P. 124 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12301-12400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fprodfac 12301*	Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
|- ( A e. NN0 -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k )
Theorem	fprodabs 12302*	The absolute value of a finite product. (Contributed by Scott Fenton, 25-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   =>    |- ( ph -> ( abs ` prod_ k e. ( M ... N ) A ) = prod_ k e. ( M ... N ) ( abs ` A ) )
Theorem	fprodeq0 12303*	Any finite product containing a zero term is itself zero. (Contributed by Scott Fenton, 27-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ( ph /\ k = N ) -> A = 0 )   =>    |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( M ... K ) A = 0 )
Theorem	fprodshft 12304*	Shift the index of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( k - K ) -> A = B )   =>    |- ( ph -> prod_ j e. ( M ... N ) A = prod_ k e. ( ( M + K ) ... ( N + K ) ) B )
Theorem	fprodrev 12305*	Reversal of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( K - k ) -> A = B )   =>    |- ( ph -> prod_ j e. ( M ... N ) A = prod_ k e. ( ( K - N ) ... ( K - M ) ) B )
Theorem	fprodconst 12306*	The product of constant terms ( k is not free in B). (Contributed by Scott Fenton, 12-Jan-2018.)
|- ( ( A e. Fin /\ B e. CC ) -> prod_ k e. A B = ( B ^ (♯ ` A ) ) )
Theorem	fprodap0 12307*	A finite product of nonzero terms is nonzero. (Contributed by Scott Fenton, 15-Jan-2018.)
|- ( 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	fprod2dlemstep 12308*	Lemma for fprod2d 12309- induction step. (Contributed by Scott Fenton, 30-Jan-2018.)
|- ( z = <. j , k >. -> D = C )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ j e. A ) -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   &    |- ( ph -> -. y e. x )   &    |- ( ph -> ( x u. { y } ) C_ A )   &    |- ( ph -> x e. Fin )   &    |- ( ps <-> prod_ j e. x prod_ k e. B C = prod_ z e. U_ j e. x ( { j } X. B ) D )   =>    |- ( ( ph /\ ps ) -> prod_ j e. ( x u. { y } ) prod_ k e. B C = prod_ z e. U_ j e. ( x u. { y } ) ( { j } X. B ) D )
Theorem	fprod2d 12309*	Write a double product as a product over a two-dimensional region. Compare fsum2d 12121. (Contributed by Scott Fenton, 30-Jan-2018.)
|- ( z = <. j , k >. -> D = C )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ j e. A ) -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> prod_ j e. A prod_ k e. B C = prod_ z e. U_ j e. A ( { j } X. B ) D )
Theorem	fprodxp 12310*	Combine two products into a single product over the cartesian product. (Contributed by Scott Fenton, 1-Feb-2018.)
|- ( z = <. j , k >. -> D = C )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> prod_ j e. A prod_ k e. B C = prod_ z e. ( A X. B ) D )
Theorem	fprodcnv 12311*	Transform a product region using the converse operation. (Contributed by Scott Fenton, 1-Feb-2018.)
|- ( x = <. j , k >. -> B = D )   &    |- ( y = <. k , j >. -> C = D )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> Rel A )   &    |- ( ( ph /\ x e. A ) -> B e. CC )   =>    |- ( ph -> prod_ x e. A B = prod_ y e. `' A C )
Theorem	fprodcom2fi 12312*	Interchange order of multiplication. Note that B ( j ) and D ( k ) are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.) (Proof shortened by JJ, 2-Aug-2021.)
|- ( ph -> A e. Fin )   &    |- ( ph -> C e. Fin )   &    |- ( ( ph /\ j e. A ) -> B e. Fin )   &    |- ( ( ph /\ k e. C ) -> D e. Fin )   &    |- ( ph -> ( ( j e. A /\ k e. B ) <-> ( k e. C /\ j e. D ) ) )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> E e. CC )   =>    |- ( ph -> prod_ j e. A prod_ k e. B E = prod_ k e. C prod_ j e. D E )
Theorem	fprodcom 12313*	Interchange product order. (Contributed by Scott Fenton, 2-Feb-2018.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> prod_ j e. A prod_ k e. B C = prod_ k e. B prod_ j e. A C )
Theorem	fprod0diagfz 12314*	Two ways to express "the product of A ( j , k ) over the triangular region M <_ j, M <_ k, j + k <_ N. Compare fisum0diag 12127. (Contributed by Scott Fenton, 2-Feb-2018.)
