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Theorem List for Metamath Proof Explorer - 18001-18100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmscl 18001 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 18002 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 18003 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 18004 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 18005 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 18006 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 18007 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 18008 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 18009 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 18010 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 18011 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 18012 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 18013 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 18014 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 18015 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 18016 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 18017 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 18018 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 18019 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 18020 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 18021 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 18022 Lemma for tmsbas 18023, tmsds 18024, and tmstopn 18025. (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 18023 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 18024 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 18025 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 18026 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 18027 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 18028 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 18029 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 18030 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 18031* 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 18045) - 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 18032* 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 18033* 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 18034* 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 18035 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 18036 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 18037 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 18038 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 18039 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 18040 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 18041* 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 18042 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 18043* 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 18044* 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 18045* 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 18046* 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 18047 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 18048* 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 18049* 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 18050* 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 18051* Lemma for metss2 18052. (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 18052* 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 18053* 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 18054* 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 18055* 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 18056* 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 18057* 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 18058* 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 18059* 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 18060 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 18061 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 18062 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 18063* 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 18064 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 18065 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 18066 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 18067 Lemma for prdsms 18071. 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 ) )
 
Theoremprdsxmslem1 18068 Lemma for prdsms 18071. 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 ) )
 
Theoremprdsxmslem2 18069* Lemma for prdsxms 18070. The topology generated by the supremum metric is the same as the product topology, when the index set is finite. (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 )   &    |-  J  =  (
 TopOpen `  Y )   &    |-  V  =  ( Base `  ( R `  k ) )   &    |-  E  =  ( ( dist `  ( R `  k ) )  |`  ( V  X.  V ) )   &    |-  K  =  (
 TopOpen `  ( R `  k ) )   &    |-  C  =  { x  |  E. g ( ( g  Fn  I  /\  A. k  e.  I  (
 g `  k )  e.  ( ( TopOpen  o.  R ) `  k )  /\  E. z  e.  Fin  A. k  e.  ( I  \  z
 ) ( g `  k )  =  U. ( ( TopOpen  o.  R ) `  k ) ) 
 /\  x  =  X_ k  e.  I  (
 g `  k )
 ) }   =>    |-  ( ph  ->  J  =  ( MetOpen `  D )
 )
 
Theoremprdsxms 18070 The indexed product structure is an extended metric space when the index set is finite. (Although the extended metric is still valid when the index set is infinite, it no longer agrees with the product topology, which is not metrizable in any case.) (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   =>    |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp ) 
 ->  Y  e.  * MetSp )
 
Theoremprdsms 18071 The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   =>    |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> MetSp )  ->  Y  e.  MetSp )
 
Theorempwsxms 18072 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  *
 MetSp  /\  I  e.  Fin )  ->  Y  e.  * MetSp )
 
Theorempwsms 18073 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  MetSp  /\  I  e.  Fin )  ->  Y  e.  MetSp )
 
Theoremxpsxms 18074 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  T  =  ( R  X.s  S )   =>    |-  ( ( R  e.  *
 MetSp  /\  S  e.  * MetSp )  ->  T  e.  *
 MetSp )
 
Theoremxpsms 18075 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  T  =  ( R  X.s  S )   =>    |-  ( ( R  e.  MetSp  /\  S  e.  MetSp )  ->  T  e.  MetSp )
 
Theoremtmsxps 18076 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  P  =  ( dist `  ( (toMetSp `  M )  X.s  (toMetSp `  N )
 ) )   &    |-  ( ph  ->  M  e.  ( * Met `  X ) )   &    |-  ( ph  ->  N  e.  ( * Met `  Y )
 )   =>    |-  ( ph  ->  P  e.  ( * Met `  ( X  X.  Y ) ) )
 
Theoremtmsxpsmopn 18077 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  P  =  ( dist `  ( (toMetSp `  M )  X.s  (toMetSp `  N )
 ) )   &    |-  ( ph  ->  M  e.  ( * Met `  X ) )   &    |-  ( ph  ->  N  e.  ( * Met `  Y )
 )   &    |-  J  =  ( MetOpen `  M )   &    |-  K  =  (
 MetOpen `  N )   &    |-  L  =  ( MetOpen `  P )   =>    |-  ( ph  ->  L  =  ( J  tX  K )
 )
 
Theoremtmsxpsval 18078 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  P  =  ( dist `  ( (toMetSp `  M )  X.s  (toMetSp `  N )
 ) )   &    |-  ( 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*
 ,  <  ) )
 
Theoremtmsxpsval2 18079 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  P  =  ( dist `  ( (toMetSp `  M )  X.s  (toMetSp `  N )
 ) )   &    |-  ( 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 >. )  =  if ( ( A M C ) 
 <_  ( B N D ) ,  ( B N D ) ,  ( A M C ) ) )
 
11.3.4  Continuity in metric spaces
 
Theoremmetcnp3 18080* Two ways to express that  F is continuous at  P for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y )  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
 ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  ( F " ( P ( ball `  C )
 z ) )  C_  ( ( F `  P ) ( ball `  D ) y ) ) ) )
 
Theoremmetcnp 18081* Two ways to say a mapping from metric  C to metric  D is continuous at point  P. (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y )  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
 ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( P C w )  <  z  ->  ( ( F `  P ) D ( F `  w ) )  <  y ) ) ) )
 
