P. 115 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11401-11500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	efgt1p2 11401	The exponential of a positive real number is greater than the sum of 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 ) )
Theorem	efgt1p 11402	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 ) )
Theorem	efgt1 11403	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 ) )
Theorem	efltim 11404	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 ) ) )
Theorem	efler 11405	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 ) )
Theorem	reef11 11406	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 ) )
Theorem	reeff1 11407	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+
Theorem	eflegeo 11408	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 ) ) )
Theorem	sinval 11409	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 ) ) )
Theorem	cosval 11410	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 ) )
Theorem	sinf 11411	Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
|- sin : CC --> CC
Theorem	cosf 11412	Domain and codomain of the cosine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
|- cos : CC --> CC
Theorem	sincl 11413	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 )
Theorem	coscl 11414	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 )
Theorem	tanvalap 11415	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 ) ) )
Theorem	tanclap 11416	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 )
Theorem	sincld 11417	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( sin ` A ) e. CC )
Theorem	coscld 11418	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( cos ` A ) e. CC )
Theorem	tanclapd 11419	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 )
Theorem	tanval2ap 11420	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 ) ) ) ) ) )
Theorem	tanval3ap 11421	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 ) ) ) )
Theorem	resinval 11422	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 ) ) ) )
Theorem	recosval 11423	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 ) ) ) )
Theorem	efi4p 11424*	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 ) ) )
Theorem	resin4p 11425*	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 ) ) ) )
Theorem	recos4p 11426*	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 ) ) ) )
Theorem	resincl 11427	The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( sin ` A ) e. RR )
Theorem	recoscl 11428	The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( cos ` A ) e. RR )
Theorem	retanclap 11429	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 )
Theorem	resincld 11430	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( sin ` A ) e. RR )
Theorem	recoscld 11431	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( cos ` A ) e. RR )
Theorem	retanclapd 11432	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 )
Theorem	sinneg 11433	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 ) )
Theorem	cosneg 11434	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 ) )
Theorem	tannegap 11435	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 ) )
Theorem	sin0 11436	Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
|- ( sin ` 0 ) = 0
Theorem	cos0 11437	Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
|- ( cos ` 0 ) = 1
Theorem	tan0 11438	The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
|- ( tan ` 0 ) = 0
Theorem	efival 11439	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 ) ) ) )
Theorem	efmival 11440	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 ) ) ) )
Theorem	efeul 11441	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 ) ) ) ) ) )
Theorem	efieq 11442	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 ) ) ) )
Theorem	sinadd 11443	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 ) ) ) )
Theorem	cosadd 11444	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 ) ) ) )
Theorem	tanaddaplem 11445	A useful intermediate step in tanaddap 11446 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 ) )
Theorem	tanaddap 11446	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 ) ) ) ) )
Theorem	sinsub 11447	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 ) ) ) )
Theorem	cossub 11448	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 ) ) ) )
Theorem	addsin 11449	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 ) ) ) ) )
Theorem	subsin 11450	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 ) ) ) ) )
Theorem	sinmul 11451	Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11444 and cossub 11448. (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 ) )
Theorem	cosmul 11452	Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 11444 and cossub 11448. (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 ) )
Theorem	addcos 11453	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 ) ) ) ) )
Theorem	subcos 11454	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 ) ) ) ) )
Theorem	sincossq 11455	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 )
Theorem	sin2t 11456	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 ) ) ) )
Theorem	cos2t 11457	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 ) )
Theorem	cos2tsin 11458	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 ) ) ) )
Theorem	sinbnd 11459	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 ) )
Theorem	cosbnd 11460	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 ) )
Theorem	sinbnd2 11461	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 ) )
Theorem	cosbnd2 11462	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 ) )
Theorem	ef01bndlem 11463*	Lemma for sin01bnd 11464 and cos01bnd 11465. (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 ) )
Theorem	sin01bnd 11464	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 ) )
Theorem	cos01bnd 11465	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 ) ) ) )
Theorem	cos1bnd 11466	Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( ( 1 / 3 ) < ( cos ` 1 ) /\ ( cos ` 1 ) < ( 2 / 3 ) )
Theorem	cos2bnd 11467	Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( -u ( 7 / 9 ) < ( cos ` 2 ) /\ ( cos ` 2 ) < -u ( 1 / 9 ) )
Theorem	sin01gt0 11468	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 ) )
Theorem	cos01gt0 11469	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 ) )
Theorem	sin02gt0 11470	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 ) )
Theorem	sincos1sgn 11471	The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( 0 < ( sin ` 1 ) /\ 0 < ( cos ` 1 ) )
Theorem	sincos2sgn 11472	The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( 0 < ( sin ` 2 ) /\ ( cos ` 2 ) < 0 )
Theorem	sin4lt0 11473	The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( sin ` 4 ) < 0
Theorem	cos12dec 11474	Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.)
|- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` B ) < ( cos ` A ) )
Theorem	absefi 11475	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 )
Theorem	absef 11476	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 ) ) )
Theorem	absefib 11477	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 ) )
Theorem	efieq1re 11478	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 )
Theorem	demoivre 11479	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 11480 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 ) ) ) ) )
Theorem	demoivreALT 11480	Alternate proof of demoivre 11479. 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 ) ) ) ) )
4.9.1.1  The circle constant (tau = 2 pi)
Syntax	ctau 11481	Extend class notation to include the constant tau, tau = 6.28318....
class tau
Definition	df-tau 11482	Define the circle constant tau, tau = 6.28318..., which is the smallest positive real number whose cosine is one. Various notations have been used or proposed for this number including tau, a three-legged variant of pi, or 2 pi. Note the difference between this constant tau and the formula variable ta. Following our convention, the constant is displayed in upright font while the variable is in italic font; furthermore, the colors are different. (Contributed by Jim Kingdon, 9-Apr-2018.) (Revised by AV, 1-Oct-2020.)
|- tau = inf ( ( RR+ i^i ( `' cos " { 1 } ) ) , RR , < )
4.9.2  _e is irrational
Theorem	eirraplem 11483*	Lemma for eirrap 11484. (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 ) )
Theorem	eirrap 11484	_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 11485. (Contributed by Jim Kingdon, 6-Jan-2023.)
|- ( Q e. QQ -> _e # Q )
Theorem	eirr 11485	_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 11484. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
|- _e e/ QQ
Theorem	egt2lt3 11486	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 )
Theorem	epos 11487	Euler's constant _e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
|- 0 < _e
Theorem	epr 11488	Euler's constant _e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
|- _e e. RR+
Theorem	ene0 11489	_e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
|- _e =/= 0
Theorem	eap0 11490	_e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
|- _e # 0
Theorem	ene1 11491	_e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
|- _e =/= 1
Theorem	eap1 11492	_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
Syntax	cdvds 11493	Extend the definition of a class to include the divides relation. See df-dvds 11494.
class ||
Definition	df-dvds 11494*	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 ) }
Theorem	divides 11495*	Define the divides relation. M || N means M divides into N with no remainder. For example, 3 || 6 (ex-dvds 12942). As proven in dvdsval3 11497, M || N <-> ( N mod M ) = 0. See divides 11495 and dvdsval2 11496 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 ) )
Theorem	dvdsval2 11496	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 ) )
Theorem	dvdsval3 11497	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 ) )
Theorem	dvdszrcl 11498	Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- ( X || Y -> ( X e. ZZ /\ Y e. ZZ ) )
Theorem	nndivdvds 11499	Strong form of dvdsval2 11496 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( B || A <-> ( A / B ) e. NN ) )
Theorem	nndivides 11500*	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 ) )