Home Metamath Proof ExplorerTheorem List (p. 155 of 454) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28704) Hilbert Space Explorer (28705-30227) Users' Mathboxes (30228-45320)

Theorem List for Metamath Proof Explorer - 15401-15500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembpolycl 15401 Closure law for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
((𝑁 ∈ ℕ0𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)

Theorembpolysum 15402* A sum for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
((𝑁 ∈ ℕ0𝑋 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁𝑘) + 1))) = (𝑋𝑁))

Theorembpolydiflem 15403* Lemma for bpolydif 15404. (Contributed by Scott Fenton, 12-Jun-2014.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝑋 ∈ ℂ)    &   ((𝜑𝑘 ∈ (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))))

Theorembpolydif 15404 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))))

Theoremfsumkthpow 15405* 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 ∈ ℕ0𝑀 ∈ ℕ0) → Σ𝑛 ∈ (0...𝑀)(𝑛𝐾) = ((((𝐾 + 1) BernPoly (𝑀 + 1)) − ((𝐾 + 1) BernPoly 0)) / (𝐾 + 1)))

Theorembpoly2 15406 The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
(𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))

Theorembpoly3 15407 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)))

Theorembpoly4 15408 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)))

Theoremfsumcube 15409* 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

Syntaxce 15410 Extend class notation to include the exponential function.
class exp

Syntaxceu 15411 Extend class notation to include Euler's constant e = 2.71828....
class e

Syntaxcsin 15412 Extend class notation to include the sine function.
class sin

Syntaxccos 15413 Extend class notation to include the cosine function.
class cos

Syntaxctan 15414 Extend class notation to include the tangent function.
class tan

Syntaxcpi 15415 Extend class notation to include the constant pi, π = 3.14159....
class π

Definitiondf-ef 15416* Define the exponential function. Its value at the complex number 𝐴 is (exp‘𝐴) and is called the "exponential of 𝐴"; see efval 15428. (Contributed by NM, 14-Mar-2005.)
exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥𝑘) / (!‘𝑘)))

Definitiondf-e 15417 Define Euler's constant e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
e = (exp‘1)

Definitiondf-sin 15418 Define the sine function. (Contributed by NM, 14-Mar-2005.)
sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))

Definitiondf-cos 15419 Define the cosine function. (Contributed by NM, 14-Mar-2005.)
cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))

Definitiondf-tan 15420 Define the tangent function. We define it this way for cmpt 5113, which requires the form (𝑥𝐴𝐵). (Contributed by Mario Carneiro, 14-Mar-2014.)
tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))

Definitiondf-pi 15421 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, < )

Theoremeftcl 15422 Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)
((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → ((𝐴𝐾) / (!‘𝐾)) ∈ ℂ)

Theoremreeftcl 15423 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) → ((𝐴𝐾) / (!‘𝐾)) ∈ ℝ)

Theoremeftabs 15424 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‘((𝐴𝐾) / (!‘𝐾))) = (((abs‘𝐴)↑𝐾) / (!‘𝐾)))

Theoremeftval 15425* 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 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝑁 ∈ ℕ0 → (𝐹𝑁) = ((𝐴𝑁) / (!‘𝑁)))

Theoremefcllem 15426* Lemma for efcl 15431. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 15234 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 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )

Theoremef0lem 15427* 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 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 = 0 → seq0( + , 𝐹) ⇝ 1)

Theoremefval 15428* Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 ((𝐴𝑘) / (!‘𝑘)))

Theoremesum 15429 Value of Euler's constant e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
e = Σ𝑘 ∈ ℕ0 (1 / (!‘𝑘))

Theoremeff 15430 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

Theoremefcl 15431 Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)

Theoremefval2 15432* Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹𝑘))

Theoremefcvg 15433* 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 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → seq0( + , 𝐹) ⇝ (exp‘𝐴))

Theoremefcvgfsum 15434* 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)((𝐴𝑘) / (!‘𝑘)))       (𝐴 ∈ ℂ → 𝐹 ⇝ (exp‘𝐴))

Theoremreefcl 15435 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)

Theoremreefcld 15436 The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (exp‘𝐴) ∈ ℝ)

Theoremere 15437 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

Theoremege2le3 15438 Lemma for egt2lt3 15554. (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)

Theoremef0 15439 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

Theoremefcj 15440 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)))

𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))    &   𝐺 = (𝑛 ∈ ℕ0 ↦ ((𝐵𝑛) / (!‘𝑛)))    &   𝐻 = (𝑛 ∈ ℕ0 ↦ (((𝐴 + 𝐵)↑𝑛) / (!‘𝑛)))    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (exp‘(𝐴 + 𝐵)) = ((exp‘𝐴) · (exp‘𝐵)))

Theoremefadd 15442 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)))

Theoremfprodefsum 15443* Move the exponential function from inside a finite product to outside a finite sum. (Contributed by Scott Fenton, 26-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)       (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴))

Theoremefcan 15444 Cancellation law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.)
(𝐴 ∈ ℂ → ((exp‘𝐴) · (exp‘-𝐴)) = 1)

Theoremefne0 15445 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.)
(𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0)

Theoremefneg 15446 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)))

Theoremeff2 15447 The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
exp:ℂ⟶(ℂ ∖ {0})

Theoremefsub 15448 Difference of exponents law for exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐴𝐵)) = ((exp‘𝐴) / (exp‘𝐵)))

Theoremefexp 15449 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))

Theoremefzval 15450 Value of the exponential function for integers. Special case of efval 15428. 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))

Theoremefgt0 15451 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))

Theoremrpefcl 15452 The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ+)

