P. 125 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12401-12500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eftlub 12401*	An upper bound on the absolute value of the infinite tail of the series expansion of the exponential function on the closed unit disk. (Contributed by Paul Chapman, 19-Jan-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   &    |- G = ( n e. NN0 |-> ( ( ( abs ` A ) ^ n ) / ( ! ` n ) ) )   &    |- H = ( n e. NN0 |-> ( ( ( ( abs ` A ) ^ M ) / ( ! ` M ) ) x. ( ( 1 / ( M + 1 ) ) ^ n ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> A e. CC )   &    |- ( ph -> ( abs ` A ) <_ 1 )   =>    |- ( ph -> ( abs ` sum_ k e. ( ZZ>= ` M ) ( F ` k ) ) <_ ( ( ( abs ` A ) ^ M ) x. ( ( M + 1 ) / ( ( ! ` M ) x. M ) ) ) )
Theorem	efsep 12402*	Separate out the next term of the power series expansion of the exponential function. The last hypothesis allows the separated terms to be rearranged as desired. (Contributed by Paul Chapman, 23-Nov-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   &    |- N = ( M + 1 )   &    |- M e. NN0   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> ( exp ` A ) = ( B + sum_ k e. ( ZZ>= ` M ) ( F ` k ) ) )   &    |- ( ph -> ( B + ( ( A ^ M ) / ( ! ` M ) ) ) = D )   =>    |- ( ph -> ( exp ` A ) = ( D + sum_ k e. ( ZZ>= ` N ) ( F ` k ) ) )
Theorem	effsumlt 12403*	The partial sums of the series expansion of the exponential function at a positive real number are bounded by the value of the function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   &    |- ( ph -> A e. RR+ )   &    |- ( ph -> N e. NN0 )   =>    |- ( ph -> ( seq 0 ( + , F ) ` N ) < ( exp ` A ) )
Theorem	eft0val 12404	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 )
Theorem	ef4p 12405*	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 ) ) )
Theorem	efgt1p2 12406	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 12407	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 12408	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 12409	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	reef11 12410	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 12411	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 12412	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 12413	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 12414	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 12415	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 12416	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 12417	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 12418	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 12419	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 12420	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 12421	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( sin ` A ) e. CC )
Theorem	coscld 12422	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( cos ` A ) e. CC )
Theorem	tanclapd 12423	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 12424	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 12425	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 12426	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 12427	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 12428*	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 12429*	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 12430*	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 12431	The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( sin ` A ) e. RR )
Theorem	recoscl 12432	The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( cos ` A ) e. RR )
Theorem	retanclap 12433	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 12434	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( sin ` A ) e. RR )
Theorem	recoscld 12435	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( cos ` A ) e. RR )
Theorem	retanclapd 12436	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 12437	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 12438	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 12439	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 12440	Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
|- ( sin ` 0 ) = 0
Theorem	cos0 12441	Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
|- ( cos ` 0 ) = 1
Theorem	tan0 12442	The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
|- ( tan ` 0 ) = 0
Theorem	efival 12443	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 12444	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 12445	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 12446	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 12447	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 12448	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 12449	A useful intermediate step in tanaddap 12450 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 12450	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 12451	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 12452	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 12453	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 12454	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 12455	Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12448 and cossub 12452. (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 12456	Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12448 and cossub 12452. (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 12457	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 12458	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 12459	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 12460	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 12461	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 12462	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 12463	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 12464	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 12465	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 12466	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 12467*	Lemma for sin01bnd 12468 and cos01bnd 12469. (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 12468	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 12469	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 12470	Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( ( 1 / 3 ) < ( cos ` 1 ) /\ ( cos ` 1 ) < ( 2 / 3 ) )
Theorem	cos2bnd 12471	Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( -u ( 7 / 9 ) < ( cos ` 2 ) /\ ( cos ` 2 ) < -u ( 1 / 9 ) )
Theorem	sinltxirr 12472*	The sine of a positive irrational number is less than its argument. Here irrational means apart from any rational number. (Contributed by Mario Carneiro, 29-Jul-2014.)
|- ( ( A e. RR+ /\ A. q e. QQ A # q ) -> ( sin ` A ) < A )
Theorem	sin01gt0 12473	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 12474	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 12475	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 12476	The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( 0 < ( sin ` 1 ) /\ 0 < ( cos ` 1 ) )
Theorem	sincos2sgn 12477	The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( 0 < ( sin ` 2 ) /\ ( cos ` 2 ) < 0 )
Theorem	sin4lt0 12478	The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( sin ` 4 ) < 0
Theorem	cos12dec 12479	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 12480	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 12481	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 12482	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 12483	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 12484	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 12485 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 12485	Alternate proof of demoivre 12484. 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 ) ) ) ) )
4.10.1.1  The circle constant (tau = 2 pi)
Syntax	ctau 12486	Extend class notation to include the constant tau, tau = 6.28318....
class tau
Definition	df-tau 12487	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 , < )
4.10.2  _e is irrational
Theorem	eirraplem 12488*	Lemma for eirrap 12489. (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 12489	_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 12490. (Contributed by Jim Kingdon, 6-Jan-2023.)
|- ( Q e. QQ -> _e # Q )
Theorem	eirr 12490	_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 12489. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
|- _e e/ QQ
Theorem	egt2lt3 12491	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 12492	Euler's constant _e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
|- 0 < _e
Theorem	epr 12493	Euler's constant _e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
|- _e e. RR+
Theorem	ene0 12494	_e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
|- _e =/= 0
Theorem	eap0 12495	_e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
|- _e # 0
Theorem	ene1 12496	_e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
|- _e =/= 1
Theorem	eap1 12497	_e is apart from 1. (Contributed by Jim Kingdon, 7-Jan-2023.)
|- _e # 1
PART 5  ELEMENTARY NUMBER THEORY This part introduces 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 12498	Extend the definition of a class to include the divides relation. See df-dvds 12499.
class ||
Definition	df-dvds 12499*	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 12500*	Define the divides relation. M || N means M divides into N with no remainder. For example, 3 || 6 (ex-dvds 16624). As proven in dvdsval3 12502, M || N <-> ( N mod M ) = 0. See divides 12500 and dvdsval2 12501 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 ) )