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Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremretopbas 18101 A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)
 |- 
 ran  (,)  e.  TopBases
 
Theoremretop 18102 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
 |-  ( topGen `  ran  (,) )  e.  Top
 
Theoremuniretop 18103 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
 |- 
 RR  =  U. ( topGen `
  ran  (,) )
 
Theoremretopon 18104 The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( topGen `  ran  (,) )  e.  (TopOn `  RR )
 
TheoremretpsOLD 18105 The standard topological space on the reals. (Contributed by NM, 10-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 <. RR ,  ( topGen `  ran  (,) ) >.  e.  TopSp OLD
 
Theoremretps 18106 The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
 |-  K  =  { <. (
 Base `  ndx ) ,  RR >. ,  <. (TopSet `  ndx ) ,  ( topGen `  ran  (,) ) >. }   =>    |-  K  e.  TopSp
 
Theoremiooretop 18107 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)
 |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
 
Theoremicccld 18108 Closed intervals are closed sets of the standard topology on  RR. (Contributed by FL, 14-Sep-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B )  e.  ( Clsd `  ( topGen `  ran  (,) )
 ) )
 
Theoremicopnfcld 18109 Right-unbounded closed intervals are closed sets of the standard topology on  RR. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  ( A  e.  RR  ->  ( A [,)  +oo )  e.  ( Clsd `  ( topGen `  ran  (,) )
 ) )
 
Theoremiocmnfcld 18110 Left-unbounded closed intervals are closed sets of the standard topology on  RR. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  ( A  e.  RR  ->  (  -oo (,] A )  e.  ( Clsd `  ( topGen `  ran  (,) )
 ) )
 
Theoremqdensere 18111  QQ is dense in the standard topology on  RR. (Contributed by NM, 1-Mar-2007.)
 |-  ( ( cls `  ( topGen `
  ran  (,) ) ) `
  QQ )  =  RR
 
Theoremcnmetdval 18112 Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  D  =  ( abs 
 o.  -  )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A D B )  =  ( abs `  ( A  -  B ) ) )
 
Theoremcnmet 18113 The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)
 |-  ( abs  o.  -  )  e.  ( Met `  CC )
 
Theoremcnxmet 18114 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( abs  o.  -  )  e.  ( * Met `  CC )
 
Theoremcnbl0 18115 Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  D  =  ( abs 
 o.  -  )   =>    |-  ( R  e.  RR* 
 ->  ( `' abs " (
 0 [,) R ) )  =  ( 0 (
 ball `  D ) R ) )
 
Theoremcnblcld 18116* Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  D  =  ( abs 
 o.  -  )   =>    |-  ( R  e.  RR* 
 ->  ( `' abs " (
 0 [,] R ) )  =  { x  e. 
 CC  |  ( 0 D x )  <_  R } )
 
Theoremcnfldms 18117 The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-fld  e.  MetSp
 
Theoremcnfldxms 18118 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-fld  e.  *
 MetSp
 
Theoremcnfldtps 18119 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-fld  e.  TopSp
 
Theoremcnfldnm 18120 The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- 
 abs  =  ( norm ` fld )
 
Theoremcnngp 18121 The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-fld  e. NrmGrp
 
Theoremcnnrg 18122 The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-fld  e. NrmRing
 
Theoremcnfldtopn 18123 The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  =  ( MetOpen `  ( abs  o. 
 -  ) )
 
Theoremcnfldtopon 18124 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e.  (TopOn `  CC )
 
Theoremcnfldtop 18125 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e.  Top
 
Theoremcnfldhaus 18126 The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e.  Haus
 
Theoremremetdval 18127 Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B ) ) )
 
Theoremremet 18128 The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  D  e.  ( Met `  RR )
 
Theoremrexmet 18129 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  D  e.  ( * Met `  RR )
 
Theorembl2ioo 18130 A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (
 ball `  D ) B )  =  ( ( A  -  B ) (,) ( A  +  B ) ) )
 
