P. 163 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fallfac1 16201	The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 1) = 𝐴)
Theorem	risefacfac 16202	Relate rising factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝑁 ∈ ℕ0 → (1 RiseFac 𝑁) = (!‘𝑁))
Theorem	fallfacfwd 16203	The forward difference of a falling factorial. (Contributed by Scott Fenton, 21-Jan-2018.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) FallFac 𝑁) − (𝐴 FallFac 𝑁)) = (𝑁 · (𝐴 FallFac (𝑁 − 1))))
Theorem	0fallfac 16204	The value of the zero falling factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.)
⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0)
Theorem	0risefac 16205	The value of the zero rising factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.)
⊢ (𝑁 ∈ ℕ → (0 RiseFac 𝑁) = 0)
Theorem	binomfallfaclem1 16206	Lemma for binomfallfac 16208. Closure law. (Contributed by Scott Fenton, 13-Mar-2018.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ ((𝜑 ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁C𝐾) · ((𝐴 FallFac (𝑁 − 𝐾)) · (𝐵 FallFac (𝐾 + 1)))) ∈ ℂ)
Theorem	binomfallfaclem2 16207*	Lemma for binomfallfac 16208. Inductive step. (Contributed by Scott Fenton, 13-Mar-2018.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜓 → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵) FallFac (𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴 FallFac ((𝑁 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))
Theorem	binomfallfac 16208*	A version of the binomial theorem using falling factorials instead of exponentials. (Contributed by Scott Fenton, 13-Mar-2018.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))
Theorem	binomrisefac 16209*	A version of the binomial theorem using rising factorials instead of exponentials. (Contributed by Scott Fenton, 16-Mar-2018.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵) RiseFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘))))
Theorem	fallfacval4 16210	Represent the falling factorial via factorials when the first argument is a natural. (Contributed by Scott Fenton, 20-Mar-2018.)
⊢ (𝑁 ∈ (0...𝐴) → (𝐴 FallFac 𝑁) = ((!‘𝐴) / (!‘(𝐴 − 𝑁))))
Theorem	bcfallfac 16211	Binomial coefficient in terms of falling factorials. (Contributed by Scott Fenton, 20-Mar-2018.)
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾)))
Theorem	fallfacfac 16212	Relate falling factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 FallFac 𝑁) = (!‘𝑁))
5.10.14  Bernoulli polynomials and sums of k-th powers
Syntax	cbp 16213	Declare the constant for the Bernoulli polynomial operator.
class BernPoly
Definition	df-bpoly 16214*	Define the Bernoulli polynomials. Here we use well-founded recursion to define the Bernoulli polynomials. This agrees with most textbook definitions, although explicit formulas do exist. (Contributed by Scott Fenton, 22-May-2014.)
⊢ BernPoly = (𝑚 ∈ ℕ0, 𝑥 ∈ ℂ ↦ (wrecs( < , ℕ0, (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))))‘𝑚))
Theorem	bpolylem 16215*	Lemma for bpolyval 16216. (Contributed by Scott Fenton, 22-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝐺 = (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))    &   ⊢ 𝐹 = wrecs( < , ℕ0, 𝐺)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
Theorem	bpolyval 16216*	The value of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
Theorem	bpoly0 16217	The value of the Bernoulli polynomials at zero. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
Theorem	bpoly1 16218	The value of the Bernoulli polynomials at one. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))
Theorem	bpolycl 16219	Closure law for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)
Theorem	bpolysum 16220*	A sum for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = (𝑋↑𝑁))
Theorem	bpolydiflem 16221*	Lemma for bpolydif 16222. (Contributed by Scott Fenton, 12-Jun-2014.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1))))    ⇒   ⊢ (𝜑 → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1))))
Theorem	bpolydif 16222	Calculate the difference between successive values of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 26-May-2014.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ ℂ) → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1))))
Theorem	fsumkthpow 16223*	A closed-form expression for the sum of 𝐾-th powers. (Contributed by Scott Fenton, 16-May-2014.) This is Metamath 100 proof #77. (Revised by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐾 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → Σ𝑛 ∈ (0...𝑀)(𝑛↑𝐾) = ((((𝐾 + 1) BernPoly (𝑀 + 1)) − ((𝐾 + 1) BernPoly 0)) / (𝐾 + 1)))
Theorem	bpoly2 16224	The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))
Theorem	bpoly3 16225	The Bernoulli polynomials at three. (Contributed by Scott Fenton, 8-Jul-2015.)
