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Theorem List for Metamath Proof Explorer - 20001-20100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremadvlog 20001 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 20002* 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 20003 Lemma for efopn 20005. (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 20004 Lemma for efopn 20005. (Contributed by Mario Carneiro, 2-May-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  (
 ( R  e.  RR+  /\  R  <  pi ) 
 ->  ( exp " (
 0 ( ball `  ( abs  o.  -  ) ) R ) )  e.  J )
 
Theoremefopn 20005 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 20006* Lemma for logtayl 20007. (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 20007* 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 20008* 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 20009* 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 20010 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 20011 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 20012 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 20013 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 20014 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 20015 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 20016 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 20017 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 20018 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( A  ^ c 
 1 )  =  A )
 
Theorem1cxp 20019 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( 1  ^ c  A )  =  1
 )
 
Theoremecxp 20020 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 20021 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 20022 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 20023 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 20024 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 20025 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 20026 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 20027 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 20028 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 20029 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 20030 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 20031 Lemma for mulcxp 20032. (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 20032 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 20033 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 20034 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 20035 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 20036 Product of exponents law for complex exponentiation. Variation on cxpmul 20035 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 20037 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 20038 Generalize cxpmul2 20036 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 20039 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 20040 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 20041 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 20042 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 20043 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 20044 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 20045 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 20046 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 20047 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 20048 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 20049 Lemma for cxpsqr 20050. (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 20050 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 20051 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  ( A  e.  RR+  ->  ( log `  ( sqr `  A ) )  =  ( ( log `  A )  /  2 ) )
 
Theoremcxp0d 20052 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 )
 
Theoremcxp1d 20053 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  1 )  =  A )
 
Theorem1cxpd 20054 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 1  ^ c  A )  =  1 )
 
Theoremcxpcld 20055 Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  B )  e.  CC )
 
Theoremcxpmul2d 20056 Product of exponents law for complex exponentiation. Variation on cxpmul 20035 with more general conditions on  A and  B when  C is an integer. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  NN0 )   =>    |-  ( ph  ->  ( A  ^ c  ( B  x.  C ) )  =  ( ( A 
 ^ c  B ) ^ C ) )
 
Theorem0cxpd 20057 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 0  ^ c  A )  =  0 )
 
Theoremcxpexpzd 20058 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  ZZ )   =>    |-  ( ph  ->  ( A  ^ c  B )  =  ( A ^ B ) )
 
Theoremcxpefd 20059 Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
 
Theoremcxpne0d 20060 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  B )  =/=  0 )
 
Theoremcxpp1d 20061 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  ( B  +  1 ) )  =  ( ( A 
 ^ c  B )  x.  A ) )
 
Theoremcxpnegd 20062 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  -u B )  =  ( 1  /  ( A  ^ c  B ) ) )
 
Theoremcxpmul2zd 20063 Generalize cxpmul2 20036 to negative integers. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  ZZ )   =>    |-  ( ph  ->  ( A  ^ c  ( B  x.  C ) )  =  ( ( A 
 ^ c  B ) ^ C ) )
 
Theoremcxpaddd 20064 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  ( B  +  C ) )  =  ( ( A 
 ^ c  B )  x.  ( A  ^ c  C ) ) )
 
Theoremcxpsubd 20065 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  ( B  -  C ) )  =  ( ( A 
 ^ c  B ) 
 /  ( A  ^ c  C ) ) )
 
Theoremcxpltd 20066 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( B  <  C  <->  ( A  ^ c  B )  <  ( A  ^ c  C ) ) )
 
Theoremcxpled 20067 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( B  <_  C  <->  ( A  ^ c  B )  <_  ( A  ^ c  C ) ) )
 
Theoremcxplead 20068 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <_  C )   =>    |-  ( ph  ->  ( A  ^ c  B ) 
 <_  ( A  ^ c  C ) )
 
