Theorem List for Intuitionistic Logic Explorer - 11301-11400 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | clim2ser2 11301* |
The limit of an infinite series with an initial segment added.
(Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario
Carneiro, 1-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ 𝐴) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (𝐴 + (seq𝑀( + , 𝐹)‘𝑁))) |
|
Theorem | iserex 11302* |
An infinite series converges, if and only if the series does with
initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.)
(Revised by Mario Carneiro, 27-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ )) |
|
Theorem | isermulc2 11303* |
Multiplication of an infinite series by a constant. (Contributed by
Paul Chapman, 14-Nov-2007.) (Revised by Jim Kingdon, 8-Apr-2023.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ (𝐶 · 𝐴)) |
|
Theorem | climlec2 11304* |
Comparison of a constant to the limit of a sequence. (Contributed by
NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
|
Theorem | iserle 11305* |
Comparison of the limits of two infinite series. (Contributed by Paul
Chapman, 12-Nov-2007.) (Revised by Mario Carneiro, 3-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)
& ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
|
Theorem | iserge0 11306* |
The limit of an infinite series of nonnegative reals is nonnegative.
(Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario
Carneiro, 3-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 0 ≤ 𝐴) |
|
Theorem | climub 11307* |
The limit of a monotonic sequence is an upper bound. (Contributed by
NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍)
& ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) ⇒ ⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝐴) |
|
Theorem | climserle 11308* |
The partial sums of a converging infinite series with nonnegative
terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.)
(Revised by Mario Carneiro, 9-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍)
& ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ 𝐴) |
|
Theorem | iser3shft 11309* |
Index shift of the limit of an infinite series. (Contributed by Mario
Carneiro, 6-Sep-2013.) (Revised by Jim Kingdon, 17-Oct-2022.)
|
⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴)) |
|
Theorem | climcau 11310* |
A converging sequence of complex numbers is a Cauchy sequence. The
converse would require excluded middle or a different definition of
Cauchy sequence (for example, fixing a rate of convergence as in
climcvg1n 11313). Theorem 12-5.3 of [Gleason] p. 180 (necessity part).
(Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro,
26-Apr-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀)
⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) |
|
Theorem | climrecvg1n 11311* |
A Cauchy sequence of real numbers converges, existence version. The
rate of convergence is fixed: all terms after the nth term must be
within 𝐶 / 𝑛 of the nth term, where 𝐶 is a
constant multiplier.
(Contributed by Jim Kingdon, 23-Aug-2021.)
|
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑛)(abs‘((𝐹‘𝑘) − (𝐹‘𝑛))) < (𝐶 / 𝑛)) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
|
Theorem | climcvg1nlem 11312* |
Lemma for climcvg1n 11313. We construct sequences of the real and
imaginary parts of each term of 𝐹, show those converge, and use
that to show that 𝐹 converges. (Contributed by Jim
Kingdon,
24-Aug-2021.)
|
⊢ (𝜑 → 𝐹:ℕ⟶ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑛)(abs‘((𝐹‘𝑘) − (𝐹‘𝑛))) < (𝐶 / 𝑛))
& ⊢ 𝐺 = (𝑥 ∈ ℕ ↦ (ℜ‘(𝐹‘𝑥))) & ⊢ 𝐻 = (𝑥 ∈ ℕ ↦
(ℑ‘(𝐹‘𝑥))) & ⊢ 𝐽 = (𝑥 ∈ ℕ ↦ (i · (𝐻‘𝑥))) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
|
Theorem | climcvg1n 11313* |
A Cauchy sequence of complex numbers converges, existence version.
The rate of convergence is fixed: all terms after the nth term must be
within 𝐶 / 𝑛 of the nth term, where 𝐶 is a
constant
multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
|
⊢ (𝜑 → 𝐹:ℕ⟶ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑛)(abs‘((𝐹‘𝑘) − (𝐹‘𝑛))) < (𝐶 / 𝑛)) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
|
Theorem | climcaucn 11314* |
A converging sequence of complex numbers is a Cauchy sequence. This is
like climcau 11310 but adds the part that (𝐹‘𝑘) is complex.
(Contributed by Jim Kingdon, 24-Aug-2021.)
|
⊢ 𝑍 = (ℤ≥‘𝑀)
⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) |
|
Theorem | serf0 11315* |
If an infinite series converges, its underlying sequence converges to
zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro,
16-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
⇒ ⊢ (𝜑 → 𝐹 ⇝ 0) |
|
4.8.2 Finite and infinite sums
|
|
Syntax | csu 11316 |
Extend class notation to include finite summations. (An underscore was
added to the ASCII token in order to facilitate set.mm text searches,
since "sum" is a commonly used word in comments.)
