HomeTheorem List (p. 464 of 507)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | ressiooinf 46301 | 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.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑆 = inf(𝐴, ℝ*, < ) & ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) & ⊢ 𝐼 = (𝑆(,)+∞) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐼) | ||
| Theorem | iocleubd 46302 | 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.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) ⇒ ⊢ (𝜑 → 𝐶 ≤ 𝐵) | ||
| Theorem | uzinico 46303 | 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.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → 𝑍 = (ℤ ∩ (𝑀[,)+∞))) | ||
| Theorem | preimaiocmnf 46304* | Preimage of a right-closed interval, unbounded below. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (◡𝐹 “ (-∞(,]𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≤ 𝐵}) | ||
| Theorem | uzinico2 46305 | 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.) |
| ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞))) | ||
| Theorem | uzinico3 46306 | 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.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) | ||
| Theorem | dmico 46307 | The domain of the closed-below, open-above interval function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ dom [,) = (ℝ* × ℝ*) | ||
| Theorem | ndmico 46308 | 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) = ∅) | ||
| Theorem | uzubioo 46309* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) | ||
| Theorem | uzubico 46310* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ (𝑋[,)+∞)𝑘 ∈ 𝑍) | ||
| Theorem | uzubioo2 46311* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥(,)+∞)𝑘 ∈ 𝑍) | ||
| Theorem | uzubico2 46312* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥[,)+∞)𝑘 ∈ 𝑍) | ||
| Theorem | iocgtlbd 46313 | An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
| Theorem | xrtgioo2 46314 | The topology on the extended reals coincides with the standard topology on the reals, when restricted to ℝ. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾t ℝ) | ||
| Theorem | fsummulc1f 46315* | Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsummulc1 15843 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 · 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶)) | ||
| Theorem | fsumnncl 46316* | Closure of a nonempty, finite sum of positive integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℕ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℕ) | ||
| Theorem | fsumge0cl 46317* | The finite sum of nonnegative reals is a nonnegative real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) | ||
| Theorem | fsumf1of 46318* | Re-index a finite sum using a bijection. Same as fsumf1o 15781, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑛𝜑 & ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷) | ||
| Theorem | fsumiunss 46319* | Sum over a disjoint indexed union, intersected with a finite set 𝐷. Similar to fsumiun 15880, but here 𝐴 and 𝐵 need not be finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ Fin) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐷)𝐶 = Σ𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐵 ∩ 𝐷) ≠ ∅}Σ𝑘 ∈ (𝐵 ∩ 𝐷)𝐶) | ||
| Theorem | fsumreclf 46320* | Closure of a finite sum of reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) | ||
| Theorem | fsumlessf 46321* | A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) | ||
| Theorem | fsumsupp0 46322* | Finite sum of function values, for a function of finite support. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝐹 supp 0)(𝐹‘𝑘) = Σ𝑘 ∈ 𝐴 (𝐹‘𝑘)) | ||
| Theorem | fsumsermpt 46323* | A finite sum expressed in terms of a partial sum of an infinite series. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴) & ⊢ 𝐺 = seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
| Theorem | fmul01 46324* | 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.) |
| ⊢ Ⅎ𝑖𝐵 & ⊢ Ⅎ𝑖𝜑 & ⊢ 𝐴 = seq𝐿( · , 𝐵) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿)) & ⊢ (𝜑 → 𝐾 ∈ (𝐿...𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) ⇒ ⊢ (𝜑 → (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1)) | ||
| Theorem | fmulcl 46325* | 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 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) & ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑁) & ⊢ (𝜑 → 𝑁 ∈ (1...𝑀)) & ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌) & ⊢ (𝜑 → 𝑇 ∈ V) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑌) | ||
| Theorem | fmuldfeqlem1 46326* | induction step for the proof of fmuldfeq 46327. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
| ⊢ Ⅎ𝑓𝜑 & ⊢ Ⅎ𝑔𝜑 & ⊢ Ⅎ𝑡𝑌 & ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))) & ⊢ (𝜑 → 𝑇 ∈ V) & ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌) & ⊢ (𝜑 → 𝑁 ∈ (1...𝑀)) & ⊢ (𝜑 → (𝑁 + 1) ∈ (1...𝑀)) & ⊢ (𝜑 → ((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑁)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) ⇒ ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘(𝑁 + 1))‘𝑡) = (seq1( · , (𝐹‘𝑡))‘(𝑁 + 1))) | ||
| Theorem | fmuldfeq 46327* | 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.) |
| ⊢ Ⅎ𝑖𝜑 & ⊢ Ⅎ𝑡𝑌 & ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) & ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀) & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))) & ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀)) & ⊢ (𝜑 → 𝑇 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌) ⇒ ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡)) | ||
| Theorem | fmul01lt1lem1 46328* | Given a finite multiplication of values between 0 and 1, a value larger than its first element is larger the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
| ⊢ Ⅎ𝑖𝐵 & ⊢ Ⅎ𝑖𝜑 & ⊢ 𝐴 = seq𝐿( · , 𝐵) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → (𝐵‘𝐿) < 𝐸) ⇒ ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸) | ||
| Theorem | fmul01lt1lem2 46329* | Given a finite multiplication of values between 0 and 1, a value 𝐸 larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
| ⊢ Ⅎ𝑖𝐵 & ⊢ Ⅎ𝑖𝜑 & ⊢ 𝐴 = seq𝐿( · , 𝐵) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑀)) & ⊢ (𝜑 → (𝐵‘𝐽) < 𝐸) ⇒ ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸) | ||
| Theorem | fmul01lt1 46330* | Given a finite multiplication of values between 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
| ⊢ Ⅎ𝑖𝐵 & ⊢ Ⅎ𝑖𝜑 & ⊢ Ⅎ𝑗𝐴 & ⊢ 𝐴 = seq1( · , 𝐵) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐵:(1...𝑀)⟶ℝ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ (𝐵‘𝑖)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ≤ 1) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → ∃𝑗 ∈ (1...𝑀)(𝐵‘𝑗) < 𝐸) ⇒ ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸) | ||
| Theorem | cncfmptss 46331* | A continuous complex function restricted to a subset is continuous, using maps-to notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵)) | ||
| Theorem | rrpsscn 46332 | The positive reals are a subset of the complex numbers. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| ⊢ ℝ+ ⊆ ℂ | ||
| Theorem | mulc1cncfg 46333* | A version of mulc1cncf 25075 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 30-Jun-2017.) |
| ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ (𝐴–cn→ℂ)) | ||
| Theorem | infrglb 46334* | 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 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ, < ) < 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑧 < 𝐵)) | ||
| Theorem | expcnfg 46335* | If 𝐹 is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 25096. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) ∈ (𝐴–cn→ℂ)) | ||
| Theorem | prodeq2ad 46336* | Equality deduction for product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶) | ||
| Theorem | fprodsplit1 46337* | Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) | ||
| Theorem | fprodexp 46338* | Positive integer exponentiation of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵↑𝑁) = (∏𝑘 ∈ 𝐴 𝐵↑𝑁)) | ||
| Theorem | fprodabs2 46339* | The absolute value of a finite product . (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵)) | ||
| Theorem | fprod0 46340* | A finite product with a zero term is zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐶 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐾 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 = 0) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0) | ||
| Theorem | mccllem 46341* | * Induction step for mccl 46342. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶)) & ⊢ (𝜑 → 𝐵 ∈ (ℕ0 ↑m (𝐶 ∪ {𝐷}))) & ⊢ (𝜑 → ∀𝑏 ∈ (ℕ0 ↑m 𝐶)((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ) ⇒ ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) ∈ ℕ) | ||
| Theorem | mccl 46342* | A multinomial coefficient, in its standard domain, is a positive integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| ⊢ Ⅎ𝑘𝐵 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ (ℕ0 ↑m 𝐴)) ⇒ ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐴 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐴 (!‘(𝐵‘𝑘))) ∈ ℕ) | ||
