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Theorem List for Metamath Proof Explorer - 23801-23900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-ana 23801* 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)))})

Theoremtaylfvallem1 23802* Lemma for taylfval 23804. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))       (((𝜑𝑋 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)) ∈ ℂ)

Theoremtaylfvallem 23803* Lemma for taylfval 23804. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))       ((𝜑𝑋 ∈ ℂ) → (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)))) ⊆ ℂ)

Theoremtaylfval 23804* 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 or ), 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 23810 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.)

(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑𝑇 = 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥𝐵)↑𝑘))))))

Theoremeltayl 23805* 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.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((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)) · ((𝑋𝐵)↑𝑘)))))))

Theoremtaylf 23806* The Taylor series defines a function on a subset of the complex numbers. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑𝑇:dom 𝑇⟶ℂ)

Theoremtayl0 23807* 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.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑 → (𝐵 ∈ dom 𝑇 ∧ (𝑇𝐵) = (𝐹𝐵)))

Theoremtaylplem1 23808* Lemma for taylpfval 23810 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))       ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))

Theoremtaylplem2 23809* Lemma for taylpfval 23810 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))       (((𝜑𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)) ∈ ℂ)

Theoremtaylpfval 23810* 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 or ), 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.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑𝑇 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥𝐵)↑𝑘))))

Theoremtaylpf 23811 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.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑𝑇:ℂ⟶ℂ)

Theoremtaylpval 23812* Value of the Taylor polynomial. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)    &   (𝜑𝑋 ∈ ℂ)       (𝜑 → (𝑇𝑋) = Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)))

Theoremtaylply2 23813* The Taylor polynomial is a polynomial of degree (at most) 𝑁. This version of taylply 23814 shows that the coefficients of 𝑇 are in a subring of the complex numbers. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)    &   (𝜑𝐷 ∈ (SubRing‘ℂfld))    &   (𝜑𝐵𝐷)    &   ((𝜑𝑘 ∈ (0...𝑁)) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ 𝐷)       (𝜑 → (𝑇 ∈ (Poly‘𝐷) ∧ (deg‘𝑇) ≤ 𝑁))

Theoremtaylply 23814 The Taylor polynomial is a polynomial of degree (at most) 𝑁. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁))

Theoremdvtaylp 23815 The derivative of the Taylor polynomial is the Taylor polynomial of the derivative of the function. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 1)))       (𝜑 → (ℂ D ((𝑁 + 1)(𝑆 Tayl 𝐹)𝐵)) = (𝑁(𝑆 Tayl (𝑆 D 𝐹))𝐵))

Theoremdvntaylp 23816 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.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))       (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = (𝑁(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))

Theoremdvntaylp0 23817 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.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑀 ∈ (0...𝑁))    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵))

Theoremtaylthlem1 23818* Lemma for taylth 23820. 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 23820 itself we must restrict to . (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) = 𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐵𝐴)    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)    &   𝑅 = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹𝑥) − (𝑇𝑥)) / ((𝑥𝐵)↑𝑁)))    &   ((𝜑 ∧ (𝑛 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D𝑛 𝐹)‘(𝑁𝑛))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁𝑛))‘𝑦)) / ((𝑦𝐵)↑𝑛))) lim 𝐵))) → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥𝐵)↑(𝑛 + 1)))) lim 𝐵))       (𝜑 → 0 ∈ (𝑅 lim 𝐵))

Theoremtaylthlem2 23819* Lemma for taylth 23820. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → dom ((ℝ D𝑛 𝐹)‘𝑁) = 𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐵𝐴)    &   𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵)    &   (𝜑𝑀 ∈ (1..^𝑁))    &   (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁𝑀))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁𝑀))‘𝑥)) / ((𝑥𝐵)↑𝑀))) lim 𝐵))       (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) / ((𝑥𝐵)↑(𝑀 + 1)))) lim 𝐵))

Theoremtaylth 23820* Taylor's theorem. The Taylor polynomial of a 𝑁-times differentiable function is such that the error term goes to zero faster than (𝑥𝐵)↑𝑁. This is Metamath 100 proof #35. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → dom ((ℝ D𝑛 𝐹)‘𝑁) = 𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐵𝐴)    &   𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵)    &   𝑅 = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹𝑥) − (𝑇𝑥)) / ((𝑥𝐵)↑𝑁)))       (𝜑 → 0 ∈ (𝑅 lim 𝐵))

14.2.2  Uniform convergence

Syntaxculm 23821 Extend class notation to include the uniform convergence predicate.
class 𝑢

