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Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremucnima 18301* An equivalent statement of the definition of uniformly continuous function. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ph  ->  U  e.  (UnifOn `  X )
 )   &    |-  ( ph  ->  V  e.  (UnifOn `  Y )
 )   &    |-  ( ph  ->  F  e.  ( U Cnu V ) )   &    |-  ( ph  ->  W  e.  V )   &    |-  G  =  ( x  e.  X ,  y  e.  X  |->  <. ( F `
  x ) ,  ( F `  y
 ) >. )   =>    |-  ( ph  ->  E. r  e.  U  ( G "
 r )  C_  W )
 
Theoremucnprima 18302* The preimage by a uniformly continuous function  F of an entourage  W of  Y is an entourage of  X. Note of the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ph  ->  U  e.  (UnifOn `  X )
 )   &    |-  ( ph  ->  V  e.  (UnifOn `  Y )
 )   &    |-  ( ph  ->  F  e.  ( U Cnu V ) )   &    |-  ( ph  ->  W  e.  V )   &    |-  G  =  ( x  e.  X ,  y  e.  X  |->  <. ( F `
  x ) ,  ( F `  y
 ) >. )   =>    |-  ( ph  ->  ( `' G " W )  e.  U )
 
Theoremiducn 18303 The identity is uniformly continuous from a uniform structure to itself. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  (  _I  |`  X )  e.  ( U Cnu U ) )
 
Theoremcstucnd 18304 A constant function is uniformly continuous. Deduction form. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( ph  ->  U  e.  (UnifOn `  X )
 )   &    |-  ( ph  ->  V  e.  (UnifOn `  Y )
 )   &    |-  ( ph  ->  A  e.  Y )   =>    |-  ( ph  ->  ( X  X.  { A }
 )  e.  ( U Cnu V ) )
 
Theoremucncn 18305 Uniform continuity implies continuity. Deduction form. Proposition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 30-Nov-2017.)
 |-  J  =  ( TopOpen `  R )   &    |-  K  =  (
 TopOpen `  S )   &    |-  ( ph  ->  R  e. UnifSp )   &    |-  ( ph  ->  S  e. UnifSp )   &    |-  ( ph  ->  R  e.  TopSp )   &    |-  ( ph  ->  S  e.  TopSp
 )   &    |-  ( ph  ->  F  e.  ( (UnifSt `  R ) Cnu (UnifSt `  S )
 ) )   =>    |-  ( ph  ->  F  e.  ( J  Cn  K ) )
 
11.3.5  Cauchy filters in uniform spaces
 
Syntaxccfilu 18306 Extend class notation with the set of Cauchy filter bases.
 class CauFilu
 
Definitiondf-cfilu 18307* Define the set of Cauchy filter bases on a uniform space. A Cauchy filter base is a filter base on the set such that for every entourage  v, there is an element  a of the filter "small enough in  v " i.e. such that every pair  { x ,  y } of points in  a is related by  v". Definition 2 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |- CauFilu  =  ( u  e.  U. ran UnifOn 
 |->  { f  e.  ( fBas `  dom  U. u )  |  A. v  e.  u  E. a  e.  f  ( a  X.  a )  C_  v }
 )
 
Theoremiscfilu 18308* The predicate " F is a Cauchy filter base on uniform space  U." (Contributed by Thierry Arnoux, 18-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  ( F  e.  (CauFilu `  U ) 
 <->  ( F  e.  ( fBas `  X )  /\  A. v  e.  U  E. a  e.  F  (
 a  X.  a )  C_  v ) ) )
 
Theoremcfilufbas 18309 A Cauchy filter base is a filter base. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  F  e.  ( fBas `  X )
 )
 
Theoremcfiluexsm 18310* For a Cauchy filter base and any entourage  V, there is an element of the filter small in  V. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U )  /\  V  e.  U )  ->  E. a  e.  F  ( a  X.  a
 )  C_  V )
 
Theoremfmucndlem 18311* Lemma for fmucnd 18312. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ( F  Fn  X  /\  A  C_  X )  ->  ( ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y ) >. )
 " ( A  X.  A ) )  =  ( ( F " A )  X.  ( F " A ) ) )
 
