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Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtsmssub 18101 The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  H : A
 --> B )   &    |-  ( ph  ->  X  e.  ( G tsums  F ) )   &    |-  ( ph  ->  Y  e.  ( G tsums  H ) )   =>    |-  ( ph  ->  ( X  .-  Y )  e.  ( G tsums  ( F  o F  .-  H ) ) )
 
Theoremtgptsmscls 18102 A sum in a topological group is uniquely determined up to a coset of  cls ( { 0 } ), which is a normal subgroup by clsnsg 18062, 0nsg 14914. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  X  e.  ( G tsums  F ) )   =>    |-  ( ph  ->  ( G tsums  F )  =  ( ( cls `  J ) `  { X } )
 )
 
Theoremtgptsmscld 18103 The set of limit points to an infinite sum in a topological group is closed. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ( G tsums  F )  e.  ( Clsd `  J ) )
 
Theoremtsmssplit 18104 Split a topological group sum into two parts. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e. TopMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  X  e.  ( G tsums  ( F  |`  C ) ) )   &    |-  ( ph  ->  Y  e.  ( G tsums  ( F  |`  D ) ) )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  A  =  ( C  u.  D ) )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  ( G tsums  F ) )
 
Theoremtsmsxplem1 18105* Lemma for tsmsxp 18107. (Contributed by Mario Carneiro, 21-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  ( ph  ->  F : ( A  X.  C ) --> B )   &    |-  ( ph  ->  H : A
 --> B )   &    |-  ( ( ph  /\  j  e.  A ) 
 ->  ( H `  j
 )  e.  ( G tsums 
 ( k  e.  C  |->  ( j F k ) ) ) )   &    |-  J  =  ( TopOpen `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  L  e.  J )   &    |-  ( ph  ->  .0.  e.  L )   &    |-  ( ph  ->  K  e.  ( ~P A  i^i  Fin ) )   &    |-  ( ph  ->  dom 
 D  C_  K )   &    |-  ( ph  ->  D  e.  ( ~P ( A  X.  C )  i^i  Fin ) )   =>    |-  ( ph  ->  E. n  e.  ( ~P C  i^i  Fin )
 ( ran  D  C_  n  /\  A. x  e.  K  ( ( H `  x )  .-  ( G 
 gsumg  ( F  |`  ( { x }  X.  n ) ) ) )  e.  L ) )
 
Theoremtsmsxplem2 18106* Lemma for tsmsxp 18107. (Contributed by Mario Carneiro, 21-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  ( ph  ->  F : ( A  X.  C ) --> B )   &    |-  ( ph  ->  H : A
 --> B )   &    |-  ( ( ph  /\  j  e.  A ) 
 ->  ( H `  j
 )  e.  ( G tsums 
 ( k  e.  C  |->  ( j F k ) ) ) )   &    |-  J  =  ( TopOpen `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  L  e.  J )   &    |-  ( ph  ->  .0.  e.  L )   &    |-  ( ph  ->  K  e.  ( ~P A  i^i  Fin ) )   &    |-  ( ph  ->  A. c  e.  S  A. d  e.  T  (
 c  .+  d )  e.  U )   &    |-  ( ph  ->  N  e.  ( ~P C  i^i  Fin ) )   &    |-  ( ph  ->  D  C_  ( K  X.  N ) )   &    |-  ( ph  ->  A. x  e.  K  ( ( H `
  x )  .-  ( G  gsumg  ( F  |`  ( { x }  X.  N ) ) ) )  e.  L )   &    |-  ( ph  ->  ( G  gsumg  ( F  |`  ( K  X.  N ) ) )  e.  S )   &    |-  ( ph  ->  A. g  e.  ( L  ^m  K ) ( G  gsumg  g )  e.  T )   =>    |-  ( ph  ->  ( G  gsumg  ( H  |`  K ) )  e.  U )
 
