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Type | Label | Description |
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Statement | ||
Theorem | taylply2 26301* | The Taylor polynomial is a polynomial of degree (at most) 𝑁. This version of taylply 26303 shows that the coefficients of 𝑇 are in a subring of the complex numbers. (Contributed by Mario Carneiro, 1-Jan-2017.) Avoid ax-mulf 11218. (Revised by GG, 30-Apr-2025.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) & ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) & ⊢ (𝜑 → 𝐷 ∈ (SubRing‘ℂfld)) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑇 ∈ (Poly‘𝐷) ∧ (deg‘𝑇) ≤ 𝑁)) | ||
Theorem | taylply2OLD 26302* | Obsolete version of taylply2 26301 as of 30-Apr-2025. (Contributed by Mario Carneiro, 1-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) & ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) & ⊢ (𝜑 → 𝐷 ∈ (SubRing‘ℂfld)) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑇 ∈ (Poly‘𝐷) ∧ (deg‘𝑇) ≤ 𝑁)) | ||
Theorem | taylply 26303 | The Taylor polynomial is a polynomial of degree (at most) 𝑁. (Contributed by Mario Carneiro, 31-Dec-2016.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) & ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) ⇒ ⊢ (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁)) | ||
Theorem | dvtaylp 26304 | The derivative of the Taylor polynomial is the Taylor polynomial of the derivative of the function. (Contributed by Mario Carneiro, 31-Dec-2016.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 1))) ⇒ ⊢ (𝜑 → (ℂ D ((𝑁 + 1)(𝑆 Tayl 𝐹)𝐵)) = (𝑁(𝑆 Tayl (𝑆 D 𝐹))𝐵)) | ||
Theorem | dvntaylp 26305 | The 𝑀-th derivative of the Taylor polynomial is the Taylor polynomial of the 𝑀-th derivative of the function. (Contributed by Mario Carneiro, 1-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀))) ⇒ ⊢ (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = (𝑁(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)) | ||
Theorem | dvntaylp0 26306 | The first 𝑁 derivatives of the Taylor polynomial at 𝐵 match the derivatives of the function from which it is derived. (Contributed by Mario Carneiro, 1-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → 𝑀 ∈ (0...𝑁)) & ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) & ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) ⇒ ⊢ (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵)) | ||
Theorem | taylthlem1 26307* | Lemma for taylth 26310. This is the main part of Taylor's theorem, except for the induction step, which is supposed to be proven using L'Hôpital's rule. However, since our proof of L'Hôpital assumes that 𝑆 = ℝ, we can only do this part generically, and for taylth 26310 itself we must restrict to ℝ. (Contributed by Mario Carneiro, 1-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) = 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) & ⊢ 𝑅 = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑥) − (𝑇‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) & ⊢ ((𝜑 ∧ (𝑛 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛))) limℂ 𝐵))) → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑛 + 1)))) limℂ 𝐵)) ⇒ ⊢ (𝜑 → 0 ∈ (𝑅 limℂ 𝐵)) | ||
Theorem | taylthlem2 26308* | Lemma for taylth 26310. (Contributed by Mario Carneiro, 1-Jan-2017.) Avoid ax-mulf 11218. (Revised by GG, 19-Apr-2025.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → dom ((ℝ D𝑛 𝐹)‘𝑁) = 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ 𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵) & ⊢ (𝜑 → 𝑀 ∈ (1..^𝑁)) & ⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀))) limℂ 𝐵)) ⇒ ⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑀 + 1)))) limℂ 𝐵)) | ||
Theorem | taylthlem2OLD 26309* | Obsolete version of taylthlem2 26308 as of 30-Apr-2025. (Contributed by Mario Carneiro, 1-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → dom ((ℝ D𝑛 𝐹)‘𝑁) = 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ 𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵) & ⊢ (𝜑 → 𝑀 ∈ (1..^𝑁)) & ⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀))) limℂ 𝐵)) ⇒ ⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑀 + 1)))) limℂ 𝐵)) | ||