|
class Σ𝑘 ∈ 𝐴 𝐵 |
|
Definition | df-sumdc 11317* |
Define the sum of a series with an index set of integers 𝐴. The
variable 𝑘 is normally a free variable in 𝐵, i.e.,
𝐵
can be
thought of as 𝐵(𝑘). This definition is the result of a
collection of discussions over the most general definition for a sum
that does not need the index set to have a specified ordering. This
definition is in two parts, one for finite sums and one for subsets of
the upper integers. When summing over a subset of the upper integers,
we extend the index set to the upper integers by adding zero outside the
domain, and then sum the set in order, setting the result to the limit
of the partial sums, if it exists. This means that conditionally
convergent sums can be evaluated meaningfully. For finite sums, we are
explicitly order-independent, by picking any bijection to a 1-based
finite sequence and summing in the induced order. In both cases we have
an if expression so that we only need 𝐵 to be
defined where
𝑘
∈ 𝐴. In the
infinite case, we also require that the indexing
set be a decidable subset of an upperset of integers (that is,
membership of integers in it is decidable). These two methods of
summation produce the same result on their common region of definition
(i.e., finite sets of integers). Examples:
Σ𝑘 ∈ {1, 2, 4}𝑘 means 1 + 2 + 4 =
7, and
Σ𝑘 ∈ ℕ(1 / (2↑𝑘)) = 1 means 1/2 + 1/4 +
1/8 + ... = 1
(geoihalfsum 11485). (Contributed by NM, 11-Dec-2005.)
(Revised by Jim
Kingdon, 21-May-2023.)
|
⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)))) |
|
Theorem | sumeq1 11318 |
Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised
by Mario Carneiro, 13-Jun-2019.)
|
⊢ (𝐴 = 𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
|
Theorem | nfsum1 11319 |
Bound-variable hypothesis builder for sum. (Contributed by NM,
11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
|
⊢ Ⅎ𝑘𝐴 ⇒ ⊢ Ⅎ𝑘Σ𝑘 ∈ 𝐴 𝐵 |
|
Theorem | nfsum 11320 |
Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not
free in 𝐴 and 𝐵, it is not free in Σ𝑘 ∈
𝐴𝐵.
(Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro,
13-Jun-2019.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵 |
|
Theorem | sumdc 11321* |
Decidability of a subset of upper integers. (Contributed by Jim
Kingdon, 1-Jan-2022.)
|
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ⊆
(ℤ≥‘𝑀)) & ⊢ (𝜑 → ∀𝑥 ∈
(ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴)
& ⊢ (𝜑 → 𝑁 ∈ ℤ)
⇒ ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴) |
|
Theorem | sumeq2 11322* |
Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised
by Mario Carneiro, 13-Jul-2013.)
|
⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
|
Theorem | cbvsum 11323 |
Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
(Revised by Mario Carneiro, 13-Jun-2019.)
|
⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
& ⊢ Ⅎ𝑘𝐴
& ⊢ Ⅎ𝑗𝐴
& ⊢ Ⅎ𝑘𝐵
& ⊢ Ⅎ𝑗𝐶 ⇒ ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
|
Theorem | cbvsumv 11324* |
Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
(Revised by Mario Carneiro, 13-Jul-2013.)
|
⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) ⇒ ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
|
Theorem | cbvsumi 11325* |
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 11326* |
Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶 |
|
Theorem | sumeq2i 11327* |
Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
|
⊢ (𝑘 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
|
Theorem | sumeq12i 11328* |
Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
|
⊢ 𝐴 = 𝐵
& ⊢ (𝑘 ∈ 𝐴 → 𝐶 = 𝐷) ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷 |
|
Theorem | sumeq1d 11329* |
Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
|
Theorem | sumeq2d 11330* |
Equality deduction for sum. Note that unlike sumeq2dv 11331, 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 11331* |
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 11332* |
Equality deduction for sum. (Contributed by Glauco Siliprandi,
5-Apr-2020.)
|
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
|
Theorem | sumeq2sdv 11333* |
Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)
|
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
|
Theorem | 2sumeq2dv 11334* |
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 11335* |
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 11336* |
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 11337* |
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 11338* |
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 11339 |
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 11340* |
Lemma for sumrbdc 11342. (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 11341* |
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 11342* |
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 11343* |
Lemma for summodc 11346. (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 11344* |
Lemma for summodc 11346. (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 11345* |
Lemma for summodc 11346. (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 11346* |
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 11347* |
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 11348* |
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 11349* |
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 11350* |
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 11351 |
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 11352* |
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 11353* |
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 11354* |
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 11355* |
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 11356* |
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 11357* |
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 11358* |
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 11359* |
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 11360* |
A finite sum expressed in terms of a partial sum of an infinite series.
The recursive definition follows as fsum1 11375 and fsump1 11383, which should
make our notation clear and from which, along with closure fsumcl 11363, 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 11361* |
- 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 11362* |
- 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 11363* |
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 11364* |
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 11365* |
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 11366* |
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 11367* |
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 11368* |
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 11369* |
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 11370* |
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 11371* |
Split a sum into two parts. A version of fsumsplit 11370 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 11372* |
A sum of a singleton is the term. A version of sumsn 11374 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 11373* |
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 11374* |
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 11375* |
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 11376* |
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 11377* |
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 11378* |
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 11379* |
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 11380* |
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 11381* |
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 11382* |
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 11383* |
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 11384* |
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 11385* |
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 11386* |
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 11387* |
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 11388* |
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 11389* |
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 11390* |
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 11391* |
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 11392* |
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 11393* |
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 11394* |
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 11395* |
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 11396* |
Optimized version of fsump1 11383 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 11397* |
Lemma for fsum2d 11398- 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 11398* |
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 11399* |
Combine two sums into a single sum over the cartesian product.
(Contributed by Mario Carneiro, 23-Apr-2014.)
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⊢ (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) |
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Theorem | fsumcnv 11400* |
Transform a region of summation by using the converse operation.
(Contributed by Mario Carneiro, 23-Apr-2014.)
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⊢ (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) |