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Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremitgulm2 24101* 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 24102* 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 24103* 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 24104* 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 24105* Lemma for radcnvlt1 24110, radcnvle 24112. 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 24106* Lemma for radcnvlt1 24110, radcnvle 24112. 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 24107* Lemma for radcnvlt1 24110, radcnvle 24112. 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 24108* 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 24109* 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 24110* 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 24111* 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 24112* 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 24113* 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 24114* 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 24115* Since by pserulm 24114 the series converges uniformly, it is also continuous by ulmcn 24091. (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 24116* Lemma for psercn 24118. (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 24117* Lemma for psercn 24118. (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 24118* 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 24119* Lemma for pserdv 24121. (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 24120* Lemma for pserdv 24121. (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 24121* 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 24122* 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 24123* Lemma for abelth 24133. (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 24124* Lemma for abelth 24133. 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 24125* Lemma for abelth 24133. (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 24126* Lemma for abelth 24133. (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 24127* Lemma for abelth 24133. (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 24128* Lemma for abelth 24133. (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 24129* Lemma for abelth 24133. (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 24130* Lemma for abelth 24133. (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 24131* Lemma for abelth 24133. (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 24132* Lemma for abelth 24133. 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 24133* 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 24118.) (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 24134* 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 24135 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
exp ∈ (ℂ–cn→ℂ)

Theoremsincn 24136 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
sin ∈ (ℂ–cn→ℂ)

Theoremcoscn 24137 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
cos ∈ (ℂ–cn→ℂ)

Theoremreeff1olem 24138* Lemma for reeff1o 24139. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)

Theoremreeff1o 24139 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 24140 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 24141 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 24142 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 24143 Lemma for pire 24148, pigt2lt4 24146 and sinpi 24147. (Contributed by Mario Carneiro, 9-May-2014.)
(𝐴 ∈ (ℝ+ ∩ (sin “ {0})) ↔ (𝐴 ∈ ℝ+ ∧ (sin‘𝐴) = 0))

Theorempilem2 24144 Lemma for pire 24148, pigt2lt4 24146 and sinpi 24147. (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 24145 Lemma for pire 24148, pigt2lt4 24146 and sinpi 24147. 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 24146 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 24147 The sine of π is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘π) = 0

Theorempire 24148 π is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
π ∈ ℝ

Theorempicn 24149 π is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
π ∈ ℂ

Theorempipos 24150 π is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
0 < π

Theorempirp 24151 π is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
π ∈ ℝ+

Theoremnegpicn 24152 is a real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-π ∈ ℂ

Theoremsinhalfpilem 24153 Lemma for sinhalfpi 24158 and coshalfpi 24159. (Contributed by Paul Chapman, 23-Jan-2008.)
((sin‘(π / 2)) = 1 ∧ (cos‘(π / 2)) = 0)

Theoremhalfpire 24154 π / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
(π / 2) ∈ ℝ

Theoremneghalfpire 24155 -π / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ

Theoremneghalfpirx 24156 -π / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ*

Theorempidiv2halves 24157 Adding π / 2 to itself is π (common case). See 2halves 11220. (Contributed by David A. Wheeler, 8-Dec-2018.)
((π / 2) + (π / 2)) = π

Theoremsinhalfpi 24158 The sine of π / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(π / 2)) = 1

Theoremcoshalfpi 24159 The cosine of π / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(π / 2)) = 0

Theoremcosneghalfpi 24160 The cosine of -π / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
(cos‘-(π / 2)) = 0

Theoremefhalfpi 24161 The exponential of iπ / 2 is i. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (π / 2))) = i

Theoremcospi 24162 The cosine of π is -1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘π) = -1

Theoremefipi 24163 The exponential of i · π. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
(exp‘(i · π)) = -1

Theoremeulerid 24164 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
((exp‘(i · π)) + 1) = 0

Theoremsin2pi 24165 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(2 · π)) = 0

Theoremcos2pi 24166 The cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(2 · π)) = 1

Theoremef2pi 24167 The exponential of 2πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (2 · π))) = 1

Theoremef2kpi 24168 The exponential of 2𝐾πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
(𝐾 ∈ ℤ → (exp‘((i · (2 · π)) · 𝐾)) = 1)

Theoremefper 24169 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 24170 Lemma for sinper 24171 and cosper 24172. (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 24171 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 24172 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 24173 If 𝐾 is an integer, 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 24174 If 𝐾 is an integer, 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 24175 Sine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (sin‘((2 · π) − 𝐴)) = -(sin‘𝐴))