Theoremtsmsxp 18107* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 15479 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 18105 for the main proof; this part mostly sets up the local assumptions. (Contributed by Mario Carneiro, 21-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  ( ph  ->  F : ( A  X.  C ) --> B )   &    |-  ( ph  ->  H : A
 --> B )   &    |-  ( ( ph  /\  j  e.  A ) 
 ->  ( H `  j
 )  e.  ( G tsums 
 ( k  e.  C  |->  ( j F k ) ) ) )   =>    |-  ( ph  ->  ( G tsums  F )  C_  ( G tsums  H ) )
 
11.2.8  Topological rings, fields, vector spaces
 
Syntaxctrg 18108 The class of all topological division rings.
 class  TopRing
 
Syntaxctdrg 18109 The class of all topological division rings.
 class TopDRing
 
Syntaxctlm 18110 The class of all topological modules.
 class TopMod
 
Syntaxctvc 18111 The class of all topological vector spaces.
 class  TopVec
 
Definitiondf-trg 18112 Define a topological ring, which is a ring such that the addition is a topological group operation and the multiplication is continuous. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  TopRing  =  { r  e.  ( TopGrp  i^i  Ring )  |  (mulGrp `  r )  e. TopMnd }
 
Definitiondf-tdrg 18113 Define a topological division ring (which differs from a topological field only in being potentially noncommutative), which is a division ring and topological ring such that the unit group of the division ring (which is the set of nonzero elements) is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- TopDRing  =  { r  e.  ( TopRing  i^i  DivRing )  |  ( (mulGrp `  r )s  (Unit `  r )
 )  e.  TopGrp }
 
Definitiondf-tlm 18114 Define a topological left module, which is just what its name suggests: instead of a group over a ring with a scalar product connecting them, it is a topological group over a topological ring with a continuous scalar product. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- TopMod  =  { w  e.  (TopMnd  i^i  LMod )  |  ( (Scalar `  w )  e.  TopRing  /\  ( .s f `  w )  e.  ( (
 ( TopOpen `  (Scalar `  w ) )  tX  ( TopOpen `  w ) )  Cn  ( TopOpen `  w )
 ) ) }
 
Definitiondf-tvc 18115 Define a topological left vector space, which is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  TopVec  =  { w  e. TopMod  |  (Scalar `  w )  e. TopDRing }
 
Theoremistrg 18116 Express the predicate " R is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  ( R  e.  TopRing  <->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )
 
Theoremtrgtmd 18117 The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  ( R  e.  TopRing  ->  M  e. TopMnd )
 
Theoremistdrg 18118 Express the predicate " R is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  <->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  U )  e.  TopGrp ) )
 
Theoremtdrgunit 18119 The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  ( Ms  U )  e.  TopGrp )
 
Theoremtrgtgp 18120 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
 
Theoremtrgtmd2 18121 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e. TopMnd )
 
Theoremtrgtps 18122 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopSp )
 
Theoremtrgrng 18123 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  Ring )
 
Theoremtrggrp 18124 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  Grp )
 
Theoremtdrgtrg 18125 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. 
 TopRing )
 
Theoremtdrgdrng 18126 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. 
 DivRing )
 
Theoremtdrgrng 18127 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e.  Ring )
 
Theoremtdrgtmd 18128 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. TopMnd )
 
Theoremtdrgtps 18129 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e.  TopSp )
 
Theoremistdrg2 18130 A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. TopDRing  <->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  ( B  \  {  .0.  } ) )  e.  TopGrp ) )
 
Theoremmulrcn 18131 The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  T  =  ( + f `  (mulGrp `  R ) )   =>    |-  ( R  e.  TopRing  ->  T  e.  ( ( J  tX  J )  Cn  J ) )
 
Theoreminvrcn2 18132 The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to itself. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  I  =  (
 invr `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  I  e.  ( ( Jt  U )  Cn  ( Jt  U ) ) )
 
Theoreminvrcn 18133 The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to the field. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  I  =  (
 invr `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  I  e.  ( ( Jt  U )  Cn  J ) )
 
