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Theorem List for Intuitionistic Logic Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrpefcl 11301 The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ+)
 
Theoremrpefcld 11302 The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (exp‘𝐴) ∈ ℝ+)
 
Theoremeftlcvg 11303* 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 11304* 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 11305* 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 11306* 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 11307* 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 11308* 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 11309 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 11310* 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 11311 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 11312 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 11313 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 11314 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)))
 
Theoremefler 11315 The exponential function on the reals is nondecreasing. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Jim Kingdon, 20-Dec-2022.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((exp‘𝐴) ≤ (exp‘𝐵) → 𝐴𝐵))
 
Theoremreef11 11316 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 11317 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 11318 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 11319 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 11320 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 11321 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 11322 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 11323 Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
(𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ)
 
Theoremcoscl 11324 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 11325 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 11326 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 11327 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (sin‘𝐴) ∈ ℂ)
 
Theoremcoscld 11328 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (cos‘𝐴) ∈ ℂ)
 
Theoremtanclapd 11329 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 11330 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 11331 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 11332 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 11333 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 11334* 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 11335* 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 11336* 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 11337 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ)
 
Theoremrecoscl 11338 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ)
 
Theoremretanclap 11339 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 11340 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (sin‘𝐴) ∈ ℝ)
 
Theoremrecoscld 11341 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (cos‘𝐴) ∈ ℝ)
 
Theoremretanclapd 11342 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → (cos‘𝐴) # 0)       (𝜑 → (tan‘𝐴) ∈ ℝ)
 
Theoremsinneg 11343 The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴))
 
Theoremcosneg 11344 The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴))
 
Theoremtannegap 11345 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 11346 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
(sin‘0) = 0
 
Theoremcos0 11347 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
(cos‘0) = 1
 
Theoremtan0 11348 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
(tan‘0) = 0
 
Theoremefival 11349 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 11350 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 11351 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 11352 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 11353 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 11354 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 11355 A useful intermediate step in tanaddap 11356 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 11356 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 11357 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 11358 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 11359 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 11360 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 11361 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11354 and cossub 11358. (Contributed by David A. Wheeler, 26-May-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (sin‘𝐵)) = (((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2))
 
Theoremcosmul 11362 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 11354 and cossub 11358. (Contributed by David A. Wheeler, 26-May-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐴) · (cos‘𝐵)) = (((cos‘(𝐴𝐵)) + (cos‘(𝐴 + 𝐵))) / 2))
 
Theoremaddcos 11363 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 11364 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 11365 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 11366 Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘(2 · 𝐴)) = (2 · ((sin‘𝐴) · (cos‘𝐴))))
 
Theoremcos2t 11367 Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1))
 
Theoremcos2tsin 11368 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 11369 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 11370 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 11371 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 11372 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 11373* Lemma for sin01bnd 11374 and cos01bnd 11375. (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 11374 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 11375 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 11376 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3))
 
Theoremcos2bnd 11377 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9))
 
Theoremsin01gt0 11378 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 11379 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 11380 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 11381 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sin‘1) ∧ 0 < (cos‘1))
 
Theoremsincos2sgn 11382 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sin‘2) ∧ (cos‘2) < 0)
 
Theoremsin4lt0 11383 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
(sin‘4) < 0
 
Theoremabsefi 11384 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 11385 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 11386 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 11387 A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)
((𝐴 ∈ ℂ ∧ (exp‘(i · 𝐴)) = 1) → 𝐴 ∈ ℝ)
 
Theoremdemoivre 11388 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 11389 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 11389 Alternate proof of demoivre 11388. 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)))))
 
4.9.2  _e is irrational
 
Theoremeirraplem 11390* Lemma for eirrap 11391. (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 11391 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 11392. (Contributed by Jim Kingdon, 6-Jan-2023.)
(𝑄 ∈ ℚ → e # 𝑄)
 
Theoremeirr 11392 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 11391. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
e ∉ ℚ
 
Theoremegt2lt3 11393 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)
 
Theoremepos 11394 Euler's constant e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
0 < e
 
Theoremepr 11395 Euler's constant e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
e ∈ ℝ+
 
Theoremene0 11396 e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
e ≠ 0
 
Theoremeap0 11397 e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
e # 0
 
Theoremene1 11398 e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
e ≠ 1
 
Theoremeap1 11399 e is apart from 1. (Contributed by Jim Kingdon, 7-Jan-2023.)
e # 1
 
PART 5  ELEMENTARY NUMBER THEORY

Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory.

 
5.1  Elementary properties of divisibility
 
5.1.1  The divides relation
 
Syntaxcdvds 11400 Extend the definition of a class to include the divides relation. See df-dvds 11401.
class
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