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Theorem List for Metamath Proof Explorer - 19901-20000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlogltb 19901 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 19902* 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 19903* 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 19904 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 19905 Natural logarithm preserves  <_. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <_  B  <->  ( log `  A )  <_  ( log `  B ) ) )
 
Theoremrplogcl 19906 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 19907 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 19908 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 19909 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 19910 The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 19909. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   =>    |-  ( ph  ->  ( cos `  ( Im `  ( log `  X ) ) )  =  ( ( Re `  X )  /  ( abs `  X ) ) )
 
Theoremcosarg0d 19911 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 19912 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 19913 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 19914 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 19915 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 19916 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 19917 The logarithm of the negative of a number with positive imaginary part is  _i pi less than the original. (Compare logneg 19889.) (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ( A  e.  CC  /\  0  <  ( Im `  A ) ) 
 ->  ( log `  -u A )  =  ( ( log `  A )  -  ( _i  x.  pi ) ) )
 
Theoremtanarg 19918 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 19919 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 19920 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 19921 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 19922 Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( log `  A )  e. 
 RR )
 
Theoremreeflogd 19923 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 19924 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 19925 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 19926 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 19927 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( log `  ( exp `  A ) )  =  A )
 
Theoremrplogcld 19928 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1  <  A )   =>    |-  ( ph  ->  ( log `  A )  e.  RR+ )
 
Theoremlogge0d 19929 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1 
 <_  A )   =>    |-  ( ph  ->  0  <_  ( log `  A ) )
 
Theoremdivlogrlim 19930 The inverse logarithm function converges to zero. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( 1  /  ( log `  x ) ) )  ~~> r  0
 
Theoremlogno1 19931 The logarithm function is not eventually bounded. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 30-May-2016.)
 |- 
 -.  ( x  e.  RR+  |->  ( log `  x ) )  e.  O ( 1 )
 
Theoremdvrelog 19932 The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( RR  _D  ( log  |`  RR+ ) )  =  ( x  e.  RR+  |->  ( 1  /  x ) )
 
Theoremrelogcn 19933 The real logarithm function is continuous. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  ( log  |`  RR+ )  e.  ( RR+ -cn-> RR )
 
Theoremellogdm 19934 Elementhood in the "continuous domain" of the complex logarithm. (Contributed by Mario Carneiro, 18-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( A  e.  D  <->  ( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ )
 ) )
 
Theoremlogdmn0 19935 A number in the continuous domain of  log is nonzero. (Contributed by Mario Carneiro, 18-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( A  e.  D  ->  A  =/=  0 )
 
Theoremlogdmnrp 19936 A number in the continuous domain of  log is not a strictly negative number. (Contributed by Mario Carneiro, 18-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( A  e.  D  ->  -.  -u A  e.  RR+ )
 
Theoremlogdmss 19937 The continuity domain of  log is a subset of the regular domain of  log. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  D  C_  ( CC  \  { 0 } )
 
Theoremlogcnlem2 19938 Lemma for logcn 19942. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  if ( A  e.  RR+ ,  A ,  ( abs `  ( Im `  A ) ) )   &    |-  T  =  ( ( abs `  A )  x.  ( R  /  ( 1  +  R ) ) )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  R  e.  RR+ )   =>    |-  ( ph  ->  if ( S  <_  T ,  S ,  T )  e.  RR+ )
 
Theoremlogcnlem3 19939 Lemma for logcn 19942. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  if ( A  e.  RR+ ,  A ,  ( abs `  ( Im `  A ) ) )   &    |-  T  =  ( ( abs `  A )  x.  ( R  /  ( 1  +  R ) ) )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <  if ( S 
 <_  T ,  S ,  T ) )   =>    |-  ( ph  ->  (
 -u pi  <  (
 ( Im `  ( log `  B ) )  -  ( Im `  ( log `  A )
 ) )  /\  (
 ( Im `  ( log `  B ) )  -  ( Im `  ( log `  A )
 ) )  <_  pi ) )
 