| Theorem | fprodcnlem 46343* | A finite product of functions to complex numbers from a common topological space is continuous. Induction step. (Contributed by Glauco Siliprandi, 8-Apr-2021.) Avoid ax-mulf 11186. (Revised by GG, 19-Apr-2025.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑍 ⊆ 𝐴) & ⊢ (𝜑 → 𝑊 ∈ (𝐴 ∖ 𝑍)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑍 𝐵) ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | fprodcn 46344* | 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.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | clim1fr1 46345* | 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))) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → 𝐹 ⇝ 1) | ||
| Theorem | isumneg 46346* | Negation of a converging sum. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 -𝐴 = -Σ𝑘 ∈ 𝑍 𝐴) | ||
| Theorem | climrec 46347* | Limit of the reciprocal of a converging sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺 ⇝ 𝐴) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0})) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (1 / (𝐺‘𝑘))) & ⊢ (𝜑 → 𝐻 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴)) | ||
| Theorem | climmulf 46348* | A version of climmul 15691 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐺 & ⊢ Ⅎ𝑘𝐻 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → 𝐺 ⇝ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵)) | ||
| Theorem | climexp 46349* | The limit of natural powers, is the natural power of the limit. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐻 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐻 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁)) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐴↑𝑁)) | ||
| Theorem | climinf 46350* | 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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < )) | ||
| Theorem | climsuselem1 46351* | 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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → (𝐼‘𝑀) ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ≥‘((𝐼‘𝑘) + 1))) ⇒ ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐼‘𝐾) ∈ (ℤ≥‘𝐾)) | ||
| Theorem | climsuse 46352* | 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.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐺 & ⊢ Ⅎ𝑘𝐼 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → (𝐼‘𝑀) ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ≥‘((𝐼‘𝑘) + 1))) & ⊢ (𝜑 → 𝐺 ∈ 𝑌) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘(𝐼‘𝑘))) ⇒ ⊢ (𝜑 → 𝐺 ⇝ 𝐴) | ||
| Theorem | climrecf 46353* | A version of climrec 46347 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐺 & ⊢ Ⅎ𝑘𝐻 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺 ⇝ 𝐴) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0})) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (1 / (𝐺‘𝑘))) & ⊢ (𝜑 → 𝐻 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴)) | ||
| Theorem | climneg 46354* | Complex limit of the negative of a sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ⇒ ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -𝐴) | ||
| Theorem | climinff 46355* | A version of climinf 46350 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < )) | ||
| Theorem | climdivf 46356* | Limit of the ratio of two converging sequences. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐺 & ⊢ Ⅎ𝑘𝐻 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → 𝐺 ⇝ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0})) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) / (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐴 / 𝐵)) | ||
| Theorem | climreeq 46357 | 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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹𝑅𝐴 ↔ 𝐹 ⇝ 𝐴)) | ||
| Theorem | ellimciota 46358* | An explicit value for the limit, when the limit exists at a limit point. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐾)‘𝐴)) & ⊢ (𝜑 → (𝐹 limℂ 𝐵) ≠ ∅) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (𝜑 → (℩𝑥𝑥 ∈ (𝐹 limℂ 𝐵)) ∈ (𝐹 limℂ 𝐵)) | ||
| Theorem | climaddf 46359* | A version of climadd 15690 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐺 & ⊢ Ⅎ𝑘𝐻 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → 𝐺 ⇝ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐴 + 𝐵)) | ||
| Theorem | mullimc 46360* | Limit of the product of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 · 𝐶)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ (𝐹 limℂ 𝐷)) & ⊢ (𝜑 → 𝑌 ∈ (𝐺 limℂ 𝐷)) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) ∈ (𝐻 limℂ 𝐷)) | ||
