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Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtlmtmd 21201 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e. TopMnd )
 
Theoremtlmtps 21202 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  TopSp )
 
Theoremtlmlmod 21203 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  LMod )
 
Theoremtlmtrg 21204 The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. TopMod  ->  F  e.  TopRing )
 
Theoremtlmscatps 21205 The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. TopMod  ->  F  e.  TopSp )
 
Theoremistvc 21206 A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  TopVec  <->  ( W  e. TopMod  /\  F  e. TopDRing ) )
 
Theoremtvctdrg 21207 The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  TopVec  ->  F  e. TopDRing )
 
Theoremcnmpt1vsca 21208* Continuity of scalar multiplication; analogue of cnmpt12f 20681 which cannot be used directly because  .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  J  =  (
 TopOpen `  W )   &    |-  K  =  ( TopOpen `  F )   &    |-  ( ph  ->  W  e. TopMod )   &    |-  ( ph  ->  L  e.  (TopOn `  X ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B ) )  e.  ( L  Cn  J ) )
 
Theoremcnmpt2vsca 21209* Continuity of scalar multiplication; analogue of cnmpt22f 20690 which cannot be used directly because  .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  J  =  (
 TopOpen `  W )   &    |-  K  =  ( TopOpen `  F )   &    |-  ( ph  ->  W  e. TopMod )   &    |-  ( ph  ->  L  e.  (TopOn `  X ) )   &    |-  ( ph  ->  M  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( L  tX  M )  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( L 
 tX  M )  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A 
 .x.  B ) )  e.  ( ( L  tX  M )  Cn  J ) )
 
Theoremtlmtgp 21210 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  TopGrp )
 
Theoremtvctlm 21211 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e. TopMod )
 
Theoremtvclmod 21212 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e.  LMod )
 
Theoremtvclvec 21213 A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e.  LVec )
 
12.3  Uniform Structures and Spaces
 
12.3.1  Uniform structures
 
Syntaxcust 21214 Extend class notation with the class function of uniform structures.
 class UnifOn
 
Definitiondf-ust 21215* Definition of a uniform structure. Definition 1 of [BourbakiTop1] p. II.1. A uniform structure is used to give a generalization of the idea of Cauchy's sequence. This definition is analogous to TopOn. Elements of an uniform structure are called entourages. (Contributed by FL, 29-May-2014.) (Revised by Thierry Arnoux, 15-Nov-2017.)
 |- UnifOn  =  ( x  e.  _V  |->  { u  |  ( u 
 C_  ~P ( x  X.  x )  /\  ( x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x ) ( v  C_  w  ->  w  e.  u )  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  (
 (  _I  |`  x ) 
 C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w ) 
 C_  v ) ) ) } )
 
Theoremustfn 21216 The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |- UnifOn  Fn  _V
 
Theoremustval 21217* The class of all uniform structures for a base  X. (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |-  ( X  e.  _V  ->  (UnifOn `  X )  =  { u  |  ( u  C_  ~P ( X  X.  X )  /\  ( X  X.  X )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( X  X.  X ) ( v  C_  w  ->  w  e.  u ) 
 /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  (
 (  _I  |`  X ) 
 C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w ) 
 C_  v ) ) ) } )
 
Theoremisust 21218* The predicate " U is a uniform structure with base  X." (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |-  ( X  e.  _V  ->  ( U  e.  (UnifOn `  X )  <->  ( U  C_  ~P ( X  X.  X )  /\  ( X  X.  X )  e.  U  /\  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X ) ( v  C_  w  ->  w  e.  U ) 
 /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  (
 (  _I  |`  X ) 
 C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
 C_  v ) ) ) ) )
 
Theoremustssxp 21219 Entourages are subsets of the Cartesian product of the base set. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X ) )
 
Theoremustssel 21220 A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X ) )  ->  ( V  C_  W  ->  W  e.  U ) )
 
Theoremustbasel 21221 The full set is always an entourage. Condition FIIb of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  ( X  X.  X )  e.  U )
 
Theoremustincl 21222 A uniform structure is closed under finite intersection. Condition FII of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  e.  U )  ->  ( V  i^i  W )  e.  U )
 
