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Theorem List for Metamath Proof Explorer - 41701-41800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremicoltubd 41701 An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,)𝐵))       (𝜑𝐶 < 𝐵)
 
Theoremqelioo 41702* The rational numbers are dense in *: any two extended real numbers have a rational between them. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑 → ∃𝑥 ∈ ℚ 𝑥 ∈ (𝐴(,)𝐵))
 
Theoremtgqioo2 41703* Every open set of reals is the (countable) union of open interval with rational bounds. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   (𝜑𝐴𝐽)       (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐴 = 𝑞))
 
Theoremiccleubd 41704 An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))       (𝜑𝐶𝐵)
 
Theoremelioored 41705 A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ (𝐵(,)𝐶))       (𝜑𝐴 ∈ ℝ)
 
Theoremioogtlbd 41706 An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))       (𝜑𝐴 < 𝐶)
 
Theoremioofun 41707 (,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Fun (,)
 
Theoremicomnfinre 41708 A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴))
 
Theoremsqrlearg 41709 The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ((𝐴↑2) ≤ 𝐴𝐴 ∈ (0[,]1)))
 
Theoremressiocsup 41710 If the supremum belongs to a set of reals, the set is a subset of the unbounded below, right-closed interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ)    &   𝑆 = sup(𝐴, ℝ*, < )    &   (𝜑𝑆𝐴)    &   𝐼 = (-∞(,]𝑆)       (𝜑𝐴𝐼)
 
Theoremressioosup 41711 If the supremum does not belong to a set of reals, the set is a subset of the unbounded below, right-open interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ)    &   𝑆 = sup(𝐴, ℝ*, < )    &   (𝜑 → ¬ 𝑆𝐴)    &   𝐼 = (-∞(,)𝑆)       (𝜑𝐴𝐼)
 
Theoremiooiinioc 41712* A left-open, right-closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵))
 
Theoremressiooinf 41713 If the infimum does not belong to a set of reals, the set is a subset of the unbounded above, left-open interval, with lower bound equal to the infimum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ)    &   𝑆 = inf(𝐴, ℝ*, < )    &   (𝜑 → ¬ 𝑆𝐴)    &   𝐼 = (𝑆(,)+∞)       (𝜑𝐴𝐼)
 
Theoremicogelbd 41714 An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,)𝐵))       (𝜑𝐴𝐶)
 
Theoremiocleubd 41715 An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴(,]𝐵))       (𝜑𝐶𝐵)
 
Theoremuzinico 41716 An upper interval of integers is the intersection of the integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)       (𝜑𝑍 = (ℤ ∩ (𝑀[,)+∞)))
 
Theorempreimaiocmnf 41717* Preimage of a right-closed interval, unbounded below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐹 “ (-∞(,]𝐵)) = {𝑥𝐴 ∣ (𝐹𝑥) ≤ 𝐵})
 
Theoremuzinico2 41718 An upper interval of integers is the intersection of a larger upper interval of integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → (ℤ𝑁) = ((ℤ𝑀) ∩ (𝑁[,)+∞)))
 
Theoremuzinico3 41719 An upper interval of integers doesn't change when it's intersected with a left-closed, unbounded above interval, with the same lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)       (𝜑𝑍 = (𝑍 ∩ (𝑀[,)+∞)))
 
Theoremicossico2 41720 Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵𝐴)       (𝜑 → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶))
 
Theoremdmico 41721 The domain of the closed-below, open-above interval function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
dom [,) = (ℝ* × ℝ*)
 
Theoremndmico 41722 The closed-below, open-above interval function's value is empty outside of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(¬ (𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = ∅)
 
Theoremuzubioo 41723* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘𝑍)
 
Theoremuzubico 41724* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → ∃𝑘 ∈ (𝑋[,)+∞)𝑘𝑍)
 
Theoremuzubioo2 41725* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)       (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥(,)+∞)𝑘𝑍)
 
Theoremuzubico2 41726* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)       (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥[,)+∞)𝑘𝑍)
 
Theoremiocgtlbd 41727 An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴(,]𝐵))       (𝜑𝐴 < 𝐶)
 
Theoremxrtgioo2 41728 The topology on the extended reals coincides with the standard topology on the reals, when restricted to . (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾t ℝ)
 
Theoremtgioo4 41729 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
 
20.36.5  Finite sums
 
Theoremfsumclf 41730* Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsumcl 15080 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℂ)
 
