| Metamath
Proof Explorer Theorem List (p. 270 of 494) | < Previous Next > | |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | (1-30937) |
(30938-32460) |
(32461-49324) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | quartlem2 26901 | Closure lemmas for quart 26904. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐸 = -(𝐴 / 4)) & ⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2)))) & ⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))) & ⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))) & ⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅))) & ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅)))) & ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3))))) ⇒ ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ)) | ||
| Theorem | quartlem3 26902 | Closure lemmas for quart 26904. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐸 = -(𝐴 / 4)) & ⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2)))) & ⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))) & ⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))) & ⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅))) & ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅)))) & ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3))))) & ⊢ (𝜑 → 𝑆 = ((√‘𝑀) / 2)) & ⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3)) & ⊢ (𝜑 → 𝑇 = (((𝑉 + 𝑊) / 2)↑𝑐(1 / 3))) & ⊢ (𝜑 → 𝑇 ≠ 0) ⇒ ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ)) | ||
| Theorem | quartlem4 26903 | Closure lemmas for quart 26904. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐸 = -(𝐴 / 4)) & ⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2)))) & ⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))) & ⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))) & ⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅))) & ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅)))) & ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3))))) & ⊢ (𝜑 → 𝑆 = ((√‘𝑀) / 2)) & ⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3)) & ⊢ (𝜑 → 𝑇 = (((𝑉 + 𝑊) / 2)↑𝑐(1 / 3))) & ⊢ (𝜑 → 𝑇 ≠ 0) & ⊢ (𝜑 → 𝑀 ≠ 0) & ⊢ (𝜑 → 𝐼 = (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))) & ⊢ (𝜑 → 𝐽 = (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))) ⇒ ⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ)) | ||
| Theorem | quart 26904 | The quartic equation, writing out all roots using square and cube root functions so that only direct substitutions remain, and we can actually claim to have a "quartic equation". Naturally, this theorem is ridiculously long (see quartfull 35170) if all the substitutions are performed. This is Metamath 100 proof #46. (Contributed by Mario Carneiro, 6-May-2015.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐸 = -(𝐴 / 4)) & ⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2)))) & ⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))) & ⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))) & ⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅))) & ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅)))) & ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3))))) & ⊢ (𝜑 → 𝑆 = ((√‘𝑀) / 2)) & ⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3)) & ⊢ (𝜑 → 𝑇 = (((𝑉 + 𝑊) / 2)↑𝑐(1 / 3))) & ⊢ (𝜑 → 𝑇 ≠ 0) & ⊢ (𝜑 → 𝑀 ≠ 0) & ⊢ (𝜑 → 𝐼 = (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))) & ⊢ (𝜑 → 𝐽 = (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))) ⇒ ⊢ (𝜑 → ((((𝑋↑4) + (𝐴 · (𝑋↑3))) + ((𝐵 · (𝑋↑2)) + ((𝐶 · 𝑋) + 𝐷))) = 0 ↔ ((𝑋 = ((𝐸 − 𝑆) + 𝐼) ∨ 𝑋 = ((𝐸 − 𝑆) − 𝐼)) ∨ (𝑋 = ((𝐸 + 𝑆) + 𝐽) ∨ 𝑋 = ((𝐸 + 𝑆) − 𝐽))))) | ||
| Syntax | casin 26905 | The arcsine function. |
| class arcsin | ||
| Syntax | cacos 26906 | The arccosine function. |
