 Home Metamath Proof ExplorerTheorem List (p. 247 of 424) < 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: Metamath Proof Explorer (1-27745) Hilbert Space Explorer (27746-29270) Users' Mathboxes (29271-42316)

Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremacoscos 24601 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)

Theoremasin1 24602 The arcsine of 1 is π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.)
(arcsin‘1) = (π / 2)

Theoremacos1 24603 The arcsine of 1 is π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.)
(arccos‘1) = 0

Theoremreasinsin 24604 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)

Theoremasinsinb 24605 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))

Theoremacoscosb 24606 Relationship between sine and arcsine. (Contributed by Mario Carneiro, 2-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (ℜ‘𝐵) ∈ (0(,)π)) → ((arccos‘𝐴) = 𝐵 ↔ (cos‘𝐵) = 𝐴))

Theoremasinbnd 24607 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)))

Theoremacosbnd 24608 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))

Theoremasinrebnd 24609 Bounds on the arcsine function. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ (-(π / 2)[,](π / 2)))

Theoremasinrecl 24610 The arcsine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ ℝ)

Theoremacosrecl 24611 The arccosine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝐴 ∈ (-1[,]1) → (arccos‘𝐴) ∈ ℝ)

Theoremcosasin 24612 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))))

Theoremsinacos 24613 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))))

Theorematandmneg 24614 The domain of the arctangent function is closed under negatives. (Contributed by Mario Carneiro, 3-Apr-2015.)
(𝐴 ∈ dom arctan → -𝐴 ∈ dom arctan)

Theorematanneg 24615 The arctangent function is odd. (Contributed by Mario Carneiro, 3-Apr-2015.)
(𝐴 ∈ dom arctan → (arctan‘-𝐴) = -(arctan‘𝐴))

Theorematan0 24616 The arctangent of zero is zero. (Contributed by Mario Carneiro, 31-Mar-2015.)
(arctan‘0) = 0

Theorematandmcj 24617 The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ dom arctan → (∗‘𝐴) ∈ dom arctan)

Theorematancj 24618 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 24615 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))))

Theorematanrecl 24619 The arctangent function is real for all real inputs. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ)

Theoremefiatan 24620 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)))))

((𝐴 ∈ dom arctan ∧ 0 ≤ (ℜ‘𝐴)) → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) ∈ ran log)

Theorematanlogadd 24622 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)

Theorematanlogsublem 24623 Lemma for atanlogsub 24624. (Contributed by Mario Carneiro, 4-Apr-2015.)
((𝐴 ∈ dom arctan ∧ 0 < (ℜ‘𝐴)) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))

Theorematanlogsub 24624 A variation on atanlogadd 24622, 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)

Theoremefiatan2 24625 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)))))

Theorem2efiatan 24626 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))

Theoremtanatan 24627 The arctangent function is an inverse to tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝐴 ∈ dom arctan → (tan‘(arctan‘𝐴)) = 𝐴)

Theorematandmtan 24628 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)

Theoremcosatan 24629 The cosine of an arctangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
(𝐴 ∈ dom arctan → (cos‘(arctan‘𝐴)) = (1 / (√‘(1 + (𝐴↑2)))))

Theoremcosatanne0 24630 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)

Theorematantan 24631 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)

Theorematantanb 24632 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))

Theorematanbndlem 24633 Lemma for atanbnd 24634. (Contributed by Mario Carneiro, 5-Apr-2015.)
(𝐴 ∈ ℝ+ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))

Theorematanbnd 24634 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)))

Theorematanord 24635 The arctangent function is strictly increasing. (Contributed by Mario Carneiro, 5-Apr-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (arctan‘𝐴) < (arctan‘𝐵)))

Theorematan1 24636 The arctangent of 1 is π / 4. (Contributed by Mario Carneiro, 2-Apr-2015.)
(arctan‘1) = (π / 4)

Theorembndatandm 24637 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)

Theorematans 24638* The "domain of continuity" of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))    &   𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷}       (𝐴𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷))

Theorematans2 24639* 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}       (𝐴𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ∈ 𝐷 ∧ (1 + (i · 𝐴)) ∈ 𝐷))