Theoremcnmpt1mulr 18134* Continuity of ring multiplication; analogue of cnmpt12f 17621 which cannot be used directly because 
.r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  TopRing )   &    |-  ( ph  ->  K  e.  (TopOn `  X ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B ) )  e.  ( K  Cn  J ) )
 
Theoremcnmpt2mulr 18135* Continuity of ring multiplication; analogue of cnmpt22f 17630 which cannot be used directly because 
.r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  TopRing )   &    |-  ( ph  ->  K  e.  (TopOn `  X ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( K  tX  L )  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( K 
 tX  L )  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A 
 .x.  B ) )  e.  ( ( K  tX  L )  Cn  J ) )
 
Theoremdvrcn 18136 The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  ./  =  (/r `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  ./  e.  ( ( J  tX  ( Jt  U ) )  Cn  J ) )
 
Theoremistlm 18137 The predicate " W is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- 
 .x.  =  ( .s f `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( TopOpen `  F )   =>    |-  ( W  e. TopMod  <->  ( ( W  e. TopMnd  /\  W  e.  LMod  /\  F  e.  TopRing )  /\  .x. 
 e.  ( ( K 
 tX  J )  Cn  J ) ) )
 
Theoremvscacn 18138 The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- 
 .x.  =  ( .s f `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( TopOpen `  F )   =>    |-  ( W  e. TopMod  ->  .x.  e.  ( ( K  tX  J )  Cn  J ) )
 
Theoremtlmtmd 18139 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e. TopMnd )
 
Theoremtlmtps 18140 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  TopSp )
 
Theoremtlmlmod 18141 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  LMod )
 
Theoremtlmtrg 18142 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 18143 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 18144 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 18145 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 18146* Continuity of scalar multiplication; analogue of cnmpt12f 17621 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 18147* Continuity of scalar multiplication; analogue of cnmpt22f 17630 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 18148 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  TopGrp )
 
Theoremtvctlm 18149 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e. TopMod )
 
Theoremtvclmod 18150 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e.  LMod )
 
Theoremtvclvec 18151 A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e.  LVec )
 
11.3  Uniform Stuctures and Spaces
 
11.3.1  Uniform structures
 
Syntaxcust 18152 Extend class notation with the class function of uniform structures.
 class UnifOn
 
Definitiondf-ust 18153* 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 defintion 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 18154 The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |- UnifOn  Fn  _V
 
Theoremustval 18155* 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 18156* 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 18157 Entourages are subsets of the cross 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 18158 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 18159 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 18160 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 18161 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 18162 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 18163* 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 18164 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 18165 A uniform structure on a non-empty 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 18166 A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  U  =/= 
 (/) )
 
Theoremustssco 18167 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 18168* 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 18169* 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 18170* 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 18171 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 18172 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 18173 The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |- 
 -.  (/)  e.  U. ran UnifOn
 
Theoremustund 18174 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 18175 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 18176 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 18177 First direction for ustbas 18180. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  U  e.  U. ran UnifOn )
 
Theoremustbas2 18178 Second direction for ustbas 18180. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  X  =  dom  U. U )
 
Theoremustuni 18179 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 18180 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 18181 Lemma for ustuqtop 18199 (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  ( V " { P } )  C_  X )
 
Theoremtrust 18182 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 18183 Extend class notation with the function inducing a topology from a uniform structure.
 class unifTop
 
Definitiondf-utop 18184* 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 18185* 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 18186* 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 18187 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 18188 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 18189 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 18190 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 18191 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 18192* Lemma for ustuqtop 18199 (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 18193* Lemma for ustuqtop 18199 (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 18194* Lemma for ustuqtop 18199, similar to ssnei2 17105 (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 18195* Lemma for ustuqtop 18199 (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 18196* Lemma for ustuqtop 18199, similar to elnei 17100 (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 18197* Lemma for ustuqtop 18199 (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 18198* Lemma for ustuqtop 18199 (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 18199* 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 18200* 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 } ) ) )
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