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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremacosf 24401 Domain and range of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arccos:ℂ⟶ℂ

Theoremacoscl 24402 Closure for the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ ℂ → (arccos‘𝐴) ∈ ℂ)

Theorematandm 24403 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))

Theorematandm2 24404 This form of atandm 24403 is a bit more useful for showing that the logarithms in df-atan 24394 are well-defined. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))

Theorematandm3 24405 A compact form of atandm 24403. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1))

Theorematandm4 24406 A compact form of atandm 24403. (Contributed by Mario Carneiro, 3-Apr-2015.)
(𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ≠ 0))

Theorematanf 24407 Domain and range of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan:(ℂ ∖ {-i, i})⟶ℂ

Theorematancl 24408 Closure for the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ dom arctan → (arctan‘𝐴) ∈ ℂ)

Theoremasinval 24409 Value of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ ℂ → (arcsin‘𝐴) = (-i · (log‘((i · 𝐴) + (√‘(1 − (𝐴↑2)))))))

Theoremacosval 24410 Value of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴)))

Theorematanval 24411 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))))))

Theorematanre 24412 A real number is in the domain of the arctangent function. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ ℝ → 𝐴 ∈ dom arctan)

Theoremasinneg 24413 The arcsine function is odd. (Contributed by Mario Carneiro, 1-Apr-2015.)
(𝐴 ∈ ℂ → (arcsin‘-𝐴) = -(arcsin‘𝐴))

Theoremacosneg 24414 The negative symmetry relation of the arccosine. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝐴 ∈ ℂ → (arccos‘-𝐴) = (π − (arccos‘𝐴)))

Theoremefiasin 24415 The exponential of the arcsine function. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ ℂ → (exp‘(i · (arcsin‘𝐴))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2)))))

Theoremsinasin 24416 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 24419 is only true under limited circumstances. (Contributed by Mario Carneiro, 1-Apr-2015.)
(𝐴 ∈ ℂ → (sin‘(arcsin‘𝐴)) = 𝐴)

Theoremcosacos 24417 The arccosine function is an inverse to cos. (Contributed by Mario Carneiro, 1-Apr-2015.)
(𝐴 ∈ ℂ → (cos‘(arccos‘𝐴)) = 𝐴)

Theoremasinsinlem 24418 Lemma for asinsin 24419. (Contributed by Mario Carneiro, 2-Apr-2015.)
((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (ℜ‘(exp‘(i · 𝐴))))

Theoremasinsin 24419 The arcsine function composed with sin is equal to the identity. This plus sinasin 24416 allow us 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 24423). (Contributed by Mario Carneiro, 2-Apr-2015.)
((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴)

Theoremacoscos 24420 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 24421 The arcsine of 1 is π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.)
(arcsin‘1) = (π / 2)

Theoremacos1 24422 The arcsine of 1 is π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.)
(arccos‘1) = 0

Theoremreasinsin 24423 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 24424 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 24425 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 24426 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 24427 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 24428 Bounds on the arcsine function. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ (-(π / 2)[,](π / 2)))

Theoremasinrecl 24429 The arcsine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ ℝ)

Theoremacosrecl 24430 The arccosine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝐴 ∈ (-1[,]1) → (arccos‘𝐴) ∈ ℝ)

Theoremcosasin 24431 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 24432 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 24433 The domain of the arctangent function is closed under negatives. (Contributed by Mario Carneiro, 3-Apr-2015.)
(𝐴 ∈ dom arctan → -𝐴 ∈ dom arctan)

Theorematanneg 24434 The arctangent function is odd. (Contributed by Mario Carneiro, 3-Apr-2015.)
(𝐴 ∈ dom arctan → (arctan‘-𝐴) = -(arctan‘𝐴))

Theorematan0 24435 The arctangent of zero is zero. (Contributed by Mario Carneiro, 31-Mar-2015.)
(arctan‘0) = 0