Theorem | taylth 26310* | Taylor's theorem. The Taylor polynomial of a 𝑁-times differentiable function is such that the error term goes to zero faster than (𝑥 − 𝐵)↑𝑁. This is Metamath 100 proof #35. (Contributed by Mario Carneiro, 1-Jan-2017.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → dom ((ℝ D𝑛 𝐹)‘𝑁) = 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ 𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵) & ⊢ 𝑅 = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑥) − (𝑇‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) ⇒ ⊢ (𝜑 → 0 ∈ (𝑅 limℂ 𝐵)) | ||
Syntax | culm 26311 | Extend class notation to include the uniform convergence predicate. |
class ⇝𝑢 | ||
Definition | df-ulm 26312* | Define the uniform convergence of a sequence of functions. Here 𝐹(⇝𝑢‘𝑆)𝐺 if 𝐹 is a sequence of functions 𝐹(𝑛), 𝑛 ∈ ℕ defined on 𝑆 and 𝐺 is a function on 𝑆, and for every 0 < 𝑥 there is a 𝑗 such that the functions 𝐹(𝑘) for 𝑗 ≤ 𝑘 are all uniformly within 𝑥 of 𝐺 on the domain 𝑆. Compare with df-clim 15464. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ ⇝𝑢 = (𝑠 ∈ V ↦ {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}) | ||
Theorem | ulmrel 26313 | The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ Rel (⇝𝑢‘𝑆) | ||
Theorem | ulmscl 26314 | Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) | ||
Theorem | ulmval 26315* | Express the predicate: The sequence of functions 𝐹 converges uniformly to 𝐺 on 𝑆. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) | ||
Theorem | ulmcl 26316 | Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | ||
Theorem | ulmf 26317* | Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) | ||
Theorem | ulmpm 26318 | Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ)) | ||
Theorem | ulmf2 26319 | Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.) |
⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | ||
Theorem | ulm2 26320* | Simplify ulmval 26315 when 𝐹 and 𝐺 are known to be functions. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) & ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = 𝐵) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴) & ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)) | ||
Theorem | ulmi 26321* | The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) & ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = 𝐵) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴) & ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶) | ||
Theorem | ulmclm 26322* | A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐻 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘)‘𝐴) = (𝐻‘𝑘)) & ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐺‘𝐴)) | ||
Theorem | ulmres 26323 | A sequence of functions converges iff the tail of the sequence converges (for any finite cutoff). (Contributed by Mario Carneiro, 24-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ (𝐹 ↾ 𝑊)(⇝𝑢‘𝑆)𝐺)) | ||
Theorem | ulmshftlem 26324* | Lemma for ulmshft 26325. (Contributed by Mario Carneiro, 24-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 = (ℤ≥‘(𝑀 + 𝐾)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) & ⊢ (𝜑 → 𝐻 = (𝑛 ∈ 𝑊 ↦ (𝐹‘(𝑛 − 𝐾)))) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐻(⇝𝑢‘𝑆)𝐺)) | ||
Theorem | ulmshft 26325* | A sequence of functions converges iff the shifted sequence converges. (Contributed by Mario Carneiro, 24-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 = (ℤ≥‘(𝑀 + 𝐾)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) & ⊢ (𝜑 → 𝐻 = (𝑛 ∈ 𝑊 ↦ (𝐹‘(𝑛 − 𝐾)))) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ 𝐻(⇝𝑢‘𝑆)𝐺)) | ||
Theorem | ulm0 26326 | Every function converges uniformly on the empty set. (Contributed by Mario Carneiro, 3-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) & ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) ⇒ ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹(⇝𝑢‘𝑆)𝐺) | ||
Theorem | ulmuni 26327 | A sequence of functions uniformly converges to at most one limit. (Contributed by Mario Carneiro, 5-Jul-2017.) |
⊢ ((𝐹(⇝𝑢‘𝑆)𝐺 ∧ 𝐹(⇝𝑢‘𝑆)𝐻) → 𝐺 = 𝐻) | ||
Theorem | ulmdm 26328 | Two ways to express that a function has a limit. (The expression ((⇝𝑢‘𝑆)‘𝐹) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Mario Carneiro, 5-Jul-2017.) |
