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Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmetdsge 18301* 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 18302* 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 18303* 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 18304* 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 18305* 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 18306* 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 18307* Lemma for metdscn 18308. (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 18308* 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 18309* 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 18310* Lemma for metnrm 18314. (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 18311* Lemma for metnrm 18314. (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 18312* Lemma for metnrm 18314. (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 18313* Lemma for metnrm 18314. (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 18314 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 18315 A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  Reg )
 
Theoremaddcnlem 18316* Lemma for addcn 18317, subcn 18318, and mulcn 18319. (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 18317 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 18318 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 18319 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 18320 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 18321 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 18322* 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 18323* Version of fsumcn 18322 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 18324* 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 18325* 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 18326* 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 18327 Extend class notation with the unit interval.
 class  II
 
Syntaxccncf 18328 Extend class notation to include the operation which returns a class of continuous complex functions.
 class  -cn->
 
Definitiondf-ii 18329 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 18330* 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 18331 The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  II  e.  (TopOn `  ( 0 [,] 1
 ) )
 
Theoremiitop 18332 The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  II  e.  Top
 
Theoremiiuni 18333 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 18334 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  II  =  ( (
 topGen `  ran  (,) )t  (
 0 [,] 1 ) )
 
Theoremdfii3 18335 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 18336 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  I  =  (flds  ( 0 [,] 1
 ) )   =>    |-  II  =  ( TopOpen `  I )
 
Theoremdfii5 18337 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 18338 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
 |-  II  e.  Comp
 
Theoremiicon 18339 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  II  e.  Con
 
Theoremcncfval 18340* 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 18341* 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 18342* Version of elcncf 18341 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 18343 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
 
Theoremcncfrss2 18344 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
 
Theoremcncff 18345 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 18346* 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 18347* 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 18348* 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 18349 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 18350 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 18351 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 18352 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 18353 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremrecncf 18354 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Re  e.  ( CC
 -cn-> RR )
 
Theoremimcncf 18355 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Im  e.  ( CC
 -cn-> RR )
 
Theoremcjcncf 18356 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  *  e.  ( CC
 -cn-> CC )
 
Theoremmulc1cncf 18357* 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 18358* 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 18359 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 18360 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 18361 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 18362 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 18363* 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 18364* 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 18365* Composition of continuous functions.  -cn-> analog of cnmpt11f 17306. (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 18366* Composition of continuous functions.  -cn-> analog of cnmpt12f 17308. (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 18367* 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 ) )
 
Theoremcdivcncf 18368* 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 18369* 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 18370* 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 18371 Absolute value is continuous. Alternate proof of abscncf 18353. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (Proof modification is discouraged.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremcncfcnvcn 18372 Rewrite cmphaushmeo 17439 for functions on the complexes. (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 ) ) )
 
Theoremcnmptre 18373* Lemma for iirevcn 18376 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 18374* 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 18375 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( X  e.  (
 0 [,] 1 )  ->  ( 1  -  X )  e.  ( 0 [,] 1 ) )
 
Theoremiirevcn 18376 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 )
 
Theoremiihalf1 18377 Map the first half of  II into  II. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( X  e.  (
 0 [,] ( 1  / 
 2 ) )  ->  ( 2  x.  X )  e.  ( 0 [,] 1 ) )
 
Theoremiihalf1cn 18378 The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  J  =  ( (
 topGen `  ran  (,) )t  (
 0 [,] ( 1  / 
 2 ) ) )   =>    |-  ( x  e.  (
 0 [,] ( 1  / 
 2 ) )  |->  ( 2  x.  x ) )  e.  ( J  Cn  II )
 
Theoremiihalf2 18379 Map the second half of  II into  II. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( X  e.  (
 ( 1  /  2
 ) [,] 1 )  ->  ( ( 2  x.  X )  -  1
 )  e.  ( 0 [,] 1 ) )
 
Theoremiihalf2cn 18380 The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  J  =  ( (
 topGen `  ran  (,) )t  (
 ( 1  /  2
 ) [,] 1 ) )   =>    |-  ( x  e.  (
 ( 1  /  2
 ) [,] 1 )  |->  ( ( 2  x.  x )  -  1 ) )  e.  ( J  Cn  II )
 
Theoremelii1 18381 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
 |-  ( X  e.  (
 0 [,] ( 1  / 
 2 ) )  <->  ( X  e.  ( 0 [,] 1
 )  /\  X  <_  ( 1  /  2 ) ) )
 
Theoremelii2 18382 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
 |-  ( ( X  e.  ( 0 [,] 1
 )  /\  -.  X  <_  ( 1  /  2 ) )  ->  X  e.  ( ( 1  / 
 2 ) [,] 1
 ) )
 
Theoremiimulcl 18383 The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  e.  ( 0 [,] 1
 )  /\  B  e.  ( 0 [,] 1
 ) )  ->  ( A  x.  B )  e.  ( 0 [,] 1
 ) )
 
Theoremiimulcn 18384* Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.)
 |-  ( x  e.  (
 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( x  x.  y ) )  e.  ( ( II  tX  II )  Cn  II )
 