⊢ (𝑋 ∈ ℂ → (3 BernPoly 𝑋) = (((𝑋↑3) − ((3 / 2) · (𝑋↑2))) + ((1 / 2) · 𝑋)))
Theorem	bpoly4 16226	The Bernoulli polynomials at four. (Contributed by Scott Fenton, 8-Jul-2015.)
⊢ (𝑋 ∈ ℂ → (4 BernPoly 𝑋) = ((((𝑋↑4) − (2 · (𝑋↑3))) + (𝑋↑2)) − (1 / ;30)))
Theorem	fsumcube 16227*	Express the sum of cubes in closed terms. (Contributed by Scott Fenton, 16-Jun-2015.)
⊢ (𝑇 ∈ ℕ0 → Σ𝑘 ∈ (0...𝑇)(𝑘↑3) = (((𝑇↑2) · ((𝑇 + 1)↑2)) / 4))
5.11  Elementary trigonometry
5.11.1  The exponential, sine, and cosine functions
Syntax	ce 16228	Extend class notation to include the exponential function.
class exp
Syntax	ceu 16229	Extend class notation to include Euler's constant e = 2.71828....
class e
Syntax	csin 16230	Extend class notation to include the sine function.
class sin
Syntax	ccos 16231	Extend class notation to include the cosine function.
class cos
Syntax	ctan 16232	Extend class notation to include the tangent function.
class tan
Syntax	cpi 16233	Extend class notation to include the constant pi, π = 3.14159....
class π
Definition	df-ef 16234*	Define the exponential function. Its value at the complex number 𝐴 is (exp‘𝐴) and is called the "exponential of 𝐴"; see efval 16246. (Contributed by NM, 14-Mar-2005.)
⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)))
Definition	df-e 16235	Define Euler's constant e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
⊢ e = (exp‘1)
Definition	df-sin 16236	Define the sine function. (Contributed by NM, 14-Mar-2005.)
⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / (2 · i)))
Definition	df-cos 16237	Define the cosine function. (Contributed by NM, 14-Mar-2005.)
⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2))
Definition	df-tan 16238	Define the tangent function. We define it this way for cmpt 5185, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by Mario Carneiro, 14-Mar-2014.)
⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
Definition	df-pi 16239	Define the constant pi, π = 3.14159..., which is the smallest positive number whose sine is zero. Definition of π in [Gleason] p. 311. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by AV, 14-Sep-2020.)
