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Theorem List for Metamath Proof Explorer - 19901-20000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremefif1olem4 19901* The exponential function of an imaginary number maps any interval of length  2 pi one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.)
 |-  F  =  ( w  e.  D  |->  ( exp `  ( _i  x.  w ) ) )   &    |-  C  =  ( `' abs " {
 1 } )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ( ph  /\  ( x  e.  D  /\  y  e.  D )
 )  ->  ( abs `  ( x  -  y
 ) )  <  (
 2  x.  pi ) )   &    |-  ( ( ph  /\  z  e.  RR )  ->  E. y  e.  D  ( ( z  -  y )  /  (
 2  x.  pi ) )  e.  ZZ )   &    |-  S  =  ( sin  |`  ( -u ( pi  /  2
 ) [,] ( pi  / 
 2 ) ) )   =>    |-  ( ph  ->  F : D
 -1-1-onto-> C )
 
Theoremefif1o 19902* The exponential function of an imaginary number maps any open-below, closed-above interval of length 
2 pi one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.)
 |-  F  =  ( w  e.  D  |->  ( exp `  ( _i  x.  w ) ) )   &    |-  C  =  ( `' abs " {
 1 } )   &    |-  D  =  ( A (,] ( A  +  ( 2  x.  pi ) ) )   =>    |-  ( A  e.  RR  ->  F : D -1-1-onto-> C )
 
Theoremefifo 19903* The exponential function of an imaginary number maps the reals onto the unit circle. (Contributed by Mario Carneiro, 13-May-2014.)
 |-  F  =  ( z  e.  RR  |->  ( exp `  ( _i  x.  z
 ) ) )   &    |-  C  =  ( `' abs " {
 1 } )   =>    |-  F : RR -onto-> C
 
Theoremeff1olem 19904* The exponential function maps the set  S, of complex numbers with imaginary part in a real interval of length  2  x.  pi, one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.)
 |-  F  =  ( w  e.  D  |->  ( exp `  ( _i  x.  w ) ) )   &    |-  S  =  ( `' Im " D )   &    |-  ( ph  ->  D 
 C_  RR )   &    |-  ( ( ph  /\  ( x  e.  D  /\  y  e.  D ) )  ->  ( abs `  ( x  -  y
 ) )  <  (
 2  x.  pi ) )   &    |-  ( ( ph  /\  z  e.  RR )  ->  E. y  e.  D  ( ( z  -  y )  /  (
 2  x.  pi ) )  e.  ZZ )   =>    |-  ( ph  ->  ( exp  |`  S ) : S -1-1-onto-> ( CC  \  {
 0 } ) )
 
Theoremeff1o 19905 The exponential function maps the set  S, of complex numbers with imaginary part in the closed-above, open-below interval from  -u pi to  pi one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
 |-  S  =  ( `' Im " ( -u pi (,] pi ) )   =>    |-  ( exp  |`  S ) : S -1-1-onto-> ( CC  \  {
 0 } )
 
13.3.4  The natural logarithm on complex numbers
 
Syntaxclog 19906 Extend class notation with the natural logarithm function on complex numbers.
 class  log
 
Syntaxccxp 19907 Extend class notation with the complex power function.
 class  ^ c
 
Definitiondf-log 19908 Define the natural logarithm function on complex numbers. See http://en.wikipedia.org/wiki/Natural_logarithm ("The natural logarithm function can also be defined as the inverse function of the exponential function"). (Contributed by Paul Chapman, 21-Apr-2008.)
 |- 
 log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
 
Definitiondf-cxp 19909* Define the power function on complex numbers. Note that the value of this function when  x  =  0 and  ( Re `  y )  <_  0 ,  y  =/=  0 should properly be undefined, but defining it by convention this way simplifies the domain. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |- 
 ^ c  =  ( x  e.  CC ,  y  e.  CC  |->  if ( x  =  0 ,  if ( y  =  0 ,  1 ,  0 ) ,  ( exp `  ( y  x.  ( log `  x ) ) ) ) )
 
