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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bpoly2 15401 | The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.) |
⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6))) | ||
Theorem | bpoly3 15402 | 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 15403 | 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 15404* | 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)) | ||
Syntax | ce 15405 | Extend class notation to include the exponential function. |
class exp | ||
Syntax | ceu 15406 | Extend class notation to include Euler's constant e = 2.71828.... |
class e | ||
Syntax | csin 15407 | Extend class notation to include the sine function. |
class sin | ||
Syntax | ccos 15408 | Extend class notation to include the cosine function. |
class cos | ||
Syntax | ctan 15409 | Extend class notation to include the tangent function. |
class tan | ||
Syntax | cpi 15410 | Extend class notation to include the constant pi, π = 3.14159.... |
class π | ||
Definition | df-ef 15411* | Define the exponential function. Its value at the complex number 𝐴 is (exp‘𝐴) and is called the "exponential of 𝐴"; see efval 15423. (Contributed by NM, 14-Mar-2005.) |
⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘))) | ||
Definition | df-e 15412 | Define Euler's constant e = 2.71828.... (Contributed by NM, 14-Mar-2005.) |
⊢ e = (exp‘1) | ||
Definition | df-sin 15413 | Define the sine function. (Contributed by NM, 14-Mar-2005.) |
⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))) | ||
Definition | df-cos 15414 | Define the cosine function. (Contributed by NM, 14-Mar-2005.) |
⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) | ||
Definition | df-tan 15415 | Define the tangent function. We define it this way for cmpt 5138, 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 15416 | 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 15417 | 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 15418 | 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 15419 | 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 15420* | 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 15421* | Lemma for efcl 15426. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 15229 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 15422* | 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 15423* | 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 15424 | Value of Euler's constant e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.) |
⊢ e = Σ𝑘 ∈ ℕ0 (1 / (!‘𝑘)) | ||
Theorem | eff 15425 | 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 15426 | 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 | efval2 15427* | 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 15428* | 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 15429* | 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 15430 | 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 15431 | 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 15432 | 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 15433 | Lemma for egt2lt3 15549. (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 15434 | 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 15435 | 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 15436* | Lemma for efadd 15437 (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 15437 | 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 15438* | 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 15439 | 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 | efne0 15440 | 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) | ||
Theorem | efneg 15441 | 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 15442 | 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 15443 | 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 15444 | 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 15445 | Value of the exponential function for integers. Special case of efval 15423. 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 15446 | 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 15447 | The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.) |
⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ+) | ||
Theorem | rpefcld 15448 | 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 15449* | 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 15450* | 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 15451* | 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 15452* | 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 15453* | 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 15454* | 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 15455 | 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 15456* | 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 15457 | 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 15458 | 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 15459 | 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 15460 | 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 15461 | 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 15462 | 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 15463 | 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 15464 | 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 15465 | 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 15466 | 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 15467 | 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 15468 | 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 15469 | 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 15470 | 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 15471 | 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 15472 | 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 15473 | Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (sin‘𝐴) ∈ ℂ) | ||
Theorem | coscld 15474 | Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ) | ||
Theorem | tancld 15475 | Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (cos‘𝐴) ≠ 0) ⇒ ⊢ (𝜑 → (tan‘𝐴) ∈ ℂ) | ||
Theorem | tanval2 15476 | 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)))))) | ||
Theorem | tanval3 15477 | 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)))) | ||
Theorem | resinval 15478 | 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)))) | ||
Theorem | recosval 15479 | 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)))) | ||
Theorem | efi4p 15480* | 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)(𝐹‘𝑘))) | ||
Theorem | resin4p 15481* | 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)(𝐹‘𝑘)))) | ||
Theorem | recos4p 15482* | 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)(𝐹‘𝑘)))) | ||
Theorem | resincl 15483 | The sine of a real number is real. (Contributed by NM, 30-Apr-2005.) |
⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ) | ||
Theorem | recoscl 15484 | The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.) |
⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ) | ||
Theorem | retancl 15485 | 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) | ||
Theorem | resincld 15486 | Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (sin‘𝐴) ∈ ℝ) | ||
Theorem | recoscld 15487 | Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (cos‘𝐴) ∈ ℝ) | ||
Theorem | retancld 15488 | Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → (cos‘𝐴) ≠ 0) ⇒ ⊢ (𝜑 → (tan‘𝐴) ∈ ℝ) | ||
Theorem | sinneg 15489 | The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.) |
⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) | ||
Theorem | cosneg 15490 | The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.) |
⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴)) | ||
Theorem | tanneg 15491 | 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)) | ||
Theorem | sin0 15492 | Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.) |
⊢ (sin‘0) = 0 | ||
Theorem | cos0 15493 | Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.) |
⊢ (cos‘0) = 1 | ||
Theorem | tan0 15494 | The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.) |
⊢ (tan‘0) = 0 | ||
Theorem | efival 15495 | 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)))) | ||
Theorem | efmival 15496 | The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.) |
⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) = ((cos‘𝐴) − (i · (sin‘𝐴)))) | ||
Theorem | sinhval 15497 | Value of the hyperbolic sine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.) |
⊢ (𝐴 ∈ ℂ → ((sin‘(i · 𝐴)) / i) = (((exp‘𝐴) − (exp‘-𝐴)) / 2)) | ||
Theorem | coshval 15498 | Value of the hyperbolic cosine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.) |
⊢ (𝐴 ∈ ℂ → (cos‘(i · 𝐴)) = (((exp‘𝐴) + (exp‘-𝐴)) / 2)) | ||
Theorem | resinhcl 15499 | The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
⊢ (𝐴 ∈ ℝ → ((sin‘(i · 𝐴)) / i) ∈ ℝ) | ||
Theorem | rpcoshcl 15500 | The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ+) |
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