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Theorem List for Metamath Proof Explorer - 18201-18300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaddcn 18201 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 18202 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 18203 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 18204 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 18205 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 18206* 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 18207* Version of fsumcn 18206 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 18208* 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 18209* 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 18210* 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 18211 Extend class notation with the unit interval.
 class  II
 
Syntaxccncf 18212 Extend class notation to include the operation which returns a class of continuous complex functions.
 class  -cn->
 
Definitiondf-ii 18213 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 18214* 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 18215 The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  II  e.  (TopOn `  ( 0 [,] 1
 ) )
 
Theoremiitop 18216 The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  II  e.  Top
 
Theoremiiuni 18217 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 18218 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  II  =  ( (
 topGen `  ran  (,) )t  (
 0 [,] 1 ) )
 
Theoremdfii3 18219 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 18220 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  I  =  (flds  ( 0 [,] 1
 ) )   =>    |-  II  =  ( TopOpen `  I )
 
Theoremdfii5 18221 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 18222 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
 |-  II  e.  Comp
 
Theoremiicon 18223 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  II  e.  Con
 
Theoremcncfval 18224* 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 18225* 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 18226* Version of elcncf 18225 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 18227 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
 
Theoremcncfrss2 18228 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
 
Theoremcncff 18229 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 18230* 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 18231* 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 18232* 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 18233 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 18234 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 18235 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 18236 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 18237 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremrecncf 18238 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Re  e.  ( CC
 -cn-> RR )
 
Theoremimcncf 18239 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Im  e.  ( CC
 -cn-> RR )
 
Theoremcjcncf 18240 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  *  e.  ( CC
 -cn-> CC )
 
Theoremmulc1cncf 18241* 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 18242* 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 18243 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 18244 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 18245 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 18246 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 18247* 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 18248* 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 18249* Composition of continuous functions.  -cn-> analog of cnmpt11f 17190. (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 18250* Composition of continuous functions.  -cn-> analog of cnmpt12f 17192. (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 18251* 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 18252* 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 18253* 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 18254* 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 18255 Absolute value is continuous. Alternate proof of abscncf 18237. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (Proof modification is discouraged.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremcncfcnvcn 18256 Rewrite cmphaushmeo 17323 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 18257* Lemma for iirevcn 18260 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 18258* 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 18259 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( X  e.  (
 0 [,] 1 )  ->  ( 1  -  X )  e.  ( 0 [,] 1 ) )
 
Theoremiirevcn 18260 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 18261 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 18262 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 18263 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 18264 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 18265 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 18266 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 18267 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 18268* 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 18269 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 18270 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 18271* 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 18272* 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 18273* 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 18274* 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 18275 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 18276* 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 18277 The extended reals are homeomorphic to the interval  [
0 ,  1 ]. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  II  ~=  (ordTop `  <_  )
 
Theoremxrcmp 18278 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 18144), this means that  RR* is a compactification of  RR. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Comp
 
Theoremxrcon 18279 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Con
 
Theoremicccvx 18280 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 18281* 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 18282* 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 18283* The canonical bijection from  ( RR  X.  RR ) to  CC described in cnref1o 10228 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 18284* Lemma for cnheibor 18285. (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 )
 
Theoremcnheibor 18285* Heine-Borel theorem for complex numbers. A subset of  CC is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  T  =  ( Jt  X )   =>    |-  ( X  C_  CC  ->  ( T  e.  Comp  <->  ( X  e.  ( Clsd `  J )  /\  E. r  e.  RR  A. x  e.  X  ( abs `  x )  <_  r ) ) )
 
Theoremcnllycmp 18286 The topology on the complexes is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e. 𝑛Locally  Comp
 
Theoremrellycmp 18287 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( topGen `  ran  (,) )  e. 𝑛Locally  Comp
 
Theorembndth 18288* The Boundedness Theorem. A continuous function from a compact topological space to the reals is bounded (above). (Boundedness below is obtained by applying this theorem to  -u F.) (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  X  =  U. J   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x )
 
Theoremevth 18289* The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  X  =  U. J   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  X  =/=  (/) )   =>    |-  ( ph  ->  E. x  e.  X  A. y  e.  X  ( F `  y )  <_  ( F `
  x ) )
 
Theoremevth2 18290* The Extreme Value Theorem, minimum version. A continuous function from a nonempty compact topological space to the reals attains its minimum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  X  =  U. J   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  X  =/=  (/) )   =>    |-  ( ph  ->  E. x  e.  X  A. y  e.  X  ( F `  x )  <_  ( F `
  y ) )
 
