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Theorem List for Intuitionistic Logic Explorer - 11701-11800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremefneg 11701 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 11702 The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
exp:ℂ⟶(ℂ ∖ {0})
 
Theoremefsub 11703 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 11704 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 11705 Value of the exponential function for integers. Special case of efval 11683. 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 11706 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 11707 The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ+)
 
Theoremrpefcld 11708 The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (exp‘𝐴) ∈ ℝ+)
 
Theoremeftlcvg 11709* 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 11710* 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 11711* 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 11712* 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 11713* 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 11714* 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 11715 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 11716* 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 11717 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 11718 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 11719 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))
 
Theoremefltim 11720 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)))
 
Theoremreef11 11721 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))
 
Theoremreeff1 11722 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 11723 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 11724 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 11725 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 11726 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 11727 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 11728 Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
(𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ)
 
Theoremcoscl 11729 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)
 
Theoremtanvalap 11730 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)))
 
Theoremtanclap 11731 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)
 
Theoremsincld 11732 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (sin‘𝐴) ∈ ℂ)
 
Theoremcoscld 11733 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (cos‘𝐴) ∈ ℂ)
 
Theoremtanclapd 11734 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (cos‘𝐴) # 0)       (𝜑 → (tan‘𝐴) ∈ ℂ)
 
Theoremtanval2ap 11735 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))))))
 
Theoremtanval3ap 11736 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))))
 
Theoremresinval 11737 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 11738 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 11739* 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 11740* 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 11741* 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 11742 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ)
 
Theoremrecoscl 11743 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ)
 
Theoremretanclap 11744 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 11745 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (sin‘𝐴) ∈ ℝ)
 
Theoremrecoscld 11746 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (cos‘𝐴) ∈ ℝ)
 
Theoremretanclapd 11747 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → (cos‘𝐴) # 0)       (𝜑 → (tan‘𝐴) ∈ ℝ)
 
Theoremsinneg 11748 The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴))
 
Theoremcosneg 11749 The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴))
 
Theoremtannegap 11750 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 11751 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
(sin‘0) = 0
 
Theoremcos0 11752 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
(cos‘0) = 1
 
Theoremtan0 11753 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
(tan‘0) = 0
 
Theoremefival 11754 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))))
 
Theoremefmival 11755 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))))
 
Theoremefeul 11756 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))))))
 
Theoremefieq 11757 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))))
 
Theoremsinadd 11758 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))))
 
Theoremcosadd 11759 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))))
 
Theoremtanaddaplem 11760 A useful intermediate step in tanaddap 11761 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))
 
Theoremtanaddap 11761 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)))))
 
Theoremsinsub 11762 Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (sin‘(𝐴𝐵)) = (((sin‘𝐴) · (cos‘𝐵)) − ((cos‘𝐴) · (sin‘𝐵))))
 
Theoremcossub 11763 Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cos‘(𝐴𝐵)) = (((cos‘𝐴) · (cos‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵))))
 
Theoremaddsin 11764 Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) + (sin‘𝐵)) = (2 · ((sin‘((𝐴 + 𝐵) / 2)) · (cos‘((𝐴𝐵) / 2)))))
 
Theoremsubsin 11765 Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) − (sin‘𝐵)) = (2 · ((cos‘((𝐴 + 𝐵) / 2)) · (sin‘((𝐴𝐵) / 2)))))
 
Theoremsinmul 11766 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11759 and cossub 11763. (Contributed by David A. Wheeler, 26-May-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (sin‘𝐵)) = (((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2))
 
Theoremcosmul 11767 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 11759 and cossub 11763. (Contributed by David A. Wheeler, 26-May-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐴) · (cos‘𝐵)) = (((cos‘(𝐴𝐵)) + (cos‘(𝐴 + 𝐵))) / 2))
 
Theoremaddcos 11768 Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐴) + (cos‘𝐵)) = (2 · ((cos‘((𝐴 + 𝐵) / 2)) · (cos‘((𝐴𝐵) / 2)))))
 
Theoremsubcos 11769 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‘((𝐴𝐵) / 2)))))
 
Theoremsincossq 11770 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)
 
Theoremsin2t 11771 Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘(2 · 𝐴)) = (2 · ((sin‘𝐴) · (cos‘𝐴))))
 
Theoremcos2t 11772 Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1))
 
Theoremcos2tsin 11773 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))))
 
Theoremsinbnd 11774 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))
 
Theoremcosbnd 11775 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))
 
Theoremsinbnd2 11776 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))
 
Theoremcosbnd2 11777 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))
 
Theoremef01bndlem 11778* Lemma for sin01bnd 11779 and cos01bnd 11780. (Contributed by Paul Chapman, 19-Jan-2008.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))       (𝐴 ∈ (0(,]1) → (abs‘Σ𝑘 ∈ (ℤ‘4)(𝐹𝑘)) < ((𝐴↑4) / 6))
 
Theoremsin01bnd 11779 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))
 
Theoremcos01bnd 11780 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))))
 
Theoremcos1bnd 11781 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3))
 
Theoremcos2bnd 11782 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9))
 
Theoremsin01gt0 11783 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))
 
Theoremcos01gt0 11784 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))
 
Theoremsin02gt0 11785 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))
 
Theoremsincos1sgn 11786 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sin‘1) ∧ 0 < (cos‘1))
 
Theoremsincos2sgn 11787 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sin‘2) ∧ (cos‘2) < 0)
 
Theoremsin4lt0 11788 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
(sin‘4) < 0
 
Theoremcos12dec 11789 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))
 
Theoremabsefi 11790 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)
 
Theoremabsef 11791 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)))
 
Theoremabsefib 11792 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))
 
Theoremefieq1re 11793 A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)
((𝐴 ∈ ℂ ∧ (exp‘(i · 𝐴)) = 1) → 𝐴 ∈ ℝ)
 
Theoremdemoivre 11794 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 11795 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)))))
 
TheoremdemoivreALT 11795 Alternate proof of demoivre 11794. 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)))))
 
4.9.1.1  The circle constant (tau = 2 pi)
 
Syntaxctau 11796 Extend class notation to include the constant tau, τ = 6.28318....
class τ
 
Definitiondf-tau 11797 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 . 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, < )
 
4.9.2  _e is irrational
 
Theoremeirraplem 11798* Lemma for eirrap 11799. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))    &   (𝜑𝑃 ∈ ℤ)    &   (𝜑𝑄 ∈ ℕ)       (𝜑 → e # (𝑃 / 𝑄))
 
Theoremeirrap 11799 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 11800. (Contributed by Jim Kingdon, 6-Jan-2023.)
(𝑄 ∈ ℚ → e # 𝑄)
 
Theoremeirr 11800 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 11799. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
e ∉ ℚ
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