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Theorem List for Metamath Proof Explorer - 18201-18300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremustref 18201 Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  A V A )
 
Theoremust0 18202 The unique uniform structure of the empty set is the empty set. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |-  (UnifOn `  (/) )  =  { { (/) } }
 
Theoremustn0 18203 The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |- 
 -.  (/)  e.  U. ran UnifOn
 
Theoremustund 18204 If two intersecting sets  A and  B are both small in  V, their union is small in  ( V ^ 2 ). Proposition 1 of [BourbakiTop1] p. II.12. This proposition actually does not require any axiom of the defintion of unifom structures. (Contributed by Thierry Arnoux, 17-Nov-2017.)
 |-  ( ph  ->  ( A  X.  A )  C_  V )   &    |-  ( ph  ->  ( B  X.  B ) 
 C_  V )   &    |-  ( ph  ->  ( A  i^i  B )  =/=  (/) )   =>    |-  ( ph  ->  ( ( A  u.  B )  X.  ( A  u.  B ) )  C_  ( V  o.  V ) )
 
Theoremustelimasn 18205 Any point  A is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  A  e.  ( V
 " { A }
 ) )
 
Theoremustneism 18206 For a point  A in  X,  ( V " { A } ) is small enough in  ( V  o.  `' V ). This proposition actually does not require any axiom of the defintion of unifom structures. (Contributed by Thierry Arnoux, 18-Nov-2017.)
 |-  ( ( V  C_  ( X  X.  X ) 
 /\  A  e.  X )  ->  ( ( V
 " { A }
 )  X.  ( V " { A } )
 )  C_  ( V  o.  `' V ) )
 
Theoremelrnust 18207 First direction for ustbas 18210. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  U  e.  U. ran UnifOn )
 
Theoremustbas2 18208 Second direction for ustbas 18210. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  X  =  dom  U. U )
 
Theoremustuni 18209 The set union of a uniform structure is the cross product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  U. U  =  ( X  X.  X ) )
 
Theoremustbas 18210 Recover the base of an uniform structure  U.  U. ran UnifOn is to UnifOn what  Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  X  =  dom  U. U   =>    |-  ( U  e.  U. ran UnifOn  <->  U  e.  (UnifOn `  X ) )
 
Theoremustimasn 18211 Lemma for ustuqtop 18229 (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  ( V " { P } )  C_  X )
 
Theoremtrust 18212 The trace of a uniform structure  U on a subset  A is a uniform structure on  A. Definition 3 of [BourbakiTop1] p. II.9. (Contributed by Thierry Arnoux, 2-Dec-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A ) )  e.  (UnifOn `  A ) )
 
11.3.2  The topology induced by an uniform structure
 
Syntaxcutop 18213 Extend class notation with the function inducing a topology from a uniform structure.
 class unifTop
 
Definitiondf-utop 18214* Definition of a topology induced by a uniform structure. Definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)
 |- unifTop  =  ( u  e.  U. ran UnifOn 
 |->  { a  e.  ~P dom  U. u  |  A. x  e.  a  E. v  e.  u  (
 v " { x }
 )  C_  a }
 )
 
Theoremutopval 18215* 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 18216* 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 18217 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 18218 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 18219 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 18220 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 18221 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 18222* Lemma for ustuqtop 18229 (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 18223* Lemma for ustuqtop 18229 (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 18224* Lemma for ustuqtop 18229, similar to ssnei2 17135 (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 18225* Lemma for ustuqtop 18229 (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 18226* Lemma for ustuqtop 18229, similar to elnei 17130 (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 18227* Lemma for ustuqtop 18229 (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 18228* Lemma for ustuqtop 18229 (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 18229* 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 neighbourhoods 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 18230* 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 18231* 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 18232 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 18233 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 18234 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 18235 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 )
 
11.3.3  Uniform Spaces
 
Syntaxcuss 18236 Extend class notation with the Uniform Structure extractor function.
 class UnifSt
 
Syntaxcusp 18237 Extend class notation with the class of uniform spaces.
 class UnifSp
 
Syntaxctus 18238 Extend class notation with the function mapping a uniform structure to a uniform space.
 class toUnifSp
 
Definitiondf-uss 18239 Define the uniform structure extractor function. Similarly with df-topn 13606 this differs from df-unif 13507 when a structure has been restricted using df-ress 13431; 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 18240 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 18241 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 18242 The uniform structure on uniform space  W. This proof uses a trick with fvprc 5681 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 18243 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 18244 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 18245  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 18246 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 18247 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 18248 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 18249 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 18250 Lemma for tusbas 18251, tusunif 18252, and tustopn 18254. (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 18251 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 18252 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 18253 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 18254 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 18255 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 18256 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 18257 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 )
 
11.3.4  Uniform continuity
 
Syntaxcucn 18258 Extend class notation with the uniform continuity operation.
 class Cnu
 
Definitiondf-ucn 18259* 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 18260* 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 18261* 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 18262* 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 18263* 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 18264* 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 18265* 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 18266 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 18267 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 18268 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 18269 Extend class notation with the set of Cauchy filter bases.
 class CauFilu
 
Definitiondf-cfilu 18270* 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 18271* 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 18272 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 18273* 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 18274* Lemma for fmucnd 18275. (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 18275* 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 18276 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 18277 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 18278 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 18279 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 18280 Extend class notation with the class of all complete uniform spaces.
 class CUnifSp
 
Definitiondf-cusp 18281* 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 18282* 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 18283 A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
 
Theoremcuspcvg 18284 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 18285* 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 18286* 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 18287 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 18288* 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 18289 Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  X  =  dom  dom  D )
 
Theorempsmetf 18290 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 18291 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 18292 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 18293 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 18294 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 18295 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 18296 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 18297 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 18298 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 18299 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 18300 Extend class notation with the class of all extended metric spaces.
 class  * MetSp
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