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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | aaliou3lem8 26401* | Lemma for aaliou3 26407. (Contributed by Stefan O'Rear, 20-Nov-2014.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℕ (2 · (2↑-(!‘(𝑥 + 1)))) ≤ (𝐵 / ((2↑(!‘𝑥))↑𝐴))) | ||
Theorem | aaliou3lem4 26402* | Lemma for aaliou3 26407. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) & ⊢ 𝐿 = Σ𝑏 ∈ ℕ (𝐹‘𝑏) & ⊢ 𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏)) ⇒ ⊢ 𝐿 ∈ ℝ | ||
Theorem | aaliou3lem5 26403* | Lemma for aaliou3 26407. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) & ⊢ 𝐿 = Σ𝑏 ∈ ℕ (𝐹‘𝑏) & ⊢ 𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏)) ⇒ ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℝ) | ||
Theorem | aaliou3lem6 26404* | Lemma for aaliou3 26407. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) & ⊢ 𝐿 = Σ𝑏 ∈ ℕ (𝐹‘𝑏) & ⊢ 𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏)) ⇒ ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) · (2↑(!‘𝐴))) ∈ ℤ) | ||
Theorem | aaliou3lem7 26405* | Lemma for aaliou3 26407. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) & ⊢ 𝐿 = Σ𝑏 ∈ ℕ (𝐹‘𝑏) & ⊢ 𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏)) ⇒ ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) ≠ 𝐿 ∧ (abs‘(𝐿 − (𝐻‘𝐴))) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) | ||
Theorem | aaliou3lem9 26406* | Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 20-Nov-2014.) |
⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) & ⊢ 𝐿 = Σ𝑏 ∈ ℕ (𝐹‘𝑏) & ⊢ 𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏)) ⇒ ⊢ ¬ 𝐿 ∈ 𝔸 | ||
Theorem | aaliou3 26407 | Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 23-Nov-2014.) |
⊢ Σ𝑘 ∈ ℕ (2↑-(!‘𝑘)) ∉ 𝔸 | ||
Syntax | ctayl 26408 | Taylor polynomial of a function. |
class Tayl | ||
Syntax | cana 26409 | The class of analytic functions. |
class Ana | ||
Definition | df-tayl 26410* | Define the Taylor polynomial or Taylor series of a function. TODO-AV: 𝑛 ∈ (ℕ0 ∪ {+∞}) should be replaced by 𝑛 ∈ ℕ0*. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ Tayl = (𝑠 ∈ {ℝ, ℂ}, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ (𝑛 ∈ (ℕ0 ∪ {+∞}), 𝑎 ∈ ∩ 𝑘 ∈ ((0[,]𝑛) ∩ ℤ)dom ((𝑠 D𝑛 𝑓)‘𝑘) ↦ ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑛) ∩ ℤ) ↦ (((((𝑠 D𝑛 𝑓)‘𝑘)‘𝑎) / (!‘𝑘)) · ((𝑥 − 𝑎)↑𝑘))))))) | ||
Definition | df-ana 26411* | Define the set of analytic functions, which are functions such that the Taylor series of the function at each point converges to the function in some neighborhood of the point. (Contributed by Mario Carneiro, 31-Dec-2016.) |
⊢ Ana = (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))}) | ||
Theorem | taylfvallem1 26412* | Lemma for taylfval 26414. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) ⇒ ⊢ (((𝜑 ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)) ∈ ℂ) | ||
Theorem | taylfvallem 26413* | Lemma for taylfval 26414. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ ℂ) → (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))) ⊆ ℂ) | ||
Theorem | taylfval 26414* |
Define the Taylor polynomial of a function. The constant Tayl is a
function of five arguments: 𝑆 is the base set with respect to
evaluate the derivatives (generally ℝ or
ℂ), 𝐹 is the
function we are approximating, at point 𝐵, to order 𝑁. The
result is a polynomial function of 𝑥.