Definitiondf-ulm 23822* Define the uniform convergence of a sequence of functions. Here 𝐹(⇝𝑢𝑆)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 𝑗𝑘 are all uniformly within 𝑥 of 𝐺 on the domain 𝑆. Compare with df-clim 13933. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝑢 = (𝑠 ∈ V ↦ {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)})

Theoremulmrel 23823 The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)
Rel (⇝𝑢𝑆)

Theoremulmscl 23824 Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(⇝𝑢𝑆)𝐺𝑆 ∈ V)

Theoremulmval 23825* Express the predicate: The sequence of functions 𝐹 converges uniformly to 𝐺 on 𝑆. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝑆𝑉 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))

Theoremulmcl 23826 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(⇝𝑢𝑆)𝐺𝐺:𝑆⟶ℂ)

Theoremulmf 23827* Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(⇝𝑢𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))

Theoremulmpm 23828 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(⇝𝑢𝑆)𝐺𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ))

Theoremulmf2 23829 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.)
((𝐹 Fn 𝑍𝐹(⇝𝑢𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))

Theoremulm2 23830* Simplify ulmval 23825 when 𝐹 and 𝐺 are known to be functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → ((𝐹𝑘)‘𝑧) = 𝐵)    &   ((𝜑𝑧𝑆) → (𝐺𝑧) = 𝐴)    &   (𝜑𝐺:𝑆⟶ℂ)    &   (𝜑𝑆𝑉)       (𝜑 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))

Theoremulmi 23831* The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → ((𝐹𝑘)‘𝑧) = 𝐵)    &   ((𝜑𝑧𝑆) → (𝐺𝑧) = 𝐴)    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝐶)

Theoremulmclm 23832* A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   (𝜑𝐴𝑆)    &   (𝜑𝐻𝑊)    &   ((𝜑𝑘𝑍) → ((𝐹𝑘)‘𝐴) = (𝐻𝑘))    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)       (𝜑𝐻 ⇝ (𝐺𝐴))

Theoremulmres 23833 A sequence of functions converges iff the tail of the sequence converges (for any finite cutoff). (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)    &   (𝜑𝑁𝑍)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))       (𝜑 → (𝐹(⇝𝑢𝑆)𝐺 ↔ (𝐹𝑊)(⇝𝑢𝑆)𝐺))

Theoremulmshftlem 23834* Lemma for ulmshft 23835. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ‘(𝑀 + 𝐾))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐾 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   (𝜑𝐻 = (𝑛𝑊 ↦ (𝐹‘(𝑛𝐾))))       (𝜑 → (𝐹(⇝𝑢𝑆)𝐺𝐻(⇝𝑢𝑆)𝐺))

Theoremulmshft 23835* A sequence of functions converges iff the shifted sequence converges. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ‘(𝑀 + 𝐾))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐾 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   (𝜑𝐻 = (𝑛𝑊 ↦ (𝐹‘(𝑛𝐾))))       (𝜑 → (𝐹(⇝𝑢𝑆)𝐺𝐻(⇝𝑢𝑆)𝐺))

Theoremulm0 23836 Every function converges uniformly on the empty set. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   (𝜑𝐺:𝑆⟶ℂ)       ((𝜑𝑆 = ∅) → 𝐹(⇝𝑢𝑆)𝐺)

Theoremulmuni 23837 An sequence of functions uniformly converges to at most one limit. (Contributed by Mario Carneiro, 5-Jul-2017.)
((𝐹(⇝𝑢𝑆)𝐺𝐹(⇝𝑢𝑆)𝐻) → 𝐺 = 𝐻)

Theoremulmdm 23838 Two ways to express that a function has a limit. (The expression ((⇝𝑢𝑆)‘𝐹) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Mario Carneiro, 5-Jul-2017.)
(𝐹 ∈ dom (⇝𝑢𝑆) ↔ 𝐹(⇝𝑢𝑆)((⇝𝑢𝑆)‘𝐹))

Theoremulmcaulem 23839* Lemma for ulmcau 23840 and ulmcau2 23841: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 13802. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))       (𝜑 → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − ((𝐹𝑗)‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑘)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − ((𝐹𝑚)‘𝑧))) < 𝑥))

Theoremulmcau 23840* 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 𝑗𝑘 the functions 𝐹(𝑘) and 𝐹(𝑗) are uniformly within 𝑥 of each other on 𝑆. This is the four-quantifier version, see ulmcau2 23841 for the more conventional five-quantifier version. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))       (𝜑 → (𝐹 ∈ dom (⇝𝑢𝑆) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − ((𝐹𝑗)‘𝑧))) < 𝑥))

