Theorem List for Intuitionistic Logic Explorer - 11001-11100 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | minclpr 11001 |
The minimum of two real numbers is one of those numbers if and only if
dichotomy (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴) holds. For example, this can be
combined with zletric 9091 if one is dealing with integers, but real
number
dichotomy in general does not follow from our axioms. (Contributed by Jim
Kingdon, 23-May-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (inf({𝐴, 𝐵}, ℝ, < ) ∈ {𝐴, 𝐵} ↔ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))) |
|
Theorem | rpmincl 11002 |
The minumum of two positive real numbers is a positive real number.
(Contributed by Jim Kingdon, 25-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+)
→ inf({𝐴, 𝐵}, ℝ, < ) ∈
ℝ+) |
|
Theorem | bdtrilem 11003 |
Lemma for bdtri 11004. (Contributed by Steven Nguyen and Jim
Kingdon,
17-May-2023.)
|
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) →
((abs‘(𝐴 −
𝐶)) + (abs‘(𝐵 − 𝐶))) ≤ (𝐶 + (abs‘((𝐴 + 𝐵) − 𝐶)))) |
|
Theorem | bdtri 11004 |
Triangle inequality for bounded values. (Contributed by Jim Kingdon,
15-May-2023.)
|
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) →
inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ) ≤ (inf({𝐴, 𝐶}, ℝ, < ) + inf({𝐵, 𝐶}, ℝ, < ))) |
|
Theorem | mul0inf 11005 |
Equality of a product with zero. A bit of a curiosity, in the sense that
theorems like abs00ap 10827 and mulap0bd 8411 may better express the ideas behind
it. (Contributed by Jim Kingdon, 31-Jul-2023.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 ↔ inf({(abs‘𝐴), (abs‘𝐵)}, ℝ, < ) = 0)) |
|
4.7.7 The maximum of two extended
reals
|
|
Theorem | xrmaxleim 11006 |
Value of maximum when we know which extended real is larger.
(Contributed by Jim Kingdon, 25-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴 ≤ 𝐵 → sup({𝐴, 𝐵}, ℝ*, < ) = 𝐵)) |
|
Theorem | xrmaxiflemcl 11007 |
Lemma for xrmaxif 11013. Closure. (Contributed by Jim Kingdon,
29-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ if(𝐵 = +∞,
+∞, if(𝐵 = -∞,
𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈
ℝ*) |
|
Theorem | xrmaxifle 11008 |
An upper bound for {𝐴, 𝐵} in the extended reals.
(Contributed by
Jim Kingdon, 26-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ 𝐴 ≤ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) |
|
Theorem | xrmaxiflemab 11009 |
Lemma for xrmaxif 11013. A variation of xrmaxleim 11006- that is, if we know
which of two real numbers is larger, we know the maximum of the two.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = 𝐵) |
|
Theorem | xrmaxiflemlub 11010 |
Lemma for xrmaxif 11013. A least upper bound for {𝐴, 𝐵}.
(Contributed by Jim Kingdon, 28-Apr-2023.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, <
)))))) ⇒ ⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)) |
|
Theorem | xrmaxiflemcom 11011 |
Lemma for xrmaxif 11013. Commutativity of an expression which we
will
later show to be the supremum. (Contributed by Jim Kingdon,
29-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ if(𝐵 = +∞,
+∞, if(𝐵 = -∞,
𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))) |
|
Theorem | xrmaxiflemval 11012* |
Lemma for xrmaxif 11013. Value of the supremum. (Contributed by
Jim
Kingdon, 29-Apr-2023.)
|
⊢ 𝑀 = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, <
))))) ⇒ ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝑀 ∈
ℝ* ∧ ∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝑀 < 𝑥 ∧ ∀𝑥 ∈ ℝ* (𝑥 < 𝑀 → ∃𝑧 ∈ {𝐴, 𝐵}𝑥 < 𝑧))) |
|
Theorem | xrmaxif 11013 |
Maximum of two extended reals in terms of if
expressions.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ sup({𝐴, 𝐵}, ℝ*, < )
= if(𝐵 = +∞,
+∞, if(𝐵 = -∞,
𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) |
|
Theorem | xrmaxcl 11014 |
The maximum of two extended reals is an extended real. (Contributed by
Jim Kingdon, 29-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ sup({𝐴, 𝐵}, ℝ*, < )
∈ ℝ*) |
|
Theorem | xrmax1sup 11015 |
An extended real is less than or equal to the maximum of it and another.
(Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
30-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ 𝐴 ≤ sup({𝐴, 𝐵}, ℝ*, <
)) |
|
Theorem | xrmax2sup 11016 |
An extended real is less than or equal to the maximum of it and another.
(Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
30-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ 𝐵 ≤ sup({𝐴, 𝐵}, ℝ*, <
)) |
|
Theorem | xrmaxrecl 11017 |
The maximum of two real numbers is the same when taken as extended reals
or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ*, < ) = sup({𝐴, 𝐵}, ℝ, < )) |
|
Theorem | xrmaxleastlt 11018 |
The maximum as a least upper bound, in terms of less than. (Contributed
by Jim Kingdon, 9-Feb-2022.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
∧ (𝐶 ∈
ℝ* ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ*, < ))) →
(𝐶 < 𝐴 ∨ 𝐶 < 𝐵)) |
|
Theorem | xrltmaxsup 11019 |
The maximum as a least upper bound. (Contributed by Jim Kingdon,
10-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (𝐶 < sup({𝐴, 𝐵}, ℝ*, < ) ↔
(𝐶 < 𝐴 ∨ 𝐶 < 𝐵))) |
|
Theorem | xrmaxltsup 11020 |
Two ways of saying the maximum of two numbers is less than a third.
(Contributed by Jim Kingdon, 30-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (sup({𝐴, 𝐵}, ℝ*, < ) < 𝐶 ↔ (𝐴 < 𝐶 ∧ 𝐵 < 𝐶))) |
|
Theorem | xrmaxlesup 11021 |
Two ways of saying the maximum of two numbers is less than or equal to a
third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim
Kingdon, 10-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (sup({𝐴, 𝐵}, ℝ*, < ) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶))) |
|
Theorem | xrmaxaddlem 11022 |
Lemma for xrmaxadd 11023. The case where 𝐴 is real. (Contributed
by
Jim Kingdon, 11-May-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)
→ sup({(𝐴
+𝑒 𝐵),
(𝐴 +𝑒
𝐶)}, ℝ*,
< ) = (𝐴
+𝑒 sup({𝐵, 𝐶}, ℝ*, <
))) |
|
Theorem | xrmaxadd 11023 |
Distributing addition over maximum. (Contributed by Jim Kingdon,
11-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → sup({(𝐴 +𝑒 𝐵), (𝐴 +𝑒 𝐶)}, ℝ*, < ) = (𝐴 +𝑒
sup({𝐵, 𝐶}, ℝ*, <
))) |
|
4.7.8 The minimum of two extended
reals
|
|
Theorem | xrnegiso 11024 |
Negation is an order anti-isomorphism of the extended reals, which is
its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
|
⊢ 𝐹 = (𝑥 ∈ ℝ* ↦
-𝑒𝑥) ⇒ ⊢ (𝐹 Isom < , ◡ < (ℝ*,
ℝ*) ∧ ◡𝐹 = 𝐹) |
|
Theorem | infxrnegsupex 11025* |
The infimum of a set of extended reals 𝐴 is the negative of the
supremum of the negatives of its elements. (Contributed by Jim Kingdon,
2-May-2023.)
|
⊢ (𝜑 → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝐴 ⊆
ℝ*) ⇒ ⊢ (𝜑 → inf(𝐴, ℝ*, < ) =
-𝑒sup({𝑧 ∈ ℝ* ∣
-𝑒𝑧
∈ 𝐴},
ℝ*, < )) |
|
Theorem | xrnegcon1d 11026 |
Contraposition law for extended real unary minus. (Contributed by Jim
Kingdon, 2-May-2023.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈
ℝ*) ⇒ ⊢ (𝜑 → (-𝑒𝐴 = 𝐵 ↔ -𝑒𝐵 = 𝐴)) |
|
Theorem | xrminmax 11027 |
Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
2-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ inf({𝐴, 𝐵}, ℝ*, < )
= -𝑒sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, <
)) |
|
Theorem | xrmincl 11028 |
The minumum of two extended reals is an extended real. (Contributed by
Jim Kingdon, 3-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ inf({𝐴, 𝐵}, ℝ*, < )
∈ ℝ*) |
|
Theorem | xrmin1inf 11029 |
The minimum of two extended reals is less than or equal to the first.
