Theorem List for Intuitionistic Logic Explorer - 11401-11500 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | cbvsumi 11401* |
Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
|
⊢ Ⅎ𝑘𝐵
& ⊢ Ⅎ𝑗𝐶
& ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) ⇒ ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
|
Theorem | sumeq1i 11402* |
Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶 |
|
Theorem | sumeq2i 11403* |
Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
|
⊢ (𝑘 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
|
Theorem | sumeq12i 11404* |
Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
|
⊢ 𝐴 = 𝐵
& ⊢ (𝑘 ∈ 𝐴 → 𝐶 = 𝐷) ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷 |
|
Theorem | sumeq1d 11405* |
Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
|
Theorem | sumeq2d 11406* |
Equality deduction for sum. Note that unlike sumeq2dv 11407, 𝑘 may
occur in 𝜑. (Contributed by NM, 1-Nov-2005.)
|
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
|
Theorem | sumeq2dv 11407* |
Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised
by Mario Carneiro, 31-Jan-2014.)
|
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
|
Theorem | sumeq2ad 11408* |
Equality deduction for sum. (Contributed by Glauco Siliprandi,
5-Apr-2020.)
|
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
|
Theorem | sumeq2sdv 11409* |
Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)
|
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
|
Theorem | 2sumeq2dv 11410* |
Equality deduction for double sum. (Contributed by NM, 3-Jan-2006.)
(Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐷) |
|
Theorem | sumeq12dv 11411* |
Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷) |
|
Theorem | sumeq12rdv 11412* |
Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷) |
|
Theorem | sumfct 11413* |
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 Jim Kingdon, 18-Sep-2022.)
|
⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → Σ𝑗 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = Σ𝑘 ∈ 𝐴 𝐵) |
|
Theorem | fz1f1o 11414* |
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-onto→𝐴))) |
|
Theorem | nnf1o 11415 |
Lemma for sum and product theorems. (Contributed by Jim Kingdon,
15-Aug-2022.)
|
⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) & ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
& ⊢ (𝜑 → 𝐺:(1...𝑁)–1-1-onto→𝐴) ⇒ ⊢ (𝜑 → 𝑁 = 𝑀) |
|
Theorem | sumrbdclem 11416* |
Lemma for sumrbdc 11418. (Contributed by Mario Carneiro,
12-Aug-2013.)
(Revised by Jim Kingdon, 8-Apr-2023.)
|
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID
𝑘 ∈ 𝐴)
& ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
⇒ ⊢ ((𝜑 ∧ 𝐴 ⊆
(ℤ≥‘𝑁)) → (seq𝑀( + , 𝐹) ↾
(ℤ≥‘𝑁)) = seq𝑁( + , 𝐹)) |
|
Theorem | fsum3cvg 11417* |
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, 12-Nov-2022.)
|
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID
𝑘 ∈ 𝐴)
& ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁)) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁)) |
|
Theorem | sumrbdc 11418* |
Rebase the starting point of a sum. (Contributed by Mario Carneiro,
14-Jul-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
|
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ⊆
(ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐴 ⊆
(ℤ≥‘𝑁)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID
𝑘 ∈ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → DECID
𝑘 ∈ 𝐴) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹) ⇝ 𝐶)) |
|
Theorem | summodclem3 11419* |
Lemma for summodc 11422. (Contributed by Mario Carneiro,
29-Mar-2014.)
(Revised by Jim Kingdon, 9-Apr-2023.)
|
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) & ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
& ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
& ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0)) ⇒ ⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁)) |
|
Theorem | summodclem2a 11420* |
Lemma for summodc 11422. (Contributed by Mario Carneiro,
3-Apr-2014.)
(Revised by Jim Kingdon, 9-Apr-2023.)
|
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID
𝑘 ∈ 𝐴)
& ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ⊆
(ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)
& ⊢ (𝜑 → 𝐾 Isom < , <
((1...(♯‘𝐴)),
𝐴)) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁)) |
|
Theorem | summodclem2 11421* |
Lemma for summodc 11422. (Contributed by Mario Carneiro,
3-Apr-2014.)
