Theorem List for Intuitionistic Logic Explorer - 11201-11300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | xrminrpcl 11201 |
The minimum of two positive reals is a positive real. (Contributed by Jim
Kingdon, 4-May-2023.)
|
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+)
→ inf({𝐴, 𝐵}, ℝ*, < )
∈ ℝ+) |
|
Theorem | xrminadd 11202 |
Distributing addition over minimum. (Contributed by Jim Kingdon,
10-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → inf({(𝐴 +𝑒 𝐵), (𝐴 +𝑒 𝐶)}, ℝ*, < ) = (𝐴 +𝑒
inf({𝐵, 𝐶}, ℝ*, <
))) |
|
Theorem | xrbdtri 11203 |
Triangle inequality for bounded values. (Contributed by Jim Kingdon,
15-May-2023.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ*
∧ 0 ≤ 𝐵) ∧
(𝐶 ∈
ℝ* ∧ 0 < 𝐶)) → inf({(𝐴 +𝑒 𝐵), 𝐶}, ℝ*, < ) ≤
(inf({𝐴, 𝐶}, ℝ*, < )
+𝑒 inf({𝐵, 𝐶}, ℝ*, <
))) |
|
Theorem | iooinsup 11204 |
Intersection of two open intervals of extended reals. (Contributed by
NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
∧ (𝐶 ∈
ℝ* ∧ 𝐷 ∈ ℝ*)) →
((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = (sup({𝐴, 𝐶}, ℝ*, < )(,)inf({𝐵, 𝐷}, ℝ*, <
))) |
|
4.8 Elementary limits and
convergence
|
|
4.8.1 Limits
|
|
Syntax | cli 11205 |
Extend class notation with convergence relation for limits.
|
class ⇝ |
|
Definition | df-clim 11206* |
Define the limit relation for complex number sequences. See clim 11208
for
its relational expression. (Contributed by NM, 28-Aug-2005.)
|
⊢ ⇝ = {〈𝑓, 𝑦〉 ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈ ℤ
∀𝑘 ∈
(ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))} |
|
Theorem | climrel 11207 |
The limit relation is a relation. (Contributed by NM, 28-Aug-2005.)
(Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ Rel ⇝ |
|
Theorem | clim 11208* |
Express the predicate: The limit of complex number sequence 𝐹 is
𝐴, or 𝐹 converges to 𝐴. This
means that for any real
𝑥, no matter how small, there always
exists an integer 𝑗 such
that the absolute difference of any later complex number in the sequence
and the limit is less than 𝑥. (Contributed by NM, 28-Aug-2005.)
(Revised by Mario Carneiro, 28-Apr-2015.)
|
⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈ ℤ
∀𝑘 ∈
(ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))) |
|
Theorem | climcl 11209 |
Closure of the limit of a sequence of complex numbers. (Contributed by
NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|
⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) |
|
Theorem | clim2 11210* |
Express the predicate: The limit of complex number sequence 𝐹 is
𝐴, or 𝐹 converges to 𝐴, with
more general quantifier
restrictions than clim 11208. (Contributed by NM, 6-Jan-2007.) (Revised
by Mario Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))) |
|
Theorem | clim2c 11211* |
Express the predicate 𝐹 converges to 𝐴. (Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
& ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 𝐴)) < 𝑥)) |
|
Theorem | clim0 11212* |
Express the predicate 𝐹 converges to 0. (Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘𝐵) < 𝑥))) |
|
Theorem | clim0c 11213* |
Express the predicate 𝐹 converges to 0. (Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝑥)) |
|
Theorem | climi 11214* |
Convergence of a sequence of complex numbers. (Contributed by NM,
11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
& ⊢ (𝜑 → 𝐹 ⇝ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶)) |
|
Theorem | climi2 11215* |
Convergence of a sequence of complex numbers. (Contributed by NM,
11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
& ⊢ (𝜑 → 𝐹 ⇝ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 𝐴)) < 𝐶) |
|
Theorem | climi0 11216* |
Convergence of a sequence of complex numbers to zero. (Contributed by
NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
& ⊢ (𝜑 → 𝐹 ⇝ 0) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝐶) |
|
Theorem | climconst 11217* |
An (eventually) constant sequence converges to its value. (Contributed
by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) ⇒ ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
|
Theorem | climconst2 11218 |
A constant sequence converges to its value. (Contributed by NM,
6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ (ℤ≥‘𝑀) ⊆ 𝑍
& ⊢ 𝑍 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴}) ⇝ 𝐴) |
|
Theorem | climz 11219 |
The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.)
(Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ (ℤ × {0}) ⇝
0 |
|
Theorem | climuni 11220 |
An infinite sequence of complex numbers converges to at most one limit.
(Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro,
31-Jan-2014.)
|
⊢ ((𝐹 ⇝ 𝐴 ∧ 𝐹 ⇝ 𝐵) → 𝐴 = 𝐵) |
|
Theorem | fclim 11221 |
The limit relation is function-like, and with range the complex numbers.
(Contributed by Mario Carneiro, 31-Jan-2014.)
|
⊢ ⇝ :dom ⇝
⟶ℂ |
|
Theorem | climdm 11222 |
Two ways to express that a function has a limit. (The expression
( ⇝ ‘𝐹) is sometimes useful as a shorthand
for "the unique limit
of the function 𝐹"). (Contributed by Mario
Carneiro,
18-Mar-2014.)
|
⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹)) |
|
Theorem | climeu 11223* |
An infinite sequence of complex numbers converges to at most one limit.
(Contributed by NM, 25-Dec-2005.)
|
⊢ (𝐹 ⇝ 𝐴 → ∃!𝑥 𝐹 ⇝ 𝑥) |
|
Theorem | climreu 11224* |
An infinite sequence of complex numbers converges to at most one limit.
(Contributed by NM, 25-Dec-2005.)
|
⊢ (𝐹 ⇝ 𝐴 → ∃!𝑥 ∈ ℂ 𝐹 ⇝ 𝑥) |
|
Theorem | climmo 11225* |
An infinite sequence of complex numbers converges to at most one limit.
(Contributed by Mario Carneiro, 13-Jul-2013.)
|
⊢ ∃*𝑥 𝐹 ⇝ 𝑥 |
|
Theorem | climeq 11226* |
Two functions that are eventually equal to one another have the same
limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario
Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
|
Theorem | climmpt 11227* |
Exhibit a function 𝐺 with the same convergence properties
as the
not-quite-function 𝐹. (Contributed by Mario Carneiro,
31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
|
Theorem | 2clim 11228* |
If two sequences converge to each other, they converge to the same
limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario
Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐺‘𝑘))) < 𝑥)
& ⊢ (𝜑 → 𝐹 ⇝ 𝐴) ⇒ ⊢ (𝜑 → 𝐺 ⇝ 𝐴) |
|
Theorem | climshftlemg 11229 |
A shifted function converges if the original function converges.
(Contributed by Mario Carneiro, 5-Nov-2013.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 → (𝐹 shift 𝑀) ⇝ 𝐴)) |
|
Theorem | climres 11230 |
A function restricted to upper integers converges iff the original
function converges. (Contributed by Mario Carneiro, 13-Jul-2013.)
(Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → ((𝐹 ↾
(ℤ≥‘𝑀)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
|
Theorem | climshft 11231 |
A shifted function converges iff the original function converges.
(Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro,
31-Jan-2014.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → ((𝐹 shift 𝑀) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
|
Theorem | serclim0 11232 |
The zero series converges to zero. (Contributed by Paul Chapman,
9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
|
⊢ (𝑀 ∈ ℤ → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝
0) |
|
Theorem | climshft2 11233* |
A shifted function converges iff the original function converges.
(Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario
Carneiro, 6-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑊)
& ⊢ (𝜑 → 𝐺 ∈ 𝑋)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 𝐾)) = (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
|
Theorem | climabs0 11234* |
Convergence to zero of the absolute value is equivalent to convergence
to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro,
31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘))) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ 𝐺 ⇝ 0)) |
|
Theorem | climcn1 11235* |
Image of a limit under a continuous map. (Contributed by Mario
Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) ∈ ℂ) & ⊢ (𝜑 → 𝐺 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐻 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℝ+ ∀𝑧 ∈ 𝐵 ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐴))) < 𝑥))
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘))) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐹‘𝐴)) |
|
Theorem | climcn2 11236* |
Image of a limit under a continuous map, two-arg version. (Contributed
by Mario Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝐶)
& ⊢ (𝜑 → 𝐵 ∈ 𝐷)
& ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐶 ∧ 𝑣 ∈ 𝐷)) → (𝑢𝐹𝑣) ∈ ℂ) & ⊢ (𝜑 → 𝐺 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐻 ⇝ 𝐵)
& ⊢ (𝜑 → 𝐾 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥))
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐶)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) ∈ 𝐷)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾‘𝑘) = ((𝐺‘𝑘)𝐹(𝐻‘𝑘))) ⇒ ⊢ (𝜑 → 𝐾 ⇝ (𝐴𝐹𝐵)) |
|
Theorem | addcn2 11237* |
Complex number addition is a continuous function. Part of Proposition
14-4.16 of [Gleason] p. 243. (We write
out the definition directly
because df-cn and df-cncf are not yet available to us. See addcncntop 13099
for the abbreviated version.) (Contributed by Mario Carneiro,
31-Jan-2014.)
|
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) →
∃𝑦 ∈
ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ
(((abs‘(𝑢 −
𝐵)) < 𝑦 ∧ (abs‘(𝑣 − 𝐶)) < 𝑧) → (abs‘((𝑢 + 𝑣) − (𝐵 + 𝐶))) < 𝐴)) |
|
Theorem | subcn2 11238* |
Complex number subtraction is a continuous function. Part of
Proposition 14-4.16 of [Gleason] p. 243.
(Contributed by Mario
Carneiro, 31-Jan-2014.)
|
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) →
∃𝑦 ∈
ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ
(((abs‘(𝑢 −
𝐵)) < 𝑦 ∧ (abs‘(𝑣 − 𝐶)) < 𝑧) → (abs‘((𝑢 − 𝑣) − (𝐵 − 𝐶))) < 𝐴)) |
|
Theorem | mulcn2 11239* |
Complex number multiplication is a continuous function. Part of
Proposition 14-4.16 of [Gleason] p. 243.
(Contributed by Mario
Carneiro, 31-Jan-2014.)
|
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) →
∃𝑦 ∈
ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ
(((abs‘(𝑢 −
𝐵)) < 𝑦 ∧ (abs‘(𝑣 − 𝐶)) < 𝑧) → (abs‘((𝑢 · 𝑣) − (𝐵 · 𝐶))) < 𝐴)) |
|
Theorem | reccn2ap 11240* |
The reciprocal function is continuous. The class 𝑇 is just for
convenience in writing the proof and typically would be passed in as an
instance of eqid 2164. (Contributed by Mario Carneiro,
9-Feb-2014.)
Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
|
⊢ 𝑇 = (inf({1, ((abs‘𝐴) · 𝐵)}, ℝ, < ) ·
((abs‘𝐴) /
2)) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝐵 ∈ ℝ+) →
∃𝑦 ∈
ℝ+ ∀𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵)) |
|
Theorem | cn1lem 11241* |
A sufficient condition for a function to be continuous. (Contributed by
Mario Carneiro, 9-Feb-2014.)