|- ( ( ph /\ ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) ) -> A e. CC )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> prod_ j e. ( 0 ... N ) prod_ k e. ( 0 ... ( N - j ) ) A = prod_ k e. ( 0 ... N ) prod_ j e. ( 0 ... ( N - k ) ) A )
Theorem	fprodrec 12315*	The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> B # 0 )   =>    |- ( ph -> prod_ k e. A ( 1 / B ) = ( 1 / prod_ k e. A B ) )
Theorem	fproddivap 12316*	The quotient of two finite products. (Contributed by Scott Fenton, 15-Jan-2018.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ( ph /\ k e. A ) -> C # 0 )   =>    |- ( ph -> prod_ k e. A ( B / C ) = ( prod_ k e. A B / prod_ k e. A C ) )
Theorem	fproddivapf 12317*	The quotient of two finite products. A version of fproddivap 12316 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 /\ k e. A ) -> C e. CC )   &    |- ( ( ph /\ k e. A ) -> C # 0 )   =>    |- ( ph -> prod_ k e. A ( B / C ) = ( prod_ k e. A B / prod_ k e. A C ) )
Theorem	fprodsplitf 12318*	Split a finite product into two parts. A version of fprodsplit 12283 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> U = ( A u. B ) )   &    |- ( ph -> U e. Fin )   &    |- ( ( ph /\ k e. U ) -> C e. CC )   =>    |- ( ph -> prod_ k e. U C = ( prod_ k e. A C x. prod_ k e. B C ) )
Theorem	fprodsplitsn 12319*	Separate out a term in a finite product. See also fprodunsn 12290 which is the same but with a distinct variable condition in place of F/ k ph. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- F/_ k D   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. V )   &    |- ( ph -> -. B e. A )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( k = B -> C = D )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> prod_ k e. ( A u. { B } ) C = ( prod_ k e. A C x. D ) )
Theorem	fprodsplit1f 12320*	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 12321*	Closure of a finite product of complex numbers. A version of fprodcl 12293 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 12322*	A finite product of terms apart from zero is apart from zero. A version of fprodap0 12307 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 12323*	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 12324*	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 12325*	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 12326*	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 12327*	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 12328	Extend class notation to include the exponential function.
class exp
Syntax	ceu 12329	Extend class notation to include Euler's constant _e = 2.71828....
class _e
Syntax	csin 12330	Extend class notation to include the sine function.
class sin
Syntax	ccos 12331	Extend class notation to include the cosine function.
class cos
Syntax	ctan 12332	Extend class notation to include the tangent function.
class tan
Syntax	cpi 12333	Extend class notation to include the constant pi, pi = 3.14159....
class pi
Definition	df-ef 12334*	Define the exponential function. Its value at the complex number A is ( exp ` A ) and is called the "exponential of A"; see efval 12347. (Contributed by NM, 14-Mar-2005.)
|- exp = ( x e. CC |-> sum_ k e. NN0 ( ( x ^ k ) / ( ! ` k ) ) )
Definition	df-e 12335	Define Euler's constant _e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
|- _e = ( exp ` 1 )
Definition	df-sin 12336	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 12337	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 12338	Define the tangent function. We define it this way for cmpt 4171, 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 12339	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 12340	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 12341	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 12342	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 12343*	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 12344*	Lemma for efcl 12350. The series that defines the exponential function converges. The ratio test cvgratgt0 12219 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 12345*	Lemma for efcl 12350. 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 12346*	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 12347*	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 12348	Value of Euler's constant _e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
|- _e = sum_ k e. NN0 ( 1 / ( ! ` k ) )
Theorem	eff 12349	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 12350	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 12351*	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 12352*	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 12353*	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 12354	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 12355	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 12356	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 12357	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 12358	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 12359	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 12360*	Lemma for efadd 12361 (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 12361	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 12362	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 12363	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 12364	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 12363 (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 12365	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 12366	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 12367	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 12368	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 12369	Value of the exponential function for integers. Special case of efval 12347. 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 12370	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 12371	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 12372	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 12373*	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 12374*	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 12375*	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 12376*	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 12377*	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 12378*	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 12379	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 12380*	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 12381	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 12382	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 12383	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 12384	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 12385	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 12386	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 12387	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 12388	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 12389	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 12390	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 12391	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 12392	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 12393	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 12394	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 12395	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 12396	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( sin ` A ) e. CC )
Theorem	coscld 12397	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( cos ` A ) e. CC )
Theorem	tanclapd 12398	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 12399	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 12400	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 ) ) ) )