Theorem List for Intuitionistic Logic Explorer - 11001-11100 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | abs2difd 11001 |
Difference of absolute values. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵))) |
|
Theorem | abs2dif2d 11002 |
Difference of absolute values. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵))) |
|
Theorem | abs2difabsd 11003 |
Absolute value of difference of absolute values. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵))) |
|
Theorem | abs3difd 11004 |
Absolute value of differences around common element. (Contributed by
Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵)))) |
|
Theorem | abs3lemd 11005 |
Lemma involving absolute value of differences. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (abs‘(𝐴 − 𝐶)) < (𝐷 / 2)) & ⊢ (𝜑 → (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) < 𝐷) |
|
Theorem | qdenre 11006* |
The rational numbers are dense in ℝ: any real
number can be
approximated with arbitrary precision by a rational number. For order
theoretic density, see qbtwnre 10065. (Contributed by BJ, 15-Oct-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) →
∃𝑥 ∈ ℚ
(abs‘(𝑥 −
𝐴)) < 𝐵) |
|
4.7.5 The maximum of two real
numbers
|
|
Theorem | maxcom 11007 |
The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10.
(Contributed by Jim Kingdon, 21-Dec-2021.)
|
⊢ sup({𝐴, 𝐵}, ℝ, < ) = sup({𝐵, 𝐴}, ℝ, < ) |
|
Theorem | maxabsle 11008 |
An upper bound for {𝐴, 𝐵}. (Contributed by Jim Kingdon,
20-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) |
|
Theorem | maxleim 11009 |
Value of maximum when we know which number is larger. (Contributed by
Jim Kingdon, 21-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 → sup({𝐴, 𝐵}, ℝ, < ) = 𝐵)) |
|
Theorem | maxabslemab 11010 |
Lemma for maxabs 11013. A variation of maxleim 11009- that is, if we know
which of two real numbers is larger, we know the maximum of the two.
(Contributed by Jim Kingdon, 21-Dec-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) = 𝐵) |
|
Theorem | maxabslemlub 11011 |
Lemma for maxabs 11013. A least upper bound for {𝐴, 𝐵}.
(Contributed by Jim Kingdon, 20-Dec-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) ⇒ ⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)) |
|
Theorem | maxabslemval 11012* |
Lemma for maxabs 11013. Value of the supremum. (Contributed by
Jim
Kingdon, 22-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) ∈ ℝ ∧ ∀𝑥 ∈ {𝐴, 𝐵} ¬ (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) < 𝑥 ∧ ∀𝑥 ∈ ℝ (𝑥 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) → ∃𝑧 ∈ {𝐴, 𝐵}𝑥 < 𝑧))) |
|
Theorem | maxabs 11013 |
Maximum of two real numbers in terms of absolute value. (Contributed by
Jim Kingdon, 20-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) |
|
Theorem | maxcl 11014 |
The maximum of two real numbers is a real number. (Contributed by Jim
Kingdon, 22-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ, < ) ∈
ℝ) |
|
Theorem | maxle1 11015 |
The maximum of two reals is no smaller than the first real. Lemma 3.10 of
[Geuvers], p. 10. (Contributed by Jim
Kingdon, 21-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ sup({𝐴, 𝐵}, ℝ, < )) |
|
Theorem | maxle2 11016 |
The maximum of two reals is no smaller than the second real. Lemma 3.10
of [Geuvers], p. 10. (Contributed by Jim
Kingdon, 21-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ sup({𝐴, 𝐵}, ℝ, < )) |
|
Theorem | maxleast 11017 |
The maximum of two reals is a least upper bound. Lemma 3.11 of
[Geuvers], p. 10. (Contributed by Jim
Kingdon, 22-Dec-2021.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶)) → sup({𝐴, 𝐵}, ℝ, < ) ≤ 𝐶) |
|
Theorem | maxleastb 11018 |
Two ways of saying the maximum of two numbers is less than or equal to a
third. (Contributed by Jim Kingdon, 31-Jan-2022.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶))) |
|
Theorem | maxleastlt 11019 |
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 | maxleb 11020 |
Equivalence of ≤ and being equal to the maximum of
two reals. Lemma
3.12 of [Geuvers], p. 10. (Contributed by
Jim Kingdon, 21-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ sup({𝐴, 𝐵}, ℝ, < ) = 𝐵)) |
|
Theorem | dfabsmax 11021 |
Absolute value of a real number in terms of maximum. Definition 3.13 of
[Geuvers], p. 11. (Contributed by BJ and
Jim Kingdon, 21-Dec-2021.)
|
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = sup({𝐴, -𝐴}, ℝ, < )) |
|
Theorem | maxltsup 11022 |
Two ways of saying the maximum of two numbers is less than a third.
(Contributed by Jim Kingdon, 10-Feb-2022.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) < 𝐶 ↔ (𝐴 < 𝐶 ∧ 𝐵 < 𝐶))) |
|
Theorem | max0addsup 11023 |
The sum of the positive and negative part functions is the absolute value
function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.)
|
⊢ (𝐴 ∈ ℝ → (sup({𝐴, 0}, ℝ, < ) +
sup({-𝐴, 0}, ℝ, <
)) = (abs‘𝐴)) |
|
Theorem | rexanre 11024* |
Combine two different upper real properties into one. (Contributed by
Mario Carneiro, 8-May-2016.)
|
⊢ (𝐴 ⊆ ℝ → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) ↔ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓)))) |
|
Theorem | rexico 11025* |
Restrict the base of an upper real quantifier to an upper real set.
