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Theorem List for Metamath Proof Explorer - 24701-24800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembndatandm 24701 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 24702* 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 24703* 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 24704* 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 24705* 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 24706* 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 24707* 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 24708* 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 24709* 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 24710* 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 24711* 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 24712 Lemma for leibpi 24714. (Contributed by Mario Carneiro, 7-Apr-2015.)
((𝑁 ∈ ℕ0 ∧ (¬ 𝑁 = 0 ∧ ¬ 2 ∥ 𝑁)) → (𝑁 ∈ ℕ ∧ ((𝑁 − 1) / 2) ∈ ℕ0))

Theoremleibpilem2 24713* 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 24714 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 14459). (2) Using leibpilem2 24713 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 24710). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 24710, Abel's theorem (abelth2 24241) 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 24708) 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 24715 The Leibniz formula for π. This version of leibpi 24714 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 24716 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 24717* Bound the error term in the series of log2cnv 24716. (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 24718 Lemma for log2ub 24721. The proof of log2ub 24721, which is simply the evaluation of log2tlbnd 24717 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 24719* Lemma for log2ub 24721. (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 24720 Lemma for log2ub 24721. 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 24721 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 24722 log2 is less than 1. This is just a weaker form of log2ub 24721 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.)
(log‘2) < 1

Theorembirthdaylem1 24723* Lemma for birthday 24726. (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 24724* 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 24725* 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 24726* 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 24727 Area of regions in the complex plane.
class area

Definitiondf-area 24728* 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 24729* 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 24730 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 24731 A measurable region is a subset of ℝ × ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.)
(𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ))

Theoremdfarea 24732* Rewrite df-area 24728 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)

Theoremareaf 24733 Area measurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.)
area:dom area⟶(0[,)+∞)

Theoremareacl 24734 The area of a measurable region is a real number. (Contributed by Mario Carneiro, 21-Jun-2015.)
(𝑆 ∈ dom area → (area‘𝑆) ∈ ℝ)

Theoremareage0 24735 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 24736* 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 24737* 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 24738* 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 24739* 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 24740* 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 24741* 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 24742). (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 24742* 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 24743* 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 24744 The inverse square root function converges to zero. (Contributed by Mario Carneiro, 18-May-2016.)
(𝑛 ∈ ℝ+ ↦ (1 / (√‘𝑛))) ⇝𝑟 0

Theoremrlimcxp 24745* 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 24746* 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 24747* 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 24748* 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 24749* 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 24750* 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 24751* Lemma for divsqrsum 24753 and divsqrtsum2 24754. (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 24752* The function 𝐹 used in divsqrsum 24753 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 24753* 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 24754* 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 24755* 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 24756* 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 24757* 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 24758* Lemma for jensen 24760. (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 24759* Lemma for jensen 24760. (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 24760* 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 24761 Lemma for amgm 24762. (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 24762 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 24763 The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)
class γ

Definitiondf-em 24764 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 24774. (Contributed by Mario Carneiro, 11-Jul-2014.)
γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))

Theoremlogdifbnd 24765 Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)
(𝐴 ∈ ℝ+ → ((log‘(𝐴 + 1)) − (log‘𝐴)) ≤ (1 / 𝐴))

Theoremlogdiflbnd 24766 Lower bound on the difference of logs. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝐴 ∈ ℝ+ → (1 / (𝐴 + 1)) ≤ ((log‘(𝐴 + 1)) − (log‘𝐴)))

Theorememcllem1 24767* Lemma for emcl 24774. 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)

Theorememcllem2 24768* Lemma for emcl 24774. 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)) ≤ (𝐹𝑁) ∧ (𝐺𝑁) ≤ (𝐺‘(𝑁 + 1))))

Theorememcllem3 24769* Lemma for emcl 24774. 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 → (𝐻𝑁) = ((𝐹𝑁) − (𝐺𝑁)))

Theorememcllem4 24770* Lemma for emcl 24774. 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

Theorememcllem5 24771* Lemma for emcl 24774. The partial sums of the series 𝑇, which is used in the definition df-em 24764, 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)

Theorememcllem6 24772* Lemma for emcl 24774. 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)

Theorememcllem7 24773* Lemma for emcl 24774 and harmonicbnd 24775. 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))

Theorememcl 24774 Closure and bounds for the Euler-Mascheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)
γ ∈ ((1 − (log‘2))[,]1)

Theoremharmonicbnd 24775* 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))

Theoremharmonicbnd2 24776* 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))

Theorememre 24777 The Euler-Mascheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)
γ ∈ ℝ

Theorememgt0 24778 The Euler-Mascheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)
0 < γ

Theoremharmonicbnd3 24779* 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))

Theoremharmoniclbnd 24780* 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))

Theoremharmonicubnd 24781* 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))

Theoremharmonicbnd4 24782* The asymptotic behavior of Σ𝑚𝐴, 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))

Theoremfsumharmonic 24783* 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.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))    &   (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))    &   ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)    &   ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐶)    &   (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛))    &   (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝐴 / 𝑛) < 𝑇) → (abs‘𝐵) ≤ 𝑅)       (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))

14.3.14  Zeta function

Syntaxczeta 24784 The Riemann zeta function.
class ζ

Definitiondf-zeta 24785* 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))) · (𝑓𝑠)) = Σ𝑛 ∈ ℕ0𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1))))

Theoremzetacvg 24786* The zeta series is convergent. (Contributed by Mario Carneiro, 18-Jul-2014.)
(𝜑𝑆 ∈ ℂ)    &   (𝜑 → 1 < (ℜ‘𝑆))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = (𝑘𝑐-𝑆))       (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )

14.3.15  Gamma function

Syntaxclgam 24787 Logarithm of the Gamma function.
class log Γ

Syntaxcgam 24788 The Gamma function.
class Γ

Syntaxcigam 24789 The inverse Gamma function.
class 1/Γ

Definitiondf-lgam 24790* 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)))

Definitiondf-gam 24791 Define the Gamma function. See df-lgam 24790 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)

Definitiondf-igam 24792 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))))

Theoremeldmgm 24793 Elementhood in the set of non-nonpositive integers. (Contributed by Mario Carneiro, 12-Jul-2014.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0))

Theoremdmgmaddn0 24794 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)

Theoremdmlogdmgm 24795 If 𝐴 is in the continuous domain of the logarithm, then it is in the domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (-∞(,]0)) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))

Theoremrpdmgm 24796 A positive real number is in the domain of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ ℝ+𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))

Theoremdmgmn0 24797 If 𝐴 is not a nonpositive integer, then 𝐴 is nonzero. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑𝐴 ≠ 0)

Theoremdmgmaddnn0 24798 If 𝐴 is not a nonpositive integer and 𝑁 is a nonnegative integer, then 𝐴 + 𝑁 is also not a nonpositive integer. (Contributed by Mario Carneiro, 6-Jul-2017.)
(𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴 + 𝑁) ∈ (ℂ ∖ (ℤ ∖ ℕ)))

Theoremdmgmdivn0 24799 Lemma for lgamf 24813. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   (𝜑𝑀 ∈ ℕ)       (𝜑 → ((𝐴 / 𝑀) + 1) ≠ 0)

Theoremlgamgulmlem1 24800* Lemma for lgamgulm 24806. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}       (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))

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