P. 247 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mbfulm 24601	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 23876.) (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 24602	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 24603*	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 24604*	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 24605*	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 24606*	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 24607*	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 24608*	Lemma for radcnvlt1 24613, radcnvle 24615. 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 24609*	Lemma for radcnvlt1 24613, radcnvle 24615. 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 24610*	Lemma for radcnvlt1 24613, radcnvle 24615. 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 24611*	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 24612*	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 24613*	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 24614*	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 24615*	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 24616*	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 24617*	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 24618*	Since by pserulm 24617 the series converges uniformly, it is also continuous by ulmcn 24594. (Contributed by Mario Carneiro, 3-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*, < )    &   ⊢ 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 24619*	Lemma for psercn 24621. (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 24620*	Lemma for psercn 24621. (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 24621*	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 24622*	Lemma for pserdv 24624. (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 24623*	Lemma for pserdv 24624. (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 24624*	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 24625*	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 24626*	Lemma for abelth 24636. (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 24627*	Lemma for abelth 24636. 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 24628*	Lemma for abelth 24636. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝑀)    &   ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑋↑𝑛)))) ∈ dom ⇝ )
Theorem	abelthlem4 24629*	Lemma for abelth 24636. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝑀)    &   ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))    ⇒   ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
Theorem	abelthlem5 24630*	Lemma for abelth 24636. (Contributed by Mario Carneiro, 1-Apr-2015.)
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝑀)    &   ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))    &   ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) → seq0( + , (𝑘 ∈ ℕ0 ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))) ∈ dom ⇝ )
Theorem	abelthlem6 24631*	Lemma for abelth 24636. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝑀)    &   ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))    &   ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑆 ∖ {1}))    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) = ((1 − 𝑋) · Σ𝑛 ∈ ℕ0 ((seq0( + , 𝐴)‘𝑛) · (𝑋↑𝑛))))
Theorem	abelthlem7a 24632*	Lemma for abelth 24636. (Contributed by Mario Carneiro, 8-May-2015.)
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝑀)    &   ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))    &   ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑆 ∖ {1}))    ⇒   ⊢ (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))
Theorem	abelthlem7 24633*	Lemma for abelth 24636. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝑀)    &   ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))    &   ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑆 ∖ {1}))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑁)(abs‘(seq0( + , 𝐴)‘𝑘)) < 𝑅)    &   ⊢ (𝜑 → (abs‘(1 − 𝑋)) < (𝑅 / (Σ𝑛 ∈ (0...(𝑁 − 1))(abs‘(seq0( + , 𝐴)‘𝑛)) + 1)))    ⇒   ⊢ (𝜑 → (abs‘(𝐹‘𝑋)) < ((𝑀 + 1) · 𝑅))
Theorem	abelthlem8 24634*	Lemma for abelth 24636. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝑀)    &   ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))    &   ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0)    ⇒   ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅))
Theorem	abelthlem9 24635*	Lemma for abelth 24636. By adjusting the constant term, we can assume that the entire series converges to 0. (Contributed by Mario Carneiro, 1-Apr-2015.)
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝑀)    &   ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))    ⇒   ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅))
Theorem	abelth 24636*	Abel's theorem. If the power series Σ𝑛 ∈ ℕ0𝐴(𝑛)(𝑥↑𝑛) is convergent at 1, then it is equal to the limit from "below", along a Stolz angle 𝑆 (note that the 𝑀 = 1 case of a Stolz angle is the real line [0, 1]). (Continuity on 𝑆 ∖ {1} follows more generally from psercn 24621.) (Contributed by Mario Carneiro, 2-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝑀)    &   ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))
Theorem	abelth2 24637*	Abel's theorem, restricted to the [0, 1] interval. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )    &   ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((0[,]1)–cn→ℂ))
14.3  Basic trigonometry
14.3.1  The exponential, sine, and cosine functions (cont.)
Theorem	efcn 24638	The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
⊢ exp ∈ (ℂ–cn→ℂ)
Theorem	sincn 24639	Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
⊢ sin ∈ (ℂ–cn→ℂ)
Theorem	coscn 24640	Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
⊢ cos ∈ (ℂ–cn→ℂ)
Theorem	reeff1olem 24641*	Lemma for reeff1o 24642. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ ((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)
Theorem	reeff1o 24642	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+
Theorem	reefiso 24643	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+)
Theorem	efcvx 24644	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))))
Theorem	reefgim 24645	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 ℝ+)    ⇒   ⊢ (exp ↾ ℝ) ∈ (ℝfld GrpIso 𝑃)
14.3.2  Properties of pi = 3.14159...
Theorem	pilem1 24646	Lemma for pire 24652, pigt2lt4 24650 and sinpi 24651. (Contributed by Mario Carneiro, 9-May-2014.)
⊢ (𝐴 ∈ (ℝ+ ∩ (◡sin “ {0})) ↔ (𝐴 ∈ ℝ+ ∧ (sin‘𝐴) = 0))
Theorem	pilem2 24647	Lemma for pire 24652, pigt2lt4 24650 and sinpi 24651. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by AV, 14-Sep-2020.)