| Theorem | ellimcabssub0 46361* | An equivalent condition for being a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) & ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐹 limℂ 𝐷) ↔ 0 ∈ (𝐺 limℂ 𝐷))) | ||
| Theorem | limcdm0 46362 | If a function has empty domain, every complex number is a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐹:∅⟶ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ℂ) | ||
| Theorem | islptre 46363* | 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 (,)) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐵 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑎 ∈ ℝ* ∀𝑏 ∈ ℝ* (𝐵 ∈ (𝑎(,)𝑏) → ((𝑎(,)𝑏) ∩ (𝐴 ∖ {𝐵})) ≠ ∅))) | ||
| Theorem | limccog 46364 | 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 𝐺 ∘ 𝐹 at 𝐴 is 𝐶. With respect to limcco 26063 and limccnp 26061, here we drop continuity assumptions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵})) & ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐴)) & ⊢ (𝜑 → 𝐶 ∈ (𝐺 limℂ 𝐵)) ⇒ ⊢ (𝜑 → 𝐶 ∈ ((𝐺 ∘ 𝐹) limℂ 𝐴)) | ||
| Theorem | limciccioolb 46365 | 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.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ) ⇒ ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐴) = (𝐹 limℂ 𝐴)) | ||
| Theorem | climf 46366* | Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 15552, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ (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 | mullimcf 46367* | Limit of the multiplication of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐺:𝐴⟶ℂ) & ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥) · (𝐺‘𝑥))) & ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐷)) & ⊢ (𝜑 → 𝐶 ∈ (𝐺 limℂ 𝐷)) ⇒ ⊢ (𝜑 → (𝐵 · 𝐶) ∈ (𝐻 limℂ 𝐷)) | ||
| Theorem | constlimc 46368* | Limit of constant function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐶)) | ||
| Theorem | rexlim2d 46369* | Inference removing two restricted quantifiers. Same as rexlimdvv 3220, but with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) | ||
| Theorem | idlimc 46370* | Limit of the identity function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑥) & ⊢ (𝜑 → 𝑋 ∈ ℂ) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐹 limℂ 𝑋)) | ||
| Theorem | divcnvg 46371* | 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 ∈ (ℤ≥‘𝑀) ↦ (𝐴 / 𝑛)) ⇝ 0) | ||
| Theorem | limcperiod 46372* | 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.) |
| ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) & ⊢ (𝜑 → 𝑇 ∈ ℂ) & ⊢ 𝐵 = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 + 𝑇)} & ⊢ (𝜑 → 𝐵 ⊆ dom 𝐹) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦)) & ⊢ (𝜑 → 𝐶 ∈ ((𝐹 ↾ 𝐴) limℂ 𝐷)) ⇒ ⊢ (𝜑 → 𝐶 ∈ ((𝐹 ↾ 𝐵) limℂ (𝐷 + 𝑇))) | ||
| Theorem | limcrecl 46373 | 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.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘𝐴)) & ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵)) ⇒ ⊢ (𝜑 → 𝐿 ∈ ℝ) | ||
| Theorem | sumnnodd 46374* | A series indexed by ℕ with only odd terms. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ (𝑘 / 2) ∈ ℕ) → (𝐹‘𝑘) = 0) & ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝐵) ⇒ ⊢ (𝜑 → (seq1( + , (𝑘 ∈ ℕ ↦ (𝐹‘((2 · 𝑘) − 1)))) ⇝ 𝐵 ∧ Σ𝑘 ∈ ℕ (𝐹‘𝑘) = Σ𝑘 ∈ ℕ (𝐹‘((2 · 𝑘) − 1)))) | ||
| Theorem | lptioo2 46375 | The upper bound of an open interval is a limit point of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) | ||
| Theorem | lptioo1 46376 | The lower bound of an open interval is a limit point of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) | ||
| Theorem | limcmptdm 46377* | The domain of a maps-to function with a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐷)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ ℂ) | ||
| Theorem | clim2f 46378* | Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 15552. Similar to clim2 15562, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ 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 | limcicciooub 46379 | 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.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ) ⇒ ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵) = (𝐹 limℂ 𝐵)) | ||
| Theorem | ltmod 46380 | A sufficient condition for a "less than" relationship for the mod operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴)) ⇒ ⊢ (𝜑 → (𝐶 mod 𝐵) < (𝐴 mod 𝐵)) | ||
| Theorem | islpcn 46381* | A characterization for a limit point for the standard topology on the complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝑃 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝑃 ∈ ((limPt‘(TopOpen‘ℂfld))‘𝑆) ↔ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)) | ||
