HomeHome Intuitionistic Logic Explorer
Theorem List (p. 134 of 141)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembl2ioo 13301 A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (
 ball `  D ) B )  =  ( ( A  -  B ) (,) ( A  +  B ) ) )
 
Theoremioo2bl 13302 An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (,) B )  =  ( ( ( A  +  B )  /  2 ) (
 ball `  D ) ( ( B  -  A )  /  2 ) ) )
 
Theoremioo2blex 13303 An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (,) B )  e.  ran  ( ball `  D ) )
 
Theoremblssioo 13304 The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |- 
 ran  ( ball `  D )  C_  ran  (,)
 
Theoremtgioo 13305 The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   &    |-  J  =  (
 MetOpen `  D )   =>    |-  ( topGen `  ran  (,) )  =  J
 
Theoremtgqioo 13306 The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  Q  =  ( topGen `  ( (,) " ( QQ 
 X.  QQ ) ) )   =>    |-  ( topGen `  ran  (,) )  =  Q
 
Theoremresubmet 13307 The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)
 |-  R  =  ( topGen `  ran  (,) )   &    |-  J  =  (
 MetOpen `  ( ( abs 
 o.  -  )  |`  ( A  X.  A ) ) )   =>    |-  ( A  C_  RR  ->  J  =  ( Rt  A ) )
 
Theoremtgioo2cntop 13308 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |-  ( topGen `  ran  (,) )  =  ( Jt  RR )
 
Theoremrerestcntop 13309 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  R  =  ( topGen `  ran  (,) )   =>    |-  ( A  C_  RR  ->  ( Jt  A )  =  ( Rt  A ) )
 
Theoremaddcncntoplem 13310* Lemma for addcncntop 13311, subcncntop 13312, and mulcncntop 13313. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |- 
 .+  : ( 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 )
 
Theoremaddcncntop 13311 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  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |- 
 +  e.  ( ( J  tX  J )  Cn  J )
 
Theoremsubcncntop 13312 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  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |- 
 -  e.  ( ( J  tX  J )  Cn  J )
 
Theoremmulcncntop 13313 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  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |- 
 x.  e.  ( ( J  tX  J )  Cn  J )
 
Theoremdivcnap 13314* Complex number division is a continuous function, when the second argument is apart from zero. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Jim Kingdon, 25-Oct-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  K  =  ( Jt  { x  e.  CC  |  x #  0 } )   =>    |-  ( y  e.  CC ,  z  e.  { x  e.  CC  |  x #  0 }  |->  ( y  /  z ) )  e.  ( ( J  tX  K )  Cn  J )
 
Theoremfsumcncntop 13315* 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  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  ( 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 ) )
 
8.2.7  Topological definitions using the reals
 
Syntaxccncf 13316 Extend class notation to include the operation which returns a class of continuous complex functions.
 class  -cn->
 
Definitiondf-cncf 13317* 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 ) } )
 
Theoremcncfval 13318* 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 13319* 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 13320* Version of elcncf 13319 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 13321 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
 
Theoremcncfrss2 13322 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
 
Theoremcncff 13323 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 13324* 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 13325* 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 13326* 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 13327 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 13328 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 13329 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 13330 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 13331 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremrecncf 13332 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Re  e.  ( CC
 -cn-> RR )
 
Theoremimcncf 13333 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Im  e.  ( CC
 -cn-> RR )
 
Theoremcjcncf 13334 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  *  e.  ( CC
 -cn-> CC )
 
Theoremmulc1cncf 13335* 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 ) )
 
Theoremdivccncfap 13336* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Jim Kingdon, 9-Jan-2023.)
 |-  F  =  ( x  e.  CC  |->  ( x 
 /  A ) )   =>    |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremcncfco 13337 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 13338 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 ) )
 
Theoremcncfcncntop 13339 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  K  =  ( Jt  A )   &    |-  L  =  ( Jt  B )   =>    |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( A -cn-> B )  =  ( K  Cn  L ) )
 
Theoremcncfcn1cntop 13340 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |-  ( CC -cn-> CC )  =  ( J  Cn  J )
 
