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Theorem List for Metamath Proof Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-cos 12601 Define the cosine function. (Contributed by NM, 14-Mar-2005.)
 |- 
 cos  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x ) )  +  ( exp `  ( -u _i  x.  x ) ) ) 
 /  2 ) )
 
Definitiondf-tan 12602 Define the tangent function. We define it this way for cmpt 4208, which requires the form  ( x  e.  A  |->  B ). (Contributed by Mario Carneiro, 14-Mar-2014.)
 |- 
 tan  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( ( sin `  x )  /  ( cos `  x ) ) )
 
Definitiondf-pi 12603 Define pi = 3.14159..., which is the smallest positive number whose sine is zero. Definition of pi in [Gleason] p. 311. (We use the inverse of less-than, " `'  <", to turn supremum into infimum; currently we don't have infimum defined separately.) (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  pi  =  sup (
 ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )
 
Theoremeftcl 12604 Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)
 |-  ( ( A  e.  CC  /\  K  e.  NN0 )  ->  ( ( A ^ K )  /  ( ! `  K ) )  e.  CC )
 
Theoremreeftcl 12605 The terms of the series expansion of the exponential function of a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)
 |-  ( ( A  e.  RR  /\  K  e.  NN0 )  ->  ( ( A ^ K )  /  ( ! `  K ) )  e.  RR )
 
Theoremeftabs 12606 The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.)
 |-  ( ( A  e.  CC  /\  K  e.  NN0 )  ->  ( abs `  (
 ( A ^ K )  /  ( ! `  K ) ) )  =  ( ( ( abs `  A ) ^ K )  /  ( ! `  K ) ) )
 
Theoremeftval 12607* The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( N  e.  NN0  ->  ( F `  N )  =  ( ( A ^ N )  /  ( ! `  N ) ) )
 
Theoremefcllem 12608* Lemma for efcl 12613. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 12588 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  e.  CC  ->  seq  0 (  +  ,  F )  e.  dom  ~~>  )
 
Theoremef0lem 12609* The series defining the exponential function converges in the (trivial) case of a zero argument. (Contributed by Steve Rodriguez, 7-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  =  0  ->  seq  0 (  +  ,  F )  ~~>  1 )
 
Theoremefval 12610* Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( A  e.  CC  ->  ( exp `  A )  =  sum_ k  e. 
 NN0  ( ( A ^ k )  /  ( ! `  k ) ) )
 
Theoremesum 12611 Value of Euler's constant  _e = 2.71828... (Contributed by Steve Rodriguez, 5-Mar-2006.)
 |-  _e  =  sum_ k  e.  NN0  ( 1  /  ( ! `  k ) )
 
Theoremeff 12612 Domain and codomain of the exponential function. (Contributed by Paul Chapman, 22-Oct-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
 |- 
 exp : CC --> CC
 
Theoremefcl 12613 Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
 
Theoremefval2 12614* Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  e.  CC  ->  ( exp `  A )  =  sum_ k  e. 
 NN0  ( F `  k ) )
 
Theoremefcvg 12615* The series that defines the exponential function converges to it. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  e.  CC  ->  seq  0 (  +  ,  F )  ~~>  ( exp `  A ) )
 
Theoremefcvgfsum 12616* Exponential function convergence in terms of a sequence of partial finite sums. (Contributed by NM, 10-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  sum_ k  e.  ( 0 ... n ) ( ( A ^ k )  /  ( ! `  k ) ) )   =>    |-  ( A  e.  CC  ->  F  ~~>  ( exp `  A ) )
 
Theoremreefcl 12617 The exponential function is real if its argument is real. (Contributed by NM, 27-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
 
Theoremreefcld 12618 The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( exp `  A )  e. 
 RR )
 
Theoremere 12619 Euler's constant  _e = 2.71828... is a real number. (Contributed by NM, 19-Mar-2005.) (Revised by Steve Rodriguez, 8-Mar-2006.)
 |-  _e  e.  RR
 