Theoremioo2bl 18131 An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (,) B )  =  ( ( ( A  +  B )  /  2 ) (
 ball `  D ) ( ( B  -  A )  /  2 ) ) )
 
Theoremioo2blex 18132 An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (,) B )  e.  ran  ( ball `  D ) )
 
Theoremblssioo 18133 The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |- 
 ran  ( ball `  D )  C_  ran  (,)
 
Theoremtgioo 18134 The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   &    |-  J  =  (
 MetOpen `  D )   =>    |-  ( topGen `  ran  (,) )  =  J
 
Theoremqdensere2 18135  QQ is dense in  RR. (Contributed by NM, 24-Aug-2007.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   &    |-  J  =  (
 MetOpen `  D )   =>    |-  ( ( cls `  J ) `  QQ )  =  RR
 
Theoremblcvx 18136 An open ball in the complex numbers is a convex set. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  S  =  ( P ( ball `  ( abs  o. 
 -  ) ) R )   =>    |-  ( ( ( P  e.  CC  /\  R  e.  RR* )  /\  ( A  e.  S  /\  B  e.  S  /\  T  e.  ( 0 [,] 1 ) ) ) 
 ->  ( ( T  x.  A )  +  (
 ( 1  -  T )  x.  B ) )  e.  S )
 
Theoremrehaus 18137 The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.)
 |-  ( topGen `  ran  (,) )  e.  Haus
 
Theoremtgqioo 18138 The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  Q  =  ( topGen `  ( (,) " ( QQ 
 X.  QQ ) ) )   =>    |-  ( topGen `  ran  (,) )  =  Q
 
Theoremre2ndc 18139 The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( topGen `  ran  (,) )  e.  2ndc
 
Theoremresubmet 18140 The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)
 |-  R  =  ( topGen `  ran  (,) )   &    |-  J  =  (
 MetOpen `  ( ( abs 
 o.  -  )  |`  ( A  X.  A ) ) )   =>    |-  ( A  C_  RR  ->  J  =  ( Rt  A ) )
 
Theoremtgioo2 18141 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  ( topGen `
  ran  (,) )  =  ( Jt  RR )
 
Theoremrerest 18142 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  R  =  ( topGen `  ran  (,) )   =>    |-  ( A  C_  RR  ->  ( Jt  A )  =  ( Rt  A ) )
 
Theoremtgioo3 18143 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  J  =  ( TopOpen `  (flds  RR )
 )   =>    |-  ( topGen `  ran  (,) )  =  J
 
Theoremxrtgioo 18144 The topology on the extended reals coincides with the standard topology on the reals, when restricted to  RR. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  J  =  ( (ordTop `  <_  )t  RR )   =>    |-  ( topGen `  ran  (,) )  =  J
 
Theoremxrrest 18145 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  (ordTop `  <_  )   &    |-  R  =  ( topGen `  ran  (,) )   =>    |-  ( A  C_  RR  ->  ( Xt  A )  =  ( Rt  A ) )
 
Theoremxrrest2 18146 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  X  =  (ordTop `  <_  )   =>    |-  ( A  C_  RR  ->  ( Jt  A )  =  ( Xt  A ) )
 
Theoremxrsxmet 18147 The metric on the extended reals is a proper extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  D  =  ( dist ` 
 RR* s )   =>    |-  D  e.  ( * Met `  RR* )
 
Theoremxrsdsre 18148 The metric on the extended reals coincides with the usual metric on the reals. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  D  =  ( dist ` 
 RR* s )   =>    |-  ( D  |`  ( RR 
 X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X. 
 RR ) )
 
Theoremxrsblre 18149 Any ball of the metric of the extended reals centered on an element of  RR is entirely contained in  RR. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  D  =  ( dist ` 
 RR* s )   =>    |-  ( ( P  e.  RR  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  C_  RR )
 