Theorematansopn 24640* 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)

Theorematansssdm 24641* 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

Theoremressatans 24642* 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

Theoremdvatan 24643* 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)) = (𝑥𝑆 ↦ (1 / (1 + (𝑥↑2))))

Theorematancn 24644* 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) ∈ (𝑆cn→ℂ)

Theorematantayl 24645* The Taylor series for arctan(𝐴). (Contributed by Mario Carneiro, 1-Apr-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴𝑛) / 𝑛)))       ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ (arctan‘𝐴))

Theorematantayl2 24646* The Taylor series for arctan(𝐴). (Contributed by Mario Carneiro, 1-Apr-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) · ((𝐴𝑛) / 𝑛))))       ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ (arctan‘𝐴))

Theorematantayl3 24647* 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))

Theoremleibpilem1 24648 Lemma for leibpi 24650. (Contributed by Mario Carneiro, 7-Apr-2015.)
((𝑁 ∈ ℕ0 ∧ (¬ 𝑁 = 0 ∧ ¬ 2 ∥ 𝑁)) → (𝑁 ∈ ℕ ∧ ((𝑁 − 1) / 2) ∈ ℕ0))

Theoremleibpilem2 24649* 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)

Theoremleibpi 24650 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 14396). (2) Using leibpilem2 24649 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 24646). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 24646, Abel's theorem (abelth2 24177) 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 24644) 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)

Theoremleibpisum 24651 The Leibniz formula for π. This version of leibpi 24650 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)

Theoremlog2cnv 24652 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)

Theoremlog2tlbnd 24653* Bound the error term in the series of log2cnv 24652. (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

Theoremlog2ublem1 24654 Lemma for log2ub 24657. The proof of log2ub 24657, which is simply the evaluation of log2tlbnd 24653 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

Theoremlog2ublem2 24655* Lemma for log2ub 24657. (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)

Theoremlog2ublem3 24656 Lemma for log2ub 24657. 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

Theoremlog2ub 24657 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)

Theoremlog2le1 24658 log2 is less than 1. This is just a weaker form of log2ub 24657 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.)
(log‘2) < 1

Theorembirthdaylem1 24659* Lemma for birthday 24662. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝑆 = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)}    &   𝑇 = {𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)}       (𝑇𝑆𝑆 ∈ Fin ∧ (𝑁 ∈ ℕ → 𝑆 ≠ ∅))

Theorembirthdaylem2 24660* 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 = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)}    &   𝑇 = {𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)}       ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((#‘𝑇) / (#‘𝑆)) = (exp‘Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁)))))

Theorembirthdaylem3 24661* 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 = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)}    &   𝑇 = {𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)}       ((𝐾 ∈ ℕ0𝑁 ∈ ℕ) → ((#‘𝑇) / (#‘𝑆)) ≤ (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁)))

Theorembirthday 24662* 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 = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)}    &   𝑇 = {𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)}    &   𝐾 = 23    &   𝑁 = 365       ((#‘𝑇) / (#‘𝑆)) < (1 / 2)

14.3.10  Areas in R^2

Syntaxcarea 24663 Area of regions in the complex plane.
class area

Definitiondf-area 24664* 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)

Theoremdmarea 24665* 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))

Theoremareambl 24666 The fibers of a measurable region are finitely measurable subsets of . (Contributed by Mario Carneiro, 21-Jun-2015.)
((𝑆 ∈ dom area ∧ 𝐴 ∈ ℝ) → ((𝑆 “ {𝐴}) ∈ dom vol ∧ (vol‘(𝑆 “ {𝐴})) ∈ ℝ))

Theoremareass 24667 A measurable region is a subset of ℝ × ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.)
(𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ))

Theoremdfarea 24668* Rewrite df-area 24664 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)

Theoremareaf 24669 Area measurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.)
area:dom area⟶(0[,)+∞)

Theoremareacl 24670 The area of a measurable region is a real number. (Contributed by Mario Carneiro, 21-Jun-2015.)
(𝑆 ∈ dom area → (area‘𝑆) ∈ ℝ)

Theoremareage0 24671 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))

Theoremareaval 24672* 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

Theoremrlimcnp 24673* 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.)
(𝜑𝐴 ⊆ (0[,)+∞))    &   (𝜑 → 0 ∈ 𝐴)    &   (𝜑𝐵 ⊆ ℝ+)    &   ((𝜑𝑥𝐴) → 𝑅 ∈ ℂ)    &   ((𝜑𝑥 ∈ ℝ+) → (𝑥𝐴 ↔ (1 / 𝑥) ∈ 𝐵))    &   (𝑥 = 0 → 𝑅 = 𝐶)    &   (𝑥 = (1 / 𝑦) → 𝑅 = 𝑆)    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝐴)       (𝜑 → ((𝑦𝐵𝑆) ⇝𝑟 𝐶 ↔ (𝑥𝐴𝑅) ∈ ((𝐾 CnP 𝐽)‘0)))

Theoremrlimcnp2 24674* 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.)
(𝜑𝐴 ⊆ (0[,)+∞))    &   (𝜑 → 0 ∈ 𝐴)    &   (𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑦𝐵) → 𝑆 ∈ ℂ)    &   ((𝜑𝑦 ∈ ℝ+) → (𝑦𝐵 ↔ (1 / 𝑦) ∈ 𝐴))    &   (𝑦 = (1 / 𝑥) → 𝑆 = 𝑅)    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝐴)       (𝜑 → ((𝑦𝐵𝑆) ⇝𝑟 𝐶 ↔ (𝑥𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))

Theoremrlimcnp3 24675* 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.)
(𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑦 ∈ ℝ+) → 𝑆 ∈ ℂ)    &   (𝑦 = (1 / 𝑥) → 𝑆 = 𝑅)    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t (0[,)+∞))       (𝜑 → ((𝑦 ∈ ℝ+𝑆) ⇝𝑟 𝐶 ↔ (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))

Theoremxrlimcnp 24676* 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.)
(𝜑𝐴 = (𝐵 ∪ {+∞}))    &   (𝜑𝐵 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝑅 ∈ ℂ)    &   (𝑥 = +∞ → 𝑅 = 𝐶)    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = ((ordTop‘ ≤ ) ↾t 𝐴)       (𝜑 → ((𝑥𝐵𝑅) ⇝𝑟 𝐶 ↔ (𝑥𝐴𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)))

Theoremefrlim 24677* 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 24678). (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑆 = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))       (𝐴 ∈ ℂ → (𝑘 ∈ ℝ+ ↦ ((1 + (𝐴 / 𝑘))↑𝑐𝑘)) ⇝𝑟 (exp‘𝐴))

Theoremdfef2 24678* 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.)
(𝜑𝐹𝑉)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = ((1 + (𝐴 / 𝑘))↑𝑘))       (𝜑𝐹 ⇝ (exp‘𝐴))

Theoremcxplim 24679* 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 / (𝑛𝑐𝐴))) ⇝𝑟 0)

Theoremsqrtlim 24680 The inverse square root function converges to zero. (Contributed by Mario Carneiro, 18-May-2016.)
(𝑛 ∈ ℝ+ ↦ (1 / (√‘𝑛))) ⇝𝑟 0

Theoremrlimcxp 24681* Any power to a positive exponent of a converging sequence also converges. (Contributed by Mario Carneiro, 18-Sep-2014.)
((𝜑𝑛𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑛𝐴𝐵) ⇝𝑟 0)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (𝑛𝐴 ↦ (𝐵𝑐𝐶)) ⇝𝑟 0)

Theoremo1cxp 24682* An eventually bounded function taken to a nonnegative power is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝜑𝐶 ∈ ℂ)    &   (𝜑 → 0 ≤ (ℜ‘𝐶))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))       (𝜑 → (𝑥𝐴 ↦ (𝐵𝑐𝐶)) ∈ 𝑂(1))

Theoremcxp2limlem 24683* A linear factor grows slower than any exponential with base greater than 1. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (𝑛 ∈ ℝ+ ↦ (𝑛 / (𝐴𝑐𝑛))) ⇝𝑟 0)

Theoremcxp2lim 24684* Any power grows slower than any exponential with base greater than 1. (Contributed by Mario Carneiro, 18-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵) → (𝑛 ∈ ℝ+ ↦ ((𝑛𝑐𝐴) / (𝐵𝑐𝑛))) ⇝𝑟 0)

Theoremcxploglim 24685* The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℝ+ → (𝑛 ∈ ℝ+ ↦ ((log‘𝑛) / (𝑛𝑐𝐴))) ⇝𝑟 0)

Theoremcxploglim2 24686* Every power of the logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 20-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+) → (𝑛 ∈ ℝ+ ↦ (((log‘𝑛)↑𝑐𝐴) / (𝑛𝑐𝐵))) ⇝𝑟 0)

Theoremdivsqrtsumlem 24687* Lemma for divsqrsum 24689 and divsqrtsum2 24690. (Contributed by Mario Carneiro, 18-May-2016.)
𝐹 = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑛)) − (2 · (√‘𝑥))))       (𝐹:ℝ+⟶ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ((𝐹𝑟 𝐿𝐴 ∈ ℝ+) → (abs‘((𝐹𝐴) − 𝐿)) ≤ (1 / (√‘𝐴))))

Theoremdivsqrsumf 24688* The function 𝐹 used in divsqrsum 24689 is a real function. (Contributed by Mario Carneiro, 12-May-2016.)
𝐹 = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑛)) − (2 · (√‘𝑥))))       𝐹:ℝ+⟶ℝ

Theoremdivsqrsum 24689* The sum Σ𝑛𝑥(1 / √𝑛) is asymptotic to 2√𝑥 + 𝐿 with a finite limit 𝐿. (In fact, this limit is ζ(1 / 2) ≈ -1.46....) (Contributed by Mario Carneiro, 9-May-2016.)
𝐹 = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑛)) − (2 · (√‘𝑥))))       𝐹 ∈ dom ⇝𝑟

Theoremdivsqrtsum2 24690* A bound on the distance of the sum Σ𝑛𝑥(1 / √𝑛) from its asymptotic value 2√𝑥 + 𝐿. (Contributed by Mario Carneiro, 18-May-2016.)
𝐹 = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑛)) − (2 · (√‘𝑥))))    &   (𝜑𝐹𝑟 𝐿)       ((𝜑𝐴 ∈ ℝ+) → (abs‘((𝐹𝐴) − 𝐿)) ≤ (1 / (√‘𝐴)))

Theoremdivsqrtsumo1 24691* The sum Σ𝑛𝑥(1 / √𝑛) has the asymptotic expansion 2√𝑥 + 𝐿 + 𝑂(1 / √𝑥), for some 𝐿. (Contributed by Mario Carneiro, 10-May-2016.)
𝐹 = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑛)) − (2 · (√‘𝑥))))    &   (𝜑𝐹𝑟 𝐿)       (𝜑 → (𝑦 ∈ ℝ+ ↦ (((𝐹𝑦) − 𝐿) · (√‘𝑦))) ∈ 𝑂(1))

14.3.12  Inequality of arithmetic and geometric means

Theoremcvxcl 24692* Closure of a 0-1 linear combination in a convex set. (Contributed by Mario Carneiro, 21-Jun-2015.)
(𝜑𝐷 ⊆ ℝ)    &   ((𝜑 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥[,]𝑦) ⊆ 𝐷)       ((𝜑 ∧ (𝑋𝐷𝑌𝐷𝑇 ∈ (0[,]1))) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)

Theoremscvxcvx 24693* A strictly convex function is convex. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑 ∧ (𝑎𝐷𝑏𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)    &   ((𝜑 ∧ (𝑥𝐷𝑦𝐷𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) < ((𝑡 · (𝐹𝑥)) + ((1 − 𝑡) · (𝐹𝑦))))       ((𝜑 ∧ (𝑋𝐷𝑌𝐷𝑇 ∈ (0[,]1))) → (𝐹‘((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌))) ≤ ((𝑇 · (𝐹𝑋)) + ((1 − 𝑇) · (𝐹𝑌))))

