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Theorem List for Metamath Proof Explorer - 19801-19900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtanabsge 19801 The tangent function is greater than or equal to its argument in absolute value. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( A  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) )  ->  ( abs `  A )  <_  ( abs `  ( tan `  A ) ) )
 
Theoremsinq12gt0 19802 The sine of a number strictly between 
0 and  pi is positive. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  (
 0 (,) pi )  -> 
 0  <  ( sin `  A ) )
 
Theoremsinq12ge0 19803 The sine of a number between  0 and  pi is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)
 |-  ( A  e.  (
 0 [,] pi )  -> 
 0  <_  ( sin `  A ) )
 
Theoremsinq34lt0t 19804 The sine of a number strictly between  pi and  2  x.  pi is negative. (Contributed by NM, 17-Aug-2008.)
 |-  ( A  e.  ( pi (,) ( 2  x.  pi ) )  ->  ( sin `  A )  <  0 )
 
Theoremcosq14gt0 19805 The cosine of a number strictly between  -u pi  /  2 and  pi  /  2 is positive. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( A  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) )  -> 
 0  <  ( cos `  A ) )
 
Theoremcosq14ge0 19806 The cosine of a number between  -u pi  /  2 and  pi  /  2 is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)
 |-  ( A  e.  ( -u ( pi  /  2
 ) [,] ( pi  / 
 2 ) )  -> 
 0  <_  ( cos `  A ) )
 
Theoremsincosq1eq 19807 Complementarity of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 25-Jan-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  +  B )  =  1 )  ->  ( sin `  ( A  x.  ( pi  / 
 2 ) ) )  =  ( cos `  ( B  x.  ( pi  / 
 2 ) ) ) )
 
Theoremsincos4thpi 19808 The sine and cosine of  pi  /  4. (Contributed by Paul Chapman, 25-Jan-2008.)
 |-  ( ( sin `  ( pi  /  4 ) )  =  ( 1  /  ( sqr `  2 )
 )  /\  ( cos `  ( pi  /  4
 ) )  =  ( 1  /  ( sqr `  2 ) ) )
 
Theoremtan4thpi 19809 The tangent of  pi  /  4. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( tan `  ( pi  /  4 ) )  =  1
 
Theoremsincos6thpi 19810 The sine and cosine of  pi  /  6. (Contributed by Paul Chapman, 25-Jan-2008.)
 |-  ( ( sin `  ( pi  /  6 ) )  =  ( 1  / 
 2 )  /\  ( cos `  ( pi  / 
 6 ) )  =  ( ( sqr `  3
 )  /  2 )
 )
 
Theoremsincos3rdpi 19811 The sine and cosine of  pi  /  3. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( ( sin `  ( pi  /  3 ) )  =  ( ( sqr `  3 )  /  2
 )  /\  ( cos `  ( pi  /  3
 ) )  =  ( 1  /  2 ) )
 
Theorempige3 19812  pi is greater or equal to 3. This proof is based on the geometric observation that a hexagon of unit side length has perimeter 6, which is less than the unit-radius circumcircle, of perimeter  2
pi. We translate this to algebra by looking at the function  _e ^ ( _i x ) as  x goes from  0 to  pi  /  3; it moves at unit speed and travels distance  1, hence  1  <_  pi 
/  3. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  3  <_  pi
 
Theoremabssinper 19813 The absolute value of sine has period  pi. (Contributed by NM, 17-Aug-2008.)
 |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( abs `  ( sin `  ( A  +  ( K  x.  pi ) ) ) )  =  ( abs `  ( sin `  A ) ) )
 
Theoremsinkpi 19814 The sine of an integer multiple of 
pi is 0. (Contributed by NM, 11-Aug-2008.)
 |-  ( K  e.  ZZ  ->  ( sin `  ( K  x.  pi ) )  =  0 )
 
Theoremcoskpi 19815 The absolute value of the cosine of an integer multiple of  pi is 1. (Contributed by NM, 19-Aug-2008.)
 |-  ( K  e.  ZZ  ->  ( abs `  ( cos `  ( K  x.  pi ) ) )  =  1 )
 