Theoremustdiag 21223 The diagonal set is included in any entourage, i.e. any point is  V -close to itself. Condition UI of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (  _I  |`  X ) 
 C_  V )
 
Theoremustinvel 21224 If  V is an entourage, so is its inverse. Condition UII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  `' V  e.  U )
 
Theoremustexhalf 21225* For each entourage  V there is an entourage  w that is "not more than half as large". Condition UIII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( w  o.  w )  C_  V )
 
Theoremustrel 21226 The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )
 
Theoremustfilxp 21227 A uniform structure on a nonempty base is a filter. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  ->  U  e.  ( Fil `  ( X  X.  X ) ) )
 
Theoremustne0 21228 A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  U  =/= 
 (/) )
 
Theoremustssco 21229 In an uniform structure, any entourage  V is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( V  o.  V ) )
 
Theoremustexsym 21230* In an uniform structure, for any entourage  V, there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  V ) )
 
Theoremustex2sym 21231* In an uniform structure, for any entourage  V, there exists a symmetrical entourage smaller than half  V. (Contributed by Thierry Arnoux, 16-Jan-2018.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  w )  C_  V ) )
 
Theoremustex3sym 21232* In an uniform structure, for any entourage  V, there exists a symmetrical entourage smaller than a third of  V. (Contributed by Thierry Arnoux, 16-Jan-2018.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  ( w  o.  w ) ) 
 C_  V ) )
 
Theoremustref 21233 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 21234 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 21235 The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |- 
 -.  (/)  e.  U. ran UnifOn
 
Theoremustund 21236 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 definition of uniform 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 21237 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 21238 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 definition of uniform 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 21239 First direction for ustbas 21242. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  U  e.  U. ran UnifOn )
 
Theoremustbas2 21240 Second direction for ustbas 21242. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  X  =  dom  U. U )
 
Theoremustuni 21241 The set union of a uniform structure is the Cartesian product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  U. U  =  ( X  X.  X ) )
 
Theoremustbas 21242 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 21243 Lemma for ustuqtop 21261. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  ( V " { P } )  C_  X )
 
Theoremtrust 21244 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 ) )
 
12.3.2  The topology induced by an uniform structure
 
Syntaxcutop 21245 Extend class notation with the function inducing a topology from a uniform structure.
 class unifTop
 
Definitiondf-utop 21246* 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 21247* 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 21248* 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 21249 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 21250 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 21251 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 21252 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 21253 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 21254* Lemma for ustuqtop 21261. (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 21255* Lemma for ustuqtop 21261. (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 21256* Lemma for ustuqtop 21261, similar to ssnei2 20132. (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 21257* Lemma for ustuqtop 21261. (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 21258* Lemma for ustuqtop 21261, similar to elnei 20127. (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 21259* Lemma for ustuqtop 21261. (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 21260* Lemma for ustuqtop 21261. (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 21261* 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 21262* 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 21263* 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 21264 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 21265 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 21266 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 21267 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 21268 Extend class notation with the Uniform Structure extractor function.
 class UnifSt
 
Syntaxcusp 21269 Extend class notation with the class of uniform spaces.
 class UnifSp
 
Syntaxctus 21270 Extend class notation with the function mapping a uniform structure to a uniform space.
 class toUnifSp
 
Definitiondf-uss 21271 Define the uniform structure extractor function. Similarly with df-topn 15322 this differs from df-unif 15213 when a structure has been restricted using df-ress 15128; 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 21272 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 21273 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 21274 The uniform structure on uniform space  W. This proof uses a trick with fvprc 5859 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 21275 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 21276 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 21277  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 21278 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 21279 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 21280 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 21281 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 21282 Lemma for tusbas 21283, tusunif 21284, and tustopn 21286. (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 21283 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 21284 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 21285 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 21286 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 21287 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 21288 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 21289 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 21290 Extend class notation with the uniform continuity operation.
 class Cnu
 
Definitiondf-ucn 21291* 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 21292* 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 21293* 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 21294* 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 21295* 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 21296* 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 21297* 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 21298 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 21299 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 21300 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 ) )
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