| class arccos | ||
| Syntax | catan 26907 | The arctangent function. |
| class arctan | ||
| Definition | df-asin 26908 | Define the arcsine function. Because sin is not a one-to-one function, the literal inverse ◡sin is not a function. Rather than attempt to find the right domain on which to restrict sin in order to get a total function, we just define it in terms of log, which we already know is total (except at 0). There are branch points at -1 and 1 (at which the function is defined), and branch cuts along the real line not between -1 and 1, which is to say (-∞, -1) ∪ (1, +∞). (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ arcsin = (𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))) | ||
| Definition | df-acos 26909 | Define the arccosine function. See also remarks for df-asin 26908. Since we define arccos in terms of arcsin, it shares the same branch points and cuts, namely (-∞, -1) ∪ (1, +∞). (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥))) | ||
| Definition | df-atan 26910 | Define the arctangent function. See also remarks for df-asin 26908. Unlike arcsin and arccos, this function is not defined everywhere, because tan(𝑧) ≠ ±i for all 𝑧 ∈ ℂ. For all other 𝑧, there is a formula for arctan(𝑧) in terms of log, and we take that as the definition. Branch points are at ±i; branch cuts are on the pure imaginary axis not between -i and i, which is to say {𝑧 ∈ ℂ ∣ (i · 𝑧) ∈ (-∞, -1) ∪ (1, +∞)}. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))) | ||
| Theorem | asinlem 26911 | The argument to the logarithm in df-asin 26908 is always nonzero. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (√‘(1 − (𝐴↑2)))) ≠ 0) | ||
| Theorem | asinlem2 26912 | The argument to the logarithm in df-asin 26908 has the property that replacing 𝐴 with -𝐴 in the expression gives the reciprocal. (Contributed by Mario Carneiro, 1-Apr-2015.) |
| ⊢ (𝐴 ∈ ℂ → (((i · 𝐴) + (√‘(1 − (𝐴↑2)))) · ((i · -𝐴) + (√‘(1 − (-𝐴↑2))))) = 1) | ||
| Theorem | asinlem3a 26913 | Lemma for asinlem3 26914. (Contributed by Mario Carneiro, 1-Apr-2015.) |
| ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) ≤ 0) → 0 ≤ (ℜ‘((i · 𝐴) + (√‘(1 − (𝐴↑2)))))) | ||
| Theorem | asinlem3 26914 | The argument to the logarithm in df-asin 26908 has nonnegative real part. (Contributed by Mario Carneiro, 1-Apr-2015.) |
| ⊢ (𝐴 ∈ ℂ → 0 ≤ (ℜ‘((i · 𝐴) + (√‘(1 − (𝐴↑2)))))) | ||
| Theorem | asinf 26915 | Domain and codomain of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ arcsin:ℂ⟶ℂ | ||
| Theorem | asincl 26916 | Closure for the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) ∈ ℂ) | ||
| Theorem | acosf 26917 | Domain and codoamin of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ arccos:ℂ⟶ℂ | ||
| Theorem | acoscl 26918 | Closure for the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ ℂ → (arccos‘𝐴) ∈ ℂ) | ||
| Theorem | atandm 26919 | Since the property is a little lengthy, we abbreviate 𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i as 𝐴 ∈ dom arctan. This is the necessary precondition for the definition of arctan to make sense. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) | ||
| Theorem | atandm2 26920 | This form of atandm 26919 is a bit more useful for showing that the logarithms in df-atan 26910 are well-defined. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0)) | ||
| Theorem | atandm3 26921 | A compact form of atandm 26919. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1)) | ||
| Theorem | atandm4 26922 | A compact form of atandm 26919. (Contributed by Mario Carneiro, 3-Apr-2015.) |
| ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ≠ 0)) | ||
| Theorem | atanf 26923 | Domain and codoamin of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ arctan:(ℂ ∖ {-i, i})⟶ℂ | ||