⊢ (𝜑 → 𝐴 ∈ (2(,)4))    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → (sin‘𝐴) = 0)    &   ⊢ (𝜑 → (sin‘𝐵) = 0)    ⇒   ⊢ (𝜑 → ((π + 𝐴) / 2) ≤ 𝐵)
Theorem	pilem3 24648	Lemma for pire 24652, pigt2lt4 24650 and sinpi 24651. 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)
Theorem	pilem3OLD 24649	Obsolete proof of pilem3 24648 as of 30-Jun-2022. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) (Revised by AV, 14-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (π ∈ (2(,)4) ∧ (sin‘π) = 0)
Theorem	pigt2lt4 24650	π is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
⊢ (2 < π ∧ π < 4)
Theorem	sinpi 24651	The sine of π is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (sin‘π) = 0
Theorem	pire 24652	π is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ π ∈ ℝ
Theorem	picn 24653	π is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ π ∈ ℂ
Theorem	pipos 24654	π is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
⊢ 0 < π
Theorem	pirp 24655	π is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ π ∈ ℝ+
Theorem	negpicn 24656	-π is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -π ∈ ℂ
Theorem	sinhalfpilem 24657	Lemma for sinhalfpi 24662 and coshalfpi 24663. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ ((sin‘(π / 2)) = 1 ∧ (cos‘(π / 2)) = 0)
Theorem	halfpire 24658	π / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
⊢ (π / 2) ∈ ℝ
Theorem	neghalfpire 24659	-π / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -(π / 2) ∈ ℝ
Theorem	neghalfpirx 24660	-π / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -(π / 2) ∈ ℝ*
Theorem	pidiv2halves 24661	Adding π / 2 to itself gives π. See 2halves 11614. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ ((π / 2) + (π / 2)) = π
Theorem	sinhalfpi 24662	The sine of π / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (sin‘(π / 2)) = 1
Theorem	coshalfpi 24663	The cosine of π / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (cos‘(π / 2)) = 0
Theorem	cosneghalfpi 24664	The cosine of -π / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
⊢ (cos‘-(π / 2)) = 0
Theorem	efhalfpi 24665	The exponential of iπ / 2 is i. (Contributed by Mario Carneiro, 9-May-2014.)
⊢ (exp‘(i · (π / 2))) = i
Theorem	cospi 24666	The cosine of π is -1. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (cos‘π) = -1
Theorem	efipi 24667	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
Theorem	eulerid 24668	Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
⊢ ((exp‘(i · π)) + 1) = 0
Theorem	sin2pi 24669	The sine of 2π is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (sin‘(2 · π)) = 0
Theorem	cos2pi 24670	The cosine of 2π is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (cos‘(2 · π)) = 1
Theorem	ef2pi 24671	The exponential of 2πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
⊢ (exp‘(i · (2 · π))) = 1
Theorem	ef2kpi 24672	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)
Theorem	efper 24673	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))
Theorem	sinperlem 24674	Lemma for sinper 24675 and cosper 24676. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
⊢ (𝐴 ∈ ℂ → (𝐹‘𝐴) = (((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)))) = (𝐹‘𝐴))
Theorem	sinper 24675	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))
Theorem	cosper 24676	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))
Theorem	sin2kpi 24677	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)
Theorem	cos2kpi 24678	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)
Theorem	sin2pim 24679	Sine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘((2 · π) − 𝐴)) = -(sin‘𝐴))
Theorem	cos2pim 24680	Cosine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘((2 · π) − 𝐴)) = (cos‘𝐴))
Theorem	sinmpi 24681	Sine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = -(sin‘𝐴))
Theorem	cosmpi 24682	Cosine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 − π)) = -(cos‘𝐴))
Theorem	sinppi 24683	Sine of a number plus π. (Contributed by NM, 10-Aug-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘(𝐴 + π)) = -(sin‘𝐴))
Theorem	cosppi 24684	Cosine of a number plus π. (Contributed by NM, 18-Aug-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = -(cos‘𝐴))
Theorem	efimpi 24685	The exponential function at i times a real number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ (𝐴 ∈ ℂ → (exp‘(i · (𝐴 − π))) = -(exp‘(i · 𝐴)))
Theorem	sinhalfpip 24686	The sine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴))
Theorem	sinhalfpim 24687	The sine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))
Theorem	coshalfpip 24688	The cosine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘((π / 2) + 𝐴)) = -(sin‘𝐴))
Theorem	coshalfpim 24689	The cosine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘((π / 2) − 𝐴)) = (sin‘𝐴))
Theorem	ptolemy 24690	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 15308, 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))))
Theorem	sincosq1lem 24691	Lemma for sincosq1sgn 24692. (Contributed by Paul Chapman, 24-Jan-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴))
Theorem	sincosq1sgn 24692	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)))
Theorem	sincosq2sgn 24693	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))
Theorem	sincosq3sgn 24694	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))
Theorem	sincosq4sgn 24695	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)))
Theorem	coseq00topi 24696	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)))
Theorem	coseq0negpitopi 24697	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)}))
Theorem	tanrpcl 24698	Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
⊢ (𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ+)
Theorem	tangtx 24699	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))
Theorem	tanabsge 24700	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)))