Theoremulmcau2 23841* 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 𝑗𝑘, 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 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))       (𝜑 → (𝐹 ∈ dom (⇝𝑢𝑆) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑘)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − ((𝐹𝑚)‘𝑧))) < 𝑥))

Theoremulmss 23842* A uniform limit of functions is still a uniform limit if restricted to a subset. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑇𝑆)    &   ((𝜑𝑥𝑍) → 𝐴𝑊)    &   (𝜑 → (𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺)       (𝜑 → (𝑥𝑍 ↦ (𝐴𝑇))(⇝𝑢𝑇)(𝐺𝑇))

Theoremulmbdd 23843* A uniform limit of bounded functions is bounded. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   ((𝜑𝑘𝑍) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘((𝐹𝑘)‘𝑧)) ≤ 𝑥)    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝐺𝑧)) ≤ 𝑥)

Theoremulmcn 23844 A uniform limit of continuous functions is continuous. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(𝑆cn→ℂ))    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)       (𝜑𝐺 ∈ (𝑆cn→ℂ))

Theoremulmdvlem1 23845* Lemma for ulmdv 23848. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑋))    &   (𝜑𝐺:𝑋⟶ℂ)    &   ((𝜑𝑧𝑋) → (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) ⇝ (𝐺𝑧))    &   (𝜑 → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻)    &   ((𝜑𝜓) → 𝐶𝑋)    &   ((𝜑𝜓) → 𝑅 ∈ ℝ+)    &   ((𝜑𝜓) → 𝑈 ∈ ℝ+)    &   ((𝜑𝜓) → 𝑊 ∈ ℝ+)    &   ((𝜑𝜓) → 𝑈 < 𝑊)    &   ((𝜑𝜓) → (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) ⊆ 𝑋)    &   ((𝜑𝜓) → (abs‘(𝑌𝐶)) < 𝑈)    &   ((𝜑𝜓) → 𝑁𝑍)    &   ((𝜑𝜓) → ∀𝑚 ∈ (ℤ𝑁)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑁))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑅 / 2) / 2))    &   ((𝜑𝜓) → (abs‘(((𝑆 D (𝐹𝑁))‘𝐶) − (𝐻𝐶))) < (𝑅 / 2))    &   ((𝜑𝜓) → 𝑌𝑋)    &   ((𝜑𝜓) → 𝑌𝐶)    &   ((𝜑𝜓) → ((abs‘(𝑌𝐶)) < 𝑊 → (abs‘(((((𝐹𝑁)‘𝑌) − ((𝐹𝑁)‘𝐶)) / (𝑌𝐶)) − ((𝑆 D (𝐹𝑁))‘𝐶))) < ((𝑅 / 2) / 2)))       ((𝜑𝜓) → (abs‘((((𝐺𝑌) − (𝐺𝐶)) / (𝑌𝐶)) − (𝐻𝐶))) < 𝑅)

Theoremulmdvlem2 23846* Lemma for ulmdv 23848. (Contributed by Mario Carneiro, 8-May-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑋))    &   (𝜑𝐺:𝑋⟶ℂ)    &   ((𝜑𝑧𝑋) → (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) ⇝ (𝐺𝑧))    &   (𝜑 → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻)       ((𝜑𝑘𝑍) → dom (𝑆 D (𝐹𝑘)) = 𝑋)

Theoremulmdvlem3 23847* Lemma for ulmdv 23848. (Contributed by Mario Carneiro, 8-May-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑋))    &   (𝜑𝐺:𝑋⟶ℂ)    &   ((𝜑𝑧𝑋) → (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) ⇝ (𝐺𝑧))    &   (𝜑 → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻)       ((𝜑𝑧𝑋) → 𝑧(𝑆 D 𝐺)(𝐻𝑧))

Theoremulmdv 23848* 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 = (ℤ𝑀)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑋))    &   (𝜑𝐺:𝑋⟶ℂ)    &   ((𝜑𝑧𝑋) → (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) ⇝ (𝐺𝑧))    &   (𝜑 → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻)       (𝜑 → (𝑆 D 𝐺) = 𝐻)