(Revised by Jim Kingdon, 4-May-2023.)
|
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)) ⇒ ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦)) |
|
Theorem | summodc 11422* |
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(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)) ⇒ ⊢ (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)))) |
|
Theorem | zsumdc 11423* |
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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑍)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 DECID 𝑥 ∈ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹))) |
|
Theorem | isum 11424* |
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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹))) |
|
Theorem | fsumgcl 11425* |
Closure for a function used to describe a sum over a nonempty finite
set. (Contributed by Jim Kingdon, 10-Oct-2022.)
|
⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)
& ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) ∈ ℂ) |
|
Theorem | fsum3 11426* |
The value of a sum over a nonempty finite set. (Contributed by Jim
Kingdon, 10-Oct-2022.)
|
⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)
& ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0)))‘𝑀)) |
|
Theorem | sum0 11427 |
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 |
|
Theorem | isumz 11428* |
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 ⊆
(ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0) |
|
Theorem | fsumf1o 11429* |
Re-index a finite sum using a bijection. (Contributed by Mario
Carneiro, 20-Apr-2014.)
|
⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
& ⊢ (𝜑 → 𝐶 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
& ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷) |
|
Theorem | isumss 11430* |
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.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) & ⊢ (𝜑 → ∀𝑗 ∈
(ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)
& ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ⊆
(ℤ≥‘𝑀)) & ⊢ (𝜑 → ∀𝑗 ∈
(ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
|
Theorem | fisumss 11431* |
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.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) & ⊢ (𝜑 → ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴)
& ⊢ (𝜑 → 𝐵 ∈ Fin) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
|
Theorem | isumss2 11432* |
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.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵)
& ⊢ (𝜑 → ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴)
& ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ) & ⊢ (𝜑 → ((𝑀 ∈ ℤ ∧ 𝐵 ⊆
(ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐵) ∨ 𝐵 ∈ Fin))
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 if(𝑘 ∈ 𝐴, 𝐶, 0)) |
|
Theorem | fsum3cvg2 11433* |
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.)
|
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID
𝑘 ∈ 𝐴)
& ⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁)) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁)) |
|
Theorem | fsumsersdc 11434* |
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.)
|
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID
𝑘 ∈ 𝐴)
& ⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁)) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq𝑀( + , 𝐹)‘𝑁)) |
|
Theorem | fsum3cvg3 11435* |
A finite sum is convergent. (Contributed by Mario Carneiro,
24-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ 𝑍)
& ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID
𝑘 ∈ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
|
Theorem | fsum3ser 11436* |
A finite sum expressed in terms of a partial sum of an infinite series.
The recursive definition follows as fsum1 11451 and fsump1 11459, which should
make our notation clear and from which, along with closure fsumcl 11439, we
will derive the basic properties of finite sums. (Contributed by NM,
11-Dec-2005.) (Revised by Jim Kingdon, 1-Oct-2022.)
|
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = 𝐴)
& ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁)) |
|
Theorem | fsumcl2lem 11437* |
- 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.)
|
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
& ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐴 ≠ ∅)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
|
Theorem | fsumcllem 11438* |
- 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.)
|
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
& ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 0 ∈ 𝑆) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
|
Theorem | fsumcl 11439* |
Closure of a finite sum of complex numbers 𝐴(𝑘). (Contributed
by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
|
Theorem | fsumrecl 11440* |
Closure of a finite sum of reals. (Contributed by NM, 9-Nov-2005.)
(Revised by Mario Carneiro, 22-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) |
|
Theorem | fsumzcl 11441* |
Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005.)
(Revised by Mario Carneiro, 22-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
|
Theorem | fsumnn0cl 11442* |
Closure of a finite sum of nonnegative integers. (Contributed by
Mario Carneiro, 23-Apr-2015.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈
ℕ0) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈
ℕ0) |
|
Theorem | fsumrpcl 11443* |
Closure of a finite sum of positive reals. (Contributed by Mario
Carneiro, 3-Jun-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈
ℝ+) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈
ℝ+) |
|
Theorem | fsumzcl2 11444* |
A finite sum with integer summands is an integer. (Contributed by
Alexander van der Vekens, 31-Aug-2018.)
|
⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
|
Theorem | fsumadd 11445* |
The sum of two finite sums. (Contributed by NM, 14-Nov-2005.) (Revised
by Mario Carneiro, 22-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶)) |
|
Theorem | fsumsplit 11446* |
Split a sum into two parts. (Contributed by Mario Carneiro,
18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.)
|
⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) & ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) & ⊢ (𝜑 → 𝑈 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
|
Theorem | fsumsplitf 11447* |
Split a sum into two parts. A version of fsumsplit 11446 using
bound-variable hypotheses instead of distinct variable conditions.