⊢ π = inf((ℝ+ ∩ (◡sin “ {0})), ℝ, < )
Theorem	eftcl 16240	Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → ((𝐴↑𝐾) / (!‘𝐾)) ∈ ℂ)
Theorem	reeftcl 16241	The terms of the series expansion of the exponential function at a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐾 ∈ ℕ0) → ((𝐴↑𝐾) / (!‘𝐾)) ∈ ℝ)
Theorem	eftabs 16242	The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘((𝐴↑𝐾) / (!‘𝐾))) = (((abs‘𝐴)↑𝐾) / (!‘𝐾)))
Theorem	eftval 16243*	The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁)))
Theorem	efcllem 16244*	Lemma for efcl 16249. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 16053 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.) (Proof shortened by AV, 9-Jul-2022.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )
Theorem	ef0lem 16245*	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 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 = 0 → seq0( + , 𝐹) ⇝ 1)
Theorem	efval 16246*	Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 ((𝐴↑𝑘) / (!‘𝑘)))
Theorem	esum 16247	Value of Euler's constant e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
⊢ e = Σ𝑘 ∈ ℕ0 (1 / (!‘𝑘))
Theorem	eff 16248	Domain and codomain of the exponential function. (Contributed by Paul Chapman, 22-Oct-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
⊢ exp:ℂ⟶ℂ
Theorem	efcl 16249	Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
Theorem	efcld 16250	Closure law for the exponential function, deduction version. (Contributed by Thierry Arnoux, 1-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (exp‘𝐴) ∈ ℂ)
Theorem	efval2 16251*	Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘))
Theorem	efcvg 16252*	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 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℂ → seq0( + , 𝐹) ⇝ (exp‘𝐴))
Theorem	efcvgfsum 16253*	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 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴↑𝑘) / (!‘𝑘)))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ⇝ (exp‘𝐴))
Theorem	reefcl 16254	The exponential function is real if its argument is real. (Contributed by NM, 27-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ)
Theorem	reefcld 16255	The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (exp‘𝐴) ∈ ℝ)
Theorem	ere 16256	Euler's constant e = 2.71828... is a real number. (Contributed by NM, 19-Mar-2005.) (Revised by Steve Rodriguez, 8-Mar-2006.)
⊢ e ∈ ℝ
Theorem	ege2le3 16257	Lemma for egt2lt3 16375. (Contributed by NM, 20-Mar-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (2 · ((1 / 2)↑𝑛)))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))    ⇒   ⊢ (2 ≤ e ∧ e ≤ 3)
Theorem	ef0 16258	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 16259	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 ∈ ℂ → (exp‘(∗‘𝐴)) = (∗‘(exp‘𝐴)))
Theorem	efaddlem 16260*	Lemma for efadd 16261 (exponential function addition law). (Contributed by Mario Carneiro, 29-Apr-2014.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ ((𝐵↑𝑛) / (!‘𝑛)))    &   ⊢ 𝐻 = (𝑛 ∈ ℕ0 ↦ (((𝐴 + 𝐵)↑𝑛) / (!‘𝑛)))    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (exp‘(𝐴 + 𝐵)) = ((exp‘𝐴) · (exp‘𝐵)))
Theorem	efadd 16261	Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐴 + 𝐵)) = ((exp‘𝐴) · (exp‘𝐵)))
Theorem	fprodefsum 16262*	Move the exponential function from inside a finite product to outside a finite sum. (Contributed by Scott Fenton, 26-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴))
Theorem	efcan 16263	Cancellation law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.)
⊢ (𝐴 ∈ ℂ → ((exp‘𝐴) · (exp‘ -𝐴)) = 1)
Theorem	efne0d 16264	The exponential of a complex number is nonzero, deduction form. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.) (Revised by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (exp‘𝐴) ≠ 0)
Theorem	efne0 16265	The exponential of a complex number is nonzero. Corollary 15-4.3 of [Gleason] p. 309. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.) (Proof shortened by TA, 14-Nov-2025.)
⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0)
Theorem	efne0OLD 16266	Obsolete version of efne0 16265 as of 14-Nov-2025. The exponential of a complex number is nonzero. Corollary 15-4.3 of [Gleason] p. 309. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0)
Theorem	efneg 16267	The exponential of the opposite is the inverse of the exponential. (Contributed by Mario Carneiro, 10-May-2014.)