Theoremcncfmptc 13341* 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 13342* 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 13343* Composition of continuous functions.  -cn-> analogue of cnmpt11f 13043. (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 ) )
 
Theoremcncfmpt2fcntop 13344* Composition of continuous functions.  -cn-> analogue of cnmpt12f 13045. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  ( 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 ) )
 
Theoremaddccncf 13345* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  F  =  ( x  e.  CC  |->  ( x  +  A ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremcdivcncfap 13346* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.)
 |-  F  =  ( x  e.  { y  e. 
 CC  |  y #  0 }  |->  ( A  /  x ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( { y  e.  CC  |  y #  0 } -cn->
 CC ) )
 
Theoremnegcncf 13347* 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 13348* 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 )
 )
 
Theoremmulcncflem 13349* Lemma for mulcncf 13350. (Contributed by Jim Kingdon, 29-May-2023.)
 |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( X
 -cn-> CC ) )   &    |-  ( ph  ->  V  e.  X )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  F  e.  RR+ )   &    |-  ( ph  ->  G  e.  RR+ )   &    |-  ( ph  ->  S  e.  RR+ )   &    |-  ( ph  ->  T  e.  RR+ )   &    |-  ( ph  ->  A. u  e.  X  ( ( abs `  ( u  -  V ) )  <  S  ->  ( abs `  ( ( ( x  e.  X  |->  A ) `  u )  -  ( ( x  e.  X  |->  A ) `
  V ) ) )  <  F ) )   &    |-  ( ph  ->  A. u  e.  X  ( ( abs `  ( u  -  V ) )  <  T  ->  ( abs `  ( ( ( x  e.  X  |->  B ) `  u )  -  ( ( x  e.  X  |->  B ) `
  V ) ) )  <  G ) )   &    |-  ( ph  ->  A. u  e.  X  ( ( ( abs `  ( [_ u  /  x ]_ A  -  [_ V  /  x ]_ A ) )  <  F  /\  ( abs `  ( [_ u  /  x ]_ B  -  [_ V  /  x ]_ B ) )  <  G )  ->  ( abs `  ( ( [_ u  /  x ]_ A  x.  [_ u  /  x ]_ B )  -  ( [_ V  /  x ]_ A  x.  [_ V  /  x ]_ B ) ) )  <  E ) )   =>    |-  ( ph  ->  E. d  e.  RR+  A. u  e.  X  ( ( abs `  ( u  -  V ) )  <  d  ->  ( abs `  ( ( ( x  e.  X  |->  ( A  x.  B ) ) `  u )  -  ( ( x  e.  X  |->  ( A  x.  B ) ) `
  V ) ) )  <  E ) )
 
Theoremmulcncf 13350* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( X
 -cn-> CC ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  x.  B ) )  e.  ( X
 -cn-> CC ) )
 
Theoremexpcncf 13351* The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( N  e.  NN0  ->  ( x  e.  CC  |->  ( x ^ N ) )  e.  ( CC
 -cn-> CC ) )
 
Theoremcnrehmeocntop 13352* The canonical bijection from  ( RR  X.  RR ) to  CC described in cnref1o 9602 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  =  ( MetOpen `  ( abs  o. 
 -  ) )   =>    |-  F  e.  (
 ( J  tX  J ) Homeo K )
 
Theoremcnopnap 13353* The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.)
 |-  ( A  e.  CC  ->  { w  e.  CC  |  w #  A }  e.  ( MetOpen `  ( abs  o. 
 -  ) ) )
 
PART 9  BASIC REAL AND COMPLEX ANALYSIS
 
9.0.1  Dedekind cuts
 
Theoremdedekindeulemuub 13354* Lemma for dedekindeu 13360. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  A. z  e.  L  z  <  A )
 
Theoremdedekindeulemub 13355* Lemma for dedekindeu 13360. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  E. x  e.  RR  A. y  e.  L  y  <  x )
 
Theoremdedekindeulemloc 13356* Lemma for dedekindeu 13360. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  A. x  e. 
 RR  A. y  e.  RR  ( x  <  y  ->  ( E. z  e.  L  x  <  z  \/  A. z  e.  L  z  <  y ) ) )
 
Theoremdedekindeulemlub 13357* Lemma for dedekindeu 13360. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  L  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  L  y  <  z
 ) ) )
 
