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Theorem List for Intuitionistic Logic Explorer - 12301-12400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmettri 12301 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 12302 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 12303 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 12304 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 12305 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 12306 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 12307 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 12308 Lemma for metres 12311. (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 12309 Lemma for metres 12311. (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 12310 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 12311 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 12312 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 12313* 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 12314* 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 12315 A ball is a set. (Contributed by Jim Kingdon, 4-May-2023.)
 |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D )  e.  _V )
 
Theoremblvalps 12316* 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 12317* 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 12318 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 12319 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 12320 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 12321 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 12322 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 12323 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 12324 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 12325 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 12326 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 12327 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 12328 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 12329 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 12330 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 12331 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 12332 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 12335 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 12333 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 12335 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 12334 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 12335 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 12336 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 12337 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 12338 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 12339* 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 12340* 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 12341 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 12342 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 12343 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 12344 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 12345* 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 12346 A ball is not empty. It is also inhabited, as seen at blcntr 12344. (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 12347 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 12348 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 12349 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 12350 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 12351 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 12352 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 12353 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 12354 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 12355* 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 12356* 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 12357* 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 ) )
 
Theoremblssex 12358* Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  ( ( D  e.  ( *Met `  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 ) )
 
Theoremssblex 12359* A nested ball exists whose radius is less than any desired amount. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  E. x  e.  RR+  ( x  <  R  /\  ( P ( ball `  D ) x )  C_  ( P ( ball `  D ) S ) ) )
 
Theoremblin2 12360* Given any two balls and a point in their intersection, there is a ball contained in the intersection with the given center point. (Contributed by Mario Carneiro, 12-Nov-2013.)
 |-  ( ( ( D  e.  ( *Met `  X )  /\  P  e.  ( B  i^i  C ) )  /\  ( B  e.  ran  ( ball `  D )  /\  C  e.  ran  ( ball `  D ) ) )  ->  E. x  e.  RR+  ( P ( ball `  D ) x )  C_  ( B  i^i  C ) )
 
Theoremblbas 12361 The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.)
 |-  ( D  e.  ( *Met `  X )  ->  ran  ( ball `  D )  e.  TopBases )
 
Theoremblres 12362 A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)
 |-  C  =  ( D  |`  ( Y  X.  Y ) )   =>    |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( P ( ball `  C ) R )  =  ( ( P ( ball `  D ) R )  i^i  Y ) )
 
Theoremxmeterval 12363 Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  ( D  e.  ( *Met `  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( A D B )  e.  RR )
 ) )
 
Theoremxmeter 12364 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  ( D  e.  ( *Met `  X )  ->  .~  Er  X )
 
Theoremxmetec 12365 The equivalence classes under the finite separation equivalence relation are infinity balls. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  (
 ( D  e.  ( *Met `  X )  /\  P  e.  X ) 
 ->  [ P ]  .~  =  ( P ( ball `  D ) +oo )
 )
 
Theoremblssec 12366 A ball centered at  P is contained in the set of points finitely separated from  P. This is just an application of ssbl 12354 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  (
 ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  ( P ( ball `  D ) S )  C_  [ P ]  .~  )
 
Theoremblpnfctr 12367 The infinity ball in an extended metric acts like an ultrametric ball in that every point in the ball is also its center. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D ) +oo ) )  ->  ( P ( ball `  D ) +oo )  =  ( A ( ball `  D ) +oo ) )
 
Theoremxmetresbl 12368 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 12365, this shows that any extended metric space can be "factored" into the disjoint union of proper metric spaces, with points in the same region measured by that region's metric, and points in different regions being distance +oo from each other. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  B  =  ( P ( ball `  D ) R )   =>    |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  e.  ( Met `  B ) )
 
6.2.4  Open sets of a metric space
 
Theoremmopnrel 12369 The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)
 |- 
 Rel  MetOpen
 
Theoremmopnval 12370 An open set is a subset of a metric space which includes a ball around each of its points. Definition 1.3-2 of [Kreyszig] p. 18. The object  ( MetOpen `  D
) is the family of all open sets in the metric space determined by the metric  D. By mopntop 12372, the open sets of a metric space form a topology 
J, whose base set is 
U. J by mopnuni 12373. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( *Met `  X )  ->  J  =  ( topGen `  ran  ( ball `  D )
 ) )
 
Theoremmopntopon 12371 The set of open sets of a metric space  X is a topology on  X. Remark in [Kreyszig] p. 19. This theorem connects the two concepts and makes available the theorems for topologies for use with metric spaces. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( *Met `  X )  ->  J  e.  (TopOn `  X ) )
 
Theoremmopntop 12372 The set of open sets of a metric space is a topology. (Contributed by NM, 28-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( *Met `  X )  ->  J  e.  Top )
 
Theoremmopnuni 12373 The union of all open sets in a metric space is its underlying set. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
 