Theoremxrsmopn 18150 The metric on the extended reals generates a topology, but this does not match the order topology on  RR*; for example  {  +oo } is open in the metric topology, but not the order topology. However, the metric topology is finer than the order topology, meaning that all open intervals are open in the metric topology. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  D  =  ( dist ` 
 RR* s )   &    |-  J  =  ( MetOpen `  D )   =>    |-  (ordTop ` 
 <_  )  C_  J
 
Theoremzcld 18151 The integers are a closed set in the topology on  RR. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |- 
 ZZ  e.  ( Clsd `  J )
 
Theoremrecld2 18152 The real numbers are a closed set in the topology on  CC. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  RR  e.  ( Clsd `  J )
 
Theoremzcld2 18153 The integers are a closed set in the topology on  CC. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  ZZ  e.  ( Clsd `  J )
 
Theoremzdis 18154 The integers are a discrete set in the topology on  CC. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  ( Jt  ZZ )  =  ~P ZZ
 
Theoremreperflem 18155* A subset of the real numbers that is closed under addition with real numbers is perfect. (Contributed by Mario Carneiro, 26-Dec-2016.)
 |-  J  =  ( TopOpen ` fld )   &    |-  (
 ( u  e.  S  /\  v  e.  RR )  ->  ( u  +  v )  e.  S )   &    |-  S  C_  CC   =>    |-  ( Jt  S )  e. Perf
 
Theoremreperf 18156 The real numbers are a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 26-Dec-2016.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  ( Jt  RR )  e. Perf
 
Theoremcnperf 18157 The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e. Perf
 
Theoremiccntr 18158 The interior of a closed interval in the standard topology on  RR is the corresponding open interval. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `  ran  (,) )
 ) `  ( A [,] B ) )  =  ( A (,) B ) )
 
Theoremicccmplem1 18159* Lemma for icccmp 18162. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  T  =  ( Jt  ( A [,] B ) )   &    |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   &    |-  S  =  { x  e.  ( A [,] B )  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  U 
 C_  J )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   =>    |-  ( ph  ->  ( A  e.  S  /\  A. y  e.  S  y  <_  B ) )
 
Theoremicccmplem2 18160* Lemma for icccmp 18162. (Contributed by Mario Carneiro, 13-Jun-2014.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  T  =  ( Jt  ( A [,] B ) )   &    |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   &    |-  S  =  { x  e.  ( A [,] B )  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  U 
 C_  J )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   &    |-  ( ph  ->  V  e.  U )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  ( G ( ball `  D ) C )  C_  V )   &    |-  G  =  sup ( S ,  RR ,  <  )   &    |-  R  =  if (
 ( G  +  ( C  /  2 ) ) 
 <_  B ,  ( G  +  ( C  / 
 2 ) ) ,  B )   =>    |-  ( ph  ->  B  e.  S )
 
Theoremicccmplem3 18161* Lemma for icccmp 18162. (Contributed by Mario Carneiro, 13-Jun-2014.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  T  =  ( Jt  ( A [,] B ) )   &    |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   &    |-  S  =  { x  e.  ( A [,] B )  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  U 
 C_  J )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   =>    |-  ( ph  ->  B  e.  S )
 
Theoremicccmp 18162 A closed interval in  RR is compact. (Contributed by Mario Carneiro, 13-Jun-2014.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  T  =  ( Jt  ( A [,] B ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  T  e.  Comp )
 
Theoremreconnlem1 18163 Lemma for reconn 18165. Connectedness in the reals-easy direction. (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( ( A 
 C_  RR  /\  ( (
 topGen `  ran  (,) )t  A )  e.  Con )  /\  ( X  e.  A  /\  Y  e.  A ) )  ->  ( X [,] Y )  C_  A )
 