Theoremmtest 23849* 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 = (ℤ𝑁)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   (𝜑𝑀𝑊)    &   ((𝜑𝑘𝑍) → (𝑀𝑘) ∈ ℝ)    &   ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))    &   (𝜑 → seq𝑁( + , 𝑀) ∈ dom ⇝ )       (𝜑 → seq𝑁( ∘𝑓 + , 𝐹) ∈ dom (⇝𝑢𝑆))

Theoremmtestbdd 23850* 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 = (ℤ𝑁)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   (𝜑𝑀𝑊)    &   ((𝜑𝑘𝑍) → (𝑀𝑘) ∈ ℝ)    &   ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))    &   (𝜑 → seq𝑁( + , 𝑀) ∈ dom ⇝ )    &   (𝜑 → seq𝑁( ∘𝑓 + , 𝐹)(⇝𝑢𝑆)𝑇)       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)

Theoremmbfulm 23851 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 23116.) (Contributed by Mario Carneiro, 18-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶MblFn)    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)       (𝜑𝐺 ∈ MblFn)

Theoremiblulm 23852 A uniform limit of integrable functions is integrable. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶𝐿1)    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)    &   (𝜑 → (vol‘𝑆) ∈ ℝ)       (𝜑𝐺 ∈ 𝐿1)

Theoremitgulm 23853* A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶𝐿1)    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)    &   (𝜑 → (vol‘𝑆) ∈ ℝ)       (𝜑 → (𝑘𝑍 ↦ ∫𝑆((𝐹𝑘)‘𝑥) d𝑥) ⇝ ∫𝑆(𝐺𝑥) d𝑥)

Theoremitgulm2 23854* A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝑥𝑆𝐴) ∈ (𝑆cn→ℂ))    &   ((𝜑𝑘𝑍) → (𝑥𝑆𝐴) ∈ 𝐿1)    &   (𝜑 → (𝑘𝑍 ↦ (𝑥𝑆𝐴))(⇝𝑢𝑆)(𝑥𝑆𝐵))    &   (𝜑 → (vol‘𝑆) ∈ ℝ)       (𝜑 → ((𝑥𝑆𝐵) ∈ 𝐿1 ∧ (𝑘𝑍 ↦ ∫𝑆𝐴 d𝑥) ⇝ ∫𝑆𝐵 d𝑥))

14.2.3  Power series

Theorempserval 23855* 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))       (𝑋 ∈ ℂ → (𝐺𝑋) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑋𝑚))))

Theorempserval2 23856* 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))       ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐺𝑋)‘𝑁) = ((𝐴𝑁) · (𝑋𝑁)))

Theorempsergf 23857* 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)       (𝜑 → (𝐺𝑋):ℕ0⟶ℂ)

Theoremradcnvlem1 23858* Lemma for radcnvlt1 23863, radcnvle 23865. 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑 → (abs‘𝑋) < (abs‘𝑌))    &   (𝜑 → seq0( + , (𝐺𝑌)) ∈ dom ⇝ )    &   𝐻 = (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))       (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )

Theoremradcnvlem2 23859* Lemma for radcnvlt1 23863, radcnvle 23865. 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑 → (abs‘𝑋) < (abs‘𝑌))    &   (𝜑 → seq0( + , (𝐺𝑌)) ∈ dom ⇝ )       (𝜑 → seq0( + , (abs ∘ (𝐺𝑋))) ∈ dom ⇝ )

Theoremradcnvlem3 23860* Lemma for radcnvlt1 23863, radcnvle 23865. 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑 → (abs‘𝑋) < (abs‘𝑌))    &   (𝜑 → seq0( + , (𝐺𝑌)) ∈ dom ⇝ )       (𝜑 → seq0( + , (𝐺𝑋)) ∈ dom ⇝ )

Theoremradcnv0 23861* Zero is always a convergent point for any power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   (𝜑𝐴:ℕ0⟶ℂ)       (𝜑 → 0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ })

Theoremradcnvcl 23862* 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )       (𝜑𝑅 ∈ (0[,]+∞))

Theoremradcnvlt1 23863* 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑 → (abs‘𝑋) < 𝑅)    &   𝐻 = (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))       (𝜑 → (seq0( + , 𝐻) ∈ dom ⇝ ∧ seq0( + , (abs ∘ (𝐺𝑋))) ∈ dom ⇝ ))

Theoremradcnvlt2 23864* 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑 → (abs‘𝑋) < 𝑅)       (𝜑 → seq0( + , (𝐺𝑋)) ∈ dom ⇝ )

Theoremradcnvle 23865* 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑 → seq0( + , (𝐺𝑋)) ∈ dom ⇝ )       (𝜑 → (abs‘𝑋) ≤ 𝑅)