⊢ (𝐹 ∈ dom (⇝𝑢‘𝑆) ↔ 𝐹(⇝𝑢‘𝑆)((⇝𝑢‘𝑆)‘𝐹)) | ||
Theorem | ulmcaulem 26329* | Lemma for ulmcau 26330 and ulmcau2 26331: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 15334. (Contributed by Mario Carneiro, 1-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) | ||
Theorem | ulmcau 26330* | A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every 0 < 𝑥 there is a 𝑗 such that for all 𝑗 ≤ 𝑘 the functions 𝐹(𝑘) and 𝐹(𝑗) are uniformly within 𝑥 of each other on 𝑆. This is the four-quantifier version, see ulmcau2 26331 for the more conventional five-quantifier version. (Contributed by Mario Carneiro, 1-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom (⇝𝑢‘𝑆) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)) | ||
Theorem | ulmcau2 26331* | A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every 0 < 𝑥 there is a 𝑗 such that for all 𝑗 ≤ 𝑘, 𝑚 the functions 𝐹(𝑘) and 𝐹(𝑚) are uniformly within 𝑥 of each other on 𝑆. (Contributed by Mario Carneiro, 1-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom (⇝𝑢‘𝑆) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) | ||
Theorem | ulmss 26332* | A uniform limit of functions is still a uniform limit if restricted to a subset. (Contributed by Mario Carneiro, 3-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑊) & ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴)(⇝𝑢‘𝑆)𝐺) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))(⇝𝑢‘𝑇)(𝐺 ↾ 𝑇)) | ||
Theorem | ulmbdd 26333* | A uniform limit of bounded functions is bounded. (Contributed by Mario Carneiro, 27-Feb-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘((𝐹‘𝑘)‘𝑧)) ≤ 𝑥) & ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘(𝐺‘𝑧)) ≤ 𝑥) | ||
Theorem | ulmcn 26334 | A uniform limit of continuous functions is continuous. (Contributed by Mario Carneiro, 27-Feb-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(𝑆–cn→ℂ)) & ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) ⇒ ⊢ (𝜑 → 𝐺 ∈ (𝑆–cn→ℂ)) | ||
Theorem | ulmdvlem1 26335* | Lemma for ulmdv 26338. (Contributed by Mario Carneiro, 3-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑋)) & ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) & ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) & ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝜓) → 𝑈 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝜓) → 𝑈 < 𝑊) & ⊢ ((𝜑 ∧ 𝜓) → (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) ⊆ 𝑋) & ⊢ ((𝜑 ∧ 𝜓) → (abs‘(𝑌 − 𝐶)) < 𝑈) & ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝜓) → ∀𝑚 ∈ (ℤ≥‘𝑁)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑅 / 2) / 2)) & ⊢ ((𝜑 ∧ 𝜓) → (abs‘(((𝑆 D (𝐹‘𝑁))‘𝐶) − (𝐻‘𝐶))) < (𝑅 / 2)) & ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ≠ 𝐶) & ⊢ ((𝜑 ∧ 𝜓) → ((abs‘(𝑌 − 𝐶)) < 𝑊 → (abs‘(((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶))) < ((𝑅 / 2) / 2))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − (𝐻‘𝐶))) < 𝑅) | ||
Theorem | ulmdvlem2 26336* | Lemma for ulmdv 26338. (Contributed by Mario Carneiro, 8-May-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑋)) & ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) & ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) ⇒ ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋) | ||
Theorem | ulmdvlem3 26337* | Lemma for ulmdv 26338. (Contributed by Mario Carneiro, 8-May-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑋)) & ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) & ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) ⇒ ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧)) | ||
Theorem | ulmdv 26338* | If 𝐹 is a sequence of differentiable functions on 𝑋 which converge pointwise to 𝐺, and the derivatives of 𝐹(𝑛) converge uniformly to 𝐻, then 𝐺 is differentiable with derivative 𝐻. (Contributed by Mario Carneiro, 27-Feb-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑋)) & ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) & ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) ⇒ ⊢ (𝜑 → (𝑆 D 𝐺) = 𝐻) | ||