Theoremdedekindeulemlu 13358* Lemma for dedekindeu 13360. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  E. x  e.  RR  ( A. q  e.  L  q  <  x  /\  A. r  e.  U  x  <  r ) )
 
Theoremdedekindeulemeu 13359* Lemma for dedekindeu 13360. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  (
 A. q  e.  L  q  <  A  /\  A. r  e.  U  A  <  r ) )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( A. q  e.  L  q  <  B  /\  A. r  e.  U  B  <  r ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  -> F.  )
 
Theoremdedekindeu 13360* A Dedekind cut identifies a unique real number. Similar to df-inp 7421 except that the the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 5-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  E! x  e.  RR  ( A. q  e.  L  q  <  x  /\  A. r  e.  U  x  <  r ) )
 
Theoremsuplociccreex 13361* An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7985 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <  C )   &    |-  ( ph  ->  A  C_  ( B [,] C ) )   &    |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  A. x  e.  ( B [,] C ) A. y  e.  ( B [,] C ) ( x  <  y  ->  ( E. z  e.  A  x  <  z  \/  A. z  e.  A  z  <  y ) ) )   =>    |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )
 
Theoremsuplociccex 13362* An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7985 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <  C )   &    |-  ( ph  ->  A  C_  ( B [,] C ) )   &    |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  A. x  e.  ( B [,] C ) A. y  e.  ( B [,] C ) ( x  <  y  ->  ( E. z  e.  A  x  <  z  \/  A. z  e.  A  z  <  y ) ) )   =>    |-  ( ph  ->  E. x  e.  ( B [,] C ) ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e.  ( B [,] C ) ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )
 
Theoremdedekindicclemuub 13363* Lemma for dedekindicc 13370. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  C  e.  U )   =>    |-  ( ph  ->  A. z  e.  L  z  <  C )
 
Theoremdedekindicclemub 13364* Lemma for dedekindicc 13370. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  E. x  e.  ( A [,] B ) A. y  e.  L  y  <  x )
 
Theoremdedekindicclemloc 13365* Lemma for dedekindicc 13370. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( E. z  e.  L  x  <  z  \/  A. z  e.  L  z  <  y ) ) )
 
Theoremdedekindicclemlub 13366* Lemma for dedekindicc 13370. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  E. x  e.  ( A [,] B ) ( A. y  e.  L  -.  x  < 
 y  /\  A. y  e.  ( A [,] B ) ( y  < 
 x  ->  E. z  e.  L  y  <  z
 ) ) )
 
Theoremdedekindicclemlu 13367* Lemma for dedekindicc 13370. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  E. x  e.  ( A [,] B ) ( A. q  e.  L  q  <  x  /\  A. r  e.  U  x  <  r ) )
 
Theoremdedekindicclemeu 13368* Lemma for dedekindicc 13370. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  C  e.  ( A [,] B ) )   &    |-  ( ph  ->  (
 A. q  e.  L  q  <  C  /\  A. r  e.  U  C  <  r ) )   &    |-  ( ph  ->  D  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( A. q  e.  L  q  <  D  /\  A. r  e.  U  D  <  r
 ) )   &    |-  ( ph  ->  C  <  D )   =>    |-  ( ph  -> F.  )
 
Theoremdedekindicclemicc 13369* Lemma for dedekindicc 13370. Same as dedekindicc 13370, except that we merely show  x to be an element of  ( A [,] B ). Later we will strengthen that to  ( A (,) B
). (Contributed by Jim Kingdon, 5-Jan-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  E! x  e.  ( A [,] B ) ( A. q  e.  L  q  <  x  /\  A. r  e.  U  x  <  r
 ) )
 
Theoremdedekindicc 13370* A Dedekind cut identifies a unique real number. Similar to df-inp 7421 except that the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 19-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  E! x  e.  ( A (,) B ) ( A. q  e.  L  q  <  x  /\  A. r  e.  U  x  <  r
 ) )
 
9.0.2  Intermediate value theorem
 
Theoremivthinclemlm 13371* Lemma for ivthinc 13380. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )
 
