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Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfsump1 14201* The addition of the next term in a finite sum of 𝐴(𝑘) is the current term plus 𝐵 i.e. 𝐴(𝑁 + 1). (Contributed by NM, 4-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀...𝑁)𝐴 + 𝐵))

Theoremisumclim 14202* An infinite sum equals the value its series converges to. (Contributed by NM, 25-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐵)       (𝜑 → Σ𝑘𝑍 𝐴 = 𝐵)

Theoremisumclim2 14203* A converging series converges to its infinite sum. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘𝑍 𝐴)

Theoremisumclim3 14204* The sequence of partial finite sums of a converging infinite series converge to the infinite sum of the series. Note that 𝑗 must not occur in 𝐴. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 ∈ dom ⇝ )    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   ((𝜑𝑗𝑍) → (𝐹𝑗) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)       (𝜑𝐹 ⇝ Σ𝑘𝑍 𝐴)

Theoremsumnul 14205* The sum of a non-convergent infinite series evaluates to the empty set. (Contributed by Paul Chapman, 4-Nov-2007.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → ¬ seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 = ∅)

Theoremisumcl 14206* The sum of a converging infinite series is a complex number. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 ∈ ℂ)

Theoremisummulc2 14207* An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐵 · Σ𝑘𝑍 𝐴) = Σ𝑘𝑍 (𝐵 · 𝐴))

Theoremisummulc1 14208* An infinite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (Σ𝑘𝑍 𝐴 · 𝐵) = Σ𝑘𝑍 (𝐴 · 𝐵))

Theoremisumdivc 14209* An infinite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (Σ𝑘𝑍 𝐴 / 𝐵) = Σ𝑘𝑍 (𝐴 / 𝐵))

Theoremisumrecl 14210* The sum of a converging infinite real series is a real number. (Contributed by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 ∈ ℝ)

Theoremisumge0 14211* An infinite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 28-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   ((𝜑𝑘𝑍) → 0 ≤ 𝐴)       (𝜑 → 0 ≤ Σ𝑘𝑍 𝐴)

Theoremisumadd 14212* Addition of infinite sums. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 (𝐴 + 𝐵) = (Σ𝑘𝑍 𝐴 + Σ𝑘𝑍 𝐵))

Theoremsumsplit 14213* Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑 → (𝐴𝐵) ⊆ 𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐶, 0))    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = if(𝑘𝐵, 𝐶, 0))    &   ((𝜑𝑘 ∈ (𝐴𝐵)) → 𝐶 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )       (𝜑 → Σ𝑘 ∈ (𝐴𝐵)𝐶 = (Σ𝑘𝐴 𝐶 + Σ𝑘𝐵 𝐶))

Theoremfsump1i 14214* Optimized version of fsump1 14201 for making sums of a concrete number of terms. (Contributed by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   𝑁 = (𝐾 + 1)    &   (𝑘 = 𝑁𝐴 = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → (𝐾𝑍 ∧ Σ𝑘 ∈ (𝑀...𝐾)𝐴 = 𝑆))    &   (𝜑 → (𝑆 + 𝐵) = 𝑇)       (𝜑 → (𝑁𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇))

Theoremfsum2dlem 14215* Lemma for fsum2d 14216- induction step. (Contributed by Mario Carneiro, 23-Apr-2014.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)    &   (𝜑 → ¬ 𝑦𝑥)    &   (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)    &   (𝜓 ↔ Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)       ((𝜑𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)

Theoremfsum2d 14216* Write a double sum as a sum over a two-dimensional region. Note that 𝐵(𝑗) is a function of 𝑗. (Contributed by Mario Carneiro, 27-Apr-2014.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)

Theoremfsumxp 14217* Combine two sums into a single sum over the cartesian product. (Contributed by Mario Carneiro, 23-Apr-2014.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 ∈ (𝐴 × 𝐵)𝐷)

Theoremfsumcnv 14218* Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)
(𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)    &   (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → Rel 𝐴)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑦 𝐴𝐶)

