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Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreconnlem1 21201 Lemma for reconn 21203. 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 21202* Lemma for reconn 21203. (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 21203* 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 21204 Corollary of reconn 21203. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
 |-  ( topGen `  ran  (,) )  e.  Con
 
Theoremiccconn 21205 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 21206 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 21207 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 21208 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 21209* 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.) (Proof shortened by AV, 26-Jul-2019.)
 |-  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 21210 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 21211 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 21212 The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 21213 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 21213 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 21214 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 21215 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 21216 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 21217* Continuity of the metric function; analogue of cnmpt12f 20037 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 21218* Continuity of the metric function; analogue of cnmpt22f 20046 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 21219 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 21220 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 21221* 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 21222* 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 21223* 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 21224* 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 21225* 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 21226* 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 21227* 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 21228* 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 21229* Lemma for metdscn 21230. (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 21230* 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 21231* 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 complex numbers. (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 21232* Lemma for metnrm 21236. (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 21233* Lemma for metnrm 21236. (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 21234* Lemma for metnrm 21236. (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 21235* Lemma for metnrm 21236. (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 21236 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 21237 A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( *Met `  X )  ->  J  e.  Reg )
 
Theoremaddcnlem 21238* Lemma for addcn 21239, subcn 21240, and mulcn 21241. (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 21239 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 21240 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 21241 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 21242 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 21243 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 21244* 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 21245* Version of fsumcn 21244 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 21246* 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 21247* 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 21248* 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 )
 
12.4.11  Topological definitions using the reals
 
Syntaxcii 21249 Extend class notation with the unit interval.
 class  II
 
Syntaxccncf 21250 Extend class notation to include the operation which returns a class of continuous complex functions.
 class  -cn->
 
Definitiondf-ii 21251 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 21252* 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 21253 The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  II  e.  (TopOn `  ( 0 [,] 1
 ) )
 
Theoremiitop 21254 The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  II  e.  Top
 
Theoremiiuni 21255 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 21256 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  II  =  ( (
 topGen `  ran  (,) )t  (
 0 [,] 1 ) )
 
Theoremdfii3 21257 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 21258 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  I  =  (flds  ( 0 [,] 1
 ) )   =>    |-  II  =  ( TopOpen `  I )
 
Theoremdfii5 21259 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 21260 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
 |-  II  e.  Comp
 
Theoremiicon 21261 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  II  e.  Con
 
Theoremcncfval 21262* 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 21263* 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 21264* Version of elcncf 21263 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 21265 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
 
Theoremcncfrss2 21266 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
 
Theoremcncff 21267 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 21268* 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 21269* 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 21270* 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 ) )
 
Theoremrescncf 21271 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( C  C_  A  ->  ( F  e.  ( A -cn-> B )  ->  ( F  |`  C )  e.  ( C -cn-> B ) ) )
 
Theoremcncffvrn 21272 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) ) 
 ->  ( F  e.  ( A -cn-> C )  <->  F : A --> C ) )
 
Theoremcncfss 21273 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
 |-  ( ( B  C_  C  /\  C  C_  CC )  ->  ( A -cn-> B )  C_  ( A -cn-> C ) )
 
Theoremclimcncf 21274 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  ( A -cn-> B ) )   &    |-  ( ph  ->  G : Z
 --> A )   &    |-  ( ph  ->  G  ~~>  D )   &    |-  ( ph  ->  D  e.  A )   =>    |-  ( ph  ->  ( F  o.  G )  ~~>  ( F `  D ) )
 
Theoremabscncf 21275 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremrecncf 21276 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Re  e.  ( CC
 -cn-> RR )
 
Theoremimcncf 21277 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Im  e.  ( CC
 -cn-> RR )
 
Theoremcjcncf 21278 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  *  e.  ( CC
 -cn-> CC )
 
Theoremmulc1cncf 21279* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  CC  |->  ( A  x.  x ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremdivccncf 21280* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
 |-  F  =  ( x  e.  CC  |->  ( x 
 /  A ) )   =>    |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  F  e.  ( CC -cn-> CC ) )
 
Theoremcncfco 21281 The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( ph  ->  F  e.  ( A -cn-> B ) )   &    |-  ( ph  ->  G  e.  ( B -cn-> C ) )   =>    |-  ( ph  ->  ( G  o.  F )  e.  ( A -cn-> C ) )
 
Theoremcncfmet 21282 Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
 |-  C  =  ( ( abs  o.  -  )  |`  ( A  X.  A ) )   &    |-  D  =  ( ( abs  o.  -  )  |`  ( B  X.  B ) )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   =>    |-  (
 ( A  C_  CC  /\  B  C_  CC )  ->  ( A -cn-> B )  =  ( J  Cn  K ) )
 