(Contributed by Mario Carneiro, 12-May-2016.)
|
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ↔ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑))) |
|
Theorem | maxclpr 11026 |
The maximum 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 9122 if one is dealing with integers, but real
number
dichotomy in general does not follow from our axioms. (Contributed by Jim
Kingdon, 1-Feb-2022.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) ∈ {𝐴, 𝐵} ↔ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))) |
|
Theorem | rpmaxcl 11027 |
The maximum of two positive real numbers is a positive real number.
(Contributed by Jim Kingdon, 10-Nov-2023.)
|
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+)
→ sup({𝐴, 𝐵}, ℝ, < ) ∈
ℝ+) |
|
Theorem | zmaxcl 11028 |
The maximum of two integers is an integer. (Contributed by Jim Kingdon,
27-Sep-2022.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → sup({𝐴, 𝐵}, ℝ, < ) ∈
ℤ) |
|
Theorem | 2zsupmax 11029 |
Two ways to express the maximum of two integers. Because order of
integers is decidable, we have more flexibility than for real numbers.
(Contributed by Jim Kingdon, 22-Jan-2023.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → sup({𝐴, 𝐵}, ℝ, < ) = if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
|
Theorem | fimaxre2 11030* |
A nonempty finite set of real numbers has an upper bound. (Contributed
by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro,
13-Feb-2014.)
|
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
|
Theorem | negfi 11031* |
The negation of a finite set of real numbers is finite. (Contributed by
AV, 9-Aug-2020.)
|
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴} ∈ Fin) |
|
4.7.6 The minimum of two real
numbers
|
|
Theorem | mincom 11032 |
The minimum of two reals is commutative. (Contributed by Jim Kingdon,
8-Feb-2021.)
|
⊢ inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < ) |
|
Theorem | minmax 11033 |
Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
8-Feb-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = -sup({-𝐴, -𝐵}, ℝ, < )) |
|
Theorem | mincl 11034 |
The minumum of two real numbers is a real number. (Contributed by Jim
Kingdon, 25-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ∈
ℝ) |
|
Theorem | min1inf 11035 |
The minimum of two numbers is less than or equal to the first.
(Contributed by Jim Kingdon, 8-Feb-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ≤ 𝐴) |
|
Theorem | min2inf 11036 |
The minimum of two numbers is less than or equal to the second.
(Contributed by Jim Kingdon, 9-Feb-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ≤ 𝐵) |
|
Theorem | lemininf 11037 |
Two ways of saying a number is less than or equal to the minimum of two
others. (Contributed by NM, 3-Aug-2007.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ, < ) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
|
Theorem | ltmininf 11038 |
Two ways of saying a number is less than the minimum of two others.
(Contributed by Jim Kingdon, 10-Feb-2022.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < inf({𝐵, 𝐶}, ℝ, < ) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶))) |
|
Theorem | minabs 11039 |
The minimum of two real numbers in terms of absolute value. (Contributed
by Jim Kingdon, 15-May-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵))) / 2)) |
|
Theorem | minclpr 11040 |
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 9122 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 11041 |
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 11042 |
Lemma for bdtri 11043. (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 11043 |
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 11044 |
Equality of a product with zero. A bit of a curiosity, in the sense that
theorems like abs00ap 10866 and mulap0bd 8442 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 11045 |
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 11046 |
Lemma for xrmaxif 11052. 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 11047 |
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 11048 |
Lemma for xrmaxif 11052. A variation of xrmaxleim 11045- 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 11049 |
Lemma for xrmaxif 11052. 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 11050 |
Lemma for xrmaxif 11052. 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 11051* |
Lemma for xrmaxif 11052. 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 11052 |
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 11053 |
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 11054 |
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 11055 |
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 11056 |
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 11057 |
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 11058 |
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 11059 |
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 11060 |
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 11061 |
Lemma for xrmaxadd 11062. 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 11062 |
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 11063 |
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 11064* |
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 11065 |
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 11066 |
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 11067 |
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 11068 |
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 11069 |
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 11070 |
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 11071 |
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 11072 |
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 11073 |
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 11074 |
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 11075 |
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 11076 |
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 11077 |
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 11078 |
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 11079 |
Extend class notation with convergence relation for limits.
|
class ⇝ |
|
Definition | df-clim 11080* |
Define the limit relation for complex number sequences. See clim 11082
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 11081 |
The limit relation is a relation. (Contributed by NM, 28-Aug-2005.)
(Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ Rel ⇝ |
|
Theorem | clim 11082* |
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 11083 |
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 11084* |
Express the predicate: The limit of complex number sequence 𝐹 is
𝐴, or 𝐹 converges to 𝐴, with
more general quantifier
restrictions than clim 11082. (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 11085* |
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 11086* |
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 11087* |
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 11088* |
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 11089* |
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 11090* |
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 11091* |
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 11092 |
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 11093 |
The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.)
(Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ (ℤ × {0}) ⇝
0 |
|
Theorem | climuni 11094 |
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 11095 |
The limit relation is function-like, and with range the complex numbers.
(Contributed by Mario Carneiro, 31-Jan-2014.)
|
⊢ ⇝ :dom ⇝
⟶ℂ |
|
Theorem | climdm 11096 |
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 11097* |
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 11098* |
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 11099* |
An infinite sequence of complex numbers converges to at most one limit.
(Contributed by Mario Carneiro, 13-Jul-2013.)
|
⊢ ∃*𝑥 𝐹 ⇝ 𝑥 |
|
Theorem | climeq 11100* |
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)) |