Theoremlogrn 19910 The range of the natural logarithm function, also the principal domain of the exponential function. This allows us to write the longer class expression as simply  ran  log. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
 |- 
 ran  log  =  ( `' Im " ( -u pi (,] pi ) )
 
Theoremellogrn 19911 Write out the property  A  e.  ran  log explicitly. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  ran  log  <->  ( A  e.  CC  /\  -u pi  <  ( Im `  A )  /\  ( Im `  A )  <_  pi ) )
 
Theoremdflog2 19912 The natural logarithm function in terms of the exponential function restricted to its principal domain. (Contributed by Paul Chapman, 21-Apr-2008.)
 |- 
 log  =  `' ( exp  |`  ran  log )
 
Theoremrelogrn 19913 The range of the natural logarithm function includes the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  RR  ->  A  e.  ran  log )
 
Theoremlogrncn 19914 The range of the natural logarithm function is a subset of the complex numbers. (Contributed by Mario Carneiro, 13-May-2014.)
 |-  ( A  e.  ran  log 
 ->  A  e.  CC )
 
Theoremeff1o2 19915 The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
 |-  ( exp  |`  ran  log ) : ran  log -1-1-onto-> ( CC  \  {
 0 } )
 
Theoremlogf1o 19916 The natural logarithm function maps the nonzero complex numbers one-to-one onto its range. (Contributed by Paul Chapman, 21-Apr-2008.)
 |- 
 log : ( CC  \  { 0 } ) -1-1-onto-> ran  log
 
Theoremdfrelog 19917 The natural logarithm function on the positive reals in terms of the real exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( log  |`  RR+ )  =  `' ( exp  |`  RR )
 
Theoremrelogf1o 19918 The natural logarithm function maps the positive reals one-to-one onto the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( log  |`  RR+ ) : RR+
 -1-1-onto-> RR
 
Theoremlogrncl 19919 Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( log `  A )  e.  ran  log )
 
Theoremlogcl 19920 Closure of the natural logarithm function. (Contributed by NM, 21-Apr-2008.) (Revised by Mario Carneiro, 23-Sep-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( log `  A )  e.  CC )
 
Theoremlogimcl 19921 Closure of the imaginary part of the logarithm function. (Contributed by Mario Carneiro, 23-Sep-2014.) (Revised by Mario Carneiro, 1-Apr-2015.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( -u pi  <  ( Im `  ( log `  A ) ) 
 /\  ( Im `  ( log `  A )
 )  <_  pi )
 )
 
Theoremlogcld 19922 The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 19920. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( log `  X )  e.  CC )
 
Theoremlogimcld 19923 The imaginary part of the logarithm is in  ( -u pi (,] pi ). Deduction form of logimcl 19921. Compare logimclad 19924. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  (
 -u pi  <  ( Im `  ( log `  X ) )  /\  ( Im
 `  ( log `  X ) )  <_  pi ) )
 
Theoremlogimclad 19924 The imaginary part of the logarithm is in  ( -u pi (,] pi ). Alternate form of logimcld 19923. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( Im `  ( log `  X ) )  e.  ( -u pi (,] pi ) )
 
Theoremlogrnaddcl 19925 The range of the natural logarithm is closed under addition with reals. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ( A  e.  ran 
 log  /\  B  e.  RR )  ->  ( A  +  B )  e.  ran  log )
 
Theoremrelogcl 19926 Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
 
Theoremeflog 19927 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( exp `  ( log `  A ) )  =  A )
 
Theoremreeflog 19928 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( A  e.  RR+  ->  ( exp `  ( log `  A ) )  =  A )
 
Theoremlogef 19929 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( A  e.  ran  log 
 ->  ( log `  ( exp `  A ) )  =  A )
 
Theoremrelogef 19930 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( A  e.  RR  ->  ( log `  ( exp `  A ) )  =  A )
 
Theoremlogeftb 19931 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  ran  log )  ->  ( ( log `  A )  =  B  <->  ( exp `  B )  =  A ) )
 
Theoremrelogeftb 19932 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR )  ->  ( ( log `  A )  =  B  <->  ( exp `  B )  =  A )
 )
 
