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Theorem List for Intuitionistic Logic Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremefgt1p 11301 The exponential 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 11302 The exponential 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 ) )
 
Theoremefltim 11303 The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B 
 ->  ( exp `  A )  <  ( exp `  B ) ) )
 
Theoremefler 11304 The exponential function on the reals is nondecreasing. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Jim Kingdon, 20-Dec-2022.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( exp `  A )  <_  ( exp `  B )  ->  A  <_  B ) )
 
Theoremreef11 11305 The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Jim Kingdon, 20-Dec-2022.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( exp `  A )  =  ( exp `  B )  <->  A  =  B ) )
 
Theoremreeff1 11306 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 11307 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 11308 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 11309 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 11310 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 11311 Domain and codomain of the cosine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |- 
 cos : CC --> CC
 
Theoremsincl 11312 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 11313 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 )
 
Theoremtanvalap 11314 Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
 |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( tan `  A )  =  ( ( sin `  A )  /  ( cos `  A ) ) )
 
Theoremtanclap 11315 The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
 |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( tan `  A )  e. 
 CC )
 
Theoremsincld 11316 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( sin `  A )  e. 
 CC )
 
Theoremcoscld 11317 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( cos `  A )  e. 
 CC )
 
Theoremtanclapd 11318 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( cos `  A ) #  0 )   =>    |-  ( ph  ->  ( tan `  A )  e. 
 CC )
 
Theoremtanval2ap 11319 Express the tangent function directly in terms of  exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
 |-  ( ( 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 ) ) ) ) ) )
 
Theoremtanval3ap 11320 Express the tangent function directly in terms of  exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
 |-  ( ( 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 11321 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 11322 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 11323* 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, 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 11324* 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 11325* 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 11326 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  RR  ->  ( sin `  A )  e.  RR )
 
Theoremrecoscl 11327 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  RR  ->  ( cos `  A )  e.  RR )
 
Theoremretanclap 11328 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 11329 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( sin `  A )  e. 
 RR )
 
Theoremrecoscld 11330 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( cos `  A )  e. 
 RR )
 
Theoremretanclapd 11331 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 11332 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 11333 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 ) )
 
Theoremtannegap 11334 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 11335 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
 |-  ( sin `  0
 )  =  0
 
Theoremcos0 11336 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
 |-  ( cos `  0
 )  =  1
 
Theoremtan0 11337 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( tan `  0
 )  =  0
 
Theoremefival 11338 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 11339 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 ) ) ) )
 
Theoremefeul 11340 Eulerian representation of the complex exponential. (Suggested by Jeff 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 11341 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 11342 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 11343 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 ) ) ) )
 
Theoremtanaddaplem 11344 A useful intermediate step in tanaddap 11345 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
 ( cos `  A ) #  0  /\  ( cos `  B ) #  0 ) )  ->  ( ( cos `  ( A  +  B )
 ) #  0  <->  ( ( tan `  A )  x.  ( tan `  B ) ) #  1 ) )
 
Theoremtanaddap 11345 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 11346 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 11347 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 11348 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 11349 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 11350 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11343 and cossub 11347. (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 11351 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 11343 and cossub 11347. (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 11352 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 11353 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 11354 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 11355 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 11356 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 11357 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 11358 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 11359 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 11360 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 11361 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 11362* Lemma for sin01bnd 11363 and cos01bnd 11364. (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 11363 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 11364 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 11365 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( ( 1  / 
 3 )  <  ( cos `  1 )  /\  ( cos `  1 )  <  ( 2  /  3
 ) )
 
Theoremcos2bnd 11366 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( -u ( 7  / 
 9 )  <  ( cos `  2 )  /\  ( cos `  2 )  < 
 -u ( 1  / 
 9 ) )
 
Theoremsin01gt0 11367 The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Wolf Lammen, 25-Sep-2020.)
 |-  ( A  e.  (
 0 (,] 1 )  -> 
 0  <  ( sin `  A ) )
 
Theoremcos01gt0 11368 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 11369 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 11370 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  1 )  /\  0  <  ( cos `  1
 ) )
 
Theoremsincos2sgn 11371 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  2 )  /\  ( cos `  2 )  <  0 )
 
Theoremsin4lt0 11372 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( sin `  4
 )  <  0
 
Theoremabsefi 11373 The absolute value of the exponential 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 11374 The absolute value of the exponential is the exponential of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)
 |-  ( A  e.  CC  ->  ( abs `  ( exp `  A ) )  =  ( exp `  ( Re `  A ) ) )
 
