Theorem List for Intuitionistic Logic Explorer - 11201-11300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | fsumcom 11201* |
Interchange order of summation. (Contributed by NM, 15-Nov-2005.)
(Revised by Mario Carneiro, 23-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑘 ∈ 𝐵 Σ𝑗 ∈ 𝐴 𝐶) |
|
Theorem | fsum0diaglem 11202* |
Lemma for fisum0diag 11203. (Contributed by Mario Carneiro,
28-Apr-2014.)
(Revised by Mario Carneiro, 8-Apr-2016.)
|
⊢ ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → (𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 𝑘)))) |
|
Theorem | fisum0diag 11203* |
Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular
region 𝑀 ≤ 𝑗, 𝑀 ≤ 𝑘, 𝑗 + 𝑘 ≤ 𝑁." (Contributed by NM,
31-Dec-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
(Revised by Mario Carneiro, 8-Apr-2016.)
|
⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗)))) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℤ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...(𝑁 − 𝑘))𝐴) |
|
Theorem | mptfzshft 11204* |
1-1 onto function in maps-to notation which shifts a finite set of
sequential integers. (Contributed by AV, 24-Aug-2019.)
|
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ)
⇒ ⊢ (𝜑 → (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑗 − 𝐾)):((𝑀 + 𝐾)...(𝑁 + 𝐾))–1-1-onto→(𝑀...𝑁)) |
|
Theorem | fsumrev 11205* |
Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised
by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑗 = (𝐾 − 𝑘) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝐾 − 𝑁)...(𝐾 − 𝑀))𝐵) |
|
Theorem | fsumshft 11206* |
Index shift of a finite sum. (Contributed by NM, 27-Nov-2005.)
(Revised by Mario Carneiro, 24-Apr-2014.) (Proof shortened by AV,
8-Sep-2019.)
|
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑗 = (𝑘 − 𝐾) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵) |
|
Theorem | fsumshftm 11207* |
Negative index shift of a finite sum. (Contributed by NM,
28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑗 = (𝑘 + 𝐾) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
|
Theorem | fisumrev2 11208* |
Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised
by Mario Carneiro, 13-Apr-2016.)
|
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑗 = ((𝑀 + 𝑁) − 𝑘) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵) |
|
Theorem | fisum0diag2 11209* |
Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular
region 0 ≤ 𝑗, 0 ≤ 𝑘, 𝑗 + 𝑘 ≤ 𝑁." (Contributed by
Mario Carneiro, 21-Jul-2014.)
|
⊢ (𝑥 = 𝑘 → 𝐵 = 𝐴)
& ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = 𝐶)
& ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗)))) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℤ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶) |
|
Theorem | fsummulc2 11210* |
A finite sum multiplied by a constant. (Contributed by NM,
12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)) |
|
Theorem | fsummulc1 11211* |
A finite sum multiplied by a constant. (Contributed by NM,
13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 · 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶)) |
|
Theorem | fsumdivapc 11212* |
A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006.)
(Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 / 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 / 𝐶)) |
|
Theorem | fsumneg 11213* |
Negation of a finite sum. (Contributed by Scott Fenton, 12-Jun-2013.)
(Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 -𝐵 = -Σ𝑘 ∈ 𝐴 𝐵) |
|
Theorem | fsumsub 11214* |
Split a finite sum over a subtraction. (Contributed by Scott Fenton,
12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 − 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 − Σ𝑘 ∈ 𝐴 𝐶)) |
|
Theorem | fsum2mul 11215* |
Separate the nested sum of the product 𝐶(𝑗) · 𝐷(𝑘).
(Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro,
24-Apr-2014.)
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⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 (𝐶 · 𝐷) = (Σ𝑗 ∈ 𝐴 𝐶 · Σ𝑘 ∈ 𝐵 𝐷)) |
|
Theorem | fsumconst 11216* |
The sum of constant terms (𝑘 is not free in 𝐵). (Contributed
by NM, 24-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵)) |
|
Theorem | fsumdifsnconst 11217* |
The sum of constant terms (𝑘 is not free in 𝐶) over an index
set excluding a singleton. (Contributed by AV, 7-Jan-2022.)