Theoremlog1 19933 The natural logarithm of  1. One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( log `  1
 )  =  0
 
Theoremloge 19934 The natural logarithm of  _e. One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( log `  _e )  =  1
 
Theoremlogneg 19935 The natural logarithm of a negative real number. (Contributed by Mario Carneiro, 13-May-2014.) (Revised by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  RR+  ->  ( log `  -u A )  =  ( ( log `  A )  +  ( _i  x.  pi ) ) )
 
Theoremlogm1 19936 The natural logarithm of negative  1. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
 |-  ( log `  -u 1
 )  =  ( _i 
 x.  pi )
 
Theoremlognegb 19937 If a number has imaginary part equal to  pi, then it is on the negative real axis and vice-versa. (Contributed by Mario Carneiro, 23-Sep-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( -u A  e.  RR+  <->  ( Im `  ( log `  A )
 )  =  pi ) )
 
Theoremrelogoprlem 19938 Lemma for relogmul 19939 and relogdiv 19940. Remark of [Cohen] p. 301 ("The proof of Property 3 is quite similar to the proof given for Property 2"). (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( ( log `  A )  e.  CC  /\  ( log `  B )  e.  CC )  ->  ( exp `  (
 ( log `  A ) F ( log `  B ) ) )  =  ( ( exp `  ( log `  A ) ) G ( exp `  ( log `  B ) ) ) )   &    |-  ( ( ( log `  A )  e.  RR  /\  ( log `  B )  e.  RR )  ->  ( ( log `  A ) F ( log `  B )
 )  e.  RR )   =>    |-  (
 ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( log `  ( A G B ) )  =  ( ( log `  A ) F ( log `  B )
 ) )
 
Theoremrelogmul 19939 The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( log `  ( A  x.  B ) )  =  ( ( log `  A )  +  ( log `  B ) ) )
 
Theoremrelogdiv 19940 The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( log `  ( A  /  B ) )  =  ( ( log `  A )  -  ( log `  B ) ) )
 
Theoremexplog 19941 Exponentiation of a nonzero complex number to an integer power. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  =  ( exp `  ( N  x.  ( log `  A ) ) ) )
 
Theoremreexplog 19942 Exponentiation of a positive real number to an integer power. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  N  e.  ZZ )  ->  ( A ^ N )  =  ( exp `  ( N  x.  ( log `  A ) ) ) )
 
Theoremrelogexp 19943 The natural logarithm of positive 
A raised to an integer power. Property 4 of [Cohen] p. 301-302, restricted to natural logarithms and integer powers  N. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  N  e.  ZZ )  ->  ( log `  ( A ^ N ) )  =  ( N  x.  ( log `  A )
 ) )
 
Theoremrelog 19944 Real part of a logarithm. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( Re `  ( log `  A ) )  =  ( log `  ( abs `  A ) ) )
 
Theoremrelogiso 19945 The natural logarithm function on positive reals determines an isomorphism from the positive reals onto the reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( log  |`  RR+ )  Isom  <  ,  <  ( RR+
 ,  RR )
 
Theoremreloggim 19946 The natural logarithm is a group isomorphism from the group of positive reals under multiplication to the group of reals under addition. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  R  =  (flds  RR )   &    |-  P  =  ( (mulGrp ` fld )s  RR+ )   =>    |-  ( log  |`  RR+ )  e.  ( P GrpIso  R )
 
Theoremlogltb 19947 The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <  B  <->  ( log `  A )  <  ( log `  B ) ) )
 
Theoremlogfac 19948* The logarithm of a factorial can be expressed as a finite sum of logs. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( N  e.  NN0  ->  ( log `  ( ! `  N ) )  = 
 sum_ k  e.  (
 1 ... N ) ( log `  k )
 )
 
Theoremeflogeq 19949* Solve an equation involving an exponential. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( ( exp `  A )  =  B  <->  E. n  e.  ZZ  A  =  ( ( log `  B )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) ) )
 