Theoremfsumcom2 14219* Interchange order of summation. Note that 𝐵(𝑗) and 𝐷(𝑘) are not necessarily constant expressions. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.) (Proof shortened by JJ, 2-Aug-2021.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑘𝐶 Σ𝑗𝐷 𝐸)

Theoremfsumcom2OLD 14220* Obsolete proof of fsumcom2 14219 as of 2-Aug-2021. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑘𝐶 Σ𝑗𝐷 𝐸)

Theoremfsumcom 14221* Interchange order of summation. (Contributed by NM, 15-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐶 = Σ𝑘𝐵 Σ𝑗𝐴 𝐶)

Theoremfsum0diaglem 14222* Lemma for fsum0diag 14223. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...(𝑁𝑘))))

Theoremfsum0diag 14223* Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular region 𝑀𝑗, 𝑀𝑘, 𝑗 + 𝑘𝑁." (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...(𝑁𝑗)))) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...(𝑁𝑘))𝐴)

Theoremmptfzshft 14224* 1-1 onto function in maps-to notation which shifts a finite set of sequential integers. Formerly part of proof for fsumshft 14226. (Contributed by AV, 24-Aug-2019.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑗𝐾)):((𝑀 + 𝐾)...(𝑁 + 𝐾))–1-1-onto→(𝑀...𝑁))

Theoremfsumrev 14225* Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑗 = (𝐾𝑘) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))𝐵)

Theoremfsumshft 14226* 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.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑗 = (𝑘𝐾) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵)

Theoremfsumshftm 14227* Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑗 = (𝑘 + 𝐾) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀𝐾)...(𝑁𝐾))𝐵)

Theoremfsumrev2 14228* Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 13-Apr-2016.)
((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑗 = ((𝑀 + 𝑁) − 𝑘) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵)

Theoremfsum0diag2 14229* Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular region 0 ≤ 𝑗, 0 ≤ 𝑘, 𝑗 + 𝑘𝑁." (Contributed by Mario Carneiro, 21-Jul-2014.)
(𝑥 = 𝑘𝐵 = 𝐴)    &   (𝑥 = (𝑘𝑗) → 𝐵 = 𝐶)    &   ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗)))) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)𝐶)

Theoremfsummulc2 14230* A finite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))

Theoremfsummulc1 14231* A finite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (Σ𝑘𝐴 𝐵 · 𝐶) = Σ𝑘𝐴 (𝐵 · 𝐶))

Theoremfsumdivc 14232* A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → (Σ𝑘𝐴 𝐵 / 𝐶) = Σ𝑘𝐴 (𝐵 / 𝐶))

Theoremfsumneg 14233* Negation of a finite sum. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 -𝐵 = -Σ𝑘𝐴 𝐵)

Theoremfsumsub 14234* Split a finite sum over a subtraction. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 (𝐵𝐶) = (Σ𝑘𝐴 𝐵 − Σ𝑘𝐴 𝐶))

Theoremfsum2mul 14235* Separate the nested sum of the product 𝐶(𝑗) · 𝐷(𝑘). (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐵) → 𝐷 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 (𝐶 · 𝐷) = (Σ𝑗𝐴 𝐶 · Σ𝑘𝐵 𝐷))

Theoremfsumconst 14236* The sum of constant terms (𝑘 is not free in 𝐴). (Contributed by NM, 24-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑘𝐴 𝐵 = ((#‘𝐴) · 𝐵))

Theoremmodfsummodslem1 14237* Lemma 1 for modfsummods 14238. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
(∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → 𝑧 / 𝑘𝐵 ∈ ℤ)

Theoremmodfsummods 14238* Induction step for modfsummod 14239. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → ((Σ𝑘𝐴 𝐵 mod 𝑁) = (Σ𝑘𝐴 (𝐵 mod 𝑁) mod 𝑁) → (Σ𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 mod 𝑁) = (Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) mod 𝑁)))

Theoremmodfsummod 14239* 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.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℤ)       (𝜑 → (Σ𝑘𝐴 𝐵 mod 𝑁) = (Σ𝑘𝐴 (𝐵 mod 𝑁) mod 𝑁))

Theoremfsumge0 14240* 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)       (𝜑 → 0 ≤ Σ𝑘𝐴 𝐵)

