P. 123 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12201-12300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	efexp 12201	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 12202	Value of the exponential function for integers. Special case of efval 12180. 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 12203	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 12204	The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ+)
Theorem	rpefcld 12205	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 12206*	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 12207*	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 12208*	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 12209*	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 12210*	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 12211*	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 12212	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 12213*	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 12214	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 12215	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 12216	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 12217	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 12218	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 12219	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 12220	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 12221	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 12222	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 12223	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 12224	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 12225	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 12226	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 12227	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 12228	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 12229	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (sin‘𝐴) ∈ ℂ)
Theorem	coscld 12230	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ)
Theorem	tanclapd 12231	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 12232	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 12233	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 12234	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 12235	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 12236*	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 12237*	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 12238*	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 12239	The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ)
Theorem	recoscl 12240	The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ)
Theorem	retanclap 12241	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 12242	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (sin‘𝐴) ∈ ℝ)
Theorem	recoscld 12243	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (cos‘𝐴) ∈ ℝ)
Theorem	retanclapd 12244	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 12245	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 12246	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 12247	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 12248	Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
⊢ (sin‘0) = 0
Theorem	cos0 12249	Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
⊢ (cos‘0) = 1
Theorem	tan0 12250	The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
⊢ (tan‘0) = 0
Theorem	efival 12251	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 12252	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	efeul 12253	Eulerian representation of the complex exponential. (Suggested by Jeff Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.)
⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((exp‘(ℜ‘𝐴)) · ((cos‘(ℑ‘𝐴)) + (i · (sin‘(ℑ‘𝐴))))))
Theorem	efieq 12254	The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((exp‘(i · 𝐴)) = (exp‘(i · 𝐵)) ↔ ((cos‘𝐴) = (cos‘𝐵) ∧ (sin‘𝐴) = (sin‘𝐵))))
Theorem	sinadd 12255	Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (sin‘(𝐴 + 𝐵)) = (((sin‘𝐴) · (cos‘𝐵)) + ((cos‘𝐴) · (sin‘𝐵))))
Theorem	cosadd 12256	Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cos‘(𝐴 + 𝐵)) = (((cos‘𝐴) · (cos‘𝐵)) − ((sin‘𝐴) · (sin‘𝐵))))
Theorem	tanaddaplem 12257	A useful intermediate step in tanaddap 12258 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((cos‘𝐴) # 0 ∧ (cos‘𝐵) # 0)) → ((cos‘(𝐴 + 𝐵)) # 0 ↔ ((tan‘𝐴) · (tan‘𝐵)) # 1))
Theorem	tanaddap 12258	Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((cos‘𝐴) # 0 ∧ (cos‘𝐵) # 0 ∧ (cos‘(𝐴 + 𝐵)) # 0)) → (tan‘(𝐴 + 𝐵)) = (((tan‘𝐴) + (tan‘𝐵)) / (1 − ((tan‘𝐴) · (tan‘𝐵)))))
Theorem	sinsub 12259	Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (sin‘(𝐴 − 𝐵)) = (((sin‘𝐴) · (cos‘𝐵)) − ((cos‘𝐴) · (sin‘𝐵))))
Theorem	cossub 12260	Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cos‘(𝐴 − 𝐵)) = (((cos‘𝐴) · (cos‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵))))
Theorem	addsin 12261	Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) + (sin‘𝐵)) = (2 · ((sin‘((𝐴 + 𝐵) / 2)) · (cos‘((𝐴 − 𝐵) / 2)))))
Theorem	subsin 12262	Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) − (sin‘𝐵)) = (2 · ((cos‘((𝐴 + 𝐵) / 2)) · (sin‘((𝐴 − 𝐵) / 2)))))
Theorem	sinmul 12263	Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12256 and cossub 12260. (Contributed by David A. Wheeler, 26-May-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (sin‘𝐵)) = (((cos‘(𝐴 − 𝐵)) − (cos‘(𝐴 + 𝐵))) / 2))
Theorem	cosmul 12264	Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12256 and cossub 12260. (Contributed by David A. Wheeler, 26-May-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐴) · (cos‘𝐵)) = (((cos‘(𝐴 − 𝐵)) + (cos‘(𝐴 + 𝐵))) / 2))
Theorem	addcos 12265	Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐴) + (cos‘𝐵)) = (2 · ((cos‘((𝐴 + 𝐵) / 2)) · (cos‘((𝐴 − 𝐵) / 2)))))
Theorem	subcos 12266	Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐵) − (cos‘𝐴)) = (2 · ((sin‘((𝐴 + 𝐵) / 2)) · (sin‘((𝐴 − 𝐵) / 2)))))
Theorem	sincossq 12267	Sine squared plus cosine squared is 1. Equation 17 of [Gleason] p. 311. Note that this holds for non-real arguments, even though individually each term is unbounded. (Contributed by NM, 15-Jan-2006.)
⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1)
Theorem	sin2t 12268	Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘(2 · 𝐴)) = (2 · ((sin‘𝐴) · (cos‘𝐴))))
Theorem	cos2t 12269	Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1))
Theorem	cos2tsin 12270	Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = (1 − (2 · ((sin‘𝐴)↑2))))
Theorem	sinbnd 12271	The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)
⊢ (𝐴 ∈ ℝ → (-1 ≤ (sin‘𝐴) ∧ (sin‘𝐴) ≤ 1))
Theorem	cosbnd 12272	The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)
⊢ (𝐴 ∈ ℝ → (-1 ≤ (cos‘𝐴) ∧ (cos‘𝐴) ≤ 1))
Theorem	sinbnd2 12273	The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)
⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ (-1[,]1))
Theorem	cosbnd2 12274	The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)
⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ (-1[,]1))
Theorem	ef01bndlem 12275*	Lemma for sin01bnd 12276 and cos01bnd 12277. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ (0(,]1) → (abs‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)) < ((𝐴↑4) / 6))
Theorem	sin01bnd 12276	Bounds on the sine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ (𝐴 ∈ (0(,]1) → ((𝐴 − ((𝐴↑3) / 3)) < (sin‘𝐴) ∧ (sin‘𝐴) < 𝐴))
Theorem	cos01bnd 12277	Bounds on the cosine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ (𝐴 ∈ (0(,]1) → ((1 − (2 · ((𝐴↑2) / 3))) < (cos‘𝐴) ∧ (cos‘𝐴) < (1 − ((𝐴↑2) / 3))))
Theorem	cos1bnd 12278	Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ ((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3))
Theorem	cos2bnd 12279	Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9))
Theorem	sinltxirr 12280*	The sine of a positive irrational number is less than its argument. Here irrational means apart from any rational number. (Contributed by Mario Carneiro, 29-Jul-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ ∀𝑞 ∈ ℚ 𝐴 # 𝑞) → (sin‘𝐴) < 𝐴)
Theorem	sin01gt0 12281	The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Wolf Lammen, 25-Sep-2020.)
⊢ (𝐴 ∈ (0(,]1) → 0 < (sin‘𝐴))
Theorem	cos01gt0 12282	The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (𝐴 ∈ (0(,]1) → 0 < (cos‘𝐴))
Theorem	sin02gt0 12283	The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (𝐴 ∈ (0(,]2) → 0 < (sin‘𝐴))
Theorem	sincos1sgn 12284	The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (0 < (sin‘1) ∧ 0 < (cos‘1))
Theorem	sincos2sgn 12285	The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (0 < (sin‘2) ∧ (cos‘2) < 0)
Theorem	sin4lt0 12286	The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (sin‘4) < 0
Theorem	cos12dec 12287	Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.)
⊢ ((𝐴 ∈ (1[,]2) ∧ 𝐵 ∈ (1[,]2) ∧ 𝐴 < 𝐵) → (cos‘𝐵) < (cos‘𝐴))
Theorem	absefi 12288	The absolute value of the exponential of an imaginary number is one. Equation 48 of [Rudin] p. 167. (Contributed by Jason Orendorff, 9-Feb-2007.)
⊢ (𝐴 ∈ ℝ → (abs‘(exp‘(i · 𝐴))) = 1)
Theorem	absef 12289	The absolute value of the exponential is the exponential of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)
⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (exp‘(ℜ‘𝐴)))
Theorem	absefib 12290	A complex number is real iff the exponential of its product with i has absolute value one. (Contributed by NM, 21-Aug-2008.)
⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (abs‘(exp‘(i · 𝐴))) = 1))
Theorem	efieq1re 12291	A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)
⊢ ((𝐴 ∈ ℂ ∧ (exp‘(i · 𝐴)) = 1) → 𝐴 ∈ ℝ)
Theorem	demoivre 12292	De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. See also demoivreALT 12293 for an alternate longer proof not using the exponential function. (Contributed by NM, 24-Jul-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))))
Theorem	demoivreALT 12293	Alternate proof of demoivre 12292. It is longer but does not use the exponential function. This is Metamath 100 proof #17. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))))
4.10.1.1  The circle constant (tau = 2 pi)
Syntax	ctau 12294	Extend class notation to include the constant tau, τ = 6.28318....
class τ
Definition	df-tau 12295	Define the circle constant tau, τ = 6.28318..., which is the smallest positive real number whose cosine is one. Various notations have been used or proposed for this number including τ, a three-legged variant of π, or 2π. Note the difference between this constant τ and the formula variable 𝜏. Following our convention, the constant is displayed in upright font while the variable is in italic font; furthermore, the colors are different. (Contributed by Jim Kingdon, 9-Apr-2018.) (Revised by AV, 1-Oct-2020.)
⊢ τ = inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < )
4.10.2  _e is irrational
Theorem	eirraplem 12296*	Lemma for eirrap 12297. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))    &   ⊢ (𝜑 → 𝑃 ∈ ℤ)    &   ⊢ (𝜑 → 𝑄 ∈ ℕ)    ⇒   ⊢ (𝜑 → e # (𝑃 / 𝑄))
Theorem	eirrap 12297	e is irrational. That is, for any rational number, e is apart from it. In the absence of excluded middle, we can distinguish between this and saying that e is not rational, which is eirr 12298. (Contributed by Jim Kingdon, 6-Jan-2023.)
⊢ (𝑄 ∈ ℚ → e # 𝑄)
Theorem	eirr 12298	e is not rational. In the absence of excluded middle, we can distinguish between this and saying that e is irrational in the sense of being apart from any rational number, which is eirrap 12297. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
⊢ e ∉ ℚ
Theorem	egt2lt3 12299	Euler's constant e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Jim Kingdon, 7-Jan-2023.)
⊢ (2 < e ∧ e < 3)
Theorem	epos 12300	Euler's constant e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
⊢ 0 < e