Theorem List for Intuitionistic Logic Explorer - 11201-11300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | rexico 11201* |
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 11202 |
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 9273 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 11203 |
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 11204 |
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 11205 |
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 11206* |
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 11207* |
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 11208 |
The minimum of two reals is commutative. (Contributed by Jim Kingdon,
8-Feb-2021.)
|
⊢ inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < ) |
|
Theorem | minmax 11209 |
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 11210 |
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 11211 |
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 11212 |
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 11213 |
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 11214 |
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 11215 |
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 11216 |
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 9273 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 11217 |
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 11218 |
Lemma for bdtri 11219. (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 11219 |
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 11220 |
Equality of a product with zero. A bit of a curiosity, in the sense that
theorems like abs00ap 11042 and mulap0bd 8590 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)) |
|
Theorem | mingeb 11221 |
Equivalence of ≤ and being equal to the minimum of
two reals.
(Contributed by Jim Kingdon, 14-Oct-2024.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ inf({𝐴, 𝐵}, ℝ, < ) = 𝐴)) |
|
Theorem | 2zinfmin 11222 |
Two ways to express the minimum of two integers. Because order of
integers is decidable, we have more flexibility than for real numbers.
(Contributed by Jim Kingdon, 14-Oct-2024.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → inf({𝐴, 𝐵}, ℝ, < ) = if(𝐴 ≤ 𝐵, 𝐴, 𝐵)) |
|
4.7.7 The maximum of two extended
reals
|
|
Theorem | xrmaxleim 11223 |
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 11224 |
Lemma for xrmaxif 11230. 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 11225 |
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 11226 |
Lemma for xrmaxif 11230. A variation of xrmaxleim 11223- 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 11227 |
Lemma for xrmaxif 11230. 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 11228 |
Lemma for xrmaxif 11230. 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 11229* |
Lemma for xrmaxif 11230. 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 11230 |
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 11231 |
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 11232 |
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 11233 |
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 11234 |
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 11235 |
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 11236 |
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 11237 |
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 11238 |
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 11239 |
Lemma for xrmaxadd 11240. 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 11240 |
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 11241 |
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 11242* |
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 11243 |
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 11244 |
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 11245 |
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 11246 |
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 11247 |
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 11248 |
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 11249 |
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 11250 |
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 11251 |
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 11252 |
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 11253 |
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 11254 |
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 11255 |
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 11256 |
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 11257 |
Extend class notation with convergence relation for limits.
|
class ⇝ |
|
Definition | df-clim 11258* |
Define the limit relation for complex number sequences. See clim 11260
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 11259 |
The limit relation is a relation. (Contributed by NM, 28-Aug-2005.)
(Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ Rel ⇝ |
|
Theorem | clim 11260* |
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 11261 |
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 11262* |
Express the predicate: The limit of complex number sequence 𝐹 is
𝐴, or 𝐹 converges to 𝐴, with
more general quantifier
restrictions than clim 11260. (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 11263* |
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 11264* |
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 11265* |
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 11266* |
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 11267* |
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 11268* |
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 11269* |
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 11270 |
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 11271 |
The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.)
(Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ (ℤ × {0}) ⇝
0 |
|
Theorem | climuni 11272 |
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 11273 |
The limit relation is function-like, and with codomian the complex
numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
|
⊢ ⇝ :dom ⇝
⟶ℂ |
|
Theorem | climdm 11274 |
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 11275* |
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 11276* |
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 11277* |
An infinite sequence of complex numbers converges to at most one limit.
(Contributed by Mario Carneiro, 13-Jul-2013.)
|
⊢ ∃*𝑥 𝐹 ⇝ 𝑥 |
|
Theorem | climeq 11278* |
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 11279* |
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 11280* |
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 11281 |
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 11282 |
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 11283 |
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 11284 |
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 11285* |
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 11286* |
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 11287* |
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 11288* |
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 11289* |
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 13685
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 11290* |
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 11291* |
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 11292* |
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 2177. (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 11293* |
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 11294* |
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 11295* |
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 11296* |
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 11297* |
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 11298* |
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 11299* |
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 11300* |
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)) |