⊢ (𝐴 ∈ ℂ → (exp‘ -𝐴) = (1 / (exp‘𝐴)))
Theorem	eff2 16268	The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
⊢ exp:ℂ⟶(ℂ ∖ {0})
Theorem	efsub 16269	Difference of exponents law for exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐴 − 𝐵)) = ((exp‘𝐴) / (exp‘𝐵)))
Theorem	efexp 16270	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 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝑁 · 𝐴)) = ((exp‘𝐴)↑𝑁))
Theorem	efzval 16271	Value of the exponential function for integers. Special case of efval 16246. Equation 30 of [Rudin] p. 164. (Contributed by Steve Rodriguez, 15-Sep-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
⊢ (𝑁 ∈ ℤ → (exp‘𝑁) = (e↑𝑁))
Theorem	efgt0 16272	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 ∈ ℝ → 0 < (exp‘𝐴))
Theorem	rpefcl 16273	The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ+)
Theorem	rpefcld 16274	The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (exp‘𝐴) ∈ ℝ+)
Theorem	eftlcvg 16275*	The tail series of the exponential function are convergent. (Contributed by Mario Carneiro, 29-Apr-2014.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → seq𝑀( + , 𝐹) ∈ dom ⇝ )
Theorem	eftlcl 16276*	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 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ ℂ)
Theorem	reeftlcl 16277*	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 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ ℝ)
Theorem	eftlub 16278*	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 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ (((abs‘𝐴)↑𝑛) / (!‘𝑛)))    &   ⊢ 𝐻 = (𝑛 ∈ ℕ0 ↦ ((((abs‘𝐴)↑𝑀) / (!‘𝑀)) · ((1 / (𝑀 + 1))↑𝑛)))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (abs‘𝐴) ≤ 1)    ⇒   ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘)) ≤ (((abs‘𝐴)↑𝑀) · ((𝑀 + 1) / ((!‘𝑀) · 𝑀))))
Theorem	efsep 16279*	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 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    &   ⊢ 𝑁 = (𝑀 + 1)    &   ⊢ 𝑀 ∈ ℕ0    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (exp‘𝐴) = (𝐵 + Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘)))    &   ⊢ (𝜑 → (𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) = 𝐷)    ⇒   ⊢ (𝜑 → (exp‘𝐴) = (𝐷 + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)))
Theorem	effsumlt 16280*	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 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) < (exp‘𝐴))
Theorem	eft0val 16281	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 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1)
Theorem	ef4p 16282*	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 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6)) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))
Theorem	efgt1p2 16283	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 ∈ ℝ+ → ((1 + 𝐴) + ((𝐴↑2) / 2)) < (exp‘𝐴))
Theorem	efgt1p 16284	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 ∈ ℝ+ → (1 + 𝐴) < (exp‘𝐴))
Theorem	efgt1 16285	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 ∈ ℝ+ → 1 < (exp‘𝐴))
Theorem	eflt 16286	The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (exp‘𝐴) < (exp‘𝐵)))
Theorem	efle 16287	The exponential function on the reals is nondecreasing. (Contributed by Mario Carneiro, 11-Mar-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (exp‘𝐴) ≤ (exp‘𝐵)))
Theorem	reef11 16288	The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Mario Carneiro, 11-Mar-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((exp‘𝐴) = (exp‘𝐵) ↔ 𝐴 = 𝐵))
Theorem	reeff1 16289	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 ↾ ℝ):ℝ–1-1→ℝ+
Theorem	eflegeo 16290	The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 < 1)    ⇒   ⊢ (𝜑 → (exp‘𝐴) ≤ (1 / (1 − 𝐴)))
Theorem	sinval 16291	Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)
⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘( -i · 𝐴))) / (2 · i)))
Theorem	cosval 16292	Value of the cosine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)
⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘( -i · 𝐴))) / 2))
Theorem	sinf 16293	Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ sin:ℂ⟶ℂ
Theorem	cosf 16294	Domain and codomain of the cosine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ cos:ℂ⟶ℂ
Theorem	sincl 16295	Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ)
Theorem	coscl 16296	Closure of the cosine function with a complex argument. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ)
Theorem	tanval 16297	Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.)
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))
Theorem	tancl 16298	The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) ∈ ℂ)
Theorem	sincld 16299	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (sin‘𝐴) ∈ ℂ)
Theorem	coscld 16300	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ)