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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | eff2 15201 | 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 15202 | 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 15203 | 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 15204 | Value of the exponential function for integers. Special case of efval 15182. 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 15205 | 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 15206 | The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.) |
⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ+) | ||
Theorem | rpefcld 15207 | 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 15208* | 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 15209* | 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 15210* | 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 15211* | 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 15212* | 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 15213* | 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 15214 | 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 15215* | 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 15216 | 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 15217 | 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 15218 | 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 | eflt 15219 | The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (exp‘𝐴) < (exp‘𝐵))) | ||
Theorem | efle 15220 | The exponential function on the reals is nondecreasing. (Contributed by Mario Carneiro, 11-Mar-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (exp‘𝐴) ≤ (exp‘𝐵))) | ||
Theorem | reef11 15221 | 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)) | ||
Theorem | reeff1 15222 | 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 15223 | 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 15224 | 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 15225 | 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 15226 | 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 15227 | 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 15228 | 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 15229 | 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 | tanval 15230 | Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) | ||
Theorem | tancl 15231 | 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) | ||
Theorem | sincld 15232 | Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (sin‘𝐴) ∈ ℂ) | ||
Theorem | coscld 15233 | Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ) | ||
Theorem | tancld 15234 | Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (cos‘𝐴) ≠ 0) ⇒ ⊢ (𝜑 → (tan‘𝐴) ∈ ℂ) | ||
Theorem | tanval2 15235 | 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)))))) | ||
Theorem | tanval3 15236 | 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)))) | ||
Theorem | resinval 15237 | 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 15238 | 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 15239* | 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 15240* | 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 15241* | 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 15242 | The sine of a real number is real. (Contributed by NM, 30-Apr-2005.) |
⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ) | ||
Theorem | recoscl 15243 | The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.) |
⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ) | ||
Theorem | retancl 15244 | 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 15245 | Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (sin‘𝐴) ∈ ℝ) | ||
Theorem | recoscld 15246 | Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (cos‘𝐴) ∈ ℝ) | ||
Theorem | retancld 15247 | 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 15248 | 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 15249 | The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.) |
⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴)) | ||
Theorem | tanneg 15250 | 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 15251 | Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.) |
⊢ (sin‘0) = 0 | ||
Theorem | cos0 15252 | Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.) |
⊢ (cos‘0) = 1 | ||
Theorem | tan0 15253 | The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.) |
⊢ (tan‘0) = 0 | ||
Theorem | efival 15254 | 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 15255 | 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 | sinhval 15256 | 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)) | ||
Theorem | coshval 15257 | 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)) | ||
Theorem | resinhcl 15258 | The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
⊢ (𝐴 ∈ ℝ → ((sin‘(i · 𝐴)) / i) ∈ ℝ) | ||
Theorem | rpcoshcl 15259 | The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ+) | ||
Theorem | recoshcl 15260 | The hyperbolic cosine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ) | ||
Theorem | retanhcl 15261 | The hyperbolic tangent of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ ℝ) | ||
Theorem | tanhlt1 15262 | 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) | ||
Theorem | tanhbnd 15263 | 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)) | ||
Theorem | efeul 15264 | 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 15265 | 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 15266 | 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 15267 | 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 | tanaddlem 15268 | A useful intermediate step in tanadd 15269 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)) | ||
Theorem | tanadd 15269 | 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 15270 | 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 15271 | 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 15272 | 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 15273 | 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 15274 | Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 15267 and cossub 15271. (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 15275 | Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 15267 and cossub 15271. (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 15276 | 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 15277 | 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 15278 | 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 15279 | Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.) |
⊢ (𝐴 ∈ ℂ → (sin‘(2 · 𝐴)) = (2 · ((sin‘𝐴) · (cos‘𝐴)))) | ||
Theorem | cos2t 15280 | Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1)) | ||
Theorem | cos2tsin 15281 | 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 15282 | 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 15283 | 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 15284 | 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 15285 | 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 15286* | Lemma for sin01bnd 15287 and cos01bnd 15288. (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 15287 | 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 15288 | 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 15289 | Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ ((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3)) | ||
Theorem | cos2bnd 15290 | Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9)) | ||
Theorem | sinltx 15291 | The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.) |
⊢ (𝐴 ∈ ℝ+ → (sin‘𝐴) < 𝐴) | ||
Theorem | sin01gt0 15292 | 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 15293 | 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 15294 | 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 15295 | The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (0 < (sin‘1) ∧ 0 < (cos‘1)) | ||
Theorem | sincos2sgn 15296 | The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) | ||
Theorem | sin4lt0 15297 | The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (sin‘4) < 0 | ||
Theorem | absefi 15298 | 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 15299 | 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 15300 | 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)) |
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