Theoremelmopn 12374* The defining property of an open set of a metric space. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( *Met `  X )  ->  ( A  e.  J  <->  ( A  C_  X  /\  A. x  e.  A  E. y  e.  ran  ( ball `  D ) ( x  e.  y  /\  y  C_  A ) ) ) )
 
Theoremmopnfss 12375 The family of open sets of a metric space is a collection of subsets of the base set. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( *Met `  X )  ->  J  C_  ~P X )
 
Theoremmopnm 12376 The base set of a metric space is open. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( *Met `  X )  ->  X  e.  J )
 
Theoremelmopn2 12377* A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( *Met `  X )  ->  ( A  e.  J  <->  ( A  C_  X  /\  A. x  e.  A  E. y  e.  RR+  ( x ( ball `  D )
 y )  C_  A ) ) )
 
Theoremmopnss 12378 An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  J )  ->  A  C_  X )
 
Theoremisxms 12379 Express the predicate "
<. X ,  D >. is an extended metric space" with underlying set  X and distance function  D. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  *MetSp  <-> 
 ( K  e.  TopSp  /\  J  =  ( MetOpen `  D ) ) )
 
Theoremisxms2 12380 Express the predicate "
<. X ,  D >. is an extended metric space" with underlying set  X and distance function  D. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  *MetSp  <-> 
 ( D  e.  ( *Met `  X )  /\  J  =  ( MetOpen `  D ) ) )
 
Theoremisms 12381 Express the predicate "
<. X ,  D >. is a metric space" with underlying set  X and distance function  D. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  MetSp  <->  ( K  e.  *MetSp  /\  D  e.  ( Met `  X ) ) )
 
Theoremisms2 12382 Express the predicate "
<. X ,  D >. is a metric space" with underlying set  X and distance function  D. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  MetSp  <->  ( D  e.  ( Met `  X )  /\  J  =  ( MetOpen `  D )
 ) )
 
Theoremxmstopn 12383 The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  *MetSp  ->  J  =  (
 MetOpen `  D ) )
 
Theoremmstopn 12384 The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  MetSp  ->  J  =  ( MetOpen `  D ) )
 
Theoremxmstps 12385 An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  *MetSp  ->  M  e.  TopSp )
 
Theoremmsxms 12386 A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  MetSp  ->  M  e.  *MetSp )
 
Theoremmstps 12387 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  MetSp  ->  M  e.  TopSp )
 
Theoremxmsxmet 12388 The distance function, suitably truncated, is an extended metric on  X. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  ( ( dist `  M )  |`  ( X  X.  X ) )   =>    |-  ( M  e.  *MetSp  ->  D  e.  ( *Met `  X )
 )
 
Theoremmsmet 12389 The distance function, suitably truncated, is a metric on  X. (Contributed by Mario Carneiro, 12-Nov-2013.)
 |-  X  =  ( Base `  M )   &    |-  D  =  ( ( dist `  M )  |`  ( X  X.  X ) )   =>    |-  ( M  e.  MetSp  ->  D  e.  ( Met `  X ) )
 
Theoremmsf 12390 The distance function of a metric space is a function into the real numbers. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  X  =  ( Base `  M )   &    |-  D  =  ( ( dist `  M )  |`  ( X  X.  X ) )   =>    |-  ( M  e.  MetSp  ->  D : ( X  X.  X ) --> RR )
 
Theoremxmsxmet2 12391 The distance function, suitably truncated, is an extended metric on  X. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( M  e.  *MetSp  ->  ( D  |`  ( X  X.  X ) )  e.  ( *Met `  X ) )
 
Theoremmsmet2 12392 The distance function, suitably truncated, is a metric on  X. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( M  e.  MetSp  ->  ( D  |`  ( X  X.  X ) )  e.  ( Met `  X ) )
 
Theoremmscl 12393 Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  MetSp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  RR )
 
Theoremxmscl 12394 Closure of the distance function of an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  *MetSp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  RR* )
 
Theoremxmsge0 12395 The distance function in an extended metric space is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  *MetSp  /\  A  e.  X  /\  B  e.  X )  ->  0  <_  ( A D B ) )
 
Theoremxmseq0 12396 The distance between two points in an extended metric space is zero iff the two points are identical. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  *MetSp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A D B )  =  0  <->  A  =  B ) )
 
Theoremxmssym 12397 The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  *MetSp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( B D A ) )
 
Theoremxmstri2 12398 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  *MetSp  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X )
 )  ->  ( A D B )  <_  (
 ( C D A ) +e ( C D B ) ) )
 
Theoremmstri2 12399 Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  MetSp  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X ) )  ->  ( A D B )  <_  ( ( C D A )  +  ( C D B ) ) )
 
Theoremxmstri 12400 Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  *MetSp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C ) +e ( C D B ) ) )
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