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Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremefeq1 23001 A complex number whose exponential is one is an integer multiple of  2 pi _i. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( A  e.  CC  ->  ( ( exp `  A )  =  1  <->  ( A  /  ( _i  x.  (
 2  x.  pi ) ) )  e.  ZZ ) )
 
Theoremcosne0 23002 The cosine function has no zeroes within the vertical strip of the complex plane between real part 
-u pi  /  2 and  pi  /  2. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) ) ) 
 ->  ( cos `  A )  =/=  0 )
 
Theoremcosordlem 23003 Lemma for cosord 23004. (Contributed by Mario Carneiro, 10-May-2014.)
 |-  ( ph  ->  A  e.  ( 0 [,] pi ) )   &    |-  ( ph  ->  B  e.  ( 0 [,]
 pi ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( cos `  B )  < 
 ( cos `  A )
 )
 
Theoremcosord 23004 Cosine is decreasing over the closed interval from  0 to  pi. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)
 |-  ( ( A  e.  ( 0 [,] pi )  /\  B  e.  (
 0 [,] pi ) ) 
 ->  ( A  <  B  <->  ( cos `  B )  <  ( cos `  A ) ) )
 
Theoremcos11 23005 Cosine is one-to-one over the closed interval from  0 to  pi. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)
 |-  ( ( A  e.  ( 0 [,] pi )  /\  B  e.  (
 0 [,] pi ) ) 
 ->  ( A  =  B  <->  ( cos `  A )  =  ( cos `  B ) ) )
 
Theoremsinord 23006 Sine is increasing over the closed interval from  -u ( pi  /  2
) to  ( pi  /  2 ). (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( ( A  e.  ( -u ( pi  / 
 2 ) [,] ( pi  /  2 ) ) 
 /\  B  e.  ( -u ( pi  /  2
 ) [,] ( pi  / 
 2 ) ) ) 
 ->  ( A  <  B  <->  ( sin `  A )  <  ( sin `  B ) ) )
 
Theoremrecosf1o 23007 The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,]
 pi ) -1-1-onto-> ( -u 1 [,] 1
 )
 
Theoremresinf1o 23008 The sine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  ( sin  |`  ( -u ( pi  /  2
 ) [,] ( pi  / 
 2 ) ) ) : ( -u ( pi  /  2 ) [,] ( pi  /  2
 ) ) -1-1-onto-> ( -u 1 [,] 1
 )
 
Theoremtanord1 23009 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 23010.) (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( ( A  e.  ( 0 [,) ( pi  /  2 ) ) 
 /\  B  e.  (
 0 [,) ( pi  / 
 2 ) ) ) 
 ->  ( A  <  B  <->  ( tan `  A )  <  ( tan `  B ) ) )
 
Theoremtanord 23010 The tangent function is strictly increasing on its principal domain. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( ( A  e.  ( -u ( pi  / 
 2 ) (,) ( pi  /  2 ) ) 
 /\  B  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) ) ) 
 ->  ( A  <  B  <->  ( tan `  A )  <  ( tan `  B ) ) )
 
Theoremtanregt0 23011 The positivity of  tan ( A ) extends to complex numbers with the same real part. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  e.  (
 0 (,) ( pi  / 
 2 ) ) ) 
 ->  0  <  ( Re
 `  ( tan `  A ) ) )
 
Theoremnegpitopissre 23012  ( -u pi (,] pi ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( -u pi (,] pi )  C_  RR
 
14.3.3  Mapping of the exponential function
 
Theoremefgh 23013* The exponential function of a scaled complex number is a group homomorphism from the group of complex numbers under addition to the set of complex numbers under multiplication. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 11-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
 |-  F  =  ( x  e.  X  |->  ( exp `  ( A  x.  x ) ) )   =>    |-  ( ( ( A  e.  CC  /\  X  e.  (SubGrp ` fld ) )  /\  B  e.  X  /\  C  e.  X )  ->  ( F `
  ( B  +  C ) )  =  ( ( F `  B )  x.  ( F `  C ) ) )
 
