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Theorem List for Intuitionistic Logic Explorer - 11101-11200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsummodclem2a 11101* Lemma for summodc 11103. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))    &   𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝑓:(1...𝑁)–1-1-onto𝐴)    &   (𝜑𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁))
 
Theoremsummodclem2 11102* Lemma for summodc 11103. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))       ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
 
Theoremsummodc 11103* A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))       (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))))
 
Theoremzsumdc 11104* Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.) (Revised by Jim Kingdon, 8-Apr-2023.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑 → ∀𝑥𝑍 DECID 𝑥𝐴)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
 
Theoremisum 11105* 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)))
 
Theoremfsumgcl 11106* Closure for a function used to describe a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
(𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)       (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
 
Theoremfsum3 11107* The value of a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
(𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀))
 
Theoremsum0 11108 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
 
Theoremisumz 11109* Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
(((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∨ 𝐴 ∈ Fin) → Σ𝑘𝐴 0 = 0)
 
Theoremfsumf1o 11110* 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)
 
Theoremisumss 11111* Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 21-Sep-2022.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑 → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐵 ⊆ (ℤ𝑀))    &   (𝜑 → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐵)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
 
Theoremfisumss 11112* Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 23-Sep-2022.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑 → ∀𝑗𝐵 DECID 𝑗𝐴)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
 
Theoremisumss2 11113* Change the index set of a sum by adding zeroes. The nonzero elements are in the contained set 𝐴 and the added zeroes compose the rest of the containing set 𝐵 which needs to be summable. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Jim Kingdon, 24-Sep-2022.)
(𝜑𝐴𝐵)    &   (𝜑 → ∀𝑗𝐵 DECID 𝑗𝐴)    &   (𝜑 → ∀𝑘𝐴 𝐶 ∈ ℂ)    &   (𝜑 → ((𝑀 ∈ ℤ ∧ 𝐵 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐵) ∨ 𝐵 ∈ Fin))       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 if(𝑘𝐴, 𝐶, 0))
 
Theoremfsum3cvg2 11114* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))
 
Theoremfsumsersdc 11115* Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Jim Kingdon, 5-May-2023.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → Σ𝑘𝐴 𝐵 = (seq𝑀( + , 𝐹)‘𝑁))
 
Theoremfsum3cvg3 11116* A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
 
Theoremfsum3ser 11117* A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 11132 and fsump1 11140, which should make our notation clear and from which, along with closure fsumcl 11120, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 1-Oct-2022.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = 𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁))
 
Theoremfsumcl2lem 11118* - 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 11119* - 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 11120* 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 11121* 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 11122* 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 11123* 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 11124* 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 11125* 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 11126* 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 11127* 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 11128* Split a sum into two parts. A version of fsumsplit 11127 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 11129* A sum of a singleton is the term. A version of sumsn 11131 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 11130* 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 11131* 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 11132* 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 11133* 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 11134* 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 11135* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
((𝑀𝑉𝑀 / 𝑘𝐴 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)
 
Theoremfsumm1 11136* 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 11137* 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 11138* 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))
 
Theoremfsumsplitsnun 11139* 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 11140* 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 11141* 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 11142* 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 11143* 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 11144* 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 11145* 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 11146* 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 11147* 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))
 
Theoremisumdivapc 11148* 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 11149* 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 11150* 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 11151* 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))
 
Theoremsumsplitdc 11152* 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) → DECID 𝑘𝐴)    &   ((𝜑𝑘𝑍) → DECID 𝑘𝐵)    &   ((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 11153* Optimized version of fsump1 11140 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))
 
Theoremfsum2dlemstep 11154* Lemma for fsum2d 11155- induction step. (Contributed by Mario Carneiro, 23-Apr-2014.) (Revised by Jim Kingdon, 8-Oct-2022.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)    &   (𝜑 → ¬ 𝑦𝑥)    &   (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)    &   (𝜑𝑥 ∈ Fin)    &   (𝜓 ↔ Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)       ((𝜑𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
 
