e ( B D C ) ) )
 
Theoremmettri3 12177 Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C )  +  ( B D C ) ) )
 
Theoremxmetrtri 12178 One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A D C ) +e  -e ( B D C ) )  <_  ( A D B ) )
 
Theoremmetrtri 12179 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 21-Apr-2023.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( abs `  ( ( A D C )  -  ( B D C ) ) )  <_  ( A D B ) )
 
Theoremmetn0 12180 A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  ( D  =/=  (/)  <->  X  =/=  (/) ) )
 
Theoremxmetres2 12181 Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( *Met `  R ) )
 
Theoremmetreslem 12182 Lemma for metres 12185. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( dom  D  =  ( X  X.  X ) 
 ->  ( D  |`  ( R  X.  R ) )  =  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R ) ) ) )
 
Theoremmetres2 12183 Lemma for metres 12185. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  R  C_  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( Met `  R ) )
 
Theoremxmetres 12184 A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( D  e.  ( *Met `  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( *Met `  ( X  i^i  R ) ) )
 
Theoremmetres 12185 A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( Met `  ( X  i^i  R ) ) )
 
Theorem0met 12186 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  (/)  e.  ( Met `  (/) )
 
6.2.3  Metric space balls
 
Theoremblfvalps 12187* The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  ( ball `  D )  =  ( x  e.  X ,  r  e.  RR*  |->  { y  e.  X  |  ( x D y )  < 
 r } ) )
 
Theoremblfval 12188* The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Proof shortened by Thierry Arnoux, 11-Feb-2018.)
 |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D )  =  ( x  e.  X ,  r  e.  RR*  |->  { y  e.  X  |  ( x D y )  < 
 r } ) )
 
Theoremblex 12189 A ball is a set. (Contributed by Jim Kingdon, 4-May-2023.)
 |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D )  e.  _V )
 
Theoremblvalps 12190* The ball around a point  P is the set of all points whose distance from  P is less than the ball's radius  R. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  =  { x  e.  X  |  ( P D x )  <  R } )
 
Theoremblval 12191* The ball around a point  P is the set of all points whose distance from  P is less than the ball's radius  R. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  =  { x  e.  X  |  ( P D x )  <  R } )
 
Theoremelblps 12192 Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( A  e.  ( P ( ball `  D ) R )  <->  ( A  e.  X  /\  ( P D A )  <  R ) ) )
 
Theoremelbl 12193 Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( A  e.  ( P ( ball `  D ) R )  <->  ( A  e.  X  /\  ( P D A )  <  R ) ) )
 
Theoremelbl2ps 12194 Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( ( D  e.  (PsMet `  X )  /\  R  e.  RR* )  /\  ( P  e.  X  /\  A  e.  X ) )  ->  ( A  e.  ( P (
 ball `  D ) R )  <->  ( P D A )  <  R ) )
 
Theoremelbl2 12195 Membership in a ball. (Contributed by NM, 9-Mar-2007.)
 |-  ( ( ( D  e.  ( *Met `  X )  /\  R  e.  RR* )  /\  ( P  e.  X  /\  A  e.  X )
 )  ->  ( A  e.  ( P ( ball `  D ) R )  <-> 
 ( P D A )  <  R ) )
 
Theoremelbl3ps 12196 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
 |-  ( ( ( D  e.  (PsMet `  X )  /\  R  e.  RR* )  /\  ( P  e.  X  /\  A  e.  X ) )  ->  ( A  e.  ( P (
 ball `  D ) R )  <->  ( A D P )  <  R ) )
 
Theoremelbl3 12197 Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
 |-  ( ( ( D  e.  ( *Met `  X )  /\  R  e.  RR* )  /\  ( P  e.  X  /\  A  e.  X )
 )  ->  ( A  e.  ( P ( ball `  D ) R )  <-> 
 ( A D P )  <  R ) )
 
Theoremblcomps 12198 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( ( D  e.  (PsMet `  X )  /\  R  e.  RR* )  /\  ( P  e.  X  /\  A  e.  X ) )  ->  ( A  e.  ( P (
 ball `  D ) R )  <->  P  e.  ( A ( ball `  D ) R ) ) )
 
Theoremblcom 12199 Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.)
 |-  ( ( ( D  e.  ( *Met `  X )  /\  R  e.  RR* )  /\  ( P  e.  X  /\  A  e.  X )
 )  ->  ( A  e.  ( P ( ball `  D ) R )  <->  P  e.  ( A ( ball `  D ) R ) ) )
 
Theoremxblpnfps 12200 The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( A  e.  ( P ( ball `  D ) +oo )  <->  ( A  e.  X  /\  ( P D A )  e.  RR ) ) )
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