Home Metamath Proof ExplorerTheorem List (p. 149 of 435) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28331) Hilbert Space Explorer (28332-29856) Users' Mathboxes (29857-43448)

Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsumeq2ii 14801* Equality theorem for sum, with the class expressions 𝐵 and 𝐶 guarded by I to be always sets. (Contributed by Mario Carneiro, 13-Jun-2019.)
(∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremsumeq2 14802* Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
(∀𝑘𝐴 𝐵 = 𝐶 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremcbvsum 14803* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
(𝑗 = 𝑘𝐵 = 𝐶)    &   𝑘𝐴    &   𝑗𝐴    &   𝑘𝐵    &   𝑗𝐶       Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶

Theoremcbvsumv 14804* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
(𝑗 = 𝑘𝐵 = 𝐶)       Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶

Theoremcbvsumi 14805* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
𝑘𝐵    &   𝑗𝐶    &   (𝑗 = 𝑘𝐵 = 𝐶)       Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶

Theoremsumeq1i 14806* Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
𝐴 = 𝐵       Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶

Theoremsumeq2i 14807* Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
(𝑘𝐴𝐵 = 𝐶)       Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶

Theoremsumeq12i 14808* Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
𝐴 = 𝐵    &   (𝑘𝐴𝐶 = 𝐷)       Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷

Theoremsumeq1d 14809* Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)

Theoremsumeq2d 14810* Equality deduction for sum. Note that unlike sumeq2dv 14811, 𝑘 may occur in 𝜑. (Contributed by NM, 1-Nov-2005.)
(𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremsumeq2dv 14811* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
((𝜑𝑘𝐴) → 𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremsumeq2ad 14812* Equality deduction for sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremsumeq2sdv 14813* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)
(𝜑𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theorem2sumeq2dv 14814* Equality deduction for double sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
((𝜑𝑗𝐴𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐶 = Σ𝑗𝐴 Σ𝑘𝐵 𝐷)

Theoremsumeq12dv 14815* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷)

Theoremsumeq12rdv 14816* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷)

Theoremsum2id 14817* The second class argument to a sum can be chosen so that it is always a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 ( I ‘𝐵)

Theoremsumfc 14818* A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
Σ𝑗𝐴 ((𝑘𝐴𝐵)‘𝑗) = Σ𝑘𝐴 𝐵

Theoremfz1f1o 14819* A lemma for working with finite sums. (Contributed by Mario Carneiro, 22-Apr-2014.)
(𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))

Theoremsumrblem 14820* Lemma for sumrb 14822. (Contributed by Mario Carneiro, 12-Aug-2013.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ (ℤ𝑀))       ((𝜑𝐴 ⊆ (ℤ𝑁)) → (seq𝑀( + , 𝐹) ↾ (ℤ𝑁)) = seq𝑁( + , 𝐹))

Theoremfsumcvg 14821* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))

Theoremsumrb 14822* Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 9-Apr-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝐴 ⊆ (ℤ𝑁))       (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹) ⇝ 𝐶))

Theoremsummolem3 14823* Lemma for summo 14826. (Contributed by Mario Carneiro, 29-Mar-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)    &   𝐻 = (𝑛 ∈ ℕ ↦ (𝐾𝑛) / 𝑘𝐵)    &   (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))    &   (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)    &   (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)       (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))

Theoremsummolem2a 14824* Lemma for summo 14826. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 20-Apr-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)    &   𝐻 = (𝑛 ∈ ℕ ↦ (𝐾𝑛) / 𝑘𝐵)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝑓:(1...𝑁)–1-1-onto𝐴)    &   (𝜑𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁))

Theoremsummolem2 14825* Lemma for summo 14826. (Contributed by Mario Carneiro, 3-Apr-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)       ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Theoremsummo 14826* A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)       (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))))

Theoremzsum 14827* Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))

Theoremisum 14828* Series sum with an upper integer index set (i.e. an infinite series). (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 7-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝑍 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))

Theoremfsum 14829* The value of a sum over a nonempty finite set. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Mario Carneiro, 13-Jun-2019.)
(𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = (seq1( + , 𝐺)‘𝑀))

Theoremsum0 14830 Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
Σ𝑘 ∈ ∅ 𝐴 = 0

Theoremsumz 14831* Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
((𝐴 ⊆ (ℤ𝑀) ∨ 𝐴 ∈ Fin) → Σ𝑘𝐴 0 = 0)

Theoremfsumf1o 14832* Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.)
(𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐶 ∈ Fin)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑛𝐶 𝐷)

Theoremsumss 14833* Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑𝐵 ⊆ (ℤ𝑀))       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)

Theoremfsumss 14834* Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)

Theoremsumss2 14835* Change the index set of a sum by adding zeroes. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
(((𝐴𝐵 ∧ ∀𝑘𝐴 𝐶 ∈ ℂ) ∧ (𝐵 ⊆ (ℤ𝑀) ∨ 𝐵 ∈ Fin)) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 if(𝑘𝐴, 𝐶, 0))

Theoremfsumcvg2 14836* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))

Theoremfsumsers 14837* Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 21-Apr-2014.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → Σ𝑘𝐴 𝐵 = (seq𝑀( + , 𝐹)‘𝑁))

Theoremfsumcvg3 14838* A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Theoremfsumser 14839* A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 14854 and fsump1i 14876, which should make our notation clear and from which, along with closure fsumcl 14842, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = 𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁))

Theoremfsumcl2lem 14840* - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by Mario Carneiro, 3-Jun-2014.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → Σ𝑘𝐴 𝐵𝑆)

