P. 270 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26901-27000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	asincl 26901	Closure for the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) ∈ ℂ)
Theorem	acosf 26902	Domain and codoamin of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ arccos:ℂ⟶ℂ
Theorem	acoscl 26903	Closure for the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ ℂ → (arccos‘𝐴) ∈ ℂ)
Theorem	atandm 26904	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 26905	This form of atandm 26904 is a bit more useful for showing that the logarithms in df-atan 26895 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 26906	A compact form of atandm 26904. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1))
Theorem	atandm4 26907	A compact form of atandm 26904. (Contributed by Mario Carneiro, 3-Apr-2015.)
⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ≠ 0))
Theorem	atanf 26908	Domain and codoamin of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ arctan:(ℂ ∖ {-i, i})⟶ℂ
Theorem	atancl 26909	Closure for the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) ∈ ℂ)
Theorem	asinval 26910	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 26911	Value of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴)))
Theorem	atanval 26912	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 26913	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 26914	The arcsine function is odd. (Contributed by Mario Carneiro, 1-Apr-2015.)
⊢ (𝐴 ∈ ℂ → (arcsin‘-𝐴) = -(arcsin‘𝐴))
Theorem	acosneg 26915	The negative symmetry relation of the arccosine. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝐴 ∈ ℂ → (arccos‘-𝐴) = (π − (arccos‘𝐴)))
Theorem	efiasin 26916	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 26917	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 26920 is only true under limited circumstances. (Contributed by Mario Carneiro, 1-Apr-2015.)
⊢ (𝐴 ∈ ℂ → (sin‘(arcsin‘𝐴)) = 𝐴)
Theorem	cosacos 26918	The arccosine function is an inverse to cos. (Contributed by Mario Carneiro, 1-Apr-2015.)
⊢ (𝐴 ∈ ℂ → (cos‘(arccos‘𝐴)) = 𝐴)
Theorem	asinsinlem 26919	Lemma for asinsin 26920. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (ℜ‘(exp‘(i · 𝐴))))
Theorem	asinsin 26920	The arcsine function composed with sin is equal to the identity. This plus sinasin 26917 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 26924). (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴)
Theorem	acoscos 26921	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 26922	The arcsine of 1 is π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (arcsin‘1) = (π / 2)
Theorem	acos1 26923	The arccosine of 1 is 0. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (arccos‘1) = 0
Theorem	reasinsin 26924	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 26925	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 26926	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 26927	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 26928	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 26929	Bounds on the arcsine function. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ (-(π / 2)[,](π / 2)))
Theorem	asinrecl 26930	The arcsine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ ℝ)
Theorem	acosrecl 26931	The arccosine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝐴 ∈ (-1[,]1) → (arccos‘𝐴) ∈ ℝ)
Theorem	cosasin 26932	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 26933	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 26934	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 26935	The arctangent function is odd. (Contributed by Mario Carneiro, 3-Apr-2015.)
⊢ (𝐴 ∈ dom arctan → (arctan‘-𝐴) = -(arctan‘𝐴))
Theorem	atan0 26936	The arctangent of zero is zero. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (arctan‘0) = 0
Theorem	atandmcj 26937	The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ dom arctan)
Theorem	atancj 26938	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 26935 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 26939	The arctangent function is real for all real inputs. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ)
Theorem	efiatan 26940	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 26941	Lemma for atanlogadd 26942. (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 26942	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 26943	Lemma for atanlogsub 26944. (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 26944	A variation on atanlogadd 26942, 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 26945	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 26946	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 26947	The arctangent function is an inverse to tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝐴 ∈ dom arctan → (tan‘(arctan‘𝐴)) = 𝐴)
Theorem	atandmtan 26948	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 26949	The cosine of an arctangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
⊢ (𝐴 ∈ dom arctan → (cos‘(arctan‘𝐴)) = (1 / (√‘(1 + (𝐴↑2)))))
Theorem	cosatanne0 26950	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 26951	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 26952	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 26953	Lemma for atanbnd 26954. (Contributed by Mario Carneiro, 5-Apr-2015.)
⊢ (𝐴 ∈ ℝ+ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
Theorem	atanbnd 26954	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 26955	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 26956	The arctangent of 1 is π / 4. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (arctan‘1) = (π / 4)
Theorem	bndatandm 26957	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 26958*	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 26959*	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 26960*	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 26961*	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 26962*	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 26963*	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 26964*	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 26965*	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 26966*	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 26967*	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 26968	Lemma for leibpi 26970. (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 26969*	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 26970	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 15689). (2) Using leibpilem2 26969 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 26966). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 26966, Abel's theorem (abelth2 26472) 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 26964) 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 26971	The Leibniz formula for π. This version of leibpi 26970 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 26972	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 26973*	Bound the error term in the series of log2cnv 26972. (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 26974	Lemma for log2ub 26977. The proof of log2ub 26977, which is simply the evaluation of log2tlbnd 26973 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 26975*	Lemma for log2ub 26977. (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 26976	Lemma for log2ub 26977. 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 26977	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 26978	log2 is less than 1. This is just a weaker form of log2ub 26977 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.)