Theorem | mtest 26339* | The Weierstrass M-test. If 𝐹 is a sequence of functions which are uniformly bounded by the convergent sequence 𝑀(𝑘), then the series generated by the sequence 𝐹 converges uniformly. (Contributed by Mario Carneiro, 3-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → (abs‘((𝐹‘𝑘)‘𝑧)) ≤ (𝑀‘𝑘)) & ⊢ (𝜑 → seq𝑁( + , 𝑀) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → seq𝑁( ∘f + , 𝐹) ∈ dom (⇝𝑢‘𝑆)) | ||
Theorem | mtestbdd 26340* | Given the hypotheses of the Weierstrass M-test, the convergent function of the sequence is uniformly bounded. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → (abs‘((𝐹‘𝑘)‘𝑧)) ≤ (𝑀‘𝑘)) & ⊢ (𝜑 → seq𝑁( + , 𝑀) ∈ dom ⇝ ) & ⊢ (𝜑 → seq𝑁( ∘f + , 𝐹)(⇝𝑢‘𝑆)𝑇) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘(𝑇‘𝑧)) ≤ 𝑥) | ||
Theorem | mbfulm 26341 | A uniform limit of measurable functions is measurable. (This is just a corollary of the fact that a pointwise limit of measurable functions is measurable, see mbflim 25596.) (Contributed by Mario Carneiro, 18-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶MblFn) & ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) ⇒ ⊢ (𝜑 → 𝐺 ∈ MblFn) | ||
Theorem | iblulm 26342 | A uniform limit of integrable functions is integrable. (Contributed by Mario Carneiro, 3-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶𝐿1) & ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) & ⊢ (𝜑 → (vol‘𝑆) ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝐿1) | ||
Theorem | itgulm 26343* | A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶𝐿1) & ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) & ⊢ (𝜑 → (vol‘𝑆) ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ ∫𝑆((𝐹‘𝑘)‘𝑥) d𝑥) ⇝ ∫𝑆(𝐺‘𝑥) d𝑥) | ||
Theorem | itgulm2 26344* | A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑥 ∈ 𝑆 ↦ 𝐴) ∈ 𝐿1) & ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑥 ∈ 𝑆 ↦ 𝐴))(⇝𝑢‘𝑆)(𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝜑 → (vol‘𝑆) ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝑆 ↦ 𝐵) ∈ 𝐿1 ∧ (𝑘 ∈ 𝑍 ↦ ∫𝑆𝐴 d𝑥) ⇝ ∫𝑆𝐵 d𝑥)) | ||
Theorem | pserval 26345* | Value of the function 𝐺 that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) ⇒ ⊢ (𝑋 ∈ ℂ → (𝐺‘𝑋) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑋↑𝑚)))) | ||
Theorem | pserval2 26346* | Value of the function 𝐺 that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) ⇒ ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐺‘𝑋)‘𝑁) = ((𝐴‘𝑁) · (𝑋↑𝑁))) | ||
Theorem | psergf 26347* | The sequence of terms in the infinite sequence defining a power series for fixed 𝑋. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐺‘𝑋):ℕ0⟶ℂ) | ||
Theorem | radcnvlem1 26348* | Lemma for radcnvlt1 26353, radcnvle 26355. If 𝑋 is a point closer to zero than 𝑌 and the power series converges at 𝑌, then it converges absolutely at 𝑋, even if the terms in the sequence are multiplied by 𝑛. (Contributed by Mario Carneiro, 31-Mar-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑌 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝑋) < (abs‘𝑌)) & ⊢ (𝜑 → seq0( + , (𝐺‘𝑌)) ∈ dom ⇝ ) & ⊢ 𝐻 = (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) ⇒ ⊢ (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ ) | ||
Theorem | radcnvlem2 26349* | Lemma for radcnvlt1 26353, radcnvle 26355. If 𝑋 is a point closer to zero than 𝑌 and the power series converges at 𝑌, then it converges absolutely at 𝑋. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑌 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝑋) < (abs‘𝑌)) & ⊢ (𝜑 → seq0( + , (𝐺‘𝑌)) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → seq0( + , (abs ∘ (𝐺‘𝑋))) ∈ dom ⇝ ) | ||
Theorem | radcnvlem3 26350* | Lemma for radcnvlt1 26353, radcnvle 26355. If 𝑋 is a point closer to zero than 𝑌 and the power series converges at 𝑌, then it converges at 𝑋. (Contributed by Mario Carneiro, 31-Mar-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑌 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝑋) < (abs‘𝑌)) & ⊢ (𝜑 → seq0( + , (𝐺‘𝑌)) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → seq0( + , (𝐺‘𝑋)) ∈ dom ⇝ ) | ||
Theorem | radcnv0 26351* | Zero is always a convergent point for any power series. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) ⇒ ⊢ (𝜑 → 0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) | ||