Theoremlogne0 19950 Logarithm of a non-1 number is not zero and thus suitable as a divisor. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( ( A  e.  RR+  /\  A  =/=  1 ) 
 ->  ( log `  A )  =/=  0 )
 
Theoremlogleb 19951 Natural logarithm preserves  <_. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <_  B  <->  ( log `  A )  <_  ( log `  B ) ) )
 
Theoremrplogcl 19952 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  ( ( A  e.  RR  /\  1  <  A )  ->  ( log `  A )  e.  RR+ )
 
Theoremlogge0 19953 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  RR  /\  1  <_  A )  ->  0  <_  ( log `  A ) )
 
Theoremlogcj 19954 The natural logarithm distributes under conjugation away from the branch cut. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( ( A  e.  CC  /\  ( Im `  A )  =/=  0
 )  ->  ( log `  ( * `  A ) )  =  ( * `  ( log `  A ) ) )
 
Theoremefiarg 19955 The exponential of the "arg" function  Im  o.  log. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( exp `  ( _i  x.  ( Im `  ( log `  A ) ) ) )  =  ( A  /  ( abs `  A )
 ) )
 
Theoremcosargd 19956 The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 19955. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( cos `  ( Im `  ( log `  X ) ) )  =  ( ( Re `  X )  /  ( abs `  X ) ) )
 
Theoremcosarg0d 19957 The cosine of the argument is zero precisely on the imaginary axis. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( ( cos `  ( Im `  ( log `  X ) ) )  =  0  <->  ( Re `  X )  =  0
 ) )
 
Theoremargregt0 19958 Closure of the argument of a complex number with positive real part. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( ( A  e.  CC  /\  0  <  ( Re `  A ) ) 
 ->  ( Im `  ( log `  A ) )  e.  ( -u ( pi  /  2 ) (,) ( pi  /  2
 ) ) )
 
Theoremargrege0 19959 Closure of the argument of a complex number with nonnegative real part. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  0  <_  ( Re
 `  A ) ) 
 ->  ( Im `  ( log `  A ) )  e.  ( -u ( pi  /  2 ) [,] ( pi  /  2
 ) ) )
 
Theoremargimgt0 19960 Closure of the argument of a complex number with positive imaginary part. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( ( A  e.  CC  /\  0  <  ( Im `  A ) ) 
 ->  ( Im `  ( log `  A ) )  e.  ( 0 (,)
 pi ) )
 
Theoremargimlt0 19961 Closure of the argument of a complex number with negative imaginary part. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( ( A  e.  CC  /\  ( Im `  A )  <  0 ) 
 ->  ( Im `  ( log `  A ) )  e.  ( -u pi (,) 0 ) )
 
Theoremlogimul 19962 Multiplying a number by  _i increases the logarithm of the number by  _i pi  / 
2. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  0  <_  ( Re
 `  A ) ) 
 ->  ( log `  ( _i  x.  A ) )  =  ( ( log `  A )  +  ( _i  x.  ( pi  / 
 2 ) ) ) )
 
Theoremlogneg2 19963 The logarithm of the negative of a number with positive imaginary part is  _i pi less than the original. (Compare logneg 19935.) (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ( A  e.  CC  /\  0  <  ( Im `  A ) ) 
 ->  ( log `  -u A )  =  ( ( log `  A )  -  ( _i  x.  pi ) ) )
 
Theoremtanarg 19964 The basic relation between the "arg" function  Im  o.  log and the arctangent. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0
 )  ->  ( tan `  ( Im `  ( log `  A ) ) )  =  ( ( Im `  A ) 
 /  ( Re `  A ) ) )
 
Theoremlogdivlti 19965 The  log x  /  x function is strictly decreasing on the reals greater than  _e. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  ( ( log `  B )  /  B )  <  ( ( log `  A )  /  A ) )
 
Theoremlogdivlt 19966 The  log x  /  x function is strictly decreasing on the reals greater than  _e. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  ( ( ( A  e.  RR  /\  _e  <_  A )  /\  ( B  e.  RR  /\  _e  <_  B ) )  ->  ( A  <  B  <->  ( ( log `  B )  /  B )  <  ( ( log `  A )  /  A ) ) )
 