Theorematandmcj 24436 The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ dom arctan → (∗‘𝐴) ∈ dom arctan)

Theorematancj 24437 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 24434 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 24438 The arctangent function is real for all real inputs. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ)

Theoremefiatan 24439 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)))))

Theorematanlogaddlem 24440 Lemma for atanlogadd 24441. (Contributed by Mario Carneiro, 3-Apr-2015.)
((𝐴 ∈ dom arctan ∧ 0 ≤ (ℜ‘𝐴)) → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) ∈ ran log)

Theorematanlogadd 24441 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 24442 Lemma for atanlogsub 24443. (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 24443 A variation on atanlogadd 24441, 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 24444 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 24445 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 24446 The arctangent function is an inverse to tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝐴 ∈ dom arctan → (tan‘(arctan‘𝐴)) = 𝐴)

Theorematandmtan 24447 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 24448 The cosine of an arctangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
(𝐴 ∈ dom arctan → (cos‘(arctan‘𝐴)) = (1 / (√‘(1 + (𝐴↑2)))))

Theoremcosatanne0 24449 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 24450 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 24451 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 24452 Lemma for atanbnd 24453. (Contributed by Mario Carneiro, 5-Apr-2015.)
(𝐴 ∈ ℝ+ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))

Theorematanbnd 24453 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 24454 The arctangent function is strictly increasing. (Contributed by Mario Carneiro, 5-Apr-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (arctan‘𝐴) < (arctan‘𝐵)))

Theorematan1 24455 The arctangent of 1 is π / 4. (Contributed by Mario Carneiro, 2-Apr-2015.)
(arctan‘1) = (π / 4)

Theorembndatandm 24456 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 24457* 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 24458* 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 24459* 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 24460* 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 24461* 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 24462* 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 24463* 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 24464* 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 24465* 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 24466* 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 24467 Lemma for leibpi 24469. (Contributed by Mario Carneiro, 7-Apr-2015.)
((𝑁 ∈ ℕ0 ∧ (¬ 𝑁 = 0 ∧ ¬ 2 ∥ 𝑁)) → (𝑁 ∈ ℕ ∧ ((𝑁 − 1) / 2) ∈ ℕ0))

Theoremleibpilem2 24468* 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 24469 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 14263). (2) Using leibpilem2 24468 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 24465). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 24465, Abel's theorem (abelth2 24000) 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 24463) 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 24470 The Leibniz formula for π. This version of leibpi 24469 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 24471 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 24472* Bound the error term in the series of log2cnv 24471. (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 24473 Lemma for log2ub 24476. The proof of log2ub 24476, which is simply the evaluation of log2tlbnd 24472 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 24474* Lemma for log2ub 24476. (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 24475 Lemma for log2ub 24476. 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 24476 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 24477 log2 is less than 1. This is just a weaker form of log2ub 24476 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.)
(log‘2) < 1

Theorembirthdaylem1 24478* Lemma for birthday 24481. (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 24479* 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 24480* 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 24481* 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 24482 Area of regions in the complex plane.
class area

Definitiondf-area 24483* 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 24484* 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 24485 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 24486 A measurable region is a subset of ℝ × ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.)
(𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ))

Theoremdfarea 24487* Rewrite df-area 24483 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)

Theoremareaf 24488 Area measurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.)
area:dom area⟶(0[,)+∞)

Theoremareacl 24489 The area of a measurable region is a real number. (Contributed by Mario Carneiro, 21-Jun-2015.)
(𝑆 ∈ dom area → (area‘𝑆) ∈ ℝ)

Theoremareage0 24490 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 24491* 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 24492* 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 24493* 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 24494* 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 24495* 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 24496* 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 24497). (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 24497* 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 24498* 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 24499 The inverse square root function converges to zero. (Contributed by Mario Carneiro, 18-May-2016.)
(𝑛 ∈ ℝ+ ↦ (1 / (√‘𝑛))) ⇝𝑟 0

Theoremrlimcxp 24500* 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)

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