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Theorem List for Metamath Proof Explorer - 14601-14700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrisefacp1d 14601 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴 RiseFac (𝑁 + 1)) = ((𝐴 RiseFac 𝑁) · (𝐴 + 𝑁)))

Theoremfallfacp1d 14602 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴 FallFac (𝑁 + 1)) = ((𝐴 FallFac 𝑁) · (𝐴𝑁)))

Theoremrisefac1 14603 The value of rising factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
(𝐴 ∈ ℂ → (𝐴 RiseFac 1) = 𝐴)

Theoremfallfac1 14604 The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
(𝐴 ∈ ℂ → (𝐴 FallFac 1) = 𝐴)

Theoremrisefacfac 14605 Relate rising factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
(𝑁 ∈ ℕ0 → (1 RiseFac 𝑁) = (!‘𝑁))

Theoremfallfacfwd 14606 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))))

Theorem0fallfac 14607 The value of the zero falling factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.)
(𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0)

Theorem0risefac 14608 The value of the zero rising factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.)
(𝑁 ∈ ℕ → (0 RiseFac 𝑁) = 0)

Theorembinomfallfaclem1 14609 Lemma for binomfallfac 14611. Closure law. (Contributed by Scott Fenton, 13-Mar-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑𝐾 ∈ (0...𝑁)) → ((𝑁C𝐾) · ((𝐴 FallFac (𝑁𝐾)) · (𝐵 FallFac (𝐾 + 1)))) ∈ ℂ)

Theorembinomfallfaclem2 14610* Lemma for binomfallfac 14611. Inductive step. (Contributed by Scott Fenton, 13-Mar-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜓 → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁𝑘)) · (𝐵 FallFac 𝑘))))       ((𝜑𝜓) → ((𝐴 + 𝐵) FallFac (𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴 FallFac ((𝑁 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))

Theorembinomfallfac 14611* 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 (𝑁𝑘)) · (𝐵 FallFac 𝑘))))

Theorembinomrisefac 14612* 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 (𝑁𝑘)) · (𝐵 RiseFac 𝑘))))

Theoremfallfacval4 14613 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) / (!‘(𝐴𝑁))))

Theorembcfallfac 14614 Binomial coefficient in terms of falling factorials. (Contributed by Scott Fenton, 20-Mar-2018.)
(𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾)))

Theoremfallfacfac 14615 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

Syntaxcbp 14616 Declare the constant for the Bernoulli polynomial operator.
class BernPoly

Definitiondf-bpoly 14617* Define the Bernoulli polynomials. Here we use well-founded recursion to define the Bernoulli polynomials. This agrees with most textbook definitions, although explicit formulae do exist. (Contributed by Scott Fenton, 22-May-2014.)
BernPoly = (𝑚 ∈ ℕ0, 𝑥 ∈ ℂ ↦ (wrecs( < , ℕ0, (𝑔 ∈ V ↦ (#‘dom 𝑔) / 𝑛((𝑥𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔𝑘) / ((𝑛𝑘) + 1))))))‘𝑚))

Theorembpolylem 14618* Lemma for bpolyval 14619. (Contributed by Scott Fenton, 22-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐺 = (𝑔 ∈ V ↦ (#‘dom 𝑔) / 𝑛((𝑋𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔𝑘) / ((𝑛𝑘) + 1)))))    &   𝐹 = wrecs( < , ℕ0, 𝐺)       ((𝑁 ∈ ℕ0𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁𝑘) + 1)))))

Theorembpolyval 14619* The value of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.)
((𝑁 ∈ ℕ0𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁𝑘) + 1)))))

Theorembpoly0 14620 The value of the Bernoulli polynomials at zero. (Contributed by Scott Fenton, 16-May-2014.)
(𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)

Theorembpoly1 14621 The value of the Bernoulli polynomials at one. (Contributed by Scott Fenton, 16-May-2014.)
(𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))

Theorembpolycl 14622 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 14623* 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 14624* Lemma for bpolydif 14625. (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 14625 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 14626* 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 14627 The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
(𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))

