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Theorem List for Metamath Proof Explorer - 19801-19900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlogneg2 19801 The logarithm of the negative of a number with positive imaginary part is  _i pi less than the original. (Compare logneg 19773.) (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ( A  e.  CC  /\  0  <  ( Im `  A ) ) 
 ->  ( log `  -u A )  =  ( ( log `  A )  -  ( _i  x.  pi ) ) )
 
Theoremtanarg 19802 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 19803 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 19804 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 19805 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 19806 Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( log `  A )  e. 
 RR )
 
Theoremreeflogd 19807 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 19808 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 19809 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 19810 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 19811 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 19812 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 19813 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 19814 The inverse logarithm function converges to zero. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( 1  /  ( log `  x ) ) )  ~~> r  0
 
Theoremlogno1 19815 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 19816 The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( RR  _D  ( log  |`  RR+ ) )  =  ( x  e.  RR+  |->  ( 1  /  x ) )
 
Theoremrelogcn 19817 The real logarithm function is continuous. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  ( log  |`  RR+ )  e.  ( RR+ -cn-> RR )
 
Theoremellogdm 19818 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 19819 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 19820 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 19821 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 19822 Lemma for logcn 19826. (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 19823 Lemma for logcn 19826. (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 19824 Lemma for logcn 19826. (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 19825* Lemma for logcn 19826. (Contributed by Mario Carneiro, 18-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( x  e.  D  |->  ( Im `  ( log `  x ) ) )  e.  ( D -cn-> RR )
 
Theoremlogcn 19826 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 19827 Lemma for dvlog 19830. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( log " D )  e.  ( TopOpen ` fld )
 
Theoremlogdmopn 19828 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 19829 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 19830* 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 19831 Lemma for dvlog2 19832. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  S  =  ( 1 ( ball `  ( abs  o. 
 -  ) ) 1 )   =>    |-  S  C_  ( CC  \  (  -oo (,] 0
 ) )
 
Theoremdvlog2 19832* 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 19830. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  S  =  ( 1 ( ball `  ( abs  o. 
 -  ) ) 1 )   =>    |-  ( CC  _D  ( log  |`  S ) )  =  ( x  e.  S  |->  ( 1  /  x ) )
 
Theoremadvlog 19833 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 19834* 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 19835 Lemma for efopn 19837. (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 19836 Lemma for efopn 19837. (Contributed by Mario Carneiro, 2-May-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  (
 ( R  e.  RR+  /\  R  <  pi ) 
 ->  ( exp " (
 0 ( ball `  ( abs  o.  -  ) ) R ) )  e.  J )
 
Theoremefopn 19837 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 19838* Lemma for logtayl 19839. (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 19839* 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 19840* 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 19841* 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 19842 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 ) ) ) )
 
Theoremangval 19843* Define the angle function, which takes two complex numbers, treated as vectors from the origin, and returns the angle between them. To convert from the geometry notation, 
m A B C, the measure of the angle with legs  A B,  C B where  C is more counterclockwise for positive angles, is represented by  ( ( C  -  B ) F ( A  -  B ) ). (Contributed by Mario Carneiro, 23-Sep-2014.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   =>    |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A F B )  =  ( Im `  ( log `  ( B  /  A ) ) ) )
 
Theoremangcan 19844* Cancel a constant multiplier in the angle function. (Contributed by Mario Carneiro, 23-Sep-2014.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   =>    |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( C  x.  A ) F ( C  x.  B ) )  =  ( A F B ) )
 
Theoremangneg 19845* Cancel a negative sign in the angle function. (Contributed by Mario Carneiro, 23-Sep-2014.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   =>    |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( -u A F -u B )  =  ( A F B ) )
 
Theoremangvald 19846* The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 19843. (Contributed by David Moews, 28-Feb-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0
 )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  Y  =/=  0 )   =>    |-  ( ph  ->  ( X F Y )  =  ( Im `  ( log `  ( Y  /  X ) ) ) )
 
Theoremangcld 19847* The (signed) angle between two vectors is in  (
-u pi (,] pi ). Deduction form. (Contributed by David Moews, 28-Feb-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0
 )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  Y  =/=  0 )   =>    |-  ( ph  ->  ( X F Y )  e.  ( -u pi (,] pi ) )
 