Theoremreconnlem2 18164* Lemma for reconn 18165. (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  U  e.  ( topGen `  ran  (,) ) )   &    |-  ( ph  ->  V  e.  ( topGen `  ran  (,) ) )   &    |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x [,] y )  C_  A )   &    |-  ( ph  ->  B  e.  ( U  i^i  A ) )   &    |-  ( ph  ->  C  e.  ( V  i^i  A ) )   &    |-  ( ph  ->  ( U  i^i  V ) 
 C_  ( RR  \  A ) )   &    |-  ( ph  ->  B  <_  C )   &    |-  S  =  sup (
 ( U  i^i  ( B [,] C ) ) ,  RR ,  <  )   =>    |-  ( ph  ->  -.  A  C_  ( U  u.  V ) )
 
Theoremreconn 18165* A subset of the reals is connected iff it has the interval property. (Contributed by Jeff Hankins, 15-Jul-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  C_  RR  ->  ( ( ( topGen `  ran  (,) )t  A )  e.  Con  <->  A. x  e.  A  A. y  e.  A  ( x [,] y )  C_  A ) )
 
Theoremretopcon 18166 Corollary of reconn 18165. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
 |-  ( topGen `  ran  (,) )  e.  Con
 
Theoremiccconn 18167 A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B ) )  e.  Con )
 
Theoremopnreen 18168 Every nonempty open set is uncountable. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 20-Feb-2015.)
 |-  ( ( A  e.  ( topGen `  ran  (,) )  /\  A  =/=  (/) )  ->  A  ~~  ~P NN )
 
Theoremrectbntr0 18169 A countable subset of the reals has empty interior. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ( A  C_  RR  /\  A  ~<_  NN )  ->  ( ( int `  ( topGen `
  ran  (,) ) ) `
  A )  =  (/) )
 
Theoremxrge0gsumle 18170 A finite sum in the nonnegative extended reals is monotonic in the support. (Contributed by Mario Carneiro, 13-Sep-2015.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo )
 )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  B  e.  ( ~P A  i^i  Fin )
 )   &    |-  ( ph  ->  C  C_  B )   =>    |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  <_  ( G  gsumg  ( F  |`  B ) ) )
 
Theoremxrge0tsms 18171* Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo )
 )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,]  +oo ) )   &    |-  S  =  sup ( ran  (  s  e.  ( ~P A  i^i  Fin )  |->  ( G 
 gsumg  ( F  |`  s ) ) ) ,  RR* ,  <  )   =>    |-  ( ph  ->  ( G tsums  F )  =  { S } )
 
Theoremxrge0tsms2 18172 Any finite or infinite sum in the nonnegative extended reals is convergent. This is a rather unique property of the set  [ 0 , 
+oo ]; a similar theorem is not true for 
RR* or  RR or  [ 0 , 
+oo ). It is true for  NN0  u.  {  +oo }, however, or more generally any additive submonoid of  [ 0 ,  +oo ) with  +oo adjoined. (Contributed by Mario Carneiro, 13-Sep-2015.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo )
 )   =>    |-  ( ( A  e.  V  /\  F : A --> ( 0 [,]  +oo ) )  ->  ( G tsums  F )  ~~  1o )
 
Theoremmetdcnlem 18173 The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  C  =  (
 dist `  RR* s )   &    |-  K  =  ( MetOpen `  C )   &    |-  ( ph  ->  D  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  Z  e.  X )   &    |-  ( ph  ->  ( A D Y )  <  ( R 
 /  2 ) )   &    |-  ( ph  ->  ( B D Z )  <  ( R  /  2 ) )   =>    |-  ( ph  ->  ( ( A D B ) C ( Y D Z ) )  <  R )
 
Theoremxmetdcn2 18174 The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 18175 we use the metric topology instead of the order topology on  RR*, which makes the theorem a bit stronger. Since  +oo is an isolated point in the metric topology, this is saying that for any points  A ,  B which are an infinite distance apart, there is a product neighborhood around 
<. A ,  B >. such that  d
( a ,  b )  =  +oo for any  a near  A and  b near  B, i.e. the distance function is locally constant  +oo. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  C  =  (
 dist `  RR* s )   &    |-  K  =  ( MetOpen `  C )   =>    |-  ( D  e.  ( * Met `  X )  ->  D  e.  ( ( J  tX  J )  Cn  K ) )
 