|
⊢ 𝐹:ℂ⟶ℂ & ⊢ ((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℂ) →
(abs‘((𝐹‘𝑧) − (𝐹‘𝐴))) ≤ (abs‘(𝑧 − 𝐴))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐴))) < 𝑥)) |
|
Theorem | abscn2 11242* |
The absolute value function is continuous. (Contributed by Mario
Carneiro, 9-Feb-2014.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((abs‘𝑧) − (abs‘𝐴))) < 𝑥)) |
|
Theorem | cjcn2 11243* |
The complex conjugate function is continuous. (Contributed by Mario
Carneiro, 9-Feb-2014.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((∗‘𝑧) − (∗‘𝐴))) < 𝑥)) |
|
Theorem | recn2 11244* |
The real part function is continuous. (Contributed by Mario Carneiro,
9-Feb-2014.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((ℜ‘𝑧) − (ℜ‘𝐴))) < 𝑥)) |
|
Theorem | imcn2 11245* |
The imaginary part function is continuous. (Contributed by Mario
Carneiro, 9-Feb-2014.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((ℑ‘𝑧) − (ℑ‘𝐴))) < 𝑥)) |
|
Theorem | climcn1lem 11246* |
The limit of a continuous function, theorem form. (Contributed by
Mario Carneiro, 9-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ 𝐻:ℂ⟶ℂ & ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+)
→ ∃𝑦 ∈
ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐻‘𝑧) − (𝐻‘𝐴))) < 𝑥))
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐻‘(𝐹‘𝑘))) ⇒ ⊢ (𝜑 → 𝐺 ⇝ (𝐻‘𝐴)) |
|
Theorem | climabs 11247* |
Limit of the absolute value of a sequence. Proposition 12-2.4(c) of
[Gleason] p. 172. (Contributed by NM,
7-Jun-2006.) (Revised by Mario
Carneiro, 9-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘))) ⇒ ⊢ (𝜑 → 𝐺 ⇝ (abs‘𝐴)) |
|
Theorem | climcj 11248* |
Limit of the complex conjugate of a sequence. Proposition 12-2.4(c)
of [Gleason] p. 172. (Contributed by
NM, 7-Jun-2006.) (Revised by
Mario Carneiro, 9-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (∗‘(𝐹‘𝑘))) ⇒ ⊢ (𝜑 → 𝐺 ⇝ (∗‘𝐴)) |
|
Theorem | climre 11249* |
Limit of the real part of a sequence. Proposition 12-2.4(c) of
[Gleason] p. 172. (Contributed by NM,
7-Jun-2006.) (Revised by Mario
Carneiro, 9-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (ℜ‘(𝐹‘𝑘))) ⇒ ⊢ (𝜑 → 𝐺 ⇝ (ℜ‘𝐴)) |
|
Theorem | climim 11250* |
Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of
[Gleason] p. 172. (Contributed by NM,
7-Jun-2006.) (Revised by Mario
Carneiro, 9-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (ℑ‘(𝐹‘𝑘))) ⇒ ⊢ (𝜑 → 𝐺 ⇝ (ℑ‘𝐴)) |
|
Theorem | climrecl 11251* |
The limit of a convergent real sequence is real. Corollary 12-2.5 of
[Gleason] p. 172. (Contributed by NM,
10-Sep-2005.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) |
|
Theorem | climge0 11252* |
A nonnegative sequence converges to a nonnegative number. (Contributed
by NM, 11-Sep-2005.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 0 ≤ 𝐴) |
|
Theorem | climadd 11253* |
Limit of the sum of two converging sequences. Proposition 12-2.1(a)
of [Gleason] p. 168. (Contributed
by NM, 24-Sep-2005.) (Proof
shortened by Mario Carneiro, 31-Jan-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐻 ∈ 𝑋)
& ⊢ (𝜑 → 𝐺 ⇝ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐴 + 𝐵)) |
|
Theorem | climmul 11254* |
Limit of the product of two converging sequences. Proposition
12-2.1(c) of [Gleason] p. 168.
(Contributed by NM, 27-Dec-2005.)
(Proof shortened by Mario Carneiro, 1-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐻 ∈ 𝑋)
& ⊢ (𝜑 → 𝐺 ⇝ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵)) |
|
Theorem | climsub 11255* |
Limit of the difference of two converging sequences. Proposition
12-2.1(b) of [Gleason] p. 168.
(Contributed by NM, 4-Aug-2007.)