Theoremfsummulc1f 41731* Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsummulc1 15130 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (Σ𝑘𝐴 𝐵 · 𝐶) = Σ𝑘𝐴 (𝐵 · 𝐶))
 
Theoremfsumnncl 41732* Closure of a nonempty, finite sum of positive integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 ≠ ∅)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℕ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℕ)
 
Theoremfsumsplit1 41733* Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   𝑘𝐷    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶𝐴)    &   (𝑘 = 𝐶𝐵 = 𝐷)       (𝜑 → Σ𝑘𝐴 𝐵 = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵))
 
Theoremfsumge0cl 41734* The finite sum of nonnegative reals is a nonnegative real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))       (𝜑 → Σ𝑘𝐴 𝐵 ∈ (0[,)+∞))
 
Theoremfsumf1of 41735* Re-index a finite sum using a bijection. Same as fsumf1o 15070, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   𝑛𝜑    &   (𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐶 ∈ Fin)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑛𝐶 𝐷)
 
Theoremfsumiunss 41736* Sum over a disjoint indexed union, intersected with a finite set 𝐷. Similar to fsumiun 15166, but here 𝐴 and 𝐵 need not be finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   ((𝜑𝑥𝐴𝑘𝐵) → 𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ Fin)       (𝜑 → Σ𝑘 𝑥𝐴 (𝐵𝐷)𝐶 = Σ𝑥 ∈ {𝑥𝐴 ∣ (𝐵𝐷) ≠ ∅}Σ𝑘 ∈ (𝐵𝐷)𝐶)
 
Theoremfsumreclf 41737* Closure of a finite sum of reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℝ)
 
Theoremfsumlessf 41738* A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → Σ𝑘𝐶 𝐵 ≤ Σ𝑘𝐴 𝐵)
 
Theoremfsumsupp0 41739* Finite sum of function values, for a function of finite support. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴⟶ℂ)       (𝜑 → Σ𝑘 ∈ (𝐹 supp 0)(𝐹𝑘) = Σ𝑘𝐴 (𝐹𝑘))
 
Theoremfsumsermpt 41740* A finite sum expressed in terms of a partial sum of an infinite series. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   𝐹 = (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴)    &   𝐺 = seq𝑀( + , (𝑘𝑍𝐴))       (𝜑𝐹 = 𝐺)
 
20.36.6  Finite multiplication of numbers and finite multiplication of functions
 
Theoremfmul01 41741* Multiplying a finite number of values in [ 0 , 1 ] , gives the final product itself a number in [ 0 , 1 ]. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑖𝐵    &   𝑖𝜑    &   𝐴 = seq𝐿( · , 𝐵)    &   (𝜑𝐿 ∈ ℤ)    &   (𝜑𝑀 ∈ (ℤ𝐿))    &   (𝜑𝐾 ∈ (𝐿...𝑀))    &   ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ∈ ℝ)    &   ((𝜑𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝑖))    &   ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ≤ 1)       (𝜑 → (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1))
 
Theoremfmulcl 41742* If ' Y ' is closed under the multiplication of two functions, then Y is closed under the multiplication ( ' X ' ) of a finite number of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))    &   𝑋 = (seq1(𝑃, 𝑈)‘𝑁)    &   (𝜑𝑁 ∈ (1...𝑀))    &   (𝜑𝑈:(1...𝑀)⟶𝑌)    &   ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)    &   (𝜑𝑇 ∈ V)       (𝜑𝑋𝑌)
 
Theoremfmuldfeqlem1 41743* induction step for the proof of fmuldfeq 41744. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑓𝜑    &   𝑔𝜑    &   𝑡𝑌    &   𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))    &   𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))    &   (𝜑𝑇 ∈ V)    &   (𝜑𝑈:(1...𝑀)⟶𝑌)    &   ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)    &   (𝜑𝑁 ∈ (1...𝑀))    &   (𝜑 → (𝑁 + 1) ∈ (1...𝑀))    &   (𝜑 → ((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑁))    &   ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ)       ((𝜑𝑡𝑇) → ((seq1(𝑃, 𝑈)‘(𝑁 + 1))‘𝑡) = (seq1( · , (𝐹𝑡))‘(𝑁 + 1)))
 