Theoremxmetdcn 18175 The metric function of an extended metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  (ordTop `  <_  )   =>    |-  ( D  e.  ( * Met `  X )  ->  D  e.  ( ( J  tX  J )  Cn  K ) )
 
Theoremmetdcn2 18176 The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  (
 topGen `  ran  (,) )   =>    |-  ( D  e.  ( Met `  X )  ->  D  e.  ( ( J  tX  J )  Cn  K ) )
 
Theoremmetdcn 18177 The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  (
 TopOpen ` fld )   =>    |-  ( D  e.  ( Met `  X )  ->  D  e.  ( ( J  tX  J )  Cn  K ) )
 
Theoremmsdcn 18178 The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   &    |-  J  =  ( TopOpen `  M )   &    |-  K  =  ( topGen `  ran  (,) )   =>    |-  ( M  e.  MetSp  ->  ( D  |`  ( X  X.  X ) )  e.  ( ( J  tX  J )  Cn  K ) )
 
Theoremcnmpt1ds 18179* Continuity of the metric function; analogue of cnmpt12f 17192 which cannot be used directly because 
D is not necessarily a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  D  =  ( dist `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  R  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  G  e.  MetSp )   &    |-  ( 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 D B ) )  e.  ( K  Cn  R ) )
 
Theoremcnmpt2ds 18180* Continuity of the metric function; analogue of cnmpt22f 17201 which cannot be used directly because  D is not necessarily a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  D  =  ( dist `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  R  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  G  e.  MetSp )   &    |-  ( 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 D B ) )  e.  ( ( K 
 tX  L )  Cn  R ) )
 
Theoremnmcn 18181 The norm of a normed group is a continuous function. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  N  =  ( norm `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  K  =  ( topGen `  ran  (,) )   =>    |-  ( G  e. NrmGrp  ->  N  e.  ( J  Cn  K ) )
 
Theoremabscn 18182 The absolute value function on complex numbers is continuous. (Contributed by NM, 22-Aug-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( topGen `  ran  (,) )   =>    |-  abs  e.  ( J  Cn  K )
 
Theoremmetdsval 18183* Value of the "distance to a set" function. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   =>    |-  ( A  e.  X  ->  ( F `  A )  =  sup ( ran  (  y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  ) )
 
Theoremmetdsf 18184* The distance from a point to a set is a nonnegative extended real number. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X )  ->  F : X --> ( 0 [,]  +oo ) )
 
Theoremmetdsge 18185* The distance from the point  A to the set  S is greater than  R iff the  R-ball around  A misses  S. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  R  e.  RR* )  ->  ( R  <_  ( F `  A ) 
 <->  ( S  i^i  ( A ( ball `  D ) R ) )  =  (/) ) )
 
Theoremmetds0 18186* If a point is in a set, its distance to the set is zero. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S ) 
 ->  ( F `  A )  =  0 )
 
Theoremmetdstri 18187* A generalization of the triangle inequality to the point-set distance function. Under the usual notation where the same symbol  d denotes the point-point and point-set distance functions, this theorem would be written  d ( a ,  S )  <_ 
d ( a ,  b )  +  d ( b ,  S
). (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X )
 )  ->  ( F `  A )  <_  (
 ( A D B ) + e ( F `
  B ) ) )
 
Theoremmetdsle 18188* The distance from a point to a set is bounded by the distance to any member of the set. (Contributed by Mario Carneiro, 5-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X )  /\  ( A  e.  S  /\  B  e.  X )
 )  ->  ( F `  B )  <_  ( A D B ) )
 
Theoremmetdsre 18189* The distance from a point to a nonempty set in a proper metric space is a real number. (Contributed by Mario Carneiro, 5-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   =>    |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F : X --> RR )
 
Theoremmetdseq0 18190* The distance from a point to a set is zero iff the point is in the closure set. (Contributed by Mario Carneiro, 14-Feb-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   &    |-  J  =  (
 MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  ->  ( ( F `  A )  =  0  <->  A  e.  (
 ( cls `  J ) `  S ) ) )
 