This "extended" version of taylpfval 26420 additionally handles the case 𝑁 = +∞, in which case this is not a polynomial but an infinite series, the Taylor series of the function. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) & ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) ⇒ ⊢ (𝜑 → 𝑇 = ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))) | ||
Theorem | eltayl 26415* | Value of the Taylor series as a relation (elementhood in the domain here expresses that the series is convergent). (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) & ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑇𝑌 ↔ (𝑋 ∈ ℂ ∧ 𝑌 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘))))))) | ||
Theorem | taylf 26416* | The Taylor series defines a function on a subset of the complex numbers. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) & ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) ⇒ ⊢ (𝜑 → 𝑇:dom 𝑇⟶ℂ) | ||
Theorem | tayl0 26417* | The Taylor series is always defined at the basepoint, with value equal to the value of the function. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) & ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) ⇒ ⊢ (𝜑 → (𝐵 ∈ dom 𝑇 ∧ (𝑇‘𝐵) = (𝐹‘𝐵))) | ||
Theorem | taylplem1 26418* | Lemma for taylpfval 26420 and similar theorems. (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)) ⇒ ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) | ||
Theorem | taylplem2 26419* | Lemma for taylpfval 26420 and similar theorems. (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)) ⇒ ⊢ (((𝜑 ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)) ∈ ℂ) | ||
Theorem | taylpfval 26420* | Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: 𝑆 is the base set with respect to evaluate the derivatives (generally ℝ or ℂ), 𝐹 is the function we are approximating, at point 𝐵, to order 𝑁. The result is a polynomial function of 𝑥. (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 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) | ||
Theorem | taylpf 26421 | The Taylor polynomial is a function on the complex numbers (even if the base set of the original function is the reals). (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:ℂ⟶ℂ) | ||
Theorem | taylpval 26422* | Value of the Taylor polynomial. (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 → 𝑋 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝑇‘𝑋) = Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘))) | ||
Theorem | taylply2 26423* | The Taylor polynomial is a polynomial of degree (at most) 𝑁. This version of taylply 26425 shows that the coefficients of 𝑇 are in a subring of the complex numbers. (Contributed by Mario Carneiro, 1-Jan-2017.) Avoid ax-mulf 11232. (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 26424* | Obsolete version of taylply2 26423 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 26425 | 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 26426 | 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 26427 | 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 26428 | 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 26429* | Lemma for taylth 26432. 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 26432 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 26430* | Lemma for taylth 26432. (Contributed by Mario Carneiro, 1-Jan-2017.) Avoid ax-mulf 11232. (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 26431* | Obsolete version of taylthlem2 26430 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 26432* | 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 26433 | Extend class notation to include the uniform convergence predicate. |
class ⇝𝑢 | ||
Definition | df-ulm 26434* | 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 15520. (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 26435 | The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ Rel (⇝𝑢‘𝑆) | ||
Theorem | ulmscl 26436 | Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) | ||
Theorem | ulmval 26437* | 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 26438 | Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | ||
Theorem | ulmf 26439* | 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 26440 | 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 26441 | 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 26442* | Simplify ulmval 26437 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 26443* | 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 26444* | 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 26445 | 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 26446* | Lemma for ulmshft 26447. (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 26447* | 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 26448 | 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 26449 | 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 26450 | 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 26451* | Lemma for ulmcau 26452 and ulmcau2 26453: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 15390. (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 26452* | 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 26453 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 26453* | 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 26454* | 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 26455* | 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 26456 | 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 26457* | Lemma for ulmdv 26460. (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 26458* | Lemma for ulmdv 26460. (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 26459* | Lemma for ulmdv 26460. (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 26460* | 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 26461* | 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 26462* | 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 26463 | 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 25716.) (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 26464 | 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 26465* | 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 26466* | 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 26467* | 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 26468* | 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 26469* | 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 26470* | Lemma for radcnvlt1 26475, radcnvle 26477. 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 26471* | Lemma for radcnvlt1 26475, radcnvle 26477. 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 26472* | Lemma for radcnvlt1 26475, radcnvle 26477. 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 26473* | 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 26474* | 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 26475* | 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 26476* | 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 26477* | 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 26478* | 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 26479* | 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 26480* | Since by pserulm 26479 the series converges uniformly, it is also continuous by ulmcn 26456. (Contributed by Mario Carneiro, 3-Mar-2015.) Avoid ax-mulf 11232. (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 26481* | Obsolete version of psercn2 26480 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 26482* | Lemma for psercn 26484. (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 26483* | Lemma for psercn 26484. (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 26484* | 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 26485* | Lemma for pserdv 26487. (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 26486* | Lemma for pserdv 26487. (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 26487* | 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 26488* | 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 26489* | Lemma for abelth 26499. (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 26490* | Lemma for abelth 26499. 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 26491* | Lemma for abelth 26499. (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 26492* | Lemma for abelth 26499. (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 26493* | Lemma for abelth 26499. (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 26494* | Lemma for abelth 26499. (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 26495* | Lemma for abelth 26499. (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 26496* | Lemma for abelth 26499. (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 26497* | Lemma for abelth 26499. (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 26498* | Lemma for abelth 26499. 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 26499* | 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 26484.) (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 26500* | 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)) |
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