Theoremfsumless 14241* A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by NM, 26-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → Σ𝑘𝐶 𝐵 ≤ Σ𝑘𝐴 𝐵)

Theoremfsumge1 14242* 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)    &   (𝑘 = 𝑀𝐵 = 𝐶)    &   (𝜑𝑀𝐴)       (𝜑𝐶 ≤ Σ𝑘𝐴 𝐵)

Theoremfsum00 14243* 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)       (𝜑 → (Σ𝑘𝐴 𝐵 = 0 ↔ ∀𝑘𝐴 𝐵 = 0))

Theoremfsumle 14244* 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐵𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 ≤ Σ𝑘𝐴 𝐶)

Theoremfsumlt 14245* 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.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐵 < 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 < Σ𝑘𝐴 𝐶)

Theoremfsumabs 14246* 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵))

Theoremtelfsumo 14247* Sum of a telescoping series, using half-open intervals. (Contributed by Mario Carneiro, 2-May-2016.)
(𝑘 = 𝑗𝐴 = 𝐵)    &   (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   (𝑘 = 𝑀𝐴 = 𝐷)    &   (𝑘 = 𝑁𝐴 = 𝐸)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵𝐶) = (𝐷𝐸))

Theoremtelfsumo2 14248* Sum of a telescoping series. (Contributed by Mario Carneiro, 2-May-2016.)
(𝑘 = 𝑗𝐴 = 𝐵)    &   (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   (𝑘 = 𝑀𝐴 = 𝐷)    &   (𝑘 = 𝑁𝐴 = 𝐸)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐶𝐵) = (𝐸𝐷))

Theoremtelfsum 14249* Sum of a telescoping series. (Contributed by Scott Fenton, 24-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)
(𝑘 = 𝑗𝐴 = 𝐵)    &   (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   (𝑘 = 𝑀𝐴 = 𝐷)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑 → (𝑁 + 1) ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)(𝐵𝐶) = (𝐷𝐸))

Theoremtelfsum2 14250* Sum of a telescoping series. (Contributed by Mario Carneiro, 15-Jun-2014.) (Revised by Mario Carneiro, 2-May-2016.)
(𝑘 = 𝑗𝐴 = 𝐵)    &   (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   (𝑘 = 𝑀𝐴 = 𝐷)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑 → (𝑁 + 1) ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)(𝐶𝐵) = (𝐸𝐷))

Theoremfsumparts 14251* Summation by parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
(𝑘 = 𝑗 → (𝐴 = 𝐵𝑉 = 𝑊))    &   (𝑘 = (𝑗 + 1) → (𝐴 = 𝐶𝑉 = 𝑋))    &   (𝑘 = 𝑀 → (𝐴 = 𝐷𝑉 = 𝑌))    &   (𝑘 = 𝑁 → (𝐴 = 𝐸𝑉 = 𝑍))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝑉 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 · (𝑋𝑊)) = (((𝐸 · 𝑍) − (𝐷 · 𝑌)) − Σ𝑗 ∈ (𝑀..^𝑁)((𝐶𝐵) · 𝑋)))

Theoremfsumrelem 14252* Lemma for fsumre 14253, fsumim 14254, and fsumcj 14255. (Contributed by Mario Carneiro, 25-Jul-2014.) (Revised by Mario Carneiro, 27-Dec-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐹:ℂ⟶ℂ    &   ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))       (𝜑 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))

Theoremfsumre 14253* The real part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (ℜ‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (ℜ‘𝐵))

Theoremfsumim 14254* The imaginary part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (ℑ‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (ℑ‘𝐵))

Theoremfsumcj 14255* The complex conjugate of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (∗‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (∗‘𝐵))

Theoremfsumrlim 14256* Limit of a finite sum of converging sequences. Note that 𝐶(𝑘) is a collection of functions with implicit parameter 𝑘, each of which converges to 𝐷(𝑘) as 𝑛 ⇝ +∞. (Contributed by Mario Carneiro, 22-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)    &   ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ⇝𝑟 𝐷)       (𝜑 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷)

