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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremulmf2 24901 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 24902* Simplify ulmval 24897 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 24903* 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 24904* 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 24905 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 24906* Lemma for ulmshft 24907. (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 24907* 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 24908 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 24909 A 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 24910 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 24911* Lemma for ulmcau 24912 and ulmcau2 24913: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 14705. (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 24912* 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 24913 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 24913* 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 24914* 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 24915* 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 24916 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 24917* Lemma for ulmdv 24920. (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 24918* Lemma for ulmdv 24920. (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 24919* Lemma for ulmdv 24920. (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 24920* 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 24921* 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 24922* 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 24923 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 24198.) (Contributed by Mario Carneiro, 18-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶MblFn)    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)       (𝜑𝐺 ∈ MblFn)
 
Theoremiblulm 24924 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 24925* 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 24926* 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) ∈ 𝐿1)    &   (𝜑 → (𝑘𝑍 ↦ (𝑥𝑆𝐴))(⇝𝑢𝑆)(𝑥𝑆𝐵))    &   (𝜑 → (vol‘𝑆) ∈ ℝ)       (𝜑 → ((𝑥𝑆𝐵) ∈ 𝐿1 ∧ (𝑘𝑍 ↦ ∫𝑆𝐴 d𝑥) ⇝ ∫𝑆𝐵 d𝑥))
 
14.2.3  Power series
 
Theorempserval 24927* 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 24928* 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 24929* 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 24930* Lemma for radcnvlt1 24935, radcnvle 24937. 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 24931* Lemma for radcnvlt1 24935, radcnvle 24937. 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 24932* Lemma for radcnvlt1 24935, radcnvle 24937. 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 24933* 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 24934* 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 24935* 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 24936* 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 24937* 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 24938* 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 24939* 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 24940* Since by pserulm 24939 the series converges uniformly, it is also continuous by ulmcn 24916. (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 24941* Lemma for psercn 24943. (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 24942* Lemma for psercn 24943. (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 24943* 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 24944* Lemma for pserdv 24946. (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 24945* Lemma for pserdv 24946. (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 24946* 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 24947* 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 24948* Lemma for abelth 24958. (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 24949* Lemma for abelth 24958. 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 24950* Lemma for abelth 24958. (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 24951* Lemma for abelth 24958. (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 24952* Lemma for abelth 24958. (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 24953* Lemma for abelth 24958. (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 24954* Lemma for abelth 24958. (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 24955* Lemma for abelth 24958. (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 24956* Lemma for abelth 24958. (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 24957* Lemma for abelth 24958. 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 24958* 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 24943.) (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 24959* 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 24960 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
exp ∈ (ℂ–cn→ℂ)
 
Theoremsincn 24961 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
sin ∈ (ℂ–cn→ℂ)
 
Theoremcoscn 24962 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
cos ∈ (ℂ–cn→ℂ)
 
Theoremreeff1olem 24963* Lemma for reeff1o 24964. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)
 
Theoremreeff1o 24964 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 24965 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 24966 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 24967 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 24968 Lemma for pire 24973, pigt2lt4 24971 and sinpi 24972. (Contributed by Mario Carneiro, 9-May-2014.)
(𝐴 ∈ (ℝ+ ∩ (sin “ {0})) ↔ (𝐴 ∈ ℝ+ ∧ (sin‘𝐴) = 0))
 
Theorempilem2 24969 Lemma for pire 24973, pigt2lt4 24971 and sinpi 24972. (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) / 2) ≤ 𝐵)
 
Theorempilem3 24970 Lemma for pire 24973, pigt2lt4 24971 and sinpi 24972. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) (Revised by AV, 14-Sep-2020.) (Proof shortened by BJ, 30-Jun-2022.)
(π ∈ (2(,)4) ∧ (sin‘π) = 0)
 
Theorempigt2lt4 24971 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 24972 The sine of π is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘π) = 0
 