Theoremege2le3 12620 Lemma for egt2lt3 12733. (Contributed by NM, 20-Mar-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN  |->  ( 2  x.  ( ( 1 
 /  2 ) ^ n ) ) )   &    |-  G  =  ( n  e.  NN0  |->  ( 1  /  ( ! `  n ) ) )   =>    |-  ( 2  <_  _e  /\  _e  <_  3 )
 
Theoremef0 12621 Value of the exponential function at 0. Equation 2 of [Gleason] p. 308. (Contributed by Steve Rodriguez, 27-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  ( exp `  0
 )  =  1
 
Theoremefcj 12622 Exponential function of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  ( A  e.  CC  ->  ( exp `  ( * `  A ) )  =  ( * `  ( exp `  A )
 ) )
 
Theoremefaddlem 12623* Lemma for efadd 12624 (exponential function addition law). (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   &    |-  G  =  ( n  e.  NN0  |->  ( ( B ^ n )  /  ( ! `  n ) ) )   &    |-  H  =  ( n  e.  NN0  |->  ( ( ( A  +  B ) ^ n )  /  ( ! `  n ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( exp `  ( A  +  B ) )  =  ( ( exp `  A )  x.  ( exp `  B ) ) )
 
Theoremefadd 12624 Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  +  B )
 )  =  ( ( exp `  A )  x.  ( exp `  B ) ) )
 
Theoremefcan 12625 Cancellation of law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.)
 |-  ( A  e.  CC  ->  ( ( exp `  A )  x.  ( exp `  -u A ) )  =  1
 )
 
Theoremefne0 12626 The exponential function never vanishes. Corollary 15-4.3 of [Gleason] p. 309. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  ( A  e.  CC  ->  ( exp `  A )  =/=  0 )
 
Theoremefneg 12627 Exponent of a negative number. (Contributed by Mario Carneiro, 10-May-2014.)
 |-  ( A  e.  CC  ->  ( exp `  -u A )  =  ( 1  /  ( exp `  A ) ) )
 
Theoremeff2 12628 The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
 |- 
 exp : CC --> ( CC  \  { 0 } )
 
Theoremefsub 12629 Difference of exponents law for exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  -  B ) )  =  ( ( exp `  A )  /  ( exp `  B ) ) )
 
Theoremefexp 12630 Exponential function to an integer power. Corollary 15-4.4 of [Gleason] p. 309, restricted to integers. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( N  x.  A ) )  =  ( ( exp `  A ) ^ N ) )
 
Theoremefzval 12631 Value of the exponential function for integers. Special case of efval 12610. Equation 30 of [Rudin] p. 164. (Contributed by Steve Rodriguez, 15-Sep-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( N  e.  ZZ  ->  ( exp `  N )  =  ( _e ^ N ) )
 
Theoremefgt0 12632 The exponential function of a real number is greater than 0. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( A  e.  RR  ->  0  <  ( exp `  A ) )
 
Theoremrpefcl 12633 The exponential function of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
 |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR+ )
 
Theoremrpefcld 12634 The exponential function of a real number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( exp `  A )  e.  RR+ )
 
Theoremeftlcvg 12635* The tail series of the exponential function are convergent. (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
 
Theoremeftlcl 12636* Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  sum_ k  e.  ( ZZ>=
 `  M ) ( F `  k )  e.  CC )
 
Theoremreeftlcl 12637* Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( ( A  e.  RR  /\  M  e.  NN0 )  ->  sum_ k  e.  ( ZZ>=
 `  M ) ( F `  k )  e.  RR )
 
Theoremeftlub 12638* An upper bound on the absolute value of the infinite tail of the series expansion of the exponential function on the closed unit disk. (Contributed by Paul Chapman, 19-Jan-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   &    |-  G  =  ( n  e.  NN0  |->  ( ( ( abs `  A ) ^ n )  /  ( ! `  n ) ) )   &    |-  H  =  ( n  e.  NN0  |->  ( ( ( ( abs `  A ) ^ M )  /  ( ! `  M ) )  x.  ( ( 1  /  ( M  +  1 ) ) ^ n ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <_ 
 1 )   =>    |-  ( ph  ->  ( abs `  sum_ k  e.  ( ZZ>=
 `  M ) ( F `  k ) )  <_  ( (
 ( abs `  A ) ^ M )  x.  (
 ( M  +  1 )  /  ( ( ! `  M )  x.  M ) ) ) )
 