Theoremfsum2d 11155* 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 11156* 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 11157* 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)
 
Theoremfisumcom2 11158* 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)    &   ((𝜑𝑘𝐶) → 𝐷 ∈ Fin)    &   (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑘𝐶 Σ𝑗𝐷 𝐸)
 
Theoremfsumcom 11159* 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 11160* Lemma for fisum0diag 11161. (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))))
 
Theoremfisum0diag 11161* 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)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...(𝑁𝑘))𝐴)
 
Theoremmptfzshft 11162* 1-1 onto function in maps-to notation which shifts a finite set of sequential integers. (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 11163* 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 11164* 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 11165* 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)
 
Theoremfisumrev2 11166* Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 13-Apr-2016.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑗 = ((𝑀 + 𝑁) − 𝑘) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵)
 
Theoremfisum0diag2 11167* 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)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)𝐶)
 
Theoremfsummulc2 11168* 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 11169* 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))
 
Theoremfsumdivapc 11170* 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 11171* 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 11172* 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 11173* 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 11174* 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 11175* 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))
 
Theoremmodfsummodlem1 11176* Lemma for modfsummod 11178. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
(∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → 𝑧 / 𝑘𝐵 ∈ ℤ)
 
Theoremmodfsummodlemstep 11177* Induction step for modfsummod 11178. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by Jim Kingdon, 12-Oct-2022.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → (Σ𝑘𝐴 𝐵 mod 𝑁) = (Σ𝑘𝐴 (𝐵 mod 𝑁) mod 𝑁))       (𝜑 → (Σ𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 mod 𝑁) = (Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) mod 𝑁))
 
Theoremmodfsummod 11178* 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 11179* 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)
 
Theoremfsumlessfi 11180* 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)    &   (𝜑𝐶𝐴)    &   (𝜑𝐶 ∈ Fin)       (𝜑 → Σ𝑘𝐶 𝐵 ≤ Σ𝑘𝐴 𝐵)
 
Theoremfsumge1 11181* 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 11182* 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 11183* 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 11184* 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 11185* 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 11186* 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 11187* 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 11188* 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 11189* 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 11190* 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 11191* Lemma for fsumre 11192, fsumim 11193, and fsumcj 11194. (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 11192* 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 11193* 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 11194* 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))
 
Theoremiserabs 11195* 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 = (ℤ𝑀)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (abs‘(𝐹𝑘)))       (𝜑 → (abs‘𝐴) ≤ 𝐵)
 
Theoremcvgcmpub 11196* 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)
 
Theoremfsumiun 11197* 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 11198* 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))
 
Theoremhash2iun 11199* The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)    &   ((𝜑𝑥𝐴𝑦𝐵) → 𝐶 ∈ Fin)    &   (𝜑Disj 𝑥𝐴 𝑦𝐵 𝐶)    &   ((𝜑𝑥𝐴) → Disj 𝑦𝐵 𝐶)       (𝜑 → (♯‘ 𝑥𝐴 𝑦𝐵 𝐶) = Σ𝑥𝐴 Σ𝑦𝐵 (♯‘𝐶))
 
Theoremhash2iun1dif1 11200* The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.)
(𝜑𝐴 ∈ Fin)    &   𝐵 = (𝐴 ∖ {𝑥})    &   ((𝜑𝑥𝐴𝑦𝐵) → 𝐶 ∈ Fin)    &   (𝜑Disj 𝑥𝐴 𝑦𝐵 𝐶)    &   ((𝜑𝑥𝐴) → Disj 𝑦𝐵 𝐶)    &   ((𝜑𝑥𝐴𝑦𝐵) → (♯‘𝐶) = 1)       (𝜑 → (♯‘ 𝑥𝐴 𝑦𝐵 𝐶) = ((♯‘𝐴) · ((♯‘𝐴) − 1)))
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