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Theorem List for Intuitionistic Logic Explorer - 13001-13100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcncfval 13001* 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 13002* 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 13003* Version of elcncf 13002 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 13004 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
 
Theoremcncfrss2 13005 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
 
Theoremcncff 13006 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 13007* 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 13008* 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 13009* 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 13010 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 13011 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 13012 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 13013 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 13014 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremrecncf 13015 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Re  e.  ( CC
 -cn-> RR )
 
Theoremimcncf 13016 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Im  e.  ( CC
 -cn-> RR )
 
Theoremcjcncf 13017 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  *  e.  ( CC
 -cn-> CC )
 
Theoremmulc1cncf 13018* 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 13019* 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 13020 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 13021 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 13022 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 13023 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 13024* 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 13025* 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 13026* Composition of continuous functions.  -cn-> analogue of cnmpt11f 12726. (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 13027* Composition of continuous functions.  -cn-> analogue of cnmpt12f 12728. (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 13028* 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 13029* 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 13030* 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 13031* 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 13032* Lemma for mulcncf 13033. (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 13033* 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 13034* 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 13035* The canonical bijection from  ( RR  X.  RR ) to  CC described in cnref1o 9560 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 13036* 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 8  BASIC REAL AND COMPLEX ANALYSIS
 
8.0.1  Dedekind cuts
 
Theoremdedekindeulemuub 13037* Lemma for dedekindeu 13043. 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 13038* Lemma for dedekindeu 13043. 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 13039* Lemma for dedekindeu 13043. 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 13040* Lemma for dedekindeu 13043. 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 13041* Lemma for dedekindeu 13043. 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 13042* Lemma for dedekindeu 13043. 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 13043* A Dedekind cut identifies a unique real number. Similar to df-inp 7387 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 13044* An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7951 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 13045* An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7951 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 13046* Lemma for dedekindicc 13053. 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 13047* Lemma for dedekindicc 13053. 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 13048* Lemma for dedekindicc 13053. 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 13049* Lemma for dedekindicc 13053. 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 13050* Lemma for dedekindicc 13053. 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 13051* Lemma for dedekindicc 13053. 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 13052* Lemma for dedekindicc 13053. Same as dedekindicc 13053, 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 13053* A Dedekind cut identifies a unique real number. Similar to df-inp 7387 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
 ) )
 
8.0.2  Intermediate value theorem
 
Theoremivthinclemlm 13054* Lemma for ivthinc 13063. 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 13055* Lemma for ivthinc 13063. 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 13056* Lemma for ivthinc 13063. 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 13057* Lemma for ivthinc 13063. 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 13058* Lemma for ivthinc 13063. 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 13059* Lemma for ivthinc 13063. 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 13060* Lemma for ivthinc 13063. 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 13061* Lemma for ivthinc 13063. 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 13062* Lemma for ivthinc 13063. 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 13063* 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 13064* 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 )
 
8.1  Derivatives
 
8.1.1  Real and complex differentiation
 
8.1.1.1  Derivatives of functions of one complex or real variable
 
Syntaxclimc 13065 The limit operator.
 class lim CC
 
Syntaxcdv 13066 The derivative operator.
 class  _D
 
Definitiondf-limced 13067* 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 13068* 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 13069 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 13070 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( F lim CC  B )  C_  CC
 
Theoremellimc3apf 13071* 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 13072* 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 13073* 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 13074* 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 13075* Lemma for limcimo 13076. (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 13076* 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 13077 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 13078 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 13079 Lemma for cnplimccntop 13081. 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 13080 Lemma for cnplimccntop 13081. 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 13081 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 13082* 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 13083*  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 ) ) ) )
 
Theoremcnlimci 13084 If  F is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  F  e.  ( A -cn-> D ) )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  ( F `  B )  e.  ( F lim CC  B ) )
 
Theoremcnmptlimc 13085* If  F is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  ( x  e.  A  |->  X )  e.  ( A -cn-> D ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( x  =  B  ->  X  =  Y )   =>    |-  ( ph  ->  Y  e.  ( ( x  e.  A  |->  X ) lim
 CC  B ) )
 
Theoremlimccnpcntop 13086 If the limit of  F at  B is  C and  G is continuous at  C, then the limit of  G  o.  F at  B is  G ( C ). (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 18-Jun-2023.)
 |-  ( ph  ->  F : A --> D )   &    |-  ( ph  ->  D  C_  CC )   &    |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( Kt  D )   &    |-  ( ph  ->  C  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  G  e.  (
 ( J  CnP  K ) `  C ) )   =>    |-  ( ph  ->  ( G `  C )  e.  (
 ( G  o.  F ) lim CC  B ) )
 
Theoremlimccnp2lem 13087* Lemma for limccnp2cntop 13088. This is most of the result, expressed in epsilon-delta form, with a large number of hypotheses so that lengthy expressions do not need to be repeated. (Contributed by Jim Kingdon, 9-Nov-2023.)
 |-  ( ( ph  /\  x  e.  A )  ->  R  e.  X )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  S  e.  Y )   &    |-  ( ph  ->  X  C_  CC )   &    |-  ( ph  ->  Y  C_ 
 CC )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   &    |-  J  =  ( ( K  tX  K )t  ( X  X.  Y ) )   &    |-  ( ph  ->  C  e.  ( ( x  e.  A  |->  R ) lim
 CC  B ) )   &    |-  ( ph  ->  D  e.  ( ( x  e.  A  |->  S ) lim CC  B ) )   &    |-  ( ph  ->  H  e.  (
 ( J  CnP  K ) `  <. C ,  D >. ) )   &    |-  F/ x ph   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  L  e.  RR+ )   &    |-  ( ph  ->  A. r  e.  X  A. s  e.  Y  (
 ( ( C ( ( abs  o.  -  )  |`  ( X  X.  X ) ) r )  <  L  /\  ( D ( ( abs 
 o.  -  )  |`  ( Y  X.  Y ) ) s )  <  L )  ->  ( ( C H D ) ( abs  o.  -  )
 ( r H s ) )  <  E ) )   &    |-  ( ph  ->  F  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  A  ( ( x #  B  /\  ( abs `  ( x  -  B ) )  <  F )  ->  ( abs `  ( R  -  C ) )  <  L ) )   &    |-  ( ph  ->  G  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  A  ( ( x #  B  /\  ( abs `  ( x  -  B ) )  <  G )  ->  ( abs `  ( S  -  D ) )  <  L ) )   =>    |-  ( ph  ->  E. d  e.  RR+  A. x  e.  A  ( ( x #  B  /\  ( abs `  ( x  -  B ) )  <  d )  ->  ( abs `  ( ( R H S )  -  ( C H D ) ) )  <  E ) )
 
