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Theorem List for Metamath Proof Explorer - 12701-12800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeft0val 12701 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 12702* 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 12703 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 12704 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 12705 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 12706 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 12707 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 12708 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 12709 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 12710 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 12711 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 12712 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 12713 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 12714 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 12715 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 12716 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 12717 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 12718 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 12719 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( sin `  A )  e. 
 CC )
 
Theoremcoscld 12720 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( cos `  A )  e. 
 CC )
 
Theoremtancld 12721 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 12722 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 12723 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 12724 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 12725 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 12726* 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 12727* 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 12728* 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 12729 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  RR  ->  ( sin `  A )  e.  RR )
 
Theoremrecoscl 12730 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  RR  ->  ( cos `  A )  e.  RR )
 
Theoremretancl 12731 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 12732 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( sin `  A )  e. 
 RR )
 
Theoremrecoscld 12733 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( cos `  A )  e. 
 RR )
 
Theoremretancld 12734 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 12735 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 12736 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 12737 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 12738 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
 |-  ( sin `  0
 )  =  0
 
Theoremcos0 12739 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
 |-  ( cos `  0
 )  =  1
 
Theoremtan0 12740 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( tan `  0
 )  =  0
 
Theoremefival 12741 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 12742 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 12743 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 12744 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 12745 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 12746 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 12747 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 12748 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 12749 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 12750 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 12751 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 12752 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 12753 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 12754 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 12755 A useful intermediate step in tanadd 12756 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 12756 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 12757 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 12758 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 12759 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 12760 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 ) ) ) ) )
 
Theoremsinmul 12761 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12754 and cossub 12758. (Contributed by David A. Wheeler, 26-May-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( sin `  A )  x.  ( sin `  B ) )  =  ( ( ( cos `  ( A  -  B ) )  -  ( cos `  ( A  +  B ) ) ) 
 /  2 ) )
 
Theoremcosmul 12762 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12754 and cossub 12758. (Contributed by David A. Wheeler, 26-May-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( cos `  A )  x.  ( cos `  B ) )  =  ( ( ( cos `  ( A  -  B ) )  +  ( cos `  ( A  +  B ) ) ) 
 /  2 ) )
 
Theoremaddcos 12763 Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( cos `  A )  +  ( cos `  B ) )  =  ( 2  x.  ( ( cos `  (
 ( A  +  B )  /  2 ) )  x.  ( cos `  (
 ( A  -  B )  /  2 ) ) ) ) )
 
Theoremsubcos 12764 Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( cos `  B )  -  ( cos `  A ) )  =  ( 2  x.  ( ( sin `  (
 ( A  +  B )  /  2 ) )  x.  ( sin `  (
 ( A  -  B )  /  2 ) ) ) ) )
 
Theoremsincossq 12765 Sine squared plus cosine squared is 1. Equation 17 of [Gleason] p. 311. Note that this holds for non-real arguments, even though individually each term is unbounded. (Contributed by NM, 15-Jan-2006.)
 |-  ( A  e.  CC  ->  ( ( ( sin `  A ) ^ 2
 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
 
Theoremsin2t 12766 Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
 |-  ( A  e.  CC  ->  ( sin `  (
 2  x.  A ) )  =  ( 2  x.  ( ( sin `  A )  x.  ( cos `  A ) ) ) )
 
Theoremcos2t 12767 Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  CC  ->  ( cos `  (
 2  x.  A ) )  =  ( ( 2  x.  ( ( cos `  A ) ^ 2 ) )  -  1 ) )
 
Theoremcos2tsin 12768 Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.)
 |-  ( A  e.  CC  ->  ( cos `  (
 2  x.  A ) )  =  ( 1  -  ( 2  x.  ( ( sin `  A ) ^ 2 ) ) ) )
 
Theoremsinbnd 12769 The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)
 |-  ( A  e.  RR  ->  ( -u 1  <_  ( sin `  A )  /\  ( sin `  A )  <_  1 ) )
 
Theoremcosbnd 12770 The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)
 |-  ( A  e.  RR  ->  ( -u 1  <_  ( cos `  A )  /\  ( cos `  A )  <_  1 ) )
 
Theoremsinbnd2 12771 The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  ( A  e.  RR  ->  ( sin `  A )  e.  ( -u 1 [,] 1 ) )
 
Theoremcosbnd2 12772 The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  ( A  e.  RR  ->  ( cos `  A )  e.  ( -u 1 [,] 1 ) )
 
Theoremef01bndlem 12773* Lemma for sin01bnd 12774 and cos01bnd 12775. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^ n )  /  ( ! `  n ) ) )   =>    |-  ( A  e.  (
 0 (,] 1 )  ->  ( abs `  sum_ k  e.  ( ZZ>= `  4 )
 ( F `  k
 ) )  <  (
 ( A ^ 4
 )  /  6 )
 )
 
Theoremsin01bnd 12774 Bounds on the sine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( A  e.  (
 0 (,] 1 )  ->  ( ( A  -  ( ( A ^
 3 )  /  3
 ) )  <  ( sin `  A )  /\  ( sin `  A )  <  A ) )
 
Theoremcos01bnd 12775 Bounds on the cosine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( A  e.  (
 0 (,] 1 )  ->  ( ( 1  -  ( 2  x.  (
 ( A ^ 2
 )  /  3 )
 ) )  <  ( cos `  A )  /\  ( cos `  A )  <  ( 1  -  (
 ( A ^ 2
 )  /  3 )
 ) ) )
 
Theoremcos1bnd 12776 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( ( 1  / 
 3 )  <  ( cos `  1 )  /\  ( cos `  1 )  <  ( 2  /  3
 ) )
 
Theoremcos2bnd 12777 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( -u ( 7  / 
 9 )  <  ( cos `  2 )  /\  ( cos `  2 )  < 
 -u ( 1  / 
 9 ) )
 
Theoremsinltx 12778 The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( A  e.  RR+  ->  ( sin `  A )  <  A )
 
Theoremsin01gt0 12779 The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( A  e.  (
 0 (,] 1 )  -> 
 0  <  ( sin `  A ) )
 
Theoremcos01gt0 12780 The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( A  e.  (
 0 (,] 1 )  -> 
 0  <  ( cos `  A ) )
 
Theoremsin02gt0 12781 The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( A  e.  (
 0 (,] 2 )  -> 
 0  <  ( sin `  A ) )
 
Theoremsincos1sgn 12782 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  1 )  /\  0  <  ( cos `  1
 ) )
 
Theoremsincos2sgn 12783 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  2 )  /\  ( cos `  2 )  <  0 )
 
Theoremsin4lt0 12784 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( sin `  4
 )  <  0
 
Theoremabsefi 12785 The absolute value of the exponential function of an imaginary number is one. Equation 48 of [Rudin] p. 167. (Contributed by Jason Orendorff, 9-Feb-2007.)
 |-  ( A  e.  RR  ->  ( abs `  ( exp `  ( _i  x.  A ) ) )  =  1 )
 
Theoremabsef 12786 The absolute value of the exponential function is the exponential function of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)
 |-  ( A  e.  CC  ->  ( abs `  ( exp `  A ) )  =  ( exp `  ( Re `  A ) ) )
 
Theoremabsefib 12787 A number is real iff its imaginary exponential has absolute value one. (Contributed by NM, 21-Aug-2008.)
 |-  ( A  e.  CC  ->  ( A  e.  RR  <->  ( abs `  ( exp `  ( _i  x.  A ) ) )  =  1 ) )
 
Theoremefieq1re 12788 A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)
 |-  ( ( A  e.  CC  /\  ( exp `  ( _i  x.  A ) )  =  1 )  ->  A  e.  RR )
 
Theoremdemoivre 12789 De Moivre's Formula. Shorter proof of demoivreALT 12790 using the exponential function. (Contributed by NM, 24-Jul-2007.)
 |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( ( ( cos `  A )  +  ( _i  x.  ( sin `  A ) ) ) ^ N )  =  ( ( cos `  ( N  x.  A ) )  +  ( _i  x.  ( sin `  ( N  x.  A ) ) ) ) )
 
TheoremdemoivreALT 12790 De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( ( ( cos `  A )  +  ( _i  x.  ( sin `  A ) ) ) ^ N )  =  ( ( cos `  ( N  x.  A ) )  +  ( _i  x.  ( sin `  ( N  x.  A ) ) ) ) )
 
5.9.2  _e is irrational
 
Theoremeirrlem 12791* Lemma for eirr 12792. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( 1 
 /  ( ! `  n ) ) )   &    |-  ( ph  ->  P  e.  ZZ )   &    |-  ( ph  ->  Q  e.  NN )   &    |-  ( ph  ->  _e  =  ( P  /  Q ) )   =>    |- 
 -.  ph
 
Theoremeirr 12792  _e is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
 |-  _e  e/  QQ
 
Theoremegt2lt3 12793 Euler's constant  _e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  ( 2  <  _e  /\  _e  <  3 )
 
Theoremepos 12794 Euler's constant  _e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
 |-  0  <  _e
 
Theoremepr 12795 Euler's constant  _e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
 |-  _e  e.  RR+
 
5.10  Cardinality of real and complex number subsets
 
5.10.1  Countability of integers and rationals
 
Theoremxpnnen 12796 The cross product of the set of natural numbers with itself is equinumerous to the set of natural numbers. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
 |-  ( NN  X.  NN )  ~~  NN
 
TheoremxpnnenOLD 12797 The cross product of the set of natural numbers with itself is equinumerous to the set of natural numbers. The key idea is to use nn0opth2 11553 to show that the mapping from natural numbers  z and  w to  ( ( z  +  w ) ^
2 )  +  w is one-to-one. (Contributed by NM, 1-Aug-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( NN  X.  NN )  ~~  NN
 
TheoremxpomenOLD 12798 The cross product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133 (which proves this with a direct, but longer, proof; ours uses instead the Schroeder-Bernstein Theorem sbth 7218 in xpnnen 12796). (Contributed by NM, 23-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( om  X.  om )  ~~  om
 
Theoremznnenlem 12799 Lemma for znnen 12800. (Contributed by NM, 31-Jul-2004.)
 |-  ( ( ( 0 
 <_  x  /\  -.  0  <_  y )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  =  y  <->  ( 2  x.  x )  =  ( ( -u 2  x.  y )  +  1 ) ) )
 
Theoremznnen 12800 The set of integers and the set of natural numbers are equinumerous. Exercise 1 of [Gleason] p. 140. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 13-Jun-2014.)
 |- 
 ZZ  ~~  NN
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