Theoremlogdivle 19967 The  log x  /  x function is strictly decreasing on the reals greater than  _e. (Contributed by Mario Carneiro, 3-May-2016.)
 |-  ( ( ( A  e.  RR  /\  _e  <_  A )  /\  ( B  e.  RR  /\  _e  <_  B ) )  ->  ( A  <_  B  <->  ( ( log `  B )  /  B )  <_  ( ( log `  A )  /  A ) ) )
 
Theoremrelogcld 19968 Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( log `  A )  e. 
 RR )
 
Theoremreeflogd 19969 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( exp `  ( log `  A ) )  =  A )
 
Theoremrelogmuld 19970 The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( log `  ( A  x.  B ) )  =  ( ( log `  A )  +  ( log `  B ) ) )
 
Theoremrelogdivd 19971 The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( log `  ( A  /  B ) )  =  ( ( log `  A )  -  ( log `  B ) ) )
 
Theoremlogled 19972 Natural logarithm preserves  <_. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( log `  A )  <_  ( log `  B ) ) )
 
Theoremrelogefd 19973 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( log `  ( exp `  A ) )  =  A )
 
Theoremrplogcld 19974 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1  <  A )   =>    |-  ( ph  ->  ( log `  A )  e.  RR+ )
 
Theoremlogge0d 19975 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1 
 <_  A )   =>    |-  ( ph  ->  0  <_  ( log `  A ) )
 
Theoremdivlogrlim 19976 The inverse logarithm function converges to zero. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( 1  /  ( log `  x ) ) )  ~~> r  0
 
Theoremlogno1 19977 The logarithm function is not eventually bounded. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 30-May-2016.)
 |- 
 -.  ( x  e.  RR+  |->  ( log `  x ) )  e.  O ( 1 )
 
Theoremdvrelog 19978 The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( RR  _D  ( log  |`  RR+ ) )  =  ( x  e.  RR+  |->  ( 1  /  x ) )
 
Theoremrelogcn 19979 The real logarithm function is continuous. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  ( log  |`  RR+ )  e.  ( RR+ -cn-> RR )
 
Theoremellogdm 19980 Elementhood in the "continuous domain" of the complex logarithm. (Contributed by Mario Carneiro, 18-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( A  e.  D  <->  ( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ )
 ) )
 
Theoremlogdmn0 19981 A number in the continuous domain of  log is nonzero. (Contributed by Mario Carneiro, 18-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( A  e.  D  ->  A  =/=  0 )
 
Theoremlogdmnrp 19982 A number in the continuous domain of  log is not a strictly negative number. (Contributed by Mario Carneiro, 18-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( A  e.  D  ->  -.  -u A  e.  RR+ )
 
Theoremlogdmss 19983 The continuity domain of  log is a subset of the regular domain of  log. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  D  C_  ( CC  \  { 0 } )
 
Theoremlogcnlem2 19984 Lemma for logcn 19988. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  if ( A  e.  RR+ ,  A ,  ( abs `  ( Im `  A ) ) )   &    |-  T  =  ( ( abs `  A )  x.  ( R  /  ( 1  +  R ) ) )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  R  e.  RR+ )   =>    |-  ( ph  ->  if ( S  <_  T ,  S ,  T )  e.  RR+ )
 
Theoremlogcnlem3 19985 Lemma for logcn 19988. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  if ( A  e.  RR+ ,  A ,  ( abs `  ( Im `  A ) ) )   &    |-  T  =  ( ( abs `  A )  x.  ( R  /  ( 1  +  R ) ) )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <  if ( S 
 <_  T ,  S ,  T ) )   =>    |-  ( ph  ->  (
 -u pi  <  (
 ( Im `  ( log `  B ) )  -  ( Im `  ( log `  A )
 ) )  /\  (
 ( Im `  ( log `  B ) )  -  ( Im `  ( log `  A )
 ) )  <_  pi ) )
 