Theoremcncfcn 21283 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  A )   &    |-  L  =  ( Jt  B )   =>    |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( A -cn-> B )  =  ( K  Cn  L ) )
 
Theoremcncfcn1 21284 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  ( CC -cn-> CC )  =  ( J  Cn  J )
 
Theoremcncfmptc 21285* A constant function is a continuous function on  CC. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
 |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  ( x  e.  S  |->  A )  e.  ( S -cn-> T ) )
 
Theoremcncfmptid 21286* The identity function is a continuous function on  CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
 |-  ( ( S  C_  T  /\  T  C_  CC )  ->  ( x  e.  S  |->  x )  e.  ( S -cn-> T ) )
 
Theoremcncfmpt1f 21287* Composition of continuous functions.  -cn-> analog of cnmpt11f 20035. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  ( ph  ->  F  e.  ( CC -cn-> CC )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> CC ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( F `
  A ) )  e.  ( X -cn-> CC ) )
 
Theoremcncfmpt2f 21288* Composition of continuous functions.  -cn-> analog of cnmpt12f 20037. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  ( ph  ->  F  e.  (
 ( J  tX  J )  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> CC )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( X -cn-> CC ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( X -cn-> CC ) )
 
Theoremcncfmpt2ss 21289* Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  J  =  ( TopOpen ` fld )   &    |-  F  e.  ( ( J  tX  J )  Cn  J )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> S ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( X
 -cn-> S ) )   &    |-  S  C_ 
 CC   &    |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( X -cn-> S ) )
 
Theoremaddccncf 21290* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  CC  |->  ( x  +  A ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremcdivcncf 21291* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } )  |->  ( A  /  x ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( ( CC  \  { 0 } ) -cn-> CC )
 )
 
Theoremnegcncf 21292* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  F  =  ( x  e.  A  |->  -u x )   =>    |-  ( A  C_  CC  ->  F  e.  ( A
 -cn-> CC ) )
 
Theoremnegfcncf 21293* The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  G  =  ( x  e.  A  |->  -u ( F `  x ) )   =>    |-  ( F  e.  ( A -cn-> CC )  ->  G  e.  ( A -cn-> CC )
 )
 
TheoremabscncfALT 21294 Absolute value is continuous. Alternate proof of abscncf 21275. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremcncfcnvcn 21295 Rewrite cmphaushmeo 20171 for functions on the complex numbers. (Contributed by Mario Carneiro, 19-Feb-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  X )   =>    |-  ( ( K  e.  Comp  /\  F  e.  ( X
 -cn-> Y ) )  ->  ( F : X -1-1-onto-> Y  <->  `' F  e.  ( Y -cn-> X ) ) )
 
Theoremexpcncf 21296* The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 21246. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( N  e.  NN0  ->  ( x  e.  CC  |->  ( x ^ N ) )  e.  ( CC
 -cn-> CC ) )
 
Theoremcnmptre 21297* Lemma for iirevcn 21300 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  R  =  ( TopOpen ` fld )   &    |-  J  =  ( ( topGen `  ran  (,) )t  A )   &    |-  K  =  ( ( topGen `  ran  (,) )t  B )   &    |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  B 
 C_  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  F  e.  B )   &    |-  ( ph  ->  ( x  e.  CC  |->  F )  e.  ( R  Cn  R ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  F )  e.  ( J  Cn  K ) )
 
Theoremcnmpt2pc 21298* Piecewise definition of a continuous function on a real interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  R  =  ( topGen `  ran  (,) )   &    |-  M  =  ( Rt  ( A [,] B ) )   &    |-  N  =  ( Rt  ( B [,] C ) )   &    |-  O  =  ( Rt  ( A [,] C ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  e.  ( A [,] C ) )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ( ph  /\  ( x  =  B  /\  y  e.  X ) )  ->  D  =  E )   &    |-  ( ph  ->  ( x  e.  ( A [,] B ) ,  y  e.  X  |->  D )  e.  ( ( M  tX  J )  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  ( B [,] C ) ,  y  e.  X  |->  E )  e.  ( ( N  tX  J )  Cn  K ) )   =>    |-  ( ph  ->  ( x  e.  ( A [,] C ) ,  y  e.  X  |->  if ( x  <_  B ,  D ,  E ) )  e.  ( ( O  tX  J )  Cn  K ) )
 
Theoremiirev 21299 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( X  e.  (
 0 [,] 1 )  ->  ( 1  -  X )  e.  ( 0 [,] 1 ) )
 
Theoremiirevcn 21300 The reversion function is a continuous map of the unit interval. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  ( x  e.  (
 0 [,] 1 )  |->  ( 1  -  x ) )  e.  ( II 
 Cn  II )
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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