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⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → Σ𝑘 ∈ (𝐴 ∖ {𝐵})𝐶 = (((♯‘𝐴) − 1) · 𝐶)) |
|
Theorem | modfsummodlem1 11218* |
Lemma for modfsummod 11220. (Contributed by Alexander van der Vekens,
1-Sep-2018.)
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⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) |
|
Theorem | modfsummodlemstep 11219* |
Induction step for modfsummod 11220. (Contributed by Alexander van der
Vekens, 1-Sep-2018.) (Revised by Jim Kingdon, 12-Oct-2022.)
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⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) & ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
& ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 mod 𝑁) = (Σ𝑘 ∈ 𝐴 (𝐵 mod 𝑁) mod 𝑁)) ⇒ ⊢ (𝜑 → (Σ𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 mod 𝑁) = (Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) mod 𝑁)) |
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Theorem | modfsummod 11220* |
A finite sum modulo a positive integer equals the finite sum of their
summands modulo the positive integer, modulo the positive integer.
(Contributed by Alexander van der Vekens, 1-Sep-2018.)
|
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ)
⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 mod 𝑁) = (Σ𝑘 ∈ 𝐴 (𝐵 mod 𝑁) mod 𝑁)) |
|
Theorem | fsumge0 11221* |
If all of the terms of a finite sum are nonnegative, so is the sum.
(Contributed by NM, 26-Dec-2005.) (Revised by Mario Carneiro,
24-Apr-2014.)
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⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
|
Theorem | fsumlessfi 11222* |
A shorter sum of nonnegative terms is no greater than a longer one.
(Contributed by NM, 26-Dec-2005.) (Revised by Jim Kingdon,
12-Oct-2022.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)
& ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
& ⊢ (𝜑 → 𝐶 ∈ Fin) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
|
Theorem | fsumge1 11223* |
A sum of nonnegative numbers is greater than or equal to any one of
its terms. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof
shortened by Mario Carneiro, 4-Jun-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)
& ⊢ (𝑘 = 𝑀 → 𝐵 = 𝐶)
& ⊢ (𝜑 → 𝑀 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐶 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
|
Theorem | fsum00 11224* |
A sum of nonnegative numbers is zero iff all terms are zero.
(Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario
Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 = 0 ↔ ∀𝑘 ∈ 𝐴 𝐵 = 0)) |
|
Theorem | fsumle 11225* |
If all of the terms of finite sums compare, so do the sums.
(Contributed by NM, 11-Dec-2005.) (Proof shortened by Mario Carneiro,
24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐶) |
|
Theorem | fsumlt 11226* |
If every term in one finite sum is less than the corresponding term in
another, then the first sum is less than the second. (Contributed by
Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 < Σ𝑘 ∈ 𝐴 𝐶) |
|
Theorem | fsumabs 11227* |
Generalized triangle inequality: the absolute value of a finite sum is
less than or equal to the sum of absolute values. (Contributed by NM,
9-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵)) |
|
Theorem | telfsumo 11228* |
Sum of a telescoping series, using half-open intervals. (Contributed by
Mario Carneiro, 2-May-2016.)
|
⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵)
& ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)
& ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷)
& ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐸)
& ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 − 𝐶) = (𝐷 − 𝐸)) |
|
Theorem | telfsumo2 11229* |
Sum of a telescoping series. (Contributed by Mario Carneiro,
2-May-2016.)
|
⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵)
& ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)
& ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷)
& ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐸)
& ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵) = (𝐸 − 𝐷)) |
|
Theorem | telfsum 11230* |
Sum of a telescoping series. (Contributed by Scott Fenton,
24-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)
|
⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵)
& ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)
& ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷)
& ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸)
& ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)(𝐵 − 𝐶) = (𝐷 − 𝐸)) |
|
Theorem | telfsum2 11231* |
Sum of a telescoping series. (Contributed by Mario Carneiro,
15-Jun-2014.) (Revised by Mario Carneiro, 2-May-2016.)
