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	fprodssdc 12301*	Change the index set to a subset in a finite sum. (Contributed by Scott Fenton, 16-Dec-2017.)
|- ( ph -> A C_ B )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ph -> A. j e. B DECID j e. A )   &    |- ( ( ph /\ k e. ( B \ A ) ) -> C = 1 )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> prod_ k e. A C = prod_ k e. B C )
Theorem	fprodmul 12302*	The product of two finite products. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   =>    |- ( ph -> prod_ k e. A ( B x. C ) = ( prod_ k e. A B x. prod_ k e. A C ) )
Theorem	prodsnf 12303*	A product of a singleton is the term. A version of prodsn 12304 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/_ k B   &    |- ( k = M -> A = B )   =>    |- ( ( M e. V /\ B e. CC ) -> prod_ k e. { M } A = B )
Theorem	prodsn 12304*	A product of a singleton is the term. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( k = M -> A = B )   =>    |- ( ( M e. V /\ B e. CC ) -> prod_ k e. { M } A = B )
Theorem	fprod1 12305*	A finite product of only one term is the term itself. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( k = M -> A = B )   =>    |- ( ( M e. ZZ /\ B e. CC ) -> prod_ k e. ( M ... M ) A = B )
Theorem	climprod1 12306	The limit of a product over one. (Contributed by Scott Fenton, 15-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   =>    |- ( ph -> seq M ( x. , ( Z X. { 1 } ) ) ~~> 1 )
Theorem	fprodsplitdc 12307*	Split a finite product into two parts. New proofs should use fprodsplit 12308 which is the same but with one fewer hypothesis. (Contributed by Scott Fenton, 16-Dec-2017.) (New usage is discouraged.)
|- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> U = ( A u. B ) )   &    |- ( ph -> U e. Fin )   &    |- ( ph -> A. j e. U DECID j e. A )   &    |- ( ( 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	fprodsplit 12308*	Split a finite product into two parts. (Contributed by Scott Fenton, 16-Dec-2017.)
|- ( 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	fprodm1 12309*	Separate out the last term in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   &    |- ( k = N -> A = B )   =>    |- ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. B ) )
Theorem	fprod1p 12310*	Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   &    |- ( k = M -> A = B )   =>    |- ( ph -> prod_ k e. ( M ... N ) A = ( B x. prod_ k e. ( ( M + 1 ) ... N ) A ) )
Theorem	fprodp1 12311*	Multiply in the last term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )   &    |- ( k = ( N + 1 ) -> A = B )   =>    |- ( ph -> prod_ k e. ( M ... ( N + 1 ) ) A = ( prod_ k e. ( M ... N ) A x. B ) )
Theorem	fprodm1s 12312*	Separate out the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   =>    |- ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) )
Theorem	fprodp1s 12313*	Multiply in the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )   =>    |- ( ph -> prod_ k e. ( M ... ( N + 1 ) ) A = ( prod_ k e. ( M ... N ) A x. [_ ( N + 1 ) / k ]_ A ) )
Theorem	prodsns 12314*	A product of the singleton is the term. (Contributed by Scott Fenton, 25-Dec-2017.)
|- ( ( M e. V /\ [_ M / k ]_ A e. CC ) -> prod_ k e. { M } A = [_ M / k ]_ A )
Theorem	fprodunsn 12315*	Multiply in an additional term in a finite product. See also fprodsplitsn 12344 which is the same but with a F/ k ph hypothesis in place of the distinct variable condition between ph and k. (Contributed by Jim Kingdon, 16-Aug-2024.)
|- F/_ k D   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. V )   &    |- ( ph -> -. B e. A )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( k = B -> C = D )   =>    |- ( ph -> prod_ k e. ( A u. { B } ) C = ( prod_ k e. A C x. D ) )
Theorem	fprodcl2lem 12316*	Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) (Revised by Jim Kingdon, 17-Aug-2024.)