| Theorem | atancl 26924 | Closure for the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) ∈ ℂ) | ||
| Theorem | asinval 26925 | Value of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) = (-i · (log‘((i · 𝐴) + (√‘(1 − (𝐴↑2))))))) | ||
| Theorem | acosval 26926 | Value of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴))) | ||
| Theorem | atanval 26927 | Value of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) | ||
| Theorem | atanre 26928 | A real number is in the domain of the arctangent function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ dom arctan) | ||
| Theorem | asinneg 26929 | The arcsine function is odd. (Contributed by Mario Carneiro, 1-Apr-2015.) |
| ⊢ (𝐴 ∈ ℂ → (arcsin‘-𝐴) = -(arcsin‘𝐴)) | ||
| Theorem | acosneg 26930 | The negative symmetry relation of the arccosine. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ ℂ → (arccos‘-𝐴) = (π − (arccos‘𝐴))) | ||
| Theorem | efiasin 26931 | The exponential of the arcsine function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ ℂ → (exp‘(i · (arcsin‘𝐴))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) | ||
| Theorem | sinasin 26932 | The arcsine function is an inverse to sin. This is the main property that justifies the notation arcsin or sin↑-1. Because sin is not an injection, the other converse identity asinsin 26935 is only true under limited circumstances. (Contributed by Mario Carneiro, 1-Apr-2015.) |
| ⊢ (𝐴 ∈ ℂ → (sin‘(arcsin‘𝐴)) = 𝐴) | ||
| Theorem | cosacos 26933 | The arccosine function is an inverse to cos. (Contributed by Mario Carneiro, 1-Apr-2015.) |
| ⊢ (𝐴 ∈ ℂ → (cos‘(arccos‘𝐴)) = 𝐴) | ||
| Theorem | asinsinlem 26934 | Lemma for asinsin 26935. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (ℜ‘(exp‘(i · 𝐴)))) | ||
| Theorem | asinsin 26935 | The arcsine function composed with sin is equal to the identity. This plus sinasin 26932 allow to view sin and arcsin as inverse operations to each other. For ease of use, we have not defined precisely the correct domain of correctness of this identity; in addition to the main region described here it is also true for some points on the branch cuts, namely when 𝐴 = (π / 2) − i𝑦 for nonnegative real 𝑦 and also symmetrically at 𝐴 = i𝑦 − (π / 2). In particular, when restricted to reals this identity extends to the closed interval [-(π / 2), (π / 2)], not just the open interval (see reasinsin 26939). (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴) | ||
| Theorem | acoscos 26936 | The arccosine function is an inverse to cos. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (0(,)π)) → (arccos‘(cos‘𝐴)) = 𝐴) | ||
| Theorem | asin1 26937 | The arcsine of 1 is π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (arcsin‘1) = (π / 2) | ||
| Theorem | acos1 26938 | The arccosine of 1 is 0. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (arccos‘1) = 0 | ||
| Theorem | reasinsin 26939 | The arcsine function composed with sin is equal to the identity. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴) | ||
| Theorem | asinsinb 26940 | Relationship between sine and arcsine. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (ℜ‘𝐵) ∈ (-(π / 2)(,)(π / 2))) → ((arcsin‘𝐴) = 𝐵 ↔ (sin‘𝐵) = 𝐴)) | ||
| Theorem | acoscosb 26941 | Relationship between cosine and arccosine. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (ℜ‘𝐵) ∈ (0(,)π)) → ((arccos‘𝐴) = 𝐵 ↔ (cos‘𝐵) = 𝐴)) | ||
| Theorem | asinbnd 26942 | The arcsine function has range within a vertical strip of the complex plane with real part between -π / 2 and π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ ℂ → (ℜ‘(arcsin‘𝐴)) ∈ (-(π / 2)[,](π / 2))) | ||
| Theorem | acosbnd 26943 | The arccosine function has range within a vertical strip of the complex plane with real part between 0 and π. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ ℂ → (ℜ‘(arccos‘𝐴)) ∈ (0[,]π)) | ||
| Theorem | asinrebnd 26944 | Bounds on the arcsine function. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ (-(π / 2)[,](π / 2))) | ||
| Theorem | asinrecl 26945 | The arcsine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ ℝ) | ||
| Theorem | acosrecl 26946 | The arccosine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ (-1[,]1) → (arccos‘𝐴) ∈ ℝ) | ||
| Theorem | cosasin 26947 | The cosine of the arcsine of 𝐴 is √(1 − 𝐴↑2). (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ ℂ → (cos‘(arcsin‘𝐴)) = (√‘(1 − (𝐴↑2)))) | ||
| Theorem | sinacos 26948 | The sine of the arccosine of 𝐴 is √(1 − 𝐴↑2). (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ ℂ → (sin‘(arccos‘𝐴)) = (√‘(1 − (𝐴↑2)))) | ||
| Theorem | atandmneg 26949 | The domain of the arctangent function is closed under negatives. (Contributed by Mario Carneiro, 3-Apr-2015.) |
| ⊢ (𝐴 ∈ dom arctan → -𝐴 ∈ dom arctan) | ||
| Theorem | atanneg 26950 | The arctangent function is odd. (Contributed by Mario Carneiro, 3-Apr-2015.) |
| ⊢ (𝐴 ∈ dom arctan → (arctan‘-𝐴) = -(arctan‘𝐴)) | ||
| Theorem | atan0 26951 | The arctangent of zero is zero. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (arctan‘0) = 0 | ||
| Theorem | atandmcj 26952 | The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ dom arctan) | ||
| Theorem | atancj 26953 | The arctangent function distributes under conjugation. (The condition that ℜ(𝐴) ≠ 0 is necessary because the branch cuts are chosen so that the negative imaginary line "agrees with" neighboring values with negative real part, while the positive imaginary line agrees with values with positive real part. This makes atanneg 26950 true unconditionally but messes up conjugation symmetry, and it is impossible to have both in a single-valued function. The claim is true on the imaginary line between -1 and 1, though.) (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (𝐴 ∈ dom arctan ∧ (∗‘(arctan‘𝐴)) = (arctan‘(∗‘𝐴)))) | ||
| Theorem | atanrecl 26954 | The arctangent function is real for all real inputs. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ) | ||
| Theorem | efiatan 26955 | Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ dom arctan → (exp‘(i · (arctan‘𝐴))) = ((√‘(1 + (i · 𝐴))) / (√‘(1 − (i · 𝐴))))) | ||
| Theorem | atanlogaddlem 26956 | Lemma for atanlogadd 26957. (Contributed by Mario Carneiro, 3-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom arctan ∧ 0 ≤ (ℜ‘𝐴)) → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) ∈ ran log) | ||
| Theorem | atanlogadd 26957 | The rule √(𝑧𝑤) = (√𝑧)(√𝑤) is not always true on the complex numbers, but it is true when the arguments of 𝑧 and 𝑤 sum to within the interval (-π, π], so there are some cases such as this one with 𝑧 = 1 + i𝐴 and 𝑤 = 1 − i𝐴 which are true unconditionally. This result can also be stated as "√(1 + 𝑧) + √(1 − 𝑧) is analytic". (Contributed by Mario Carneiro, 3-Apr-2015.) |
| ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) ∈ ran log) | ||
| Theorem | atanlogsublem 26958 | Lemma for atanlogsub 26959. (Contributed by Mario Carneiro, 4-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom arctan ∧ 0 < (ℜ‘𝐴)) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π)) | ||
| Theorem | atanlogsub 26959 | A variation on atanlogadd 26957, to show that √(1 + i𝑧) / √(1 − i𝑧) = √((1 + i𝑧) / (1 − i𝑧)) under more limited conditions. (Contributed by Mario Carneiro, 4-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ran log) | ||
| Theorem | efiatan2 26960 | Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 3-Apr-2015.) |
| ⊢ (𝐴 ∈ dom arctan → (exp‘(i · (arctan‘𝐴))) = ((1 + (i · 𝐴)) / (√‘(1 + (𝐴↑2))))) | ||
| Theorem | 2efiatan 26961 | Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ dom arctan → (exp‘(2 · (i · (arctan‘𝐴)))) = (((2 · i) / (𝐴 + i)) − 1)) | ||
| Theorem | tanatan 26962 | The arctangent function is an inverse to tan. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (𝐴 ∈ dom arctan → (tan‘(arctan‘𝐴)) = 𝐴) | ||
| Theorem | atandmtan 26963 | The tangent function has range contained in the domain of the arctangent. (Contributed by Mario Carneiro, 31-Mar-2015.) |
| ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) ∈ dom arctan) | ||
| Theorem | cosatan 26964 | The cosine of an arctangent. (Contributed by Mario Carneiro, 3-Apr-2015.) |
| ⊢ (𝐴 ∈ dom arctan → (cos‘(arctan‘𝐴)) = (1 / (√‘(1 + (𝐴↑2))))) | ||
| Theorem | cosatanne0 26965 | The arctangent function has range contained in the domain of the tangent. (Contributed by Mario Carneiro, 3-Apr-2015.) |
| ⊢ (𝐴 ∈ dom arctan → (cos‘(arctan‘𝐴)) ≠ 0) | ||
| Theorem | atantan 26966 | The arctangent function is an inverse to tan. (Contributed by Mario Carneiro, 5-Apr-2015.) |
| ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arctan‘(tan‘𝐴)) = 𝐴) | ||
| Theorem | atantanb 26967 | Relationship between tangent and arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom arctan ∧ 𝐵 ∈ ℂ ∧ (ℜ‘𝐵) ∈ (-(π / 2)(,)(π / 2))) → ((arctan‘𝐴) = 𝐵 ↔ (tan‘𝐵) = 𝐴)) | ||
| Theorem | atanbndlem 26968 | Lemma for atanbnd 26969. (Contributed by Mario Carneiro, 5-Apr-2015.) |
| ⊢ (𝐴 ∈ ℝ+ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2))) | ||
| Theorem | atanbnd 26969 | The arctangent function is bounded by π / 2 on the reals. (Contributed by Mario Carneiro, 5-Apr-2015.) |
| ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2))) | ||
| Theorem | atanord 26970 | The arctangent function is strictly increasing. (Contributed by Mario Carneiro, 5-Apr-2015.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (arctan‘𝐴) < (arctan‘𝐵))) | ||
| Theorem | atan1 26971 | The arctangent of 1 is π / 4. (Contributed by Mario Carneiro, 2-Apr-2015.) |
| ⊢ (arctan‘1) = (π / 4) | ||
| Theorem | bndatandm 26972 | A point in the open unit disk is in the domain of the arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.) |
| ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ dom arctan) | ||
| Theorem | atans 26973* | The "domain of continuity" of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) & ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ⇒ ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷)) | ||
| Theorem | atans2 26974* | It suffices to show that 1 − i𝐴 and 1 + i𝐴 are in the continuity domain of log to show that 𝐴 is in the continuity domain of arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) & ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ⇒ ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ∈ 𝐷 ∧ (1 + (i · 𝐴)) ∈ 𝐷)) | ||
| Theorem | atansopn 26975* | The domain of continuity of the arctangent is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) & ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ⇒ ⊢ 𝑆 ∈ (TopOpen‘ℂfld) | ||
| Theorem | atansssdm 26976* | The domain of continuity of the arctangent is a subset of the actual domain of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) & ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ⇒ ⊢ 𝑆 ⊆ dom arctan | ||
| Theorem | ressatans 26977* | The real number line is a subset of the domain of continuity of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) & ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ⇒ ⊢ ℝ ⊆ 𝑆 | ||
| Theorem | dvatan 26978* | The derivative of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) & ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ⇒ ⊢ (ℂ D (arctan ↾ 𝑆)) = (𝑥 ∈ 𝑆 ↦ (1 / (1 + (𝑥↑2)))) | ||
| Theorem | atancn 26979* | The arctangent is a continuous function. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) & ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ⇒ ⊢ (arctan ↾ 𝑆) ∈ (𝑆–cn→ℂ) | ||
| Theorem | atantayl 26980* | The Taylor series for arctan(𝐴). (Contributed by Mario Carneiro, 1-Apr-2015.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ (arctan‘𝐴)) | ||
| Theorem | atantayl2 26981* | The Taylor series for arctan(𝐴). (Contributed by Mario Carneiro, 1-Apr-2015.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) · ((𝐴↑𝑛) / 𝑛)))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ (arctan‘𝐴)) | ||
| Theorem | atantayl3 26982* | The Taylor series for arctan(𝐴). (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) · ((𝐴↑((2 · 𝑛) + 1)) / ((2 · 𝑛) + 1)))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , 𝐹) ⇝ (arctan‘𝐴)) | ||
| Theorem | leibpilem1 26983 | Lemma for leibpi 26985. (Contributed by Mario Carneiro, 7-Apr-2015.) (Proof shortened by Steven Nguyen, 23-Mar-2023.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ (¬ 𝑁 = 0 ∧ ¬ 2 ∥ 𝑁)) → (𝑁 ∈ ℕ ∧ ((𝑁 − 1) / 2) ∈ ℕ0)) | ||
| Theorem | leibpilem2 26984* | The Leibniz formula for π. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))) & ⊢ 𝐺 = (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))) & ⊢ 𝐴 ∈ V ⇒ ⊢ (seq0( + , 𝐹) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴) | ||
| Theorem | leibpi 26985 | The Leibniz formula for π. This proof depends on three main facts: (1) the series 𝐹 is convergent, because it is an alternating series (iseralt 15721). (2) Using leibpilem2 26984 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 26981). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 26981, Abel's theorem (abelth2 26486) says that the limit along any sequence converging to 1, such as 1 − 1 / 𝑛, of the power series converges to the power series extended to 1, and then since arctan is continuous at 1 (atancn 26979) we get the desired result. This is Metamath 100 proof #26. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))) ⇒ ⊢ seq0( + , 𝐹) ⇝ (π / 4) | ||
| Theorem | leibpisum 26986 | The Leibniz formula for π. This version of leibpi 26985 looks nicer but does not assert that the series is convergent so is not as practically useful. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ Σ𝑛 ∈ ℕ0 ((-1↑𝑛) / ((2 · 𝑛) + 1)) = (π / 4) | ||
| Theorem | log2cnv 26987 | Using the Taylor series for arctan(i / 3), produce a rapidly convergent series for log2. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (2 / ((3 · ((2 · 𝑛) + 1)) · (9↑𝑛)))) ⇒ ⊢ seq0( + , 𝐹) ⇝ (log‘2) | ||
| Theorem | log2tlbnd 26988* | Bound the error term in the series of log2cnv 26987. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ (𝑁 ∈ ℕ0 → ((log‘2) − Σ𝑛 ∈ (0...(𝑁 − 1))(2 / ((3 · ((2 · 𝑛) + 1)) · (9↑𝑛)))) ∈ (0[,](3 / ((4 · ((2 · 𝑁) + 1)) · (9↑𝑁))))) | ||
| Theorem | log2ublem1 26989 | Lemma for log2ub 26992. The proof of log2ub 26992, which is simply the evaluation of log2tlbnd 26988 for 𝑁 = 4, takes the form of the addition of five fractions and showing this is less than another fraction. We could just perform exact arithmetic on these fractions, get a large rational number, and just multiply everything to verify the claim, but as anyone who uses decimal numbers for this task knows, it is often better to pick a common denominator 𝑑 (usually a large power of 10) and work with the closest approximations of the form 𝑛 / 𝑑 for some integer 𝑛 instead. It turns out that for our purposes it is sufficient to take 𝑑 = (3↑7) · 5 · 7, which is also nice because it shares many factors in common with the fractions in question. (Contributed by Mario Carneiro, 17-Apr-2015.) |
