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Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremblssioo 18301 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 18302 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 18303  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 18304 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 18305 The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.)
 |-  ( topGen `  ran  (,) )  e.  Haus
 
Theoremtgqioo 18306 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 18307 The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( topGen `  ran  (,) )  e.  2ndc
 
Theoremresubmet 18308 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 18309 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 18310 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 18311 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 18312 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 18313 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 18314 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 18315 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 18316 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 18317 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 18318 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 18319 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 18320 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 18321 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 18322 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 18323* 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 18324 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 18325 The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e. Perf
 
Theoremiccntr 18326 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 18327* Lemma for icccmp 18330. (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 18328* Lemma for icccmp 18330. (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 18329* Lemma for icccmp 18330. (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 18330 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 18331 Lemma for reconn 18333. 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 18332* Lemma for reconn 18333. (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 18333* 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 18334 Corollary of reconn 18333. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
 |-  ( topGen `  ran  (,) )  e.  Con
 
Theoremiccconn 18335 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 18336 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 18337 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 18338 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 18339* 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 18340 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 18341 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 18342 The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 18343 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 18343 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 18344 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 18345 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 18346 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 18347* Continuity of the metric function; analogue of cnmpt12f 17360 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 18348* Continuity of the metric function; analogue of cnmpt22f 17369 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 18349 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 18350 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 18351* 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 18352* 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 18353* 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 18354* 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 18355* 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 18356* 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 18357* 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 18358* 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 18359* Lemma for metdscn 18360. (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 18360* 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 18361* 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 18362* Lemma for metnrm 18366. (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 18363* Lemma for metnrm 18366. (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 18364* Lemma for metnrm 18366. (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 18365* Lemma for metnrm 18366. (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 18366 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 18367 A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  Reg )
 
Theoremaddcnlem 18368* Lemma for addcn 18369, subcn 18370, and mulcn 18371. (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 )
 
Theoremaddcn 18369 Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  +  e.  ( ( J  tX  J )  Cn  J )
 
Theoremsubcn 18370 Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  -  e.  ( ( J  tX  J )  Cn  J )
 
Theoremmulcn 18371 Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  x.  e.  ( ( J  tX  J )  Cn  J )
 
Theoremdivcn 18372 Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  ( CC  \  {
 0 } ) )   =>    |-  /  e.  ( ( J 
 tX  K )  Cn  J )
 
Theoremcnfldtgp 18373 The complex numbers form a topological group under addition, with the standard topology induced by the absolute value metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-fld  e.  TopGrp
 
Theoremfsumcn 18374* A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for  B normally contains free variables  k and  x to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  K  =  ( TopOpen ` fld )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  sum_ k  e.  A  B )  e.  ( J  Cn  K ) )
 
Theoremfsum2cn 18375* Version of fsumcn 18374 for two-argument mappings. (Contributed by Mario Carneiro, 6-May-2014.)
 |-  K  =  ( TopOpen ` fld )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  L  e.  (TopOn `  Y )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J 
 tX  L )  Cn  K ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  sum_ k  e.  A  B )  e.  ( ( J  tX  L )  Cn  K ) )
 
Theoremexpcn 18376* The power function on complex numbers, for fixed exponent  N, is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  ( N  e.  NN0  ->  ( x  e.  CC  |->  ( x ^ N ) )  e.  ( J  Cn  J ) )
 
Theoremdivccn 18377* Division by a nonzero constant is a continuous operation. (Contributed by Mario Carneiro, 5-May-2014.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  (
 ( A  e.  CC  /\  A  =/=  0 ) 
 ->  ( x  e.  CC  |->  ( x  /  A ) )  e.  ( J  Cn  J ) )
 
Theoremsqcn 18378* The square function on complex numbers is continuous. (Contributed by NM, 13-Jun-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  ( x  e.  CC  |->  ( x ^ 2 ) )  e.  ( J  Cn  J )
 
