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Theorem List for Intuitionistic Logic Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempsmettri 14101 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 14102 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 14103 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 14104 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 14105 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 14106 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 ) ) ) )
 
8.2.2  Basic metric space properties
 
Syntaxcxms 14107 Extend class notation with the class of extended metric spaces.
 class  *MetSp
 
Syntaxcms 14108 Extend class notation with the class of metric spaces.
 class  MetSp
 
Syntaxctms 14109 Extend class notation with the function mapping a metric to the metric space it defines.
 class toMetSp
 
Definitiondf-xms 14110 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 14111 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 14112 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 14113 The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
 |- 
 Rel  Met
 
Theoremxmetrel 14114 The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
 |- 
 Rel  *Met
 
Theoremismet 14115* 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 14116* 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 14117* 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 14118* 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 14119* 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 14120* Lemma for metf 14122 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 14121 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 14122 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 14123 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 14124 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 14125 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 14126 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  D  e.  ( *Met `  X ) )
 
Theoremxmetdmdm 14127 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( *Met `  X )  ->  X  =  dom  dom  D )
 
Theoremmetdmdm 14128 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  X  =  dom  dom  D )
 
Theoremxmetunirn 14129 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 14130 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 14131 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 14132 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 14133 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 14134 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 14135 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 14136 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 14137 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 14138 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 14139 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 14140 An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( D  e.  ( *Met `  X )  ->  D  e.  (PsMet `  X ) )
 
Theoremxmettpos 14141 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 14142 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 14143 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 14144 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 14145 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 14146 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 14147 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 14148 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 14149 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 14150 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 14151 Lemma for metres 14154. (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 14152 Lemma for metres 14154. (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 14153 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 14154 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 14155 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  (/)  e.  ( Met `  (/) )
 
8.2.3  Metric space balls
 
Theoremblfvalps 14156* 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 14157* 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 14158 A ball is a set. (Contributed by Jim Kingdon, 4-May-2023.)
 |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D )  e.  _V )
 
Theoremblvalps 14159* 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 14160* 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 14161 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 14162 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 14163 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 14164 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 14165 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 14166 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 14167 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 14168 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 14169 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 ) ) )
 
Theoremxblpnf 14170 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.)
 |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  ->  ( A  e.  ( P ( ball `  D ) +oo )  <->  ( A  e.  X  /\  ( P D A )  e.  RR ) ) )
 
Theoremblpnf 14171 The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  P  e.  X ) 
 ->  ( P ( ball `  D ) +oo )  =  X )
 
Theorembldisj 14172 Two balls are disjoint if the center-to-center distance is more than the sum of the radii. (Contributed by Mario Carneiro, 30-Dec-2013.)
 |-  ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  Q  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR*  /\  ( R +e S )  <_  ( P D Q ) ) )  ->  ( ( P ( ball `  D ) R )  i^i  ( Q ( ball `  D ) S ) )  =  (/) )
 
Theoremblgt0 14173 A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  /\  A  e.  ( P ( ball `  D ) R ) )  -> 
 0  <  R )
 
Theorembl2in 14174 Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X ) 
 /\  ( R  e.  RR  /\  R  <_  (
 ( P D Q )  /  2 ) ) )  ->  ( ( P ( ball `  D ) R )  i^i  ( Q ( ball `  D ) R ) )  =  (/) )
 
Theoremxblss2ps 14175 One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 14178 for extended metrics, we have to assume the balls are a finite distance apart, or else  P will not even be in the infinity ball around  Q. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ph  ->  D  e.  (PsMet `  X )
 )   &    |-  ( ph  ->  P  e.  X )   &    |-  ( ph  ->  Q  e.  X )   &    |-  ( ph  ->  R  e.  RR* )   &    |-  ( ph  ->  S  e.  RR* )   &    |-  ( ph  ->  ( P D Q )  e.  RR )   &    |-  ( ph  ->  ( P D Q )  <_  ( S +e  -e R ) )   =>    |-  ( ph  ->  ( P ( ball `  D ) R )  C_  ( Q ( ball `  D ) S ) )
 
Theoremxblss2 14176 One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 14178 for extended metrics, we have to assume the balls are a finite distance apart, or else  P will not even be in the infinity ball around  Q. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  D  e.  ( *Met `  X ) )   &    |-  ( ph  ->  P  e.  X )   &    |-  ( ph  ->  Q  e.  X )   &    |-  ( ph  ->  R  e.  RR* )   &    |-  ( ph  ->  S  e.  RR* )   &    |-  ( ph  ->  ( P D Q )  e.  RR )   &    |-  ( ph  ->  ( P D Q )  <_  ( S +e  -e R ) )   =>    |-  ( ph  ->  ( P ( ball `  D ) R )  C_  ( Q ( ball `  D ) S ) )
 