|
⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵)
& ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)
& ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷)
& ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸)
& ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)(𝐶 − 𝐵) = (𝐸 − 𝐷)) |
|
Theorem | fsumparts 11232* |
Summation by parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
|
⊢ (𝑘 = 𝑗 → (𝐴 = 𝐵 ∧ 𝑉 = 𝑊)) & ⊢ (𝑘 = (𝑗 + 1) → (𝐴 = 𝐶 ∧ 𝑉 = 𝑋)) & ⊢ (𝑘 = 𝑀 → (𝐴 = 𝐷 ∧ 𝑉 = 𝑌)) & ⊢ (𝑘 = 𝑁 → (𝐴 = 𝐸 ∧ 𝑉 = 𝑍)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑉 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 · (𝑋 − 𝑊)) = (((𝐸 · 𝑍) − (𝐷 · 𝑌)) − Σ𝑗 ∈ (𝑀..^𝑁)((𝐶 − 𝐵) · 𝑋))) |
|
Theorem | fsumrelem 11233* |
Lemma for fsumre 11234, fsumim 11235, and fsumcj 11236. (Contributed by Mario
Carneiro, 25-Jul-2014.) (Revised by Mario Carneiro, 27-Dec-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ 𝐹:ℂ⟶ℂ & ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦))) ⇒ ⊢ (𝜑 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)) |
|
Theorem | fsumre 11234* |
The real part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.)
(Revised by Mario Carneiro, 25-Jul-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (ℜ‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (ℜ‘𝐵)) |
|
Theorem | fsumim 11235* |
The imaginary part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.)
(Revised by Mario Carneiro, 25-Jul-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (ℑ‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (ℑ‘𝐵)) |
|
Theorem | fsumcj 11236* |
The complex conjugate of a sum. (Contributed by Paul Chapman,
9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (∗‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (∗‘𝐵)) |
|
Theorem | iserabs 11237* |
Generalized triangle inequality: the absolute value of an infinite sum
is less than or equal to the sum of absolute values. (Contributed by
Paul Chapman, 10-Sep-2007.) (Revised by Jim Kingdon, 14-Dec-2022.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)
& ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)
& ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘))) ⇒ ⊢ (𝜑 → (abs‘𝐴) ≤ 𝐵) |
|
Theorem | cvgcmpub 11238* |
An upper bound for the limit of a real infinite series. This theorem
can also be used to compare two infinite series. (Contributed by Mario
Carneiro, 24-Mar-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)
& ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 𝐵 ≤ 𝐴) |
|
Theorem | fsumiun 11239* |
Sum over a disjoint indexed union. (Contributed by Mario Carneiro,
1-Jul-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵𝐶 = Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶) |
|
Theorem | hashiun 11240* |
The cardinality of a disjoint indexed union. (Contributed by Mario
Carneiro, 24-Jan-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) ⇒ ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ𝑥 ∈ 𝐴 (♯‘𝐵)) |
|
Theorem | hash2iun 11241* |
The cardinality of a nested disjoint indexed union. (Contributed by AV,
9-Jan-2022.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Fin) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Disj 𝑦 ∈ 𝐵 𝐶) ⇒ ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 (♯‘𝐶)) |
|
Theorem | hash2iun1dif1 11242* |
The cardinality of a nested disjoint indexed union. (Contributed by AV,
9-Jan-2022.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ 𝐵 = (𝐴 ∖ {𝑥})
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Fin) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Disj 𝑦 ∈ 𝐵 𝐶)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (♯‘𝐶) = 1) ⇒ ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = ((♯‘𝐴) · ((♯‘𝐴) − 1))) |
|
Theorem | hashrabrex 11243* |
The number of elements in a class abstraction with a restricted
existential quantification. (Contributed by Alexander van der Vekens,
29-Jul-2018.)