| ⊢ (((3↑7) · (5 · 7)) · 𝐴) ≤ 𝐵 & ⊢ 𝐴 ∈ ℝ & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐶 = (𝐴 + (𝐷 / 𝐸)) & ⊢ (𝐵 + 𝐹) = 𝐺 & ⊢ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹) ⇒ ⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺 | ||
| Theorem | log2ublem2 26990* | Lemma for log2ub 26992. (Contributed by Mario Carneiro, 17-Apr-2015.) |
| ⊢ (((3↑7) · (5 · 7)) · Σ𝑛 ∈ (0...𝐾)(2 / ((3 · ((2 · 𝑛) + 1)) · (9↑𝑛)))) ≤ (2 · 𝐵) & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ (𝑁 − 1) = 𝐾 & ⊢ (𝐵 + 𝐹) = 𝐺 & ⊢ 𝑀 ∈ ℕ0 & ⊢ (𝑀 + 𝑁) = 3 & ⊢ ((5 · 7) · (9↑𝑀)) = (((2 · 𝑁) + 1) · 𝐹) ⇒ ⊢ (((3↑7) · (5 · 7)) · Σ𝑛 ∈ (0...𝑁)(2 / ((3 · ((2 · 𝑛) + 1)) · (9↑𝑛)))) ≤ (2 · 𝐺) | ||
| Theorem | log2ublem3 26991 | Lemma for log2ub 26992. In decimal, this is a proof that the first four terms of the series for log2 is less than 53056 / 76545. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.) |
| ⊢ (((3↑7) · (5 · 7)) · Σ𝑛 ∈ (0...3)(2 / ((3 · ((2 · 𝑛) + 1)) · (9↑𝑛)))) ≤ ;;;;53056 | ||
| Theorem | log2ub 26992 | log2 is less than 253 / 365. If written in decimal, this is because log2 = 0.693147... is less than 253/365 = 0.693151... , so this is a very tight bound, at five decimal places. (Contributed by Mario Carneiro, 7-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
| ⊢ (log‘2) < (;;253 / ;;365) | ||
| Theorem | log2le1 26993 | log2 is less than 1. This is just a weaker form of log2ub 26992 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
| ⊢ (log‘2) < 1 | ||
| Theorem | birthdaylem1 26994* | Lemma for birthday 26997. (Contributed by Mario Carneiro, 17-Apr-2015.) |
| ⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} & ⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)} ⇒ ⊢ (𝑇 ⊆ 𝑆 ∧ 𝑆 ∈ Fin ∧ (𝑁 ∈ ℕ → 𝑆 ≠ ∅)) | ||
| Theorem | birthdaylem2 26995* | For general 𝑁 and 𝐾, count the fraction of injective functions from 1...𝐾 to 1...𝑁. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} & ⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)} ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑇) / (♯‘𝑆)) = (exp‘Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))) | ||
| Theorem | birthdaylem3 26996* | For general 𝑁 and 𝐾, upper-bound the fraction of injective functions from 1...𝐾 to 1...𝑁. (Contributed by Mario Carneiro, 17-Apr-2015.) |
| ⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} & ⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)} ⇒ ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → ((♯‘𝑇) / (♯‘𝑆)) ≤ (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁))) | ||
| Theorem | birthday 26997* | The Birthday Problem. There is a more than even chance that out of 23 people in a room, at least two of them have the same birthday. Mathematically, this is asserting that for 𝐾 = 23 and 𝑁 = 365, fewer than half of the set of all functions from 1...𝐾 to 1...𝑁 are injective. This is Metamath 100 proof #93. (Contributed by Mario Carneiro, 17-Apr-2015.) |
| ⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} & ⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)} & ⊢ 𝐾 = ;23 & ⊢ 𝑁 = ;;365 ⇒ ⊢ ((♯‘𝑇) / (♯‘𝑆)) < (1 / 2) | ||
| Syntax | carea 26998 | Area of regions in the complex plane. |
| class area | ||
| Definition | df-area 26999* | Define the area of a subset of ℝ × ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.) |
| ⊢ area = (𝑠 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) | ||
| Theorem | dmarea 27000* | The domain of the area function is the set of finitely measurable subsets of ℝ × ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.) |
| ⊢ (𝐴 ∈ dom area ↔ (𝐴 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1)) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |