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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | atantan 25501 | 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 25502 | 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 25503 | Lemma for atanbnd 25504. (Contributed by Mario Carneiro, 5-Apr-2015.) |
⊢ (𝐴 ∈ ℝ+ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2))) | ||
Theorem | atanbnd 25504 | 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 25505 | 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 25506 | The arctangent of 1 is π / 4. (Contributed by Mario Carneiro, 2-Apr-2015.) |
⊢ (arctan‘1) = (π / 4) | ||
Theorem | bndatandm 25507 | 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 25508* | 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 25509* | 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 25510* | 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 25511* | 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 25512* | 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 25513* | 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 25514* | 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 25515* | 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 25516* | 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 25517* | 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 25518 | Lemma for leibpi 25520. (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 25519* | 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 25520 | 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 15041). (2) Using leibpilem2 25519 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 25516). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 25516, Abel's theorem (abelth2 25030) 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 25514) 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 25521 | The Leibniz formula for π. This version of leibpi 25520 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 25522 | 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 25523* | Bound the error term in the series of log2cnv 25522. (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 25524 | Lemma for log2ub 25527. The proof of log2ub 25527, which is simply the evaluation of log2tlbnd 25523 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 25525* | Lemma for log2ub 25527. (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 25526 | Lemma for log2ub 25527. 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 25527 | 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 25528 | log2 is less than 1. This is just a weaker form of log2ub 25527 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
⊢ (log‘2) < 1 | ||
Theorem | birthdaylem1 25529* | Lemma for birthday 25532. (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 25530* | 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 25531* | 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 25532* | 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 25533 | Area of regions in the complex plane. |
class area | ||
Definition | df-area 25534* | 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 25535* | 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 25536 | 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 25537 | A measurable region is a subset of ℝ × ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.) |
⊢ (𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ)) | ||
Theorem | dfarea 25538* | Rewrite df-area 25534 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.) |
⊢ area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) | ||
Theorem | areaf 25539 | Area measurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.) |
⊢ area:dom area⟶(0[,)+∞) | ||
Theorem | areacl 25540 | 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 25541 | 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 25542* | 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) | ||
Theorem | rlimcnp 25543* | 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 25544* | 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 25545* | 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 25546* | 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 25547* | 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 25548). (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)) | ||
Theorem | dfef2 25548* | 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 25549* | 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) | ||
Theorem | sqrtlim 25550 | The inverse square root function converges to zero. (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ (𝑛 ∈ ℝ+ ↦ (1 / (√‘𝑛))) ⇝𝑟 0 | ||
Theorem | rlimcxp 25551* | Any power to a positive exponent of a converging sequence also converges. (Contributed by Mario Carneiro, 18-Sep-2014.) |
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 0) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ (𝐵↑𝑐𝐶)) ⇝𝑟 0) | ||
Theorem | o1cxp 25552* | An eventually bounded function taken to a nonnegative power is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 0 ≤ (ℜ‘𝐶)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵↑𝑐𝐶)) ∈ 𝑂(1)) | ||
Theorem | cxp2limlem 25553* | 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 / (𝐴↑𝑐𝑛))) ⇝𝑟 0) | ||
Theorem | cxp2lim 25554* | 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+ ↦ ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))) ⇝𝑟 0) | ||
Theorem | cxploglim 25555* | The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.) |
⊢ (𝐴 ∈ ℝ+ → (𝑛 ∈ ℝ+ ↦ ((log‘𝑛) / (𝑛↑𝑐𝐴))) ⇝𝑟 0) | ||
Theorem | cxploglim2 25556* | 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) / (𝑛↑𝑐𝐵))) ⇝𝑟 0) | ||
Theorem | divsqrtsumlem 25557* | Lemma for divsqrsum 25559 and divsqrtsum2 25560. (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ 𝐹 = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑛)) − (2 · (√‘𝑥)))) ⇒ ⊢ (𝐹:ℝ+⟶ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ((𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ (1 / (√‘𝐴)))) | ||
Theorem | divsqrsumf 25558* | The function 𝐹 used in divsqrsum 25559 is a real function. (Contributed by Mario Carneiro, 12-May-2016.) |
⊢ 𝐹 = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑛)) − (2 · (√‘𝑥)))) ⇒ ⊢ 𝐹:ℝ+⟶ℝ | ||
Theorem | divsqrsum 25559* | The sum Σ𝑛 ≤ 𝑥(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 | ||
Theorem | divsqrtsum2 25560* | A bound on the distance of the sum Σ𝑛 ≤ 𝑥(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)))) & ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ (1 / (√‘𝐴))) | ||
Theorem | divsqrtsumo1 25561* | The sum Σ𝑛 ≤ 𝑥(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)))) & ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿) ⇒ ⊢ (𝜑 → (𝑦 ∈ ℝ+ ↦ (((𝐹‘𝑦) − 𝐿) · (√‘𝑦))) ∈ 𝑂(1)) | ||
Theorem | cvxcl 25562* | Closure of a 0-1 linear combination in a convex set. (Contributed by Mario Carneiro, 21-Jun-2015.) |
⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥[,]𝑦) ⊆ 𝐷) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷) | ||
Theorem | scvxcvx 25563* | A strictly convex function is convex. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) < ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝐹‘((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌))) ≤ ((𝑇 · (𝐹‘𝑋)) + ((1 − 𝑇) · (𝐹‘𝑌)))) | ||
Theorem | jensenlem1 25564* | Lemma for jensen 25566. (Contributed by Mario Carneiro, 4-Jun-2016.) |
⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑋:𝐴⟶𝐷) & ⊢ (𝜑 → 0 < (ℂfld Σg 𝑇)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))) & ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵) & ⊢ (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴) & ⊢ 𝑆 = (ℂfld Σg (𝑇 ↾ 𝐵)) & ⊢ 𝐿 = (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) ⇒ ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧))) | ||
Theorem | jensenlem2 25565* | Lemma for jensen 25566. (Contributed by Mario Carneiro, 21-Jun-2015.) |
⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑋:𝐴⟶𝐷) & ⊢ (𝜑 → 0 < (ℂfld Σg 𝑇)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))) & ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵) & ⊢ (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴) & ⊢ 𝑆 = (ℂfld Σg (𝑇 ↾ 𝐵)) & ⊢ 𝐿 = (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ∈ ℝ+) & ⊢ (𝜑 → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷) & ⊢ (𝜑 → (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) ⇒ ⊢ (𝜑 → (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿))) | ||
Theorem | jensen 25566* | 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.) |
⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑋:𝐴⟶𝐷) & ⊢ (𝜑 → 0 < (ℂfld Σg 𝑇)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))) ⇒ ⊢ (𝜑 → (((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇))) ≤ ((ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld Σg 𝑇)))) | ||
Theorem | amgmlem 25567 | Lemma for amgm 25568. (Contributed by Mario Carneiro, 21-Jun-2015.) |
⊢ 𝑀 = (mulGrp‘ℂfld) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ+) ⇒ ⊢ (𝜑 → ((𝑀 Σg 𝐹)↑𝑐(1 / (♯‘𝐴))) ≤ ((ℂfld Σg 𝐹) / (♯‘𝐴))) | ||
Theorem | amgm 25568 | 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), 𝑥 ∈ 𝐴 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))) | ||
Syntax | cem 25569 | The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.) |
class γ | ||
Definition | df-em 25570 | 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 25580. (Contributed by Mario Carneiro, 11-Jul-2014.) |
⊢ γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 / 𝑘)))) | ||
Theorem | logdifbnd 25571 | Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.) |
⊢ (𝐴 ∈ ℝ+ → ((log‘(𝐴 + 1)) − (log‘𝐴)) ≤ (1 / 𝐴)) | ||
Theorem | logdiflbnd 25572 | Lower bound on the difference of logs. (Contributed by Mario Carneiro, 3-Jul-2017.) |
⊢ (𝐴 ∈ ℝ+ → (1 / (𝐴 + 1)) ≤ ((log‘(𝐴 + 1)) − (log‘𝐴))) | ||
Theorem | emcllem1 25573* | Lemma for emcl 25580. The series 𝐹 and 𝐺 are sequences of real numbers that approach γ from above and below, respectively. (Contributed by Mario Carneiro, 11-Jul-2014.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛))) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1)))) ⇒ ⊢ (𝐹:ℕ⟶ℝ ∧ 𝐺:ℕ⟶ℝ) | ||
Theorem | emcllem2 25574* | Lemma for emcl 25580. 𝐹 is increasing, and 𝐺 is decreasing. (Contributed by Mario Carneiro, 11-Jul-2014.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛))) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1)))) ⇒ ⊢ (𝑁 ∈ ℕ → ((𝐹‘(𝑁 + 1)) ≤ (𝐹‘𝑁) ∧ (𝐺‘𝑁) ≤ (𝐺‘(𝑁 + 1)))) | ||
Theorem | emcllem3 25575* | Lemma for emcl 25580. The function 𝐻 is the difference between 𝐹 and 𝐺. (Contributed by Mario Carneiro, 11-Jul-2014.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛))) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1)))) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛)))) ⇒ ⊢ (𝑁 ∈ ℕ → (𝐻‘𝑁) = ((𝐹‘𝑁) − (𝐺‘𝑁))) | ||
Theorem | emcllem4 25576* | Lemma for emcl 25580. The difference between series 𝐹 and 𝐺 tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛))) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1)))) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛)))) ⇒ ⊢ 𝐻 ⇝ 0 | ||
Theorem | emcllem5 25577* | Lemma for emcl 25580. The partial sums of the series 𝑇, which is used in the definition df-em 25570, is in fact the same as 𝐺. (Contributed by Mario Carneiro, 11-Jul-2014.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛))) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1)))) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛)))) & ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ ((1 / 𝑛) − (log‘(1 + (1 / 𝑛))))) ⇒ ⊢ 𝐺 = seq1( + , 𝑇) | ||
Theorem | emcllem6 25578* | Lemma for emcl 25580. By the previous lemmas, 𝐹 and 𝐺 must approach a common limit, which is γ by definition. (Contributed by Mario Carneiro, 11-Jul-2014.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛))) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1)))) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛)))) & ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ ((1 / 𝑛) − (log‘(1 + (1 / 𝑛))))) ⇒ ⊢ (𝐹 ⇝ γ ∧ 𝐺 ⇝ γ) | ||
Theorem | emcllem7 25579* | Lemma for emcl 25580 and harmonicbnd 25581. Derive bounds on γ as 𝐹(1) and 𝐺(1). (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛))) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1)))) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛)))) & ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ ((1 / 𝑛) − (log‘(1 + (1 / 𝑛))))) ⇒ ⊢ (γ ∈ ((1 − (log‘2))[,]1) ∧ 𝐹:ℕ⟶(γ[,]1) ∧ 𝐺:ℕ⟶((1 − (log‘2))[,]γ)) | ||
Theorem | emcl 25580 | Closure and bounds for the Euler-Mascheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.) |
⊢ γ ∈ ((1 − (log‘2))[,]1) | ||
Theorem | harmonicbnd 25581* | A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 9-Apr-2016.) |
⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ (γ[,]1)) | ||