Theoremfmucnd 18312* The image of a Cauchy filter base by an uniformly continuous function is a Cauchy filter base. Deduction form. Proposition 3 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 18-Nov-2017.)
 |-  ( ph  ->  U  e.  (UnifOn `  X )
 )   &    |-  ( ph  ->  V  e.  (UnifOn `  Y )
 )   &    |-  ( ph  ->  F  e.  ( U Cnu V ) )   &    |-  ( ph  ->  C  e.  (CauFilu `  U ) )   &    |-  D  =  ran  ( a  e.  C  |->  ( F "
 a ) )   =>    |-  ( ph  ->  D  e.  (CauFilu `
  V ) )
 
Theoremcfilufg 18313 The filter generated by a Cauchy filter base is still a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  ( X
 filGen F )  e.  (CauFilu `  U ) )
 
Theoremtrcfilu 18314 Condition for the trace of a Cauchy filter base to be a Cauchy filter base for the restricted uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U )  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  ( Ft  A )  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )
 
Theoremcfiluweak 18315 A Cauchy filter base is also a Cauchy filter base on any coarser uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `
  ( Ut  ( A  X.  A ) ) ) )  ->  F  e.  (CauFilu `
  U ) )
 
Theoremneipcfilu 18316 In an uniform space, a neighboring filter is a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)
 |-  X  =  ( Base `  W )   &    |-  J  =  (
 TopOpen `  W )   &    |-  U  =  (UnifSt `  W )   =>    |-  (
 ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  ->  ( ( nei `  J ) `  { P } )  e.  (CauFilu `
  U ) )
 
11.3.6  Complete uniform spaces
 
Syntaxccusp 18317 Extend class notation with the class of all complete uniform spaces.
 class CUnifSp
 
Definitiondf-cusp 18318* Define the class of all complete uniform spaces. Definition 3 of [BourbakiTop1] p. II.15. (Contributed by Thierry Arnoux, 1-Dec-2017.)
 |- CUnifSp  =  { w  e. UnifSp  |  A. c  e.  ( Fil `  ( Base `  w )
 ) ( c  e.  (CauFilu `
  (UnifSt `  w ) )  ->  ( (
 TopOpen `  w )  fLim  c )  =/=  (/) ) }
 
Theoremiscusp 18319* The predicate " W is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |-  ( W  e. CUnifSp  <->  ( W  e. UnifSp  /\ 
 A. c  e.  ( Fil `  ( Base `  W ) ) ( c  e.  (CauFilu `
  (UnifSt `  W ) )  ->  ( (
 TopOpen `  W )  fLim  c )  =/=  (/) ) ) )
 
Theoremcuspusp 18320 A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
 
Theoremcuspcvg 18321 In a complete uniform space, any Cauchy filter  C has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (
 TopOpen `  W )   =>    |-  ( ( W  e. CUnifSp  /\  C  e.  (CauFilu `  (UnifSt `  W ) ) 
 /\  C  e.  ( Fil `  B ) ) 
 ->  ( J  fLim  C )  =/=  (/) )
 
Theoremiscusp2 18322* The predicate " W is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.)
 |-  B  =  ( Base `  W )   &    |-  U  =  (UnifSt `  W )   &    |-  J  =  (
 TopOpen `  W )   =>    |-  ( W  e. CUnifSp  <->  ( W  e. UnifSp  /\  A. c  e.  ( Fil `  B ) ( c  e.  (CauFilu `
  U )  ->  ( J  fLim  c )  =/=  (/) ) ) )
 
