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Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-xms 17901 Define the (proper) class of all 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 17902 Define the (proper) class of all 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 17903 Define the function mapping a metric to a metric space. (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 ) >. ) )
 
Theoremismet 17904* 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 17905* 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 17906* 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 17907* 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 17908* 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 17909* Lemma for metf 17911 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 17910 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 17911 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 17912 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 17913 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 17914 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 17915 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  D  e.  ( * Met `  X ) )
 
Theoremxmetdmdm 17916 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  X  =  dom  dom  D )
 
Theoremmetdmdm 17917 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  X  =  dom  dom  D )
 
Theoremxmetunirn 17918 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 17919 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 17920 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 17921 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 17922 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 17923 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 17924 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 17925 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 17926 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 17927 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 17928 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 ) )
 
Theoremxmettpos 17929 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 17930 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 17931 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 17932 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 17933 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 17934 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 17935 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 ) )
 
Theoremxmetrtri2 17936 The reverse triangle inequality for the distance function of an extended metric. In order to express the "extended absolute value function", we use the distance function xrsdsval 16431 defined on the extended real structure. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  K  =  ( dist ` 
 RR* s )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D C ) K ( B D C ) )  <_  ( A D B ) )
 
Theoremmetrtri 17937 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( abs `  ( ( A D C )  -  ( B D C ) ) )  <_  ( A D B ) )
 
Theoremxmetgt0 17938 The distance function of an extended metric space is positive for unequal points. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A  =/=  B  <->  0  <  ( A D B ) ) )
 
Theoremmetgt0 17939 The distance function of a metric space is positive for unequal points. Definition 14-1.1(b) of [Gleason] p. 223 and its converse. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A  =/=  B  <->  0  <  ( A D B ) ) )
 
Theoremmetn0 17940 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 17941 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 17942 Lemma for metres 17945. (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 17943 Lemma for metres 17945. (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 17944 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 17945 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 17946 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  (/)  e.  ( Met `  (/) )
 
Theoremprdsdsf 17947* The product metric is a function into the nonnegative extended reals. In general this means that it is not a metric but rather an *extended* metric (even when all the factors are metrics), but it will be a metric when restricted to regions where it does not take infinite values. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( * Met `  V ) )   =>    |-  ( ph  ->  D : ( B  X.  B ) --> ( 0 [,]  +oo ) )
 
Theoremprdsxmetlem 17948* The product metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( * Met `  V ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  B ) )
 
Theoremprdsxmet 17949* The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 17948. (Contributed by Mario Carneiro, 26-Sep-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( * Met `  V ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  B ) )
 
Theoremprdsmet 17950* The product metric is a metric when the index set is finite. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( Met `  V ) )   =>    |-  ( ph  ->  D  e.  ( Met `  B ) )
 
Theoremressprdsds 17951* Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( ph  ->  Y  =  ( S X_s ( x  e.  I  |->  R ) ) )   &    |-  ( ph  ->  H  =  ( T X_s ( x  e.  I  |->  ( Rs  A ) ) ) )   &    |-  B  =  (
 Base `  H )   &    |-  D  =  ( dist `  Y )   &    |-  E  =  ( dist `  H )   &    |-  ( ph  ->  S  e.  U )   &    |-  ( ph  ->  T  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  (
 ( ph  /\  x  e.  I )  ->  R  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  A  e.  Z )   =>    |-  ( ph  ->  E  =  ( D  |`  ( B  X.  B ) ) )
 
Theoremresspwsds 17952 Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( ph  ->  Y  =  ( R  ^s  I )
 )   &    |-  ( ph  ->  H  =  ( ( Rs  A ) 
 ^s  I ) )   &    |-  B  =  ( Base `  H )   &    |-  D  =  ( dist `  Y )   &    |-  E  =  ( dist `  H )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  E  =  ( D  |`  ( B  X.  B ) ) )
 
Theoremimasdsf1olem 17953* Lemma for imasdsf1o 17954. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  E  e.  ( * Met `  V )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  W  =  ( RR* ss  ( RR*  \  {  -oo } ) )   &    |-  S  =  { h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  (
 ( F `  ( 1st `  ( h `  1 ) ) )  =  ( F `  X )  /\  ( F `
  ( 2nd `  ( h `  n ) ) )  =  ( F `
  Y )  /\  A. i  e.  ( 1
 ... ( n  -  1 ) ) ( F `  ( 2nd `  ( h `  i
 ) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1
 ) ) ) ) ) }   &    |-  T  =  U_ n  e.  NN  ran  (
 g  e.  S  |->  (
 RR* s  gsumg  ( E  o.  g
 ) ) )   =>    |-  ( ph  ->  ( ( F `  X ) D ( F `  Y ) )  =  ( X E Y ) )
 
Theoremimasdsf1o 17954 The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  E  e.  ( * Met `  V )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( ( F `  X ) D ( F `  Y ) )  =  ( X E Y ) )
 
Theoremimasf1oxmet 17955 The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  E  e.  ( * Met `  V )
 )   =>    |-  ( ph  ->  D  e.  ( * Met `  B ) )
 
Theoremimasf1omet 17956 The image of a metric is a metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  E  e.  ( Met `  V ) )   =>    |-  ( ph  ->  D  e.  ( Met `  B )
 )
 
Theoremxpsdsfn 17957 Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  P  =  (
 dist `  T )   =>    |-  ( ph  ->  P  Fn  ( ( X  X.  Y )  X.  ( X  X.  Y ) ) )
 
Theoremxpsdsfn2 17958 Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  P  =  (
 dist `  T )   =>    |-  ( ph  ->  P  Fn  ( ( Base `  T )  X.  ( Base `  T ) ) )
 
Theoremxpsxmetlem 17959* Lemma for xpsxmet 17960. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  P  =  (
 dist `  T )   &    |-  M  =  ( ( dist `  R )  |`  ( X  X.  X ) )   &    |-  N  =  ( ( dist `  S )  |`  ( Y  X.  Y ) )   &    |-  ( ph  ->  M  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  N  e.  ( * Met `  Y ) )   =>    |-  ( ph  ->  ( dist `  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } )
 ) )  e.  ( * Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } )
 ) ) )
 
Theoremxpsxmet 17960 A product metric of extended metrics is an extended metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  P  =  (
 dist `  T )   &    |-  M  =  ( ( dist `  R )  |`  ( X  X.  X ) )   &    |-  N  =  ( ( dist `  S )  |`  ( Y  X.  Y ) )   &    |-  ( ph  ->  M  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  N  e.  ( * Met `  Y ) )   =>    |-  ( ph  ->  P  e.  ( * Met `  ( X  X.  Y ) ) )
 
Theoremxpsdsval 17961 Value of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  P  =  (
 dist `  T )   &    |-  M  =  ( ( dist `  R )  |`  ( X  X.  X ) )   &    |-  N  =  ( ( dist `  S )  |`  ( Y  X.  Y ) )   &    |-  ( ph  ->  M  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  N  e.  ( * Met `  Y ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   =>    |-  ( ph  ->  (
 <. A ,  B >. P
 <. C ,  D >. )  =  sup ( {
 ( A M C ) ,  ( B N D ) } ,  RR*
 ,  <  ) )
 
Theoremxpsmet 17962 The direct product of two metric spaces. Definition 14-1.5 of [Gleason] p. 225. (Contributed by NM, 20-Jun-2007.) (Revised by Mario Carneiro, 20-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  P  =  (
 dist `  T )   &    |-  M  =  ( ( dist `  R )  |`  ( X  X.  X ) )   &    |-  N  =  ( ( dist `  S )  |`  ( Y  X.  Y ) )   &    |-  ( ph  ->  M  e.  ( Met `  X ) )   &    |-  ( ph  ->  N  e.  ( Met `  Y )
 )   =>    |-  ( ph  ->  P  e.  ( Met `  ( X  X.  Y ) ) )
 
11.3.2  Metric space balls
 
Theoremblfval 17963* The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  ( D  e.  ( * Met `  X )  ->  ( ball `  D )  =  ( x  e.  X ,  r  e.  RR*  |->  { y  e.  X  |  ( x D y )  < 
 r } ) )
 
Theoremblval 17964* 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 } )
 
Theoremelbl 17965 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 ) ) )
 
Theoremelbl2 17966 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 ) )
 
Theoremelbl3 17967 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 ) )
 
Theoremblcom 17968 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 ) ) )
 
Theoremxblpnf 17969 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 17970 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 17971 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 17972 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 17973 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 ) )  =  (/) )
 
Theoremxblss2 17974 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 17975 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 ) )
 
Theoremblss2 17975 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 17976 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 ) )
 
Theoremblf 17977 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 )
 
Theoremblrn 17978* 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 ) ) )
 
Theoremxblcntr 17979 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 ) )
 
Theoremblcntr 17980 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 ) )
 
Theoremxbln0 17981 A ball is nonempty iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ( P (
 ball `  D ) R )  =/=  (/)  <->  0  <  R ) )
 
Theorembln0 17982 A ball is not empty. (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 )  =/=  (/) )
 
Theoremblelrn 17983 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 17984 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 )
 
Theoremunirnbl 17985 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 )
 
Theoremblin 17986 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 ) if ( R  <_  S ,  R ,  S ) ) )
 
Theoremssbl 17987 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 ) )
 
Theoremblss 17988* 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 )
 
Theoremblssex 17989* 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 17990* 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 17991* 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 17992 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 17993 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 17994 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 17995 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  ( D  e.  ( * Met `  X )  ->  .~  Er  X )
 
Theoremxmetec 17996 The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 6723, infinity balls are either identical or disjoint, quite unlike the usual situation with Euclidean balls which admit many kinds of overlap. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  (
 ( D  e.  ( * Met `  X )  /\  P  e.  X ) 
 ->  [ P ]  .~  =  ( P ( ball `  D )  +oo )
 )
 
Theoremblssec 17997 A ball centered at  P is contained in the set of points finitely separated from  P. This is just an application of ssbl 17987 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 17998 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 17999 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 17996, 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 ) )
 
11.3.3  Open sets of a metric space
 
Theoremmopnval 18000 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 18002, the open sets of a metric space form a topology 
J, whose base set is 
U. J by mopnuni 18003. (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 )
 ) )
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