Theoremfsumo1 14257* The finite sum of eventually bounded functions (where the index set 𝐵 does not depend on 𝑥) is eventually bounded. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 22-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)    &   ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ 𝑂(1))       (𝜑 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1))

Theoremo1fsum 14258* If 𝐴(𝑘) is O(1), then Σ𝑘𝑥, 𝐴(𝑘) is O(𝑥). (Contributed by Mario Carneiro, 23-May-2016.)
((𝜑𝑘 ∈ ℕ) → 𝐴𝑉)    &   (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1))       (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))

Theoremseqabs 14259* Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by Mario Carneiro, 26-Mar-2014.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) = (abs‘(𝐹𝑘)))       (𝜑 → (abs‘(seq𝑀( + , 𝐹)‘𝑁)) ≤ (seq𝑀( + , 𝐺)‘𝑁))

Theoremiserabs 14260* 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 Mario Carneiro, 27-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (abs‘(𝐹𝑘)))       (𝜑 → (abs‘𝐴) ≤ 𝐵)

Theoremcvgcmp 14261* A comparison test for convergence of a real infinite series. Exercise 3 of [Gleason] p. 182. (Contributed by NM, 1-May-2005.) (Revised by Mario Carneiro, 24-Mar-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ (𝐺𝑘))    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐺𝑘) ≤ (𝐹𝑘))       (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Theoremcvgcmpub 14262* 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 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ≤ (𝐹𝑘))       (𝜑𝐵𝐴)

Theoremcvgcmpce 14263* A comparison test for convergence of a complex infinite series. (Contributed by NM, 25-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐺𝑘)) ≤ (𝐶 · (𝐹𝑘)))       (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Theoremabscvgcvg 14264* An absolutely convergent series is convergent. (Contributed by Mario Carneiro, 28-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (abs‘(𝐺𝑘)))    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Theoremclimfsum 14265* Limit of a finite sum of converging sequences. Note that 𝐹(𝑘) is a collection of functions with implicit parameter 𝑘, each of which converges to 𝐵(𝑘) as 𝑛 ⇝ +∞. (Contributed by Mario Carneiro, 22-Jul-2014.) (Proof shortened by Mario Carneiro, 22-May-2016.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐹𝐵)    &   (𝜑𝐻𝑊)    &   ((𝜑 ∧ (𝑘𝐴𝑛𝑍)) → (𝐹𝑛) ∈ ℂ)    &   ((𝜑𝑛𝑍) → (𝐻𝑛) = Σ𝑘𝐴 (𝐹𝑛))       (𝜑𝐻 ⇝ Σ𝑘𝐴 𝐵)

Theoremfsumiun 14266* Sum over a disjoint indexed union. (Contributed by Mario Carneiro, 1-Jul-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘 𝑥𝐴 𝐵𝐶 = Σ𝑥𝐴 Σ𝑘𝐵 𝐶)

Theoremhashiun 14267* The cardinality of a disjoint indexed union. (Contributed by Mario Carneiro, 24-Jan-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)    &   (𝜑Disj 𝑥𝐴 𝐵)       (𝜑 → (#‘ 𝑥𝐴 𝐵) = Σ𝑥𝐴 (#‘𝐵))

Theoremhashrabrex 14268* The number of elements in a class abstraction with a restricted existential quantification. (Contributed by Alexander van der Vekens, 29-Jul-2018.)
(𝜑𝑌 ∈ Fin)    &   ((𝜑𝑦𝑌) → {𝑥𝑋𝜓} ∈ Fin)    &   (𝜑Disj 𝑦𝑌 {𝑥𝑋𝜓})       (𝜑 → (#‘{𝑥𝑋 ∣ ∃𝑦𝑌 𝜓}) = Σ𝑦𝑌 (#‘{𝑥𝑋𝜓}))

Theoremhashuni 14269* The cardinality of a disjoint union. (Contributed by Mario Carneiro, 24-Jan-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ Fin)    &   (𝜑Disj 𝑥𝐴 𝑥)       (𝜑 → (#‘ 𝐴) = Σ𝑥𝐴 (#‘𝑥))

