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Theorem List for Metamath Proof Explorer - 18801-18900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnmoeq0 18801 The operator norm is zero only for the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( ( N `  F )  =  0  <->  F  =  ( V  X.  {  .0.  }
 ) ) )
 
Theoremnmoco 18802 An upper bound on the operator norm of a composition. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  N  =  ( S
 normOp U )   &    |-  L  =  ( T normOp U )   &    |-  M  =  ( S normOp T )   =>    |-  ( ( F  e.  ( T NGHom  U )  /\  G  e.  ( S NGHom  T ) )  ->  ( N `  ( F  o.  G ) )  <_  ( ( L `  F )  x.  ( M `  G ) ) )
 
Theoremnghmco 18803 The composition of normed group homomorphisms is a normed group homomorphism. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  ( ( F  e.  ( T NGHom  U )  /\  G  e.  ( S NGHom  T ) )  ->  ( F  o.  G )  e.  ( S NGHom  U ) )
 
Theoremnmotri 18804 Triangle inequality for the operator norm. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  .+  =  ( +g  `  T )   =>    |-  ( ( T  e.  Abel  /\  F  e.  ( S NGHom  T )  /\  G  e.  ( S NGHom  T ) )  ->  ( N `  ( F  o F  .+  G ) ) 
 <_  ( ( N `  F )  +  ( N `  G ) ) )
 
Theoremnghmplusg 18805 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |- 
 .+  =  ( +g  `  T )   =>    |-  ( ( T  e.  Abel  /\  F  e.  ( S NGHom  T )  /\  G  e.  ( S NGHom  T ) ) 
 ->  ( F  o F  .+  G )  e.  ( S NGHom  T ) )
 
Theorem0nghm 18806 The zero operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  V  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( V  X.  {  .0.  }
 )  e.  ( S NGHom  T ) )
 
Theoremnmoid 18807 The operator norm of the identity function on a nontrivial group. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  N  =  ( S
 normOp S )   &    |-  V  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ( S  e. NrmGrp  /\ 
 {  .0.  }  C.  V )  ->  ( N `  (  _I  |`  V )
 )  =  1 )
 
Theoremidnghm 18808 The identity operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  V  =  ( Base `  S )   =>    |-  ( S  e. NrmGrp  ->  (  _I  |`  V )  e.  ( S NGHom  S ) )
 
Theoremnmods 18809 Upper bound for the distance between the values of a bounded linear operator. (Contributed by Mario Carneiro, 22-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  C  =  ( dist `  S )   &    |-  D  =  ( dist `  T )   =>    |-  (
 ( F  e.  ( S NGHom  T )  /\  A  e.  V  /\  B  e.  V )  ->  ( ( F `  A ) D ( F `  B ) )  <_  ( ( N `  F )  x.  ( A C B ) ) )
 
Theoremnghmcn 18810 A normed group homomorphism is a continuous function. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  J  =  ( TopOpen `  S )   &    |-  K  =  (
 TopOpen `  T )   =>    |-  ( F  e.  ( S NGHom  T )  ->  F  e.  ( J  Cn  K ) )
 
Theoremisnmhm 18811 A normed module homomorphism is a left module homomorphism which is also a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  <->  ( ( S  e. NrmMod  /\  T  e. NrmMod )  /\  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) ) )
 
Theoremnmhmrcl1 18812 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  S  e. NrmMod )
 
Theoremnmhmrcl2 18813 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  T  e. NrmMod )
 
Theoremnmhmlmhm 18814 A normed module homomorphism is a left module homomorphism (a linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  F  e.  ( S LMHom  T ) )
 
Theoremnmhmnghm 18815 A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  F  e.  ( S NGHom  T ) )
 
Theoremnmhmghm 18816 A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  F  e.  ( S  GrpHom  T ) )
 
Theoremisnmhm2 18817 A normed module homomorphism is a left module homomorphism with bounded norm (a bounded linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmMod  /\  T  e. NrmMod  /\  F  e.  ( S LMHom  T ) ) 
 ->  ( F  e.  ( S NMHom  T )  <->  ( N `  F )  e.  RR ) )
 
Theoremnmhmcl 18818 A normed module homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( F  e.  ( S NMHom  T )  ->  ( N `  F )  e. 
 RR )
 