Theoremangrteqvd 19848* Two vectors are at a right angle iff their quotient is purely imaginary. (Contributed by David Moews, 28-Feb-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0
 )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  Y  =/=  0 )   =>    |-  ( ph  ->  ( ( X F Y )  e.  { ( pi  /  2 ) ,  -u ( pi  /  2
 ) }  <->  ( Re `  ( Y  /  X ) )  =  0 ) )
 
Theoremcosangneg2d 19849* The cosine of the angle between  X and  -u Y is the negative of that between  X and  Y. If A, B and C are collinear points, this implies that the cosines of DBA and DBC sum to zero, i.e., that DBA and DBC are supplementary. (Contributed by David Moews, 28-Feb-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  0
 )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  Y  =/=  0 )   =>    |-  ( ph  ->  ( cos `  ( X F -u Y ) )  =  -u ( cos `  ( X F Y ) ) )
 
Theoremangrtmuld 19850* Perpendicularity of two vectors does not change under rescaling the second. (Contributed by David Moews, 28-Feb-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  Z  e.  CC )   &    |-  ( ph  ->  X  =/=  0 )   &    |-  ( ph  ->  Y  =/=  0
 )   &    |-  ( ph  ->  Z  =/=  0 )   &    |-  ( ph  ->  ( Z  /  Y )  e.  RR )   =>    |-  ( ph  ->  ( ( X F Y )  e.  { ( pi  /  2 ) ,  -u ( pi  /  2
 ) }  <->  ( X F Z )  e.  { ( pi  /  2 ) ,  -u ( pi  /  2
 ) } ) )
 
Theoremang180lem1 19851* Lemma for ang180 19856. Show that the "revolution number"  N is an integer, using efeq1 19723 to show that since the product of the three arguments  A ,  1  / 
( 1  -  A
) ,  ( A  -  1 )  /  A is  -u 1, the sum of the logarithms must be an integer multiple of  2
pi _i away from  pi _i  =  log ( -u 1 ). (Contributed by Mario Carneiro, 23-Sep-2014.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  T  =  ( ( ( log `  (
 1  /  ( 1  -  A ) ) )  +  ( log `  (
 ( A  -  1
 )  /  A )
 ) )  +  ( log `  A ) )   &    |-  N  =  ( (
 ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
 ) )   =>    |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 ) 
 ->  ( N  e.  ZZ  /\  ( T  /  _i )  e.  RR )
 )
 
Theoremang180lem2 19852* Lemma for ang180 19856. Show that the revolution number  N is strictly between  -u 2 and  1. Both bounds are established by iterating using the bounds on the imaginary part of the logarithm, logimcl 19759, but the resulting bound gives only  N  <_ 
1 for the upper bound. The case  N  =  1 is not ruled out here, but it is in some sense an "edge case" that can only happen under very specific conditions; in particular we show that all the angle arguments  A ,  1  /  ( 1  -  A ) ,  ( A  -  1 )  /  A must lie on the negative real axis, which is a contradiction because clearly if  A is negative then the other two are positive real. (Contributed by Mario Carneiro, 23-Sep-2014.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  T  =  ( ( ( log `  (
 1  /  ( 1  -  A ) ) )  +  ( log `  (
 ( A  -  1
 )  /  A )
 ) )  +  ( log `  A ) )   &    |-  N  =  ( (
 ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
 ) )   =>    |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 ) 
 ->  ( -u 2  <  N  /\  N  <  1 ) )
 
Theoremang180lem3 19853* Lemma for ang180 19856. Since ang180lem1 19851 shows that  N is an integer and ang180lem2 19852 shows that  N is strictly between  -u 2 and  1, it follows that  N  e.  { -u 1 ,  0 }, and these two cases correspond to the two possible values for  T. (Contributed by Mario Carneiro, 23-Sep-2014.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  T  =  ( ( ( log `  (
 1  /  ( 1  -  A ) ) )  +  ( log `  (
 ( A  -  1
 )  /  A )
 ) )  +  ( log `  A ) )   &    |-  N  =  ( (
 ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
 ) )   =>    |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 ) 
 ->  T  e.  { -u ( _i  x.  pi ) ,  ( _i  x.  pi ) } )
 