Theoremblss2ps 14177 One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  Q  e.  X ) 
 /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_  ( S  -  R ) ) ) 
 ->  ( P ( ball `  D ) R ) 
 C_  ( Q (
 ball `  D ) S ) )
 
Theoremblss2 14178 One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  Q  e.  X )  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_  ( S  -  R ) ) )  ->  ( P ( ball `  D ) R )  C_  ( Q ( ball `  D ) S ) )
 
Theoremblhalf 14179 A ball of radius  R  /  2 is contained in a ball of radius  R centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.)
 |-  ( ( ( M  e.  ( *Met `  X )  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R 
 /  2 ) ) ) )  ->  ( Y ( ball `  M ) ( R  / 
 2 ) )  C_  ( Z ( ball `  M ) R ) )
 
Theoremblfps 14180 Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
 --> ~P X )
 
Theoremblf 14181 Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* ) --> ~P X )
 
Theoremblrnps 14182* Membership in the range of the ball function. Note that  ran  ( ball `  D ) is the collection of all balls for metric 
D. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  ( A  e.  ran  ( ball `  D )  <->  E. x  e.  X  E. r  e.  RR*  A  =  ( x ( ball `  D ) r ) ) )
 
Theoremblrn 14183* Membership in the range of the ball function. Note that  ran  ( ball `  D ) is the collection of all balls for metric 
D. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  ( D  e.  ( *Met `  X )  ->  ( A  e.  ran  ( ball `  D )  <->  E. x  e.  X  E. r  e.  RR*  A  =  ( x ( ball `  D ) r ) ) )
 
Theoremxblcntrps 14184 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  ( P ( ball `  D ) R ) )
 
Theoremxblcntr 14185 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  ( P ( ball `  D ) R ) )
 
Theoremblcntrps 14186 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e.  RR+ )  ->  P  e.  ( P ( ball `  D ) R ) )
 
Theoremblcntr 14187 A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR+ )  ->  P  e.  ( P ( ball `  D ) R ) )
 
Theoremxblm 14188* A ball is inhabited iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( E. x  x  e.  ( P (
 ball `  D ) R )  <->  0  <  R ) )
 
Theorembln0 14189 A ball is not empty. It is also inhabited, as seen at blcntr 14187. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR+ )  ->  ( P ( ball `  D ) R )  =/=  (/) )
 
Theoremblelrnps 14190 A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  e.  ran  ( ball `  D )
 )
 
Theoremblelrn 14191 A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  e.  ran  ( ball `  D ) )
 
Theoremblssm 14192 A ball is a subset of the base set of a metric space. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R ) 
 C_  X )
 
Theoremunirnblps 14193 The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  U. ran  ( ball `  D )  =  X )
 
Theoremunirnbl 14194 The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  ( D  e.  ( *Met `  X )  ->  U. ran  ( ball `  D )  =  X )
 
Theoremblininf 14195 The intersection of two balls with the same center is the smaller of them. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  ->  ( ( P (
 ball `  D ) R )  i^i  ( P ( ball `  D ) S ) )  =  ( P ( ball `  D )inf ( { R ,  S } ,  RR* ,  <  )
 ) )
 
Theoremssblps 14196 The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* )  /\  R  <_  S )  ->  ( P ( ball `  D ) R ) 
 C_  ( P (
 ball `  D ) S ) )
 
Theoremssbl 14197 The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* )  /\  R  <_  S )  ->  ( P ( ball `  D ) R )  C_  ( P ( ball `  D ) S ) )
 
Theoremblssps 14198* Any point  P in a ball  B can be centered in another ball that is a subset of  B. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  B  e.  ran  ( ball `  D )  /\  P  e.  B )  ->  E. x  e.  RR+  ( P (
 ball `  D ) x )  C_  B )
 
Theoremblss 14199* Any point  P in a ball  B can be centered in another ball that is a subset of  B. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  ran  ( ball `  D )  /\  P  e.  B ) 
 ->  E. x  e.  RR+  ( P ( ball `  D ) x )  C_  B )
 
Theoremblssexps 14200* Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( E. x  e. 
 ran  ( ball `  D ) ( P  e.  x  /\  x  C_  A ) 
 <-> 
 E. r  e.  RR+  ( P ( ball `  D ) r )  C_  A ) )
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