|
⊢ (𝜑 → 𝑌 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → {𝑥 ∈ 𝑋 ∣ 𝜓} ∈ Fin) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜓}) ⇒ ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 𝜓}) = Σ𝑦 ∈ 𝑌 (♯‘{𝑥 ∈ 𝑋 ∣ 𝜓})) |
|
Theorem | hashuni 11244* |
The cardinality of a disjoint union. (Contributed by Mario Carneiro,
24-Jan-2015.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ Fin) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥) ⇒ ⊢ (𝜑 → (♯‘∪ 𝐴)
= Σ𝑥 ∈ 𝐴 (♯‘𝑥)) |
|
4.8.3 The binomial theorem
|
|
Theorem | binomlem 11245* |
Lemma for binom 11246 (binomial theorem). Inductive step.
(Contributed by
NM, 6-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜓 → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑(𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
|
Theorem | binom 11246* |
The binomial theorem: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to
𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)). Theorem
15-2.8 of [Gleason] p. 296. This part
of the proof sets up the
induction and does the base case, with the bulk of the work (the
induction step) in binomlem 11245. This is Metamath 100 proof #44.
(Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro,
24-Apr-2014.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))) |
|
Theorem | binom1p 11247* |
Special case of the binomial theorem for (1 + 𝐴)↑𝑁.
(Contributed by Paul Chapman, 10-May-2007.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((1 +
𝐴)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (𝐴↑𝑘))) |
|
Theorem | binom11 11248* |
Special case of the binomial theorem for 2↑𝑁. (Contributed by
Mario Carneiro, 13-Mar-2014.)
|
⊢ (𝑁 ∈ ℕ0 →
(2↑𝑁) = Σ𝑘 ∈ (0...𝑁)(𝑁C𝑘)) |
|
Theorem | binom1dif 11249* |
A summation for the difference between ((𝐴 + 1)↑𝑁) and
(𝐴↑𝑁). (Contributed by Scott Fenton,
9-Apr-2014.) (Revised by
Mario Carneiro, 22-May-2014.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) →
(((𝐴 + 1)↑𝑁) − (𝐴↑𝑁)) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · (𝐴↑𝑘))) |
|
Theorem | bcxmaslem1 11250 |
Lemma for bcxmas 11251. (Contributed by Paul Chapman,
18-May-2007.)
|
⊢ (𝐴 = 𝐵 → ((𝑁 + 𝐴)C𝐴) = ((𝑁 + 𝐵)C𝐵)) |
|
Theorem | bcxmas 11251* |
Parallel summation (Christmas Stocking) theorem for Pascal's Triangle.
(Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro,
24-Apr-2014.)
|
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)
→ (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)) |
|
4.8.4 Infinite sums (cont.)
|
|
Theorem | isumshft 11252* |
Index shift of an infinite sum. (Contributed by Paul Chapman,
31-Oct-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 =
(ℤ≥‘(𝑀 + 𝐾)) & ⊢ (𝑗 = (𝐾 + 𝑘) → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑊) → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝑊 𝐴 = Σ𝑘 ∈ 𝑍 𝐵) |
|
Theorem | isumsplit 11253* |
Split off the first 𝑁 terms of an infinite sum.
