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Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremege2le3 14801 Lemma for egt2lt3 14915. (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 14802 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 14803 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 14805 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 14806* 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 14807 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 14808 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 14809 Exponent of a negative number. (Contributed by Mario Carneiro, 10-May-2014.)
(𝐴 ∈ ℂ → (exp‘-𝐴) = (1 / (exp‘𝐴)))

Theoremeff2 14810 The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
exp:ℂ⟶(ℂ ∖ {0})

Theoremefsub 14811 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 14812 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 14813 Value of the exponential function for integers. Special case of efval 14791. 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 14814 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 14815 The exponential function of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ+)

Theoremrpefcld 14816 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 14817* 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 14818* 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 14819* 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 14820* 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 14821* 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 14822* 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 14823 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 14824* 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 14825 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 14826 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 14827 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 14828 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 14829 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 14830 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 14831 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 14832 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 14833 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 14834 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 14835 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 14836 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 14837 Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
(𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ)

Theoremcoscl 14838 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 14839 Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))

Theoremtancl 14840 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 14841 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (sin‘𝐴) ∈ ℂ)

Theoremcoscld 14842 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (cos‘𝐴) ∈ ℂ)

Theoremtancld 14843 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (cos‘𝐴) ≠ 0)       (𝜑 → (tan‘𝐴) ∈ ℂ)

Theoremtanval2 14844 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))))))

Theoremtanval3 14845 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))))

Theoremresinval 14846 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 14847 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 14848* Separate out the first four terms of the infinite series expansion of the exponential function of an imaginary number. (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 14849* 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 14850* 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 14851 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ)

Theoremrecoscl 14852 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ)

Theoremretancl 14853 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 14854 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (sin‘𝐴) ∈ ℝ)

Theoremrecoscld 14855 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (cos‘𝐴) ∈ ℝ)

Theoremretancld 14856 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → (cos‘𝐴) ≠ 0)       (𝜑 → (tan‘𝐴) ∈ ℝ)

Theoremsinneg 14857 The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴))

Theoremcosneg 14858 The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴))

Theoremtanneg 14859 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 14860 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
(sin‘0) = 0

Theoremcos0 14861 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
(cos‘0) = 1

Theoremtan0 14862 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
(tan‘0) = 0

Theoremefival 14863 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 14864 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))))

Theoremsinhval 14865 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))

Theoremcoshval 14866 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))

Theoremresinhcl 14867 The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ → ((sin‘(i · 𝐴)) / i) ∈ ℝ)

Theoremrpcoshcl 14868 The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ+)

Theoremrecoshcl 14869 The hyperbolic cosine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ)

Theoremretanhcl 14870 The hyperbolic tangent of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ ℝ)

Theoremtanhlt1 14871 The hyperbolic tangent of a real number is upper bounded by 1. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) < 1)

Theoremtanhbnd 14872 The hyperbolic tangent of a real number is bounded by 1. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ (-1(,)1))

Theoremefeul 14873 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 14874 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 14875 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 14876 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))))

Theoremtanaddlem 14877 A useful intermediate step in tanadd 14878 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((cos‘𝐴) ≠ 0 ∧ (cos‘𝐵) ≠ 0)) → ((cos‘(𝐴 + 𝐵)) ≠ 0 ↔ ((tan‘𝐴) · (tan‘𝐵)) ≠ 1))

(((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 14879 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 14880 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 14881 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 14882 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 14883 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 14876 and cossub 14880. (Contributed by David A. Wheeler, 26-May-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (sin‘𝐵)) = (((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2))

Theoremcosmul 14884 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 14876 and cossub 14880. (Contributed by David A. Wheeler, 26-May-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐴) · (cos‘𝐵)) = (((cos‘(𝐴𝐵)) + (cos‘(𝐴 + 𝐵))) / 2))

Theoremaddcos 14885 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 14886 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 14887 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 14888 Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘(2 · 𝐴)) = (2 · ((sin‘𝐴) · (cos‘𝐴))))

Theoremcos2t 14889 Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1))

Theoremcos2tsin 14890 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 14891 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 14892 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 14893 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 14894 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 14895* Lemma for sin01bnd 14896 and cos01bnd 14897. (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 14896 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 14897 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 14898 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3))

Theoremcos2bnd 14899 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9))

Theoremsinltx 14900 The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.)
(𝐴 ∈ ℝ+ → (sin‘𝐴) < 𝐴)

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