Theoremang180lem4 19854* Lemma for ang180 19856. Reduce the statement to one variable. (Contributed by Mario Carneiro, 23-Sep-2014.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   =>    |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 ) 
 ->  ( ( ( ( 1  -  A ) F 1 )  +  ( A F ( A  -  1 ) ) )  +  ( 1 F A ) )  e.  { -u pi ,  pi } )
 
Theoremang180lem5 19855* Lemma for ang180 19856: Reduce the statement to two variables. (Contributed by Mario Carneiro, 23-Sep-2014.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   =>    |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
 ( ( ( A  -  B ) F A )  +  ( B F ( B  -  A ) ) )  +  ( A F B ) )  e. 
 { -u pi ,  pi } )
 
Theoremang180 19856* The sum of angles  m A B C  +  m B C A  +  m C A B in a triangle adds up to either  pi or  -u pi, i.e. 180 degrees. (The sign is due to the two possible orientations of vertex arrangement and our signed notion of angle). (Contributed by Mario Carneiro, 23-Sep-2014.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   =>    |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C ) )  ->  (
 ( ( ( C  -  B ) F ( A  -  B ) )  +  (
 ( A  -  C ) F ( B  -  C ) ) )  +  ( ( B  -  A ) F ( C  -  A ) ) )  e. 
 { -u pi ,  pi } )
 
Theoremlawcoslem1 19857 Lemma for Law of Cosines lawcos 19858. Here we prove the law for a point at the origin and two distinct points U and V, using an expanded version of the signed angle expression on the complex plane. (Contributed by David A. Wheeler, 11-Jun-2015.)
 |-  ( ph  ->  U  e.  CC )   &    |-  ( ph  ->  V  e.  CC )   &    |-  ( ph  ->  U  =/=  0
 )   &    |-  ( ph  ->  V  =/=  0 )   =>    |-  ( ph  ->  (
 ( abs `  ( U  -  V ) ) ^
 2 )  =  ( ( ( ( abs `  U ) ^ 2
 )  +  ( ( abs `  V ) ^ 2 ) )  -  ( 2  x.  ( ( ( abs `  U )  x.  ( abs `  V ) )  x.  ( ( Re
 `  ( U  /  V ) )  /  ( abs `  ( U  /  V ) ) ) ) ) ) )
 
Theoremlawcos 19858* Law of Cosines. Given three distinct points A, B, and C, prove a relationship between their segment lengths. This theorem is expressed using the complex number plane as a plane, where  F is the signed angle construct (as used in ang180 19856),  X is the distance of line segment BC,  Y is the distance of line segment AC,  Z is the distance of line segment AB, and  O is the distinguished (signed) angle m/_ BCA on the complex plane. We translate triangle ABC to move C to the origin (C-C), B to U=(B-C), and A to V=(A-C), then use lemma lawcoslem1 19857 to prove this algebraically simpler case. The metamath convention is to use a signed angle; in this case the sign doesn't matter because we use the cosine of the angle (see cosneg 12301). The Pythagorean Theorem pythag 19859 is a special case of the law of cosines. The theorem's expression and approach were suggested by Mario Carneiro. (Contributed by David A. Wheeler, 12-Jun-2015.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  X  =  ( abs `  ( B  -  C ) )   &    |-  Y  =  ( abs `  ( A  -  C ) )   &    |-  Z  =  ( abs `  ( A  -  B ) )   &    |-  O  =  ( ( B  -  C ) F ( A  -  C ) )   =>    |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C ) )  ->  ( Z ^ 2 )  =  ( ( ( X ^ 2 )  +  ( Y ^
 2 ) )  -  ( 2  x.  (
 ( X  x.  Y )  x.  ( cos `  O ) ) ) ) )
 
