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Theorem List for Metamath Proof Explorer - 39601-39700   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.31.5  Finite sums

Theoremfsumclf 39601* Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsumcl 14445 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 39602* Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsummulc1 14498 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 39603* 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 39604* 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 39605* 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 39606* Re-index a finite sum using a bijection. Same as fsumf1o 14435, 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 39607* Sum over a disjoint indexed union, intersected with a finite set 𝐷. Similar to fsumiun 14534, 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 39608* 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 39609* 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 39610* 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 39611* 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)

20.31.6  Finite multiplication of numbers and finite multiplication of functions

Theoremfmul01 39612* 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 39613* 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 39614* induction step for the proof of fmuldfeq 39615. (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 39615* 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 39616* 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 39617* 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 39618* 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 39619* 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 39620 The positive reals are a subset of the complex numbers. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
+ ⊆ ℂ

Theoremmulc1cncfg 39621* A version of mulc1cncf 22689 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 39622* 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 39623* If 𝐹 is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 22706. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑥𝐹    &   (𝜑𝐹 ∈ (𝐴cn→ℂ))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑥𝐴 ↦ ((𝐹𝑥)↑𝑁)) ∈ (𝐴cn→ℂ))

Theoremprodeq2ad 39624* Equality deduction for product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theoremfprodsplit1 39625* 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 39626* 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 39627* 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 39628* 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 39629* * Induction step for mccl 39630. (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 39630* 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 39631* 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 39632* 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))

20.31.7  Limits

Theoremclim1fr1 39633* 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 39634* 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 39635* 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 39636* A version of climmul 14344 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 39637* 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 39638* 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 39639* 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 39640* 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 39641* A version of climrec 39635 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 39642* 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 39643* A version of climinf 39638 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 39644* 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 39645 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 39646* 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 39647* A version of climadd 14343 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 39648* 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 39649* 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 39650 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 39651* 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 39652 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 23638 and limccnp 23636, 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 39653 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 39654* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 14206, 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 39655* 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 39656* Limit of constant function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑𝐵 ∈ (𝐹 lim 𝐶))

Theoremrexlim2d 39657* Inference removing two restricted quantifiers. Same as rexlimdvv 3033, 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 39658* Limit of the identity function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   𝐹 = (𝑥𝐴𝑥)    &   (𝜑𝑋 ∈ ℂ)       (𝜑𝑋 ∈ (𝐹 lim 𝑋))

Theoremdivcnvg 39659* 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 39660* 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 39661 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 39662* 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 39663 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 39664 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 39665 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 39666 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 39667* The domain of a map-to function with a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ (𝐹 lim 𝐷))       (𝜑𝐴 ⊆ ℂ)

Theoremclim2f 39668* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 14206. Similar to clim2 14216, 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 39669 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 39670 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 39671* 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))

Theoremlptre2pt 39672* If a set in the real line has a limit point than it contains two distinct points that are closer than a given distance. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐽 = (topGen‘ran (,))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → ((limPt‘𝐽)‘𝐴) ≠ ∅)    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ (abs‘(𝑥𝑦)) < 𝐸))

Theoremlimsupre 39673* If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑 → sup(𝐵, ℝ*, < ) = +∞)    &   (𝜑𝐹:𝐵⟶ℝ)    &   (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))       (𝜑 → (lim sup‘𝐹) ∈ ℝ)

Theoremlimcresiooub 39674 The left limit doesn't change if the function is restricted to a smaller open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 < 𝐶)    &   (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴)    &   (𝜑𝐷 ∈ ℝ*)    &   (𝜑𝐷𝐵)       (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) lim 𝐶) = ((𝐹 ↾ (𝐷(,)𝐶)) lim 𝐶))

Theoremlimcresioolb 39675 The right limit doesn't change if the function is restricted to a smaller open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 < 𝐶)    &   (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴)    &   (𝜑𝐷 ∈ ℝ*)    &   (𝜑𝐶𝐷)       (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) lim 𝐵) = ((𝐹 ↾ (𝐵(,)𝐷)) lim 𝐵))

Theoremlimcleqr 39676 If the left and the right limits are equal, the limit of the function exits and the three limits coincide. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐽 = (topGen‘ran (,))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) lim 𝐵))    &   (𝜑𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) lim 𝐵))    &   (𝜑𝐿 = 𝑅)       (𝜑𝐿 ∈ (𝐹 lim 𝐵))

Theoremlptioo2cn 39677 The upper bound of an open interval is a limit point of the interval, wirth respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐽 = (TopOpen‘ℂfld)    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))

Theoremlptioo1cn 39678 The lower bound of an open interval is a limit point of the interval, wirth respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐽 = (TopOpen‘ℂfld)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))

Theoremneglimc 39679* Limit of the negative function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴 ↦ -𝐵)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ (𝐹 lim 𝐷))       (𝜑 → -𝐶 ∈ (𝐺 lim 𝐷))

Theoremaddlimc 39680* Sum of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ (𝐵 + 𝐶))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)    &   (𝜑𝐸 ∈ (𝐹 lim 𝐷))    &   (𝜑𝐼 ∈ (𝐺 lim 𝐷))       (𝜑 → (𝐸 + 𝐼) ∈ (𝐻 lim 𝐷))

