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Theorem List for Metamath Proof Explorer - 19801-19900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcosneghalfpi 19801 The cosine of  -u pi  /  2 is zero. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( cos `  -u ( pi  /  2 ) )  =  0
 
Theoremefhalfpi 19802 The exponential of  _i pi  /  2 is  _i. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( exp `  ( _i  x.  ( pi  / 
 2 ) ) )  =  _i
 
Theoremcospi 19803 The cosine of  pi is  -u 1. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( cos `  pi )  =  -u 1
 
Theoremefipi 19804 The exponential of  _i pi. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( exp `  ( _i  x.  pi ) )  =  -u 1
 
Theoremeulerid 19805 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  ( ( exp `  ( _i  x.  pi ) )  +  1 )  =  0
 
Theoremsin2pi 19806 The sine of  2 pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( sin `  (
 2  x.  pi ) )  =  0
 
Theoremcos2pi 19807 The cosine of  2 pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( cos `  (
 2  x.  pi ) )  =  1
 
Theoremef2pi 19808 The exponential of  2 pi _i is  1. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( exp `  ( _i  x.  ( 2  x.  pi ) ) )  =  1
 
Theoremef2kpi 19809 The exponential of  2 K pi _i is  1. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( K  e.  ZZ  ->  ( exp `  (
 ( _i  x.  (
 2  x.  pi ) )  x.  K ) )  =  1 )
 
Theoremefper 19810 The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)
 |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  ( A  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) ) )  =  ( exp `  A ) )
 
Theoremsinperlem 19811 Lemma for sinper 19812 and cosper 19813. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( A  e.  CC  ->  ( F `  A )  =  ( (
 ( exp `  ( _i  x.  A ) ) O ( exp `  ( -u _i  x.  A ) ) )  /  D ) )   &    |-  ( ( A  +  ( K  x.  ( 2  x.  pi ) ) )  e. 
 CC  ->  ( F `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( ( ( exp `  ( _i  x.  ( A  +  ( K  x.  (
 2  x.  pi ) ) ) ) ) O ( exp `  ( -u _i  x.  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) ) ) )  /  D ) )   =>    |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( F `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( F `
  A ) )
 
Theoremsinper 19812 The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( sin `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( sin `  A ) )
 
Theoremcosper 19813 The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( cos `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( cos `  A ) )
 
Theoremsin2kpi 19814 If  K is an integer, the sine of  2 K pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( K  e.  ZZ  ->  ( sin `  ( K  x.  ( 2  x.  pi ) ) )  =  0 )
 
Theoremcos2kpi 19815 If  K is an integer, the cosine of  2 K pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( K  e.  ZZ  ->  ( cos `  ( K  x.  ( 2  x.  pi ) ) )  =  1 )
 
Theoremsin2pim 19816 Sine of a number subtracted from  2  x.  pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( sin `  (
 ( 2  x.  pi )  -  A ) )  =  -u ( sin `  A ) )
 
Theoremcos2pim 19817 Cosine of a number subtracted from  2  x.  pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( cos `  (
 ( 2  x.  pi )  -  A ) )  =  ( cos `  A ) )
 
Theoremsinmpi 19818 Sine of a number less  pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( sin `  ( A  -  pi ) )  =  -u ( sin `  A ) )
 
Theoremcosmpi 19819 Cosine of a number less  pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( cos `  ( A  -  pi ) )  =  -u ( cos `  A ) )
 
Theoremsinppi 19820 Sine of a number plus  pi. (Contributed by NM, 10-Aug-2008.)
 |-  ( A  e.  CC  ->  ( sin `  ( A  +  pi )
 )  =  -u ( sin `  A ) )
 
Theoremcosppi 19821 Cosine of a complex number plus  pi. (Contributed by NM, 18-Aug-2008.)
 |-  ( A  e.  CC  ->  ( cos `  ( A  +  pi )
 )  =  -u ( cos `  A ) )
 
Theoremefimpi 19822 The exponential function of  _i times a real number less 
pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( A  -  pi ) ) )  =  -u ( exp `  ( _i  x.  A ) ) )
 
Theoremsinhalfpip 19823 The sine of  pi  /  2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  CC  ->  ( sin `  (
 ( pi  /  2
 )  +  A ) )  =  ( cos `  A ) )
 
Theoremsinhalfpim 19824 The sine of  pi  /  2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  CC  ->  ( sin `  (
 ( pi  /  2
 )  -  A ) )  =  ( cos `  A ) )
 