| Theorem | lptre2pt 46382* | 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 (,)) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → ((limPt‘𝐽)‘𝐴) ≠ ∅) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (abs‘(𝑥 − 𝑦)) < 𝐸)) | ||
| Theorem | limsupre 46383* | If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.) |
| ⊢ (𝜑 → 𝐵 ⊆ ℝ) & ⊢ (𝜑 → sup(𝐵, ℝ*, < ) = +∞) & ⊢ (𝜑 → 𝐹:𝐵⟶ℝ) & ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) | ||
| Theorem | limcresiooub 46384 | The left limit doesn't change if the function is restricted to a smaller open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 < 𝐶) & ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ ℝ*) & ⊢ (𝜑 → 𝐷 ≤ 𝐵) ⇒ ⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) limℂ 𝐶) = ((𝐹 ↾ (𝐷(,)𝐶)) limℂ 𝐶)) | ||
| Theorem | limcresioolb 46385 | The right limit doesn't change if the function is restricted to a smaller open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 < 𝐶) & ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ≤ 𝐷) ⇒ ⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) limℂ 𝐵) = ((𝐹 ↾ (𝐵(,)𝐷)) limℂ 𝐵)) | ||
| Theorem | limcleqr 46386 | 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) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) & ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) & ⊢ (𝜑 → 𝐿 = 𝑅) ⇒ ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵)) | ||
| Theorem | lptioo2cn 46387 | 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) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) | ||
| Theorem | lptioo1cn 46388 | 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) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) | ||
| Theorem | neglimc 46389* | Limit of the negative function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ -𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐷)) ⇒ ⊢ (𝜑 → -𝐶 ∈ (𝐺 limℂ 𝐷)) | ||
| Theorem | addlimc 46390* | Sum of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐸 ∈ (𝐹 limℂ 𝐷)) & ⊢ (𝜑 → 𝐼 ∈ (𝐺 limℂ 𝐷)) ⇒ ⊢ (𝜑 → (𝐸 + 𝐼) ∈ (𝐻 limℂ 𝐷)) | ||
| Theorem | 0ellimcdiv 46391* | If the numerator converges to 0 and the denominator converges to a nonzero number, then the fraction converges to 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (ℂ ∖ {0})) & ⊢ (𝜑 → 0 ∈ (𝐹 limℂ 𝐸)) & ⊢ (𝜑 → 𝐷 ∈ (𝐺 limℂ 𝐸)) & ⊢ (𝜑 → 𝐷 ≠ 0) ⇒ ⊢ (𝜑 → 0 ∈ (𝐻 limℂ 𝐸)) | ||
| Theorem | clim2cf 46392* | Express the predicate 𝐹 converges to 𝐴. Similar to clim2 15562, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ 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 | limclner 46393 | 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) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵)))) & ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞)))) & ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) & ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) & ⊢ (𝜑 → 𝐿 ≠ 𝑅) ⇒ ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ∅) | ||
| Theorem | sublimc 46394* | Subtraction of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐸 ∈ (𝐹 limℂ 𝐷)) & ⊢ (𝜑 → 𝐼 ∈ (𝐺 limℂ 𝐷)) ⇒ ⊢ (𝜑 → (𝐸 − 𝐼) ∈ (𝐻 limℂ 𝐷)) | ||
| Theorem | reclimc 46395* | Limit of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (1 / 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (ℂ ∖ {0})) & ⊢ (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐷)) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → (1 / 𝐶) ∈ (𝐺 limℂ 𝐷)) | ||
| Theorem | clim0cf 46396* | Express the predicate 𝐹 converges to 0. Similar to clim 15552, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ 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 | limclr 46397 | 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) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵)))) & ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞)))) & ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) & ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) ⇒ ⊢ (𝜑 → (((𝐹 limℂ 𝐵) ≠ ∅ ↔ 𝐿 = 𝑅) ∧ (𝐿 = 𝑅 → 𝐿 ∈ (𝐹 limℂ 𝐵)))) | ||
| Theorem | divlimc 46398* | Limit of the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (ℂ ∖ {0})) & ⊢ (𝜑 → 𝑋 ∈ (𝐹 limℂ 𝐷)) & ⊢ (𝜑 → 𝑌 ∈ (𝐺 limℂ 𝐷)) & ⊢ (𝜑 → 𝑌 ≠ 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → (𝑋 / 𝑌) ∈ (𝐻 limℂ 𝐷)) | ||
| Theorem | expfac 46399* | Factorial grows faster than exponential. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) ⇒ ⊢ (𝐴 ∈ ℂ → 𝐹 ⇝ 0) | ||
| Theorem | climconstmpt 46400* | A constant sequence converges to its value. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴) ⇝ 𝐴) | ||
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