Theoremcnextucn 18323* Extension by continuity. Proposition 11 of [BourbakiTop1] p. II.20. Given a topology  J on  X, a subset  A dense in  X, this states a condition for  F from  A to a space  Y Hausdorff and complete to be extensible by continuity (Contributed by Thierry Arnoux, 4-Dec-2017.)
 |-  X  =  ( Base `  V )   &    |-  Y  =  (
 Base `  W )   &    |-  J  =  ( TopOpen `  V )   &    |-  K  =  ( TopOpen `  W )   &    |-  U  =  (UnifSt `  W )   &    |-  ( ph  ->  V  e.  TopSp )   &    |-  ( ph  ->  W  e.  TopSp
 )   &    |-  ( ph  ->  W  e. CUnifSp )   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  A 
 C_  X )   &    |-  ( ph  ->  F : A --> Y )   &    |-  ( ph  ->  ( ( cls `  J ) `  A )  =  X )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  ( ( Y  FilMap  F ) `  ( ( ( nei `  J ) `  { x }
 )t 
 A ) )  e.  (CauFilu `
  U ) )   =>    |-  ( ph  ->  ( ( JCnExt K ) `  F )  e.  ( J  Cn  K ) )
 
Theoremucnextcn 18324 Extension by continuity. Theorem 2 of [BourbakiTop1] p. II.20. Given an uniform space on a set  X, a subset  A dense in  X, and a function  F uniformly continuous from  A to  Y, that function can be extended by continuity to the whole  X, and its extension is uniformly continuous. (Contributed by Thierry Arnoux, 25-Jan-2018.)
 |-  X  =  ( Base `  V )   &    |-  Y  =  (
 Base `  W )   &    |-  J  =  ( TopOpen `  V )   &    |-  K  =  ( TopOpen `  W )   &    |-  S  =  (UnifSt `  V )   &    |-  T  =  (UnifSt `  ( Vs  A ) )   &    |-  U  =  (UnifSt `  W )   &    |-  ( ph  ->  V  e.  TopSp )   &    |-  ( ph  ->  V  e. UnifSp )   &    |-  ( ph  ->  W  e.  TopSp )   &    |-  ( ph  ->  W  e. CUnifSp )   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  A 
 C_  X )   &    |-  ( ph  ->  F  e.  ( T Cnu
 U ) )   &    |-  ( ph  ->  ( ( cls `  J ) `  A )  =  X )   =>    |-  ( ph  ->  ( ( JCnExt
 K ) `  F )  e.  ( J  Cn  K ) )
 
11.4  Metric spaces
 
11.4.1  Pseudometric spaces
 
Theoremispsmet 18325* Express the predicate " D is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( X  e.  V  ->  ( D  e.  (PsMet `  X )  <->  ( D :
 ( X  X.  X )
 --> RR*  /\  A. x  e.  X  ( ( x D x )  =  0  /\  A. y  e.  X  A. z  e.  X  ( x D y )  <_  (
 ( z D x ) + e ( z D y ) ) ) ) ) )
 
Theorempsmetdmdm 18326 Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  X  =  dom  dom  D )
 
Theorempsmetf 18327 The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  D : ( X  X.  X ) --> RR* )
 
Theorempsmetcl 18328 Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  RR* )
 
Theorempsmet0 18329 The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X )  ->  ( A D A )  =  0 )
 
Theorempsmettri2 18330 Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X )
 )  ->  ( A D B )  <_  (
 ( C D A ) + e ( C D B ) ) )
 
Theorempsmetsym 18331 The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( B D A ) )
 
Theorempsmettri 18332 Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C ) + e ( C D B ) ) )
 
Theorempsmetge0 18333 The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  0  <_  ( A D B ) )
 
Theorempsmetxrge0 18334 The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  D : ( X  X.  X ) --> ( 0 [,]  +oo ) )
 
Theorempsmetres2 18335 Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  ( D  |`  ( R  X.  R ) )  e.  (PsMet `  R ) )
 
Theorempsmetlecl 18336 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  RR  /\  ( A D B )  <_  C ) )  ->  ( A D B )  e.  RR )
 
11.4.2  Basic metric space properties
 
Syntaxcxme 18337 Extend class notation with the class of all extended metric spaces.
 class  * MetSp
 
Syntaxcmt 18338 Extend class notation with the class of all metric spaces.
 class  MetSp
 
Syntaxctmt 18339 Extend class notation with the function mapping a metric to a metric space.
 class toMetSp
 
Definitiondf-xms 18340 Define the (proper) class of all extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- 
 * MetSp  =  { f  e.  TopSp  |  ( TopOpen `  f )  =  ( MetOpen `  ( ( dist `  f
 )  |`  ( ( Base `  f )  X.  ( Base `  f ) ) ) ) }
 