Theorem | harmonicbnd2 25582* | A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ((1 − (log‘2))[,]γ)) | ||
Theorem | emre 25583 | The Euler-Mascheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.) |
⊢ γ ∈ ℝ | ||
Theorem | emgt0 25584 | The Euler-Mascheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.) |
⊢ 0 < γ | ||
Theorem | harmonicbnd3 25585* | A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ (𝑁 ∈ ℕ0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ)) | ||
Theorem | harmoniclbnd 25586* | A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) | ||
Theorem | harmonicubnd 25587* | A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ≤ ((log‘𝐴) + 1)) | ||
Theorem | harmonicbnd4 25588* | The asymptotic behavior of Σ𝑚 ≤ 𝐴, 1 / 𝑚 = log𝐴 + γ + 𝑂(1 / 𝐴). (Contributed by Mario Carneiro, 14-May-2016.) |
⊢ (𝐴 ∈ ℝ+ → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴)) | ||
Theorem | fsumharmonic 25589* | Bound a finite sum based on the harmonic series, where the "strong" bound 𝐶 only applies asymptotically, and there is a "weak" bound 𝑅 for the remaining values. (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇)) & ⊢ (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅)) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐶) & ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛)) & ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝐴 / 𝑛) < 𝑇) → (abs‘𝐵) ≤ 𝑅) ⇒ ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1)))) | ||
Syntax | czeta 25590 | The Riemann zeta function. |
class ζ | ||
Definition | df-zeta 25591* | Define the Riemann zeta function. This definition uses a series expansion of the alternating zeta function ~? zetaalt that is convergent everywhere except 1, but going from the alternating zeta function to the regular zeta function requires dividing by 1 − 2↑(1 − 𝑠), which has zeroes other than 1. To extract the correct value of the zeta function at these points, we extend the divided alternating zeta function by continuity. (Contributed by Mario Carneiro, 18-Jul-2014.) |
⊢ ζ = (℩𝑓 ∈ ((ℂ ∖ {1})–cn→ℂ)∀𝑠 ∈ (ℂ ∖ {1})((1 − (2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠)) = Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))) | ||
Theorem | zetacvg 25592* | The zeta series is convergent. (Contributed by Mario Carneiro, 18-Jul-2014.) |
⊢ (𝜑 → 𝑆 ∈ ℂ) & ⊢ (𝜑 → 1 < (ℜ‘𝑆)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝑘↑𝑐-𝑆)) ⇒ ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ ) | ||
Syntax | clgam 25593 | Logarithm of the Gamma function. |
class log Γ | ||
Syntax | cgam 25594 | The Gamma function. |
class Γ | ||
Syntax | cigam 25595 | The inverse Gamma function. |
class 1/Γ | ||
Definition | df-lgam 25596* | Define the log-Gamma function. We can work with this form of the gamma function a bit easier than the equivalent expression for the gamma function itself, and moreover this function is not actually equal to log(Γ(𝑥)) because the branch cuts are placed differently (we do have exp(log Γ(𝑥)) = Γ(𝑥), though). This definition is attributed to Euler, and unlike the usual integral definition is defined on the entire complex plane except the nonpositive integers ℤ ∖ ℕ, where the function has simple poles. (Contributed by Mario Carneiro, 12-Jul-2014.) |
⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))) | ||
Definition | df-gam 25597 | Define the Gamma function. See df-lgam 25596 for more information about the reason for this definition in terms of the log-gamma function. (Contributed by Mario Carneiro, 12-Jul-2014.) |
⊢ Γ = (exp ∘ log Γ) | ||
Definition | df-igam 25598 | Define the inverse Gamma function, which is defined everywhere, unlike the Gamma function itself. (Contributed by Mario Carneiro, 16-Jul-2017.) |
⊢ 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥)))) | ||
Theorem | eldmgm 25599 | Elementhood in the set of non-nonpositive integers. (Contributed by Mario Carneiro, 12-Jul-2014.) |
⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) | ||
Theorem | dmgmaddn0 25600 | If 𝐴 is not a nonpositive integer, then 𝐴 + 𝑁 is nonzero for any nonnegative integer 𝑁. (Contributed by Mario Carneiro, 12-Jul-2014.) |
⊢ ((𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ∧ 𝑁 ∈ ℕ0) → (𝐴 + 𝑁) ≠ 0) |
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