Theorempire 24973 π is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
π ∈ ℝ
 
Theorempicn 24974 π is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
π ∈ ℂ
 
Theorempipos 24975 π is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
0 < π
 
Theorempirp 24976 π is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
π ∈ ℝ+
 
Theoremnegpicn 24977 is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
-π ∈ ℂ
 
Theoremsinhalfpilem 24978 Lemma for sinhalfpi 24983 and coshalfpi 24984. (Contributed by Paul Chapman, 23-Jan-2008.)
((sin‘(π / 2)) = 1 ∧ (cos‘(π / 2)) = 0)
 
Theoremhalfpire 24979 π / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
(π / 2) ∈ ℝ
 
Theoremneghalfpire 24980 -π / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ
 
Theoremneghalfpirx 24981 -π / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ*
 
Theorempidiv2halves 24982 Adding π / 2 to itself gives π. See 2halves 11854. (Contributed by David A. Wheeler, 8-Dec-2018.)
((π / 2) + (π / 2)) = π
 
Theoremsinhalfpi 24983 The sine of π / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(π / 2)) = 1
 
Theoremcoshalfpi 24984 The cosine of π / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(π / 2)) = 0
 
Theoremcosneghalfpi 24985 The cosine of -π / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
(cos‘-(π / 2)) = 0
 
Theoremefhalfpi 24986 The exponential of iπ / 2 is i. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (π / 2))) = i
 
Theoremcospi 24987 The cosine of π is -1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘π) = -1
 
Theoremefipi 24988 The exponential of i · π is -1. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
(exp‘(i · π)) = -1
 
Theoremeulerid 24989 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
((exp‘(i · π)) + 1) = 0
 
Theoremsin2pi 24990 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(2 · π)) = 0
 
Theoremcos2pi 24991 The cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(2 · π)) = 1
 
Theoremef2pi 24992 The exponential of 2πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (2 · π))) = 1
 
Theoremef2kpi 24993 If 𝐾 is an integer, then the exponential of 2𝐾πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
(𝐾 ∈ ℤ → (exp‘((i · (2 · π)) · 𝐾)) = 1)
 
Theoremefper 24994 The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)
((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℤ) → (exp‘(𝐴 + ((i · (2 · π)) · 𝐾))) = (exp‘𝐴))
 
Theoremsinperlem 24995 Lemma for sinper 24996 and cosper 24997. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
(𝐴 ∈ ℂ → (𝐹𝐴) = (((exp‘(i · 𝐴))𝑂(exp‘(-i · 𝐴))) / 𝐷))    &   ((𝐴 + (𝐾 · (2 · π))) ∈ ℂ → (𝐹‘(𝐴 + (𝐾 · (2 · π)))) = (((exp‘(i · (𝐴 + (𝐾 · (2 · π)))))𝑂(exp‘(-i · (𝐴 + (𝐾 · (2 · π)))))) / 𝐷))       ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℤ) → (𝐹‘(𝐴 + (𝐾 · (2 · π)))) = (𝐹𝐴))
 
Theoremsinper 24996 The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℤ) → (sin‘(𝐴 + (𝐾 · (2 · π)))) = (sin‘𝐴))
 
Theoremcosper 24997 The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℤ) → (cos‘(𝐴 + (𝐾 · (2 · π)))) = (cos‘𝐴))
 
Theoremsin2kpi 24998 If 𝐾 is an integer, then the sine of 2𝐾π is 0. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
(𝐾 ∈ ℤ → (sin‘(𝐾 · (2 · π))) = 0)
 
Theoremcos2kpi 24999 If 𝐾 is an integer, then the cosine of 2𝐾π is 1. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
(𝐾 ∈ ℤ → (cos‘(𝐾 · (2 · π))) = 1)
 
Theoremsin2pim 25000 Sine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (sin‘((2 · π) − 𝐴)) = -(sin‘𝐴))
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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