Definitiondf-ms 18341 Define the (proper) class of all metric spaces. (Contributed by NM, 27-Aug-2006.)
 |- 
 MetSp  =  { f  e.  * MetSp  |  (
 ( dist `  f )  |`  ( ( Base `  f
 )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f ) ) }
 
Definitiondf-tms 18342 Define the function mapping a metric to a metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- toMetSp  =  ( d  e.  U. ran  * Met  |->  ( { <. ( Base `  ndx ) , 
 dom  dom  d >. ,  <. (
 dist `  ndx ) ,  d >. } sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >. ) )
 
Theoremismet 18343* Express the predicate " D is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( X  e.  A  ->  ( D  e.  ( Met `  X )  <->  ( D :
 ( X  X.  X )
 --> RR  /\  A. x  e.  X  A. y  e.  X  ( ( ( x D y )  =  0  <->  x  =  y
 )  /\  A. z  e.  X  ( x D y )  <_  (
 ( z D x )  +  ( z D y ) ) ) ) ) )
 
Theoremisxmet 18344* Express the predicate " D is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( X  e.  A  ->  ( D  e.  ( * Met `  X )  <->  ( D : ( X  X.  X ) --> RR*  /\  A. x  e.  X  A. y  e.  X  ( ( ( x D y )  =  0  <->  x  =  y
 )  /\  A. z  e.  X  ( x D y )  <_  (
 ( z D x ) + e ( z D y ) ) ) ) ) )
 
Theoremismeti 18345* Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  X  e.  _V   &    |-  D : ( X  X.  X ) --> RR   &    |-  (
 ( x  e.  X  /\  y  e.  X )  ->  ( ( x D y )  =  0  <->  x  =  y
 ) )   &    |-  ( ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( x D y )  <_  ( ( z D x )  +  (
 z D y ) ) )   =>    |-  D  e.  ( Met `  X )
 
Theoremisxmetd 18346* Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  D : ( X  X.  X ) --> RR* )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( ( x D y )  =  0  <-> 
 x  =  y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x D y )  <_  ( ( z D x ) + e
 ( z D y ) ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  X ) )
 
Theoremisxmet2d 18347* It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample:  D ( x ,  y )  =  if ( x  =  y ,  0 , 
-oo ) satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  D : ( X  X.  X ) --> RR* )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
 0  <_  ( x D y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( ( x D y )  <_ 
 0 
 <->  x  =  y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  /\  ( ( z D x )  e. 
 RR  /\  ( z D y )  e. 
 RR ) )  ->  ( x D y ) 
 <_  ( ( z D x )  +  (
 z D y ) ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  X ) )
 
Theoremmetflem 18348* Lemma for metf 18350 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  ( D : ( X  X.  X ) --> RR  /\  A. x  e.  X  A. y  e.  X  (
 ( ( x D y )  =  0  <-> 
 x  =  y ) 
 /\  A. z  e.  X  ( x D y ) 
 <_  ( ( z D x )  +  (
 z D y ) ) ) ) )
 
Theoremxmetf 18349 Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
 
Theoremmetf 18350 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)
 |-  ( D  e.  ( Met `  X )  ->  D : ( X  X.  X ) --> RR )
 
Theoremxmetcl 18351 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A D B )  e.  RR* )
 
Theoremmetcl 18352 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  RR )
 
Theoremismet2 18353 An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  <->  ( D  e.  ( * Met `  X )  /\  D : ( X  X.  X ) --> RR ) )
 
Theoremmetxmet 18354 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  D  e.  ( * Met `  X ) )
 
Theoremxmetdmdm 18355 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  X  =  dom  dom  D )
 
Theoremmetdmdm 18356 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  X  =  dom  dom  D )
 
Theoremxmetunirn 18357 Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( D  e.  U. ran  * Met  <->  D  e.  ( * Met `  dom  dom  D ) )
 
Theoremxmeteq0 18358 The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( ( A D B )  =  0  <->  A  =  B ) )
 
Theoremmeteq0 18359 The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A D B )  =  0  <->  A  =  B ) )
 