⊢ (log‘2) < 1
Theorem	birthdaylem1 26979*	Lemma for birthday 26982. (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 26980*	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 26981*	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 26982*	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 26983	Area of regions in the complex plane.
class area
Definition	df-area 26984*	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 26985*	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 26986	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 26987	A measurable region is a subset of ℝ × ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ (𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ))
Theorem	dfarea 26988*	Rewrite df-area 26984 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
Theorem	areaf 26989	Area measurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ area:dom area⟶(0[,)+∞)
Theorem	areacl 26990	The area of a measurable region is a real number. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ (𝑆 ∈ dom area → (area‘𝑆) ∈ ℝ)
Theorem	areage0 26991	The area of a measurable region is greater than or equal to zero. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ (𝑆 ∈ dom area → 0 ≤ (area‘𝑆))
Theorem	areaval 26992*	The area of a measurable region is greater than or equal to zero. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ (𝑆 ∈ dom area → (area‘𝑆) = ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥)
14.3.11  More miscellaneous converging sequences
Theorem	rlimcnp 26993*	Relate a limit of a real-valued sequence at infinity to the continuity of the function 𝑆(𝑦) = 𝑅(1 / 𝑦) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
⊢ (𝜑 → 𝐴 ⊆ (0[,)+∞))    &   ⊢ (𝜑 → 0 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥 ∈ 𝐴 ↔ (1 / 𝑥) ∈ 𝐵))    &   ⊢ (𝑥 = 0 → 𝑅 = 𝐶)    &   ⊢ (𝑥 = (1 / 𝑦) → 𝑅 = 𝑆)    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝐴)    ⇒   ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝𝑟 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘0)))
Theorem	rlimcnp2 26994*	Relate a limit of a real-valued sequence at infinity to the continuity of the function 𝑆(𝑦) = 𝑅(1 / 𝑦) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
⊢ (𝜑 → 𝐴 ⊆ (0[,)+∞))    &   ⊢ (𝜑 → 0 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))    &   ⊢ (𝑦 = (1 / 𝑥) → 𝑆 = 𝑅)    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝐴)    ⇒   ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝𝑟 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))
Theorem	rlimcnp3 26995*	Relate a limit of a real-valued sequence at infinity to the continuity of the function 𝑆(𝑦) = 𝑅(1 / 𝑦) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑆 ∈ ℂ)    &   ⊢ (𝑦 = (1 / 𝑥) → 𝑆 = 𝑅)    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t (0[,)+∞))    ⇒   ⊢ (𝜑 → ((𝑦 ∈ ℝ+ ↦ 𝑆) ⇝𝑟 𝐶 ↔ (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))
Theorem	xrlimcnp 26996*	Relate a limit of a real-valued sequence at infinity to the continuity of the corresponding extended real function at +∞. Since any ⇝𝑟 limit can be written in the form on the left side of the implication, this shows that real limits are a special case of topological continuity at a point. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ (𝜑 → 𝐴 = (𝐵 ∪ {+∞}))    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ ℂ)    &   ⊢ (𝑥 = +∞ → 𝑅 = 𝐶)    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = ((ordTop‘ ≤ ) ↾t 𝐴)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝑅) ⇝𝑟 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)))
Theorem	efrlim 26997*	The limit of the sequence (1 + 𝐴 / 𝑘)↑𝑘 is the exponential function. This is often taken as an alternate definition of the exponential function (see also dfef2 26999). (Contributed by Mario Carneiro, 1-Mar-2015.) Avoid ax-mulf 11238. (Revised by GG, 19-Apr-2025.)
⊢ 𝑆 = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))    ⇒   ⊢ (𝐴 ∈ ℂ → (𝑘 ∈ ℝ+ ↦ ((1 + (𝐴 / 𝑘))↑𝑐𝑘)) ⇝𝑟 (exp‘𝐴))
Theorem	efrlimOLD 26998*	Obsolete version of efrlim 26997 as of 19-Apr-2025. (Contributed by Mario Carneiro, 1-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑆 = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))    ⇒   ⊢ (𝐴 ∈ ℂ → (𝑘 ∈ ℝ+ ↦ ((1 + (𝐴 / 𝑘))↑𝑐𝑘)) ⇝𝑟 (exp‘𝐴))
Theorem	dfef2 26999*	The limit of the sequence (1 + 𝐴 / 𝑘)↑𝑘 as 𝑘 goes to +∞ is (exp‘𝐴). This is another common definition of e. (Contributed by Mario Carneiro, 1-Mar-2015.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = ((1 + (𝐴 / 𝑘))↑𝑘))    ⇒   ⊢ (𝜑 → 𝐹 ⇝ (exp‘𝐴))
Theorem	cxplim 27000*	A power to a negative exponent goes to zero as the base becomes large. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Mario Carneiro, 18-May-2016.)
⊢ (𝐴 ∈ ℝ+ → (𝑛 ∈ ℝ+ ↦ (1 / (𝑛↑𝑐𝐴))) ⇝𝑟 0)