Theorempythag 19859* Pythagorean Theorem. Given three distinct points A, B, and C that form a right triangle (with the right angle at C), prove a relationship between their segment lengths. This theorem is expressed using the complex number plane as a plane, where  F is the signed angle construct (as used in ang180 19856),  X is the distance of line segment BC,  Y is the distance of line segment AC,  Z is the distance of line segment AB (the hypotenuse), and  O is the distinguished (signed) right angle m/_ BCA. We use the law of cosines lawcos 19858 to prove this, along with simple trig facts like coshalfpi 19669 and cosneg 12301. (Contributed by David A. Wheeler, 13-Jun-2015.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  X  =  ( abs `  ( B  -  C ) )   &    |-  Y  =  ( abs `  ( A  -  C ) )   &    |-  Z  =  ( abs `  ( A  -  B ) )   &    |-  O  =  ( ( B  -  C ) F ( A  -  C ) )   =>    |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C )  /\  O  e.  { ( pi  / 
 2 ) ,  -u ( pi  /  2 ) }
 )  ->  ( Z ^ 2 )  =  ( ( X ^
 2 )  +  ( Y ^ 2 ) ) )
 
Theoremlogreclem 19860 Symmetry of the natural logarithm range by negation. Lemma for logrec 19861. (Contributed by Saveliy Skresanov, 27-Dec-2016.)
 |-  ( ( A  e.  ran 
 log  /\  -.  ( Im
 `  A )  =  pi )  ->  -u A  e.  ran  log )
 
Theoremlogrec 19861 Logarithm of a reciprocal changes sign. (Contributed by Saveliy Skresanov, 28-Dec-2016.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  ( Im `  ( log `  A ) )  =/=  pi )  ->  ( log `  A )  =  -u ( log `  (
 1  /  A )
 ) )
 
Theoremisosctrlem1 19862 Lemma for isosctr 19865. (Contributed by Saveliy Skresanov, 30-Dec-2016.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A ) 
 ->  ( Im `  ( log `  ( 1  -  A ) ) )  =/=  pi )
 
Theoremisosctrlem2 19863 Lemma for isosctr 19865. Corresponds to the case where one vertex is at 0, another at 1 and the third lies on the unit circle. (Contributed by Saveliy Skresanov, 31-Dec-2016.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A ) 
 ->  ( Im `  ( log `  ( 1  -  A ) ) )  =  ( Im `  ( log `  ( -u A  /  ( 1  -  A ) ) ) ) )
 
Theoremisosctrlem3 19864* Lemma for isosctr 19865. Corresponds to the case where one vertex is at 0. (Contributed by Saveliy Skresanov, 1-Jan-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   =>    |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/=  B )  /\  ( abs `  A )  =  ( abs `  B )
 )  ->  ( -u A F ( B  -  A ) )  =  ( ( A  -  B ) F -u B ) )
 
Theoremisosctr 19865* Isosceles triangle theorem. (Contributed by Saveliy Skresanov, 1-Jan-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   =>    |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B )  /\  ( abs `  ( A  -  C ) )  =  ( abs `  ( B  -  C ) ) ) 
 ->  ( ( C  -  A ) F ( B  -  A ) )  =  ( ( A  -  B ) F ( C  -  B ) ) )
 
Theoremssscongptld 19866* If two triangles have equal sides, one angle in one triangle has the same cosine as the corresponding angle in the other triangle. This is a partial form of the SSS congruence theorem.

This theorem is proven by using lawcos 19858 on both triangles to express one side in terms of the other two, and then equating these expressions and reducing this algebraically to get an equality of cosines of angles. (Contributed by David Moews, 28-Feb-2017.)

 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  E  e.  CC )   &    |-  ( ph  ->  G  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  =/=  C )   &    |-  ( ph  ->  D  =/=  E )   &    |-  ( ph  ->  E  =/=  G )   &    |-  ( ph  ->  ( abs `  ( A  -  B ) )  =  ( abs `  ( D  -  E ) ) )   &    |-  ( ph  ->  ( abs `  ( B  -  C ) )  =  ( abs `  ( E  -  G ) ) )   &    |-  ( ph  ->  ( abs `  ( C  -  A ) )  =  ( abs `  ( G  -  D ) ) )   =>    |-  ( ph  ->  ( cos `  ( ( A  -  B ) F ( C  -  B ) ) )  =  ( cos `  (
 ( D  -  E ) F ( G  -  E ) ) ) )
 
Theoremaffineequiv 19867 Equivalence between two ways of expressing  B as an affine combination of  A and  C. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  ( B  =  ( ( D  x.  A )  +  ( ( 1  -  D )  x.  C ) )  <->  ( C  -  B )  =  ( D  x.  ( C  -  A ) ) ) )
 