Theoremlimccnp2cntop 13088* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Nov-2023.)
 |-  ( ( ph  /\  x  e.  A )  ->  R  e.  X )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  S  e.  Y )   &    |-  ( ph  ->  X  C_  CC )   &    |-  ( ph  ->  Y  C_ 
 CC )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   &    |-  J  =  ( ( K  tX  K )t  ( X  X.  Y ) )   &    |-  ( ph  ->  C  e.  ( ( x  e.  A  |->  R ) lim
 CC  B ) )   &    |-  ( ph  ->  D  e.  ( ( x  e.  A  |->  S ) lim CC  B ) )   &    |-  ( ph  ->  H  e.  (
 ( J  CnP  K ) `  <. C ,  D >. ) )   =>    |-  ( ph  ->  ( C H D )  e.  ( ( x  e.  A  |->  ( R H S ) ) lim CC  B ) )
 
Theoremlimccoap 13089* Composition of two limits. This theorem is only usable in the case where  x #  X implies R(x) #  C so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)
 |-  ( ( ph  /\  x  e.  { w  e.  A  |  w #  X }
 )  ->  R  e.  { w  e.  B  |  w #  C } )   &    |-  (
 ( ph  /\  y  e. 
 { w  e.  B  |  w #  C }
 )  ->  S  e.  CC )   &    |-  ( ph  ->  C  e.  ( ( x  e.  { w  e.  A  |  w #  X }  |->  R ) lim CC  X ) )   &    |-  ( ph  ->  D  e.  (
 ( y  e.  { w  e.  B  |  w #  C }  |->  S ) lim
 CC  C ) )   &    |-  ( y  =  R  ->  S  =  T )   =>    |-  ( ph  ->  D  e.  ( ( x  e. 
 { w  e.  A  |  w #  X }  |->  T ) lim CC  X ) )
 
Theoremreldvg 13090 The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
 |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S ) ) 
 ->  Rel  ( S  _D  F ) )
 
Theoremdvlemap 13091* Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
 |-  ( ph  ->  F : D --> CC )   &    |-  ( ph  ->  D  C_  CC )   &    |-  ( ph  ->  B  e.  D )   =>    |-  ( ( ph  /\  A  e.  { w  e.  D  |  w #  B }
 )  ->  ( (
 ( F `  A )  -  ( F `  B ) )  /  ( A  -  B ) )  e.  CC )
 
Theoremdvfvalap 13092* Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
 |-  T  =  ( Kt  S )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  ( ( S  _D  F )  = 
 U_ x  e.  (
 ( int `  T ) `  A ) ( { x }  X.  (
 ( z  e.  { w  e.  A  |  w #  x }  |->  ( ( ( F `  z
 )  -  ( F `
  x ) ) 
 /  ( z  -  x ) ) ) lim
 CC  x ) ) 
 /\  ( S  _D  F )  C_  ( ( ( int `  T ) `  A )  X.  CC ) ) )
 
Theoremeldvap 13093* The differentiable predicate. A function  F is differentiable at  B with derivative  C iff  F is defined in a neighborhood of  B and the difference quotient has limit  C at  B. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
 |-  T  =  ( Kt  S )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   &    |-  G  =  ( z  e.  { w  e.  A  |  w #  B }  |->  ( ( ( F `  z )  -  ( F `  B ) )  /  ( z  -  B ) ) )   &    |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   =>    |-  ( ph  ->  ( B ( S  _D  F ) C  <->  ( B  e.  ( ( int `  T ) `  A )  /\  C  e.  ( G lim CC  B ) ) ) )
 
Theoremdvcl 13094 The derivative function takes values in the complex numbers. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   =>    |-  ( ( ph  /\  B ( S  _D  F ) C )  ->  C  e.  CC )
 
Theoremdvbssntrcntop 13095 The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  J  =  ( Kt  S )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ph  ->  dom  ( S  _D  F )  C_  ( ( int `  J ) `  A ) )
 
Theoremdvbss 13096 The set of differentiable points is a subset of the domain of the function. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   =>    |-  ( ph  ->  dom  ( S  _D  F )  C_  A )
 
Theoremdvbsssg 13097 The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
 |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S ) ) 
 ->  dom  ( S  _D  F )  C_  S )
 
Theoremrecnprss 13098 Both  RR and  CC are subsets of  CC. (Contributed by Mario Carneiro, 10-Feb-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  S 
 C_  CC )
 
Theoremdvfgg 13099 Explicitly write out the functionality condition on derivative for  S  =  RR and 
CC. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
 |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC 
 ^pm  S ) )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
 
Theoremdvfpm 13100 The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 28-Jul-2023.)
 |-  ( F  e.  ( CC  ^pm  RR )  ->  ( RR  _D  F ) : dom  ( RR 
 _D  F ) --> CC )
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