P. 257 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25601-25700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	abelthlem3 25601*	Lemma for abelth 25609. (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 25602*	Lemma for abelth 25609. (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 25603*	Lemma for abelth 25609. (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 25604*	Lemma for abelth 25609. (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 25605*	Lemma for abelth 25609. (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 25606*	Lemma for abelth 25609. (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 25607*	Lemma for abelth 25609. (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 25608*	Lemma for abelth 25609. 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 25609*	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 25594.) (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 25610*	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 25611	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 25612	Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
⊢ sin ∈ (ℂ–cn→ℂ)
Theorem	coscn 25613	Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
⊢ cos ∈ (ℂ–cn→ℂ)
Theorem	reeff1olem 25614*	Lemma for reeff1o 25615. (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 25615	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 25616	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 25617	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 25618	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 25619	Lemma for pire 25624, pigt2lt4 25622 and sinpi 25623. (Contributed by Mario Carneiro, 9-May-2014.)
⊢ (𝐴 ∈ (ℝ+ ∩ (◡sin “ {0})) ↔ (𝐴 ∈ ℝ+ ∧ (sin‘𝐴) = 0))
Theorem	pilem2 25620	Lemma for pire 25624, pigt2lt4 25622 and sinpi 25623. (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 25621	Lemma for pire 25624, pigt2lt4 25622 and sinpi 25623. 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	pigt2lt4 25622	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 25623	The sine of π is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (sin‘π) = 0
Theorem	pire 25624	π is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ π ∈ ℝ
Theorem	picn 25625	π is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ π ∈ ℂ
Theorem	pipos 25626	π is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
⊢ 0 < π
Theorem	pirp 25627	π is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ π ∈ ℝ+
Theorem	negpicn 25628	-π is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -π ∈ ℂ
Theorem	sinhalfpilem 25629	Lemma for sinhalfpi 25634 and coshalfpi 25635. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ ((sin‘(π / 2)) = 1 ∧ (cos‘(π / 2)) = 0)
Theorem	halfpire 25630	π / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
⊢ (π / 2) ∈ ℝ
Theorem	neghalfpire 25631	-π / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -(π / 2) ∈ ℝ
Theorem	neghalfpirx 25632	-π / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -(π / 2) ∈ ℝ*
Theorem	pidiv2halves 25633	Adding π / 2 to itself gives π. See 2halves 12210. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ ((π / 2) + (π / 2)) = π
Theorem	sinhalfpi 25634	The sine of π / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (sin‘(π / 2)) = 1
Theorem	coshalfpi 25635	The cosine of π / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (cos‘(π / 2)) = 0
Theorem	cosneghalfpi 25636	The cosine of -π / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
⊢ (cos‘-(π / 2)) = 0
Theorem	efhalfpi 25637	The exponential of iπ / 2 is i. (Contributed by Mario Carneiro, 9-May-2014.)
⊢ (exp‘(i · (π / 2))) = i
Theorem	cospi 25638	The cosine of π is -1. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (cos‘π) = -1
Theorem	efipi 25639	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 25640	Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
⊢ ((exp‘(i · π)) + 1) = 0
Theorem	sin2pi 25641	The sine of 2π is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (sin‘(2 · π)) = 0
Theorem	cos2pi 25642	The cosine of 2π is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (cos‘(2 · π)) = 1
Theorem	ef2pi 25643	The exponential of 2πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
⊢ (exp‘(i · (2 · π))) = 1
Theorem	ef2kpi 25644	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 25645	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 25646	Lemma for sinper 25647 and cosper 25648. (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 25647	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 25648	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 25649	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 25650	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 25651	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 25652	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 25653	Sine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = -(sin‘𝐴))
Theorem	cosmpi 25654	Cosine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 − π)) = -(cos‘𝐴))
Theorem	sinppi 25655	Sine of a number plus π. (Contributed by NM, 10-Aug-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘(𝐴 + π)) = -(sin‘𝐴))
Theorem	cosppi 25656	Cosine of a number plus π. (Contributed by NM, 18-Aug-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = -(cos‘𝐴))
Theorem	efimpi 25657	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 25658	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 25659	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 25660	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 25661	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 25662	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 15890, 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 25663	Lemma for sincosq1sgn 25664. (Contributed by Paul Chapman, 24-Jan-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴))
Theorem	sincosq1sgn 25664	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 25665	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 25666	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 25667	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 25668	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 25669	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 25670	Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
⊢ (𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ+)
Theorem	tangtx 25671	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 25672	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)))
Theorem	sinq12gt0 25673	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))
Theorem	sinq12ge0 25674	The sine of a number between 0 and π is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)
⊢ (𝐴 ∈ (0[,]π) → 0 ≤ (sin‘𝐴))
Theorem	sinq34lt0t 25675	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)
Theorem	cosq14gt0 25676	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))
Theorem	cosq14ge0 25677	The cosine of a number between -π / 2 and π / 2 is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)
⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) → 0 ≤ (cos‘𝐴))
Theorem	sincosq1eq 25678	Complementarity of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 25-Jan-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 + 𝐵) = 1) → (sin‘(𝐴 · (π / 2))) = (cos‘(𝐵 · (π / 2))))
Theorem	sincos4thpi 25679	The sine and cosine of π / 4. (Contributed by Paul Chapman, 25-Jan-2008.)
