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Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsincosq4sgn 20401 The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  (
 ( 3  x.  ( pi  /  2 ) ) (,) ( 2  x.  pi ) )  ->  ( ( sin `  A )  <  0  /\  0  <  ( cos `  A ) ) )
 
Theoremcoseq00topi 20402 Location of the zeroes of cosine in 
( 0 [,] pi ). (Contributed by David Moews, 28-Feb-2017.)
 |-  ( A  e.  (
 0 [,] pi )  ->  ( ( cos `  A )  =  0  <->  A  =  ( pi  /  2 ) ) )
 
Theoremcoseq0negpitopi 20403 Location of the zeroes of cosine in 
( -u pi (,] pi ). (Contributed by David Moews, 28-Feb-2017.)
 |-  ( A  e.  ( -u pi (,] pi ) 
 ->  ( ( cos `  A )  =  0  <->  A  e.  { ( pi  /  2 ) ,  -u ( pi  /  2
 ) } ) )
 
Theoremtanrpcl 20404 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( A  e.  (
 0 (,) ( pi  / 
 2 ) )  ->  ( tan `  A )  e.  RR+ )
 
Theoremtangtx 20405 The tangent function is greater than its argument on positive reals in its principal domain. (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( A  e.  (
 0 (,) ( pi  / 
 2 ) )  ->  A  <  ( tan `  A ) )
 
Theoremtanabsge 20406 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 20407 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 20408 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 20409 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 20410 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 20411 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 20412 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 20413 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 20414 The tangent of  pi  /  4. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( tan `  ( pi  /  4 ) )  =  1
 
Theoremsincos6thpi 20415 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 20416 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 20417  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 20418 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 20419 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 20420 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 20421 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 20422 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 20423 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 20424 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 20425 Lemma for cosord 20426. (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 20426 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 20427 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 20428 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 20429 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 20430 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 20431 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 20432.) (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( ( A  e.  ( 0 [,) ( pi  /  2 ) ) 
 /\  B  e.  (
 0 [,) ( pi  / 
 2 ) ) ) 
 ->  ( A  <  B  <->  ( tan `  A )  <  ( tan `  B ) ) )
 
Theoremtanord 20432 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 20433 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 20434  ( -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 20435* 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 20436* Lemma for efif1o 20440. (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 20437* Lemma for efif1o 20440. (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 20438* Lemma for efif1o 20440. (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 20439* 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 20440* 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 20441* 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 20442* 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 20443 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 20444 Extend class notation with the natural logarithm function on complex numbers.
 class  log
 
Syntaxccxp 20445 Extend class notation with the complex power function.
 class  ^ c
 
Definitiondf-log 20446 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 20447* 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 20448 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 20449 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 20450 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 20451 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 20452 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 20453 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 20454 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 20455 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 20456 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 20457 Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( log `  A )  e.  ran  log )
 
Theoremlogcl 20458 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 20459 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 20460 The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 20458. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( log `  X )  e.  CC )
 
Theoremlogimcld 20461 The imaginary part of the logarithm is in  ( -u pi (,] pi ). Deduction form of logimcl 20459. Compare logimclad 20462. (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 20462 The imaginary part of the logarithm is in  ( -u pi (,] pi ). Alternate form of logimcld 20461. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( Im `  ( log `  X ) )  e.  ( -u pi (,] pi ) )
 
Theoremabslogimle 20463 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 20464 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 20465 Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
 
Theoremeflog 20466 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 20467 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 20468 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 20469 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 20470 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 20471 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 20472 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 20473 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 20474 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 20475 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 20476 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 20477 Lemma for relogmul 20478 and relogdiv 20479. 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 20478 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 20479 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 20480 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 20481 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 20482 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 20483 Real part of a logarithm. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( Re `  ( log `  A ) )  =  ( log `  ( abs `  A ) ) )
 
Theoremrelogiso 20484 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 20485 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 20486 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 20487* 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 20488* 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 20489 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 20490 Natural logarithm preserves  <_. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <_  B  <->  ( log `  A )  <_  ( log `  B ) ) )
 
Theoremrplogcl 20491 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 20492 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 20493 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 20494 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 20495 The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 20494. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( cos `  ( Im `  ( log `  X ) ) )  =  ( ( Re `  X )  /  ( abs `  X ) ) )
 
Theoremcosarg0d 20496 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 20497 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 20498 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 20499 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 20500 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 ) )
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