Theoremxmettri2 18360 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X ) )  ->  ( A D B )  <_  (
 ( C D A ) + e ( C D B ) ) )
 
Theoremmettri2 18361 Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X )
 )  ->  ( A D B )  <_  (
 ( C D A )  +  ( C D B ) ) )
 
Theoremxmet0 18362 The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X )  ->  ( A D A )  =  0
 )
 
Theoremmet0 18363 The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X ) 
 ->  ( A D A )  =  0 )
 
Theoremxmetge0 18364 The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  0  <_  ( A D B ) )
 
Theoremmetge0 18365 The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  0  <_  ( A D B ) )
 
Theoremxmetlecl 18366 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  RR  /\  ( A D B )  <_  C ) )  ->  ( A D B )  e.  RR )
 
Theoremxmetsym 18367 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A D B )  =  ( B D A ) )
 
Theoremxmetpsmet 18368 An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( D  e.  ( * Met `  X )  ->  D  e.  (PsMet `  X ) )
 
Theoremxmettpos 18369 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( * Met `  X )  -> tpos 
 D  =  D )
 
Theoremmetsym 18370 The distance function of a metric space is symmetric. Definition 14-1.1(c) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( B D A ) )
 
Theoremxmettri 18371 Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A D B )  <_  (
 ( A D C ) + e ( C D B ) ) )
 
Theoremmettri 18372 Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C )  +  ( C D B ) ) )
 
Theoremxmettri3 18373 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A D B )  <_  (
 ( A D C ) + e ( B D C ) ) )
 
Theoremmettri3 18374 Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C )  +  ( B D C ) ) )
 
Theoremxmetrtri 18375 One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A D C ) + e  - e ( B D C ) )  <_  ( A D B ) )
 
Theoremxmetrtri2 18376 The reverse triangle inequality for the distance function of an extended metric. In order to express the "extended absolute value function", we use the distance function xrsdsval 16732 defined on the extended real structure. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  K  =  ( dist ` 
 RR* s )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D C ) K ( B D C ) )  <_  ( A D B ) )
 
Theoremmetrtri 18377 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.)
 |-  ( ( D  e.  ( Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( abs `  ( ( A D C )  -  ( B D C ) ) )  <_  ( A D B ) )
 
Theoremxmetgt0 18378 The distance function of an extended metric space is positive for unequal points. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A  =/=  B  <->  0  <  ( A D B ) ) )
 
Theoremmetgt0 18379 The distance function of a metric space is positive for unequal points. Definition 14-1.1(b) of [Gleason] p. 223 and its converse. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A  =/=  B  <->  0  <  ( A D B ) ) )
 
Theoremmetn0 18380 A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  ( D  =/=  (/)  <->  X  =/=  (/) ) )
 
Theoremxmetres2 18381 Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  R  C_  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( * Met `  R ) )
 
Theoremmetreslem 18382 Lemma for metres 18385. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( dom  D  =  ( X  X.  X ) 
 ->  ( D  |`  ( R  X.  R ) )  =  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R ) ) ) )
 
Theoremmetres2 18383 Lemma for metres 18385. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( D  e.  ( Met `  X )  /\  R  C_  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( Met `  R ) )
 
Theoremxmetres 18384 A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( * Met `  ( X  i^i  R ) ) )
 
Theoremmetres 18385 A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( Met `  ( X  i^i  R ) ) )
 
Theorem0met 18386 The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  (/)  e.  ( Met `  (/) )
 
Theoremprdsdsf 18387* The product metric is a function into the nonnegative extended reals. In general this means that it is not a metric but rather an *extended* metric (even when all the factors are metrics), but it will be a metric when restricted to regions where it does not take infinite values. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( * Met `  V ) )   =>    |-  ( ph  ->  D : ( B  X.  B ) --> ( 0 [,]  +oo ) )
 
Theoremprdsxmetlem 18388* The product metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( * Met `  V ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  B ) )
 
Theoremprdsxmet 18389* The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 18388. (Contributed by Mario Carneiro, 26-Sep-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( * Met `  V ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  B ) )
 