Theoremfmuldfeq 41744* X and Z are two equivalent definitions of the finite product of real functions. Y is a set of real functions from a common domain T, Y is closed under function multiplication and U is a finite sequence of functions in Y. M is the number of functions multiplied together. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑖𝜑    &   𝑡𝑌    &   𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))    &   𝑋 = (seq1(𝑃, 𝑈)‘𝑀)    &   𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))    &   𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))    &   (𝜑𝑇 ∈ V)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑈:(1...𝑀)⟶𝑌)    &   ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)       ((𝜑𝑡𝑇) → (𝑋𝑡) = (𝑍𝑡))
 
Theoremfmul01lt1lem1 41745* Given a finite multiplication of values betweeen 0 and 1, a value larger than its first element is larger the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑖𝐵    &   𝑖𝜑    &   𝐴 = seq𝐿( · , 𝐵)    &   (𝜑𝐿 ∈ ℤ)    &   (𝜑𝑀 ∈ (ℤ𝐿))    &   ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ∈ ℝ)    &   ((𝜑𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝑖))    &   ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ≤ 1)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (𝐵𝐿) < 𝐸)       (𝜑 → (𝐴𝑀) < 𝐸)
 
Theoremfmul01lt1lem2 41746* Given a finite multiplication of values betweeen 0 and 1, a value 𝐸 larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑖𝐵    &   𝑖𝜑    &   𝐴 = seq𝐿( · , 𝐵)    &   (𝜑𝐿 ∈ ℤ)    &   (𝜑𝑀 ∈ (ℤ𝐿))    &   ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ∈ ℝ)    &   ((𝜑𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝑖))    &   ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ≤ 1)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐽 ∈ (𝐿...𝑀))    &   (𝜑 → (𝐵𝐽) < 𝐸)       (𝜑 → (𝐴𝑀) < 𝐸)
 
Theoremfmul01lt1 41747* Given a finite multiplication of values betweeen 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑖𝐵    &   𝑖𝜑    &   𝑗𝐴    &   𝐴 = seq1( · , 𝐵)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐵:(1...𝑀)⟶ℝ)    &   ((𝜑𝑖 ∈ (1...𝑀)) → 0 ≤ (𝐵𝑖))    &   ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵𝑖) ≤ 1)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → ∃𝑗 ∈ (1...𝑀)(𝐵𝑗) < 𝐸)       (𝜑 → (𝐴𝑀) < 𝐸)
 
Theoremcncfmptss 41748* A continuous complex function restricted to a subset is continuous, using maps-to notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑥𝐹    &   (𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐶𝐴)       (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) ∈ (𝐶cn𝐵))
 
Theoremrrpsscn 41749 The positive reals are a subset of the complex numbers. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
+ ⊆ ℂ
 
Theoremmulc1cncfg 41750* A version of mulc1cncf 23442 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
𝑥𝐹    &   𝑥𝜑    &   (𝜑𝐹 ∈ (𝐴cn→ℂ))    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝑥𝐴 ↦ (𝐵 · (𝐹𝑥))) ∈ (𝐴cn→ℂ))
 
Theoreminfrglb 41751* The infimum of a nonempty bounded set of reals is the greatest lower bound. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.)
(((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ, < ) < 𝐵 ↔ ∃𝑧𝐴 𝑧 < 𝐵))
 
Theoremexpcnfg 41752* If 𝐹 is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 23459. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑥𝐹    &   (𝜑𝐹 ∈ (𝐴cn→ℂ))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑥𝐴 ↦ ((𝐹𝑥)↑𝑁)) ∈ (𝐴cn→ℂ))
 
Theoremprodeq2ad 41753* Equality deduction for product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)
 
Theoremfprodsplit1 41754* Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶𝐴)    &   ((𝜑𝑘 = 𝐶) → 𝐵 = 𝐷)       (𝜑 → ∏𝑘𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵))
 
Theoremfprodexp 41755* Positive integer exponentiation of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 (𝐵𝑁) = (∏𝑘𝐴 𝐵𝑁))
 
Theoremfprodabs2 41756* The absolute value of a finite product . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵))
 
Theoremfprod0 41757* A finite product with a zero term is zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   𝑘𝐶    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝑘 = 𝐾𝐵 = 𝐶)    &   (𝜑𝐾𝐴)    &   (𝜑𝐶 = 0)       (𝜑 → ∏𝑘𝐴 𝐵 = 0)
 