Theoremdivcxpd 20069 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( ( A  /  B )  ^ c  C )  =  ( ( A 
 ^ c  C ) 
 /  ( B  ^ c  C ) ) )
 
Theoremrecxpcld 20070 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  ^ c  B )  e.  RR )
 
Theoremcxpge0d 20071 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  0 
 <_  ( A  ^ c  B ) )
 
Theoremcxple2ad 20072 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( A  ^ c  C ) 
 <_  ( B  ^ c  C ) )
 
Theoremcxplt2d 20073 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  ^ c  C )  <  ( B  ^ c  C ) ) )
 
Theoremcxple2d 20074 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  ^ c  C )  <_  ( B  ^ c  C ) ) )
 
Theoremmulcxpd 20075 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  ^ c  C )  =  ( ( A 
 ^ c  C )  x.  ( B  ^ c  C ) ) )
 
Theoremcxprecd 20076 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( 1  /  A )  ^ c  B )  =  ( 1  /  ( A  ^ c  B ) ) )
 
Theoremrpcxpcld 20077 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  ^ c  B )  e.  RR+ )
 
Theoremlogcxpd 20078 Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( log `  ( A  ^ c  B ) )  =  ( B  x.  ( log `  A )
 ) )
 
Theoremcxplt3d 20079 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  1
 )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( B  <  C  <->  ( A  ^ c  C )  <  ( A  ^ c  B ) ) )
 
Theoremcxple3d 20080 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  1
 )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( B  <_  C  <->  ( A  ^ c  C )  <_  ( A  ^ c  B ) ) )
 
Theoremcxpmuld 20081 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  ( B  x.  C ) )  =  ( ( A 
 ^ c  B ) 
 ^ c  C ) )
 
Theoremdvcxp1 20082* The derivative of a complex power with respect to the first argument. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( x 
 ^ c  A ) ) )  =  ( x  e.  RR+  |->  ( A  x.  ( x  ^ c  ( A  -  1
 ) ) ) ) )
 
Theoremdvcxp2 20083* The derivative of a complex power with respect to the second argument. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( A  e.  RR+  ->  ( CC  _D  ( x  e.  CC  |->  ( A 
 ^ c  x ) ) )  =  ( x  e.  CC  |->  ( ( log `  A )  x.  ( A  ^ c  x ) ) ) )
 
Theoremdvsqr 20084 The derivative of the real square root function. (Contributed by Mario Carneiro, 1-May-2016.)
 |-  ( RR  _D  ( x  e.  RR+  |->  ( sqr `  x ) ) )  =  ( x  e.  RR+  |->  ( 1  /  ( 2  x.  ( sqr `  x ) ) ) )
 
Theoremcxpcn 20085* Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  J  =  (
 TopOpen ` fld )   &    |-  K  =  ( Jt  D )   =>    |-  ( x  e.  D ,  y  e.  CC  |->  ( x  ^ c  y ) )  e.  (
 ( K  tX  J )  Cn  J )
 
Theoremcxpcn2 20086* Continuity of the complex power function, when the base is real. (Contributed by Mario Carneiro, 1-May-2016.)
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  RR+ )   =>    |-  ( x  e.  RR+ ,  y  e.  CC  |->  ( x  ^ c  y ) )  e.  (
 ( K  tX  J )  Cn  J )
 
Theoremcxpcn3lem 20087* Lemma for cxpcn3 20088. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  D  =  ( `' Re " RR+ )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  ( 0 [,)  +oo ) )   &    |-  L  =  ( Jt  D )   &    |-  U  =  ( if ( ( Re
 `  A )  <_ 
 1 ,  ( Re
 `  A ) ,  1 )  /  2
 )   &    |-  T  =  if ( U  <_  ( E  ^ c  ( 1  /  U ) ) ,  U ,  ( E  ^ c  ( 1  /  U ) ) )   =>    |-  ( ( A  e.  D  /\  E  e.  RR+ )  ->  E. d  e.  RR+  A. a  e.  (
 0 [,)  +oo ) A. b  e.  D  (
 ( ( abs `  a
 )  <  d  /\  ( abs `  ( A  -  b ) )  < 
 d )  ->  ( abs `  ( a  ^ c  b ) )  <  E ) )
 