Theoremlogcnlem4 19940 Lemma for logcn 19942. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  if ( A  e.  RR+ ,  A ,  ( abs `  ( Im `  A ) ) )   &    |-  T  =  ( ( abs `  A )  x.  ( R  /  ( 1  +  R ) ) )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <  if ( S 
 <_  T ,  S ,  T ) )   =>    |-  ( ph  ->  ( abs `  ( ( Im `  ( log `  A ) )  -  ( Im `  ( log `  B ) ) ) )  <  R )
 
Theoremlogcnlem5 19941* Lemma for logcn 19942. (Contributed by Mario Carneiro, 18-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( x  e.  D  |->  ( Im `  ( log `  x ) ) )  e.  ( D -cn-> RR )
 
Theoremlogcn 19942 The logarithm function is continuous away from the branch cut at negative reals. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( log  |`  D )  e.  ( D -cn-> CC )
 
Theoremdvloglem 19943 Lemma for dvlog 19946. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( log " D )  e.  ( TopOpen ` fld )
 
Theoremlogdmopn 19944 The "continuous domain" of  log is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  D  e.  ( TopOpen ` fld )
 
Theoremlogf1o2 19945 The logarithm maps its continuous domain bijectively onto the set of numbers with imaginary part  -u pi  <  Im ( z )  <  pi. The negative reals are mapped to the numbers with imaginary part equal to  pi. (Contributed by Mario Carneiro, 2-May-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( log  |`  D ) : D -1-1-onto-> ( `' Im "
 ( -u pi (,) pi ) )
 
Theoremdvlog 19946* The derivative of the complex logarithm function. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( CC  _D  ( log  |`  D ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
 
Theoremdvlog2lem 19947 Lemma for dvlog2 19948. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  S  =  ( 1 ( ball `  ( abs  o. 
 -  ) ) 1 )   =>    |-  S  C_  ( CC  \  (  -oo (,] 0
 ) )
 
Theoremdvlog2 19948* The derivative of the complex logarithm function on the open unit ball centered at  1, a sometimes easier region to work with than the  CC  \  (  -oo ,  0 ] of dvlog 19946. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  S  =  ( 1 ( ball `  ( abs  o. 
 -  ) ) 1 )   =>    |-  ( CC  _D  ( log  |`  S ) )  =  ( x  e.  S  |->  ( 1  /  x ) )
 
Theoremadvlog 19949 The antiderivative of the logarithm. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( RR  _D  ( x  e.  RR+  |->  ( x  x.  ( ( log `  x )  -  1
 ) ) ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
 
Theoremadvlogexp 19950* The antiderivative of a power of the logarithm. (Set  A  =  1 and multiply by  ( -u 1
) ^ N  x.  N ! to get the antiderivative of  log ( x ) ^ N itself.) (Contributed by Mario Carneiro, 22-May-2016.)
 |-  ( ( A  e.  RR+  /\  N  e.  NN0 )  ->  ( RR  _D  ( x  e.  RR+  |->  ( x  x.  sum_ k  e.  (
 0 ... N ) ( ( ( log `  ( A  /  x ) ) ^ k )  /  ( ! `  k ) ) ) ) )  =  ( x  e.  RR+  |->  ( ( ( log `  ( A  /  x ) ) ^ N )  /  ( ! `  N ) ) ) )
 
Theoremefopnlem1 19951 Lemma for efopn 19953. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  A  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  ( abs `  ( Im `  A ) )  <  pi )
 
Theoremefopnlem2 19952 Lemma for efopn 19953. (Contributed by Mario Carneiro, 2-May-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  (
 ( R  e.  RR+  /\  R  <  pi ) 
 ->  ( exp " (
 0 ( ball `  ( abs  o.  -  ) ) R ) )  e.  J )
 
Theoremefopn 19953 The exponential map is an open map. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  ( S  e.  J  ->  ( exp " S )  e.  J )
 
Theoremlogtayllem 19954* Lemma for logtayl 19955. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  ,  ( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  (
 1  /  n )
 )  x.  ( A ^ n ) ) ) )  e.  dom  ~~>  )
 
