P. 264 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	aalioulem5 26301*	Lemma for aaliou 26303. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ 𝑁 = (deg‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 0)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
Theorem	aalioulem6 26302*	Lemma for aaliou 26303. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ 𝑁 = (deg‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 0)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
Theorem	aaliou 26303*	Liouville's theorem on diophantine approximation: Any algebraic number, being a root of a polynomial 𝐹 in integer coefficients, is not approximable beyond order 𝑁 = deg(𝐹) by rational numbers. In this form, it also applies to rational numbers themselves, which are not well approximable by other rational numbers. This is Metamath 100 proof #18. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ 𝑁 = (deg‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 0)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
Theorem	geolim3 26304*	Geometric series convergence with arbitrary shift, radix, and multiplicative constant. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (abs‘𝐵) < 1)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑘 ∈ (ℤ≥‘𝐴) ↦ (𝐶 · (𝐵↑(𝑘 − 𝐴))))    ⇒   ⊢ (𝜑 → seq𝐴( + , 𝐹) ⇝ (𝐶 / (1 − 𝐵)))
Theorem	aaliou2 26305*	Liouville's approximation theorem for algebraic numbers per se. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐴 ∈ (𝔸 ∩ ℝ) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
Theorem	aaliou2b 26306*	Liouville's approximation theorem extended to complex 𝐴. (Contributed by Stefan O'Rear, 20-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
Theorem	aaliou3lem1 26307*	Lemma for aaliou3 26316. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ 𝐺 = (𝑐 ∈ (ℤ≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))))    ⇒   ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐺‘𝐵) ∈ ℝ)
Theorem	aaliou3lem2 26308*	Lemma for aaliou3 26316. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ 𝐺 = (𝑐 ∈ (ℤ≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))))    &   ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    ⇒   ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝐵) ∈ (0(,](𝐺‘𝐵)))
Theorem	aaliou3lem3 26309*	Lemma for aaliou3 26316. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ 𝐺 = (𝑐 ∈ (ℤ≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))))    &   ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    ⇒   ⊢ (𝐴 ∈ ℕ → (seq𝐴( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ≥‘𝐴)(𝐹‘𝑏) ∈ ℝ+ ∧ Σ𝑏 ∈ (ℤ≥‘𝐴)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘𝐴)))))
Theorem	aaliou3lem8 26310*	Lemma for aaliou3 26316. (Contributed by Stefan O'Rear, 20-Nov-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℕ (2 · (2↑-(!‘(𝑥 + 1)))) ≤ (𝐵 / ((2↑(!‘𝑥))↑𝐴)))
Theorem	aaliou3lem4 26311*	Lemma for aaliou3 26316. (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 26312*	Lemma for aaliou3 26316. (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 26313*	Lemma for aaliou3 26316. (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 26314*	Lemma for aaliou3 26316. (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 26315*	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 26316	Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 23-Nov-2014.)
⊢ Σ𝑘 ∈ ℕ (2↑-(!‘𝑘)) ∉ 𝔸
14.2  Sequences and series
14.2.1  Taylor polynomials and Taylor's theorem
Syntax	ctayl 26317	Taylor polynomial of a function.
class Tayl
Syntax	cana 26318	The class of analytic functions.
class Ana
Definition	df-tayl 26319*	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 26320*	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 26321*	Lemma for taylfval 26323. (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 26322*	Lemma for taylfval 26323. (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 26323*	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 26329 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 26324*	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 26325*	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 26326*	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 26327*	Lemma for taylpfval 26329 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 26328*	Lemma for taylpfval 26329 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 26329*	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 26330	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 26331*	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 26332*	The Taylor polynomial is a polynomial of degree (at most) 𝑁. This version of taylply 26334 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 26333*	Obsolete version of taylply2 26332 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 26334	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 26335	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 26336	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 26337	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 26338*	Lemma for taylth 26341. 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 26341 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 26339*	Lemma for taylth 26341. (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 26340*	Obsolete version of taylthlem2 26339 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 26341*	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))
14.2.2  Uniform convergence
Syntax	culm 26342	Extend class notation to include the uniform convergence predicate.
class ⇝𝑢
Definition	df-ulm 26343*	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 26344	The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)
⊢ Rel (⇝𝑢‘𝑆)
Theorem	ulmscl 26345	Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.)
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V)
Theorem	ulmval 26346*	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 26347	Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ)
Theorem	ulmf 26348*	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 26349	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 26350	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 26351*	Simplify ulmval 26346 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 26352*	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 26353*	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 26354	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 26355*	Lemma for ulmshft 26356. (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 26356*	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 26357	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 26358	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 26359	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 26360*	Lemma for ulmcau 26361 and ulmcau2 26362: 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 26361*	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 26362 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 26362*	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 26363*	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 26364*	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 26365	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 26366*	Lemma for ulmdv 26369. (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 26367*	Lemma for ulmdv 26369. (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 26368*	Lemma for ulmdv 26369. (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 26369*	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 26370*	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 26371*	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 26372	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 25627.) (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 26373	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 26374*	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 26375*	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))
14.2.3  Power series
Theorem	pserval 26376*	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 26377*	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 26378*	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 26379*	Lemma for radcnvlt1 26384, radcnvle 26386. 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 26380*	Lemma for radcnvlt1 26384, radcnvle 26386. 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 26381*	Lemma for radcnvlt1 26384, radcnvle 26386. 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 26382*	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 26383*	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 26384*	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 26385*	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 26386*	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 26387*	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 26388*	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 26389*	Since by pserulm 26388 the series converges uniformly, it is also continuous by ulmcn 26365. (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 26390*	Obsolete version of psercn2 26389 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 26391*	Lemma for psercn 26393. (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 26392*	Lemma for psercn 26393. (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 26393*	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 26394*	Lemma for pserdv 26396. (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 26395*	Lemma for pserdv 26396. (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 26396*	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 26397*	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 26398*	Lemma for abelth 26408. (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 26399*	Lemma for abelth 26408. 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 26400*	Lemma for abelth 26408. (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 ⇝ )