Theorem0ellimcdiv 39681* If the numerator converges to 0 and the denominator converges to non zero then the fraction converges to 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ (𝐵 / 𝐶))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ (ℂ ∖ {0}))    &   (𝜑 → 0 ∈ (𝐹 lim 𝐸))    &   (𝜑𝐷 ∈ (𝐺 lim 𝐸))    &   (𝜑𝐷 ≠ 0)       (𝜑 → 0 ∈ (𝐻 lim 𝐸))

Theoremclim2cf 39682* Express the predicate 𝐹 converges to 𝐴. Similar to clim2 14216, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → (𝐹𝐴 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝑥))

Theoremlimclner 39683 For a limit point, both from the left and from the right, of the domain, the limit of the function exits only if the left and the right limits are equal. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐽 = (topGen‘ran (,))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))))    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞))))    &   (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) lim 𝐵))    &   (𝜑𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) lim 𝐵))    &   (𝜑𝐿𝑅)       (𝜑 → (𝐹 lim 𝐵) = ∅)

Theoremsublimc 39684* Subtraction of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ (𝐵𝐶))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)    &   (𝜑𝐸 ∈ (𝐹 lim 𝐷))    &   (𝜑𝐼 ∈ (𝐺 lim 𝐷))       (𝜑 → (𝐸𝐼) ∈ (𝐻 lim 𝐷))

Theoremreclimc 39685* Limit of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴 ↦ (1 / 𝐵))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (ℂ ∖ {0}))    &   (𝜑𝐶 ∈ (𝐹 lim 𝐷))    &   (𝜑𝐶 ≠ 0)       (𝜑 → (1 / 𝐶) ∈ (𝐺 lim 𝐷))

Theoremclim0cf 39686* Express the predicate 𝐹 converges to 0. Similar to clim 14206, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘𝐵) < 𝑥))

Theoremlimclr 39687 For a limit point, both from the left and from the right, of the domain, the limit of the function exits only if the left and the right limits are equal. In this case, the three limits coincide. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐽 = (topGen‘ran (,))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))))    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞))))    &   (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) lim 𝐵))    &   (𝜑𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) lim 𝐵))       (𝜑 → (((𝐹 lim 𝐵) ≠ ∅ ↔ 𝐿 = 𝑅) ∧ (𝐿 = 𝑅𝐿 ∈ (𝐹 lim 𝐵))))

Theoremdivlimc 39688* Limit of the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ (𝐵 / 𝐶))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ (ℂ ∖ {0}))    &   (𝜑𝑋 ∈ (𝐹 lim 𝐷))    &   (𝜑𝑌 ∈ (𝐺 lim 𝐷))    &   (𝜑𝑌 ≠ 0)    &   ((𝜑𝑥𝐴) → 𝐶 ≠ 0)       (𝜑 → (𝑋 / 𝑌) ∈ (𝐻 lim 𝐷))

Theoremexpfac 39689* Factorial grows faster than exponential. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → 𝐹 ⇝ 0)

Theoremclimconstmpt 39690* A constant sequence converges to its value. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴 ∈ ℂ)       (𝜑 → (𝑥𝑍𝐴) ⇝ 𝐴)

Theoremclimresmpt 39691* A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑍 = (ℤ𝑀)    &   𝐹 = (𝑥𝑍𝐴)    &   (𝜑𝑁𝑍)    &   𝐺 = (𝑥 ∈ (ℤ𝑁) ↦ 𝐴)       (𝜑 → (𝐺𝐵𝐹𝐵))

Theoremclimsubmpt 39692* Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑘𝑍𝐴) ⇝ 𝐶)    &   (𝜑 → (𝑘𝑍𝐵) ⇝ 𝐷)       (𝜑 → (𝑘𝑍 ↦ (𝐴𝐵)) ⇝ (𝐶𝐷))

Theoremclimsubc2mpt 39693* Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → (𝑘𝑍𝐴) ⇝ 𝐶)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝑘𝑍 ↦ (𝐴𝐵)) ⇝ (𝐶𝐵))

Theoremclimsubc1mpt 39694* Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑘𝑍𝐵) ⇝ 𝐶)       (𝜑 → (𝑘𝑍 ↦ (𝐴𝐵)) ⇝ (𝐴𝐶))

Theoremfnlimfv 39695* The value of the limit function 𝐺 at any point of its domain 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐷    &   𝑥𝐹    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝑋𝐷)       (𝜑 → (𝐺𝑋) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))

Theoremclimreclf 39696* The limit of a convergent real sequence is real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)       (𝜑𝐴 ∈ ℝ)

Theoremclimeldmeq 39697* Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ))

Theoremclimf2 39698* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 14206, but without the disjoint var constraint 𝜑𝑘 and 𝐹𝑘. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))

Theoremfnlimcnv 39699* The sequence of function values converges to the value of the limit function 𝐺 at any point of its domain 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝑋𝐷)       (𝜑 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋)) ⇝ (𝐺𝑋))

Theoremclimeldmeqmpt 39700* Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑅)    &   (𝜑𝑍𝐴)    &   ((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑𝐶𝑆)    &   (𝜑𝑍𝐶)    &   ((𝜑𝑘𝐶) → 𝐷𝑊)    &   ((𝜑𝑘𝑍) → 𝐵 = 𝐷)       (𝜑 → ((𝑘𝐴𝐵) ∈ dom ⇝ ↔ (𝑘𝐶𝐷) ∈ dom ⇝ ))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
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