Theoremdvradcnv 23866* 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝐻 = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 1) · (𝐴‘(𝑛 + 1))) · (𝑋𝑛)))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑 → (abs‘𝑋) < 𝑅)       (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )

Theorempserulm 23867* 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝐻 = (𝑖 ∈ ℕ0 ↦ (𝑦𝑆 ↦ (seq0( + , (𝐺𝑦))‘𝑖)))    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑𝑀 < 𝑅)    &   (𝜑𝑆 ⊆ (abs “ (0[,]𝑀)))       (𝜑𝐻(⇝𝑢𝑆)𝐹)

Theorempsercn2 23868* Since by pserulm 23867 the series converges uniformly, it is also continuous by ulmcn 23844. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝐻 = (𝑖 ∈ ℕ0 ↦ (𝑦𝑆 ↦ (seq0( + , (𝐺𝑦))‘𝑖)))    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑𝑀 < 𝑅)    &   (𝜑𝑆 ⊆ (abs “ (0[,]𝑀)))       (𝜑𝐹 ∈ (𝑆cn→ℂ))

Theorempsercnlem2 23869* Lemma for psercn 23871. (Contributed by Mario Carneiro, 18-Mar-2015.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝑆 = (abs “ (0[,)𝑅))    &   ((𝜑𝑎𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀𝑀 < 𝑅))       ((𝜑𝑎𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (abs “ (0[,]𝑀)) ∧ (abs “ (0[,]𝑀)) ⊆ 𝑆))

Theorempsercnlem1 23870* Lemma for psercn 23871. (Contributed by Mario Carneiro, 18-Mar-2015.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝑆 = (abs “ (0[,)𝑅))    &   𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))       ((𝜑𝑎𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀𝑀 < 𝑅))

Theorempsercn 23871* 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝑆 = (abs “ (0[,)𝑅))    &   𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))       (𝜑𝐹 ∈ (𝑆cn→ℂ))

Theorempserdvlem1 23872* Lemma for pserdv 23874. (Contributed by Mario Carneiro, 7-May-2015.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝑆 = (abs “ (0[,)𝑅))    &   𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))       ((𝜑𝑎𝑆) → ((((abs‘𝑎) + 𝑀) / 2) ∈ ℝ+ ∧ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2) ∧ (((abs‘𝑎) + 𝑀) / 2) < 𝑅))

Theorempserdvlem2 23873* Lemma for pserdv 23874. (Contributed by Mario Carneiro, 7-May-2015.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ 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))       ((𝜑𝑎𝑆) → (ℂ D (𝐹𝐵)) = (𝑦𝐵 ↦ Σ𝑘 ∈ ℕ0 (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑦𝑘))))

Theorempserdv 23874* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ 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) = (𝑦𝑆 ↦ Σ𝑘 ∈ ℕ0 (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑦𝑘))))

Theorempserdv2 23875* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ 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) = (𝑦𝑆 ↦ Σ𝑘 ∈ ℕ ((𝑘 · (𝐴𝑘)) · (𝑦↑(𝑘 − 1)))))

Theoremabelthlem1 23876* Lemma for abelth 23886. (Contributed by Mario Carneiro, 1-Apr-2015.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )       (𝜑 → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑧𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))

Theoremabelthlem2 23877* Lemma for abelth 23886. 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.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝑀)    &   𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}       (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))

Theoremabelthlem3 23878* Lemma for abelth 23886. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝑀)    &   𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}       ((𝜑𝑋𝑆) → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑋𝑛)))) ∈ dom ⇝ )

Theoremabelthlem4 23879* Lemma for abelth 23886. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝑀)    &   𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   𝐹 = (𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))       (𝜑𝐹:𝑆⟶ℂ)

Theoremabelthlem5 23880* Lemma for abelth 23886. (Contributed by Mario Carneiro, 1-Apr-2015.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝑀)    &   𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   𝐹 = (𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))    &   (𝜑 → seq0( + , 𝐴) ⇝ 0)       ((𝜑𝑋 ∈ (0(ball‘(abs ∘ − ))1)) → seq0( + , (𝑘 ∈ ℕ0 ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋𝑘)))) ∈ dom ⇝ )

Theoremabelthlem6 23881* Lemma for abelth 23886. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝑀)    &   𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   𝐹 = (𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))    &   (𝜑 → seq0( + , 𝐴) ⇝ 0)    &   (𝜑𝑋 ∈ (𝑆 ∖ {1}))       (𝜑 → (𝐹𝑋) = ((1 − 𝑋) · Σ𝑛 ∈ ℕ0 ((seq0( + , 𝐴)‘𝑛) · (𝑋𝑛))))