Theoremprdsmet 18390* The product metric is a metric when the index set is finite. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( Met `  V ) )   =>    |-  ( ph  ->  D  e.  ( Met `  B ) )
 
Theoremressprdsds 18391* Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( ph  ->  Y  =  ( S X_s ( x  e.  I  |->  R ) ) )   &    |-  ( ph  ->  H  =  ( T X_s ( x  e.  I  |->  ( Rs  A ) ) ) )   &    |-  B  =  (
 Base `  H )   &    |-  D  =  ( dist `  Y )   &    |-  E  =  ( dist `  H )   &    |-  ( ph  ->  S  e.  U )   &    |-  ( ph  ->  T  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  (
 ( ph  /\  x  e.  I )  ->  R  e.  X )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  A  e.  Z )   =>    |-  ( ph  ->  E  =  ( D  |`  ( B  X.  B ) ) )
 
Theoremresspwsds 18392 Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( ph  ->  Y  =  ( R  ^s  I )
 )   &    |-  ( ph  ->  H  =  ( ( Rs  A ) 
 ^s  I ) )   &    |-  B  =  ( Base `  H )   &    |-  D  =  ( dist `  Y )   &    |-  E  =  ( dist `  H )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  E  =  ( D  |`  ( B  X.  B ) ) )
 
Theoremimasdsf1olem 18393* Lemma for imasdsf1o 18394. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  E  e.  ( * Met `  V )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  W  =  ( RR* ss  ( RR*  \  {  -oo } ) )   &    |-  S  =  { h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  (
 ( F `  ( 1st `  ( h `  1 ) ) )  =  ( F `  X )  /\  ( F `
  ( 2nd `  ( h `  n ) ) )  =  ( F `
  Y )  /\  A. i  e.  ( 1
 ... ( n  -  1 ) ) ( F `  ( 2nd `  ( h `  i
 ) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1
 ) ) ) ) ) }   &    |-  T  =  U_ n  e.  NN  ran  (
 g  e.  S  |->  (
 RR* s  gsumg  ( E  o.  g
 ) ) )   =>    |-  ( ph  ->  ( ( F `  X ) D ( F `  Y ) )  =  ( X E Y ) )
 
Theoremimasdsf1o 18394 The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  E  e.  ( * Met `  V )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( ( F `  X ) D ( F `  Y ) )  =  ( X E Y ) )
 
Theoremimasf1oxmet 18395 The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  E  e.  ( * Met `  V )
 )   =>    |-  ( ph  ->  D  e.  ( * Met `  B ) )
 
Theoremimasf1omet 18396 The image of a metric is a metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  E  e.  ( Met `  V ) )   =>    |-  ( ph  ->  D  e.  ( Met `  B )
 )
 
Theoremxpsdsfn 18397 Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  P  =  (
 dist `  T )   =>    |-  ( ph  ->  P  Fn  ( ( X  X.  Y )  X.  ( X  X.  Y ) ) )
 
Theoremxpsdsfn2 18398 Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  P  =  (
 dist `  T )   =>    |-  ( ph  ->  P  Fn  ( ( Base `  T )  X.  ( Base `  T ) ) )
 
Theoremxpsxmetlem 18399* Lemma for xpsxmet 18400. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  P  =  (
 dist `  T )   &    |-  M  =  ( ( dist `  R )  |`  ( X  X.  X ) )   &    |-  N  =  ( ( dist `  S )  |`  ( Y  X.  Y ) )   &    |-  ( ph  ->  M  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  N  e.  ( * Met `  Y ) )   =>    |-  ( ph  ->  ( dist `  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } )
 ) )  e.  ( * Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } )
 ) ) )
 
Theoremxpsxmet 18400 A product metric of extended metrics is an extended metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  P  =  (
 dist `  T )   &    |-  M  =  ( ( dist `  R )  |`  ( X  X.  X ) )   &    |-  N  =  ( ( dist `  S )  |`  ( Y  X.  Y ) )   &    |-  ( ph  ->  M  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  N  e.  ( * Met `  Y ) )   =>    |-  ( ph  ->  P  e.  ( * Met `  ( X  X.  Y ) ) )
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