Theoremlogtayl 19955* The Taylor series for  -u log ( 1  -  A ). (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  1 (  +  ,  ( k  e.  NN  |->  ( ( A ^
 k )  /  k
 ) ) )  ~~>  -u ( log `  ( 1  -  A ) ) )
 
Theoremlogtaylsum 19956* The Taylor series for  -u log ( 1  -  A ), as an infinite sum. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  sum_ k  e.  NN  (
 ( A ^ k
 )  /  k )  =  -u ( log `  (
 1  -  A ) ) )
 
Theoremlogtayl2 19957* Power series expression for the logarithm. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  S  =  ( 1 ( ball `  ( abs  o. 
 -  ) ) 1 )   =>    |-  ( A  e.  S  ->  seq  1 (  +  ,  ( k  e.  NN  |->  ( ( ( -u 1 ^ ( k  -  1 ) )  /  k )  x.  (
 ( A  -  1
 ) ^ k ) ) ) )  ~~>  ( log `  A ) )
 
Theoremlogccv 19958 The natural logarithm function on the reals is a strictly concave function. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B ) 
 /\  T  e.  (
 0 (,) 1 ) ) 
 ->  ( ( T  x.  ( log `  A )
 )  +  ( ( 1  -  T )  x.  ( log `  B ) ) )  < 
 ( log `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) ) ) )
 
Theoremcxpval 19959 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 , 
 0 ) ,  ( exp `  ( B  x.  ( log `  A )
 ) ) ) )
 
Theoremcxpef 19960 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A )
 ) ) )
 
Theorem0cxp 19961 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( 0  ^ c  A )  =  0 )
 
Theoremcxpexpz 19962 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  ZZ )  ->  ( A  ^ c  B )  =  ( A ^ B ) )
 
Theoremcxpexp 19963 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  NN0 )  ->  ( A  ^ c  B )  =  ( A ^ B ) )
 
Theoremlogcxp 19964 Logarithm of a complex power. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR )  ->  ( log `  ( A  ^ c  B ) )  =  ( B  x.  ( log `  A ) ) )
 
Theoremcxp0 19965 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( A  ^ c 
 0 )  =  1 )
 
Theoremcxp1 19966 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( A  ^ c 
 1 )  =  A )
 
Theorem1cxp 19967 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( 1  ^ c  A )  =  1
 )
 
Theoremecxp 19968 Write the exponential function as an exponent to the power  _e. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( _e  ^ c  A )  =  ( exp `  A ) )
 
Theoremcxpcl 19969 Closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c  B )  e.  CC )
 
Theoremrecxpcl 19970 Real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  RR  /\  0  <_  A  /\  B  e.  RR )  ->  ( A  ^ c  B )  e.  RR )
 
Theoremrpcxpcl 19971 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR )  ->  ( A  ^ c  B )  e.  RR+ )
 
Theoremcxpne0 19972 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  B )  =/=  0
 )
 
Theoremcxpeq0 19973 Complex exponentiation is zero iff the mantissa is zero and the exponent is nonzero. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A 
 ^ c  B )  =  0  <->  ( A  =  0  /\  B  =/=  0
 ) ) )
 
Theoremcxpadd 19974 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A 
 ^ c  ( B  +  C ) )  =  ( ( A 
 ^ c  B )  x.  ( A  ^ c  C ) ) )
 
Theoremcxpp1 19975 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  ( B  +  1
 ) )  =  ( ( A  ^ c  B )  x.  A ) )
 
Theoremcxpneg 19976 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  -u B )  =  ( 1  /  ( A 
 ^ c  B ) ) )
 
Theoremcxpsub 19977 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A 
 ^ c  ( B  -  C ) )  =  ( ( A 
 ^ c  B ) 
 /  ( A  ^ c  C ) ) )
 
Theoremcxpge0 19978 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  RR  /\  0  <_  A  /\  B  e.  RR )  ->  0  <_  ( A  ^ c  B ) )
 
Theoremmulcxplem 19979 Lemma for mulcxp 19980. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( 0  ^ c  C )  =  ( ( A  ^ c  C )  x.  ( 0  ^ c  C ) ) )
 