Theoremlogcnlem4 19986 Lemma for logcn 19988. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  if ( A  e.  RR+ ,  A ,  ( abs `  ( Im `  A ) ) )   &    |-  T  =  ( ( abs `  A )  x.  ( R  /  ( 1  +  R ) ) )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <  if ( S 
 <_  T ,  S ,  T ) )   =>    |-  ( ph  ->  ( abs `  ( ( Im `  ( log `  A ) )  -  ( Im `  ( log `  B ) ) ) )  <  R )
 
Theoremlogcnlem5 19987* Lemma for logcn 19988. (Contributed by Mario Carneiro, 18-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( x  e.  D  |->  ( Im `  ( log `  x ) ) )  e.  ( D -cn-> RR )
 
Theoremlogcn 19988 The logarithm function is continuous away from the branch cut at negative reals. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( log  |`  D )  e.  ( D -cn-> CC )
 
Theoremdvloglem 19989 Lemma for dvlog 19992. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( log " D )  e.  ( TopOpen ` fld )
 
Theoremlogdmopn 19990 The "continuous domain" of  log is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  D  e.  ( TopOpen ` fld )
 
Theoremlogf1o2 19991 The logarithm maps its continuous domain bijectively onto the set of numbers with imaginary part  -u pi  <  Im ( z )  <  pi. The negative reals are mapped to the numbers with imaginary part equal to  pi. (Contributed by Mario Carneiro, 2-May-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( log  |`  D ) : D -1-1-onto-> ( `' Im "
 ( -u pi (,) pi ) )
 
Theoremdvlog 19992* The derivative of the complex logarithm function. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( CC  _D  ( log  |`  D ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
 
Theoremdvlog2lem 19993 Lemma for dvlog2 19994. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  S  =  ( 1 ( ball `  ( abs  o. 
 -  ) ) 1 )   =>    |-  S  C_  ( CC  \  (  -oo (,] 0
 ) )
 
Theoremdvlog2 19994* The derivative of the complex logarithm function on the open unit ball centered at  1, a sometimes easier region to work with than the  CC  \  (  -oo ,  0 ] of dvlog 19992. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  S  =  ( 1 ( ball `  ( abs  o. 
 -  ) ) 1 )   =>    |-  ( CC  _D  ( log  |`  S ) )  =  ( x  e.  S  |->  ( 1  /  x ) )
 
Theoremadvlog 19995 The antiderivative of the logarithm. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( RR  _D  ( x  e.  RR+  |->  ( x  x.  ( ( log `  x )  -  1
 ) ) ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
 
Theoremadvlogexp 19996* The antiderivative of a power of the logarithm. (Set  A  =  1 and multiply by  ( -u 1
) ^ N  x.  N ! to get the antiderivative of  log ( x ) ^ N itself.) (Contributed by Mario Carneiro, 22-May-2016.)
 |-  ( ( A  e.  RR+  /\  N  e.  NN0 )  ->  ( RR  _D  ( x  e.  RR+  |->  ( x  x.  sum_ k  e.  (
 0 ... N ) ( ( ( log `  ( A  /  x ) ) ^ k )  /  ( ! `  k ) ) ) ) )  =  ( x  e.  RR+  |->  ( ( ( log `  ( A  /  x ) ) ^ N )  /  ( ! `  N ) ) ) )
 
Theoremefopnlem1 19997 Lemma for efopn 19999. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  A  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  ( abs `  ( Im `  A ) )  <  pi )
 
Theoremefopnlem2 19998 Lemma for efopn 19999. (Contributed by Mario Carneiro, 2-May-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  (
 ( R  e.  RR+  /\  R  <  pi ) 
 ->  ( exp " (
 0 ( ball `  ( abs  o.  -  ) ) R ) )  e.  J )
 
Theoremefopn 19999 The exponential map is an open map. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  ( S  e.  J  ->  ( exp " S )  e.  J )
 
Theoremlogtayllem 20000* Lemma for logtayl 20001. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  ,  ( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  (
 1  /  n )
 )  x.  ( A ^ n ) ) ) )  e.  dom  ~~>  )
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