11.3.9  Topological definitions using the reals
 
Syntaxcii 18379 Extend class notation with the unit interval.
 class  II
 
Syntaxccncf 18380 Extend class notation to include the operation which returns a class of continuous complex functions.
 class  -cn->
 
Definitiondf-ii 18381 Define the unit interval with the Euclidean topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  II  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( (
 0 [,] 1 )  X.  ( 0 [,] 1
 ) ) ) )
 
Definitiondf-cncf 18382* Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)
 |- 
 -cn->  =  ( a  e. 
 ~P CC ,  b  e.  ~P CC  |->  { f  e.  ( b  ^m  a
 )  |  A. x  e.  a  A. e  e.  RR+  E. d  e.  RR+  A. y  e.  a  ( ( abs `  ( x  -  y ) )  <  d  ->  ( abs `  ( ( f `
  x )  -  ( f `  y
 ) ) )  < 
 e ) } )
 
Theoremiitopon 18383 The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  II  e.  (TopOn `  ( 0 [,] 1
 ) )
 
Theoremiitop 18384 The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  II  e.  Top
 
Theoremiiuni 18385 The base set of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Jan-2014.)
 |-  ( 0 [,] 1
 )  =  U. II
 
Theoremdfii2 18386 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  II  =  ( (
 topGen `  ran  (,) )t  (
 0 [,] 1 ) )
 
Theoremdfii3 18387 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  II  =  ( Jt  ( 0 [,] 1
 ) )
 
Theoremdfii4 18388 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  I  =  (flds  ( 0 [,] 1
 ) )   =>    |-  II  =  ( TopOpen `  I )
 
Theoremdfii5 18389 The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  II  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
 ) ) ) )
 
Theoremiicmp 18390 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
 |-  II  e.  Comp
 
Theoremiicon 18391 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  II  e.  Con
 
Theoremcncfval 18392* The value of the continuous complex function operation is the set of continuous functions from  A to  B. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
 |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( A -cn-> B )  =  { f  e.  ( B  ^m  A )  |  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w ) )  <  z  ->  ( abs `  ( ( f `
  x )  -  ( f `  w ) ) )  < 
 y ) } )
 
Theoremelcncf 18393* Membership in the set of continuous complex functions from  A to  B. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
 |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w ) )  <  z  ->  ( abs `  ( ( F `
  x )  -  ( F `  w ) ) )  <  y
 ) ) ) )
 
Theoremelcncf2 18394* Version of elcncf 18393 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
 |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x ) )  <  z  ->  ( abs `  ( ( F `
  w )  -  ( F `  x ) ) )  <  y
 ) ) ) )
 
Theoremcncfrss 18395 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
 
Theoremcncfrss2 18396 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
 
Theoremcncff 18397 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  F : A --> B )
 
Theoremcncfi 18398* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( F  e.  ( A -cn-> B )  /\  C  e.  A  /\  R  e.  RR+ )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  C ) )  <  z  ->  ( abs `  ( ( F `  w )  -  ( F `  C ) ) )  <  R ) )
 
Theoremelcncf1di 18399* Membership in the set of continuous complex functions from  A to  B. (Contributed by Paul Chapman, 26-Nov-2007.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ ) )   &    |-  ( ph  ->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  ( ( abs `  ( x  -  w ) )  <  Z  ->  ( abs `  ( ( F `
  x )  -  ( F `  w ) ) )  <  y
 ) ) )   =>    |-  ( ph  ->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A
 -cn-> B ) ) )
 
Theoremelcncf1ii 18400* Membership in the set of continuous complex functions from  A to  B. (Contributed by Paul Chapman, 26-Nov-2007.)
 |-  F : A --> B   &    |-  (
 ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ )   &    |-  (
 ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
 ( abs `  ( x  -  w ) )  <  Z  ->  ( abs `  (
 ( F `  x )  -  ( F `  w ) ) )  <  y ) )   =>    |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) )
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