Theoremefif1olem1 23014* Lemma for efif1o 23018. (Contributed by Mario Carneiro, 13-May-2014.)
 |-  D  =  ( A (,] ( A  +  ( 2  x.  pi ) ) )   =>    |-  ( ( A  e.  RR  /\  ( x  e.  D  /\  y  e.  D )
 )  ->  ( abs `  ( x  -  y
 ) )  <  (
 2  x.  pi ) )
 
Theoremefif1olem2 23015* Lemma for efif1o 23018. (Contributed by Mario Carneiro, 13-May-2014.)
 |-  D  =  ( A (,] ( A  +  ( 2  x.  pi ) ) )   =>    |-  ( ( A  e.  RR  /\  z  e.  RR )  ->  E. y  e.  D  ( ( z  -  y )  /  ( 2  x.  pi ) )  e.  ZZ )
 
Theoremefif1olem3 23016* Lemma for efif1o 23018. (Contributed by Mario Carneiro, 8-May-2015.)
 |-  F  =  ( w  e.  D  |->  ( exp `  ( _i  x.  w ) ) )   &    |-  C  =  ( `' abs " {
 1 } )   =>    |-  ( ( ph  /\  x  e.  C ) 
 ->  ( Im `  ( sqr `  x ) )  e.  ( -u 1 [,] 1 ) )
 
Theoremefif1olem4 23017* 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 23018* 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 23019* 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 23020* 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 23021 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 } )
 
Theoremefabl 23022* The image of a subgroup of the group  +, under the exponential function of a scaled complex number, is an Abelian group. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
 |-  F  =  ( x  e.  X  |->  ( exp `  ( A  x.  x ) ) )   &    |-  G  =  ( (mulGrp ` fld )s  ran  F )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  X  e.  (SubGrp ` fld ) )   =>    |-  ( ph  ->  G  e.  Abel )
 
Theoremefsubm 23023* The image of a subgroup of the group  +, under the exponential function of a scaled complex number is a submonoid of the multiplicative group of ℂfld. (Contributed by Thierry Arnoux, 26-Jan-2020.)
 |-  F  =  ( x  e.  X  |->  ( exp `  ( A  x.  x ) ) )   &    |-  G  =  ( (mulGrp ` fld )s  ran  F )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  X  e.  (SubGrp ` fld ) )   =>    |-  ( ph  ->  ran  F  e.  (SubMnd `  (mulGrp ` fld ) ) )
 
Theoremcircgrp 23024 The circle group  T is an Abelian group. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
 |-  C  =  ( `'
 abs " { 1 } )   &    |-  T  =  ( (mulGrp ` fld )s  C )   =>    |-  T  e.  Abel
 
Theoremcircsubm 23025 The circle group  T is a submonoid of the multiplicative group of ℂfld. (Contributed by Thierry Arnoux, 26-Jan-2020.)
 |-  C  =  ( `'
 abs " { 1 } )   &    |-  T  =  ( (mulGrp ` fld )s  C )   =>    |-  C  e.  (SubMnd `  (mulGrp ` fld ) )
 
Theoremrzgrp 23026 The quotient group R/Z is a group. (Contributed by Thierry Arnoux, 26-Jan-2020.)
 |-  R  =  (RRfld  /.s  (RRfld ~QG  ZZ )
 )   =>    |-  R  e.  Grp
 
14.3.4  The natural logarithm on complex numbers
 
Syntaxclog 23027 Extend class notation with the natural logarithm function on complex numbers.
 class  log
 
Syntaxccxp 23028 Extend class notation with the complex power function.
 class  ^c
 
Definitiondf-log 23029 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"). To obtain a function, only the principle value of the multivalued inverses of the exponential function, i.e. the inverse whose imaginary part lies in the interval (-pi, pi], see https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Paul Chapman, 21-Apr-2008.)
 |- 
 log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
 
Definitiondf-cxp 23030* 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 23031 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 23032 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 23033 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 23034 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 23035 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 23036 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 23037 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 23038 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 23039 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 23040 Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( log `  A )  e.  ran  log )
 