Theoremidnmhm 18819 The identity operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  V  =  ( Base `  S )   =>    |-  ( S  e. NrmMod  ->  (  _I  |`  V )  e.  ( S NMHom  S ) )
 
Theorem0nmhm 18820 The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  V  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  F  =  (Scalar `  S )   &    |-  G  =  (Scalar `  T )   =>    |-  ( ( S  e. NrmMod  /\  T  e. NrmMod  /\  F  =  G )  ->  ( V  X.  {  .0.  }
 )  e.  ( S NMHom  T ) )
 
Theoremnmhmco 18821 The composition of bounded linear operators is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  ( ( F  e.  ( T NMHom  U )  /\  G  e.  ( S NMHom  T ) )  ->  ( F  o.  G )  e.  ( S NMHom  U ) )
 
Theoremnmhmplusg 18822 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |- 
 .+  =  ( +g  `  T )   =>    |-  ( ( F  e.  ( S NMHom  T )  /\  G  e.  ( S NMHom  T ) )  ->  ( F  o F  .+  G )  e.  ( S NMHom  T ) )
 
11.4.10  Topology on the reals
 
Theoremqtopbaslem 18823 The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  S  C_  RR*   =>    |-  ( (,) " ( S  X.  S ) )  e.  TopBases
 
Theoremqtopbas 18824 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
 |-  ( (,) " ( QQ  X.  QQ ) )  e.  TopBases
 
Theoremretopbas 18825 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 18826 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
 |-  ( topGen `  ran  (,) )  e.  Top
 
Theoremuniretop 18827 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
 |- 
 RR  =  U. ( topGen `
  ran  (,) )
 
Theoremretopon 18828 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 18829 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 18830 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 18831 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)
 |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
 
Theoremicccld 18832 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 18833 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 18834 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 18835  QQ is dense in the standard topology on  RR. (Contributed by NM, 1-Mar-2007.)
 |-  ( ( cls `  ( topGen `
  ran  (,) ) ) `
  QQ )  =  RR
 
Theoremcnmetdval 18836 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 18837 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 18838 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( abs  o.  -  )  e.  ( * Met `  CC )
 
Theoremcnbl0 18839 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 18840* 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 18841 The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-fld  e.  MetSp
 
Theoremcnfldxms 18842 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-fld  e.  *
 MetSp
 
Theoremcnfldtps 18843 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-fld  e.  TopSp
 
Theoremcnfldnm 18844 The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- 
 abs  =  ( norm ` fld )
 
Theoremcnngp 18845 The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-fld  e. NrmGrp
 
Theoremcnnrg 18846 The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-fld  e. NrmRing
 
Theoremcnfldtopn 18847 The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  =  ( MetOpen `  ( abs  o. 
 -  ) )
 
Theoremcnfldtopon 18848 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e.  (TopOn `  CC )
 
Theoremcnfldtop 18849 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e.  Top
 
Theoremcnfldhaus 18850 The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e.  Haus
 
Theoremremetdval 18851 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 18852 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 18853 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 18854 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 18855 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 18856 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 18857 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 18858 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 18859  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 18860 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 18861 The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.)
 |-  ( topGen `  ran  (,) )  e.  Haus
 
Theoremtgqioo 18862 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 18863 The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( topGen `  ran  (,) )  e.  2ndc
 
Theoremresubmet 18864 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 18865 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 18866 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 18867 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 18868 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 18869 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 18870 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 18871 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 18872 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 18873 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 18874 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 18875 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 18876 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 18877 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 18878 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
 
Theoremsszcld 18879 Every subset of the integers are closed in the topology on  CC. (Contributed by Mario Carneiro, 6-Jul-2017.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  ( A  C_  ZZ  ->  A  e.  ( Clsd `  J )
 )
 
Theoremreperflem 18880* 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 18881 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 18882 The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e. Perf
 
Theoremiccntr 18883 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 18884* Lemma for icccmp 18887. (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 18885* Lemma for icccmp 18887. (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 18886* Lemma for icccmp 18887. (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 18887 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 18888 Lemma for reconn 18890. 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 18889* Lemma for reconn 18890. (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 18890* 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 18891 Corollary of reconn 18890. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
 |-  ( topGen `  ran  (,) )  e.  Con
 
Theoremiccconn 18892 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 18893 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 18894 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 18895 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 18896* 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 18897 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 18898 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 18899 The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 18900 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 18900 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 ) )
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