Theoremcos2pim 24176 Cosine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (cos‘((2 · π) − 𝐴)) = (cos‘𝐴))

Theoremsinmpi 24177 Sine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = -(sin‘𝐴))

Theoremcosmpi 24178 Cosine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (cos‘(𝐴 − π)) = -(cos‘𝐴))

Theoremsinppi 24179 Sine of a number plus π. (Contributed by NM, 10-Aug-2008.)
(𝐴 ∈ ℂ → (sin‘(𝐴 + π)) = -(sin‘𝐴))

Theoremcosppi 24180 Cosine of a complex number plus π. (Contributed by NM, 18-Aug-2008.)
(𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = -(cos‘𝐴))

Theoremefimpi 24181 The exponential function of i times a real number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (exp‘(i · (𝐴 − π))) = -(exp‘(i · 𝐴)))

Theoremsinhalfpip 24182 The sine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴))

Theoremsinhalfpim 24183 The sine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))

Theoremcoshalfpip 24184 The cosine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘((π / 2) + 𝐴)) = -(sin‘𝐴))

Theoremcoshalfpim 24185 The cosine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘((π / 2) − 𝐴)) = (sin‘𝐴))

Theoremptolemy 24186 Ptolemy's Theorem. This theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). This particular version is expressed using the sine function. It is proved by expanding all the multiplication of sines to a product of cosines of differences using sinmul 14846, then using algebraic simplification to show that both sides are equal. This formalization is based on the proof in "Trigonometry" by Gelfand and Saul. This is Metamath 100 proof #95. (Contributed by David A. Wheeler, 31-May-2015.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (((sin‘𝐴) · (sin‘𝐵)) + ((sin‘𝐶) · (sin‘𝐷))) = ((sin‘(𝐵 + 𝐶)) · (sin‘(𝐴 + 𝐶))))

Theoremsincosq1lem 24187 Lemma for sincosq1sgn 24188. (Contributed by Paul Chapman, 24-Jan-2008.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴𝐴 < (π / 2)) → 0 < (sin‘𝐴))

Theoremsincosq1sgn 24188 The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))

Theoremsincosq2sgn 24189 The signs of the sine and cosine functions in the second quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ((π / 2)(,)π) → (0 < (sin‘𝐴) ∧ (cos‘𝐴) < 0))

Theoremsincosq3sgn 24190 The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ (π(,)(3 · (π / 2))) → ((sin‘𝐴) < 0 ∧ (cos‘𝐴) < 0))

Theoremsincosq4sgn 24191 The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ((3 · (π / 2))(,)(2 · π)) → ((sin‘𝐴) < 0 ∧ 0 < (cos‘𝐴)))

Theoremcoseq00topi 24192 Location of the zeroes of cosine in (0[,]π). (Contributed by David Moews, 28-Feb-2017.)
(𝐴 ∈ (0[,]π) → ((cos‘𝐴) = 0 ↔ 𝐴 = (π / 2)))

Theoremcoseq0negpitopi 24193 Location of the zeroes of cosine in (-π(,]π). (Contributed by David Moews, 28-Feb-2017.)
(𝐴 ∈ (-π(,]π) → ((cos‘𝐴) = 0 ↔ 𝐴 ∈ {(π / 2), -(π / 2)}))

Theoremtanrpcl 24194 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
(𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ+)

Theoremtangtx 24195 The tangent function is greater than its argument on positive reals in its principal domain. (Contributed by Mario Carneiro, 29-Jul-2014.)
(𝐴 ∈ (0(,)(π / 2)) → 𝐴 < (tan‘𝐴))

Theoremtanabsge 24196 The tangent function is greater than or equal to its argument in absolute value. (Contributed by Mario Carneiro, 25-Feb-2015.)
(𝐴 ∈ (-(π / 2)(,)(π / 2)) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))

Theoremsinq12gt0 24197 The sine of a number strictly between 0 and π is positive. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ (0(,)π) → 0 < (sin‘𝐴))

Theoremsinq12ge0 24198 The sine of a number between 0 and π is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)
(𝐴 ∈ (0[,]π) → 0 ≤ (sin‘𝐴))

Theoremsinq34lt0t 24199 The sine of a number strictly between π and 2 · π is negative. (Contributed by NM, 17-Aug-2008.)
(𝐴 ∈ (π(,)(2 · π)) → (sin‘𝐴) < 0)

Theoremcosq14gt0 24200 The cosine of a number strictly between -π / 2 and π / 2 is positive. (Contributed by Mario Carneiro, 25-Feb-2015.)
(𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (cos‘𝐴))

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