Theoremlogcl 23041 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 23042 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 23043 The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 23041. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( log `  X )  e.  CC )
 
Theoremlogimcld 23044 The imaginary part of the logarithm is in  ( -u pi (,] pi ). Deduction form of logimcl 23042. Compare logimclad 23045. (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 23045 The imaginary part of the logarithm is in  ( -u pi (,] pi ). Alternate form of logimcld 23044. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( Im `  ( log `  X ) )  e.  ( -u pi (,] pi ) )
 
Theoremabslogimle 23046 The imaginary part of the logarithm function has absolute value less than pi. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( abs `  ( Im `  ( log `  A ) ) )  <_  pi )
 
Theoremlogrnaddcl 23047 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 23048 Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
 
Theoremeflog 23049 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 )
 
Theoremlogeq0im1 23050 If the logarithm of a number is 0, the number must be 1. (Contributed by David A. Wheeler, 22-Jul-2017.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  ( log `  A )  =  0 )  ->  A  =  1 )
 
Theoremlogccne0 23051 The logarithm isn't 0 if its argument isn't 0 or 1. (Contributed by David A. Wheeler, 17-Jul-2017.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 ) 
 ->  ( log `  A )  =/=  0 )
 
Theoremlogne0 23052 Logarithm of a non-1 positive real number is not zero and thus suitable as a divisor. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 14-Jun-2020.)
 |-  ( ( A  e.  RR+  /\  A  =/=  1 ) 
 ->  ( log `  A )  =/=  0 )
 
Theoremreeflog 23053 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 23054 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 23055 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 23056 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 23057 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 23058 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 23059 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 23060 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 23061 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 23062 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 23063 Lemma for relogmul 23064 and relogdiv 23065. 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 23064 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 23065 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 23066 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 23067 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 23068 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 23069 Real part of a logarithm. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( Re `  ( log `  A ) )  =  ( log `  ( abs `  A ) ) )
 
Theoremrelogiso 23070 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 23071 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.) (Revised by Thierry Arnoux, 30-Jun-2019.)
 |-  P  =  ( (mulGrp ` fld )s  RR+ )   =>    |-  ( log  |`  RR+ )  e.  ( P GrpIso RRfld )
 
Theoremlogltb 23072 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 23073* 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 23074* 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 ) ) ) )
 
Theoremlogleb 23075 Natural logarithm preserves  <_. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <_  B  <->  ( log `  A )  <_  ( log `  B ) ) )
 
Theoremrplogcl 23076 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 23077 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 23078 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 23079 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 23080 The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 23079. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( cos `  ( Im `  ( log `  X ) ) )  =  ( ( Re `  X )  /  ( abs `  X ) ) )
 
Theoremcosarg0d 23081 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 23082 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 23083 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 23084 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 23085 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 23086 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 23087 The logarithm of the negative of a number with positive imaginary part is  _i  x.  pi less than the original. (Compare logneg 23060.) (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ( A  e.  CC  /\  0  <  ( Im `  A ) ) 
 ->  ( log `  -u A )  =  ( ( log `  A )  -  ( _i  x.  pi ) ) )
 
Theoremlogmul2 23088 Generalization of relogmul 23064 to a complex left argument. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  RR+ )  ->  ( log `  ( A  x.  B ) )  =  ( ( log `  A )  +  ( log `  B ) ) )
 
Theoremlogdiv2 23089 Generalization of relogdiv 23065 to a complex left argument. (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  RR+ )  ->  ( log `  ( A  /  B ) )  =  ( ( log `  A )  -  ( log `  B ) ) )
 
Theoremabslogle 23090 Bound on the magnitude of the complex logarithm function. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( abs `  ( log `  A ) )  <_  ( ( abs `  ( log `  ( abs `  A ) ) )  +  pi ) )
 
Theoremtanarg 23091 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 23092 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 23093 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 23094 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 23095 Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( log `  A )  e. 
 RR )
 
Theoremreeflogd 23096 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 23097 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 23098 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 23099 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 23100 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 )
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