Theoremefsep 12639* Separate out the next term of the power series expansion of the exponential function. The last hypothesis allows the separated terms to be rearranged as desired. (Contributed by Paul Chapman, 23-Nov-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   &    |-  N  =  ( M  +  1 )   &    |-  M  e.  NN0   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( exp `  A )  =  ( B  +  sum_ k  e.  ( ZZ>=
 `  M ) ( F `  k ) ) )   &    |-  ( ph  ->  ( B  +  ( ( A ^ M ) 
 /  ( ! `  M ) ) )  =  D )   =>    |-  ( ph  ->  ( exp `  A )  =  ( D  +  sum_ k  e.  ( ZZ>= `  N ) ( F `  k ) ) )
 
Theoremeffsumlt 12640* The partial sums of the series expansion of the exponential function of a positive real number are bounded by the value of the function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (  seq  0 (  +  ,  F ) `  N )  <  ( exp `  A ) )
 
Theoremeft0val 12641 The value of the first term of the series expansion of the exponential function is 1. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  ( A  e.  CC  ->  ( ( A ^
 0 )  /  ( ! `  0 ) )  =  1 )
 
Theoremef4p 12642* Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  e.  CC  ->  ( exp `  A )  =  ( (
 ( ( 1  +  A )  +  (
 ( A ^ 2
 )  /  2 )
 )  +  ( ( A ^ 3 ) 
 /  6 ) )  +  sum_ k  e.  ( ZZ>=
 `  4 ) ( F `  k ) ) )
 
Theoremefgt1p2 12643 The exponential function of a positive real number is greater than the first three terms of the series expansion. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( A  e.  RR+  ->  ( ( 1  +  A )  +  (
 ( A ^ 2
 )  /  2 )
 )  <  ( exp `  A ) )
 
Theoremefgt1p 12644 The exponential function of a positive real number is greater than 1 plus that number. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( A  e.  RR+  ->  ( 1  +  A )  <  ( exp `  A ) )
 
Theoremefgt1 12645 The exponential function of a positive real number is greater than 1. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( A  e.  RR+  -> 
 1  <  ( exp `  A ) )
 
Theoremeflt 12646 The exponential function on the reals is strictly monotonic. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( exp `  A )  <  ( exp `  B ) ) )
 
Theoremefle 12647 The exponential function on the reals is strictly monotonic. (Contributed by Mario Carneiro, 11-Mar-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <-> 
 ( exp `  A )  <_  ( exp `  B ) ) )
 
Theoremreef11 12648 The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Mario Carneiro, 11-Mar-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( exp `  A )  =  ( exp `  B )  <->  A  =  B ) )
 
Theoremreeff1 12649 The exponential function maps real arguments one-to-one to positive reals. (Contributed by Steve Rodriguez, 25-Aug-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( exp  |`  RR ) : RR -1-1-> RR+
 
Theoremeflegeo 12650 The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A  <  1 )   =>    |-  ( ph  ->  ( exp `  A )  <_  ( 1  /  (
 1  -  A ) ) )
 
Theoremsinval 12651 Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( A  e.  CC  ->  ( sin `  A )  =  ( (
 ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) 
 /  ( 2  x.  _i ) ) )
 
Theoremcosval 12652 Value of the cosine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( A  e.  CC  ->  ( cos `  A )  =  ( (
 ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) ) 
 /  2 ) )
 
Theoremsinf 12653 Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |- 
 sin : CC --> CC
 
Theoremcosf 12654 Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |- 
 cos : CC --> CC
 
Theoremsincl 12655 Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
 
Theoremcoscl 12656 Closure of the cosine function with a complex argument. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
 
Theoremtanval 12657 Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  =  ( ( sin `  A )  /  ( cos `  A ) ) )
 
Theoremtancl 12658 The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  e.  CC )
 