Theorem | radcnvcl 26352* | The radius of convergence 𝑅 of an infinite series is a nonnegative extended real number. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ⇒ ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞)) | ||
Theorem | radcnvlt1 26353* | If 𝑋 is within the open disk of radius 𝑅 centered at zero, then the infinite series converges absolutely at 𝑋, and also converges when the series is multiplied by 𝑛. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝑋) < 𝑅) & ⊢ 𝐻 = (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) ⇒ ⊢ (𝜑 → (seq0( + , 𝐻) ∈ dom ⇝ ∧ seq0( + , (abs ∘ (𝐺‘𝑋))) ∈ dom ⇝ )) | ||
Theorem | radcnvlt2 26354* | If 𝑋 is within the open disk of radius 𝑅 centered at zero, then the infinite series converges at 𝑋. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝑋) < 𝑅) ⇒ ⊢ (𝜑 → seq0( + , (𝐺‘𝑋)) ∈ dom ⇝ ) | ||
Theorem | radcnvle 26355* | If 𝑋 is a convergent point of the infinite series, then 𝑋 is within the closed disk of radius 𝑅 centered at zero. Or, by contraposition, the series diverges at any point strictly more than 𝑅 from the origin. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → seq0( + , (𝐺‘𝑋)) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → (abs‘𝑋) ≤ 𝑅) | ||
Theorem | dvradcnv 26356* | The radius of convergence of the (formal) derivative 𝐻 of the power series 𝐺 is at least as large as the radius of convergence of 𝐺. (In fact they are equal, but we don't have as much use for the negative side of this claim.) (Contributed by Mario Carneiro, 31-Mar-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ 𝐻 = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 1) · (𝐴‘(𝑛 + 1))) · (𝑋↑𝑛))) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝑋) < 𝑅) ⇒ ⊢ (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ ) | ||
Theorem | pserulm 26357* | If 𝑆 is a region contained in a circle of radius 𝑀 < 𝑅, then the sequence of partial sums of the infinite series converges uniformly on 𝑆. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ 𝐻 = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 𝑀 < 𝑅) & ⊢ (𝜑 → 𝑆 ⊆ (◡abs “ (0[,]𝑀))) ⇒ ⊢ (𝜑 → 𝐻(⇝𝑢‘𝑆)𝐹) | ||
Theorem | psercn2 26358* | Since by pserulm 26357 the series converges uniformly, it is also continuous by ulmcn 26334. (Contributed by Mario Carneiro, 3-Mar-2015.) Avoid ax-mulf 11218. (Revised by GG, 16-Mar-2025.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ 𝐻 = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 𝑀 < 𝑅) & ⊢ (𝜑 → 𝑆 ⊆ (◡abs “ (0[,]𝑀))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ)) | ||
Theorem | psercn2OLD 26359* | Obsolete version of psercn2 26358 as of 16-Apr-2025. (Contributed by Mario Carneiro, 3-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ 𝐻 = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 𝑀 < 𝑅) & ⊢ (𝜑 → 𝑆 ⊆ (◡abs “ (0[,]𝑀))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ)) | ||
Theorem | psercnlem2 26360* | Lemma for psercn 26362. (Contributed by Mario Carneiro, 18-Mar-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅)) ⇒ ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀)) ∧ (◡abs “ (0[,]𝑀)) ⊆ 𝑆)) | ||
Theorem | psercnlem1 26361* | Lemma for psercn 26362. (Contributed by Mario Carneiro, 18-Mar-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) & ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) ⇒ ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅)) | ||
Theorem | psercn 26362* | An infinite series converges to a continuous function on the open disk of radius 𝑅, where 𝑅 is the radius of convergence of the series. (Contributed by Mario Carneiro, 4-Mar-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) & ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ)) | ||
Theorem | pserdvlem1 26363* | Lemma for pserdv 26365. (Contributed by Mario Carneiro, 7-May-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) & ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) ⇒ ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((((abs‘𝑎) + 𝑀) / 2) ∈ ℝ+ ∧ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2) ∧ (((abs‘𝑎) + 𝑀) / 2) < 𝑅)) | ||