Theoremsineq0 19816 A complex number whose sine is zero is an integer multiple of  pi. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( A  e.  CC  ->  ( ( sin `  A )  =  0  <->  ( A  /  pi )  e.  ZZ ) )
 
Theoremcoseq1 19817 A complex number whose cosine is one is an integer multiple of  2
pi. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  ( A  e.  CC  ->  ( ( cos `  A )  =  1  <->  ( A  /  ( 2  x.  pi ) )  e.  ZZ ) )
 
Theoremefeq1 19818 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 19819 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 19820 Lemma for cosord 19821. (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 19821 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 19822 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 19823 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 19824 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 19825 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 19826 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 19827.) (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( ( A  e.  ( 0 [,) ( pi  /  2 ) ) 
 /\  B  e.  (
 0 [,) ( pi  / 
 2 ) ) ) 
 ->  ( A  <  B  <->  ( tan `  A )  <  ( tan `  B ) ) )
 
Theoremtanord 19827 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 19828 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 19829  ( -u pi (,] pi ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( -u pi (,] pi )  C_  RR
 
13.3.3  Mapping of the exponential function
 
Theoremefgh 19830* 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.)
 |-  F  =  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( F `
  ( B  +  C ) )  =  ( ( F `  B )  x.  ( F `  C ) ) )
 
Theoremefif1olem1 19831* Lemma for efif1o 19835. (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 19832* Lemma for efif1o 19835. (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 19833* Lemma for efif1o 19835. (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 19834* 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 19835* 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 19836* 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 19837* 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 19838 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 19839 Extend class notation with the natural logarithm function on complex numbers.
 class  log
 
Syntaxccxp 19840 Extend class notation with the complex power function.
 class  ^ c
 
Definitiondf-log 19841 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 19842* 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 19843 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 19844 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 19845 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 19846 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 19847 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 19848 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 19849 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 19850 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 19851 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 19852 Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( log `  A )  e.  ran  log )
 
Theoremlogcl 19853 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 19854 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 19855 The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 19853. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( log `  X )  e.  CC )
 
Theoremlogimcld 19856 The imaginary part of the logarithm is in  ( -u pi (,] pi ). Deduction form of logimcl 19854. Compare logimclad 19857. (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 19857 The imaginary part of the logarithm is in  ( -u pi (,] pi ). Alternate form of logimcld 19856. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( Im `  ( log `  X ) )  e.  ( -u pi (,] pi ) )
 
Theoremlogrnaddcl 19858 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 19859 Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
 
Theoremeflog 19860 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 19861 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 19862 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 19863 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 19864 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 19865 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 19866 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 19867 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 19868 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 19869 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 19870 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 19871 Lemma for relogmul 19872 and relogdiv 19873. 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 19872 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 19873 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 19874 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 19875 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 19876 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 19877 Real part of a logarithm. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( Re `  ( log `  A ) )  =  ( log `  ( abs `  A ) ) )
 
Theoremrelogiso 19878 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 19879 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 19880 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 19881* 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 19882* 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 19883 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 19884 Natural logarithm preserves  <_. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <_  B  <->  ( log `  A )  <_  ( log `  B ) ) )
 
Theoremrplogcl 19885 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 19886 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 19887 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 19888 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 19889 The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 19888. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( cos `  ( Im `  ( log `  X ) ) )  =  ( ( Re `  X )  /  ( abs `  X ) ) )
 
Theoremcosarg0d 19890 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 19891 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 19892 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 19893 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 19894 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 19895 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 19896 The logarithm of the negative of a number with positive imaginary part is  _i pi less than the original. (Compare logneg 19868.) (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ( A  e.  CC  /\  0  <  ( Im `  A ) ) 
 ->  ( log `  -u A )  =  ( ( log `  A )  -  ( _i  x.  pi ) ) )
 
Theoremtanarg 19897 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 19898 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 19899 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 19900 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 ) ) )
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