Theoremabelthlem7a 23882* Lemma for abelth 23886. (Contributed by Mario Carneiro, 8-May-2015.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝑀)    &   𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   𝐹 = (𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))    &   (𝜑 → seq0( + , 𝐴) ⇝ 0)    &   (𝜑𝑋 ∈ (𝑆 ∖ {1}))       (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))

Theoremabelthlem7 23883* Lemma for abelth 23886. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝑀)    &   𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   𝐹 = (𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))    &   (𝜑 → seq0( + , 𝐴) ⇝ 0)    &   (𝜑𝑋 ∈ (𝑆 ∖ {1}))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ (ℤ𝑁)(abs‘(seq0( + , 𝐴)‘𝑘)) < 𝑅)    &   (𝜑 → (abs‘(1 − 𝑋)) < (𝑅 / (Σ𝑛 ∈ (0...(𝑁 − 1))(abs‘(seq0( + , 𝐴)‘𝑛)) + 1)))       (𝜑 → (abs‘(𝐹𝑋)) < ((𝑀 + 1) · 𝑅))

Theoremabelthlem8 23884* Lemma for abelth 23886. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝑀)    &   𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   𝐹 = (𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))    &   (𝜑 → seq0( + , 𝐴) ⇝ 0)       ((𝜑𝑅 ∈ ℝ+) → ∃𝑤 ∈ ℝ+𝑦𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑅))

Theoremabelthlem9 23885* Lemma for abelth 23886. By adjusting the constant term, we can assume that the entire series converges to 0. (Contributed by Mario Carneiro, 1-Apr-2015.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝑀)    &   𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   𝐹 = (𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))       ((𝜑𝑅 ∈ ℝ+) → ∃𝑤 ∈ ℝ+𝑦𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑅))

Theoremabelth 23886* Abel's theorem. If the power series Σ𝑛 ∈ ℕ0𝐴(𝑛)(𝑥𝑛) 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 23871.) (Contributed by Mario Carneiro, 2-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝑀)    &   𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   𝐹 = (𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))       (𝜑𝐹 ∈ (𝑆cn→ℂ))

Theoremabelth2 23887* Abel's theorem, restricted to the [0, 1] interval. (Contributed by Mario Carneiro, 2-Apr-2015.)
(𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   𝐹 = (𝑥 ∈ (0[,]1) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))       (𝜑𝐹 ∈ ((0[,]1)–cn→ℂ))

14.3  Basic trigonometry

14.3.1  The exponential, sine, and cosine functions (cont.)

Theoremefcn 23888 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
exp ∈ (ℂ–cn→ℂ)

Theoremsincn 23889 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
sin ∈ (ℂ–cn→ℂ)

Theoremcoscn 23890 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
cos ∈ (ℂ–cn→ℂ)

Theoremreeff1olem 23891* Lemma for reeff1o 23892. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)

Theoremreeff1o 23892 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+

Theoremreefiso 23893 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+)

Theoremefcvx 23894 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))))

Theoremreefgim 23895 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+)       (exp ↾ ℝ) ∈ (ℝfld GrpIso 𝑃)

14.3.2  Properties of pi = 3.14159...

Theorempilem1 23896 Lemma for pire 23901, pigt2lt4 23899 and sinpi 23900. (Contributed by Mario Carneiro, 9-May-2014.)
(𝐴 ∈ (ℝ+ ∩ (sin “ {0})) ↔ (𝐴 ∈ ℝ+ ∧ (sin‘𝐴) = 0))

Theorempilem2 23897 Lemma for pire 23901, pigt2lt4 23899 and sinpi 23900. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by AV, 14-Sep-2020.)
(𝜑𝐴 ∈ (2(,)4))    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → (sin‘𝐴) = 0)    &   (𝜑 → (sin‘𝐵) = 0)    &   (𝜑 → π < 𝐴)       (𝜑 → ((π + 𝐴) / 2) ≤ 𝐵)

Theorempilem3 23898 Lemma for pire 23901, pigt2lt4 23899 and sinpi 23900. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) (Revised by AV, 14-Sep-2020.)
(π ∈ (2(,)4) ∧ (sin‘π) = 0)

Theorempigt2lt4 23899 π is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
(2 < π ∧ π < 4)

Theoremsinpi 23900 The sine of π is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘π) = 0

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