Theoremmetcnp2 18082* Two ways to say a mapping from metric  C to metric  D is continuous at point  P. The distance arguments are swapped compared to metcnp 18081 (and Munkres' metcn 18083) for compatibility with df-lm 16953. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y )  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
 ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( w C P )  <  z  ->  (
 ( F `  w ) D ( F `  P ) )  < 
 y ) ) ) )
 
Theoremmetcn 18083* Two ways to say a mapping from metric  C to metric  D is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon"  y there is a positive "delta"  z such that a distance less than delta in  C maps to a distance less than epsilon in  D. (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. x  e.  X  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( x C w )  <  z  ->  ( ( F `  x ) D ( F `  w ) )  <  y ) ) ) )
 
Theoremmetcnpi 18084* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 18081. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y ) ) 
 /\  ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  <  x  ->  ( ( F `  P ) D ( F `  y ) )  <  A ) )
 
Theoremmetcnpi2 18085* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 18082. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y ) ) 
 /\  ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  <  x  ->  ( ( F `  y ) D ( F `  P ) )  <  A ) )
 
Theoremmetcnpi3 18086* Epsilon-delta property of a metric space function continuous at  P. A variation of metcnpi2 18085 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y ) ) 
 /\  ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  <_  x  ->  ( ( F `  y ) D ( F `  P ) )  <_  A )
 )
 
Theoremtxmetcnp 18087* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  L  =  ( MetOpen `  E )   =>    |-  (
 ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y )  /\  E  e.  ( * Met `  Z ) ) 
 /\  ( A  e.  X  /\  B  e.  Y ) )  ->  ( F  e.  ( ( ( J  tX  K )  CnP  L ) `  <. A ,  B >. )  <->  ( F :
 ( X  X.  Y )
 --> Z  /\  A. z  e.  RR+  E. w  e.  RR+  A. u  e.  X  A. v  e.  Y  ( ( ( A C u )  <  w  /\  ( B D v )  <  w )  ->  ( ( A F B ) E ( u F v ) )  <  z ) ) ) )
 
Theoremtxmetcn 18088* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  L  =  ( MetOpen `  E )   =>    |-  (
 ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y )  /\  E  e.  ( * Met `  Z ) ) 
 ->  ( F  e.  (
 ( J  tX  K )  Cn  L )  <->  ( F :
 ( X  X.  Y )
 --> Z  /\  A. x  e.  X  A. y  e.  Y  A. z  e.  RR+  E. w  e.  RR+  A. u  e.  X  A. v  e.  Y  (
 ( ( x C u )  <  w  /\  ( y D v )  <  w ) 
 ->  ( ( x F y ) E ( u F v ) )  <  z ) ) ) )
 
11.3.5  Examples of metric spaces
 
Theoremdscmet 18089* The discrete metric on any set  X. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( x  =  y ,  0 ,  1 ) )   =>    |-  ( X  e.  V  ->  D  e.  ( Met `  X ) )
 
Theoremdscopn 18090* The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( x  =  y ,  0 ,  1 ) )   =>    |-  ( X  e.  V  ->  ( MetOpen `  D )  =  ~P X )
 
Theoremnrmmetd 18091* Show that a group norm generates a metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  F : X --> RR )   &    |-  (
 ( ph  /\  x  e.  X )  ->  (
 ( F `  x )  =  0  <->  x  =  .0.  ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( F `
  ( x  .-  y ) )  <_  ( ( F `  x )  +  ( F `  y ) ) )   =>    |-  ( ph  ->  ( F  o.  .-  )  e.  ( Met `  X )
 )
 
Theoremabvmet 18092 An absolute value  F generates a metric defined by  d (
x ,  y )  =  F ( x  -  y ), analogously to cnmet 18275. (In fact, the ring structure is not needed at all; the group properties abveq0 15585 and abvtri 15589, abvneg 15593 are sufficient.) (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  A  =  (AbsVal `  R )   &    |-  .-  =  ( -g `  R )   =>    |-  ( F  e.  A  ->  ( F  o.  .-  )  e.  ( Met `  X ) )
 
11.3.6  Normed algebraic structures
 
Syntaxcnm 18093 Norm of a normed ring.
 class  norm
 
Syntaxcngp 18094 The class of all normed groups.
 class NrmGrp
 
Syntaxctng 18095 Make a normed group from a norm and a group.
 class toNrmGrp
 
Syntaxcnrg 18096 Normed ring.
 class NrmRing
 
Syntaxcnlm 18097 Normed module.
 class NrmMod
 
Syntaxcnvc 18098 Normed vector space.
 class NrmVec
 
Definitiondf-nm 18099* Define the norm on a group or ring (when it makes sense) in terms of the distance to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |- 
 norm  =  ( w  e.  _V  |->  ( x  e.  ( Base `  w )  |->  ( x ( dist `  w ) ( 0g
 `  w ) ) ) )
 
Definitiondf-ngp 18100 Define a normed group, which is a group with a right-translation-invariant metric. This is not a standard notion, but is helpful as the most general context in which a metric-like norm makes sense. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |- NrmGrp  =  { g  e.  ( Grp  i^i  MetSp )  |  ( ( norm `  g )  o.  ( -g `  g
 ) )  C_  ( dist `  g ) }
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