Theoremmccllem 41758* * Induction step for mccl 41759. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷 ∈ (𝐴𝐶))    &   (𝜑𝐵 ∈ (ℕ0m (𝐶 ∪ {𝐷})))    &   (𝜑 → ∀𝑏 ∈ (ℕ0m 𝐶)((!‘Σ𝑘𝐶 (𝑏𝑘)) / ∏𝑘𝐶 (!‘(𝑏𝑘))) ∈ ℕ)       (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵𝑘))) ∈ ℕ)
 
Theoremmccl 41759* A multinomial coefficient, in its standard domain, is a positive integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝐵    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ (ℕ0m 𝐴))       (𝜑 → ((!‘Σ𝑘𝐴 (𝐵𝑘)) / ∏𝑘𝐴 (!‘(𝐵𝑘))) ∈ ℕ)
 
Theoremfprodcnlem 41760* A finite product of functions to complex numbers from a common topological space is continuous. Induction step. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑍𝐴)    &   (𝜑𝑊 ∈ (𝐴𝑍))    &   (𝜑 → (𝑥𝑋 ↦ ∏𝑘𝑍 𝐵) ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾))
 
Theoremfprodcn 41761* A finite product of functions to complex numbers from a common topological space is continuous. The class expression for 𝐵 normally contains free variables 𝑘 and 𝑥 to index it. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ ∏𝑘𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
 
20.36.7  Limits
 
Theoremclim1fr1 41762* A class of sequences of fractions that converge to 1. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴 · 𝑛) + 𝐵) / (𝐴 · 𝑛)))    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ∈ ℂ)       (𝜑𝐹 ⇝ 1)
 
Theoremisumneg 41763* Negation of a converging sum. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → Σ𝑘𝑍 𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 -𝐴 = -Σ𝑘𝑍 𝐴)
 
Theoremclimrec 41764* Limit of the reciprocal of a converging sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺𝐴)    &   (𝜑𝐴 ≠ 0)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ (ℂ ∖ {0}))    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = (1 / (𝐺𝑘)))    &   (𝜑𝐻𝑊)       (𝜑𝐻 ⇝ (1 / 𝐴))
 
Theoremclimmulf 41765* A version of climmul 14979 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑘𝜑    &   𝑘𝐹    &   𝑘𝐺    &   𝑘𝐻    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐻𝑋)    &   (𝜑𝐺𝐵)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = ((𝐹𝑘) · (𝐺𝑘)))       (𝜑𝐻 ⇝ (𝐴 · 𝐵))
 
Theoremclimexp 41766* The limit of natural powers, is the natural power of the limit. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑘𝜑    &   𝑘𝐹    &   𝑘𝐻    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶ℂ)    &   (𝜑𝐹𝐴)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐻𝑉)    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = ((𝐹𝑘)↑𝑁))       (𝜑𝐻 ⇝ (𝐴𝑁))
 
Theoremcliminf 41767* A bounded monotonic nonincreasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶ℝ)    &   ((𝜑𝑘𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘))       (𝜑𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
 
Theoremclimsuselem1 41768* The subsequence index 𝐼 has the expected properties: it belongs to the same upper integers as the original index, and it is always greater than or equal to the original index. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → (𝐼𝑀) ∈ 𝑍)    &   ((𝜑𝑘𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ‘((𝐼𝑘) + 1)))       ((𝜑𝐾𝑍) → (𝐼𝐾) ∈ (ℤ𝐾))
 
Theoremclimsuse 41769* A subsequence 𝐺 of a converging sequence 𝐹, converges to the same limit. 𝐼 is the strictly increasing and it is used to index the subsequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑘𝜑    &   𝑘𝐹    &   𝑘𝐺    &   𝑘𝐼    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑋)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   (𝜑𝐹𝐴)    &   (𝜑 → (𝐼𝑀) ∈ 𝑍)    &   ((𝜑𝑘𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ‘((𝐼𝑘) + 1)))    &   (𝜑𝐺𝑌)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐹‘(𝐼𝑘)))       (𝜑𝐺𝐴)
 
Theoremclimrecf 41770* A version of climrec 41764 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑘𝜑    &   𝑘𝐺    &   𝑘𝐻    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺𝐴)    &   (𝜑𝐴 ≠ 0)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ (ℂ ∖ {0}))    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = (1 / (𝐺𝑘)))    &   (𝜑𝐻𝑊)       (𝜑𝐻 ⇝ (1 / 𝐴))
 
Theoremclimneg 41771* Complex limit of the negative of a sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑 → (𝑘𝑍 ↦ -(𝐹𝑘)) ⇝ -𝐴)
 
Theoremcliminff 41772* A version of climinf 41767 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶ℝ)    &   ((𝜑𝑘𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘))       (𝜑𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
 
Theoremclimdivf 41773* Limit of the ratio of two converging sequences. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑘𝜑    &   𝑘𝐹    &   𝑘𝐺    &   𝑘𝐻    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐻𝑋)    &   (𝜑𝐺𝐵)    &   (𝜑𝐵 ≠ 0)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ (ℂ ∖ {0}))    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = ((𝐹𝑘) / (𝐺𝑘)))       (𝜑𝐻 ⇝ (𝐴 / 𝐵))
 
Theoremclimreeq 41774 If 𝐹 is a real function, then 𝐹 converges to 𝐴 with respect to the standard topology on the reals if and only if it converges to 𝐴 with respect to the standard topology on complex numbers. In the theorem, 𝑅 is defined to be convergence w.r.t. the standard topology on the reals and then 𝐹𝑅𝐴 represents the statement "𝐹 converges to 𝐴, with respect to the standard topology on the reals". Notice that there is no need for the hypothesis that 𝐴 is a real number. (Contributed by Glauco Siliprandi, 2-Jul-2017.)
𝑅 = (⇝𝑡‘(topGen‘ran (,)))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹𝑅𝐴𝐹𝐴))
 
Theoremellimciota 41775* An explicit value for the limit, when the limit exists at a limit point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ((limPt‘𝐾)‘𝐴))    &   (𝜑 → (𝐹 lim 𝐵) ≠ ∅)    &   𝐾 = (TopOpen‘ℂfld)       (𝜑 → (℩𝑥𝑥 ∈ (𝐹 lim 𝐵)) ∈ (𝐹 lim 𝐵))
 
Theoremclimaddf 41776* A version of climadd 14978 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑘𝜑    &   𝑘𝐹    &   𝑘𝐺    &   𝑘𝐻    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐻𝑋)    &   (𝜑𝐺𝐵)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = ((𝐹𝑘) + (𝐺𝑘)))       (𝜑𝐻 ⇝ (𝐴 + 𝐵))
 
Theoremmullimc 41777* Limit of the product of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ (𝐵 · 𝐶))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)    &   (𝜑𝑋 ∈ (𝐹 lim 𝐷))    &   (𝜑𝑌 ∈ (𝐺 lim 𝐷))       (𝜑 → (𝑋 · 𝑌) ∈ (𝐻 lim 𝐷))
 
Theoremellimcabssub0 41778* An equivalent condition for being a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴 ↦ (𝐵𝐶))    &   (𝜑𝐴 ⊆ ℂ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐶 ∈ (𝐹 lim 𝐷) ↔ 0 ∈ (𝐺 lim 𝐷)))
 
Theoremlimcdm0 41779 If a function has empty domain, every complex number is a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:∅⟶ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐹 lim 𝐵) = ℂ)
 
Theoremislptre 41780* An equivalence condition for a limit point w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐽 = (topGen‘ran (,))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐵 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑎 ∈ ℝ*𝑏 ∈ ℝ* (𝐵 ∈ (𝑎(,)𝑏) → ((𝑎(,)𝑏) ∩ (𝐴 ∖ {𝐵})) ≠ ∅)))
 
Theoremlimccog 41781 Limit of the composition of two functions. If the limit of 𝐹 at 𝐴 is 𝐵 and the limit of 𝐺 at 𝐵 is 𝐶, then the limit of 𝐺𝐹 at 𝐴 is 𝐶. With respect to limcco 24420 and limccnp 24418, here we drop continuity assumptions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))    &   (𝜑𝐵 ∈ (𝐹 lim 𝐴))    &   (𝜑𝐶 ∈ (𝐺 lim 𝐵))       (𝜑𝐶 ∈ ((𝐺𝐹) lim 𝐴))
 
Theoremlimciccioolb 41782 The limit of a function at the lower bound of a closed interval only depends on the values in the inner open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)       (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim 𝐴) = (𝐹 lim 𝐴))
 
Theoremclimf 41783* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 14841, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑘𝐹    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
 