Theoremrpefcld 15453 The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (exp‘𝐴) ∈ ℝ+)

Theoremeftlcvg 15454* The tail series of the exponential function are convergent. (Contributed by Mario Carneiro, 29-Apr-2014.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Theoremeftlcl 15455* 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 ↦ ((𝐴𝑛) / (!‘𝑛)))       ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → Σ𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ ℂ)

Theoremreeftlcl 15456* 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 ↦ ((𝐴𝑛) / (!‘𝑛)))       ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → Σ𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ ℝ)

Theoremeftlub 15457* 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 ↦ ((𝐴𝑛) / (!‘𝑛)))    &   𝐺 = (𝑛 ∈ ℕ0 ↦ (((abs‘𝐴)↑𝑛) / (!‘𝑛)))    &   𝐻 = (𝑛 ∈ ℕ0 ↦ ((((abs‘𝐴)↑𝑀) / (!‘𝑀)) · ((1 / (𝑀 + 1))↑𝑛)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑 → (abs‘𝐴) ≤ 1)       (𝜑 → (abs‘Σ𝑘 ∈ (ℤ𝑀)(𝐹𝑘)) ≤ (((abs‘𝐴)↑𝑀) · ((𝑀 + 1) / ((!‘𝑀) · 𝑀))))

Theoremefsep 15458* 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 ↦ ((𝐴𝑛) / (!‘𝑛)))    &   𝑁 = (𝑀 + 1)    &   𝑀 ∈ ℕ0    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (exp‘𝐴) = (𝐵 + Σ𝑘 ∈ (ℤ𝑀)(𝐹𝑘)))    &   (𝜑 → (𝐵 + ((𝐴𝑀) / (!‘𝑀))) = 𝐷)       (𝜑 → (exp‘𝐴) = (𝐷 + Σ𝑘 ∈ (ℤ𝑁)(𝐹𝑘)))

Theoremeffsumlt 15459* 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 ↦ ((𝐴𝑛) / (!‘𝑛)))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (seq0( + , 𝐹)‘𝑁) < (exp‘𝐴))

Theoremeft0val 15460 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)

Theoremef4p 15461* 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 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → (exp‘𝐴) = ((((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6)) + Σ𝑘 ∈ (ℤ‘4)(𝐹𝑘)))

Theoremefgt1p2 15462 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))

Theoremefgt1p 15463 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))

Theoremefgt1 15464 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))

Theoremeflt 15465 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)))

Theoremefle 15466 The exponential function on the reals is nondecreasing. (Contributed by Mario Carneiro, 11-Mar-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ (exp‘𝐴) ≤ (exp‘𝐵)))

Theoremreef11 15467 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))

Theoremreeff1 15468 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+

Theoremeflegeo 15469 The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 < 1)       (𝜑 → (exp‘𝐴) ≤ (1 / (1 − 𝐴)))

Theoremsinval 15470 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)))

Theoremcosval 15471 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))

Theoremsinf 15472 Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
sin:ℂ⟶ℂ

Theoremcosf 15473 Domain and codomain of the cosine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
cos:ℂ⟶ℂ

Theoremsincl 15474 Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
(𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ)

Theoremcoscl 15475 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)

Theoremtanval 15476 Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))

Theoremtancl 15477 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)

Theoremsincld 15478 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (sin‘𝐴) ∈ ℂ)

Theoremcoscld 15479 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (cos‘𝐴) ∈ ℂ)

Theoremtancld 15480 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (cos‘𝐴) ≠ 0)       (𝜑 → (tan‘𝐴) ∈ ℂ)

Theoremtanval2 15481 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (i · ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))))

Theoremtanval3 15482 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℂ ∧ ((exp‘(2 · (i · 𝐴))) + 1) ≠ 0) → (tan‘𝐴) = (((exp‘(2 · (i · 𝐴))) − 1) / (i · ((exp‘(2 · (i · 𝐴))) + 1))))

Theoremresinval 15483 The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ → (sin‘𝐴) = (ℑ‘(exp‘(i · 𝐴))))

Theoremrecosval 15484 The cosine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ → (cos‘𝐴) = (ℜ‘(exp‘(i · 𝐴))))

Theoremefi4p 15485* 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 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = (((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ‘4)(𝐹𝑘)))

Theoremresin4p 15486* 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 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))       (𝐴 ∈ ℝ → (sin‘𝐴) = ((𝐴 − ((𝐴↑3) / 6)) + (ℑ‘Σ𝑘 ∈ (ℤ‘4)(𝐹𝑘))))

Theoremrecos4p 15487* 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 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))       (𝐴 ∈ ℝ → (cos‘𝐴) = ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ‘4)(𝐹𝑘))))

Theoremresincl 15488 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ)

Theoremrecoscl 15489 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ)

Theoremretancl 15490 The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
((𝐴 ∈ ℝ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) ∈ ℝ)

Theoremresincld 15491 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (sin‘𝐴) ∈ ℝ)

Theoremrecoscld 15492 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (cos‘𝐴) ∈ ℝ)

Theoremretancld 15493 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → (cos‘𝐴) ≠ 0)       (𝜑 → (tan‘𝐴) ∈ ℝ)

Theoremsinneg 15494 The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴))

Theoremcosneg 15495 The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴))

Theoremtanneg 15496 The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘-𝐴) = -(tan‘𝐴))

Theoremsin0 15497 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
(sin‘0) = 0

Theoremcos0 15498 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
(cos‘0) = 1

Theoremtan0 15499 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
(tan‘0) = 0

Theoremefival 15500 The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = ((cos‘𝐴) + (i · (sin‘𝐴))))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45320
 Copyright terms: Public domain < Previous  Next >