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Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremimasdsf1o 17901 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 17902 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 17903 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 17904 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 17905 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 17906* Lemma for xpsxmet 17907. (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 17907 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 17908 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 17909 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 17910* 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 17911* 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 17912 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 17913 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 17914 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 17915 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 17916 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 17917 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 17918 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 17919 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 17920 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 17921 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 17922 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 17922 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 17923 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 17924 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 17925* 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 17926 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 17927 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 17928 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 17929 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 17930 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 17931 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 17932 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 17933 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 17934 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 17935* 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 17936* 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 17937* 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 17938* 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 17939 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 17940 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 17941 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 17942 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  ( D  e.  ( * Met `  X )  ->  .~  Er  X )
 
Theoremxmetec 17943 The equivalence classes under the finite separation equivalence relation are infinity balls. Thus (by erdisj 6675) 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 17944 A ball centered at  P is contained in the set of points finitely separated from  P. This is just an application of ssbl 17934 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 17945 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 17946 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 17943, 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 17947 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 17949, the open sets of a metric space form a topology 
J, whose base set is 
U. J by mopnuni 17950. (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 17948 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 17949 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 17950 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 17951* 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 17952 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 17953 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 17954* 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 17955 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 17956 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 17957 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 17958 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 17959 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 17960 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 ) )
 
Theoremmstopn 17961 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 17962 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  * MetSp  ->  M  e.  TopSp )
 
Theoremmsxms 17963 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  MetSp  ->  M  e.  * MetSp )
 
Theoremmstps 17964 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  MetSp  ->  M  e.  TopSp )
 
Theoremxmsxmet 17965 The distance function, suitably truncated, is a 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 17966 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 17967 Mapping of the distance function of a metric space. (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 17968 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 ) )
 
Theoremmsmet2 17969 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 17970 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 17971 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 17972 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 17973 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 )  =  0  <->  A  =  B ) )
 
Theoremxmssym 17974 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 17975 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 17976 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 17977 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 ) ) )
 
Theoremmstri 17978 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 )  +  ( C D B ) ) )
 
Theoremxmstri3 17979 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  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C ) + e ( B D C ) ) )
 
Theoremmstri3 17980 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  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A D B )  <_  ( ( A D C )  +  ( B D C ) ) )
 
Theoremmsrtri 17981 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  MetSp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( abs `  ( ( A D C )  -  ( B D C ) ) )  <_  ( A D B ) )
 
Theoremxmspropd 17982 Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  (
 ( dist `  K )  |`  ( B  X.  B ) )  =  (
 ( dist `  L )  |`  ( B  X.  B ) ) )   &    |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L ) )   =>    |-  ( ph  ->  ( K  e.  * MetSp  <->  L  e.  * MetSp ) )
 
Theoremmspropd 17983 Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  (
 ( dist `  K )  |`  ( B  X.  B ) )  =  (
 ( dist `  L )  |`  ( B  X.  B ) ) )   &    |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L ) )   =>    |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
 
Theoremsetsmsbas 17984 The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   =>    |-  ( ph  ->  X  =  ( Base `  K )
 )
 
Theoremsetsmsds 17985 The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   =>    |-  ( ph  ->  ( dist `  M )  =  (
 dist `  K ) )
 
Theoremsetsmstset 17986 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   &    |-  ( ph  ->  M  e.  V )   =>    |-  ( ph  ->  ( MetOpen `  D )  =  (TopSet `  K ) )
 
Theoremsetsmstopn 17987 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   &    |-  ( ph  ->  M  e.  V )   =>    |-  ( ph  ->  ( MetOpen `  D )  =  (
 TopOpen `  K ) )
 
Theoremsetsxms 17988 The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   &    |-  ( ph  ->  M  e.  V )   =>    |-  ( ph  ->  ( K  e.  * MetSp  <->  D  e.  ( * Met `  X )
 ) )
 
Theoremsetsms 17989 The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   &    |-  ( ph  ->  M  e.  V )   =>    |-  ( ph  ->  ( K  e.  MetSp  <->  D  e.  ( Met `  X ) ) )
 
Theoremtmsval 17990 For any metric there is an associated metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  M  =  { <. (
 Base `  ndx ) ,  X >. ,  <. ( dist ` 
 ndx ) ,  D >. }   &    |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )
 
Theoremtmslem 17991 Lemma for tmsbas 17992, tmsds 17993, and tmstopn 17994. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  M  =  { <. (
 Base `  ndx ) ,  X >. ,  <. ( dist ` 
 ndx ) ,  D >. }   &    |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  ( X  =  (
 Base `  K )  /\  D  =  ( dist `  K )  /\  ( MetOpen `  D )  =  (
 TopOpen `  K ) ) )
 
Theoremtmsbas 17992 The base set of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  X  =  ( Base `  K ) )
 
Theoremtmsds 17993 The metric of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  D  =  ( dist `  K ) )
 
Theoremtmstopn 17994 The topology of a constructed metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  K  =  (toMetSp `  D )   &    |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  =  ( TopOpen `  K ) )
 
Theoremtmsxms 17995 The constructed metric space is an extended metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  K  e.  * MetSp )
 
Theoremtmsms 17996 The constructed metric space is a metric space given a metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( Met `  X )  ->  K  e.  MetSp )
 
Theoremimasf1obl 17997 The image of a metric space ball. (Contributed by Mario Carneiro, 28-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  ->  P  e.  V )   &    |-  ( ph  ->  S  e.  RR* )   =>    |-  ( ph  ->  (
 ( F `  P ) ( ball `  D ) S )  =  ( F " ( P ( ball `  E ) S ) ) )
 
Theoremimasf1oxms 17998 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  * MetSp )   =>    |-  ( ph  ->  U  e.  * MetSp )
 
Theoremimasf1oms 17999 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  MetSp )   =>    |-  ( ph  ->  U  e.  MetSp )
 
Theoremprdsbl 18000* A ball in the product metric for finite index set is the Cartesian product of balls in all coordinates. For infinite index set this is no longer true; instead the correct statement is that a *closed ball* is the product of closed balls in each coordinate (where closed ball means a set of the form in blcld 18014) - for a counterexample the point  p in  RR ^ NN whose  n-th coordinate is  1  -  1  /  n is in  X_ n  e.  NN ball ( 0 ,  1 ) but is not in the  1-ball of the product (since  d ( 0 ,  p )  =  1).

The last assumption,  0  <  A, is needed only in the case  I  =  (/), when the right side evaluates to  { (/) } and the left evaluates to  (/) if  A  <_  0 and  {
(/) } if  0  <  A. (Contributed by Mario Carneiro, 28-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  ->  P  e.  B )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  0  <  A )   =>    |-  ( ph  ->  ( P ( ball `  D ) A )  =  X_ x  e.  I  (
 ( P `  x ) ( ball `  E ) A ) )
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