Theoremaffineequiv2 19868 Equivalence between two ways of expressing  B as an affine combination of  A and  C. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  ( B  =  ( ( D  x.  A )  +  ( ( 1  -  D )  x.  C ) )  <->  ( B  -  A )  =  (
 ( 1  -  D )  x.  ( C  -  A ) ) ) )
 
Theoremangpieqvdlem 19869 Equivalence used in the proof of angpieqvd 19872. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  (
 -u ( ( C  -  B )  /  ( A  -  B ) )  e.  RR+  <->  ( ( C  -  B )  /  ( C  -  A ) )  e.  (
 0 (,) 1 ) ) )
 
Theoremangpieqvdlem2 19870* Equivalence used in angpieqvd 19872. (Contributed by David Moews, 28-Feb-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  e.  RR+  <->  ( ( A  -  B ) F ( C  -  B ) )  =  pi ) )
 
Theoremangpined 19871* If the angle at ABC is  pi, then A is not equal to C. (Contributed by David Moews, 28-Feb-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  (
 ( ( A  -  B ) F ( C  -  B ) )  =  pi  ->  A  =/=  C ) )
 
Theoremangpieqvd 19872* The angle ABC is  pi iff B is a nontrivial convex combination of A and C, i.e., iff B is in the interior of the segment AC. (Contributed by David Moews, 28-Feb-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  (
 ( ( A  -  B ) F ( C  -  B ) )  =  pi  <->  E. w  e.  (
 0 (,) 1 ) B  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  C ) ) ) )
 
Theoremchordthmlem 19873* If M is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 19866 to observe that, since AMQ and BMQ have equal sides, the angles QMB and QMA must be equal. Since they are supplementary, both must be right. (Contributed by David Moews, 28-Feb-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  Q  e.  CC )   &    |-  ( ph  ->  M  =  ( ( A  +  B )  / 
 2 ) )   &    |-  ( ph  ->  ( abs `  ( A  -  Q ) )  =  ( abs `  ( B  -  Q ) ) )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  Q  =/=  M )   =>    |-  ( ph  ->  (
 ( Q  -  M ) F ( B  -  M ) )  e. 
 { ( pi  / 
 2 ) ,  -u ( pi  /  2 ) }
 )
 
Theoremchordthmlem2 19874* If M is the midpoint of AB, AQ = BQ, and P is on the line AB, then QMP is a right angle. This is proven by reduction to the special case chordthmlem 19873, where P = B, and using angrtmuld 19850 to observe that QMP is right iff QMB is. (Contributed by David Moews, 28-Feb-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  Q  e.  CC )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  M  =  ( ( A  +  B )  /  2 ) )   &    |-  ( ph  ->  P  =  ( ( X  x.  A )  +  (
 ( 1  -  X )  x.  B ) ) )   &    |-  ( ph  ->  ( abs `  ( A  -  Q ) )  =  ( abs `  ( B  -  Q ) ) )   &    |-  ( ph  ->  P  =/=  M )   &    |-  ( ph  ->  Q  =/=  M )   =>    |-  ( ph  ->  (
 ( Q  -  M ) F ( P  -  M ) )  e. 
 { ( pi  / 
 2 ) ,  -u ( pi  /  2 ) }
 )
 
Theoremchordthmlem3 19875 If M is the midpoint of AB, AQ = BQ, and P is on the line AB, then PQ2 = QM2  + PM2. This follows from chordthmlem2 19874 and the Pythagorean theorem (pythag 19859) in the case where P and Q are unequal to M. If either P or Q equals M, the result is trivial. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  Q  e.  CC )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  M  =  ( ( A  +  B )  / 
 2 ) )   &    |-  ( ph  ->  P  =  ( ( X  x.  A )  +  ( (
 1  -  X )  x.  B ) ) )   &    |-  ( ph  ->  ( abs `  ( A  -  Q ) )  =  ( abs `  ( B  -  Q ) ) )   =>    |-  ( ph  ->  (
 ( abs `  ( P  -  Q ) ) ^
 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M ) ) ^ 2 ) ) )
 