Theoremmetdscnlem 18191* Lemma for metdscn 18192. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   &    |-  J  =  (
 MetOpen `  D )   &    |-  C  =  ( dist `  RR* s )   &    |-  K  =  ( MetOpen `  C )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  S  C_  X )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  ( A D B )  <  R )   =>    |-  ( ph  ->  (
 ( F `  A ) + e  - e
 ( F `  B ) )  <  R )
 
Theoremmetdscn 18192* The function  F which gives the distance from a point to a set is a continuous function into the metric topology of the extended reals. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   &    |-  J  =  (
 MetOpen `  D )   &    |-  C  =  ( dist `  RR* s )   &    |-  K  =  ( MetOpen `  C )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X )  ->  F  e.  ( J  Cn  K ) )
 
Theoremmetdscn2 18193* The function  F which gives the distance from a point to a nonempty set in a metric space is a continuous function into the topology of the complexes. (Contributed by Mario Carneiro, 5-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   &    |-  J  =  (
 MetOpen `  D )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F  e.  ( J  Cn  K ) )
 
Theoremmetnrmlem1a 18194* Lemma for metnrm 18198. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  S  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  T  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  ( S  i^i  T )  =  (/) )   =>    |-  ( ( ph  /\  A  e.  T )  ->  (
 0  <  ( F `  A )  /\  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `
  A ) )  e.  RR+ ) )
 
Theoremmetnrmlem1 18195* Lemma for metnrm 18198. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  S  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  T  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  ( S  i^i  T )  =  (/) )   =>    |-  ( ( ph  /\  ( A  e.  S  /\  B  e.  T )
 )  ->  if (
 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( A D B ) )
 
Theoremmetnrmlem2 18196* Lemma for metnrm 18198. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  S  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  T  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  ( S  i^i  T )  =  (/) )   &    |-  U  =  U_ t  e.  T  (
 t ( ball `  D ) ( if (
 1  <_  ( F `  t ) ,  1 ,  ( F `  t ) )  / 
 2 ) )   =>    |-  ( ph  ->  ( U  e.  J  /\  T  C_  U ) )
 
Theoremmetnrmlem3 18197* Lemma for metnrm 18198. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  sup ( ran  (  y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  S  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  T  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  ( S  i^i  T )  =  (/) )   &    |-  U  =  U_ t  e.  T  (
 t ( ball `  D ) ( if (
 1  <_  ( F `  t ) ,  1 ,  ( F `  t ) )  / 
 2 ) )   &    |-  G  =  ( x  e.  X  |->  sup ( ran  (  y  e.  T  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )   &    |-  V  =  U_ s  e.  S  (
 s ( ball `  D ) ( if (
 1  <_  ( G `  s ) ,  1 ,  ( G `  s ) )  / 
 2 ) )   =>    |-  ( ph  ->  E. z  e.  J  E. w  e.  J  ( S  C_  z  /\  T  C_  w  /\  ( z  i^i  w )  =  (/) ) )
 
Theoremmetnrm 18198 A metric space is normal. (Contributed by Jeff Hankins, 31-Aug-2013.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  Nrm )
 
Theoremmetreg 18199 A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  Reg )
 
Theoremaddcnlem 18200* Lemma for addcn 18201, subcn 18202, and mulcn 18203. (Contributed by Mario Carneiro, 5-May-2014.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  .+  :
 ( CC  X.  CC )
 --> CC   &    |-  ( ( a  e.  RR+  /\  b  e. 
 CC  /\  c  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
 ( ( abs `  ( u  -  b ) )  <  y  /\  ( abs `  ( v  -  c ) )  < 
 z )  ->  ( abs `  ( ( u 
 .+  v )  -  ( b  .+  c ) ) )  <  a
 ) )   =>    |- 
 .+  e.  ( ( J  tX  J )  Cn  J )
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