(Proof shortened by Mario Carneiro, 1-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐻 ∈ 𝑋)
& ⊢ (𝜑 → 𝐺 ⇝ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐴 − 𝐵)) |
|
Theorem | climaddc1 11256* |
Limit of a constant 𝐶 added to each term of a sequence.
(Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro,
3-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) + 𝐶)) ⇒ ⊢ (𝜑 → 𝐺 ⇝ (𝐴 + 𝐶)) |
|
Theorem | climaddc2 11257* |
Limit of a constant 𝐶 added to each term of a sequence.
(Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro,
3-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 + (𝐹‘𝑘))) ⇒ ⊢ (𝜑 → 𝐺 ⇝ (𝐶 + 𝐴)) |
|
Theorem | climmulc2 11258* |
Limit of a sequence multiplied by a constant 𝐶. Corollary
12-2.2 of [Gleason] p. 171.
(Contributed by NM, 24-Sep-2005.)
(Revised by Mario Carneiro, 3-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) ⇒ ⊢ (𝜑 → 𝐺 ⇝ (𝐶 · 𝐴)) |
|
Theorem | climsubc1 11259* |
Limit of a constant 𝐶 subtracted from each term of a
sequence.
(Contributed by Mario Carneiro, 9-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) − 𝐶)) ⇒ ⊢ (𝜑 → 𝐺 ⇝ (𝐴 − 𝐶)) |
|
Theorem | climsubc2 11260* |
Limit of a constant 𝐶 minus each term of a sequence.
(Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro,
9-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 − (𝐹‘𝑘))) ⇒ ⊢ (𝜑 → 𝐺 ⇝ (𝐶 − 𝐴)) |
|
Theorem | climle 11261* |
Comparison of the limits of two sequences. (Contributed by Paul
Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 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 | climsqz 11262* |
Convergence of a sequence sandwiched between another converging
sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by
Mario Carneiro, 3-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≤ 𝐴) ⇒ ⊢ (𝜑 → 𝐺 ⇝ 𝐴) |
|
Theorem | climsqz2 11263* |
Convergence of a sequence sandwiched between another converging
sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised
by Mario Carneiro, 3-Feb-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
& ⊢ (𝜑 → 𝐺 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≤ (𝐹‘𝑘))
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → 𝐺 ⇝ 𝐴) |
|
Theorem | clim2ser 11264* |
The limit of an infinite series with an initial segment removed.
(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 → seq𝑀( + , 𝐹) ⇝ 𝐴) ⇒ ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ (𝐴 − (seq𝑀( + , 𝐹)‘𝑁))) |
|
Theorem | clim2ser2 11265* |
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 11266* |
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 11267* |
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 11268* |
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 11269* |
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 11270* |
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 11271* |
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 11272* |
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 11273* |
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 11274* |
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 11277). 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 11275* |
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 11276* |
Lemma for climcvg1n 11277. 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 11277* |
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 11278* |
A converging sequence of complex numbers is a Cauchy sequence. This is
like climcau 11274 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 11279* |
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 11280 |
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 11281* |
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 11449). (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 11282 |
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 11283 |
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 11284 |
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 11285* |
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 11286* |
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 11287 |
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 11288* |
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 11289* |
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 11290* |
Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶 |
|
Theorem | sumeq2i 11291* |
Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
|
⊢ (𝑘 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
|
Theorem | sumeq12i 11292* |
Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
|
⊢ 𝐴 = 𝐵
& ⊢ (𝑘 ∈ 𝐴 → 𝐶 = 𝐷) ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷 |
|
Theorem | sumeq1d 11293* |
Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
|
Theorem | sumeq2d 11294* |
Equality deduction for sum. Note that unlike sumeq2dv 11295, 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 11295* |
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 11296* |
Equality deduction for sum. (Contributed by Glauco Siliprandi,
5-Apr-2020.)
|
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
|
Theorem | sumeq2sdv 11297* |
Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)
|
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
|
Theorem | 2sumeq2dv 11298* |
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 11299* |
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 11300* |
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) |