Theoremsincld 12659 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( sin `  A )  e. 
 CC )
 
Theoremcoscld 12660 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( cos `  A )  e. 
 CC )
 
Theoremtancld 12661 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( cos `  A )  =/=  0 )   =>    |-  ( ph  ->  ( tan `  A )  e. 
 CC )
 
Theoremtanval2 12662 Express the tangent function directly in terms of  exp. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  =  ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) 
 /  ( _i  x.  ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) )
 
Theoremtanval3 12663 Express the tangent function directly in terms of  exp. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( ( A  e.  CC  /\  ( ( exp `  ( 2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  ->  ( tan `  A )  =  ( ( ( exp `  ( 2  x.  ( _i  x.  A ) ) )  -  1 ) 
 /  ( _i  x.  ( ( exp `  (
 2  x.  ( _i 
 x.  A ) ) )  +  1 ) ) ) )
 
Theoremresinval 12664 The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  RR  ->  ( sin `  A )  =  ( Im `  ( exp `  ( _i  x.  A ) ) ) )
 
Theoremrecosval 12665 The cosine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  RR  ->  ( cos `  A )  =  ( Re `  ( exp `  ( _i  x.  A ) ) ) )
 
Theoremefi4p 12666* Separate out the first four terms of the infinite series expansion of the exponential function of an imaginary number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^ n )  /  ( ! `  n ) ) )   =>    |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( ( 1  -  ( ( A ^ 2 ) 
 /  2 ) )  +  ( _i  x.  ( A  -  (
 ( A ^ 3
 )  /  6 )
 ) ) )  +  sum_
 k  e.  ( ZZ>= `  4 ) ( F `
  k ) ) )
 
Theoremresin4p 12667* Separate out the first four terms of the infinite series expansion of the sine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^ n )  /  ( ! `  n ) ) )   =>    |-  ( A  e.  RR  ->  ( sin `  A )  =  ( ( A  -  ( ( A ^ 3 )  / 
 6 ) )  +  ( Im `  sum_ k  e.  ( ZZ>= `  4 )
 ( F `  k
 ) ) ) )
 
Theoremrecos4p 12668* Separate out the first four terms of the infinite series expansion of the cosine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^ n )  /  ( ! `  n ) ) )   =>    |-  ( A  e.  RR  ->  ( cos `  A )  =  ( (
 1  -  ( ( A ^ 2 ) 
 /  2 ) )  +  ( Re `  sum_
 k  e.  ( ZZ>= `  4 ) ( F `
  k ) ) ) )
 
Theoremresincl 12669 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  RR  ->  ( sin `  A )  e.  RR )
 
Theoremrecoscl 12670 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  RR  ->  ( cos `  A )  e.  RR )
 
Theoremretancl 12671 The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  ( ( A  e.  RR  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  e.  RR )
 
Theoremresincld 12672 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( sin `  A )  e. 
 RR )
 
Theoremrecoscld 12673 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( cos `  A )  e. 
 RR )
 
Theoremretancld 12674 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  ( cos `  A )  =/=  0 )   =>    |-  ( ph  ->  ( tan `  A )  e. 
 RR )
 
Theoremsinneg 12675 The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  CC  ->  ( sin `  -u A )  =  -u ( sin `  A ) )
 
Theoremcosneg 12676 The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  CC  ->  ( cos `  -u A )  =  ( cos `  A ) )
 
Theoremtanneg 12677 The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.)
 |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  -u A )  =  -u ( tan `  A ) )
 
Theoremsin0 12678 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
 |-  ( sin `  0
 )  =  0
 
Theoremcos0 12679 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
 |-  ( cos `  0
 )  =  1
 
Theoremtan0 12680 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( tan `  0
 )  =  0
 
Theoremefival 12681 The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( cos `  A )  +  ( _i  x.  ( sin `  A ) ) ) )
 
Theoremefmival 12682 The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.)
 |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  =  ( ( cos `  A )  -  ( _i  x.  ( sin `  A ) ) ) )
 
