HomeHome Metamath Proof Explorer
Theorem List (p. 201 of 328)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21514)
  Hilbert Space Explorer  Hilbert Space Explorer
(21515-23037)
  Users' Mathboxes  Users' Mathboxes
(23038-32776)
 

Theorem List for Metamath Proof Explorer - 20001-20100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrelogcn 20001 The real logarithm function is continuous. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  ( log  |`  RR+ )  e.  ( RR+ -cn-> RR )
 
Theoremellogdm 20002 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 20003 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 20004 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 20005 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 20006 Lemma for logcn 20010. (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 20007 Lemma for logcn 20010. (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 20008 Lemma for logcn 20010. (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 20009* Lemma for logcn 20010. (Contributed by Mario Carneiro, 18-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( x  e.  D  |->  ( Im `  ( log `  x ) ) )  e.  ( D -cn-> RR )
 
Theoremlogcn 20010 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 20011 Lemma for dvlog 20014. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( log " D )  e.  ( TopOpen ` fld )
 
Theoremlogdmopn 20012 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 20013 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 20014* 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 20015 Lemma for dvlog2 20016. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  S  =  ( 1 ( ball `  ( abs  o. 
 -  ) ) 1 )   =>    |-  S  C_  ( CC  \  (  -oo (,] 0
 ) )
 
Theoremdvlog2 20016* 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 20014. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  S  =  ( 1 ( ball `  ( abs  o. 
 -  ) ) 1 )   =>    |-  ( CC  _D  ( log  |`  S ) )  =  ( x  e.  S  |->  ( 1  /  x ) )
 
Theoremadvlog 20017 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 20018* 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 20019 Lemma for efopn 20021. (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 20020 Lemma for efopn 20021. (Contributed by Mario Carneiro, 2-May-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  (
 ( R  e.  RR+  /\  R  <  pi ) 
 ->  ( exp " (
 0 ( ball `  ( abs  o.  -  ) ) R ) )  e.  J )
 
Theoremefopn 20021 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 20022* Lemma for logtayl 20023. (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 20023* 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 20024* 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 20025* 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 20026 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 20027 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 20028 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 20029 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 20030 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 20031 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 20032 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 20033 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 20034 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( A  ^ c 
 1 )  =  A )
 
Theorem1cxp 20035 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( 1  ^ c  A )  =  1
 )
 
Theoremecxp 20036 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 20037 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 20038 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 20039 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 20040 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 20041 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 20042 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 20043 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 20044 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 20045 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 20046 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 20047 Lemma for mulcxp 20048. (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 20048 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 20049 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 20050 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 20051 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 20052 Product of exponents law for complex exponentiation. Variation on cxpmul 20051 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 20053 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 20054 Generalize cxpmul2 20052 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 20055 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 20056 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 20057 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 20058 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 20059 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 20060 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 20061 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 20062 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 20063 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 20064 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 20065 Lemma for cxpsqr 20066. (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 20066 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 20067 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  ( A  e.  RR+  ->  ( log `  ( sqr `  A ) )  =  ( ( log `  A )  /  2 ) )
 
Theoremcxp0d 20068 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 20069 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 20070 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 20071 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 20072 Product of exponents law for complex exponentiation. Variation on cxpmul 20051 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 20073 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 20074 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 20075 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 20076 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 20077 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 20078 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 20079 Generalize cxpmul2 20052 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 20080 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 20081 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 20082 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 20083 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 20084 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 20085 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 20086 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 20087 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 20088 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 20089 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 20090 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 20091 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 20092 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 20093 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 20094 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 20095 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 20096 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 20097 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 20098* 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 20099* 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 20100 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 ) ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32776
  Copyright terms: Public domain < Previous  Next >