|- ( ph -> S C_ CC )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. S )   &    |- ( ph -> A =/= (/) )   =>    |- ( ph -> prod_ k e. A B e. S )
Theorem	fprodcllem 12317*	Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> S C_ CC )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. S )   &    |- ( ph -> 1 e. S )   =>    |- ( ph -> prod_ k e. A B e. S )
Theorem	fprodcl 12318*	Closure of a finite product of complex numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> prod_ k e. A B e. CC )
Theorem	fprodrecl 12319*	Closure of a finite product of real numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   =>    |- ( ph -> prod_ k e. A B e. RR )
Theorem	fprodzcl 12320*	Closure of a finite product of integers. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. ZZ )   =>    |- ( ph -> prod_ k e. A B e. ZZ )
Theorem	fprodnncl 12321*	Closure of a finite product of positive integers. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. NN )   =>    |- ( ph -> prod_ k e. A B e. NN )
Theorem	fprodrpcl 12322*	Closure of a finite product of positive reals. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR+ )   =>    |- ( ph -> prod_ k e. A B e. RR+ )
Theorem	fprodnn0cl 12323*	Closure of a finite product of nonnegative integers. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. NN0 )   =>    |- ( ph -> prod_ k e. A B e. NN0 )
Theorem	fprodcllemf 12324*	Finite product closure lemma. A version of fprodcllem 12317 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> S C_ CC )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. S )   &    |- ( ph -> 1 e. S )   =>    |- ( ph -> prod_ k e. A B e. S )
Theorem	fprodreclf 12325*	Closure of a finite product of real numbers. A version of fprodrecl 12319 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. RR )   =>    |- ( ph -> prod_ k e. A B e. RR )
Theorem	fprodfac 12326*	Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
|- ( A e. NN0 -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k )
Theorem	fprodabs 12327*	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 12328*	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 12329*	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 12330*	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 12331*	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 12332*	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 12333*	Lemma for fprod2d 12334- 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 12334*	Write a double product as a product over a two-dimensional region. Compare fsum2d 12146. (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 12335*	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 12336*	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 12337*	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 12338*	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 12339*	Two ways to express "the product of A ( j , k ) over the triangular region M <_ j, M <_ k, j + k <_ N. Compare fisum0diag 12152. (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 12340*	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 12341*	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 12342*	The quotient of two finite products. A version of fproddivap 12341 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 12343*	Split a finite product into two parts. A version of fprodsplit 12308 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 12344*	Separate out a term in a finite product. See also fprodunsn 12315 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 12345*	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 12346*	Closure of a finite product of complex numbers. A version of fprodcl 12318 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 12347*	A finite product of terms apart from zero is apart from zero. A version of fprodap0 12332 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 12348*	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 12349*	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 12350*	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 12351*	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 12352*	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 12353	Extend class notation to include the exponential function.
class exp
Syntax	ceu 12354	Extend class notation to include Euler's constant _e = 2.71828....
class _e
Syntax	csin 12355	Extend class notation to include the sine function.
class sin
Syntax	ccos 12356	Extend class notation to include the cosine function.
class cos
Syntax	ctan 12357	Extend class notation to include the tangent function.
class tan
Syntax	cpi 12358	Extend class notation to include the constant pi, pi = 3.14159....
class pi
Definition	df-ef 12359*	Define the exponential function. Its value at the complex number A is ( exp ` A ) and is called the "exponential of A"; see efval 12372. (Contributed by NM, 14-Mar-2005.)
|- exp = ( x e. CC |-> sum_ k e. NN0 ( ( x ^ k ) / ( ! ` k ) ) )
Definition	df-e 12360	Define Euler's constant _e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
|- _e = ( exp ` 1 )
Definition	df-sin 12361	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 12362	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 12363	Define the tangent function. We define it this way for cmpt 4176, 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 12364	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 12365	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 12366	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 12367	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 12368*	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 12369*	Lemma for efcl 12375. The series that defines the exponential function converges. The ratio test cvgratgt0 12244 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 12370*	Lemma for efcl 12375. 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 12371*	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 12372*	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 12373	Value of Euler's constant _e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
|- _e = sum_ k e. NN0 ( 1 / ( ! ` k ) )
Theorem	eff 12374	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 12375	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 12376*	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 12377*	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 12378*	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 12379	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 12380	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 12381	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 12382	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 12383	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 12384	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 12385*	Lemma for efadd 12386 (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 12386	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 12387	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 12388	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 12389	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 12388 (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 12390	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 12391	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 12392	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 12393	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 12394	Value of the exponential function for integers. Special case of efval 12372. 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 12395	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 12396	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 12397	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 12398*	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 12399*	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 12400*	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 )