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Theorem List for Metamath Proof Explorer - 20901-21000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremutopval 20901* The topology induced by a uniform structure  U. (Contributed by Thierry Arnoux, 30-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  (unifTop `  U )  =  {
 a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " { x } )  C_  a } )
 
Theoremelutop 20902* Open sets in the topology induced by an uniform structure  U on  X (Contributed by Thierry Arnoux, 30-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  ( A  e.  (unifTop `  U ) 
 <->  ( A  C_  X  /\  A. x  e.  A  E. v  e.  U  ( v " { x } )  C_  A ) ) )
 
Theoremutoptop 20903 The topology induced by a uniform structure  U is a topology. (Contributed by Thierry Arnoux, 30-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  (unifTop `  U )  e.  Top )
 
Theoremutopbas 20904 The base of the topology induced by a uniform structure  U. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  X  =  U. (unifTop `  U ) )
 
Theoremutoptopon 20905 Topology induced by a uniform structure  U with its base set. (Contributed by Thierry Arnoux, 5-Jan-2018.)
 |-  ( U  e.  (UnifOn `  X )  ->  (unifTop `  U )  e.  (TopOn `  X ) )
 
Theoremrestutop 20906 Restriction of a topology induced by an uniform structure (Contributed by Thierry Arnoux, 12-Dec-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( (unifTop `  U )t  A )  C_  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
 
Theoremrestutopopn 20907 The restriction of the topology induced by an uniform structure to an open set. (Contributed by Thierry Arnoux, 16-Dec-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U ) )  ->  ( (unifTop `  U )t  A )  =  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
 
Theoremustuqtoplem 20908* Lemma for ustuqtop 20915 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  (
 v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  A  e.  V )  ->  ( A  e.  ( N `  P )  <->  E. w  e.  U  A  =  ( w " { P } )
 ) )
 
Theoremustuqtop0 20909* Lemma for ustuqtop 20915 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  (
 v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( U  e.  (UnifOn `  X )  ->  N : X --> ~P ~P X )
 
Theoremustuqtop1 20910* Lemma for ustuqtop 20915, similar to ssnei2 19784 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  (
 v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X ) 
 /\  a  e.  ( N `  p ) ) 
 ->  b  e.  ( N `  p ) )
 
Theoremustuqtop2 20911* Lemma for ustuqtop 20915 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  (
 v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) ) 
 C_  ( N `  p ) )
 
Theoremustuqtop3 20912* Lemma for ustuqtop 20915, similar to elnei 19779 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  (
 v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p ) ) 
 ->  p  e.  a
 )
 
Theoremustuqtop4 20913* Lemma for ustuqtop 20915 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  (
 v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p ) ) 
 ->  E. b  e.  ( N `  p ) A. q  e.  b  a  e.  ( N `  q
 ) )
 
Theoremustuqtop5 20914* Lemma for ustuqtop 20915 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  (
 v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `
  p ) )
 
Theoremustuqtop 20915* For a given uniform structure  U on a set  X, there is a unique topology  j such that the set  ran  ( v  e.  U  |->  ( v
" { p }
) ) is the filter of the neighborhoods of  p for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  (
 v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( U  e.  (UnifOn `  X )  ->  E! j  e.  (TopOn `  X ) A. p  e.  X  ( N `  p )  =  ( ( nei `  j ) `  { p } ) )
 
Theoremutopsnneiplem 20916* The neighborhoods of a point  P for the topology induced by an uniform space  U. (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  J  =  (unifTop `  U )   &    |-  K  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) }   &    |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( ( nei `  J ) `  { P }
 )  =  ran  (
 v  e.  U  |->  ( v " { P } ) ) )
 
Theoremutopsnneip 20917* The neighborhoods of a point  P for the topology induced by an uniform space  U. (Contributed by Thierry Arnoux, 13-Jan-2018.)
 |-  J  =  (unifTop `  U )   =>    |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( ( nei `  J ) `  { P }
 )  =  ran  (
 v  e.  U  |->  ( v " { P } ) ) )
 
