P. 119 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11801-11900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	efcllemp 11801*	Lemma for efcl 11807. The series that defines the exponential function converges. The ratio test cvgratgt0 11676 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> K e. NN )   &    |- ( ph -> ( 2 x. ( abs ` A ) ) < K )   =>    |- ( ph -> seq 0 ( + , F ) e. dom ~~> )
Theorem	efcllem 11802*	Lemma for efcl 11807. The series that defines the exponential function converges. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( A e. CC -> seq 0 ( + , F ) e. dom ~~> )
Theorem	ef0lem 11803*	The series defining the exponential function converges in the (trivial) case of a zero argument. (Contributed by Steve Rodriguez, 7-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )
Theorem	efval 11804*	Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
|- ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( ( A ^ k ) / ( ! ` k ) ) )
Theorem	esum 11805	Value of Euler's constant _e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
|- _e = sum_ k e. NN0 ( 1 / ( ! ` k ) )
Theorem	eff 11806	Domain and codomain of the exponential function. (Contributed by Paul Chapman, 22-Oct-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
|- exp : CC --> CC
Theorem	efcl 11807	Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
|- ( A e. CC -> ( exp ` A ) e. CC )
Theorem	efval2 11808*	Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( F ` k ) )
Theorem	efcvg 11809*	The series that defines the exponential function converges to it. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( A e. CC -> seq 0 ( + , F ) ~~> ( exp ` A ) )
Theorem	efcvgfsum 11810*	Exponential function convergence in terms of a sequence of partial finite sums. (Contributed by NM, 10-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- F = ( n e. NN0 |-> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) )   =>    |- ( A e. CC -> F ~~> ( exp ` A ) )
Theorem	reefcl 11811	The exponential function is real if its argument is real. (Contributed by NM, 27-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- ( A e. RR -> ( exp ` A ) e. RR )
Theorem	reefcld 11812	The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( exp ` A ) e. RR )
Theorem	ere 11813	Euler's constant _e = 2.71828... is a real number. (Contributed by NM, 19-Mar-2005.) (Revised by Steve Rodriguez, 8-Mar-2006.)
|- _e e. RR
Theorem	ege2le3 11814	Euler's constant _e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 20-Mar-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
|- F = ( n e. NN |-> ( 2 x. ( ( 1 / 2 ) ^ n ) ) )   &    |- G = ( n e. NN0 |-> ( 1 / ( ! ` n ) ) )   =>    |- ( 2 <_ _e /\ _e <_ 3 )
Theorem	ef0 11815	Value of the exponential function at 0. Equation 2 of [Gleason] p. 308. (Contributed by Steve Rodriguez, 27-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- ( exp ` 0 ) = 1
Theorem	efcj 11816	The exponential of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- ( A e. CC -> ( exp ` ( * ` A ) ) = ( * ` ( exp ` A ) ) )
Theorem	efaddlem 11817*	Lemma for efadd 11818 (exponential function addition law). (Contributed by Mario Carneiro, 29-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   &    |- G = ( n e. NN0 |-> ( ( B ^ n ) / ( ! ` n ) ) )   &    |- H = ( n e. NN0 |-> ( ( ( A + B ) ^ n ) / ( ! ` n ) ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( exp ` ( A + B ) ) = ( ( exp ` A ) x. ( exp ` B ) ) )
Theorem	efadd 11818	Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( exp ` ( A + B ) ) = ( ( exp ` A ) x. ( exp ` B ) ) )
Theorem	efcan 11819	Cancellation law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.)
|- ( A e. CC -> ( ( exp ` A ) x. ( exp ` -u A ) ) = 1 )
Theorem	efap0 11820	The exponential of a complex number is apart from zero. (Contributed by Jim Kingdon, 12-Dec-2022.)
|- ( A e. CC -> ( exp ` A ) # 0 )
Theorem	efne0 11821	The exponential of a complex number is nonzero. Corollary 15-4.3 of [Gleason] p. 309. The same result also holds with not equal replaced by apart, as seen at efap0 11820 (which will be more useful in most contexts). (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.)
|- ( A e. CC -> ( exp ` A ) =/= 0 )
Theorem	efneg 11822	The exponential of the opposite is the inverse of the exponential. (Contributed by Mario Carneiro, 10-May-2014.)
|- ( A e. CC -> ( exp ` -u A ) = ( 1 / ( exp ` A ) ) )
Theorem	eff2 11823	The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
|- exp : CC --> ( CC \ { 0 } )
Theorem	efsub 11824	Difference of exponents law for exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( exp ` ( A - B ) ) = ( ( exp ` A ) / ( exp ` B ) ) )
Theorem	efexp 11825	The exponential of an integer power. Corollary 15-4.4 of [Gleason] p. 309, restricted to integers. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
|- ( ( A e. CC /\ N e. ZZ ) -> ( exp ` ( N x. A ) ) = ( ( exp ` A ) ^ N ) )
Theorem	efzval 11826	Value of the exponential function for integers. Special case of efval 11804. Equation 30 of [Rudin] p. 164. (Contributed by Steve Rodriguez, 15-Sep-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
|- ( N e. ZZ -> ( exp ` N ) = ( _e ^ N ) )
Theorem	efgt0 11827	The exponential of a real number is greater than 0. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
|- ( A e. RR -> 0 < ( exp ` A ) )
Theorem	rpefcl 11828	The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
|- ( A e. RR -> ( exp ` A ) e. RR+ )
Theorem	rpefcld 11829	The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( exp ` A ) e. RR+ )
Theorem	eftlcvg 11830*	The tail series of the exponential function are convergent. (Contributed by Mario Carneiro, 29-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( ( A e. CC /\ M e. NN0 ) -> seq M ( + , F ) e. dom ~~> )
Theorem	eftlcl 11831*	Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( ( A e. CC /\ M e. NN0 ) -> sum_ k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
Theorem	reeftlcl 11832*	Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( ( A e. RR /\ M e. NN0 ) -> sum_ k e. ( ZZ>= ` M ) ( F ` k ) e. RR )
Theorem	eftlub 11833*	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 11834*	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 11835*	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 11836	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 11837*	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 11838	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 11839	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 11840	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 11841	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 11842	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 11843	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 11844	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 11845	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 11846	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 11847	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 11848	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 11849	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 11850	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 11851	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 11852	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 11853	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( sin ` A ) e. CC )
Theorem	coscld 11854	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( cos ` A ) e. CC )
Theorem	tanclapd 11855	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 11856	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 11857	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 11858	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 11859	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 11860*	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 11861*	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 11862*	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 11863	The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( sin ` A ) e. RR )
Theorem	recoscl 11864	The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( cos ` A ) e. RR )
Theorem	retanclap 11865	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 11866	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( sin ` A ) e. RR )
Theorem	recoscld 11867	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( cos ` A ) e. RR )
Theorem	retanclapd 11868	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 11869	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 11870	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 11871	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 11872	Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
|- ( sin ` 0 ) = 0
Theorem	cos0 11873	Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
|- ( cos ` 0 ) = 1
Theorem	tan0 11874	The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
|- ( tan ` 0 ) = 0
Theorem	efival 11875	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 11876	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 11877	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 11878	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 11879	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 11880	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 11881	A useful intermediate step in tanaddap 11882 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 11882	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 11883	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 11884	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 11885	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 11886	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 11887	Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11880 and cossub 11884. (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 11888	Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 11880 and cossub 11884. (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 11889	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 11890	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 11891	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 11892	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 11893	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 11894	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 11895	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 11896	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 11897	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 11898	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 11899*	Lemma for sin01bnd 11900 and cos01bnd 11901. (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 11900	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 ) )