(Contributed by
Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 21-Oct-2022.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 =
(ℤ≥‘𝑁)
& ⊢ (𝜑 → 𝑁 ∈ 𝑍)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ 𝑊 𝐴)) |
|
Theorem | isum1p 11254* |
The infinite sum of a converging infinite series equals the first term
plus the infinite sum of the rest of it. (Contributed by NM,
2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴)) |
|
Theorem | isumnn0nn 11255* |
Sum from 0 to infinity in terms of sum from 1 to infinity. (Contributed
by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝑘 = 0 → 𝐴 = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
|
Theorem | isumrpcl 11256* |
The infinite sum of positive reals is positive. (Contributed by Paul
Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 =
(ℤ≥‘𝑁)
& ⊢ (𝜑 → 𝑁 ∈ 𝑍)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈
ℝ+) |
|
Theorem | isumle 11257* |
Comparison of two infinite sums. (Contributed by Paul Chapman,
13-Nov-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ 𝐵)
& ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) & ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ≤ Σ𝑘 ∈ 𝑍 𝐵) |
|
Theorem | isumlessdc 11258* |
A finite sum of nonnegative numbers is less than or equal to its limit.
(Contributed by Mario Carneiro, 24-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ 𝑍)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
& ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 DECID 𝑘 ∈ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵)
& ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝑍 𝐵) |
|
4.8.5 Miscellaneous converging and diverging
sequences
|
|
Theorem | divcnv 11259* |
The sequence of reciprocals of positive integers, multiplied by the
factor 𝐴, converges to zero. (Contributed by
NM, 6-Feb-2008.)
(Revised by Jim Kingdon, 22-Oct-2022.)
|
⊢ (𝐴 ∈ ℂ → (𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)) ⇝ 0) |
|
4.8.6 Arithmetic series
|
|
Theorem | arisum 11260* |
Arithmetic series sum of the first 𝑁 positive integers. This is
Metamath 100 proof #68. (Contributed by FL, 16-Nov-2006.) (Proof
shortened by Mario Carneiro, 22-May-2014.)
|
⊢ (𝑁 ∈ ℕ0 →
Σ𝑘 ∈ (1...𝑁)𝑘 = (((𝑁↑2) + 𝑁) / 2)) |
|
Theorem | arisum2 11261* |
Arithmetic series sum of the first 𝑁 nonnegative integers.
(Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV,
2-Aug-2021.)
|
⊢ (𝑁 ∈ ℕ0 →
Σ𝑘 ∈
(0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2)) |
|
Theorem | trireciplem 11262 |
Lemma for trirecip 11263. Show that the sum converges. (Contributed
by
Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro,
22-May-2014.)
|
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) ⇒ ⊢ seq1( + , 𝐹) ⇝ 1 |
|
Theorem | trirecip 11263 |
The sum of the reciprocals of the triangle numbers converge to two.
This is Metamath 100 proof #42. (Contributed by Scott Fenton,
23-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
|
⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = 2 |
|
4.8.7 Geometric series
|
|
Theorem | expcnvap0 11264* |
A sequence of powers of a complex number 𝐴 with absolute value
smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.)
(Revised by Jim Kingdon, 23-Oct-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐴) < 1) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) ⇝ 0) |
|
Theorem | expcnvre 11265* |
A sequence of powers of a nonnegative real number less than one
converges to zero. (Contributed by Jim Kingdon, 28-Oct-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) ⇝ 0) |
|
Theorem | expcnv 11266* |
A sequence of powers of a complex number 𝐴 with absolute value
smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.)
(Revised by Jim Kingdon, 28-Oct-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐴) <
1) ⇒ ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) ⇝ 0) |
|
Theorem | explecnv 11267* |
A sequence of terms converges to zero when it is less than powers of a
number 𝐴 whose absolute value is smaller than
1. (Contributed by
NM, 19-Jul-2008.) (Revised by Mario Carneiro, 26-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐴) < 1) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ≤ (𝐴↑𝑘)) ⇒ ⊢ (𝜑 → 𝐹 ⇝ 0) |
|
Theorem | geosergap 11268* |
The value of the finite geometric series 𝐴↑𝑀 + 𝐴↑(𝑀 + 1) +...
+ 𝐴↑(𝑁 − 1). (Contributed by Mario
Carneiro, 2-May-2016.)
(Revised by Jim Kingdon, 24-Oct-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 1) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) = (((𝐴↑𝑀) − (𝐴↑𝑁)) / (1 − 𝐴))) |
|
Theorem | geoserap 11269* |
The value of the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +...