Theoremutopsnnei 20918 Images of singletons by entourages 
V are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018.)
 |-  J  =  (unifTop `  U )   =>    |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  ( V " { P } )  e.  (
 ( nei `  J ) `  { P } )
 )
 
Theoremutop2nei 20919 For any symmetrical entourage  V and any relation  M, build a neighborhood of  M. First part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 14-Jan-2018.)
 |-  J  =  (unifTop `  U )   =>    |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V )  /\  M  C_  ( X  X.  X ) ) 
 ->  ( V  o.  ( M  o.  V ) )  e.  ( ( nei `  ( J  tX  J ) ) `  M ) )
 
Theoremutop3cls 20920 Relation between a topological closure and a symmetric entourage in an uniform space. Second part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Jan-2018.)
 |-  J  =  (unifTop `  U )   =>    |-  ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) ) 
 /\  ( V  e.  U  /\  `' V  =  V ) )  ->  ( ( cls `  ( J  tX  J ) ) `
  M )  C_  ( V  o.  ( M  o.  V ) ) )
 
Theoremutopreg 20921 All Hausdorff uniform spaces are regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 16-Jan-2018.)
 |-  J  =  (unifTop `  U )   =>    |-  ( ( U  e.  (UnifOn `  X )  /\  J  e.  Haus )  ->  J  e.  Reg )
 
12.3.3  Uniform Spaces
 
Syntaxcuss 20922 Extend class notation with the Uniform Structure extractor function.
 class UnifSt
 
Syntaxcusp 20923 Extend class notation with the class of uniform spaces.
 class UnifSp
 
Syntaxctus 20924 Extend class notation with the function mapping a uniform structure to a uniform space.
 class toUnifSp
 
Definitiondf-uss 20925 Define the uniform structure extractor function. Similarly with df-topn 14913 this differs from df-unif 14807 when a structure has been restricted using df-ress 14723; in this case the  UnifSet component will still have a uniform set over the larger set, and this function fixes this by restricting the uniform set as well. (Contributed by Thierry Arnoux, 1-Dec-2017.)
 |- UnifSt  =  ( f  e.  _V  |->  ( ( UnifSet `  f
 )t  ( ( Base `  f
 )  X.  ( Base `  f ) ) ) )
 
Definitiondf-usp 20926 Definition of a uniform space, i.e. a base set with an uniform structure and its induced topology. Derived from definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)
 |- UnifSp  =  { f  |  ( (UnifSt `  f )  e.  (UnifOn `  ( Base `  f ) )  /\  ( TopOpen `  f )  =  (unifTop `  (UnifSt `  f
 ) ) ) }
 
Definitiondf-tus 20927 Define the function mapping a uniform structure to a uniform space. (Contributed by Thierry Arnoux, 17-Nov-2017.)
 |- toUnifSp  =  ( u  e.  U. ran UnifOn 
 |->  ( { <. ( Base ` 
 ndx ) ,  dom  U. u >. ,  <. ( UnifSet `  ndx ) ,  u >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  u ) >. ) )
 
Theoremussval 20928 The uniform structure on uniform space  W. This proof uses a trick with fvprc 5842 to avoid requiring  W to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |-  B  =  ( Base `  W )   &    |-  U  =  (
 UnifSet `  W )   =>    |-  ( Ut  ( B  X.  B ) )  =  (UnifSt `  W )
 
Theoremussid 20929 In case the base of the UnifSt element of the uniform space is the base of its element structure, then UnifSt does not restrict it further. (Contributed by Thierry Arnoux, 4-Dec-2017.)
 |-  B  =  ( Base `  W )   &    |-  U  =  (
 UnifSet `  W )   =>    |-  ( ( B  X.  B )  = 
 U. U  ->  U  =  (UnifSt `  W )
 )
 