(Contributed by Glauco Siliprandi, 5-Apr-2020.)
|
⊢ Ⅎ𝑘𝜑
& ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) & ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) & ⊢ (𝜑 → 𝑈 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
|
Theorem | sumsnf 11448* |
A sum of a singleton is the term. A version of sumsn 11450 using
bound-variable hypotheses instead of distinct variable conditions.
(Contributed by Glauco Siliprandi, 5-Apr-2020.)
|
⊢ Ⅎ𝑘𝐵
& ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) ⇒ ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
|
Theorem | fsumsplitsn 11449* |
Separate out a term in a finite sum. (Contributed by Glauco Siliprandi,
5-Apr-2020.)
|
⊢ Ⅎ𝑘𝜑
& ⊢ Ⅎ𝑘𝐷
& ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷)
& ⊢ (𝜑 → 𝐷 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + 𝐷)) |
|
Theorem | sumsn 11450* |
A sum of a singleton is the term. (Contributed by Mario Carneiro,
22-Apr-2014.)
|
⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) ⇒ ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
|
Theorem | fsum1 11451* |
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 = 𝑀 → 𝐴 = 𝐵) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (𝑀...𝑀)𝐴 = 𝐵) |
|
Theorem | sumpr 11452* |
A sum over a pair is the sum of the elements. (Contributed by Thierry
Arnoux, 12-Dec-2016.)
|
⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷)
& ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸)
& ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ)) & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸)) |
|
Theorem | sumtp 11453* |
A sum over a triple is the sum of the elements. (Contributed by AV,
24-Jul-2020.)
|
⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸)
& ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹)
& ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺)
& ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ)) & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) & ⊢ (𝜑 → 𝐴 ≠ 𝐵)
& ⊢ (𝜑 → 𝐴 ≠ 𝐶)
& ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺)) |
|
Theorem | sumsns 11454* |
A sum of a singleton is the term. (Contributed by Mario Carneiro,
22-Apr-2014.)
|
⊢ ((𝑀 ∈ 𝑉 ∧ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
|
Theorem | fsumm1 11455* |
Separate out the last term in a finite sum. (Contributed by Mario
Carneiro, 26-Apr-2014.)
|
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵)) |
|
Theorem | fzosump1 11456* |
Separate out the last term in a finite sum. (Contributed by Mario
Carneiro, 13-Apr-2016.)
|
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀..^𝑁)𝐴 + 𝐵)) |
|
Theorem | fsum1p 11457* |
Separate out the first term in a finite sum. (Contributed by NM,
3-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
|
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
|
Theorem | fsumsplitsnun 11458* |
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 ∧ (𝑍 ∈ 𝑉 ∧ 𝑍 ∉ 𝐴) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 = (Σ𝑘 ∈ 𝐴 𝐵 + ⦋𝑍 / 𝑘⦌𝐵)) |
|
Theorem | fsump1 11459* |
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.)
|
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ) & ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀...𝑁)𝐴 + 𝐵)) |
|
Theorem | isumclim 11460* |
An infinite sum equals the value its series converges to.
(Contributed by NM, 25-Dec-2005.) (Revised by Mario Carneiro,
23-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = 𝐵) |
|
Theorem | isumclim2 11461* |
A converging series converges to its infinite sum. (Contributed by NM,
2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑍 𝐴) |
|
Theorem | isumclim3 11462* |
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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = Σ𝑘 ∈ (𝑀...𝑗)𝐴) ⇒ ⊢ (𝜑 → 𝐹 ⇝ Σ𝑘 ∈ 𝑍 𝐴) |
|
Theorem | sumnul 11463* |
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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → ¬ seq𝑀( + , 𝐹) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ∅) |
|
Theorem | isumcl 11464* |
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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ∈ ℂ) |
|
Theorem | isummulc2 11465* |
An infinite sum multiplied by a constant. (Contributed by NM,
12-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐵 · Σ𝑘 ∈ 𝑍 𝐴) = Σ𝑘 ∈ 𝑍 (𝐵 · 𝐴)) |
|
Theorem | isummulc1 11466* |
An infinite sum multiplied by a constant. (Contributed by NM,
13-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝑍 𝐴 · 𝐵) = Σ𝑘 ∈ 𝑍 (𝐴 · 𝐵)) |
|
Theorem | isumdivapc 11467* |
An infinite sum divided by a constant. (Contributed by NM, 2-Jan-2006.)