Theoremivthinclemum 13372* Lemma for ivthinc 13380. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  R )
 
Theoremivthinclemlopn 13373* Lemma for ivthinc 13380. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   &    |-  ( ph  ->  Q  e.  L )   =>    |-  ( ph  ->  E. r  e.  L  Q  <  r
 )
 
Theoremivthinclemlr 13374* Lemma for ivthinc 13380. The lower cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
 
Theoremivthinclemuopn 13375* Lemma for ivthinc 13380. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   &    |-  ( ph  ->  S  e.  R )   =>    |-  ( ph  ->  E. q  e.  R  q  <  S )
 
Theoremivthinclemur 13376* Lemma for ivthinc 13380. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  R  <->  E. q  e.  R  q  <  r ) )
 
Theoremivthinclemdisj 13377* Lemma for ivthinc 13380. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  ( L  i^i  R )  =  (/) )
 
Theoremivthinclemloc 13378* Lemma for ivthinc 13380. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  (
 q  e.  L  \/  r  e.  R )
 ) )
 
Theoremivthinclemex 13379* Lemma for ivthinc 13380. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  E! z  e.  ( A (,) B ) ( A. q  e.  L  q  <  z  /\  A. r  e.  R  z  <  r ) )
 
Theoremivthinc 13380* The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   =>    |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
 
Theoremivthdec 13381* The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  B )  <  U  /\  U  <  ( F `  A ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  y )  <  ( F `  x ) )   =>    |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
 
9.1  Derivatives
 
9.1.1  Real and complex differentiation
 
9.1.1.1  Derivatives of functions of one complex or real variable
 
Syntaxclimc 13382 The limit operator.
 class lim CC
 
Syntaxcdv 13383 The derivative operator.
 class  _D
 
Definitiondf-limced 13384* Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
 |- lim
 CC  =  ( f  e.  ( CC  ^pm  CC ) ,  x  e. 
 CC  |->  { y  e.  CC  |  ( ( f : dom  f --> CC  /\  dom  f  C_  CC )  /\  ( x  e.  CC  /\ 
 A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  f ( ( z #  x  /\  ( abs `  ( z  -  x ) )  < 
 d )  ->  ( abs `  ( ( f `
  z )  -  y ) )  < 
 e ) ) ) } )
 
Definitiondf-dvap 13385* Define the derivative operator. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set  s here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of  CC and is well-behaved when  s contains no isolated points, we will restrict our attention to the cases  s  =  RR or  s  =  CC for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
 |- 
 _D  =  ( s  e.  ~P CC ,  f  e.  ( CC  ^pm  s )  |->  U_ x  e.  ( ( int `  (
 ( MetOpen `  ( abs  o. 
 -  ) )t  s ) ) `  dom  f
 ) ( { x }  X.  ( ( z  e.  { w  e. 
 dom  f  |  w #  x }  |->  ( ( ( f `  z
 )  -  ( f `
  x ) ) 
 /  ( z  -  x ) ) ) lim
 CC  x ) ) )
 
Theoremlimcrcl 13386 Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( C  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC )
 )
 
Theoremlimccl 13387 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( F lim CC  B )  C_  CC
 
Theoremellimc3apf 13388* Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  F/_ z F   =>    |-  ( ph  ->  ( C  e.  ( F lim
 CC  B )  <->  ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
 y )  ->  ( abs `  ( ( F `
  z )  -  C ) )  < 
 x ) ) ) )
 
Theoremellimc3ap 13389* Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) Use apartness. (Revised by Jim Kingdon, 3-Jun-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( C  e.  ( F lim CC  B )  <->  ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
 y )  ->  ( abs `  ( ( F `
  z )  -  C ) )  < 
 x ) ) ) )
 
Theoremlimcdifap 13390* It suffices to consider functions which are not defined at  B to define the limit of a function. In particular, the value of the original function  F at  B does not affect the limit of  F. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   =>    |-  ( ph  ->  ( F lim CC  B )  =  ( ( F  |`  { x  e.  A  |  x #  B } ) lim CC  B ) )
 