Theoremfsumcllem 14841* - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 3-Jun-2014.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑 → 0 ∈ 𝑆)       (𝜑 → Σ𝑘𝐴 𝐵𝑆)

Theoremfsumcl 14842* Closure of a finite sum of complex numbers 𝐴(𝑘). (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℂ)

Theoremfsumrecl 14843* Closure of a finite sum of reals. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℝ)

Theoremfsumzcl 14844* Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℤ)

Theoremfsumnn0cl 14845* Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℕ0)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℕ0)

Theoremfsumrpcl 14846* Closure of a finite sum of positive reals. (Contributed by Mario Carneiro, 3-Jun-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ+)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℝ+)

Theoremfsumzcl2 14847* A finite sum with integer summands is an integer. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘𝐴 𝐵 ∈ ℤ)

Theoremfsumadd 14848* The sum of two finite sums. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))

Theoremfsumsplit 14849* Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈 = (𝐴𝐵))    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑𝑘𝑈) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝑈 𝐶 = (Σ𝑘𝐴 𝐶 + Σ𝑘𝐵 𝐶))

Theoremfsumsplitf 14850* Split a sum into two parts. A version of fsumsplit 14849 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈 = (𝐴𝐵))    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑𝑘𝑈) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝑈 𝐶 = (Σ𝑘𝐴 𝐶 + Σ𝑘𝐵 𝐶))

Theoremsumsnf 14851* A sum of a singleton is the term. A version of sumsn 14853 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝐵    &   (𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀𝑉𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)

Theoremfsumsplitsn 14852* Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   𝑘𝐷    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝑉)    &   (𝜑 → ¬ 𝐵𝐴)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   (𝑘 = 𝐵𝐶 = 𝐷)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘𝐴 𝐶 + 𝐷))

Theoremsumsn 14853* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
(𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀𝑉𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)

Theoremfsum1 14854* The finite sum of 𝐴(𝑘) from 𝑘 = 𝑀 to 𝑀 (i.e. a sum with only one term) is 𝐵 i.e. 𝐴(𝑀). (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
(𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (𝑀...𝑀)𝐴 = 𝐵)

Theoremsumpr 14855* A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.)
(𝑘 = 𝐴𝐶 = 𝐷)    &   (𝑘 = 𝐵𝐶 = 𝐸)    &   (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))    &   (𝜑 → (𝐴𝑉𝐵𝑊))    &   (𝜑𝐴𝐵)       (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))

Theoremsumtp 14856* A sum over a triple is the sum of the elements. (Contributed by AV, 24-Jul-2020.)
(𝑘 = 𝐴𝐷 = 𝐸)    &   (𝑘 = 𝐵𝐷 = 𝐹)    &   (𝑘 = 𝐶𝐷 = 𝐺)    &   (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ))    &   (𝜑 → (𝐴𝑉𝐵𝑊𝐶𝑋))    &   (𝜑𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺))

Theoremsumsns 14857* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
((𝑀𝑉𝑀 / 𝑘𝐴 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)

Theoremfsumm1 14858* Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = 𝑁𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))

Theoremfzosump1 14859* Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 13-Apr-2016.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = 𝑁𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀..^(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀..^𝑁)𝐴 + 𝐵))

Theoremfsum1p 14860* Separate out the first term in a finite sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = 𝑀𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))

Theoremfsummsnunz 14861* A finite sum all of whose summands are integers is itself an integer (case where the summation set is the union of a finite set and a singleton). (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 17-Dec-2021.)
((𝐴 ∈ Fin ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ)

Theoremfsumsplitsnun 14862* Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 17-Dec-2021.)
((𝐴 ∈ Fin ∧ (𝑍𝑉𝑍𝐴) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 = (Σ𝑘𝐴 𝐵 + 𝑍 / 𝑘𝐵))

Theoremfsump1 14863* 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 14864* 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 14865* 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 14866* The sequence of partial finite sums of a converging infinite series converges 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 14867* 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 14868* 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 14869* 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 14870* 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 14871* 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 14872* 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 14873* 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 14874* 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 14875* 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 14876* Optimized version of fsump1 14863 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 14877* Lemma for fsum2d 14878- 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 14878* 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 14879* 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 14880* 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 14881* 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)

Theoremfsumcom 14882* 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 14883* Lemma for fsum0diag 14884. (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 14884* 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 14885* 1-1 onto function in maps-to notation which shifts a finite set of sequential integers. Formerly part of proof for fsumshft 14887. (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 14886* 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 14887* 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 14888* 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 14889* 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 14890* 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 14891* 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 14892* 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 14893* 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 14894* 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 14895* 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 14896* 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 14897* 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) → Σ𝑘𝐴 𝐵 = ((♯‘𝐴) · 𝐵))

Theoremfsumdifsnconst 14898* The sum of constant terms (𝑘 is not free in 𝐶) over an index set excluding a singleton. (Contributed by AV, 7-Jan-2022.)
((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶 ∈ ℂ) → Σ𝑘 ∈ (𝐴 ∖ {𝐵})𝐶 = (((♯‘𝐴) − 1) · 𝐶))

Theoremmodfsummodslem1 14899* Lemma 1 for modfsummods 14900. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
(∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → 𝑧 / 𝑘𝐵 ∈ ℤ)

Theoremmodfsummods 14900* Induction step for modfsummod 14901. (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)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43448
 Copyright terms: Public domain < Previous  Next >