(Revised by Mario Carneiro, 23-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝑍 𝐴 / 𝐵) = Σ𝑘 ∈ 𝑍 (𝐴 / 𝐵)) |
|
Theorem | isumrecl 11468* |
The sum of a converging infinite real series is a real number.
(Contributed by Mario Carneiro, 24-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ∈ ℝ) |
|
Theorem | isumge0 11469* |
An infinite sum of nonnegative terms is nonnegative. (Contributed by
Mario Carneiro, 28-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝑍 𝐴) |
|
Theorem | isumadd 11470* |
Addition of infinite sums. (Contributed by Mario Carneiro,
18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) & ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 (𝐴 + 𝐵) = (Σ𝑘 ∈ 𝑍 𝐴 + Σ𝑘 ∈ 𝑍 𝐵)) |
|
Theorem | sumsplitdc 11471* |
Split a sum into two parts. (Contributed by Mario Carneiro,
18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) & ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝑍)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → DECID 𝑘 ∈ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → DECID 𝑘 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐶, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = if(𝑘 ∈ 𝐵, 𝐶, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ 𝐵)) → 𝐶 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) & ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝
) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∪ 𝐵)𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
|
Theorem | fsump1i 11472* |
Optimized version of fsump1 11459 for making sums of a concrete number of
terms. (Contributed by Mario Carneiro, 23-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑁 = (𝐾 + 1) & ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (𝐾 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝐾)𝐴 = 𝑆)) & ⊢ (𝜑 → (𝑆 + 𝐵) = 𝑇) ⇒ ⊢ (𝜑 → (𝑁 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇)) |
|
Theorem | fsum2dlemstep 11473* |
Lemma for fsum2d 11474- induction step. (Contributed by Mario
Carneiro,
23-Apr-2014.) (Revised by Jim Kingdon, 8-Oct-2022.)
|
⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝐷 = 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ) & ⊢ (𝜑 → ¬ 𝑦 ∈ 𝑥)
& ⊢ (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
& ⊢ (𝜑 → 𝑥 ∈ Fin) & ⊢ (𝜓 ↔ Σ𝑗 ∈ 𝑥 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷) |
|
Theorem | fsum2d 11474* |
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)
& ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷) |
|
Theorem | fsumxp 11475* |
Combine two sums into a single sum over the cartesian product.
(Contributed by Mario Carneiro, 23-Apr-2014.)
|
⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝐷 = 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ (𝐴 × 𝐵)𝐷) |
|
Theorem | fsumcnv 11476* |
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)
& ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → Rel 𝐴)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ◡ 𝐴𝐶) |
|
Theorem | fisumcom2 11477* |
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.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐷 ∈ Fin) & ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))) & ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑘 ∈ 𝐶 Σ𝑗 ∈ 𝐷 𝐸) |
|
Theorem | fsumcom 11478* |
Interchange order of summation. (Contributed by NM, 15-Nov-2005.)
(Revised by Mario Carneiro, 23-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑘 ∈ 𝐵 Σ𝑗 ∈ 𝐴 𝐶) |
|
Theorem | fsum0diaglem 11479* |
Lemma for fisum0diag 11480. (Contributed by Mario Carneiro,
28-Apr-2014.)
(Revised by Mario Carneiro, 8-Apr-2016.)
|
⊢ ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → (𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 𝑘)))) |
|
Theorem | fisum0diag 11480* |
Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular
region 𝑀 ≤ 𝑗, 𝑀 ≤ 𝑘, 𝑗 + 𝑘 ≤ 𝑁". (Contributed by NM,
31-Dec-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
(Revised by Mario Carneiro, 8-Apr-2016.)
|
⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗)))) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℤ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...(𝑁 − 𝑘))𝐴) |
|
Theorem | mptfzshft 11481* |
1-1 onto function in maps-to notation which shifts a finite set of
sequential integers. (Contributed by AV, 24-Aug-2019.)
|
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ)
⇒ ⊢ (𝜑 → (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑗 − 𝐾)):((𝑀 + 𝐾)...(𝑁 + 𝐾))–1-1-onto→(𝑀...𝑁)) |
|
Theorem | fsumrev 11482* |
Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised
by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑗 = (𝐾 − 𝑘) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝐾 − 𝑁)...(𝐾 − 𝑀))𝐵) |
|
Theorem | fsumshft 11483* |
Index shift of a finite sum. (Contributed by NM, 27-Nov-2005.)
