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Theorem List for Metamath Proof Explorer - 18001-18100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelbl 18001 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 18002 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 18003 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 18004 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 18005 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 18006 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 18007 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 18008 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 18009 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 18010 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 18011 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 18011 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 18012 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 18013 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 18014* 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 18015 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 18016 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 18017 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 18018 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 18019 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 18020 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 18021 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 18022 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 18023 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 18024* 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 18025* 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 18026* 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 18027* 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 18028 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 18029 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 18030 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 18031 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  ( D  e.  ( * Met `  X )  ->  .~  Er  X )
 
Theoremxmetec 18032 The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 6749, 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 18033 A ball centered at  P is contained in the set of points finitely separated from  P. This is just an application of ssbl 18023 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 18034 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 18035 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 18032, 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 18036 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 18038, the open sets of a metric space form a topology 
J, whose base set is 
U. J by mopnuni 18039. (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 18037 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 18038 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 18039 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 18040* 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 18041 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 18042 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 18043* 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 18044 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 18045 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 18046 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 18047 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 18048 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 18049 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 18050 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 18051 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  * MetSp  ->  M  e.  TopSp )
 
Theoremmsxms 18052 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  MetSp  ->  M  e.  * MetSp )
 
Theoremmstps 18053 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  MetSp  ->  M  e.  TopSp )
 
Theoremxmsxmet 18054 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 18055 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 18056 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 18057 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 18058 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 18059 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 18060 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 18061 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 18062 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 18063 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 18064 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 18065 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 18066 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 18067 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 18068 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 18069 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 18070 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 18071 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 18072 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 18073 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 18074 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 18075 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 18076 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 18077 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 18078 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 18079 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 18080 Lemma for tmsbas 18081, tmsds 18082, and tmstopn 18083. (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 18081 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 18082 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 18083 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 18084 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 18085 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 18086 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 18087 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 18088 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 18089* 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 18103) - 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 ) )
 
Theoremmopni 18090* An open set of a metric space includes a ball around each of its points. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  P  e.  A ) 
 ->  E. x  e.  ran  ( ball `  D )
 ( P  e.  x  /\  x  C_  A ) )
 
Theoremmopni2 18091* An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  P  e.  A ) 
 ->  E. x  e.  RR+  ( P ( ball `  D ) x )  C_  A )
 
Theoremmopni3 18092* An open set of a metric space includes an arbitrarily small ball around each of its points. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  P  e.  A )  /\  R  e.  RR+ )  ->  E. x  e.  RR+  ( x  <  R  /\  ( P (
 ball `  D ) x )  C_  A )
 )
 
Theoremblssopn 18093 The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  ran  ( ball `  D )  C_  J )
 
Theoremunimopn 18094 The union of a collection of open sets of a metric space is open. Theorem T2 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  C_  J )  ->  U. A  e.  J )
 
Theoremmopnin 18095 The intersection of two open sets of a metric space is open. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  B  e.  J ) 
 ->  ( A  i^i  B )  e.  J )
 
Theoremmopn0 18096 The empty set is an open set of a metric space. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  -> 
 (/)  e.  J )
 
Theoremrnblopn 18097 A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  B  e.  ran  ( ball `  D )
 )  ->  B  e.  J )
 
Theoremblopn 18098 A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  e.  J )
 
Theoremneibl 18099* The neighborhoods around a point  P of a metric space are those subsets containing a ball around  P. Definition of neighborhood in [Kreyszig] p. 19. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X )  ->  ( N  e.  ( ( nei `  J ) `  { P }
 ) 
 <->  ( N  C_  X  /\  E. r  e.  RR+  ( P ( ball `  D ) r )  C_  N ) ) )
 
Theoremblnei 18100 A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR+ )  ->  ( P ( ball `  D ) R )  e.  ( ( nei `  J ) `  { P } ) )
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