Theoremmullimcf 41784* Limit of the multiplication of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐺:𝐴⟶ℂ)    &   𝐻 = (𝑥𝐴 ↦ ((𝐹𝑥) · (𝐺𝑥)))    &   (𝜑𝐵 ∈ (𝐹 lim 𝐷))    &   (𝜑𝐶 ∈ (𝐺 lim 𝐷))       (𝜑 → (𝐵 · 𝐶) ∈ (𝐻 lim 𝐷))
 
Theoremconstlimc 41785* Limit of constant function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑𝐵 ∈ (𝐹 lim 𝐶))
 
Theoremrexlim2d 41786* Inference removing two restricted quantifiers. Same as rexlimdvv 3293, but with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
 
Theoremidlimc 41787* Limit of the identity function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   𝐹 = (𝑥𝐴𝑥)    &   (𝜑𝑋 ∈ ℂ)       (𝜑𝑋 ∈ (𝐹 lim 𝑋))
 
Theoremdivcnvg 41788* The sequence of reciprocals of positive integers, multiplied by the factor 𝐴, converges to zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ) → (𝑛 ∈ (ℤ𝑀) ↦ (𝐴 / 𝑛)) ⇝ 0)
 
Theoremlimcperiod 41789* If 𝐹 is a periodic function with period 𝑇, the limit doesn't change if we shift the limiting point by 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:dom 𝐹⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐴 ⊆ dom 𝐹)    &   (𝜑𝑇 ∈ ℂ)    &   𝐵 = {𝑥 ∈ ℂ ∣ ∃𝑦𝐴 𝑥 = (𝑦 + 𝑇)}    &   (𝜑𝐵 ⊆ dom 𝐹)    &   ((𝜑𝑦𝐴) → (𝐹‘(𝑦 + 𝑇)) = (𝐹𝑦))    &   (𝜑𝐶 ∈ ((𝐹𝐴) lim 𝐷))       (𝜑𝐶 ∈ ((𝐹𝐵) lim (𝐷 + 𝑇)))
 
Theoremlimcrecl 41790 If 𝐹 is a real-valued function, 𝐵 is a limit point of its domain, and the limit of 𝐹 at 𝐵 exists, then this limit is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘𝐴))    &   (𝜑𝐿 ∈ (𝐹 lim 𝐵))       (𝜑𝐿 ∈ ℝ)
 
Theoremsumnnodd 41791* A series indexed by with only odd terms. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℕ⟶ℂ)    &   ((𝜑𝑘 ∈ ℕ ∧ (𝑘 / 2) ∈ ℕ) → (𝐹𝑘) = 0)    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝐵)       (𝜑 → (seq1( + , (𝑘 ∈ ℕ ↦ (𝐹‘((2 · 𝑘) − 1)))) ⇝ 𝐵 ∧ Σ𝑘 ∈ ℕ (𝐹𝑘) = Σ𝑘 ∈ ℕ (𝐹‘((2 · 𝑘) − 1))))
 
Theoremlptioo2 41792 The upper bound of an open interval is a limit point of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐽 = (topGen‘ran (,))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))
 
Theoremlptioo1 41793 The lower bound of an open interval is a limit point of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐽 = (topGen‘ran (,))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))
 
Theoremelprn1 41794 A member of an unordered pair that is not the "first", must be the "second". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → 𝐴 = 𝐶)
 
Theoremelprn2 41795 A member of an unordered pair that is not the "second", must be the "first". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → 𝐴 = 𝐵)
 
Theoremlimcmptdm 41796* The domain of a maps-to function with a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ (𝐹 lim 𝐷))       (𝜑𝐴 ⊆ ℂ)
 
Theoremclim2f 41797* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 14841. Similar to clim2 14851, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
 
Theoremlimcicciooub 41798 The limit of a function at the upper bound of a closed interval only depends on the values in the inner open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)       (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim 𝐵) = (𝐹 lim 𝐵))
 
Theoremltmod 41799 A sufficient condition for a "less than" relationship for the mod operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴))       (𝜑 → (𝐶 mod 𝐵) < (𝐴 mod 𝐵))
 
Theoremislpcn 41800* A characterization for a limit point for the standard topology on the complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑃 ∈ ℂ)       (𝜑 → (𝑃 ∈ ((limPt‘(TopOpen‘ℂfld))‘𝑆) ↔ ∀𝑒 ∈ ℝ+𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥𝑃)) < 𝑒))
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