(Revised by Mario Carneiro, 24-Apr-2014.) (Proof shortened by AV,
8-Sep-2019.)
|
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑗 = (𝑘 − 𝐾) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵) |
|
Theorem | fsumshftm 11484* |
Negative index shift of a finite sum. (Contributed by NM,
28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑗 = (𝑘 + 𝐾) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
|
Theorem | fisumrev2 11485* |
Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised
by Mario Carneiro, 13-Apr-2016.)
|
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑗 = ((𝑀 + 𝑁) − 𝑘) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵) |
|
Theorem | fisum0diag2 11486* |
Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular
region 0 ≤ 𝑗, 0 ≤ 𝑘, 𝑗 + 𝑘 ≤ 𝑁". (Contributed by
Mario Carneiro, 21-Jul-2014.)
|
⊢ (𝑥 = 𝑘 → 𝐵 = 𝐴)
& ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = 𝐶)
& ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗)))) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℤ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶) |
|
Theorem | fsummulc2 11487* |
A finite sum multiplied by a constant. (Contributed by NM,
12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)) |
|
Theorem | fsummulc1 11488* |
A finite sum multiplied by a constant. (Contributed by NM,
13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 · 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶)) |
|
Theorem | fsumdivapc 11489* |
A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006.)
(Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 / 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 / 𝐶)) |
|
Theorem | fsumneg 11490* |
Negation of a finite sum. (Contributed by Scott Fenton, 12-Jun-2013.)
(Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 -𝐵 = -Σ𝑘 ∈ 𝐴 𝐵) |
|
Theorem | fsumsub 11491* |
Split a finite sum over a subtraction. (Contributed by Scott Fenton,
12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 − 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 − Σ𝑘 ∈ 𝐴 𝐶)) |
|
Theorem | fsum2mul 11492* |
Separate the nested sum of the product 𝐶(𝑗) · 𝐷(𝑘).
(Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro,
24-Apr-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 (𝐶 · 𝐷) = (Σ𝑗 ∈ 𝐴 𝐶 · Σ𝑘 ∈ 𝐵 𝐷)) |
|
Theorem | fsumconst 11493* |
The sum of constant terms (𝑘 is not free in 𝐵). (Contributed
by NM, 24-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵)) |
|
Theorem | fsumdifsnconst 11494* |
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 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → Σ𝑘 ∈ (𝐴 ∖ {𝐵})𝐶 = (((♯‘𝐴) − 1) · 𝐶)) |
|
Theorem | modfsummodlem1 11495* |
Lemma for modfsummod 11497. (Contributed by Alexander van der Vekens,
1-Sep-2018.)
|
⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) |
|
Theorem | modfsummodlemstep 11496* |
Induction step for modfsummod 11497. (Contributed by Alexander van der
Vekens, 1-Sep-2018.) (Revised by Jim Kingdon, 12-Oct-2022.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) & ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
& ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 mod 𝑁) = (Σ𝑘 ∈ 𝐴 (𝐵 mod 𝑁) mod 𝑁)) ⇒ ⊢ (𝜑 → (Σ𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 mod 𝑁) = (Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) mod 𝑁)) |
|
Theorem | modfsummod 11497* |
A finite sum modulo a positive integer equals the finite sum of their
summands modulo the positive integer, modulo the positive integer.
(Contributed by Alexander van der Vekens, 1-Sep-2018.)
|
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ)
⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 mod 𝑁) = (Σ𝑘 ∈ 𝐴 (𝐵 mod 𝑁) mod 𝑁)) |
|
Theorem | fsumge0 11498* |
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.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
|
Theorem | fsumlessfi 11499* |
A shorter sum of nonnegative terms is no greater than a longer one.
(Contributed by NM, 26-Dec-2005.) (Revised by Jim Kingdon,
12-Oct-2022.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)
& ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
& ⊢ (𝜑 → 𝐶 ∈ Fin) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
|
Theorem | fsumge1 11500* |
A sum of nonnegative numbers is greater than or equal to any one of
its terms. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof
shortened by Mario Carneiro, 4-Jun-2014.)
|
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)
& ⊢ (𝑘 = 𝑀 → 𝐵 = 𝐶)
& ⊢ (𝜑 → 𝑀 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐶 ≤ Σ𝑘 ∈ 𝐴 𝐵) |