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Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssblex 17901* 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 17902* 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 17903 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 17904 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 17905 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 17906 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  ( D  e.  ( * Met `  X )  ->  .~  Er  X )
 
Theoremxmetec 17907 The equivalence classes under the finite separation equivalence relation are infinity balls. Thus (by erdisj 6640) 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 17908 A ball centered at  P is contained in the set of points finitely separated from  P. This is just an application of ssbl 17898 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 17909 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 17910 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 17907, 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 17911 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 17913, the open sets of a metric space form a topology 
J, whose base set is 
U. J by mopnuni 17914. (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 17912 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 17913 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 17914 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 17915* 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 17916 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 17917 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 17918* 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 17919 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 17920 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 17921 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 17922 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 17923 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 17924 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 17925 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 17926 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  * MetSp  ->  M  e.  TopSp )
 
Theoremmsxms 17927 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  MetSp  ->  M  e.  * MetSp )
 
Theoremmstps 17928 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  MetSp  ->  M  e.  TopSp )
 
Theoremxmsxmet 17929 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 17930 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 17931 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 17932 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 17933 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 17934 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 17935 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 17936 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 17937 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 17938 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 17939 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 17940 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 17941 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 17942 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 17943 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 17944 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 17945 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 17946 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 17947 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 17948 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 17949 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 17950 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 17951 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 17952 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 17953 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 17954 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 17955 Lemma for tmsbas 17956, tmsds 17957, and tmstopn 17958. (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 17956 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 17957 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 17958 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 17959 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 17960 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 17961 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 17962 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 17963 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 17964* 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 17978) - 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 17965* 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 17966* 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 17967* 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 17968 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 17969 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 17970 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 17971 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 17972 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 17973 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 17974* 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 17975 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 } ) )
 
Theoremlpbl 17976* Every ball around a limit point  P of a subset  S includes a member of  S (even if  P  e/  S). (Contributed by NM, 9-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  P  e.  ( ( limPt `  J ) `  S ) ) 
 /\  R  e.  RR+ )  ->  E. x  e.  S  x  e.  ( P ( ball `  D ) R ) )
 
Theoremblsscls2 17977* A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  S  =  {
 z  e.  X  |  ( P D z ) 
 <_  R }   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  <  T ) )  ->  S  C_  ( P ( ball `  D ) T ) )
 
Theoremblcld 17978* A "closed ball" in a metric space is actually closed. (Contributed by Mario Carneiro, 31-Dec-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  S  =  {
 z  e.  X  |  ( P D z ) 
 <_  R }   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  S  e.  ( Clsd `  J ) )
 
Theoremblcls 17979* The closure of an open ball in a metric space is contained in the corresponding closed ball. (The converse is not, in general, true; for example, with the discrete metric, the closed ball of radius 1 is the whole space, but the open ball of radius 1 is just a point, whose closure is also a point.) (Contributed by Mario Carneiro, 31-Dec-2013.)
 |-  J  =  ( MetOpen `  D )   &    |-  S  =  {
 z  e.  X  |  ( P D z ) 
 <_  R }   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ( cls `  J ) `  ( P (
 ball `  D ) R ) )  C_  S )
 
Theoremblsscls 17980 If two concentric balls have different radii, the closure of the smaller one is contained in the larger one. (Contributed by Mario Carneiro, 5-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR*  /\  R  <  S ) )  ->  (
 ( cls `  J ) `  ( P ( ball `  D ) R ) )  C_  ( P ( ball `  D ) S ) )
 
Theoremmetss 17981* Two ways of saying that metric  D generates a finer topology than metric  C. (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X ) )  ->  ( J 
 C_  K  <->  A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( x (
 ball `  D ) s )  C_  ( x ( ball `  C )
 r ) ) )
 
Theoremmetequiv 17982* Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeffrey Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X ) )  ->  ( J  =  K  <->  A. x  e.  X  ( A. r  e.  RR+  E. s  e.  RR+  ( x ( ball `  D )
 s )  C_  ( x ( ball `  C ) r )  /\  A. a  e.  RR+  E. b  e.  RR+  ( x (
 ball `  C ) b )  C_  ( x ( ball `  D )
 a ) ) ) )
 
Theoremmetequiv2 17983* If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X ) )  ->  ( A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
 ball `  D ) s ) )  ->  J  =  K ) )
 