Theoremicoopnst 18385 A half-open interval starting at  A is open in the closed interval from  A to  B. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( A [,] B )  X.  ( A [,] B ) ) ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  ->  ( A [,) C )  e.  J ) )
 
Theoremiocopnst 18386 A half-open interval ending at  B is open in the closed interval from  A to  B. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( A [,] B )  X.  ( A [,] B ) ) ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A [,) B )  ->  ( C (,] B )  e.  J ) )
 
Theoremicchmeo 18387* The natural bijection from  [ 0 ,  1 ] to an arbitrary nontrivial closed interval  [ A ,  B ] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  ( ( x  x.  B )  +  ( (
 1  -  x )  x.  A ) ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) 
 ->  F  e.  ( II 
 Homeo  ( Jt  ( A [,] B ) ) ) )
 
Theoremicopnfcnv 18388* Define a bijection from  [ 0 ,  1 ) to  [ 0 , 
+oo ). (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,) 1 )  |->  ( x 
 /  ( 1  -  x ) ) )   =>    |-  ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,)  +oo )  /\  `' F  =  ( y  e.  ( 0 [,)  +oo )  |->  ( y  /  ( 1  +  y
 ) ) ) )
 
Theoremicopnfhmeo 18389* The defined bijection from  [ 0 ,  1 ) to  [ 0 , 
+oo ) is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,) 1 )  |->  ( x 
 /  ( 1  -  x ) ) )   &    |-  J  =  ( TopOpen ` fld )   =>    |-  ( F  Isom  <  ,  <  ( ( 0 [,) 1
 ) ,  ( 0 [,)  +oo ) )  /\  F  e.  ( ( Jt  ( 0 [,) 1
 ) )  Homeo  ( Jt  ( 0 [,)  +oo )
 ) ) )
 
Theoremiccpnfcnv 18390* Define a bijection from  [ 0 ,  1 ] to  [ 0 , 
+oo ]. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) ) )   =>    |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  ( y  e.  ( 0 [,]  +oo )  |->  if ( y  = 
 +oo ,  1 ,  ( y  /  (
 1  +  y ) ) ) ) )
 
Theoremiccpnfhmeo 18391 The defined bijection from  [ 0 ,  1 ] to  [ 0 , 
+oo ] is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) ) )   &    |-  K  =  ( (ordTop ` 
 <_  )t  ( 0 [,]  +oo ) )   =>    |-  ( F  Isom  <  ,  <  ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) )  /\  F  e.  ( II  Homeo  K ) )
 
Theoremxrhmeo 18392* The bijection from  [ -u 1 ,  1 ] to the extended reals is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) ) )   &    |-  G  =  ( y  e.  ( -u 1 [,] 1
 )  |->  if ( 0  <_  y ,  ( F `  y ) ,  - e
 ( F `  -u y
 ) ) )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  (ordTop `  <_  )   =>    |-  ( G  Isom  <  ,  <  ( ( -u 1 [,] 1 ) , 
 RR* )  /\  G  e.  ( ( Jt  ( -u 1 [,] 1 ) ) 
 Homeo  (ordTop `  <_  ) ) )
 
Theoremxrhmph 18393 The extended reals are homeomorphic to the interval  [
0 ,  1 ]. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  II  ~=  (ordTop `  <_  )
 
Theoremxrcmp 18394 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 18260), this means that  RR* is a compactification of  RR. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Comp
 
Theoremxrcon 18395 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Con
 
Theoremicccvx 18396 A linear combination of two reals lies in the interval between them. Equivalently, a closed interval is a convex set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) 
 /\  T  e.  (
 0 [,] 1 ) ) 
 ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( A [,] B ) ) )
 
Theoremoprpiece1res1 18397* Restriction to the first part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  A  <_  B   &    |-  R  e.  _V   &    |-  S  e.  _V   &    |-  K  e.  ( A [,] B )   &    |-  F  =  ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )   &    |-  G  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  R )   =>    |-  ( F  |`  ( ( A [,] K )  X.  C ) )  =  G
 
Theoremoprpiece1res2 18398* Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  A  <_  B   &    |-  R  e.  _V   &    |-  S  e.  _V   &    |-  K  e.  ( A [,] B )   &    |-  F  =  ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )   &    |-  ( x  =  K  ->  R  =  P )   &    |-  ( x  =  K  ->  S  =  Q )   &    |-  (
 y  e.  C  ->  P  =  Q )   &    |-  G  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  S )   =>    |-  ( F  |`  ( ( K [,] B )  X.  C ) )  =  G
 
Theoremcnrehmeo 18399* The canonical bijection from  ( RR  X.  RR ) to  CC described in cnref1o 10302 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if  ( RR  X.  RR ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y
 ) ) )   &    |-  J  =  ( topGen `  ran  (,) )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  F  e.  ( ( J  tX  J )  Homeo  K )
 
Theoremcnheiborlem 18400* Lemma for cnheibor 18401. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  T  =  ( Jt  X )   &    |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y ) ) )   &    |-  Y  =  ( F " ( ( -u R [,] R )  X.  ( -u R [,] R ) ) )   =>    |-  ( ( X  e.  ( Clsd `  J )  /\  ( R  e.  RR  /\ 
 A. z  e.  X  ( abs `  z )  <_  R ) )  ->  T  e.  Comp )
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