⊢ ((sin‘(π / 4)) = (1 / (√‘2)) ∧ (cos‘(π / 4)) = (1 / (√‘2)))
Theorem	tan4thpi 25680	The tangent of π / 4. (Contributed by Mario Carneiro, 5-Apr-2015.)
⊢ (tan‘(π / 4)) = 1
Theorem	sincos6thpi 25681	The sine and cosine of π / 6. (Contributed by Paul Chapman, 25-Jan-2008.) (Revised by Wolf Lammen, 24-Sep-2020.)
⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
Theorem	sincos3rdpi 25682	The sine and cosine of π / 3. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ ((sin‘(π / 3)) = ((√‘3) / 2) ∧ (cos‘(π / 3)) = (1 / 2))
Theorem	pigt3 25683	π is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ 3 < π
Theorem	pige3 25684	π is greater than or equal to 3. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ 3 ≤ π
Theorem	pige3ALT 25685	Alternate proof of pige3 25684. This proof is based on the geometric observation that a hexagon of unit side length has perimeter 6, which is less than the unit-radius circumcircle, of perimeter 2π. We translate this to algebra by looking at the function e↑(i𝑥) as 𝑥 goes from 0 to π / 3; it moves at unit speed and travels distance 1, hence 1 ≤ π / 3. (Contributed by Mario Carneiro, 21-May-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 3 ≤ π
Theorem	abssinper 25686	The absolute value of sine has period π. (Contributed by NM, 17-Aug-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℤ) → (abs‘(sin‘(𝐴 + (𝐾 · π)))) = (abs‘(sin‘𝐴)))
Theorem	sinkpi 25687	The sine of an integer multiple of π is 0. (Contributed by NM, 11-Aug-2008.)
⊢ (𝐾 ∈ ℤ → (sin‘(𝐾 · π)) = 0)
Theorem	coskpi 25688	The absolute value of the cosine of an integer multiple of π is 1. (Contributed by NM, 19-Aug-2008.)
⊢ (𝐾 ∈ ℤ → (abs‘(cos‘(𝐾 · π))) = 1)
Theorem	sineq0 25689	A complex number whose sine is zero is an integer multiple of π. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.)
⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) = 0 ↔ (𝐴 / π) ∈ ℤ))
Theorem	coseq1 25690	A complex number whose cosine is one is an integer multiple of 2π. (Contributed by Mario Carneiro, 12-May-2014.)
⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) = 1 ↔ (𝐴 / (2 · π)) ∈ ℤ))
Theorem	cos02pilt1 25691	Cosine is less than one between zero and 2 · π. (Contributed by Jim Kingdon, 23-Mar-2024.)
⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1)
Theorem	cosq34lt1 25692	Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 23-Mar-2024.)
⊢ (𝐴 ∈ (π[,)(2 · π)) → (cos‘𝐴) < 1)
Theorem	efeq1 25693	A complex number whose exponential is one is an integer multiple of 2πi. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.)
⊢ (𝐴 ∈ ℂ → ((exp‘𝐴) = 1 ↔ (𝐴 / (i · (2 · π))) ∈ ℤ))
Theorem	cosne0 25694	The cosine function has no zeroes within the vertical strip of the complex plane between real part -π / 2 and π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝐴) ≠ 0)
Theorem	cosordlem 25695	Lemma for cosord 25696. (Contributed by Mario Carneiro, 10-May-2014.)
⊢ (𝜑 → 𝐴 ∈ (0[,]π))    &   ⊢ (𝜑 → 𝐵 ∈ (0[,]π))    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (cos‘𝐵) < (cos‘𝐴))
Theorem	cosord 25696	Cosine is decreasing over the closed interval from 0 to π. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)
⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (𝐴 < 𝐵 ↔ (cos‘𝐵) < (cos‘𝐴)))
Theorem	cos0pilt1 25697	Cosine is between minus one and one on the open interval between zero and π. (Contributed by Jim Kingdon, 7-May-2024.)
⊢ (𝐴 ∈ (0(,)π) → (cos‘𝐴) ∈ (-1(,)1))
Theorem	cos11 25698	Cosine is one-to-one over the closed interval from 0 to π. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)
⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (𝐴 = 𝐵 ↔ (cos‘𝐴) = (cos‘𝐵)))
Theorem	sinord 25699	Sine is increasing over the closed interval from -(π / 2) to (π / 2). (Contributed by Mario Carneiro, 29-Jul-2014.)
⊢ ((𝐴 ∈ (-(π / 2)[,](π / 2)) ∧ 𝐵 ∈ (-(π / 2)[,](π / 2))) → (𝐴 < 𝐵 ↔ (sin‘𝐴) < (sin‘𝐵)))
Theorem	recosf1o 25700	The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)
⊢ (cos ↾ (0[,]π)):(0[,]π)–1-1-onto→(-1[,]1)