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	fprodsplit1f 12401*	Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> F/_ k D )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> C e. A )   &    |- ( ( ph /\ k = C ) -> B = D )   =>    |- ( ph -> prod_ k e. A B = ( D x. prod_ k e. ( A \ { C } ) B ) )
Theorem	fprodclf 12402*	Closure of a finite product of complex numbers. A version of fprodcl 12374 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> prod_ k e. A B e. CC )
Theorem	fprodap0f 12403*	A finite product of terms apart from zero is apart from zero. A version of fprodap0 12388 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> B # 0 )   =>    |- ( ph -> prod_ k e. A B # 0 )
Theorem	fprodge0 12404*	If all the terms of a finite product are nonnegative, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   =>    |- ( ph -> 0 <_ prod_ k e. A B )
Theorem	fprodeq0g 12405*	Any finite product containing a zero term is itself zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> C e. A )   &    |- ( ( ph /\ k = C ) -> B = 0 )   =>    |- ( ph -> prod_ k e. A B = 0 )
Theorem	fprodge1 12406*	If all of the terms of a finite product are greater than or equal to 1, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 1 <_ B )   =>    |- ( ph -> 1 <_ prod_ k e. A B )
Theorem	fprodle 12407*	If all the terms of two finite products are nonnegative and compare, so do the two products. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   &    |- ( ( ph /\ k e. A ) -> C e. RR )   &    |- ( ( ph /\ k e. A ) -> B <_ C )   =>    |- ( ph -> prod_ k e. A B <_ prod_ k e. A C )
Theorem	fprodmodd 12408*	If all factors of two finite products are equal modulo M, the products are equal modulo M. (Contributed by AV, 7-Jul-2021.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. ZZ )   &    |- ( ( ph /\ k e. A ) -> C e. ZZ )   &    |- ( ph -> M e. NN )   &    |- ( ( ph /\ k e. A ) -> ( B mod M ) = ( C mod M ) )   =>    |- ( ph -> ( prod_ k e. A B mod M ) = ( prod_ k e. A C mod M ) )
4.10  Elementary trigonometry
4.10.1  The exponential, sine, and cosine functions
Syntax	ce 12409	Extend class notation to include the exponential function.
class exp
Syntax	ceu 12410	Extend class notation to include Euler's constant _e = 2.71828....
class _e
Syntax	csin 12411	Extend class notation to include the sine function.
class sin
Syntax	ccos 12412	Extend class notation to include the cosine function.
class cos
Syntax	ctan 12413	Extend class notation to include the tangent function.
class tan
Syntax	cpi 12414	Extend class notation to include the constant pi, pi = 3.14159....
class pi
Definition	df-ef 12415*	Define the exponential function. Its value at the complex number A is ( exp ` A ) and is called the "exponential of A"; see efval 12428. (Contributed by NM, 14-Mar-2005.)
|- exp = ( x e. CC |-> sum_ k e. NN0 ( ( x ^ k ) / ( ! ` k ) ) )
Definition	df-e 12416	Define Euler's constant _e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
|- _e = ( exp ` 1 )
Definition	df-sin 12417	Define the sine function. (Contributed by NM, 14-Mar-2005.)
|- sin = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) )
Definition	df-cos 12418	Define the cosine function. (Contributed by NM, 14-Mar-2005.)
|- cos = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) )
Definition	df-tan 12419	Define the tangent function. We define it this way for cmpt 4192, which requires the form ( x e. A |-> B ). (Contributed by Mario Carneiro, 14-Mar-2014.)
|- tan = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) / ( cos ` x ) ) )
Definition	df-pi 12420	Define the constant pi, pi = 3.14159..., which is the smallest positive number whose sine is zero. Definition of pi in [Gleason] p. 311. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by AV, 14-Sep-2020.)
|- pi = inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < )
Theorem	eftcl 12421	Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)
|- ( ( A e. CC /\ K e. NN0 ) -> ( ( A ^ K ) / ( ! ` K ) ) e. CC )
Theorem	reeftcl 12422	The terms of the series expansion of the exponential function at a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)
|- ( ( A e. RR /\ K e. NN0 ) -> ( ( A ^ K ) / ( ! ` K ) ) e. RR )
Theorem	eftabs 12423	The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.)
|- ( ( A e. CC /\ K e. NN0 ) -> ( abs ` ( ( A ^ K ) / ( ! ` K ) ) ) = ( ( ( abs ` A ) ^ K ) / ( ! ` K ) ) )
Theorem	eftvalcn 12424*	The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 8-Dec-2022.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( ( A e. CC /\ N e. NN0 ) -> ( F ` N ) = ( ( A ^ N ) / ( ! ` N ) ) )
Theorem	efcllemp 12425*	Lemma for efcl 12431. The series that defines the exponential function converges. The ratio test cvgratgt0 12300 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 12426*	Lemma for efcl 12431. 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 12427*	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 12428*	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 12429	Value of Euler's constant _e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
|- _e = sum_ k e. NN0 ( 1 / ( ! ` k ) )
Theorem	eff 12430	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 12431	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 12432*	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 12433*	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 12434*	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 12435	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 12436	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 12437	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 12438	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 12439	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 12440	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 12441*	Lemma for efadd 12442 (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 12442	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 12443	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 12444	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 12445	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 12444 (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 12446	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 12447	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 12448	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 12449	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 12450	Value of the exponential function for integers. Special case of efval 12428. 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 12451	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 12452	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 12453	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 12454*	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 12455*	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 12456*	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 12457*	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 12458*	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 12459*	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 12460	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 12461*	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 12462	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 12463	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 12464	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 12465	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 12466	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 12467	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 12468	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 12469	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 12470	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 12471	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 12472	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 12473	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 12474	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 12475	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 12476	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 12477	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( sin ` A ) e. CC )
Theorem	coscld 12478	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( cos ` A ) e. CC )
Theorem	tanclapd 12479	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 12480	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 12481	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 12482	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 12483	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 12484*	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 12485*	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 12486*	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 12487	The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( sin ` A ) e. RR )
Theorem	recoscl 12488	The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( cos ` A ) e. RR )
Theorem	retanclap 12489	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 12490	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( sin ` A ) e. RR )
Theorem	recoscld 12491	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( cos ` A ) e. RR )
Theorem	retanclapd 12492	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 12493	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 12494	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 12495	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 12496	Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
|- ( sin ` 0 ) = 0
Theorem	cos0 12497	Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
|- ( cos ` 0 ) = 1
Theorem	tan0 12498	The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
|- ( tan ` 0 ) = 0
Theorem	efival 12499	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 12500	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 ) ) ) )