Theoremmetss2lem 17984* Lemma for metss2 17985. (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  ( ph  ->  C  e.  ( Met `  X ) )   &    |-  ( ph  ->  D  e.  ( Met `  X )
 )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x C y )  <_  ( R  x.  ( x D y ) ) )   =>    |-  ( ( ph  /\  ( x  e.  X  /\  S  e.  RR+ ) ) 
 ->  ( x ( ball `  D ) ( S 
 /  R ) ) 
 C_  ( x (
 ball `  C ) S ) )
 
Theoremmetss2 17985* If the metric  D is "strongly finer" than  C (meaning that there is a positive real constant 
R such that  C ( x ,  y )  <_  R  x.  D (
x ,  y )), then  D generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  ( ph  ->  C  e.  ( Met `  X ) )   &    |-  ( ph  ->  D  e.  ( Met `  X )
 )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x C y )  <_  ( R  x.  ( x D y ) ) )   =>    |-  ( ph  ->  J  C_  K )
 
Theoremcomet 17986* The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  F : ( 0 [,]  +oo ) --> RR* )   &    |-  ( ( ph  /\  x  e.  ( 0 [,]  +oo ) )  ->  ( ( F `  x )  =  0  <->  x  =  0 ) )   &    |-  ( ( ph  /\  ( x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo ) ) )  ->  ( x  <_  y  ->  ( F `  x ) 
 <_  ( F `  y
 ) ) )   &    |-  (
 ( ph  /\  ( x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo ) ) )  ->  ( F `  ( x + e y ) )  <_  ( ( F `  x ) + e ( F `  y ) ) )   =>    |-  ( ph  ->  ( F  o.  D )  e.  ( * Met `  X )
 )
 
Theoremstdbdmetval 17987* Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( R  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  if ( ( A C B )  <_  R ,  ( A C B ) ,  R ) )
 
Theoremstdbdxmet 17988* The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( * Met `  X ) )
 
Theoremstdbdmet 17989* The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  R  e.  RR+ )  ->  D  e.  ( Met `  X ) )
 
Theoremstdbdbl 17990* The standard bounded metric corresponding to  C generates the same balls as  C for radii less than  R. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( ( C  e.  ( * Met `  X )  /\  R  e.  RR*  /\  0  <  R )  /\  ( P  e.  X  /\  S  e.  RR*  /\  S  <_  R ) )  ->  ( P ( ball `  D ) S )  =  ( P ( ball `  C ) S ) )
 
Theoremstdbdmopn 17991* The standard bounded metric corresponding to  C generates the same topology as  C. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   &    |-  J  =  ( MetOpen `  C )   =>    |-  (
 ( C  e.  ( * Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  J  =  ( MetOpen `  D )
 )
 
Theoremmopnex 17992* The topology generated by an extended metric can also be generated by a true metric. Thus "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  E. d  e.  ( Met `  X ) J  =  ( MetOpen `  d
 ) )
 
Theoremmethaus 17993 The topology generated by a metric space is Hausdorff. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  Haus )
 
Theoremmet1stc 17994 The topology generated by a metric space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  1stc )
 
Theoremmet2ndci 17995 A separable metric space (a metric space with a countable dense subset) is second-countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  ( A  C_  X  /\  A  ~<_  om  /\  ( ( cls `  J ) `  A )  =  X ) )  ->  J  e.  2ndc )
 
Theoremmet2ndc 17996* A metric space is second-countable iff it is separable (has a countable dense subset). (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  ( J  e.  2ndc  <->  E. x  e.  ~P  X ( x  ~<_  om  /\  ( ( cls `  J ) `  x )  =  X ) ) )
 
Theoremmetrest 17997 Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)
 |-  D  =  ( C  |`  ( Y  X.  Y ) )   &    |-  J  =  (
 MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   =>    |-  (
 ( C  e.  ( * Met `  X )  /\  Y  C_  X )  ->  ( Jt  Y )  =  K )
 
Theoremressxms 17998 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( K  e.  *
 MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  * MetSp )
 
Theoremressms 17999 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( K  e.  MetSp  /\  A  e.  V ) 
 ->  ( Ks  A )  e.  MetSp )
 
Theoremprdsmslem1 18000 Lemma for prdsms 18004. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  D  =  ( dist `  Y )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  R : I --> MetSp
 )   =>    |-  ( ph  ->  D  e.  ( Met `  B ) )
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