Theoremjensenlem1 24694* Lemma for jensen 24696. (Contributed by Mario Carneiro, 4-Jun-2016.)
(𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑 ∧ (𝑎𝐷𝑏𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑇:𝐴⟶(0[,)+∞))    &   (𝜑𝑋:𝐴𝐷)    &   (𝜑 → 0 < (ℂfld Σg 𝑇))    &   ((𝜑 ∧ (𝑥𝐷𝑦𝐷𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹𝑥)) + ((1 − 𝑡) · (𝐹𝑦))))    &   (𝜑 → ¬ 𝑧𝐵)    &   (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴)    &   𝑆 = (ℂfld Σg (𝑇𝐵))    &   𝐿 = (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧})))       (𝜑𝐿 = (𝑆 + (𝑇𝑧)))

Theoremjensenlem2 24695* Lemma for jensen 24696. (Contributed by Mario Carneiro, 21-Jun-2015.)
(𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑 ∧ (𝑎𝐷𝑏𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑇:𝐴⟶(0[,)+∞))    &   (𝜑𝑋:𝐴𝐷)    &   (𝜑 → 0 < (ℂfld Σg 𝑇))    &   ((𝜑 ∧ (𝑥𝐷𝑦𝐷𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹𝑥)) + ((1 − 𝑡) · (𝐹𝑦))))    &   (𝜑 → ¬ 𝑧𝐵)    &   (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴)    &   𝑆 = (ℂfld Σg (𝑇𝐵))    &   𝐿 = (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧})))    &   (𝜑𝑆 ∈ ℝ+)    &   (𝜑 → ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷)    &   (𝜑 → (𝐹‘((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝐵)) / 𝑆)) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐵)) / 𝑆))       (𝜑 → (((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)))

Theoremjensen 24696* Jensen's inequality, a finite extension of the definition of convexity (the last hypothesis). (Contributed by Mario Carneiro, 21-Jun-2015.) (Proof shortened by AV, 27-Jul-2019.)
(𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑 ∧ (𝑎𝐷𝑏𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑇:𝐴⟶(0[,)+∞))    &   (𝜑𝑋:𝐴𝐷)    &   (𝜑 → 0 < (ℂfld Σg 𝑇))    &   ((𝜑 ∧ (𝑥𝐷𝑦𝐷𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹𝑥)) + ((1 − 𝑡) · (𝐹𝑦))))       (𝜑 → (((ℂfld Σg (𝑇𝑓 · 𝑋)) / (ℂfld Σg 𝑇)) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg (𝑇𝑓 · 𝑋)) / (ℂfld Σg 𝑇))) ≤ ((ℂfld Σg (𝑇𝑓 · (𝐹𝑋))) / (ℂfld Σg 𝑇))))

Theoremamgmlem 24697 Lemma for amgm 24698. (Contributed by Mario Carneiro, 21-Jun-2015.)
𝑀 = (mulGrp‘ℂfld)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐹:𝐴⟶ℝ+)       (𝜑 → ((𝑀 Σg 𝐹)↑𝑐(1 / (#‘𝐴))) ≤ ((ℂfld Σg 𝐹) / (#‘𝐴)))

Theoremamgm 24698 Inequality of arithmetic and geometric means. Here (𝑀 Σg 𝐹) calculates the group sum within the multiplicative monoid of the complex numbers (or in other words, it multiplies the elements 𝐹(𝑥), 𝑥𝐴 together), and (ℂfld Σg 𝐹) calculates the group sum in the additive group (i.e. the sum of the elements). This is Metamath 100 proof #38. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑀 = (mulGrp‘ℂfld)       ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(0[,)+∞)) → ((𝑀 Σg 𝐹)↑𝑐(1 / (#‘𝐴))) ≤ ((ℂfld Σg 𝐹) / (#‘𝐴)))

14.3.13  Euler-Mascheroni constant

Syntaxcem 24699 The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)
class γ

Definitiondf-em 24700 Define the Euler-Mascheroni constant, γ = 0.577... . This is the limit of the series Σ𝑘 ∈ (1...𝑚)(1 / 𝑘) − (log‘𝑚), with a proof that the limit exists in emcl 24710. (Contributed by Mario Carneiro, 11-Jul-2014.)
γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
 Copyright terms: Public domain < Previous  Next >