Theoremqshash 14270* The cardinality of a set with an equivalence relation is the sum of the cardinalities of its equivalence classes. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝜑 Er 𝐴)    &   (𝜑𝐴 ∈ Fin)       (𝜑 → (#‘𝐴) = Σ𝑥 ∈ (𝐴 / )(#‘𝑥))

Theoremackbijnn 14271* Translate the Ackermann bijection ackbij1 8823 onto the positive integers. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝐹 = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦𝑥 (2↑𝑦))       𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0

5.10.4  The binomial theorem

Theorembinomlem 14272* Lemma for binom 14273 (binomial theorem). Inductive step. (Contributed by NM, 6-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜓 → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))       ((𝜑𝜓) → ((𝐴 + 𝐵)↑(𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵𝑘))))

Theorembinom 14273* The binomial theorem: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴𝑘) · (𝐵↑(𝑁𝑘)). 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 14272. 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↑(𝑁𝑘)) · (𝐵𝑘))))

Theorembinom1p 14274* 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) · (𝐴𝑘)))

Theorembinom11 14275* Special case of the binomial theorem for 2↑𝑁. (Contributed by Mario Carneiro, 13-Mar-2014.)
(𝑁 ∈ ℕ0 → (2↑𝑁) = Σ𝑘 ∈ (0...𝑁)(𝑁C𝑘))

Theorembinom1dif 14276* A summation for the difference between ((𝐴 + 1)↑𝑁) and (𝐴𝑁). (Contributed by Scott Fenton, 9-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((𝐴 + 1)↑𝑁) − (𝐴𝑁)) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · (𝐴𝑘)))

Theorembcxmaslem1 14277 Lemma for bcxmas 14278. (Contributed by Paul Chapman, 18-May-2007.)
(𝐴 = 𝐵 → ((𝑁 + 𝐴)C𝐴) = ((𝑁 + 𝐵)C𝐵))

Theorembcxmas 14278* Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))

5.10.5  The inclusion/exclusion principle

Theoremincexclem 14279* Lemma for incexc 14280. (Contributed by Mario Carneiro, 7-Aug-2017.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐵) − (#‘(𝐵 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴((-1↑(#‘𝑠)) · (#‘(𝐵 𝑠))))

Theoremincexc 14280* The inclusion/exclusion principle for counting the elements of a finite union of finite sets. This is Metamath 100 proof #96. (Contributed by Mario Carneiro, 7-Aug-2017.)
((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘ 𝑠)))

Theoremincexc2 14281* The inclusion/exclusion principle for counting the elements of a finite union of finite sets. (Contributed by Mario Carneiro, 7-Aug-2017.)
((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘ 𝐴) = Σ𝑛 ∈ (1...(#‘𝐴))((-1↑(𝑛 − 1)) · Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} (#‘ 𝑠)))

5.10.6  Infinite sums (cont.)

Theoremisumshft 14282* Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ‘(𝑀 + 𝐾))    &   (𝑗 = (𝐾 + 𝑘) → 𝐴 = 𝐵)    &   (𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑗𝑊) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗𝑊 𝐴 = Σ𝑘𝑍 𝐵)

Theoremisumsplit 14283* Split off the first 𝑁 terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘𝑊 𝐴))

Theoremisum1p 14284* 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 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 = ((𝐹𝑀) + Σ𝑘 ∈ (ℤ‘(𝑀 + 1))𝐴))

Theoremisumnn0nn 14285* 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)    &   ((𝜑𝑘 ∈ ℕ0) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ)    &   (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴))

Theoremisumrpcl 14286* The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ+)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑊 𝐴 ∈ ℝ+)

Theoremisumle 14287* Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝑍) → 𝐴𝐵)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 ≤ Σ𝑘𝑍 𝐵)

Theoremisumless 14288* A finite sum of nonnegative numbers is less or equal to its limit. (Contributed by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝑍) → 0 ≤ 𝐵)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝐴 𝐵 ≤ Σ𝑘𝑍 𝐵)