Theoremlimcmpted 13391* Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)
 |-  ( ph  ->  A  C_ 
 CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  (
 ( ph  /\  z  e.  A )  ->  D  e.  CC )   =>    |-  ( ph  ->  ( C  e.  ( (
 z  e.  A  |->  D ) lim CC  B )  <-> 
 ( C  e.  CC  /\ 
 A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  ( z  -  B ) )  <  y ) 
 ->  ( abs `  ( D  -  C ) )  <  x ) ) ) )
 
Theoremlimcimolemlt 13392* Lemma for limcimo 13393. (Contributed by Jim Kingdon, 3-Jul-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  ( Kt  S ) )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  { q  e.  C  |  q #  B }  C_  A )   &    |-  K  =  ( MetOpen `  ( abs  o. 
 -  ) )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  X  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  Y  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
 z  -  B ) )  <  D ) 
 ->  ( abs `  (
 ( F `  z
 )  -  X ) )  <  ( ( abs `  ( X  -  Y ) )  / 
 2 ) ) )   &    |-  ( ph  ->  G  e.  RR+ )   &    |-  ( ph  ->  A. w  e.  A  ( ( w #  B  /\  ( abs `  ( w  -  B ) )  <  G )  ->  ( abs `  ( ( F `  w )  -  Y ) )  <  ( ( abs `  ( X  -  Y ) )  / 
 2 ) ) )   =>    |-  ( ph  ->  ( abs `  ( X  -  Y ) )  <  ( abs `  ( X  -  Y ) ) )
 
Theoremlimcimo 13393* Conditions which ensure there is at most one limit value of  F at  B. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 8-Jul-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  ( Kt  S ) )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  { q  e.  C  |  q #  B }  C_  A )   &    |-  K  =  ( MetOpen `  ( abs  o. 
 -  ) )   =>    |-  ( ph  ->  E* x  x  e.  ( F lim CC  B ) )
 
Theoremlimcresi 13394 Any limit of  F is also a limit of the restriction of  F. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( F lim CC  B )  C_  ( ( F  |`  C ) lim CC  B )
 
Theoremcnplimcim 13395 If a function is continuous at  B, its limit at  B equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Jun-2023.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( Kt  A )   =>    |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( F  e.  ( ( J  CnP  K ) `  B ) 
 ->  ( F : A --> CC  /\  ( F `  B )  e.  ( F lim CC  B ) ) ) )
 
Theoremcnplimclemle 13396 Lemma for cnplimccntop 13398. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( Kt  A )   &    |-  ( ph  ->  A 
 C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  ( F `  B )  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  Z  e.  A )   &    |-  (
 ( ph  /\  Z #  B  /\  ( abs `  ( Z  -  B ) )  <  D )  ->  ( abs `  ( ( F `  Z )  -  ( F `  B ) ) )  <  ( E  /  2 ) )   &    |-  ( ph  ->  ( abs `  ( Z  -  B ) )  <  D )   =>    |-  ( ph  ->  ( abs `  ( ( F `  Z )  -  ( F `  B ) ) )  <  E )
 
Theoremcnplimclemr 13397 Lemma for cnplimccntop 13398. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( Kt  A )   &    |-  ( ph  ->  A 
 C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  ( F `  B )  e.  ( F lim CC  B ) )   =>    |-  ( ph  ->  F  e.  ( ( J  CnP  K ) `  B ) )
 
Theoremcnplimccntop 13398 A function is continuous at  B iff its limit at  B equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( Kt  A )   =>    |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( F  e.  ( ( J  CnP  K ) `  B )  <-> 
 ( F : A --> CC  /\  ( F `  B )  e.  ( F lim CC  B ) ) ) )
 
Theoremcnlimcim 13399* If  F is a continuous function, the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 16-Jun-2023.)
 |-  ( A  C_  CC  ->  ( F  e.  ( A -cn-> CC )  ->  ( F : A --> CC  /\  A. x  e.  A  ( F `  x )  e.  ( F lim CC  x ) ) ) )
 
Theoremcnlimc 13400*  F is a continuous function iff the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( A  C_  CC  ->  ( F  e.  ( A -cn-> CC )  <->  ( F : A
 --> CC  /\  A. x  e.  A  ( F `  x )  e.  ( F lim CC  x ) ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14078
  Copyright terms: Public domain < Previous  Next >