Theoremabsefib 11375 A complex number is real iff the exponential of its product with  _i has absolute value one. (Contributed by NM, 21-Aug-2008.)
 |-  ( A  e.  CC  ->  ( A  e.  RR  <->  ( abs `  ( exp `  ( _i  x.  A ) ) )  =  1 ) )
 
Theoremefieq1re 11376 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 11377 De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. See also demoivreALT 11378 for an alternate longer proof not 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 11378 Alternate proof of demoivre 11377. It is longer but does not use the exponential function. This is Metamath 100 proof #17. (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 ) ) ) ) )
 
4.9.2  _e is irrational
 
Theoremeirraplem 11379* Lemma for eirrap 11380. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.)
 |-  F  =  ( n  e.  NN0  |->  ( 1 
 /  ( ! `  n ) ) )   &    |-  ( ph  ->  P  e.  ZZ )   &    |-  ( ph  ->  Q  e.  NN )   =>    |-  ( ph  ->  _e #  ( P  /  Q ) )
 
Theoremeirrap 11380  _e is irrational. That is, for any rational number,  _e is apart from it. In the absence of excluded middle, we can distinguish between this and saying that  _e is not rational, which is eirr 11381. (Contributed by Jim Kingdon, 6-Jan-2023.)
 |-  ( Q  e.  QQ  ->  _e #  Q )
 
Theoremeirr 11381  _e is not rational. In the absence of excluded middle, we can distinguish between this and saying that  _e is irrational in the sense of being apart from any rational number, which is eirrap 11380. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
 |-  _e  e/  QQ
 
Theoremegt2lt3 11382 Euler's constant  _e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Jim Kingdon, 7-Jan-2023.)
 |-  ( 2  <  _e  /\  _e  <  3 )
 
Theoremepos 11383 Euler's constant  _e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
 |-  0  <  _e
 
Theoremepr 11384 Euler's constant  _e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
 |-  _e  e.  RR+
 
Theoremene0 11385  _e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  0
 
Theoremeap0 11386  _e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
 |-  _e #  0
 
Theoremene1 11387  _e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  1
 
Theoremeap1 11388  _e is apart from 1. (Contributed by Jim Kingdon, 7-Jan-2023.)
 |-  _e #  1
 
PART 5  ELEMENTARY NUMBER THEORY

Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory.

 
5.1  Elementary properties of divisibility
 
5.1.1  The divides relation
 
Syntaxcdvds 11389 Extend the definition of a class to include the divides relation. See df-dvds 11390.
 class  ||
 
Definitiondf-dvds 11390* Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ||  =  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x )  =  y ) }
 
Theoremdivides 11391* Define the divides relation.  M  ||  N means  M divides into  N with no remainder. For example,  3  ||  6 (ex-dvds 12753). As proven in dvdsval3 11393, 
M  ||  N  <->  ( N  mod  M )  =  0. See divides 11391 and dvdsval2 11392 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <-> 
 E. n  e.  ZZ  ( n  x.  M )  =  N )
 )
 
Theoremdvdsval2 11392 One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )
 
Theoremdvdsval3 11393 One nonzero integer divides another integer if and only if the remainder upon division is zero, see remark in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.)
 |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  ( N  mod  M )  =  0 )
 )
 
Theoremdvdszrcl 11394 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  ( X  ||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ )
 )
 
Theoremnndivdvds 11395 Strong form of dvdsval2 11392 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( B  ||  A 
 <->  ( A  /  B )  e.  NN )
 )
 
Theoremnndivides 11396* Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  ||  N 
 <-> 
 E. n  e.  NN  ( n  x.  M )  =  N )
 )
 
Theoremdvdsdc 11397 Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
 |-  ( ( M  e.  NN  /\  N  e.  ZZ )  -> DECID  M  ||  N )
 
Theoremmoddvds 11398 Two ways to say  A  ==  B (mod  N), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  mod  N )  =  ( B 
 mod  N )  <->  N  ||  ( A  -  B ) ) )
 
Theoremdvds0lem 11399 A lemma to assist theorems of 
|| with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  x.  M )  =  N )  ->  M  ||  N )
 
Theoremdvds1lem 11400* A lemma to assist theorems of 
|| with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ph  ->  ( J  e.  ZZ  /\  K  e.  ZZ ) )   &    |-  ( ph  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )   &    |-  ( ( ph  /\  x  e.  ZZ )  ->  Z  e.  ZZ )   &    |-  (
 ( ph  /\  x  e. 
 ZZ )  ->  (
 ( x  x.  J )  =  K  ->  ( Z  x.  M )  =  N ) )   =>    |-  ( ph  ->  ( J  ||  K  ->  M  ||  N ) )
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