P. 125 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12401-12500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fprodsplitf 12401*	Split a finite product into two parts. A version of fprodsplit 12366 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝑈 𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ 𝐵 𝐶))
Theorem	fprodsplitsn 12402*	Separate out a term in a finite product. See also fprodunsn 12373 which is the same but with a distinct variable condition in place of Ⅎ𝑘𝜑. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐷    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))
Theorem	fprodsplit1f 12403*	Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → Ⅎ𝑘𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵))
Theorem	fprodclf 12404*	Closure of a finite product of complex numbers. A version of fprodcl 12376 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
Theorem	fprodap0f 12405*	A finite product of terms apart from zero is apart from zero. A version of fprodap0 12390 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 # 0)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 # 0)
Theorem	fprodge0 12406*	If all the terms of a finite product are nonnegative, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵)
Theorem	fprodeq0g 12407*	Any finite product containing a zero term is itself zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 0)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0)
Theorem	fprodge1 12408*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵)
Theorem	fprodle 12409*	If all the terms of two finite products are nonnegative and compare, so do the two products. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≤ ∏𝑘 ∈ 𝐴 𝐶)
Theorem	fprodmodd 12410*	If all factors of two finite products are equal modulo 𝑀, the products are equal modulo 𝑀. (Contributed by AV, 7-Jul-2021.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 mod 𝑀) = (𝐶 mod 𝑀))    ⇒   ⊢ (𝜑 → (∏𝑘 ∈ 𝐴 𝐵 mod 𝑀) = (∏𝑘 ∈ 𝐴 𝐶 mod 𝑀))
4.10  Elementary trigonometry
4.10.1  The exponential, sine, and cosine functions
Syntax	ce 12411	Extend class notation to include the exponential function.
class exp
Syntax	ceu 12412	Extend class notation to include Euler's constant e = 2.71828....
class e
Syntax	csin 12413	Extend class notation to include the sine function.
class sin
Syntax	ccos 12414	Extend class notation to include the cosine function.
class cos
Syntax	ctan 12415	Extend class notation to include the tangent function.
class tan
Syntax	cpi 12416	Extend class notation to include the constant pi, π = 3.14159....
class π
Definition	df-ef 12417*	Define the exponential function. Its value at the complex number 𝐴 is (exp‘𝐴) and is called the "exponential of 𝐴"; see efval 12430. (Contributed by NM, 14-Mar-2005.)
⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)))
Definition	df-e 12418	Define Euler's constant e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
⊢ e = (exp‘1)
Definition	df-sin 12419	Define the sine function. (Contributed by NM, 14-Mar-2005.)
⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
Definition	df-cos 12420	Define the cosine function. (Contributed by NM, 14-Mar-2005.)
⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
Definition	df-tan 12421	Define the tangent function. We define it this way for cmpt 4192, 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 12422	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 12423	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 12424	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 12425	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	eftvalcn 12426*	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 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁)))
Theorem	efcllemp 12427*	Lemma for efcl 12433. The series that defines the exponential function converges. The ratio test cvgratgt0 12302 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → (2 · (abs‘𝐴)) < 𝐾)    ⇒   ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
Theorem	efcllem 12428*	Lemma for efcl 12433. The series that defines the exponential function converges. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )
Theorem	ef0lem 12429*	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 12430*	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 12431	Value of Euler's constant e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
⊢ e = Σ𝑘 ∈ ℕ0 (1 / (!‘𝑘))
Theorem	eff 12432	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 12433	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 12434*	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 12435*	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 12436*	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 12437	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 12438	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 12439	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 12440	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 ∈ ℕ ↦ (2 · ((1 / 2)↑𝑛)))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))    ⇒   ⊢ (2 ≤ e ∧ e ≤ 3)
Theorem	ef0 12441	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 12442	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 12443*	Lemma for efadd 12444 (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 12444	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	efcan 12445	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	efap0 12446	The exponential of a complex number is apart from zero. (Contributed by Jim Kingdon, 12-Dec-2022.)
⊢ (𝐴 ∈ ℂ → (exp‘𝐴) # 0)
Theorem	efne0 12447	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 12446 (which will be more useful in most contexts). (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.)
⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0)
Theorem	efneg 12448	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 12449	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 12450	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 12451	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 12452	Value of the exponential function for integers. Special case of efval 12430. 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 12453	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 12454	The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ+)
Theorem	rpefcld 12455	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 12456*	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 12457*	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 12458*	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 12459*	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 12460*	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 12461*	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 12462	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 12463*	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 12464	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 12465	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 12466	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	efltim 12467	The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (exp‘𝐴) < (exp‘𝐵)))
Theorem	reef11 12468	The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Jim Kingdon, 20-Dec-2022.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((exp‘𝐴) = (exp‘𝐵) ↔ 𝐴 = 𝐵))
Theorem	reeff1 12469	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 12470	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 12471	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 12472	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 12473	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 12474	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 12475	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 12476	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	tanvalap 12477	Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))
Theorem	tanclap 12478	The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) ∈ ℂ)
Theorem	sincld 12479	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (sin‘𝐴) ∈ ℂ)
Theorem	coscld 12480	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ)
Theorem	tanclapd 12481	Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (cos‘𝐴) # 0)    ⇒   ⊢ (𝜑 → (tan‘𝐴) ∈ ℂ)
Theorem	tanval2ap 12482	Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (i · ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))))
Theorem	tanval3ap 12483	Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
⊢ ((𝐴 ∈ ℂ ∧ ((exp‘(2 · (i · 𝐴))) + 1) # 0) → (tan‘𝐴) = (((exp‘(2 · (i · 𝐴))) − 1) / (i · ((exp‘(2 · (i · 𝐴))) + 1))))
Theorem	resinval 12484	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 12485	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 12486*	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 12487*	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 12488*	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 12489	The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ)
Theorem	recoscl 12490	The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ)
Theorem	retanclap 12491	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 12492	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (sin‘𝐴) ∈ ℝ)
Theorem	recoscld 12493	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (cos‘𝐴) ∈ ℝ)
Theorem	retanclapd 12494	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 12495	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 12496	The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴))
Theorem	tannegap 12497	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 12498	Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
⊢ (sin‘0) = 0
Theorem	cos0 12499	Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
⊢ (cos‘0) = 1
Theorem	tan0 12500	The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
⊢ (tan‘0) = 0