Theorem | pserdvlem2 26364* | Lemma for pserdv 26365. (Contributed by Mario Carneiro, 7-May-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) & ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) & ⊢ 𝐵 = (0(ball‘(abs ∘ − ))(((abs‘𝑎) + 𝑀) / 2)) ⇒ ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (ℂ D (𝐹 ↾ 𝐵)) = (𝑦 ∈ 𝐵 ↦ Σ𝑘 ∈ ℕ0 (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑦↑𝑘)))) | ||
Theorem | pserdv 26365* | The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) & ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) & ⊢ 𝐵 = (0(ball‘(abs ∘ − ))(((abs‘𝑎) + 𝑀) / 2)) ⇒ ⊢ (𝜑 → (ℂ D 𝐹) = (𝑦 ∈ 𝑆 ↦ Σ𝑘 ∈ ℕ0 (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑦↑𝑘)))) | ||
Theorem | pserdv2 26366* | The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.) |
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) & ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) & ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) & ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) & ⊢ 𝐵 = (0(ball‘(abs ∘ − ))(((abs‘𝑎) + 𝑀) / 2)) ⇒ ⊢ (𝜑 → (ℂ D 𝐹) = (𝑦 ∈ 𝑆 ↦ Σ𝑘 ∈ ℕ ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))))) | ||
Theorem | abelthlem1 26367* | Lemma for abelth 26377. (Contributed by Mario Carneiro, 1-Apr-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) | ||
Theorem | abelthlem2 26368* | Lemma for abelth 26377. The peculiar region 𝑆, known as a Stolz angle , is a teardrop-shaped subset of the closed unit ball containing 1. Indeed, except for 1 itself, the rest of the Stolz angle is enclosed in the open unit ball. (Contributed by Mario Carneiro, 31-Mar-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝑀) & ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} ⇒ ⊢ (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1))) | ||
Theorem | abelthlem3 26369* | Lemma for abelth 26377. (Contributed by Mario Carneiro, 31-Mar-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝑀) & ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑋↑𝑛)))) ∈ dom ⇝ ) | ||
Theorem | abelthlem4 26370* | Lemma for abelth 26377. (Contributed by Mario Carneiro, 31-Mar-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝑀) & ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ⇒ ⊢ (𝜑 → 𝐹:𝑆⟶ℂ) | ||
Theorem | abelthlem5 26371* | Lemma for abelth 26377. (Contributed by Mario Carneiro, 1-Apr-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝑀) & ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) & ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) → seq0( + , (𝑘 ∈ ℕ0 ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))) ∈ dom ⇝ ) | ||
Theorem | abelthlem6 26372* | Lemma for abelth 26377. (Contributed by Mario Carneiro, 2-Apr-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝑀) & ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) & ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0) & ⊢ (𝜑 → 𝑋 ∈ (𝑆 ∖ {1})) ⇒ ⊢ (𝜑 → (𝐹‘𝑋) = ((1 − 𝑋) · Σ𝑛 ∈ ℕ0 ((seq0( + , 𝐴)‘𝑛) · (𝑋↑𝑛)))) | ||
Theorem | abelthlem7a 26373* | Lemma for abelth 26377. (Contributed by Mario Carneiro, 8-May-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝑀) & ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) & ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0) & ⊢ (𝜑 → 𝑋 ∈ (𝑆 ∖ {1})) ⇒ ⊢ (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋))))) | ||
Theorem | abelthlem7 26374* | Lemma for abelth 26377. (Contributed by Mario Carneiro, 2-Apr-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝑀) & ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) & ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0) & ⊢ (𝜑 → 𝑋 ∈ (𝑆 ∖ {1})) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑁)(abs‘(seq0( + , 𝐴)‘𝑘)) < 𝑅) & ⊢ (𝜑 → (abs‘(1 − 𝑋)) < (𝑅 / (Σ𝑛 ∈ (0...(𝑁 − 1))(abs‘(seq0( + , 𝐴)‘𝑛)) + 1))) ⇒ ⊢ (𝜑 → (abs‘(𝐹‘𝑋)) < ((𝑀 + 1) · 𝑅)) | ||
Theorem | abelthlem8 26375* | Lemma for abelth 26377. (Contributed by Mario Carneiro, 2-Apr-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝑀) & ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) & ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0) ⇒ ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅)) | ||
Theorem | abelthlem9 26376* | Lemma for abelth 26377. By adjusting the constant term, we can assume that the entire series converges to 0. (Contributed by Mario Carneiro, 1-Apr-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝑀) & ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ⇒ ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅)) | ||
