P. 267 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26601-26700   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	cacos 26601	The arccosine function.
class arccos
Syntax	catan 26602	The arctangent function.
class arctan
Definition	df-asin 26603	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 26604	Define the arccosine function. See also remarks for df-asin 26603. 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 26605	Define the arctangent function. See also remarks for df-asin 26603. 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 26606	The argument to the logarithm in df-asin 26603 is always nonzero. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (√‘(1 − (𝐴↑2)))) ≠ 0)
Theorem	asinlem2 26607	The argument to the logarithm in df-asin 26603 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 26608	Lemma for asinlem3 26609. (Contributed by Mario Carneiro, 1-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) ≤ 0) → 0 ≤ (ℜ‘((i · 𝐴) + (√‘(1 − (𝐴↑2))))))
Theorem	asinlem3 26609	The argument to the logarithm in df-asin 26603 has nonnegative real part. (Contributed by Mario Carneiro, 1-Apr-2015.)
⊢ (𝐴 ∈ ℂ → 0 ≤ (ℜ‘((i · 𝐴) + (√‘(1 − (𝐴↑2))))))
Theorem	asinf 26610	Domain and codomain of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ arcsin:ℂ⟶ℂ
Theorem	asincl 26611	Closure for the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) ∈ ℂ)
Theorem	acosf 26612	Domain and codoamin of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ arccos:ℂ⟶ℂ
Theorem	acoscl 26613	Closure for the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ ℂ → (arccos‘𝐴) ∈ ℂ)
Theorem	atandm 26614	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 26615	This form of atandm 26614 is a bit more useful for showing that the logarithms in df-atan 26605 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 26616	A compact form of atandm 26614. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1))
Theorem	atandm4 26617	A compact form of atandm 26614. (Contributed by Mario Carneiro, 3-Apr-2015.)
⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ≠ 0))
Theorem	atanf 26618	Domain and codoamin of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ arctan:(ℂ ∖ {-i, i})⟶ℂ
Theorem	atancl 26619	Closure for the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) ∈ ℂ)
Theorem	asinval 26620	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 26621	Value of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴)))
Theorem	atanval 26622	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 26623	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 26624	The arcsine function is odd. (Contributed by Mario Carneiro, 1-Apr-2015.)
⊢ (𝐴 ∈ ℂ → (arcsin‘-𝐴) = -(arcsin‘𝐴))
Theorem	acosneg 26625	The negative symmetry relation of the arccosine. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝐴 ∈ ℂ → (arccos‘-𝐴) = (π − (arccos‘𝐴)))
Theorem	efiasin 26626	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 26627	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 26630 is only true under limited circumstances. (Contributed by Mario Carneiro, 1-Apr-2015.)
⊢ (𝐴 ∈ ℂ → (sin‘(arcsin‘𝐴)) = 𝐴)
Theorem	cosacos 26628	The arccosine function is an inverse to cos. (Contributed by Mario Carneiro, 1-Apr-2015.)
⊢ (𝐴 ∈ ℂ → (cos‘(arccos‘𝐴)) = 𝐴)
Theorem	asinsinlem 26629	Lemma for asinsin 26630. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (ℜ‘(exp‘(i · 𝐴))))
Theorem	asinsin 26630	The arcsine function composed with sin is equal to the identity. This plus sinasin 26627 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 26634). (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴)
Theorem	acoscos 26631	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 26632	The arcsine of 1 is π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (arcsin‘1) = (π / 2)
Theorem	acos1 26633	The arccosine of 1 is 0. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (arccos‘1) = 0
Theorem	reasinsin 26634	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 26635	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 26636	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 26637	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 26638	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 26639	Bounds on the arcsine function. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ (-(π / 2)[,](π / 2)))
Theorem	asinrecl 26640	The arcsine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ ℝ)
Theorem	acosrecl 26641	The arccosine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝐴 ∈ (-1[,]1) → (arccos‘𝐴) ∈ ℝ)
Theorem	cosasin 26642	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 26643	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 26644	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 26645	The arctangent function is odd. (Contributed by Mario Carneiro, 3-Apr-2015.)
⊢ (𝐴 ∈ dom arctan → (arctan‘-𝐴) = -(arctan‘𝐴))
Theorem	atan0 26646	The arctangent of zero is zero. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (arctan‘0) = 0
Theorem	atandmcj 26647	The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ dom arctan)
Theorem	atancj 26648	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 26645 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 26649	The arctangent function is real for all real inputs. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ)
Theorem	efiatan 26650	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 26651	Lemma for atanlogadd 26652. (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 26652	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 26653	Lemma for atanlogsub 26654. (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 26654	A variation on atanlogadd 26652, 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 26655	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 26656	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 26657	The arctangent function is an inverse to tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝐴 ∈ dom arctan → (tan‘(arctan‘𝐴)) = 𝐴)
Theorem	atandmtan 26658	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 26659	The cosine of an arctangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
⊢ (𝐴 ∈ dom arctan → (cos‘(arctan‘𝐴)) = (1 / (√‘(1 + (𝐴↑2)))))
Theorem	cosatanne0 26660	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 26661	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 26662	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 26663	Lemma for atanbnd 26664. (Contributed by Mario Carneiro, 5-Apr-2015.)
⊢ (𝐴 ∈ ℝ+ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
Theorem	atanbnd 26664	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 26665	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 26666	The arctangent of 1 is π / 4. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (arctan‘1) = (π / 4)
Theorem	bndatandm 26667	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 26668*	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 26669*	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 26670*	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 26671*	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 26672*	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 26673*	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 26674*	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 26675*	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 26676*	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 26677*	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 26678	Lemma for leibpi 26680. (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 26679*	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 26680	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 15636). (2) Using leibpilem2 26679 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 26676). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 26676, Abel's theorem (abelth2 26187) 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 26674) 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 26681	The Leibniz formula for π. This version of leibpi 26680 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 26682	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 26683*	Bound the error term in the series of log2cnv 26682. (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)))))
14.3.9  The Birthday Problem
Theorem	log2ublem1 26684	Lemma for log2ub 26687. The proof of log2ub 26687, which is simply the evaluation of log2tlbnd 26683 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 26685*	Lemma for log2ub 26687. (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 26686	Lemma for log2ub 26687. 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 26687	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 26688	log2 is less than 1. This is just a weaker form of log2ub 26687 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.)
⊢ (log‘2) < 1
Theorem	birthdaylem1 26689*	Lemma for birthday 26692. (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 26690*	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 26691*	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 26692*	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)
14.3.10  Areas in R^2
Syntax	carea 26693	Area of regions in the complex plane.
class area
Definition	df-area 26694*	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 26695*	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))
Theorem	areambl 26696	The fibers of a measurable region are finitely measurable subsets of ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ ((𝑆 ∈ dom area ∧ 𝐴 ∈ ℝ) → ((𝑆 “ {𝐴}) ∈ dom vol ∧ (vol‘(𝑆 “ {𝐴})) ∈ ℝ))
Theorem	areass 26697	A measurable region is a subset of ℝ × ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ (𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ))
Theorem	dfarea 26698*	Rewrite df-area 26694 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
Theorem	areaf 26699	Area measurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ area:dom area⟶(0[,)+∞)
Theorem	areacl 26700	The area of a measurable region is a real number. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ (𝑆 ∈ dom area → (area‘𝑆) ∈ ℝ)