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Theorem List for Metamath Proof Explorer - 12401-12500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeflegeo 12401 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 12402 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 12403 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 12404 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 12405 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 12406 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 12407 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 12408 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 12409 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 12410 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( sin `  A )  e. 
 CC )
 
Theoremcoscld 12411 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( cos `  A )  e. 
 CC )
 
Theoremtancld 12412 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 12413 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 12414 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 12415 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 12416 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 12417* 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 12418* 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 12419* 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 12420 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  RR  ->  ( sin `  A )  e.  RR )
 
Theoremrecoscl 12421 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  RR  ->  ( cos `  A )  e.  RR )
 
Theoremretancl 12422 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 12423 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( sin `  A )  e. 
 RR )
 
Theoremrecoscld 12424 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( cos `  A )  e. 
 RR )
 
Theoremretancld 12425 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 12426 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 12427 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 12428 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 12429 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
 |-  ( sin `  0
 )  =  0
 
Theoremcos0 12430 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
 |-  ( cos `  0
 )  =  1
 
Theoremtan0 12431 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( tan `  0
 )  =  0
 
Theoremefival 12432 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 12433 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 12434 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 12435 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 12436 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 12437 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 12438 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 12439 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 12440 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 12441 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 12442 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 12443 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 12444 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 12445 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 12446 A useful intermediate step in tanadd 12447 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 12447 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 12448 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 12449 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 12450 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 12451 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 12452 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12445 and cossub 12449. (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 12453 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12445 and cossub 12449. (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 12454 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 12455 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 12456 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 12457 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 12458 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 12459 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 12460 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 12461 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 12462 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 12463 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 12464* Lemma for sin01bnd 12465 and cos01bnd 12466. (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 12465 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 12466 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 12467 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( ( 1  / 
 3 )  <  ( cos `  1 )  /\  ( cos `  1 )  <  ( 2  /  3
 ) )
 
Theoremcos2bnd 12468 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( -u ( 7  / 
 9 )  <  ( cos `  2 )  /\  ( cos `  2 )  < 
 -u ( 1  / 
 9 ) )
 
Theoremsinltx 12469 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 12470 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 12471 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 12472 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 12473 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  1 )  /\  0  <  ( cos `  1
 ) )
 
Theoremsincos2sgn 12474 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  2 )  /\  ( cos `  2 )  <  0 )
 
Theoremsin4lt0 12475 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( sin `  4
 )  <  0
 
Theoremabsefi 12476 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 12477 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 12478 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 12479 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 12480 De Moivre's Formula. Shorter proof of demoivreALT 12481 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 12481 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 12482* Lemma for eirr 12483. (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 12483  _e is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
 |-  _e  e/  QQ
 
Theoremegt2lt3 12484 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 12485 Euler's constant  _e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
 |-  0  <  _e
 
Theoremepr 12486 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 12487 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 12488 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 11287 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 12489 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 6981 in xpnnen 12487). (Contributed by NM, 23-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( om  X.  om )  ~~  om
 
Theoremznnenlem 12490 Lemma for znnen 12491. (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 12491 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
 
Theoremqnnen 12492 The rational numbers are countable. This proof does not use the Axiom of Choice, even though it uses an onto function, because the base set  ( ZZ  X.  NN ) is numerable. Exercise 2 of [Enderton] p. 133. (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 3-Mar-2013.)
 |- 
 QQ  ~~  NN
 
5.10.2  The reals are uncountable
 
Theoremrpnnen2lem1 12493* Lemma for rpnnen2 12504. (Contributed by Mario Carneiro, 13-May-2013.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  NN  /\  N  e.  NN )  ->  ( ( F `
  A ) `  N )  =  if ( N  e.  A ,  ( ( 1  / 
 3 ) ^ N ) ,  0 )
 )
 
Theoremrpnnen2lem2 12494* Lemma for rpnnen2 12504. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( A  C_  NN  ->  ( F `  A ) : NN --> RR )
 
Theoremrpnnen2lem3 12495* Lemma for rpnnen2 12504. (Contributed by Mario Carneiro, 13-May-2013.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |- 
 seq  1 (  +  ,  ( F `  NN ) )  ~~>  ( 1  / 
 2 )
 
Theoremrpnnen2lem4 12496* Lemma for rpnnen2 12504. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 31-Aug-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  B  /\  B  C_  NN  /\  k  e.  NN )  ->  ( 0  <_  (
 ( F `  A ) `  k )  /\  ( ( F `  A ) `  k
 )  <_  ( ( F `  B ) `  k ) ) )
 
Theoremrpnnen2lem5 12497* Lemma for rpnnen2 12504. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  NN  /\  M  e.  NN )  ->  seq  M (  +  ,  ( F `  A ) )  e. 
 dom 
 ~~>  )
 
Theoremrpnnen2lem6 12498* Lemma for rpnnen2 12504. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  NN  /\  M  e.  NN )  ->  sum_ k  e.  ( ZZ>=
 `  M ) ( ( F `  A ) `  k )  e. 
 RR )
 
Theoremrpnnen2lem7 12499* Lemma for rpnnen2 12504. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  -> 
 sum_ k  e.  ( ZZ>=
 `  M ) ( ( F `  A ) `  k )  <_  sum_ k  e.  ( ZZ>= `  M ) ( ( F `  B ) `
  k ) )
 
Theoremrpnnen2lem8 12500* Lemma for rpnnen2 12504. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  NN  /\  M  e.  NN )  ->  sum_ k  e.  NN  ( ( F `  A ) `  k
 )  =  ( sum_ k  e.  ( 1 ... ( M  -  1
 ) ) ( ( F `  A ) `
  k )  +  sum_
 k  e.  ( ZZ>= `  M ) ( ( F `  A ) `
  k ) ) )
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