+ 𝐴↑(𝑁 − 1). This is Metamath 100
proof #66. (Contributed by
NM, 12-May-2006.) (Revised by Jim Kingdon, 24-Oct-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 1) & ⊢ (𝜑 → 𝑁 ∈
ℕ0) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴))) |
|
Theorem | pwm1geoserap1 11270* |
The n-th power of a number decreased by 1 expressed by the finite
geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1).
(Contributed by AV, 14-Aug-2021.) (Revised by Jim Kingdon,
24-Oct-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 # 1) ⇒ ⊢ (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))) |
|
Theorem | absltap 11271 |
Less-than of absolute value implies apartness. (Contributed by Jim
Kingdon, 29-Oct-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐴) < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 # 𝐵) |
|
Theorem | absgtap 11272 |
Greater-than of absolute value implies apartness. (Contributed by Jim
Kingdon, 29-Oct-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 < (abs‘𝐴)) ⇒ ⊢ (𝜑 → 𝐴 # 𝐵) |
|
Theorem | geolim 11273* |
The partial sums in the infinite series 1 + 𝐴↑1 + 𝐴↑2...
converge to (1 / (1 − 𝐴)). (Contributed by NM,
15-May-2006.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐴) < 1) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (𝐴↑𝑘)) ⇒ ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − 𝐴))) |
|
Theorem | geolim2 11274* |
The partial sums in the geometric series 𝐴↑𝑀 + 𝐴↑(𝑀 + 1)...
converge to ((𝐴↑𝑀) / (1 − 𝐴)). (Contributed by NM,
6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐴) < 1) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐴↑𝑘)) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ((𝐴↑𝑀) / (1 − 𝐴))) |
|
Theorem | georeclim 11275* |
The limit of a geometric series of reciprocals. (Contributed by Paul
Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 1 < (abs‘𝐴)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘)) ⇒ ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1))) |
|
Theorem | geo2sum 11276* |
The value of the finite geometric series 2↑-1 + 2↑-2
+...
+ 2↑-𝑁, multiplied by a constant.
(Contributed by Mario
Carneiro, 17-Mar-2014.) (Revised by Mario Carneiro, 26-Apr-2014.)
|
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℂ) → Σ𝑘 ∈ (1...𝑁)(𝐴 / (2↑𝑘)) = (𝐴 − (𝐴 / (2↑𝑁)))) |
|
Theorem | geo2sum2 11277* |
The value of the finite geometric series 1 + 2 + 4 + 8
+...
+ 2↑(𝑁 − 1). (Contributed by Mario
Carneiro, 7-Sep-2016.)
|
⊢ (𝑁 ∈ ℕ0 →
Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = ((2↑𝑁) − 1)) |
|
Theorem | geo2lim 11278* |
The value of the infinite geometric series
2↑-1 + 2↑-2 +... , multiplied by a
constant. (Contributed
by Mario Carneiro, 15-Jun-2014.)
|
⊢ 𝐹 = (𝑘 ∈ ℕ ↦ (𝐴 / (2↑𝑘))) ⇒ ⊢ (𝐴 ∈ ℂ → seq1( + , 𝐹) ⇝ 𝐴) |
|
Theorem | geoisum 11279* |
The infinite sum of 1 + 𝐴↑1 + 𝐴↑2... is (1 /
(1 − 𝐴)).
(Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro,
26-Apr-2014.)
|
⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → Σ𝑘 ∈ ℕ0
(𝐴↑𝑘) = (1 / (1 − 𝐴))) |
|
Theorem | geoisumr 11280* |
The infinite sum of reciprocals
1 + (1 / 𝐴)↑1 + (1 / 𝐴)↑2... is 𝐴 / (𝐴 − 1).
(Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro,
26-Apr-2014.)
|
⊢ ((𝐴 ∈ ℂ ∧ 1 <
(abs‘𝐴)) →
Σ𝑘 ∈
ℕ0 ((1 / 𝐴)↑𝑘) = (𝐴 / (𝐴 − 1))) |
|
Theorem | geoisum1 11281* |
The infinite sum of 𝐴↑1 + 𝐴↑2... is (𝐴 / (1 − 𝐴)).
(Contributed by NM, 1-Nov-2007.) (Revised by Mario Carneiro,
26-Apr-2014.)
|
⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → Σ𝑘 ∈ ℕ (𝐴↑𝑘) = (𝐴 / (1 − 𝐴))) |
|
Theorem | geoisum1c 11282* |
The infinite sum of 𝐴 · (𝑅↑1) + 𝐴 · (𝑅↑2)... is
(𝐴
· 𝑅) / (1 −
𝑅). (Contributed by
NM, 2-Nov-2007.) (Revised
by Mario Carneiro, 26-Apr-2014.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝐴 · (𝑅↑𝑘)) = ((𝐴 · 𝑅) / (1 − 𝑅))) |
|
Theorem | 0.999... 11283 |
The recurring decimal 0.999..., which is defined as the infinite sum 0.9 +
0.09 + 0.009 + ... i.e. 9 / 10↑1 + 9 / 10↑2 + 9
/ 10↑3
+ ..., is exactly equal to 1. (Contributed by NM,
2-Nov-2007.)
(Revised by AV, 8-Sep-2021.)
|
⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
|
Theorem | geoihalfsum 11284 |
Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... =
1. Uses geoisum1 11281. This is a representation of .111... in
binary with
an infinite number of 1's. Theorem 0.999... 11283 proves a similar claim for
.999... in base 10. (Contributed by David A. Wheeler, 4-Jan-2017.)
(Proof shortened by AV, 9-Jul-2022.)
|
⊢ Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 |
|
4.8.8 Ratio test for infinite series
convergence
|
|
Theorem | cvgratnnlembern 11285 |
Lemma for cvgratnn 11293. Upper bound for a geometric progression of
positive ratio less than one. (Contributed by Jim Kingdon,
24-Nov-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ (𝜑 → 𝑀 ∈ ℕ)
⇒ ⊢ (𝜑 → (𝐴↑𝑀) < ((1 / ((1 / 𝐴) − 1)) / 𝑀)) |
|
Theorem | cvgratnnlemnexp 11286* |
Lemma for cvgratnn 11293. (Contributed by Jim Kingdon, 15-Nov-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) & ⊢ (𝜑 → 𝑁 ∈ ℕ)
⇒ ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1)))) |
|
Theorem | cvgratnnlemmn 11287* |
Lemma for cvgratnn 11293. (Contributed by Jim Kingdon,
15-Nov-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
⇒ ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀)))) |
|
Theorem | cvgratnnlemseq 11288* |
Lemma for cvgratnn 11293. (Contributed by Jim Kingdon,
21-Nov-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
⇒ ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) = Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖)) |
|
Theorem | cvgratnnlemabsle 11289* |
Lemma for cvgratnn 11293. (Contributed by Jim Kingdon,
21-Nov-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
⇒ ⊢ (𝜑 → (abs‘Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖)) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)))) |
|
Theorem | cvgratnnlemsumlt 11290* |
Lemma for cvgratnn 11293. (Contributed by Jim Kingdon,
23-Nov-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
⇒ ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) < (𝐴 / (1 − 𝐴))) |
|
Theorem | cvgratnnlemfm 11291* |
Lemma for cvgratnn 11293. (Contributed by Jim Kingdon, 23-Nov-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) & ⊢ (𝜑 → 𝑀 ∈ ℕ)
⇒ ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) < ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀)) |
|
Theorem | cvgratnnlemrate 11292* |
Lemma for cvgratnn 11293. (Contributed by Jim Kingdon, 21-Nov-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
⇒ ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) < (((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) · (𝐴 / (1 − 𝐴))) / 𝑀)) |
|
Theorem | cvgratnn 11293* |
Ratio test for convergence of a complex infinite series. If the ratio
𝐴 of the absolute values of successive
terms in an infinite
sequence 𝐹 is less than 1 for all terms, then
the infinite sum of
the terms of 𝐹 converges to a complex number.