Theorem | abelth 26377* | Abel's theorem. If the power series Σ𝑛 ∈ ℕ0𝐴(𝑛)(𝑥↑𝑛) is convergent at 1, then it is equal to the limit from "below", along a Stolz angle 𝑆 (note that the 𝑀 = 1 case of a Stolz angle is the real line [0, 1]). (Continuity on 𝑆 ∖ {1} follows more generally from psercn 26362.) (Contributed by Mario Carneiro, 2-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝑀) & ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ)) | ||
Theorem | abelth2 26378* | Abel's theorem, restricted to the [0, 1] interval. (Contributed by Mario Carneiro, 2-Apr-2015.) |
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) & ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ⇒ ⊢ (𝜑 → 𝐹 ∈ ((0[,]1)–cn→ℂ)) | ||
Theorem | efcn 26379 | The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.) |
⊢ exp ∈ (ℂ–cn→ℂ) | ||
Theorem | sincn 26380 | Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.) |
⊢ sin ∈ (ℂ–cn→ℂ) | ||
Theorem | coscn 26381 | Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.) |
⊢ cos ∈ (ℂ–cn→ℂ) | ||
Theorem | reeff1olem 26382* | Lemma for reeff1o 26383. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈) | ||
Theorem | reeff1o 26383 | The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.) |
⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+ | ||
Theorem | reefiso 26384 | The exponential function on the reals determines an isomorphism from reals onto positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) (Revised by Mario Carneiro, 11-Mar-2014.) |
⊢ (exp ↾ ℝ) Isom < , < (ℝ, ℝ+) | ||
Theorem | efcvx 26385 | The exponential function on the reals is a strictly convex function. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (exp‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) < ((𝑇 · (exp‘𝐴)) + ((1 − 𝑇) · (exp‘𝐵)))) | ||
Theorem | reefgim 26386 | The exponential function is a group isomorphism from the group of reals under addition to the group of positive reals under multiplication. (Contributed by Mario Carneiro, 21-Jun-2015.) (Revised by Thierry Arnoux, 30-Jun-2019.) |
⊢ 𝑃 = ((mulGrp‘ℂfld) ↾s ℝ+) ⇒ ⊢ (exp ↾ ℝ) ∈ (ℝfld GrpIso 𝑃) | ||
Theorem | pilem1 26387 | Lemma for pire 26392, pigt2lt4 26390 and sinpi 26391. (Contributed by Mario Carneiro, 9-May-2014.) |
⊢ (𝐴 ∈ (ℝ+ ∩ (◡sin “ {0})) ↔ (𝐴 ∈ ℝ+ ∧ (sin‘𝐴) = 0)) | ||
Theorem | pilem2 26388 | Lemma for pire 26392, pigt2lt4 26390 and sinpi 26391. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by AV, 14-Sep-2020.) |
⊢ (𝜑 → 𝐴 ∈ (2(,)4)) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → (sin‘𝐴) = 0) & ⊢ (𝜑 → (sin‘𝐵) = 0) ⇒ ⊢ (𝜑 → ((π + 𝐴) / 2) ≤ 𝐵) | ||
Theorem | pilem3 26389 | Lemma for pire 26392, pigt2lt4 26390 and sinpi 26391. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) (Revised by AV, 14-Sep-2020.) (Proof shortened by BJ, 30-Jun-2022.) |
⊢ (π ∈ (2(,)4) ∧ (sin‘π) = 0) | ||
Theorem | pigt2lt4 26390 | π is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.) |
⊢ (2 < π ∧ π < 4) | ||
Theorem | sinpi 26391 | The sine of π is 0. (Contributed by Paul Chapman, 23-Jan-2008.) |
⊢ (sin‘π) = 0 | ||
Theorem | pire 26392 | π is a real number. (Contributed by Paul Chapman, 23-Jan-2008.) |
⊢ π ∈ ℝ | ||
Theorem | picn 26393 | π is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ π ∈ ℂ | ||
Theorem | pipos 26394 | π is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.) |
⊢ 0 < π | ||
Theorem | pirp 26395 | π is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ π ∈ ℝ+ | ||
Theorem | negpicn 26396 | -π is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -π ∈ ℂ | ||
Theorem | sinhalfpilem 26397 | Lemma for sinhalfpi 26402 and coshalfpi 26403. (Contributed by Paul Chapman, 23-Jan-2008.) |
⊢ ((sin‘(π / 2)) = 1 ∧ (cos‘(π / 2)) = 0) | ||
Theorem | halfpire 26398 | π / 2 is real. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (π / 2) ∈ ℝ | ||
Theorem | neghalfpire 26399 | -π / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -(π / 2) ∈ ℝ | ||
Theorem | neghalfpirx 26400 | -π / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -(π / 2) ∈ ℝ* |
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