Theoremcoshalfpip 19825 The cosine of  pi  /  2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  CC  ->  ( cos `  (
 ( pi  /  2
 )  +  A ) )  =  -u ( sin `  A ) )
 
Theoremcoshalfpim 19826 The cosine of  pi  /  2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  CC  ->  ( cos `  (
 ( pi  /  2
 )  -  A ) )  =  ( sin `  A ) )
 
Theoremptolemy 19827 Ptolemy's Theorem. This theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). This particular version is expressed using the sine function. It is proved by expanding all the multiplication of sines to a product of cosines of differences using sinmul 12415, then using algebraic simplification to show that both sides are equal. This formalization is based on the proof in "Trigonometry" by Gelfand and Saul. (Contributed by David A. Wheeler, 31-May-2015.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC )  /\  (
 ( A  +  B )  +  ( C  +  D ) )  =  pi )  ->  (
 ( ( sin `  A )  x.  ( sin `  B ) )  +  (
 ( sin `  C )  x.  ( sin `  D ) ) )  =  ( ( sin `  ( B  +  C )
 )  x.  ( sin `  ( A  +  C ) ) ) )
 
Theoremsincosq1lem 19828 Lemma for sincosq1sgn 19829. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  ( pi 
 /  2 ) ) 
 ->  0  <  ( sin `  A ) )
 
Theoremsincosq1sgn 19829 The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  (
 0 (,) ( pi  / 
 2 ) )  ->  ( 0  <  ( sin `  A )  /\  0  <  ( cos `  A ) ) )
 
Theoremsincosq2sgn 19830 The signs of the sine and cosine functions in the second quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  (
 ( pi  /  2
 ) (,) pi )  ->  ( 0  <  ( sin `  A )  /\  ( cos `  A )  <  0 ) )
 
Theoremsincosq3sgn 19831 The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  ( pi (,) ( 3  x.  ( pi  /  2
 ) ) )  ->  ( ( sin `  A )  <  0  /\  ( cos `  A )  < 
 0 ) )
 
Theoremsincosq4sgn 19832 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 19833 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 19834 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 19835 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( A  e.  (
 0 (,) ( pi  / 
 2 ) )  ->  ( tan `  A )  e.  RR+ )
 
Theoremtangtx 19836 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 19837 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 19838 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 19839 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 19840 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 19841 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 19842 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 19843 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 19844 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 19845 The tangent of  pi  /  4. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( tan `  ( pi  /  4 ) )  =  1
 
Theoremsincos6thpi 19846 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 19847 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 19848  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 19849 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 19850 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 19851 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 19852 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 19853 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 19854 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 19855 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 19856 Lemma for cosord 19857. (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 19857 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 19858 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 19859 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 19860 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 19861 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 19862 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 19863.) (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( ( A  e.  ( 0 [,) ( pi  /  2 ) ) 
 /\  B  e.  (
 0 [,) ( pi  / 
 2 ) ) ) 
 ->  ( A  <  B  <->  ( tan `  A )  <  ( tan `  B ) ) )
 
Theoremtanord 19863 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 19864 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 19865  ( -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 19866* 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 19867* Lemma for efif1o 19871. (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 19868* Lemma for efif1o 19871. (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 19869* Lemma for efif1o 19871. (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 19870* 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 19871* 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 19872* 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 19873* 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 19874 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 19875 Extend class notation with the natural logarithm function on complex numbers.
 class  log
 
Syntaxccxp 19876 Extend class notation with the complex power function.
 class  ^ c
 
Definitiondf-log 19877 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 19878* 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 19879 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 19880 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 19881 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 19882 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 19883 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 19884 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 19885 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 19886 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 19887 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 19888 Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( log `  A )  e.  ran  log )
 
Theoremlogcl 19889 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 19890 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 19891 The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 19889. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( log `  X )  e.  CC )
 
Theoremlogimcld 19892 The imaginary part of the logarithm is in  ( -u pi (,] pi ). Deduction form of logimcl 19890. Compare logimclad 19893. (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 19893 The imaginary part of the logarithm is in  ( -u pi (,] pi ). Alternate form of logimcld 19892. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( Im `  ( log `  X ) )  e.  ( -u pi (,] pi ) )
 
Theoremlogrnaddcl 19894 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 19895 Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
 
Theoremeflog 19896 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 19897 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 19898 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 19899 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 19900 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 ) )
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