Theoremchordthmlem4 19876 If P is on the segment AB and M is the midpoint of AB, then PA  x. PB = BM2  - PM2. If all lengths are reexpressed as fractions of AB, this reduces to the identity  X  x.  (
1  -  X )  =  ( 1  / 
2 )2  -  ( ( 1  /  2 )  -  X )2. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  X  e.  (
 0 [,] 1 ) )   &    |-  ( ph  ->  M  =  ( ( A  +  B )  /  2
 ) )   &    |-  ( ph  ->  P  =  ( ( X  x.  A )  +  ( ( 1  -  X )  x.  B ) ) )   =>    |-  ( ph  ->  ( ( abs `  ( P  -  A ) )  x.  ( abs `  ( P  -  B ) ) )  =  ( ( ( abs `  ( B  -  M ) ) ^ 2 )  -  ( ( abs `  ( P  -  M ) ) ^ 2 ) ) )
 
Theoremchordthmlem5 19877 If P is on the segment AB and AQ = BQ, then PA  x. PB = BQ2  - PQ2. This follows from two uses of chordthmlem3 19875 to show that PQ2 = QM2  + PM2 and BQ2 = QM2  + BM2, so BQ2  - PQ2 = (QM2  + BM2)  - (QM2  + PM2) = BM2  - PM2, which equals PA  x. PB by chordthmlem4 19876. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  Q  e.  CC )   &    |-  ( ph  ->  X  e.  ( 0 [,] 1
 ) )   &    |-  ( ph  ->  P  =  ( ( X  x.  A )  +  ( ( 1  -  X )  x.  B ) ) )   &    |-  ( ph  ->  ( abs `  ( A  -  Q ) )  =  ( abs `  ( B  -  Q ) ) )   =>    |-  ( ph  ->  (
 ( abs `  ( P  -  A ) )  x.  ( abs `  ( P  -  B ) ) )  =  ( ( ( abs `  ( B  -  Q ) ) ^ 2 )  -  ( ( abs `  ( P  -  Q ) ) ^ 2 ) ) )
 
Theoremchordthm 19878* The intersecting chords theorem. If points A, B, C, and D lie on a circle (with center Q, say), and the point P is on the interior of the segments AB and CD, then the two products of lengths PA  x. PB and PC  x. PD are equal. The Euclidean plane is identified with the complex plane, and the fact that P is on AB and on CD is expressed by the hypothesis that the angles APB and CPD are equal to  pi. The result is proven by using chordthmlem5 19877 twice to show that PA  x. PB and PC  x. PD both equal BQ2  - PQ2. This is similar to the proof of the theorem given in Euclid's Elements, where it is Proposition III.35. (Contributed by David Moews, 28-Feb-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  P  e.  CC )   &    |-  ( ph  ->  A  =/=  P )   &    |-  ( ph  ->  B  =/=  P )   &    |-  ( ph  ->  C  =/=  P )   &    |-  ( ph  ->  D  =/=  P )   &    |-  ( ph  ->  ( ( A  -  P ) F ( B  -  P ) )  =  pi )   &    |-  ( ph  ->  ( ( C  -  P ) F ( D  -  P ) )  =  pi )   &    |-  ( ph  ->  Q  e.  CC )   &    |-  ( ph  ->  ( abs `  ( A  -  Q ) )  =  ( abs `  ( B  -  Q ) ) )   &    |-  ( ph  ->  ( abs `  ( A  -  Q ) )  =  ( abs `  ( C  -  Q ) ) )   &    |-  ( ph  ->  ( abs `  ( A  -  Q ) )  =  ( abs `  ( D  -  Q ) ) )   =>    |-  ( ph  ->  (
 ( abs `  ( P  -  A ) )  x.  ( abs `  ( P  -  B ) ) )  =  ( ( abs `  ( P  -  C ) )  x.  ( abs `  ( P  -  D ) ) ) )
 
Theoremcxpval 19879 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 19880 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 19881 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 19882 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 19883 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 19884 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 19885 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 19886 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( A  ^ c 
 1 )  =  A )
 
Theorem1cxp 19887 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( 1  ^ c  A )  =  1
 )
 
Theoremecxp 19888 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 19889 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 19890 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 19891 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 19892 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 19893 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 19894 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 19895 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 19896 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 19897 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 19898 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 19899 Lemma for mulcxp 19900. (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 19900 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 ) ) )
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