Theoremlebnumlem1 18291* Lemma for lebnum 18294. The function  F measures the sum of all of the distances to escape the sets of the cover. Since by assumption it is a cover, there is at least one set which covers a given point, and since it is open, the point is a positive distance from the edge of the set. Thus the sum is a strictly positive number. (Contributed by Mario Carneiro, 14-Feb-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  U 
 C_  J )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  -.  X  e.  U )   &    |-  F  =  ( y  e.  X  |->  sum_ k  e.  U  sup ( ran  (  z  e.  ( X  \  k )  |->  ( y D z ) ) ,  RR* ,  `'  <  ) )   =>    |-  ( ph  ->  F : X --> RR+ )
 
Theoremlebnumlem2 18292* Lemma for lebnum 18294. As a finite sum of point-to-set distance functions, which are continuous by metdscn 18192, the function  F is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  U 
 C_  J )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  -.  X  e.  U )   &    |-  F  =  ( y  e.  X  |->  sum_ k  e.  U  sup ( ran  (  z  e.  ( X  \  k )  |->  ( y D z ) ) ,  RR* ,  `'  <  ) )   &    |-  K  =  (
 topGen `  ran  (,) )   =>    |-  ( ph  ->  F  e.  ( J  Cn  K ) )
 
Theoremlebnumlem3 18293* Lemma for lebnum 18294. By the previous lemmas,  F is continuous and positive on a compact set, so it has a positive minimum  r. Then setting  d  =  r  /  # ( U ), since for each  u  e.  U we have  ball ( x ,  d )  C_  u iff  d  <_  d ( x ,  X  \  u ), if  -.  ball (
x ,  d ) 
C_  u for all  u then summing over  u yields  sum_ u  e.  U
d ( x ,  X  \  u )  =  F ( x )  <  sum_ u  e.  U d  =  r, in contradiction to the assumption that  r is the minimum of  F. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  U 
 C_  J )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  -.  X  e.  U )   &    |-  F  =  ( y  e.  X  |->  sum_ k  e.  U  sup ( ran  (  z  e.  ( X  \  k )  |->  ( y D z ) ) ,  RR* ,  `'  <  ) )   &    |-  K  =  (
 topGen `  ran  (,) )   =>    |-  ( ph  ->  E. d  e.  RR+  A. x  e.  X  E. u  e.  U  ( x ( ball `  D ) d )  C_  u )
 
Theoremlebnum 18294* The Lebesgue number lemma, or Lebesgue covering lemma. If  X is a compact metric space and  U is an open cover of  X, then there exists a positive real number 
d such that every ball of size  d (and every subset of a ball of size  d, including every subset of diameter less than  d) is a subset of some member of the cover. (Contributed by Mario Carneiro, 14-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  U 
 C_  J )   &    |-  ( ph  ->  X  =  U. U )   =>    |-  ( ph  ->  E. d  e.  RR+  A. x  e.  X  E. u  e.  U  ( x ( ball `  D ) d )  C_  u )
 
Theoremxlebnum 18295* Generalize lebnum 18294 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  U  C_  J )   &    |-  ( ph  ->  X  =  U. U )   =>    |-  ( ph  ->  E. d  e.  RR+  A. x  e.  X  E. u  e.  U  ( x (
 ball `  D ) d )  C_  u )
 
Theoremlebnumii 18296* Specialize the Lebesgue number lemma lebnum 18294 to the unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)
 |-  ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  ->  E. n  e.  NN  A. k  e.  ( 1 ... n ) E. u  e.  U  ( ( ( k  -  1 )  /  n ) [,] (
 k  /  n )
 )  C_  u )
 
11.3.10  Path homotopy
 
Syntaxchtpy 18297 Extend class notation with the class of homotopies between two continuous functions.
 class Htpy
 
Syntaxcphtpy 18298 Extend class notation with the class of path homotopies between two continuous functions.
 class  PHtpy
 
Syntaxcphtpc 18299 Extend class notation with the path homotopy relation.
 class  ~=ph
 
Definitiondf-htpy 18300* Define the function which takes topological spaces  X ,  Y and two continuous functions  F ,  G : X
--> Y and returns the class of homotopies from  F to  G. (Contributed by Mario Carneiro, 22-Feb-2015.)
 |- Htpy  =  ( x  e.  Top ,  y  e.  Top  |->  ( f  e.  ( x  Cn  y ) ,  g  e.  ( x  Cn  y
 )  |->  { h  e.  (
 ( x  tX  II )  Cn  y )  | 
 A. s  e.  U. x ( ( s h 0 )  =  ( f `  s
 )  /\  ( s h 1 )  =  ( g `  s
 ) ) } )
 )
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