Theoremisusp 20930 The predicate  W is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
 |-  B  =  ( Base `  W )   &    |-  U  =  (UnifSt `  W )   &    |-  J  =  (
 TopOpen `  W )   =>    |-  ( W  e. UnifSp  <->  ( U  e.  (UnifOn `  B )  /\  J  =  (unifTop `  U ) ) )
 
Theoremressunif 20931  UnifSet is unaffected by restriction. (Contributed by Thierry Arnoux, 7-Dec-2017.)
 |-  H  =  ( Gs  A )   &    |-  U  =  (
 UnifSet `  G )   =>    |-  ( A  e.  V  ->  U  =  (
 UnifSet `  H ) )
 
Theoremressuss 20932 Value of the uniform structure of a restricted space. (Contributed by Thierry Arnoux, 12-Dec-2017.)
 |-  ( A  e.  V  ->  (UnifSt `  ( Ws  A ) )  =  (
 (UnifSt `  W )t  ( A  X.  A ) ) )
 
Theoremressust 20933 The uniform structure of a restricted space. (Contributed by Thierry Arnoux, 22-Jan-2018.)
 |-  X  =  ( Base `  W )   &    |-  T  =  (UnifSt `  ( Ws  A ) )   =>    |-  ( ( W  e. UnifSp  /\  A  C_  X )  ->  T  e.  (UnifOn `  A ) )
 
Theoremressusp 20934 The restriction of a uniform topological space to an open set is a uniform space. (Contributed by Thierry Arnoux, 16-Dec-2017.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (
 TopOpen `  W )   =>    |-  ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  A  e.  J ) 
 ->  ( Ws  A )  e. UnifSp )
 
Theoremtusval 20935 The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  (toUnifSp `  U )  =  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
 
Theoremtuslem 20936 Lemma for tusbas 20937, tusunif 20938, and tustopn 20940. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  ( X  =  ( Base `  K )  /\  U  =  ( UnifSet `  K )  /\  (unifTop `  U )  =  ( TopOpen `  K )
 ) )
 
Theoremtusbas 20937 The base set of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  X  =  ( Base `  K )
 )
 
Theoremtusunif 20938 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  U  =  ( UnifSet `  K )
 )
 
Theoremtususs 20939 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  U  =  (UnifSt `  K )
 )
 
Theoremtustopn 20940 The topology induced by a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   &    |-  J  =  (unifTop `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  J  =  ( TopOpen `  K )
 )
 
Theoremtususp 20941 A constructed uniform space is an uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  K  e. UnifSp )
 
Theoremtustps 20942 A constructed uniform space is a topological space. (Contributed by Thierry Arnoux, 25-Jan-2018.)
 |-  K  =  (toUnifSp `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  K  e.  TopSp )
 
Theoremuspreg 20943 If a uniform space is Hausdorff, it is regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 4-Jan-2018.)
 |-  J  =  ( TopOpen `  W )   =>    |-  ( ( W  e. UnifSp  /\  J  e.  Haus )  ->  J  e.  Reg )
 
12.3.4  Uniform continuity
 
Syntaxcucn 20944 Extend class notation with the uniform continuity operation.
 class Cnu
 
Definitiondf-ucn 20945* Define a function on two uniform structures which value is the set of uniformly continuous functions from the first uniform structure to the second. A function  f is uniformly continuous if, roughly speaking, it is possible to guarantee that  ( f `  x
) and  ( f `  y ) be as close to each other as we please by requiring only that  x and  y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between  ( f `  x
) and  ( f `  y ) cannot depend on  x and  y themselves. This formulation is the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |- Cnu  =  ( u  e.  U. ran UnifOn ,  v  e.  U. ran UnifOn 
 |->  { f  e.  ( dom  U. v  ^m  dom  U. u )  |  A. s  e.  v  E. r  e.  u  A. x  e.  dom  U. u A. y  e.  dom  U. u ( x r y  ->  ( f `  x ) s ( f `  y ) ) } )
 