Theoremmulcxp 19980 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  (
 ( A  x.  B )  ^ c  C )  =  ( ( A 
 ^ c  C )  x.  ( B  ^ c  C ) ) )
 
Theoremcxprec 19981 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  CC )  ->  ( ( 1  /  A )  ^ c  B )  =  ( 1  /  ( A  ^ c  B ) ) )
 
Theoremdivcxp 19982 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR+  /\  C  e.  CC )  ->  ( ( A  /  B ) 
 ^ c  C )  =  ( ( A 
 ^ c  C ) 
 /  ( B  ^ c  C ) ) )
 
Theoremcxpmul 19983 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  C  e.  CC )  ->  ( A  ^ c  ( B  x.  C ) )  =  (
 ( A  ^ c  B )  ^ c  C ) )
 
Theoremcxpmul2 19984 Product of exponents law for complex exponentiation. Variation on cxpmul 19983 with more general conditions on  A and  B when  C is an integer. (Contributed by Mario Carneiro, 9-Aug-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  NN0 )  ->  ( A  ^ c  ( B  x.  C ) )  =  (
 ( A  ^ c  B ) ^ C ) )
 
Theoremcxproot 19985 The complex power function allows us to write n-th roots via the idiom  A  ^ c 
( 1  /  N
). (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A 
 ^ c  ( 1 
 /  N ) ) ^ N )  =  A )
 
Theoremcxpmul2z 19986 Generalize cxpmul2 19984 to negative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  C  e.  ZZ ) )  ->  ( A  ^ c  ( B  x.  C ) )  =  ( ( A  ^ c  B ) ^ C ) )
 
Theoremabscxp 19987 Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  CC )  ->  ( abs `  ( A  ^ c  B ) )  =  ( A 
 ^ c  ( Re
 `  B ) ) )
 
Theoremabscxp2 19988 Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  RR )  ->  ( abs `  ( A  ^ c  B ) )  =  ( ( abs `  A )  ^ c  B )
 )
 
Theoremcxplt 19989 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B  <  C  <->  ( A  ^ c  B )  <  ( A  ^ c  C ) ) )
 
Theoremcxple 19990 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B  <_  C  <->  ( A  ^ c  B )  <_  ( A  ^ c  C ) ) )
 
Theoremcxplea 19991 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  ( B  e.  RR  /\  C  e.  RR )  /\  B  <_  C )  ->  ( A  ^ c  B ) 
 <_  ( A  ^ c  C ) )
 
Theoremcxple2 19992 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( A  ^ c  C )  <_  ( B  ^ c  C ) ) )
 
Theoremcxplt2 19993 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( A  <  B  <->  ( A  ^ c  C )  <  ( B  ^ c  C ) ) )
 
Theoremcxple2a 19994 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( 0 
 <_  A  /\  0  <_  C )  /\  A  <_  B )  ->  ( A  ^ c  C )  <_  ( B  ^ c  C ) )
 
Theoremcxplt3 19995 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( ( ( A  e.  RR+  /\  A  <  1 )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B  <  C  <->  ( A  ^ c  C )  <  ( A  ^ c  B ) ) )
 
Theoremcxple3 19996 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( ( ( A  e.  RR+  /\  A  <  1 )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B  <_  C  <->  ( A  ^ c  C )  <_  ( A  ^ c  B ) ) )
 
Theoremcxpsqrlem 19997 Lemma for cxpsqr 19998. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1 
 /  2 ) )  =  -u ( sqr `  A ) )  ->  ( _i 
 x.  ( sqr `  A ) )  e.  RR )
 
Theoremcxpsqr 19998 The complex exponential function with exponent  1  /  2 exactly matches the complex square root function (the branch cut is in the same place for both functions), and thus serves as a suitable generalization to other  n-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( A  ^ c  ( 1  /  2
 ) )  =  ( sqr `  A )
 )
 
Theoremlogsqr 19999 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  ( A  e.  RR+  ->  ( log `  ( sqr `  A ) )  =  ( ( log `  A )  /  2 ) )
 
Theoremcxp0d 20000 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  0 )  =  1 )
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