Theoremsinhval 12683 Value of the hyperbolic sine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  CC  ->  ( ( sin `  ( _i  x.  A ) ) 
 /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A ) ) 
 /  2 ) )
 
Theoremcoshval 12684 Value of the hyperbolic cosine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  CC  ->  ( cos `  ( _i  x.  A ) )  =  ( ( ( exp `  A )  +  ( exp `  -u A ) )  /  2
 ) )
 
Theoremresinhcl 12685 The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( ( sin `  ( _i  x.  A ) ) 
 /  _i )  e. 
 RR )
 
Theoremrpcoshcl 12686 The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR+ )
 
Theoremrecoshcl 12687 The hyperbolic cosine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR )
 
Theoremretanhcl 12688 The hyperbolic tangent of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( ( tan `  ( _i  x.  A ) ) 
 /  _i )  e. 
 RR )
 
Theoremtanhlt1 12689 The hyperbolic tangent of a real number is upper bounded by  1. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( ( tan `  ( _i  x.  A ) ) 
 /  _i )  < 
 1 )
 
Theoremtanhbnd 12690 The hyperbolic tangent of a real number is bounded by  1. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( ( tan `  ( _i  x.  A ) ) 
 /  _i )  e.  ( -u 1 (,) 1
 ) )
 
Theoremefeul 12691 Eulerian representation of the complex exponential. (Suggested by Jeffrey Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.)
 |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( exp `  ( Re `  A ) )  x.  ( ( cos `  ( Im `  A ) )  +  ( _i  x.  ( sin `  ( Im `  A ) ) ) ) ) )
 
Theoremefieq 12692 The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( exp `  ( _i  x.  A ) )  =  ( exp `  ( _i  x.  B ) )  <->  ( ( cos `  A )  =  ( cos `  B )  /\  ( sin `  A )  =  ( sin `  B ) ) ) )
 
Theoremsinadd 12693 Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( sin `  ( A  +  B )
 )  =  ( ( ( sin `  A )  x.  ( cos `  B ) )  +  (
 ( cos `  A )  x.  ( sin `  B ) ) ) )
 
Theoremcosadd 12694 Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  +  B )
 )  =  ( ( ( cos `  A )  x.  ( cos `  B ) )  -  (
 ( sin `  A )  x.  ( sin `  B ) ) ) )
 
Theoremtanaddlem 12695 A useful intermediate step in tanadd 12696 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
 ( cos `  A )  =/=  0  /\  ( cos `  B )  =/=  0
 ) )  ->  (
 ( cos `  ( A  +  B ) )  =/=  0  <->  ( ( tan `  A )  x.  ( tan `  B ) )  =/=  1 ) )
 
Theoremtanadd 12696 Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
 ( cos `  A )  =/=  0  /\  ( cos `  B )  =/=  0  /\  ( cos `  ( A  +  B )
 )  =/=  0 )
 )  ->  ( tan `  ( A  +  B ) )  =  (
 ( ( tan `  A )  +  ( tan `  B ) )  /  ( 1  -  (
 ( tan `  A )  x.  ( tan `  B ) ) ) ) )
 
Theoremsinsub 12697 Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( sin `  ( A  -  B ) )  =  ( ( ( sin `  A )  x.  ( cos `  B ) )  -  (
 ( cos `  A )  x.  ( sin `  B ) ) ) )
 
Theoremcossub 12698 Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  -  B ) )  =  ( ( ( cos `  A )  x.  ( cos `  B ) )  +  (
 ( sin `  A )  x.  ( sin `  B ) ) ) )
 
Theoremaddsin 12699 Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( sin `  A )  +  ( sin `  B ) )  =  ( 2  x.  ( ( sin `  (
 ( A  +  B )  /  2 ) )  x.  ( cos `  (
 ( A  -  B )  /  2 ) ) ) ) )
 
Theoremsubsin 12700 Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( sin `  A )  -  ( sin `  B ) )  =  ( 2  x.  ( ( cos `  (
 ( A  +  B )  /  2 ) )  x.  ( sin `  (
 ( A  -  B )  /  2 ) ) ) ) )
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