Theoremcxpcn3 20088* Extend continuity of the complex power function to a base of zero, as long as the exponent has strictly positive real part. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  D  =  ( `' Re " RR+ )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  ( 0 [,)  +oo ) )   &    |-  L  =  ( Jt  D )   =>    |-  ( x  e.  (
 0 [,)  +oo ) ,  y  e.  D  |->  ( x  ^ c  y ) )  e.  (
 ( K  tX  L )  Cn  J )
 
Theoremresqrcn 20089 Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,)  +oo ) -cn-> RR )
 
Theoremsqrcn 20090 Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( sqr  |`  D )  e.  ( D -cn-> CC )
 
Theoremcxpaddlelem 20091 Lemma for cxpaddle 20092. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A 
 <_  1 )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  B 
 <_  1 )   =>    |-  ( ph  ->  A  <_  ( A  ^ c  B ) )
 
Theoremcxpaddle 20092 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  C 
 <_  1 )   =>    |-  ( ph  ->  (
 ( A  +  B )  ^ c  C ) 
 <_  ( ( A  ^ c  C )  +  ( B  ^ c  C ) ) )
 
Theoremabscxpbnd 20093 Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  0  <_  ( Re `  B ) )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  ( abs `  A )  <_  M )   =>    |-  ( ph  ->  ( abs `  ( A  ^ c  B ) )  <_  ( ( M  ^ c  ( Re `  B ) )  x.  ( exp `  ( ( abs `  B )  x.  pi ) ) ) )
 
Theoremroot1id 20094 Property of an  N-th root of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( N  e.  NN  ->  ( ( -u 1  ^ c  ( 2  /  N ) ) ^ N )  =  1
 )
 
Theoremroot1eq1 20095 The only powers of an  N-th root of unity that equal 
1 are the multiples of  N. In other words,  -u 1  ^ c 
( 2  /  N
) has order  N in the multiplicative group of nonzero complex numbers. (In fact, these and their powers are the only elements of finite order in the complexes.) (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  ( ( N  e.  NN  /\  K  e.  ZZ )  ->  ( ( (
 -u 1  ^ c  ( 2  /  N ) ) ^ K )  =  1  <->  N  ||  K ) )
 
Theoremroot1cj 20096 Within the  N-th roots of unity, the conjugate of the  K-th root is the  N  -  K-th root. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( N  e.  NN  /\  K  e.  ZZ )  ->  ( * `  ( ( -u 1  ^ c  ( 2  /  N ) ) ^ K ) )  =  ( ( -u 1  ^ c  ( 2  /  N ) ) ^
 ( N  -  K ) ) )
 
Theoremcxpeq 20097* Solve an equation involving an  N-th power. The expression  -u 1  ^ c  ( 2  /  N )  =  exp ( 2 pi _i 
/  N ) is a way to write the primitive  N-th root of unity with smallest positive argument. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( A  e.  CC  /\  N  e.  NN  /\  B  e.  CC )  ->  ( ( A ^ N )  =  B  <->  E. n  e.  ( 0
 ... ( N  -  1 ) ) A  =  ( ( B 
 ^ c  ( 1 
 /  N ) )  x.  ( ( -u 1  ^ c  ( 2 
 /  N ) ) ^ n ) ) ) )
 
Theoremloglesqr 20098 An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( log `  ( A  +  1 )
 )  <_  ( sqr `  A ) )
 
13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords
 
Theoremangval 20099* Define the angle function, which takes two complex numbers, treated as vectors from the origin, and returns the angle between them, in the range  (  -  pi ,  pi ]. 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 20100* 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 ) )
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