Theorembpoly3 14628 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 14629 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 14630* 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 14631 Extend class notation to include the exponential function.
class exp

Syntaxceu 14632 Extend class notation to include Euler's constant = 2.7182818....
class e

Syntaxcsin 14633 Extend class notation to include the sine function.
class sin

Syntaxccos 14634 Extend class notation to include the cosine function.
class cos

Syntaxctan 14635 Extend class notation to include the tangent function.
class tan

Syntaxcpi 14636 Extend class notation to include pi = 3.14159....
class π

Definitiondf-ef 14637* Define the exponential function. (Contributed by NM, 14-Mar-2005.)
exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥𝑘) / (!‘𝑘)))

Definitiondf-e 14638 Define Euler's constant 2.71828.... (Contributed by NM, 14-Mar-2005.)
e = (exp‘1)

Definitiondf-sin 14639 Define the sine function. (Contributed by NM, 14-Mar-2005.)
sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))

Definitiondf-cos 14640 Define the cosine function. (Contributed by NM, 14-Mar-2005.)
cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))

Definitiondf-tan 14641 Define the tangent function. We define it this way for cmpt 4643, which requires the form (𝑥𝐴𝐵). (Contributed by Mario Carneiro, 14-Mar-2014.)
tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))

Definitiondf-pi 14642 Define 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.)
π = inf((ℝ+ ∩ (sin “ {0})), ℝ, < )

Theoremeftcl 14643 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 14644 The terms of the series expansion of the exponential function of a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)
((𝐴 ∈ ℝ ∧ 𝐾 ∈ ℕ0) → ((𝐴𝐾) / (!‘𝐾)) ∈ ℝ)

Theoremeftabs 14645 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 14646* 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 14647* Lemma for efcl 14652. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 14454 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )

Theoremef0lem 14648* 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 14649* 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 14650 Value of Euler's constant e = 2.71828... (Contributed by Steve Rodriguez, 5-Mar-2006.)
e = Σ𝑘 ∈ ℕ0 (1 / (!‘𝑘))

Theoremeff 14651 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 14652 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 14653* 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 14654* 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 14655* 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 14656 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 14657 The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (exp‘𝐴) ∈ ℝ)

Theoremere 14658 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 14659 Lemma for egt2lt3 14773. (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 14660 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 14661 Exponential function 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 14663 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 14664* 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 14665 Cancellation of law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.)
(𝐴 ∈ ℂ → ((exp‘𝐴) · (exp‘-𝐴)) = 1)

Theoremefne0 14666 The exponential function never vanishes. 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 14667 Exponent of a negative number. (Contributed by Mario Carneiro, 10-May-2014.)
(𝐴 ∈ ℂ → (exp‘-𝐴) = (1 / (exp‘𝐴)))

Theoremeff2 14668 The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
exp:ℂ⟶(ℂ ∖ {0})

Theoremefsub 14669 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 14670 Exponential function to 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 14671 Value of the exponential function for integers. Special case of efval 14649. 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 14672 The exponential function 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 14673 The exponential function of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ+)

Theoremrpefcld 14674 The exponential function of a real number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (exp‘𝐴) ∈ ℝ+)

Theoremeftlcvg 14675* 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 14676* 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 14677* 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 14678* 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 14679* 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 14680* The partial sums of the series expansion of the exponential function of 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 14681 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 14682* 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 14683 The exponential function of a positive real number is greater than 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 14684 The exponential function 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 14685 The exponential function 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 14686 The exponential function on the reals is strictly monotonic. (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 14687 The exponential function on the reals is strictly monotonic. (Contributed by Mario Carneiro, 11-Mar-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ (exp‘𝐴) ≤ (exp‘𝐵)))

Theoremreef11 14688 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 14689 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 14690 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 14691 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 14692 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 14693 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 14694 Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
cos:ℂ⟶ℂ

Theoremsincl 14695 Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
(𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ)

Theoremcoscl 14696 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 14697 Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))

Theoremtancl 14698 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 14699 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (sin‘𝐴) ∈ ℂ)

Theoremcoscld 14700 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (cos‘𝐴) ∈ ℂ)

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