Theoremucnval 20946* The set of all uniformly continuous function from uniform space  U to uniform space  V. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  (UnifOn `  Y ) )  ->  ( U Cnu V )  =  { f  e.  ( Y  ^m  X )  |  A. s  e.  V  E. r  e.  U  A. x  e.  X  A. y  e.  X  ( x r y  ->  ( f `  x ) s ( f `  y ) ) } )
 
Theoremisucn 20947* The predicate " F is a uniformly continuous function from uniform space  U to uniform space  V." (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  (UnifOn `  Y ) )  ->  ( F  e.  ( U Cnu V )  <-> 
 ( F : X --> Y  /\  A. s  e.  V  E. r  e.  U  A. x  e.  X  A. y  e.  X  ( x r y  ->  ( F `  x ) s ( F `  y ) ) ) ) )
 
Theoremisucn2 20948* The predicate " F is a uniformly continuous function from uniform space  U to uniform space  V." , expressed with filter bases for the entourages. (Contributed by Thierry Arnoux, 26-Jan-2018.)
 |-  U  =  ( ( X  X.  X )
 filGen R )   &    |-  V  =  ( ( Y  X.  Y ) filGen S )   &    |-  ( ph  ->  U  e.  (UnifOn `  X ) )   &    |-  ( ph  ->  V  e.  (UnifOn `  Y ) )   &    |-  ( ph  ->  R  e.  ( fBas `  ( X  X.  X ) ) )   &    |-  ( ph  ->  S  e.  ( fBas `  ( Y  X.  Y ) ) )   =>    |-  ( ph  ->  ( F  e.  ( U Cnu V )  <->  ( F : X
 --> Y  /\  A. s  e.  S  E. r  e.  R  A. x  e.  X  A. y  e.  X  ( x r y  ->  ( F `  x ) s ( F `  y ) ) ) ) )
 
Theoremucnimalem 20949* Reformulate the  G function as a mapping with one variable. (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
 ) >. )   =>    |-  G  =  ( p  e.  ( X  X.  X )  |->  <. ( F `
  ( 1st `  p ) ) ,  ( F `  ( 2nd `  p ) ) >. )
 
Theoremucnima 20950* 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 20951* 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 20952 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 20953 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 20954 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 ) )
 
12.3.5  Cauchy filters in uniform spaces
 
Syntaxccfilu 20955 Extend class notation with the set of Cauchy filter bases.
 class CauFilu
 
Definitiondf-cfilu 20956* 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 20957* 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 20958 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 20959* 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 20960* Lemma for fmucnd 20961. (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 20961* 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 20962 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 20963 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 20964 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 20965 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 ) )
 
12.3.6  Complete uniform spaces
 
Syntaxccusp 20966 Extend class notation with the class of all complete uniform spaces.
 class CUnifSp
 
Definitiondf-cusp 20967* 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 20968* 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 20969 A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
 
Theoremcuspcvg 20970 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 20971* 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 20972* 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 20973 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 ) )
 
12.4  Metric spaces
 
12.4.1  Pseudometric spaces
 
Theoremispsmet 20974* 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 20975 Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  X  =  dom  dom  D )
 
Theorempsmetf 20976 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 20977 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 20978 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 20979 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 20980 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 20981 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 20982 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 20983 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 20984 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 20985 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 )
 
12.4.2  Basic metric space properties
 
Syntaxcxme 20986 Extend class notation with the class of all extended metric spaces.
 class  *MetSp
 
Syntaxcmt 20987 Extend class notation with the class of all metric spaces.
 class  MetSp
 
Syntaxctmt 20988 Extend class notation with the function mapping a metric to a metric space.
 class toMetSp
 
Definitiondf-xms 20989 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 20990 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 20991 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 20992* 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 20993* 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 20994* 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 20995* 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 20996* 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 20997* Lemma for metf 20999 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 20998 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 20999 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 21000 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* )
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38473
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