Although this
theorem is similar to cvgratz 11294 and cvgratgt0 11295, the decision to
index starting at one is not merely cosmetic, as proving convergence
using climcvg1n 11112 is sensitive to how a sequence is indexed.
(Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon,
12-Nov-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) ⇒ ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ ) |
|
Theorem | cvgratz 11294* |
Ratio test for convergence of a complex infinite series. If the ratio
𝐴 of the absolute values of successive
terms in an infinite sequence
𝐹 is less than 1 for all terms, then
the infinite sum of the terms
of 𝐹 converges to a complex number.
(Contributed by NM,
26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
|
Theorem | cvgratgt0 11295* |
Ratio test for convergence of a complex infinite series. If the ratio
𝐴 of the absolute values of successive
terms in an infinite sequence
𝐹 is less than 1 for all terms beyond
some index 𝐵, then the
infinite sum of the terms of 𝐹 converges to a complex number.
(Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon,
11-Nov-2022.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 =
(ℤ≥‘𝑁)
& ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ (𝜑 → 𝑁 ∈ 𝑍)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
|
4.8.9 Mertens' theorem
|
|
Theorem | mertenslemub 11296* |
Lemma for mertensabs 11299. An upper bound for 𝑇. (Contributed by
Jim Kingdon, 3-Dec-2022.)
|
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝
)
& ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑆 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))} & ⊢ (𝜑 → 𝑋 ∈ 𝑇)
& ⊢ (𝜑 → 𝑆 ∈ ℕ)
⇒ ⊢ (𝜑 → 𝑋 ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈
(ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))) |
|
Theorem | mertenslemi1 11297* |
Lemma for mertensabs 11299. (Contributed by Mario Carneiro,
29-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
|
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴)) & ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) & ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝
)
& ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))} & ⊢ (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈
(ℤ≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)))) & ⊢ (𝜑 → 𝑃 ∈ ℝ) & ⊢ (𝜑 → (𝜓 ∧ (𝑡 ∈ ℕ0 ∧
∀𝑚 ∈
(ℤ≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (𝑃 + 1))))) & ⊢ (𝜑 → 0 ≤ 𝑃)
& ⊢ (𝜑 → ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑃) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈
(ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈
(ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸) |
|
Theorem | mertenslem2 11298* |
Lemma for mertensabs 11299. (Contributed by Mario Carneiro,
28-Apr-2014.)
|
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴)) & ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) & ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝
)
& ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))} & ⊢ (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈
(ℤ≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈
(ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈
(ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸) |
|
Theorem | mertensabs 11299* |
Mertens' theorem. If 𝐴(𝑗) is an absolutely convergent series
and
𝐵(𝑘) is convergent, then
(Σ𝑗 ∈ ℕ0𝐴(𝑗) · Σ𝑘 ∈ ℕ0𝐵(𝑘)) =
Σ𝑘 ∈ ℕ0Σ𝑗 ∈ (0...𝑘)(𝐴(𝑗) · 𝐵(𝑘 − 𝑗)) (and
this latter series is convergent). This latter sum is commonly known as
the Cauchy product of the sequences. The proof follows the outline at
http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem.
(Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon,
8-Dec-2022.)
|
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴)) & ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) & ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝
)
& ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ ) & ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0
𝐵)) |
|
4.8.10 Finite and infinite
products
|
|
4.8.10.1 Product sequences
|
|
Theorem | prodf 11300* |
An infinite product of complex terms is a function from an upper set of
integers to ℂ. (Contributed by Scott
Fenton, 4-Dec-2017.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
⇒ ⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ) |