(Contributed by Jim Kingdon, 3-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ inf({𝐴, 𝐵}, ℝ*, < )
≤ 𝐴) |
|
Theorem | xrmin2inf 11030 |
The minimum of two extended reals is less than or equal to the second.
(Contributed by Jim Kingdon, 3-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ inf({𝐴, 𝐵}, ℝ*, < )
≤ 𝐵) |
|
Theorem | xrmineqinf 11031 |
The minimum of two extended reals is equal to the second if the first is
bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim
Kingdon, 3-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐵 ≤ 𝐴) → inf({𝐴, 𝐵}, ℝ*, < ) = 𝐵) |
|
Theorem | xrltmininf 11032 |
Two ways of saying an extended real is less than the minimum of two
others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
3-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (𝐴 < inf({𝐵, 𝐶}, ℝ*, < ) ↔
(𝐴 < 𝐵 ∧ 𝐴 < 𝐶))) |
|
Theorem | xrlemininf 11033 |
Two ways of saying a number is less than or equal to the minimum of two
others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim
Kingdon, 4-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ*, < ) ↔
(𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
|
Theorem | xrminltinf 11034 |
Two ways of saying an extended real is greater than the minimum of two
others. (Contributed by Jim Kingdon, 19-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (inf({𝐵, 𝐶}, ℝ*, < ) < 𝐴 ↔ (𝐵 < 𝐴 ∨ 𝐶 < 𝐴))) |
|
Theorem | xrminrecl 11035 |
The minimum of two real numbers is the same when taken as extended reals
or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ*, < ) = inf({𝐴, 𝐵}, ℝ, < )) |
|
Theorem | xrminrpcl 11036 |
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 11037 |
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 11038 |
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 11039 |
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 11040 |
Extend class notation with convergence relation for limits.
|
class ⇝ |
|
Definition | df-clim 11041* |
Define the limit relation for complex number sequences. See clim 11043
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 11042 |
The limit relation is a relation. (Contributed by NM, 28-Aug-2005.)
(Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ Rel ⇝ |
|
Theorem | clim 11043* |
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 11044 |
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 11045* |
Express the predicate: The limit of complex number sequence 𝐹 is
𝐴, or 𝐹 converges to 𝐴, with
more general quantifier
restrictions than clim 11043. (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 11046* |
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 11047* |
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 11048* |
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 11049* |
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 11050* |
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 11051* |
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 11052* |
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 11053 |
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 11054 |
The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.)
(Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ (ℤ × {0}) ⇝
0 |
|
Theorem | climuni 11055 |
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 11056 |
The limit relation is function-like, and with range the complex numbers.
(Contributed by Mario Carneiro, 31-Jan-2014.)
|
⊢ ⇝ :dom ⇝
⟶ℂ |
|
Theorem | climdm 11057 |
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 11058* |
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 11059* |
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 11060* |
An infinite sequence of complex numbers converges to at most one limit.
(Contributed by Mario Carneiro, 13-Jul-2013.)
|
⊢ ∃*𝑥 𝐹 ⇝ 𝑥 |
|
Theorem | climeq 11061* |
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 11062* |
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 11063* |
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 11064 |
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 11065 |
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 11066 |
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 11067 |
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 11068* |
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 11069* |
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 11070* |
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 11071* |
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 11072* |
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 12710
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 11073* |
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 11074* |
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 11075* |
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 2137. (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 11076* |
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 11077* |
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 11078* |
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 11079* |
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 11080* |
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 11081* |
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 11082* |
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 11083* |
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 11084* |
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 11085* |
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 11086* |
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 11087* |
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 11088* |
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 11089* |
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 11090* |
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 11091* |
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 11092* |
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 11093* |
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 11094* |
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 11095* |
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 11096* |
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 11097* |
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 11098* |
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 11099* |
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 11100* |
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))) |