Theoremisumsup2 14289* An infinite sum of nonnegative terms is equal to the supremum of the partial sums. (Contributed by Mario Carneiro, 12-Jun-2014.)
𝑍 = (ℤ𝑀)    &   𝐺 = seq𝑀( + , 𝐹)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑍) → 0 ≤ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐺𝑗) ≤ 𝑥)       (𝜑𝐺 ⇝ sup(ran 𝐺, ℝ, < ))

Theoremisumsup 14290* An infinite sum of nonnegative terms is equal to the supremum of the partial sums. (Contributed by Mario Carneiro, 12-Jun-2014.)
𝑍 = (ℤ𝑀)    &   𝐺 = seq𝑀( + , 𝐹)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑍) → 0 ≤ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐺𝑗) ≤ 𝑥)       (𝜑 → Σ𝑘𝑍 𝐴 = sup(ran 𝐺, ℝ, < ))

Theoremisumltss 14291* A partial sum of a series with positive terms is less than the infinite sum. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ+)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝐴 𝐵 < Σ𝑘𝑍 𝐵)

Theoremclimcndslem1 14292* Lemma for climcnds 14294: bound the original series by the condensed series. (Contributed by Mario Carneiro, 18-Jul-2014.)
((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))    &   ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))       ((𝜑𝑁 ∈ ℕ0) → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁))

Theoremclimcndslem2 14293* Lemma for climcnds 14294: bound the condensed series by the original series. (Contributed by Mario Carneiro, 18-Jul-2014.)
((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))    &   ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))       ((𝜑𝑁 ∈ ℕ) → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))

Theoremclimcnds 14294* The Cauchy condensation test. If 𝑎(𝑘) is a decreasing sequence of nonnegative terms, then Σ𝑘 ∈ ℕ𝑎(𝑘) converges iff Σ𝑛 ∈ ℕ02↑𝑛 · 𝑎(2↑𝑛) converges. (Contributed by Mario Carneiro, 18-Jul-2014.)
((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))    &   ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))       (𝜑 → (seq1( + , 𝐹) ∈ dom ⇝ ↔ seq0( + , 𝐺) ∈ dom ⇝ ))

5.10.7  Miscellaneous converging and diverging sequences

Theoremdivrcnv 14295* The sequence of reciprocals of real numbers, multiplied by the factor 𝐴, converges to zero. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℂ → (𝑛 ∈ ℝ+ ↦ (𝐴 / 𝑛)) ⇝𝑟 0)

Theoremdivcnv 14296* The sequence of reciprocals of positive integers, multiplied by the factor 𝐴, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℂ → (𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)) ⇝ 0)

Theoremflo1 14297 The floor function satisfies ⌊(𝑥) = 𝑥 + 𝑂(1). (Contributed by Mario Carneiro, 21-May-2016.)
(𝑥 ∈ ℝ ↦ (𝑥 − (⌊‘𝑥))) ∈ 𝑂(1)

Theoremdivcnvshft 14298* Limit of a ratio function. (Contributed by Scott Fenton, 16-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐴 / (𝑘 + 𝐵)))       (𝜑𝐹 ⇝ 0)

Theoremsupcvg 14299* Extract a sequence 𝑓 in 𝑋 such that the image of the points in the bounded set 𝐴 converges to the supremum 𝑆 of the set. Similar to Equation 4 of [Kreyszig] p. 144. The proof uses countable choice ax-cc 9020. (Contributed by Mario Carneiro, 15-Feb-2013.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
𝑋 ∈ V    &   𝑆 = sup(𝐴, ℝ, < )    &   𝑅 = (𝑛 ∈ ℕ ↦ (𝑆 − (1 / 𝑛)))    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐹:𝑋onto𝐴)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)       (𝜑 → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ (𝐹𝑓) ⇝ 𝑆))

Theoreminfcvgaux1i 14300* Auxiliary theorem for applications of supcvg 14299. Hypothesis for several supremum theorems. (Contributed by NM, 8-Feb-2008.)
𝑅 = {𝑥 ∣ ∃𝑦𝑋 𝑥 = -𝐴}    &   (𝑦𝑋𝐴 ∈ ℝ)    &   𝑍𝑋    &   𝑧 ∈ ℝ ∀𝑤𝑅 𝑤𝑧       (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤𝑅 𝑤𝑧)

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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