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Theorem List for Metamath Proof Explorer - 38601-38700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempnfel0pnf 38601 +∞ is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
+∞ ∈ (0[,]+∞)

Theoremge0nemnf2 38602 A nonnegative extended real is not -∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞)

Theoremeliccnelico 38603 An element of a closed interval that is not a member of the left closed right open interval, must be the upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))    &   (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵))       (𝜑𝐶 = 𝐵)

Theoremeliccelicod 38604 A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))    &   (𝜑𝐶𝐵)       (𝜑𝐶 ∈ (𝐴[,)𝐵))

Theoremge0xrre 38605 A nonnegative extended real that is not +∞ is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ)

Theoremge0lere 38606 A nonnegative extended Real number smaller than or equal to a Real number, is a Real number. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ (0[,]+∞))    &   (𝜑𝐵𝐴)       (𝜑𝐵 ∈ ℝ)

Theoremelicores 38607* Membership in a left-closed, right-open interval with real bounds. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦))

Theoreminficc 38608 The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝑆 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑆 ≠ ∅)       (𝜑 → inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐵))

Theoremqinioo 38609 The rational numbers are dense in . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → ((ℚ ∩ (𝐴(,)𝐵)) = ∅ ↔ 𝐵𝐴))

Theoremlenelioc 38610 A real number smaller than or equal to the lower bound of a left open right closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐶𝐴)       (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))

Theoremioonct 38611 C non empty open interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   𝐶 = (𝐴(,)𝐵)       (𝜑 → ¬ 𝐶 ≼ ω)

Theoremxrgtnelicc 38612 A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 < 𝐶)       (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Theoremiccdificc 38613 The difference of two closed intervals with the same lower bound. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝐴[,]𝐶) ∖ (𝐴[,]𝐵)) = (𝐵(,]𝐶))

Theoremiocnct 38614 A non empty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   𝐶 = (𝐴(,]𝐵)       (𝜑 → ¬ 𝐶 ≼ ω)

Theoremiccnct 38615 A closed interval, with more than one element is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   𝐶 = (𝐴[,]𝐵)       (𝜑 → ¬ 𝐶 ≼ ω)

Theoremiooiinicc 38616* A closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 𝑛 ∈ ℕ ((𝐴 − (1 / 𝑛))(,)(𝐵 + (1 / 𝑛))) = (𝐴[,]𝐵))

Theoremiccgelbd 38617 An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))       (𝜑𝐴𝐶)

Theoremiooltubd 38618 An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))       (𝜑𝐶 < 𝐵)

Theoremicoltubd 38619 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 38620* 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 38621* 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 38622 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 38623 A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ (𝐵(,)𝐶))       (𝜑𝐴 ∈ ℝ)

Theoremioogtlbd 38624 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 38625 (,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Fun (,)

Theoremicomnfinre 38626 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 38627 The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ((𝐴↑2) ≤ 𝐴𝐴 ∈ (0[,]1)))

Theoremressiocsup 38628 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 38629 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 38630* 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 38631 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)

21.31.5  Finite sums

Theoremsumeq2ad 38632* Equality deduction for sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremfsumclf 38633* Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsumcl 14311 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)

Theoremfsumsplitf 38634* Split a sum into two parts. A version of fsumsplit 14318 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈 = (𝐴𝐵))    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑𝑘𝑈) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝑈 𝐶 = (Σ𝑘𝐴 𝐶 + Σ𝑘𝐵 𝐶))

Theoremfsummulc1f 38635* Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsummulc1 14359 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))

Theoremsumsnf 38636* A sum of a singleton is the term. A version of sumsn 14319 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝐵    &   (𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀𝑉𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)

Theoremfsumsplitsn 38637* Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   𝑘𝐷    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝑉)    &   (𝜑 → ¬ 𝐵𝐴)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   (𝑘 = 𝐵𝐶 = 𝐷)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘𝐴 𝐶 + 𝐷))

Theoremfsumnncl 38638* Closure of a non empty, finite sum of positive integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 ≠ ∅)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℕ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℕ)

Theoremfsumsplit1 38639* 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 38640* 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 38641* Re-index a finite sum using a bijection. Same as fsumf1o 14301, 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 38642* Sum over a disjoint indexed union, intersected with a finite set 𝐷. Similar to fsumiun 14394, 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 38643* 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 38644* 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 38645* 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 38646* 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)

21.31.6  Finite multiplication of numbers and finite multiplication of functions

Theoremfmul01 38647* 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 38648* 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 38649* induction step for the proof of fmuldfeq 38650. (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 38650* 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 38651* Given a finite multiplication of values betweeen 0 and 1, a value larger than its frist 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 38652* 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 38653* 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 38654* A continuous complex function restricted to a subset is continuous, using "map to" notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑥𝐹    &   (𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐶𝐴)       (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) ∈ (𝐶cn𝐵))

Theoremrrpsscn 38655 The positive reals are a subset of the complex numbers. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
+ ⊆ ℂ

Theoremmulc1cncfg 38656* A version of mulc1cncf 22516 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 38657* 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 38658* If 𝐹 is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 22533. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑥𝐹    &   (𝜑𝐹 ∈ (𝐴cn→ℂ))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑥𝐴 ↦ ((𝐹𝑥)↑𝑁)) ∈ (𝐴cn→ℂ))

Theoremprodeq2ad 38659* Equality deduction for product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theoremfprodsplit1 38660* 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 38661* 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 38662* 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 38663* 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 38664* * Induction step for mccl 38665. (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 38665* 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 38666* 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 38667* 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))

21.31.7  Limits

Theoremclim1fr1 38668* 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 38669* 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 38670* 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 38671* A version of climmul 14211 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 38672* 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 38673* A bounded monotonic non increasing 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 38674* The subsequence index 𝐼 has the expected properties: it belongs to the same upper integers as the original index, and it is always larger or equal than 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 38675* 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 38676* A version of climrec 38670 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 38677* 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 38678* A version of climinf 38673 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 38679* 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 38680 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 38681* 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 38682* A version of climadd 14210 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 38683* 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 38684* 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 38685 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 38686* 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 38687 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 23463 and limccnp 23461, 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 38688 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 38689* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 14073, 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 38690* 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 38691* Limit of constant function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑𝐵 ∈ (𝐹 lim 𝐶))

Theoremrexlim2d 38692* Inference removing two restricted quantifiers. Same as rexlimdvv 3019, 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 38693* Limit of the identity function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   𝐹 = (𝑥𝐴𝑥)    &   (𝜑𝑋 ∈ ℂ)       (𝜑𝑋 ∈ (𝐹 lim 𝑋))

Theoremdivcnvg 38694* 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 38695* 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 38696 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 38697* 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 38698 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 38699 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 38700 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)

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